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Baia de Criș – gmina w Rumunii, w okręgu Hunedoara. Obejmuje miejscowości Baia de Criș, Baldovin, Căraci, Cărăstău, Lunca, Rișca, Rișculița, Țebea i Văleni. W 2011 roku liczyła 2611 mieszkańców.
Przypisy
Gminy w Rumunii
Okręg Hunedoara | {
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The Notre Dame of Dadiangas University (NDDU), also known as Notre Dame, is a private Catholic basic and higher education institution run by the Marist Brothers of the Schools or FMS (Fratres Maristae a Scholis) in General Santos, South Cotabato, Philippines. It was founded by the Marist Brothers in 1953 and offers undergraduate and graduate courses as well as primary and secondary education. Furthermore, it offers a broad spectrum of academic programs through its colleges. It consists of three campuses in General Santos — the main campus along Marist Avenue, the Lagao Campus where the Notre Dame of Dadiangas University-Integrated Basic Education Department (NDDU-IBED) is located and the Espina Campus. Today, NDDU is the only private university in the city of General Santos.
Under the CHED Memorandum Order No. 48, the school was granted university status on June 26, 2006.
Affiliations
As a Marist institution, Notre Dame of Dadiangas University is affiliated with the Marist Schools in the Philippines and linked internationally with University of Alcalá, Spain, Universidad Marista, A.C. Mexico City, Universidad Marista, Guadalajara, Mexico, Pontifícia Universidade Católica do Paraná and Universidad Marista, De San Luis Potosí, Mexico.
The university is a member of Notre Dame Educational Association, Inc. (NDEA), a network of schools named Notre Dame in the Philippines, which is under the patronage of the Blessed Virgin Mary,
Colleges
Business College
College of Education
College of Health Sciences
College of Arts and Sciences
College of Engineering, Architecture, and Technology
Notable alumni
Manny Pacquiao (First Year College only), boxer, politician, basketball coach
Bo Perasol (High School Graduate), basketball coach
Carlos Isagani Zarate (College Graduate), politician
Other Notre Dame Schools in the Philippines
Notre Dame University - Cotabato City (NDU)
Notre Dame of Marbel University (NDMU)
Notre Dame of Jolo College (NDJC)
Notre Dame of Kidapawan College (NDKC)
Notre Dame of Greater Manila (NDGM)
Notre Dame of Tacurong College (NDTC)
Notre Dame of Midsayap College (NDMC)
See also
List of Marist Brothers schools
References
External links
Marist Brothers schools
Catholic universities and colleges in the Philippines
Catholic elementary schools in the Philippines
Catholic secondary schools in the Philippines
Educational institutions established in 1953
Notre Dame Educational Association
1953 establishments in the Philippines
Universities and colleges in General Santos | {
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{"url":"http:\/\/mathhelpforum.com\/geometry\/43456-equation-circle.html","text":"1. Equation of a circle\n\nA(4,3) and B(-4,-3) are the endpoints of the diameter of a circle. Determin the center of the circle and the equation of the circle.\nI found the center as (0,0) by using the midpoint formula.\nwhere do i go from here?\n\n2. Recall the equation of a circle: $\\displaystyle (x - h)^{2} + (x - k)^{2} = r^{2}$\n\nwhere $\\displaystyle (h,k)$ is the coordinate of circle's centre and $\\displaystyle r$ (not $\\displaystyle r^2$) is your radius.\n\nYou have your centre and you can easily find the radius via the pythagorean theorem (draw a line to the coordinate (4,3) and a right-angled triangle can be constructed) or the distance formula. Whichever.","date":"2018-03-24 11:04:53","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.7451713681221008, \"perplexity\": 276.8187222863252}, \"config\": {\"markdown_headings\": false, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 5, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2018-13\/segments\/1521257650188.31\/warc\/CC-MAIN-20180324093251-20180324113251-00551.warc.gz\"}"} | null | null |
Der Elefantenfuß (Beaucarnea recurvata) ist eine Pflanzenart der Gattung Beaucarnea in der Familie der Spargelgewächse (Asparagaceae). Das Artepitheton recurvata bedeutet "zurückgekrümmt". Trivialnamen sind "Ponytail Palm", "Monja" und "Palma culona".
Beschreibung
Beaucarnea recurvata wächst als sukkulenter Baum und erreicht Wuchshöhen von bis zu 9 Metern. Der an der Basis nur mäßig geschwollene Stamm ist darüber schlank und nur wenig verzweigt. Der im Jugendstadium fast kugelförmige Caudex wird später 4 bis 6 Meter lang und erreicht an der Basis Durchmesser von bis zu 50 Zentimetern und mehr. Die Borke ist glatt. Die grünen linealischen, leicht verjüngten und zurückgebogenen Laubblätter sind dünn, flach oder leicht gefurcht. Sie sind 90 bis 180 Zentimeter lang und 15 bis 20 Millimeter breit.
Der fast sitzende, breit eiförmige bis rispige Blütenstand erreicht eine Länge von mehr als 1 Meter. Seine Zweige erster Ordnung sind bis zu 30 Zentimeter, die unteren Seitenzweige bis zu 15 Zentimeter und die Verästelungen bis zu 5 Zentimeter lang. Die 3 bis 4 Millimeter langen kreisrunden Kapselfrüchte sind an ihrer Spitze und der Basis ausgerandet.
Verbreitung und Systematik
Beaucarnea recurvata ist in Mexiko im Bundesstaat Vera Cruz in Xerophyten-Regionen verbreitet. Die Art wächst vergesellschaftet mit Neobuxbaumia scoparia und Selenicereus testudo.
Die Erstbeschreibung erfolgte 1861 durch Charles Lemaire. Ein nomenklatorisches Synonym ist Nolina recurvata (Lem.) Hemsl. (1884).
Beaucarnea recurvata gehört zu Sektion Beaucarnea. Sie wächst in trockenen, tropischen Wäldern. Charakteristisch sind die unregelmäßig verzweigten Bäume mit dem basal verdickten Caudex. Typisch sind variable, herabfallende, gedrehte Blätter, während die verwandte Beaucarnea stricta steife Blätter besitzt.
Literatur
Mary & Gary Irish: Agaves, yuccas, and related plants: A gardener's guide. Timber Press, 2000, ISBN 0881924423. S. 186–188, Pl 54.
Einzelnachweise
Weblinks
Nolinoideen | {
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Ohio Weekly Roundup
Ill Mannered Brewing cut the ribbon on their new brewery this week, opening a brand new facility in the shadow of their previous home in Powell. The new taproom is considerably larger than the prior iteration and features a spacious patio. The added brewery space will allow for future capacity increases. Check it out now or pop in for their third anniversary party on Saturday, August 18. Read details about Ill Mannered's new digs at craftbrewingbusiness.com
Fans of Columbus Brewing Company's Insane Wanderer IPA, a Mosaic hop heavy brew released in June of this year, can now pick up the second entry in the series. Insane Wanderer Vol. II brings Ohio-grown Cashmere hops to the fore, lending bold flavors of ripe apricot and peach. Available in six-packs where your favorite CBC beers are sold.
There are still a couple of days to pop into Smokehouse Brewing and wish them a happy 20th anniversary this week. The brewery has brought back Ol' 33 Tavern Ale, an English-style bitter first brewed in August of 1998, and is offering many retro-priced food and drink specials through the weekend. For details, visit facebook.com/smokehousebrewing
GREATER CLEVELAND
The new Fat Head's Brewery & Beer Hall in Middleburg Heights will officially open to the public on Monday, August 20 at 11 a.m. The massive new brewery, which is already cranking out large volumes of Fat Head's flagship beers, will also feature a "small" 15 barrel pilot brewery (compared to the 68 barrel production rig) for experimental, taproom-exclusive beers. Read Douglas Trattner's preview at clevescene.com
Terrestrial Brewing has two new brews coming out today: C.C. Rider hazy IPA and Figaro Vienna Lager. C.C. Rider takes resinous, piney Chinook and Centennial hops and unlocks layers of fresh tropical fruit. Figaro is a brilliant and clean amber lager: malt forward with a hint of hop bitterness to round it out. Tapping today at Terrestrial's Battery Park tasting room. Details at facebook.com/terrestrialbrewing/
Some time ago, Double Wing Brewing resurrected the old P.O.C. brand (Pride Of Cleveland, as dubbed by its fans) from the long-defunct Pilsener Brewing Company. This past week, Double Wing released a tequila barrel aged version of the P.O.C. pilsner called WOodbine 1600, which was the phone number for Pilsener Brewing. Look for this very limited release on Cleveland-area store shelves before it's gone.
The lager fan who writes this blog is not at all concerned about the late July/early August influx of Festbier and Märzen styles. So, no surprise, I'm happy to report that six-packs of Lager Heads Oktoberfest hit store shelves this week. If you're still in search of the refreshing, fruity brews of summer, Lager Heads also just tapped Freedom of Peach gose at the brewery tasting room in Medina. Learn more at facebook.com/lagerheadsbrewingcompany
Akron can look forward to some new craft beers next year: Eighty-Three Brewery will open in one of the historic buildings on Akron's East End that was part of Goodyear Tire & Rubber's former world headquarters. Brewer Chris Surak – a graphic designer by trade – hopes to bring his creative approach to the art and science of brewing. Check out Rick Armon's full write-up of Eighty-Three Brewery on ohio.com
Hop Tree Brewing will host a Karma Keg event at their brewery in Hudson on Thursday, August 16, sponsored by Cutler Real Estate. Patrons of the event suggest a charity and make a donation to be able to drink from the keg. At the end of the evening, one patron's name is drawn and all the evening's donations are given to the winner's charity. For event info, visit facebook.com/hoptreebrewing
Moeller Brew Barn announced this week that they will open a second location in Troy next spring. The new brewery, taproom and restaurant will occupy a historic church built in 1919. The plans for a second location coincide with an expansion already underway at the original brewery in Maria Stein. That project will add 3,200 square feet of production space onto the existing building. Lisa Grigsby has further coverage at mostmetro.com
In the summer of 2014, toxic bacteria in Lake Erie forced some 400,000 people in the Toledo area to stop bathing in or drinking their tap water. Farm runoff and other factors have led to toxic cyanobacteria blooms in western Lake Erie every summer since then. Michigan Radio will bring their Issues & Ale series to Black Cloister Brewing Company in Toledo to talk about the safety of Lake Erie's water. Register to attend the discussion at michiganradio.org
Cheers to a year of beer from Y Bridge Brewing in Zanesville! The brewery recently received their A1A permit, which allows the sale of wine, liquor, and beers made by other breweries. The anniversary celebration kicks off today at 3 p.m. with a re-release tapping of Y Bridge's Downtown Holiday Brown ale. Details available at facebook.com/YBridgeBrewing
Double Edge Brewing, Outerbelt Brewing and Combustion Brewery have teamed up to make a session NEIPA for this year's Lancaster Brewfest. You can get your first few pints of it (because it's a session beer, you know) at Combustion's Pickerington taproom on Tuesday, August 14. For details, visit facebook.com/combustionbrewing
The hottest ticket in Cincinnati is for Listermann Brewing Company's 10 year anniversary party on Friday, September 7. The brewery is curating a beer list of over 20 distinguished breweries from around the state and country, including many beers that have never been poured in Ohio. Tickets include 30 four ounce samples and complimentary food, plus first access to bottle sales of Listermann's anniversary brew. Get tickets now at cincyticket.com/listermann10
Feeling Good, the first release in a new series of hazy IPAs from Rhinegest, is canned and available at their Over-the-Rhine taproom. Born out of a collaborative endeavor with Castle Island Brewing Co. in Norwood, Massachusetts, Feeling Good is festooned with apricot, mango, and citrus notes. Available in four-packs of 16 ounce cans starting today at 3 p.m. For more details about Feeling Good and the new Hazy series, check out the blog at rhinegeist.com⠀
Christian Moerlein's Das Über Fest Lager is back for the approaching Oktoberfest season. This Wiesn Lager is the same style sold at the Munich Oktoberfest; lighter in color and body than a Märzen, it has some bready sweetness with a crisp, clean finish. Celebrate all things Oktoberfest at the Christian Moerlein Malt House Taproom on Saturday, August 11. Details at facebook.com/christianmoerlein
Ale-O-Ween, the Ohio Craft Brewers Association's Halloween-themed beer festival, will again haunt the Steam Plant in Dayton on Saturday, October 20. This year, Ale-O-Ween is taking over the outdoor space around the venue and will boast 40 Ohio craft breweries at the event. Ohio On Tap users in attendance will receive five bonus passport stamps redeemable for prizes. Tickets go on sale Monday, August 20: purchase on the first day and save $10 each! For tickets and a list of participating breweries, visit ohiocraftbeer.org/ale-o-ween
Yellow Springs Brewery is adding more tours of their brew house. Tours will be given each Saturday and Sunday at 5 p.m. for $10 per-person. Your tour ticket includes an exclusive look into the brewery, a beer or soda of your choice and a pint glass to take home. To purchase tour tickets, go to yellowspringsbrewery.com/brewery/tours/
Ohio Weekly Roundup 6.29.2018 | {
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\section{Introduction}
\label{sec:intro}
Acute ischaemic stroke is the most common type of stroke and a leading cause of disability and mortality worldwide \cite{neethi2022stroke}. It is a condition caused by the formation of clots, following interruption of blood flow to the brain. If the blockage is not resolved, the extent of dead tissue increases and the irreversible ischaemic core expands over time. As Saver \cite{saver2006time} stated, "Time is brain" for stroke diagnosis and treatment, and it is essential to carry out the appropriate treatment in a timely manner. Although thrombectomy is the most effective treatment for ischaemic stroke cases, there is a risk of brain haemorrhage and death. Therefore, determining if a patient just admitted can benefit from mechanical thrombectomy leading to a good functional outcome, is an important step towards reducing risk and improving the quality of life for stroke patients.
Methods for automatic outcome prediction of stroke treatment have been proposed using logistic regression \cite{venema2017selection,ramos2020predicting}, random forests \cite{vanOs2018,Nawabi2021}, support vector machines \cite{ramos2020predicting,hofmeister2020clot}, and recently, convolutional neural networks (CNNs) \cite{hilbert2019data,Bacchi2019DeepStudy,SAMAK2022102089}. Some use clinical records \cite{venema2017selection,vanOs2018,ramos2020predicting}, imaging information \cite{hilbert2019data,hofmeister2020clot,Nawabi2021}, or a combination of both \cite{Bacchi2019DeepStudy,samak2020prediction,kappelhof2021evolutionary}. The CNN-based models have been applied to various imaging modalities, \eg magnetic resonance imaging (MRI), NCCT and CT angiography (CTA). While such deep learning models perform well in medical image analysis, 3D CNN models that exploit 3D brain volumes require numerous parameters and computational resources. Furthermore, they cannot learn long-range relationships due to their limited receptive field. In contrast, more recently, transformers have achieved outstanding results in various applications thanks to their big data and model size scalability and better longer-range attention-based modelling capability \cite{dosovitskiy2020image,khan2021transformers}. However, pure transformer-based methods have not been widely applied in medical image classification due to their limited performance on small datasets \cite{jang2022m3t}.
\begin{figure}[t]
\centering
\includegraphics[width=\linewidth]{intro_net.png}
\caption{TranSOP predicts functional outcome of ischaemic stroke treatment leveraging only the baseline NCCT scan and clinical records available on hospital admission.}
\label{FIG:intro_fig}
\end{figure}
In this paper, we introduce TranSOP, a transformer-based multimodal architecture to predict functional outcomes of ischaemic stroke patients 90 days after treatment (see Fig \ref{FIG:intro_fig}). We combine clinical metadata (\eg gender, age, hypertension, glucose level) and 3D NCCT obtained at the point of hospital admission for 500 ischaemic stroke patients. We also explore different strategies for this multimodal fusion and conduct extensive experiments on various architectures, including ViT, ViT with CNN, pre-trained ViT (from DeiT \cite{touvron2021training}) and Swin transformer (SwinT) \cite{liu2021swin} in our TranSOP model.
\begin{figure*}[t]
\centering
\includegraphics[width=\textwidth]{transformer_network7.png}
\caption{Overview of our proposed transformer-based multimodal architecture, TranSOP. PE: positional encoding, CLS: a token/vector that represents the input volume for classification, MHSA: Multi-head self-attention, MLP: multi-layer perceptron, FC: fully connected layer.}
\label{figc6:trans_modal}
\end{figure*}
\section{Related Works}
\label{sec:related}
{There are only a few studies that have employed CNN-based multimodal networks to predict the functional outcome of stroke treatments, \eg for thrombolysis \cite{Bacchi2019DeepStudy} and for thrombectomy \cite{samak2020prediction,SAMAK2022102089}. Bacchi \etal \cite{Bacchi2019DeepStudy} applied a CNN model to 3D NCCT images and clinical records of patients who underwent thrombolysis treatment. {Samak \etal \cite{samak2020prediction} also proposed a multimodal CNN architecture with channel-wise and spatial attentional blocks to predict dichotomised mRS scores from baseline 3D NCCT scans and clinical records of MR CLEAN \cite{fransen2014mr} dataset}. {Further, in \cite{SAMAK2022102089}, Samak \etal }additionally incorporate 1-week follow-up scans during their model training to encode stroke changes over time for better mRS score prediction.}
Transformers have shown significant success in natural language processing, \eg machine translation \cite{vaswani2017attention}, and computer vision, \eg medical imaging tasks \cite{dai2021transmed,hatamizadeh2022swin}. They facilitate a mechanism of self-attention that can model the long-range dependency of sequences and focus on important features. Dosovitskiy \etal \cite{dosovitskiy2020image} proposed the first pure vision transformer (ViT), applied directly to sequences of image patches for image classification. ViTs have obtained comparable and even better results in some tasks than CNNs, \eg for object detection \cite{carion2020end,wang2022fpdetr}.
Since its introduction ViT has been deployed in medical image segmentation using different imaging modalities. UNETR \cite{hatamizadeh2022unetr} adapts the commonly deployed and successful U-Net architecture \cite{cciccek20163d}, by replacing its convolutional encoder with a transformer encoder and modifying its convolutional decoder based on the output of the transformer encoder for image segmentation. Similarly, other studies \cite{hatamizadeh2022unetformer,hatamizadeh2022swin,li2022transiam} also replace the convolutional encoder with a transformer encoder, while some integrate the transformer encoder into the bottleneck of a U-Net-like model \cite{luo2021ucatr,wang2021transbts,ji2021multi,wang2022metrans} or use hybrid blocks that combine the convolutional and transformer layers \cite{gao2021utnet,liu2022transfusion}. Such works have been applied to NCCT \cite{luo2021ucatr}, MRI \cite{wang2021transbts,gao2021utnet,hatamizadeh2022swin,liu2022transfusion} and microscope \cite{ji2021multi} images. In another recent work, Amador \etal \cite{amador2022hybrid} propose a hybrid model that performs segmentation of the final lesion outcome {of ischaemic stroke} from {baseline} spatio-temporal CT perfusion (CTP) images using a transformer encoder embedded in the U-Net bottleneck.
Although most transformer-based models in medical image analysis are in the {\it segmentation} domain, there are some studies that have employed them on medical image {\it classification}, \eg for COVID-19 \cite{mondal2021xvitcos,park2022multi}, retinal disease \cite{wu2021vision}, cell analysis \cite{wang2021transpath,gao2021instance}, brain tumour \cite{dai2021transmed,jun2021medical}, Alzheimer's disease \cite{li2022trans,jang2022m3t} classification and age estimation \cite{jun2021medical,he2021global}. These methods are based on a pure transformer \cite{mondal2021xvitcos,yu2021mil,matsoukas2021time,gheflati2022vision,bhattacharya2022radiotransformer} or a hybrid model that uses ResNet \cite{dai2021transmed,li2022trans,wu2021vision}, DenseNet \cite{park2022multi} or a CNN module \cite{wang2021transpath,sriram2021covid,gao2021instance,he2021global,jang2022m3t} followed by a transformer encoder on 2D imaging modalities like X-Rays \cite{bhattacharya2022radiotransformer,jacenkow2022indication}, microscope images \cite{wang2021transpath,gao2021instance} and 3D MRI volumes \cite{dai2021transmed,li2022trans,jang2022m3t,jun2021medical}. To the best of our knowledge, there are no studies using the transformer in 3D NCCT classification and prediction of functional stroke outcomes from unimodal or multimodal data.
\section{Proposed Method}
An overview of the proposed architecture, TranSOP, is shown in Fig. \ref{figc6:trans_modal}, which includes a transformer encoder and a multimodal fusion module to predict mRS scores.
Transformers can process 1D input sequences, as originally used in the NLP domain where each word is embedded in a 1D vector as a token. Similarly, we split a 3D NCCT volume, $X^{ncct}_i \in \mathbb{R}^{1 \times D \times W \times H}$, into 1D vectors via patch embedding where $D$, $W$, and $H$ are depth, width and height, and a volume is divided into non-overlapping patches of size $P^3$, which generate a sequence of 1D patch vectors of length $L=[\frac{D}{P}] \times [\frac{W}{P}] \times [\frac{H}{P}]$.
We use a convolutional layer to project each patch into a $K$ dimensional embedding space \cite{touvron2021training,hatamizadeh2022swin}. We add a learnable parameter $[CLS] \in \mathbb{R}^{1 \times K}$, to the patch embedding sequence to represent the entire volume for classification. In addition, a learnable positional encoding, ($PE \in \mathbb{R}^{(L+1) \times K}$) is added to the sequences, so that the spatial information of the patches can be preserved (see Fig. \ref{figc6:trans_modal}). Next, a series of transformer blocks, each including a normalisation layer followed by multi-head self-attention (MHSA), a normalisation layer, and a multi-layer perceptron (MLP) head are utilised in the {transformer encoder}. Then, an MLP head is applied to the classification token to extract NCCT volume features $z^{ncct}$ for the fusion process. Clinical metadata features $z^{clinic}$ are computed by a fully connected layer (FC) (orange box in Fig. \ref{figc6:trans_modal}).
{In the multimodal fusion module, a stack of two FCs with a dropout layer in-between prepare the input scan's $z^{ncct}$ and $z^{clinic}$ for fusion (see right box in Fig. \ref{figc6:trans_modal}).} We use two methods for the fusion of these image volume and clinical features, (i) concatenation where both features are joined to make a larger 1D vector and (ii) addition where both features are added element-wise with each feature vector multiplied by a learnable weight. Finally, another stack of FC, Dropout, and FC layers is applied to the fused features before being passed to a \textit{Softmax} layer for final predictions. These predictions are dichotomised mRS scores, where mRS $\leq$ 2 indicates a good outcome and mRS $>$ 2 expresses a bad outcome. Note that, dropout layers are deactivated during inference.
\section{Experiments \& Results}
\label{sec:exp}
{\bfseries Dataset --} We used the MR CLEAN Trial dataset\footnote{https://www.mrclean-trial.org/home.html}, collected from a multi-centre study, which is one of the most comprehensive datasets of patients who underwent ischaemic stroke treatment.
Five hundred patients (233 assigned to mechanical thrombectomy and 267 to usual care) were treated in 16 medical centres in the Netherlands. We refer the reader to the MR CLEAN study protocol \cite{Berkhemer2015AStroke,fransen2014mr} for more detailed information on the dataset.
Through pre-processing, some of the apparent variations due to various acquisition protocols at different clinical centres were reduced to allow our model to deal with more similar standard input. First, all scans were re-sampled to the same voxel size of 3x1x1$mm^3$, followed by clipping the intensity range of 0-80HU. The skull structure was then removed in the NCCT scans and the volumes were cropped to $32\times192\times128$ from the centre.
Data augmentations, such as horizontal/vertical flips and Gaussian noise, were applied to increase the variation and number of input samples to help improve the robustness of the network. Finally, the voxels of the NCCT scans were normalised to zero mean and one standard deviation.
{\bfseries Implementation Details --} We split the dataset into three subsets, training (70\%, 350 patients), validation (15\%, 75 patients) and testing (15\%, 75 patients). The proposed model was trained for 500 epochs using an Adam optimiser with a weight decay of 0.0001, a learning rate of 0.0003, and a batch size of 24. A cosine learning rate scheduler was used. The experiments were implemented in PyTorch and MONAI \cite{monai2020} on a single NVIDIA P100 16GB GPU.
{\bfseries Details of Experiments --} We evaluated the performance of our proposed approach against two existing methods and various transformer architectures that also operate on 3D NCCT volumes and predict the functional outcome of stroke treatment. The methods of Bacchi \etal \cite{Bacchi2019DeepStudy} and {Samak \etal \cite{samak2020prediction}} {which both use imaging and clinical information,} were re-trained on the registered MR CLEAN dataset and our data split from scratch. {Although, the {FeMA \cite{SAMAK2022102089}} model performs a similar task, it additionally uses 1-week follow-up scans that contain information on stroke changes after treatment during model training. Hence, in the interest of direct comparability, we do not include that work in the present evaluation.}
{We also evaluated our TranSOP approach using different transformer architectures for its encoder part. These are referred to as TranSOP$_{ViT}$, TranSOP$_{DeiT}$, TranSOP$_{ConViT}$ and TranSOP$_{SwinT}$. TranSOP$_{ViT}$ uses the ViT network and is trained from scratch, TranSOP$_{DeiT}$ utilises the ImageNet pre-trained DeiT model to demonstrate the effect of transfer learning, and TranSOP$_{ConViT}$ uses the first three layers of convolutional blocks before the input is fed into the ViT model to explore the performance of a hybrid model. These three models have the same ViT network which consist of 12 layers of transformer blocks, 12 heads, a hidden MLP feature size of 768 and 3072. In TranSOP$_{SwinT}$, four stages each consisting of two Swin transformer blocks and $N$ MHSA heads, where $N=\{3, 6, 12, 24\}$ for each stage respectively, were used. The {ClinicDNN} model only consumed clinical information to show the expected benefit from imaging information.} {Note, the multimodal fusion step is the same for all the models.}
\begin{table*}
\caption{Results of the models with and without clinical records. The best and second best results are shown in bold and underlined respectively. The second and third rows are convolutional-based models. CI is confidence interval.}
\label{tablec6:init_result_end2end2}
\centering
\scalebox{0.91}{
\begingroup
\renewcommand{\arraystretch}{1.}
\begin{tabular}{lcccclccc}
\toprule
& \multicolumn{3}{c}{\textbf{w/o Clinical Records}} & \phantom{a}& \multirow{2}{*}{\textbf{Fusion}} & \multicolumn{3}{c}{\textbf{with Clinical Records}} \\
\cmidrule{2-4} \cmidrule{7-9}
\textbf{Method} & \textbf{ACC {\scriptsize (95\% CI)}} & \textbf{F1-score {\scriptsize (95\% CI)}} & \textbf{AUC {\scriptsize (95\% CI)}} & & & \textbf{ACC {\scriptsize (95\% CI)}} & \textbf{F1-score {\scriptsize (95\% CI)}} & \textbf{AUC {\scriptsize (95\% CI)}} \\
\midrule
ClinicDNN\textsuperscript{\textasteriskcentered} &-&-&-&& -& 0.75 {\scriptsize (0.65-0.85)} & 0.44 {\scriptsize (0.19-0.64)} & 0.73 {\scriptsize (0.57-0.86)} \\
\midrule
\multirow{2}{*}{{Samak \etal \cite{samak2020prediction}}} &
\multirow{2}{*}{\underline{0.72} {\scriptsize (0.62-0.82)}} &
\multirow{2}{*}{{0.33} {\scriptsize (0.09-0.53)}} &
\multirow{2}{*}{{0.63} {\scriptsize (0.44-0.81) }} &
\multirow{2}{*}{} & concat&
\multirow{1}{*}{{0.77} {\scriptsize (0.66-0.87)}} &
\multirow{1}{*}{{0.47} {\scriptsize (0.18-0.67)}} &
\multirow{1}{*}{{0.78} {\scriptsize (0.63-0.91)}} \\
& & & &&add& \underline{0.79} {\scriptsize (0.69-0.89)} & {0.44} {\scriptsize (0.17-0.67)} & {0.71} {\scriptsize (0.51-0.88)} \\
\midrule
\multirow{2}{*}{{Bacchi \etal \cite{Bacchi2019DeepStudy}}} &
\multirow{2}{*}{\textbf{0.75} {\scriptsize (0.65-0.85)}} &
\multirow{2}{*}{{0.40} {\scriptsize (0.16-0.60)}} &
\multirow{2}{*}{\underline{0.66} {\scriptsize (0.48-0.80)}} &
\multirow{2}{*}{} & concat&
\multirow{1}{*}{{0.73} {\scriptsize (0.62-0.83)}} &
\multirow{1}{*}{{0.51} {\scriptsize (0.29-0.68)}} &
\multirow{1}{*}{{0.78} {\scriptsize (0.62-0.90)}} \\
& & & &&add& {0.73} {\scriptsize (0.62-0.83)} & {0.51} {\scriptsize (0.29-0.68)} & {0.78} {\scriptsize (0.62-0.90)} \\
\midrule
\multirow{2}{*}{{TranSOP$_{ConViT}$}} &
\multirow{2}{*}{{0.58} {\scriptsize (0.46-0.69)}} &
\multirow{2}{*}{{0.40} {\scriptsize (0.21-0.56)}} &
\multirow{2}{*}{\textbf{0.67} {\scriptsize (0.46-0.85)}} &
\multirow{2}{*}{} & concat&
\multirow{1}{*}{{0.77} {\scriptsize (0.68-0.87)}} &
\multirow{1}{*}{\underline{0.58} {\scriptsize (0.36-0.74)}} &
\multirow{1}{*}{{0.83} {\scriptsize (0.72-0.93)}} \\
& & & &&add& {0.77} {\scriptsize (0.68-0.87)} & \underline{0.58} {\scriptsize (0.36-0.74)} & {0.82} {\scriptsize (0.71-0.92)} \\
\midrule
\multirow{2}{*}{{TranSOP$_{DeiT}$}} &
\multirow{2}{*}{{0.58} {\scriptsize (0.46-0.69)}} &
\multirow{2}{*}{{0.40} {\scriptsize (0.21-0.56)}} &
\multirow{2}{*}{{0.63} {\scriptsize (0.44-0.80)}} &
\multirow{2}{*}{} & concat&
\multirow{1}{*}{{0.77} {\scriptsize (0.68-0.86)}} &
\multirow{1}{*}{{0.53} {\scriptsize 0.30-0.71)}} &
\multirow{1}{*}{{0.82} {\scriptsize (0.68-0.93)}} \\
& & & &&add& \underline{0.79} {\scriptsize (0.69-0.89)} & {0.52} {\scriptsize (0.27-0.71)} & \underline{0.84} {\scriptsize (0.71-0.94)} \\
\midrule
\multirow{2}{*}{{TranSOP$_{ViT}$}} &
\multirow{2}{*}{{0.58} {\scriptsize (0.46-0.69)}} &
\multirow{2}{*}{{0.40} {\scriptsize (0.21-0.56)}} &
\multirow{2}{*}{{0.60} {\scriptsize (0.40-0.78)}} &
\multirow{2}{*}{} & concat&
\multirow{1}{*}{\textbf{0.80} {\scriptsize (0.70-0.89)}} &
\multirow{1}{*}{{0.53} {\scriptsize (0.28-0.74)}} &
\multirow{1}{*}{\underline{0.84} {\scriptsize (0.72-0.94)}} \\
& & & &&add& \textbf{0.80} {\scriptsize (0.70-0.89)} & \textbf{0.59} {\scriptsize (0.35-0.76)} & {0.83} {\scriptsize (0.71-0.93)} \\
\midrule
\multirow{2}{*}{{TranSOP$_{SwinT}$}} &
\multirow{2}{*}{{0.58} {\scriptsize (0.46-0.69)}} &
\multirow{2}{*}{{0.40} {\scriptsize (0.21-0.56)}} &
\multirow{2}{*}{{0.64} {\scriptsize (0.44-0.82) }} &
\multirow{2}{*}{} & concat&
\multirow{1}{*}{{0.76} {\scriptsize (0.66-0.86)}} &
\multirow{1}{*}{{0.54} {\scriptsize (0.32-0.71)}} &
\multirow{1}{*}{{0.83} {\scriptsize (0.71-0.93)}} \\
& & & &&add& \underline{0.79} {\scriptsize (0.69-0.89)} & {0.55} {\scriptsize (0.31-0.73)}& \textbf{0.85} {\scriptsize (0.75-0.94)} \\
\bottomrule
\multicolumn{9}{l}{{\textsuperscript{\textasteriskcentered} A method that uses only clinical metadata information.}} \\
\end{tabular}
\endgroup
}
\end{table*}
We evaluated the classification performance of the models with three commonly used metrics, Accuracy, {\it F1-score} and Area Under ROC Curve (AUC).
{Table \ref{tablec6:init_result_end2end2} reports the evaluations of the transformer-based and convolution-based networks, along with confidence intervals, for two fusion methods. Broadly, the CNN-based state of the art works \cite{Bacchi2019DeepStudy,samak2020prediction} outperformed the transformer methods when only imaging information was used, for example, \cite{Bacchi2019DeepStudy} and \cite{samak2020prediction} performed best and second best in accuracy at 0.75 and 0.72 respectively. On the other hand, transformer-based methods exceeded Bacchi \etal \cite{Bacchi2019DeepStudy} and Samak \etal \cite{samak2020prediction} when clinical records were included for multimodal analysis, with the best result obtained by TranSOP$_{SwinT}$ at 0.85 AUC. These variations in performance by the transformer could be attributed to both the transformer's appetite for larger datasets (see \cite{khan2021transformers}), and its already established superiority in handling 1D natural language data.} {As TranSOP$_{SwinT}$ achieved the best AUC score, and it is more efficient thanks to its hierarchical architecture and shifted windowing, it can be a more preferable approach.}
{The results on the use of fusion methods (concat and addition) are inconclusive and further investigation on more efficient fusion methods is necessary.}
\section{Conclusion}
\label{sec:conclusion}
In this work, we investigated the performance of various networks in predicting the functional outcome of ischaemic stroke treatment based on 3D NCCT scans and clinical information, such as age, sex, and demographic data from the patient's medical history records. Transformer models outperformed convolutional architectures in multimodal settings. This suggests that transformer models, although not performing as well on only imaging data, can learn better complementary imaging information when combined with clinical metadata. In future work, we plan to investigate and explore a data-efficient transformer model for small image datasets. In addition, we would like to extend the proposed architecture to use follow-up scans, such as used in the FeMA \cite{SAMAK2022102089} method during model training.
\section{Acknowledgments}
\label{sec:acknowledgments}
The authors would like to thank the Principal Investigators of the MR CLEAN trial: Profs Aad van der Lugt, Diederik W.J. Dippel, Charles B.L.M. Majoie, Yvo B. W.E.M. Roos, Wim H. van Zwam and Robert J. van Oostenbrugge for providing the data.
Z.A. Samak is funded by the Ministry of Education (1416/YLSY), the Republic of Türkiye.
{\small
\bibliographystyle{IEEEbib}
| {
"redpajama_set_name": "RedPajamaArXiv"
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Spotblue.com is delighted to offer this fantastic range of properties which are located within a grand scale project opposite the marina and the Marmara Sea in Beylikduzu between Buyukcekmece and Esenyurt.
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Two bedroom apartments, from 72 square metres, prices from 850,000 TL.
One bedroom apartments, from 50 square metres, prices from 580,000 TL.
Three bedroom apartments, from 124 square metres, prices from 1,299,000 TL.
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Four bedroom semi detached villas, from 454 square metres, prices from 1,022,000 USD.
Five detached bedroom villas, from 572 square metres, prices from 1,585,000 USD.
The estimated completion date for this town project is 2020. | {
"redpajama_set_name": "RedPajamaC4"
} | 1,498 |
{"url":"https:\/\/oj.uz\/problem\/view\/POI11_met","text":"## View problem - Meteors (POI11_met)\n\nTime limit Memory limit # of submissions # of accepted Ratio\n6000 ms 64 MiB 57 11 19.3%\n\nByteotian Interstellar Union (BIU) has recently discovered a new planet in a nearby galaxy. The planet is unsuitable for colonisation due to strange meteor showers, which on the other hand make it an exceptionally interesting object of study.\n\nThe member states of BIU have already placed space stations close to the planet's orbit. The stations' goal is to take samples of the rocks flying by. The BIU Commission has partitioned the orbit into $m$ sectors, numbered from $1$ to $m$, where the sectors $1$ and $m$ are adjacent. In each sector there is a single space station, belonging to one of the $n$ member states.\n\nEach state has declared a number of meteor samples it intends to gather before the mission ends. Your task is to determine, for each state, when it can stop taking samples, based on the meteor shower predictions for the years to come.\n\n### Input\n\nThe first line of the standard input gives two integers, $n$ and $m$ ($1 \\le n,m \\le 300\\,000$), separated by a single space, that denote, respectively, the number of BIU member states and the number of sectors the orbit has been partitioned into.\n\nIn the second line there are $m$ integers $o_i$ ($1 \\le o_i \\le n$), separated by single spaces, that denote the states owning stations in successive sectors.\n\nIn the third line there are $n$ integers $p_i$ ($1 \\le p_i \\le 10^9$), separated by single spaces, that denote the numbers of meteor samples that the successive states intend to gather.\n\nIn the fourth line there is a single integer $k$ ($1 \\le k \\le 300\\,000$) that denotes the number of meteor showers predictions. The following $k$ lines specify the (predicted) meteor showers chronologically. The $i$-th of these lines holds three integers $l_i, r_i, a_i$ (separated by single spaces), which denote that a meteor shower is expected in sectors $l_i,l_{i+1},\\ldots,r_i$ (if $l_i \\le r_i$) or sectors $l_i,l_{i+1},\\ldots,m,1,\\ldots,r_i$ (if $l_i > r_i$), which should provide each station in those sectors with $a_i$ meteor samples ($1 \\le a_i \\le 10^9$).\n\nIn tests worth at least 20% of the points it additionally holds that $n,m,k \\le 1\\,000$.\n\n### Output\n\nYour program should print $n$ lines on the standard output. The $i$-th of them should contain a single integer $w_i$, denoting the number of shower after which the stations belonging to the $i$-th state are expected to gather at least $p_i$ samples, or the word NIE (Polish for no) if that state is not expected to gather enough samples in the foreseeable future.\n\n### Example\n\nFor the input data:\n\n3 5\n1 3 2 1 3\n10 5 7\n3\n4 2 4\n1 3 1\n3 5 2\n\n\nthe correct result is:\n\n3\nNIE\n1\n\n\nTask authors: Pawe\u0142 Mechlinski and Jakub Pachocki.\n\nEdit problem statement on github","date":"2020-02-29 05:45:10","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 1, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.7741764783859253, \"perplexity\": 1197.448819745011}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2020-10\/segments\/1581875148671.99\/warc\/CC-MAIN-20200229053151-20200229083151-00340.warc.gz\"}"} | null | null |
{"url":"https:\/\/hilbertthm90.wordpress.com\/2013\/07\/10\/p-divisible-groups-revisited-1\/","text":"# p-Divisible Groups Revisited\u00a01\n\nI\u2019ve posted about ${p}$-divisible groups all over the place over the past few years (see: here, here, and here). I\u2019ll just do a quick recap here on the \u201cclassical setting\u201d to remind you of what we know so far. This will kick-start a series on some more subtle aspects I\u2019d like to discuss which are kind of scary at first.\n\nSuppose ${G}$ is a ${p}$-divisible group over ${k}$, a perfect field of characteristic ${p>0}$. We can be extremely explicit in classifying all such objects. Recall that ${G}$ is just an injective limit of group schemes ${G=\\varinjlim G_\\nu}$ where we have an exact sequence ${0\\rightarrow G_\\nu \\rightarrow G_{\\nu+1}\\stackrel{p^\\nu}{\\rightarrow} G_{\\nu+1}}$ and there is a fixed integer ${h}$ such that group schemes ${G_{\\nu}}$ are finite of rank ${p^{\\nu h}}$.\n\nAs a corollary to the standard connected-\u00e9tale sequence for group schemes we get a canonical decomposition called the connected-\u00e9tale sequence:\n\n$\\displaystyle 0\\rightarrow G^0 \\rightarrow G \\rightarrow G^{et} \\rightarrow 0$\n\nwhere ${G^0}$ is connected and ${G^{et}}$ is \u00e9tale. Since ${k}$ was assumed to be perfect, this sequence actually splits. Thus ${G}$ is a semi-direct product of an \u00e9tale ${p}$-divisible group and a connected ${p}$-divisible group. If you\u2019ve seen the theory for finite, flat group schemes, then you\u2019ll know that we usually decompose these two categories even further so that we get a piece that is connected with connected dual, connected with \u00e9tale dual, \u00e9tale with connected dual, and \u00e9tale with \u00e9tale dual.\n\nThe standard examples to keep in mind for these four categories are ${\\alpha_p}$, ${\\mu_p}$, ${\\mathbb{Z}\/p}$, and ${\\mathbb{Z}\/\\ell}$ for ${\\ell\\neq p}$ respectively. When we restrict ourselves to ${p}$-divisible groups the last category can\u2019t appear in the decomposition of ${G_\\nu}$ (since \u00e9tale things are dimension 0, if something and its dual are both \u00e9tale, then it would have to have height 0). I think it is not a priori clear, but the four category decomposition is a direct sum decomposition, and hence in this case we get that ${G\\simeq G^0\\oplus G^{et}}$ giving us a really clear idea of what these things look like.\n\nAs usual we can describe \u00e9tale group schemes in a nice way because they are just constant after base change. Thus the functor ${G^{et}\\mapsto G^{et}(\\overline{k})}$ is an equivalence of categories between \u00e9tale ${p}$-divisible groups and the category of inverse systems of ${Gal(\\overline{k}\/k)}$-sets of order ${p^{\\nu h}}$. Thus, after sufficient base change, we get an abstract isomorphism with the constant group scheme ${\\prod \\mathbb{Q}_p\/\\mathbb{Z}_p}$ for some product (for the ${p}$-divisible group case it will be a finite direct sum).\n\nAll we have left now is to describe the possibilities for ${G^0}$, but this is a classical result as well. There is an equivalence of categories between the category of divisible, commutative, formal Lie groups and connected ${p}$-divisible groups given simply by taking the colimit of the ${p^n}$-torsion ${A\\mapsto \\varinjlim A[p^n]}$. The canonical example to keep in mind is ${\\varinjlim \\mathbb{G}_m[p^n]=\\mu_{p^\\infty}}$. This is connected only because in characteristic ${p}$ we have ${(x^p-1)=(x-1)^p}$, so ${\\mu_{p^n}=Spec(k[x]\/(x-1)^{p^n})}$. In any other characteristic this group scheme would be \u00e9tale and totally disconnected.\n\nThis brings us to the first subtlety which can cause a lot of confusion because of the abuse of notation. A few times ago we talked about the fact that ${E[p]}$ for an elliptic curve was either ${\\mathbb{Z}\/p}$ or ${0}$ depending on whether or not it was ordinary or supersingular (respectively). It is dangerous to write this, because here we mean ${E}$ as a group (really ${E(\\overline{k})}$) and ${E[p]}$ the ${p}$-torsion in this group.\n\nWhen talking about the ${p}$-divisible group ${E[p^\\infty]=\\varinjlim E[p^n]}$ we are referring to ${E\/k}$ as a group scheme and ${E[p^n]}$ as the (always!) non-trivial, finite, flat group scheme which is the kernel of the isogeny ${p^n: E\\rightarrow E}$. The first way kills off the infinitesimal part so that we are just left with some nice reduced thing, and that\u2019s why we can get ${0}$, because for a supersingular elliptic curve the group scheme ${E[p^n]}$ is purely infinitesimal, i.e. has trivial \u00e9tale part.\n\nRecall also that we pointed out that ${E[p]\\simeq \\mathbb{Z}\/p}$ for an ordinary elliptic curve by using some flat cohomology trick. But this trick is only telling us that the reduced group is cyclic of order ${p}$, but it does not tell us the scheme structure. In fact, in this case ${E[p^n]\\simeq \\mu_{p^n}\\oplus \\mathbb{Z}\/p^n}$ giving us ${E[p^\\infty]\\simeq \\mu_{p^\\infty}\\oplus \\mathbb{Q}_p\/\\mathbb{Z}_p}$. So this is a word of warning that when working these things out you need to be very careful that you understand whether or not you are figuring out the full group scheme structure or just reduced part. It can be hard to tell sometimes.\n\n### Author: hilbertthm90\n\nI write about math, philosophy, literature, music, science, computer science, gaming or whatever strikes my fancy that day.\n\n### One thought on \u201cp-Divisible Groups Revisited\u00a01\u201d\n\n1. Reblogged this on Observer.","date":"2015-07-31 19:24:09","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 58, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.8949058651924133, \"perplexity\": 193.30073813122686}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2015-32\/segments\/1438042988311.72\/warc\/CC-MAIN-20150728002308-00189-ip-10-236-191-2.ec2.internal.warc.gz\"}"} | null | null |
Q: Convert integer to week interval How one can convert integer to week interval?
CREATE TABLE integers( i integer);
INSERT INTO integers VALUES ('10');
Output would be table with one column indicating 10 weeks interval.
http://sqlfiddle.com/#!17/4b404/5/0
A: One take would be to create constant interval of 1 week and multiply it by integer.
I would prefer function to do it directly, but I am not aware of it.
SELECT interval '1 week' * i AS weeks_interval FROM integers;
A: Your solution is well accepted.
If you don't want to keep the "1" in the string you could write this instead
SELECT (i || 'week')::interval FROM intervals
demo: db<>fiddle
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 7,402 |
Q: Quoting variable in application call I'm trying to run an application that needs a parameter passed in single or double quotes:
bwa mem -R '@RG\tID:foo\tLB=foo\tPL=illumina\tPU=1234.1\tSM=bar' ...
I'm trying to run this in a script:
#!/usr/bin/bash
var=foo_bar
first=$(echo $var | cut -d '_' -f 1)
second=$(echo $var | cut -d '_' -f 2)
readgroup="@RG\tID:$first\tLB=HUM\tPL=illumina\tPU=1234.1\tSM=$second"
echo \'$readgroup\'
'@RG\tID:foo\tLB=HUM\tPL=illumina\tPU=1234.1\tSM=bar'
However, when I use this as a parameter in my application call:
var=foo_bar
first=$(echo $var | cut -d '_' -f 1)
second=$(echo $var | cut -d '_' -f 2)
readgroup="@RG\tID:$first\tLB=HUM\tPL=illumina\tPU=1234.1\tSM=$second"
bwa mem -R \'$readgroup\' ...
it's interpreted as
\'@RG\tID:HUM-7\tLB=HUM\tPL=illumina\tPU=1234.1\tSM=HUM-7\'
What am I doing wrong?
A: The application doesn't care about the quotes; they are only used to protect the string from the shell.
var=foo_bar
IFS=_ read first second <<< "$var"
readgroup="@RG\tID:$first\tLB=HUM\tPL=illumina\tPU=1234.1\tSM=$second"
bwa mem -R "$readgroup"
If the \t are actually supposed to be literal tab characters, consider using printf instead of an assignment statement.
printf -v readgroup '@RG\tID:%s\tLB=HUM\tPL=illumina\tPU=1234.1\tSM=%s' "$first" "$second"
A: Should be like:
readgroup="'@RG\tID:$first\tLB=HUM\tPL=illumina\tPU=1234.1\tSM=$second'" [with '' wrapping]
readgroup="@RG\tID:$first\tLB=HUM\tPL=illumina\tPU=1234.1\tSM=$second" [without]
Important thing is to use -e option for echo [ See man echo ]
echo -e $readgroup
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 9,363 |
Here's a quick digital sketch of what we're thinking for the Miami poster. The idea is to have a braves batter swinging a marlin instead of a bat, I think a clean and simple illustration style could be cool and then fill in the gaps with the type in various sizes/shapes. Just wanted to run this by y'all before we pushed any further on it. | {
"redpajama_set_name": "RedPajamaC4"
} | 9,528 |
By fundraising or making a donation you are enabling us to provide vital resources, information and advice that have the power to save lives and prevent accidents! You are also ensuring that we can lobby for legislative change and raise awareness of life-threatening issues.
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Thank you for your interest in fundraising for the Royal Society for the Prevention of Accidents (RoSPA). On the roads, at work, at home or at play, we've spent nearly 100 years working to make life safer. And, as a charity, we couldn't have done it without the help of those who share our vision of life, free from serious accidental injury. Together we can stop the millions of accidents that are waiting to happen! If you don't think you'll have time to organise your own fundraising event, you can still show your support by making a donation here! | {
"redpajama_set_name": "RedPajamaC4"
} | 2,213 |
class Recipe < ApplicationRecord
belongs_to :creator, class_name: "User"
has_many :ratings
has_many :ingredients, dependent: :destroy
accepts_nested_attributes_for :ingredients, :allow_destroy => true
def average_rating
ratings = self.ratings.to_a
if ratings.length > 0
value = 0
ratings.each do |rating|
value += rating.value
end
return value / ratings.length
else
return 0
end
end
def self.sort_by_rating_type (args) # will receive course_type in string "all" if you neglect the course_type
recipes_array = args[:array]
type = args[:type]
if recipes_array.length> 0
if type == "all"
recipes_array.sort_by {|x| x.average_rating}
else
return recipes_array.sort_by {|x| x.average_rating }.select!{|x| x.course_type == type}
end
else
return []
end
####return the whole list of recipe, need to limit the number or randomize as necessary
end
def self.search_method (args) # will receive the hash {search_args:"something" , type:"ingredients" }
recipes_array = args[:array]
type = args[:type]
search = args[:search_args]
if type == "name"
return recipes_array.sort_by {|x| x.average_rating}.select!{|x| x.name.include? search }
elsif type == "ingredients"
return recipes_array.sort_by {|x| x.average_rating}.select! {|x| x.has_ingredients?(search)}
elsif type == "description"
return recipes_array.sort_by {|x| x.average_rating}.select!{|x| x.description.include? search }
end
########### return the whole list of recipe
end
def has_ingredients? (ingredient)
self.ingredients.each do |i|
if i.item == ingredient
true
end
end
false
end
end
| {
"redpajama_set_name": "RedPajamaGithub"
} | 6,421 |
Chebzí (něm. Hollunder) je drobná vesnice (osada), která je částí obce Písečná. Nachází se 3 km jižně od Písečné, v údolí na horním toku potoka Chebzí pod stejnojmenným kopcem (549 m n. m.).
Historie
Chebzí vzniklo na frývaldovském panství patřícím vratislavským biskupům roku 1772 parcelací panských pastvin. Jeho jméno odkazuje na hojný výskyt černého bezu.
Při zavedení obecního zřízení v roce 1850 se Chebzí stalo osadou Širokého Brodu, avšak roku 1924 přešlo k Písečné.
Vývoj počtu obyvatel
Počet obyvatel Chebzí podle sčítání nebo jiných úředních záznamů:
V Chebzí je evidováno 11 adres : 7 čísel popisných (trvalé objekty) a 4 čísla evidenční (dočasné či rekreační objekty). Při sčítání lidu roku 2001 zde bylo napočteno 7 domů, z toho 6 trvale obydlených.
Zajímavosti
přírodní památka Chebzí, louky s výskytem řady zvláště chráněných druhů rostlin (zejména vstavačovitých)
okolí Chebzí patří do Chráněné krajinné oblasti Jeseníky
Fotogalerie
Reference
Externí odkazy
Mj. o Chebzí na stránkách Písečné
Vesnice v okrese Jeseník
Sídla ve Zlatohorské vrchovině
Písečná (okres Jeseník) | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 4,977 |
We are so happy that we had the opportunity to see the first show of Les Misérables School Edition because I want to encourage you all to see it, as well.
The Gateway puts on the most incredible performances with their Performing Arts School. All of the young actors in the show are INCREDIBLE! The entire crowd was on their feet at the end of the show cheering for what was a stellar 2 1/2 hour performance by amazing local talent.
My oldest daughter said to me , "Mommy I felt the emotion in this play and the singing gave me goosebumps." I felt those goosebumps , too. Not just from one actor but numerous ones. These kids can sing and act and you can tell they are passionate about what they are doing and have worked so hard.
My daughters love to sit and watch theatre productions but, I think this particular show is more suited for older kids, teens and adults.
Audiences can enjoy the show during various showtimes between now and Dec 9th, 2018 at The Gateway Playhouse in Bellport. | {
"redpajama_set_name": "RedPajamaC4"
} | 6,711 |
Garmin GPS is a navigation system.The working of GPS technology is amazing. This technology works any weather condition like rainy, winter, snow, summer etc. They work any climate, anywhere in the world, round the clock, their setup is absolutely free. | {
"redpajama_set_name": "RedPajamaC4"
} | 4,716 |
Q: Removing Flickr account from Photos app of Windows 8 I am using Windows 8 which comes with a "Photos" application capable of connecting to different cloud services like Flickr, SkyDrive, Facebook, etc. to display your photos.
I've entered my Flickr account information to this app and let it display access my account to display my photos, now I want to disconnect it. How can I do it?
A: You can simply revoke access from Windows 8 in your Flickr account settings:
*
*Go to settings of Flickr account.
*Click Sharing & Extending tab.
*Click Edit in the Account links section.
*You will see the list of the applications you've given access to your account.
One of them is Microsoft, http://aka.ms/flickr.
*Click Remove permission? next to Microsoft to revoke the access.
Alternatively you can go to http://aka.ms/flickr, clear the View your photos and videos from Flickr check box, and click Save.
This page is displayed in Russian for me, and I translated the titles, so the real interface text could slightly differ.
A: To disconnect flickr account from the photos application, one must follow the steps below.
*
*Open Photos Application
*Press Windows + C and select "Settings"
*Select "Options"
*Select "Options" next to Flickr item (Web page opens, profile.live.com)
*login to your live account if needed
*Remove flickr account int the page
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 1,880 |
No announcements have been posted yet for U10 Olmsted Falls Bulldogs.
Welcome to the U10 Olmsted Falls Bulldogs Website! Here is where you can post and find your team schedules, news, scores, photos, documents, and message boards. It should make it easier to know what's going on with the team.
Thank you for stopping by, We are working on a new and more user friendly web site also including a updated mobile app for your smartphone. Please hang on and excuse the dust , Please stop back often as we will be making new changes and up dates all the time. Our scheduled launch date is January 8th. With open early bird registration starting on January 15th. | {
"redpajama_set_name": "RedPajamaC4"
} | 4,698 |
Ebbe l'ordine dall'imperatore di deporre o uccidere papa Gregorio II, reo di non aver voluto pagare le tasse all'Impero. Inviò il cartulario Giordane, il subdiacono Giovanni Lurion, e l'ex duca di Roma Basilio ad assassinare il Papa, ma la congiura fu scoperta e i primi due vennero giustiziati, il terzo fu costretto a entrare in monastero. Visto il fallimento della congiura, l'esarca ordinò all'esercito dell'esarcato di marciare verso Roma, per deporre con la forza il Pontefice; grazie però all'alleanza con i longobardi di Spoleto e della Tuscia, l'esercito del ducato romano, favorevole al Papa, sconfisse l'esercito bizantino presso il ponte Salario.
Nel 727 l'Esarcato scoppiò in rivolta contro il divieto imperiale del culto delle immagini (iconoclastia). A Ravenna scoppiarono lotte tra una fazione filo-papale e una filo-bizantina, nella quale rimase vittima l'esarca Paolo, ucciso in circostanze ignote. Per vendicare il suo assassinio, l'imperatore bizantino Leone III inviò una spedizione punitiva contro Ravenna, ma il corpo di spedizione bizantino fu sconfitto dai ribelli ravennati. Il successore di Paolo, Eutichio, riuscì a riprendere il controllo di Ravenna, anche se ebbe inizialmente delle difficoltà a causa del mancato appoggio dell'esercito.
Secondo lo storico John Julius Norwich, Paolo e il primo leggendario doge di Venezia, Paolo Lucio Anafesto, sarebbero la stessa persona. Tra l'altro il magister militum di Paolo per coincidenza era chiamato Marcello, lo stesso nome del successore reputato del Paoluccio come doge, Marcello Tegalliano.
Note
Bibliografia
John Julius Norwich, A History of Venice, Alfred A. Knopf, New York, 1982
Ravegnani, I Bizantini in Italia, Mulino, Bologna, 2004. | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 9,198 |
\section{Introduction}
The study of the finite temperature phase transition in QCD with Wilson
fermions is much more complicated than in the staggered fermion formulation,
because of the absence of an order parameter due to explicit chiral symmetry
breaking. This means e.g. that the existence of a massless pion at finite
lattice spacing is not at all obvious.
In recent years a detailed picture of the finite temperature phase diagramm of
QCD with Wilson fermions has emerged \cite{Phdiag}. This picture is based on
the idea of spontaneous breakdown of parity and flavour symmetry \cite{Aoki-I}
and has been investigated analytically as well as numerically.
The features of this phase diagramm are
(i) the critical line $\kappa_c(\beta)$ defined by a vanishing pion screening
mass for finite temporal lattice size turns back towards strong coupling
forming a cusp,
(ii) the region bounded by the critical line represent a phase of spontaneously
broken parity and flavour symmetry,
(iii) the finite temperature phase transition line $\kappa_t(\beta)$ presumably
does not cross the critical line, but runs past it towards larger values of
the hopping parameter\cite{Aoki-II}.
From an analysis of the Gross-Neveu model in two dimensions, where three cusps
connected to doublers develop, one expects the critical line for QCD in four
dimensions to form five cusps moving towards weak coupling with increasing
temporal lattice size $N_\tau$. Simulations with the standard Wilson
formulation for quarks and gluons have shown, that at $N_\tau=1/(aT)=4$ the
tip of the cusp lies in the strong coupling regime at $\beta \approx 3.9$ and
moves only slowly towards weak coupling as $N_\tau$ is increased.
Since one
expects the same features to hold for a wider class of actions including the
clover action, which tends to reduce cutoff dependencies, a study of
the phase diagramm using improved actions is of practical as well as
theoretical importance\cite{improv}.
\section{Results}
We have conducted a simulation of 2 flavour QCD on an $8^3 \times 4$ lattice
using tree level Symanzik improved actions for quarks and gluons. In the gauge
sector this amounts to adding a $2 \times 1$-loop to the standard plaquette
action and for the fermions in adding the clover term with $c_{SW}=1$.
We have used
a Hybrid Monte Carlo algorithm with a timestep $\delta\tau=0.01$ and the number
of molecular dynamics steps $N_{MD}=20$, which so far amounts to rather short
trajectories. We simulated several $\kappa$ values
for each $\beta=3.00, 3.50, 3.75$ and $4.00$. We have measured the Polyakov
loop, the pion norm and the average number of iterations it takes to invert
the fermion matrix. Each observable is now discussed in detail.
\subsection{Locating the critical line}
Although not a physical observable in its own right, the average number of
iterations it takes to invert the fermion matrix is a very good indicator for
criticality.
The simulations where done in the following way. At each $\beta$-value we
started a simulation at $\kappa=0.12$ and used a thermalized configuration from
this run as a start configuration at a higher value of $\kappa$. We continued
to do so for higher and higher $\kappa$-values. At those $\beta$-values, where
we saw a drastic increase in the number of iterations, we started a simulation
at a much higher $\kappa$-value and continued towards smaller $\kappa$-values.
Our findings are summarized in {Fig.\,1}. At $\beta=3.00$ we were not
able to simulate the system for $0.1770<\kappa<0.1825$. At $\beta=3.50$ we
saw an initial increase of iterations, even after switching from minimal
residual to conjugate gradient, which usually decreased the number of
iterations. With increasing $\kappa$ the number of iterations decreased
again, only to increase again at fairly high $\kappa$-values. At $\beta=3.75$
and $4.00$ we saw a similar behaviour as at $\beta=3.50$ but not as pronounced.
We experimented with different inversion routines and our conclusions are, that
close to the critical line conjugate gradient is superiour to overrelaxed
minimal residual and BiCGstab1.
\begin{figure}[t]
\begin{center}
\epsfig{file=fig1.eps, width=7.75cm}
\end{center}
{\small Fig.\,1 Average number of iterations as a function of
$\kappa$ for $\beta=3.00$ (triangles), $\beta=3.50$ (circles),
$\beta=3.75$ (diamonds) and $\beta=4.00$ (boxes). The inverter used are
conjugate gradient (lines), overrelaxed minimal residual (dashed),
BiCGstab1 (dotted)}
\end{figure}
\subsection{Pion Norm}
Since it is not possible to reliably extract the pion screening mass on small
lattices, we use the pion norm instead to indicate the existence of a critical
line of vanishing pion screening mass.
The pion norm is the integrated pion correlator and is defined as
follows:
\bq
\Pi= \frac{1}{4 {N_{\sigma}}^3 {N_{\tau}}} \cdot {\rm Tr~}
\left[{\cal M}^{-1} \gamma_5 {\cal M}^{-1} \gamma_5\right]
\end{equation}
Here ${\cal M}$ is the fermion matrix on a particular gauge configuration.
Near the critical line the pion norm behaves as $\Pi \approx 1/m_{\pi}^2$,
hence a diverging pion norm indicates the existence of the critical line.
Our results are displayed in {Fig.\,2}. At $\beta=3.00$ we find a clear signal
for two critical lines close to $\kappa=0.1770$ and $0.1825$.
The difference in
$\kappa$ is already quite small, so we are near the tip of the cusp. At
$\beta=3.50$ the pion norm develops a small peak at $\kappa=0.1550$, which is
located where the crossover from the low temperature to the high temperature
phase starts (see section on Polyakov loop). No divergent behaviour is observed
and the critical line ceases to exist in this coupling region. The simulations
at $\beta=3.75$ and $4.00$ do not even find a peak for the pion norm, but
rather a smooth behaviour as a function of hopping parameter. We conclude, that
the critical line turns back towards strong coupling and develops a cusp
between $\beta=3.00$ and $3.50$.
\begin{figure}[t]
\begin{center}
\epsfig{file=fig2.eps, width=7.75cm}
\end{center}
{\small Fig.\,2 Pion norm as a function of $\kappa$ for $\beta=3.00$
(triangles), $\beta=3.50$ (circles), $\beta=3.75$ (diamonds) and
$\beta=4.00$ (boxes).}
\vspace{-1ex}
\end{figure}
\subsection{Polyakov Loop}
The Polyakov loop is defined as follows:
\bq
L = \frac{1}{{N_{\sigma}}^3} \sum_{\vec{x}} \frac{1}{N_c} {\rm Tr~} \prod_{\tau=1}^{{N_{\tau}}}
U_4(\vec{x},\tau)
\end{equation}
This observable is sensitive to the finite temperature phase transition
although it is no order parameter in the full theory. Our results are displayed
in {Fig.\,3}. At $\beta=3.00$ we find
a confined phase for $\kappa \le 0.1770$. For for $\kappa > 0.1825$, when one
approaches the critical line from above, the Polyakov loop decreases to
$|L| \approx 0.1$. This indicates that the system develops a low temperature
behaviour in the vicinity of the critical line. On the other hand we do
not see a sharp crossover, so we cannot conclude that one crosses the thermal
line as one lowers $\kappa$ towards $\kappa_c(\beta)$.
At $\beta=3.50$ the Polyakov loop displays a sharp crossover phenomenon, which
means that the system crosses the thermal line for
{$0.1550 < \kappa < 0.1600$}.
At $\beta=3.75$ and $4.00$ the system is in the high temperature phase down to
$\kappa=0.12$. This is not unexpected, since the finite temperature phase
transition in the quenched theory for our choice of action occurs at
$\beta_c=4.07$.
\begin{figure}[t]
\begin{center}
\epsfig{file=fig3.eps, width=7.75cm}
\end{center}
{\small Fig.\,3 Polyakov loop as a function of $\kappa$ for
$\beta=3.00$ (triangles), $\beta=3.50$ (circles), $\beta=3.75$ (diamonds)
and $\beta=4.00$ (boxes).}
\end{figure}
\section{Summary and Conclusions}
From measuring the pion norm, Polyakov loop and the average number of
iterations to invert the fermion matrix we conlude, that the finite temperature
phase structure observed with the standard Wilson formulation is preserved for
the tree level Symanzik improved formulation of quarks and gluons. We note that
the difference in $\beta$ between the location of the cusp and the quenched
$\beta_c$ is considerably reduced. What this means in terms of physical scales
remains to be investigated by measurements of the lattice spacing. After this
preparatory simulation on a small lattice, we will investigate the phase
diagramm on larger lattices including a precise measurement of the
pion screening mass and quark mass, the latter enabling us to study the chiral
condensate as well \cite{ChWI}.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 586 |
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I can not wait Sting ray cityisland Cruise in January. | {
"redpajama_set_name": "RedPajamaC4"
} | 7,388 |
{"url":"https:\/\/en.wikipedia.org\/wiki\/Happiness_economics","text":"# Happiness economics\n\nJump to: navigation, search\n\nThe economics of happiness or happiness economics is the quantitative and theoretical study of happiness, positive and negative affect, well-being, quality of life, life satisfaction and related concepts, typically combining economics with other fields such as psychology, health and sociology. It typically treats such happiness-related measures, rather than wealth, income or profit, as something to be maximized. The field has grown substantially since the late 20th century, for example by the development of methods, surveys and indices to measure happiness and related concepts.[1] Its findings have been described as a challenge to the economics profession.[2]\n\n## Subject classifications\n\nThe subject may be categorized in various ways, depending on specificity, intersection, and cross-classification. For example, within the Journal of Economic Literature classification codes, it has been categorized under:\n\n## Metrology\n\nGiven its very nature, reported happiness is subjective.[5] It is difficult to compare one person's happiness with another's.[1] It can be especially difficult to compare happiness across cultures.[1] However, many happiness economists believe they have solved this comparison problem. Cross-sections of large data samples across nations and time demonstrate consistent patterns in the determinants of happiness.[1]\n\nHappiness is typically measured using subjective measures \u2013 e.g. self-reported surveys \u2013 and\/or objective measures. One concern has always been the accuracy and reliability of people\u2019s responses to happiness surveys.[6] Objective measures such as lifespan, income, and education are often used as well as or instead of subjectively reported happiness, though this assumes that they generally produce happiness, which while plausible may not necessarily be the case. The terms quality of life or well-being are often used to encompass these more objective measures.\n\nSome scientists claim that happiness can be measured both subjectively and objectively by observing the joy center of the brain lit up with advanced imaging,[6] although this raises philosophical issues, for example about whether this can be treated as more reliable than reported subjective happiness.\n\nMicro-econometric happiness equations have the standard form: ${\\displaystyle W_{it}=\\alpha +\\beta {x_{it}}+\\epsilon _{it}}$.[1] In this equation ${\\displaystyle W}$ is the reported well-being of individual ${\\displaystyle i}$ at time ${\\displaystyle t}$, and ${\\displaystyle x}$ is a vector of known variables, which include socio-demographic and socioeconomic characteristics.[1]\n\n## Determinants\n\nTypically national financial measures, such as gross domestic product (GDP) and gross national product (GNP), have been used as a measure of successful policy. There is a significant association between GDP and happiness, with citizens in wealthier nations being happier than those in poorer nations.[citation needed] It has been argued that this relationship extends only to an average GDP per capita of about $15,000.[7] Conclusions in this are controversial.[8] Other economists have disputed the accuracy of these studies, finding a logarithmic correlation between GDP per capita and self-reported happiness extending without boundary.[9] It has also been noted that since life expectancy has continued to increase in nations wealthier than this, often partly attributed to economic growth, happy life years have continued to increase.[9][10] ### Individual income Historically, economists have said that well-being is a simple function of income. However, it has been found that once wealth reaches a subsistence level, its effectiveness as a generator of well-being is greatly diminished.[11] Happiness economists hope to change the way governments view well-being and how to most effectively govern and allocate resources given this paradox.[12] In 2010, Daniel Kahneman and Angus Deaton found that higher earners generally reported better life satisfaction, but people's day-to-day emotional well-being only rose with earnings until a threshold annual income of$75,000.[13]\n\nOther factors have been suggested as making people happier than money.[6] A short term course of psychological therapy is 32 times more cost effective at increasing happiness than simply increasing income.[14][15]\n\nScholars at the University of Virginia, University of British Columbia and Harvard University released a study in 2011 after examining numerous academic paper in response to an apparent contradiction: \u201cWhen asked to take stock of their lives, people with more money report being a good deal more satisfied. But when asked how happy they are at the moment, people with more money are barely different than those with less.\u201d Published in the Journal of Consumer Psychology, the study is entitled \u201cIf Money Doesn\u2019t Make You Happy, Then You Probably Aren\u2019t Spending It Right\u201d and included the following eight general recommendations:\n\n\u2022 Spend money on \u201cexperiences\u201d rather than goods.\n\u2022 Donate money to others, including charities, rather than spending it solely on oneself.\n\u2022 Spend small amounts of money on many small, temporary pleasures rather than less often on larger ones.\n\u2022 Don\u2019t spend money on \u201cextended warranties and other forms of overpriced insurance.\"\n\u2022 Adjust one's mindset to \u201cpay now, consume later,\u201d instead of \u201cconsume now, pay later.\u201d\n\u2022 Exercise circumspection about the day-to-day consequences of a purchase beforehand.\n\u2022 Rather than buying products that provide the \"best deal,\" make purchases based on what will facilitate well-being.\n\u2022 Seek out the opinions of other people who have prior experience of a product before purchasing it.[16]\n\nIn their \"Unhappy Cities\" working paper, published in July 2014 by the National Bureau of Economic Research (NBER), Edward Glaeser, Joshua Gottlieb and Oren Ziv examined the self-reported subjective well-being of people living in American metropolitan areas, particularly in relation to the notion that \"individuals make trade-offs among competing objectives, including but not limited to happiness.\" The researchers findings revealed that people living in metropolitan areas where lower levels of happiness are reported are receiving higher real wages, and they suggest in their conclusion that \"humans are quite understandably willing to sacrifice both happiness and life satisfaction if the price is right.\"[17]\n\n### Social security\n\nProfessor Ruut Veenhoven showed that social security payments do not seem to add to happiness. This may be due to the fact that non-self-earned income (e.g., from a lottery) does not add to happiness in general either. Happiness may be the mind's reward to a useful action. However, Johan Norberg of CIS, a free enterprise economy think tank, presents a hypothesis that as people who think that they themselves control their lives are more happy, paternalist institutions may decrease happiness.[18][19]\n\nAn alternative perspective focuses on the role of the welfare state as an institution that improves quality of life not only by increasing the extent to which basic human needs are met, but also by promoting greater control of one's life by limiting the degree to which individuals find themselves at the mercy of impersonal market forces that are indifferent to the fate of individuals. This is the argument suggested by the U.S. political scientist Benjamin Radcliff, who has presented a series of papers in peer reviewed scholarly journals demonstrating that a more generous welfare state contributes to higher levels of life satisfaction, and does so to rich and poor alike.[20][21][22]\n\n### Employment\n\nWork is important to happiness. It creates a sense of purpose, beneficial relationships with co-workers, and also earns money. Losing one's job can be a great source of unhappiness.[12]\n\n### Relationships and children\n\nRelative declines in female happiness have eroded a gender gap in happiness in which women in the 1970s typically reported higher subjective well-being than did men.[23]\n\nIn rich societies, where a rise in income doesn't equate to an increase in levels of subjective well-being, personal relationships are the determining factors of happiness.[24]\n\nGlaeser, Gottlieb and Ziv suggest in their conclusion that the happiness trade-offs that individuals seem willing to make aligns with the tendency of parents to report less happiness, as they sacrifice their personal well-being for the \"price\" of having children.[17]\n\n### Freedom and control\n\nThere is a significant correlation between feeling in control of one's own life and happiness levels.\n\nA study conducted at the University of Zurich suggested that democracy and federalism bring well-being to individuals.[25] It concluded that the more direct political participation possibilities available to citizens raises their subjective well-being.[25] Two reasons were given for this finding. First, a more active role for citizens enables better monitoring of professional politicians by citizens, which leads to greater satisfaction with government output.[25] Second, the ability for citizens to get involved in and have control over the political process, independently increases well-being.[25]\n\nHigher economic freedom, as measured by both the Heritage and the Fraser indices, correlates strongly with higher self-reported happiness.[9][10]\n\nAmerican psychologist Barry Schwartz argues in his book The Paradox of Choice that too many consumer and lifestyle choices can produce anxiety and unhappiness due to analysis paralysis and raised expectations of satisfaction.\n\n### Religious diversity\n\nNational cross-sectional data suggest an inverse relation between religious diversity and happiness, possibly by facilitating more bonding (and less bridging) social capital.[26]\n\n### Leisure\n\nThe amount of spare time people have, as well as their control over how much spare time they have, correlates with happiness.[27]\n\nWhereas leisure pursuits increase happiness, watching television is an anomaly, as it seems to correlate with lower happiness. This may be because people who watch a lot of television are lacking in better sources of happiness, such as relationships and other leisure pursuits; that is, people watch television if they don't have anything better to do.[citation needed]\n\n## Happiness economics and indices timeline\n\nThe idea that happiness is important to a society is not new. Many other prominent intellectuals, philosophers and political leaders throughout history, including Aristotle, Confucius, and Plato, incorporated happiness into their work.[1]\n\nThomas Jefferson put the \"pursuit of happiness\" on the same level as life and liberty in the United States' Declaration of Independence.[28] Jeremy Bentham believed that public policy should attempt to maximize happiness, and he even attempted to estimate a \"hedonic calculus\".[6] However, the American ruling philosophy protects the right of individuals to seek their own happiness, but does not place an equal responsibility for the citizens' happiness on the government. In the United States, there is no explicit policy that requires the rulers to develop the physical and mental well-being of the citizens or hold the government agencies accountable for their performance against specific measures or metrics of well-being. Until the 1972 there was no formal government policy, anywhere in the world, that placed happiness and well-being as a main criterion for public policy decision making.\n\nThe following is a chronological list of happiness economics and well-being indices:\n\n\"Gross National Happiness is more important than Gross National Product\" by Jigme Singye Wangchuck, King of Bhutan. Slogan on a wall in Thimphu's School of Traditional Arts\n\n1972 \u2013 Bhutan's former king, Jigme Singye Wangchuck, introduced the Gross National Happiness (GNH) philosophy and its four development pillars at an international conference.[29]\n\n2005 \u2013 Med Jones of the International Institute of Management introduced the first GNH Index and Global GNH Index Survey. The GNH Index, also known as Gross National Well-being (GNW) Index framework served as the first integrated objective (economic) and subjective (happiness) socioeconomic development framework. Prior the GNH Index, there were few development indices that improved upon the gross domestic product (GDP), but did not measure happiness. For example, the Genuine Progress Indicator was focused on the environmental cost of economic development, then later (in 2006) it was updated to include similar measures to the GNH Index. Another development index is the Human Development Index (HDI) that originally focused on literacy and education but also did not measure happiness.[30] The HDI now measures three basic dimensions of human development, health (as measured by life expectancy at birth), overall knowledge level (as measured by the literacy rate), and standard of living (as measured by GDP per capita for a given year). Among the criticisms of the HDI is the complaint that it is a mixture of stock measures (life expectancy at birth and literacy rate) and a flow measure (GDP per capita for a given year). To overcome this criticism, Hou, Walsh, and Zhang (2015) proposed a new index called HDIF (Human Development Index Flow), in which they replaced life expectancy at birth by the under-five mortality rate (for a given year), and they also replaced the literacy rate by the gross primary school enrollment ratio for a given year). They calculated both the HDI and the HDIF for many countries and found that \u201cthe HDIF and the HDI tend to converge for wealthy countries and diverge for poor countries, especially those with low HDI rankings.\u201d The development performance of poor countries improved using the HDIF while the performance of the wealthy countries declined.[31]\n\n2006 \u2013 The Genuine Progress Indicator was updated from a green measurement system to a broader concept that included quantitative measurement of well-being and happiness.[32] The new measure is motivated by the philosophy of the GNH and the same notion of that subjective measures like well-being are more relevant and important than more objective measures like consumption. It is not measured directly, but only the factors which are believed to lead to it.\n\n2007 \u2013 Thailand releases Green and Happiness Index (GHI).[33]\n\n2008 \u2013 French President Nicolas Sarkozy launched a Happiness Initiative similar to GNH, calling for the inclusion of happiness and well-being among the criteria for national governance policies. He commissioned three prominent economists, Joseph Stiglitz (USA), Amartya Sen (India), Jean-Paul Fitoussi (France), to publish a report calling for a global \"statistical system which goes beyond commercial activity to measure personal well-being.\" Later it was described as gross domestic happiness (GDH).[34] The GDH Index is similar to the GNH Index of 2005.\n\n2009 \u2013 In the United States, the Gallup poll system launched the happiness survey collecting data on national scale.[35] The Gallup Well-Being Index was modeled after the GNH Index framework of 2005. The Well-Being Index score is an average of six sub-indexes that measures life evaluation, emotional health, work environment, physical health, healthy behaviors, and access to basic necessities. In October 2009, the USA scored 66.1\/100.\n\n2010 \u2013 The concept was taken seriously, as the Centre for Bhutan Studies, under the leadership of Karma Ura, developed a sophisticated survey instrument to measure the population's general level of well-being.[36] Two Canadians, Michael and Martha Pennock played a major role in developing the Bhutanese survey, which took a six- to seven-hour interview to complete. They developed a shorter international version of the survey which has been used in their home region of Victoria BC as well as in Brazil. The Pennocks also collaborated with Ura in the production of a policy lens which is used by the Bhutanese GNH Commission for anticipating the impact of policy initiatives upon the levels of GNH in Bhutan[37]\n\n2010 \u2013 The Center for Bhutan Studies further defined the original four pillars with greater specificity into eight general contributors to happiness\u2014physical, mental and spiritual health; time-balance; social and community vitality; cultural vitality; education; living standards; good governance; and ecological vitality. The Bhutan GNH Index.[38]\n\n2010 \u2013 The Oxford Poverty and Human Development Initiative OPHI at the University of Oxford in UK, launched the Multidimensional Poverty Index (MPI) for the United Nations Development Programme, (UNDP). Similar to the GNH Index of 2005, OPHI promotes collection and analysis of data on five dimensions including Quality of work, Empowerment, Physical safety, Ability to go about without shame, Psychological wellbeing.[39]\n\n2011 \u2013 UN General Assembly Resolution 65\/309, titled \"Happiness: towards a holistic approach to development\"[40]\n\n2011 \u2013 The Organization for Economic Co-operation and Development (OECD) launched \"Better Life Index\" (BLI).[41]\n\n2011 \u2013 The United Nations released the World Happiness Report\n\n2011 \u2013 Canadian Index of Wellbeing Network (CIW Network) released The Canadian Index of Wellbeing (CIW).[42]\n\n2011 - The Israeli newspaper Haaretz published an article suggesting that western GDP economics is an incomplete development model and called for the adoption of Bhutan's GNH philosophy and Jones' GNH Index in Israel.[43]\n\n2011 - Chuluun Togtokh criticized the HDI in an article published in Nature, calling for a revised HDI, writing that \u201cThe revised index should include each nation\u2019s per capita carbon emissions, and so become a Human Sustainable Development Index (HSDI).\u201d[44] Bravo (2014) provided details of how the HSDI was computed and proposed an amended HSDI by including the proportion of forested area in each country. He argued that this proposed indicator \u201crepresents an important measure of the capacity of natural system to provide fundamental ecological services.\u201d[45]\n\n2012 \u2013 In a report prepared for the US Congressman Hansen Clarke, R, researchers Ben Beachy and Juston Zorn, at John F. Kennedy School of Government in Harvard University, recommended that \"the Congress should prescribe the broad parameters of new, carefully designed supplemental national indicators; it should launch a bipartisan commission of experts to address unresolved methodological issues, and include alternative indicators.\" They proposed that the government can use the survey results to see which well-being dimensions are least satisfied and which districts and demographic groups are most deficient, so as to allocate resources accordingly. The report list the Gross National Happiness Index and its seven measurement area as one of the main frameworks to consider.[46]\n\n2012 \u2013 Professor Peter T. Coleman, a director of the International Center for Cooperation and Conflict Resolution at Columbia University, suggested that Jones' GNH Index initiative could inform the Global Peace Index Initiative GPI.[47]\n\n2012 \u2013 South Korea launched Happiness Index citing the GNH Index framework.[48]\n\n2012 \u2013 The government of Goa, India, published a strategy for socioeconomic development citing the GNH Index as a model for measuring happiness.[49]\n\n2012 \u2013 The city of Seattle in Washington, launched its own happiness index initiative, emphasizing measures similar to the GNH Index.[50]\n\n2013 \u2013 The Social Progress Index SPI was launched by Michael Porter\n\n2013 \u2013 The president of Singapore, Tony Tan, proposed that in addition to building up substantial financial reserves, Singapore needed to focus on building up its \"social reserves\", a concept that appears to have parallels to GNH.[51]\n\n2013 \u2013 Economist Karol Jan Borowiecki motivates that well-being indices can be obtained from the way people communicate, as is established in psychology, and compiles the first well-being indices covering the life-time of a person.[52]\n\n2014 \u2013 The government of Dubai launched its localized Happiness Index to measure the public\u2019s contentment and satisfaction with different government services.[53]\n\n2014 \u2013 The United Kingdom launched its own well-being and happiness statistics.[54]\n\n## Related Studies\n\nThe Satisfaction with Life Index. Blue through red represent most to least happy respectively; grey areas have no reliable data available.\n\nThe Satisfaction with Life Index is an attempt to show the average self-reported happiness in different nations. This is an example of a recent trend to use direct measures of happiness, such as surveys asking people how happy they are, as an alternative to traditional measures of policy success such as GDP or GNP. Some studies suggest that happiness can be measured effectively.[55][56] The Inter-American Development Bank (IDB), published in November 2008 a major study on happiness economics in Latin America and the Caribbean.[57]\n\nIn 2013, John Helliwell, Richard Layard and Jeffery Sachs compiled a treatise under the title \u201cWorld Happiness report 2013\u201d to elaborate on the measurement of popular happiness in different countries thereby adding to the wealth of happiness data available while specifically discussing the issues of measurement, explanation and policy. Global and Regional Happiness Levels are explained in terms of 10 regional groupings of countries based on happiness data available for the year 2010-2012. The happiness level is explained as a function of GDP per capita, social support, and healthy life expectancy, freedom to make life choices, generosity and perceptions of corruption.[58]\n\nThere are also several examples of measures that includes self-reported happiness as one variable. Happy Life Years, a concept brought by Dutch sociologist Ruut Veenhoven, combines self-reported happiness with life expectancy. The Happy Planet Index combines it with life expectancy and ecological footprint.\n\nGross National Happiness (GNH) is a concept introduced by the King of Bhutan in 1972 as an alternative to GDP. Several countries have already developed or are in the process of developing such an index.[6][59] Bhutan\u2019s index has led that country to limit the amount of deforestation it will allow and to require that all tourists to its nation must spend US\\$200[6] Allegedly, low-budget tourism and deforestation lead to unhappiness.[6]\n\nAfter the military coup of 2006, Thailand also instituted an index.[6] The stated promise of the new Prime Minister Surayud Chulanont is to make the Thai people not only richer but happier as well.[6] Much like GDP results, Thailand releases monthly GNH data.[60] The Thai GNH index is based on a 1\u201310 scale with 10 being the most happy.[60] As of May 13, 2007, the Thai GNH measured 5.1 points.[60] The index uses poll data from the population surveying various satisfaction factors such as, security, public utilities, good governance, trade, social justice, allocation of resources, education and community problems.[60]\n\nAustralia,[59] China, France[61] and the United Kingdom[62] are also coming up with indexes to measure national happiness.[6] The UK began to measure national wellbeing in 2012.[63] North Korea also announced an international Happiness Index in 2011 through Korean Central Television. North Korea itself came in second, behind #1 China.[64] Canada released the Canadian Index of Wellbeing (CIW) in 2011 to track changes in wellbeing. The CIW has adopted the following working definition of wellbeing: The presence of the highest possible quality of life in its full breadth of expression focused on but not necessarily exclusive to: good living standards, robust health, a sustainable environment, vital communities, an educated populace, balanced time use, high levels of democratic participation, and access to and participation in leisure and culture[65]\n\nEcuador's and Bolivia's new constitutions state the indigenous concept of \"good life\" (\"buen vivir\" in Spanish, \"sumak kawsay\" in Quichua, and \"suma qama\u00f1a\" in Aymara) as the goal of sustainable development.\n\n## Neoclassical economics\n\nNeoclassical, as well as classical economics, are not subsumed under the term happiness economics although the original goal was to increase the happiness of the people. Classical and neoclassical economics are stages in the development of welfare economics and are characterized by mathematical modeling. Happiness economics represents a radical break with this tradition. The measurement of subjective happiness respectively life satisfaction by means of survey research across nations and time (in addition to objective measures like lifespan, wealth, security etc.) marks the beginning of happiness economics.\n\n## Criticism\n\nSome have suggested that establishing happiness as a metric is only meant to serve political goals.[6] Recently there has been concern that happiness research could be used to advance authoritarian aims.[6] As a result, some participants at a happiness conference in Rome have suggested that happiness research should not be used as a matter of public policy but rather used to inform individuals.[6]\n\nIn addition, survey findings can lead to ambiguous interpretations. For example, a happiness study conducted in Russia during the 1990s[1] indicated that as unemployment grew, the well-being of both those employed and unemployed rose. The interpretation of this could be that it resulted from diminished expectations and respondents who were less critical of their own situation when many around them were unemployed,[1] or it could be interpreted as being the result of everyone benefitting from the unpaid work that the unemployed were able to do for their families and communities with their increased time resource.\n\nBooks\n\nArticles\n\n## References and notes\n\n1. Carol Graham, 2008. \"happiness, economics of,\" The New Palgrave Dictionary of Economics, 2nd Edition. Abstract. Prepublication copy.\n\u2022 _____, 2005. \"The Economics of Happiness: Insights on Globalization from a Novel Approach,\" World Economics, 6(3), pp. 41-58 (indicated there as adapted from previous source).\n\u2022 David G. Blanchflower, 2008. \"Happiness Economics,\" NBER Reporter Online, (2), pp. 7-10. Abstract-linked-footnotes version.\n2. ^ Richard Layard, 2006. \"Happiness and Public Policy: A Challenge to the Profession,\" Economic Journal, 116(510), Conference Papers, pp. C24-C33.\n3. ^ Ulf-G, Gerdtham; Magnus. \"The Relationship Between Happiness, Health, and Socio-economic Factors: Results Based on Swedish Microdata\". Journal of Socio-Economics. 30 (6): 553\u2013557. doi:10.1016\/S1053-5357(01)00118-4.\n4. ^ Carol Graham, 2010. \"The Challenges of Incorporating Empowerment into the HDI: Some Lessons from Happiness Economics and Quality of Life Research,\" 54 pages, Human Development Reports Research Paper, 2010\/13, United Nations.\n5. ^ Ruut Veenhoven, World Database of Happiness, 2007\n6. Rana Foroohar, \"Money v. Happiness: Nations Rethink Priorities\", Newsweek, April 5, 2007.\n7. ^ Bruno S. Frey and Alois Stutzer, 2002. Happiness and Economics: How the Economy and Institutions Affect Human Well-Being, Description and preview. Princeton University Press, &, in the UK, John Wiley & Sons.\n8. ^ See Easterlin paradox for details.\n9. ^ a b c In Pursuit of Happiness Research. Is It Reliable? What Does It Imply for Policy? The Cato Institute. April 11, 2007\n10. ^ a b Cato Institute. \"About Cato\". Retrieved 2010-03-22.\n11. ^ Explaining Economics\n12. ^ a b Andrew Oswald, A Non-Technical Introduction to the Economics of Happiness, 1999\n13. ^ Holmes, Bob (7 September 2010). \"Money can buy you happiness \u2013 up to a point\". New Scientist. Retrieved 11 September 2010.\n14. ^ Money does not make you happy 'but therapy does', The Telegraph, November 20th 2009]\n15. ^\n16. ^ Rozanne Larsen (15 September 2011). \"If money doesn't make you happy, then you probably aren't spending it rightf\". Journalist's Resource. Harvard Kennedy School's Shorenstein Center and the Carnegie-Knight Initiative. Retrieved 17 August 2014.\n17. ^ a b Edward L. Glaeser; Joshua D. Gottlieb; Oren Ziv (July 2014). \"NBER Working Paper Series: \"Unhappy Cities\"\" (PDF). The National Bureau of Economic Research (NBER). The National Bureau of Economic Research (NBER). Retrieved 17 August 2014.\n18. ^ The Scientist's Pursuit of Happiness, Policy, Spring 2005.\n19. ^ The Centre for Independent Studies. \"About CIS\". Retrieved 2010-03-22.\n20. ^ Radcliff, Benjamin (2001). \"Politics, Markets, and Life Satisfaction\". American Political Science Review. 95 (4): 939\u2013952.\n21. ^ Radcliff, Benjamin; Pacek, Alexander (2008). \"Assessing the Welfare State: the Politics of Happiness\". Perspectives on Politics. 6: 267\u2013277.\n22. ^ Alvarez-Diaz, A.; Gonzalez, L.; Radcliff, B. (2010). \"The Politics of Happiness: On the Political Determinants of Quality of Life in the American States\". The Journal of Politics. 72 (3): 894\u2013905. doi:10.1017\/s0022381610000241.\n23. ^ Federal Reserve Bank of San Francisco, The Paradox of Declining Female Happiness, May 2009\n24. ^ Layard, R (2007). \"Setting happiness as a national goal\". The Futurist. 41 (4): 37.\n25. ^ a b c d Bruno S. Frey & Alois Stutzer, Happiness, Economy and Institutions, 4-5, 1999\n26. ^ Okulicz-Kozaryn, Adam (2011). \"Does Religious Diversity Make Us Unhappy?\". Mental Health, Religion & Culture. 14 (10): 1063\u20131076. doi:10.1080\/13674676.2010.550277.\n27. ^ Eriksson, Lina; Mahmud Rice, James; Goodin, Robert E. (2007). \"Temporal Aspects of Life Satisfaction\" (PDF). Social Indicators Research. 80 (3): 511\u2013533. doi:10.1007\/s11205-006-0005-z.\n28. ^ \"Thomas Jefferson. 1743-1826. John Bartlett, comp. 1919. Familiar Quotations, 10th ed.\". Retrieved February 18, 2016.\n29. ^\n30. ^\n31. ^ Hou, Jack; Walsh, Patrick (September 2015). Jing Zhang, 3d name. \"The dynamics of Human Development Index\". The Social Science Journal. 52 (3): 331\u2013347. doi:10.1016\/j.soscij.2014.07.003.\n32. ^ \"Beyond GDP Paper | Page 2 | John Talberth, Center for Sustainable Economy 2012 at The European Commission Website\"\n33. ^ \"Green and Happiness Index (GHI\"\n34. ^ \"Sarkozy proposes the joie de vivre index\". 14 September 2009.\n35. ^\n36. ^ \"Gross National Happiness\". The Centre for Bhutan Studies. Retrieved 24 March 2011.\n37. ^ Pennock, Michael; Ura, Karma (1 January 2011). \"Gross national happiness as a framework for health impact assessment\". Environmental Impact Assessment Review. 31 (1): 61\u201365. doi:10.1016\/j.eiar.2010.04.003 \u2013 via ScienceDirect.\n38. ^ \"Bhutan GNH Index\"\n39. ^ \"A wealth of data\" \u2013 via The Economist.\n40. ^ UN Happiness Resolution\"\n41. ^\n42. ^\n43. ^ \"Why Ordinary Economic Indicators Do Not Tell The Whole Truth\", Haaretz, Israel\n44. ^ Togtokh, Chuluun (17 November 2011). \"Time to Stop Celebrating the Polluters\". Nature.\n45. ^ Bravo, Giangiacomo (February 2014). \"The Human Sustainable Development Index: New Calculations and a First Critical Analysis\". Ecological Indicators. 37: 149. doi:10.1016\/j.ecolind.2013.10.020.\n46. ^ \"National Indicators for a New Era, Ben, Beachy and Juston Zorn, of the John F Kennedy School of Government at Harvard University\"\n47. ^\n48. ^ Nam, Joo-Ha; \uae40\uc0c1\ubd09 (1 January 2012). \"The Measurement of the EconomicHappy Index in South Korea\". 18 (2).\n49. ^ \"Goa 2035 Vision, page 111\"\n50. ^ \"Seattle Happiness Initiative\"\n51. ^ \"Leong Wai Kit, \"S'pore needs both financial and 'social' reserves to thrive: President Tony Tan\", 'Today', 6 November 2013.\". TODAYonline. Retrieved 23 October 2014.\n52. ^ Borowiecki, Karol J. \" How Are You, My Dearest Mozart? Well-being and Creativity of Three Famous Composers Based on their Letters\" \" Review of Economics and Statistics \", 2016, forthcoming\n53. ^\n54. ^\n55. ^ The True Measure of Success - Wired.com\n56. ^ \"Happiness\" is not enough- Samuel Brittan: Templeton Lecture Inst. of Economic Affairs 22\/11\/01\n57. ^\n58. ^\n59. ^ a b Andrew Revkin, \"A New Measure of Well-Being From a Happy Little Kingdom\", The New York Times, October 4, 2005, [hereinafter \"New Measure\"].[clarification needed]\n60. ^ a b c d Thailand's Gross Domestic Happiness Index Falls, Monsters and Critics, 2007\n61. ^ Vandore, Emma (2008-01-14). \"Inventing the 'Glad Domestic Product'\". The Star. Toronto. Retrieved 2010-05-04.\n62. ^\n63. ^ Foundation, Internet Memory. \"[ARCHIVED CONTENT] UK Government Web Archive \u2013 The National Archives\".\n64. ^ \"\"\ubd81\ud55c\uc758 \ud589\ubcf5\uc9c0\uc218\ub294 \uc138\uacc4 2\uc704\u2026 \ub0a8\ud55c \ud589\ubcf5\uc9c0\uc218\ub294 152\uc704\"\ub77c\uace0?\", Chosun Ilbo, 2011-05-27, retrieved 2011-05-28\n65. ^ 2012 CIW composite index reveals Canadian wellbeing is on the decline, 2012-03-16, retrieved 2013-05-31","date":"2016-08-29 06:29:19","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 5, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.35605713725090027, \"perplexity\": 8638.040131795606}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": false}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2016-36\/segments\/1471982952852.53\/warc\/CC-MAIN-20160823200912-00178-ip-10-153-172-175.ec2.internal.warc.gz\"}"} | null | null |
Pearls and gold are a favorite combination at JewelryCult. This month, we are craving for a double hoop earring by Delfina Delettrez, adorned with suspended freshwater pearls, to face the new season with character and sophistication.
The 360º Double Hoop Earring is part of the designer's Autumn-Winter Collection, which draws inspiration from both the ancient and the hypermodern.
The pearls and precious stones are not set with the traditional claw technique. Instead, they're pierced through the gold bands, giving the illusion of being suspended in space.
As technology becomes further embedded in our lives, new forms of communication and expression stand as the new norm.
Emojis, for example, are featured in films, and there is even a World Emoji Day. If you use them to express your emotions in a text, why not wear them?
They crept effortlessly into our lives, and now they are everywhere. Emojis convey emotions in a way that words often can't and help us express humor or irony in a more spontaneous way. They became the symbol of a new generation, marked by fast communications and permanent online presence.
The word emoji comes from Japanese, meaning picture or character. The cute faces we know so well came after the text-based emoticons, originating in in the late 1900s on Japanese mobile phones.
When they were internationally included in Apple's iPhone, and later in Android and other devices, they became popular worldwide.
Fiery and radiant, diamonds are fascinating. They are probably the most prized and popular gemstone in the world, symbolizing status and true, everlasting love. But there are also some curious facts and myths surrounding them.
Diamonds are known not only for their beautiful appearance but also for their remarkable physical qualities. Their name derives from the word "adamas," meaning unconquerable or indestructible, as a reference to the gem's exceptional hardness.
The million years old pure carbon crystals are formed deep in the planet's mantle, bellow continental masses, where the perfect conditions of depth, pressure, and temperature are reunited to form them to occur. Diamonds are an ancient gift from the Earth's core, brought to the surface by deep-source volcanic eruptions.
5 stylish dresses and jewelry for summer 2017
June marks the ending of spring and the arrival of the first summer days. We paired five beautiful jewelry pieces and designer dresses to celebrate the arrival of the new season, and get you from spring to summer in the most fashionable way.
Pairing jewelry and clothing isn't always easy. They should go seamlessly together, without overpowering each other, and the combination should look natural, flattering and effortless. With this in mind, we selected five of our favorite dresses, and carefully matched them with pieces we love right now.
Color, shape, and materials play a great deal in creating the perfect outfit for your jewelry. The five looks fit different personalities and contexts and include combinations for all tastes.
Cult Pieces: Nautilus Pendant, by Aurélie Bidermann
At Jewelrycult.com, every day we are inspired by beautiful jewelry designs that appeal to different moods and imageries. As summer begins to arrive, we are drawn to a piece that brings the freshness of the ocean to the city, in a very elegant way.
The Nautilus Pendant by Aurélie Bidermann is part of the French designer's fine jewelry collection. The delicate 18-karat yellow gold pendant features 35 multicolored sapphires, seven tsavorites and six diamonds.
The iconic paperclip clasp makes it the perfect match for the brand's hammered 18-karat gold bracelet or necklace. | {
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Haven Riviera Cancun is an adults-only, all-inclusive, luxury beachfront resort located between Cancun and Riviera Maya. Our exclusive resort features 333 contemporary suites, a full service spa, a variety of restaurants and bars and sophisticated venues for meetings, weddings and private events.
Haven Riviera Cancun is a member of Hipotels, a family-owned Spanish hotel collection founded in 1970 in Mallorca, Spain. Hipotels fully owns and operates 29 hotels with close to 13,000 guest rooms, located exclusively in the Balearic Islands, the Canary Islands, and Andalusia. In 2015, Hipotels decided to develop its first Mexico/Caribbean hotel complex, which will include 4 hotels and 2,000 rooms. | {
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{"url":"https:\/\/electronics.stackexchange.com\/questions\/485419\/how-to-calculate-the-gain-of-a-bjt-differential-amplifier-long-tailed-pair","text":"# How to calculate the gain of a BJT Differential Amplifier (Long Tailed Pair)\n\nI'm currently reading Designing Audio Power Amplifiers by Bob Cordell. After seeing the \"Basic 50W Amplifier\" that he shows in the beginning of chapter 2, I now am trying to use the same building blocks to design an amp to specifications I've chosen to help me understand things.\n\nI've chosen an output power of 5W through an 8ohm load (Which should at least be audible I hope), this means I am going to need a peak output voltage of about 9V. I'm assuming my input signal is coming from something like a phone, and I'm making an educated guess that its peak voltage is 1v. This means I'm going to need a total gain of about 9.\n\nThe building blocks of his circuit are as follows:\n\nStage 1 of 3 (Input Amplification Stage): Constant Current Driven Differential Long Tailed Pair\n\n\u2022 Rejects noise well, and negative feedback is easy to achieve from the final output of the amplifier with a simple voltage divider\n\nStage 2 of 3 (Voltage Amplification Stage): Constant Current Driven Common Emitter Amplifier\n\n\u2022 Easy to get a really high gain from what I understand\n\nStage 3 of 3 (Output Amplification Stage): Triple Emitter Follower\n\n\u2022 High input impedance, with a decent amount of output capability.\n\nThis is basically an op-amp I think, but with higher power capabilities. What I'm really struggling with is the input stage.\n\nWhat I've been taught in my Electronics 1 class so far about BJTs is that for these amplifier configurations, the gain usually is a ratio of the resistors used (and maybe includes the base-emitter resistance). I have no idea where the gain is coming from in this schematic. I basically just fiddled with the resistor values until I got an output that looked alright-ish.\n\nIs a source out there that goes through the design process of this type of configuration, or can anyone can help me out? Thanks!\n\n\u2022 But in the Bob's book, you can find all the equations you need to use. And in this case, the gain is $A_V = \\frac{R_C}{2(re + RE)}$ \u2013\u00a0G36 Mar 10 at 6:58\n\u2022 R2 is way too big (the LED current is too small). Also,electronics.stackexchange.com\/questions\/347501\/\u2026 try read this electronics.stackexchange.com\/questions\/343833\/\u2026 \u2013\u00a0G36 Mar 10 at 7:02\n\u2022 So your circuit the gain cannot be larger than: $A_V = \\frac{5k\\Omega}{2 * 300\\Omega} = 8.3 V\/V$ \u2013\u00a0G36 Mar 10 at 7:06\n\u2022 Thank you for such a fast reply! I guess I should read more closely. I just went back to the diff-amp part of chapter 1 and that equation is definitely there, I need more sleep! I do have one more question though, how do you choose how much current to source\/sink through the LTP? Regarding the constant current source I have up top, are you sure that resistance is too high? I'm getting right around my target current value in the simulation. The schematic doesn't say it but its a green LED with a 2.1V drop, so 1.4V across R1 gives about 3.5mA. \u2013\u00a0Michael Beckwith Mar 10 at 7:21\n\u2022 Nevermind you are right about the resistor, I'm only getting 2.9mA out, which is actually kind of far off from 3.5. Thank you for that advice. \u2013\u00a0Michael Beckwith Mar 10 at 7:33","date":"2020-05-25 02:52:52","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.6476710438728333, \"perplexity\": 1070.2086686008784}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2020-24\/segments\/1590347387155.10\/warc\/CC-MAIN-20200525001747-20200525031747-00130.warc.gz\"}"} | null | null |
Pretreatment with coenzyme Q10 improves ovarian response and embryo quality in low-prognosis young women with decreased ovarian reserve: a randomized controlled trial
Yangying Xu1,2,3,
Victoria Nisenblat2,3,
Cuiling Lu2,3,
Rong Li2,3,
Jie Qiao2,3,
Xiumei Zhen2,3 &
Shuyu Wang1
Management of women with reduced ovarian reserve or poor ovarian response (POR) to stimulation is one of the major challenges in reproductive medicine. The primary causes of POR remain elusive and oxidative stress was proposed as one of the important contributors. It has been suggested that focus on the specific subpopulations within heterogeneous group of poor responders could assist in evaluating optimal management strategies for these patients. This study investigated the effect of anti-oxidant treatment with coenzyme Q10 (CoQ10) on ovarian response and embryo quality in young low-prognosis patients with POR.
This prospective, randomized controlled study included 186 consecutive patients with POR stratified according to the POSEIDON classification group 3 (age < 35, poor ovarian reserve parameters). The participants were randomized to the CoQ10 pre-treatment for 60 days preceding IVF-ICSI cycle or no pre-treatment. The number of high quality embryos was a primary outcome measure.
A total of 169 participants were evaluated (76 treated with CoQ10 and 93 controls); 17 women were excluded due to low compliance with CoQ10 administration. The baseline demographic and clinical characteristics were comparable between the groups. CoQ10 pretreatment resulted in significantly lower gonadotrophin requirements and higher peak E2 levels. Women in CoQ10 group had increased number of retrieved oocytes (4, IQR 2–5), higher fertilization rate (67.49%) and more high-quality embryos (1, IQR 0–2); p < 0.05. Significantly less women treated with CoQ10 had cancelled embryo transfer because of poor embryo development than controls (8.33% vs. 22.89%, p = 0.04) and more women from treatment group had available cryopreserved embryos (18.42% vs. 4.3%, p = 0.012). The clinical pregnancy and live birth rates per embryo transfer and per one complete stimulation cycle tended to be higher in CoQ10 group but did not achieve statistical significance.
Pretreatment with CoQ10 improves ovarian response to stimulation and embryological parameters in young women with poor ovarian reserve in IVF-ICSI cycles. Further work is required to determine whether there is an effect on clinical treatment endpoints.
Poor response to controlled ovarian hyperstimulation (COH) remains one of the main challenges of the assisted reproductive technology (ART) treatments. Despite impressive advances in the field, many women exhibit inadequate response to gonadotrophins, referred to as 'poor or low responders' and have higher odds of cycle cancellation, fewer oocytes at retrieval, lower oocyte quality and reduced number of embryos for transfer. Collectively, this results in serial failure of the ART cycles and is frustrating for both patients and their caregivers. The exact incidence of the condition is hard to establish owing to variable definitions in literature with the estimates ranging from 5.6 to 35.1% of ART cycles [1]. Multiple interventions have been proposed to improve reproductive outcomes in women with poor ovarian response (POR), but the randomized intervention studies and meta-analyses of these studies reveal conflicting results [2, 3]. Currently, the evidence-based therapeutic strategies to improve ovarian response and reproductive outcomes in women with POR are lacking, and treating clinicians often offer empirical treatments with little clinical evidence to support their use [4]. Furthermore, it has been increasingly acknowledged that the available ovarian reserve tests are not reliable to predict pregnancy after assisted conception [5]. We do not have universally accepted tests to predict response to treatment, which is of important value for counseling couples regarding their treatment pathways and for setting patients' expectations.
It has been proposed that a heterogeneity of the included population is the main barrier in evaluating the interventions and the factors that guide prognosis for POR [6]. An internationally-agreed consensus on the definition of POR reached by an ESHRE Campus Workshop held in Bologna in 2010 suggests that at least 2 out of 3 features must be present: (1) advanced maternal age or any other risk factor for POR; (2) previous POR; (3) abnormal ovarian reserve test [7]. This uniform definition, however, implies that POR constitutes heterogeneous group of women with respect to age, previous reproductive experience and ovarian reserve tests that may have different response to the interventions [6]. While age-dependent decline in ovarian reserve and oocyte quality accounts for poor response in older women, an underlying etiology for its occurrence earlier in life is less clear. It is possible that younger women with compromised ovarian reserve represent a distinct subpopulation within POR group, and their fertility prognosis may differ from that of older women with low ovarian reserve markers or from similar age women with adequate ovarian reserve parameters but suboptimal response to ovarian stimulation [8].
Taking the above considerations into account, the recently established POSEIDON group (Patient-Oriented Strategies Encompassing Individualize Oocyte Number) proposed a new stratification of women with POR undergoing ART treatments, which includes 4 subgroups based on women's age, ovarian reserve parameters and previous response to ovarian stimulation [9]. The POSEIDON concept introduces personalized medicine approach to the POR population and is expected to be more effective in identifying the subsets of patients who could benefit from specific interventions [10].
The physiology of poor ovarian response is not fully understood and the molecular events underlying POR remain unknown. Oxidative stress and mitochondrial dysfunction are among the most investigated possible mechanisms [11]. Mitochondria are the most abundant organelles in oocytes and early embryos that generate approximately 90% of reactive oxygen species (ROS), the end products of oxygen metabolism, and convert ROS into an inactive state via antioxidant defense mechanisms [12]. Higher levels of ROS accumulating in mitochondria during multiple physiological conditions contribute to mitochondrial dysfunction and increase in oxidative stress. This, in turn, leads to oxidative damage to DNA and other intra-cellular aberrations, which are similar to the age-related changes [12, 13]. Thus, improving mitochondrial function by supplementing antioxidants has been proposed as one of the important strategies to enhance reproductive performance [11, 14].
Coenzyme Q10 (CoQ10) is a lipid-soluble coenzyme and is an essential component of the inner mitochondrial membrane. CoQ10 is primarily involved in electron transport in the mitochondrial respiratory chain and oxidative phosphorylation to produce adenosine triphosphate (ATP). CoQ10 acts as an antioxidant by inhibiting lipid peroxidation and DNA oxidation, thus is capable of strengthening endogenous antioxidant system within a cell [15]. CoQ10 supplementation has been shown to improve cardiovascular function and male fertility [16,17,18]. Reduced concentrations of CoQ10 in plasma have been associated with hypogonadism and altered levels of other steroid hormones [19]. Decrease in CoQ10 level is commonly observed in individuals in late 30th and appears to co-occur with the age-related decline in fertility and increased rate of embryo aneuploidy, suggesting a contribution of the reduced expression of CoQ10 to ovarian ageing [20]. Several animal studies have demonstrated that CoQ10 protects ovarian reserve, counteracts physiological ovarian ageing by restoring mitochondrial function and increases the rate of embryo cleavage and blastocyst formation [21,22,23]. In the clinical setting, CoQ10 supplementation led to better response to ovulation induction and decreased odds of fetal aneuploidy in 35–43-year-old women [24, 25]. To date, however, no study has investigated whether CoQ10 pretreatment could improve the ART treatment outcomes in young subpopulation of poor responders in a randomized setting.
On the above evidence, this study focused on investigating the effect of CoQ10 supplementation on response to ovarian stimulation in the group of young women with diminished ovarian reserve, corresponding to the Poseidon's stratification group 3 [9]. We hypothesized that increased oxidative stress has a prominent effect on premature decline of ovarian function in these women, which could be amenable to anti-oxidant therapy.
Study design and randomization
This was a prospective randomized controlled study, conducted at the Reproductive Medical Center of the Peking University Third Hospital, a tertiary university hospital and a center of excellence in Reproductive Medicine in China. The study is reported according to the CONSORT guidelines. The flow of the patients in this study is presented in Fig. 1.
Flow of the patients through the trial
All the participants were randomized 1:1 to either CoQ10 treatment (study group) or no treatment (control group) followed by an ART cycle. The randomization was performed over the period of 14 months (between June 2, 2015 and July 31, 2016) by using the computer-generated randomization codes, which were then placed in the sealed, opaque sequentially numbered envelopes by a third party (nurse practitioner) who was not directly involved in the patient management or in the randomization process. The envelopes were handed out to the participants upon completing the informed consent. The study participants and the investigators were not blinded to the patient grouping. The participants were followed through one completed ART cycle until all frozen embryos generated from the index cycle were used or until delivery in those who achieved pregnancy.
All consecutive women who were found to have POR and were referred to IVF-ET cycle in our institution were approached. POR was defined according to the ESHRE Bologna criteria [7]. The study inclusion criteria were: age < 35 years, anti-Mullerian hormone (AMH) < 1.2 ng/ml, and antral follicle count (AFC) < 5, the parameters that corresponded to a low prognosis group 3 as per the POSEIDON stratification [9]. Exclusion criteria were: age ≥ 35 years, history of ovarian surgery, endocrine or autoimmune disease (e.g. diabetes, thyroid disease or presence of anti-thyroid antibodies or PCOS), chromosomal abnormality, uterine malformations, more than 3 previous IVF cycles, treatment with cholesterol-lowering drugs, previous treatment with anti-oxidants (last 5 years) or known allergy to CoQ10 or ubiquinol (the water-soluble isoform of CoQ10). All the participants completed the questionnaire with demographic, medical and reproductive information and underwent clinical examination, pelvic ultrasound, chromosome analysis, AMH test, reproductive endocrine profile and thyroid studies. All the included women were specifically asked about any previous treatment with anti-oxidants such as CoQ10, ubiquinol, vitamin A, vitamin E, vitamin C, beta-carotene or selenium, including the duration and time of treatment.
The intervention in the study group included oral administration of CoQ10 (GNC Holdings Inc., Pittsburg, PA, USA) 200 mg three times a day, for a period of 60 days in an open label fashion. The ART treatment (in vitro fertilization (IVF) or intracytoplasmic sperm injection (ICSI)) was commenced in the first menstrual cycle upon completion of CoQ10 treatment. The control group commenced ART (IVF or ICSI) after enrollment without any additional treatment.
Ovarian stimulation and oocyte retrieval
All participants underwent ovarian stimulation with the short GnRH-antagonist protocol. A combination of recombinant follicle stimulating hormone (FSH) (Gonal-F, 225 IU/day, Merck Serono SA Aubonne Branch) and human menopausal gonadotrophin (Menotropins for injection FSH 75 IU: LH 75 IU, 225 IU/day, Livzon Pharmaceutical Group Inc.) in a fixed-dose was started on Day 2 of the menstrual cycle with the option to adjust dose according to response after 4 days of stimulation (Day 6 of menstrual cycle). GnRH antagonist (Cetrorelix 250 μg/day, Merck Serono, Darmstadt, Germany) was started when a leading follicle of 12 mm was achieved. Recombinant human chorionic gonadotrophin (hCG) trigger (Ovidrele 250 μg; Merck Serono S.p.A, Rome, Italy) was administered when at least one follicle was above 18 mm. The cycle was cancelled when there were no follicles with diameter ≥ 14 mm after 8–9 days of gonadotrophin therapy or when peak E2 level was below 250 pmol/l.
Ultrasound-guided transvaginal oocyte retrieval was performed 36–38 h after the trigger injection by using a 17-gauge double-lumen needle (Cook Medical) and a vacuum pump (Cook Medical) under pressure at 125 mmHg. Each follicle sized above 12–14 mm was drained, and follicle flushing was not performed. The cumulus-oocyte complexes (COCs) were removed from the collection fluid using a sterile glass pipette and washed in G-IVF Plus media (Vitrolife, Sweden) and transported to the laboratory.
Oocyte insemination and embryo culture
Oocytes were inseminated either by conventional IVF or by ICSI depending on sperm quality. Oocytes undergoing IVF insemination were placed into a dish with G-IVF (Vitrolife) covered in mineral oil. Oocytes undergoing ICSI were denuded and injected if maturation status was confirmed by the presence of the first polar body (PB). Fertilization was assessed 17–19 h after insemination and was defined by the presence of two pronuclears (2PN) and two PBs. All embryos were transferred to GM medium (G-M, Life Global, CT, USA) for a further 48 h of culture. Embryo development and quality were assessed 68–72 h (day 3) after insemination, based on the number of blastomeres, blastomere symmetry, percentage of fragmentation, and quality of cytoplasm according to the criteria established by the Istanbul Consensus Workshop on Embryo Assessment [26]. All supernumerary day-3 embryos were cryopreserved by vitrification (JIEYING laboratory Inc., Canada) for future use.
Endometrial preparation and embryo transfer
All patients underwent transfer of day-3 embryos in a fresh cycle and subsequent frozen embryo transfer (FET) when the cryopreserved embryos generated from the index stimulation cycle were available. The embryos with the best morphological grade were selected for transfer. In absence of high-quality embryos, transfer of any embryo quality was considered after careful patient counselling.
In a fresh cycle, the luteal phase was supported with progesterone intravaginal gel (Crinone 8% 90 mg/day, Merck-Serono) commenced on the day of oocyte retrieval until 14 days after embryo transfer. In women with positive pregnancy test, luteal support was continued until 8 weeks gestation. The protocols used for FET utilized either natural cycle or artificial estradiol and progesterone endometrium priming in normo-ovulatory and oligo-ovulatory women, respectively. In natural cycle, ovulation was tracked with transvaginal ultrasound and urine LH kit. Oral dydrogesterone (Duphaston, 20 mg daily for 7 days; Abbott Biologicals B.V.) was commenced for luteal phase support 3 days after LH surge on the day of embryo transfer until 8 weeks gestation. In artificial FET protocol, oral estradiol valerate (Progynova 6 mg/day, Schering, Berlin, Germany) was initiated on the third day of the menstrual cycle and endometrial thickness was monitored with transvaginal ultrasonography. When the endometrial thickness exceeded 8 mm, luteal support with progesterone intravaginal gel (Crinone 8% 90 mg, daily; Merck-Serono), combined with oral dydrogesterone (Duphaston, 20 mg daily for 7 days; Abbott Biologicals B.V.) was added and embryo transfer was performed after 5 days. Hormonal treatment was stopped if pregnancy test was negative or continued until 11 weeks gestation with tapering off after 10 weeks. Single or double cleavage-stage embryo transfer were performed by using a soft catheter (K-Soft 5100; Cook, Queensland, Australia) without ultrasound guidance. Serum hCG was measured 14 days after embryo transfer and was considered positive for hCG level ≥ 10 IU. Transvaginal ultrasonography at 30 days after transfer was used to confirm clinical pregnancy.
Hormone assay procedures
All the hormonal assays were performed at the endocrine laboratory of the Peking University Third Hospital Reproductive Centre by using commercially available kits. Serum concentrations of hCG were determined by using the commercially available ELISA kit (Beckman DXI800, Beckman, USA) according to the manufacturer's instructions. Serum levels of anti-Mullerian hormone (AMH) were measured by automated assays using commercially available kit (Ashlab, USA). Serum luteinizing hormone (LH), FSH, estradiol (E2), and Progesterone (P) were tested using the Immulite 1000 assay based on chemiluminescence (DPC, Poway, CA).
The lower detection limit of the hCG and the AMH assays was 0.5 IU/L and 0.06 ng/ml respectively. The intra- and inter-assay coefficient of variation (CV) for hCG activity was 5% and for AMH was 8%. The lower detection limit of LH, FSH, E2 and P was 0.05 IU/L, 0.12 IU/L, 73.4 pmol/L, 0.64 nmol/L, respectively. The CV of LH and FSH was 6% and of E2 and P was 10%.
The primary outcome measure was the number of high quality day-3 embryos generated from one stimulation cycle. High quality embryos were defined as embryos that reached 6 to 8-cell stage with cytoplasmic fragmentation occupying less than 10% of the embryo surface and had equal size blastomeres.
The secondary outcomes included ovarian response parameters (duration of stimulation, total dose of gonadotrophins, peak E2 level and endometrial thickness on the day of hCG trigger), embryological parameters (number of oocytes retrieved, fertilization rate, number of patients with frozen embryos and number of patients who did not achieve embryo transfer) and clinical parameters (miscarriage, clinical pregnancy and live birth rate). Fertilization rate was defined as the number of 2PN embryos divided by the number of inseminated oocytes. Clinical pregnancy was defined as a presence of intrauterine gestational sac observed on ultrasound after 30 days of embryo transfer. Miscarriage was defined as a loss of clinical pregnancy before 24 weeks of gestation. Live birth was defined as the birth of at least one living child, irrespective of the duration of gestation. Clinical pregnancy and live birth rate were calculated per embryo transfer cycle as number of pregnancies/ live births divided per number of women who had transfer. Cumulative pregnancy and live birth rate were defined as the number of clinical pregnancies/live births generated from the index ART cycle following fresh or frozen embryo transfer divided by all women who received treatment. In addition, markers of ovarian reserve, including AMH, day 3 FSH and AFC were evaluated before and after CoQ10 treatment in the participants from the intervention (study) group.
Sample size calculation
The sample size calculation for this study was based on the number of high quality embryos as primary outcome. In our center women with poor response have an average 0.6–0.8 high quality embryos per woman. To detect a difference of 50% in primary outcome measure (from 0.6–0.8 to 1.0–1.2 embryos per woman) with alpha 0.05 and power 0.80, the required sample size was estimated at 76 women in each arm. When accounted for a drop out rate of 20%, each arm required 92 women.
The Student's t-test or Mann-Whitney U test were used for comparisons of continuous variables between the groups depending on the distribution of the data. The chi-squared test or Fisher's exact test, where appropriate, were used for comparisons of categorical variables. Results are presented as mean ± standard deviation (SD), median and interquartile range (IQR) or as percentages. Statistical significance was set at a probability (p) value < 0.05. All statistical analyses were performed using SPSS 22.0 software (IBM Corp., Armonk, NY, USA).
A total of 436 women met inclusion criteria. Of them, 186 women agreed to participate and were enrolled in the study, 93 women in each arm. Among the participants who were randomized to the intervention (CoQ10 treatment) group, 17 women were excluded from the analysis for the following reasons: one woman changed her mind to undergo ART and 16 women discontinued CoQ10 treatment due to the compliance issues. Overall, 76 women were retained in the study group and 93 women comprised the control group. All the participants shared the features of POSEIDON group 3, i.e. low prognosis patients younger than 35 years old with poor ovarian reserve pre-stimulation parameters. Baseline characteristics were comparable between the two groups with respect to age, BMI, duration of infertility, parity, ovarian reserve tests and causes of infertility (Table 1). Most participants were diagnosed with primary infertility and were ART treatment-naïve.
Table 1 Baseline characteristics of the study population
In the treatment group, no local or systemic side effects related to the use of oral CoQ10 were reported. Sequential measurements of ovarian reserve markers before and after CoQ10 treatment are presented in Table 2. The levels of basal day-3 FSH were significantly lower after 60 days supplementation of CoQ10 compared to the pre-treatment levels in the same group of women. In contrast, the levels of AMH and AFC were almost identical before and after CoQ10 treatment (Table 2).
Table 2 Ovarian reserve markers before and after CoQ10 treatment in the study group
The parameters of ovarian response to stimulation and the embryology outcomes of ART cycles in the study population are summarized in Table 3. The amount of gonadotrophin used was significantly lower in CoQ10 treatment group than in controls (p = 0.03). The duration of gonadotrophin therapy tended to be shorter in the participants treated with CoQ10, but the difference did not reach statistical significance (p = 0.08). Peak E2 serum concentrations were significantly higher in the CoQ10 group, but there was no difference in the mean endometrial thickness on the day of hCG trigger between the two groups. In the CoQ10 treatment group there were fewer cancelled cases due to suboptimal ovarian response (5.23%, 4/76) compared to the control group (10.75%, 10/93) although this difference failed to achieve statistical significance, p = 0.27. Overall, 94.74% (72/76) women from the CoQ10 group and 89.25% (83/93) women from the control group received hCG and underwent oocyte retrieval. The median number of retrieved oocytes was significantly higher after CoQ10 pre-treatment (4, IQR 2–5), than in controls (2, IQR 1–2), p = 0.002. Most women had conventional IVF and the number of ICSI cycles was comparable between the groups. The median number of fertilized oocytes and fertilization rate were significantly higher in women treated with CoQ10 than in controls, p < 0.05. The median number of high quality day-3 embryos available per patient in the CoQ10 group was 1 (IQR 0–2) and in control group was 0 (IQR 0–1.75), with significant difference in favor of CoQ10 treatment, p = 0.03.
Table 3 ART cycle stimulation parameters and embryology outcomes
Among the patients in CoQ10 group who underwent oocyte retrieval, there was significantly lower number of women who did not achieve embryo transfer because of failure to retrieve oocytes or due to the absence of useable embryos (8.33%, 6/72) compared to women from the control group (22.89%, 19/83), p = 0.04 (Table 4). Collectively, embryos were available for 66 women in the CoQ10 group and 64 women in the control group, all of whom underwent fresh embryo transfer. The number of fresh embryo transfer cycles in the CoQ10 groups was comparable to that in controls. More patients in the CoQ10 group had cryopreserved embryos (18.42%, 14/76 vs. 4.3%, 4/93, respectively, p = 0.02) and the number of frozen-thaw embryo transfers from the index stimulation cycle was significantly higher, p = 0.01 (Table 4). In 14.29%, 2/14 women from the CoQ10 group with available cryopreserved embryos and in 25%, 1/4 controls, embryos did not recover after thawing. One to two embryos were replaced into the uterus in each transfer cycle with higher median number in the CoQ10 group (2, IQR 1–2) than in controls (1, IQR 1–2), p = 0.04.
Table 4 Clinical reproductive outcomes
In the CoQ10 group there were 23 clinical pregnancies following fresh embryo transfer and one additional pregnancy following frozen-thaw embryo transfer. In the control group there were 16 clinical pregnancies after fresh embryo transfer and no pregnancies after frozen-thaw transfer. The were no spontaneously conceived pregnancies in either group. Successful live birth was achieved in 22 women from the CoQ10 (21 after fresh and 1 after frozen-thaw transfer) and in 14 women from the control group. Clinical pregnancy rate and live birth rate per fresh embryo transfer cycle were 34.85% and 31.82% in women treated with CoQ10, and 25% and 21.88% in controls, respectively. The clinical estimates for frozen-thaw embryo transfer were not calculated due to the paucity of the available data. When the transfers of all embryos originating from the complete ART cycle were considered, in women treated with CoQ10 the cumulative clinical pregnancy rate after one complete cycle was 31.58%, 24/76 and the cumulative live birth rate was 28.95%, 22/76. In the control group, the cumulative clinical pregnancy rate was 17.20%, 16/93 and the cumulative live birth rate was 15.54%, 14/93, respectively. Miscarriage rate was 8.67% in women from the CoQ10 group and 12.5% in controls. Although women from the CoQ10 group had higher clinical pregnancy and live birth rates with lower occurrence of pregnancy loss, the difference between the treatment and control groups failed to achieve statistical significance for each of these outcomes.
In this study we demonstrated potential benefit of CoQ10 treatment in improving ovarian response to gonadotrophin stimulation in young women with low ovarian reserve. To the best of our knowledge, this is the first study that evaluated an effect of anti-oxidant treatment in specific phenotypic subgroup of women with POR.
Our results demonstrate that pre-treatment with CoQ10 resulted in significant decrease in the total amount of gonadotrophin needed to achieve ovarian response, shorter duration of stimulation, higher peak E2 levels and the number of oocytes retrieved. CoQ10 treatment led to significant increase in fertilization rate and in the number of high quality embryos. There was significantly lower rate of cancelled cycles because of no response to stimulation, less cancelled embryo transfers because of failed embryo development and larger number of cycles with cryopreserved embryos in the CoQ10 treated group than in controls. The clinical pregnancy and live birth rates were higher after CoQ10 treatment then in controls, but these differences failed to achieve significance, presumably due to insufficient sample size. Taken together, our data suggest that CoQ10 administration enhances ovarian response to stimulation and improves oocyte and embryo quality.
The findings of this study are approximately in line with previous reports that linked CoQ10 with improved reproductive outcomes. Small randomized placebo-controlled study in 24 participants (10 women in CoQ10 and 14 in placebo group) have demonstrated higher peak concentration of E2, increased number of high quality cleavage embryos, and a trend towards decreased aneuploidy and higher clinical pregnancy rate after 60 days treatment with 600 mg CoQ10 [25]. However, the study was underpowered and failed to demonstrate significant difference in clinical outcomes between the groups [25]. Another randomized controlled study in 101 young women with PCOS demonstrated that addition of CoQ10 in a dose of 180 mg during ovulation induction with clomiphene citrate improved ovarian response in clomiphene-resistant women and resulted in higher clinical pregnancy rate [24]. Retrospective analysis in 797 IUI and 253 IVF cycles in women older than 36–37 years revealed that addition of 600 mg CoQ10 to dehydroepiandrosterone (DHEA) over the period longer than 1 month resulted in lower dose of gonadotrophins and higher number of mature follicles than in women treated with DHEA alone [27]. The authors did not demonstrate significant difference in the embryological or clinical outcomes, and the comparisons with untreated controls were not available [27].
The plausible effect of CoQ10 on reproductive function is attributed to its effect on the antioxidative capacity and energy production in the oocyte [10, 28, 29]. CoQ10, the only synthesized lipid soluble antioxidant in humans, is an essential component of the mitochondrial respiratory chain, serving an important antioxidant function both in mitochondria and in lipid membranes [15]. ROS-induced DNA damage in ovary leads to genomic instability, mutations and apoptosis of oocytes, and is thought to be ameliorated by an antioxidant activity of CoQ10 [22]. CoQ10 has been also shown to improve mitochondrial function and restore energy production by mitochondria [23]. Mitochondrial dysfunction in oocytes results in decreased oxidative phosphorylation and suboptimal levels of mitochondria-generated ATP, which has been strongly associated with poor reproductive performance, including diminished ovarian reserve, poor oocyte quality, abnormal fertilization and deranged preimplantation embryo development [29, 30]. Energy production by mitochondria is important for steroid hormone biosynthesis, oocyte maturation, fertilization, and early embryonic development [31, 32]. It has been demonstrated that CoQ10 supplemented in aged animal model has improved mitochondrial membrane potential, mitochondrial ATP production and mitotic spindle orientation [21]. Treatment with CoQ10 increased the number of ovulated oocytes and reduced ROS in oocytes to the levels observed in young animals, indicating this is an effective strategy to reverse the effect of reproductive ageing [33]. In humans, levels of CoQ10 in the follicular fluid positively correlated with oocyte maturation, embryo grade and pregnancy rate in women undergoing ART [34, 35]. While oocyte appears to be the main target of CoQ10, it remains unclear whether anti-oxidant treatment also improves uterine environment. We did not demonstrate any differences in endometrial thickness, but there were no data to confidently comment on the effect of CoQ10 in intra-uterine milieu.
CoQ10 has been also associated with improved ovarian reserve. In rodents, CoQ10 administration reversed ovarian toxicity of cisplatin, leading to increase in the serum AMH concentrations, improved AMH-positive follicle count and lower number of atretic follicles [22]. Exposure to CoQ10 restored ovarian reserve in mice with induced accelerated oocyte loss [21]. Currently, however, there is relative paucity of information concerning the exact mechanism by which CoQ10 influences ovarian reserve in humans and it is difficult to conclude whether CoQ10 rescues follicles from apoptosis or enhances primordial follicle activation. In this study there was significant decrease in baseline FSH levels after 60 days of CoQ10 administration. It is possible that a change in FSH levels could also have occurred without CoQ10 treatment over a period of two-three months, but this seems unlikely considering that previous study in 287 infertile men showed 14% decline in FSH levels after 3 months of CoQ10 supplementation with continuing decrease throughout 12 months therapy [18]. In contrast, we did not observe improvement in other ovarian reserve markers, namely AMH and AFC and such discrepancy between our and animal studies could be explained by different treatment protocols and variation in physiological parameters between species. In rodents, 8–12 weeks of CoQ10 exposure corresponds to about ¼ of the life span, which is considerably longer interval in relation to a reproductive cycle when compared to analogous treatment period in humans. It has been supposed that two months exposure to CoQ10 could improve energy production in the ovary but might not be long enough to restore prolonged effect of oxidative damage [25]. It should be noted that it takes about three months for a primordial follicle to reach the preovulatory stage [36]. AMH is predominantly produced upon transition from the primordial to primary follicles when they are recruited from the dormant pool and represents early stages of growth [37]. Thus, short duration of CoQ10 administration is likely to influence late events of follicle maturation but may not be sufficient to improve follicle recruitment evidenced by AMH levels. Indeed, the study that reported significant increase in antral follicles in CoQ10, included women who were treated with CoQ10 for an average of 8.8 ± 6.2 months [27].
The optimal timing, duration and dose of CoQ10 supplementation remain unclear. In this study, the duration of treatment was selected arbitrary based on previous study in IVF population [25]. It could be argued that CoQ10 treatment implies a delay in initiation of ART cycle and thus longer pretreatment period may be less acceptable to the patients. It has been demonstrated that CoQ10 is well tolerated and safe for healthy adults at intake of up to 900 mg/day [38]. We were guided by previous experience in selecting the dose of CoQ10, although this was rather intuitive choice [18, 24, 25, 27].
The main strength of this study is that it focused on a specific phenotype within a broad heterogeneous group of women with POR. All the participants shared similar demographic and clinical characteristics and had comparable pre-treatment markers of ovarian activity. In addition, we utilized an unbiased randomization process and applied the similar laboratory and clinical protocols to all the participants.
The important limitation of our study was its small sample size and we were unable to detect significant differences in clinical outcomes. Live birth is an ultimate outcome of infertility treatment and is more appropriate estimate for patient counseling. The POSEIDON group has recently suggested that the number of oocytes needed to obtain at least one euploid embryo per patient is a more practical treatment endpoint for the studies in women with POR and helps to define the short-term goals for management [10]. In adopting this approach, we chose the number of high quality embryos as a primary outcome measure and calculated the sample size accordingly. High drop-out rate in the study group due to CoQ10 discontinuation was additional limiting factor that should be considered in future studies. In line with the reported by others, CoQ10 administration did not cause any adverse reactions or side effects in this study [39], but all women who discontinued treatment reported difficulty to comply with the CoQ10 regime requiring three times a day administration. Finally, in this study we did not evaluate the levels of oxidative stress markers before or after treatment and did not assess the influence of other lifestyle factors that may pose women at higher risk. A threshold effect of CoQ10 may vary on individual level due to interference with other environmental exposures leading to oxidative stress and this should be considered in the design of future studies.
In summary, pretreatment with CoQ10 increases ovarian response to stimulation and improves oocyte and embryo quality in young low prognosis patients with diminished ovarian reserve. There is a possible beneficial effect on clinical pregnancy and live birth rates, but this needs to be confirmed in larger randomized controlled studies. Further work is required to establish the optimal length, timing and dosage of treatment and to evaluate the therapeutic effect of CoQ10 supplementation in other subgroups of low prognosis women with POR.
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We sincerely thank the women who participated in the study. We also thank the medical and nursing staff of the Reproductive Medical Center of Peking University Third Hospital for their assistance in patient recruitment and management.
This study was supported by National key research and development project (2016YFC1000302) and the scientific research foundation for the returned overseas Ministry of Education (A70538–3).
Department of Reproduction, Beijing Obstetrics and Gynecology Hospital, Capital Medical University, Beijing, 100026, China
Yangying Xu & Shuyu Wang
Reproductive Medical Center, Department of Obstetrics and Gynecology, Peking University Third Hospital, Beijing, 100123, China
Yangying Xu, Victoria Nisenblat, Cuiling Lu, Rong Li, Jie Qiao & Xiumei Zhen
Key Laboratory of Assisted Reproduction, Ministry of Education, Beijing, China
Yangying Xu
Victoria Nisenblat
Cuiling Lu
Jie Qiao
Xiumei Zhen
Shuyu Wang
YYX took part in the patient enrolment, management and follow-up, performed data analysis and prepared the first draft of the manuscript; VN contributed to study design, data analysis and preparation of the manuscript; CL was involved in embryological experiments and contributed to study design; RL and JQ contributed to study design and were involved in critical discussions; XZ contributed to the concept, design and preparation of the manuscript and was involved in patient enrolment and management; SW coordinated the research and contributed to study design and critical discussions. All authors read and approved the final manuscript.
Correspondence to Xiumei Zhen or Shuyu Wang.
This study was approved by the Ethics Committee of Peking University Third Hospital (#2014091). The clinical trial registration ID on the Chinese Clinical Trial Registry is ChiCTR-IPR-17010945, http://www.chictr.org.cn/index.aspx. All the participants provided written consent to participate in the study.
Xu, Y., Nisenblat, V., Lu, C. et al. Pretreatment with coenzyme Q10 improves ovarian response and embryo quality in low-prognosis young women with decreased ovarian reserve: a randomized controlled trial. Reprod Biol Endocrinol 16, 29 (2018). https://doi.org/10.1186/s12958-018-0343-0
Poor ovarian response
POSEIDON stratification
High-quality embryos | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 1,655 |
\section*{Introduction}
Brain actively \textit{untangles} the input sensory data and fits them in behaviorally relevant dimensions that enables an organism to perform recognition effortlessly, in spite of variations \cite{dicarlo2012does, thorpe1996speed, dicarlo2007untangling}. For instance, in visual data, object translation, rotation, lighting changes and so forth cause complex nonlinear changes in the original input space. However, the brain still extracts high-level behaviorally relevant constructs from these varying input conditions and recognizes the objects accurately. What remains unknown is how brain accomplishes this untangling.
Here, we introduce the concept of chaos-guided input structuring in a reservoir computing network that provides an avenue to untangle stimuli in the input space and improve the ability of a stimulus to entrain neural dynamics. Specifically, we show that the complex dynamics arising from the recurrent structure of a randomly connected reservoir \cite{rajan2006eigenvalue, kadmon2015transition, stern2014dynamics} can be used to extract an explicit phase relationship between the input stimulus and the spontaneous chaotic neuronal response. Then, aligning the input phase along the dominant projections determining the intrinsic chaotic activity, causes the random chaotic fluctuations or trajectories of the network to become locally stable channels or dynamic attractor states that, in turn, improve its' inference capability. In fact, using mean field analysis, we derive the effect of introducing varying phase association between the input and the network's spontaneous activity. Our results demonstrate that successful formation of stable attractors is strongly determined from the input alignment.
\section*{Model Description}
We describe the effect of chaos guided input structuring on a standard firing-rate based reservoir model of \textit{N} interconnected neurons. Specifically, each neuron in the network is described by an activation variable $x_i$ $\forall i= 1,2,...N$, satisfying
\begin{equation}
\tau dx_i/dt = -x_i + \sum_{j=1}^{N} W_{ij} r_j + W_{Input}I;\hspace{1mm}
z = \sum_{j=1}^{N} W_{Out} r_j
\end{equation}
where $r_i(t) = \phi(x_i(t))$ represents the firing rate of each neuron characterized by the nonlinear response function, $\phi(x) = tanh(x)$ and $\tau =10 ms$ is the neuron time constant. $W$ represents a sparse $N\times N$ recurrent weight matrix (with $W_{ij}$ equal to the strength of the synapse connecting unit $j$ to unit $i$) chosen randomly and independently from a Gaussian distribution with $0$ mean and variance, $g^2/p_c N$ \cite{van1996chaos, van1998chaotic}, where $g$ is the synaptic gain parameter and $p_c$ is the connection probability between units.
The output unit $z$ reads out the activity of the network through the connectivity matrix, $W_{Out}$, with initial values drawn from a Gaussian distribution with 0 mean and variance $1/N$. The readout weights are trained using Recursive Least Square (RLS) algorithm \cite{laje2013robust, sussillo2009generating, jaeger2004harnessing}. The input weight matrix, $W_{Input}$, is drawn from a Gaussian distribution with zero mean and unit variance. The external input, $I$, is an oscillatory sinusoidal signal, $I = I_0 cos(2\pi ft + \chi)$, with amplitude $I_0$, frequency $f$, that is the same for each unit $i$. Here, we use a phase factor $\chi$ chosen randomly and independently from a uniform distribution between $0$ and $2\pi$. This ensures that the spatial pattern of input is not correlated with the recurrent connectivity initially. Through input alignment analysis, we then obtain the optimal phases to project the inputs in the preferred direction of the network's spontaneous activity. In all our simulations (without loss of generality), throughout the paper we have assumed, $p_c = 0.1, N=800, g=1.5, f= 10 Hz$.
\begin{figure*}
\includegraphics[width=\linewidth]{fig1_2_comb1.pdf
\caption{\label{fig2} (a) Cartoon depicting the angle between subspace defined by the first two PCs of chaotic activity (blue) and input driven activity (red). (b) [Left] Projections of the reservoir activity in the 3D space of PC Vectors 1, 2, 3 - / [Right] Trajectories of 5 reservoir neurons across 10 different trials - driven by input rotated by $\theta_{rotate}$. (c) Relationship between input temporal phase ($\Theta$) and the orientation of driven activity with respect to chaotic subspace ($\theta_{rotate}$) for varying input amplitude and frequency. (d) A reservoir framework [Left] stimulated by a brief sinusoidal input ($t=0 - 50 ms$) trained to generate a timed output response shown in [Right]. (e) Cartoon showing the different angles in the chaotic subspace along which the inputs can be aligned. (f) Euclidean distance between trajectories of same (and different) inputs plotted for different orientation of $I_1, I_2$ in the chaotic subspace. (g) Projections of the reservoir activity in the 3D space of PC Vectors 1, 2, 3 corresponding to the two inputs $I_1, I_2$ for different alignment conditions.}
\vspace{-4mm}
\end{figure*}
\section*{Subspace Alignment}
First, we ask the question how is the subspace of input driven activity aligned with respect to the subspace of spontaneous activity of a recurrent network. Using Principal Component Analysis, we observed that the input-driven trajectory converges to a uniform shape becoming more circular with increasing input amplitude \cite{S1ab}. We utilize the concept of principal angles, introduced in \cite{rajan2010inferring, ipsen1995angle}, to visualize the relationship between the chaotic and input driven (circular) subspace. Specifically, for two subspaces of dimension $D_1$ and $D_2$ defined by unit principal component vectors (that are mutually orthogonal) ${V_1}^a$, for $a = 1,2,...D_1$ and ${V_2}^b$, for $b = 1,2,...D_2$, the angle between them is
\begin{equation}
\theta = arccos(min(SingularValueOf({V_1}^a . {V_2}^b)))
\label{eq3}
\end{equation}
Fig. \ref{fig2} (a) schematically represents the angle between the circular input driven network activity and the irregular spontaneous chaotic activity. Here, $\theta_{chaos}$ (and $\theta_{driven}$) refers to the subspace defined by the first two Principal Components (PCs) of the intrinsic chaotic activity (and periodic driven activity). It is evident that rotating the circular orbit by $\theta_{rotate}$ will align it along the chaotic trajectory projection.
We observe that aligning the inputs in directions (along dominant PCs) that account for maximal variance in the chaotic spontaneous activity facilitates intrinsic noise suppression at relatively low input amplitudes, thereby, allowing the network to produce stable trajectories. For instance, instead of using random phase input, we set $I = I_0 cos(2\pi ft + \Theta)$ and visualize the network activity as shown in Fig. \ref{fig2} (b). Even at lower amplitude of $I_0 =1.5$, we observe a uniform circular orbit (in the PC subspace) for the network activity that is characteristic of reduction in intrinsic noise and input sensitization. In fact, even after the input is turned off after $t=50 ms$, the neural units yield stable and synchronized trajectories with minimal variation across different trials (Fig. \ref{fig2} (b, Right)) in comparison to the random phase input driven network (of higher amplitude) in \cite{S1ab}. This shows the effectiveness of subspace alignment for intrinsic noise suppression. In addition, working in low input-amplitude regimes offers an additional advantage of higher network dimensionality \cite{S1c}, that in turn improves the overall discriminative ability of the network. Note, previous work \cite{rajan2010inferring, rajan2010stimulus} have shown that spatial structure of the input does not have a keen influence on the spatial structure of the network response. Here, we bring in this association explicitly with subspace alignment.
$\Theta$, in the above analysis, is the input phase that corresponds to a subspace rotation of driven activity toward spontaneous chaotic activity. We observe that the temporal phase of the input contributes to the neuronal activity in a recurrent network. Fig. \ref{fig2} (c) illustrates this correlation wherein the input phase determines the orientation of the input-driven circular orbit with respect to the dominant subspace of intrinsic chaotic activity. For a given input frequency ($f =10 Hz$), input phase, $\Theta = 83.2^{\circ}$, aligns the driven activity ($\theta_{driven}$) along the chaotic activity ($\theta_{chaos}$) resulting in $\theta_{rotate} =0^{\circ}$ for varying input amplitude ($I_0 = 1.5, 3$).
An interesting observation here is that the frequency of the input modifies the orientation of the evoked response that yields different input phases at which $\theta_{chaos}$ and $\theta_{driven}$ are aligned (refer to Fig. \ref{fig2} (c, Right)). We also observe that the subspace alignment is extremely sensitive toward the input phase in certain regions with abrupt jumps and non-smooth correlation. This non-linear behavior is a consequence of the recurrent connectivity that overall shapes the complex interaction between the driving input and the intrinsic dynamics. While this correlation yields several important implications for network modeling experiments, we utilize this behavior for subspace alignment. Consequently, in all our experiments, for a given $\theta_{rotate}$, we find a corresponding input phase $\Theta$ that approximately aligns the input in the preferred direction.
\section*{Impact of Input Structuring on Discriminative Capability}
Next, we describe the implication of input alignment along the chaotic projections on the overall learning ability of the network.
First, we trained a recurrent network with two output units to generate a timed response at $t= 1s$ as shown in Fig. \ref{fig2} (d, Right). Two distinct and brief sinusoidal inputs (of 50 ms duration and amplitude $I_0 = 1.5$) were used to stimulate the recurrent network. The network trajectories produced were then mapped to the output units using RLS training (to learn the weights $W_{Out}$). Here, the network (after readout training) is expected to produce timed output dynamics at readout unit 1 or 2 in response to input $I_1$ or $I_2$, respectively. The network is reliable if it generates consistent response at the readout units across repeated presentations of the inputs during testing, across different trials. This simple experiment utilizes the fact that neural dynamics in a recurrent network implicitly encode timing that is fundamental to the processing and generation of complex spatio-temporal patterns. Note, in such cases of multiple inputs, values of both inputs are zero, except for a timing window during which one input is briefly turned on in a given trial.
Since both the inputs, in the above experiment, have same amplitude and frequency dynamics, the circular orbit describing the network activity in the input-driven state (for both inputs) is almost similar giving rise to one principal angle ($\theta_{{driven}_{1,2}}$ in Fig. \ref{fig2} (e)) for the input subspace. To discriminate between the output responses for the two inputs, it is apparent that the inputs have to be aligned in different directions. One obvious choice is to align each input along two different principal angles defining the chaotic spontaneous activity (i.e. $I_1$ along $\angle PC1.PC2$ and $I_2$ along $\angle PC3.PC4$). Note, $\angle PC1.PC2$ denotes the angle $\theta$ calculated using Eqn. \ref{eq3}. Another approach is to align $I_1$ along $\angle PC1.PC2 \equiv \theta_{chaos}$ and $I_2$ along $\angle PC1.PC2 + 90^\circ \equiv \theta_{chaos,90^\circ}$ as shown in Fig. \ref{fig2} (e). We analyze the latter in detail as it involves input phase rotation in one subspace that makes it easier for formal theoretical analysis.
To characterize the discriminative performance of the network, we evaluated the Euclidean distances (measured as $\sqrt{1/N\sum_{i=1}^{N}(r_{i,1}(t)-r_{i,2}(t))^2}$ , where $r_1(t)$ ($r_2(t)$) is the firing rate activity of the network corresponding to $I_1$ ($I_2$)) between the inter-/intra-input trajectories in response to different inputs ($I_1, I_2$), and to a slightly varied version of the same input (for instance, $I_{1,2} = (I_0 +\epsilon)cos(2\pi ft + \Theta_{1,2})$ where $\epsilon$ is a random number between $[0, 0.5]$) and $\Theta_1 (\Theta_2)$ is the input phase that aligns $I_1$ ($I_2$) along $\theta_{chaos}$ ($\theta_{chaos, 90^\circ}$). The inter-/intra-input trajectory distances are plotted in Fig. \ref{fig2} (f) for both scenarios-with and without input alignment. It is desirable to have larger inter-trajectory distance and small intra-trajectory distance such that the network easily distinguishes between two inputs while being able to reproduce the required output response even when a particular input is slightly perturbed. We observe that aligning the inputs in direction parallel and perpendicular to the dominant projections (Fig. \ref{fig2} (f, Middle)) increases the inter-trajectory distance compared to the non-aligned case (Fig. \ref{fig2} (f, Top)) while decreasing the intra-input trajectory separation. This further ascertains the fact that subspace alignment reduces intrinsic fluctuations within a network thereby enhancing its prediction capability. Note, without input alignment, the intrinsic fluctuations cannot be overcome with low-amplitude inputs ($I_0=1.5$). Hence, for fair comparison and to obtain stable readout-trainable trajectory in the non-aligned case, we use a higher input amplitude of $I_0 =3$.
We hypothesize that intrinsic noise suppression occurs as input subspace alignment along dominant projections (that account for maximal variance such as $PC1, PC2$) causes chaotic trajectories along different directions (in this case, along $\theta_{chaos}$, $\theta_{chaos,90^\circ}$) to become locally stable channels or \textit{attractor states}. These attractors behave as potential wells (or local minima from an optimization standpoint) toward which the network activity converges for different inputs. Thus, the successful formation of stable yet distinctive attractors for different inputs are strongly influenced by the orientation along which the inputs are aligned. As a consequence of our hypothesis, depending upon the orientation of the input with respect to the dominant chaotic activity ($\theta_{chaos}$ in Fig. \ref{fig2} (e)), the extent of noise suppression will vary, for a particular trajectory, that will eventually alter the stability of the attractor states. To test this, we rotated $I_2$ (from $\theta_{chaos,90^\circ}$) further by $90^\circ$ ($\theta_{chaos,180^\circ}$ in Fig. \ref{fig2} (e)) and monitored the intra-trajectory distance. Note, $I_1$ and $I_2$ are anti-phase correlated in the chaotic subspace. In Fig. \ref{fig2} (f, Middle) corresponding to $90^\circ$ phase difference between ($I_1, I_2$), $I_2$ corresponds to a more stable attractor than $I_1$ since the intra-distance for the former is lower. In contrast, in Fig. \ref{fig2} (f, Right) corresponding to $180^\circ$ phase difference, $I_1$ turns out be more stable than $I_2$. Note, the $90^\circ, 180^\circ$ phase difference between $I_1, I_2$ (mentioned above and in the remainder of the paper) refers to the phase difference between the inputs in the chaotic subspace after subspace alignment using $\Theta$. For our analysis, $\Theta_1 = 83.2^\circ, \Theta_2 = 111^\circ$ yields $\sim 90^\circ$ phase between $I_1, I_2$ in chaotic subspace, while $\Theta_1 = 83.2^\circ, \Theta_2 = 263.2^\circ$ yields $\sim 180^\circ$ phase.
In addition to the trajectory distance, visualizing the network activity in the 3-D PC space (Fig. \ref{fig2} (g)), also, shows the influence of input orientation (and hence the phase correlation) toward formation of distinct attractor states. Since $I_1, I_2$ are aligned in the subspace defined by $\angle PC1.PC2$, the 2D projection of the circular orbit onto PC1 and PC2 in both input aligned scenarios ($90^\circ$, $180^\circ$ phase) are comparable. However, the third dimension, PC3, marks the difference between the two input projections. In fact, the progress of the network activity as time evolves (shown by dashed arrows in Fig. \ref{fig2} (g)) follows a completely different cycle for the input aligned scenarios. The change in the overall rotation cycle from anti-clockwise ($I_2$ with $90^\circ$ phase, Fig. \ref{fig2} (g, Middle)) to clockwise ($I_2$ with $180^\circ$ phase, Fig. \ref{fig2} (g, Bottom)) can be viewed as an indication toward the altering of the attractor state stability. On the other hand, the non-aligned case with $I_0 =3$ yields incoherent and more random trajectory (Fig. \ref{fig2} (g, Top) representative of intrinsic noise. In order to get more coherent activity and to suppress the noise further, we need to increase the input amplitude to $I_0 \ge 5$ as shown in \cite{S1ab}.
\section*{Mean Field Analysis}
To explain the above results analytically, we use mean-field methods developed to evaluate the properties of random network models in the limit $N \rightarrow \infty$ \cite{rajan2010stimulus, sompolinsky1988chaos}. A key quantity in Mean Field Theory (MFT) is the average autocorrelation function that characterizes the interaction within the network as
\begin{equation}
C(\tau) = 1/N \sum_{i=1}^{N} <\phi (x_i(t)) \phi (x_i(t+\tau))>
\label{eq4}
\end{equation}
where $<>$ denotes the time average. The main idea of MFT is to replace the network interaction term in Eqn. 1 by Gaussian noise $\eta$ such that $\frac{d{x_i}^1}{dt} = -{x_i}^1 +\eta$, where $x_i = {x_i}^0 + {x_i}^1$ and ${x_i}^0 (t) = A cos(2\pi ft + \zeta)$ with $A = I_0/\sqrt{1+(2\pi ft)^2}$. Here, $\zeta$ incorporates the averaged temporal phase relationship between the reservoir neurons and the input induced by input subspace alignment, $\zeta(\theta_{rotate}) = \Theta$. The temporal correlation of $\eta$ is calculated self-consistently from $C(\tau)$. For self-consistence, the first and second moment of $\eta$ must match the moments of the network interaction term. Thus, we get $<\eta_i (t)> = 0$ as mean of the recurrent synaptic matrix $<W_{ij}> = 0$. For calculating the second moment , we use the identity $<W_{ij}W_{kl}> = g^2 \delta_{ij} \delta_{kl}/N$ and obtain $<\eta_i (t) \eta_j(t+\tau)> = g^2C(\tau)$. Combining this result with the MFT noise-interaction based network equation yields
\begin{equation}
\frac{d^2 \Delta (\tau)}{d \tau^2} = \Delta(\tau) - g^2C(\tau)
\label{eq5}
\end{equation}
where $\Delta(\tau) = < {x_i}^1 (t) {x_i}^1 (t+\tau)>$. Eqn. \ref{eq5} resembles the Newtonian motion equation of a classical particle moving under the influence of force given by the right hand side of the equation. This force depends on $C$ that, in turn, depends on the input subspace alignment ($\zeta$) which directs the initial position of the particle (or state of the network $\Delta(0)$). From this analogy, it is evident that analyzing the overall potential energy function of the particle (or network) will be equivalent to visualizing the different attractor states formed in a network in response to a particular input stimulus. Thus, we formulated an expression for the correlation function (with certain constraints) using Taylor series expansion, that allows us to derive the force and hence the dynamics of the network under various input alignment conditions.
The non-linear firing rate function $r(x) = \phi(x) = tanh(gx)$ can be expanded with Taylor series for small values of $g$, i.e. $g = 1 + \delta$, where $\delta$ denotes a small increment in $g$ beyond $1$. Note, $g=1+\delta$ satisfies the criterion, $g >1$ \cite{rajan2006eigenvalue, sompolinsky1988chaos}, to operate the networks in chaotic regime. Also, the overall network statistics does not change with $g$ being expressed as a gain factor in the firing-rate function instead of overall synaptic strength. Using $tanh(gx) \simeq gx-1/3g^3x^3 +2/15g^5x^5$, we can express $C(\tau)$ from Eqn. \ref{eq4} as $C = 1/2g^2A^2cos(\omega \tau +2\zeta) + l(g^2-2g^4m) + 2/3g^6l^3$, where $m=\Delta(0), l=\Delta(\tau), \omega = 2\pi f$. Now, we can express Eqn. \ref{eq5} as $\frac{d^2l}{d \tau^2} = l- C$. Writing $l=k\delta$ due to the small limit of $g$, Eqn. \ref{eq5} simplifies to
\begin{equation}
\frac{d^2k}{dt^2} = -G + n k -2/3k^3
\label{eq10}
\end{equation}
where $G = g^2A^2cos(\omega \tau +2\zeta)/(2\delta^3) $ and $n$ is a parameter defined in terms of $m, \delta$. Note, supplementary section \cite{S2} provides a detailed derivation of Eqn. \ref{eq10} and comments about the assumptions on initial conditions. Note, Eqn. \ref{eq10} is an approximate version of Eqn. \ref{eq5} that depicts network activity in the manner of Newtonian motion independent of all intrinsic time (or averaging parameters) while taking into account the influence of input alignment. Now, we can express the potential of the network driven by a force, $F$, equivalent to the right hand side of Eqn. \ref{eq10} as
\begin{equation}
V= -\int{Fdk} = Gk-n/2k^2+k^4/6
\label{eq11}
\end{equation}
We solve Eqn. \ref{eq10}, \ref{eq11} with initial conditions $k(0) =1$, $\dot{k}(0) =0$ and monitor the change in force, $F$, and potential, $V$, for different values of $G$. First, let us examine the attractor state formation when there is no input stimulus (i.e. $G=0$) by visualizing the potential $V$. For $G=0$, the expressions for force and potential become
\begin{equation}
F(k) = nk-2/3k^3 ;\hspace{2mm} V(k) = -nk^2/2+k^4/6
\label{eq12}
\end{equation}
\begin{figure}
\centering
\includegraphics[width=0.6\linewidth]{fig4_1.pdf}
\vspace{-4mm}
\caption{\label{fig4} Evolution of potential energy (and hence attractor state formation) by varying the input $G$ as a function of $\zeta$}
\vspace{-4mm}
\end{figure}
Fig. \ref{fig4} shows the evolution of potential energy as $k$ varies for different $G$. Since external input $G=0$, the network dynamics is chaotic that results in the formation of potential wells that are both equally stable. The network activity will thus converge to any one of these wells (that can be interpreted as attractor states) depending upon the initial state or starting point. This supports the observation in \cite{S1ab} that a network with no input yields chaotic activity with incoherent and irregular trajectory for every trial. For nonzero G, the force (and potential) equation will be dependent on $\zeta$ since $G \simeq cos(\omega \tau +2\zeta)$. For different values of $\zeta$, we solved for $V$ (Eqn. \ref{eq11}) numerically and plotted the potential evolution as shown in Fig. \ref{fig4}. For $\zeta =\pi/4$, the potential well is more attractive on the left end. This validates the fact that intrinsic fluctuations are suppressed in the presence of an input. For $\zeta =\pi /2$, the left attractor becomes more stable. Changing $\zeta$ further shows that the potential well on the right end becomes more stable. This result confirms that input subspace alignment with respect to the initial chaotic state influences the overall stability and convergence capability of a recurrent network.
The fact that stability corresponding to different attractor states ($\zeta =\pi /2, \pi$) arises, qualifies our earlier hypothesis that input orientation with respect to the chaotic subspace alters the attractor state stability, corroborating the result of Fig. \ref{fig2} (f). Finally, we illustrate the effectiveness of input alignment on a complex motor pattern generation task with reliable generation of learnt handwritten patterns over multiple trials, even in presence of perturbations \cite{laje2013robust}. The detailed analysis and results are shown in the supplementary \cite{S3}.
Note, we solved Eqn. \ref{eq11} by setting some initial and boundary value conditions on $k$ and by iterating over different $n$ until we reached a steady state solution. Changing these conditions will result in a completely new set of $\zeta$ values (different from those in Fig. \ref{fig4}). Nevertheless, we will observe a similar evolution of the potential well and change in attractor state stability as Fig. \ref{fig4}. Furthermore, the MFT calculations use $\zeta$ to denote a functional relationship between subspace alignment and input phase that eventually affects the attractor state stability. In the future, we will examine the real-time evaluation of $\zeta$ and its' impact on the analytical studies. Finally, the constraint under which we derive the potential energy functions and show the altering of attractor state is $g = 1 + \delta$. We expect all our results to be valid for large $g$ as well since Eqn. \ref{eq5} (that was simplified with Taylor expansion) still remains unchanged.
\section*{Conclusion}
Models of cortical networks often use diverse plasticity mechanisms for effective tuning of recurrent connections to suppress the intrinsic chaos (or fluctuations) \cite{laje2013robust,panda2017learning}. We show that input alignment alone produces stable and repeatable trajectories, even, in presence of variable internal neuronal dynamics for dynamical computations. Combining input alignment with recurrent synaptic plasticity mechanism can further enable learning of stable correlated network activity at the output (or readout layer) that is resistant to external perturbation to a large extent. Furthermore, since input subspace alignment allows us to operate networks at low amplitude while maintaining a stable network activity, it provides an additional advantage of higher dimensionality. A network of higher dimensionality offers larger number of disassociated principal chaotic projections along which different inputs can be aligned \cite{S1c}. Thus, for a classification task, wherein the network has to discriminate between 10 different inputs (of varying frequencies and underlying statistics), our notion of \textit{untangling} with chaos-guided input structuring can, thus, serve as a foundation for building robust recurrent networks with improved inference ability. Further investigation is required to examine which orientations specifically improve the discrimination capability of the network and the impact of a given alignment on the stability of the readout dynamics around an output target. In summary, the analyses we present suggest that input alignment in the chaotic subspace has a large impact on the network dynamics and eventually determines the stability of an attractor state. In fact, we can control the network's convergence toward different stable attractor channels during its voyage in the neural state space by regulating the input orientation. This indicates that, besides synaptic strength variance \cite{rajan2006eigenvalue}, a critical quantity that might be modified by modulatory and plasticity mechanisms controlling neural circuit dynamics is the input stimulus alignment.
\subsection*{Acknowledgments}
P.P. and K.R. are supported in part by Center for Brain-inspired Computing (C-BRIC), an SRC and DoD sponsored center, the Semiconductor Research Corporation, the NSF, Intel Corporation and by the DoD Vannevar Bush Fellowship.
\section*{Supplementary Material}
\section{PCA for examining Network Activity}
\begin{figure}
\includegraphics[width = \linewidth]{fig1_1.pdf
\caption*{\textbf{Figure S1} (a) [Left] For zero input, projections of the chaotic spontaneous activity onto PC vectors 1, 5, 50 , [Middle] Visualization of the chaotic trajectory in 3D subspace composed of dominant PC vectors 1,2,3 that account for significant variance in network activity, [Right] Trajectories of 5 reservoir neurons across 10 different trials. (b) Same as panel (a), but for non-chaotic input driven activity with input amplitude $I_0 =5$. (c) Effective dimensionality of the network at different input amplitudes for g=1.5, g=2.5.}
\end{figure}
To examine the structure of the recurrent network's representations, we visualize and compare the neural trajectories in response to varying inputs using Principal Component Analysis (PCA) \cite{rajan2010inferring}. The network state at any given time instant can be described by a point in the \textit{N}-dimensional space with coordinates corresponding to the firing rates of the \textit{N} neuronal units. With time, the network activity traverses a trajectory in this N-dimensional space and we use PCA to outline the subspace in which this trajectory lies. To conduct PCA, we diagonalize the equal-time cross-correlation matrix of the firing rates of the \textit{N} units as
\begin{equation}
D_{ij} = <(r_i(t)-<r_i>)(r_j(t)-<r_j>)>
\end{equation}
where the angle brackets, $<>$, denote time average and $r(t)$ denotes the firing rate activity of the neuron. The eigenvalues of the matrix $D$ (specifically, $\lambda_a/\sum_{a=1}^{N}\lambda_a$, where $\lambda_a$ is the eigenvalue corresponding to principal component $a$) indicate the contribution of different Principal Components (PCs) toward the fluctuations/total variance in the spontaneous activity of the network.
Fig. S1 shows the impact of varying input amplitude ($I_0$) on the spontaneous chaotic activity of the network. For $I_0 = 0$, the network is completely chaotic as is evident from the highly variable projections of the network activity onto different Principal Components (PCs) as shown in Fig. S1 (a, Left)
Generally, the leading $10-15\%$ (depending upon the value of $g$) of the PCs account for $\sim 95\%$ of the network's chaotic activity \cite{rajan2010inferring}. Visualizing the network activity in a 3D space composed of the dominant principal components (PC1, 2, 3) shows a random and irregular trajectory characteristic of chaos (Fig. S1 (a, Middle). In fact, plotting the trajectories (firing rate $r(t)$ of the neuron as time evolves) of 5 recurrent units in the network (Fig. S1 (a, Right) shows diverging and incoherent activity across 10 different trials, also, representative of intrinsic chaos. In addition, the projections of the network activity onto components with smaller variances fluctuate more rapidly and irregularly (Fig. S1 (a, Left)). This further corroborates the fact that the leading PCs define a network's spontaneous chaotic activity.
Driving the recurrent network with a sinusoidal input of high amplitude (Fig. S1 (b)) sensitizes the network toward the input, thereby, suppressing the intrinsic chaotic fluctuations. The PC projections of the network activity are relatively periodic.
A noteworthy observation here is that the trajectories of the recurrent units (Fig. S1 (b, Right) become more stable and consistent across 10 different presentations of the input pattern with increasing amplitude. A readout layer appended to a recurrent network can be easily trained on these stable trajectories for a particular task. Thus, the input amplitude determines the network's encoding trajectories and in turn, its' inference ability. In fact, the chaotic intrinsic activity is completely suppressed for larger inputs. However, this is not preferred as input dominance drastically declines the discriminative ability of a network that can be justified by dimensionality measurements. The effective dimensionality of a reservoir is calculated as $N_{eff} = \sum_{a=1}^{N}(\lambda_a^2)^{-1}$ that provides a measure of the effective number of PCs describing a network's activity for a given input stimulus condition. Fig. S1 (c) illustrates how the effective dimensionality decreases with increasing input amplitude for different $g$ values. It is, hence, critical that input drive be strong enough to influence network activity while not overriding the intrinsic chaotic dynamics to enable the network to operate at the \textit{edge of chaos}. Note, higher $g$ in Fig. S1 (c) yields a larger dimensionality due to richer chaotic activity.
In our simulations in Fig. S1 (b), the input is shown for $50 ms$ starting at $t =0$. Thus, we observe that the trajectories of the recurrent units are chaotic until the input is turned on. Although the network returns to spontaneous chaotic fluctuations when the input is turned off (at $t = 50 ms$), we observe that the network trajectories are stable and non-chaotic that is in coherence with the previous findings from \cite{bertschinger2004real,rajan2010stimulus}. From the visualization of network activity in the dominant PC space, we see that the input-driven trajectory converges to a uniform shape becoming more circular (along PC1 and PC2 dimensions) with higher input amplitude (Fig. S1 (b)). This informs us that the orbit describing the network activity in the input-driven state consists of a circle in a two-dimensional subspace of the full N-dimensional hyperspace of the neuronal activities. Note, all simulations in supplementary are conducted with similar parameters mentioned in the manuscript, i.e., $ N=800, f= 10Hz, p_c =0.1$.
\section{Mean Field Derivation with Taylor Series}
First, let us solve for ${x_i}^1$ such that we can get an expression for the correlation function, $C(\tau)$ in Eqn. 3 of main manuscript. Noting that, ${x_i}^1$ is driven by Gaussian noise (as indicated by the MFT noise-interaction equation: $\frac{d{x_i}^1}{dt} = -{x_i}^1 +\eta$), we can assume their moments as $<{x_i}^1(t)> = <{x_i}^1(t + \tau)> = 0$, $<{x_i}^1(t) {x_i}^1(t)> = <{x_i}^1(t + \tau) {x_i}^1(t + \tau)> = \Delta(0)$ and $<{x_i}^1(t) {x_i}^1(t + \tau)> =\Delta(\tau)$. $x^1(t)$ (dropping index $i$ as all neuronal variables have similar statistics) can then be written as
\begin{equation}
x^1(t) = \alpha z_1 + \beta z_3; \hspace{3mm}
x^1(t+\tau) = \alpha z_2 + \gamma z_3
\label{eq6}
\end{equation}
where $z_1, z_2, z_3$ are Gaussian random variables with 0 mean/unit variance and $\alpha = \sqrt{\Delta(0) - |\Delta(\tau)|}, \beta = sgn(\Delta(\tau))\sqrt{|\Delta(\tau)|},\\ \gamma= \sqrt{|\Delta(\tau)|}$. Now, writing $x=x^0 + x^1$, $C$ is computed by integrating over $z_1, z_2, z_3$ as
\begin{eqnarray}
C(\tau) = && 1/N \sum_{i=1}^{N} <<\phi ({x_i}^0(t)
+ \alpha z_1 + \beta z_3)>_{z_1}\nonumber \\
&& <\phi ({x_i}^0(t+\tau)) + \alpha z_2
+ \gamma z_3>_{z_2}>_{z_3}
\label{eq7}
\end{eqnarray}
where $<f(z)>_z = \int_{- \infty}^{\infty} dz \frac{f(z) exp(-z^2/2)}{\sqrt{2\pi}} $ for $z = z_1, z_2, z_3$. Now, ${x_i}^0 (t) = A cos(2\pi ft + \zeta)$, where $A = I_0/\sqrt{1+(2\pi ft)^2}$ (solve $\frac{d{x_i}^0}{dt} = -{x_i}^0 +I_0 cos(2 \pi ft + \zeta)$ for ${x_i}^0$) and $\zeta$ incorporates the averaged temporal phase relationship between the individual neurons and the input induced by input subspace alignment. Replacing the value of ${x_i}^0$ in Eqn. \ref{eq7}, we get
\begin{eqnarray}
C(\tau) = && 1/N \sum_{i=1}^{N} <<<\phi (Acos(\zeta)+ \alpha z_1 + \beta z_3)>_{z_1}) \nonumber \\
&& <\phi(Acos(\omega \tau + \zeta) + \alpha z_2 + \gamma z_3>_{z_2}>_{z_3})>_\zeta
\label{eq8}
\end{eqnarray}
The above correlation function also satisfies Eqn. 4 of main manuscript. Note, $\omega = 2\pi f$ in Eqn. \ref{eq8}.
Now we solve Eqn. \ref{eq8} using Taylor series approximation for $tanh(gx) = gx-1/3g^3x^3 +2/15g^5x^5$. We have
\begin{eqnarray}
\phi(Acos(\zeta) + \alpha z_1 +\beta z_3)= &&g(Acos(\zeta) + \alpha z_1 +\beta z_3)-1/3g^3(\alpha z_1+\beta z_3)^3+2/15g^5(\alpha z_1 + \beta z_3)^5 \nonumber \\
=&& g(Acos(\zeta) + \alpha z_1 + \beta z_3) -1/3g^3(\alpha^3 {z_1}^3 + 3 \alpha^2 {z_1}^2 \beta z_3 + 3 \alpha z_1\beta^2 {z_3}^2 + \beta^3 {z_3}^3) \nonumber\\
&& +2/15 g^5(\alpha^5 {z_1}^5 + 5 \alpha^4 {z_1}^4 \beta z_3 + 10 \alpha^3 {z_1}^3 \beta^2 {z_3}^2 +10\alpha^2 {z_1}^2 \beta^3 {z_3}^3 \nonumber\\
&&+5 \alpha z_1 \beta^4{z_3}^4+\beta^5{z_3}^5
\label{eq9}
\end{eqnarray}
Now using $<{z_1}^2>_{z_1} =1, <{z_1}^4>_{z_1} =3, <{z_1}^6>_{z_1} =15$ and noting that averages over odd powers of $z_1$ are zero, we get
\begin{eqnarray}
<\phi(Acos(\zeta) + \alpha z_1 +\beta z_3)>_{z_1}= &&g(Acos(\zeta) +\beta z_3)-1/3g^3(3 \alpha^2 \gamma z_3 +\gamma^3 {z_3}^3)\nonumber \\
&&+ 2/15g^5(15\alpha^4 \beta z_3 + 10 \alpha^2\beta^3 {z_3}^3 +\beta^5 {z_3}^5)
\label{eq10}
\end{eqnarray}
Similarly,
\begin{eqnarray}
<\phi(Acos(\zeta +\omega \tau) + \alpha z_2 +\gamma z_3)>_{z_2}= &&g(Acos(\zeta +\omega \tau) +\gamma z_3)-1/3g^3(3 \alpha^2 \gamma z_3 +\gamma^3 {z_3}^3)\nonumber \\
&&+ 2/15g^5(15\alpha^4 \gamma z_3 + 10 \alpha^2\gamma^3 {z_3}^3 +\gamma^5 {z_3}^5)
\label{eq11}
\end{eqnarray}
Multiplying Eqn. \ref{eq10} with Eqn. \ref{eq11} upto the sixth order, we can simplify Eqn. \ref{eq8} as
\begin{eqnarray}
C =&&<<g^2 A^2 cos(\zeta) cos(\zeta +\omega \tau) + g^2 \beta \gamma {z_2}^2 -2 g^4\alpha^2 \beta \gamma {z_3}^2 -1/3 g^4(\beta^3 \gamma +\beta \gamma^3){z_3}^2\nonumber \\
&&+ 4g^6 \alpha^4 \beta \gamma {z_3}^2 + 4/3 g^6 \alpha^2(\beta^3\gamma + \beta \gamma^3){z_3}^4 +2/15 g^6(\beta^5 \gamma +\beta \gamma^5){z_3}^6 \nonumber \\
&& +g^6\alpha^4 \beta \gamma {z_3}^2+1/3g^6\alpha^2(\beta^3\gamma+\beta\gamma^3){z_3}^4+1/9g^6\beta^3\gamma^3{z_3}^6>_{z_3}>_\zeta
\label{eq12}
\end{eqnarray}
Now averaging Eqn. \ref{eq12} over $\zeta$ will still retain the $\zeta$ term unlike general MFT calculations where $\zeta$ gets washed out since there is no fixed relationship between the input and recurrent activity. Here, we use yet another approximation i.e. $Cos(A). Cos(B) \simeq Cos(A+B)$ and finally get $C$ as
\begin{eqnarray}
C =&&1/2g^2 A^2 cos(2\zeta +\omega \tau) +g^2 \beta \gamma - 2 g^4\alpha^2 \beta \gamma - g^4 (\beta^3\gamma + \beta \gamma^3) +5g^6\alpha^4\beta\gamma \nonumber\\
&&+5g^6\alpha^2(\beta^3\gamma + \beta \gamma^3) +2g^6(\beta^5 \gamma +\beta\gamma^5)+5/3 g^6 \beta^3 \gamma^3
\label{eq13}
\end{eqnarray}
Using the definition of $\alpha, \beta, \gamma$ in Eqn. \ref{eq6} and denoting $m=\Delta(0), l=\Delta(\tau)$, we get
\begin{eqnarray}
\alpha^2=m-|l|;\hspace{1mm} \beta\gamma=l;\hspace{1mm} \beta^3\gamma=\beta\gamma^2=|l|l; \hspace{1mm} \beta^5\gamma=\beta\gamma^5=\beta^3\gamma^3=l^3
\label{eq14_1}
\end{eqnarray}
Substituting $\alpha, \beta, \gamma$ values from Eqn. \ref{eq14_1} in Eqn. \ref{eq13}, we get
\begin{eqnarray}
C =&&1/2g^2 A^2 cos(2\zeta +\omega \tau) +g^2 l-2g^4(m-|l|)l-2g^4|l|l+5g^6(m-|l|^2l +10g^6(m-|l|)|l|l+17/3g^6l^3 \nonumber\\
=&& 1/2g^2A^2cos(\omega \tau +2\zeta) + l(g^2-2g^4m) + 2/3g^6l^3
\label{eq14}
\end{eqnarray}
Now, $l$ satisfies $\frac{d^2l}{dt^2}=l-C$. By putting the value of $C$ from Eqn. \ref{eq14} we get
\begin{eqnarray}
\frac{d^2l}{dt^2}=&& 1/2g^2A^2cos(\omega \tau +2\zeta) + (1-g^2-2g^4m)l + 2/3g^6l^3
\label{eq15_1}
\end{eqnarray}
For deriving Eqn.5 from the main manuscript, we use $l=k\delta$ and define $1/2g^2A^2cos(\omega \tau +2\zeta) = G\delta^3$. Then, Eqn. \ref{eq15_1} becomes
\begin{eqnarray}
\delta^3\frac{d^2k}{dt^2}=&& -\delta^3G+\delta(-2\delta+2(1+4\delta)m)k-2/3\delta^3k^3
\label{eq15}
\end{eqnarray}
Now, we introduce the parameter $n$, where $m = \delta +(n/2-4)\delta^2$ to further simplify Eqn. \ref{eq15} as
\begin{eqnarray}
\frac{d^2k}{dt^2} = -G + n k -2/3k^3
\label{eq16}
\end{eqnarray}
Eqn. \ref{eq16} is the force equation (Eqn. 5 in the manuscript) from which we derived the potential energy. Note, the MFT approximate equations are derived with respect to a single input $G$ driving the entire network. Multiple inputs ($G_{in}$ where $in=1,2...$) and corresponding alignment of the inputs along different projections (as in Fig. 1 of main manuscript) will result in a potential well that can be roughly interpreted as a linear combination of $V_{in}$ observed for each input, $G_{in}$. The linear combination will follow a similar evolution profile as shown in Fig. 2 of main manuscript. Thus, the change in intra-trajectory distance for varying alignment of the two inputs in the chaotic subspace, shown in Fig. 1 (f) of main manuscript, is justified from the given analysis.
\section{Handwriting Generation}
To further elucidate the effectiveness of input alignment for complex pattern generation, we trained a recurrent network to generate the handwritten words ``chaos'' and ``neuron'' in response to two different inputs \cite{laje2013robust, S5}. After obtaining the principal angle of the chaotic spontaneous activity, we aligned the input $I_1$ corresponding to ``chaos'' along $\angle PC1.PC2$ using optimal input phase $\Theta$. Then, we monitored the output activity for different orientation (i.e. $90^\circ, 180^\circ$) of input $I_2$, corresponding to ``neuron'', with respect to $I_1$ in the chaotic subspace. The two output units (representing the $x$ and $y$ axes) were trained using RLS to trace the original target locations ($x(t), y(t)$) of the handwritten patterns at each time instant. Fig. S2 (a) shows the handwritten patterns generated by the network across 10 test trials for the scenario when inputs are aligned at $90^\circ$ in the chaotic subspace. We observe similar robust patterns generated for $180^\circ$ phase as well. The notable feature of input alignment is that the chaotic trajectories become locally stable channels and function as dynamic attractor states. This can be visualized from the stable and synchronized trajectories observed for different neural units in Fig. 1 (b, Right) of main manuscript (and in Fig. S1). However, external perturbation can induce more chaos in the reservoir that will overwhelm the stable patterns of activity.
To test the susceptibility of the dynamic attractor states formed with input structuring to external perturbation, we introduced random Gaussian noise onto a trained model along with the standard intrinsic chaos-aligned inputs during testing. The injection of noise alters the net external current received by the neuronal units ($I = \Sigma_i[W_{{Input}_i} I + N_0 rand(i)]$, where $N_0$ is the noise amplitude, $i$ denotes a neural unit in the reservoir and $rand$ is the random Gaussian distribution). Fig. S2 (b) shows the mean squared error (calculated as the average Euclidean distance between the target ($x, y$) and the actual output produced at different time instants, averaged across 20 test trials) of the network for varying levels of noise. As $N_0$ increases, we observe a steady increase in the error value implying degradation in the prediction capability of the network. However, for moderate noise (with $N_0 <0.01$), the network exhibits high robustness with negligible degradation in prediction capability for both the words. Interestingly, for $90^\circ$ phase difference, ``neuron'' is more stable than ``chaos'' with increased reproducibility across different trials even with more noise ($N_0 =0.2$). In contrast, for $180^\circ$ phase, ``chaos'' is less sensitive to noise (Fig. S2 (b)). On the other hand, for $45^\circ$ phase alignment between $I_1, I_2$ in the chaotic subspace, we observe that the network is sensitive even toward slight perturbation ($N_0 = 0.001$). This implies that the attractor states formed, in this case, are very unstable. This further corroborates the fact that the extent of noise suppression and hence the attractor state stability varies based upon the input alignment.
\begin{figure}
\centering
\includegraphics[width=0.8\linewidth]{fig5_1.pdf
\caption*{\textbf{Figure S2} (a) Handwriting patterns generated across 10 test trials in response to $I_1$ [Top] and $I_2$ [Bottom] in absence of external noise (b) Variation of performance (measured as mean squared error) with different noise amplitude shown for each output pattern in different input alignment scenarios. $Chaos_{90,180,45}$ represents the error value obtained during ``chaos'' generation when $I_1, I_2$ have $90^\circ, 180^\circ, 45^\circ$ phase difference in the chaotic subspace, respectively.}
\vspace{-4mm}
\end{figure}
\begin{figure}
\centering
\includegraphics[width = 0.9\linewidth]{fig_supp.pdf
\caption*{\textbf{Figure S3} Generation of handwriting patterns corresponding to two brief sinusoidal inputs $I_1, I_2$ for different phase difference between the inputs in the chaotic subspace. [Left column] shows the patterns generated in absence of external perturbation in one test trial, [Middle Column] shows the trajectories of 5 recurrent units in the reservoir across 10 test trials corresponding to each output pattern, [Right Column] shows the patterns generated in presence external perturbation for one test trial.}
\end{figure}
Fig. S3 shows the handwritten pattern generated in one test trial for different phase alignment between $I_1, I_2$, when $I_1$ is aligned along the principal angle defining the spontaneous chaotic activity of the network. It is noteworthy to mention that the neural trajectories of the recurrent units corresponding to both cases are stable. In fact, we observe in the $90^\circ$ case, the trajectories of neurons responding to $I_1$ that corresponds to output ``chaos'' become slightly divergent and incoherent beyond $1000ms$. In contrast, the trajectories of units responding to the word ``neuron'' are more synergized and coherent throughout the $1500 ms$ time period of simulation. This indicates that the network activity for ``neuron'' converges to a more stable attractor state than ``chaos''. As a result, we see that the network is more robust while reproducing ``neuron'' even in presence of external perturbation ($N_0$, noise amplitude is 0.2). In the $180^\circ$ phase difference case, we see exactly opposite stability phenomena with ``chaos'' converging to more stable attractor.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 713 |
class UserCohort < ApplicationRecord
belongs_to :cohort
belongs_to :user
validates_uniqueness_of :user_id, scope: [:cohort_id]
end
| {
"redpajama_set_name": "RedPajamaGithub"
} | 4,794 |
Die Dampfbierbrauerei Essen-Borbeck, kurz Dampfe genannt, ist eine Brauerei für Dampfbier im Essener Stadtteil Borbeck mit Gaststätte und Biergarten.
Geschichte
Gegründet zum Jahreswechsel 1895/1896 als Schloßbrauerei Marx & Co. wurde, nach Errichtung des alten Sudhauses, des Lagerkellers und des Malzturms, bald die erste Dampfmaschine aufgestellt. Im Jahr 1903 wurde die Brauerei umbenannt in Borbecker Brauereigesellschaft mit beschränkter Haftung, Borbeck (Rheinland). Zwei Jahre später wurden ein neues Sudhaus und neue Kellergebäude gebaut sowie eine leistungsfähigere Dampfmaschine beschafft. Im Jahr 1908 produzierte die Dampf-Bierbrauerei als Rheinisch-Westfälische Brauerei-Aktiengesellschaft Essen, bevor im Jahr 1928 die Essener Bürger-Bräu Aktiengesellschaft mit der Kronen-Brauerei AG Essen-Borbeck fusionierte.
Viele Jahre später, ab 1982, begann eine behutsame Renovierung der Anlage nach alten Plänen. Die alten Sudwerke wurden instand gesetzt und eine moderne Gastronomie in die alten Lagerkeller integriert. Auch der Biergarten mit Bierhalle kam hinzu und wurde 1984 eröffnet.
Seitdem wurde die Anlage um die Braustube der Stern-Brauerei erweitert und Festsaal, Biergarten, Terrassenkomplex, Fassabfüllung und Bürgerstube wurden modernisiert. Im Jahr 2009 feierte die Dampfbierbrauerei ihr 25-jähriges Jubiläum.
Biersorten
Die Brauerei vertreibt ihr klassisches untergäriges helles Dampfbier mit einem Alkoholgehalt von 4,8 %, das naturtrübe Borbecker Zwickelbier mit 5,0 % sowie mit 5,1 % das dunkle Salonbier. Saisonabhängig gibt es nach Karneval das dunkelmalzige Borbecker Fastenbier, etwas später im Jahr den Borbecker Maibock, im Herbst das Borbecker Erntedankbier und schließlich den dunklen Winterbock mit 7 % Alkohol. Ein naturtrübes Borbecker Radler ergänzt das Angebot.
Weblinks
Homepage der Dampfbierbrauerei
Die Dampfe bei Borbeck.de
Einzelnachweise
Produzierendes Unternehmen (Essen)
Brauerei (Nordrhein-Westfalen)
Gegründet 1895
Borbeck-Mitte
Bier (Essen) | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 7,803 |
German news
TheHealthGuild
Home Medical News According to experts, it is time to reevaluate the subtype of Parkinson's...
According to experts, it is time to reevaluate the subtype of Parkinson's disease
TheHealthReporter
An international working group says it is time to re-evaluate the current subtype Parkinson's disease (PD), which he says has dubious clinical applicability and biological relevance.
The analysis of the studies underlying the current PD subtype has multiple limitations, including the lack of longitudinal data.
"Most of the studies were, for example, cross-sectional," said study author and working group member Tiago A. Mestre, MD, PhD, associate professor of neurology at the University of Ottawa, Ontario , Canada. Medscape Medical News.
He also noted that the vast majority of studies focused on clinical biomarkers rather than targets.
The researchers noted that future research on PD subtypes should use longitudinal data and include participants whose cases had the same duration of the predefined disease.
The conclusions were published online April 13 a Parkinson's Disease Journal.
The international working group is investigating
PD is characterized by clinical and pathological heterogeneity. Previously, researchers have attempted to identify subtypes of the disease to understand its heterogeneity, clarify its pathophysiology, and develop better and more specific treatments. It is unclear to what extent research on disease subtypes has achieved these goals.
The International Parkinson's Disease and Movement Disorder Society established the working group for Parkinson's disease subtypes in 2018. The goals of the working group were to evaluate the subtype studies conducted so far and indicate areas for future studies in this field.
For the analysis, the working group sought PubMed and MEDLINE to identify studies on PD subtypes. A standardized approach was used to assess the methodological quality and clinical applicability of each study.
The researchers classified the studies included into two groups according to the date of publication. One group included studies from 1980 to 2014 and the other included studies from 2015 to 2019. They compared the average methodological quality of the studies during these two periods.
In total, 38 studies were included in the analysis. Twenty-six (68.4%) were transversal and 32 (84.2%) were performed in tertiary care centers. The average duration of the disease was approximately 60 months. A significant number of studies did not report common patient descriptors, such as unified Parkinson's disease score scale scores.
Approximately two-thirds of the studies used a data-based approach for subtyping; 21% used a hypothesis-based approach. Most (64%) of the data-based studies identified subtypes using at least three phenotypic domains; 7 of 8 hypothesis-based studies used only one domain.
The motor domain was used most frequently in both types of study, followed by non-motor domains, such as cognitive, emotional, or autonomous domains. Only five studies, all based on data, used nonclinical biomarkers.
Most studies did not achieve high scores on a methodological quality checklist. Weaknesses included recruiting patients from clinics rather than the community, lack of justification for sample size, and failure to match baseline characteristics.
The four highest quality studies were published after 2016. They were longitudinal, data-based studies conducted at various centers. Two of these studies included patients whose cases were of the same stage or duration of the disease. Although these studies incorporated similar clinical domains into their analyzes, there were no clear similarities between the subtypes identified by these studies.
The clinical significance of differences between subtypes was assessed as poor for most studies. The same was true of the possible implications and applicability of the treatment of most studies to the wider population of PD patients. Due to the small number of longitudinal studies, the stability of subtypes over time was unclear, as was their prognostic value.
"Mandatory" biomarker target
Although non-clinical biomarkers are desirable, they are not easy to obtain. "The field of biomarkers has been expanding and I think it is not yet mature, for the most part, in terms of having valid biomarkers that we can use consistently in Parkinson's disease," Mestre said. "It's a need we established after reviewing the available data."
Two possible goals of subtyping patients with PD are to establish a prognosis and choose treatment.
"If we consider that, any classification we have of the subtypes of Parkinson's disease will have to be easily applied to clinical practice. I think that was our main consideration to really evaluate the clinical applicability," he said. Master.
The results suggest several areas for future research on possible PD subtypes. For example, the inclusion of non-clinical domains in the definition of PD subtypes would offer "a way to recognize the biological foundations that would allow us to identify more valid subtypes of Parkinson's disease," he added.
Similarly, there is a need for longitudinal studies, which clarify the prognostic relevance of subtypes. Neurologists also need methods to translate group-level data into information relevant to individual patients.
Valuable contribution
Commenting on the research of Medscape Medical News, Raul Martínez Fernández, MD, PhD, neurologist and clinical researcher at HM Puerta del Sur University Hospital in Madrid, Spain, said the study makes a valuable contribution.
"The main strengths of the study are its methodological accuracy and the fact that it addresses an issue that has important clinical and research implications in the field," he said.
The main weakness of the study is the heterogeneity of the results it analyzed, he added. There are few suitable studies available in the literature and most of them differ in important ways in methodology. This points to the need for greater methodological homogeneity in this research, said Martínez Fernández.
The predominance of cross-sectional studies over longitudinal studies is likely to result from the fact that a cross-sectional study design is less demanding and cheaper, and that these studies can be executed more quickly than longitudinal studies, he noted.
It was important for future research to follow patients before the disease and for a longer period, he added. In addition to clinical presentation and evolution, objective biological markers are needed to define PD subtypes. These markers may include functional neuroimaging studies, cerebrospinal fluid analysis, and genetic testing.
The authors are right to point to the need to establish formal guidelines to focus research in this field, Martinez Fernandez said.
"It is mandatory to define prodromal populations using objective biomarkers and determine measures to objectively assess disease progression," he added. "Specifically, image markers and neurodegeneration of alpha-synuclein tracers would be essential to understand the progression of Parkinson's disease, both in general terms and for each subtype."
The study was funded by the International Parkinson and Movement Disorder Society. Mestre and Martínez Fernández have not revealed any relevant financial relationship.
J Parkinsons Dis. Published online April 13, 2021. Full text
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"redpajama_set_name": "RedPajamaCommonCrawl"
} | 1,618 |
using Microsoft.VisualStudio.Threading;
using System;
using System.Threading;
using System.Threading.Tasks;
using YetiCommon;
using YetiVSI.Util;
using Task = System.Threading.Tasks.Task;
namespace YetiVSI
{
/// <summary>
/// An exception thrown by tasks that have been explicitly aborted.
/// The inner exception contains the abort reason provided to Abort.
/// </summary>
public class TaskAbortedException : Exception
{
public TaskAbortedException(Exception reason) : base("Task aborted", reason)
{
}
}
/// <summary>
/// The handle to a cancelable operation within the operation itself.
/// </summary>
public interface ICancelable
{
/// <summary>
/// Indicates if this operation has been canceled - via ICancelableTask.Cancel,
/// ICancelabeTask.Abort, or by the user through a UI dialog.
/// </summary>
bool IsCanceled { get; }
/// <summary>
/// Returns the interface which is used to report progress of this operation to the user.
/// </summary>
IProgress<string> Progress { get; }
/// <summary>
/// Returns cancelation token which allows the client subscribe to cancel event.
/// </summary>
CancellationToken Token { get; }
/// <summary>
/// If this operation is canceled, then stop its execution.
/// </summary>
///
/// <exception cref="OperationCanceledException">
/// Thrown if the task was canceled via ICancelableTask.Cancel or a UI dialog.
/// </exception>
/// <exception cref="TaskAbortedException">
/// Thrown if the task was aborted via ICancelableTask.Abort.
/// </exception>
///
/// <remarks>
/// Use IsCanceled to check for canceled/aborted state without throwing.
/// </remarks>
void ThrowIfCancellationRequested();
}
public class NothingToCancel : ICancelable
{
public bool IsCanceled => false;
public IProgress<string> Progress { get; } = new Progress<string>();
public CancellationToken Token => CancellationToken.None;
public void ThrowIfCancellationRequested()
{
}
}
/// <summary>
/// Encapsulates a task that runs on another thread, while blocking the current thread with a
/// UI dialog that can cancel the operation.
/// </summary>
public interface ICancelableTask : ICancelable
{
/// <summary>
/// Runs the encapsulated action on a background thread and blocks the UI with a dialog.
/// </summary>
///
/// <returns>
/// 'true' is returned if the operation completes, and 'false' is returned if it was
/// canceled by the user, by calling Cancel, or by throwing OperationCanceledException
/// inside the task.
/// </returns>
///
/// <exception cref="InvalidOperationException">
/// Thrown if Run is called more than once on this task.
/// </exception>
/// <exception cref="TaskAbortedException">
/// Thrown if the task was aborted via Abort.
/// </exception>
/// <exception cref="Exception">
/// Any exception thrown by the task (except OperationCanceledException) is re-thrown.
/// </exception>
bool Run();
/// <summary>
/// Cancels the operation. Doesn't actually stop execution, but dismisses the UI dialog and
/// returns control to the call site where the operation was started.
/// </summary>
///
/// <remarks>
/// This method is intended to be used outside the task itself. The task can cancel itself
/// by throwing an OperationCanceledException.
///
/// Calling Cancel() when IsCancelled is true results in a no-op.
/// </remarks>
void Cancel();
/// <summary>
/// Similar to Cancel except that Run throws a TaskAbortedException at the call site where
/// the operation was started.
/// </summary>
///
/// <param name="reason">Used as the InnerException of TaskAbortedException</param>
///
/// <remarks>
/// This method is intended to be used outside the task itself. The task can abort itself by
/// throwing the exception directly.
///
/// Calling Abort when IsCanceled is true results in a no-op.
/// </remarks>
void Abort(Exception reason);
}
/// <summary>
/// Encapsulates a cancelable task that runs on another thread and returns a value.
/// </summary>
/// <typeparam name="T">The type of the value returned by the operation</typeparam>
public interface ICancelableTask<T> : ICancelableTask
{
/// <summary>
/// Contains the result of running the encapsulated action, if it completed successfully.
/// If the action failed or the task was canceled, the result is default(T).
/// </summary>
T Result { get; }
}
// Implements cancelable tasks. Use CancelableTask.Factory to instantiate a new task.
public abstract class CancelableTask : ICancelableTask, IProgress<string>
{
// Creates cancelable tasks. This class is substituted for testing.
public class Factory
{
// Default delay before showing the progress dialog, chosen for consistency with the
// previously used CommonMessagePump-based wait dialog.
static readonly TimeSpan defaultDelay = TimeSpan.FromSeconds(2);
static readonly TimeSpan defaultReportPeriod = TimeSpan.FromMilliseconds(100);
readonly JoinableTaskContext taskContext;
readonly ProgressDialog.Factory progressDialogFactory;
readonly string title;
readonly TimeSpan delay;
readonly TimeSpan reportPeriod;
// Instantiate a factory, setting YetiConstants.Title for the progress dialog title text
// and using a default delay.
public Factory(JoinableTaskContext taskContext,
ProgressDialog.Factory waitDialogFactory) : this(
taskContext, waitDialogFactory, YetiConstants.YetiTitle, defaultDelay,
defaultReportPeriod)
{
taskContext.ThrowIfNotOnMainThread();
}
public Factory(JoinableTaskContext taskContext,
ProgressDialog.Factory progressDialogFactory, string title,
TimeSpan delay, TimeSpan reportPeriod)
{
this.taskContext = taskContext;
this.progressDialogFactory = progressDialogFactory;
this.title = title;
this.delay = delay;
this.reportPeriod = reportPeriod;
}
[Obsolete("This constructor only exists to support mocking libraries.", error: true)]
protected Factory()
{
}
// Create a task that runs the given synchronous action on a background thread,
// and waits on the resulting async task.
public virtual ICancelableTask Create(string text, Func<ICancelable, Task> action)
{
return new NoResult(taskContext, progressDialogFactory.Create(title, text), action,
delay, reportPeriod);
}
// Create a task that runs the given synchronous action on a background thread,
// and waits on the resulting async task.
public virtual ICancelableTask Create(string text, string title,
Func<ICancelable, Task> action)
{
return new NoResult(taskContext, progressDialogFactory.Create(title, text),
action, delay, reportPeriod);
}
// Create a task that runs the given synchronous action on a background thread,
// and waits on the resulting async task, and captures its result.
public virtual ICancelableTask<T> Create<T>(string text,
Func<ICancelable, Task<T>> action)
{
return new WithResult<T>(taskContext, progressDialogFactory.Create(title, text),
action, delay, reportPeriod);
}
}
public bool IsCanceled => cancellationSource.IsCancellationRequested;
public IProgress<string> Progress => this;
public CancellationToken Token => cancellationSource.Token;
readonly JoinableTaskContext taskContext;
readonly IProgressDialog progressDialog;
readonly TimeSpan delay;
readonly CancellationTokenSource cancellationSource = new CancellationTokenSource();
readonly TimeSpan reportPeriod;
string reportValue = string.Empty;
bool isStarted = false;
bool isCompleted = false;
Exception abortReason;
CancelableTask(JoinableTaskContext taskContext, IProgressDialog progressDialog,
TimeSpan delay, TimeSpan reportPeriod)
{
this.taskContext = taskContext;
this.progressDialog = progressDialog;
this.delay = delay;
this.reportPeriod = reportPeriod;
}
public bool Run()
{
taskContext.ThrowIfNotOnMainThread();
if (isStarted)
{
throw new InvalidOperationException("Task can only be started once");
}
isStarted = true;
if (!cancellationSource.IsCancellationRequested)
{
try
{
var joinableTask = taskContext.Factory.RunAsync(
() => RunTaskAsync().WithCancellation(cancellationSource.Token));
if (reportPeriod > TimeSpan.Zero)
{
#pragma warning disable VSTHRD110 // Observe result of async calls
taskContext.Factory.RunAsync(() => GetReportTaskAsync());
#pragma warning restore VSTHRD110 // Observe result of async calls
}
if (!TryJoin(joinableTask, delay))
{
JoinWithProgressDialog(joinableTask);
}
}
catch (OperationCanceledException)
{
cancellationSource.Cancel();
}
finally
{
isCompleted = true;
}
}
// Handle aborted tasks. Aborted tasks are also canceled, so check abort first.
if (abortReason != null)
{
throw new TaskAbortedException(abortReason);
}
// Report external Cancel() events.
return !cancellationSource.IsCancellationRequested;
}
public void Cancel() => cancellationSource.Cancel();
public void Abort(Exception reason)
{
abortReason = reason;
Cancel();
}
void IProgress<string>.Report(string value)
{
reportValue = value;
if (reportPeriod == TimeSpan.Zero)
{
ReportProgress();
}
}
async Task GetReportTaskAsync()
{
while (!IsProgressCompletedOrCancelled())
{
await Task.Delay(reportPeriod, Token);
ReportProgress();
}
}
void ReportProgress()
{
if (!IsProgressCompletedOrCancelled())
{
#pragma warning disable VSTHRD110 // Observe result of async calls
taskContext.Factory.RunAsync(async () =>
#pragma warning restore VSTHRD110 // Observe result of async calls
{
await taskContext.Factory.SwitchToMainThreadAsync();
progressDialog.Message = reportValue;
});
}
}
bool IsProgressCompletedOrCancelled()
{
return Token.IsCancellationRequested || isCompleted;
}
public void ThrowIfCancellationRequested()
{
if (IsCanceled)
{
if (abortReason != null)
{
throw new TaskAbortedException(abortReason);
}
else
{
throw new OperationCanceledException(cancellationSource.Token);
}
}
}
// Derived classes implement this method to start executing the task on a background thread.
protected abstract Task RunTaskAsync();
/// <summary>
/// Try to join a task, with a timeout.
/// </summary>
/// <returns>
/// True if the task ran to completion in the time allotted, false if the task is still
/// running.
/// </returns>
bool TryJoin(JoinableTask joinableTask, TimeSpan timeout)
{
// Task.Delay is used instead of a CancellationToken because `Task.Delay(TimeSpan.Zero)`
// always completes synchronously, whereas `new CancellationTokenSource(TimeSpan.Zero)`
// may or may not be canceled by the time it returns.
return taskContext.Factory.Run(async () =>
{
var task = joinableTask.JoinAsync();
if (await Task.WhenAny(Task.Delay(delay), task) == task)
{
await task; // Propagate exceptions
return true;
}
return false;
});
}
/// <summary>
/// Joins a task while displaying a progress dialog. If the dialog is closed by the user
/// the entire CancelableTask is canceled.
/// </summary>
void JoinWithProgressDialog(JoinableTask joinableTask)
{
taskContext.ThrowIfNotOnMainThread();
// Kick off a task that will mark the dialog as complete after joinableTask completes.
var dialogTask = taskContext.Factory.RunAsync(async () =>
{
await taskContext.Factory.SwitchToMainThreadAsync();
try
{
await joinableTask.JoinAsync();
}
finally
{
progressDialog.Complete();
}
});
// Display the dialog and block until it's closed. The dialog spins up its own message
// pump, which allows tasks to continue executing continuations on the main thread
// despite the fact that we're making a blocking call.
// If dialogTask succeeds in marking the dialog as complete, it closes and ShowModal
// returns true.
// If the dialog is canceled by the user, it closes, ShowModal returns false, and we
// cancel the task using the cancellation source.
if (!progressDialog.ShowModal())
{
cancellationSource.Cancel();
}
dialogTask.Join();
}
// Implement a task that runs actions on a background thread.
class NoResult : CancelableTask
{
readonly Func<Task> task;
public NoResult(JoinableTaskContext taskContext, IProgressDialog dialog,
Func<ICancelable, Task> action, TimeSpan delay, TimeSpan reportPeriod)
: base(taskContext, dialog, delay, reportPeriod)
{
task = () => action(this);
}
protected override Task RunTaskAsync()
{
return Task.Run(task); // Runs the task on a background thread
}
}
// Implements a task that runs actions on a background thread and captures the results.
class WithResult<T> : CancelableTask, ICancelableTask<T>
{
readonly Func<Task> task;
public T Result { get; private set; }
public WithResult(JoinableTaskContext taskContext, IProgressDialog dialog,
Func<ICancelable, Task<T>> func, TimeSpan delay,
TimeSpan reportPeriod) : base(taskContext, dialog, delay,
reportPeriod)
{
task = async () =>
{
var result = await func(this);
if (!cancellationSource.IsCancellationRequested)
{
Result = result;
}
};
}
protected override Task RunTaskAsync()
{
return Task.Run(task); // Runs the task on a background thread
}
}
}
public static class CancelableTaskFactoryExtensions
{
// Create a task that runs the given synchronous action on a background thread.
public static ICancelableTask Create(this CancelableTask.Factory factory, string text,
Action<ICancelable> action) => factory.Create(
text, t =>
{
action(t);
return Task.CompletedTask;
});
// Create a task that runs the given synchronous action on a background thread and returns
// a value.
public static ICancelableTask<T> Create<T>(this CancelableTask.Factory factory, string text,
Func<ICancelable, T> action) =>
factory.Create(text, t => { return Task.FromResult(action(t)); });
// Create a task that runs the given synchronous action on a background thread,
// waits on the resulting async task, and captures its result.
public static ICancelableTask<T> Create<T>(this CancelableTask.Factory factory, string text,
Func<Task<T>> action) =>
factory.Create(text, _ => action());
}
} | {
"redpajama_set_name": "RedPajamaGithub"
} | 4,412 |
Overview of Anti-SLAPP Laws
By Austin Vining and Sarah Matthews, with special thanks to Maya Gandhi
Anti-SLAPP laws provide defendants a way to quickly dismiss meritless lawsuits—known as "SLAPPs" or "Strategic Lawsuits Against Public Participation"—filed against them for exercising their First Amendment rights. These laws aim to discourage the filing of SLAPP suits and prevent them from imposing significant litigation costs and chilling protected speech.
In recent years, several states have adopted or amended their anti-SLAPP laws. As of June 2021, 31 states and the District of Columbia have anti-SLAPP laws, including Arizona, Arkansas, California, Colorado, Connecticut, Delaware, Florida, Georgia, Hawaii, Illinois, Indiana, Kansas, Louisiana, Maine, Maryland, Massachusetts, Missouri, Nebraska, Nevada, New Mexico, New York, Oklahoma, Oregon, Pennsylvania, Rhode Island, Tennessee, Texas, Utah, Vermont, Virginia, and Washington. Minnesota previously passed an anti-SLAPP law, but it was struck down as unconstitutional, as discussed below.
Anti-SLAPP protections vary significantly from state to state. For example, in some states, like Arizona, they only protect defendants from cases brought in retaliation for petitioning the government. In others, such as California, the laws broadly protect speech made in connection with a public issue. For the most part, anti-SLAPP laws are broad enough to cover SLAPP suits aimed at silencing or retaliating against journalists or news outlets for critical reporting. These laws typically provide critical protections to the news media—allowing defendants to secure a quick dismissal before the costly discovery process begins, permitting defendants who win their anti-SLAPP motions to recover attorney's fees and costs, automatically staying discovery once the defendant has filed an anti-SLAPP motion, and allowing defendants to immediately appeal a trial court's denial of an anti-SLAPP motion.
Recent changes in state anti-SLAPP laws
Recent legislation has trended toward more robust protections for defendants in SLAPP suits.
In May 2021, Washington became the first state to pass a version of the Uniform Public Expression Protection Act ("UPEPA"), a model anti-SLAPP law drafted by the Uniform Law Commission that has been introduced in several state legislatures. The new anti-SLAPP protection applies to suits based on a person's exercise of speech, press, assembly, petition, and association rights "on a matter of public concern." UPEPA, 2021 Wash. Legis. Serv., ch. 259, § 2(2)(c). Washington previously adopted an anti-SLAPP law in 2010, but the state supreme court struck it down in 2015, finding that it violated the state constitutional right to a trial by jury, by authorizing judges to adjudicate factual questions in non-frivolous cases before trial. Davis v. Cox, 351 P.3d 862, 864 (Wash. 2015), abrogated on other grounds by Maytown Sand & Gravel, LLC v. Thurston Cnty., 423 P.3d 223 (Wash. 2018). The new law avoids this problem by adopting language that tracks the existing summary judgment and dismissal standards. It thus essentially allows defendants to bring motions for summary judgment much earlier in the proceedings rather than after a lengthy and expensive discovery period.
In November 2020, New York significantly expanded its anti-SLAPP law, which had previously only covered cases brought by plaintiffs seeking public permits, zoning changes, or other entitlements from a government body. The 2020 amendments broadened the anti-SLAPP law to cover cases involving "any communication in a place open to the public or a public forum in connection with an issue of public interest" or "any other lawful conduct in furtherance of the exercise of the constitutional right of free speech in connection with an issue of public interest . . . ." N.Y. Civ. Rights Law § 76-a(1)(a)(1)-(2) (McKinney). Accordingly, New York's revised anti-SLAPP law should apply to news reporting generally. The new law also requires courts to stay discovery pending resolution of an anti-SLAPP motion and entitles a prevailing SLAPP defendant to attorney's fees and costs.
In June 2019, Colorado became the newest state to adopt anti-SLAPP protections. The law allows a defendant to file a special motion to dismiss claims arising from the exercise of the right of petition or free speech in connection with a public issue. Colo. Rev. Stat. § 13-20-1101(3)(a) (2019). The Centennial State's new law follows similarly strong laws passed in Connecticut and Kansas in recent years. Conn. Gen. Stat. Ann. § 52-196a (2019) (adopted in 2017); Kan. Stat. Ann. § 60-5320 (2019) (adopted in 2016).
Tennessee significantly improved its anti-SLAPP protections in 2019 to protect people from lawsuits "filed in response to [their] exercise of the right of free speech, right to petition, or right of association." Tenn. Code Ann. § 20-17-104(a) (2019). The law permits defendants to file a motion to dismiss a SLAPP suit before the costly discovery process begins, immediately appeal the denial of an anti-SLAPP motion, and recover attorney's fees if a court rules in their favor. Tenn. Code Ann. § 20-17-104 (2019). Previously, Tennessee's anti-SLAPP law only protected statements made to governmental agencies. § 4-21-1003.
In 2017, Virginia amended its anti-SLAPP law to include actions based on "matters of public concern that would be protected under the First Amendment" and to permit successful defendants to recover attorney's fees and costs. Va. Code Ann. § 8.01-223.2 (2019). However, unlike most anti-SLAPP laws, the Virginia law still fails to identify any special procedures allowing a defendant to invoke these protections at an early stage of the proceedings.
Not all changes in recent years have strengthened anti-SLAPP protections. In 2019, Texas amended its anti-SLAPP law to limit the types of statements that could receive protection. While the previous version of the Texas law allowed defendants to seek dismissal of lawsuits broadly "related to" a person's exercise of the right of free speech, petition, or association, the new statute requires the claim to be more narrowly "based on" or "in response to" the exercise of one of those rights. Tex. Civ. Prac. & Rem. § 27.003(a) (2019). Legislators also abandoned anti-SLAPP protections for speech regarding trade secrets or non-compete agreements, potentially allowing employers to intimidate whistleblowers with employment-related suits.
Courts struck down anti-SLAPP laws in Washington and Minnesota, and Washington enacted an updated law
Courts in Washington and Minnesota struck down their states' anti-SLAPP laws, finding them unconstitutional under their respective state constitutions. As discussed above, however, Washington enacted an updated anti-SLAPP law in 2021 that addressed the concerns of the state supreme court.
In 2016, a Minnesota appellate court similarly found that state's anti-SLAPP law unconstitutional, finding that the law "deprive[s] the non-moving party of the right to a jury trial by requiring a court to make pretrial factual findings to determine whether the moving party is immune from liability." Mobile Diagnostic Imaging v. Hooten, 889 N.W.2d 27, 35 (Minn. Ct. App. 2016). The following year, the Minnesota Supreme Court agreed, finding that state's anti-SLAPP law unconstitutional as applied to claims alleging torts because it requires a district court to make pretrial factual finding in violation of the plaintiff's right to a trial by jury under the Minnesota constitution. Leiendecker v. Asian Women United of Minn., 895 N.W.2d 623, 637–38 (Minn. 2017). These decisions raise concerns that courts in other states that recognize a plaintiff's right to a trial by jury may follow suit.
Courts disagree on whether anti-SLAPP protections apply in federal court
If a plaintiff sues a journalist in federal court for a state law tort, such as libel, it is not always clear whether the journalist can invoke the protections of the state's anti-SLAPP law, assuming one exists. Congress has never passed a federal anti-SLAPP law, and courts across the country disagree about whether state anti-SLAPP provisions apply in federal court. The analysis turns on whether a state's anti-SLAPP law creates substantive rights and does not conflict with federal rules. Some federal courts of appeals have found both requirements satisfied and allowed defendants to invoke these protections in federal court. See, e.g., Adelson v. Harris, 774 F.3d 803, 809 (2d Cir. 2014) (finding application of Nevada's anti-SLAPP provisions in federal court "unproblematic"); Liberty Synergistics Inc. v. Microflo Ltd., 718 F.3d 138, 144 (2d Cir. 2013) (applying California's anti-SLAPP law in federal court); Godin v. Schencks, 629 F.3d 79, 81 (1st Cir. 2010) (finding that Maine's anti-SLAPP law applied in federal court); United States ex rel. Newsham v. Lockheed Missiles & Space Co., 190 F.3d 963, 973 (9th Cir. 1999) (applying certain provisions of the California anti-SLAPP law in federal court).
But other federal appellate courts, particularly in recent years, have disagreed. See, e.g., Klocke v. Watson, 936 F.3d 240, 245 (5th Cir. 2019), as revised (Aug. 29, 2019) (finding that Texas anti-SLAPP law's burden-shifting framework could not apply in federal court because it imposed additional requirements beyond those found in the Federal Rules of Civil Procedure); Carbone v. Cable News Network, Inc., 910 F.3d 1345, 1351 (11th Cir. 2018) (finding that motion-to-strike procedure in Georgia anti-SLAPP law conflicted with federal rules and could not apply in federal court); Los Lobos Renewable Power, LLC v. Americulture, Inc., 885 F.3d 659, 662 (10th Cir. 2018), cert. denied, 139 S. Ct. 591 (2018) (finding that New Mexico's anti-SLAPP law does not apply in federal court); Abbas v. Foreign Policy Grp., LLC, 783 F.3d 1328, 1332 (D.C. Cir. 2015) (finding that D.C. anti-SLAPP law does not apply in federal court).
The Supreme Court has not yet weighed in on the matter. | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 3,795 |
summermusicandarts.com
Question: How Many Hours Of Gaming Is Healthy?
How many hours of gaming a day is healthy?
Is 5 hours of gaming bad?
Is gaming a mental illness?
Is 10 hours of gaming bad?
Do video games rot your brain?
Does gaming make you angry?
Is playing video games 3 hours a day bad?
How long should you play fortnite a day?
How many hours of video games is too much for adults?
How many hours a week should you play video games?
Is it unhealthy to play videogames all day?
Is gaming a waste of time?
How many hours of gaming is too much?
What is the disadvantages of playing video games?
How often should you take a break from gaming?
Do video games affect relationships?
You can play anywhere from 1–4 hours a day without worry.
If you play 5 hours or more, then you're probably sacrificing something else… and that's getting dangerously close to video game addiction.
Health-wise, you should be able to play more than 5 hours a day – that is if you take regular breaks..
It really depends on your reason for playing that much. Playing 5 hours and not doing things you have to do, very bad. Playing 5 hours after you got everything you "have" to do but passing on more rewarding real world things you "could" be doing, that's neutral to bad depending on where you are in life.
The World Health Organization (WHO) says yes. Recently, the WHO officially recognized "gaming disorder" as a mental health condition — adding the disorder to the International Classification of Diseases, or the ICD-11, the organization's official diagnostic manual, according to CBS News.
The amount of time spent playing videogames is not, per se, good or bad. It depends on the things that you are not doing while spending time on games what matters; let's imagine a situation. You're an adult, working from 9:00 to 17:00. … In this case, it probably won't be a problem to spend 10 hours on videogames.
Violent shooter video games really DO rot your brain: Frequent players have less gray matter, study reveals. Playing violent 'shooter' video games can damage the brain and may even increase the risk of Alzheimer's disease, brain scans suggest.
A study finds that video games may cause aggressive behaviour, but the trigger isn't violent content, it's frustration.
Three hours per day doesn't take all of it. You can choose the amount of time you will spend on gaming, and as long as you don't feel that it's invading too much on your free time, three hours a day is perfectly fine.
"Fortnite" games are short — typically less than 20 to 25 minutes. So after getting close to winning, it's highly tempting to give it another try, similar to a slot machine. In general, a healthy gaming diet would include no more than 40 minutes a night on school nights and no more than an hour a day on weekends.
While there isn't a consensus on how many hours of video games (and general screen time) is too much, the finding by Twenge and her colleagues that more than five hours per day is excessive seems reasonable.
GAMERS SPEND MORE THAN SIX HOURS EACH WEEK PLAYING Video gamers spend an average of six hours, 20 minutes each week playing games. This is a decrease of 11 percent in the last year. Nearly one-third (32 percent) play more than seven hours each week, with 17 percent playing more than 12 hours a week.
But too much video game playing may cause problems. It's hard to get enough active play and exercise if you're always inside playing video games. And without enough exercise, kids can become overweight. Overdoing video games also could affect other important stuff, like friendships and how well a kid does in school.
Video games are designed for relaxation. They give people a chance to escape from their busy, stressful and routine lives. When people say playing video games is a waste of time, they are referring to those who play all day or all night long. This is a waste of time.
The bottom-line: "One to nine hours per week seems to be safe, but playing more than nine hours — one hour on weekdays and two hours on weekend days — may be not recommended for children 7 to 11 years old," said study author Dr. Jesus Pujol.
Other difficulties also arise through prolonged periods of gaming or when "addiction" creeps in; indeed, excessive internet use or gameplay can lead to adverse consequences on behaviour such as social withdrawal, loneliness, depression, or even heightened irritation and trouble focusing on less stimulating work or …
every 50 to 90 minutesAim to take breaks every 50 to 90 minutes of gaming. Breaks should be at least five minutes long, but 15 to 20 minutes is ideal. If you're experiencing a lot of stress while gaming, then the break should be longer.
Young adults who spend a lot of time playing video games have poorer relationships with their families and friends, the results of a new study indicate. US researchers looked at over 800 college students and found that as the amount of playing time went up, the quality of relationships with parents and peers went down.
How Can I Start Investing With $500?
How much money should you start investing with?
Question: Which Stock Pays The Highest Dividend?
Which oil company pays the highest dividend?
Are Wash Sales Reported To The IRS?
Why do I need 25k to day trade? Since day traders hold
Question: What Other IPhone Is The Same Size As The 11?
Is the iPhone 7 worth buying in 2020? Best answer
Quick Answer: How Many GB Is GTA 5?
How many GB is GTA 5 PC? 94 GBSize of GTA 5 game The
Question: Who Is The Strongest Flash?
Is Nora faster than Barry? A c-4 explosion explodes
Question: Who Has 5g Right Now?
Do I have 5g in my area? To track 5G with Ookla'
What'S All The Hype About 5g?
Is there 5g in USA? There are actually two 5G networks
Is Qualcomm A Good Buy?
Who is Qualcomm biggest competitor? Qualcomm'
Question: How Much Will Starlink Be Worth?
How much will SpaceX stock be worth? If SpaceX did
Quick Answer: How Much RAM Is Too Much RAM?
Is 64gb of RAM too much? In all honesty today'
Question: Does Warren Buffett Like Bitcoin?
Does Bitcoin have a future? Bitcoin Price in 2030 –
Quick Answer: How Do We Get Money From YouTube?
How many views do you need to get paid on YouTube?
© 2021 summermusicandarts.com | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 1,453 |
{"url":"https:\/\/socratic.org\/questions\/what-is-the-standard-form-of-y-11x-1-11x-1","text":"# What is the standard form of y=(11x - 1)(11x - 1) ?\n\n$121 {x}^{2} - 22 x + 1$\n${\\left(a + b\\right)}^{2} = {a}^{2} + 2 a b + {b}^{2}$","date":"2022-05-26 23:33:36","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 2, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 1, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.8345129489898682, \"perplexity\": 3140.032601333255}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2022-21\/segments\/1652662627464.60\/warc\/CC-MAIN-20220526224902-20220527014902-00515.warc.gz\"}"} | null | null |
package prometheus
import (
"fmt"
"net/http"
"testing"
"time"
"github.com/stretchr/testify/require"
)
const validUniqueGauge = `# HELP cadvisor_version_info A metric with a constant '1' value labeled by kernel version, OS version, docker version, cadvisor version & cadvisor revision.
# TYPE cadvisor_version_info gauge
cadvisor_version_info{cadvisorRevision="",cadvisorVersion="",dockerVersion="1.8.2",kernelVersion="3.10.0-229.20.1.el7.x86_64",osVersion="CentOS Linux 7 (Core)"} 1
`
const validUniqueCounter = `# HELP get_token_fail_count Counter of failed Token() requests to the alternate token source
# TYPE get_token_fail_count counter
get_token_fail_count 0
`
const validUniqueSummary = `# HELP http_request_duration_microseconds The HTTP request latencies in microseconds.
# TYPE http_request_duration_microseconds summary
http_request_duration_microseconds{handler="prometheus",quantile="0.5"} 552048.506
http_request_duration_microseconds{handler="prometheus",quantile="0.9"} 5.876804288e+06
http_request_duration_microseconds{handler="prometheus",quantile="0.99"} 5.876804288e+06
http_request_duration_microseconds_sum{handler="prometheus"} 1.8909097205e+07
http_request_duration_microseconds_count{handler="prometheus"} 9
`
const validUniqueHistogram = `# HELP apiserver_request_latencies Response latency distribution in microseconds for each verb, resource and client.
# TYPE apiserver_request_latencies histogram
apiserver_request_latencies_bucket{resource="bindings",verb="POST",le="125000"} 1994
apiserver_request_latencies_bucket{resource="bindings",verb="POST",le="250000"} 1997
apiserver_request_latencies_bucket{resource="bindings",verb="POST",le="500000"} 2000
apiserver_request_latencies_bucket{resource="bindings",verb="POST",le="1e+06"} 2005
apiserver_request_latencies_bucket{resource="bindings",verb="POST",le="2e+06"} 2012
apiserver_request_latencies_bucket{resource="bindings",verb="POST",le="4e+06"} 2017
apiserver_request_latencies_bucket{resource="bindings",verb="POST",le="8e+06"} 2024
apiserver_request_latencies_bucket{resource="bindings",verb="POST",le="+Inf"} 2025
apiserver_request_latencies_sum{resource="bindings",verb="POST"} 1.02726334e+08
apiserver_request_latencies_count{resource="bindings",verb="POST"} 2025
`
func TestParseValidPrometheus(t *testing.T) {
// Gauge value
metrics, err := Parse([]byte(validUniqueGauge), http.Header{}, false)
require.NoError(t, err)
require.Len(t, metrics, 1)
require.Equal(t, "cadvisor_version_info", metrics[0].Name())
require.Equal(t, map[string]interface{}{
"gauge": float64(1),
}, metrics[0].Fields())
require.Equal(t, map[string]string{
"osVersion": "CentOS Linux 7 (Core)",
"cadvisorRevision": "",
"cadvisorVersion": "",
"dockerVersion": "1.8.2",
"kernelVersion": "3.10.0-229.20.1.el7.x86_64",
}, metrics[0].Tags())
// Counter value
metrics, err = Parse([]byte(validUniqueCounter), http.Header{}, false)
require.NoError(t, err)
require.Len(t, metrics, 1)
require.Equal(t, "get_token_fail_count", metrics[0].Name())
require.Equal(t, map[string]interface{}{
"counter": float64(0),
}, metrics[0].Fields())
require.Equal(t, map[string]string{}, metrics[0].Tags())
// Summary data
//SetDefaultTags(map[string]string{})
metrics, err = Parse([]byte(validUniqueSummary), http.Header{}, false)
require.NoError(t, err)
require.Len(t, metrics, 1)
require.Equal(t, "http_request_duration_microseconds", metrics[0].Name())
require.Equal(t, map[string]interface{}{
"0.5": 552048.506,
"0.9": 5.876804288e+06,
"0.99": 5.876804288e+06,
"count": 9.0,
"sum": 1.8909097205e+07,
}, metrics[0].Fields())
require.Equal(t, map[string]string{"handler": "prometheus"}, metrics[0].Tags())
// histogram data
metrics, err = Parse([]byte(validUniqueHistogram), http.Header{}, false)
require.NoError(t, err)
require.Len(t, metrics, 1)
require.Equal(t, "apiserver_request_latencies", metrics[0].Name())
require.Equal(t, map[string]interface{}{
"500000": 2000.0,
"count": 2025.0,
"sum": 1.02726334e+08,
"250000": 1997.0,
"2e+06": 2012.0,
"4e+06": 2017.0,
"8e+06": 2024.0,
"+Inf": 2025.0,
"125000": 1994.0,
"1e+06": 2005.0,
}, metrics[0].Fields())
require.Equal(t,
map[string]string{"verb": "POST", "resource": "bindings"},
metrics[0].Tags())
}
func TestMetricsWithTimestamp(t *testing.T) {
testTime := time.Date(2020, time.October, 4, 17, 0, 0, 0, time.UTC)
testTimeUnix := testTime.UnixNano() / int64(time.Millisecond)
metricsWithTimestamps := fmt.Sprintf(`
# TYPE test_counter counter
test_counter{label="test"} 1 %d
`, testTimeUnix)
// IgnoreTimestamp is false
metrics, err := Parse([]byte(metricsWithTimestamps), http.Header{}, false)
require.NoError(t, err)
require.Len(t, metrics, 1)
require.Equal(t, "test_counter", metrics[0].Name())
require.Equal(t, map[string]interface{}{
"counter": float64(1),
}, metrics[0].Fields())
require.Equal(t, map[string]string{
"label": "test",
}, metrics[0].Tags())
require.Equal(t, testTime, metrics[0].Time().UTC())
// IgnoreTimestamp is true
metrics, err = Parse([]byte(metricsWithTimestamps), http.Header{}, true)
require.NoError(t, err)
require.Len(t, metrics, 1)
require.Equal(t, "test_counter", metrics[0].Name())
require.Equal(t, map[string]interface{}{
"counter": float64(1),
}, metrics[0].Fields())
require.Equal(t, map[string]string{
"label": "test",
}, metrics[0].Tags())
require.WithinDuration(t, time.Now(), metrics[0].Time().UTC(), 5*time.Second)
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 6,567 |
\section{Introduction}
The suppression of charmonium states was proposed a long time ago as a signature of the production of a deconfined state in
nucleus-nucleus collisions \cite{Sat86}. However, it was very soon realized that nuclear effects not related to deconfinement or,
more generally, to the production of a hot medium may influence the observed charmonia suppression~\cite{Ger88,Kha97}. Such effects were investigated
through the study of charmonia production in \mbox{p-A} collisions. In these reactions, the produced $c\overline c$ pair may interact
with the cold nuclear medium of the target nucleus, hindering the formation of a bound state.
Several \mbox{p-A} data samples exist today for \rm J/$\psi$\ production at fixed target energies, in particular from NA50 at the
SPS~\cite{Ale03,Ale04,Ale06}, from E866 at FNAL~\cite{Lei00} and from the HERA-B Collaboration at HERA~\cite{Abt09}.
Nuclear effects are usually
parametrized by comparing the yields for various nuclear targets in a certain kinematical region, and then fitting their
$A$ dependence in terms of the simple power law $A^\alpha$. Alternatively, the data are analyzed in the framework of the Glauber
model~\cite{Gla59},
and their $A$ dependence expressed as an effective ``absorption'' cross-section $\sigma_{\rm J/\psi}^{eff}$.
The interpretation of \rm J/$\psi$\ results
in \mbox{p-A} collisions is the object of a rather strong theory effort. Nowadays, it is clear that not only the nuclear
dissociation of the $c\overline c$ pair plays a role, but also effects like shadowing, initial- and final-state parton energy loss, and
possibly the intrinsic charm component of the projectile should be taken into account in a realistic description of the
results~\cite{Vog00}.
When studying \rm J/$\psi$\ suppression in \mbox{A-A} collisions, a precise knowledge of nuclear effects is an essential requisite to
disentangle genuine hot-medium effects. In the approach commonly used up to now~\cite{Kha97}, the effective quantity
$\sigma_{\rm J/\psi}^{eff}$ is
obtained analyzing \mbox{p-A} data taken in the same kinematic domain of \mbox{A-A} collisions under study. Then, it is assumed that
in both \mbox{p-A} and \mbox{A-A} collisions nuclear effects, parametrized through the quantity $\sigma_{\rm J/\psi}^{eff}$, scale with $L$,
the mean thickness of nuclear matter seen by the $c\overline c$ pair in its way through the projectile and target nuclei.
In this way it is possible to determine the expected \rm J/$\psi$\ yield for nuclear collisions as a function of centrality~\cite{Ale04}.
With this approach, a significant anomalous suppression (i.e., a suppression which goes beyond the estimated contribution from nuclear
effects) has been detected at SPS energies~\cite{Ale05,Arn07}.
In this article, we go a step further by taking explicitly into account parton shadowing in the determination of the
cold nuclear matter effects in \mbox{A-A} collisions. Shadowing of partons in nuclei is a depletion of their population at
small momentum fraction of the nucleon, $x$, compared to that in a free nucleon, with a corresponding enhancement at moderate $x$
(anti-shadowing). Contrarily to final state dissociation of the $c\overline c$ pair, shadowing
is not expected to scale with $L$, because in \mbox{p-A} collisions only partons in the target are affected by shadowing,
whereas in \mbox{A-A} the projectile is also involved. Therefore, a different approach to the evaluation of nuclear effects is required.
\section{Shadowing effects: \lowercase{p}-A collisions}
To give an estimate of shadowing effects on \rm J/$\psi$\ production, we use the Color Evaporation Model (CEM) at leading order
(LO)~\cite{Glu78}.
In this approach the charmonium production cross section for \mbox{p-A} collisions is obtained by integrating the free $c\overline c$
cross section from energy threshold to the open charm threshold. In absence of final-state interactions of the produced \rm J/$\psi$\ one
has~\cite{Vog99}
\begin{equation}
\frac{1}{A}\frac{d\sigma_{\rm J/\psi}}{dx_F}=2F\int_{2m_c}^{2m_D}mdm\frac{H_{pA}(x_1,x_2,m^2)}{\sqrt{s}\sqrt{x_F^2s+4m^2}}
\label{eq:1}
\end{equation}
where $F$ is the fraction of $c\overline c$ pairs which gives a \rm J/$\psi$\ in the final state and $H_{pA}$ is given by
\begin{equation}
\begin{split}
H_{pA}(x_1,x_2,m^2)=f_g^p(x_1,m^2)f_g^A(x_2,m^2)\sigma_{gg}(m^2)+\\ +\sum_{q=u,d,s}f_q^p(x_1,m^2)f_{\overline q}^A(x_2,m^2)\sigma_{q\overline
q}(m^2)
\end{split}
\label{eq:2}
\end{equation}
$H_{pA}$ is the sum of two terms, corresponding to the elementary $c\overline{c}$ production by gluon fusion and $q\overline{q}$
annihilation, convoluted with the parton densities $f_i^A(x,Q^2)$ and $f_i^p(x,Q^2)$ in the target and projectile nucleons, evaluated at $Q=m$.
$x_1$ ($x_2$) is the fraction of the nucleon momentum carried by the parton of the projectile (target) which interacts.
We have used in our calculation the GRV98LO~\cite{Glu98} set of parton distribution functions (PDF). The PDF for
nucleons inside nuclei are modified with respect to the free ones according to the following expression:
\begin{equation}
f_i^A(x,Q^2)=R_i^A(x,Q^2)f_i^N(x,Q^2)
\label{eq:3}
\end{equation}
Various parametrizations of the PDF modifications from nuclear effects exist. We have used
the EKS98~\cite{Esk99} and EPS08~\cite{Esk08} sets, which are available for all mass numbers and have been implemented in the frame of
the commonly used LHAPDF interface~\cite{Wha05}.
The calculation has been performed for several A values, corresponding to Be, Al, Cu, In, W, Pb and U target nuclei.
For each target the ratio between the \rm J/$\psi$\ production cross section (per nucleon-nucleon collision) in \mbox{p-A} and \mbox{p-p}, which we
refer to as shadowing factor in the following, is given by
\begin{equation}
S_{pA}^{\rm J/\psi}(x_F)=\frac{1}{A}\frac{d\sigma_{\rm J/\psi}^{pA}/dx_F}{d\sigma_{\rm
J/\psi}^{pp}/dx_F}
\label{eq:4}
\end{equation}
In Fig.~\ref{fig:1} we present the result of the calculation of the shadowing factors for \mbox{p-A} collisions at 158 GeV, the energy
used in the \mbox{A-A} data taking at the SPS. The plots refer to midrapidity, and we separately show the shadowing factors for
the $gg$ and $q\overline q$ fraction of the production cross section. We plot the results as a function of the $L$ variable~\cite{Sha01},
computed for each nucleus using the Glauber model with realistic nuclear density distributions~\cite{DeV87}.
\begin{figure}[h]
\centering
\resizebox{0.48\textwidth}{!}
{\includegraphics*[bb=0 0 530 530]{fig1.eps}}
\caption{Shadowing factors for pA collisions at 158 GeV, at midrapidity. Open circles refer to the $gg$ fraction of the cross section,
open squares to $q\overline q$, solid circles to the total cross section. The top plot has been obtained with the EKS98 set, the bottom
plot with EPS08.}
\label{fig:1}
\end{figure}
It can be noted that both EKS98 and EPS08 sets give shadowing factors larger than 1, and that such an anti-shadowing is more pronounced
when using EPS08. The $q\overline q$ fraction of the cross section exhibits a small shadowing, which is more than counterbalanced by the
dominant $gg$ production process. We have also verified, by choosing another
set of PDFs (MRST2001LO~\cite{Mar02}) that our results do not depend, within at most 2\%, on the specific choice of the PDF set. The flattening
of $S_{pA}^{\rm J/\psi}$ for heavy nuclei when using EPS08 is because of the fact that a constant shadowing is assumed beyond A=208~\cite{Sal09}.
In Fig.~\ref{fig:2} we present the results of a similar calculation at slightly forward ($y=0.5$, $y=1$) and backward ($y=-0.5$, $y=-1$)
rapidity. We also plot the $y=0$ result for comparison. We recall that the region $0<y<1$ is the one where \rm J/$\psi$\ results are
available at SPS energies for \mbox{A-A} collisions. The shadowing factors are found to depend on $y$, as expected,
since $x_2$ is directly related to this quantity. We recall that $x_2=(m_{\rm J/\psi}/\sqrt{s})\exp(-y)$ for \rm J/$\psi$\
production in the CEM at LO. In Fig.~\ref{fig:2new} we plot the EKS98 and EPS08 parametrizations of nuclear effects on PDFs for the Pb
nucleus, relative to $Q=m_{\rm J/\psi}$.
The arrows indicate the $x_2$ values corresponding to the rapidities where we have performed the shadowing factor
calculation of Fig.~\ref{fig:2}. It can be clearly seen that from $y=-1$ towards $y=1$ one goes from a shadowing to an anti-shadowing regime,
reaching a maximum of the anti-shadowing effect at $x_2\sim$0.10 for the EKS98 set ($\sim$0.13 for EPS08), which then decreases going towards
smaller $x_2$ (corresponding to more forward rapidities). Such an evolution can clearly be identified looking, at fixed $L$, to the shadowing
factors shown in Fig.~\ref{fig:2}, which are $<$1 at $y=-1$, then increase and finally decrease at forward rapidity.
We note that the shadowing factors calculated for $y=-0.5, -1$ are identical, for symmetry reasons, to those for \mbox{A-p} collisions
at $y=0.5, 1$. Results on \mbox{A-p} collisions are interesting in order to study effects
from PDF modifications in the projectile, which will be important when studying \mbox{A-A} interactions. We see, for example, that at
$y=1$ the shadowing effects in \mbox{A-p} are very different with respect to \mbox{p-A}.
\begin{figure}[h]
\centering
\resizebox{0.48\textwidth}{!}
{\includegraphics*[bb=0 0 530 530]{fig2.eps}}
\caption{Shadowing factors for \rm J/$\psi$\ production in \mbox{p-A} collisions at 158 GeV, using EKS98 (top plot) and EPS08 (bottom plot).
Solid and open triangles refer to $y=1$ and $y=-1$ respectively, solid and open squares to $y=0.5$ and $y=-0.5$, circles to $y=0$.}
\label{fig:2}
\end{figure}
\begin{figure}[h]
\centering
\resizebox{0.48\textwidth}{!}
{\includegraphics*[bb=0 0 530 530]{fig3.eps}}
\caption{The EKS98 (continuous line) and EPS08 (dashed line) parametrizations of nuclear modifications to the PDFs for the Pb nucleus,
calculated at $Q=m_{\rm J/\psi}$.
The arrows correspond to the rapidity values where the calculation of Fig.~\ref{fig:2} has been performed.}
\label{fig:2new}
\end{figure}
\section{Shadowing effects: \mbox{A-A} collisions}
When moving to \mbox{A-A} collisions, the cross section and the shadowing factors, integrated over centrality, can be calculated with
an expression similar to Eq.~\ref{eq:1}, by replacing $H_{pA}(x_1,x_2,m^2)$ with $H_{AA}(x_1,x_2,m^2)$, taking now into account that
shadowing affects both projectile and target nuclei. The problem becomes more complicate if shadowing factors have to be calculated for
various centrality intervals. Clearly, for various geometries of the collision, either the halo or the core of the nuclei will be mainly
involved, and the shadowing effects will be more important in the core than in the halo. Various parametrizations of the local shadowing
inside the nucleus have been proposed. We have used two of them~\cite{Kle03}:
\begin{equation}
R_{i,\rho}^A(x,Q^2,\vec{r},z)=1+N^A_\rho(R_i^A(x,Q^2)-1)\frac{\rho_A(\vec{r},z)}{\rho_0}
\label{eq:5}
\end{equation}
\noindent{and}
\begin{equation}
R_{i,L}^A(x,Q^2,\vec{r},z)=1+N^A_L(R_i^A(x,Q^2)-1)\frac{\int dz\rho_A(\vec{r},z)}{\int dz\rho_A(0,z)}
\label{eq:6}
\end{equation}
In the first one, shadowing in a certain location $(\vec{r},z)$ inside the nucleus is proportional to the local nuclear density
$\rho_A(\vec{r},z)$, while in the second it is proportional to the length $L$ of nuclear matter crossed by the parton on its way
through the nucleus. The normalization $N^A_\rho$ is fixed to ensure that
$\int d\vec{r}dz R_{i,\rho}^A(x,Q^2,\vec{r},z)=R_i^A(x,Q^2)$ (and similarly for $N^A_L$).
The study of the shadowing factors in \mbox{A-A} collisions as a function of centrality has been performed for \mbox{In-In} and
\mbox{Pb-Pb} at 158 GeV/nucleon (see also~\cite{Eme99} for a previous investigation of the influence of shadowing on the centrality dependence
of J/$\psi$ and Drell-Yan cross sections).
A large number of events has been generated for centrality values in the interval 0$<b<$12 fm for
\mbox{In-In} and 0$<b<$16 fm for \mbox{Pb-Pb}, in steps of 2 fm, with a Glauber Monte-Carlo approach. We have used
$\sigma_{inel}^{pp}$=30 mb and the measured nuclear density distributions for In and Pb~\cite{DeV87}. For every \mbox{N-N} collision in each
\mbox{A-A} interaction, we calculate $\rho_A(\vec{r},z)$ and $L(\vec{r})$ for the two colliding nucleons, and the product of the two
corresponding shadowing factors, according to Eq.~\ref{eq:5} and~\ref{eq:6}. By averaging the shadowing factors over all the \mbox{N-N}
collisions in each \mbox{A-A} interaction, we get the centrality dependence of the shadowing factors. In Fig.~\ref{fig:3} we show, as a
function of $L$, the calculated shadowing factors for \mbox{In-In} and \mbox{Pb-Pb} collisions
at $y$=0.5. For the SPS data at 158 GeV, this is the rapidity where the acceptance reaches its maximum. The symbols, connected by the
continuous lines, have been calculated using Eq.~\ref{eq:5} for the local dependence of shadowing inside the nucleus, whereas the dashed
line has been obtained using the parameterization of Eq.~\ref{eq:6}.
We notice that the two parametrizations give similar results, their difference not exceeding 2-3\%. In absence of other nuclear effects,
this result implies a $\sim$10\% anti-shadowing effect for central nucleus-nucleus collisions. When using EPS08, the effect increases
up to $\sim$25\%.
\begin{figure}[h]
\centering
\resizebox{0.48\textwidth}{!}
{\includegraphics*[bb=0 0 530 530]{fig4.eps}}
\caption{Shadowing factors for \mbox{In-In} (top) and \mbox{Pb-Pb} collisions at 158 GeV, at $y$=0.5, obtained with EKS98. Solid circles
refer to the EKS98 parametrization, open circles to EPS08. The symbols connected by a
continuous line have been calculated using Eq.~\ref{eq:5}, whereas the dashed lines have
been obtained using the parametrization of Eq.~\ref{eq:6} for the local dependence of shadowing.}
\label{fig:3}
\end{figure}
\section{Comparison of cold nuclear matter effects in \lowercase{p}-A and \mbox{A-A} collisions}
We now want to discuss the extrapolation of cold nuclear matter effects measured in \mbox{p-A} to \mbox{A-A} collisions, taking into
account shadowing effects together with final state absorption of the created $c\overline c$ pair. To do that, we model final state
absorption effects using the simple law $\exp(-\rho\sigma^{abs}_{\rm J/\psi}L)$, which has been shown to reproduce, at first order,
data on \rm J/$\psi$\ production at fixed target energies. In this approach, the \rm J/$\psi$\ production cross section per \mbox{N-N} collision
in \mbox{p-A} is given by
\begin{equation}
\frac{1}{A}\sigma^{pA}_{\rm J/\psi}=\sigma^{NN}_{\rm J/\psi}\times S_{pA}^{\rm J/\psi}\times\exp(-\rho\sigma^{abs}_{\rm J/\psi}L)
\label{eq:7}
\end{equation}
We then fit the \mbox{p-A} cross-section per \mbox{N-N} collision, obtained through Eq.~\ref{eq:7} with the simple
$\exp(-\rho\sigma^{eff}_{\rm J/\psi}L)$ law, i.e. neglecting the existence of shadowing. This is the procedure usually followed at SPS
energies to extract the so-called ``normal nuclear absorption''. Clearly this approach, when shadowing factors are not negligible, leads
to two main consequences. On one hand the obtained $\sigma^{eff}_{\rm J/\psi}$ values do not represent anymore the size of final-state
absorption, but an effective quantity which, due to the presence of
shadowing~\cite{Lou09,Arl09}, may be quite different from $\sigma^{abs}_{\rm J/\psi}$.
On the other hand, because shadowing effects do not necessarily have the same $L$-dependence of final state absorption,
$\exp(-\rho\sigma^{eff}_{\rm J/\psi}L)$ may not
give anymore a reasonable fit of the \mbox{p-A} data. However, given the size of the shadowing corrections, their deviation from the
exponential behavior is difficult to observe in the existing data samples.
The \rm J/$\psi$\ cross section per \mbox{N-N} collision in \mbox{A-A}, as a function of centrality, is then obtained using Eq.~\ref{eq:7},
replacing $S_{pA}^{\rm J/\psi}$ with $S_{AA}^{\rm J/\psi}$ and $1/A$ with $1/N_{\rm coll}$, where the shadowing factor and the
number of \mbox{N-N} collisions are calculated for the various centrality bins.
We can now compare the extrapolation of the exponential fit of the \mbox{p-A} calculation with what has been obtained for \mbox{A-A}
collisions. When doing that, we may expect significant deviations from
such an extrapolation. First of all, in \mbox{A-A} shadowing affects not only the target nucleus but also the projectile, leading to an
extra-effect that is clearly not present in the naive extrapolation of the exponential fit of \mbox{p-A}. Furthermore, these deviations
may heavily depend on rapidity, because when moving away from $y$=0 the $x$-region probed in the projectile becomes different from that of
the target.
To illustrate the procedure, we show in Fig.~\ref{fig:4}, for 158 GeV energy and $y$=0.5, the expected behavior for \mbox{p-A}
collisions, including shadowing and having assumed $\sigma^{abs}_{\rm J/\psi}$=4 mb. We also show the result of an exponential fit to
\mbox{p-A} data, which gives $\sigma^{eff}_{\rm J/\psi}$=3.0 mb, a value smaller than $\sigma^{abs}_{\rm J/\psi}$, because of the presence of an
anti-shadowing effect. In the same plot we compare this exponential fit with the calculated cross section values per \mbox{N-N}
collision for \mbox{In-In} and \mbox{Pb-Pb}, as a function of centrality. We clearly see that \mbox{A-A} cross sections deviate from the
extrapolation of \mbox{p-A} results.
In particular, at fixed $L$, the \rm J/$\psi$\ cross section per \mbox{N-N} collision is systematically lower in \mbox{A-A} with respect to
\mbox{p-A}. However, we note that the relative behavior of \mbox{A-A} results with respect to \mbox{p-A}, at a certain $L$, cannot be easily
deduced by a simple inspection of the \mbox{p-A} (and \mbox{A-p}) shadowing factors. In fact, from the geometry of the interaction, it can be
shown that the same $L$ for \mbox{p-A}~(\mbox{A-p}) and \mbox{A-A} corresponds to very different average nuclear densities probed in the
collision and therefore to a different average strength of the shadowing effects. (To give a numerical example, the $L$ value
corresponding to \mbox{p-Pb} collisions is obtained with \mbox{Pb-Pb} collisions at $b=12$ fm. The average nuclear densities probed are
0.76$\rho_0$ and 0.48$\rho_0$ respectively, where $\rho_0$ is the core nuclear density).
Therefore, the \mbox{A-A} shadowing factors cannot be obtained as a simple product of \mbox{p-A} and \mbox{A-p} shadowing at the same $L$.
\begin{figure}[htbp]
\centering
\resizebox{0.45\textwidth}{!}
{\includegraphics*[bb=0 0 530 530]{fig5.eps}}
\caption{Comparison of \mbox{p-A} and \mbox {A-A} cross sections for \rm J/$\psi$\ at 158 GeV, for $y$=0.5, including shadowing and final state
absorption, in arbitrary units. The \mbox{p-A} cross sections per \mbox{N-N} collisions are shown as triangles. The line represents an
exponential fit to \mbox{p-A} data. \mbox{In-In} and \mbox{Pb-Pb} cross sections are shown as circles and squares, respectively.}
\label{fig:4}
\end{figure}
A deviation of \mbox{A-A} results from the \mbox{p-A} extrapolations was indeed found at SPS energies, and it was called
``anomalous \rm J/$\psi$\ suppression''~\cite{Ale05,Arn07}. Usually this effect was connected to hot nuclear matter effects, including the production of a
deconfined state~\cite{Gra02}. The result of Fig.~\ref{fig:4} shows that at least a fraction of this effect can be attributed to having
neglected the influence of shadowing in the determination of the ``nuclear absorption'' reference. In Fig.~\ref{fig:5} we present the ratio
between the \rm J/$\psi$\ cross sections for \mbox{In-In} and \mbox{Pb-Pb} and the exponential extrapolation of \mbox{p-A} results, for three
rapidity values ($y$=0, 0.5 and 1), using EKS98 and Eq.~\ref{eq:5} for the local dependence of shadowing.
This kinematical range corresponds to the region where \rm J/$\psi$\ production has been
studied by the NA50/NA60 experiments in nuclear collisions. The values of these ratios do not depend on the specific
$\sigma^{abs}_{\rm J/\psi}$ value used in the calculation, because, at fixed $L$, the factors $\exp(-\rho\sigma^{abs}_{\rm J/\psi}L)$
cancel out in the ratio of the cross sections between \mbox{A-A} and \mbox{p-A}.
\begin{figure}[htbp]
\centering
\resizebox{0.45\textwidth}{!}
{\includegraphics*[bb=0 0 530 530]{fig6.eps}}
\caption{Ratio between the \mbox{A-A} cross section per \mbox{N-N} collision and the extrapolation of \mbox{p-A} results, for three
rapidity values, using the EKS98 parametrization. The results for \mbox{In-In} and \mbox{Pb-Pb} are shown as circles and squares,
respectively.}
\label{fig:5}
\end{figure}
The values plotted in Fig.~\ref{fig:5} show that a simple extrapolation of \mbox{p-A} results, obtained through a fit of the
$A$-dependence that parametrizes all nuclear effects under a single parameter $\sigma^{eff}_{\rm J/\psi}$, is not in agreement
with \mbox{A-A} \rm J/$\psi$\ cross sections as a function of centrality. This is basically because of the presence in \mbox{A-A} of shadowing
effects in the projectile that are of course not present in \mbox{p-A}. Although the effect remains rather small at midrapidity, it
increases fast when moving away from $y$=0, reaching a discrepancy of $\sim$20\% between the extrapolation of \mbox{p-A} data and the
central \mbox{A-A} results for $y$=1. This effect must clearly be taken into account when looking for hot nuclear matter effects in
\mbox{A-A} data. In particular, the anomalous suppression values observed at the SPS must be rescaled by the values shown in
Fig.~\ref{fig:5} at the corresponding rapidities, resulting in a $\sim$10\% average reduction of this effect in the interval 0$<y<$1
where data were taken. When using the EPS08 parametrization, similar values to those of Fig.~\ref{fig:5} were obtained, with
relative discrepancies not larger than $\sim$2\%.
However, it is well known that the uncertainties on the modification of gluon PDFs are
quite large, and very recent analysis (EPS09) are now starting to systematically address this issue~\cite{Esk09}. The LO set of
EPS09 gluon PDFs modification has an average value, in the $x$-region corresponding to SPS data, quite similar to the EKS98 one, with
an error of the order of $\pm$15\%. By injecting such an uncertainty in our calculation, it turns out that the values shown in
Fig.~\ref{fig:5} vary by about 5\%. Finally, other sets of nuclear modifications to gluon PDFs exist in the literature that exhibit either
no (or little) anti-shadowing in our $x$-region (nDS/nDSg~\cite{deF04}) or an anti-shadowing very strongly increasing with $x$
(HKN~\cite{Hir07}). The use of such sets in our analysis gives almost no difference between
the extrapolation of \mbox{p-A} results and \mbox{A-A} for nDS/nDSg (ratios $\sim$1 in Fig.~\ref{fig:5}), or higher values,
increasing with $y$, for \mbox{A-A} with respect to \mbox{p-A} for HKN (i.e. ratios larger than 1 in Fig.~\ref{fig:5}).
However, it was pointed out~\cite{Esk09} that such analyses might be less constrained in the $x$-region under study, since
they do not make use of data from high-p$_{\rm T}$ $\pi$ production in \mbox{d-Au} collisions from the BNL Relativistic Heavy-Ion Collider
(RHIC)~\cite{Adl07,Ada06}, which are
relevant for the determination of the large-$x$ region gluon contribution.
\section{Conclusions}
We have investigated in this article the role of shadowing for \rm J/$\psi$\ production in \mbox{p-A} and \mbox{A-A} collisions at
SPS energies. In particular, we have shown that an extrapolation of cold nuclear matter effects measured in \mbox{p-A} which does not take explicitly
into account shadowing, cannot reproduce in a correct way such effects for \mbox{A-A}. In the frame of an LO Color Evaporation Model calculation,
performed using the EKS98 and EPS08 parametrizations, we have shown that neglecting shadowing, the \mbox{p-A} extrapolation is biased by
$\sim$10\% at $y$=0.5 for central \mbox{In-In} and \mbox{Pb-Pb} collisions. Such a bias must be taken into account in analysis that aim
at determining effects from hot nuclear matter in \rm J/$\psi$\ production in \mbox{A-A} collisions at SPS energies
(the so-called ``anomalous suppression''). In particular it may be quantitatively important for lighter systems such as \mbox{In-In}, where
the deviations from \mbox{p-A} extrapolations are relatively small.
\begin{acknowledgments}
The authors wish to thank F.~Arleo, C.A.~Salgado and R.~Vogt for useful discussions on the topic covered in this article and for their comments
on the manuscript.
We also gratefully acknowledge the help of D.~Berzano and F.~Prino for some of the computational aspects.
\end{acknowledgments}
| {
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Drummondita es un género con ocho especies de plantas de flores perteneciente a la familia Rutaceae.
Especies seleccionadas
Drummondita calida
Drummondita ericoides
Drummondita hasellii
Drummondita longifolia
Drummondita microphylla
Drummondita miniata
Drummondita pheblium
Drummondita wilsonii
Referencias
Rutaceae | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 5,493 |
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MediaSarah PalinEat The PressRepublican Convention
Buck Up, Media. If You're The Enemy, You're Finally Doing Something Right
Sarah Palin, who has spent less than a week in the media spotlight and has thus far made herself completely unavailable for interviews, is blasting the media with both barrels, deriding them for daring to cover her.
Rachel Sklar
Founder, TheLi.st
Nov 8, 2008, 06:42 AM EST | Updated May 25, 2011
Sarah Palin isn't even finished her speech and already the "blame the media" meme has been pushed explicitly to the forefront — this woman who has spent less than a week in the media spotlight, who has thus far made herself completely unavailable for interviews, has blasted the media with both barrels, deriding them for daring to cover her and the revelations that continue to emerge about this untested, unknown and unvetted candidate.
Here's how she characterized the media's coverage of her for these past few days, since it's been revealed that John McCain met her once and didn't investigate her political past:
I'm not a member of the permanent political establishment. And I've learned quickly, these past few days, that if you're not a member in good standing of the Washington elite, then some in the media consider a candidate unqualified for that reason alone.
Really? Care to qualify that with any examples? Because I haven't seen you submit to any interviews, sit for a grilling on "Meet The Press" (oh, to have Russert here right now!), or open yourself up willingly to the scrutiny of the fourth estate, whose importance as a check on the political process is so critical that it was enshrined in the U.S. Constitution by the founders.
I don't think the press has any problem with mavericks — hell, John McCain has gotten by on that one with the help of his 'base' for eight years now. And I don't think that the press requires any candidate to be party of the "Washington elite" or the "permanent political establishment" — otherwise a certain former community organizer (new dirty word!) wouldn't have had a hope in hell of those glowing magazine covers. But it's not unreasonable to expect that a candidate for the second-highest office in the land make herself available to the media — the representative of the people, at least in terms of asking the hard questions that a governor might, say, hire a lawyer to consult with before answering. Just by way of example.
So the McCain campaign was so miffed by Campbell Brown's tough questioning of their surrogate — a trained, smooth, competent, TV-ready surrogate well-versed in talking points — who still couldn't muster up proof of Palin's leadership mettle in the Alaska National Guard — that they pulled McCain from an interview? So they're accusing the press of being "on a mission to destroy" Sarah Palin? That's crazy. No one has had time yet to form an opinion — let alone enough information. And a campaign with nothing to fear would have no problem throwing open the doors and saying, come on in, we've got nothing to hide...and we know that because we actually, you know, checked. You need a whole lot less bluster when the facts are on your side.
There's no reason for Sarah Palin — or the McCain campaign — to be so shocked that the media might want to actually know something about a VP nominee. My God, how many weeks of speculation and discussions about vetting did we endure between the end of the primaries and this point? Palin was a wild-card candidate, so far better known for her penchant for moose-burgers, aerial wolf hunting, and — yes — her pregnant and unmarried 17 year old daughter than she is for her actual qualifications that being a heartbeat away from the presidency. There is absolutely nothing wrong with the media's desire to even that score. That is its job, plain and simple — and if it doesn't always thrill the guy on the Straight Talk Express, well, so be it.
So — here's a little newsflash for Sarah Palin, to paraphrase her speech: The media isn't writing about you to seek your good opinion — they're writing about you to serve the people of this country. Americans expect the media to investigate their candidates for office for the right reasons, not just to get the right access. If you really want to serve the people — as opposed to just your party, or yourself — then you'll do well to remember that.
Update: Quoting Joe Klein from earlier today:
There is a tendency in the media to kick ourselves, cringe and withdraw, when we are criticized. But I hope my colleagues stand strong in this case: it is important for the public to know that Palin raised taxes as governor, supported the Bridge to Nowhere before she opposed it, pursued pork-barrel projects as mayor, tried to ban books at the local library and thinks the war in Iraq is "a task from God." The attempts by the McCain campaign to bully us into not reporting such things are not only stupidly aggressive, but unprofessional in the extreme.
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Q: Generalized modulo arithmetic This question (and especially this answer and the comments on it) actually made me think about a sort of generalized modulo arithmetic that would deal with all modulos at once and would basically make all the notations equally valid.
Here's the formal definition:
Consider the set $\mathbb Z\times\mathbb N$. Now on that set define the equivalence relation
$$(a,b) \sim (c,d) \text{ iff } b=d \text{ and } b|(a-c)$$
where the convention is used that $0|n$ iff $n=0$, so $(a,0)=(b,0)$ iff $a=b$. Be $M$ the set of equivalence classes under this relation. In the following I write the members of $M$ with square brackets, that is,
$$[a,b] = \{x \in \mathbb Z\times\mathbb N: x \sim (a,b)\}$$
Now define addition as
$$[a,b] + [c,d] = [a + c, \gcd(b,d)]$$
and
$$[a,b]\,[c,d] = [ac, \gcd(b,d)]$$
where $\operatorname{lcd}(0,n)=n$.
Clearly the set $\{[z,0]: z\in Z\}$ is equivalent to the integers (and will in the following be identified with them, so $z=[z,0]$), and the set $\{(z,n): z\in Z\}$ is equivalent to the classes modulo $n$ (with the extreme case $n=1$ which contains only a single element).
Now we define an unary operator $\mod: \mathbb N\to M, z\mapsto [1,z]$. Moreover, in a slight abuse of notation, we define
$$[a,b] \equiv [c,d] \text{ iff } [a,\gcd(b,d)] = [c,\gcd(b,d)]$$
This is a slight abuse in notation because here $\equiv$ is of course no equivalence relation (but then, the use of $\equiv$ for equivalence is a convention; we also normally use $+$ for commutative operations, yet on the ordinals $+$ is not commutative).
Now with those definitions, in $m \equiv n \pmod k$ the left hand side in an integer, identified with $[m,0]$, and the right hand side is the product of $[n,0]$ with $[1,k]$ (the result of the unary $\mod{}$ operator), which gives $[n,k]$. Now according to above definition of $\equiv$, $[m,0]\equiv[n,k]$ just means $[m,k] = [n,k]$ which is exactly what the ordinary interpretation of the above expression says. On the other hand, in this interpretation it would be equally correct to write $m \pmod k\equiv n\pmod k$ or even $m\pmod k = n\pmod k$ (however not $m = n \pmod k$).
Now my question is twofold:
*
*Does the above actually make sense? Or is there some fundamental problem with this that I didn't notice?
*Has such a construction (not necessarily the same, but in the same spirit, and also allowing mixing of different modulos in the same equation) already been considered? Does it possibly even have some practical use?
| {
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\section{Introduction}
We consider a general model of sender-receiver games.\ The specific feature
of our games is that the sender has an outside option. After the cheap talk
phase, the receiver proposes a decision to the sender; if the sender
approves it, the decision is made; otherwise, the sender chooses his outside
option, which can be interpreted as \textquotedblleft exit\textquotedblright
. Under complete information, the game reduces to an ultimatum game, in
which one player makes a \textquotedblleft take it or leave
it\textquotedblright\ offer to the other.\ In our framework, this other
player has private information and can send a costless message to the
receiver before getting an offer.
We are interested in situations in which the sender's approval is crucial to
the receiver.\ We thus assume that the receiver's utility in case of exit is
very low, as compared to what he can expect if the sender accepts his
proposal. It is not difficult to find examples in which a decision-maker
consults with an informed party before making a proposal that can be
ultimately rejected and in which rejection has unvaluable, damaging
consequences for the decision-maker. For instance, firms try to figure out
workers' requirements in order to avoid strikes and boycotts.\ Governments
discuss with kidnappers, hoping that hostages will not be killed. As a third
example, analyzed in Matthews (1989), the U.S.\ Congress may worry about the
President's veto.
We focus on equilibria in which the sender does not make uncredible threats
at the approval stage, namely, accepts a proposal if and only if it gives
him at least the utility of his outside option.\footnote{%
Our solution concept is basically subgame perfect Nash equilibrium. Except
for the approval stage, our model behaves as a standard cheap talk game, in
which Perfect Bayesian equilibrium is not restrictive.} We ask whether our
sender-receiver game has an equilibrium in which exit does not occur.
To answer this question, we introduce an auxiliary \textquotedblleft limit
game\textquotedblright\ $\Gamma $, in which equilibria are necessarily
without exit. The equilibria of $\Gamma $ are easily characterized by two
sets of conditions: incentive compatibility and constrained optimization.
Both sets of conditions are tractable but satisfying them jointly is
demanding. Existence of an equilibrium in the game $\Gamma $ is not
obvious.\ For instance, as opposed to standard sender-receiver games, $%
\Gamma $ may not have any nonrevealing equilibrium. This means that, in
absence of information transmission, the receiver cannot make any decision
that would give at least his reservation utility to the sender, whatever his
type. We identify various assumptions which guarantee that, in a situation
like this, the sender can credibly reveal some information to the receiver,
in such a way that exit will never happen.
We maintain the following assumptions on the game $\Gamma $: the sender has
finitely many types (which can be multidimensional, e.g., belong to $\mathbb{%
R}^{n_{1}}$, for some $n_{1}$), the receiver has a compact set of decisions
(typically, a closed, bounded set in $\mathbb{R}^{n_{2}}$, for some $n_{2}$)
and both players' utility functions are continuous.\footnote{%
This covers the particular case where the receiver has finitely many
actions, over which he can randomize.} We also make the \textit{sine qua non}
assumption that under complete information, i.e., when the receiver knows
the sender's type, there exists a decision that gives the sender at least
his reservation utility. We then consider the subsets of all types such that
there is a decision inducing them to participate and we call
\textquotedblleft participation structure\textquotedblright\ the maximal
subsets (with respect to set inclusion).\ For instance, if the sender has
only two possible types, 1 and 2, the participation structure is either $%
\left\{ \left\{ 1\right\} ,\left\{ 2\right\} \right\} $ or $\left\{
1,2\right\} $.
We establish that the game $\Gamma $ has a \textit{partitional} equilibrium,
namely, an equilibrium in which the sender's strategy is pure, in the
following cases:
\begin{description}
\item[(i)] the sender has two types;
\item[(ii)] the participation structure is a partition of the type set;
\item[(iii)] the decision set is a real interval and for every type, the
sender's utility function is monotonic in the receiver's decision;
\item[(iv)] the receiver's utility function -- when the sender participates
-- does not depend on the sender's type.
\end{description}
Cases (i) and (ii) are rather straightforward, with (ii) generalizing (i).
Case (iii) applies in particular when the receiver has only two actions,
over which he can randomize.\ Case (iv) has the most important scope.\ It
applies as soon as the receiver knows his own preferences over decisions,
but is eager to make a choice that will ensure the -- type-dependent --
informed player's participation. Existence of a partitional equilibrium
under (iv) is the main result of the paper (Theorem 8).
The previous assumptions may look restrictive, but, without them, existence
of an equilibrium in $\Gamma $ cannot be guaranteed, even if the sender is
allowed to use a mixed strategy.\ We indeed propose an example, in which the
sender has three types, the receiver has three actions, the participation
structure is not a partition and the utility functions are type-dependent.
In this example, there is no \textit{mixed} equilibrium.\ However, an
equilibrium does exist if the information transmission stage is handled by a
mediator.
Finally, we propose a complete analysis when the sender has three possible
types. We identify two kinds of \textquotedblleft participation
structures\textquotedblright\ beyond the straightforward case (ii) above.
The first one arises in the example mentioned in the previous paragraph.\
Existence of a mediated equilibrium can then be established.\ In the other
case, another example shows that there may not be any partitional
equilibrium.\ However we prove that there always exists a mixed equilibrium
if the decision set is convex and the utility functions are affine
(Proposition 9).
Here is a description of the paper.\ We discuss the related literature
below. In Section 2, we make the sender-receiver game $\Gamma $ and the
solution concept fully precise.\ Propositions 3 and 4 (which are established
in Section 6.1) allow us to argue that the game $\Gamma $ is relevant to our
study. In Section 3, we establish existence of a partitional equilibrium in $%
\Gamma $ under assumptions (i), (ii), (iii) or (iv) above. Our main result,
Theorem 8, is associated with case (iv).\ Section 4 is devoted to examples.\
Sections 4.1 and 4.2 propose a family of kidnapping games.\ Section 4.1
illustrates partitional equilibrium.\ Section 4.2 proposes a game that does
not have any partitional equilibrium but has a mixed equilibrium. Section
4.3 goes on with a game that does not have any mixed equilibrium but has a
mediated equilibrium. The three type case, including Proposition 9, is the
topic of Section 5. Section 6 is an appendix containing the proofs of
Propositions 3, 4 (Section 6.1) and 9 (Section 6.2).
\bigskip
\noindent \textbf{Related papers}
Shimizu (2013, 2017) adds an approval stage to Crawford and Sobel (1982)'s
sender-receiver game, in the popular case where the prior is uniform over
the unit interval and the utility functions are quadratic.\ He assumes, as
we do, that exit is damaging for the receiver but the setup is otherwise
quite different from ours. He shows that, in his particular model, credible
exit possibilities can make cheap talk informative even when the players'
conflict of interest is relatively large.
Matthews (1989) studies a sender-receiver game motivated by a specific
application, in which the sender is the U.S. President, the receiver is the
Congress and the decision is about a practical matter, like the level of
military expenditures. The President can veto the Congress' proposal.
Preferences are unimodal, as in Shimizu (2013, 2017), but the receiver's
utility does not depend on the sender's type (as in the current paper,
Theorem 8, Section 3.4).\ More importantly, in Matthews (1989)'s model, the
sender's rejection leads to status quo, rather than to exit, and does not
necessarily yield a very low utility to the receiver. Matthews (1989)'s
point is to show that thanks to incomplete information on the President's
type, veto can happen at equilibrium, i.e., without relying on uncredible
threats.
Our model can be viewed as a principal-agent problem in which the principal
-- alias the receiver -- cannot commit to a mechanism at the ex ante stage.\
This is an extreme case of Bester and Strausz (2001)'s principal-agent
problem with limited commitment.\ In this context, it makes sense to allow
the agent -- alias the sender -- to veto the principal's decision.\ Under a
mechanism design perspective, the principal looks for an equilibrium that
gives him the best ex ante expected utility, which amounts to an equilibrium
in which all types accept the principal's proposal if the principal's
utility, when the agent chooses his outside option, is sufficiently low.
This leads us to impose individual rationality conditions for the agent at
the \textquotedblleft posterior\textquotedblright\ stage, i.e., after the
principal has made a proposal. The relevance of posterior individual
rationality and its impact on incentive compatibility have been stressed in
a number of papers, e.g., Gresik (1991), Compte and Jehiel (2007, 2009)
Forges (1990, 1999) and Matthews and Postlewaite (1989).
Finally, Forges and Horst (2018)'s concept \textquotedblleft talk and
cooperate (perfect Bayesian) equilibrium\textquotedblright\ (TCE, Section
5.3) is motivated by the same questions as the present paper, but is defined
in a different model: the sender also has to make a decision, which is
relevant to his own payoff only.\ At a TCE, the receiver (who can be
interpreted as a principal) proposes a joint decision, which the sender
accepts whatever his type.\ Should player 1 reject player 2's proposal, both
players would choose an action, independently of each other. By contrast, in
the present paper, the sender just chooses an outside option.\ Forges and
Horst (2018) establish an existence result for another solution concept --
\textquotedblleft cooperate and talk (perfect Bayesian)
equilibrium\textquotedblright\ (CTE) -- but just indicate that their
methodology does not apply to TCE.
\section{Model}
\subsection{Sender-receiver games}
We start with a family of games\textbf{\ }$\Gamma (v_{0})$, $v_{0}\in
\mathbb{R}$, between a sender (player 1) and a receiver (player 2).\ $\Gamma
(v_{0})$ is described as follows:
\begin{itemize}
\item A type $k\in K$ is chosen according to a prior probability $p\in
\Delta (K)$.
\item Player 1 is informed of $k$.
\item Player 1 sends a message $m\in M$ to player 2.
\item Player 2 proposes a decision $x\in X$ to player 1.
\item If player 1 accepts player 2's proposal, the decision $x$ is enforced,
player 1 gets $U^{k}(x)$ and player 2 gets $V^{k}(x)$.
\item If player 1 rejects player 2's proposal, player 1 chooses an outside
option and gets $u_{0}^{k}$.\ Player 2 gets $v_{0}$.
\end{itemize}
\bigskip
We assume that:
\begin{itemize}
\item The set of types $K$ is finite\footnote{%
We do not make any assumption beyond the fact that there are finitely many
types; in particular, types can be \textquotedblleft
multidimensional,\textquotedblright\ with $K\subset \mathbb{R}^{n_{1}}$, for
some $n_{1}\geq 1$.} and $p^{k}>0$ $\forall k\in K$.
\item The set of messages $M$ is finite, such that $\mid M\mid \geq \mid
K\mid $.
\item The set of decisions $X$ is compact.\footnote{%
We will indicate explicitly when $X$ will be required to be convex.} As a
typical example, $X\subset \mathbb{R}^{n_{2}}$, for some $n_{2}\geq 1$; for
instance, player 2 has a finite set of actions $A$ and $X=\Delta (A)$
corresponds to the set of mixed strategies of player 2.
\item The utility functions $U^{k}:X\rightarrow \mathbb{R}$ and $%
V^{k}:X\rightarrow \mathbb{R}$ are continuous; for instance, if $X=\Delta
(A) $, $U^{k}$ and $V^{k}$ may correspond to expected utility.
\end{itemize}
We further assume that:
\begin{itemize}
\item For every $k\in K$, there exists$\ x\in X\ $such that$\ U^{k}(x)\geq
u_{0}^{k}$.
\item For every $k\in K$, for every $x\in X$, $V^{k}(x)\geq v_{0}$, namely,
\begin{description}
\item $v_{0}\leq \min_{k\in K}\min_{x\in X}V^{k}(x)$.
\end{description}
\end{itemize}
\bigskip
We are interested in situations in which the sender's approval is crucial to
the receiver, namely, in which $v_{0}$ can be arbitrarily low.\ Let us
denote as $\Gamma $ the \textquotedblleft limit game,\textquotedblright\ in
which $v_{0}=-\infty $. We will show that $\Gamma $ is a tractable tool,
which is appropriate to study $\Gamma (v_{0})$ when $v_{0}$ is small enough.
\bigskip
Let us set, for every $L\subseteq K$%
\begin{equation}
X(L)=\left\{ x\in X:U^{k}(x)\geq u_{0}^{k}\text{, }k\in L\right\} \text{.}
\label{X(L)}
\end{equation}%
Given a subset of types $L$, $X(L)$ is the set of decisions that are
acceptable by all types in $L$. We write $X(k)$ for $X(\left\{ k\right\} )$,
so that $X(L)=\bigcap\limits_{k\in L}X(k)$.
\subsection{Equilibria}
Our solution concept, in $\Gamma (v_{0})$ and $\Gamma $, is basically
subgame perfect Nash equilibrium, but perfect Bayesian equilibrium would not
be more demanding: as in standard sender-receiver games, finding beliefs
rationalizing player 2's choices is not an issue. What is crucial here is to
avoid non-credible threats from player 1. In the sequel, we simply refer to
\textquotedblleft equilibrium.\textquotedblright
At a subgame perfect Nash equilibrium, player 1 of type $k$ accepts (resp.,
rejects) player 2's proposal $x$ when $U^{k}(x)>u_{0}^{k}$ (resp., $%
U^{k}(x)<u_{0}^{k}$). We further assume that player 1 accepts the proposal
when he is indifferent, which is consistent with our interest in situations
in which player 2 strictly prefers that player 1 participates. By proceeding
backwards, $\Gamma (v_{0})$ amounts to a standard sender-receiver game, with
the following utility functions (in which $I$ denotes the indicator
function):%
\begin{equation}
U_{+}^{k}(x)=U^{k}(x)I(U^{k}(x)\geq
u_{0}^{k})+u_{0}^{k}I(U^{k}(x)<u_{0}^{k})=\max \left\{
U^{k}(x),u_{0}^{k}\right\} \label{PayU+}
\end{equation}%
for player 1 of type $k$ and%
\begin{equation}
W^{k}(v_{0},x)=V^{k}(x)I(U^{k}(x)\geq u_{0}^{k})+v_{0}I(U^{k}(x)<u_{0}^{k})%
\text{.} \label{Payv0}
\end{equation}%
for player 2, when player 1 is of type $k$.\footnote{%
This observation is made in Chen, Nartik and Sobel (2008), in their account
of Matthews (1989).} In the latter sender-receiver game, the receiver's
utility function is not necessarily continuous, but it is
upper-semi-continuous.
\begin{lem}\label{prop1}
For every $k\in K$ and $v_{0}\in \mathbb{R}$, the utility function $%
W^{k}(v_{0},\cdot )$ defined by (\ref{Payv0}) is upper-semi-continuous.
\end{lem}
\noindent{\bf Proof:} Let $x_{n}\in X$, $x_{n}\rightarrow x$.\ The only
possibly delicate case is when $U^{k}(x_{n})<u_{0}^{k}$ for every $n$ and $%
U^{k}(x)=u_{0}^{k}$. Then $W^{k}(v_{0},x_{n})=v_{0}\leq
V^{k}(x)=W^{k}(v_{0},x)$, using our assumption. $\blacksquare $
\bigskip
Having determined player 1's behavior at the approval stage, we can define a
strategy for player 1 (in $\Gamma (v_{0})$ and $\Gamma $) as a mapping $%
\sigma :K\rightarrow \Delta (M)$.\ We interpret $\sigma (k)(m)$ as the
probability that player 1 sends message $m$ when his type is $k$, and denote
it as $\sigma (m\mid k)$. We adopt the following notations:%
\begin{equation}
\text{For every }m\in M\text{, }P_{\sigma }(m)=\sum\limits_{k}p^{k}\sigma
(m\mid k)\text{.} \label{prob(m)}
\end{equation}%
\begin{equation}
\text{For every }k\in K\text{ and }m\in M\text{ s.t. }P_{\sigma }(m)>0\text{%
, }p_{m}^{k}(\sigma )=\frac{p^{k}\sigma (m\mid k)}{P_{\sigma }(m)}\text{.}
\label{posterior}
\end{equation}%
$p_{m}^{k}(\sigma )$ is thus the posterior probability of type $k$ computed
from $p$ and $\sigma $; let $p_{m}(\sigma )=(p_{m}^{k}(\sigma ))_{k\in K}$
denote the corresponding posterior probability distribution over $K$. We
have $\sum\limits_{m}P_{\sigma }(m)p_{m}(\sigma )=p$.
We say that $\sigma $ is nonrevealing\ if player 1 sends his message in a
type-independent way, namely, if $\sigma (m\mid k)=\sigma (m\mid k^{\prime
}) $ for every $m\in M$, $k$, $k^{\prime }\in K$.\ In this case, $%
p_{m}(\sigma )=p$ for every $m$ s.t. $P_{\sigma }(m)>0$.
For player 2, a strategy is a mapping $\tau :M\rightarrow X$, namely, a
\textquotedblleft pure\textquotedblright\ strategy with respect to the set $%
X $ (but as indicated above, $X=\Delta (A)$ for a finite set of actions $A$
is a particular case).\footnote{%
Restriction to pure strategies of player 2 is justified by the fact that
these will be enough to establish existence of equilibria in $\Gamma $.}
We say that $(\sigma ,\tau )$ is \textquotedblleft without
exit\textquotedblright\ if%
\begin{equation}
U^{k}(\tau (m))\geq u_{0}^{k}\text{ }\forall k\in K\text{, }\forall m\in M%
\text{ s.t. }\sigma (m\mid k)>0\text{,} \label{Noexit}
\end{equation}%
namely, if%
\begin{equation*}
U_{+}^{k}(\tau (m))=U^{k}(\tau (m))\text{ }\forall k\in K\text{, }\forall
m\in M\text{ s.t. }\sigma (m\mid k)>0\text{.}
\end{equation*}%
Recalling (\ref{X(L)}) and denoting by supp $q$ the support of a probability
distribution $q\in \Delta (K)$, condition (\ref{Noexit}) is equivalent to%
\begin{equation}
\tau (m)\in X(\text{supp }p_{m}(\sigma ))\text{ }\forall m\in M\text{ s.t. }%
P_{\sigma }(m)>0\text{.} \label{Noexitbis}
\end{equation}
At an equilibrium of $\Gamma $, we require that player 2's expected utility
be finite ($>-\infty $). Hence an equilibrium of $\Gamma $ is necessarily
without exit. By contrast, an equilibrium of $\Gamma (v_{0})$ may involve
exit of some types.
\begin{proposition}\label{prop2}
For every $v_{0}\in \mathbb{R}$, the game $\Gamma (v_{0})$ has a
nonrevealing\ equilibrium (possibly with exit). The game $\Gamma $ has a
nonrevealing equilibrium if and only if $X(K)\neq \emptyset $. Hence $\Gamma
$ may not have any nonrevealing equilibrium.
\end{proposition}
\noindent \textbf{Proof:} The following strategies define a nonrevealing
equilibrium in $\Gamma (v_{0})$: player 1 sends the same message $m\in M$
whatever his type and then accepts $x$ if and only if $U^{k}(x)\geq
u_{0}^{k} $; whatever the message, player 2 chooses $x^{\ast }\in X$ to
maximize $\sum_{k}p^{k}W^{k}(v_{0},x)$, which is well-defined thanks to
Lemma 1.
In $\Gamma $, if $X(K)\neq \emptyset $, a nonrevealing equilibrium can be
achieved as above, provided that player 2 chooses $x^{\ast }\in X$ to
maximize $\sum_{k}p^{k}V^{k}(x)$ subject to $x\in X(K)$. If $X(K)=\emptyset $
and player 1's message is type-independent, condition (\ref{Noexitbis})
cannot be satisfied. $\blacksquare $
\bigskip
The next two propositions (which are established in Section 6.1) give us
some foundations to study the equilibria of the limit game $\Gamma $ by
making precise relationships between the latter and the equilibria without
exit of $\Gamma (v_{0})$.
\begin{proposition}\label{prop3}
Let $(\sigma ,\tau )$ be an equilibrium without exit in $\Gamma (v_{0})$,
for some $v_{0}\in \mathbb{R}$.\ Then $(\sigma ,\tau )$ is an equilibrium
without exit in $\Gamma (z_{0})$ for every $z_{0}\in \mathbb{R}$ such that $%
z_{0}\leq v_{0}$ and is also an equilibrium in $\Gamma $, with the same
interim expected utility as in $\Gamma (v_{0})$ for both players.
\end{proposition}
In other words, if $\Gamma $\ has no a equilibrium (which indeed may happen,
see Section 4.3), then, \textit{whatever }$v_{0}\in \mathbb{R}$\textit{, }$%
\Gamma (v_{0})$\textit{\ has no equilibrium without exit, that is, all
equilibria of }$\Gamma (v_{0})$\textit{\ must involve non-participation of
at least one type.}
\begin{proposition}\label{prop4}
Let $(\sigma ,\tau )$ be an equilibrium in $\Gamma $. Then there exists $%
v_{0}\in \mathbb{R}$ such that, for every $z_{0}\leq v_{0}$, $(\sigma ,\tau
) $ is an equilibrium without exit of $\Gamma (z_{0})$, with the same
interim expected utility as in $\Gamma $ for both players.
\end{proposition}
The previous properties are useful under a \textit{mechanism design}
perspective.\ Assume player 2 is a \textquotedblleft
principal\textquotedblright\ who cannot commit to a mechanism $\mu
:K\rightarrow X$ but receives a message from the agent (player 1) and then,
makes a decision in $X$ subject to the agent's participation constraints.\
With this interpretation, which turns out to be an extreme case of Bester
and Strausz (2001)'s model, an optimal mechanism amounts to an equilibrium
of $\Gamma (v_{0})$ which maximizes player 2's ex ante expected utility. At
a given $v_{0}$, the equilibrium of $\Gamma (v_{0})$ that is best for player
2 may involve the exit of some types of player 1.\ By contrast, player 2's
best equilibrium payoff in the limit game $\Gamma $, when it exists, is
achieved at an equilibrium $(\sigma ^{\ast },\tau ^{\ast })$ without exit.
Let us denote player 2's corresponding payoff as $v_{NE}^{\ast }$. By
Proposition 4, if $v_{0}$ is small enough, $(\sigma ^{\ast },\tau ^{\ast })$
is an equilibrium \textit{without exit} in $\Gamma (v_{0})$, giving the same
expected utility $v_{NE}^{\ast }$ to player 2. By Proposition 3, $%
v_{NE}^{\ast }$ is the \textit{best} equilibrium payoff player 2 can achieve
at an equilibrium \textit{without exit} in $\Gamma (v_{0})$. Moreover, as we
show in details in Section 6.1, if $v_{0}$ is sufficiently low, player 2
cannot expect an expected utility higher than $v_{NE}^{\ast }$ at an
equilibrium of $\Gamma (v_{0})$ \textit{with exit} of some types.\footnote{%
The observation that the principal's ex ante expected utility is maximized
when all types of the agent participate is also made in Bester and Strausz
(2001), footnote 8.} Summing up, if the limit game $\Gamma $ has an
equilibrium, an optimal mechanism, when the principal's utility $v_{0}$ in
case of exit is sufficiently low, can be found by maximizing player 2's
utility over the equilibria of $\Gamma $, without worrying about the precise
level $v_{0}$.
\section{Existence of a partitional equilibrium in $\Gamma $}
In this section, we focus on the game $\Gamma $.\ We identify various
sufficient conditions for the existence of a \textit{partitional}
equilibrium in $\Gamma $, in which player 1 uses a pure strategy, namely a
mapping $\sigma :K\rightarrow M$, to send his message to player 2. The
strategy $\sigma $ then induces the partition $\left\{ K_{m}\text{, }m\in
\sigma (K)\right\} $ of $K$, with $K_{m}=\sigma ^{-1}(m)=\left\{ k\in
K:\sigma (k)=m\right\} $.\ In this case, (\ref{prob(m)}) and (\ref{posterior}%
) become respectively:%
\begin{equation}
\text{For every }m\in M\text{, }P_{\sigma }(m)=\sum\limits_{k\in K_{m}}p^{k}%
\text{.} \label{prob(m) pure}
\end{equation}%
\begin{equation}
\text{For every }k\in K\text{ and }m\in M\text{ s.t. }P_{\sigma }(m)>0\text{%
, }p_{m}^{k}(\sigma )=\frac{p^{k}I(k\in K_{m})}{P_{\sigma }(m)}\text{.}
\label{posterior pure}
\end{equation}
As in Section 2, for player 2, we focus on strategies of the form $\tau
:M\rightarrow X$. At an equilibrium of $\Gamma $, given player 1's strategy $%
\sigma $ and the message $m$ he receives, player 2 updates his belief over $%
K $ into $p_{m}(\sigma )$.$\ $To avoid exit, we must have
\begin{equation}
X(\text{supp }p_{m}(\sigma ))\neq \emptyset \label{Xsupp}
\end{equation}%
Player 2's strategy $\tau $ is then a best response to $\sigma $ in $\Gamma $
if and only if%
\begin{equation}
\forall m\in M\text{ s.t. }P_{\sigma }(m)>0\text{, }\tau (m)\in \arg
\max_{x\in X(\text{supp }p_{m}(\sigma ))}\sum\limits_{k}p_{m}^{k}(\sigma
)V^{k}(x)\text{.} \label{Const Opt}
\end{equation}%
We refer to these conditions as to constrained optimization.
Player 1's equilibrium conditions reduce to incentive compatibility
conditions expressing that given player 2's strategy $\tau $, player 1 of
type $k$ prefers to send $\sigma (k)$ than any other message $m$, namely,
\begin{equation}
U^{k}\left( \tau (\sigma (k))\right) )\geq U^{k}\left( \tau (m)\right) \text{
for every }k\in K\text{ and }m\in M\text{.} \label{IC}
\end{equation}
\subsection{Two types}
If only two types are possible, we show that either there is a decision
giving both types at least their reservation utility or full revelation of
information is credible and allows to avoid exit.
\begin{proposition}\label{prop5}
Let us assume that $\mid K\mid =2$.\ Then $\Gamma $ has a partitional
equilibrium.
\end{proposition}
\noindent \textbf{Proof: }If\textbf{\ }$X(1)\cap X(2)\neq \emptyset $, let%
\begin{equation*}
x^{\ast }\in \arg \max_{x\in X(1)\cap X(2)}\left[ p^{1}V^{1}(x)+p^{2}V^{2}(x)%
\right]
\end{equation*}%
and let $m^{\ast }$ be an arbitrary element of $M$.\ Then $\sigma (1)=\sigma
(2)=m^{\ast }$ and $\tau (m)=x^{\ast }$ for every $m\in M$ defines a
nonrevealing equilibrium of $\Gamma $.
Otherwise, if $X(1)\cap X(2)=\emptyset $, let%
\begin{equation*}
x_{k}\in \arg \max_{x\in X(k)}V^{k}(x)\text{\quad }k=1,2
\end{equation*}%
and let $m_{1}\neq m_{2}$ be two distinct elements of $M$.\ Then $\sigma
(k)=m_{k}$, $\tau (m_{k})=x_{k}$, $k=1,2$, defines a fully revealing
equilibrium of $\Gamma $. Indeed, constrained optimization (\ref{Const Opt})
holds by construction; to see that incentive compatibility (\ref{IC}) also
holds, observe that $x_{k}\in X(k)$.\ Hence $x_{k}\notin X(\ell )$ for $\ell
\neq k$, since $X(1)\cap X(2)=\emptyset $.\ In other words, $U^{\ell
}(x_{k})<u_{0}^{\ell }\leq U^{\ell }(x_{\ell })$ for $\ell \neq k$. $%
\blacksquare $
\subsection{Straightforward partitional equilibria}
In this section, we propose an easy generalization of Proposition 5 when the
sender has an arbitrary number of types. Recalling the definition of $X(L)$
(see (\ref{X(L)})), let us set%
\begin{equation*}
\mathcal{T}=\left\{ \emptyset \neq L\subseteq K:X(L)\neq \emptyset \right\}
\end{equation*}%
and define $\mathcal{T}^{\ast }$ as the set of maximal elements of $\mathcal{%
T}$ for set inclusion, namely,%
\begin{equation}
\mathcal{T}^{\ast }=\left\{ L\in \mathcal{T}:\left[ L^{\prime }\in \mathcal{T%
}\text{ and }L\subseteq L^{\prime }\right] \Rightarrow L^{\prime }=L\right\}
\text{.} \label{Maximal elements}
\end{equation}%
We refer to $\mathcal{T}^{\ast }$ as to the \textquotedblleft participation
structure\textquotedblright\ of the game $\Gamma $.
\begin{proposition}\label{prop6}
If the participation structure of $\Gamma \mathcal{\ }$is a partition of $K$%
, $\Gamma $ has a partitional equilibrium.
\end{proposition}
\noindent \textbf{Proof:}
Let $\mathcal{T}^{\ast }=\left\{ K_{r}\right\} $.\ Consider the strategy of
player 1 consisting of revealing the cell $K_{r}$ containing his type. Let $%
x_{r}^{\ast }\in X(K_{r})$ be an optimal decision of player 2 when he learns
that player 1's type belongs $K_{r}$, namely,%
\begin{equation*}
x_{r}^{\ast }\in \arg \max_{x\in X(K_{r})}\sum_{k\in K_{r}}\frac{p^{k}}{%
\sum_{j\in K_{r}}p^{j}}V^{k}(x)\text{.}
\end{equation*}%
Constrained optimization (\ref{Const Opt}) holds by construction.\ Incentive
compatibility (\ref{IC}) is also immediate, because if $k\in $ $K_{r}$, $%
x_{r}^{\ast }\in X(K_{r})$ while for $j\neq r$, $x_{j}^{\ast }\notin
X(K_{r}) $. $\blacksquare $
\subsection{Decision in \textbf{a real interval and monotonic utility
function for the sender}}
The next result holds in particular when player 2's decision can be
interpreted as a probability distribution over two possible actions (i.e., $%
X=\Delta (A)$,$\mid A\mid =2$) and the utility $U^{k}(x)$ of player 1 of
type $k$ is expected utility with respect to\ $x$.
\begin{proposition}\label{prop7}
Let us assume that \textit{the decision set }$X$ \textit{is} \textit{a real
interval and every utility function }$U^{k}$, $k\in K$, \textit{is monotonic
over} $X$.\ Then $\Gamma $ has a partitional equilibrium.
\end{proposition}
\noindent \textbf{Proof:}
Let us take, without loss of generality, $X=\left[ 0,1\right] $; define%
\begin{eqnarray*}
K_{-} &=&\left\{ k\in K:U^{k}\text{ is weakly decreasing and not constant}%
\right\} \\
K_{+} &=&\left\{ k\in K:U^{k}\text{ is weakly increasing or constant}\right\}
\end{eqnarray*}%
We can find $x_{0}^{k}$, $k\in K$, such that if $k\in K_{-}$, $U^{k}(x)\geq
u_{0}^{k}\Leftrightarrow x\leq x_{0}^{k}$ and if $k\in K_{+}$, $U^{k}(x)\geq
u_{0}^{k}\Leftrightarrow x\geq x_{0}^{k}$. We define next
\begin{center}
$%
\begin{array}{cccc}
x_{-}=\min_{k\in K_{-}}x_{0}^{k} & \text{if }K_{-}\neq \emptyset \text{ ;} &
x_{-}=1 & \text{if }K_{-}=\emptyset \text{.} \\
x_{+}=\max_{k\in K_{+}}x_{0}^{k} & \text{if }K_{+}\neq \emptyset \text{ ;} &
x_{+}=0 & \text{if }K_{+}=\emptyset \text{.}%
\end{array}%
$
\end{center}
\noindent If $x_{+}\leq x_{-}$ (in particular, if $K_{-}$ or $K_{+}$ $%
=\emptyset $), let%
\begin{equation*}
x^{\ast }\in \arg \max_{\left[ x_{+},x_{-}\right] }\sum_{k\in K}p^{k}V^{k}(x)
\end{equation*}%
and let $m^{\ast }$ be an arbitrary element of $M$.\ Then $\sigma
(k)=m^{\ast }$ for every $k\in K$ and $\tau (m)=x^{\ast }$ for every $m\in M$
defines a nonrevealing equilibrium of $\Gamma $.
\noindent If $x_{+}>x_{-}$, let $m_{-}^{\ast }\neq m_{+}^{\ast }$ be two
distinct elements of $M$.\ Take $\sigma (k)=m_{-}^{\ast }$ if $k\in K_{-}$, $%
\sigma (k)=m_{+}^{\ast }$ if $k\in K_{+}$, namely, $\sigma $ induces the
partition $\left\{ K_{-},K_{+}\right\} $.\ Player 2's corresponding
posterior probability distribution on $K$ can be computed as in (\ref%
{posterior pure}):%
\begin{equation*}
p_{m_{-}^{\ast }}^{k}=\frac{p^{k}I(k\in K_{-})}{\sum_{j\in K_{-}}p^{j}}\text{
and }p_{m_{+}^{\ast }}^{k}=\frac{p^{k}I(k\in K_{+})}{\sum_{j\in K_{+}}p^{j}}%
\text{.}
\end{equation*}%
Let then%
\begin{equation*}
x_{-}^{\ast }\in \arg \max_{\left[ 0,x_{-}\right] }\sum\limits_{k\in
K_{-}}p_{m_{-}^{\ast }}^{k}V^{k}(x)\text{ and }x_{+}^{\ast }\in \arg \max_{%
\left[ x_{+},1\right] }\sum\limits_{k\in K_{+}}p_{m_{+}^{\ast }}^{k}V^{k}(x)%
\text{.}
\end{equation*}%
Constrained optimization (\ref{Const Opt}) holds by construction.\ There
remains to check incentive compatibility (\ref{IC}). Observe that $%
x_{-}^{\ast }<x_{+}^{\ast }$; for $k\in K_{-}$, $U^{k}$ is decreasing, hence
$U^{k}(x_{-}^{\ast })\geq U^{k}(x_{+}^{\ast })$. Similarly for $k\in K_{+}$,
$U^{k}$ is increasing so that $U^{k}(x_{-}^{\ast })\leq U^{k}(x_{+}^{\ast })$%
. $\blacksquare $
\subsection{Type-independent utility function for the receiver}
In this section, we assume that, when the sender accepts the receiver's
proposal, the utility function of the receiver does not depend on the
sender's type, namely, that $V^{k}(x)=V(x)$ for every $k$ and $x$. This
assumption is sometimes referred to as \textquotedblleft private
values\textquotedblright\ or \textquotedblleft known-own
payoff.\textquotedblright\ Matthews (1989) formulates it in the context of a
game of information transmission with sender's approval.
\begin{theorem}\label{prop8}
\textit{Let us assume that player 2's utility function does not depend on
player 1's type, namely, that there exists a continuous function }$%
V:X\rightarrow \mathbb{R}$ such that\textit{\ }$V^{k}=V$ \textit{for every} $%
k\in K$\textit{.\ }Then $\Gamma $ has a partitional equilibrium.
\end{theorem}
The proof consists of an algorithm, which constructs a partitional
equilibrium that is as revealing as possible, given the incentive
compatibility constraints to be fulfilled.\ More precisely, the initial
candidate is the fully revealing equilibrium. Imagine that type $k$ would
envy type $\ell $ if one tried to implement fully revealing strategies,
while type $\ell $ would not envy any type. By merging type $\ell $ and type
$k$, one reduces the incentives problem. A key property is that, if player
2's utility function is independent of player 1's type, then player 2's
optimal decision $x_{\ell }$ when facing type $\ell $ remains optimal when
facing type $\ell $ \textit{or} type $k$.\ Before making use of it, we first
show, by relying on the same kind of argument, that the envy relation cannot
have any cycle.
\bigskip
\noindent \textbf{Proof:}
Let us fix, for every $k\in K$,%
\begin{equation}
x_{k}\in \arg \max_{x\in X(k)}V(x)\text{.} \label{opti k}
\end{equation}%
The existence of such $x_{k}$'s is guaranteed by our assumptions. If the
previous optimization problem has several solutions, we take $x_{k}$ to
maximize $U^{k}(x)$.
For every pair of types\textit{\ }$j,k\in K$, we say that \textquotedblleft
type $k$ envies type $j$\textquotedblright\ -- and write $k\mathcal{R}j$ --
if $U^{k}(x_{j})>U^{k}(x_{k})$. An immediate property is that%
\begin{equation}
\text{for every }j,k\in K\text{, }k\mathcal{R}j\Rightarrow V(x_{k})>V(x_{j})%
\text{.} \label{propR}
\end{equation}
To show this, observe that, by definition, $x_{k}\in X(k)$, i.e., $%
U^{k}(x_{k})\geq u_{0}^{k}$.\ Hence, if $k\mathcal{R}j$, we must have $%
U^{k}(x_{j})>u_{0}^{k}$, which implies $x_{j}\in X(k)$ (so that $x_{j}\in
X(j)\cap X(k)$) and $V(x_{k})\geq V(x_{j})$.\ But $V(x_{k})=V(x_{j})$ cannot
arise, because $U^{k}(x_{j})>U^{k}(x_{k})$ and, in case of multiple
solutions to $\max_{x\in X(k)}V(x)$, we choose $x_{k}$ to maximize $U^{k}(x)$%
.
The previous property implies that the envy relation $\mathcal{R}$ has no
cycle.
We will gradually construct a subset $L\subseteq K$ of \textit{leader types}
which do not envy any other type in $L$ and a subset $F=K\setminus L$ of
\textit{follower types }which envy a type in $L$.
We start with $L=F=\emptyset $. Let us denote as $\alpha _{1}<\cdots <\alpha
_{n}$ the distinct values among $V(x_{k})$, $k\in K$. Necessarily, $n\leq
\mid K\mid $.\ Define then%
\begin{equation}
K_{j}=\left\{ k\in K:V(x_{k})=\alpha _{j}\right\} \text{\quad }j=1,...,n%
\text{.} \label{Kj}
\end{equation}
\begin{description}
\item[Step 1] Consider every type $k\in K_{1}$: $V(x_{k})=\alpha _{1}$ is
strictly below any other $\alpha _{j}$.\ By (\ref{propR}), type $k$ cannot
envy any other type. We modify $L$ into $L=K_{1}$, while $F$ does not change
($F=\emptyset $).
\item[Step 2] Consider every type $k\in K_{2}$. If $k$ does not envy any
type, put $k$ in $L$. Otherwise, again by (\ref{propR}), $k$ can only envy a
type in $L$ (as defined at the end of step 1, namely, $K_{1}$), put $k$ in $%
F $.
\item[$\cdots $]
\item[Step \textit{j}] Let $L$ and $F$ be the sets of leaders and followers
constructed so far. $L\cup F=K_{1}\cup \cdots \cup K_{j-1}$ so that by (\ref%
{propR}) and (\ref{Kj}), types in $L\cup F$ cannot envy types in $K_{j}$.
Consider every such type $k\in K_{j}$. If $k$ envies a type in $L$, put $k$
in $F$. Otherwise, put $k$ in $L$. $L$ and $F$ are thus updated at the end
of step \textit{j}.
\item[$\cdots $]
\item[Step\textit{\ n}] Proceed as for step \textit{j}.\ Deduce the final
sets of leaders and followers.
\end{description}
For instance, if $\mid K\mid =3$ and $\mathcal{R}$ is fully described by $3%
\mathcal{R}2$ and $2\mathcal{R}1$, the previous construction results in $%
K_{1}=\left\{ 1\right\} $, $K_{2}=\left\{ 2\right\} $, $K_{3}=\left\{
3\right\} $, $L=\left\{ 1,3\right\} $.
Using the $x_{k}$'s defined by (\ref{opti k}) and the set $L$, we construct
an equilibrium $(\sigma ,\tau )$ of $\Gamma $. For simplicity, we rename the
messages in $M$ so that $L\subseteq M$. Player 1's strategy is such that $%
\sigma (K)=L$.\ More precisely, $\sigma :K\rightarrow L$ is defined by
\begin{center}
$%
\begin{array}{lll}
\sigma (k)= & k & \text{if }k\in L \\
& \arg \max_{j\in L,k\mathcal{R}j}U^{k}(x_{j}) & \text{if }k\in K\setminus L%
\text{.}%
\end{array}%
$
\end{center}
\noindent In other words, leader types announce themselves, while non leader
types report the leader type they most envy.\ Player 2's strategy is defined
by $\tau :L\rightarrow X:\tau (\ell )=x_{\ell }$, with $x_{\ell }$ defined
by (\ref{opti k}).
Incentive compatibility (\ref{IC}) follows from the fact that player 2's
strategy $\tau $ restricts his decisions to the subset $\left\{ x_{\ell
},\ell \in L\right\} $. Hence types in $L$, who cannot envy any other type
in $L$, are truthful.\ Types in $K\setminus L$ behave as well as they can
given the player 2's restricted decision set.
If player 1 follows $\sigma $, then, given message $\ell \in L$, player 2
deduces that player 1's type $k\in \sigma ^{-1}(\ell )$.\ The set $\sigma
^{-1}(\ell )$ contains $\ell $, $x_{\ell }\in X(\ell )$ by (\ref{opti k})
and all other types in $\sigma ^{-1}(\ell )$ envy $\ell $, so that $x_{\ell
}\in \bigcap_{k\in \sigma ^{-1}(\ell )}X(k)$.\ Since $x_{\ell }$ is a
maximizer of $V(x)$ over $X(\ell )$, it is also a maximizer of $V(x)$ over $%
\bigcap_{k\in \sigma ^{-1}(\ell )}X(k)$. $\blacksquare $
\bigskip
\noindent \textbf{Remarks:}
\begin{description}
\item[-] A main feature of the proof of Theorem 8 is that, in the
partitional equilibrium that is constructed, the receiver makes a decision
in a subset of $\left\{ x_{k},k\in K\right\} $ where $x_{k}$ is the optimal
decision he would make if he were sure to face type $k$. The receiver's
private values guarantee that if type $k$ envies type $\ell $, then $x_{\ell
}$, the receiver's optimal choice when he faces type $\ell $ for sure (i.e.,
under the constraint $x\in X(\ell )$), is still optimal when he faces type $%
k $ or type $\ell $ (i.e., under the constraint $x\in X(\ell )\cap X(k)$).\
This property may no longer hold when player 2's utility is type-dependent.
\item[-] Theorem 8 does not depend on the underlying utility representation:
the result holds if the receiver's von Neumann-Morgenstern \textit{%
preferences} over $X$ given type $k$ are equivalent for every $k\in K$.
\end{description}
\section{Examples (including a counter-example)}
\subsection{Partitional equilibrium}
Let the informed player have three possible types, i.e., $K=\left\{
1,2,3\right\} $ and let the uninformed player's decision set be%
\begin{equation*}
X=\left\{ (x_{a},x_{b}):x_{a}\geq 0,x_{b}\geq 0,x_{a}+x_{b}\leq 100\right\}
\text{.}
\end{equation*}%
Let the utility function and reservation utility of the informed player be
\bigskip
\begin{center}
$%
\begin{array}[t]{cc}
U^{1}(x)=x_{a}-x_{b} & u_{0}^{1}=30\text{,} \\
U^{2}(x)=x_{b}-x_{a} & u_{0}^{2}=40\text{,} \\
U^{3}(x)=x_{a}+2x_{b} & u_{0}^{3}=20\text{.}%
\end{array}%
$
\bigskip
\end{center}
\noindent Let the uninformed player's utility function be type-independent:%
\begin{equation*}
V^{k}(x)=V(x)=-(x_{a}+x_{b})\text{, }k=1,2,3\text{.}
\end{equation*}
There are two goods, $a$ and $b$, $X$ accounts for the decision-maker's
resource constraints.\ Type 1 likes good $a$, dislikes good $b$; type 2 has
symmetric preferences; type 3 likes both goods, and likes good $b$ more than
good $a$.
As a possible interpretation, player 1 is a kidnapper who can have political
motivations (type 1), just look for a monetary ransom (type 2) or be
opportunistic (type 3).\ Good $a$ stands for political prisoners who can be
released while good $b$ stands for money.\ If player 1 does not accept
player 2's offer, the hostage is killed, leading to an invaluable loss for
player 2.
Recalling (\ref{X(L)}) and using \textquotedblleft $Co$\textquotedblright\
for convex hull, we have here%
\begin{eqnarray*}
X(\left\{ 1\right\} ) &=&X(\left\{ 1,3\right\} )=Co\left\{
(30,0),(100,0),(65,35)\right\} \text{,} \\
X(\left\{ 2\right\} ) &=&X(\left\{ 2,3\right\} )=Co\left\{
(0,40),(0,100),(30,70)\right\} \text{,} \\
X(\left\{ 3\right\} ) &=&Co\left\{ (20,0),(100,0),(0,100),(0,10)\right\}
\text{,} \\
X(\left\{ 1,2\right\} ) &=&X(\left\{ 1,2,3\right\} )=\emptyset \text{.}
\end{eqnarray*}
Assume first that player 1's type $k$ is known, namely, that $p^{k}=1$. Let
then $x_{k}^{\ast }$ be the uninformed player's optimal decision (in $X$)
when he faces type $k$:%
\begin{equation}
x_{1}^{\ast }=(30,0)\text{, }x_{2}^{\ast }=(0,40)\text{, }x_{3}^{\ast
}=(0,10)\text{.} \label{CR ex}
\end{equation}
Suppose next that only two types are possible.\ If $p^{1}=0$, given that $%
X(\left\{ 2,3\right\} )\neq \emptyset $, there is a nonrevealing
equilibrium, $x_{2}^{\ast }=(0,40)$.\ Similarly for $p^{2}=0$, with $%
x_{1}^{\ast }=(30,0)$.\ If $p^{3}=0$, there is no way to satisfy type 1 and
type 2 at the same time.\ But there is a completely revealing equilibrium: $%
x_{1}^{\ast }=(30,0)$ to type 1, $x_{2}^{\ast }=(0,40)$ to type 2 is
incentive compatible. This illustrates Proposition 5.
Let the three types be possible, namely $p^{k}>0$ for $k=1,2,3$. There is no
nonrevealing equilibrium, since $X(\left\{ 1,2,3\right\} )=\emptyset $.\
There is no completely revealing equilibrium either: (\ref{CR ex}) implies
that type 3 would pretend to be type 2 (type 3 envies type 1 and type 2 even
more).
As expected from Theorem 8, there exists a partitional equilibrium.\ The
informed player is invited to report whether his type is 1 or not. If he
reports type 1, the uninformed player proposes $x_{1}^{\ast }=(30,0)$.\ If
the informed player reports that his type is not 1, the uninformed player
proposes%
\begin{equation*}
\arg \min_{x\in X(\left\{ 2,3\right\} )}(x_{a}+x_{b})=x_{2}^{\ast }=(0,40)%
\text{.}
\end{equation*}%
As in the proof of Theorem 8, incentive compatibility is ensured by the fact
that the decision proposed to type 3 is the one he most envies among $%
x_{1}^{\ast }$ and $x_{2}^{\ast }$.
\subsection{Mixed equilibrium}
Let us modify the uninformed player's utility function in the previous
example, to make it depend on the informed player's type:%
\begin{eqnarray*}
V^{1}(x) &=&\frac{x_{a}}{3}\text{,} \\
V^{2}(x) &=&\frac{x_{b}}{3}\text{,} \\
V^{3}(x) &=&-(x_{a}+x_{b})\text{.}
\end{eqnarray*}%
A possible interpretation is that the decision-maker is happy to pay\ when
the kidnapper has sharp preferences.
Let as above $x_{k}^{\ast }$ denote the uninformed player's optimal decision
(in $X$) when he faces type $k$; we have now%
\begin{equation}
x_{1}^{\ast }=(100,0)\text{, }x_{2}^{\ast }=(0,100)\text{, }x_{3}^{\ast
}=(0,10)\text{.} \label{CR exbis}
\end{equation}
Let us take $p=(\frac{1}{3},\frac{1}{3},\frac{1}{3})$.\ There is no\textit{\
}nonrevealing equilibrium, since $X(\left\{ 1,2,3\right\} )=\emptyset $.
There is no\textit{\ }completely revealing equilibrium: given (\ref{CR exbis}%
), type 3 would pretend to be type 2. More generally, there is no
partitional equilibrium.\ Given the above description of the sets $X(L)$,
two possible partitions must still be considered: $\left\{ \left\{ 1\right\}
,\left\{ 2,3\right\} \right\} $ and $\left\{ \left\{ 1,3\right\} ,\left\{
2\right\} \right\} $.
\bigskip
\noindent $\left\{ \left\{ 1\right\} ,\left\{ 2,3\right\} \right\} $: if the
uninformed player believes he faces type 1 (posterior $(1,0,0)$), his
optimal choice is $x_{1}^{\ast }=(100,0)$; if he believes he faces type 2 or
type 3 (posterior $(0,\frac{1}{2},\frac{1}{2})$), his optimal choice is
\begin{equation*}
x_{23}^{\ast }=\arg \min_{x\in X(\left\{ 2,3\right\} )}\left[ x_{a}+\frac{2}{%
3}x_{b}\right] =(0,40)\text{.}
\end{equation*}%
This cannot be incentive compatible for type 3:%
\begin{equation*}
100=U^{3}(x_{1}^{\ast })>U^{3}(x_{23}^{\ast })=80\text{.}
\end{equation*}
\bigskip
\noindent $\left\{ \left\{ 1,3\right\} ,\left\{ 2\right\} \right\} $: if the
uninformed player believes he faces type 2 (posterior $(0,1,0)$), his
optimal choice is $x_{2}^{\ast }=(0,100)$; if he believes he faces type 1 or
type 3 (posterior $(\frac{1}{2},0,\frac{1}{2})$), his optimal choice is
\begin{equation*}
x_{13}^{\ast }=\arg \min_{x\in X(\left\{ 1,3\right\} )}\left[ \frac{2}{3}%
x_{a}+x_{b}\right] =(100,0)\text{.}
\end{equation*}%
Again, this cannot be incentive compatible for type 3:%
\begin{equation*}
200=U^{3}(x_{2}^{\ast })>U^{3}(x_{13}^{\ast })=100\text{.}
\end{equation*}%
This illustrates that \textit{Theorem 8 does not extend to the case where
player 2's utility function depends on player 1's type.}
Let us show that if player 1 uses a \textit{mixed} strategy, a partially
revealing equilibrium exists in this example: type 1 reports that his type
belongs to $\left\{ 1,3\right\} $, type 2 reports that his type belongs to $%
\left\{ 2,3\right\} $, type 3 reports that his type belongs to $\left\{
1,3\right\} $ (resp., $\left\{ 2,3\right\} $) with probability $\frac{1}{3}$
(resp., $\frac{2}{3}$). If the informed player follows this reporting
strategy, the uninformed player's posterior upon receiving $\left\{
1,3\right\} $ is $(\frac{3}{4},0,\frac{1}{4})$ while upon receiving $\left\{
2,3\right\} $, it is $(0,\frac{3}{5},\frac{2}{5})$. Given $\left\{
1,3\right\} $, the uninformed player's problem reduces to $\min_{x\in
X(\left\{ 1,3\right\} )}x_{b}$.\ Every $x=(x_{a},0)$ with $x_{a}\in \left[
30,100\right] $ is optimal.\ Let us take $x_{13}^{\ast }=(80,0)$. Given $%
\left\{ 2,3\right\} $, the uninformed player's optimal choice is%
\begin{equation*}
x_{23}^{\ast }=\arg \min_{x\in X(\left\{ 2,3\right\} )}\left[ 2x_{a}+x_{b}%
\right] =(0,40)\text{.}
\end{equation*}%
There remains to check incentive compatibility.\ Type 1 prefers $%
x_{13}^{\ast }=(80,0)$ to $x_{23}^{\ast }=(0,40)$, and vice-versa for type
2. Type 3 must be indifferent between sending $\left\{ 1,3\right\} $ or $%
\left\{ 2,3\right\} $, because he must randomize between these two
outcomes.\ Indeed we have $U^{3}(x_{13}^{\ast })=U^{3}(x_{23}^{\ast })=80$.
Proposition 9 in Section 5 states that the previous construction can be
generalized.
\subsection{No equilibrium at all}
In the following example, none of the existence results of Section 3 can be
applied. We will show that there is no equilibrium, even if player 1 makes
use of a mixed strategy.$\ $The game is described by:$\mid K\mid =3$, $%
X=\Delta (A)$, where $A=\left\{ a,b,c\right\} $, $u_{0}^{k}=0$, $k=1,2,3$.\
The following payoff matrices describe $(U^{k}(\alpha ),V^{k}(\alpha ))$ for
every $\alpha \in A$:
\bigskip
\begin{center}
$%
\begin{array}{cccc}
& a & b & c \\
& & & \\
k=1 & 0,2 & -2,0 & 1,1 \\
& & & \\
k=2 & 1,1 & 0,2 & -2,0 \\
& & & \\
k=3 & -2,0 & 1,1 & 0,2%
\end{array}%
$
\bigskip
\end{center}
The utility functions over $X=\Delta (A)$ are obtained as expected utilities
with respect to $x=(x_{a},x_{b},x_{c})$.
If player 2 knows that he faces type $k$ (i.e., $p^{k}=1$), he gets his
first best by choosing $a$ if $k=1$, $b$ if $k=2$, $c$ if $k=3$. But if $%
p^{k}>0$ for every $k$, there is no nonrevealing equilibrium ($%
\sum_{k}U^{k}(x)<0$ for every $x\in \Delta (A)$) and no fully revealing
equilibrium (incentive compatibility is violated).
Looking for a partially revealing equilibrium, we first check that there is
a unique, nonrevealing equilibrium, as soon as only two types are possible.\
Take, e.g., $p^{3}=0$. Then%
\begin{eqnarray*}
X(\left\{ 1,2\right\} ) &=&\left\{ x\in X:-2x_{b}+x_{c}\geq 0\text{ and }%
x_{a}-2x_{c}\geq 0\right\} \\
&=&Co\left\{ (1,0,0),(\frac{2}{3},0,\frac{1}{3}),(\frac{4}{7},\frac{1}{7},%
\frac{2}{7})\right\}
\end{eqnarray*}%
and player 2's optimization problem is:%
\begin{equation*}
\max p^{1}(2x_{a}+x_{c})+p^{2}(x_{a}+2x_{b})\text{ s.t. }x\in X(\left\{
1,2\right\} )\text{.}
\end{equation*}%
For every $p$ such that $p^{1}>0$ and $p^{2}>0$, the unique solution is
achieved at $x=(1,0,0)$, namely, action $a$ with probability 1. Similarly,
action $b$ is the only solution if $p^{2}>0$ and $p^{3}>0$, action $c$ is
the only solution if $p^{1}>0$ and $p^{3}>0$.
Let us start with $p$ such that $p^{k}>0$ for every $k$.\ By sending his
message according to a mixed strategy $\sigma $, player 1 \textquotedblleft
splits\textquotedblright\ the prior belief $p$ into posteriors $p_{m}(\sigma
)$ such that $\sum\limits_{m}P_{\sigma }(m)p_{m}(\sigma )=p$ (see (\ref%
{posterior})).\ Taking account of incentive compatibility, $p$ cannot be
split (only) extreme points, because there is no fully revealing
equilibrium.\ At least one of the posteriors $p_{m}$ must be on an edge, say
$p_{m}^{3}=0$, so that $\tau (m)=a$. There should be another posterior $%
p_{m^{\prime }}$ with $p_{m^{\prime }}^{3}>0$, with $\tau (m^{\prime })=b$
or $c$. To achieve the posteriors $p_{m}$ and $p_{m^{\prime }}$, message $m$
must be sent with positive probability by types 1 and 2, while message $%
m^{\prime }$ must be sent with positive probability by at least type 3. If $%
\tau (m^{\prime })=b$, type 2 strictly prefers $m^{\prime }$ to $m$.\ If $%
x(m^{\prime })=c$, type 1 strictly prefers $m^{\prime }$ to $m$. Hence there
is no incentive compatible splitting and thus no equilibrium at all, even if
player 1 can use a mixed strategy.
A mediated equilibrium, in which information transmission is monitored by a
mediator, can nevertheless be achieved in the previous example.\ Consider
the following three lotteries over $A$: $\delta ^{1}=(\frac{1}{2},0,\frac{1}{%
2})$, $\delta ^{2}=(\frac{1}{2},\frac{1}{2},0)$, $\delta ^{3}=(0,\frac{1}{2},%
\frac{1}{2})$.\ Assume that, instead of selecting a message by himself,
player 1 can just choose among these three lotteries. If player 1 expects
player 2 to pick the action selected by the lottery, player 1 prefers $%
\delta ^{k}$ over the other two lotteries when his type is $k$.\ Similarly,
player 2 is happy to choose the action recommended by the lottery if he
believes that player 1 reveals his type truthfully to the mediator. This
procedure will be generalized in the next section.
\section{Equilibrium in the case of three types}
In this section, we propose a thorough analysis of the equilibria of $\Gamma
$ when player 1 has three possible types\textit{\ }($\mid K\mid =3$). Recall
that $\mathcal{T}^{\ast }$ denotes the participation structure of $\Gamma $
(see (\ref{X(L)}) and (\ref{Maximal elements})).
\bigskip
When $\mid K\mid =3$, there are three typical cases:
\begin{enumerate}
\item $\mathcal{T}^{\ast }$ is a partition of $K$.
\item $\mathcal{T}^{\ast }=\left\{ \left\{ 1,2\right\} ,\left\{ 1,3\right\}
,\left\{ 2,3\right\} \right\} $.
\item $\mathcal{T}^{\ast }=\left\{ \left\{ 1,3\right\} ,\left\{ 2,3\right\}
\right\} $.
\end{enumerate}
In case 1, by Proposition 6, $\Gamma $ has\textit{\ }a partitional
equilibrium. Case 2 means that player 2 is able to obtain the approval of
every pair of types but cannot ensure the participation of the three types
simultaneously.\ In this case, as illustrated in Section 4.3, $\Gamma $ may
have no mixed equilibrium.\ We will show below that $\Gamma $ always has a
\textit{mediated} equilibrium. In case 3, which has been illustrated in
Section 4.2., player 2 can only guarantee the approval of two pairs of
types.\ We will establish that under further assumptions, $\Gamma $ always
has then a mixed (possibly not pure) equilibrium.
\subsection{$\mathcal{T}^{\ast }=\left\{ \left\{ 1,2\right\} ,\left\{
1,3\right\} ,\left\{ 2,3\right\} \right\} $}
Let us enrich the description of $\Gamma $ by adding a mediator who invites
player 1 to report a type (in $K$) and then selects a decision (in $X$) that
he recommends player 2.\ At a mediated equilibrium, player 1 truthfully
reveals his type to the mediator, player 2 proposes to player 1 the decision
$x$ that is recommended by the mediator and finally, player 1 of type $k$
accepts player 2's proposal $x^{\prime }$ provided that $U^{k}(x^{\prime
})\geq u_{0}^{k}$.
Let us construct a mediated equilibrium in $\Gamma $ when $K=\left\{
1,2,3\right\} $ and $\mathcal{T}^{\ast }=\left\{ \left\{ 1,2\right\}
,\left\{ 1,3\right\} ,\left\{ 2,3\right\} \right\} $.\ Recall that $p^{k}>0$
for every $k$. For every pair $(j,k)$ of types, let $x_{jk}^{\ast }\in
X(\left\{ j,k\right\} )$ be an optimal decision for player 2 when he learns
that player 1 is of type $j$ or $k$, namely,%
\begin{equation}
x_{jk}^{\ast }\in \arg \max \left[ \frac{p^{j}}{p^{j}+p^{k}}V^{j}(x)+\frac{%
p^{k}}{p^{j}+p^{k}}V^{k}(x)\right] \text{.} \label{Const opt ex}
\end{equation}%
Consider the following mediator: for every $k=1,2,3$, if player 1 reports
type $k$, he selects $x_{ik}^{\ast }$ or $x_{jk}^{\ast }$, $i,j\neq k$, $%
i\neq j$, with equal probability $\frac{1}{2}$ and recommends it to player
2.\ If player 1 reports his type $k$ truthfully, player 2 learns, with equal
probability, that player 1's type is in $\left\{ i,k\right\} $ or in $%
\left\{ j,k\right\} $ for $i,j\neq k$, $i\neq j$.\ Condition (\ref{Const opt
ex}) guarantees that player 2 follows the mediator's recommendation.
For player 1, let us consider $k=1$.\ By reporting his type truthfully,
player 1 obtains%
\begin{equation*}
\frac{1}{2}U^{1}(x_{12}^{\ast })+\frac{1}{2}U^{1}(x_{13}^{\ast })\text{.}
\end{equation*}%
If he lies by, say, pretending to be of type $2$, he obtains\footnote{%
Player 1's incentive compatibility condition reflects the fact that this
player can veto player 2's offer, namely, can lie about his type and/or
reject player 2's proposal. This is a \textquotedblleft veto-incentive
compatibility condition\textquotedblright\ (see, e.g., Forges (1999)), which
implies posterior individual rationality.}%
\begin{equation*}
U_{2}^{1}=\frac{1}{2}\max \left\{ U^{1}(x_{12}^{\ast }),u_{0}^{1}\right\} +%
\frac{1}{2}\max \left\{ U^{1}(x_{23}^{\ast }),u_{0}^{1}\right\} \text{.}
\end{equation*}%
By construction, $x_{12}^{\ast }\in X(\left\{ 1,2\right\} )$ and $%
x_{13}^{\ast }\in X(\left\{ 1,3\right\} )$.\ Hence $U^{1}(x_{12}^{\ast
})\geq u_{0}^{1}$ and $U^{1}(x_{13}^{\ast })\geq u_{0}^{1}$. But $%
x_{23}^{\ast }\notin X(1)$ because $\left\{ 2,3\right\} $ is maximal. Hence,
$U^{1}(x_{23}^{\ast })<u_{0}^{1}$.\
\begin{equation*}
U_{2}^{1}=\frac{1}{2}U^{1}(x_{12}^{\ast })+\frac{1}{2}u_{0}^{1}\leq \frac{1}{%
2}U^{1}(x_{12}^{\ast })+\frac{1}{2}U^{1}(x_{13}^{\ast })\text{.}
\end{equation*}%
The other incentive compatibility conditions of player 1 can be checked in a
symmetric way. $\blacksquare $
\subsection{$\mathcal{T}^{\ast }=\left\{ \left\{ 1,3\right\} ,\left\{
2,3\right\} \right\} $}
\begin{proposition}\label{prop9}
Let us assume that $K=\left\{ 1,2,3\right\} $, the participation structure $%
\mathcal{T}^{\ast }=\left\{ \left\{ 1,3\right\} ,\left\{ 2,3\right\}
\right\} $, \textit{the decision set }$X$ \textit{is compact and convex} and
\textit{the utility functions }$U^{k}$ and $V^{k}$, $k\in K$, \textit{are
affine}.\ Then $\Gamma $ \textit{has a partially revealing equilibrium}, in
which player 1 uses a mixed, possibly not pure, strategy.
\end{proposition}
\noindent{\bf Proof:} See Section 6.2.\ We establish that there must exist an
equilibrium in which type 1 reports that his type belongs to $\left\{
1,3\right\} $, type 2 reports that his type belongs to $\left\{ 2,3\right\} $
and type 3 reports that his type belongs to $\left\{ 1,3\right\} $ (resp., $%
\left\{ 2,3\right\} $) with some probability $\delta \in (0,1)$ (resp., $%
1-\delta $). Incentive compatibility requires that type 3 be indifferent
between reporting $\left\{ 1,3\right\} $ or $\left\{ 2,3\right\} $. Such an
equilibrium is shown to be the only possible one in the example of Section
4.2.
\section{Appendix}
\subsection{Proof of Propositions 3 and 4}
For the sake of completeness, we first explicitly recall the conditions to
be satisfied by an equilibrium of $\Gamma (v_{0})$ whether they involve exit
of some types on equilibrium path or not.
Let us fix a pair of strategies $\sigma :K\rightarrow \Delta (M)$ and $\tau
:M\rightarrow X$. Player 1's equilibrium conditions\textbf{\ }can be written
as%
\begin{equation}
U_{+}^{k}(\tau (m))\geq U_{+}^{k}(\tau (m^{\prime }))\text{ }\forall k\in K%
\text{, }\forall m\in M\text{ s.t. }\sigma (m\mid k)>0,\forall m^{\prime
}\in M\text{.} \label{EqP1}
\end{equation}
\noindent Player 2's equilibrium conditions can be written as%
\begin{equation*}
\sum\limits_{k}p_{m}^{k}(\sigma )W^{k}(v_{0},\tau (m))\geq
\sum\limits_{k}p_{m}^{k}(\sigma )W^{k}(v_{0},x)\text{ }\forall m\in M\text{
s.t. }P_{\sigma }(m)>0\text{, }\forall x\in X\text{.}
\end{equation*}
We deduce that\textbf{\ }the necessary and sufficient conditions for $%
(\sigma ,\tau )$ to be an equilibrium without exit are:
\bigskip
For player 1:%
\begin{equation}
U^{k}(\tau (m))\geq U^{k}(\tau (m^{\prime }))\text{ }\forall k\in K\text{, }%
\forall m\in M\text{ s.t. }\sigma (m\mid k)>0,\forall m^{\prime }\in M\text{,%
} \label{EqP1noexit}
\end{equation}
implying that%
\begin{equation*}
U^{k}(\tau (m))=U^{k}(\tau (m^{\prime }))\text{ }\forall m,m^{\prime }\in M%
\text{ s.t. }\sigma (m\mid k)>0\text{ and }\sigma (m^{\prime }\mid k)>0\text{%
.}
\end{equation*}
For player 2:%
\begin{equation}
\tau (m)\in \left[ \arg \max_{x\in X}\sum\limits_{k}p_{m}^{k}(\sigma
)W^{k}(v_{0},x)\right] \cap X(\text{supp }p_{m}(\sigma ))\text{,}
\label{OptiNoexit}
\end{equation}%
implying constrained optimization, namely (\ref{Const Opt}).
\bigskip
Recall that by definition, an equilibrium of $\Gamma $ cannot involve exit,
so that the conditions for $(\sigma ,\tau )$ to be an equilibrium in $\Gamma
$ are thus just (\ref{EqP1noexit}) and (\ref{Const Opt}).
\bigskip
\noindent \textbf{Remarks on} (\ref{EqP1noexit}):
\begin{itemize}
\item As a refinement of subgame perfect equilibrium, Matthews (1989)
strengthens (\ref{EqP1}) to (\ref{EqP1noexit}) in the case of equilibria
which typically involve exit on path.
\item For an equilibrium without exit, player 1's equilibrium conditions
take the simple form (\ref{EqP1noexit}) because player 2's strategy $\tau $
is pure.\ In an equilibrium without exit, for every $k\in K$, player 1's
equilibrium strategy consists of sending $m$, selected with probability $%
\sigma (m\mid k)$, and then accept player 2's proposal, namely, $\tau (m)$,
which is fully anticipated at the time to choose $m$.\ A deviation consists
of sending $m^{\prime }\in M$ (possibly such that $\sigma (m^{\prime }\mid
k)=0$) and then, accept or reject player 2's proposal $\tau (m^{\prime })$.\
Player 1's equilibrium conditions thus take the form%
\begin{equation*}
U^{k}(\tau (m))\geq \max \left\{ U^{k}(\tau (m^{\prime })),u_{0}^{k}\right\}
\text{ }\forall k\in K\text{, }\forall m\in M\text{ s.t. }\sigma (m\mid
k)>0,\forall m^{\prime }\in M\text{.}
\end{equation*}%
These are equivalent to (\ref{EqP1noexit}), since (\ref{Noexit}) holds at an
equilibrium without exit. If player 2's strategy $\tau $ were mixed ($\tau
:M\rightarrow \Delta (X)$), we would have to write\footnote{%
The expression is similar to \textquotedblleft veto-incentive
compatibility\textquotedblright\ (see, e.g., Forges (1999)) and implies
posterior individual rationality.}%
\begin{equation*}
\sum\limits_{x}\tau (x\mid m)U^{k}(x)\geq \sum\limits_{x}\tau (x\mid
m^{\prime })\max \left\{ U^{k}(x)),u_{0}^{k}\right\} \text{.}
\end{equation*}
\end{itemize}
\bigskip
\noindent \textbf{Proposition 3.} \textit{Let }$(\sigma ,\tau )$\textit{\ be
an equilibrium without exit in }$\Gamma (v_{0})$\textit{, for some }$%
v_{0}\in \mathbb{R}$\textit{.\ Then }$(\sigma ,\tau )$\textit{\ is an
equilibrium without exit in }$\Gamma (z_{0})$\textit{\ for every }$z_{0}\in
R $\textit{\ such that }$z_{0}\leq v_{0}$\textit{\ and is also an
equilibrium in }$\Gamma $\textit{, with the same interim expected utility as
in }$\Gamma (v_{0})$\textit{\ for both players.}
\bigskip
\noindent \textbf{Proof:} If $(\sigma ,\tau )$ satisfies (\ref{OptiNoexit})
in $\Gamma (v_{0})$, the same holds in $\Gamma (z_{0})$ since for every $%
x\in X$,%
\begin{equation*}
\sum\limits_{k}p_{m}^{k}(\sigma )W^{k}(z_{0},x)\leq
\sum\limits_{k}p_{m}^{k}(\sigma )W^{k}(v_{0},x)
\end{equation*}%
and for every $x\in X($supp $p_{m}(\sigma ))$,%
\begin{equation*}
\sum\limits_{k}p_{m}^{k}(\sigma
)W^{k}(z_{0},x)=\sum\limits_{k}p_{m}^{k}(\sigma
)W^{k}(v_{0},x)=\sum\limits_{k}p_{m}^{k}(\sigma )V^{k}(x)\text{.}
\end{equation*}
Furthermore, (\ref{Const Opt}) must hold and player 1's equilibrium
conditions (\ref{EqP1noexit}) are the same in $\Gamma (v_{0})$ and $\Gamma
(z_{0})$ or $\Gamma $ as long as player 2's strategy remains unchanged. $%
\blacksquare $
\bigskip
\noindent \textbf{Proposition 4.} \textit{Let }$(\sigma ,\tau )$\textit{\ be
an equilibrium in }$\Gamma $\textit{. Then there exists }$v_{0}\in \mathbb{R}
$\textit{\ such that, for every }$z_{0}\leq v_{0}$\textit{, }$(\sigma ,\tau
) $\textit{\ is an equilibrium without exit of }$\Gamma (z_{0})$\textit{,
with the same interim expected utility as in }$\Gamma $\textit{\ for both
players.}
\bigskip
\noindent \textbf{Proof:} Let $(\sigma ,\tau )$ be an equilibrium without
exit in $\Gamma $.\ By definition, constrained optimization (\ref{Const Opt}%
) holds, so that in particular $\tau (m)\in X($supp $p_{m}(\sigma ))$ for
every $m$ such that $P_{\sigma }(m)>0$. Let us keep player 1's strategy, $%
\sigma $, fixed. Player 2's strategy $\tau $ remains a best reply to $\sigma
$ in $\Gamma (v_{0})$, with $v_{0}\leq \min_{k\in K}\min_{x\in X}V^{k}(x)$,
provided that $v_{0}$ is such that optimality of no exit holds, namely,
recalling (\ref{OptiNoexit}),%
\begin{equation*}
\forall m\in M\text{ s.t. }P_{\sigma }(m)>0,\sum\limits_{k}p_{m}^{k}(\sigma
)V^{k}(\tau (m))\geq \max_{x\in X}\sum\limits_{k}p_{m}^{k}(\sigma
)W^{k}(v_{0},x)\text{.}
\end{equation*}%
These can be viewed as finitely many inequalities over $v_{0}$,which have a
solution in $\mathbb{R}$, since the RHS are well-defined, for every $%
v_{0}\leq \min_{k\in K}\min_{x\in X}V^{k}(x)$, by Lemma 1.\footnote{%
The RHS of the inequalities can be rewritten as%
\begin{equation*}
\max_{L\varsubsetneq \text{supp}(p_{m}(\sigma ))}\max_{x\in X(L)}\left\{
\sum\limits_{k\in L}p_{m}^{k}(\sigma )V^{k}(x)+v_{0}\sum\limits_{k\in \text{%
supp}(p_{m}(\sigma ))\diagdown L}p_{m}^{k}(\sigma )\right\} \text{.}
\end{equation*}%
} Hence there exists $v_{0}$ such that $(\sigma ,\tau )$ is an equilibrium
without exit of $\Gamma (v_{0})$ and from Proposition 3, in $\Gamma (z_{0})$%
, for every $z_{0}\leq v_{0}$. $\blacksquare $
\bigskip
\noindent \textbf{Application to a mechanism design problem}
In Section 3, we have established that, under various reasonable
assumptions, $\Gamma $ has a partitional equilibrium $(\sigma ,\tau )$, in
which both $\sigma $ and $\tau $ are pure.\ In this case, we can easily
compute the highest ex ante expected utility that player 2, interpreted here
as the principal, can obtain at a partitional equilibrium of $\Gamma $.%
\footnote{%
The number of pure equilibrium payoffs is finite, in the same way as the
number of partitions of $K$.\ Hence as soon as there is a pure equilibrium
in $\Gamma $, there is an equilibrium achieving the highest expected payoff
for the receiver.} As explained in Section 2, this application is inspired
by a particular case of Bester and Strausz (2001)'s model.
Let us show that there exists $v_{0}\in \mathbb{R}$ such that, for every $%
z_{0}\leq v_{0}$, the highest ex ante expected utility player 2 can obtain
at an arbitrary partitional equilibrium of $\Gamma (z_{0})$ (which can
involve exit or not) is the same as in $\Gamma $.\ More precisely, there
exists $v_{0}\in \mathbb{R}$ such that, for every $z_{0}\leq v_{0}$, the
best partitional equilibrium for player 2 in $\Gamma $ remains the best
partitional equilibrium for player 2 in $\Gamma (z_{0})$.
Let $v_{NE}^{\ast }$ be the highest ex ante expected utility player 2 can
obtain at a partitional equilibrium of $\Gamma $.\ This number is
well-defined if $\Gamma $ has a partitional equilibrium.\ Let $(\sigma
^{\ast },\tau ^{\ast })$ achieve the expected utility $v_{NE}^{\ast }$ for
player 2. Using Proposition 4, there exists $v_{0}$ sufficiently small such
that for every $z_{0}\leq v_{0}$, $(\sigma ^{\ast },\tau ^{\ast })$ is an
equilibrium without exit of $\Gamma (z_{0})$ with the same expected utility $%
v_{NE}^{\ast }$ for player 2. By Proposition 3, for every such $z_{0}$,
there does not exist any equilibrium \textit{without exit} giving a higher
expected utility to player 2 (because such an equilibrium would still be an
equilibrium of $\Gamma $, with the same expected utilities).
Let us consider the partitional equilibria $(\sigma ,\tau )$ of $\Gamma
(v_{0})$ in which exit possibly occurs, i.e., in which the set%
\begin{equation*}
K_{E}=\left\{ k\in K:U^{k}(\tau \circ \sigma (k))<u_{0}^{k}\right\} \neq
\emptyset \text{,}
\end{equation*}%
i.e., $p_{E}=_{def}\sum_{k\in K_{E}}p^{k}>0$. The highest expected utility
player 2 can achieve at such an equilibrium is%
\begin{equation*}
p_{E}v_{0}+(1-p_{E})\overline{v}
\end{equation*}%
where%
\begin{equation*}
\overline{v}=\max_{k\in K}\max_{x\in X}V^{k}(x)\text{.}
\end{equation*}%
If $v_{0}$ is such that, for every $p_{E}$ that can arise given the prior $p$%
,%
\begin{equation*}
p_{E}v_{0}+(1-p_{E})\overline{v}\leq v_{NE}^{\ast }\text{,}
\end{equation*}%
\begin{equation}
\text{namely, }v_{0}\leq \frac{1}{p_{E}}\left[ v_{NE}^{\ast }-(1-p_{E})%
\overline{v}\right] \label{highest ut}
\end{equation}%
then $(\sigma ^{\ast },\tau ^{\ast })$ will guarantee the highest possible
equilibrium utility to player 2, in every game $\Gamma (z_{0})$ with $%
z_{0}\leq v_{0}$.
Let $\underline{k}$ be the type with the smallest prior probability, namely,
such that $p^{\underline{k}}=\min \left\{ p^{1},\cdots ,p^{K}\right\} $. The
inequality (\ref{highest ut}) will hold at every $p_{E}$ that can arise
given the prior $p$ as soon as it holds at $p_{E}=p^{\underline{k}}$: we
just have to require%
\begin{equation*}
v_{0}\leq \frac{1}{p^{\underline{k}}}\left[ v_{NE}^{\ast }-(1-p^{\underline{k%
}})\overline{v}\right] \text{.}
\end{equation*}
The previous result is quite intuitive: an upper bound on the receiver's
expected utility at an equilibrium of $\Gamma (v_{0})$ with exit is obtained
when the receiver's proposal is rejected by only the least likely type,
while the best possible utility is achieved at all the other types. If $%
v_{0} $ is sufficiently low, the best equilibrium utility for the receiver
in $\Gamma (v_{0})$ will be not be achieved at an equilibrium with exit, but
rather at an equilibrium without exit, which is in turn is necessarily an
equilibrium of $\Gamma $.
\subsection{Proof of Proposition 9}
\noindent \textbf{Proposition 9.} \textit{Let us assume that} $K=\{1,2,3\},$ ${\cal T}^*=\{\{1,3\},\{2,3\}\}$ \textit{the decision set} $X$ \textit{is compact and convex and the utility functions} $U^k$ \textit{and} $V^k, k \in K,$ \textit{are affine. Then} $\Gamma$ \textit{has a partially revealing} {\it equilibrium}.
\smallskip
For simplicity we assume here that $u_0^k=0$ for each $k$. This is w.l.o.g. since we can translate the payoffs of each type of the sender. We start with preliminaries.
\subsubsection{Mappings and multi-valued mappings}
\noindent Define for each $p$ in $\Delta(K)$:
$f(p)=\sup\{\sum_{k \in K} p^k V^k(x), x \in X( {\rm supp}\; p)\}\in I\!\!R\cup\{-\infty\}$,
$Y(p)= {\rm Argmax} _{x \in X( {\rm supp}\; p)} \sum_k p^kV^k(x)\subset X( {\rm supp}\; p)$, and
$\Phi(p)=\{(U^k(x))_{k \in K}, x \in Y(p)\} \; \subset I\!\!R^K.$\\
The sets $Y(p)$ and $\Phi(p)$ are convex compact subsets of $I\!\!R$ and $I\!\!R^K$, respectively.
If $X( {\rm supp}\; p)\neq \emptyset$, then $f(p)\in I\!\!R$, $Y(p)\neq \emptyset$ and $\Phi(p)\neq \emptyset$. For each $u\in \Phi(p)$, we have $u^k\geq 0$ for each $k\in {\rm supp}\; p$.
At an equilibrium of $\Gamma$, if the belief of the receiver (after having received the message of the sender) is $p$, then he has to propose a decision in $Y(p)$, inducing a vector payoff in $\Phi(p)$ for the different types of player 1.
We will use in the sequel the following three lemmas (Lemma \ref{lem4} is a simple mean-value theorem for correspondences).
\begin{lem} \label{lem2} The mapping $f$ is u.s.c. and convex.
If $p_n\xrightarrow[n \to \infty]{} p \in \Delta(K)$ with $ {\rm supp}\; p_n= {\rm supp}\; p$ for each $n$, then $f(p_n)\xrightarrow[n \to \infty]{} f(p)$.
\end{lem}
\noindent{\bf Proof:} Suppose $p_n\xrightarrow[n \to \infty]{} p$. Then for $n$ large enough, $ {\rm supp}\; p_n\supset {\rm supp}\; p$ so $X( {\rm supp}\; p_n)\subset X( {\rm supp}\; p)$. It follows that $\limsup_n f(p_n)\leq f(p)$. (whether $f(p)=-\infty$ or not)
If $ {\rm supp}\; p_n= {\rm supp}\; p$ for each $n$, then $|f(p_n)-f(p)|\leq \sup_{x \in X} \sum_{k \in K} |p_n^k-p^k| |V^k(x)|$, and $f(p_n)\xrightarrow[n \to \infty]{} f(p)$.
If $p=\lambda p_1+(1-\lambda) p_2$ with $\lambda\in (0,1)$, then $ {\rm supp}\; p= {\rm supp}\; p_1 \cup \; {\rm supp}\; p_2$ and $X( {\rm supp}\; p)= X( {\rm supp}\; p_1)\cap X( {\rm supp}\; p_2)$. If $f(p)=-\infty$ then $\lambda f(p_1)+(1-\lambda) f(p_2)\geq f(p)$. Consider $x$ in $X( {\rm supp}\; p)$, we have $f(p_1)\geq \sum_{k \in K} p_1^k V^k(x)$ and $f(p_2)\geq \sum_{k \in K} p_2^k V^k(x)$, so $\lambda f(p_1)+(1-\lambda) f(p_2)\geq \sum_{k \in K} p^k V^k(x)$, and taking the supremum for $x$ in $X( {\rm supp}\; p)$ we get $\lambda f(p_1)+(1-\lambda) f(p_2)\geq f(p)$. Hence $f$ is convex. $\blacksquare $
\begin{lem} \label{lem3} Consider a converging sequence $p_n\xrightarrow[n \to \infty]{} p \in \Delta(K)$.
a) Assume $\limsup_n f(p_n)=f(p)$. Then if $u_n\xrightarrow[n \to \infty]{} u \in I\!\!R^K$, with $u_n\in \Phi(p_n)$ for each $n$, we have
$u\in \Phi(p)$,
b) Otherwise $\limsup_n f(p_n)<f(p)$. Then there exists $n_0$ such that for each $u\in \Phi(p)$ and $n \geq n_0$, one can find $k\in {\rm supp}\; p_n\backslash \{ {\rm supp}\; p\}$ such that $u^k<0$.
\end{lem}
\noindent{\bf Proof:} a) Without loss of generality we assume that $f(p_n)\xrightarrow[n \to \infty]{} f(p)$. Write $u_n=(U^k(x_n))_{k \in K}$ with $x_n$ in $Y(p_n)$ for each $n$. By taking a converging subsequence we can assume that $x_n$ converges to some $x$ in $X$. Since $p_n\xrightarrow[n \to \infty]{} p $, for $n$ large enough $ {\rm supp}\; p_n\supset {\rm supp}\; p$ so $x\in X( {\rm supp}\; p)$. And $\sum_{k \in K}p_n^k V^k(x_n)=f(p_n)\xrightarrow[n \to \infty]{} f(p)$, so $\sum_{k \in K}p^k V^k(x)=f(p)$. Then $x$ belongs to $Y(p)$, and $u \in \Phi(p)$.
b) Assume that $\limsup_n f(p_n)<f(p)$. We first claim that for $n$ large enough, $Y(p)\cap X( {\rm supp}\; (p_n))=\emptyset$. Otherwise, we can find $x$ in $Y(p)\cap X( {\rm supp}\; p_n))$ for infinitely many $n$'s, we have $f(p)=\sum_k p^k V^k(x)$ and $f(p_n)\geq \sum_k p_n^k V^k(x)$ for infinitely many $n$'s, so $\limsup_n f(p_n)\geq f(p)$ which is a contradiction. We have shown that there exists $n_0$ such that for $n\geq n_0$, $Y(p)\cap X( {\rm supp}\; (p_n))=\emptyset$. If $x\in Y(p)$ and $n\geq n_0$, then $x\notin X( {\rm supp}\; p_n)$. So if $u\in \Phi(p)$ and $n\geq n_0$, there exists $k\in {\rm supp}\; p_n\backslash \{ {\rm supp}\; p\}$ such that $u^k<0$.$\blacksquare $
\begin{lem} \label{lem4} Let $F:[0,1] \rightrightarrows I\!\!R$ be a correspondence with non empty convex values and compact graph. If $F(0)\subset \{x \in I\!\!R, x< 0\}$ and $F(1)\subset \{x \in I\!\!R, x> 0\}$, there exists $t$ in $(0,1)$ such that $0\in F(t)$.\end{lem}
\noindent{\bf Proof:} The sets $C_+=\{t\in [0,1], F(t)\cap I\!\!R_+\neq \emptyset\}$ and $C_-=\{t\in [0,1], F(t)\cap I\!\!R_-\neq \emptyset\}$ are closed because $F$ is u.s.c. Since $F$ has non empty values, $C_+$ and $C_-$ are non empty, and $C_+ \cup C_-=[0,1]$. By connexity of $[0,1]$, one can find $t$ in both sets, that is such that $F(t)$ intersects both $I\!\!R_+$ and $I\!\!R_-$. Since $F(t)$ is convex, it contains 0. $\blacksquare $
\subsubsection{\bf Existence of an equilibrium}
For $k=2,3$, define $\delta_k$ as the Dirac measure on the state $k$, and $p_{-k}$ as the conditional probability on $K$ knowing the state is not $k$:
$$\delta_2=(0,1,0),\;\; \delta_3=(0,0,1), \;\;p_{-2}=(\frac{p_1}{p_1+p_3},0, \frac{p_3}{p_1+p_3})\;\; {\rm and} \;\; p_{-3}=(\frac{p_1}{p_1+p_2}, \frac{p_2}{p_1+p_2},0).$$
Choose $u_2$ in $\Phi(\delta_2)$, $u_3$ in $\Phi(\delta_3)$, $u_{1,2}$ in $\Phi(p_{-3})$ and $u_{1,3}$ in $\Phi(p_{-2})$. These are vectors in $I\!\!R^3$, and to simply notations we write:
$$u_2=\left(
\begin{array}{c}
a\\
+ \\
- \\
\end{array}
\right), u_3=\left(
\begin{array}{c}
b\\
- \\
+ \\
\end{array}
\right), u_{1,2}=\left(
\begin{array}{c}
c\geq 0\\
+\\
- \\
\end{array}
\right), u_{1,3}=\left(
\begin{array}{c}
d\geq 0\\
-\\
+\\
\end{array}
\right).$$
\noindent with $a=u_2^1$, $b=u_3^1$, $c=u_{1,2}^1$ and $d=u_{1,3}^1$.
Here $+$ means $\geq 0$, and $-$ means $<0$. We have $u_2^2\geq 0$, $u_3^3\geq 0$, $c\geq 0$, $u_{1,2}^2\geq 0$, $d\geq 0$ and $u_{1,3}^3\geq 0$ since for each $p$ and $u\in \Phi(p)$, we have $u^k\geq 0$ for each $k\in {\rm supp}\; p$. The subset $\{2,3\}$ is not in ${\cal T}$, this gives $u_2^3<0$, $u_3^2<0$, $u_{1,2}^3< 0$ and $u_{1,3}^2< 0$.
Suppose $a\leq d$. Then a simple equilibrium exists. Player 1 uses the partition $\{\{2\}, \{1,3\}\}$ to communicate: he sends the message $m=2$ if the state is 2, and the message $m=\{1,3\}$ if the state is 1 or 3. Player 2 proposes $x_2$ in $Y(\delta_2)$ such that $u_2=(U^k(x_2))_{k \in K}$ after receiving $m=2$, and proposes $x_{1,3}$ in $Y(p_{-2})$ such that $u_{1,3}=(U^k(x_{1,3}))_{k \in K}$ after receiving $m=\{1,3\}$. By definition of $Y(\delta_2)$ and $Y(p_{-2})$, player 2 is in best reply. And no type of player 1 has an incentive to deviate, so we have an equilibrium where player 1 plays pure. If we suppose $b\leq c$, we have a similar equilibrium where player 1 uses the partition $\{\{3\}, \{1,2\}\}$.\\
From now on, we assume that $a>d\geq 0$ and $b>c\geq 0$. Then $a\geq 0$. Consider any sequence $(p_n)$ converging to the Dirac measure on state 2 such that $ {\rm supp}\; p_n =\{1,2\}$ for each $n$. By Lemma \ref{lem3} part b), we must have $\limsup_n f(p_n)\geq f(p)$, and since $f$ is u.s.c., $\limsup_n f(p_n)= f(p)$. This being true for any such sequence, $f(p_n)\xrightarrow[n \to \infty]{} f(p)$. That is, the restriction of $f$ to the set $\{p, {\rm supp}\; p\subset\{1,2\}\}$ is continuous at $\delta_2$.
And by Lemma \ref{lem3} part a), the restriction of $\Phi$ to the segment $[p_{-3},\delta_2]$ has a closed graph. Similarly we have $b\geq 0$, and we can prove that the restriction of $f$ to the set $\{p, {\rm supp}\; p\subset\{1,3\}\}$ is continuous at $\delta_3$, and
the restriction of $\Phi$ to the segment $[p_{-2},\delta_3]$ has a closed graph. \\
The initial probability $p$ is on the segment $[\delta_2,p_{-2}]$, and also on the segment $[\delta_3,p_{-3}]$. For $t\in [0,1]$, define $q_t=t \delta_2+(1-t)p_{-3}$ and $q'_t$ in $[p_{-2}, \delta_3]$ such that $p$ belongs to the segment $[q_t,q'_t]$. $q'_t$ is uniquely defined for each $t$, $q'_0=\delta_3$ and $q'_1=p_{-2}$. We are going to construct an equilibrium with posteriors $q_t$ and $q'_t$ for some appropriate $t$. We need player 1 of type 1 to be indifferent between splitting to $q_t$ and $q'_t$.
\begin{center}
\setlength{\unitlength}{1mm}
\begin{picture}(100,60)
\put(20,0){\line(1,0){60}}
\thicklines
{ \qbezier(20,0)(35,26)(50,52)}
{\bf { \qbezier(80,0)(65,26)(50,52)}}
\thinlines
{ \qbezier(35,26)(35,26)(80,0)}
{ \qbezier(65,26)(65,26)(20,0)}
{ \qbezier(25,9)(50,17)(67,23)}
\put(15,0){$\delta_2$}
\put(82,0){$\delta_3$}
\put(50,54){$\delta_1$}
\put(28,26){$p_{-3}$}
\put(66,26){$p_{-2}$}
\put(20,9){$q_t$}
\put(68,22){$q'_t$}
\put(50,14){$p$}
\end{picture}
\end{center}
\vspace{0,5cm}
Define the correspondence $F:[0,1] \rightrightarrows I\!\!R$, with for each $t$ in $[0,1]$:
$$F(t)=\{ u^1-v^1, u\in \Phi( q_t), v\in \Phi(q'_t)\}.$$
$F$ clearly has non empty convex compact values. We have seen that the restrictions of $\Phi$ to the segments $[p_{-3},\delta_2]$ and $[p_{-2},\delta_3]$ have closed graphs, moreover $q_t$ and $q'_t$ are continuous in $t$, hence $F$ has a closed graph.
$F(0)=\{ u^1-v^1, u\in \Phi( p_{-3}), v\in \Phi(\delta_3)\}$. If $F(0)\cap I\!\!R_+\neq \emptyset$, there exists a pure equilibrium where player 1 uses the partition $\{\{3\}, \{1,2\}\}$, so we assume that $F(0)$ is a subset of $\{x \in I\!\!R, x<0\}$.
Similarly, we assume that $F(1)=\{ u^1-v^1, u\in \Phi(\delta_2 ), v\in \Phi(p_{-2})\}$ is a subset of $\{x \in I\!\!R, x>0\}$ (otherwise there exists an equilibrium where player 1 uses the partition $\{\{2\}, \{1,3\}\}$). Then by Lemma \ref{lem4} we can find $t^*$ in $[0,1]$ such that $0\in F(t^*)$.
We can now conclude the proof. We can find $x$ in $Y(q_{t^*})$, $y$ in $Y(q'_{t^*})$, $u=(U^k(x))_k\in \Phi(q_{t^*})$ and $u'=(U^k(y))_{k}\in \Phi(q'_{t^*})$ such that for some $e\geq 0$:
$$u=\left(
\begin{array}{c}
e\\
+ \\
- \\
\end{array}
\right), u'=\left(
\begin{array}{c}
e\\
- \\
+\\
\end{array}
\right)$$
We have an equilibrium as follows. Player 1 sends a message so as to induce the posteriors $q_{t^*}$ and $q'_{t^*}$ (type 2 sends the message 2, type 3 sends the message 3, and type 1 randomizes between the messages 2 and 3 so that the posteriors are $q_{t^*}$ after $m=2$ and $q'_{t^*}$ after $m=3$). Player 2 then proposes $x$ at $q_{t^*}$, and $y$ at $q'_{t^*}$. Player 2 is in best reply by construction. Type 1 of player 1 is indifferent. If type 2 of player 1 deviates and sends $q'_{t^*}$, player 2 will propose $y$ and type 2 will reject it, having the reserve payoff of 0, which is not better than the payoff without deviating. Similarly, player 1 of type 3 has no profitable deviation, and we have an equilibrium. $\blacksquare $
\newpage
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 2,478 |
Cellaria anceps är en mossdjursart som beskrevs av Harmer 1926. Cellaria anceps ingår i släktet Cellaria och familjen Cellariidae. Inga underarter finns listade i Catalogue of Life.
Källor
Mossdjur
anceps | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 4,516 |
Motor Symptoms
Non-Motor Symptoms
Living With Parkinson's
Experimental Treatments
Non-drug Treatments
Safe Exercise
Cannabis and Parkinson's Symptoms
Speech Difficulties
Caring for Mental Health
Managing Sex and Intimacy Issues
The Bright Side – Jamie Askari
Embrace the Shake — Samantha Felder
Life, Lemons, and Lemonade — Lori DePorter
Possibilities With Parkinson's — Dr. C
Shaking Things Up — Mary Beth Skylis
Journeying Through Parkinson's Disease — Sherri Woodbridge
Resilient: Living Relentlessly — Jo Gambosi
Slow Is the New Fast — Jean Mellano
Sidekicks Program to Connect Parkinson's Patients and Youth
by Ana de Barros, PhD | April 9, 2018
Two Parkinson's organizations are joining forces with the Davis Phinney Foundation and Lundbeck to start a program that connects people with Parkinson's disease and school-aged youth through fun and creative story-sharing.
A key goal of Sidekicks is to let Parkinson's patients and youths share stories about each other in ways that promote mutual understanding and foster connection.
Other Sidekicks partners are the American Parkinson's Disease Association, Parkinson's Foundation and the Michael J. Fox Foundation for Parkinson's Research.
The program includes four interactive sessions designed to facilitate creative expression and help participants gain insights into one another's experiences. During the sessions, the participants will work together to create projects like handprint art, imaginative ideascapes, and rock art.
Youth participants will be able to deepen their understanding of Parkinson's, learn how to have a positive outlook, and meet people who have different experiences from their own.
"We're thrilled to launch Sidekicks with our partners as a new way to help people with Parkinson's live well today," Polly Dawkins, executive director of the Davis Phinney Foundation, said in a press release. "This program harnesses the power of intergenerational relationships while capturing the life stories of people with Parkinson's in a creative and unexpected way."
The Parkinson's and youth participants will be surveyed before and after the Sidekicks program to see whether it produces any benefits.
The surveys will look at whether Sidekicks can reduce Parkinson's patients' feelings of isolation and improve their self-esteem. They will also evaluate whether Sidekicks helps young people learn more about Parkinson's, whether it helps them develop a more positive attitudes towards people with the disease, and where it gives them a more positive outlook on dealing with adversity and life in general.
"We strive to inspire progress in Parkinson's through partnerships and programs that make a difference in the community," said Lorena Di Carlo, Lundbeck's vice president and general manager of neurology. "There is a common feeling of isolation among people with Parkinson's, and intergenerational programs like Sidekicks can help inspire positivity in living with Parkinson's while creating greater understanding for youth of what it's like to live with this condition."
The program will start in Chicago, Denver and Tampa in April, in conjunction with Parkinson's Awareness Month. It will roll out in seven additional U.S. cities in the fall.
You can join the Parkinson's Sidekicks Facebook community here.
Ana de Barros, PhD Ana holds a PhD in immunology from the University of Lisbon and worked as a postdoctoral researcher at Instituto de Medicina Molecular (iMM) in Lisbon, Portugal. Ana was awarded two FCT fellowships and has won the Portuguese Immunology Society Best Paper and Best Poster award in 2009 and 2010, as well as the CESPU International Research Award in 2010. After leaving the lab to pursue a career in science communication, she served as the director of science communication at iMM Lisbon.
American Parkinson's Disease Association, Davis Phinney Foundation, intergenerational program, Lundbeck, MJFF, Parkinson's Awareness Month, Parkinson's Foundation, Parkinson's organizations, Sidekicks
Setting the Stage for Insight Meditation With Parkinson's Disease
New Video-based Algorithm May Help Predict Tremor in Parkinson's
NOS2 Exhibits Therapeutic Potential in Mouse Study
FDA Lifts Hold on IkT-148009; Trial to Resume Soon
Interim Analysis of Buntanetap Phase 3 Trial Expected This Year
Your Parkinson's Community
Visit the Parkinson's News Today forums to connect with others in the Parkinson's community.
January 27, 2023 News by Lindsey Shapiro, PhD
January 27, 2023 News by Marisa Wexler, MS
Get regular updates to your inbox.
BioNews, Inc.
3 W Garden St
Website: bionews.com
Disable Notifications
This site is strictly a news and information website about the disease. It does not provide medical advice, diagnosis or treatment. This content is not intended to be a substitute for professional medical advice, diagnosis, or treatment. Always seek the advice of your physician or other qualified health provider with any questions you may have regarding a medical condition. Never disregard professional medical advice or delay in seeking it because of something you have read on this website. | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 6,559 |
\section{Introduction}
Tidal tails, resulting from the interactions of galaxies, can create
self-gravitating structures which are the seeds of dwarf galaxies. In
these galaxies, commonly known as {\it Tidal Dwarf Galaxies} (TDGs),
the Dark Matter (DM) content is very small, which make them more
vulnerable to stellar feedback. 1-D DM-free models of dwarf galaxies
have already demonstrated that they can remain bound even with a large
fraction of supernova-expelled gas (Hensler et al. \cite{htg04}).
\section{The model}
We perform 2-D chemodynamical simulations of DM-poor dwarf galaxies in
order to study the impact of feedback from the ongoing star formation
(SF) to such structures characterized by a low binding energy (Kroupa
\cite{k98}). SF depends on the density and temperature of the
gas. Self-gravity and a treatment of the chemical evolution of the
stellar and gaseous component are considered. We present here models
with an initial total gas mass of 4 $\times$ 10$^8$ M$_\odot$ and we
have modified further parameters as SF efficiency $\epsilon_{\rm SF}$,
temperature threshold T$_{\rm thr}$ for SF, initial mass distribution
(spherical vs. aspherical), and gas infall (no, subsonic, supersonic).
\section{Results}
In a model with $\epsilon_{\rm SF}=0.2$, T$_{\rm thr}$ =10$^4$ K, type
II supernovae produce a large-scale galactic wind able to unbind most
of the gas after $\sim$ 100 Myr and SF can not proceed further. If we
reduce T$_{\rm thr}$ to 10$^3$ K, a smaller amount of gas fulfills the
conditions for the SF, therefore the galaxy does not experience a
burst but a continuous episode of SF of mild intensity lasting several
hundreds of Myr. Also a reduction of $\epsilon_{\rm SF}$ to 0.1 leads
to a SF rate milder compared to the standard model and no disruptive
bursts occur within the first 100 Myr. An aspherical initial gas
distribution favors the development of a large-scale outflow in the
polar direction. However, the transport of material along the R-axis
is not significant and stars still can form inside the cavity walls
(see Fig. 1). Models with gas infall reveal that in case of a
supersonic infall a shock wave propagates towards the galaxy center
and triggers a very powerful disruptive central starburst. A subsonic
infall provides instead a continuous reservoir of gas for the SF and
does not trigger starbursts. Also in this case, the SF lasts several
hundreds of Myr.
\begin{figure}
\vspace{-1.2cm}\includegraphics[height=0.7\columnwidth]
{recchi_fig1}
\label{gs}
\vspace{-3.6cm}
\caption{ Density
contours of the gaseous (upper panels) and stellar (lower panels)
phase at 4 evolutionary times for the aspherical model.}
\end{figure}
\section{Conclusions}
From our models we found that TDGs are not necessarily destroyed
rapidly and can sustain SF for times longer than 200 Myr and that
galactic winds do not always suppress SF. Further work is necessary to
sharpen our understanding of these issues.
\begin{acknowledgements}
This project is supported by the German Science Foundation (DFG) as part
of the priority programme 1177.
\end{acknowledgements}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 3,353 |
http://www.judgemikemorgan.com/wp-login.php?action=logout&redirect_to=http%3A%2F%2Fwww.judgemikemorgan.com%2F&_wpnonce=e89bacfc37
US President Barack Obama Endorses
Judge Mike Morgan for N.C. Supreme Court
Full Scale Video Link | Official Press Release | Youtube Link
Najuma Thorpe, Ember PR + Communications
(301) 806-4500 | najuma@emberpr.com
Barack Obama Endorses Judge Mike Morgan for N.C. Supreme Court
(Washington, DC) – Barack Obama endorses Judge Mike Morgan for N.C. Supreme Court. Judge Morgan has been a judge for more than 27 years spanning three different judgeships, including five years as a North Carolina Administrative Law Judge, and a decade as a District Court Judge for Wake County. He has served as a North Carolina Superior Court Judge since 2005.
"Judge Morgan is a fair, experienced judge who is more than qualified to serve on the N.C. Supreme Court," said Obama. "The Supreme Court race in North Carolina is pivotal. Key issues including gerrymandering, civil rights, coal ash, school vouchers and many others have come before this court. We need judges with a track record of reviewing cases and acting in a consistent, impartial manner. Judge Mike Morgan will serve North Carolina well on the N.C. Supreme Court."
Judge Morgan has a unique blend of veteran judicial experience in the courtroom, lengthy judicial teaching in the classroom, proven leadership in the legal and wider community and steadfast commitment to society — experiences that eminently qualify him to be an Associate Justice of the North Carolina Supreme Court.
Public service is at Judge Morgan's core. He has been a member of a host of professional organizations and has worked with many community-based organizations. He has worked to assist with the successful reentry of formerly-incarcerated individuals, to educate the community about domestic violence issues and to foster stronger relationships between the law enforcement community and the communities they serve. Judge Morgan is also a member of the N.C. Equal Access to Justice Commission, the Chief Justice's Media and the Courts Commission, and is the state's representative on the Racial and Ethnic Bias Consortium. In addition, he is a veteran faculty member of the National Judicial College, where he has taught and trained judges from all over the United States for the past 24 years.
Judge Morgan is a graduate of Duke University, where he obtained a Bachelor of Arts Degree in History and Sociology. He furthered his education at North Carolina Central University School of Law, distinguishing himself by serving as student body president in his final year of law school and earning his Juris Doctorate with Honors.
Judge Mike Morgan Looks To Shift Political Balance Of NC Supreme Court
By LAURA PELLICER & FRANK STASIO • OCT 20, 2016
Host Frank Stasio speaks with Wake County Superior Court Judge Mike Morgan.
http://www.judgemikemorgan.com/wp-content/uploads/2016/10/sot102016segB.mp3
© 2016 Paid for by The Judge Mike Morgan Committee Bruce E. Anderson, Treasurer. Web Design and Video by timothybowser.com .
Judge Michael Morgan | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 9,447 |
// Copyright (c) 2007-2018 ppy Pty Ltd <contact@ppy.sh>.
// Licensed under the MIT Licence - https://raw.githubusercontent.com/ppy/osu/master/LICENCE
using System;
using System.Collections.Generic;
using System.Linq;
using NUnit.Framework;
using osu.Framework.Graphics.Containers;
using osu.Game.Overlays.Toolbar;
namespace osu.Game.Tests.Visual
{
[TestFixture]
public class TestCaseToolbar : OsuTestCase
{
public override IReadOnlyList<Type> RequiredTypes => new[]
{
typeof(ToolbarButton),
typeof(ToolbarModeSelector),
typeof(ToolbarModeButton),
typeof(ToolbarNotificationButton),
};
public TestCaseToolbar()
{
var toolbar = new Toolbar { State = Visibility.Visible };
Add(toolbar);
var notificationButton = toolbar.Children.OfType<FillFlowContainer>().Last().Children.OfType<ToolbarNotificationButton>().First();
void setNotifications(int count) => AddStep($"set notification count to {count}", () => notificationButton.NotificationCount.Value = count);
setNotifications(1);
setNotifications(2);
setNotifications(3);
setNotifications(0);
setNotifications(144);
}
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 7,626 |
classdef OlsOlaProcess3d < matlab.System
%OLSOLAPROCESS3D OLS/OLA wrapper for 3-D analysis and synthesis system
%
% Reference:
% Shogo Muramatsu and Hitoshi Kiya,
% ''Parallel Processing Techniques for Multidimensional Sampling
% Lattice Alteration Based on Overlap-Add and Overlap-Save Methods,''
% IEICE Trans. on Fundamentals, Vol.E78-A, No.8, pp.939-943, Aug. 1995
%
% Requirements: MATLAB R2018a
%
% Copyright (c) 2018-2020, Shogo MURAMATSU
%
% All rights reserved.
%
% Contact address: Shogo MURAMATSU,
% Faculty of Engineering, Niigata University,
% 8050 2-no-cho Ikarashi, Nishi-ku,
% Niigata, 950-2181, JAPAN
%
% http://msiplab.eng.niigata-u.ac.jp/
%
properties (Nontunable)
Analyzer
Synthesizer
BoundaryOperation
PadSize = [0 0 0]
SplitFactor = []
CoefsManipulator = []
end
properties (Logical)
UseParallel = false
IsIntegrityTest = true
end
properties (Nontunable, PositiveInteger, Hidden)
VerticalSplitFactor = 1
HorizontalSplitFactor = 1
DepthSplitFactor = 1
end
properties (Hidden, Transient)
BoundaryOperationSet = ...
matlab.system.StringSet({'Circular'});
end
properties (Access = private, Nontunable)
synthesizers
analyzers
coefsmanipulators
refSize
refSubSize
refScales
subPadSize
subPadArrays
nWorkers
end
methods
% Constractor
function obj = OlsOlaProcess3d(varargin)
import saivdr.dictionary.utility.Direction
import saivdr.dictionary.olaols.*
setProperties(obj,nargin,varargin{:})
if ~isempty(obj.Analyzer)
obj.BoundaryOperation = obj.Analyzer.BoundaryOperation;
end
if ~isempty(obj.SplitFactor)
obj.VerticalSplitFactor = obj.SplitFactor(Direction.VERTICAL);
obj.HorizontalSplitFactor = obj.SplitFactor(Direction.HORIZONTAL);
obj.DepthSplitFactor = obj.SplitFactor(Direction.DEPTH);
end
if isempty(obj.CoefsManipulator)
import saivdr.utility.CoefsManipulator
obj.CoefsManipulator = CoefsManipulator();
end
end
end
methods(Access = protected)
function flag = isInactivePropertyImpl(obj,propertyName)
if strcmp(propertyName,'VerticalSplitFactor') || ...
strcmp(propertyName,'HorizontalSplitFactor') || ...
strcmp(propertyName,'DepthSplitFactor')
flag = ~isempty(obj.SplitFactor);
else
flag = false;
end
end
function s = saveObjectImpl(obj)
s = saveObjectImpl@matlab.System(obj);
s.Synthesizer = matlab.System.saveObject(obj.Synthesizer);
s.Analyzer = matlab.System.saveObject(obj.Analyzer);
s.nWorkers = obj.nWorkers;
s.refScales = obj.refScales;
s.subPadSize = obj.subPadSize;
s.subPadArrays = obj.subPadArrays;
end
function loadObjectImpl(obj,s,wasLocked)
obj.subPadArrays = s.subPadArrays;
obj.subPadSize = s.subPadSize;
obj.refScales = s.refScales;
obj.nWorkers = s.nWorkers;
obj.Synthesizer = matlab.System.loadObject(s.Synthesizer);
obj.Analyzer = matlab.System.loadObject(s.Analyzer);
loadObjectImpl@matlab.System(obj,s,wasLocked);
end
function setupImpl(obj,srcImg,nLevels)
% Preperation
verticalSplitFactor = obj.VerticalSplitFactor;
horizontalSplitFactor = obj.HorizontalSplitFactor;
depthSplitFactor = obj.DepthSplitFactor;
nSplit = verticalSplitFactor*horizontalSplitFactor*depthSplitFactor;
% Analyzers
obj.Analyzer.release();
refAnalyzer = obj.Analyzer.clone();
[refCoefs,refScales_] = refAnalyzer.step(srcImg,nLevels);
% Synthesizers
obj.Synthesizer.release();
refSynthesizer = obj.Synthesizer.clone();
% Manipulators
obj.CoefsManipulator.release();
refCoefsManipulator = obj.CoefsManipulator.clone();
% Parameters
obj.refSize = size(srcImg);
obj.refSubSize = obj.refSize*diag(1./[...
verticalSplitFactor,...
horizontalSplitFactor,...
depthSplitFactor]);
scaleRatio = refScales_*diag(1./obj.refSize);
obj.subPadSize = scaleRatio*diag(obj.PadSize);
obj.refScales = refScales_;
% Clone
obj.synthesizers = cell(nSplit,1);
obj.analyzers = cell(nSplit,1);
if obj.UseParallel
obj.nWorkers = Inf;
for iSplit=1:nSplit
obj.analyzers{iSplit} = clone(obj.Analyzer);
obj.synthesizers{iSplit} = clone(obj.Synthesizer);
obj.coefsmanipulators{iSplit} = clone(obj.CoefsManipulator);
end
else
obj.nWorkers = 0;
for iSplit=1:nSplit
obj.analyzers{iSplit} = obj.Analyzer;
obj.synthesizers{iSplit} = obj.Synthesizer;
obj.coefsmanipulators{iSplit} = clone(obj.CoefsManipulator);
end
end
%Evaluate
% Check if srcImg is divisible by split factors
exceptionId = 'SaivDr:IllegalSplitFactorException';
message = 'Split factor must be a divisor of array size.';
if sum(mod(obj.refSubSize,1)) ~= 0
throw(MException(exceptionId,message))
end
% Check if scales are divisible by split factors
if sum(mod(obj.subPadSize,1)) ~= 0
throw(MException('SaivDr','Illegal Pad Size.'))
end
% Allocate memory for zero padding of arrays
nChs = size(refScales_,1);
obj.subPadArrays = cell(nChs,1);
nCoefs = 0;
for iCh = 1:nChs
subScale = refScales_(iCh,:)*diag(1./[...
verticalSplitFactor,...
horizontalSplitFactor,...
depthSplitFactor]);
nDim = subScale+2*obj.subPadSize(iCh,:);
obj.subPadArrays{iCh} = zeros(nDim);
nCoefs = nCoefs + prod(nDim);
end
% Check integrity
if obj.IsIntegrityTest
exceptionId = 'SaivDr:ReconstructionFailureException';
message = 'Failure occurs in reconstruction. Please check the split and padding size.';
%
refCoefsOut = refCoefsManipulator.step(refCoefs);
imgExpctd = refSynthesizer.step(refCoefsOut,refScales_);
%
imgActual = obj.stepImpl(srcImg,nLevels);
diffImg = imgExpctd - imgActual;
if norm(diffImg(:))/numel(diffImg) > 1e-6
throw(MException(exceptionId,message))
end
end
% Delete reference analyzer and synthesizer
refAnalyzer.delete()
refSynthesizer.delete()
refCoefsManipulator.delete()
end
function recImg = stepImpl(obj,srcImg,nLevels)
% Parameters
nWorkers_ = obj.nWorkers;
analyzers_ = obj.analyzers;
synthesizers_ = obj.synthesizers;
coefsmanipulators_ = obj.coefsmanipulators;
% Define support functions
extract_ols = @(c,s) obj.extract_ols_(c,s);
padding_ola = @(c) obj.padding_ola_(c);
arr2vec = @(a) obj.arr2vec_(a);
% Circular global padding
srcImg_ = padarray(srcImg,obj.PadSize,'circular');
% Overlap save split
subImgs = obj.split_ols_(srcImg_);
% Parallel processing
nSplit = length(subImgs);
subRecImg = cell(nSplit,1);
parfor (iSplit=1:nSplit,nWorkers_)
% Analyze
[subCoefs, subScales] = ...
analyzers_{iSplit}.step(subImgs{iSplit},nLevels);
% Extract significant coefs.
coefspre = extract_ols(subCoefs,subScales);
% Process for coefficients
coefspost = ...
coefsmanipulators_{iSplit}.step(coefspre);
% Zero padding for convolution
subCoefArray = padding_ola(coefspost);
% Synthesis
[subCoefs,subScales] = arr2vec(subCoefArray);
subRecImg{iSplit} = ...
step(synthesizers_{iSplit},subCoefs,subScales);
end
% 4. Overlap add (Circular)
recImg = circular_ola_(obj,subRecImg);
end
end
methods (Access = private)
function recImg = circular_ola_(obj,subRecImg)
import saivdr.dictionary.utility.Direction
verticalSplitFactor = obj.VerticalSplitFactor;
horizontalSplitFactor = obj.HorizontalSplitFactor;
depthSplitFactor = obj.DepthSplitFactor;
stepsize = obj.refSubSize;
overlap = size(subRecImg{1})-stepsize;
recImg = zeros(obj.refSize+overlap);
% Overlap add
iSplit = 0;
tIdxDep = 0;
for iDepSplit = 1:depthSplitFactor
sIdxDep = tIdxDep + 1;
tIdxDep = sIdxDep + stepsize(Direction.DEPTH) - 1;
eIdxDep = tIdxDep + overlap(Direction.DEPTH);
tIdxHor = 0;
for iHorSplit = 1:horizontalSplitFactor
sIdxHor = tIdxHor + 1;
tIdxHor = sIdxHor + stepsize(Direction.HORIZONTAL) - 1;
eIdxHor = tIdxHor + overlap(Direction.HORIZONTAL);
tIdxVer = 0;
for iVerSplit = 1:verticalSplitFactor
iSplit = iSplit + 1;
sIdxVer = tIdxVer + 1;
tIdxVer = sIdxVer + stepsize(Direction.VERTICAL) - 1;
eIdxVer = tIdxVer + overlap(Direction.VERTICAL);
recImg(sIdxVer:eIdxVer,sIdxHor:eIdxHor,sIdxDep:eIdxDep) = ...
recImg(sIdxVer:eIdxVer,sIdxHor:eIdxHor,sIdxDep:eIdxDep) + ...
subRecImg{iSplit};
end
end
end
% Folding
recImg(1:overlap(Direction.VERTICAL),:,:) = ...
recImg(1:overlap(Direction.VERTICAL),:,:) + ...
recImg(end-overlap(Direction.VERTICAL)+1:end,:,:);
recImg(:,1:overlap(Direction.HORIZONTAL),:) = ...
recImg(:,1:overlap(Direction.HORIZONTAL),:) + ...
recImg(:,end-overlap(Direction.HORIZONTAL)+1:end,:);
recImg(:,:,1:overlap(Direction.DEPTH)) = ...
recImg(:,:,1:overlap(Direction.DEPTH)) + ...
recImg(:,:,end-overlap(Direction.DEPTH)+1:end);
% Cropping & circular shift
recImg = circshift(recImg(...
1:obj.refSize(Direction.VERTICAL),...
1:obj.refSize(Direction.HORIZONTAL),...
1:obj.refSize(Direction.DEPTH)),-overlap/2);
end
function subCoefArray = padding_ola_(obj,subCoefArray)
import saivdr.dictionary.utility.Direction
nChs = size(subCoefArray,2);
subPadSize_ = obj.subPadSize;
subPadArrays_ = obj.subPadArrays;
for iCh = 1:nChs
sRowIdx = subPadSize_(iCh,Direction.VERTICAL)+1;
eRowIdx = sRowIdx + size(subCoefArray{iCh},Direction.VERTICAL)-1;
sColIdx = subPadSize_(iCh,Direction.HORIZONTAL)+1;
eColIdx = sColIdx + size(subCoefArray{iCh},Direction.HORIZONTAL)-1;
sLayIdx = subPadSize_(iCh,Direction.DEPTH)+1;
eLayIdx = sLayIdx + size(subCoefArray{iCh},Direction.DEPTH)-1;
tmpArray = subPadArrays_{iCh};
tmpArray(sRowIdx:eRowIdx,sColIdx:eColIdx,sLayIdx:eLayIdx) ...
= subCoefArray{iCh};
subCoefArray{iCh} = tmpArray;
end
end
function [coefsCrop,scalesCrop] = ...
extract_ols_(obj,coefsSplit,scalesSplit)
import saivdr.dictionary.utility.Direction
verticalSplitFactor = obj.VerticalSplitFactor;
horizontalSplitFactor = obj.HorizontalSplitFactor;
depthSplitFactor = obj.DepthSplitFactor;
refSubScales = obj.refScales*diag(...
1./[verticalSplitFactor,horizontalSplitFactor,depthSplitFactor]);
nChs = size(refSubScales,1);
%
coefsCrop = cell(1,nChs);
eIdx = 0;
for iCh = 1:nChs
stepsize = refSubScales(iCh,:);
sIdx = eIdx + 1;
eIdx = sIdx + prod(scalesSplit(iCh,:)) - 1;
tmpVec = coefsSplit(sIdx:eIdx);
tmpArray = reshape(tmpVec,scalesSplit(iCh,:));
%
offset = (scalesSplit(iCh,:) - refSubScales(iCh,:))/2;
sRowIdx = offset(Direction.VERTICAL) + 1;
eRowIdx = sRowIdx + stepsize(Direction.VERTICAL) - 1;
sColIdx = offset(Direction.HORIZONTAL) + 1;
eColIdx = sColIdx + stepsize(Direction.HORIZONTAL) - 1;
sLayIdx = offset(Direction.DEPTH) + 1;
eLayIdx = sLayIdx + stepsize(Direction.DEPTH) - 1;
%
tmpArrayCrop = tmpArray(sRowIdx:eRowIdx,sColIdx:eColIdx,sLayIdx:eLayIdx);
coefsCrop{iCh} = tmpArrayCrop;
end
if nargout > 1
scalesCrop = refSubScales;
end
end
function subImgs = split_ols_(obj,srcImg)
import saivdr.dictionary.utility.Direction
verticalSplitFactor = obj.VerticalSplitFactor;
horizontalSplitFactor = obj.HorizontalSplitFactor;
depthSplitFactor = obj.DepthSplitFactor;
nSplit = verticalSplitFactor*...
horizontalSplitFactor*...
depthSplitFactor;
stepsize = obj.refSubSize;
overlap = 2*obj.PadSize;
%
subImgs = cell(nSplit,1);
idx = 0;
for iLaySplit = 1:depthSplitFactor
sLayIdx = (iLaySplit-1)*stepsize(Direction.DEPTH) + 1;
eLayIdx = iLaySplit*stepsize(Direction.DEPTH) + ...
overlap(Direction.DEPTH);
for iHorSplit = 1:horizontalSplitFactor
sColIdx = (iHorSplit-1)*stepsize(Direction.HORIZONTAL) + 1;
eColIdx = iHorSplit*stepsize(Direction.HORIZONTAL) + ...
overlap(Direction.HORIZONTAL);
for iVerSplit = 1:verticalSplitFactor
idx = idx + 1;
sRowIdx = (iVerSplit-1)*stepsize(Direction.VERTICAL) + 1;
eRowIdx = iVerSplit*stepsize(Direction.VERTICAL) + ...
overlap(Direction.VERTICAL);
subImgs{idx} = srcImg(sRowIdx:eRowIdx,...
sColIdx:eColIdx,...
sLayIdx:eLayIdx);
end
end
end
end
end
methods (Access = private, Static)
function [subCoefs,subScales] = arr2vec_(subCoefArray)
nChs = size(subCoefArray,2);
subScales = zeros(nChs,3);
tmpCoefs_ = cell(1,nChs);
for iCh = 1:nChs
tmpArray = subCoefArray{iCh};
subScales(iCh,:) = size(tmpArray);
tmpCoefs_{iCh} = tmpArray(:).';
end
subCoefs = cell2mat(tmpCoefs_);
end
end
end
| {
"redpajama_set_name": "RedPajamaGithub"
} | 5,502 |
{"url":"http:\/\/www.koreascience.or.kr\/article\/JAKO201010250392674.page","text":"# CHARACTERIZATIONS ON CHAIN RECURRENCES\n\n\u2022 Published : 2010.03.31\n\u2022 56 5\n\n#### Abstract\n\nIt is well known that there is a residual subset J of the space of $C^1$-diffeomorphisms on a compact Riemannian manifold M such that the maps f $\\mapsto$ chain recurrent set of f and f $\\mapsto$ number of chain components of f are continuous on J. In this paper we get the flow version of the above results on diffeomorphisms.\n\n#### Keywords\n\nchain recurrence;residual set;flow\n\n#### References\n\n1. L. Block and John E. Franke, The chain recurrent set for maps of the interval, Proc. Amer. Math. Soc. 87 (1983), no. 4, 723\u2013727. https:\/\/doi.org\/10.2307\/2043368\n2. I. U. Bronstein and A. Ya. Kopanskii, Chain recurrence in dynamical systems without uniqueness, Nonlinear Anal. 12 (1988), no. 2, 147\u2013154. https:\/\/doi.org\/10.1016\/0362-546X(88)90031-4\n3. C. Conley, The gradient structure of a flow, I. B. M. Res. RC 3932. Yorktown Heights, N.Y., 1972.\n4. C. Conley, Isolated Invariant Sets and the Morse Index, CBMS Regional Conference Series in Mathematics, 38. American Mathematical Society, Providence, R.I., 1978.\n5. H. Chu, Chain recurrence for multi-valued dynamical systems on noncompact spaces, Nonlinear Anal. 61 (2005), no. 5, 715\u2013723. https:\/\/doi.org\/10.1016\/j.na.2005.01.024\n6. H. Chu and J. Park, Attractors for relations in ${\\delta}$-compact spaces, Topology Appl. 148 (2005), no. 1-3, 201\u2013212. https:\/\/doi.org\/10.1016\/j.topol.2003.05.009\n7. M. Hurley, Bifurcation and chain recurrence, Ergodic Theory Dynam. Systems 3 (1983), no. 2, 231\u2013240.\n8. M. Hurley, Fixed points of topologically stable flows, Trans. Amer. Math. Soc. 294 (1986), no. 2, 625\u2013633. https:\/\/doi.org\/10.2307\/2000204\n9. M. Hurley, Chain recurrence and attraction in noncompact spaces, Ergodic Theory Dynam. Systems 11 (1991), no. 4, 709\u2013729.\n10. M. Hurley, Noncompact chain recurrence and attraction, Proc. Amer. Math. Soc. 115 (1992), no. 4, 1139\u20131148. https:\/\/doi.org\/10.2307\/2159367\n11. M. Hurley, Chain recurrence, semiflows, and gradients, J. Dynam. Differential Equations 7 (1995), no. 3, 437\u2013456. https:\/\/doi.org\/10.1007\/BF02219371\n12. S. Newhouse, Diffeomorphisms with infinitely many sinks, Topology 13 (1974), 9\u201318. https:\/\/doi.org\/10.1016\/0040-9383(74)90034-2\n13. P. Oprocha, Chain recurrence in multidimensional time discrete dynamical systems, Discrete Contin. Dyn. Syst. 20 (2008), no. 4, 1039\u20131056. https:\/\/doi.org\/10.3934\/dcds.2008.20.1039\n14. J. Park, D. Kang, and H. Chu, Stabilities in multi-valued dynamical systems, Nonlinear Anal. 67 (2007), no. 7, 2050\u20132059. https:\/\/doi.org\/10.1016\/j.na.2006.06.057\n15. P. Koscielniak, Generic properties of $Z^2$-actions on the interval, Topology Appl. 154 (2007), no. 14, 2672\u20132677. https:\/\/doi.org\/10.1016\/j.topol.2007.05.001\n\n#### Cited by\n\n1. SOME REMARKS ON CHAIN PROLONGATIONS IN DYNAMICAL SYSTEMS vol.26, pp.2, 2013, https:\/\/doi.org\/10.14403\/jcms.2013.26.2.351\n2. A TOPOLOGICAL CHARACTERIZATION OF \u00ad\u2126-LIMIT SETS ON DYNAMICAL SYSTEMS vol.27, pp.3, 2014, https:\/\/doi.org\/10.14403\/jcms.2014.27.3.523","date":"2019-06-16 14:45:18","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.6983209848403931, \"perplexity\": 1313.573750835292}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 20, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2019-26\/segments\/1560627998250.13\/warc\/CC-MAIN-20190616142725-20190616164725-00130.warc.gz\"}"} | null | null |
<?php
if (empty($lang) || !is_array($lang)) {
$lang = array();
}
$lang = array_merge($lang, array(
'ONELOGIN_SAML_METADATA_LINK' => 'Go to the metadata of this SP',
'ONELOGIN_SAML_SETTINGS_TITLE' => 'Onelogin SAML Settings',
'ONELOGIN_SAML_IDP_SECTION_TITLE' => 'IDENTITY PROVIDER SETTINGS',
'ONELOGIN_SAML_IDP_SECTION_DESC' => "Set here some info related to the IdP that will be connected with our phpBB. You can find this values at the Onelogin's platform in the phpBB App at the Single Sign-On tab",
'ONELOGIN_SAML_IDP_ENTITY_TITLE' => 'IdP Entity Id * ',
'ONELOGIN_SAML_IDP_ENTITY_DESC' => 'Identifier of the IdP entity. ("Issuer URL")',
'ONELOGIN_SAML_IDP_SSO_TITLE' => 'Single Sign On Service Url * ',
'ONELOGIN_SAML_IDP_SSO_DESC' => 'SSO endpoint info of the IdP. URL target of the IdP where the SP will send the Authentication Request. ("SAML 2.0 Endpoint (HTTP)")',
'ONELOGIN_SAML_IDP_SLO_TITLE' => 'Single Log Out Service Url',
'ONELOGIN_SAML_IDP_SLO_DESC' => 'SLO endpoint info of the IdP. URL target of the IdP where the SP will send the SLO Request. ("SLO Endpoint (HTTP)")',
'ONELOGIN_SAML_IDP_X509CERT_TITLE' => 'X.509 Certificate',
'ONELOGIN_SAML_IDP_X509CERT_DESC' => 'Public x509 certificate of the IdP. ("X.509 certificate")',
'ONELOGIN_SAML_OPTIONS_SECTION_TITLE' => 'OPTIONS',
'ONELOGIN_SAML_OPTIONS_SECTION_DESC' => 'In this section the behavior of the plugin is set.',
'ONELOGIN_SAML_AUTOCREATE_TITLE' => 'Create user if not exists',
'ONELOGIN_SAML_AUTOCREATE_DESC' => 'Auto-provisioning. If user not exists, phpBB will create a new user with the data provided by the IdP.
Review the Mapping section.',
'ONELOGIN_SAML_UPDATEUSER_TITLE' => 'Update user data',
'ONELOGIN_SAML_UPDATEUSER_DESC' => 'Auto-update. phpBB will update the account of the user with the data provided by the IdP.
Review the Mapping section.',
'ONELOGIN_SAML_FORCELOGIN_TITLE' => 'Force SAML login',
'ONELOGIN_SAML_FORCELOGIN_DESC' => 'Protect phpBB and force the user to authenticate at the IdP in order to access',
'ONELOGIN_SAML_SECOND_AUTH_METHOD_TITLE' => 'Alternative auth method',
'ONELOGIN_SAML_SECOND_AUTH_METHOD_DESC' => "When Force SAML login is disable, you can access to the login page adding a '&normal' parameter in order to log using an alternative method: db or ldap, providing the user credentials",
'ONELOGIN_SAML_SLO_TITLE' => 'Single Log Out',
'ONELOGIN_SAML_SLO_DESC' => 'Enable/disable Single Log Out. SLO is a complex functionality, the most common SLO implementation is based on front-channel (redirections), sometimes if the SLO workflow fails a user can be blocked in an unhandled view. If the admin does not controls the set of apps involved in the SLO process maybe is better to disable this functionality due could carry more problems than benefits.',
'ONELOGIN_SAML_ACCOUNT_MATCHER_TITLE' => 'Match phpBB account by',
'ONELOGIN_SAML_ACCOUNT_MATCHER_DESC' => "Select what field will be used in order to find the user account. If you select the 'email' fieldname the plugin will prevent that the user can change his mail in his profile.",
'USERNAME' => 'username',
'ONELOGIN_SAML_ATTR_MAPPING_SECTION_TITLE' => 'ATTRIBUTE MAPPING',
'ONELOGIN_SAML_ATTR_MAPPING_SECTION_DESC' => "Sometimes the names of the attributes sent by the IdP not match the names used by phpBB for the user accounts. In this section we can set the mapping between IdP fields and phpBB fields. Notice that this mapping could be also set at Onelogin's IdP",
'ONELOGIN_SAML_ATTR_MAPPING_USERNAME_TITLE' => 'Username',
'ONELOGIN_SAML_ATTR_MAPPING_USERNAME_DESC' => '',
'ONELOGIN_SAML_ATTR_MAPPING_MAIL_TITLE' => 'E-mail',
'ONELOGIN_SAML_ATTR_MAPPING_MAIL_DESC' => '',
'ONELOGIN_SAML_ATTR_MAPPING_GROUPS_TITLE' => 'Group',
'ONELOGIN_SAML_ATTR_MAPPING_GROUPS_DESC' => '',
'ONELOGIN_SAML_GROUP_MAPPING_SECTION_TITLE' => 'GROUP MAPPING',
'ONELOGIN_SAML_GROUP_MAPPING_SECTION_DESC' => "The IdP can use it's own groups. Set in this section the mapping between IdP and phpBB groups. Accepts multiple valued comma separated. Example: Administrators,Guests,Registered users <br>You can map pre-defined groups or map your custom groups adding the name and its related map",
'ONELOGIN_SAML_GROUP_MAPPING_ADMINISTRATORS_TITLE' => 'Administrators',
'ONELOGIN_SAML_GROUP_MAPPING_BOTS_TITLE' => 'Bots',
'ONELOGIN_SAML_GROUP_MAPPING_GLOBAL_MODERATORS_TITLE' => 'Global moderators',
'ONELOGIN_SAML_GROUP_MAPPING_GUESTS_TITLE' => 'Guests',
'ONELOGIN_SAML_GROUP_MAPPING_NEWLY_REGISTERED_USERS_TITLE' => 'Newly registered users',
'ONELOGIN_SAML_GROUP_MAPPING_REGISTERED_USERS_TITLE' => 'Registered users',
'ONELOGIN_SAML_GROUP_MAPPING_REGISTERED_COPPA_USERS_TITLE' => 'Registered COPPA users',
'ONELOGIN_SAML_GROUP_MAPPING_CUSTOM1_TITLE' => 'Custom group 1',
'ONELOGIN_SAML_GROUP_MAPPING_CUSTOM2_TITLE' => 'Custom group 2',
'ONELOGIN_SAML_GROUP_MAPPING_CUSTOM3_TITLE' => 'Custom group 3',
'ONELOGIN_SAML_GROUP_MAPPING_CUSTOM4_TITLE' => 'Custom group 4',
'ONELOGIN_SAML_GROUP_MAPPING_CUSTOM5_TITLE' => 'Custom group 5',
'ONELOGIN_SAML_ADVANCED_SETTINGS_SECTION_TITLE' => 'ADVANCED SETTINGS',
'ONELOGIN_SAML_ADVANCED_SETTINGS_SECTION_DESC' => "Handle some other parameters related to customizations and security issues.<br>If sign/encryption is enabled, then x509 cert and private key for the SP must be provided. There are 2 ways:<br>1. Store them as files named sp.key and sp.crt on the 'certs' folder of the plugin. (be sure that the folder is protected and not exposed to internet)<br>2. Store them at the database, filling the corresponding textareas. (take care of security issues)",
'ONELOGIN_SAML_ADVANCED_SETTINGS_DEBUG_TITLE' => 'Debug Mode',
'ONELOGIN_SAML_ADVANCED_SETTINGS_DEBUG_DESC' => 'Enable it when your are debugging the SAML workflow. Errors and Warnigs will be showed.',
'ONELOGIN_SAML_ADVANCED_SETTINGS_STRICT_MODE_TITLE' => 'Strict Mode',
'ONELOGIN_SAML_ADVANCED_SETTINGS_STRICT_MODE_DESC' => 'If Strict mode is Enabled, then phpBB will reject unsigned or unencrypted messages if it expects them signed or encrypted. Also will reject the messages if not strictly follow the SAML standard: Destination, NameId, Conditions ... are validated too.',
'ONELOGIN_SAML_ADVANCED_SETTINGS_SP_ENTITY_ID_TITLE' => 'Service Provider Entity Id',
'ONELOGIN_SAML_ADVANCED_SETTINGS_SP_ENTITY_ID_DESC' => "Set the Entity ID for the Service Provider. If not provided, 'php-saml' will be used.",
'ONELOGIN_SAML_ADVANCED_SETTINGS_NAMEID_ENCRYPTED_TITLE' => 'Encrypt nameID',
'ONELOGIN_SAML_ADVANCED_SETTINGS_NAMEID_ENCRYPTED_DESC' => 'The nameID sent by this SP will be encrypted.',
'ONELOGIN_SAML_ADVANCED_SETTINGS_AUTHN_REQUEST_SIGNED_TITLE' => 'Sign AuthnRequest',
'ONELOGIN_SAML_ADVANCED_SETTINGS_AUTHN_REQUEST_SIGNED_DESC' => 'The samlp:AuthnRequest messages sent by this SP will be signed.',
'ONELOGIN_SAML_ADVANCED_SETTINGS_LOGOUT_REQUEST_SIGNED_TITLE' => 'Sign LogoutRequest',
'ONELOGIN_SAML_ADVANCED_SETTINGS_LOGOUT_REQUEST_SIGNED_DESC' => 'The samlp:logoutRequest messages sent by this SP will be signed.',
'ONELOGIN_SAML_ADVANCED_SETTINGS_LOGOUT_RESPONSE_SIGNED_TITLE' => 'Sign LogoutResponse',
'ONELOGIN_SAML_ADVANCED_SETTINGS_LOGOUT_RESPONSE_SIGNED_DESC' => 'The samlp:logoutResponse messages sent by this SP will be signed.',
'ONELOGIN_SAML_ADVANCED_SETTINGS_WANT_MESSAGE_SIGNED_TITLE' => 'Reject Unsigned Messages',
'ONELOGIN_SAML_ADVANCED_SETTINGS_WANT_MESSAGE_SIGNED_DESC' => 'Reject unsigned samlp:Response, samlp:LogoutRequest and samlp:LogoutResponse received',
'ONELOGIN_SAML_ADVANCED_SETTINGS_WANT_ASSERTION_SIGNED_TITLE' => 'Reject Unsigned Assertions',
'ONELOGIN_SAML_ADVANCED_SETTINGS_WANT_ASSERTION_SIGNED_DESC' => 'Reject unsigned saml:Assertion received',
'ONELOGIN_SAML_ADVANCED_SETTINGS_WANT_ASSERTION_ENCRYPTED_TITLE' => 'Reject Unencrypted Assertions',
'ONELOGIN_SAML_ADVANCED_SETTINGS_WANT_ASSERTION_ENCRYPTED_DESC' => 'Reject unencrypted saml:Assertion received',
'ONELOGIN_SAML_ADVANCED_SETTINGS_SP_X509CERT_TITLE' => 'Service Provider X.509 Certificate',
'ONELOGIN_SAML_ADVANCED_SETTINGS_SP_X509CERT_DESC' => 'Public x509 certificate of the SP. Leave this field empty if you gonna provide the cert by the sp.crt',
'ONELOGIN_SAML_ADVANCED_SETTINGS_SP_PRIVATEKEY_TITLE' => 'Service Provider Private Key',
'ONELOGIN_SAML_ADVANCED_SETTINGS_SP_PRIVATEKEY_DESC' => 'Private Key of the SP. Leave this field empty if you gonna provide the private key by the sp.key',
'ONELOGIN_SAML_CANNOT_INCLUDE_TOOLKIT' => "Unable to load Onelogin's PHP Toolkit. Did you installed it in the right directory?",
'ONELOGIN_SAML_CANNOT_INCLUDE_SETTINGS' => "Unable to load Onelogin's PHP Settings. Did you installed it in the right directory?",
'ONELOGIN_SAML_IDP_ENTITY_ID_NOT_DEFINED' => 'Review the SAML settings, the IdP entityid is required',
'ONELOGIN_SAML_IDP_SSO_URL_NOT_DEFINED' => 'Review the SAML settings, the IdP SSO URL is required',
'ONELOGIN_SAML_NOT_CONFIGURED' => 'The Onelogin SSO/SAML plugin is not correctly configured',
'NO_AUTHENTICATED' => 'SAML failed, user was not authenticated',
'NO_USERNAME' => 'Login requires a username and none could be obtained from the IdP.',
'NO_EMAIL' => 'Login requires a mail and none could be obtained from the IdP.',
'USER_DOES_NOT_EXIST_AND_NO_DATA' => 'The user authenticated in the IdP does not exists in phpBB, tried to create an account but username or mail were not provided',
'USER_DOES_NOT_EXIST_AND_NOT_ALLOWED_TO_CREATE' => 'The user authenticated in the IdP does not exists in phpBB and is not allowed to create an account',
'SLO PROCESS FAILED' => 'SLO process failed'
));
| {
"redpajama_set_name": "RedPajamaGithub"
} | 9,617 |
Public remains skeptical of church's vow to reform
BY TERRIE MORGAN-BESECKER / Published: August 19, 2018
Catholic bishops across Pennsylvania vowed to increase transparency and accountability in the wake of the scathing grand jury report on child sexual abuse, but the public remains skeptical.
Each of the six dioceses targeted in the report voluntarily identified predator priests in addition to those named in the report. Church officials say it reflects their sincere effort to protect the public and atone for the cover-ups that allowed the abuse to continue for decades.
Critics question why it took until now for the dioceses to release the names. They contend the church is more interested in damage control than reforming its ways.
"This is something we've been asking for for years," said Becky Ianni, president of the Virginia chapter of Survivors Network of those Abused by Priests. "I don't take it as them being proactive as much as they are trying to look better after this horrible report."
The sentiment exemplifies the immense challenge the church faces in restoring the faith and trust of the more than 3 million parishioners in Pennsylvania, religious experts say.
"The church is the one that brings us healing and peace," said Sandra Yocum, Ph.D., a professor of faith and culture at the University of Dayton, Ohio. "How is the church going to be able to fulfill that mission given all these allegations and the leadership's seeming inability, at least in the past, to take this seriously."
The church has taken steps over the years to ensure the safety of children. In 2002, it implemented reforms that dictate how dioceses must handle cases, including mandates that the accused priest be removed from duties and the cases be referred to law enforcement.
"I think they had hoped that once they started fixing the problem and got good procedures in place to ensure there are no abusive priests that they could say we solved the problem, let's move on," said the Rev. Thomas Reese, a Jesuit priest who has studied the Catholic Church for decades. "That does not seem to satisfy people."
Yocum and Reese said the real test will come in the ensuing months and years as the public watches how the church handles the crisis, particularly any new allegations that surface against a clergy member.
"When there is an incident in a local community, someone should come and speak to the community directly," Yocum said. "Here is what we know, here is what we've done and here is what happens next."
The Allentown diocese already has taken steps to address new abuse allegations proactively, said spokesman Matt Kerr. Its list of predator priests, published on the diocese website, includes information on Francis Nave, a priest from Bath who recently was accused in a lawsuit of masturbating while chatting online with a youth. The lawsuit, filed in June, says the incident occurred in 2012. The diocese notes Nave was removed from ministry pending an investigation and the case was referred to law enforcement.
"We felt it was important to put him on there," Kerr said. "He was already publicly named, so we thought it was the proper thing to do."
While he strongly advocates for transparency, Reese said he has some concerns about publicly naming clergy members before any investigation is done to determine the credibility of the allegations.
"People have a right to their reputations in any profession," Reese said.
He likened the situation to a newspaper receiving a letter accusing a reporter of violating journalistic ethics.
"Does the newspaper have to report that the next day in the paper, or would it do an examination to see if the allegation was credible?" he said. "The church faces exactly the same problem."
However the church opts to handle cases, it's crucial that they be consistent, Reese and Yocum said.
"There will be a lot of suspicion around priests and church leadership," Yocum said. "People are going to be watching if there is real change. ... If there is consistency in the response to credible allegations, that will be critical in the next decade to help restore trust in the church."
tbesecker@timesshamrock.com; 570-348-9137
@tmbeseckerTT | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 7,250 |
Q: icpc error #10104: unable to open 'tbb_placeholder' BOOST + ICPC + TBB + C++11 I was trying to compile a boost+Intel TBB code using intel compiler (ICC-13.0.1) The TBB library is latest (4.2) and boost is 1.55.0. The code incorporates C++11 features and so I use the std=c++11 flag
I am getting this error:
icpc: error #10104: unable to open 'tbb_pLaCeHoLdEr'
make: *** [driver.o] Error 1
My Makefile contents are as follows:
INCLUDE_PROJECT = /home/aniketnp/multiproc/
INCLUDE_BOOST = /home/aniketnp/boost_1_55_0/
TBB_L = /home/aniketnp/tbb42_20140122oss/lib/intel64
TBB_I = /home/aniketnp/tbb42_20140122oss/include
CXXFLAGS = -I$(INCLUDE_PROJECT) -I$(INCLUDE_BOOST) -I$(TBB_I) -L$(TBB_L) -tbb -lpthread -std=c++11
Could any body speculate what could be the possible error?
*
*The boost and TBB are in my home directory.
*The ICC13.1 is loaded through a module in our cluster.
*The LD_LIBRARY_PATH variable correctly shows the TBB_L variable in makefile (properly exported in the .bashrc file.
A: looks like installation error of something that uses TBB. TBB has similar place in tbbvars.sh scripts but uses 'SUBSTITUTE_INSTALL_DIR_HERE' instead. So, search for tbb_pLaCeHoLdEr in your project and its components and replace accordingly
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{"url":"http:\/\/physics.stackexchange.com\/questions\/51587\/spinning-spheres-colliding?answertab=active","text":"# Spinning spheres colliding\n\nIn an ideal environment with no friction, in a vacuum, what happens to the velocity of the spin of two spheres spinning in perfect parity at two different velocities when they come into contact?\n\n-\n\nI think it would be more interesting if you allowed friction between the spheres. In that case we can still use conservation of angular momentum. So for example if the balls had equal and opposite angular momentum before collision they could both have zero angular momentum after collision.\n\n-\nTwo balls with equal and opposite angular momentum would be achiral, correct? How about if you have two balls which are chiral \u2014 spinning in opposite directions, but with the same velocity? \u2013\u00a0 Luke Burns Jan 19 '13 at 18:01\nI am not sure it is enlightening to bring chirality into this. I think the result you get just has to be consistent with the total angular momentum, obtained by vector addition, being conserved. \u2013\u00a0 physicsphile Jan 20 '13 at 22:26\n\nThe rotating surfaces of the spheres would just slide over each other at the instant of contact: no forces perpendicular to the line connecting the centers of the two spheres would exist (i.e. no torque would exist). They would undergo a perfectly elastic collision (no loss of energy, thus no friction), thus conserving angular (they just keep spinning) and linear (elastic collision) momentum.\n\nThe angular velocity of each sphere remains constant: the only 'usual' things which can change the angular velocity is exerting torque or changing the mass of the sphere. At contact, colliding spinning spheres which have no friction just don't care about the spinning.\n\n-","date":"2015-03-29 22:37:06","metadata":"{\"extraction_info\": {\"found_math\": false, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.9144733548164368, \"perplexity\": 352.20327598260815}, \"config\": {\"markdown_headings\": true, \"markdown_code\": false, \"boilerplate_config\": {\"ratio_threshold\": 0.3, \"absolute_threshold\": 10, \"end_threshold\": 5, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2015-14\/segments\/1427131298755.8\/warc\/CC-MAIN-20150323172138-00268-ip-10-168-14-71.ec2.internal.warc.gz\"}"} | null | null |
Vizzit provide statistical and monitoring tools that are suitable for both simple and advance analysis of visitor behavior on your website.
Want to download the full Vizzit list? Use the search to the left! | {
"redpajama_set_name": "RedPajamaC4"
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\subsubsection{Tensor Notation.}
In the following, we denote tensor contractions by means of the Einstein
notation;\footnote{%
For the sake of simplicity and without any loss of generality, we ignore any
distinction between covariant and contravariant vectors; this means we treat
any index as a subscript.
}
let us briefly explain said notation by means of an example. In the contraction
$C_{abc} = A_{ai} B_{ibc}$, the entries {\tt$C$[a,b,c]} of the resulting
three-dimensional tensor $C \in \mathbb R^{a \times b \times c}$ are computed as
$\forall \texttt a \forall \texttt b \forall \texttt c. C\texttt{[a,b,c]} =
\sum_\texttt i A\texttt{[a,i]} B\texttt{[i,b,c]}$. (In this notation, a
matrix-matrix product is denoted by $C_{ab} = A_{ai} B_{ib}$.) The indices that
appear in both tensors $A$ and $B$ --- the summation indices $i, j, \ldots$ ---
are called \emph{contracted}, while those that only appear in either $A$ or $B$
(and thus in $C$) --- $a, b, c, \ldots$ --- are called \emph{free} or
\emph{uncontracted}. W.l.o.g., we assume that tensors are stored as
Fortran-style contiguous multidimensional arrays: vectors (1D tensors) are
stored contiguously, matrices (2D tensors) are stored as sequence of column
vectors, 3D tensors (visualized as cubes) are stored as a sequence of matrices
(planes of the cube), and so on.
\subsubsection{Related Work.}
The most prominent project targeting the efficient computation of tensor
contractions is probably the Tensor Contraction Engine, a compiler built
specifically for multi-tensor multi-index contractions to be executed within
memory constraints~\cite{tce}; in light of the wide diffusion and nearly
optimal efficiency of the BLAS library, an extension to TCE was proposed to
compute contractions via BLAS operations~\cite{lu}.
In the same spirit, we provided simple rules to build a taxonomy for all
contractions between two tensors, identifying which BLAS routines are usable and
how to best exploit them~\cite{DiNapoli2014:210}.
\parsum{(performance prediction)}
There also exists a variety of work in the field of performance prediction in
the context of dense linear algebra. A notable example is
Iakymchuk~et~al.~\cite{roman,roman2}, where the authors model the performance of
dense linear algebra algorithms analytically based on very detailed models of
the occurring cache-misses.
Also, in~\cite{modeling}, we use measurement-based performance models to predict
the behavior of blocked algorithms.
However, none of these works target or address high-performance tensor
contractions and their peculiarities, i.e., very regular patterns in routine
invocation and memory access, but highly skewed dimensionality (tiny sizes for
at least one of the dimensions).
\subsubsection{Structure of the Paper.}
The rest of this paper is structured as follows. The systematic generation of
BLAS-based algorithms for tensor contractions is discussed in
\autoref{sec:alggen}. Our performance prediction framework is introduced in
\autoref{sec:pred}, and experimental results for a range of contractions are
presented \autoref{sec:results}.
\section{Introduction}
\label{sec:intro}
\input{intro}
\section{Algorithm Generation}
\label{sec:alggen}
\input{alggen}
\section{Performance Prediction}
\label{sec:pred}
\input{pred}
\section{Results}
\label{sec:results}
\input{results}
\section{Conclusion}
\input{conclusion}
\bibliographystyle{splncs}
\subsection{Repeated Execution}
\parsum{starting point: repeated execution}
The first, most intuitive, attempt to predict the performance of an algorithm
relies on the isolated and repeated measurement of its BLAS kernel. We
implemented this approach by executing each kernel ten times and extracting the
median execution time; the corresponding estimate is then obtained by
multiplying the median by the number of kernel invocations within the algorithm.
In our example, this boils down to multiplying the kernel execution time with
the product of all loop lengths.
\parsum{insufficient accuracy <= assumed all in cache}
The performance profiles predicted by this first, rough approach are shown in
\autoref{fig:pred_step}. By comparing this figure with the reference
\autoref{fig:pred_meas}, it becomes apparent that while the two top algorithms
are already correctly identified, the performance of almost all algorithms is
consistently overestimated. In other words, when executed as part of the
algorithms, the BLAS kernels take longer to complete than in the isolated
micro-benchmarks. The reason for this discrepancy is that the micro-benchmarks
invoke the kernels repeatedly, with the same memory regions as operands, i.e.,
they operate on warm data (the operands remain in the CPU's cache). Within the
algorithm, by contrast, at least one, and potentially even all of the operands,
vary from one invocation to the next, i.e., the kernels operate at least
partially on cold data.
\subsection{Operand Access Distance}
\parsum{access distance}
In order to improve the accuracy of the predictions, the idea is to first
identify the status of the cache in the algorithm prior to the invocation of the
BLAS kernel (``precondition''), and then to replicate such a status in the
micro-benchmark. For this purpose, each algorithm is symbolically analyzed to
reconstruct the order of memory accesses involving the kernel's operands. For
each operand, we determine the set of memory regions $M$ that were loaded into
cache since its last access, and define the {\em access distance} as the sum of
the size of these regions $M$.
\parsum{cache setup}
Once the access distances for all operands of a kernel are determined, we can
create an artificial sequence of memory accesses to reconstruct the cache
precondition. Based on this cache setup, the BLAS kernels are timed in a
micro-benchmark that closely resembles the actual execution of the algorithm.
As before, these micro-benchmarks are repeated and timed ten times to yield a
stable median. From the median, the performance of the algorithms is again
obtained based on the number of kernel invocations per algorithm execution.
\parsum{assumption}
To predict which memory regions are in cache, we assume a fully associative
Least Recently Used (LRU) cache replacement policy\footnote{%
Due to the regular storage format and memory access strides of dense linear
algebra operations such as the considered tensor contractions, this
simplifying assumption does not affect the reliability of the results.
} and sum up the size of all memory regions accessed since an operand's last
use, yielding the access distance. In first instance, we also assume that all
loops surrounding the kernel are somewhere in the middle of their traversal
(i.e., not in their first iteration); this assumption will be lifted later.
\parsum{one kernel only case}
We now describe how to obtain the access distance for each of the operands. The
presented method is general and allows for any combinations of loops and
multiple kernels within the abstract syntax tree (AST), however for the sake of
clarity, we limit the discussion to ASTs that only consist of a series of loops
with a single call to a BLAS kernel at their innermost loop.
\parsum{back-traversal through AST}
For each operand $\mathcal Op$, we examine the algorithm's AST (see
\autoref{sec:alggen}) with the kernel of interest as a starting point. The AST
is traversed backwards until the previous access to $\mathcal Op$ (or the AST's
root) is found, thereby collecting all other operands involved in kernels in the
initially empty set $M$. Going up the AST, three different cases can be
encountered.
\begin{enumerate}
\item {\bf $\mathcal{O}p$ does not vary across the surrounding loop.}
\begin{example}
\sloppy
In algorithm \algname{ca}{gemv}~(\ref*{plt:ai_ibc:ca_gemv}),
repeated below, the operand {\tt$B$[:,:,c]} does not depend on the
surrounding loop's iterator {\tt a}. Hence, $M = \emptyset$ and the
operand's access distance is 0.
\vspace{.5ex}
\centering\scriptsize
\newcommand\contraction{ai_ibc}
\begin{algorithm}{ca}{gemv}
\begin{lstlisting}
for c = 1:$c$
for a = 1:$a$
$C$[a,:,c] += $A$[a,:] $B$[:,:,c]!\hfill!(gemv)
\end{lstlisting}
\end{algorithm}%
\begin{tikzpicture}[scale=.8]
\begin{drawcube}
\sliceA[outer]
\sliceC[outer]
\sliceAC
\end{drawcube}
\node at (1,0,0) {$\pluseq$};
\begin{drawsquare}[(2,0,0)]
\sliceA
\end{drawsquare}
\begin{drawcube}[(3.5,0,0)]
\sliceC
\end{drawcube}
\end{tikzpicture}
\end{example}
$\mathcal{O}p$ refers to the same memory region as in the previous
iteration of the surrounding loop. The back-traversal therefore
terminates and the memory regions collected in $M$ so far determine the
access distance.
\item {\bf $\mathcal Op$ varies across the surrounding loop.}
\begin{example}
In algorithm \algname{ca}{gemv}~(\ref*{plt:ai_ibc:ca_gemv}), the
operand {\tt$A$[a,:]} depends on the surrounding loop's iterator
{\tt a}.
\end{example}
$\mathcal Op$ referred to a different memory region in the previous
iteration of the loop. As a result, it is safe to assume that at least
all memory regions covered by all kernel operands throughout these loops
were accessed since the last access to $\mathcal Op$. Hence, all
operands are added to $M$ and the memory regions are symbolically joined
along the dimensions the loop iterated over.
\addtocounter{example}{-1}
\begin{example}[continued]
\sloppy
The algorithm's kernel operates on {\tt$A$[a,:]}, {\tt$B$[:,:,c]},
and {\tt$C$[a,:,c]}. Joining these operands across the index {\tt a}
yields the memory regions {\tt$M = \{A\text{[:,:]}, B\text{[:,:,c]},
C\text{[:,:,c]}\}$}.
\end{example}
Since a previous access to $\mathcal Op$ was not yet detected, the
traversal proceeds by going up one level in the AST, and applying the
method recursively: the surrounding loop now takes the role of the
starting node and we look for a previous access $\mathcal Op$ joined
across this loop.
\addtocounter{example}{-1}
\begin{example}[continued]
The back-traversal now looks for a previous access to {\tt$A$[:,:]}
({\tt$A$[a,:]} joint across {\tt a}) on the second-innermost loop.
This time, the region is independent of the surrounding loop's
iterator {\tt c}; therefore, in this second step, case 1. above
applies and the access distance is computed from the previously
collected set {\tt $M = \{A\text{[:,:]}, B\text{[:,:,c]},
C\text{[:,:,c]}\}$}.
\end{example}
\item {\bf The parent node is the AST's root.}
\begin{example}
In algorithm \algname{ca}{gemv}~(\ref*{plt:ai_ibc:ca_gemv}), the
operand {\tt$C$[a,:,c]} depends on both of the surrounding loops'
iterators {\tt a} and {\tt c}. Therefore, the back-traversal
encounters case 2. above in both its first and second step, joining
the kernel's operands {\tt$A$[a,:]}, {\tt$B$[:,:,c]}, and
{\tt$C$[a,:,c]} across first {\tt a} and then {\tt c}, yielding
{\tt$M = \{A\text{[:,:]}, B\text{[:,:,:]}, C\text{[:,:,:]}\}$}. In
the third step of the back-traversal, the outermost loop is already
the starting point --- the algorithm's root is reached.
\end{example}
In this case, the considered region is accessed only once (and for the
first time). Since we do not know how the contraction is used (within a
surrounding program), we can generally not make any assertions on the
access distance. For the purpose of this paper, in which we execute the
contraction repeatedly to measure its performance, however, we assume
that no further memory regions were loaded since the last invocation of
the contraction --- i.e., we compute the access distance from the
previously collected memory regions in $M$.
\end{enumerate}
\parsum{generate setup}
Based on the such obtained access distances for each operand of an algorithm's
kernel, we now construct a list of memory accesses that emulates the accesses
within the algorithm prior to the kernel's execution. This list consists of
accesses to the kernel's operands, interleaved with accesses to remote memory
regions, in order to flush portions of the cache corresponding to the access
distances: First, we access the operand with the largest access distance, then
a remote region that accounts for the difference to the next smaller access
distance, followed by the next operand, and so on until the operands with the
smallest access distance followed by a remote access of this size. If the
access distances to the first operands in this list are larger than $\frac54$
times the cache size, the list is truncated down to this limit at the front.
\begin{example}
For algorithm \algname{ca}{gemv}~(\ref*{plt:ai_ibc:ca_gemv}), the following
table summarizes the operands $O$, their sizes $s$, the corresponding
collections $M$ and the implicated access distances $d$ for contraction
sizes $a = b = c = 400$ and $i = 8$ (all sizes in doubles = 8 bytes):
\begin{center}
\small
\vspace{-1ex}
\setlength{\tabcolsep}{1em}
\begin{tabular}{lrlr}
\toprule
$O$ &$s$ &$M$ &$d$ \\
\midrule
{\tt$B$[:,:,c]} &$3{,}200$ &$\emptyset$ &$0$\\
{\tt$A$[a,:]} &$8$ &{\tt$\{A\text{[:,:]}, B\text{[:,:,c]}, C\text{[:,:,c]}\}$} &$166{,}400$\\
{\tt$C$[a,:,c]} &$400$ &{\tt$\{A\text{[:,:]}, B\text{[:,:,:]}, C\text{[:,:,:]}\}$} &$65{,}283{,}200$\\
\bottomrule
\end{tabular}
\end{center}
From these distances, we get the following list of memory accesses as a
setup for the \gemv-kernel, where $[s]$ correspond to remote memory accesses
of size $s$:
\begin{center}
{\tt$C$[a,:,c]},
$[65{,}116{,}792]$,
{\tt$A$[a,:]},
$[163{,}200]$,
{\tt$B$[:,:,c]}.
\end{center}
Note, that remote accesses are not directly of the access distance's sizes;
however, this size is reached for each operand as the sum of the sizes of
all accesses to its right in this list. (e.g., the access distances of
{\tt$A$[a,:]} is reached as {\tt$163{,}200 +
\mathrm{sizeof}(B\text{[:,:,c]}) = 166{,}400$}).
Now, the largest access distance is at $65{,}283{,}200$ considerably larger
than $983{,}040$ ($\frac54$ times the cache size of $\frac{6\mathrm{MB}}8 =
786{,}432$ doubles). Hence, the list is cut at this size, yielding the
final setup for this algorithm's micro-benchmark:
\begin{center}
$[816{,}632]$,
{\tt$A$[a,:]},
$[163{,}200]$,
{\tt$B$[:,:,c]}.
\end{center}
\end{example}
The thus obtained benchmark, consisting of the setup followed by the kernel
invocation, is once more executed ten times. The median of the kernel run-times
of these ten benchmarks is then used to compute our second execution time
estimate.
\parsum{improvements and shortcomings of the new estimates}
In \autoref{fig:pred_step+s}, we present the flops/cycle performance of our new
estimates. These predictions are much closer to the measured performance
(\autoref{fig:pred_meas}) than the first rough estimates
(\autoref{fig:pred_step}). For several algorithms (such as
\algname{ic}{ger}~(\ref*{plt:ai_ibc:ic_ger}), \autoref{algs:ai_ibc}), the error
is already within a few percent; for many others instead, the predictions are
still off. In particular, the performance of some algorithms --- for instance,
\algname{bi}{ger}~(\ref*{plt:ai_ibc:bi_ger}) (see \autoref{algs:ai_ibc}) --- is
underestimated; this is due to the fact that based on the access distance,
certain operands are placed out of cache, while in practice they are (partially)
brought into cache through either prefetching or because they share cache-lines
across loop iterations. We address this discrepancy by further refining our
micro-benchmarks.
\subsection{Cache Prefetching}
\parsum{explanation prefetching}
In the considered type of tensor contraction algorithms, prefetching of operands
or sharing of cache lines across loop iterations occur frequently.
\begin{example}
In algorithm \algname{bi}{ger}~(\ref*{plt:ai_ibc:bi_ger}), the operand
{\tt$A$[:,i]} points to a different memory location in each iteration of the
inner loop across {\tt i}. However, these vectors-operands are consecutive
in memory; thus, when reaching the end of {\tt$A$[:,i]}, the prefetcher will
likely already load the next memory elements, which constitute {\tt$A$[:,i]}
in the next iteration. Likewise, operand {\tt$B$[i,b,:]} varies across
inner loop iterations; however, since this loop iterates over the region's
first dimension {\tt i}, 8 consecutive operands\footnote{%
The cache-line size is $64\mathrm{B} = 8$ doubles.
} {\tt$B$[i,b,:]} will occupy the same cache-line.
\end{example}
\parsum{detect prefetching}
Such prefetching situations occur when a certain set of conditions are met,
namely:
\begin{enumerate}
\item the operand varies across the directly surrounding loop, and
\item the iterator of this loop indexes
\begin{itemize}
\item either the first dimension of the operand,
\item or its second dimension, while the first is accessed entirely,
or fits in a single cache-line.
\end{itemize}
\end{enumerate}
As part of our AST-based algorithm analysis, such conditions are tested; when
both of them are met, we can use a slight modification of the previously
introduced method to compute the {\em prefetch distance}, i.e., how long ago the
prefetching occurred. These prefetch distances are then integrated into the
micro-bench\-mark's setup list just like the access distances, only that for
prefetch accesses the access is limited to one cache-line along an operand's
first dimension.
\begin{example}
In algorithm \algname{ca}{gemv}~(\ref*{plt:ai_ibc:ca_gemv}), for which we
explicitly constructed the setup list in the previous section, both operands
{\tt$A$[a,:]} and {\tt$C$[a,:,b]} meet both of the prefetching conditions:
1. they vary across the surrounding loop iterator {\tt a} and 2. {\tt a}
indexes their first dimensions (sharing of cache-lines). As a result, their
prefetch distances are $0$ and the prefetching access will access the entire
operands since their extension along the first, contiguously stored
dimension is $1$. Since the remaining operand {\tt$B$[:,:,c]} has an access
distance of $0$, all operands are now accessed immediately before the kernel
invocation; the setup list is reduced to
\begin{center}
{\tt$C$[a,:,c]},
{\tt$A$[a,:]},
{\tt$B$[:,:,c]}.
\end{center}
(Since this setup consists only of accesses to the operands, it becomes
redundant in out micro-benchmarks, because each of the ten repetitions will
already touch all operands for the next repetition; hence, in such a case,
we omit the setup altogether.)
\end{example}
\parsum{prefetching results}
Now accounting for prefetching, we obtain the performance estimates shown in
\autoref{fig:pred_step+sp}. Here, several algorithms,
e.g.~\algname{ba}{gemv}~(\ref*{plt:ai_ibc:ba_gemv}), are estimated closer to
their measured performance. However, several other algorithms, including
\algname{ca}{gemv}~(\ref*{plt:ai_ibc:ca_gemv}) are overestimated in performance
(i.e., underestimated in execution time). There are two separate causes for
this discrepancy.
\begin{itemize}
\item In several algorithms, such as
\algname{ca}{gemv}~(\ref*{plt:ai_ibc:ca_gemv}), where prefetching
implicitly happens due to sharing of cache-lines, the prefetcher fails
once a new cache-line is reached.
\item In other algorithms, such as
\algname{bi}{ger}~(\ref*{plt:ai_ibc:bi_ger}), the innermost loop is so
short (here: 8 iterations) that each first iteration of the loop
significantly impacts performance.
\end{itemize}
These two causes are teated in separately in the following sections.
\subsection{Prefetching Failures}
\parsum{prefetching along first tensor dimension fails every 8}
For those algorithms in which certain operands are identified as prefetched
because they share cache lines across iterations (i.e., the surrounding loop
indexes their first dimension), the CPU would need to prefetch the next
cache-line every 8 iterations (1 cache-line = 8 doubles). However, as a
detailed analysis of hand-instrumented algorithms has shown, the CPU fails to do
so. As a result, in every 8th iteration of the innermost loop, the operand is
not available and the kernel may take significantly longer.
\parsum{new, separate benchmark}
We account for this prefetching-artifact by performing two separate
micro-benchmarks: one simulating the 7 iterations, in which the operand is
available in cache, as before, and one for the 8th iteration, where we account
for the failure to prefetch and eliminate the emulated prefetching from our
setup-list. The prediction for the total execution time is now obtained from
weighting these two benchmark timings according to their number of occurrences
in the algorithm and summing them up.
\begin{example}
In algorithm \algname{ca}{gemv}~(\ref*{plt:ai_ibc:ca_gemv}), the memory
regions of both {\tt$A$[a,:]} and {\tt$C$[a,:,c]}, respectively, share
cache-lines across iterations of the innermost loops over {\tt a}. Hence,
affecting not one but two of the kernel's operands, every 8th iteration the
kernel execution time increases drastically by a factor of about $4.5$. To
account for these ``prefetching failures'', we introduce a second set of
micro-benchmarks, where the prefetching emulating accesses are removed from
the setup list, resulting for $a = b = c = 400$ and $i = 8$, as without
prefetching, in:
\begin{center}
$[816{,}632]$,
{\tt$A$[a,:]},
$[163{,}200]$,
{\tt$B$[:,:,c]}.
\end{center}
\end{example}
\parsum{prefetching failure results}
\autoref{fig:pred_step+spx} shows the predictions obtained after this
improvement. Most noticeably in \algname{ca}{gemv}~(\ref*{plt:ai_ibc:ca_gemv}),
the overestimation of algorithms where iterations share cache-lines are now
corrected.
\subsection{First Loop Iterations}
\parsum{first iteration way off}
The predictions for several algorithms, such as
\algname{ci}{ger}~(\ref*{plt:ai_ibc:ci_ger}), are still severely off, because
the innermost loop of these algorithms is very short (in our example 8
iterations long). In such a case, the predictions are very accurate for all but
the first iteration. Due to vastly different cache preconditions for this first
iteration, however, its performance can be significantly different (in our case,
up to $10\times$ slower). Combined with the low total iteration count, this
results in predictions that are off by a factor of up to $2$.
\parsum{micro-benchmark first iteration separately}
To treat such situations, we introduce separate benchmarks to predict the
performance of the first iteration of the innermost loop (and further loops if
their first iterations account for more than 1\% of the total kernel
invocations). For this purpose, the access distance evaluation method is
slightly modified: instead of the kernel itself, the starting point is now the
loop whose first iteration is considered, and the set $M$ already contains all
of the kernel's memory regions joined across this loop.
\begin{example}
In algorithm \algname{ci}{ger}~(\ref*{plt:ai_ibc:ci_ger}), the innermost
loop over $i$ is in our example only $8$ iterations long. For all but the
first iteration, the operand {\tt$C$[:,:,c]} stays the same, while
{\tt$A$[:,i]} and {\tt$B$[i,:,c]} are prefetched, leading to optimal
conditions for performance. In the first iteration (i.e., the next {\tt c}
iteration) however, {\tt$C$[:,:,c]} refers to a different memory location
and prefetching fails for {\tt$A$[:,i]}, leading to severely lower
performance.
\end{example}
\parsum{weight and sum separate timings}
From these improved access distances, the cache setup and micro-benchmark are
performed just as before. As for the ``prefetching failures'', the prediction
for the total execution time is now obtained from weighting of all relevant
benchmark timings with the corresponding number of occurrences within the
algorithm.
\parsum{results}
In \autoref{fig:pred_pred}, we present the improved performance predictions
obtained from this modification. The performance of all algorithms is now
predicted with satisfying accuracy.
\subsection{Test 1: $C_{abc} = A_{ai} B_{ibc}$, Different Setup}
\label{sec:ai_ibc2}
We commence with the same contraction used as case study in the previous
section, yet with an entirely different setup: the sizes of $a$, $b$, and $c$
are now fixed to 128, while the value of $i$ ranges from 8 to 1{,}024. As
experimental environment, we use a 10-core Intel Ivy Bridge-EP E5-2680 v2
processor running at a frequency of 3.6 GHz (Turbo) and 25 MB of L3 cache. Each
core can execute 8 double precision flops/cycle. The routines for both the
actual measurements and the micro-benchmarks were linked to the Intel Math
Kernel Library (MKL, version 11.0) BLAS implementation.
\hyperref[fig:ai_ibc2]{Figure~\ref*{fig:ai_ibc2}}
contains the performance measurements and the corresponding predictions for all
36 algorithms (see \autoref{algs:ai_ibc}). Although everything, ranging from
the problem size to the machine and BLAS library was changed in this setup, the
predictions are of equivalent quality and our tool correclty determines that the
\gemm-based algorithms (\algname{c}{gemm}~(\ref*{plt:ai_ibc:c_gemm}) and
\algname{b}{gemm}~(\ref*{plt:ai_ibc:b_gemm})) perform best and equally well.
\input{figures/ai_ibc2}
\subsection{Test 2: $C_a = A_{iaj} B_{ji}$, only BLAS-1 and BLAS-2}
\label{sec:noblas3}
For certain contractions (e.g., those involving 1D tensors), \gemm cannot be
used as a compute kernel, and only algorithms based on BLAS-2 or BLAS-1 are
possible. One such scenario is encontered in the contraction $C_a = A_{iaj}
B_{ji}$, for which our generator yields 8 BLAS-based algorithms:
\begin{itemize}
\item 4 \dot-based:
\algname{aj}{dot}~(\ref*{plt:iaj_ji:aj_dot}),
\algname{ja}{dot}~(\ref*{plt:iaj_ji:ja_dot}),
\algname{ai}{dot}~(\ref*{plt:iaj_ji:ai_dot}),
\algname{ia}{dot}~(\ref*{plt:iaj_ji:ia_dot});
\item 2 \axpy-based:
\algname{ij}{axpy}~(\ref*{plt:iaj_ji:ij_axpy}),
\algname{ji}{axpy}~(\ref*{plt:iaj_ji:ji_axpy});
\item 2 \gemv-based (see \autoref{alg:iaj_ji}):
\algname{j}{gemv}~(\ref*{plt:iaj_ji:j_gemv}),
\algname{i'}{gemv}~(\ref*{plt:iaj_ji:i'_gemv}).
\end{itemize}
The measured and predicted performance for these algorithms is shown in
\autoref{fig:iaj_ji}. Our predictions clearly discriminate the fastest
algorithm \algname{j}{gemv}~(\ref*{plt:iaj_ji:j_gemv}) across the board.
Furthermore, the next group of four algorithms is also correctly identified and
the low performance (due to the overhead of the involved matrix-copy operation)
of the second \gemv-based algorithm
\algname{i'}{gemv}~(\ref*{plt:iaj_ji:i'_gemv}) is predicted too.
\input{algs/iaj_ji}
\input{figures/iaj_ji}
\subsection{Test 3: $C_{abc} = A_{ija} B_{jbic}$, Challenging Contraction}
\label{sec:ijb_jcid}
We now turn to a more complex example: $C_{abc} = A_{ija} B_{jbic}$. For this
contraction, we look at a total of 176 different algorithms:
\begin{itemize}
\item 48 \dot-based,
\item 72 \axpy-based,
\item 36 \gemv-based,
\item 12 \ger-based, and
\item 8 \gemm-based:\\
\algname{cj'}{gemm}~(\ref*{plt:ijb_jcid:cj'_gemm}),
\algname{jc'}{gemm}~(\ref*{plt:ijb_jcid:jc'_gemm}),
\algname{ci'}{gemm}~(\ref*{plt:ijb_jcid:ci'_gemm}),
\algname{i'c}{gemm}~(\ref*{plt:ijb_jcid:i'c_gemm}),\\
\algname{bj'}{gemm}~(\ref*{plt:ijb_jcid:bj'_gemm}),
\algname{jb'}{gemm}~(\ref*{plt:ijb_jcid:jb'_gemm}),
\algname{bi'}{gemm}~(\ref*{plt:ijb_jcid:bi'_gemm}),
\algname{i'b}{gemm}~(\ref*{plt:ijb_jcid:i'b_gemm}).
\end{itemize}
All \gemm-based (see \autoref{algs:ijb_jcid}) and several of the \gemv-based
algorithms involve copy operations to ensure that each matrix has a contiguously
stored dimension, as required by the BLAS interface. Once again, we consider a
very challenging scenario where both contracted indices are of size $i = j = 8$
and the free indices $a = b = c$ vary together.
Starting with the predictions, in \autoref{fig:ijb_jcid_pred}, we present the
expected flops/cycle of the 176 algorithms, where BLAS-1 and BLAS-2 algorithms
are grouped by kernel. Even with the copy operations, the \gemm-based
algorithms are the fastest ones. However, within these 8 algorithms, the
performance differs by more than 20\%. Focusing on the \gemm-algorithms, we
compare with corresponding performance measurements\footnote{%
Slow tensor contraction algorithms were stopped before reaching the largest
test-cases by limiting the total measurement time per algorithm to 15
minutes.
} in \autoref{fig:ijb_jcid_meas}. The comparison shows that our predictions
clearly separate the bulk of fast algorithms from the slightly less efficient
ones.
\input{algs/ijb_jcid}
\input{figures/ijb_jcid}
\subsubsection{Multithreading.}
The algorithms considered here can make use of shared memory parallelism by
employing multithreaded BLAS kernels. To focus on the impact of parallelism, we
increase the contracted tensor dimension sizes to $i = j = 32$ and use all 10
cores of the Ivy Bridge-EP CPU with {\sc OpenBLAS}. Performance predictions and
measurements for this setup are presented in \autoref{fig:ijb_jcid10}. Our
predictions correctly separate the three groups of \gemm-based implementations;
moreover, algorithms \algname{i'c}{gemm}~(\ref*{plt:ijb_jcid:i'c_gemm}) and
\algname{i'b}{gemm}~(\ref*{plt:ijb_jcid:i'b_gemm}) (see
\autoref{algs:ijb_jcid}), which reach 60 flops/cycle,\footnote{%
Using 10 cores, the theoretical peak performance is 80 flops/cycle.
} are identified as the fastest. The slowest algorithm
(\algname{jb'}{gemm}~(\ref*{plt:ijb_jcid:jb'_gemm})) on the other hand merely
reaches 20 flops/per cycle. This $3\times$~difference in performance among
\gemm-based algorithms emphasizes the importance of selecting the right
algorithm.
\input{figures/ijb_jcid10}
\subsection{Efficiency Study}
\parsum{compare total prediction time with measurement time}
The ultimate goal of this work is to automatically and quickly select the
fastest algorithm for a given tensor contraction. The experiments presented so
far provide evidence that our automated approach successfully identifies the
fastest algorithm(s). With this last experiment, we investigate the efficiency
of our micro-benchmark-based approach. For this purpose, we use again the
contraction $C_{abc} = A_{ai} B_{ibc}$, with $i = 8$ and varying $a = b = c$.
\hyperref[fig:eff]{Figure~\ref*{fig:eff}} displays the ratio of how much faster
our micro-benchmark is compared to executing the corresponding algorithm. In
general, our prediction proves to be several orders of magnitude faster than the
algorithm itself. At $a = b = c = 1{,}000$, this relative improvement is
smallest for the \gemm-based algorithms~(\ref*{plt:eff:gemm}) at $10^3\times$,
since each \gemm performs a significant portion of the computation; for the
\ger-based algorithms~(\ref*{plt:eff:ger}), it lies between $6\cdot10^3$ and
$10^4\times$ and for the \gemv-based algorithms~(\ref*{plt:eff:gemv}) the gain
is $5\cdot10^5$ to $10^6\times$; finally, the gain for both BLAS-1-based
algorithms~(\ref*{plt:eff:axpy}, \ref*{plt:eff:dot}), where each BLAS-call only
performs a tiny fraction of the contraction, our prediction is between $6$ and
$9$ orders of magnitude faster than the execution.
\input{figures/eff}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 439 |
Monterrico y Hawaii es un espacio natural de Guatemala situado en la costa del Pacífico, en el departamento de Santa Rosa. Aunque forma una unidad natural está compuesto administrativamente por la Reserva Natural Monterrico y el parque nacional Hawaii cada uno con una gestión independiente. Ambas entidades fueron creadas en 1956 y tienen una superficie conjunta de 1000 hectáreas. Este espacio es la reserva de manglares mejor conservada sobre la costa del Pacífico guatemalteco.
Fauna
Entre los mangles anidan muchas aves migratorias como los pelícanos. La laguna de la Palmilla, cerca da la aldea de Monterrico es un lugar ideal para la observación de estas aves. Los manglares son refugio, entre otras especies, para la iguana verde y el cocodrilo, de los cuales existen programas de reproducción.
En las playas de origen volcánico con arenas de color gris oscuro y café anidan las tortugas marinas, parlama y baulas, cuya época de cría se extiende principalmente entre los meses de mayo a septiembre. Las tortugas marinas han progresado notablemente, pues el centro de reproducción de la aldea Hawaii las cuida y protege.
Referencias
Hoteles en Monterrico
Mar y ecología en Monterrico
Áreas protegidas de Guatemala | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 5,884 |
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<td class="colFirst"><code>void</code></td>
<td class="colLast"><span class="typeNameLabel">DefaultRuntimeBuilder.</span><code><span class="memberNameLink"><a href="../../../../../../com/communote/server/core/application/DefaultRuntimeBuilder.html#setApplicationPreparedCallback-com.communote.server.core.bootstrap.ApplicationPreparedCallback-">setApplicationPreparedCallback</a></span>(<a href="../../../../../../com/communote/server/core/bootstrap/ApplicationPreparedCallback.html" title="interface in com.communote.server.core.bootstrap">ApplicationPreparedCallback</a> applicationPreparedCallback)</code>
<div class="block">Set the callback to invoke after basic preparations like reading configurations, setting up
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<td class="colFirst"><code>void</code></td>
<td class="colLast"><span class="typeNameLabel">ApplicationInitializer.</span><code><span class="memberNameLink"><a href="../../../../../../com/communote/server/core/bootstrap/ApplicationInitializer.html#setApplicationPreparedCallback-com.communote.server.core.bootstrap.ApplicationPreparedCallback-">setApplicationPreparedCallback</a></span>(<a href="../../../../../../com/communote/server/core/bootstrap/ApplicationPreparedCallback.html" title="interface in com.communote.server.core.bootstrap">ApplicationPreparedCallback</a> applicationPreparedCallback)</code>
<div class="block">Set the callback to invoke after building the Spring ApplicationContext is completed.</div>
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<td class="colFirst"><code>class </code></td>
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| {
"redpajama_set_name": "RedPajamaGithub"
} | 3,281 |
Home Newswire Will Congress Face Down the Deep State? – Consortiumnews
Will Congress Face Down the Deep State? – Consortiumnews
Jan 31, 2018: 1:48 am
The House Intelligence Committee's vote on Monday to release a memorandum describing alleged malfeasance at the DOJ and the FBI could test constitutional principles, writes Ray McGovern.
By Ray McGovern
With the House Intelligence Committee vote yesterday to release its four-page memorandum reportedly based on documentary evidence of possible crimes by top Justice Department and FBI leaders, the die is cast. Russia-gate and FBI-gate are now joined at the hip.
The coming weeks will show whether the U.S. intelligence establishment (the FBI/CIA/NSA, AKA the "Deep State") will be able to prevent its leaders from being held to account. Past precedent suggests that the cabal that conjured up Russia-gate will not have to pick up a "go-to-jail" card. This, despite the widespread guilt suggested by the abrupt way that several senior-echelon DOJ and FBI rats have already jumped ship. Not to mention the manner in which FBI Deputy Director Andrew McCabe, was unceremoniously pushed overboard yesterday, after Director Christopher Wray was given a look at the extra-legal capers described in the House Intelligence Committee memorandum.
Granted, at first glance Deep State's efforts to undercut candidate Donald Trump at first seem so risky and audacious as to be unbelievable. By now, though, Americans should be able to wrap their heads around, one, the dire threat that outsider Trump was seen to be posing to the Deep State and to the ease with which it held sway under President Barack Obama; and, two, expected immunity from prosecution if Deep State crimes were eventually discovered after the election, since "everybody knew" Hillary Clinton was going to win. Oops.
Accountability This Time?
There seems to be an outside chance, this time, that the culprits who did actually interfere in the 2016 presidential election in an effort to make sure Trump could not win, and then did all in their power to sabotage him after he his electoral victory, will be held to account by unusually feisty members of the House. It is abundantly clear that members of the House Intelligence and House Judiciary Committees are now in possession of the kind of unambiguous, first-hand… | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 3,465 |
Bharat Road Network Ltd (BRNL) – Financial and Strategic SWOT Analysis Review provides you an in-depth strategic SWOT analysis of the company's businesses and operations. The profile has been compiled by GlobalData to bring to you a clear and an unbiased view of the company's key strengths and weaknesses and the potential opportunities and threats. The profile helps you formulate strategies that augment your business by enabling you to understand your partners, customers and competitors better.
Bharat Road Network, Ltd. (BRNL) is a provider of transportation infrastructure construction services. The company's business operation is based on build–operate–transfer (BOT) model. It focuses on development, implementation, operation, and maintenance of roads and highways. The company undertakes highway construction projects, including construction of highways, roads, bridges and express freeways. BRNL also offers project building, design and development, valuation, financial modeling, claim management, debt syndication, concession management and project advisory services to its clients. The company caters is services to various state governments. It conducts its business operations primarily in India. BRNL is headquartered in Kolkata, West Bengal, India. | {
"redpajama_set_name": "RedPajamaC4"
} | 5,640 |
Шато дьо Куси () представлява останалите до днес руини от един от най-значимите средновековни феодални замъци в Европа. Намира се в община Куси льо Шато Офрик, департамент Ен, Франция. По време на Първата световна война през 1917 г. огромният кръгъл донжон на шатото понася големи щети, когато германски военни сили го взривяват. По онова време се е издигал на 60 метра височина и е бил най-високият донжон във Франция.
Външни препратки
Писмено описание на замъка
Датабазата за шато дьо Куси на Министерството на културата във Франция
Снимки от Министреството на културата
Колекция стари пощенски картички от Куси
Замъци във Франция
О дьо Франс | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 6,876 |
In the case of Frank Underwood, his goals and objectives are very easy to define – increase and build power. Everything he does, every action he takes, is evaluated against that defined goal. There's nothing ambiguous about it. In marketing, we often let ambiguity or subjectivity control decision making. How many times have you heard your boss say, "I don't like blue (or pick a color)." There's nothing measureable about that. In fact, there's nothing actionable about it; it's driven by personal preferences. With clear defined goals you can remove the ambiguity and work toward something tangible.
Whether it's Zoe Barnes or Raymond Tusk, our power-loving "protagonist" had a strategy in how he used these relationships to further his objectives. "If you want to earn my loyalty, then you have to offer yours in return. And if we can agree to that – well, you're a man with imagination." - Frank Underwood to Raymond Tusk.
Prior to making any decisions, there was thought and evaluation – does this fit with the strategy? Again, this helps alleviate subjectivity and, just as important, keeps concepts and tactics focused on the prize because everything is evaluated and measured against the strategy. Strategies may need to change, but they always serve as the guide.
While the strategy serves as a guide, there needs to be freedom to explore different solutions and designs in order to achieve the desired outcomes. Frank will try something different if he thinks it will achieve his goal. From his strong-armed tactics in the Senate to his back-channel conversations with Feng, he doesn't shy away from the unconventional. From a marketing perspective, you need to be open to exploring different designs, media channels, etc. The great thing is that nothing is permanent today.
Try a display ad; if no one clicks after an appropriate test, change it. Want to experiment with a new home page design? Run a small A/B test. You can even use online efforts to help "experiment" for what will ultimately guide print campaigns. The great thing about analytics and testing is that we no longer have to rely on "instincts." The user can tell us directly what they like and what they don't like based on clicking, not clicking, bouncing, etc. You may think that home page design doesn't work, but you may find out the user's higher engagement metrics will tell you something different.
No one would accuse Frank of subscribing to Ron Popeil's "set and forget it" approach. Just ask Ms. Barnes. When that relationship no longer produced results, and in fact started to become a negative, he … shall we say … shoved it aside. Evaluate campaigns and other efforts regularly. If they're working, continue moving forward and maybe even increasing the spend for the impactful ROI. If they're not, don't wait to see how it goes. Don't be afraid to pull the plug and reallocate dollars to efforts that are producing measureable and positive impactful results. After all, as Frank once said, "Treading water is the same as drowning for people like you and me." Now, just imagine if Frank Underwood were your director of sales and marketing. | {
"redpajama_set_name": "RedPajamaC4"
} | 6,562 |
Q: Can we record screen in ionic I am new to ionic and I want to create something like screen view recorder for both iOS and android platform. I searched a lot but was unable to find any cordova plugin for screen recording. I just want to know, what I am asking is feasible or not??
A: For Android use ionic-screen-recorder.
For Ios use cordova-replay.
Use these two Cordova plugins and you can record the screen.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 7,818 |
package es.upm;
/**
* @author Alberto Sánchez
* Antonio San Agustín
* Alberto Campos
*/
public class Lanzamiento {
//Se crean los lanzadores con son respectivas variables para calculos posteriores
//Añadir Lanzadores
Lanzador1 lanzador1 = new Lanzador1();
Lanzador2 lanzador2 = new Lanzador2();
Lanzador3 lanzador3 = new Lanzador3();
Lanzador4 lanzador4 = new Lanzador4();
Lanzador5 lanzador5 = new Lanzador5();
Lanzador6 lanzador6 = new Lanzador6();
Lanzador7 lanzador7 = new Lanzador7();
Lanzador8 lanzador8 = new Lanzador8();
double aLanzador1 = 0;
double aLanzador2 = 0;
double aLanzador3 = 0;
double aLanzador4 = 0;
double aLanzador5 = 0;
double aLanzador6 = 0;
double aLanzador7 = 0;
double aLanzador8 = 0;
double contador1 = 0;
double contador2 = 0;
double contador3 = 0;
double contador4 = 0;
double contador5 = 0;
double contador6 = 0;
double contador7 = 0;
double contador8 = 0;
/**
* Este metodo se encarga de redondear a 2 decimales
* @param numero Numero que queremos redondear
* @return
*/
public double Redondear(double numero){
return Math.rint(numero*100)/100;
}
/**
* Este metodo se encarga de imprimir por pantalla los alcances de cada lanzador
* ademas de el jugador que ha hecho el mejor lanzamiento
* @throws InterruptedException
*/
public void getAlcances() throws InterruptedException{
double mayorDistancia = 0;
double segundo=0;
double tercero=0;
//Se inicia el lanzamiento y aportamos las distancias de tiro
//Añadir Lanzadores
aLanzador1 = lanzador1.medirAlcance(lanzador1.getVLanzador(), lanzador1.getAltura());
if(aLanzador1>60.0){
System.out.println("Vitezslay Vesely: " + Redondear(aLanzador1) + " m");}
else{System.out.println("Vitezslay Vesely: " + "Nulo" );}
Thread.sleep(1000);
aLanzador2 = lanzador2.medirAlcance(lanzador2.getVLanzador(), lanzador2.getAltura());
if(aLanzador2>60.0){
System.out.println("Tero Pitkamaki: " + Redondear(aLanzador2) + " m");}
else{System.out.println("Tero Pitkamaki: " + "Nulo" );}
Thread.sleep(1000);
aLanzador3 = lanzador3.medirAlcance(lanzador3.getVLanzador(), lanzador3.getAltura());
if(aLanzador3>60.0){
System.out.println("Dmitri Tarabin: " + Redondear(aLanzador3) + " m");}
else{System.out.println("Dmitri Tarabin: " + "Nulo" );}
Thread.sleep(1000);
aLanzador4 = lanzador4.medirAlcance(lanzador4.getVLanzador(), lanzador4.getAltura());
if(aLanzador4>60.0){
System.out.println("Julius Yego: " + Redondear(aLanzador4) + " m");}
else{System.out.println("Julius Yego: " + "Nulo" );}
Thread.sleep(1000);
aLanzador5 = lanzador5.medirAlcance(lanzador5.getVLanzador(), lanzador5.getAltura());
if(aLanzador5>60.0){
System.out.println("Roman Avramenko: " + Redondear(aLanzador5) + " m");}
else{System.out.println("Roman Avramenko: " + "Nulo" );}
Thread.sleep(1000);
aLanzador6 = lanzador6.medirAlcance(lanzador6.getVLanzador(), lanzador6.getAltura());
if(aLanzador6>60.0){
System.out.println("Antti Ruuskanen: " + Redondear(aLanzador6) + " m");}
else{System.out.println("Antti Ruuskanen: " + "Nulo" );}
Thread.sleep(1000);
aLanzador7 = lanzador7.medirAlcance(lanzador7.getVLanzador(), lanzador7.getAltura());
if(aLanzador7>60.0){
System.out.println("Andreas Thorkildsen: " + Redondear(aLanzador7) + " m");}
else{System.out.println("Andreas Thorkildsen: " + "Nulo" );}
Thread.sleep(1000);
aLanzador8 = lanzador8.medirAlcance(lanzador8.getVLanzador(), lanzador8.getAltura());
if(aLanzador8>60.0){
System.out.println("Ihab Abdelrahman El Sayed: " + Redondear(aLanzador8) + " m");}
else{System.out.println("Ihab Abdelrahman El Sayed: " + "Nulo" );}
Thread.sleep(1000);
double [] alcances = {aLanzador1, aLanzador2, aLanzador3,
aLanzador4, aLanzador5, aLanzador6
, aLanzador7, aLanzador8}; //Array que contiene las distancias de lanzamiento
//Elige el mayor alcance del array
int izquierda=1;
int derecha=alcances.length;
do{
for(int i=alcances.length-1;i>0;i--){
if(alcances[i-1]>alcances[i]){
double aux=alcances[i];
alcances[i]=alcances[i-1];
alcances[i-1]=aux;
}
}
izquierda=izquierda+1;
for(int j=1;j<alcances.length;j++){
if(alcances[j-1]>alcances[j]){
double a=alcances[j];
alcances[j]=alcances[j-1];
alcances[j-1]=a;
}
}
derecha = derecha-1;
}while(derecha >= izquierda);
for(int i=alcances.length-1;i>0;i--){
mayorDistancia=alcances[7];
segundo=alcances[6];
tercero=alcances[5];
}
System.out.println("");
//Elegimos el vencedor
if(aLanzador1 == mayorDistancia){
System.out.println("Vitezslay Vesely ha conseguido la medalla de oro con una distancia de: " + Redondear(aLanzador1) + " m");
}
if(aLanzador1 == segundo){
System.out.println("Vitezslay Vesely ha conseguido la medalla de plata con una distancia de: " + Redondear(aLanzador1) + " m");
}
if(aLanzador1 == tercero){
System.out.println("Vitezslay Vesely ha conseguido la medalla de bronce con una distancia de: " + Redondear(aLanzador1) + " m");
}
if(aLanzador2 == mayorDistancia){
System.out.println("Tero Pitkamaki ha conseguido la medalla de oro con una distancia de: " + Redondear(aLanzador2) + " m");
}
if(aLanzador2 == segundo){
System.out.println("Tero Pitkamaki ha conseguido la medalla de plata con una distancia de: " + Redondear(aLanzador2) + " m");
}
if(aLanzador2 == tercero){
System.out.println("Tero Pitkamaki ha conseguido la medalla de bronce con una distancia de: " + Redondear(aLanzador2) + " m");
}
if(aLanzador3 == segundo){
System.out.println("Dmitri Tarabin ha conseguido la medalla de plata con una distancia de: " + Redondear(aLanzador2) + " m");
}
if(aLanzador3 == mayorDistancia){
System.out.println("Dmitri Tarabin ha conseguido la medalla de oro con una distancia de: " + Redondear(aLanzador3) + " m");
}
if(aLanzador3 == tercero){
System.out.println("Dmitri Tarabin ha conseguido la medalla de bronce con una distancia de: " + Redondear(aLanzador2) + " m");
}
if(aLanzador4 == mayorDistancia){
System.out.println("Julius Yego ha conseguido la medalla de oro con una distancia de: " + Redondear(aLanzador4) + " m");
}
if(aLanzador4 == segundo){
System.out.println("Julius Yego ha conseguido la medalla de plata con una distancia de: " + Redondear(aLanzador4) + " m");
}
if(aLanzador4 == tercero){
System.out.println("Julius Yego ha conseguido la medalla de bronce con una distancia de: " + Redondear(aLanzador4) + " m");
}
if(aLanzador5 == mayorDistancia){
System.out.println("Roman Avramenko ha conseguido la medalla de oro con una distancia de: " + Redondear(aLanzador5) + " m");
}
if(aLanzador5 == segundo){
System.out.println("Roman Avramenko ha conseguido la medalla de plata con una distancia de: " + Redondear(aLanzador5) + " m");
}
if(aLanzador5 == tercero){
System.out.println("Roman Avramenko ha conseguido la medalla de bronce con una distancia de: " + Redondear(aLanzador5) + " m");
}
if(aLanzador6 == mayorDistancia){
System.out.println("Antti Ruuskanen ha ha conseguido la medalla de oro con una distancia de: " + Redondear(aLanzador6) + " m");
}
if(aLanzador6 == segundo){
System.out.println("Antti Ruuskanen ha ha conseguido la medalla de plata con una distancia de: " + Redondear(aLanzador6) + " m");
}
if(aLanzador6 == tercero){
System.out.println("Antti Ruuskanen ha ha conseguido la medalla de bronce con una distancia de: " + Redondear(aLanzador6) + " m");
}
if(aLanzador7 == mayorDistancia){
System.out.println("Andreas Thorkildsen ha conseguido la medalla de oro con una distancia de: " + Redondear(aLanzador7) + " m");
}
if(aLanzador7 == segundo){
System.out.println("Andreas Thorkildsen ha conseguido la medalla de plata con una distancia de: " + Redondear(aLanzador7) + " m");
}
if(aLanzador7 == tercero){
System.out.println("Andreas Thorkildsen ha conseguido la medalla de bronce con una distancia de: " + Redondear(aLanzador7) + " m");
}
if(aLanzador8 == mayorDistancia){
System.out.println("Ihab Abdelrahman El Sayed ha conseguido la medalla de oro con una distancia de: " + Redondear(aLanzador8) + " m");
}
if(aLanzador8 == segundo){
System.out.println("Ihab Abdelrahman El Sayed ha conseguido la medalla de plata con una distancia de: " + Redondear(aLanzador8) + " m");
}
if(aLanzador8 == tercero){
System.out.println("Ihab Abdelrahman El Sayed ha conseguido la medalla de bronce con una distancia de: " + Redondear(aLanzador8) + " m");
}
}
/**
* Este metodo junto con lanzar1000 se encarga de calcular el porcentaje de
* victorias entre los jugadores, en particular, añade el numero de victorias a un
* contador para que luego con el metodo lanzar1000 podamos ejecutarlo las veces
* que queramos y calcular asi el porcentaje que tienen los lanzadores de vencer
*/
public void getVencedores(){
double mayorDistancia = 0;
aLanzador1 = lanzador1.medirAlcance(lanzador1.getVLanzador(), lanzador1.getAltura());
aLanzador2 = lanzador2.medirAlcance(lanzador2.getVLanzador(), lanzador2.getAltura());
aLanzador3 = lanzador3.medirAlcance(lanzador3.getVLanzador(), lanzador3.getAltura());
aLanzador4 = lanzador4.medirAlcance(lanzador4.getVLanzador(), lanzador4.getAltura());
aLanzador5 = lanzador5.medirAlcance(lanzador5.getVLanzador(), lanzador5.getAltura());
aLanzador6 = lanzador6.medirAlcance(lanzador6.getVLanzador(), lanzador6.getAltura());
aLanzador7 = lanzador7.medirAlcance(lanzador7.getVLanzador(), lanzador7.getAltura());
aLanzador8 = lanzador8.medirAlcance(lanzador8.getVLanzador(), lanzador8.getAltura());
double [] alcances = {aLanzador1, aLanzador2, aLanzador3,
aLanzador4, aLanzador5, aLanzador6
, aLanzador7, aLanzador8}; //Array que contiene las distancias de lanzamiento
//Elige el mayor alcance del array
for (int i=0; i<alcances.length; i++){
if(alcances[i] > mayorDistancia){
mayorDistancia = alcances[i];
}
}
//Añadimos victoria a su respectivo contador
if(aLanzador1 == mayorDistancia){
contador1++;
}
if(aLanzador2 == mayorDistancia){
contador2++;
}
if(aLanzador3 == mayorDistancia){
contador3++;
}
if(aLanzador4 == mayorDistancia){
contador4++;
}
if(aLanzador5 == mayorDistancia){
contador5++;
}
if(aLanzador6 == mayorDistancia){
contador6++;
}
if(aLanzador7 == mayorDistancia){
contador7++;
}
if(aLanzador8 == mayorDistancia){
contador8++;
}
}
/*
/**
* Ejecutamos el metodo getVencedores tantas veces como queramos,
* en este caso 1000, para calcular la probabilidad en porcentaje de
* vencer sobre los demas
*/
public void lanzar1000(){
for (int i=0; i<1000; i++){
getVencedores();
}
System.out.println("Vitezslay Vesely: " + Redondear((contador1/1000)*100) + "% (opciones de ganar)");
System.out.println("Tero Pitkamaki: " + Redondear((contador2/1000)*100) + "% (opciones de ganar)");
System.out.println("Dmitri Tarabin: " + Redondear((contador3/1000)*100) + "% (opciones de ganar)");
System.out.println("Julius Yego: " + Redondear((contador4/1000)*100) + "% (opciones de ganar)");
System.out.println("Roman Avramenko: " + Redondear((contador5/1000)*100) + "% (opciones de ganar)");
System.out.println("Antti Ruuskanen: " + Redondear((contador6/1000)*100) + "% (opciones de ganar)");
System.out.println("Andreas Thorkildsen: " + Redondear((contador7/1000)*100) + "% (opciones de ganar)");
System.out.println("Ihab Abdelrahman El Sayed: " + Redondear((contador8/1000)*100) + "% (opciones de ganar)");
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 8,960 |
An 'Ugly' UCD building explained - The Restaurant
Image Credit: Doireann de Courcy Mac Donnell
Sophia Finucane tells us why Walker's 1969 Restaurant Building is worth a second look.
By Sophia Finucane | Oct 5 2020
The UCD restaurant building, completed in 1969 by architect Ronnie Walker of Scott Tallon Walker, was awarded the RIAI Gold Medal for 1968-1970.
Officially named the Gerard Manley Hopkins building, the Restaurant building is often seen as ugly converse to the stark white and silver curves found in newer campus buildings. Its exposed grey concrete is often described as 'oppressive,' and geometric shapes perceived as dated. Despite comments that it is 'nothing special' or even a little depressing, it has a fascinating history and an understated beauty both inside and out.
the idealism behind the design… was meant to give students freedom to interact and share ideas
In the documentary Talking to My Father, Walker's son Simon, an architect himself teaching in the UCD School of Architecture, explains how the restaurant building "originally had a completely open floor plan," and that "the idealism behind the design… was meant to give students freedom to interact and share ideas". At a squint, the original skeleton with its large, surrounding windows and wall-less chamber is still visible. Being split into private franchises has completely eradicated this feeling, but also the political sentiment behind it of community and collaboration. Now, garishly coloured signs compete for students' commerce and distract from the structure underneath. Notably, the recent ownership of these franchises by the controversial Aramark, and now Gather & Gather fosters the sense of dislike towards the restaurant as a 'place', and seems to be the nail in the coffin of Walker's vision.
Those less in love with brutalism than its remaining fans like Simon Walker, a supporter of DOCOMOMO (International Committee for Documentation and Conservation of Buildings, Sites and Neighbourhoods of the Modern Movement), may argue that the building was never beautiful, even pre-franchise, but a consideration of why it is successful architecture may sway some. The building was built to function, and that is so clear in its structure. The exposed concrete bears no façade. This in itself speaks to a certain mindset those in university may possess: a desire not to waste materials in such a wasteful world, or a desire for a blank canvas for which to place ideas on. Frankly, there is no paint to strip off in the Irish climate and require labour to fix. Outside, large overhangs protect students from rain to move freely around the vast campus. The inner structure creates a magnificent uninterrupted space, which accommodates eating, working space and also has the potential to be turned into a makeshift lecture theatre at a moment's notice. Beautifully crafted sliding screens were originally installed to allow the space to adapt easily, which have since been removed. Surely there is little more a university needs.
The restaurant building in UCD was built for a modern, politically-forward university campus in a quickly developing independent and equal Ireland. The beauty lies therein, and the architectural style would no doubt suit a new building required for a similar function today.
More Art & Architecture
Ugly UCD building: The James Joyce Library
Interior Design in 2020
The Royal Institute of the Architects of Ireland Conference 2020: a success
An 'Ugly' UCD Building Explained: Tierney | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 9,475 |
Zielona Góra – ośrodek przemysłu włókienniczego (głównie wełnianego, tkanin dekoracyjnych – wykładziny podłogowe, tkanin techn.), ponadto przemysł precyzyjny, środków transportu (wagony), spożywczy, meblarski, elektrociepłownia; węzeł drogowy i kolejowy; port lotniczy.
Żagań – przemysł włókienniczy, wyrobów metalowych powszechnego użytku lodówki (zamrażarki), spożywczy, drzewny; węzeł kolejowy.
Żary – przemysł włókienniczy (m.in. dywany), odzieżowy, maszynowy, elektrotechniczny, drzewny, materiałów budowlanych; huta szkła (szyby samochodowe), browar, węzeł kolejowy.
Sulęcin – przemysł włókienniczy, spożywczy, maszynowy, ceramiczny, drzewny; węzeł kolejowy i drogowy.
Nowa Sól – przemysł metalurgiczny, lniarski, maszynowy, spożywczy, wytwórnia nici, rzeczna stocznia remontowa, port rzeczny (Odra), węzeł kolejowy.
Lubsko – zakłady produkcji materiałów: budowlanych, obuwniczych, spożywczych, drzewnych.
Jasień – fabryka maszyn budowlanych, wytwórnia mebli.
Gospodarka i przemysł w Żarach
Żary są znaczącym ośrodkiem przemysłowym. Położenie w pasie przygranicznym ma duży wpływ na rozwój gospodarczy miasta. Bogate tradycje przemysłowe wraz z korzystnymi warunkami rozwoju, stworzonymi przez władze Żar, sprawiły, że żarskie firmy szybko dostosowały się do nowych warunków ekonomicznych. Coraz większą rolę odgrywa w Żarach przemysł drzewny, elektryczny, metalowy oraz materiałów budowlanych. W wiele żarskich firm zainwestowali obcokrajowcy, korzystając z wykwalifikowanej kadry pracowniczej. Zainwestowany kapitał zagraniczny to do tej pory około 350 mln dolarów. Żary zajmują 26. miejsce wśród najbardziej atrakcyjnych dla inwestorów miast powiatowych w rankingu sporządzonym przez Instytut Badań nad Gospodarką Rynkową. W 2005 r. zajęły 15 miejsce w rankingu gmin miejskich ogłoszonym przez gazetę "Rzeczpospolita".
Najwięksi inwestorzy zagraniczni w Żarach:
Kronopol Sp. z o.o. – płyty wiórowe OSB, panele, balaty itp. – kapitał szwajcarski
Sekurit Saint-Gobain HanGlas Sp. z o.o.- producent szyb samochodowych – kapitał francuski
Pol-Orsa i APO – Tessile Sp. z o.o. – materiały motoryzacyjne – kapitał włoski
Probet-Dasag Sp. z o.o. – wyroby terazzo i z kamienia naturalnego – kapitał niemiecki
MK Systemy Kominowe Sp. z o.o. – kominy z blachy kwasoodpornej – kapitał niemiecki
Spomasz S.A. – konstrukcje stalowe – maszyny rolnicze – kapitał holenderski
Magnaplast Sp. z o.o. – rury plastikowe – kapitał niemiecki
Mozaikę firm zagranicznych uzupełniają polskie firmy, takie jak: Relpol S.A. – producent przekaźników – notowany na warszawskiej giełdzie, HART SM – producent szkła hartowanego dla AGD, kabin prysznicowych, firma odzieżowa LUSATIA NOBLE MODE, DEKORA – producent tkanin obiciowych, mebli, okien plastikowych, Poli-Eco – producent listew plastikowych, stolarki plastikowej, aluminiowej i drewnianej, MAGOREX- producent wyrobów ze stali kwasoodpornej i aluminium, oraz wiele innych.
Żary są znaczącym ośrodkiem przemysłowym. Silne tradycje przemysłu tekstylnego sprawiły, że po wojnie miasto zyskało przydomek "Manchesteru zachodniej Polski". Dzisiaj o obliczu Żar decyduje położenie w pasie przygranicznym. W pobliżu miasta (20–40 km) znajdują się polsko-niemieckie drogowe przejścia graniczne w Olszynie, Łęknicy i Przewozie oraz kolejowe przejście w Forście.
"Kręgosłupem" żarskiej gospodarki są firmy małe i średnie. W 1999 r. w mieście było zarejestrowanych 3,8 tys. podmiotów gospodarczych, w tym ponad 120 spółek z ograniczoną odpowiedzialnością i spółek akcyjnych. Większość to firmy usługowe i handlowe.
Na terenie gminy Żagań najbardziej rozwinięty jest przemysł ceramiczno-budowlany i włókienniczy.
Największe zakłady przemysłowe regionu to (podano w porządku alfabetycznym, uwzględniając lokalizację):
GNiAZWRSP – Ferma Tuczu Trzody w Chotkowie,
Gozdnickie Zakłady Ceramiki Budowlanej w Gozdnicy,
VITROSILICON S.A. w Iłowej,
Zakłady Tkanin Technicznych "ESKORD" S.A. w Iłowej,
Przedsiębiorstwo Wielobranżowe "INSTEX" Sp. z o.o. w Trzebowie,
Huta Szkła w Wymiarkach,
Fabryka Wyrobów Dzianych "TEXTIL – DEKOR" Sp. z o.o., w Żaganiu,
Fabryka Wyrobów Wełnianych "POLTOPS" Sp. z o.o., w Żaganiu,
LUSATIA NOBLE MODE Oddział C w Żaganiu (dawne Zakłady Przemysłu Odzieżowego).
Zakład Wielobranżowy – Galwanizernia w Żaganiu,
Zakład Przemysłu Wełnianego SUNSET TEXTIL w Żaganiu.
Szprotawa (aglomeracja szprotawska) – przemysł metalurgiczny, maszynowy, spożywczy, zakłady produkcji i przetwórstwa metali, tworzyw sztucznych. Największe zakłady przemysłowe regionu to (podano w porządku alfabetycznym):
BADER Polska Sp. z o.o. – produkcja tapicerek skórzanych do ekskluzywnych marek samochodów,
BEWA Sp. z o.o. S.K. – produkcja elementów stosowanych przy budowie systemów kanalizacji sanitarnej i deszczowej
Consalnet – produkcja fototapet, obrazów, naklejek i szablonów malarskich,
Przedsiębiorstwo Rolno Produkcyjno Handlowe "EKOPOL" Sp. z o.o. w Przecławiu –gorzelnia,
Hoszman Schody i drzwi Sp. z o.o. – produkcja schodów, drzwi oraz mebli drewnianych,
Lakos Sp. z o.o. – produkcja konstrukcji stalowych, aluminiowych i ze stali nierdzewnej,
PPHU Martpol – produkcja kołder i poduszek,
M-S PICO Rusztowania sp. z o.o. w Borowinie – produkcja rusztowań budowlanych szybkiego montażu,
Nord Systemy Napędowe Sp. z o.o. – produkcja silników elektrycznych oraz elementów napędowych,
Peter Schmidt Okna i drzwi PVC Sp. z o.o. – produkcja okien i drzwi PCV,
POLMETAL S.A. w Małomicach –fabryka Wyrobów Blaszanych,
Wiązar System Sp. z o.o. – produkcja wiązarów dachowych oraz prefabrykowanych konstrukcji kratowych z litego drewna łączonego płytkami kolczastymi w systemie MiTek,
Valuepack – producent opakowań.
Zielona Góra – elektrociepłownia stanowi centralne źródło ciepła dla Zielonej Góry w zakresie pokrycia potrzeb na ogrzewanie i ciepłą wodę użytkową oraz jest źródłem energii elektrycznej.
Energia elektryczna i ciepło są wytwarzane sprzedawane są bezpośrednio dystrybutorom energii. Energia elektryczna wytwarzana w pełnym skojarzeniu z produkcją ciepła z bloku węglowego sprzedawana jest do Enea S.A. w Poznaniu, a energia elektryczna produkowana w bloku gazowo – parowym (BGP) sprzedawana jest Polskim Sieciom Elektroenergetycznym S.A. w ramach kontraktu długoterminowego (KDT).
Ciepło produkowane w bloku węglowym i gazowo-parowym sprzedawane jest Dalkii Zielona Góra Sp. z o.o.
Okręgi przemysłowe
Gospodarka w województwie lubuskim | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 8,955 |
Q: C#: A general question about order of conditions in an if-statement So I have the following code:
public bool TryGetItem(Item searchItem, out Item listItem)
{
listItem = ItemList.Find(x => x.Blueprint == searchItem.Blueprint);
if (listItem != null) return true;
else return false;
}
public bool TryAddItem(ItemInstance item)
{
Item listItem;
if (item.isStackable && TryGetItem(item, out listItem)) // the is the line of interest
{
listItem.amount += item.amount;
return true;
}
else if (...)
...
}
This gives me an "Use of unassigned local variable" error for "listItem.amount"
However if I switch the order of the conditions in the if-statement, the error goes away?
why is that?
public bool TryAddItem(ItemInstance item)
{
Item listItem;
if (TryGetItem(item, out listItem) && item.isStackable) // the is the line of interest
{
listItem.amount += item.amount;
return true;
}
else if (...)
...
}
I want it to be the first way, since there is no reason to search for something if the isStackable-bool is false anyway, any comments?
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 6,516 |
Arm Champs 2
(thing) by Servo5678 Tue Feb 04 2003 at 0:47:07
Ever want to play a game that can break your arm? Well, here's your chance. 1992's Arm Champs 2 features a mechanical arm that can arm wrestle with a player at the drop of a quarter. Unfortunately the game could be configured to unleash the ultimate power of machines and break players' arms in mere seconds, snapping bones into pieces. In China, for example, one store that offered the game as a diversion reported over a dozen injuries in the first few months the game was available.
If you should have a death wish one can find the game for sale (typically used) for around $2,000. It was originally manufactured in Japan for the Asian market by Jaleco with warning labels about the game's strength printed in English.
http://www.coinopexpress.com/products/machines/dedicated_machines/Arm_Champs_2_1933.html
Dumb, Dumber, and Dumbest
Chilly Bang! Bang! Juice Video Games (A) Jaleco warning label
Death Wish China Japan
Roman sexuality
How to Take Group Photos of Children
For Translation into Latin
Record Shop Amnesia syndrome | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 1,655 |
package migrations
import "github.com/BurntSushi/migration"
func DropLocks(tx migration.LimitedTx) error {
_, err := tx.Exec("DROP TABLE locks")
if err != nil {
return err
}
return nil
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 1,732 |
About SHA
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ANTHONY CAPUANO '87
Anthony Capuano '87
Executive Vice President Global Chief Development Officer
Anthony Capuano is Marriott's executive vice president and global chief development officer. He is responsible for the global development of all Marriott lodging brands, and supervises offices in Bangkok, Barcelona, Beijing, Cape Town, Delhi, Dubai, Ho Chi Minh City, Hong Kong, Istanbul, Jakarta, London, Mexico City, Moscow, Paris, Rome, São Paulo, Shanghai, Tokyo, Zagreb, Zurich, and multiple offices across North America.
Capuano began his Marriott International career in 1995 as part of the Market Planning and Feasibility team. Between 1997 and 2005, he led Marriott's full-service development efforts in the Western U.S. and Canada. In early 2008, his responsibilities expanded to include North America, the Caribbean, and Latin America. In 2009, he assumed global development responsibility for the company.
Capuano began his professional career in Laventhol and Horwath's Boston-based Leisure Time Advisory Group. He then joined Kenneth Leventhal and Company's hospitality consulting group in Los Angeles, CA.
Capuano earned his bachelor's degree in hotel administration from Cornell University. He is an active member of the Cornell Hotel Society and a member of the Cornell School of Hotel Administration (SHA) Dean's Advisory Board. Capuano is also a member of the American Hotel and Lodging Association's IREFAC Council.
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Back to top of page. | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 761 |
Contents
Irreverent Litanies
Copyright © 2019 by Zack Rogow. All rights reserved.
Dedication
The Big Questions Are Making a Comeback
Spiritually Challenged
Credo
Remembering Robin Siegel
Idol Conjectures
Kind of Green
Fossils and Fuels: The Evening Commute
If Wordsworth Were Alive Today
Even Shakespeare
You Have Someone Else's Name
People Keep Having Babies
What the World Needs Now
Notes I Pass to Myself
A Small Knowledge
The Sticky Blues
Reunion in Paris
One in the Other :—A Guessing Game
Questions about Kissing
Upside-down Sonnet
Going the Distance
On Hold
L'dor V'dor: From Generation to Generation
What Do You Think?
Labor Coach
Reading to the Kids
Deep-Sea Highway
Sleep, City
The Life Expectancy of Shirts
Asylum Street, etc.
With a Wide-Angle Lens
Europe in the Nineteenth Century:
Images for a Stereopticon
4,000 Pianos
based on a topic suggested by Miranda Rogow Sachs
There Should Be Unicorns
Film Noir
Borscht: A Recipe
Buying Challah in Silicon Valley
Against Forgetting
Legacies
For Erin Hyman (1972-2014)
Rebbetzin of Temple Beth Sholom, San Francisco
For a Dear Cousin
Visions of SoMa
Blankets and the Lack
Man Sitting in Alley with Coat Covering His Head
Urban Tanka
Psalmodic
The Bar Mitzvah Boy
Solids
Why I Can't Say Your Name
Psalm Minus One
In memory of Eva Saulitis
Dayenu
Acknowledgments
Irreverent Litanies
Zack Rogow
Regal House Publishing
Copyright © 2019 by Zack Rogow. All rights reserved.
Published by
Regal House Publishing, LLC
Raleigh, NC 27612
All rights reserved
ISBN -13 (paperback): 978-1-947548-63-3
ISBN -13 (epub): 978-1-947548-64-0
ISBN -13 (Mobi): 9781947548770
Library of Congress Control Number: 2019930302
All efforts were made to determine the copyright holders and obtain their permissions in any circumstance where copyrighted material was used. The publisher apologizes if any errors were made during this process, or if any omissions occurred. If noted, please contact the publisher and all efforts will be made to incorporate permissions in future editions.
Interior and cover design by Lafayette & Greene
lafayetteandgreene.com
Front cover artwork "Step Out of Line II" © by Marie Bourget.
Back cover author drawing by Heidi Noriko.
Regal House Publishing, LLC
https://regalhousepublishing.com
The following is a work of fiction created by the author. All names, individuals, characters, places, items, brands, events, etc. were either the product of the author or were used fictitiously. Any name, place, event, person, brand, or item, current or past, is entirely coincidental.
All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior permission of Regal House Publishing.
Printed in the United States of America
Dedication
To my son Gabriel, whose questions
always make me question my ideas
The Big Questions Are Making a Comeback
Spiritually Challenged
Ten days sitting Zen and not one word
slipped past her lips.
My friend can only say, "Amazing!" I just wonder
why she wanted to bury all those thoughts.
With a language faster than words
the born-agains in the dorm room next to mine
woke me at dawn with glossolalia.
I thought, They really need to get laid.
Visiting Chartres, the sculpture
on the cathedral façade
breathes, the stained glass is like candies
of light. But the arches only stretch the emptiness.
Faith seems like an excuse
for a potluck, or for not learning how to dance.
Isn't religion like the United Nations?
Agree on the principles, then ignore them.
The saints on their Corinthian columns,
in the Sinai and Sahara,
the lotused bodhisattvas—
their eyes don't look down from my walls.
But when a fist of cells
roots into Deena's womb
and the chemo shaves her black tresses,
I think about her warm cackle;
her Isabelle and Harry,
just eleven and four;
and her work calls, all for others,
and I hear myself say, "Please, God..."
Credo
I believe that gravity is a temporary condition
I believe that all forms of blue cheese are sacred—why else would they call it gorgonzola?
I believe that dental assistants get so moralistic about flossing
because their work brings them terribly close to God
I believe there are such people as spiritual healers and they
are overpaid
I believe in a radical democratic equality where your cousin is
entitled to listen to Neil Diamond
I believe that anyone who contributes to the extinction of a
species should spend ten years in the jungle surviving
on gathered plants
I believe the designated hitter is an abomination
I believe that there are alternate realities where Paul Klee
would be considered a photorealist
I believe that all nations and ethnic groups have a right to
self-determination in order to make their own
disastrous mistakes
I believe in a Universal Treaty of Human Population
Reduction that every country will participate in
proportionately—easy enough to negotiate, right?
I believe that women were put on Earth to satisfy men and
other women and that men were put on Earth to
satisfy women and other men yes I believe
That the universe does not necessarily have a purpose but if
it does it might be hazelnut gelato
I believe that five days a year people should be allowed to
come into work late just because they stayed up till
dawn
I believe that paying for memberships in gyms and pools
creates a mysterious barrier to exercise
I believe that those who obey every rule should hike to the
North Pole in stiletto heels
I believe that love is the most perfect thing and therefore not
practical for humans Wait I don't believe the second
part
I believe that music is the most supreme speech and that
speech is the most supreme music
I believe all languages are spattered full of moonstones
jaspers star rubies and jade
I believe that poems should end before their readers start to
think about their next meal
And
I believe in the power of beauty to redeem all things
Especially broken snowmobiles highway entrance ramps and
airport bathroom sinks that sense your hands praying
for the water to flow
Remembering Robin Siegel
She was my friend's girlfriend, not mine.
But once they'd broken up
I knocked on Robin's dorm room door.
She gave me Dreamsongs and Ariel,
pulled by the black holes of poetry.
Robin had eyes like dark comets
and a way of winching her mouth
if she doubted what I said.
She braided her Yiddish-brown locks
or swept them up
like Audrey Hepburn.
The weirdness in people made her laugh,
not in mockery,
but in sympathy.
Robin doodled tapestries
of closely woven faces,
mouths and noses stretched like gum.
One day we tried to get lost,
held hands all the way
to the football stadium, picked anonymous
wildflowers we saved in tumblers.
We spent the night in my one-room apartment
and the morning after
at the student health center
waiting for the morning-after.
Was graduation even two months gone
when her sister phoned?
Though she was calling everyone
in Robin's address book she couldn't help crying
one more time as she spooled out the auto accident.
In our literature classes Robin and I read
the famous elegies: "To an Athlete Dying Young":
"Smart lad, to slip betimes away..."
and Milton's "weep no more" for drowned Lycidas,
serenaded by angels.
Great poems, great lies.
Don't block the grief
by talking of an afterlife with smeared edges
or newborns incorporating Himalayan lamas.
Do weep for Robin.
The world weighs less without her.
She had a habit of hugging goodbye
and then, when my back was turned,
following me down the street
just to see how long it would take
for me to realize she was still right behind me.
Idol Conjectures
What if we ate round tofu on the night of the full moon
What if everyone just refused to drive cars unless they ran on
art nouveau solar panels
What if we responded to all work emails after 5 pm with a
cake recipe or a movie suggestion What if
Robots started writing poetry and forming exclusive literary
cliques for robot neosurrealism robot confessionalism
What if Elvis really is alive on the island of Elba about to
claim he's the rightful Emperor of the Frenchwearing
an outfit halfway between his spangled white jumpsuit
and Napoleon's dress uniform would you be a
groupie for his Hundred Days concert tour till his
final gig at Waterloo
What if they cloned Mozart and he decided to become a
professional wrestler
What if all the animals are planning a bloody uprising
masterminded by ostriches with Gothic moths as
couriers
What if legislatures only drafted laws and the entire
population voted yes or no online
What if national borders just gradually disintegrated from
neglect
What if to be or not to be really were the question for our
whole species
What if we launched all nukes into space in the direction of
the Horsehead Nebula
What if another version of you could stroll the Earth
experiencing different things but fundamentally
similar
What if there were a hundred of those beings
Billions of them
What if there already are
Kind of Green
Fossils and Fuels: The Evening Commute
The traffic on 280 laps up gas
as it lashes around the hills
above the Crystal Springs Reservoir.
The oncoming headlights form
a Nile of fireflies.
Ahead, taillights
brighten like blown embers.
Each bubble of music
chases the one in front.
Double digits of lanes, and all of them
saturated.
As the coastal mountains lean
toward the setting sun, the rays
transform the water
into a matador's suit of lights.
It's beautiful, the evening commute: beautiful
as a bullfight.
If Wordsworth Were Alive Today
Wordsworth, if you were here alive today
in cities of dittoed apartment blocks
identical as soldiers' grave-white rocks
what would you think of us? What could you say
of glaciers turned to ghosts, herons robed in oil,
and gyred tides of polyethylene
that leatherbacks mistake for jellies' sheen?
I wonder if you'd fly from all this spoil.
Or would you dive into the present mode,
denounce steam coal, fracked veins, the dollar trap?
Take a day job as an art custodian
and maybe teach yourself software to code
a write-your-own-Romantic-poem app,
while nights you learn to cook Cambodian.
Even Shakespeare
On some days even Shakespeare must have thought
There's too much gore at Hamlet's denouement,
or he regretted jokes crowds hadn't bought
and Romeo's ruffly speeches made him yawn.
Metaphysical grunts of Lear seemed trite
and Will wished his love in her plumed bonnet
would spread her knees without his having to write
yet another motley-minded sonnet.
Then he'd exit the Globe and slip away,
mayhap in the midst of Act III or IV,
and follow Thames' tide at close of day,
watching an egret on the nearer shore
suddenly take flight above the flood:
a white shadow blazing against the mud.
You Have Someone Else's Name
Manet and Monet—
how unlikely is that?
But it's only yeasted from there:
one blues great called Etta James,
another named Etta Jones.
And they both sang
"Don't Go to Strangers."
Not to mention one of my favorite actors, Maggie Smith,
with exactly the same name as the poet Maggie Smith.
As for my name,
there's already a Zach Rogowski on Facebook,
a handsome devil
with his arm around his beaming wife.
It's only a matter of time
before I Google my moniker
and a total stranger stares back.
A name used to make a body stand apart,
to last longer than this machine of flesh.
Now it's just something we share,
like our favorite salted caramel whatever.
Everything we hold dear is borrowed,
even our names,
those rented ice skates,
and one day,
when our ankles are sore and the cold
has warmed our cheeks,
we will have to give them back.
People Keep Having Babies
There are already more humans on earth than seconds in a century
but enough children are born every day
to populate Miami or Belfast
The oceans fill up like forgotten bathtubs
and it won't be long
before Siberian tigers live only inside rectangles
but those in love still make babies
There are cities you've never even heard of
with over a million people
and parents snug their newborns in soft folded cotton
Block after block
of apartment towers are smashed together
with no thought where all the glass and steel will end up
Goodnight mittens Goodnight moon
We treat this planet like teenage boys joyriding a stolen Chevy
and parents still lift their babies gently
from their cribs and press
their cheeks to their cheeks
What the World Needs Now
for Hal David, Burt Bacharach, and Jackie DeShannon
What the world needs now is an electric car with fins
What the world needs now is a loosey-goosey interpretation
of the most sacred books
What the world needs now is a way to live many lives
simultaneously
What the world needs now is a whole country just for
elephants and their trusted companions
What the world needs now is a villanelle you can rock out to
Freedom
What the world needs now is freedom ringing from Badaling
Mountain in China to Kilimanjaro in Tanzania to
Denali in Alaska to Mount Elbrus in Russia
What the world needs now is a tofu recipe that tastes as good
as a medium-rare burger just off the grill
What the world needs now is a glue that can fasten Monday
onto Saturday and Sunday
What the world needs now is a cigarette that cures cancer
What the world needs now is a Frisbee that can turn 90
degrees
What the world needs now is for everyone to stop running
the water when they brush their teeth—it would
saves lives—I'm not sure how but I know it would
What the world needs now is Marilyn Monroe in a flannel
shirt and construction worker boots
What the world needs now is a way to play back our dreams
What the world needs now is love sweet love
At least that's what I need
Notes I Pass to Myself
A Small Knowledge
I stole a bag of soldiers
when I was nine.
Little monochromatic men,
invulnerable, with palms
already smoothed into cylinders
to grab weapons.
My father
had died six years before. My mother
had secretly gone back to her first husband,
now remarried and with a son
almost my age.
A phone call exposed the transgression.
Then the disappointment in my mother's voice—
"We've shopped there since you were born,
we've got a charge account at that store!
You could've had whatever
you wanted. Why did you steal?"
I didn't know. I returned the booty
and apologized to the owner,
the sweetest woman.
I only knew
when I took what didn't belong to me,
the blood shook
in all my veins.
The Sticky Blues
Catch the 8:19 to my desk
Down in Silicon Valley
Would I rather be blind drunk
In some blind alley?
Now you know I've got those
Got those Baudelaire blues
What's with all these jobs
That drain your hourglass of time
The work I want to do
It doesn't earn a dime
I let it slip I got those
Got those Baudelaire blues
My body craves that music
Slinky notes of Sarah Vaughan
Gonna stay up till the sun
Forgets how to dawn
And I got a bad case
Bad case of Baudelaire blues
Drink a flight of blue wine
From an alabaster cup
Carve myself some lilies
When I sit down to sup
Wish I could eat away
Eat away these Charles Baudelaire blues
Puedo escribir esta noche
Los versos mas tristes
Sinking fast like a ship
Tossed in holy water these days
These days I even got those
Got those Pablo Neruda blues
I just wanna be done
With this sticky state of mind
Can I ever shuck this mood
And throw it far behind
Tell me how do I shed
Gotta shed these Baudelaire blues
Reunion in Paris
for Anie Montigny
Once I wrote you bad poems
while we hitchhiked
from the dancing tiles of the Alhambra
to alleys in Fez
where potters worked waist-deep in clay pits,
the two of us shouldering backpacks
across starlit Atlas Mountains
to a loop of oases in the Sahara
where the July sun was so strong it
slapped our faces.
Twenty-two hours we waited for a ride
on a back road in Algeria,
shaking our heads at the only traffic—
a herd of sheep.
When we parted in Tunis
I sat down on the cathedral steps
and wept into my hands.
Now, forty years later,
could we really be meeting in your Paris?
It's like seeing you in another life
where we've been reincarnated
as a middle-aged man and woman—
you, a professor of anthropology,
with an apartment full
of wooden trunks carved for bridal trousseaus,
brocaded pillows,
and ceramic flasks glazed
with deer and fish;
and me, surprised father
to four children.
We follow the bends in the Seine all afternoon
with the sun sprinkling sequins on the river.
Later you feed me
confitures de maison
you've preserved in thick glass jars,
apricots with cardamom,
and little green Reine Claude plums so sweet
they barely needed sugar.
Is any part of us still left, Anie,
after forty years
when the very cells of our body
have been replaced five times over,
when the towering justice our generation envisioned
has fractured
into enclaves
fenced with electricity
and our faces show the wear
of four decades
of friends and family
now gone? What's left,
if not your eyes
like September Paris skies
and your way of laughing at a moment
just because it is what it is.
One in the Other :—A Guessing Game
In Paris in 1953, members of the Surrealist group made up a game based on the alchemical concept of "One in the Other"—the idea that all things are connected. In this game, someone thinks of an object and then has to describe it using language associated with a completely different thing. For example, the poet André Breton thought of a lion, but he was asked to describe a lit match. Breton said, "I'm thinking of a match with a mane of fire." Can you guess the second object for each of these? Answers at the end.
1.
I'm thinking of a cloud you can put on a chain and carry in
your pocket
I'm thinking of a cloud of shining steel
I'm thinking how no two clouds are identical
I'm thinking of a cloud that slides over a box and rips its seal
I'm thinking of a cloud with jagged edges that can open a door
2.
I'm a rose you can plug into a socket
I'm a rose with metal petals and serrated edges
I'm a rose with thorns that slash through wood
I'm a rose that makes a deafening noise
I'm a rose that could slice off your hand if you're not careful
3.
I'm a wish that starts as a tiny egg
I'm a wish that crawls first and then grows wings
I'm a wish you can only see at night
I'm a wish that leaves trails on the darkness
4.
I'm thinking of lips you see in the street
I'm thinking of lips that form a perfect circle
I'm thinking of lips with a name molded into them
I'm thinking of lips poured from molten steel
I'm thinking of lips so heavy it takes a strong person to open
them
I'm thinking of lips that reveal a world beneath our world
5.
I'm a cigarette cold enough to freeze your fingers
I'm a cigarette flying upward in the wind
I'm lots of cigarettes that form a man with eyes of coal
I'm a cigarette that melts in warm weather
I'm millions of cigarettes falling from the sky
I'm a cigarette so white it can blind you
6.
I'm a diamond ring in blue and red and yellow and green
a diamond ring with a stone as large as a head
a diamond ring that gives a little when you squeeze it
a diamond ring that bursts if you press too hard
I'm a diamond ring lighter than air
a diamond ring that floats away if you don't hold it tight
Answers: 1. cloud/key 2. rose/electric saw
3. wish/firefly 4. lips/manhole cover
5. cigarette/snow 6. diamond ring/helium balloon
Questions about Kissing
When does a kiss end?
How far does a kiss travel?
Are kisses the food of the tongue?
How many people are in one kiss?
How many anemones, orangutans, lynxes?
What parts of your own body have you kissed?
What is the opposite of kissing?
Which kisses do you remember best?
How do kisses break into prisons?
Which of the Seven Wonders of the World was built for a kiss?
What do tongues say when they meet?
Through what tent flap do kisses enter dreams?
Who kissed Marco Polo in China?
What poems are printed with lipstick?
What musical instruments are cousins of teeth?
Which is the best wing of the body to kiss?
When does a kiss begin?
Upside-down Sonnet
So, should we step on toe now, like dancers?
I, for one, am very short on answers.
We know love can dry, a roast overcooked;
smell sour-sweet, like lilies kept too long;
or frizz with memories overbooked.
No more coincidences, no more song.
But once this love was the livest I've ever seen,
so bullish, so fierce, with such prancing.
Could that person glazed with love have been
me, and was that other, you, advancing
into our own made-up mambo fairytale,
both eager to combo our lips and thighs?
I saw it all with new-glasses detail
as I gazed into your nautical eyes.
Going the Distance
Today I drove by the house where we first lived
with the orange poppies in the garden
that loosened their fists at sunrise
and the pipe-cleaner purple blooms of the Mexican sage.
At night we sprung up the bedroom shade
to see the curves of the hills
polished by moonlight.
My girls were so unsettled
by your sudden presence
I had to put them to sleep in our bed
and when they dropped off
I snuck out to join you
on a mattress we threw onto the floor
in the back hallway by the utility sink
and even so we thought we were the luckiest
creatures on this planet.
And look at us now, two boxers
in the fifteenth round—
puffy-eyed, split-lipped,
stuck in a clench.
Neither of us can knock out the other,
but neither wants to throw in the towel.
On Hold
So, you're not half of the greatest love
since Bacall showed Bogart
she could whistle,
no lost aunt leaves you
her nude portrait
that Diego painted during their affair,
no callback after an interview
for the dream job
cut exactly to your measurements.
Instead, night after night
you just watch old movies
with those incredibly fake backgrounds
behind cars that float
down LA boulevards after dark,
or you purchase all the ingredients
for grilled swordfish with mango salsa
and then settle for a rerun
of last night's spaghetti.
Maybe you plan to hear a Brazilian bossa singer
but end up staying home,
listening to your team on the radio
as they blow the lead late in the game
again
and the announcer somehow manages
to put a brave face on it.
L'dor V'dor: From Generation to Generation
What Do You Think?
What do you think about people who act like 15 Items or Less doesn't apply to them?
What do you think about drivers who muscle into the turn lane without waiting in line—in a hybrid?
What do you think about scholars who write arm-length footnotes with words like "heuristic" and "counter-hegemonic" and then tell you how they're changing the world?
What do you think about countries that can't go five years without starting a war—countries like the one I live in and love?
What do you think about people who say their religion is the best? They can't all be right, but they could all be wrong—I once heard the Zen master Suzuki Roshi describe with joy tasting the wafer when he took communion in St. Patrick's
How do you feel about automatic weapons in private hands—they're needed to repel space invaders, right?
What do you think about the idea that everything you get in life is deserved—Genocide? Bad Karma. See ya later!
How do you react to tearing up the body in the name of exercise and health?
What do you think about people who never have a problem with anything and always answer the question "How are you?" with a chirpy "Great!"?
What do you think about artists who believe it's perfectly fine if their work only makes sense to their friends?
I don't know what I think about poems that start every line
with the same words
And I'm not sure what I think about growing old and looking more and more each day like my Grandpa Saul with my spectacles and with my gray hair melting away like a glacier
And I don't want to think about death on those days with an argent sky and newborn grass on the hills and I imagine all my memories evaporating like a sudden lake formed by a rainstorm in the Sahara where I hitchhiked in the violent heat of July at age eighteen and I can't bear for all this ever to end
Labor Coach
Birth is an ugly thing.
The labor grinds like a stuck clock.
Pain immolates the person you love most,
you can't do anything
except remind her to breathe,
and she screams into your eyes
as if you caused it.
Masked faces gawk at intimate moments.
Instruments are smeared with numbers.
Poop snakes out, and the episiotomy gushes like red oil.
Birth is an ugly thing
until her tightest push
jellies the baby past the pelvis,
and that new creature appears
fresh as the day
that water first caught the sun's flash.
Then you tuck that little joey under your chin—
its tiny fingers already grappling yours—
and hold that beating body,
and suddenly you feel so strange—
as if you actually belong to this world.
Reading to the Kids
Wizards wield brittle light,
fencing to the death.
Dragons incinerate
with sneezes. Gryphons
flex those scimitar nails.
Shlubby ogres lug their clubs.
In the soft arms
of the sofa,
parent and child
lean into each other,
reading together.
Deep-Sea Highway
Driving to pick up my son from school, I glance at the odometer: 25,319 miles. If I'd headed in one direction instead of back and forth to my son's school and to work, by now I could have motored all the way around the globe along the equator. I would have passed through the Congo with the windshield wipers shoving green rain, laced up the Andes right into Ecuador, and slipped through villages in India under a moon like a glass of milk. Of course, most of my route would be underwater, watching the dolphins' playtime from below, the sunlight prisming through the kelp forests, the schools of fish performing their silver ballets. But it would also be a pain to drive in the ocean with chameleon squids suctioned to the side of my car, turning the exact gold of my Toyota Corolla, forcing me to don scuba gear and peel them off the windshield. Yes, driving that undersea Route 66 would be quite a kick, except that once I traversed the entire equator, I would just be back where I started, picking up my son from school, kissing him hello on top of his redolent head of hair, asking him where he left his dinosaur camouflage jacket, since he finds a new way to lose it almost every day, just like his dad.
Sleep, City
At the
time of night
when each house turns
into a crate of darkness,
and even the traffic lights
have a difficult time
keeping their eyes open,
this city sits glazed and empty
as a raku bowl, except for one man
in the only open laundromat,
gazing at his shirts and pants
while they leap in the dryer
like a bunch of schoolboys,
wondering how his life
arrived at this spot,
as the moon sinks
to the horizon
and clinks
into its
slot.
The Life Expectancy of Shirts
We outlive them, our shirts. Too easily they get
snagged by chain link fences,
or pockmarked by sauces
twirled with golden oils.
Do they wait for us to choose them
like dance partners, impatient
with stringent hangers
and jostling closets?
Do they long to pump their arms
along boulevards past midnight,
to rub their cuffs against light
blue cashmere?
Only a few years they survive,
as long as eels, or foxes.
We outlast even the shirts we're fondest of,
the daring silk, the thick cotton
hefty as an old sail,
or the paisley
with its vegetal teardrops.
But when we finally are dressed
in our very last shirt—
sleeves yanked onto rigored arms—
some of our favorites actually
survive us—a fact rarely mentioned
in obituaries.
Asylum Street, etc.
In the sophomore apartment muraled
with wildly colored parrots
we penned phone numbers
on the kitchen wall.
Poems dripped down the halls.
One day the ceiling avalanched
my roommate's empty bed.
Where is he now
and the ones who shared the house on Asylum Street,
headquarters for a tribe of drag queens.
I had a brass bed
missing some rails
and wrote my first poems there
about my own waste land.
In the flat on Park Street
where hearses unloaded next door,
the former tenants
stayed on with a dog named Boris
who wore a red bandanna
and tripped with his owner.
We harvested the Concord grapes
that laced the fence
till they translated
into a cloud of fruit flies in the fridge.
My first place alone
the sunsets simmered on the Hudson.
I hotplated brown rice,
while the man I shared a bathroom with wept
each time his kids and ex left.
In the apartment near Avenue A
the kitchen had a bathtub.
I raced the cockroaches for my dinner.
My brand new stereo was kidnapped.
After the gun mugging, I fled
to the pad in San Francisco
that looked out on Angel Island.
My roommate Fran and I cruised the discos
where gay and straight boogied together.
One night after dancing till two
Fran and I stood in the hallway
outside our adjacent bedrooms.
"Your place or mine?" she asked.
At the commune in Brooklyn
two house members were no longer speaking:
"Would you ask Sharon to pass me the salt,"
said the man who had caressed her
only a few weeks before.
Another roommate moved from our house
right to a ward in Bellevue.
The kids didn't know
who their fathers were.
Then the married house
with linoleum
like a Seurat painting viewed too close,
the perfect three-bedroom.
From outside, you could see a chimney
but the fireplace
was buried inside a wall.
And now the house with shades
of green, inside and out,
and a garage packed to the ceiling
with all the other houses.
This time maybe,
maybe a different ending.
With a Wide-Angle Lens
Europe in the Nineteenth Century:
Images for a Stereopticon
Wrecked on opium, Thomas DeQuincey
sees Coleridge's foot as a goldfish
in a basin missing its walls.
The young Alfred de Musset is touching
George Sand's left nipple
with a sable watercolor brush.
"Tu m'aimes?" he asks her.
"Et toi?" She turns her head
and blows cigar smoke in the face
of the Paris dawn.
Beethoven is having a bad hair day.
Darwin, bent over his bird's-eye maple escritoire,
collaborates with a dead mackerel
to prove that God left no forwarding address.
As he inhales a blur of lavender
by the moon bridge,
the aging Claude Monet paints so furiously
he erases the garden he loves.
Meanwhile Bismarck ascends
to the base camp of the Schneefernerkopf,
so he can survey
from the blue jewel of the glacier
all the placid little duchies
waiting for his strong arm.
Under the pontifical dome
of the British Museum Library
Karl Marx claps shut a frayed ledger
scored with statistics
on English textile production,
realizing he has just found the key
that unlocks all history, all thought.
A tiny map of ink
has formed on the side of his hand,
blood-red.
4,000 Pianos
based on a topic suggested by Miranda Rogow Sachs
When the Nazis cattled away the Paris Jews,
using the butts of their Mausers to pack
the starred men, women, and children
into boxcars darker than sleep,
those families had to leave behind
their armoires, carved chess sets,
glazed casserole dishes,
and their pianos,
the instrument a daughter used
to practice every evening
Mendelssohn's Songs without Words.
The soldiers swaddled the pianos,
angling them down snail-shell stairways.
They numbered each one,
and constructed piles of them
in the Palais de Chaillot,
two giant art deco arms
in the shadow of the Eiffel Tower,
where only an occasional rodent
plinked their keys.
A few of the best instruments they nestled
in the villas of Nazi officers—
a Bösendorfer, or the hidden
harp of a Steinway. Partygoers
thumped on them
during soirées of forced champagne
run up from the cellar with arms full.
Or was there an officer with the stripes of an Oberleutnant
who tried late at night
to recover a piece he could once play by ear—
a Mendelssohn melody interrupted
only by a few staccato blasts
in the distance?
There Should Be Unicorns
There should be unicorns with flirty eyes
and polished horns that flash like saxophones.
There should be hobbits seeking ancient prize,
who tote their satchels stuffed with gin and scones.
There should be fairies zooming all the night
to mirror constellations—just for kicks—
while raspy dwarves parade beneath their light,
then march to twinkling mines with lamps and picks.
There should be satyrs, gryphons, centaurs, djinns,
fierce goddesses with those bonus arms, elves,
gargoyles, golems, mermaids with blingy fins,
and squadrons of angels who fan themselves.
There should be a God who judges our ends,
loves all, rights wrongs. Yes, there should be, but friends...
Film Noir
On a street jeweled with shadows and trembling with light,
Deep eyes over a trench coat narrow with shame.
In a world big enough to hide in, no one does right.
A tainted blond sings lamé ballads every night.
Her wit, her power—and she knows so well the game
Of a street jeweled with shadows and veiled by the light.
A bullet—a body that he covers from sight.
This city! Every match here can become a flame.
In a world big enough to hide in—how to do right?
The trench coat solves the puzzle—no, maybe not quite.
He makes one false move, he's a fall guy for the blame
On a street jeweled with shadows and trembling with light.
Is the blonde on the level? They had that one night.
He thinks the redhead pictures him inside a frame.
The world's big enough to hide in. Should he do right,
And will he crack the case and get the girl? He might
Or might not win the race to rinse his name.
On a street jeweled with shadows, and trembling with light
The world's big enough to hide in—still, he does right.
Borscht: A Recipe
based on Mollie Katzen's soup
Take two potatoes
with their grandparent eyes.
Slice a fistful of beets
till their blood stains the cutting board.
Simmer in chicken broth
the color of Shabbat candlelight.
Chop two yellow onions as the family stories
rush down your cheeks.
Sauté till the onions turn to old glass.
Sprinkle with caraway—
the strokes of Torah scribes.
Peel away layers of cabbage.
Slice two carrots into kopecks.
Sprinkle in salt and vinegar,
flavors of exile.
Mix all together and cook
on the stove's liberty torch.
Dollop in some wild honey.
Serve hot off the stove
garnished with dill and sour cream—
evergreen against snow.
Stir till the soup in your bowl turns
tongue-pink, roots once
hard and tough
now sweet, now tender, now bitter.
Buying Challah in Silicon Valley
In this immaculate suburb
where I earn my daily bread
the saleslady in the French soap store
wears a white lab coat
as if dispensing life-saving medications.
Another boutique is completely dedicated
to sachets of lavender and anise,
and even the thrift shop is pricey.
Thanks be to Google,
the omniscient, the omnipotent,
where many of the locals work,
I discover that a bakery
hiding in this town
sells challah.
On a Friday afternoon I hunt
for this establishment, squished
between Safeway and Starbucks.
The kind woman who helps me is Mary Anne,
named for both the mother and grandmother
of you-know-who.
And even though
Mary Anne asks me if I'd like the challah sliced,
and even though
the baker hasn't quite grasped
that there's no such thing
as too many poppy seeds,
still, when I chauffeur the loaf home
and its fragrance floods my car,
when our family gathers
under the eyes of the candles
to say the brachot,
we can still touch
in the sweet-soft folds of yellow grain
the warm hands
of the Miriams and Abrahams
who came before us.
Against Forgetting
Legacies
for Eva Saulitis—Alaskan poet, essayist, and marine biologist
You sketched your mother-in-law
so clearly in your poem:
her fearless mouth and hairstyle,
and the list you made
of what she left you:
"Hiroshige prints, a 1950s lamp,
and a volume of bad Hawaiian poetry."
I'll never get to tell you now, Eva, how much
I like that poem. But still your voice permeates the page,
the voice I came to know
when I tried to keep up with you
jogging around the lake of weeping loons
and when you brewed hibiscus blackberry tea
in the dorm room you turned into a home
the two weeks a year we taught together.
I only knew your poems, your laugh,
and those deep-set Latvian blue eyes.
I didn't know your meticulous recordings
of the croonings of killer whales
and how you and your husband showed
that the private conversations of the pod
spelled their genetic isolation.
The two of you theorized
that the oil that bled
from the Exxon Valdez
doomed this generation
to be their last.
Then you discovered the tumor
you called your "intruder," your "fugitive."
How to fathom such a life
as huge as a statue,
though pretension never
had the smallest niche
among your thoughts.
I'm still trying to catch up to you
in your poems
with their sampler of birdsongs,
Polynesian trade winds,
and mountains that hold onto the dark
even after the sky has begun to glow.
For Erin Hyman (1972-2014)
Rebbetzin of Temple Beth Sholom, San Francisco
Glass knives. Splinters. Clattering
fragments. That's the first month
without you.
How could the universe allow it—
you hadn't even tasted the frosting
on your son's eighth cake
and then—
snapped in half by illness.
Not youth, not beauty, not brilliance
could protect you. Not medicine, not faith,
not laughter. Not even love.
But with your deep eyes for art
you declined to see
only scars. Dying, you wrote
about your hundred-year-old Grandpa Johnny
who felt he still had paintings
that wanted to leap out of him,
about a black bronze of Lilith
hanging from a wall like Spiderman,
and the healing power of chocolate cassis scones.
I believe you would want us to thank
the God who took you
for giving us even this day, this month
of deafening sunshine.
The sculpture of Lilith, you wrote,
like the morphing stories that have evolved
around her name, remains
wild, suspended, resisting
any attempt to pin her down.
For a Dear Cousin
It started
harmlessly enough
as a way to
break the pain
but then your husband
gradually let himself
splinter into shards
first his fingers
then his forehead and knees
finally his snare-drum heart
It's so hard
to see the person you love most
change into ash
You don't know how
you'll ever stand up again
but some moments
you feel
weightless
like when the sadness follows you
as you walk
in the precise rain
or at dinner parties
when the laughter clatters
like dishes
In your new home
where the lines meet
clean
as an art gallery
your old furniture
feels chilly
You ask yourself
why are you the one who has to
sign for all of this
But could it be
because you realize
we live in a universe
whose shape
we don't even know yet
saddle
sphere
or membrane
What are the chances
that we washed up alive
here on this blue beach
Visions of SoMa
SoMa, or South of Market, is a district in San Francisco that has a large population of homeless and street people. I worked in the neighborhood from 2009 to 2013.
Blankets and the Lack
With eyes closed
he looks right up into the sun
in the middle
of a crowded downtown sidewalk.
This man sleeps with no cover,
flat on his back, his pillow
only two sneakers
covered by a square of cardboard.
I can't imagine the fatigue
that allows him this oblivion
in broad daylight
and in plain sight
of San Francisco City Hall
with its Fabergé egg of a dome,
the lantern and longitudes
brushed with gold.
Man Sitting in Alley with Coat Covering His Head
This man wears his coat
backwards, so it completely covers his face.
Where was he ten, twenty,
forty years ago?
Did they fly him to a target
in clothes made to look like sand?
Was there once a mother
who woke him with soft lips?
Now he sits in this alleyway,
his coat hugging the dark,
his back against the wall.
Urban Tanka
1/
Sitting cross-legged head
bowed right to the sidewalk
homeless man
please move please
show me you're still alive
2/
She has her whole life
ahead of her—the young woman
sitting half-dressed on the sidewalk
probing the back of her hand for a vein
with a loaded needle
3/
His dream makes him smile
under a pastel-
yellow blanket
his head leaning
against the foot of a parking meter
4/
Twin sisters with identical
gray buns on Mission Street
push together
a shopping cart
draped with cloudy plastic bags
5/
Rousting the homeless
from their concrete beds
the police officer
walks the sun-sliced alleys
as slowly as possible
6/
The couple asleep
on the sidewalk
with a wheelchair at their feet
snuggle closer than you and I
in our warm bed last night
7/
On the sidewalk by the man
cocooned in his sleeping bag
I place a plastic container
with blueberries and toast
left over from my son's breakfast
8/
He sleeps on this well-traveled sidewalk
his hand tightly wrapped
around the leash of a Weimeraner
whose gaze questions
each passerby
Psalmodic
The Bar Mitzvah Boy
His sudden body fits like a suit
two sizes too large.
While he chants the prayer
his voice is a rock
scratched against glass.
Could there be a ritual more awkward
than a thirteen-year-old boy singing
an ancient language
in front of a crowd of people?
Will somebody please let this kid go home
and kill space aliens on a screen?
Then the bar mitzvah boy
lifts the torah by the rimonim
and nestles the scroll against his shoulder
like a parent carrying a sleeping child.
He tours the parchment around the congregation,
each person touching a fringed tallis
to the coiled words, as if completing
an electrical circuit. The congregants
close their prayer books with a little kiss
to the binding, like saying goodbye
to a loved one they know
they'll see again soon.
The bar mitzvah boy sounds out his parashah
about decorating the first temple:
And thou shalt bring in the candlestick
and light the lamps thereof
And thou shalt set the altar of gold for the incense...
and the altar of the burnt offering before the door.
Veheveta et-hamenorah
Veha'aleyta et-neroteiha
venatata et mizbach hazahav liktoret...
et mizbach ha-olah lifnei petach
And now those musty
syllables
coupled together
millennia ago
seem like skipping stones.
Solids
I wonder about
celestial bodies, why they're all spheres. Howcome
there are no star-shaped stars
and why no pyramid planets, cubes,
or dodecahedra for that matter?
The universe has a way
of whittling down our sharper edges.
Plato
said we're born knowing all shapes.
We just have to wake up
our memories.
He believed
the soul was a perfect sphere
and yet each one could be distinguished
as a particular person,
maybe by its color and rotation,
like subatomic particles.
It's possible
all knowledge has already been stated
and we're just
rephrasing previous theories,
alternating them like statues
in a wax museum:
Elvis, the Beatles, Madonna.
What happens to
wax museums in a fire?
Hidden shapes
emerge from and return to
light-infused plasma.
Like my father
making candles when I was a kid,
heating the raw, whitish wax in large pots
he poured into cylindrical tennis ball cans.
When the wax cooled, he decorated the candles
by melting onto them my multicolored crayons,
Vermillion, Ochre, and Burnt Sienna.
In his early thirties then, he couldn't have known
his life was almost over.
And those candles
lit the whole house
during the hurricanes that spun onto Fire Island
where we spent summers
running barefoot on the torching sand,
once taking refuge
from an angry tempest named Hazel
in the only solid building, a church.
I was three, it was the first time
I'd entered a house of worship—
such a huge, rectangular space—and I said,
"I don't like this restaurant."
Flickers of the past
when they thought
you could sometimes eavesdrop
on the Music of the Spheres,
that the universe leaked harmony,
something between a morning raga,
a Bach fugue, and Coleman Hawkins'
sax solo on "Body and Soul."
Personally, I believe that all of us—
you, me, Elvis, Madonna—are
Variations of the same bright filaments
and that we all share the moments
when we're impenetrably alone.
Why I Can't Say Your Name
No not for the reasons of my Old World ancestors
afraid of uttering those almighty syllables
casually or in a curse
Scared of
accidentally tossing out the most sacred letters
with the chicken bones and potato peels
No I won't say your name
because those who led the pogroms against my blood
displayed your enameled face
Because priests invoked you
against lovers who honored their desires
while pushing altar boys under their cassocks
Because so many Josephs and Marys
dragged long bags of cotton down row after row
hungering for one good meal
with Jesus in the afterlife
Because the Inquisitors showed Galileo Galilei
the iron instruments
that could make him look away
from the ellipses of the earth
and yet he grumbled E pur si muove
Because motorcycle assassins machete secular bloggers
and even their own beautiful Sufis
who tell stories of Rumi's teacher Shams-i-Tabrizi
selling girdles from town to town
while he saw the divine in a cat licking her kitten
Because my mother's atheist ashes would shatter their urn
No I can't say your name
though I covet the serenity of those who do
Psalm Minus One
In memory of Eva Saulitis
Praise the galaxies those sprinklers of stars
Praise the twine of light we call the sun
Praise the aquarelle canyons and the torn silhouettes of
the sierras
Praise the claws of lightning and the machines of hurricanes
Praise the rivers and their names the Blind River the Rosetta Branch of the Nile the Zezë La Lézarde Rivière Lorca's Guadalquivir that hooks through Seville
Praise the cities with their lingerie bridges and their
glass polyhedra
Praise our shelters and homes the circular yurts and
buttercream chateaus
Praise the wildflower smell of babies' heads
Praise the children who have feet in order to skip
Praise the customer service representatives the ungroomed heretics the person who locks up the bowling alley after everyone else has driven home in the snow
Praise the lovers and the dead who exist only in their bodies
Praise the artist nearsighted and far
Praise orange peppers purple cabbages and the Fibonacci
patterns of Romanesco
Praise the four thousand species of frogs
Praise dust motes like the last zizz of a firework
Praise our unicellular cousins in a tiny paperweight of water
Praise all the molecules their infinitesimal mosques
Praise the electrons for their uncertain orbits
Praise the transuranium elements with their millisecond lives ununpentium ununseptium ununoctium
Dayenu
Dayenu: The refrain from a song in the Passover seder that means, "It would have been enough."
If there had been only stars with their stitches of light and no planets—that would have been enough
If there had been only lifeless planets with strands of quartz and gold—Dayenu
If there had been just the breathing oceans vaulting against boulders
Only microorganisms with their endless division and multiplication
If there had been only forests of coral
Merely the minnows their tiny hearts flicking inside glass bodies
If there had been only the firefish the butterfly fish and seahorses disguised as kelp
Only the desert and its runnels of sand
Just green beards of grass and no redwoods
If there had been only black onyx no lemon serpentine no azurite
Only the insects with their x-ray wings and flashing songs
Just the geometry of snake skins no mammals
Only gray mice their hearts valving six hundred times a minute
If there had been nothing but a hummingbird pinning itself in midair
Only the elephant with its vault of bones only the zebra so unaware of its own beauty
Only monkeys chattering their no language—that would have been enough
Only the first humans weighing their young in their arms—Dayenu
Only that and not the peacock domes of Isfahan
Not the "Mood Indigo" of Ella's Ellington album
Or the white stone garden of Ryoanji Temple where no viewer can view every dark rock
No broken torso of Apollo
Or acts of love beyond number
Would it really have been enough?
Acknowledgments
The author wishes to thank the editors and judges who selected poems from this collection for the following award and publications:
Celestine Award in Poetry, Holy Names University, 2015: "Fossils and Fuels: The Evening Commute," "There Should Be Unicorns," "What Do You Think," "What the World Needs Now;" also published in Between the Lines.
Impact: An Anthology of Short Memoirs, Telling Our Stories Press: "A Small Knowledge."
Proceedings of the First Asian Literature Festival, Gwangju, South Korea, November 2017: "Dayenu," "What the World Needs Now," also translated into Korean.
Aji Magazine: "The Sticky Blues"; Blast Furnace: "Credo"; Catamaran Literary Reader: "Europe in the Nineteenth Century: Images for a Stereopticon"; Cirque: "Legacies"; Dime Show Review: "Dayenu," "Why I Can't Say Your Name"; The Dreaming Machine: "Dayenu," "Idol Conjectures," "What the World Needs Now," "You Have Someone Else's Name"; Foliate Oak: "For Erin Hyman (1972–2014)"; Fredericksburg Literary and Art Review: "Asylum Street, etc.," "Going the Distance," "Spiritually Challenged"; Hamilton Stone Review: "Labor Coach," "People Keep Having Babies"; Hawai'i Pacific Review: "The Life Expectancy of Shirts"; Illuminations: "For a Dear Cousin," "Upside-down Sonnet"; Indian River Review: "Sleep, City"; Inner Art Journal: "Urban Tanka 3/, 4/, 5/, and 7/"; J.: The Jewish News Weekly of Northern California: "Making Borscht"; Kestrel: "On Hold," "Solids"; La Macchina sognante (bilingual Italian-English publication): "Credo," "Fossils and Fuels: The Evening Commute," "The Life Expectancy of Shirts"; Mission at Tenth: "Blankets and the Lack"; Recours au poème (bilingual French-English publication): "Blankets and the Lack," "Questions about Kissing," "Reunion in Paris"; Santa Cruz Good Times: "Questions about Kissing," "Reading to the Kids"; Scapegoat Review: "4,000 Pianos"; South Florida Poetry Journal: "Psalm Minus One," "The Bar Mitzvah Boy"; Spillway: "Urban Tanka 2/ and 8/"; Star 82 Review: "Deep-Sea Highway"; Switched-on Gutenberg: "If Wordsworth Were Alive Today"; Tiger's Eye: "One in the Other: A Guessing Game," "Remembering Robin Siegel"; The Transnational (bilingual German-English publication): "Credo"; Transparent Words: "Urban Tanka 1/ and 6/"; Valparaiso Poetry Review: "Film Noir"; Wandering Cities: "Man Sitting in Alley with Coat Covering His Head"; Le Voci della Luna (bilingual Italian-English publication): "Blankets and the Lack"; West Texas Literary Review: "Even Shakespeare."
## Contents
1. Irreverent Litanies
2. Copyright © 2019 by Zack Rogow. All rights reserved.
3. Dedication
4. The Big Questions Are Making a Comeback
1. Spiritually Challenged
2. Credo
3. Remembering Robin Siegel
4. Idol Conjectures
5. Kind of Green
1. Fossils and Fuels: The Evening Commute
2. If Wordsworth Were Alive Today
3. Even Shakespeare
4. You Have Someone Else's Name
5. People Keep Having Babies
6. What the World Needs Now
6. Notes I Pass to Myself
1. A Small Knowledge
2. The Sticky Blues
3. Reunion in Paris
4. One in the Other :—A Guessing Game
5. Questions about Kissing
6. Upside-down Sonnet
7. Going the Distance
8. On Hold
7. L'dor V'dor: From Generation to Generation
1. What Do You Think?
2. Labor Coach
3. Reading to the Kids
4. Deep-Sea Highway
5. Sleep, City
6. The Life Expectancy of Shirts
7. Asylum Street, etc.
8. With a Wide-Angle Lens
1. Europe in the Nineteenth Century:
2. Images for a Stereopticon
3. 4,000 Pianos
4. based on a topic suggested by Miranda Rogow Sachs
5. There Should Be Unicorns
6. Film Noir
7. Borscht: A Recipe
8. Buying Challah in Silicon Valley
9. Against Forgetting
1. Legacies
2. For Erin Hyman (1972-2014)
3. Rebbetzin of Temple Beth Sholom, San Francisco
4. For a Dear Cousin
10. Visions of SoMa
1. Blankets and the Lack
2. Man Sitting in Alley with Coat Covering His Head
3. Urban Tanka
11. Psalmodic
1. The Bar Mitzvah Boy
2. Solids
3. Why I Can't Say Your Name
4. Psalm Minus One
5. In memory of Eva Saulitis
6. Dayenu
7. Acknowledgments
## Landmarks
1. Cover
2. Table of Contents
| {
"redpajama_set_name": "RedPajamaBook"
} | 88 |
using Mosa.Compiler.Framework;
namespace Mosa.Platform.ARMv6.Instructions
{
/// <summary>
/// Asr instruction: Arithmetic Shift Right
/// </summary>
public class Asr : ARMv6Instruction
{
#region Construction
/// <summary>
/// Initializes a new instance of <see cref="Asr"/>.
/// </summary>
public Asr() :
base(1, 3)
{
}
#endregion Construction
#region Methods
/// <summary>
/// Emits the specified platform instruction.
/// </summary>
/// <param name="node">The node.</param>
/// <param name="emitter">The emitter.</param>
protected override void Emit(InstructionNode node, MachineCodeEmitter emitter)
{
// TODO
}
#endregion Methods
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 5,034 |
Q: How many classes and packages do we have in java 6? I would like to know the number of built in classes and packages that are available with Java 6.
Please provide me the url from where this information is available.
Thanks
A: I have counted it : 203 packages and 3792 classes
A: 3793 after a line count of http://java.sun.com/javase/6/docs/api/allclasses-frame.html
(less one for the heading)
EDIT: 203 packages in http://java.sun.com/javase/6/docs/api/overview-frame.html
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 1,107 |
\section{Introduction}
There are three types of the well-known neutrino mixing matrices; tribimaximal, bimaximal, and democratic. These three neutrino mixing matrices patterns predict the mixing angle $\theta_{13}=0$. Recently, the evidence of nonzero $\theta_{13}$ due to the achievement of experimental methods and tools, the assumption that the value of mixing angle $\theta_{13}$ is very small and tend to zero must be corrected or even ruled out. Concerning with the well-known mixing matrix, especially tribimaximal neutrino mixing matrix, Ishimori and Ma \cite{Ishimori} stated explicitly that the tribimaximal mixing matrix may be dead due to the experimental fact that mixing angle $\theta_{13}$ is not zero. The nonzero and relatively large mixing angle $\theta_{13}$ have already been reported by MINOS \cite{Minos}, Double Chooz \cite{Double}, T2K \cite{T2K}, Daya Bay \cite{Daya}, and RENO \cite{RENO} collaborations.
The evidence of nonzero and relatively large $\theta_{13}$ as reported by many collaborations, several authors have already proposed some methods and models in order to explain the existence of nonzero $\theta_{13}$. The simple way to accommodate a nonzero $\theta_{13}$ is to modify the neutrino mixing matrix by introducing a perturbation matrix into known mixing matrix such that it can produces a nonzero $\theta_{13}$ \cite{He11, Damanik, Brahmachari}, breaking the scaling ansatz \cite{Biswajit}, and the other is to build the model by using some discrete symmetries \cite{Cao, Luca, Ge1, Ge2}.
In this paper we modify the tribimaximal mixing matrix by introducung a simple perturbation matrix and calculate the mixing angle $\theta_{13}$ by using the advantages of the mixing angles $\theta_{21}$ and $\theta_{32}$ from the experimental results. This paper is organized as follow: in section 2, we modify tribimaximal mixing matrix by introducing a simple perturbation matrix. In section 3, we determine the neutrino mass spectrum from modified tribimaximal mixing matrix. Finally, section 4 is devoted to conclusion.
\section{Nonzero $\theta_{13}$ from the modified tribimaximal mixing matrix}
The tribimaximal neutrino mixing matrix existence is due to the experimental facts that mixing of flavors do exist in the leptonic sector especially in neutrino sector as well as in the quarks sector. The neutrino eigenstates in flavor basis ($\nu_{e}, \nu_{\mu}, \nu_{\tau}$) relate to the eigenstates of neutrino in mass basis ($\nu_{1}, \nu_{2}, \nu_{3}$) as follow:
\begin{eqnarray}
\nu_{i}=V_{ij}\nu_{j},
\end{eqnarray}
where $V_{ij} ( i = e, \mu, \tau; j = 1,2,3)$ are the elements of neutrino mixing matrix. The mixing matrix $V$ can be parameterized as follow:
\begin{eqnarray}
V=\bordermatrix{& & &\cr
&c_{12}c_{13} &s_{12}c_{13} &s_{13}e^{-i\phi}\cr
&-s_{12}c_{23}-c_{12}s_{23}s_{13}e^{i\phi} &c_{12}c_{23}-s_{12} s_{23}s_{13}e^{i\phi}&s_{23}c_{13}\cr
&s_{12}s_{23}-c_{12}c_{23}s_{13}e^{i\phi} &-c_{12}s_{23}-s_{12}c_{23}s_{13}e^{i\phi} &c_{23}c_{13}}
\label{V}
\end{eqnarray}
where $c_{ij}$ is the $\cos\theta_{ij}$, $s_{ij}$ is the $\sin\theta_{ij}$, and $\theta_{ij}$ are the mixing angles.
One of the well-known neutrino mixing matrix ($V$) is the tribimaximal neutrino mixing matrix ($V_{TB}$) which given by \cite{Harrison, Harrisona, Xing, Harrisonb, Harrisonc, He}:
\begin{eqnarray}
V_{TB}=\bordermatrix{& & &\cr
&\sqrt{\frac{2}{3}} &\frac{1}{\sqrt{3}} &0\cr
&-\frac{1}{\sqrt{6}} &\frac{1}{\sqrt{3}} &\frac{1}{\sqrt{2}}\cr
&-\frac{1}{\sqrt{6}} &\frac{1}{\sqrt{3}} &-\frac{1}{\sqrt{2}}}.
\label{tb}
\end{eqnarray}
As one can see from Eq. (\ref{tb}) that the entry $V_{e3}=0$ which imply that the mixing angle $\theta_{13}$ must be zero in the tribimaximal mixing matrix. However, the latest result from long baseline neutrino oscillation experiment T2K indicates that $\theta_{13}$ is relatively large. For a vanishing Dirac CP-violating phase, the T2K collaboration reported that the values of $\theta_{13}$ are \cite{T2K}:
\begin{equation}
5.0^{o}\leq\theta_{13}\leq 16.0^{o}, {\rm and}~5.8^{o}\leq\theta_{13}\leq 17.8^{o},
\end{equation}
for neutrino mass in norma (NH)l and inverted (IH) hierarchies respectively, and the current combined world data\cite{Gonzales-Carcia}-\cite{Fogli}:
\begin{equation}
\Delta m_{21}^{2}=7.59\pm0.20 (_{-0.69}^{+0.61}) \times 10^{-5}~\rm{eV^{2}},\label{21}
\end{equation}
\begin{equation}
\Delta m_{32}^{2}=2.46\pm0.12(\pm0.37) \times 10^{-3}~\rm{eV^{2}},~\rm(for~ NH)\label{32}
\end{equation}
\begin{equation}
\Delta m_{32}^{2}=-2.36\pm0.11(\pm0.37) \times 10^{-3}~\rm{eV^{2}},~\rm(for~ IH)\label{321}
\end{equation}
\begin{equation}
\theta_{12}=34.5\pm1.0 (_{-2.8}^{3.2})^{o},~~\theta_{23}=42.8_{-2.9}^{+4.5}(_{-7.3}^{+10.7})^{o},~~\theta_{13}=5.1_{-3.3}^{+3.0}(\leq 12.0)^{o},
\label{GD}
\end{equation}
at $1\sigma~(3\sigma)$ level. The latest experimental result on $\theta_{13}$ is reported by Daya Bay Collaboration which gives \cite{Daya}:
\begin{equation}
\sin^{2}2\theta_{13}=0.092\pm 0.016 (\rm{stat}.)\pm 0.005 (\rm{syst.}),
\end{equation}
and RENO Collaboration reported that \cite{RENO}:
\begin{equation}
\sin^{2}2\theta_{13}=0.113\pm 0.013 (\rm{stat.})\pm 0.014 (\rm{syst.}).
\end{equation}
Modification of neutrino mixing matrix, by introducing a perturbation matrices into neutrino mixing matrices in Eq. (\ref{tb}), is the easiest way to obtain the nonzero $\theta_{13}$. The value of $\theta_{13}$ can be obtained in some parameters that can be fitted from experimental results. In this paper, the modified neutrino mixing matrices to be considered are given by:
\begin{equation}
V_{{\rm TB}}^{'}=V_{{\rm TB}}V_{y},\label{Modi1}
\end{equation}
where $V_{y}$ is the perturbation matrices to the neutrino mixing matrices. We take the form of the perturbation matrices as follow:
\begin{equation}
V_{y}=\bordermatrix{& & &\cr
&1 &0 &0\cr
&0 &c_{y} &s_{y}\cr
&0 &-s_{y} &c_{y}\cr}.
\label{xy}
\end{equation}
where $c_{y}$ is the $\cos{y}$, and $s_{y}$ is the $\sin{y}$.
By inserting Eqs. (\ref{tb}) and (\ref{xy}) into Eqs. (\ref{Modi1}), we then have the modified neutrino mixing matrices as follow:
\begin{equation}
V_{{\rm TB}}^{'}=\bordermatrix{& & &\cr
&\frac{\sqrt{6}}{3} &\frac{\sqrt{3}}{3}c_{y} &\frac{\sqrt{3}}{3}s_{y}\cr
&-\frac{\sqrt{6}}{6} &\frac{\sqrt{3}}{3}c_{y}-\frac{\sqrt{2}}{2}s_{y} &\frac{\sqrt{3}}{3}s_{y}+\frac{\sqrt{2}}{2}c_{y}\cr
&-\frac{\sqrt{6}}{6} &\frac{\sqrt{3}}{3}c_{y}+\frac{\sqrt{2}}{2}s_{y} &\frac{\sqrt{3}}{3}s_{y}-\frac{\sqrt{2}}{2}c_{y}},\label{Mo1}
\end{equation}
By comparing Eqs. (\ref{Mo1}) with the neutrino mixing in standard parameterization form as shown in Eq. (\ref{V}) with $\varphi=0$, then we obtain:
\begin{equation}
\tan\theta_{12}=\left|\frac{\sqrt{2}c_{y}}{2}\right|,~~
\tan\theta_{23}=\left|\frac{\frac{\sqrt{3}}{3}s_{y}+\frac{\sqrt{2}}{2}c_{y}}{\frac{\sqrt{3}}{3}s_{y}-\frac{\sqrt{2}}{2}c_{y}}\right|,~~
\sin\theta_{13}=\left|\frac{\sqrt{3}}{3}s_{y}\right|.
\label{1}
\end{equation}
From Eq. (\ref{1}) it is apparent that for $y\rightarrow 0$, the value of $\tan\theta_{12}\rightarrow \sqrt{2}/2$ and $\tan\theta_{23}\rightarrow 1$ which imply that $\theta_{12}\rightarrow 35.264^{o}$ and $\theta_{23}\rightarrow 45^{o}$. From Eq. (\ref{1}), one can see that it is possible to determine the value $y$ and therefore the value of $\theta_{13}$ by using the experimental values of $\theta_{12}$ and $\theta_{23}$ in Eq. (\ref{GD}).
By inserting the experimental values of $\theta_{12}$ and $\theta_{23}$ in Eq. (\ref{GD}) into Eq. (\ref{1}), we obtain the relations:
\begin{equation}
c_{y}=-0.03167630078 s_{y},\label{c1}
\end{equation}
when we use $\theta_{23}$, and
\begin{equation}
c_{y}=0.9713265692,\label{c2}
\end{equation}
when we use $\theta_{12}$. From both Eqs. (\ref{c1}) and (\ref{c2}), we can see that the realistic value for $c_{y}$ is the value $c_{y}$ in Eq. (\ref{c2}) that is $y=13.7537^{o}$. It means that in this modification scenario, only the experimental mixing angle $\theta_{12}$ related to the mixing angle $\theta_{13}$. From Eq. (\ref{c2}), we have:
\begin{equation}
\sin\theta_{13}=0.137265,
\end{equation}
that imply the mixing angle $\theta_{13}=7.89^{o}$ which is in agreement with the T2K \cite{T2K} and Daya Bay experimental results \cite{Daya}.
\section{Neutrino masses from modified tribimaximal mixing matrix}
We construct the neutrino mass matrix $M_{\nu}$ in flavor eigenstates basis (where the charged lepton mass matrix is diagonal). In this basis, the neutrino mass matrix can be diagonalized by a unitary matrix $V$ as follow:
\begin{equation}
M_{\nu}=VMV^{T},\label{aa}
\end{equation}
where the diagonal neutrino mass matrix $M=diag(m_{1},m_{2},m_{3})$.
If we put $V$ is the modified neutrino mixing matrix in Eq. (\ref{Mo1}), then Eq. (\ref{aa}) gives the neutrino mass matrix:
\begin{equation}
M_{\nu}=\bordermatrix{& & &\cr
&A &B &C\cr
&B &D &E\cr
& C&E &F}=\bordermatrix{& & &\cr
&(M_{\nu})_{11} &(M_{\nu})_{12} &(M_{\nu})_{13}\cr
&(M_{\nu})_{21} &(M_{\nu})_{22} &(M_{\nu})_{23}\cr
&(M_{\nu})_{31} &(M_{\nu})_{32} &(M_{\nu})_{33}},\label{Mv}
\end{equation}
where:
\begin{equation}
(M_{\nu})_{11}=\frac{2m_{1}}{3}+\frac{m_{2}}{3}c_{y}^{2}+\frac{m_{3}}{3}s_{y}^{2},\label{M11}
\end{equation}
\begin{equation}
(M_{\nu})_{12}=(M_{\nu})_{21}=-\frac{m_{1}}{3}+m_{2}\left(\frac{1}{3}c_{y}^{2}-\frac{\sqrt{6}}{6}c_{y}s_{y}\right)+m_{3}\left(\frac{1}{3}s_{y}^{2}+\frac{\sqrt{6}}{6}s_{y}c_{y}\right),\label{M12}
\end{equation}
\begin{equation}
(M_{\nu})_{13}=(M_{\nu})_{31}=-\frac{m_{1}}{3}+m_{2}\left(\frac{1}{3}c_{y}^{2}+\frac{\sqrt{6}}{6}c_{y}s_{y}\right)+m_{3}\left(\frac{1}{3}s_{y}^{2}-\frac{\sqrt{6}}{6}s_{y}c_{y}\right),\label{M13}
\end{equation}
\begin{equation}
(M_{\nu})_{22}=\frac{m_{1}}{6}+m_{2}\left(\frac{\sqrt{3}}{3}c_{y}-\frac{\sqrt{2}}{2}s_{y}\right)^{2}+m_{3}\left(\frac{\sqrt{3}}{3}s_{y}+\frac{\sqrt{2}}{2}c_{y}\right)^{2},\label{M14}
\end{equation}
\begin{equation}
(M_{\nu})_{23}=(M_{\nu})_{32}=\frac{m_{1}}{6}+m_{2}\left(\frac{1}{3}c_{y}^{2}-\frac{1}{2}s_{y}^{2}\right)+m_{3}\left(\frac{1}{3}s_{y}^{2}-\frac{1}{2}c_{y}^{2}\right),\label{M15}
\end{equation}
\begin{equation}
(M_{\nu})_{33}=\frac{m_{1}}{6}+m_{2}\left(\frac{\sqrt{3}}{3}c_{y}+\frac{\sqrt{2}}{2}s_{y}\right)^{2}+m_{3}\left(\frac{\sqrt{3}}{3}s_{y}-\frac{\sqrt{2}}{2}c_{y}\right)^{2}.\label{M16}
\end{equation}
To simplify the problem such that we can determine the neutrino masses, which can correctly predict the neutrino mass spectrum, we impose texture zero into neutrino mass matrix in Eq. (\ref{Mv}). Texture zero of neutrino mass matrix indicates the existence of additional symmetries beyond the Standard Model Particle Physics \cite {Damanik2007, Fritzsch2011}. By imposing some possibilities texture zero into Eq. (\ref{Mv}), we then find that only one texture zero: $(M_{\nu})_{11}=(M_{\nu})_{13}=0$ can correctly predict the nuetrino mass spectrum. From this texture zero pattern, we have:
\begin{equation}
m_{2}=-1.400444385 m_{1},~m_{3}=-12.00741191 m_{1},\label{mm}
\end{equation}
that implies that the neutrino mass hierarchy is normal hierarchy: $\left|m_{1}\right|<\left|m_{2}\right|<\left|m_{3}\right|$.
If we use the experimental value of the solar neutrino squared-mass difference ($\Delta m_{21}^{2}$) in Eq. (\ref{21}) to determine the neutrino masses in Eq. (\ref{mm}), then we have:
\begin{equation}
m_{1}=0.00888595~{\rm eV},~m_{2}=0.01244428~{\rm eV},~m_{3}=0.10669729~{\rm eV}. \label{m}
\end{equation}
The obtained neutrino masses in Eq. (\ref{m}) cannot give correctly the squared-mass difference for atmospheric neutrino ($\Delta m_{32}^{2}$) in Eq. (\ref{32}). Conversely, if we use the experimental value of $\Delta m_{32}^{2}$ in Eq. (\ref{32}) to determine the value of neutrino masses in Eq. (\ref{mm}), then the obtained neutrino masses cannot correctly predict the squared-mass difference for solar neutrino in Eq. (\ref{21}).
\section{Conclusion}
By introducing a simple perturbation matrix into tribimaximal mixing matrix, we then have the modified tribimaximal neutrino mixing matrix that can give nonzero $\theta_{13}=7.89$ which is in agreement with the present experimental results. The neutrino mass matrix from the modified tribimaximal neutrino mixing matrix with two zeros texture predict the neutrino mass spectrum in normal hierarchy: $\left|m_{1}\right|<\left|m_{2}\right|<\left|m_{3}\right|$. If we use the solar neutrino squared-mass difference to determine the values of neutrino masses, then we cannot have the correct value for the atmospheric squared-mass difference. Conversely, if we use the experimental valeu of the squared-mass difference to determine the neutrino masses, then we cannot have the correct value for the solar neutrino squared-mass difference.
\section*{Acknowledgment}
Author thank to DP2M Dikti Kemendiknas and Sanata Dharma University Yogyakarta for financial support to attend the $24^{th}$ Rencontres de Blois 2012, Chateau France.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 5,034 |
namespace Nighthawk {
MockMetricsEvaluator::MockMetricsEvaluator() = default;
} // namespace Nighthawk
| {
"redpajama_set_name": "RedPajamaGithub"
} | 6,689 |
Espagnac (Espanhac en occitan) est une commune française située dans le département de la Corrèze et la région Nouvelle-Aquitaine. Ses habitants sont les Spaniacois(ses).
Géographie
Commune du Massif central arrosée par la Saint-Bonnette.
Localisation
Climat
Le climat qui caractérise la commune est qualifié, en 2010, de « climat océanique altéré », selon la typologie des climats de la France qui compte alors huit grands types de climats en métropole. En 2020, la commune ressort du même type de climat dans la classification établie par Météo-France, qui ne compte désormais, en première approche, que cinq grands types de climats en métropole. Il s'agit d'une zone de transition entre le climat océanique, le climat de montagne et le climat semi-continental. Les écarts de température entre hiver et été augmentent avec l'éloignement de la mer. La pluviométrie est plus faible qu'en bord de mer, sauf aux abords des reliefs.
Les paramètres climatiques qui ont permis d'établir la typologie de 2010 comportent six variables pour les températures et huit pour les précipitations, dont les valeurs correspondent à la normale 1971-2000. Les sept principales variables caractérisant la commune sont présentées dans l'encadré ci-après.
Avec le changement climatique, ces variables ont évolué. Une étude réalisée en 2014 par la Direction générale de l'Énergie et du Climat complétée par des études régionales prévoit en effet que la température moyenne devrait croître et la pluviométrie moyenne baisser, avec toutefois de fortes variations régionales. Ces changements peuvent être constatés sur la station météorologique de Météo-France la plus proche, « Tulle », sur la commune de Tulle, mise en service en 1957 et qui se trouve à à vol d'oiseau, où la température moyenne annuelle est de et la hauteur de précipitations de pour la période 1981-2010.
Sur la station météorologique historique la plus proche, « Brive », sur la commune de Brive-la-Gaillarde, mise en service en 1987 et à , la température moyenne annuelle évolue de pour la période 1971-2000, à pour 1981-2010, puis à pour 1991-2020.
Urbanisme
Typologie
Espagnac est une commune rurale. Elle fait en effet partie des communes peu ou très peu denses, au sens de la grille communale de densité de l'Insee.
Par ailleurs la commune fait partie de l'aire d'attraction de Tulle, dont elle est une commune de la couronne. Cette aire, qui regroupe , est catégorisée dans les aires de moins de .
Occupation des sols
L'occupation des sols de la commune, telle qu'elle ressort de la base de données européenne d'occupation biophysique des sols Corine Land Cover (CLC), est marquée par l'importance des forêts et milieux semi-naturels (62,9 % en 2018), une proportion sensiblement équivalente à celle de 1990 (63,8 %). La répartition détaillée en 2018 est la suivante :
forêts (61,4 %), prairies (29 %), zones agricoles hétérogènes (4,3 %), milieux à végétation arbustive et/ou herbacée (1,5 %), eaux continentales (1,4 %), espaces verts artificialisés, non agricoles (1,3 %), zones urbanisées (1,1 %).
L'IGN met par ailleurs à disposition un outil en ligne permettant de comparer l'évolution dans le temps de l'occupation des sols de la commune (ou de territoires à des échelles différentes). Plusieurs époques sont accessibles sous forme de cartes ou photos aériennes : la carte de Cassini (), la carte d'état-major (1820-1866) et la période actuelle (1950 à aujourd'hui).
Risques majeurs
Le territoire de la commune d'Espagnac est vulnérable à différents aléas naturels : météorologiques (tempête, orage, neige, grand froid, canicule ou sécheresse), inondations et séisme (sismicité très faible). Il est également exposé à un risque particulier : le risque de radon. Un site publié par le BRGM permet d'évaluer simplement et rapidement les risques d'un bien localisé soit par son adresse soit par le numéro de sa parcelle.
Risques naturels
Certaines parties du territoire communal sont susceptibles d'être affectées par le risque d'inondation par débordement de cours d'eau, notamment la Saint-Bonnette. La commune a été reconnue en état de catastrophe naturelle au titre des dommages causés par les inondations et coulées de boue survenues en 1982, 1983, 1999, 2001, 2016 et 2018. Le risque inondation est pris en compte dans l'aménagement du territoire de la commune par le biais du plan de prévention des risques (PPR) inondation « Vézère », approuvé le .
Le retrait-gonflement des sols argileux est susceptible d'engendrer des dommages importants aux bâtiments en cas d'alternance de périodes de sécheresse et de pluie. 52 % de la superficie communale est en aléa moyen ou fort (26,8 % au niveau départemental et 48,5 % au niveau national). Sur les dénombrés sur la commune en 2019, sont en en aléa moyen ou fort, soit 48 %, à comparer aux 36 % au niveau départemental et 54 % au niveau national. Une cartographie de l'exposition du territoire national au retrait gonflement des sols argileux est disponible sur le site du BRGM.
Par ailleurs, afin de mieux appréhender le risque d'affaissement de terrain, l'inventaire national des cavités souterraines permet de localiser celles situées sur la commune.
Concernant les mouvements de terrains, la commune a été reconnue en état de catastrophe naturelle au titre des dommages causés par la sécheresse en 2018, 2019 et 2020 et par des mouvements de terrain en 1999.
Risque particulier
Dans plusieurs parties du territoire national, le radon, accumulé dans certains logements ou autres locaux, peut constituer une source significative d'exposition de la population aux rayonnements ionisants. Certaines communes du département sont concernées par le risque radon à un niveau plus ou moins élevé. Selon la classification de 2018, la commune d'Espagnac est classée en zone 3, à savoir zone à potentiel radon significatif.
Économie
Histoire
En dépit de plusieurs indices d'une occupation néolithique, puis celtique, sur le territoire de la commune, le village lui-même semble n'avoir été fondé qu'au VIIème siècle, sur le site d'une ancienne villa gallo-romaine. Celle-ci n'a fait, jusqu'à présent, l'objet d'aucune fouille. On trouve également sur la commune les traces d'autres villae et d'une tuilerie de même époque.
C'est durant le Haut Moyen Âge, à la période mérovingienne, qu' Espagnac semble avoir acquis une importance locale, attestée par l'existence d'un atelier monétaire ( trois Triens d'or mérovingiens portant la mention "Spaniaco" y ont été découverts ), d'un cimetière étendu d'où de nombreux sarcophages ont été exhumés, et d'une abbaye. Espagnac devint ensuite le siège (chef-lieu) d'une Vicairie civile très étendue sous les Carolingiens.
La première mention écrite de son nom apparaît vers 930, dans le testament d' Adhémar des Échelles. Il est repris ensuite dans différents cartulaires. Devenue dépendance de l'abbaye de Tulle, Espagnac déclina durant le bas Moyen Âge, période où de nombreuses exploitations agricoles y sont reprises en charge par le monastère d' Aubazine (XIV-XVème siècle).
La tour du Prieur, une église et une chapelle dédiées au culte marial furent brûlées durant les guerres de Religion par les troupes de l'amiral de Coligny, au retour de leur défaite à Moncontour. L'église principale, initialement fondée au VIIème siècle, fut plusieurs fois reconstruite et modifiée. Son bâtit actuel garde des traces du XIVème siècle, mais fut largement remanié depuis le XVIème siècle. Ses vitraux ont été restaurés récemment. Une petite chapelle carolingienne restaurée et le château médiéval de Puy-de-Val (propriété privée), inscrit à l'inventaire des monuments historiques, subsistent sur le territoire de la commune.
Héraldique
Politique et administration
Démographie
Environnement
La commune bénéficie du label "Village étoilé" pour la protection du ciel et l'environnement nocturnes.
Lieux et monuments
L'église paroissiale Saint-Gervais-Saint-Protais d'Espagnac.
L'ancienne gare de "Le Mortier-Gumond", sur la ligne du Transcorrézien tronçon Le Mortier-Gumond / Laroche-Canillac faisant partie des Tramways de la Corrèze, qui fonctionna de 1913 à 1938. Cette gare, aujourd'hui en mauvais état, est située sur le territoire de la commune dans le hameau du Mortier à équidistance entre les deux communes d'Espagnac et de Gumond.
Personnalités liées à la commune
Jean-Baptiste Joseph de Sahuguet d'Amarzit d'Espagnac (1713-1783), général français.
Marc René Marie d'Amarzit de Sahuguet d'Espagnac (1752-1794), spéculateur financier français.
Charles-Antoine Léonard de Sahuguet, baron d'Espagnac et comte de Sancerre (1758-1837).
Notes et références
Notes et cartes
Notes
Cartes
Voir aussi
Articles connexes
Liste des communes de la Corrèze
Anciennes communes de la Corrèze
Communauté d'agglomération Tulle Agglo
Liste des monuments historiques de la Corrèze
Liens externes
Commune en Corrèze
Commune dans l'arrondissement de Tulle
Commune dans la communauté d'agglomération Tulle Agglo
Aire urbaine de Tulle
Aire d'attraction de Tulle | {
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Discussion in 'rFactor' started by St3fan, Dec 8, 2015.
1. I see HistorX is the most discussed here. Is there a non-p2p way to download the mod? I find even the FTP download link on HistorX's official website leads to a torrent file. I can't do and P2P download on my computer because I think the protocol is prohibited by my ISP.
2. Is there a way to improve FFB? I've downloaded mods according to people's suggestions, including a F1 2005 mod, Le Mans Mod, Enduseries by Enduracers, Virtua_LM, Shift Streets, etc. All of them give me a very bad feeling in FFB even after I put the realfeel plugin settings according to the modder. All I feel from my G27 is just pure resistance without any information about the road (curbs can be felt of course but that's pretty much it for me) and how my front/back wheels are doing. I assume there is a way to vastly improve the FFB in RF1 because I love how the cars feel in GSC a lot. Could you guys give me some suggestions?
I have GT Legends. But for some reason after I upgrade my video card it doesn't run any more.
1. HistorX was hosted on drivingitalia but i just checked, now you can only grab the torrent there too, not really aware of a different mirror at the moment.
2. I don't know much about rf1 vanilla ffb but HistorX comes with realfeel by default, it's important to change the ffb text and set it up like explained in the manual though.
You do not need to have GTL installed, a disc (or digital download somewhere on you pc) will be checked before HistorX installs, that's all.
And anything you need to know about our club you can find in Stig's videos . | {
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{"url":"https:\/\/www.os2world.com\/wiki\/index.php\/EmTeX_and_the_WorkPlace_Shell","text":"# EmTeX and the WorkPlace Shell\n\nDate: 11-Aug-1992\n\nYou don't need an integrated TeX environment to use emTeX under OS\/2 2.0. An integrated environment comes with OS\/2 2.0: the WorkPlace Shell. Here's a tutorial.\n\nNote: texit.cmd can also be used from the command prompt under OS\/2 1.3.\n\nThis has to be done only once:\n\nCreate an emTeX folder and put there program icon for texit.cmd. (see your OS\/2 documentation for details). texit.cmd is available in the directory where you've found this file.\n\nEnter \"%**n\" (omit the quotes) in the \"Parameters\" field of the program objects and leave the \"Working directory\" field empty.\n\nYou can also create program objects for vp.cmd (using the same settings) and prthplj.cmd, for instance. Then, you can print a DVI file by moving the icon of the DVI file onto the icon for vp.cmd or prthplj.cmd.\n\nMake a copy of the \"Data File\" icon in the \"Templates\" folder (use Ctrl and the right mouse button) and rename it \"TeX File.tex\". (You'll get asked whether you want to keep the extension. Answer \"No\".)\n\nThis has to be done once per document:\n\nCreate a folder for your document: Drag a Folder icon from the Templates folder to somewhere. Rename the folder \"textest\", for instance. Drag the \"TeX File.tex\" icon from the Templates folder to the \"textest\" folder to create a new TeX document. Rename it \"textest.tex\".\n\nThis is how to use the icons:\n\nDouble-click on the icon to invoke the editor.\n\n\u00a0%Format: lplain\n\\documentstyle{article}\n\\begin{document}\nBlah blah bla\n\\end{document}\n\n\nand save the file. Move the icon onto the texit.cmd icon. LaTeX will be invoked to compile the text. If the dvitest.dvi etc. icons don't show up automatically, select \"Refresh\" for the textest folder. Move the textest.dvi icon onto the dvipm icon to preview. Move the textest.dvi icon onto the prthplj icon to print.\n\nPut texit2.cmd (available in the same directory as this file) into a directory listed in the PATH environment variable, c:\\emtex for instance. Change the \"TeX File.tex\" object in the Templates folder by opening the Settings notebook, choosing the Menu tab, selecting the \"~Open\" entry in the upper list box, selecting the lower \"Create another...\" button, filling the dialog box with \"TeX\" (Menu item name) and \"texit2.cmd\" (Program name).\n\nAfter doing that, new \"TeX File.tex\" objects will have an additional item in the Open menu, TeX. Selecting that item will run texit2.cmd, which will call texit.cmd (which will call emTeX) to compile your TeX file (the \"TeX file.tex\" object).\n\ntexit.cmd details\n\ntexit.cmd is a batch file (written in REXX - it will run under OS\/2 1.2 extended edition, OS\/2 1.3 and OS\/2 2.0) for running emTeX. It reads the first line of the input file to find out what format file to use for the input file. For instance, place\n\n%format lplain\n\n\nin the first line of your file to automatically choose lplain.fmt. You can also use\n\n%Format: lplain\n%format: lplain\n\n\nand\n\n%format lplain\n\n\nText after the name of the format file will be ignored. You can insert any number of blanks between \"%\" and \"format\". You can insert any number of blanks between \"format\" (or \"format:\") and the name of the format.\n\nThe batch file also runs an editor if you correctly set the EMTEXED environment variable and exit TeX by typing e (see texware.doc or tex.doc). texit.cmd automatically chooses a unique temporary file name for the \/a option.\n\ntexit.cmd calls tex386.exe if you're using OS\/2 2.0 and if tex386.exe is present. Otherwise, texp.exe is used.\n\nIcon for dvipm\n\nHere's how to set the dvipm icon for vp.cmd: Open the settings notebook for dvipm.exe. Select the \"General\" page. Edit the icon. Select \"Save as\" and store the icon into dvipm.ico, for instance. Quit Icon Editor. Now you can select dvipm.ico as icon for vp.cmd.\n\nThis does not yet work\n\nTo make dvipm be called when you double-click a DVI file icon, open the settings for vp.cmd, select the Association tab, type \"*.dvi\" in the \"New Name\" field and press the \"Add\" button.\n\nAs the name of the Desktop folder contains special characters (a blank, for instance), this works only for DVI files in 'sane' directories, that is, folders that are not subfolders of the desktop folder.","date":"2018-06-18 09:42:34","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 1, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.5927246809005737, \"perplexity\": 11391.508523358845}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2018-26\/segments\/1529267860168.62\/warc\/CC-MAIN-20180618090026-20180618110026-00208.warc.gz\"}"} | null | null |
Q: How to use magneticHeading in Swift? I'm trying to make a simple by feet navigator, using a arrow image, rotating to indicate the direction to follow with a constant refresh.
Using magneticHeading i found error unexpectedly found nil while unwrapping an Optional value, maybe i forgot something.
How i can use correctly magneticHeading?
var northLatitude :CLLocationDegrees
var latitude :CLLocationDegrees = 46.0
var longitude :CLLocationDegrees = 7.0
func locationManager(manager: CLLocationManager!, didUpdateLocations locations: [AnyObject]!) {
longitude = manager.location.coordinate.longitude
latitude = manager.location.coordinate.latitude
var buildingLocation:CLLocationCoordinate2D = CLLocationCoordinate2DMake(latitude, longitude)
var regionOfInterest:MKCoordinateRegion = MKCoordinateRegionMake(buildingLocation, span)
self.mapView.setRegion(regionOfInterest, animated: true)
self.mapView.addAnnotation(buildingAnnotation)
northLatitude = manager.heading.magneticHeading //ERROR
}
A: You can calculate the angle at which the arrow should point and then use the transform property of the image view to change direction. Pseudo code will be something like this.
func onPositionChange()
{
UIView.animateWithDuration(1.0, animations:
{
var angle_in_degrees:CGFloat = 10 // Calculated Angle.
var angle_in_radians = (angle_in_degrees * 3.1415) / 180.0
self.imgView.transform = CGAffineTransformMakeRotation(angle_in_radians)
})
}
| {
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package org.knowm.xchange.coindirect.dto.marketdata;
import static org.assertj.core.api.Assertions.assertThat;
import com.fasterxml.jackson.databind.ObjectMapper;
import java.io.IOException;
import java.io.InputStream;
import org.junit.Test;
public class CoindirectTradesTest {
@Test
public void testUnmarshal() throws IOException {
// Read in the JSON from the example resources
InputStream is =
CoindirectTradesTest.class.getResourceAsStream(
"/org/knowm/xchange/coindirect/dto/marketdata/example-trade-history.json");
ObjectMapper mapper = new ObjectMapper();
CoindirectTrades coindirectTrades = mapper.readValue(is, CoindirectTrades.class);
// Verify that the example data was unmarshalled correctly
assertThat(coindirectTrades.data.size()).isEqualTo(2);
assertThat(coindirectTrades.metaData.market).isEqualTo("ETH-BTC");
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 8,426 |
Q: Kubernetes on google cloud with hostPath mount I've been developing an app on my local laptop (Mac) with Minikube. Instead of packaging the code and files into the docker image, I use hostPath and volumeMount that points to the code/file directory on my Mac, so that I can avoid rebuilding the image every time.
Now I would like to do the same iterative testing with google cloud. What's the best way to "mount" my local code/file directory and run pods remotely on the cloud? I don't want to package the code into a docker image, push to dockerhub, and then pull from dockerhub on gcloud. My dockerhub is a free account and would expose my code.
A: You want:
You want to mount your local file system into your remote Kubernetes cluster.
Answer:
As far I know, you can't do this. Its possible in minikube, because, you can mount your local directory with minikube.
Solution:
I can tell you an alternative way. May be this is not what you want. But it can help you.
Do you use git? If your answer is yes and also if you have no problem to keep your files into git repository, following process will help you.
spec:
containers:
- image: nginx
name: nginx
volumeMounts:
- mountPath: /mypath
name: git-volume
volumes:
- name: git-volume
gitRepo:
repository: "git@somewhere:me/my-git-repository.git"
revision: "22f1d8406d464b0c0874075539c1f2e96c253775"
When you will create this Pod, my-git-repository will be mounted into your directory /mypath inside your Pod container.
Basically, you can tell your Pod to pull this git from specific branch. So every time, you change your code, push it. Then create Pod again.
Read volumes/#gitrepo
A: Easiest method to replicate your setup would be to use a storage bucket for the mount point.
For your setup, just pull the code to the local host when needing to build from the storage bucket. I am assuming you have a build script to do the configuration part.
However as per the other comment, you could just use gcr to host your config files and use deployment manager to build.
A: Steps for using the Google Cloud Registry:
Build Docker Image
docker build -t <image-name>:<tag> <path-to-dockerfile>
Tag for GCloud Container Registry
docker tag <image-name>:<tag> us.gcr.io/<gcloud-project-id>/<image-name>:<tag>
Container Registry
gcloud docker -- push us.gcr.io/<gcloud-project-id>/<image-name>:<tag>
Your spec will then point to the container registry path:
spec:
containers:
- name: hello-world
image: us.gcr.io/<gcloud-project-id>/<image-name>:<tag>
ports:
- name: http
containerPort: 8080
| {
"redpajama_set_name": "RedPajamaStackExchange"
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Вну́трішня торгі́вля — це торгівля, яка ведеться виключно в межах певної країни. Розділена така торгівля на дві категорії — гуртова та роздрібна. Гуртова торгівля пов'язана з купівлею товарів у виробників або дилерів у великій кількості і продаж в невеликій кількості для тих, хто може купити в роздріб.
Роздрібна торгівля пов'язана з продажем товарів у невеликих кількостях для споживачів. На практиці, однак, виробники й гуртовики можуть також проводити роздрібну торгівлю товарів в обхід посередника роздрібної торгівлі, за допомогою яких вони отримують більш високий прибуток.
Значення і роль
Важливість внутрішньої торгівлі в країні полягає в тому, що вона полегшує обмін товарів всередині країни. Роблячи це, вона також гарантує, що фактори виробництва досягають потрібних місць, щоб економіка країни могла вирости. Надаючи різні види товарів та послуг для доставки у всі частини країни покращує рівень життя жителів країни, а також рівень зайнятості в країні. І це сприяє зростанню галузі за рахунок забезпечення доступності сировини.
Гуртова торгівля
Гуртова торгівля здійснюється за цінами, які нижчі від роздрібних цін, але вони повинні забезпечувати нормально працюючому промисловому та торговельному підприємству відшкодування витрат, внесення платежів у бюджет і створення фондів економічного стимулювання та прибуток. Гуртова торгівля стимулює збут товарів, формує їхній асортимент, складає і транспортує товари.
Найвпливовішими гуртовими торговельними структурами є великі торговельні фірми, торгові доми, дистриб'юторські фірми тощо.
Роздрібна торгівля
В Україні на практиці існує недосконалість цього терміну. Наприклад, чи можна визначити, яку ліцензію повинний мати продавець у випадку придбання платником єдиного податку кількох ящиків горілки? Якщо цей платник бере товар для перепродажу, — то гуртову, а якщо для власних потреб — то роздрібну. А тому, якщо відсутня відповідна ліцензія, є всі підстави застосувати до продавця санкції, передбачені ст. 156 КУпАП.
Мета купівлі товару є вагомою для маркетології. Купівельні мотиви є визначним чинником для сегментації ринку та визначення цільової групи споживачів.
Див. також
трансфертне ціноутворення
Примітки
Посилання
Сутність та місце гуртової торгівлі в ринкових умовах
Джерела
Внутрішня торгівля: регіональні аспекти розвитку : монографія / [Азарян О. М., Антонюк Я. М., Апопій В. В. та ін.] ; за заг. наук. ред. : О. О. Шубіна, Я. А. Гончарука ; Донец. нац. ун-т економіки і торгівлі ім. Михайла Туган-Барановського, Укоопспілка, Львів. комерц. акад. — Донецьк : ДонНУЕТ ; Львів : [б. в.], 2007. — 404 с. : іл., табл. — Бібліогр.: с. 369—373 (79 назв). — 600 пр. — ISBN 978-966-385-065-8
Внутрішня торгівля України: проблеми і перспективи розвитку : монографія / за ред. В. В. Апопія, П. Ю. Балабана. – Львів : Новий Світ-2000, 2014. – 565 с. : іл. – Бібліогр.: с. 545-564 (273 назви). – ISBN 978-966-418-271-0
Торгівля | {
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Q: Fill null values with next non-null value bigquery I have the following table with data from GA in BigQuery
userid visitid purchase_date
GH8932 12345 2017-04-09
GH8932 12346 null
GH8932 12347 null
GH8932 12348 null
GH8932 12349 2017-05-30
GH8932 12350 null
GH8932 12351 null
GH8932 12352 2017-06-07
GH8932 12353 null
GH8932 12354 2017-06-30
And I want the resulting table to be
userid visitid purchase_date
GH8932 12345 2017-04-09
GH8932 12346 2017-05-30
GH8932 12347 2017-05-30
GH8932 12348 2017-05-30
GH8932 12349 2017-05-30
GH8932 12350 2017-06-07
GH8932 12351 2017-06-07
GH8932 12352 2017-06-07
GH8932 12353 2017-06-30
GH8932 12354 2017-06-30
I tried the following
select
a.userid,
a.visitid,
b.purchase_date
from x
left join
(
select
userid,
visitid,
purchase_data
from x
where purchase_date is not null) as b
on x.userid = b.userid
where x.visitid <= b.visitid
But, this doesn't provide the solution I'm looking for.
Appreciate the help for a BQ rookie
A: Use IGNORE NULLS with FIRST_VALUE to find a non null date in the desired window:
SELECT
userid,
visitid,
FIRST_VALUE(purchase_date IGNORE NULLS) OVER (
PARTITION BY userid ORDER BY visitid
ROWS BETWEEN CURRENT ROW AND
UNBOUNDED FOLLOWING) AS purchase_date
FROM x;
| {
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Joseph Michael Manganiello (født 28. december 1976) er en amerikansk skuespiller. Hans professionelle filmkarriere begyndte, da han spillede Flash Thompson i Sam Raimis Spider-Man. Hans gennembrudsrolle var som varulven Alcide Herveaux i fem sæsoner af HBO-serien True Blood.
Manganiello er også kendt for sine roller i film som Magic Mike, Magic Mike XXL, Pee-wees Big Holiday, What to Expect When You're Expecting, og Sabotage. Han er også kendt for sin rolle i How I Met Your Mother som Marshalls studiekammerat Brad. I slutningen af 2013 blev han en udgivet forfatter, da hans første bog, Evolution, blev udgivet af Simon & Schusters Gallery Books. Hans instruktørdebut kom i 2014 med dokumentarfilmen La Bare, som han også producerede og finansierede. I 2017 vandt han en Mid-Atlantic Emmy for sin fortæller-rolle i dokumentarfilmen Pittsburgh Is Home: The Penguins Story, der dokumenterede ishockeyholdet Pittsburgh Penguins' første 50 års historie. Han er del af flere velgørende organisationer, hovedsageligt UPMC Children's Hospital i Pittsburgh, hvor han sidder som del af bestyrelsen. I 2016 blev Manganiello castet som Slade Wilson / Deathstroke i DC Extended Universe, hvor han havde sin første optræden i Justice League i 2017.
Opvækst
Manganiello blev født i Pittsburgh, Pennsylvania, som søn af Susan (født Brachanow; ) og Charles John Manganiello. Hans far er født og opvokset uden for Boston, er af italiensk afstamning fra Napoli, og hans mor er af armensk, østrigsk og kroatisk herkomst. Hans oldemor var en overlever af det armenske folkedrab, hvor hendes mand og syv af hendes børn blev myrdet, og hendes ottende barn druknede under flugten.
Manganiello voksede op i Mt. Lebanon, Pennsylvania. Han har en yngre bror, Nicholas. Han gik på på St. Bernard, en romersk-katolsk folkeskole i Mt. Lebanon og gik derefter på Mt. Lebanon high school, hvorfra han dimitterede med "honors" i 1995 og vandt skolens Great Alumni Award i 2011. I skoletiden var han kaptajn for sine football-, basketball- og volleyballhold og fortsatte med at spille på universitetsniveau i alle tre sportsgrene. I hans sidste skoleår fik han rollen som Jud Fry i skolens opsætning af Oklahoma! og var involveret i skolens tv-studie. Han lånte udstyr til at lave film med sine venner og blev interesseret i at spille skuespil for at blive en bedre filmskaber.
Efter en række sportsskader, herunder et overrevet ledbånd i knæet efter at have sparket et kickoff i en footballkamp mod Ringgold High School, gik Manganiello til optagelsesprøve på Carnegie Mellon School of Drama i løbet af hans sidste år i high school. Han kom ikke ind, men meldte sig så ind på University of Pittsburgh og fik arbejde på et teater. Han ansøgte til Carnegie Mellon et år senere og blev en af 17 studerende, der blev optaget til skuespilprogrammet. Han optrådte i teaterproduktioner og skrev, producerede og spillede i en studenterfilm med titlen Out of Courage 2: Out for Vengeance. Han tog afgangseksamen i 2000 med en BFA i skuespil. Han rejste til New York og Los Angeles med hjælp fra universitet for at deltage i gruppeauditioner, som gav ham kontakter i underholdningsbranchen.
Karriere
Tidlig karriere
Mens han læste ved Carnegie Mellon University, optrådte Manganiello i talrige opsætninger på Pittsburghs teaterscener, bl.a. Ulfheim i Henrik Ibsens When We Dead Awaken, Lorenzo i Shakespeares The Merchant of Venice på Quantum Theatre og som Joe i Pittsburgh-premieren på The Last Night of Ballyhoo. Han flyttede til Los Angeles, Californien efter eksamen fra Carnegie Mellon. Han blev hurtigt opskrevet hos en talentagent, og tre dage senere gik han til audition til rollen som Peter Parker i Sam Raimi-instruerede filmen Spider -Man (2002). Han fik rollen som Eugene "Flash" Thompson, Peter Parkers ærkefjende, som sit første skuespilsjob efter college. Han genoptog rollen flere år senere og havde en kort cameo i slutningen af Spider-Man 3 (2007).
Manganiello begyndte at finde arbejde indenfor tv, hvor han spillede Tori Spellings kæreste på VH1's So Notorious i 2006, og gæsteoptrådte i Las Vegas, Jake in Progress og Close to Home. Samme år spillede han også John Leguizamos Anonyme Alkoholikere-sponsor i CBS-tv-piloten Edison. I 2007 optrådte han i Scrubs-afsnittet " My No Good Reason " og i MyNetwork TV's senaftenssæbeopera American Heiress. Han spillede også Officer Litchman, Linda Cardellinis karakters flirt, i fire afsnit af NBC's ER. Han vendte tilbage til teatret og spilledeThe Chick Magnet i maj 2007 ved premieren i New York City for Skirts & Flirts, et monologshow af Gloria Calderon Kellett, en rolle der gjorde ham til finalist ved HBO's "Aspen Comedy Festival". Han spillede derefter Stanley Kowalski i A Streetcar Named Desire på West Virginia Public Theatre i 2008, som blev instrueret af hans tidligere Carnegie Mellon-professor Geoffrey Hitch.
Han spillede hovedrollen som Leo Belraggio, en jazzmusiker fra New York, i vestkystens premiere af Terrence McNallys Unusual Acts of Devotion i juni 2009. Stykket blev opført på La Jolla Playhouse ved University of California i San Diego. Sommeren før arbejdede Manganiello sammen med McNally og instruktør Leonard Foglia om at skabe rollen til Ojai Playwrights Conference. Han spillede Brad Morris gennem flere sæsoner af CBS-komedien How I Met Your Mother. I 2008 sluttede han sig til rollelisten i The CW-dramaet One Tree Hills femte sæson, hvor han spillede bartender Owen Morello. Scenerne blev optage i Wilmington, North Carolina, og han vendte tilbage til rollen i seriens sjette og syvende sæson.
Manganiello spillede Stu på Fox-sitcommen 'Til Death i to afsnit og medvirkede det år også i kortfilmen Wounded, som han vandt "Best Short Film" ved Big Island Film Festival 2011 for. Han medvirkede i direct-to-video krigsfilmen Behind Enemy Lines: Colombia i 2009, hvor han spillede løjtnant Sean Macklin, en Navy SEAL-gruppeleder. For at tilføre produktionen ægthed, trænede han i flere måneder med en tidligere Navy SEAL, som han betalte for at være på settet og bo på skuespillernes hotel. Filmen blev optaget i Puerto Rico. Han optrådte i et afsnit af Medium i 2009 og har gæsteoptrådt i alle tre serier af CBS's CSI-tv-serier (CSI: Crime Scene Investigation, CSI: Miami og CSI: NY). Han havde en rolle i den independent film Irene in Time. I 2010 optrådte han i tv-reklamer for Taco Bell. Han optog tv-pilotafsnit til 100 Questions og til Pittsburgh-sitcommen Livin 'on a Prayer.
True Blood
I slutningen af 2009 blev Manganiello castet som varulv Alcide Herveaux i tredje sæson af HBO's True Blood. Hans arbejde med serien gennem de næste fem år gjorde ham til et kendt ansigt og han modtog flere anerkendelser, bl.a. i 2011 med en Scream Award for "Breakout Performance - Male" og en delt pris for "Best Ensemble", samt en Saturn Award for "Best Guest Starring Role in Television" samt en NewNowNext Award. True Blood var hans yndlingstv-serie, før han sluttede sig til rollelisten. Han blev oprindeligt castet til rollen som Coot, en anden varulv i serien, men blev bedt af producenterne om at læse rollen som Alcide i stedet. Han læste bøgerne bag serien, mens han lod sit hår og skæg gro ud. Han trænede to gange om dagen i flere måneder for at få en muskelmasse, der matchede beskrivelsen af karakteren i bøgerne. Manganiello gik også i solarie for at kunne skille sig ud fra vampyrerne i serien og brugte tid på at studere ægte levende ulve.
I begyndelsen af 2011 blev Manganiello inviteret til audition som Superman i Man of Steel. På grund af planlægningsproblemer med True Blood blev han tvunget til at springe fra de sidste auditionsprocesser. Han fortalte Access Hollywood i et interview: "De ville have mig til at prøvekøre rollen og de bad faktisk om mine mål til en dragt og alt det ... deres optagelsesdatoer ændrede sig, og det ville have taget 11 uger ud af mit True Blood-skema. Mine skemaer ville ikke kunne hænge sammen i sidste ende ... så desværre kom jeg aldrig til at prøve rollen af på skærmen, jeg fik aldrig prøvet dragten. " Efter afslutningen af optagelserne til sæson fire af True Blood, medvirkede han et afsnit af den amerikanske tv-serie White Collar med sin tidligere dramaskole klassekammerat Matt Bomer og optrådte herefter i filmatiseringen af bestselleren What To Expect When You're Expecting, før han vendte tilbage til Los Angeles for at medvirke i et afsnit af Two and a Half Men overfor Ashton Kutcher.
Magic Mike
Manganiello blev castet som Big Dick Richie i Steven Soderberghs Magic Mike, i historien om en ung stripper i Tampa, Florida, spillet af skuespilleren Channing Tatum. Magic Mike ville ende med at blive et kulturelt og box office-fænomen, der affødte efterfølgeren Magic Mike XXL i 2015.
Imellem Magic Mike-filmene og afslutningen af sine sidste to sæsoner af True Blood forgrenede Manganiello sig på en lang række forskellige måder. Han medvirkede i David Ayers film Sabotage med sit barndomsidol Arnold Schwarzenegger, der endte med at blive en ven og mentor for Manganiello og skrev forordet til hans bog Evolution, der blev udgivet gennem Simon & Schuster's Gallery Books i efteråret 2013. I det efterår havde Manganiello også en tilbagevenden til teaterscenen som Stanley Kowalski i A Streetcar Named Desire, men denne gang var det ved det prestigefyldte Yale Repertory Theatre i New Haven, Connecticut.
Et par måneder senere tog Manganiello sin dokumentarfilm La Bare, som han finansierede, instruerede og producerede under sit nye produktionsselskab "3:59" med sin bror Nick, til Park City, Utah til Slamdance Film Festival. Han modtog flere tilbud fra distributører i lobbyen efter den første screening og solgte inden for 24 timer de internationale og indenlandske rettigheder, bl.a. til Showtime og Netflix. Manganiello modtog "Triple Threat Award" fra Maui International Film Festival senere samme år for sit arbejde med filmen.
Efter True Blood
I august 2016 blev testoptagelser af Slade Wilson/Deathstroke på settet af en endnu ikke navngivet film lagt ud på Twitter af Ben Affleck. I september 2016 bekræftede Geoff Johns, DC Comics' formand og CCO der medstyrede DCEU og medforfattede/medproducerede Batman-filmen med Affleck, at Deathstroke vil optræde i det fælles filmunivers, hvor Manganiello portrætterer karakteren. I 2019 blev han bekræftet at dukke op i Ben Afflecks Batman-filmen.
I 2017 vandt Manganiello en Emmy for sin fortæller-rolle i dokumentaren Pittsburgh is Home: The Penguins Story , der dokumenterede ishockeyholdet Pittsburgh Penguins hockeyholds første 50 års historie.
Privatliv
Manganiello er fan af Pittsburgh Steelers og rugby league-holdet Wests Tigers. Han instruerede og producerede den korte dokumentarfilm DieHardz fra 2007 om Steelers-fans, der mødes på barer i Los Angeles, Californien . Manganiello er også en kendt Pittsburgh Penguins-fan og var vært for NHL Awards 2017 og NHL Expansion Draft 2017.
Manganiello var engang en roadie for bandet Goldfinger og er ven med forsanger John Feldmann; han turnerede internationalt med gruppen som del af deres sikkerhedshold.
Manganiello blev forlovet med skuespillerinden Sofía Vergara juledag 2014 efter seks måneders forhold. De giftede sig i Palm Beach, Florida den 21. november 2015.
Han har også været en livslang Dungeons & Dragons-fan og -spiller, der har optrådt i afsnit af The Big Bang Theory, Nerd Poker, CelebriD & D, webshowet Critical Role og Force Gray, hvor han spiller spillet. Han var involveret i Hasbros nye udgave af eventyrbrætspillet Hero Quest ved at skrive en opgavebog som et stretch goal. Selvom stretch goalet ikke blev nået, meddelte han senere, at han arbejdede sammen med Hasbro for at få opgavebogen og de yderligere spilstykker inkluderet alligevel.
Manganiello er udøver af transcendental meditation .
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\section{Introduction}
\label{sec1}
A unique feature of cluster randomized trials (CRTs) is that intact clusters, such as schools or clinics, are randomized to intervention arms \cite{Murray1998,Donner2000}. Randomization at the cluster level often carries pragmatic considerations, for example, administrative convenience, political reasons and prevention of treatment contamination \cite{Turner2017a}. A stepped wedge cluster randomized trial is a type of longitudinal design that sequentially switches clusters to intervention during the course of the study until all clusters are treated \cite{Hussey2007}. Such designs have become increasingly popular due to their logistical flexibility and perceived ethical benefits. Because individual outcomes within the same cluster tend be more similar than those in different clusters, the intraclass correlation coefficient (ICC) plays a central role in designing CRTs. While the traditional posttest-only parallel design (i.e. parallel cluster randomized design without a baseline period as defined in Murray \cite{Murray1998}) requires adjustment for a single ICC, longitudinal cluster randomized design such as the crossover or stepped wedge design allows multiple outcome measurements from the same cluster and so naturally requires additional correlation parameters to characterize the within-cluster structure \cite{LiTurnerPreisser2018,Li2019}. Correspondingly, sample size and power calculations for stepped wedge designs necessitate the specification of more than one correlation parameters. For example, Hemming et al. \cite{Hemming2015} considered both the within-period and between-period ICCs in their sample size procedure for a cross-sectional design. Hooper et al. \cite{Hooper2016} and Li et al. \cite{LiTurnerPreisser2018} examined a three-correlation structure that additionally accounts for the within-individual repeated measurements in a closed-cohort design.
Despite existing development of multi-parameter correlation structures for designing stepped wedge trials, most of them assumed a constant between-period ICC with a few exceptions. For example, in a cross-sectional design where outcome data are obtained from a different set of participants in each cluster-period \cite{Murray1998,Barker2016}, Hemming et al. \cite{Hemming2015} allowed the between-period ICC to be different from the within-period ICC, but restricted the between-period ICC to be constant irrespective of the distance between periods. Relaxing the constant between-period ICC assumption for a cross-sectional design, Kasza et al. \cite{Kasza2017,Kasza2018} studied a non-uniform correlation structure with a decay parameter and proposed a sample size procedure that accounts for the exponential correlation decay. Grantham et al. \cite{Grantham2019} further extended their sample size procedure to allow for continuous-time correlation decay in multiple-periods CRTs with continuous recruitment. From a trial planning standpoint, if correlation decay is present, Kasza et al. \cite{Kasza2017} indicated that omitting the correlation decay in a cross-sectional design would either underestimate or overestimate the true variance of the intervention effect, which led to inaccurate sample size determination. As we demonstrate in Section \ref{sec:impact}, similar considerations carry over to closed-cohort designs, where outcome data are collected from the same set of participants in each cluster-period \cite{Murray1998,Copas2015,Barker2016}. Particularly, the constant between-period ICC assumption in Hooper et al. \cite{Hooper2016} and Li et al. \cite{LiTurnerPreisser2018} may not always be realistic and it is therefore necessary to develop alternative design and analysis strategies accounting for correlation decay in cohort stepped wedge studies.
\begin{table}[htbp]
\centering
\caption{Example correlation structures in the literature on designing stepped wedge cluster randomized trials.} \label{Table1}
\begin{tabular}{clll}
\toprule
Decay & Design feature & Correlation structure & Example references \\
\midrule
\multirow{2}{*}{No} & Cross-sectional & Nested exchangeable & Hemming et al. \cite{Hemming2015}; Hooper et al. \cite{Hooper2016}; Li et al. \cite{LiTurnerPreisser2018}\\
& Closed-cohort & Block exchangeable & Hooper et al. \cite{Hooper2016}; Li et al. \cite{LiTurnerPreisser2018,Li2018}\\
\multirow{2}{*}{Yes} & Cross-sectional & Exponential decay & Kasza et al. \cite{Kasza2017}; Kasza and Forbes \cite{Kasza2018}; Grantham et al. \cite{Grantham2019}\\
& Closed-cohort & Proportional decay & This article \\
\bottomrule
\end{tabular}
\end{table}
Two popular modeling approaches for stepped wedge designs are cluster-specific models (e.g., random-effects models) and population-averaged models \cite{Preisser2003}. The parameter estimates from a population-averaged model can be interpreted as the marginal intervention effect for the participating individuals combined over all cluster-periods, and may be preferred over the cluster-specific models for trials conducted in the health policy or health services settings \cite{LiTurnerPreisser2018}. In this article, we consider a population-averaged model with a decay correlation structure. As indicated in Table \ref{Table1}, the first contribution of this article is to fill in the gap and characterize a proportional decay structure appropriate for cohort stepped wedge designs. Such a proportional decay structure has been previously introduced in analyzing clustered longitudinal data; see, for example, Lefkopoulou et al. \cite{Lefkopoulou1989}; Shults and Morrow \cite{Shults2002} and Liu et al. \cite{Liu2002}, but has not been exemplified in CRTs with a staggered randomization. Based on the proportional decay structure, we derive a new closed-form variance expression to facilitate sample size and power determination. Based on the derived variance expression, we additionally obtain a simple-to-use design effect and study how the power depends on the correlation parameters. Since the sample size procedure requires input for the correlation parameters, accurate estimation of the correlations is instrumental for future planning of stepped wedge trials. Therefore, a second contribution of this article is to introduce a modified generalized estimating equations (GEE) approach to accurately estimate the decay correlation structure along with the marginal mean parameters. The traditional GEE \cite{Liang1986} are modified in the two ways. (i) As simple moment estimators for the decay correlation structure may not be easy to obtain, we estimate the marginal mean and correlation parameters by quasi-least squares (QLS) \cite{Chaganty1997,Chaganty1999}. The QLS approach shares the same estimating equations with GEE regarding the marginal mean parameters, but is flexible enough to accommodate non-standard correlation structures. Similar to the traditional GEE estimator, the QLS estimator is also robust to correlation misspecification. (ii) Since stepped wedge cluster randomized design frequently includes a small number of clusters, we refine the QLS approach by incorporating appropriate finite-sample bias-corrections to both the estimation of correlation parameters as well as the variance of the intervention effect.
The remainder of this article is organized as follows. The notations of cohort stepped wedge designs are introduced in Section \ref{sec:note}. In Section \ref{sec2}, we provide the proportional decay correlation structure and discuss the QLS estimators to estimate the correlation structure. In Section \ref{sec3}, we develop closed-form procedures for sample size and power calculations based on the population-averaged model coupled with the proportional decay structure. We conduct a simulation study in Section \ref{sec4} to examine the accuracy of the proposed power procedure when the trials are analyzed by the QLS approach. Section \ref{sec5} provides two illustrative examples of cohort stepped wedge designs and Section \ref{sec6} concludes.
\section{Notations and Basic Setup}\label{sec:note}
We consider a cohort stepped wedge design, where a closed cohort of individuals are enrolled at each of the $I$ participating clusters at the start of the trial. We mainly focus on cohort designs to inform the applications in Section \ref{sec5}, and will defer the discussion of cross-sectional designs to Section \ref{sec6}. We assume the trial involves a total of $T$ time periods. All clusters start from the control condition, and may be randomly chosen to switch to intervention during the course of the study, until all clusters are treated at the end of the $T$th period. Individual participants will be scheduled for outcome measurement during each period, and so each individual has a total of $T$ repeated measurements. Denote $y_{ijt}$ as the outcome for individual $j$ ($j=1,\ldots,N_i$) from cluster $i$ ($i=1,\ldots,I$) at period $t$ ($t=1,\ldots,T$). A step is defined as the pre-planned time point when at least one cluster crosses over from control to intervention. We denote the total number of steps by $S$ ($2\leq S\leq T-1$), and assume that $m_s$ clusters cross over at step $s$ such that $\sum_{s=1}^S m_s=I$. We assume a complete design in the terminology of Hemming et al. \cite{Hemming2015} such that outcome measurements are taken for all individuals during each period. Following Woertman et al. \cite{Woertman2013}, we define the baseline measurements as those taken before any cluster is randomized to intervention, and follow-up measurements as those taken after at least one cluster is randomized to intervention. We assume there are $b\geq 1$ baseline measurements planned under the control condition, and $c_s\geq 1$ follow-up measurements planned between step $s$ and step $s+1$ (or end of study). Each measurement time point is associated with a distinct time period and the total number of periods $T=b+\sum_{s=1}^S c_s$. A standard stepped wedge design is given by $b=c_s=1$ for all $s$, and $T=S+1$ ($T\geq 3$). A schematic illustration of a standard design with $I=6$ clusters and $T=4$ periods can be found in Figure \ref{fig:swd1}. We also provide a schematic illustration of an alternative design with $I=6$ clusters and $T=6$ periods in Web Figure 1.
\begin{figure}[h]
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\caption{A schematic illustration of a standard stepped wedge design with $I=6$ clusters and $T=4$ periods. Each cell with a zero entry indicates a control cluster-period and each cell with a one entry indicates an intervention cluster-period.}
\label{fig:swd1}
\end{figure}
\section{Analysis considerations: Models and Statistical Inference}\label{sec2}
\subsection{Population-averaged models}\label{sec:model}
The population-averaged model relates the marginal mean, $\mu_{ijt}$, to the time trend and the intervention effect by
\begin{equation}\label{eq:glm}
g(\mu_{ijt})=\beta_t+X_{it}\delta,
\end{equation}
where $g$ is the link function and $\beta_t$ is the $t$th period effect \cite{LiTurnerPreisser2018}. Further, $X_{it}$ is the intervention status, which equals $1$ or $0$ depending on whether cluster $i$ receives intervention during period $t$, and $\delta$ describes the intervention effect on the link function scale. Model \eqref{eq:glm} can be regarded as the marginal counterpart of a number of existing random-effects models, such as those proposed in Hemming et al. \cite{Hemming2015}, Hooper et al. \cite{Hooper2016} and Kasza et al. \cite{Kasza2017,Kasza2018}. Like these random-effects models, our marginal model \eqref{eq:glm} does not specify treatment by period interaction and so $\delta$ should be interpreted as the average intervention effect across periods. We write the collection of model parameters as $\theta=(\beta_1,\ldots,\beta_T,\delta)'$, the collection of intervention status for cluster $i$ (a sequence of ones preceded by zeros) as $X_i=(X_{i1},\ldots,X_{iT})'$, and define $v(\mu_{ijt})$ as a known variance function. To allow for potential correlation decay over time, we define the proportional decay correlation structure similar to Lefkopoulou et al. \cite{Lefkopoulou1989}. Specifically, we define the within-period correlation as the correlation between outcomes for two distinct individuals from the same cluster during the same period, i.e. $\text{corr}(y_{ijt},y_{ij't})=\tau$ for $j\neq j'$. The same definition is prevalently used in the posttest-only parallel designs \cite{Murray1998}. We then assume a first-order autoregressive structure for the set of within-individual repeated measurements. Therefore the within-individual correlation, which describes the association between outcomes measured at time $t$ and $t'$ of the same individual, is $\text{corr}(y_{ijt},y_{ijt'})=\rho^{|t-t'|}$, $t\neq t'$. Finally, we define the between-period correlation as the correlation between outcome measured at time $t$ for individual $j$ and outcome measured at time $t'$ for individual $j'$, and assumes a decay structure as $\text{corr}(y_{ijt},y_{ij't'})=\tau\rho^{|t-t'|}$ for $j\neq j'$, $t\neq t'$.
\begin{table}[htbp]
\centering
\caption{Examples of the proportional decay structure for cohort designs and the exponential decay structure for cross-sectional designs. The illustration is based on a stepped wedge trial with $T=3$ periods and $N_i=2$ measurements per cluster-period. Define $y_i=(y_{i11},y_{i12},y_{i13},y_{i21},y_{i22},y_{i23})'$.} \label{Table3}
\begin{tabular}{cll}
\toprule
& Proportional decay structure & Exponential decay structure \\
\midrule
$\text{corr}(y_i)$ & $\left(\begin{array}{@{}ccc|ccc@{}}
1 & \rho & \rho^2 & \tau & \tau\rho & \tau\rho^2\\
\rho & 1 & \rho & \tau\rho & \tau &\tau\rho\\
\rho^2 & \rho & 1 & \tau\rho^2 & \tau\rho & \tau\\ \hline
\tau & \tau\rho & \tau\rho^2 & 1 & \rho & \rho^2\\
\tau\rho & \tau & \tau\rho & \rho & 1 &\rho\\
\tau\rho^2 & \tau\rho & \tau & \rho^2 & \rho & 1\\
\end{array}\right)$ &
$\left(\begin{array}{@{}ccc|ccc@{}}
1 & \tau\rho & \tau\rho^2 & \tau & \tau\rho & \tau\rho^2\\
\tau\rho & 1 & \tau\rho & \tau\rho & \tau &\tau\rho\\
\tau\rho^2 & \tau\rho & 1 & \tau\rho^2 & \tau\rho & \tau\\ \hline
\tau & \tau\rho & \tau\rho^2 & 1 & \tau\rho & \tau\rho^2\\
\tau\rho & \tau & \tau\rho & \tau\rho & 1 &\tau\rho\\
\tau\rho^2 & \tau\rho & \tau & \tau\rho^2 & \tau\rho & 1\\
\end{array}\right)$\\
\bottomrule
\end{tabular}
\end{table}
In a closed-cohort design, typically the between-period correlation is much smaller than the within-individual correlation for any two fixed periods, since the former is defined for measurements from two distinct individuals while the latter is defined for those from the same individual. This ordering of magnitude is reflected in the proportional decay structure because $\tau<1$ and $\text{corr}(y_{ijt},y_{ij't'})=\tau\rho^{|t-t'|}<\text{corr}(y_{ijt},y_{ijt'})=\rho^{|t-t'|}$. By comparison, the exponential decay structure by Kasza et al. \cite{Kasza2017} is based on cross-sectional designs and therefore obviates the need for modeling the within-individual correlation from repeated measurements. In fact, the exponential decay structure assumes $\text{corr}(y_{ijt},y_{ij't'})=\tau\rho^{|t-t'|}$ irrespective of whether $j=j'$ since two different sets of participants are included in two different periods for a cross-sectional design. In this respect, parameters $\tau$ and $\rho$ have similar interpretations in both structures. We provide a visual comparison of these two structures in Table \ref{Table3}. In summary, the proportional decay correlation structure is defined through two parameters, $\tau$ and $\rho$, with the former resembling the traditional ICC definition in a parallel design and the latter controlling for the degree of correlation decay. Of note, Shults and Morrow \cite{Shults2002} and Liu et al. \cite{Liu2002} also adopted the same proportional decay structure in longitudinal CRTs with a parallel assignment, and we extend the discussion of this decay structure to CRTs with a staggered assignment.
\subsection{Quasi-least squares analysis}
We use the QLS approach introduced by Shults and Morrow \cite{Shults2002} to simultaneously estimate the intervention effect in model \eqref{eq:glm} and the correlation parameters in a cohort stepped wedge design. In particular, the QLS approach and the traditional GEE approach share the same estimating equations for the marginal mean parameters, whereas the former provides a flexible and convenient way to estimate non-standard correlation structures. Furthermore, both the QLS and GEE approaches are robust to correlation misspecification, namely estimators for the marginal intervention effect remain consistent even if the working correlation model is misspecified. In sufficiently large samples (usually $I\geq 30$), the robust sandwich variance could also be used to adequately quantify the uncertainty of the intervention effect estimate even under correlation misspecification. We refer the readers to the textbook of Shults and Hilbe \cite{Shults2014} for a full exposition on the advantages of QLS over the traditional GEE.
Write $y_{ij}=(y_{ij1},\ldots,y_{ijT})'$, $\mu_{ij}=(\mu_{ij1},\ldots,\mu_{ijT})'$, $y_i=(y_{i1}',\ldots,y_{iN_i}')'$ and $\mu_i=(\mu_{i1}',\ldots,\mu_{iN_i}')'$. Further define $D_i(\theta)=\partial\mu_i/\partial\theta'$, and let the working covariance of $y_i$ be $V_i=\phi A_i^{1/2}(\theta)R_i(\alpha_0,\alpha_1)A_i^{1/2}(\theta)$, where $\phi>0$ is the dispersion parameter, $A_i(\theta)=\text{diag}\{A_{i1}(\theta),\ldots,A_{iN_i}(\theta)\}$, $A_{ij}(\theta)=\text{diag}\{v(\mu_{ij1}),\ldots,v(\mu_{ijT})\}$, and $R_i(\alpha_0,\alpha_1)$ is a positive definite working correlation parameterized by $\alpha_0$ and $\alpha_1$. We assume the true correlation structure among elements of $y_i$ is the proportional decay structure, denoted by $R_i(\tau,\rho)$. In matrix notations, we can verify that the proportional decay structure induces separability between $\tau$ and $\rho$ in that $R_i(\tau,\rho)=G_i(\tau)\otimes F(\rho)$, where $G_i(\tau)$ is a $N_i\times N_i$ exchangeable correlation matrix (with respect to $\tau$) and $F(\rho)$ is a $T\times T$ first-order autoregressive correlation matrix, as given below
\begin{eqnarray*}
G_i(\tau)=\left(
\begin{array}{ccccc}
1 & \tau & \tau & \ldots & \tau \\
\tau & 1 & \tau & \ldots & \tau \\
\vdots & \vdots & \vdots & \ddots & \vdots \\
\tau & \tau & \tau & \ldots & 1 \\
\end{array}
\right),~~~~~~~~~~~F(\rho)=\left(
\begin{array}{ccccc}
1 & \rho & \rho^2 & \ldots & \rho^{T-1} \\
\rho & 1 & \rho & \ldots & \rho^{T-2} \\
\vdots & \vdots & \vdots & \ddots & \vdots \\
\rho^{T-1} & \rho^{T-2} & \rho^{T-3} & \ldots & 1 \\
\end{array}
\right).
\end{eqnarray*}
We could verify that the determinant $$\det\{R_i(\tau,\rho)\}=\det\{G_i(\tau)\}^T\det\{F(\rho)\}^{N_i}=
(1-\tau)^{T(N_i-1)}\{1+(N_i-1)\tau\}^T(1-\rho^2)^{(T-1)N_i}.$$
Therefore, valid correlation values that ensure positive definite $R_i(\tau,\rho)$ should be contained in the triangular region
\begin{eqnarray}\label{eq:range}
\mathcal{S}=\left\{(\tau,\rho):-\frac{1}{\max\{N_1,\ldots,N_I\}-1}<\tau<1,-1<\rho< 1\right\}.
\end{eqnarray}
Finally, the inverse of the $R_i$ also exists in closed form and is given by $R_i^{-1}(\tau,\rho)=G_i^{-1}(\tau)\otimes F_i^{-1}(\rho)$, where
$$G_i^{-1}(\tau)=\frac{1}{1-\tau}I_i
-\frac{\tau}{(1-\tau)\{(1+(N_i-1)\tau)\}}J_i,~~~ F^{-1}(\rho)=\frac{1}{1-\rho^2}\{I+\rho^2 C_2-\rho C_1\},$$
$I_i$ is a $N_i\times N_i$ identity matrix, $J_i$ is a $N_i\times N_i$ matrix of ones, $C_2=\text{diag}(0,1,\ldots,1,0)$, and $C_1$ is a $T\times T$ tridiagonal matrix with zeros on the main diagonal and ones on the two sub-diagonals.
To introduce the QLS estimating equations, we further define $r_{ij}(\theta)=A_{ij}^{-1/2}(\theta)(y_{ij}-\mu_{ij})$, and write $r_i(\theta)=(r_{i1}'(\theta),\ldots,r_{iN_i}'(\theta))'$. The first-stage QLS estimates for $\theta$, $\alpha_0$ and $\alpha_1$ are obtained by alternating between the following estimating equations until convergence
\begin{eqnarray}
&&\sum_{i=1}^I D_i'(\theta)A_i^{-1/2}(\theta)R_i^{-1}(\alpha_0,\alpha_1)r_i(\theta)=0,\label{eq:gee}\\
&&\sum_{i=1}^I \frac{\partial}{\partial\alpha_0} \left\{r_i'(\theta)
R_i^{-1}(\alpha_0,\alpha_1)r_i(\theta)\right\}=0,\label{eq:alpha0}\\
&&\sum_{i=1}^I \frac{\partial}{\partial\alpha_1}\left\{r_i'(\theta)
R_i^{-1}(\alpha_0,\alpha_1)r_i(\theta)\right\}=0.\label{eq:alpha1}
\end{eqnarray}
In particular, \eqref{eq:gee} is the usual GEE coupled with the proportional decay structure, and \eqref{eq:alpha0}, \eqref{eq:alpha1} are scalar equations for the first-stage correlation estimates. Further, closed-form solutions exist for $\hat{\alpha}_0$ and $\hat{\alpha}_1$ (within an iterative step) and are provided in Web Appendix A. Chaganty and Shults \cite{Chaganty1999} showed that $\hat{\alpha}_0$ and $\hat{\alpha}_1$ are asymptotically biased for $\tau$ and $\rho$. To eliminate the large-sample bias in the first-stage correlation estimates, Chaganty and Shults \cite{Chaganty1999} provided the following second-stage estimating equations to obtain $\hat{\tau}$, $\hat{\rho}$
\begin{eqnarray}
\sum_{i=1}^I \text{tr}\left\{\frac{\partial G_i^{-1}(\hat{\alpha}_0)}{\partial\alpha_0}G_i(\tau)\right\}&=&0\label{eq:secondalpha0}\\
\text{tr}\left\{\frac{\partial F^{-1}(\hat{\alpha}_1)}{\partial\alpha_1}F(\rho)\right\}&=&0.\label{eq:secondalpha1}
\end{eqnarray}
The closed-form solution for \eqref{eq:secondalpha0} and \eqref{eq:secondalpha1} are provided by Shults and Morrow \cite{Shults2002} as
\begin{equation*}
\hat{\tau}=\sum_{i=1}^I\frac{N_i(N_i-1)\hat{\alpha}_0\{2+(N_i-2)\hat{\alpha}_0\}}
{\{1+(N_i-1)\hat{\alpha}_0\}^2}\Big/\sum_{i=1}^I\frac{N_i(N_i-1)\{1+(N_i-1)\hat{\alpha}_0^2\}}
{\{1+(N_i-1)\hat{\alpha}_0\}^2},
\end{equation*}
and $\hat{\rho}=2\hat{\alpha}_1/(1+\hat{\alpha}_1^2)$.
\subsection{Bias-corrected correlation estimation}
Although the correlation estimates obtained from the second-stage QLS estimating equations are unbiased in large samples, they could be subject to finite-sample bias. The finite-sample bias is a typical consideration in stepped wedge cluster randomized trials \cite{LiTurnerPreisser2018}, as they usually include a small number of clusters ($I\leq 30$). We refine the QLS approach with finite-sample bias-corrections to the correlation estimating equations by utilizing the cluster-leverage \cite{Preisser1996}, defined as $H_i=D_i(\theta)(\sum_{i=1}^I D_i'(\theta)V_iD_i(\theta))^{-1}D_i'(\theta)V_i$. Specifically when $I$ is small, the estimated residual $y_i-\hat{\mu}_i$ tends to be biased towards zero, and following Preisser et al. \cite{Preisser2008}, we have $E[(y_i-\hat{\mu}_i)(y_i-\hat{\mu}_i)']\approx (I-H_i)\text{cov}(y_i)=\phi(I-H_i)A_i^{1/2}\text{corr}(y_i)A_i^{1/2}$ and therefore $E[r_i(\hat{\theta})r_i'(\hat{\theta})]\approx \phi A_i^{-1/2}(I-H_i)A_i^{1/2}\text{corr}(y_i)$. This last equation suggests that $\phi^{-1}A_i^{-1/2}(I-H_i)^{-1}A_i^{1/2}r_i(\hat{\theta})r_i'(\hat{\theta})$ is a better estimator for $\text{corr}(y_i)$ compared to the simple cross-product $\phi^{-1}r_i(\hat{\theta})r_i'(\hat{\theta})$, since the former accounts for finite-sample bias in a multiplicative fashion. Further observe that
\begin{eqnarray*}
r_i'(\theta)
R_i^{-1}(\alpha_0,\alpha_1)r_i(\theta)&=&\text{tr}\{R_i^{-1}(\alpha_0,\alpha_1)r_i(\theta)r_i'(\theta)\}
\propto \phi^{-1}\text{tr}\{R_i^{-1}(\alpha_0,\alpha_1)r_i(\theta)r_i'(\theta)\}
\end{eqnarray*}
for all values of $\alpha_0$, $\alpha_1$ and $\theta$, we propose to replace the first-stage estimating equations \eqref{eq:alpha0} and \eqref{eq:alpha1} by
\begin{eqnarray}
&&\sum_{i=1}^I \frac{\partial}{\partial\alpha_0} \text{tr}\left\{
R_i^{-1}(\alpha_0,\alpha_1)\tilde{R}_i(\theta)\right\}
=\sum_{i=1}^I \frac{\partial}{\partial\alpha_0} \text{tr}\left[
\left\{G_i^{-1}(\alpha_0)\otimes F^{-1}(\alpha_1)\right\}\tilde{R}_i(\theta)\right]=0\label{eq:refalpha0}\\
&&\sum_{i=1}^I \frac{\partial}{\partial\alpha_1} \text{tr}\left\{
R_i^{-1}(\alpha_0,\alpha_1)\tilde{R}_i(\theta)\right\}=
\sum_{i=1}^I \frac{\partial}{\partial\alpha_1} \text{tr}\left[
\left\{G_i^{-1}(\alpha_0)\otimes F^{-1}(\alpha_1)\right\}\tilde{R}_i(\theta)\right]=0\label{eq:refalpha1}
\end{eqnarray}
where $\tilde{R}_i(\theta)=\phi^{-1}A_i^{-1/2}(I-H_i)^{-1}A_i^{1/2}r_i(\theta)r_i'(\theta)$ represents the matrix-adjusted estimator for the correlation structure. The solutions obtained from \eqref{eq:refalpha0} and \eqref{eq:refalpha1} could effectively reduce the finite-sample bias in $\hat{\alpha}_0$ and $\hat{\alpha}_1$, which would in turn decrease the finite-sample bias in the QLS estimators for $\tau$ and $\rho$. Of note, similar finite-sample matrix adjustment was developed by Preisser et al. \cite{Preisser2008} for the Prentice-type GEE \cite{Prentice1988}, and we extend this finite-sample bias-correction approach to the QLS estimating equations. Accurately estimating the correlation parameters in the analysis stage has practical implications since these estimates could be used to guide the planning of future trials \cite{Murray1998}. Additional details of the matrix-adjusted estimating equations, \eqref{eq:refalpha0} and \eqref{eq:refalpha1}, along with the closed-form updates are provided in Web Appendix B.
\subsection{Bias-corrected covariance estimation}
The availability of a small number of clusters may also have implications for estimating the variance using GEE-based approaches \citep{Turner2017b}. In general, the variance of the marginal mean model parameter $\hat{\theta}$ can be estimated using the model-based variance $\Omega_1^{-1}=(\sum_{i=1}^I D_i'V_iD_i)^{-1}$ or the sandwich variance $\Omega_1^{-1}\Omega_0\Omega_1^{-1}$, where
\begin{equation}\label{eq:sand}
\Omega_0=\sum_{i=1}^I C_iD_i'V_i^{-1}B_i(y_i-\mu_i)(y_i-\mu_i)'B_i'V_i^{-1}D_iC_i,
\end{equation}
and both ${\Omega}_0$, ${\Omega}_1$ are evaluated at $(\hat{\theta},\hat{\tau},\hat{\rho})$. When both $C_i$ and $B_i$ are identity matrices, equation (\ref{eq:sand}) reduces to the uncorrected sandwich estimator of Liang and Zeger \cite{Liang1986}, which we denote as BC0. BC0 provides valid inference regardless of the correct specification of the working correlation $R_i$, as long as the number of clusters is sufficiently large ($I\geq 30$), while the consistency of the model-based variance requires the correct specification of the correlation structure. As $r_i(\hat{\theta})$ is biased towards zero with a limited number of clusters, BC0 is likely to underestimate the variance and alternative choices of matrices $C_i$ and $B_i$ may be necessary to provide a partial correction to the finite-sample bias \cite{Turner2017b}. We consider three popular approaches for bias-corrections summarized below and also in Table \ref{Table4}: the finite-sample correction due to Kauermann and Carroll \cite{Kauermann2001}, or BC1, is given by $C_i=I$ and $B_i=(I-H_i)^{-1/2}$; the finite-sample correction due to Mancl and DeRouen \cite{Mancl2001}, or BC2, is given by $C_i=I$ and $B_i=(I-H_i)^{-1}$; the finite-sample correction due to Fay and Graubard \cite{Fay2001}, or BC3, given by $C_i=\text{diag}\{(1-\min\{\zeta,[{D_i'V^{-1}_iD_i\Omega_1^{-1}}]_{jj}\})^{-1/2}\}$ and $B_i=I$, where the bound parameter $\zeta$ is a user-defined constant ($<1$) with a default value $0.75$. Because the matrix elements of the cluster leverage are between $0$ and $1$, we generally have $\text{BC0}<\text{BC1}<\text{BC2}$ in terms of amount of correction \cite{Preisser2008}. Further, Scott et al. \cite{Scott2014} have shown that BC3 tends to be close to BC1. Of note, the estimation of dispersion parameter should only affect the model-based variance. Similar to Liang and Zeger \cite{Liang1986}, we propose to consistently update the dispersion parameter from iteration $s$ to $s+1$ by $\hat{\phi}^{(s+1)}=\hat{\phi}^{(s)}{\sum_{i=1}^I \text{tr}(\tilde{R}_i)}/\{\sum_{i=1}^I TN_i-(T+1)\}$.
\begin{table}[htbp]
\centering
\caption{Summary of bias-corrected sandwich variance estimators for $\hat{\theta}$.} \label{Table4}
\begin{tabular}{llccl}
\toprule
Label & Correction & $C_i$ & $B_i$ & References \\ \midrule
BC0 & none & $I$ & $I$ & Liang and Zeger\cite{Liang1986} \\
BC1 & less & $I$ & $(I-H_i)^{-1/2}$ & Kauermann and Carroll\cite{Kauermann2001} \\
BC2 & more & $I$ & $(I-H_i)^{-1}$ & Mancl and DeRouen\cite{Mancl2001} \\
BC3 & less & $\text{diag}\{(1-\min\{\zeta,[{D_i'V^{-1}_iD_i\Omega_1^{-1}}]_{jj}\})^{-1/2}\}$
& $I$ & Fay and Graubard\cite{Fay2001} \\
\bottomrule
\end{tabular}
\end{table}
\section{Design Considerations: Sample Size and Power Calculations}
\label{sec3}
\subsection{Closed-form variance of the intervention effect}
Under the null hypothesis $H_0$: $\delta=\delta_0$, the large-sample variance of $\sqrt{n}(\hat{\delta}-\delta_0)$ is provided by the $(T+1,T+1)$ element of the large-sample covariance matrix of $\sqrt{n}(\hat{\theta}-\theta_0)$. Since the QLS estimator $\hat{\delta}$ is asymptotically normal, we could use the $z$-test statistic $\hat{\delta}/\sqrt{\text{var}(\hat{\delta})}$ to test the null of no intervention effect $H_0$: $\delta=0$, and the power to detect an intervention effect of size $\delta\neq 0$ with a prescribed type I error rate $\alpha$ is approximately
\begin{equation}\label{eq:powerz}
\text{power}=\Phi\left(z_{\alpha/2}+\frac{|\delta|}{\sqrt{\text{var}(\hat{\delta})}}\right),
\end{equation}
where $\Phi$ is the standard normal cumulative distribution function and $z_{\alpha/2}$ is the normal quantile such that $\Phi(z_{\alpha/2})=1-\alpha/2$. Because there is uncertainty in estimating the asymptotic variance $\text{var}(\hat{\delta})$, an alternative two-sided test uses the same statistic but refers to the $t$-distribution. We consider two choices of degrees of freedom (DoF). The first DoF dates back to Mancl and DeRouen \cite{Mancl2001} and equals the number of clusters minus the number of regression parameters; this DoF has been previously used in the GEE analyses of parallel CRTs \cite{Li2017}, three-level CRTs\cite{Teerenstra2010}, crossover CRTs\cite{Li2019}, stepped wedge CRTs\cite{LiTurnerPreisser2018,Grayling2017a}, and shown to have test size not exceeding the nominal level. The second DoF was suggested in the PhD dissertation of Ford \cite{Ford2018} and specifies $\text{DoF}=I-2$. This choice of DoF was found to provide excellent control of type I error rate for GEE analyses of both parallel CRTs and stepped wedge CRTs in Ford and Westgate \cite{Ford2017,Ford2018}. With the same effect size $\delta$ and prescribed type I error rate $\alpha$, the power of the $t$-test is approximately
\begin{equation}\label{eq:powert}
\text{power}=\Psi_{\text{DoF}}\left(t_{\alpha/2,\text{DoF}}+\frac{|\delta|}{\sqrt{\text{var}(\hat{\delta})}}\right),
\end{equation}
where $\Psi_{\text{DoF}}$ is the $t$ distribution function and the quantile $t_{\alpha/2}$ is chosen such that $\Psi_{\text{DoF}}(t_{\alpha/2})=1-\alpha/2$. We notice that because the $t$-distribution has a heavier tail compared with the standard normal distribution, the QLS $z$-test is more likely to result in a inflated type I error rate with the use of BC0 than the corresponding QLS $t$-test. As the bias-corrected sandwich variance estimators (BC1, BC2, and BC3) provide different degrees of inflation relative to the uncorrected variance BC0, an investigation of both the $z$- and $t$-tests coupled with the collection of alternative variance estimators could help inform the practical choice among the analytical options for stepped wedge CRTs.
To assist the design of stepped wedge trials allowing for correlation decay, we derive a new closed-form variance expression for $\hat{\delta}$ assuming the outcome is continuous and $g$ is the identity link function. We will return to categorical outcomes and nonlinear link functions in Section \ref{sec6}. To do so, we follow Shih \citep{Shih1997} and assume the covariance of $Y_i$ to be known as $\text{var}(Y_i)=V_i$. Therefore, $\text{var}(\hat{\delta})$ is the $(T+1,T+1)$ element of the model-based variance ${\Omega}_1^{-1}$. We further assume a balanced design such that an equal number of participants will be recruited in each cluster prior to the first period, so that $N_i=N$. Such a simplification assumption is routinely made in designing stepped wedge trials.
Under a balanced design, we could write the design matrix corresponding to cluster $i$ as $Z_i=1_N\otimes (I_T, X_i)$, where $1_N$ is a $N$-vector of ones. Then the variance of the intervention effect $\hat{\delta}$ equals to the lower-right element of $\phi\{\sum_{i=1}^I Z_i'R_i^{-1}(\tau,\rho)Z_i\}^{-1}$, where $\phi$ is the marginal variance. We show in Web Appendix C that a closed-form variance expression for $\hat{\delta}$ is
\begin{equation}\label{eq:vardelta}
\text{var}(\hat{\delta})=\frac{(\phi I/N)(1-\rho^2)\{1+(N-1)\tau\}}{(IU-W)(1+\rho^2)-2(IV-Q)\rho},
\end{equation}
where the design constant $U=\sum_{i=1}^I\sum_{t=1}^{T}X_{it}$ is the total number of cluster-periods exposed under the intervention condition, $W=\sum_{t=1}^{T}\left(\sum_{i=1}^I X_{it}\right)^2$ is the squared number of clusters receiving the intervention summed across periods, $V=\sum_{i=1}^I\sum_{t=1}^{T-1}X_{it}X_{i,t+1}$ and $Q=\sum_{t=1}^{T-1}\left(\sum_{i=1}^I X_{it}\right)\left(\sum_{i=1}^I X_{i,t+1}\right)$ are cross-product terms resulting from the decay correlation structure. It is interesting to see that this variance expression does not depend on the magnitude of the period effect $\beta_t$ as long as they are controlled for in the marginal mean model. Noticeably, the QLS-based variance \eqref{eq:vardelta} extends the formula due to Liu et al. \cite{Liu2002} to longitudinal cluster designs with staggered randomization. Further, as the cohort size $N$ becomes large, the variance expression converges to
\begin{equation}\label{eq:varlim}
\lim_{N\rightarrow\infty} \text{var}(\hat{\delta})=\frac{\phi I(1-\rho^2)\tau}{(IU-W)(1+\rho^2)-2(IV-Q)\rho},
\end{equation}
which is a finite constant since $|\rho|<1$ and $\tau>0$ for large $N$, according to \eqref{eq:range}. Therefore, the limit of the variance is a positive constant determined by available design resources $I$, $T$ and two correlation values $\tau$ ,$\rho$, and cannot be made arbitrarily small. In other words, the power of the stepped wedge design may not be increased to one by solely increasing the cohort size, which is consistent with the known results for parallel cluster randomized designs \citep{Murray1998}. For this reason, when $N$ is large, variance \eqref{eq:varlim} could be used in the design stage to approximate the variance \eqref{eq:vardelta}. Finally, given hypothesized values for $I$, $N$, $T$ and correlation parameters, variance expression \eqref{eq:vardelta} or \eqref{eq:varlim} can be used in equations \eqref{eq:powerz} and \eqref{eq:powert} to obtain the predicted power.
\subsection{The design effect}
For determining the required sample size based on equation \eqref{eq:powerz} and \eqref{eq:powert}, it is straightforward to solve $N$ by fixing the required number of clusters $I$ but not the other way around. However, with a pre-determined cohort size $N$ for each cluster, we could postulate a series of values for $I$ and find the smallest value such that the estimated power is at least equal to the prescribed level. Additionally, in the following case studied by Woertman et al. \cite{Woertman2013}, we could derive a simple expression for the design effect (DE) relative to an individually randomized trial to simplify sample size calculation. Specifically, we assume that an equal number of clusters switch to intervention at each step so that $m_s=m$, and further an equal number of measurements are taken between steps such that $c_s=s$ for all $s=1,\ldots,S$. We then write the total number of clusters $I=Sm$ and total number of periods $T=b+Sc$, and the design constants become
\begin{equation*}
\begin{split}
U&=\frac{1}{2}S(S+1)mc,~~W=\Big(\frac{1}{3}S^3+\frac{1}{2}S^2+\frac{1}{6}S\Big)m^2c,\\
V&=U-Sm,~~~~~~~~~~~~Q=W-\frac{1}{2}S(S+1)m^2.
\end{split}
\end{equation*}
Plugging the design constants back into the variance formula \eqref{eq:vardelta}, and dividing by the variance of the two-sample mean difference $4\phi/(NSm)$, we obtain
\begin{equation}\label{eq:de}
\text{DE}=\frac{3S}{2(S-1)}\frac{(1-\rho^2)}{\{(S+1)c(1-\rho)^2+6\rho\}}\{1+(N-1)\tau\}.
\end{equation}
The above design effect allows us to easily study how the design resources affect the statistical efficiency relative to individual randomization and how the correlation parameters affect the statistical power. For example, since the design effect is free of $b$, the relative design efficiency does not change according to the number of baseline periods. However, for fixed values of the correlation parameters, increasing the number of steps $S$ and number of measurements between steps $c$ decreases the design effect and increases the efficiency. On the other hand, for fixed design resources, larger values of the within-period correlation $\tau$ increases the design effect, confirming that $\tau$ functions as the traditional ICC of a parallel cluster randomized trial. By contrast, the role of correlation parameter $\rho$ is characterized by $f(p)=(1-\rho^2)/[(S+1)c(1-\rho)^2+6\rho]$, which is monotonically increasing on $(-1,r)$ and decreasing on $(r,1)$, where $$r=1+\frac{\sqrt{3}\{\sqrt{3}-\sqrt{2(S+1)c-3}\}}{(S+1)c-3}\in(0,1).$$
For convenience, we could define the decay parameter $d=1-\rho$ so that $d=0$ and $d=1$ correspond to no decay and total decay respectively. Since it is more plausible that $\rho\in(0,1)$, the above result suggests that with an increasing level of correlation decay, the design effect first increases to its largest value and then decreases, with the maximum design effect obtained at $\rho=r$. A numerical illustration of the design effect as a function of the decay parameter is provided in Web Figure 2.
\section{A Simulation Study}\label{sec4}
\subsection{Simulation design}
We carry out a simulation study (i) to compare the correlation estimators from the uncorrected QLS and the proposed matrix-adjusted QLS (MAQLS), and (ii) to evaluate the utility of the proposed power formula for QLS-based analyses of stepped wedge CRTs. For the second objective, we first determine the empirical type I error rates for the QLS-based tests coupled with alternative variance estimators, and then identify valid tests (those with a close-to-nominal type I error rate) whose empirical power corresponds well with the predicted power from the proposed formula. Findings specific to the second objective are informative for practical data analysis since we prefer tests that maintain a valid size and meanwhile demonstrate empirical power that is at least the magnitude of the analytical prediction.
\begin{table}[htbp]
\centering
\caption{Percent relative bias of the correlation parameters based on uncorrected quasi-least squares (QLS) and matrix-adjusted quasi-least squares (MAQLS) for each simulation scenario when the treatment effect is zero.}\label{Table5}
\begin{tabular}{cccrrcrrrr}
\toprule
\multicolumn{2}{c}{Correlations} & \multicolumn{1}{c}{Effect size} &
\multicolumn{3}{c}{Design resource} & \multicolumn{2}{c}{Percent bias for $\tau$} &
\multicolumn{2}{c}{Percent bias for $\rho$}\\
\midrule
$\tau$ & $\rho$ & $\delta$ & $I$ & $N$ & $T$ & QLS & MAQLS & QLS & MAQLS\\
\midrule
0.03 & 0.2 & 0 & 18 & 10 & 7 & -26.6 & 3.5 & -0.6 & 0.2 \\
0.03 & 0.2 & 0 & 18 & 24 & 4 & -16.0 & 5.5 & -0.3 & 0.0 \\
0.03 & 0.2 & 0 & 20 & 14 & 5 & -20.2 & 3.4 & -0.5 & 0.0 \\
0.03 & 0.2 & 0 & 21 & 8 & 4 & -29.4 & 3.2 & -0.4 & 0.2 \\\smallskip
0.03 & 0.2 & 0 & 15 & 8 & 4 & -41.3 & 4.7 & -0.6 & 0.3 \\
0.03 & 0.8 & 0 & 16 & 12 & 5 & -26.2 & 4.4 & -0.6 & 0.0 \\
0.03 & 0.8 & 0 & 24 & 7 & 5 & -27.7 & 0.6 & -0.6 & -0.1 \\
0.03 & 0.8 & 0 & 12 & 8 & 5 & -49.4 & 8.2 & -2.1 & -0.4 \\
0.03 & 0.8 & 0 & 12 & 5 & 4 & -74.2 & 6.6 & -2.3 & -0.5 \\\smallskip
0.03 & 0.8 & 0 & 10 & 5 & 3 & -91.6 & -0.4 & -1.5 & -0.5 \\
0.10 & 0.2 & 0 & 21 & 11 & 8 & -9.2 & 3.9 & -0.6 & 0.6 \\
0.10 & 0.2 & 0 & 24 & 11 & 7 & -8.1 & 3.4 & -0.4 & 0.6 \\
0.10 & 0.2 & 0 & 15 & 16 & 6 & -11.8 & 7.1 & -0.6 & 1.1 \\
0.10 & 0.2 & 0 & 18 & 8 & 7 & -12.7 & 3.9 & -0.8 & 0.7 \\\smallskip
0.10 & 0.2 & 0 & 16 & 7 & 5 & -16.7 & 3.7 & -0.7 & 1.0 \\
0.10 & 0.8 & 0 & 20 & 18 & 5 & -8.0 & 5.0 & -0.1 & 0.2 \\
0.10 & 0.8 & 0 & 15 & 9 & 4 & -15.2 & 3.1 & -0.2 & 0.2 \\
0.10 & 0.8 & 0 & 10 & 20 & 3 & -16.9 & 6.5 & -0.1 & 0.3 \\
0.10 & 0.8 & 0 & 12 & 5 & 5 & -27.0 & 2.6 & -1.5 & 0.0 \\
0.10 & 0.8 & 0 & 9 & 7 & 4 & -29.7 & 3.8 & -1.0 & 0.1 \\
\bottomrule
\end{tabular}
\end{table}
Within-cluster correlated continuous outcomes were generated from a multivariate normal distribution with mean given by $\mu_{ijt}=\beta_t+X_{it}\delta$ and covariance $\phi R(\tau,\rho)$, where $R(\tau,\rho)$ is the proportional decay structure defined in Section \ref{sec:model}. We set the marginal variance $\phi=1$ and assumed a gently increasing period effect such that $\beta_1=0$ and $\beta_{t+1}-\beta_t=0.1\times (0.5)^{t-1}$ for $t\geq 1$. As discussed before, the predicted power should be insensitive to the magnitude of the period effects as long as they are accounted for in the QLS analyses. We fix the effect size $\delta/\phi^{1/2}$ at zero for studying test size and choose $\delta/\phi^{1/2}$ from $\{0.2,0.3,0.4,0.5\}$ for studying power. We choose the within-period correlation $\tau\in\{0.03,0.1\}$, which represent typical ICC values reported in the parallel CRTs \citep{Murray1998}. We further chose $\rho\in\{0.2,0.8\}$, representing large and moderate degree of correlation decay over time. The number of clusters are varied from $9$ to $24$ as stepped wedge CRTs usually include a limited number of clusters. We specify the number of periods $3\leq T\leq 8$ as these values are frequently reported in practice, according to recent reviews by Martin et al. \cite{Martin2016} and Grayling et al. \cite{Grayling2017b}. Finally, the cohort size are chosen as $5\leq N\leq 24$ to ensure that the predicted power is at least $80\%$. For illustration, we focus on standard stepped wedge designs so that there is only one baseline period and the number of steps $S=T-1$. In other words, an equal number of $I/S$ clusters cross over to intervention during each step, and the outcome is measured only once for each individual between consecutive steps. For each scenario, $10000$ data replications were generated and analyzed using both QLS and MAQLS. For the first objective, we report the percent relative bias in estimating $\tau$ and $\rho$. In general, an unbiased approach for estimating the correlation parameters is preferred since accurate reporting of correlations is critical for planning future trials. Web Tables 1 and 2 provide a summary of the simulation scenarios along with the convergence rates. The convergence rates are similar between QLS and MAQLS, and all exceed $97\%$. For the second objective, we consider both the $z$-tests and the $t$-tests for testing the null hypothesis of no intervention effect, coupled with five different variance estimators for $\hat{\delta}$, namely, the model-based variance, BC0, BC1, BC2 and BC3. The nominal type I error rate is held fixed at $5\%$, and we consider an empirical type I error rate between $4.5\%$ and $5.5\%$ to be acceptable based on the margin of error from a binomial model with $10000$ replications. By a similar reasoning, since the predicted power in each scenario is at least $80\%$, we consider an empirical power that differs by no more than $0.8\%$ from the predicted value to be acceptable.
\subsection{Results}
\begin{table}[htbp]
\centering
\caption{Simulation scenarios, nominal size, along with the empirical type I error rates corresponding to the MAQLS $z$-test and $t$-test ($\text{DoF}=I-2$) with different variance estimators. Empirical type I error rates between $4.5\%$ and $5.5\%$ are highlighted in boldface and considered acceptable based on the margin of error from a binomial model with $10000$ Monte Carlo replications. Pred: nominal type I error rate; MB: model-based variance; BC0: uncorrected sandwich variance of Liang and Zeger (1986); BC1: bias-corrected sandwich variance of Kauermann and Carroll (2001); BC2: bias-corrected sandwich variance of Mancl and DeRouen (2001); BC3: bias-corrected sandwich variance of Fay and Graubard (2001).}\label{tb:main1}
\begin{tabular}{cccrrrrrrrrrrrrrrr}
\toprule
& & & & & & \multicolumn{6}{c}{$z$-test} & \multicolumn{6}{c}{$t$-test}\\
\cmidrule(lr){7-12}\cmidrule(lr){13-18}
$\tau$ & $\rho$ & $\delta$ & $I$ & $N$ & $T$
& Pred & MB & BC0 & BC1 & BC2 & BC3 & Pred & MB & BC0 & BC1 & BC2 & BC3\\
\midrule
0.03 & 0.2 & 0 & 18 & 10 & 7 & 5.0 & \bf{4.8} & 8.4 & 6.8 & \bf{5.1} & 7.1 & 5.0 & 3.4 & 6.3 & \bf{4.7} & 3.8 & \bf{4.9} \\
0.03 & 0.2 & 0 & 18 & 24 & 4 & 5.0 & \bf{4.9} & 8.1 & 6.5 & \bf{5.0} & 6.5 & 5.0 & 3.4 & 6.0 & \bf{4.7} & 3.6 & \bf{4.7} \\
0.03 & 0.2 & 0 & 20 & 14 & 5 & 5.0 & \bf{5.1} & 8.3 & 6.9 & 5.8 & 7.0 & 5.0 & 3.8 & 6.5 & \bf{5.4} & 4.4 & \bf{5.5} \\
0.03 & 0.2 & 0 & 21 & 8 & 4 & 5.0 & \bf{5.2} & 7.9 & 6.6 & \bf{5.3} & 6.6 & 5.0 & 3.8 & 6.2 & \bf{5.0} & 4.0 & \bf{4.9} \\\smallskip
0.03 & 0.2 & 0 & 15 & 8 & 4 & 5.0 & 5.7 & 9.7 & 7.7 & 5.8 & 7.7 & 5.0 & 3.7 & 7.0 & \bf{5.3} & 4.1 & \bf{5.3} \\
0.03 & 0.8 & 0 & 16 & 12 & 5 & 5.0 & 5.6 & 9.0 & 7.3 & 5.6 & 7.3 & 5.0 & 3.8 & 6.7 & \bf{5.1} & 3.8 & \bf{5.1} \\
0.03 & 0.8 & 0 & 24 & 7 & 5 & 5.0 & \bf{5.3} & 7.7 & 6.6 & \bf{5.5} & 6.6 & 5.0 & 4.3 & 6.2 & \bf{5.2} & 4.3 & \bf{5.3} \\
0.03 & 0.8 & 0 & 12 & 8 & 5 & 5.0 & 6.0 & 10.6 & 7.9 & 5.8 & 7.9 & 5.0 & 3.5 & 7.0 & \bf{4.9} & 3.2 & \bf{5.0} \\
0.03 & 0.8 & 0 & 12 & 5 & 4 & 5.0 & 5.7 & 10.3 & 7.8 & 5.7 & 7.5 & 5.0 & 3.4 & 6.9 & \bf{5.0} & 3.5 & \bf{4.9} \\\smallskip
0.03 & 0.8 & 0 & 10 & 5 & 3 & 5.0 & 6.3 & 11.2 & 8.2 & 5.6 & 7.1 & 5.0 & 3.0 & 7.0 & \bf{4.6} & 3.1 & 3.8 \\
0.10 & 0.2 & 0 & 21 & 11 & 8 & 5.0 & \bf{5.0} & 8.1 & 6.9 & 5.7 & 7.0 & 5.0 & 3.8 & 6.6 & \bf{5.3} & 4.4 & 5.6 \\
0.10 & 0.2 & 0 & 24 & 11 & 7 & 5.0 & \bf{5.1} & 7.7 & 6.4 & \bf{5.3} & 6.5 & 5.0 & 3.8 & 6.1 & \bf{5.1} & 4.3 & \bf{5.2} \\
0.10 & 0.2 & 0 & 15 & 16 & 6 & 5.0 & \bf{4.8} & 9.7 & 7.5 & 5.7 & 7.8 & 5.0 & 3.1 & 6.8 & \bf{5.2} & 4.0 & \bf{5.3} \\
0.10 & 0.2 & 0 & 18 & 8 & 7 & 5.0 & \bf{4.8} & 8.8 & 7.1 & 5.6 & 7.3 & 5.0 & 3.4 & 6.7 & \bf{5.2} & 4.0 & \bf{5.4} \\\smallskip
0.10 & 0.2 & 0 & 16 & 7 & 5 & 5.0 & \bf{5.2} & 9.1 & 7.2 & \bf{5.5} & 7.3 & 5.0 & 3.4 & 6.6 & \bf{5.1} & 3.8 & \bf{5.2} \\
0.10 & 0.8 & 0 & 20 & 18 & 5 & 5.0 & \bf{5.3} & 7.9 & 6.4 & \bf{5.2} & 6.5 & 5.0 & 3.8 & 6.1 & \bf{4.8} & 3.9 & \bf{4.8} \\
0.10 & 0.8 & 0 & 15 & 9 & 4 & 5.0 & \bf{5.5} & 9.1 & 7.2 & \bf{5.3} & 7.0 & 5.0 & 3.7 & 6.6 & \bf{4.8} & 3.7 & \bf{4.6} \\
0.10 & 0.8 & 0 & 10 & 20 & 3 & 5.0 & 5.6 & 11.7 & 8.5 & 6.0 & 7.4 & 5.0 & 2.8 & 7.3 & \bf{4.9} & 3.0 & 4.0 \\
0.10 & 0.8 & 0 & 12 & 5 & 5 & 5.0 & 5.8 & 10.5 & 7.9 & 5.7 & 7.9 & 5.0 & 3.3 & 6.9 & \bf{4.9} & 3.3 & \bf{4.9} \\
0.10 & 0.8 & 0 & 9 & 7 & 4 & 5.0 & 6.0 & 12.0 & 8.6 & 5.7 & 8.3 & 5.0 & 2.8 & 7.2 & \bf{4.7} & 2.9 & \bf{4.5} \\
\bottomrule
\end{tabular}
\end{table}
For the first objective, we summarize in Table \ref{Table5} and Web Table 3 the percent relative bias in estimating the correlations with QLS and MAQLS. It is evident that the percent bias in estimating the within-period correlation $\tau$ is much larger than that in estimating the correlation parameter $\rho$, without respect to the incorporation of matrix adjustment to the first-stage estimating equations (\ref{eq:alpha0}) and (\ref{eq:alpha1}). However, the QLS estimator for $\tau$ exhibits noticeable downward bias, especially when the number of clusters is not large. By contrast, MAQLS substantially reduces such finite-sample bias and improves the estimation of $\tau$. On the other hand, the QLS estimator for the parameter $\rho$ seems more accurate in that the absolute percent bias only occasionally exceeds one. Nevertheless, MAQLS still mildly improves the estimation of $\rho$ in that the absolute percent bias is always maintained under one. The comparative findings between QLS and MAQLS are consistent regardless of the magnitude of intervention effect $\delta$. Therefore, MAQLS is the preferred approach because it provides much less biased estimates for the correlation parameters; these more accurate correlation estimates will eventually facilitate accurate estimation of sample size and power for future cohort stepped wedge trials.
\begin{table}[htbp]
\centering
\caption{Simulation scenarios, predicted power, along with the difference between empirical and predicted power corresponding to the MAQLS $z$-test and $t$-test ($\text{DoF}=I-2$) with different variance estimators. Differences from the prediction within $0.8\%$ are highlighted in boldface and considered acceptable. Pred: predicted power; MB: model-based variance; BC0: uncorrected sandwich variance of Liang and Zeger (1986); BC1: bias-corrected sandwich variance of Kauermann and Carroll (2001); BC2: bias-corrected sandwich variance of Mancl and DeRouen (2001); BC3: bias-corrected sandwich variance of Fay and Graubard (2001).}\label{tb:main2}
\begin{tabular}{cccrrrrrrrrrrrrrrr}
\toprule
& & & & & & \multicolumn{6}{c}{$z$-test} & \multicolumn{6}{c}{$t$-test}\\
\cmidrule(lr){7-12}\cmidrule(lr){13-18}
$\tau$ & $\rho$ & $\delta$ & $I$ & $N$ & $T$
& Pred & MB & BC0 & BC1 & BC2 & BC3 & Pred & MB & BC0 & BC1 & BC2 & BC3\\
\midrule
0.03 & 0.2 & 0.3 & 18 & 10 & 7 & 89.9 & \bf{-0.5} & 1.1 & -1.0 & -3.4 & -0.9 & 86.0 & \bf{0.2} & 2.4 & \bf{-0.1} & -3.4 & \bf{0.2} \\
0.03 & 0.2 & 0.3 & 18 & 24 & 4 & 88.6 & \bf{-0.4} & 1.5 & \bf{-0.4} & -2.9 & \bf{-0.5} & 84.4 & \bf{0.5} & 3.2 & \bf{0.5} & -2.5 & \bf{0.4} \\
0.03 & 0.2 & 0.3 & 20 & 14 & 5 & 89.7 & \bf{0.0} & 1.5 & \bf{-0.5} & -2.6 & \bf{-0.3} & 86.2 & \bf{0.3} & 2.6 & \bf{0.2} & -2.5 & \bf{0.4} \\
0.03 & 0.2 & 0.4 & 21 & 8 & 4 & 87.5 & \bf{-0.5} & 1.3 & \bf{-0.8} & -3.4 & -0.9 & 83.9 & \bf{-0.1} & 2.2 & \bf{-0.5} & -3.2 & \bf{-0.6} \\\smallskip
0.03 & 0.2 & 0.5 & 15 & 8 & 4 & 90.7 & -1.1 & 1.0 & -1.2 & -3.9 & -1.1 & 85.9 & \bf{-0.3} & 3.0 & \bf{-0.2} & -4.3 & \bf{-0.3} \\
0.03 & 0.8 & 0.2 & 16 & 12 & 5 & 88.6 & \bf{-0.5} & 1.2 & -1.4 & -4.4 & -1.3 & 83.8 & \bf{0.1} & 2.5 & \bf{-0.6} & -4.4 & \bf{-0.6} \\
0.03 & 0.8 & 0.2 & 24 & 7 & 5 & 88.2 & \bf{-0.5} & 1.1 & \bf{-0.6} & -2.4 & \bf{-0.7} & 85.2 & \bf{0.2} & 1.8 & \bf{0.0} & -2.3 & \bf{0.0} \\
0.03 & 0.8 & 0.3 & 12 & 8 & 5 & 94.1 & -1.1 & \bf{0.4} & -1.8 & -4.6 & -1.9 & 88.7 & \bf{0.3} & 2.7 & \bf{-0.4} & -4.8 & \bf{-0.3} \\
0.03 & 0.8 & 0.4 & 12 & 5 & 4 & 95.2 & \bf{-0.8} & \bf{0.7} & -1.2 & -3.8 & -1.5 & 90.3 & \bf{0.5} & 2.8 & \bf{-0.3} & -4.1 & \bf{-0.6} \\\smallskip
0.03 & 0.8 & 0.5 & 10 & 5 & 3 & 94.6 & \bf{-0.7} & 1.1 & -1.1 & -4.8 & -2.3 & 87.8 & \bf{0.8} & 4.3 & \bf{-0.3} & -6.0 & -2.4 \\
0.10 & 0.2 & 0.3 & 21 & 11 & 8 & 87.8 & \bf{-0.6} & 1.5 & \bf{-0.4} & -2.8 & \bf{-0.1} & 84.3 & \bf{-0.1} & 2.6 & \bf{0.1} & -2.8 & \bf{0.5} \\
0.10 & 0.2 & 0.3 & 24 & 11 & 7 & 87.8 & \bf{-0.6} & 1.2 & \bf{-0.3} & -2.5 & \bf{-0.1} & 84.8 & \bf{0.0} & 2.1 & \bf{0.2} & -2.2 & \bf{0.4} \\
0.10 & 0.2 & 0.4 & 15 & 16 & 6 & 90.6 & -1.3 & 1.2 & -1.0 & -4.3 & \bf{-0.8} & 85.8 & \bf{-0.8} & 2.9 & \bf{-0.3} & -4.1 & \bf{0.1} \\
0.10 & 0.2 & 0.4 & 18 & 8 & 7 & 91.6 & \bf{-0.1} & 1.5 & \bf{-0.4} & -2.7 & \bf{-0.1} & 87.9 & \bf{0.4} & 2.7 & \bf{0.3} & -2.5 & \bf{0.6} \\\smallskip
0.10 & 0.2 & 0.5 & 16 & 7 & 5 & 88.6 & \bf{-0.8} & 1.9 & \bf{-0.6} & -3.5 & \bf{-0.4} & 83.8 & \bf{-0.1} & 3.4 & \bf{0.3} & -3.5 & \bf{0.6} \\
0.10 & 0.8 & 0.2 & 20 & 18 & 5 & 86.1 & \bf{-0.5} & 1.8 & \bf{-0.5} & -3.0 & \bf{-0.6} & 82.1 & \bf{-0.3} & 2.7 & \bf{0.0} & -3.2 & \bf{0.0} \\
0.10 & 0.8 & 0.3 & 15 & 9 & 4 & 89.5 & -1.0 & 1.0 & -1.5 & -4.6 & -1.8 & 84.5 & \bf{-0.1} & 2.6 & \bf{-0.8} & -4.8 & -1.1 \\
0.10 & 0.8 & 0.4 & 10 & 20 & 3 & 94.4 & \bf{-0.6} & 1.6 & \bf{-0.8} & -4.3 & -2.1 & 87.5 & \bf{0.4} & 4.6 & \bf{0.6} & -5.3 & -1.8 \\
0.10 & 0.8 & 0.4 & 12 & 5 & 5 & 93.2 & \bf{-0.5} & 0.9 & -1.3 & -4.4 & -1.4 & 87.5 & \bf{0.7} & 3.2 & \bf{-0.3} & -4.8 & \bf{-0.3} \\
0.10 & 0.8 & 0.5 & 9 & 7 & 4 & 97.3 & \bf{-0.6} & \bf{0.4} & -1.1 & -4.2 & -1.4 & 91.4 & 1.1 & 3.5 & \bf{-0.2} & -5.7 & \bf{-0.7} \\
\bottomrule
\end{tabular}
\end{table}
For the second objective, we present the empirical type I error rates of the $z$-tests and $t$-tests for the QLS and MAQLS analyses in Web Tables 4-6 and Web Figure 3. Overall, we observe that the matrix adjustment to the correlation estimation mildly affects the tests with the model-based variance but has little impact on the tests with the sandwich variance. This is in accordance with Lu et al. \cite{Lu2007}, who observed the same results for the GEE analyses of pretest-posttest CRTs. Since MAQLS provides more accurate estimation of the correlations, we will focus on this approach. Table \ref{tb:main1} summarizes the empirical type I error rates of the MAQLS $z$-tests and $t$-tests with $\text{DoF}=I-2$. We leave the results for $t$-tests with $\text{DoF}=I-(T+1)$ to Web Table 6 as these tests are conservative in many cases. From Table \ref{tb:main1}, we observe that MAQLS $z$-tests are more liberal than the corresponding MAQLS $t$-tests. The type I error rates of the MAQLS $z$-tests coupled with the model-based variance or BC2 are close to nominal when $I\geq 20$, while the MAQLS $z$-tests coupled with BC0, BC1 or BC3 are always liberal. By contrast, only the MAQLS $t$-tests with BC0 remain liberal, while MAQLS $t$-tests with BC1 or BC3 maintain close-to-nominal size and MAQLS $t$-tests with model-based variance or BC2 are conservative. Overall, the $t$-tests with $\text{DoF}=I-2$ and BC1 demonstrate test sizes that consistently agree with the nominal level.
Web Tables 7-9 and Web Figure 4 present the predicted and empirical power for each simulation scenario. Because we are only interested in tests that maintain close-to-nominal sizes, we summarize in Table \ref{tb:main2} the differences between the empirical and the predicted power only for the MAQLS $z$-tests and $t$-tests with $\text{DoF}=I-2$. Among the $z$-tests, only the choice of BC2 provides substantially lower power than predicted. While the choices of model-based variance, BC1 and BC3 provide adequate power for the $z$-tests in a number of scenarios, one should be cautious in adopting these tests with a small number of clusters since they may carry an inflated test size. On the other hand, the empirical power for MAQLS $t$-tests coupled with the model-based variance, BC1 or BC3 corresponds reasonably well with the analytical prediction from the proposed formula even for as few as $9$ clusters, while the empirical power for MAQLS $t$-tests with BC2 still tends to be substantially lower than predicted. Interestingly, the MAQLS $t$-test with $\text{DoF}=I-(T+1)$ coupled with model-based variance, BC1 or BC3 also demonstrates empirical power fairly close to prediction, even though these tests are more conservative under the null. These tests may not be preferred over the $t$-tests with $\text{DoF}=I-2$ since they are less powerful.
\section{Numerical Illustrations}
\label{sec5}
\subsection{The AEP study}
We illustrate the proposed sample size procedure to design a cohort stepped wedge CRT that aims to study the effect of an exercise intervention on the physical function of patients with end-stage renal disease \cite{Bennett2013}. The intervention was an accredited exercise physiologist (AEP) coordinated resistance exercise program, offered at hemodialysis clinics to improve the quality of life for dialysis patients. During the planning phase, it was determined that $I=15$ clinics (clusters) were available, and would be randomized over $T=4$ periods evenly spaced across $48$ weeks. At baseline, no exercise programs were offered to any clinic. At week 12, 36 and 48, a random subset of $5$ clinics cross over from control to intervention. A closed cohort of patients were recruited at baseline, and would be followed up during the study period. The primary patient-level outcome was the 30-second sit-to-stand (STS) test, recording the number of times a patient could rise from and return to a seated position in a 30-second time frame. The 30-second STS test was conducted at the end of each period, resulting in $4$ outcome measurements per patient. Based on a prior study within a similar context, a conservative estimate of the effect size was given by $\delta/\phi=0.325$ \cite{Cappy1999}, and the within-period correlation was estimated to be $\tau=0.03$ \cite{Littenberg2006}. With $I=15$ clusters, the simulations suggest the MAQLS $t$-test with $\text{DoF}=I-2=13$ could maintain nominal size and adequate power; we illustrate the power calculation based on the $t$-test statistic.
Given this is a standard stepped wedge design where an equal number of clinics switch to intervention at each step, we can show that $U=IT/2$, $W=I^2T(2T-1)/\{6(T-1)\}$, $V=I(T-2)/2$ and $Q=I^2T(T-2)/\{3(T-1)\}$. The variance expression \eqref{eq:vardelta} is then simplified to
\begin{eqnarray}\label{eq:varsimp}
\text{var}(\hat{\delta})=\frac{6(\phi/N)(T-1)(1-\rho^2)\{1+(N-1)\tau\}}
{I(T-2)\{T(1-\rho)^2+6\rho\}}.
\end{eqnarray}
If we anticipate large correlation decay so that $\rho=0.2$, the power is estimated using equation \eqref{eq:powerz} and \eqref{eq:varsimp} to be $79.4\%$ if $N=21$ and $80.5\%$ if $N=22$. Therefore at least $N=22$ patients should be recruited in each clinic to achieve $80\%$ power under the proportional decay structure. On the other hand, we could arrive at the same results by using the design effect \eqref{eq:de}. For example, in an individual randomized study, $348$ patients would be required for the hypothesized effect size. Assuming $21$ patients will be included in each clinic, the design effect is approximately $0.92$, indicating a total of $320$ patients in approximately $15.2$ clinics would be required. Since the study affords to randomize only $15$ clinics, we increase the cohort size to $N=22$, resulting in a design effect $0.94$. Therefore, $326$ patients are required for a total of $326/22\approx 14.8$ clinics, and we conclude that $22$ patients in $15$ clinics ensured $80\%$ power.
While the within-period correlation estimate was available from prior studies, published estimates of the correlation parameter $\rho$ (or decay parameter $d=1-\rho$) are currently rare. For this reason, we carry out a sensitivity analysis on the power and present the results in panel (a) of Figure \ref{Fig3}, where we fix the design resources but vary $\tau\in (0.03,0.06)$ and $\rho\in (0,1)$. Note that the upper bound of the within-period correlation $\tau$ was reported by Littenberg and MacLean \cite{Littenberg2006} and is used in this assessment. As expected, larger values of the within-period correlation reduce the study power, and further, given a certain value of the within-period correlation, a greater magnitude of decay (smaller $\rho$ or larger $d$) generally reduces the study power unless $\rho\approx 0$ (or $d\approx 1$). For the hypothesized $\tau=0.03$, the study power remains at least close to $80\%$ regardless of the correlation decay. On the other hand, the amount of correlation decay could result in further power loss if the within-period correlation $\tau$ increases. Nevertheless, the power loss is at most around $10\%$ even if the within-period correlation $\tau$ approximates the upper bound $0.06$.
\begin{figure}[htbp]
\centering\includegraphics[scale=0.4]{Fig3_EG.pdf}
\caption{Sensitivity analysis of study power for (a) the AEP study with $I=15$ clusters and $N=22$ individuals within each cluster, (b) the CORE study with $I=11$ clusters and $N=9$ individuals within each cluster, and (c) the CORE study with $I=11$ clusters and $N=30$ individuals within each cluster.}
\label{Fig3}
\end{figure}
\subsection{The CORE study}
We next illustrate the proposed sample size procedure by designing the CORE stepped wedge trial. The CORE study is a cluster randomized trial which aims to evaluate the patient-centered service design in health providers to improve the psychosocial recovery outcomes for people with severe mental illness in Australia \cite{Palmer2015}. The new service design intervention adopted the Experience Based Co-Design (EBCD) to identify users' positive and negative experiences of the service, and involved patients' participation to co-design solutions to the negative experiences. A total of $I=11$ teams from four health service providers would be participating the study; each team involved a number of service users who will be affected by the intervention. A stepped wedge design was considered appropriate for the study due to logistical constraint in simultaneously introducing the intervention to more than a few teams. The EBCD intervention will be delivered to the clusters in three waves, each with a duration of 9 months. Four teams will start the intervention in wave 1 and wave 2, respectively, while the remaining three teams receive the intervention in the final wave. In other words, the study includes four periods, with a baseline period lasting about 6 months.
The outcome of interest is the improvement in psychosocial recovery measured by the Recovery Assessment Scale Revised \citep[(RAS-R)][]{Lusczakoski2014}, and was measured for each user at the end of baseline period and each of the three follow-up period. The standardized effect size on the psychosocial recovery outcome was estimated to be $0.35$, and the within-period correlation was assumed to be $\tau=0.1$ \citep{Palmer2015}. Since the study affords to randomize only $11$ clusters, there may be a risk of inflated type I error rate with a $z$-test. As the $t$-test with $\text{DoF}=I-2=9$ performs best with respect to empirical size and power in the simulations, we determine the required cohort size based on a that test using expressions \eqref{eq:powert} and \eqref{eq:vardelta}. Assuming the correlation decay is only moderate so that $\rho=0.8$, power is estimated to be $0.79$ for $N=8$ and $0.81$ for $N=9$, barring drop out. Therefore $N=9$ is required to ensure $80\%$ power given a $5\%$ test size. We further conducted a sensitivity analysis to see how power changes according to the degree of correlation decay, and presented the power contour in panel (b) of Figure \ref{Fig3}. Due to the small sample size and the heavy tail of the $t$ distribution, the study is sensitive to correlation decay when $\tau=0.1$, and remains so even if $\tau$ approaches zero. On the other hand, the actual study planned to recruit $N=30$ users in each team. With this larger cohort size, panel (c) of Figure \ref{Fig3} suggests that the power becomes less sensitive to the correlation decay, especially as the within-period correlation approaches zero. For example, if $\tau\leq 0.02$, the study power remains at least around $80\%$ regardless of the amount of correlation decay.
\subsection{Consequences of specifying a non-decay correlation structure}
\label{sec:impact}
For GEE analyses of cohort stepped wedge designs, Li et al. \cite{LiTurnerPreisser2018} recently developed the block exchangeable correlation structure for estimating the sample size and power. With a continuous outcome and identity link function, the block exchangeable correlation structure is also implied by the linear mixed effects model for cohort studies discussed in Hooper et al. \cite{Hooper2016}, and Girling and Hemming \cite{Girling2016}. To understand the implications of alternative correlation models, we compare the variance of the intervention effect estimator obtained under the proportional decay correlation structure to that obtained under the block exchangeable correlation structure. Specifically, both the proportional decay structure and the block exchangeable correlation structure assume a constant within-period correlation, i.e. $\text{corr}(y_{ijt},y_{ij't})=\tau$ for $j\neq j'$. However, the latter correlation model also assumes constant between-period and within-individual correlations such that $\text{corr}(y_{ijt},y_{ij't'})=\alpha_1^{\text{BE}}$ for $j\neq j'$, $t\neq t'$ and $\text{corr}(y_{ijt},y_{ijt'})=\alpha_2^{\text{BE}}$, $t\neq t'$. To focus ideas, we carry out the comparisons based on the standard stepped wedge designs with a single baseline period and an equal number of clinics switching to intervention at each step. We refer to the variance of $\hat{\delta}$ obtained under the proportional decay structure as $\text{var}^{\text{PD}}(\hat{\delta})$, whose expression is provided in \eqref{eq:varsimp}. We refer to the variance of $\hat{\delta}$ obtained under the block exchangeable structure as $\text{var}^{\text{BE}}(\hat{\delta})$, whose expression is derived in Li et al. \cite{LiTurnerPreisser2018} as
\begin{eqnarray}\label{eq:varBE}
\text{var}^{\text{BE}}(\hat{\delta})=\frac{12(\phi/N)(T-1)\lambda_3\lambda_4}
{I(T-2)\{(T-1)\lambda_3+(T+1)\lambda_4\}},
\end{eqnarray}
with $\lambda_3=1+(N-1)(\tau-\alpha_1^{\text{BE}})-\alpha_2^{\text{BE}}$ and $\lambda_4=1+(N-1)\tau+(T-1)(N-1)\alpha_1^{\text{BE}}+(T-1)\alpha_2^{\text{BE}}$ as the two distinct eigenvalues of the block exchangeable matrix. If we define function $h(\alpha_1^{\text{BE}},\alpha_2^{\text{BE}})=(N-1)\alpha_1^{\text{BE}}+
\alpha_2^{\text{BE}}$, we can write the relative variance as
\begin{eqnarray}\label{eq:vr}
\frac{\text{var}^{\text{PD}}(\hat{\delta})}{\text{var}^{\text{BE}}(\hat{\delta})}
=\frac{(1-\rho^2)}
{2\{T(1-\rho)^2+6\rho\}}\left[\frac{(T-1)\{1+(N-1)\tau\}}{1+(N-1)\tau+(T-1)
h(\alpha_1^{\text{BE}},\alpha_2^{\text{BE}})}+\frac{(T+1)\{1+(N-1)\tau\}}
{1+(N-1)\tau-
h(\alpha_1^{\text{BE}},\alpha_2^{\text{BE}})}\right].
\end{eqnarray}
For each value of the within-period correlation $\tau$ and each value of the decay parameter $d=1-\rho$ in the proportional decay model, there may exist pairs of values for $(\alpha_1^{\text{BE}},\alpha_2^{\text{BE}})$ that result in the same variance of the intervention effect. Obtaining the equal variance correspondence is equivalent to finding the straight line $h(\alpha_1^{\text{BE}},\alpha_2^{\text{BE}})=\eta$ that solves $\text{var}^{\text{PD}}(\hat{\delta})/\text{var}^{\text{BE}}(\hat{\delta})=1$. Because the relative variance is quadratic in $h(\alpha_1^{\text{BE}},\alpha_2^{\text{BE}})$, such a line may not always exist within the plausible range of $(\alpha_1^{\text{BE}},\alpha_2^{\text{BE}})$ that ensures a positive definite block exchangeable correlation matrix. We confirm this observation by plotting the relative variance for varying correlation parameters. Fixing $\tau=0.03$, $T=4$ and $N=20$ similar to the AEP study, we present in Figure \ref{FigS1} the contour of $\text{var}^{\text{PD}}(\hat{\delta})/\text{var}^{\text{BE}}(\hat{\delta})$ over the the $(\alpha_1^{\text{BE}},\alpha_2^{\text{BE}})$ plane for each level of decay $d\in\{0,1,\ldots,0.9\}$. The dashed thick line indicates the equal variance correspondence. We also present the contour plots by specifying $N=100$, $T=8$ and $\tau=0.1$ in Web Figures 5-7. It is evident that the existence and location of the equal variance line on the $(\alpha_1^{\text{BE}},\alpha_2^{\text{BE}})$ plane depends on the value of correlation decay, cohort size and number of periods. If the equal variance line exists, the value of $\tau$ and number of periods only affects its location while cohort size $N$ further affects its slope, as expected from inspecting expression \eqref{eq:vr}. Apart from the equal variance correspondence, the two variances are generally different. Depending on the gray shades (colored shades in the online version), either the proportional decay or the block exchangeable structure will lead to a larger variance of the intervention effect and require a larger sample size. As a result, using the block exchangeable correlation structure in the presence of correlation decay could lead to an overestimation or underestimation of the true intervention effect variance, and vice versa. Particularly, the variances returned from these two correlation models become close when $\alpha_1^{\text{BE}}$, $\alpha_2^{\text{BE}}$ approximate zero and when the decay parameter, $d=1-\rho$, approximates one. These two restrictions result in many nearly-zero entries in the block exchangeable and proportional decay correlation matrices, increasing the dependence of sample size estimation on the within-period correlation $\tau$. Therefore, it is anticipated that the variances from the two models become similar in this particular scenario, even though in general there could be large differences between $\text{var}^{\text{PD}}(\hat{\delta})$ and $\text{var}^{\text{BE}}(\hat{\delta})$. To summarize, our key message for cohort stepped wedge designs echo the principal findings in Kasza et al. \cite{Kasza2017} for cross-sectional designs: it is possible to grossly overestimate or underestimate the variance of the intervention effect if the correlation model is misspecified, except in restrictive scenarios. In practice, researchers could investigate the sensitivity of sample size estimates to misspecification of the correlation structure when there is limited preliminary data at the design stage of the trial.
\begin{figure}[htbp]
\centering\includegraphics[scale=0.42]{FigVar_New_S1_basic.pdf}
\caption{Contour plots for the relative variance obtained under the proportional decay correlation model and the block exchangeable correlation model, for varying values of the proportional decay model decay parameter $d$ and the block exchangeable model correlation parameters $\alpha_1^{\text{BE}}$, $\alpha_2^{\text{BE}}$. In all panels, the within-period correlation $\tau=0.03$, the number of periods $T=4$ and the cohort size $N=20$. The dashed thick line indicates the equality of variances.}
\label{FigS1}
\end{figure}
\section{Discussion}
\label{sec6}
This article expanded on the design and analysis considerations for cohort stepped wedge CRTs in the presence of correlation decay. Since a cohort design involves repeated outcome assessments for fixed sets of individuals, we adopted the proportional decay structure of Lefkopoulou et al. \cite{Lefkopoulou1989} to characterize the within-cluster correlations among the outcome measurements. Based on a marginal mean model accounting for the treatment and period effects, we developed a new sample size and power procedure to design stepped wedge CRTs accounting for such correlation decay. To apply this procedure, a key step is to obtain reasonable values for the correlation parameters. The within-period correlation, $\tau$, is similar to the traditional ICC in a parallel cluster randomized trial, and may often be found in previous studies with a similar endpoint. By contrast, the correlation parameter, $\rho$ (or decay parameter $d=1-\rho$), is not as commonly reported in the literature, and therefore the sensitivity of power should be investigated across a range of values for $\rho$, as illustrated in Section \ref{sec5}. Given that accurate reporting of correlations is vitally important for designing future stepped wedge trials, we also provided an improved, matrix-adjusted quasi-least squares approach to estimate the correlation parameters along with the marginal mean parameters. The MAQLS has little impact on the estimation of the marginal mean parameters and the associated statistical tests coupled with the sandwich variance, but it substantially reduces the bias in estimating the within-period correlation $\tau$ and mildly improves the estimation of $\rho$, as confirmed in our simulation study.
In our simulation study with a small number of clusters, we found that, regardless of choice of variance estimators, the $t$-tests provide better control of the type I error rates compared to the $z$-tests, which are often liberal. Regardless of the two choices of DoF, the $t$-tests coupled with the model-based variance, BC1 and BC3 preserve the nominal size and demonstrate empirical power that agrees well with analytical prediction. Since the $t$ distribution with a smaller DoF has a heavier tail, which implies a smaller power under the alternative, we prefer the $t$-tests with $\text{DoF}=I-2$ for the design and analysis of cohort stepped wedge CRTs under the proportional correlation decay structure. An additional piece of evidence that favors $\text{DoF}=I-2$ is found in the recent simulation study by Ford \cite{Ford2018}, who showed that the GEE $t$-tests with $\text{DoF}=I-2$ provide satisfactory control of type I error rates even for $I=6$ but in the absence of correlation decay. Additional work is needed to investigate whether these extremely small sample sizes could provide adequate power under the proportional decay structure. On the other hand, although the $t$-tests with $\text{DoF}=I-(T+1)$ could provide adequate power using the model-based variance, BC1 or BC3, they are frequently conservative under the null with a small number of clusters. In fact, one needs a minimum of $T+2$ clusters to provide at least one DoF, which further limits the applications of such a $t$-test in the design and analysis of small stepped wedge trials.
Recent reviews of stepped wedge CRTs \cite{Barker2016,Martin2016,Grayling2017b} indicated that both the cross-sectional and cohort designs were common in practice. Although we have developed the design and analysis strategies specifically for cohort stepped wedge CRTs, a parallel discussion for cross-sectional stepped wedge CRTs could be equally informative. As discussed in Section \ref{sec:model}, the exponential decay structure is often used to model correlation decay in multi-period cluster randomized trials with repeated cross-sectional samples \cite{Kasza2017}. Assuming there are $N_i$ observations in each period for each cluster, we could write the exponential decay structure as $L_i(\tau,\rho)=(1-\tau)I_{TN_i}+\tau J_N\otimes F(\rho)$ without changing the interpretation of $\tau$ and $\rho$ from the cohort setting. To estimate the intervention effect and correlation parameters, the MAQLS approach could still be applied once we replace the second-stage estimating equations \eqref{eq:secondalpha0} and \eqref{eq:secondalpha1} by
\begin{eqnarray*}
\sum_{i=1}^I \text{tr}\left\{\frac{\partial L_i^{-1}(\hat{\alpha}_0,\hat{\alpha}_1)}{\partial\alpha_0}L_i(\tau,\rho)\right\}&=&0,\\
\sum_{i=1}^I \text{tr}\left\{\frac{\partial L_i^{-1}(\hat{\alpha}_0,\hat{\alpha}_1)}{\partial\alpha_1}L_i(\tau,\rho)\right\}&=&0.
\end{eqnarray*}
Such modifications are necessary because $\tau$ and $\rho$ are no longer separable in $L_i(\tau,\rho)$, and hence updates for $\tau$ and $\rho$ do not come in closed forms. Correspondingly, the inseparability between $\tau$ and $\rho$ also precludes the derivation of an analytical inverse $L_i^{-1}(\tau,\rho)$, and therefore one may not be able to obtain a simple algebraic expression for $\text{var}(\hat{\delta})$. As a result, sample size and power calculation requires numerically inverting the correlation matrix $L_i(\tau,\rho)$. In fact, with a continuous outcome and the identity link, it is straightforward to show that the QLS-based sample size procedure with $L_i(\tau,\rho)$ coincides with the mixed-effects model-based sample size procedure developed in Kasza et al.\cite{Kasza2017} with exponential correlation decay.
We have assumed that each individual has complete follow-up during the study. In reality, individual drop-out may be anticipated and could be accounted for in the design phase. Given an expected attrition rate $\gamma$, a simple and commonly-used strategy is to inflate the required total sample size by $1/(1-\gamma)$, so that a complete-trajectory GEE analysis may provide adequate power if the drop-out or missingness is completely at random \cite{little2002}. More sophisticated approaches that deal with monotone missingness have been studied for repeated-measure randomized trials and may be adapted to the stepped wedge context by considering staggered treatment assignments and appropriate levels of clustering \cite{Rochon1998,Jung2003}. In any event, trial implementation methodologies to prevent attrition bias in longitudinal cluster randomized trials merit further investigation \cite{Prost2015}.
One simplification we made in the sample size and power calculations was to assume equal cluster (cohort) sizes. It has been shown that cluster size imbalance leads to reduced power in parallel CRTs and therefore may be accounted for in the design phase \citep{Eldridge2006}. For a stepped wedge trial, Girling \cite{Girling2018} computed the relative efficiency of unequal versus equal cluster sizes by assuming a linear mixed-effects model without correlation decay. It was concluded that the efficiency loss due to unequal cluster sizes is unlikely to exceed $12\%$ across a wide range design of resources and correlation values. Nevertheless, a corresponding expression for the relative efficiency accounting for correlation decay is currently not available and should merit additional study. The availability of such expressions for relative efficiency could inform the amount of additional design resources required to compensate the efficiency loss due to unequal cluster sizes. Another limitation of our design strategy is that we have assumed the proportional decay correlation is the correctly specified within-cluster dependency structure. However, both the QLS or MAQLS estimators for the intervention effect remain consistent even if the correlation structure is misspecified. If it is anticipated in the design phase that the working correlation may be misspecified, one could follow the general idea of Rochon \cite{Rochon1998} to develop a modified sample size procedure based on the sandwich variance.
Finally, we have assumed a continuous outcome and an identity link function, corresponding to the scenarios of the illustrative examples. In practice, categorical outcomes could be collected in cohort stepped wedge designs. Under correlation decay, we could extend the QLS-based sample size procedure to accommodate binary and count outcomes by following the steps outlined in Section 3.2 of Li et al. \cite{LiTurnerPreisser2018}. In those cases, a further complication is that the variance is an explicit function of the marginal mean, and so the magnitude of the period effects necessarily affects the variance for the intervention effect. We plan to carry out future work to investigate the operating characteristics of such sample size procedures for binary and count outcomes.
\section*{Acknowledgments}
This work is supported within the National Institutes of Health (NIH) Health Care Systems Research Collaboratory by the NIH Common Fund through cooperative agreement U24AT009676 from the Office of Strategic Coordination within the Office of the NIH Director and cooperative agreement (UH3DA047003) from the National Institute on Drug Abuse. The content is solely the responsibility of the author and does not necessarily represent the official views of the National Institutes of Health. The author thanks Dr. John S. Preisser at the University of North Carolina at Chapel Hill for providing thoughtful comments to an earlier version of this article. The author is also grateful for the Associate Editor and two anonymous reviewers for providing helpful comments, which improved the exposition of this work.
\section*{Conflict of interest}
The author declares no potential conflict of interests.
\section*{Supporting information}
Additional supporting information including Web Appendices A--F, Web Tables and Figures, and R code for fitting the matrix-adjusted quasi-least squares with continuous outcomes may be found online in the supporting information tab for this article.
\bibliographystyle{wileyNJD-AMA}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 4,802 |
\section{Introduction}
\label{sec: introduction}
Video super-resolution (SR) aims to estimate high-resolution (HR) frames from a low-resolution (LR) sequence.
It is a fundamental problem in the vision and graphics communities and has received active research efforts within the last decade as high-definition devices have been widely used in our daily lives.
As the HR sequences are usually contaminated by unknown blur, it is quite challenging to restore HR images from low-resolution sequences.
Since video SR is an ill-posed problem, conventional methods usually estimate underlying motion and latent images simultaneously in
a variational approach~\cite{Milanfar/tip04,prior/eccv96,prior/tip01,DBLP:conf/cvpr/ShaharFI11,Bayesian/vsr/tpami14,maziyang/vsr/cvpr15}.
To improve the performance, kinds of hand-crafted priors on the latent images and motion fields have been widely used in these methods.
In spite of achieving decent results, these algorithms usually need to complex energy functions or complex matching processes and the performance is limited by the hand-crafted priors.
In addition, most of these algorithms usually use known blur kernels (e.g., Gaussian blur kernel, Bicubic kernel) and do not model blur kernels in the restoration, which cannot effectively capture the intrinsic characteristics of video SR~\cite{Bayesian/vsr/tpami14}.
Motivated by the first end-to-end trainable network for single image SR~\cite{SRCNN/tpmai}, lots of methods based on deep convolutional neural networks (CNNs) have been proposed~\cite{VDSR,Accelerating/dong,DBPN,RCAN,edsr,SRGAN}.
These algorithms achieve decent results in single image SR. However, directly using these algorithms cannot solve the video SR problem well.
To overcome this problem, most existing algorithms focus on developing effective motion fields and alignment estimation methods.
For example, the subpixel motion compensation based on optical flow~\cite{xintao/iccv17}, deformable alignment networks~\cite{TDAN,edvr}, and spatial alignment networks~\cite{liu/iccv17,vespcn,tof}.
To better restore latent images, the recurrent approaches and Generative Adversarial Networks (GANs) have been developed~\cite{TecoGAN,GAN/video/tip19}.
These methods significantly promote the progress of video SR.
However, they usually assume the blur kernel is known (e.g., Bicubic kernel). Therefore, without modeling the blur kernel usually leads to over-smoothed results (Figure~\ref{fig: teaser}).
To overcome this problem, several algorithms explicitly estimate blur kernels for SR~\cite{Tomer/blindsr/iccv13,Gu/cvpr19,Irani/cvpr18,iGAN/kernel/nips19}.
These algorithms show that using the estimated blur kernels for image SR is able to improve the results significantly~\cite{Tomer/blindsr/iccv13,iGAN/kernel/nips19}.
However, these algorithms are mainly developed for single image SR which cannot be extended to video SR directly.
The methods by~\cite{Bayesian/vsr/tpami14,maziyang/vsr/cvpr15} simultaneously estimate underlying motion and blur kernels for image restoration. However, the performance is limited by the hand-crafted image priors.
To overcome the above problems, we propose an effective video SR algorithm that simultaneously estimates underlying motion blur, motion field, and latent image by deep CNN models so that our method can not only avoid the hand-crafted priors and but also effectively estimate blur kernels and motion fields for better image restoration.
The proposed algorithm mainly consists of motion blur estimation, motion field estimation, and latent image restoration modules.
The motion blur estimation generates blur kernels based on the image formation of video SR and is able to provide intermediate latent images with sharp contents.
The motion field estimation is used to explore the spatio-temporal information from adjacent frames so that it can guide the deep CNN model for better image restoration.
By training the proposed algorithm in an end-to-end manner, it is able to generate clearer images with finer structural details (Figure~\ref{fig: teaser}).
The main contributions are summarized as follows:
\begin{itemize}
\item We propose an effective video SR algorithm that simultaneously estimates blur kernels, motion fileds, and latent images by deep CNN models.
\item We develop an effective kernel estimation method and image deconvolution algorithm based on the image formation of video SR.
To restore high-quality images, we explore the spatio-temporal information from the adjacent frames so that it can guide the deep CNN model for better image restoration.
\item We both quantitatively and qualitatively evaluate the proposed algorithm on benchmark datasets and real-world videos and show that it performs favorably against state-of-the-art methods.
\end{itemize}
\section{Related Work}
\label{sec: Related-Work}
We briefly discuss methods most relevant to this work and put this work in proper context.
\vspace{-2mm}
{\flushleft \bf{Variational approach.}}
Since video SR is highly ill-posed, early approaches mainly focus on developing effective priors~\cite{Milanfar/tip04,prior/eccv96,prior/tip01,DBLP:conf/cvpr/ShaharFI11} on the HR images to solve this problem.
As these methods usually use known blur kernels to approximate the real ones which will lead to over-smoothed results. Several methods~\cite{Bayesian/vsr/tpami14,maziyang/vsr/cvpr15} simultaneously estimate motion fileds, blur kernels, and latent images in a Maximum a posteriori (MAP) framework.
In~\cite{Bayesian/vsr/tpami14}, Liu and Sun solve video SR by a Bayesian framework, where the motion fileds, blur kernels, latent images, and noise levels are estimated simultaneously.
Ma et al.~\cite{maziyang/vsr/cvpr15} propose an effective Expectation Maximization (EM) framework to jointly solve video SR and blur estimation.
Although promising results have been achieved, these algorithms require solving complex optimization problems. In addition, the performance is limited by the hand-crafted priors.
\vspace{-2mm}
{\flushleft \bf{Deep learning approach.}}
Motivated by the success of deep learning-based single image SR~\cite{SRCNN/tpmai,VDSR,Accelerating/dong,DBPN,RCAN,edsr,SRGAN},
several methods~\cite{Huang/nips15,liao/iccv15,Kappeler/tci16,vespcn,liu/iccv17,xintao/iccv17,tof,jo/cvpr18,VDBPN/cvpr19,edvr,TDAN} explore the spatio-temporal information for video SR.
Huang et al.~\cite{Huang/nips15} develop an effective bidirectional recurrent convolutional network to model the long-term contextual information.
Some algorithms~\cite{liao/iccv15,Kappeler/tci16} first estimate motion fields based on the hand-crafted priors and then use a deep CNN model to restore high-quality images.
In~\cite{vespcn}, Caballero et al. develop an effective motion compensation and explore the spatio-temporal information for video SR.
Liu et al.~\cite{liu/iccv17} develop a temporal adaptive neural network and a spatial alignment network to better explore the temporal information.
In~\cite{xintao/iccv17}, Tao et al. propose an effective subpixel motion compensation layer based on the estimated motion fields for video SR.
Xue et al.~\cite{tof} demonstrate the effect of optical flow on video image restoration and propose an effective video restoration framework to solve general video restoration problems.
Instead of explicitly using optical flow for alignment, Jo et al.~\cite{jo/cvpr18} dynamically estimate upsampling filters.
In~\cite{VDBPN/cvpr19}, Haris et al. extend the deep back-projection method~\cite{DBPN} by a recurrent network.
Wang et al.~\cite{edvr} improve the deformable convolution~\cite{TDAN} and develop an effective temporal and spatial attention to solve video restoration.
This algorithm wins the champions in the NTIRE19 video restoration~\cite{REDS}.
To generate more realistic images, GANs have been used to solve the both single~\cite{SRGAN,Sajjadi_2017_ICCV,GAN/data/eccv18}
and video~\cite{TecoGAN,GAN/video/tip19} SR problems.
These algorithms generate decent results on video SR. However, these algorithms either explicitly or implicitly assume that the blur kernels are known and do not model the blur kernels for SR, which accordingly leads to over-smoothed results.
Estimating blur kernels has been demonstrated effective for image SR, especially for the details restoration~\cite{Tomer/blindsr/iccv13,Gu/cvpr19,Irani/cvpr18,iGAN/kernel/nips19,blindsr/sa09,zhangkai/cvpr19/sr}.
However, these algorithms are designed for single image SR. Few of them have been developed for video SR.
Different from these methods, we propose a deep CNN model to simultaneously estimate blur kernels, motion fields, and latent frames so that high-quality videos can be better-restored.
\begin{figure*}[!t]\footnotesize
\begin{center}
\begin{tabular}{c}
\includegraphics[width = 0.98\linewidth]{figures/flowchart_v4}\\
\end{tabular}
\end{center}
\vspace{-4mm}
\caption{An overview of the proposed method. The proposed algorithm takes three adjacent frames and initialized input kernel as the input and super-resolves the center image (i.e., $L_i$).
First, we use $\mathcal{N}_k$ to estimate blur kernels from initialized input kernel, where the Gaussian blur kernel is used as the initialized kernel. Then, we generate an intermediate HR image ($\tilde{I}_{i}^{*}$) based on an image deconvolution method with the estimated blur kernels.
To remove the artifacts in $\tilde{I}_{i}^{*}$, we compute the optical flow based on the Bicubic usampling results of three adjacent frames and generate the warped images (i.e., $\tilde{I}_{i+1}^{b}$, $\tilde{I}_{i-1}^{b}$) to guide the restoration of $\tilde{I}_{i}^{*}$ based on $\mathcal{N}_I$.
The proposed algorithm is jointly trained in an end-to-end manner and generates better high-quality images.
The mathematical operators are detailed in main contents.
}
\label{fig: flow-chart}
\vspace{-4mm}
\end{figure*}
\section{Revisiting Variational Methods}
\label{sec: motivation}
The proposed algorithm is motivated by the variational methods~\cite{Milanfar/tip04,Bayesian/vsr/tpami14,maziyang/vsr/cvpr15} for video SR.
In this section, we first revisit how these variational methods~\cite{Milanfar/tip04,Bayesian/vsr/tpami14,maziyang/vsr/cvpr15} solve video SR and then introduce the proposed algorithm.
Following the definitions of~\cite{Milanfar/tip04,Bayesian/vsr/tpami14,maziyang/vsr/cvpr15}, the degradation model for video SR is:
\begin{equation}
L_j = \mathbf{SK}\mathbf{F}_{\mathrm{u}_{i\to j}}I_i + n_j,
\label{eq: video-sr-model}
\end{equation}
where $\{L_{j}\}_{j =i-N}^{i+N}$ denote a set of LR images with $2N+1$ frames; $I_i$ denotes the HR image; $n_j$ denotes image noise,
$\mathbf{S}$ and $\mathbf{K}$ denote the matrix form of down-sampling and blur kernel; $\mathbf{F}_{\mathrm{u}_{i\to j}}$ denotes the warping matrix w.r.t. optical flow $\mathrm{u}_{i\to j}$,
and $\mathrm{u}_{i\to j}$ denotes the optical flow from $I_i$ to $I_j$.
Based on the degradation model~\eqref{eq: video-sr-model}, the HR image ${I_i}$, optical flow $\mathrm{u}_{i\to j}$, and blur kernel ${K}$ can be estimated by a Maximum a posteriori (MAP):
\begin{equation}
\begin{split}
\{I_i^*, K^*, \{\mathrm{u}^*_{i\to j}\}\} &= \arg\max_{{I_i}, {K}, \{\mathrm{u}_{i\to j}\}}p({I_i}, {K}, \{\mathrm{u}_{i\to j}\}|\{{L}_j\}), \\
& = p({I_i})p({K})\prod_jp(\mathrm{u}_{i\to j}) p({{L}_i}|{I_i}, {K})\\
&\qquad\qquad \prod_{j\neq i}p(\{{L}_j\}|{I_i}, {K}, \{\mathrm{u}_{i\to j}\})
\label{eq: video-sr-model-map}
\end{split}
\end{equation}
Using hand-crafted image priors $\rho({I_i})$, $\varphi(\mathrm{u}_{i\to j})$, and $\phi(K)$ on the HR image ${I_i}$, optical flow $\mathrm{u}_{i\to j}$, and blur kernel ${K}$, respectively, the video SR process can be achieved by alternatively minimizing~\cite{Bayesian/vsr/tpami14}:
\begin{equation}
\begin{split}
I_i^* = &\arg\min_{I_i}\|\mathbf{SK}I_i - L_i\| + \\
&\sum_{j=i-N, j\neq i}^{i+N}\|\mathbf{SK}\mathbf{F}_{\mathrm{u}_{i\to j}}I_i - L_j\| + \rho({I_i}),
\label{eq: video-sr-model-latent-frame}
\end{split}
\end{equation}
\begin{equation}
\begin{split}
\mathrm{u}_{i\to j}^* = \arg\min_{\mathrm{u}_{i\to j}}\|\mathbf{SK}\mathbf{F}_{\mathrm{u}_{i\to j}}I_i - L_j\| + \varphi(\mathrm{u}_{i\to j}),
\label{eq: video-sr-model-optical-flow}
\end{split}
\end{equation}
and
\begin{equation}
\begin{split}
K^* = \arg\min_K\|\mathbf{S}\mathbf{T}_{I_i}K - L_i\| + \phi(K),
\label{eq: video-sr-model-kernel}
\end{split}
\end{equation}
where $\mathbf{T}_{I_i}$ is a matrix of latent HR image $I_i$ w.r.t. $K$~\cite{Bayesian/vsr/tpami14}.
Although the video SR algorithms~\cite{Bayesian/vsr/tpami14,maziyang/vsr/cvpr15} based on above model have been demonstrated effective in both benchmark datasets and real-world videos,
they need to define the hand-crafted image priors $\rho({I_i})$, $\varphi(\mathrm{u}_{i\to j})$, and $\phi(K)$ which usually lead to highly non-convex objective function~\eqref{eq: video-sr-model-map}.
This makes the video SR problem more difficult to solve. In addition, the performance of video SR is limited by the hand-crafted image priors.
We further note that most existing deep learning-based methods usually employ deep CNN models to solve video SR problem.
Although these methods do not need to define hand-crafted priors, they cannot capture the intrinsic characteristics of video SR as the blur kernel is assumed to be known (e.g., Bicubic~\cite{edvr}, Gaussian~\cite{gaussian/kernel/cvpr18}) and do not model it in the SR process, which accordingly lead to over-smoothed results.
To overcome these problems, we develop an effective deep CNN model which consists of motion blur estimation, motion field estimation, and latent image restoration for video SR.
The proposed model does not need the hand-crafted priors and is able to capture the intrinsic characteristics of degradation process in video SR by modeling blur kernels. Thus, it can generate much better super-resolved videos with clearer structural details.
\section{Proposed Algorithm}
The overview of the proposed method is shown in Figure~\ref{fig: flow-chart}.
In the following, we explain the main ideas for each component in details.
\subsection{Motion blur estimation}
\label{ssec: Optical flow estimation}
We note that the motion blur estimation based on~\eqref{eq: video-sr-model-kernel} needs to define a hand-crafted prior $\phi(K)$ which usually leads to a complex optimization process.
To avoid hand-crafted priors and the complex optimization process, we develop a deep CNN model $\mathcal{N}_k$ to effectively estimate motion blur kernels.
The network $\mathcal{N}_k$ takes initialized Gaussian kernels as input and refine it based on the degradation model~\eqref{eq: video-sr-model}.
Given the HR images $\{I_i\}$ and the corresponding LR images $\{L_i\}$, we use the first term of~\eqref{eq: video-sr-model-kernel}
(which is related to the degradation model~\eqref{eq: video-sr-model}) to constrain the deep CNN model $\mathcal{N}_k$:
\begin{equation}
\mathcal{L}_k = \|\mathbf{S}\tilde{K}I_i - L_i\|_1,
\label{eq: kernel-estimation-loss}
\end{equation}
where $\tilde{K}$ denotes the output of the deep CNN model $\mathcal{N}_k$ and $\ell_1$ norm is used.
Similar to~\cite{dongwei/deepdeblur}, the motion blur estimation network $\mathcal{N}_k$ consists of two fully connected layers, where the first fully connected layer is followed by a ReLU activation function
and the second one is follow by a Softmax function to ensure that each element of the blur kernel is nonnegative and the summation of all elements is $1$.
Figure~\ref{fig: intermediate-result-kernel}(c) shows the estimated blur kernel from bicubic downsampling LR images. We note that the shape of the blur kernel is quite similar to that of Bicubic blur kernel~\cite{levin/etal/blindsr}. We will demonstrate the effectiveness of the motion blur estimation in Section~\ref{sec: Analysis and Discussions}.
\begin{figure}[!t]\footnotesize
\centering
\begin{tabular}{ccc}
\includegraphics[width=0.32\linewidth]{figures/kernel/00000001_near_crop_re} &\hspace{-4mm}
\includegraphics[width=0.32\linewidth]{figures/kernel/1_bicubic_trim} &\hspace{-4mm}
\includegraphics[width=0.32\linewidth]{figures/kernel/1_deblur_trim} \\
(a) LR &\hspace{-4mm} (b) Bicubic &\hspace{-4mm} (c) Deblurred \& kernel\\
\end{tabular}
\caption{Effect of the intermediate latent image restoration and motion blur estimation. Using the estimated blur kernel to deblur LR images generates sharper images (c).}%
\label{fig: intermediate-result-kernel}
\vspace{-4mm}
\end{figure}
\subsection{Intermediate latent image restoration}
\label{sec: Latent frame restoration}
With the blur kernel $K$, we can estimate HR image from input LR image $L_i$ according to~\eqref{eq: video-sr-model-latent-frame}.
However, solving~\eqref{eq: video-sr-model-latent-frame} needs the optical flow and image prior.
Recent algorithms~\cite{jiawei/cvpr17/deblur,kaizhang/cvpr17/deblur} show that the image prior can be efficiently learned by deep CNN models
so that the restoration process can be achieved by alternatively solving a simple model to restore intermediate latent images and using deep CNN models to remove the noise and artifacts in the intermediate latent images.
Motivated by the success of~\cite{jiawei/cvpr17/deblur,kaizhang/cvpr17/deblur}, we first estimate an intermediate HR image by a simple image deconvolution model and then explore the information of adjacent frames and deep CNN models to restore high-quality images.
To obtain the intermediate HR image efficiently, we propose an image deconvolution model based on the image formation~\eqref{eq: video-sr-model} by:
\begin{equation}
\vspace{-1mm}
\tilde{I}_i^* = \arg\min_{I_i}\|\mathbf{S}\mathbf{\tilde{K}}I_i - L_i\|^2 + \gamma\|\nabla I_i\|^2,
\label{eq: deconvolution-step-1}
\vspace{-1mm}
\end{equation}
where $\|\nabla I_i\|^2$ is used to make the problem well-posed and $L_2$ norm is used to make the problem be efficiently solved, and $\nabla$ denotes the gradient operator.
Note that~\eqref{eq: deconvolution-step-1} is a least square problem. We can get the closed-form solution by:
\begin{equation}
\vspace{-1mm}
\tilde{I}_i^* = \left(\mathbf{\tilde{K}}^{\top}\mathbf{S}^{\top}\mathbf{S}\mathbf{\tilde{K}}^{\top} + \gamma (\mathbf{D}_v^{\top}\mathbf{D}_v + \mathbf{D}_h^{\top}\mathbf{D}_h)\right)^{-1}{\mathbf{\tilde{K}}^{\top}\mathbf{S}^{\top}L_i},
\label{eq: deconvolution-step-1}
\end{equation}
where $\mathbf{D}_h$ and $\mathbf{D}_v$ denote the matrices of derivative filters in horizontal and vertical directions.
Figure~\ref{fig: intermediate-result-kernel}(c) shows the estimated intermediate HR image $\tilde{I}_i^*$.
Note that though $\tilde{I}_i^*$ contains noise and artifacts, it also contains some clear contents which facilitate the following image restoration, especially for the structural details restoration (Figure~\ref{fig: teaser}(i)).
In the following, we will use the adjacent frames and deep CNN models to remove noise and artifacts in $\tilde{I}_i^*$.
\subsection{Optical flow estimation}
\label{ssec: Optical flow estimation}
The optical flow is used to warp adjacent frames to the reference frame and provide more reliable information for the reference frame restoration.
In this work, we use the PWC-Net~\cite{pwcnent/deqing} as the proposed optical flow estimation algorithm given its small model size and decent performance.
We note that the intermediate HR image contains artifacts and noise which may interfere the optical flow estimation.
Thus, we use the Bicubic upsampling result of each LR image to compute the initial optical flow.
Given any three adjacent frames $L_{i-1}$, $L_{i}$, and $L_{i+1}$, we first use the Bicubic upsampling to obtain $I_{i-1}^{b}$, $I_{i}^{b}$, and $I_{i+1}^{b}$, respectively. Then, the PWC-Net (denoted as $\mathcal{N}_o$ in Figure~\ref{fig: flow-chart}) is used to compute optical flow $\mathrm{u}_{i-1\to i}$ and $\mathrm{u}_{i+1\to i}$ based on the Bicubic upsampling results, where the PWC-Net for the computations of $u_{i-1\to i}$ and $u_{i+1\to i}$ shares the same parameters.
Based on the estimated optical flow, we use the bilinear interpolation method to obtain the warped images $I_{i-1}^{b}(\mathrm{x} + \mathrm{u}_{i-1\to i})$
and $I_{i+1}^{b}(\mathrm{x} + \mathrm{u}_{i+1\to i})$ according to~\cite{pwcnent/deqing} (i.e., $\tilde{I}_{i+1}^{b}$, $\tilde{I}_{i-1}^{b}$ in Figure~\ref{fig: flow-chart}).
\subsection{Guided image restoration}
\label{sec: High-quality HR image restoration}
Using the warped images $\tilde{I}_{i-1}^{b}$ and $\tilde{I}_{i+1}^{b}$ as the guidance,
we can employ existing deep CNN models for image restoration to estimate a high-quality image from $\tilde{I}_i^*$.
In this paper, we use the deep CNN model by~\cite{RCAN} to restore high-quality images,
where we change the network input as the concatenation of $\tilde{I}_{i-1}^{b}$, $\tilde{I}_{i+1}^{b}$, and $\tilde{I}_i^{*}$.
However, as $\tilde{I}_{i-1}^{b}$, $\tilde{I}_{i+1}^{b}$, and $\tilde{I}_i^{*}$ are HR images, they will increase the computational cost.
To overcome this problem, we adopt the space-to-depth transformation~\cite{gaussian/kernel/cvpr18} to divide these HR images into LR ones.
Thus, the high-quality image can be obtained by
\begin{equation}
I_i^* = \mathcal{N}_{I}(\mathcal{C}(\mathcal{S}(\tilde{I}_{i+1}^{b}; \mathcal{S}(\tilde{I}_i^*); \mathcal{S}(\tilde{I}_{i-1}^{b}))),
\label{eq: image-restoration-final}
\end{equation}
where $\mathcal{N}_{I}$ denotes the restoration network, $\mathcal{C}$ denotes the concatenation operation, and $\mathcal{S}$ denotes the space-to-depth transformation.
We use the following loss function to constrain the network $\mathcal{N}_{I}$:
\begin{equation}
\vspace{-1mm}
\mathcal{L} = \|I_i^* - I_i\|_1.
\label{eq: loss-function-whole}
\vspace{-1mm}
\end{equation}
\vspace{-1mm}
\subsection{Implementation details}
\label{ssec: Parameter settings and training data}
\vspace{-1mm}
{\flushleft \bf{Training datasets.}}
We train the proposed algorithm using the REDS dataset~\cite{REDS}, where the REDS dataset contains 300 videos, each video contains 100 frames with an image size of $720\times 1280$ pixels.
Among 300 videos, 240 videos are used for training, 30 videos are used for validation, and the remaining 30 videos are used for test.
During the training, we randomly choose 45 consecutive frames from each video in the training dataset to train the proposed algorithm.
\vspace{-5mm}
{\flushleft \bf{Parameter settings and training details.}}
We empirically set $\gamma = 0.02$.
We use the similar data augmentation method to~\cite{edvr} to generate training data. The batch size is set to be $8$, and the size of each image patch is $64\times 64$ pixels.
In the training process, we use the ADAM optimizer~\cite{adam} with parameters $\beta_1 = 0.9$, $\beta_2 = 0.999$, and $\epsilon = 10^{-8}$.
The motion blur estimation network $\mathcal{N}_k$ takes the Gaussian kernels as the input, where the settings of the Gaussian kernels are the same as~\cite{shan/sr/sa08}.
The size of Gaussian kernel is empirically set to be $15\times 15$ pixels.
The sizes of the two fully connected layers are set to be $1000$ and $225$, respectively. The output size of $\mathcal{N}_k$ is set to be $15\times 15$ pixels.
The optical flow estimation network $\mathcal{N}_o$ is initialized by the pre-trained model~\cite{pwcnent/deqing}.
Both the kernel estimation network $\mathcal{N}_k$ and the image restoration network $\mathcal{N}_I$ use the random initialization and are trained from scratch.
The learning rates for both kernel estimation network $\mathcal{N}_k$ and image restoration network $\mathcal{N}_I$ are initialized to be $10^{-4}$.
As we use the pre-trained model~\cite{pwcnent/deqing} to initialize the optical flow estimation network, the learning rate for this network is initialized to be $10^{-6}$.
All the learning rates decrease to ${0.2}$ times after every 50 epochs.
During the training process, we first train $\mathcal{N}_k$ and then jointly train $\mathcal{N}_o$ and $\mathcal{N}_I$ in an end-to-end manner.
The algorithm is implemented based on the PyTorch. More experimental results are included in the supplemental material.
The training code and test model are available at~\url{https://github.com/jspan/blindvsr}.
\begin{table*}[!t]
\vspace{-1mm}
\caption{Quantitative evaluations on the REDS dataset~\cite{REDS} in terms of PSNR and SSIM. All the results are generated according to the published models for fair comparisons. The best two results are shown in \textcolor[rgb]{1.00,0.00,0.00}{\textbf{red}} and \textcolor[rgb]{0.00,0.00,1.00}{\underline{blue}}.
}
\vspace{1mm}
\label{tab: results-reds}
\footnotesize
\centering
\begin{tabular}{lcccccccccccc}
\toprule
%
Methods &Bicubic &RCAN~\cite{RCAN} & SPMC~\cite{xintao/iccv17} &DUF~\cite{jo/cvpr18} &TOF~\cite{tof} &RBPN~\cite{VDBPN/cvpr19} &EDVR-M~\cite{edvr} &EDVR~\cite{edvr} &Ours \\
\hline
Avg. PSNRs &25.59 &28.71 &27.74 &28.60 &27.77 &29.82 & 30.51 &\textcolor[rgb]{1.00,0.00,0.00}{\textbf{31.07 }} &\textcolor[rgb]{0.00,0.00,1.00}{\underline{30.51}} \\
Avg. SSIMs &0.7077 &0.8184 &0.7915 &0.8254 &0.7949 &0.8537 &0.8699 & \textcolor[rgb]{1.00,0.00,0.00}{\textbf{0.8802}} &\textcolor[rgb]{0.00,0.00,1.00}{\underline{0.8674}} \\
\bottomrule
\end{tabular}
\vspace{-3mm}
\end{table*}
\begin{figure*}[!t]\footnotesize
\begin{center}
\begin{tabular}{cccccccc}
\multicolumn{3}{c}{\multirow{5}*[74pt]{\includegraphics[width=0.368\linewidth, height = 0.352\linewidth]{figures/reds20/00000010_mark}}}&\hspace{-4.5mm}
\includegraphics[width=0.15\linewidth]{figures/reds20/00000010_x4} &\hspace{-4.5mm}
\includegraphics[width=0.15\linewidth]{figures/reds20/00000010_bic_crop_x4} &\hspace{-4.5mm}
\includegraphics[width=0.15\linewidth]{figures/reds20/00000010_rcan_crop_x4} &\hspace{-4.5mm}
\includegraphics[width=0.15\linewidth]{figures/reds20/00000010_rcan_crop_x4} \\
\multicolumn{3}{c}{~} &\hspace{-4.5mm} (b) HR patch &\hspace{-4.5mm} (c) Bicubic &\hspace{-4.5mm} (d) RCAN~\cite{RCAN} &\hspace{-4.5mm} (e) SPMC~\cite{xintao/iccv17}\\
\multicolumn{3}{c}{~} & \hspace{-4.5mm}
\includegraphics[width=0.15\linewidth]{figures/reds20/00000010_duf_crop_x4} & \hspace{-4.5mm}
\includegraphics[width=0.15\linewidth]{figures/reds20/00000010_toflow_crop_x4} & \hspace{-4.5mm}
\includegraphics[width=0.15\linewidth]{figures/reds20/00000010_rbpn_crop_x4} & \hspace{-4.5mm}
\includegraphics[width=0.15\linewidth]{figures/reds20/00000010_ours_crop_x4} \\
\multicolumn{3}{c}{\hspace{-4.5mm} (a) Ground truth HR image} & \hspace{-4.5mm} (f) DUF~\cite{jo/cvpr18} &\hspace{-4.5mm} (g) TOFlow~\cite{tof}&\hspace{-4.5mm} (h) RBPN~\cite{VDBPN/cvpr19} & \hspace{-4.5mm} (i) Ours\\
\end{tabular}
\end{center}
\vspace{-3mm}
\caption{Video SR result ($\times 4$) on the REDS dataset~\cite{REDS}. The proposed algorithm recovers high-quality images with clearer structures.
}%
\label{fig: results-reds}
\vspace{-0mm}
\end{figure*}
\begin{table*}[!t]
\vspace{-1mm}
\caption{Quantitative evaluations on the Vid4 dataset~\cite{Bayesian/vsr/tpami14} and SPMCS dataset~\cite{xintao/iccv17} in terms of PSNR and SSIM.
All the results are generated according to the published models for fair comparisons. * means the values from the reported results~\cite{tof}. The best two results are shown in \textcolor[rgb]{1.00,0.00,0.00}{\textbf{red}} and \textcolor[rgb]{0.00,0.00,1.00}{\underline{blue}}.
}
\vspace{1mm}
\label{tab: result-datasets-vid4-spmc}
\footnotesize
\centering
\begin{tabular}{l|ccccccccccccc}
\toprule
%
& Methods &Bicubic & BayesianSR~\cite{maziyang/vsr/cvpr15}* &RCAN~\cite{RCAN} & SPMC~\cite{xintao/iccv17} &DUF~\cite{jo/cvpr18} &TOFlow~\cite{tof} &Ours\\
\hline
\multirow{2}{*}{Vid4 dataset~\cite{Bayesian/vsr/tpami14}} &Avg. PSNRs &21.91 & 21.95 &24.03 &24.39 &\textcolor[rgb]{1.00,0.00,0.00}{\bf{25.84}} &24.22 &\textcolor[rgb]{0.00,0.00,1.00}{\underline{25.35}} \\
&Avg. SSIMs &0.5825 &0.7369 &0.7206 &0.7534 &\textcolor[rgb]{1.00,0.00,0.00}{\bf{0.8151}} &0.7396 &\textcolor[rgb]{0.00,0.00,1.00}{\underline{0.7868}} \\
\hline
\hline
\multirow{2}{*}{SPMCS dataset~\cite{xintao/iccv17}} &Avg. PSNRs &25.16 &- &28.60 &28.19 &\textcolor[rgb]{0.00,0.00,1.00}{\underline{29.31}} &27.62 &\textcolor[rgb]{1.00,0.00,0.00}{\bf{29.54}} \\
&Avg. SSIMs &0.6962 &- &0.8253 &0.8164 &\textcolor[rgb]{1.00,0.00,0.00}{\textbf{0.8554}} &0.8048 &\textcolor[rgb]{0.00,0.00,1.00}{\underline{0.8532}} \\
\bottomrule
\end{tabular}
\vspace{-5mm}
\end{table*}
\begin{figure*}[!t]\footnotesize
\begin{center}
\begin{tabular}{cccccccc}
\multicolumn{3}{c}{\multirow{5}*[52pt]{\includegraphics[width=0.368\linewidth, height = 0.263\linewidth]{figures/vid4/c/00000035_mark}}}&\hspace{-4.5mm}
\includegraphics[width=0.15\linewidth]{figures/vid4/c/00000001_gt_x4} &\hspace{-4.5mm}
\includegraphics[width=0.15\linewidth]{figures/vid4/c/00000001_gt_bic_crop_x4} &\hspace{-4.5mm}
\includegraphics[width=0.15\linewidth]{figures/vid4/c/00000001_rcan_crop_x4} &\hspace{-4.5mm}
\includegraphics[width=0.15\linewidth]{figures/vid4/c/00000001_spmc_crop_x4} \\
\multicolumn{3}{c}{~} &\hspace{-4.5mm} (b1) HR patch &\hspace{-4.5mm} (c1) Bicubic &\hspace{-4.5mm} (d1) RCAN~\cite{RCAN} &\hspace{-4.5mm} (e1) SPMC~\cite{xintao/iccv17}\\
\multicolumn{3}{c}{~} & \hspace{-4.5mm}
\includegraphics[width=0.15\linewidth]{figures/vid4/c/00000001_duf_crop_x4} & \hspace{-4.5mm}
\includegraphics[width=0.15\linewidth]{figures/vid4/c/00000001_toflow_crop_x4} & \hspace{-4.5mm}
\includegraphics[width=0.15\linewidth]{figures/vid4/c/00000001_rbpn_crop_x4} & \hspace{-4.5mm}
\includegraphics[width=0.15\linewidth]{figures/vid4/c/00000001_ours_vimeo_crop_x4} \\
\multicolumn{3}{c}{\hspace{-4.5mm} (a1) Ground truth HR image} & \hspace{-4.5mm} (f1) DUF~\cite{jo/cvpr18} &\hspace{-4.5mm} (g1) TOFlow~\cite{tof} &\hspace{-4.5mm} (h1) RBPN~\cite{VDBPN/cvpr19} & \hspace{-4.5mm} (i1) Ours\\
\multicolumn{3}{c}{\multirow{5}*[64pt]{\includegraphics[width=0.368\linewidth, height = 0.31\linewidth]{figures/spmc/a/00000035_mark}}}&\hspace{-4.5mm}
\includegraphics[width=0.15\linewidth]{figures/spmc/a/0013_gt_x4} &\hspace{-4.5mm}
\includegraphics[width=0.15\linewidth]{figures/spmc/a/0013_gt_bic_crop_x4} &\hspace{-4.5mm}
\includegraphics[width=0.15\linewidth]{figures/spmc/a/0013_rcan_crop_x4} &\hspace{-4.5mm}
\includegraphics[width=0.15\linewidth]{figures/spmc/a/013_spmc_crop_x4} \\
\multicolumn{3}{c}{~} &\hspace{-4.5mm} (b1) HR patch &\hspace{-4.5mm} (c2) Bicubic &\hspace{-4.5mm} (d2) RCAN~\cite{RCAN} &\hspace{-4.5mm} (e2) SPMC~\cite{xintao/iccv17}\\
\multicolumn{3}{c}{~} & \hspace{-4.5mm}
\includegraphics[width=0.15\linewidth]{figures/spmc/a/0013_duf_crop_x4} & \hspace{-4.5mm}
\includegraphics[width=0.15\linewidth]{figures/spmc/a/0013_toflow_crop_x4} & \hspace{-4.5mm}
\includegraphics[width=0.15\linewidth]{figures/spmc/a/0013_rbpn_crop_x4} & \hspace{-4.5mm}
\includegraphics[width=0.15\linewidth]{figures/spmc/a/0013_ours_vimeo_crop_x4} \\
\multicolumn{3}{c}{\hspace{-4.5mm} (a2) Ground truth HR image} & \hspace{-4.5mm} (f2) DUF~\cite{jo/cvpr18} &\hspace{-4.5mm} (g2) TOFlow~\cite{tof} &\hspace{-4.5mm} (h2) RBPN~\cite{VDBPN/cvpr19} & \hspace{-4.5mm} (i2) Ours\\
\end{tabular}
\end{center}
\vspace{-2mm}
\caption{Video SR results ($\times 4$) on the Vid4~\cite{Bayesian/vsr/tpami14} and SPMCS~\cite{xintao/iccv17} datasets. The proposed algorithm generates much clearer images.
}%
\label{fig: results-vid4-spmc}
\vspace{-5mm}
\end{figure*}
\section{Experimental Results}
In this section, we compare the proposed algorithm against state-of-the-art methods using publicly available benchmark datasets.
\vspace{-2mm}
{\flushleft \bf{Quantitative evaluations.}}
We compare the proposed algorithm against state-of-the-art methods including the variational methods~\cite{maziyang/vsr/cvpr15}
and deep CNN-based methods including DUF~\cite{jo/cvpr18}, TOFlow~\cite{tof}, RBPN~\cite{VDBPN/cvpr19}, EDVR~\cite{edvr}.
In addition, we compare the proposed method with state-of-the-art deep CNNs-based single image SR~\cite{RCAN} (RCAN).
We use the PSNR and SSIM as the evaluation metrics to evaluate the quality of each restored image on synthetic datasets.
The PSNR and SSIM values of each restored image are calculated using RGB channels based on the script by~\cite{edvr}.
Table~\ref{tab: results-reds} shows the quantitative evaluation results on 4 videos from the REDS test dataset~\cite{REDS}, where these 4 videos are also used in~\cite{edvr} for test.
Overall, the proposed method achieves comparable results compared to the EDVR algorithm and outperforms other algorithms by a large margin.
Figure~\ref{fig: results-reds} shows some results with a scale factor of $4$ by the top-performing methods on the REDS dataset~\cite{REDS}.
We note that the state-of-the-art single image SR method~\cite{RCAN} does not recover the structural details well as shown in Figure~\ref{fig: results-reds}(d).
Tao et al.~\cite{xintao/iccv17} develop an effective warping layer for video SR. However, this structural details in the super-resolved image are not sharp.
Xue et al.~\cite{tof} explicitly use optical flow and warping operations for video restoration.
However, this algorithm assumes the blur kernel is known and takes the Bicubic upsampled images as inputs which are blurry (Figure~\ref{fig: intermediate-result-kernel}(b)) and thus accordingly affect the details restoration (Figure~\ref{fig: results-reds}(g)).
Jo et al.~\cite{jo/cvpr18} develop an effective algorithm to dynamically estimate upsampling filters and residual images for video SR. However, the structural details are not restored well due to the inaccurate upsampling filters (Figure~\ref{fig: results-reds}(h)).
As the proposed algorithm develops a motion blur estimation which provides an intermediate latent HR image with sharp contents, it generates much clearer images with finer details (Figure~\ref{fig: results-reds}(i)).
We then evaluate the proposed algorithm on the test dataset by Liu et al.~\cite{Bayesian/vsr/tpami14} (Vid4) and Tao et al.~\cite{xintao/iccv17} (SPMCS).
Table~\ref{tab: result-datasets-vid4-spmc} shows the quantitative results on the Vid4 and SPMCS datasets.
We note that the variational model-based method~\cite{Bayesian/vsr/tpami14} estimates blur kernels from video sequences to restore images. However, the performance of these methods is limited by the hand-crafted image priors.
The deep CNN-based methods~\cite{xintao/iccv17,jo/cvpr18,tof,VDBPN/cvpr19} generate the results with higher PSNR and SSIM values than the variational model-based methods.
In contrast, our algorithm generates favorable results in terms of PSNR and SSIM due to the use of motion blur estimation.
Figure~\ref{fig: results-vid4-spmc} shows some SR results with a scale factor of 4 by the top-performing methods on the Vid4~\cite{Bayesian/vsr/tpami14} and SPMCS~\cite{xintao/iccv17} datasets.
State-of-the-art methods do not recover the structural details well.
In contrast, the proposed method jointly estimates motion blur, motion fields, and latent images. The motion blur estimation is able to generate intermediate HR image with clear details, which thus lead to much clearer images with finer structural details.
\begin{figure*}[!t]\footnotesize
\begin{center}
\begin{tabular}{cccccccc}
\multicolumn{3}{c}{\multirow{5}*[64pt]{\includegraphics[width=0.368\linewidth, height = 0.31\linewidth]{figures/real/0027_mark}}}&\hspace{-4.5mm}
\includegraphics[width=0.15\linewidth]{figures/real/0027_bic_crop_x4} &\hspace{-4.5mm}
\includegraphics[width=0.15\linewidth]{figures/real/0027_rcan_crop_x4} &\hspace{-4.5mm}
\includegraphics[width=0.15\linewidth]{figures/real/0027_spmc_crop_x4} &\hspace{-4.5mm}
\includegraphics[width=0.15\linewidth]{figures/real/0027_duf_crop_x4} \\
\multicolumn{3}{c}{~} &\hspace{-4.5mm} (b1) Bicubic &\hspace{-4.5mm} (c1) RCAN~\cite{RCAN} &\hspace{-4.5mm} (d1) SPMC~\cite{xintao/iccv17} &\hspace{-4.5mm} (e1) DUF~\cite{jo/cvpr18} \\
\multicolumn{3}{c}{~} & \hspace{-4.5mm}
\includegraphics[width=0.15\linewidth]{figures/real/0027_toflow_crop_x4} & \hspace{-4.5mm}
\includegraphics[width=0.15\linewidth]{figures/real/0027_rbpn_crop_x4} & \hspace{-4.5mm}
\includegraphics[width=0.15\linewidth]{figures/real/0027_edvr_reds_crop_x4} & \hspace{-4.5mm}
\includegraphics[width=0.15\linewidth]{figures/real/0027_ours_crop_x4} \\
\multicolumn{3}{c}{\hspace{-4.5mm} (a1) Input} & \hspace{-4.5mm} (f1) TOF~\cite{tof} &\hspace{-4.5mm} (g1) RBPN~\cite{VDBPN/cvpr19} &\hspace{-4.5mm} (h1) EDVR~\cite{edvr} & \hspace{-4.5mm} (i1) Ours\\
\multicolumn{3}{c}{\multirow{5}*[59pt]{\includegraphics[width=0.368\linewidth, height = 0.288\linewidth]{figures/real2/0004_mark}}}&\hspace{-4.5mm}
\includegraphics[width=0.15\linewidth]{figures/real2/0004_bic_crop_x4} &\hspace{-4.5mm}
\includegraphics[width=0.15\linewidth]{figures/real2/0004_rcan_crop_x4} &\hspace{-4.5mm}
\includegraphics[width=0.15\linewidth]{figures/real2/0004_spmc_crop_x4} &\hspace{-4.5mm}
\includegraphics[width=0.15\linewidth]{figures/real2/0004_duf_crop_x4} \\
\multicolumn{3}{c}{~} &\hspace{-4.5mm} (b2) Bicubic &\hspace{-4.5mm} (c2) RCAN~\cite{RCAN} &\hspace{-4.5mm} (d2) SPMC~\cite{xintao/iccv17} &\hspace{-4.5mm} (e2) DUF~\cite{jo/cvpr18} \\
\multicolumn{3}{c}{~} & \hspace{-4.5mm}
\includegraphics[width=0.15\linewidth]{figures/real2/0004_toflow_crop_x4} & \hspace{-4.5mm}
\includegraphics[width=0.15\linewidth]{figures/real2/0004_rbpn_crop_x4} & \hspace{-4.5mm}
\includegraphics[width=0.15\linewidth]{figures/real2/0004_edvr_reds_crop_x4} & \hspace{-4.5mm}
\includegraphics[width=0.15\linewidth]{figures/real2/0004_ours_crop_x4} \\
\multicolumn{3}{c}{\hspace{-4.5mm} (a2) Input} & \hspace{-4.5mm} (f2) TOFlow~\cite{tof} &\hspace{-4.5mm} (g2) RBPN~\cite{VDBPN/cvpr19} &\hspace{-4.5mm} (h2) EDVR~\cite{edvr} & \hspace{-4.5mm} (i2) Ours\\
\end{tabular}
\end{center}
\vspace{-3mm}
\caption{Video SR results on real videos ($\times 4$). The proposed algorithm generates much clearer images.
}%
\label{fig: results-real-videos}
\vspace{-3mm}
\end{figure*}
\vspace{-2mm}
{\flushleft \bf{Qualitative evaluations.}}
We further qualitatively evaluate the proposed algorithm against state-of-the-art methods on real videos.
Figure~\ref{fig: results-real-videos} shows a real example from~\cite{liao/iccv15}.
The results by state-of-the-art methods~\cite{jo/cvpr18,VDBPN/cvpr19,tof,edvr} are still blurry.
In contrast, our algorithm generates the images with clearer detailed structures, which demonstrate that the proposed algorithm generalizes well.
\vspace{-1mm}
\section{Analysis and Discussions}
\label{sec: Analysis and Discussions}
We have shown that using the motion blur estimation is able to help details restoration in video SR.
In this section, we further analyze the effect of the proposed algorithm.
\begin{table}[!t]
\caption{Effectiveness of the motion blur estimation on video SR ($\times 4$). The results are obtained from the REDS dataset~\cite{REDS}.
}
\label{tab: baselines-kernel}
\footnotesize
\centering
\begin{tabular}{lcccccccccc}
\toprule
Methods &BaselineLR &BaselineHR &Ours \\
\hline
Avg. PSNRs &30.18 &30.48 &\textcolor[rgb]{1.00,0.00,0.00}{\textbf{30.51}} \\
Avg. SSIMs &0.8609 &0.8669 &\textcolor[rgb]{1.00,0.00,0.00}{\textbf{0.8674}} \\
\bottomrule
\end{tabular}
\vspace{-4.0mm}
\end{table}
\begin{figure}[!t]\footnotesize
\centering
\begin{tabular}{ccc}
\includegraphics[width=0.32\linewidth]{figures/ana/00000004_x4} &\hspace{-4mm}
\includegraphics[width=0.32\linewidth]{figures/ana/00000004_bic_crop_x4} &\hspace{-4mm}
\includegraphics[width=0.32\linewidth]{figures/ana/00000004_duf_crop_x4} \\
(a) HR patch &\hspace{-4mm} (b) Bicubic & \hspace{-4mm} (c) DUF~\cite{jo/cvpr18} \\
\includegraphics[width=0.32\linewidth]{figures/ana/00000004_pwc_rcan_crop_x4} &\hspace{-4mm}
\includegraphics[width=0.32\linewidth]{figures/ana/00000004_ours_woDeconv_crop_x4} &\hspace{-4mm}
\includegraphics[width=0.32\linewidth]{figures/ana/00000004_ours_crop_x4} \\
(d) BaselineLR &\hspace{-4mm} (e) BaselineHR &\hspace{-4mm} (f) Ours \\
\end{tabular}
\caption{Effectiveness of the motion blur estimation on video SR ($\times$4). Using motion blur estimation is able to generate the results with clearer structural details.}%
\label{fig: baseline-analysis}
\vspace{-4mm}
\end{figure}
\vspace{-3mm}
{\flushleft \bf{Effectiveness of the motion blur estimation.}}
The proposed motion blur estimation process provides blur kernels which thus leads to the intermediate latent images with clear contents for better details restoration.
To demonstrate the effectiveness of this method, we disable this step in the proposed algorithm for fair comparisons.
Thus, the proposed method reduces to the method without using the motion blur estimation and intermediate latent image restoration.
For this case, the inputs of this baseline method can be either the bicubic upsampling results (i.e., $I_i^{b}$, $I_{i+1}^{b}$, and $I_{i-1}^b$) (BaselineHR for short) or the original LR images (BaselineLR for short).
Table~\ref{tab: baselines-kernel} shows the quantitative evaluations on the REDS dataset~\cite{REDS}.
The average PSNR by our method is 0.33dB higher than that by BaselineLR, which demonstrates that using motion blur estimation is able to generate much better results.
The visualizations in Figure~\ref{fig: baseline-analysis} further demonstrate that directly estimating the HR images using deep CNN models without motion blur estimation does not generates the results with clearer structural details, while the proposed method generates much clearer images.
\vspace{-2mm}
\begin{table}[!t]
\caption{Accuracy of the estimated motion blur kernels.
}
\label{tab: kernel-accuracy}
\footnotesize
\centering
\begin{tabular}{lcccccccccc}
\toprule
Methods &Ma et al.~\cite{maziyang/vsr/cvpr15} &Gaussian kernel &Ours\\%&Ours
\hline
Avg. PSNRs &31.24 &25.48 &\textcolor[rgb]{1.00,0.00,0.00}{\textbf{42.06}} \\
Avg. SSIMs &0.9272 &0.8726 &\textcolor[rgb]{1.00,0.00,0.00}{\textbf{0.9962}} \\
\bottomrule
\end{tabular}
\vspace{-5mm}
\end{table}
\vspace{-2mm}
{\flushleft \bf{Analysis on motion blur kernels.}}
Different from existing video SR methods that use known blur kernels (e.g., Gaussian blur kernel~\cite{gaussian/kernel/cvpr18}) and do not model them in the video SR process,
we develop a motion blur estimation method to estimate blur kernels for video SR.
To examine accuracy of the estimated blur kernels, we apply the estimated blur kernels and downsampling operation to the ground truth HR images to generate LR images.
We apply the Bicubic downsampling to the HR images to obtain the LR ones as the ground truth LR images.
The quality of the regenerated LR images is used to measure the accuracy of the estimated blur kernels.
Table~\ref{tab: kernel-accuracy} shows that regenerated LR images by the proposed method are closed to the ground truth LR images, which indicates the proposed algorithm is able to estimate more accurate blur kernels than those conventional variational model-based method~\cite{maziyang/vsr/cvpr15}.
\begin{figure}[!t]\footnotesize
\centering
\begin{tabular}{ccc}
\hspace{-2mm}
\includegraphics[width=0.32\linewidth]{figures/kernel/kernel_gaussian_re}
\includegraphics[width=0.32\linewidth]{figures/kernel/kernel_without_finetune_re} \\
(a) Initial kernel & (b) Estimated kernel\\
\end{tabular}
\caption{Visualizations of the estimated motion blur kernels generated by the network $\mathcal{N}_k$. We use the Bicubic downampling as the degradation process of video SR for test.}
\label{fig: intermediate-kernels}
\vspace{-4mm}
\end{figure}
Figure~\ref{fig: intermediate-kernels} shows the visualizations of estimated blur kernels by $\mathcal{N}_k$ when the degradation process is the Bicubic downampling.
We note that shape of the estimated blur kernels are similar to that of Bicubic kernel as demonstrated in~\cite{levin/etal/blindsr}. Thus, both the quantitative and qualitative results demonstrate that the proposed algorithm is able to capture the degradation process well.
\vspace{-3mm}
{\flushleft \bf{Video SR with motion blur.}}
The proposed algorithm is able to super-resolve videos containing small motion blur to some extent.
Figure~\ref{fig: results-real-videos}(a) shows a LR frame from a LR video in~\cite{maziyang/vsr/cvpr15}, where the LR video contains small motion blur.
As most video SR methods~\cite{xintao/iccv17,jo/cvpr18,tof,VDBPN/cvpr19,edvr} do not model the motion blur, the generated results are blurry.
Although the method~\cite{maziyang/vsr/cvpr15} is able to solve video SR with motion blur, it is limited to the hand-crafted priors.
In contrast, the proposed method generates much clearer images.
\section{Concluding Remarks}
%
We have proposed an effective video super-resolution algorithm. The proposed algorithm consists of motion blur estimation, motion field estimation, and latent image restoration modules.
%
The motion blur estimation module is able to provide reliable blur kernels.
%
With the estimated blur kernel, we develop an image deconvolution method based on the image formation model of video super-resolution to generate intermediate latent images
so that some sharp image contents can be restored well.
%
To generate high-quality images, we use the motion estimation module to explore the information from adjacent frames to constrain the deep CNN model for better image restoration.
We have shown that the proposed method is able to generate much clearer images with finer structural details due to the use of motion blur kernel estimation. Both quantitative evaluations and qualitative evaluations show that the proposed algorithm performs favorably against state-of-the-art methods.
\clearpage
{\small
\bibliographystyle{ieee_fullname}
| {
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A bill to legalize the use of medical marijuana for qualifying patients and to create a statewide system of "Compassion Centers" has been introduced in the Keystone State.
Senator Daylin Leach brought SB 1003 forward on April 25th with Senators Larry Farnese, James Ferlo and Wayne Fontana as the initial co-sponsors. The legislation has been referred to the Senate Public Health and Welfare Committee.
"I'm happy to have re-introduced Senate Bill 1003 and am hopeful it will be brought up for consideration by the Legislature. It is a common-sense bill that would simply give sick people access to medication so they feel better," said the bill's prime sponsor, Senator Daylin Leach (D). "Countless studies show marijuana can alleviate the side effects of many diseases. It's time we give Pennsylvanians access to the treatment they need and deserve," Leach went on to say.
Advocates at Pennsylvanians for Medical Marijuana PA4MMJ are pushing for several changes to the bill when it gets to committee this session. These include re-naming the bill to The Governor Raymond Shafer Compassionate Use Medical Marijuana Act.
To speak with advocates, medical experts or cannabis patients in Pennsylvania please contact Chris Goldstein, media coordinator at PA4MMJ. [email protected]. | {
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Federation Style Homes Floor Plans Elegant Federation Style 4 Bedroom House Plan 180 Clm Australian Dream Home – Make the most of your space with these decorating ideas for Federation Style Homes Floor Plans Elegant Federation Style 4 Bedroom House Plan 180 Clm Australian Dream Home from leading designers. Copy these fashionable Federation Style Homes Floor Plans Elegant Federation Style 4 Bedroom House Plan 180 Clm Australian Dream Home wall decor ideas to deliver your empty walls . To get you motivated, here's our pick of ideas out of homes we've search. Just because you live in a Federation Style Homes Floor Plans Elegant Federation Style 4 Bedroom House Plan 180 Clm Australian Dream Home apartment doesn't indicate you need to settle for boring, pint-sized furniture. And if you've discovered the perfect Federation Style Homes Floor Plans Elegant Federation Style 4 Bedroom House Plan 180 Clm Australian Dream Home of your fantasies, minute spaces could fold down and up and a number of other space saving furniture examples and ideas. Federation Style Homes Floor Plans Elegant Federation Style 4 Bedroom House Plan 180 Clm Australian Dream Home may have a large impact with the right design. Decorating a Federation Style Homes Floor Plans Elegant Federation Style 4 Bedroom House Plan 180 Clm Australian Dream Home can get additional catchy with strange nooks and corners. Utilize light, crisp colours to combat unique attributes like reduced ceilings. Bright whites seem luxe in almost any sized room, and even turn this slanted-ceiling children' room to a spacious play zone. Place the'large' to a Federation Style Homes Floor Plans Elegant Federation Style 4 Bedroom House Plan 180 Clm Australian Dream Home with bold pieces and strong patterns. When decorating with artwork that wows, neutral bedding is essential. Your Federation Style Homes Floor Plans Elegant Federation Style 4 Bedroom House Plan 180 Clm Australian Dream Home needs to make a statement without being overcrowded. A little background, paint, or a few colorful accents can make a huge impact in your home. You can tackle every one of these ideas in one day, even though the outcomes will seem like it took way more time to pull away. Shade is a powerful design tool in decoration, as well as in interior design which is the art of composing, and coordinating colors together to create a stylish scheme on the inside design of the space. It's essential to interior designers to acquire a profound encounter with colours, understand their emotional effects, and comprehend the meaning of each color in various locations and situations in order to produce appropriate combinations for each place.
Mixing colours together could result in making a state of mind as seen by the audience, and could eventually lead to negative or positive effects on them. Color mix make a tiny room look larger or smaller.So it is the Interior designer profession to select appropriate colors to get a place in a way people want to look and feel in the area.
Among the most important things when planning home plan would be the traffic patterns.You need to thinking more detail concerning the way to creating a strategy for each days activity from begin to finish. That is all about finding which way is the fastest route, what places to hit first to avoid traffic and delays, so we can achived efficiency. While that could be a stretch from home design and planning, the idea of traffic pattern planning is your underlining attention that coincides with creating the efficient flow or traffic movement in your home. House plan traffic patterns should be carefully considered in the design of each room and the floor plan design with relation to the adjoining room. One of the best method to efficiently ascertain house plan traffic patterns is, you need to imagine yourself actually moving through the home, multiple posibbility can be created via the visualitation. Do not forget to devote a minimum amount of space to traffic areas. Planning is take the important function, so precious square footage could be lost if you do not plan accordingly. The most efficient design entails producing the illusion of hallway traffic passages or patterns with furniture placement. This is often seen in spacious floor plans. They use a minimum amount of interior walls to different chambers. It involves a bit of forethought by the designer and homeowner, but pro active preparation eliminates traffic pattern issues and furniture structure for the homeowner. You should keep the size of hallways and corridors should to a minimal. Be certain you also keep hallways short in distance. | {
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Лари́са Дми́трівна То́карєва (, Красноярськ, РРФСР, СРСР) — радянський і український художник кіно.
Життєпис
Народилась 1947 р. в Красноярську в родині службовця.
Закінчила Красноярське художнє училище ім. В.Сурикова (1967), художній факультет Всесоюзного державного інституту кінематографії (1974).
З 1974 р. — художник-постановник Одеської кіностудії художніх фільмів. Оформила понад двадцять кінокартин.
Член Національної спілки кінематографістів України.
Фільмографія
Художник-постановник:
«Хлопчаки їхали на фронт» (1975, у співавт. з Євгенією Ліодт)
«Туфлі з золотими пряжками» (1976, т/ф, 2 с)
«Свідоцтво про бідність» (1977)
«Д'Артаньян та три мушкетери» (1978, т/ф, 3 а)
«Петля Оріона» (1980, у співавт. з Олександром Токарєвим)
«Куди він дінеться!» (1981)
«Весільний подарунок» (1982)
«На мить озирнутися...» (1984)
«Точка повернення» (1987)
«В Криму не завжди літо» (1987)
«Гу-га» (1989)
«Каталажка» (1990)
«Невстановлена особа» (1990)
«Мушкетери двадцять років по тому» (1992)
«Дафніс і Хлоя» (1993)
«Зроби мені боляче» (1993)
«Таємниця королеви Анни, або Мушкетери тридцять років по тому» (1993)
«Одружити Казанову» (2009) та ін.
Література
Спілка кінематографістів України. К., 1985. — С. 156.
Посилання
Токарєва Лариса Дмитрівна (фільмографія) (kino-teatr.ru)
Уродженці Красноярська
Художники кіно СРСР
Українські художники кіно
Члени Національної спілки кінематографістів України
Випускники ВДІК
Художники кіно XX століття
Художники кіно XXI століття
Працівники Одеської кіностудії | {
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Moussa Guindo (born 25 January 1991) is a retired Ivorian-Malian footballer who played as a defender.
Club career
Born in Adjamé, Ivory Coast, Guindo began his career by Académie de Sol Beni, Guindo was promoted 2008 and was one of ASEC Mimosas's youngest players but a regular in defence as he enjoyed a first choice berth under coach Patrick Liewig. In January 2009 he joined Charlton Athletic. and returned to ASEC Mimosas on loan in July 2009. He moved to the land of his forefathers, Mali, and signed with Stade Malien.
In 2012, he joined the Tunisian League club CA Bizertin.
International career
Guindo represented the Under-20 national team from Mali at 2009 African Youth Championship in Rwanda, and was member of the U-17 at CAN 2006, lastly the 2011 FIFA U-20 World Cup.
References
1991 births
Living people
Footballers from Abidjan
Ivorian people of Malian descent
Malian footballers
Association football defenders
Association football midfielders
ASEC Mimosas players
Stade Malien players
CA Bizertin players
Mali under-20 international footballers
Malian expatriate footballers
Expatriate footballers in England
Malian expatriate sportspeople in England
Expatriate footballers in Tunisia
Malian expatriate sportspeople in Tunisia
Tunisian Ligue Professionnelle 1 players | {
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{"url":"https:\/\/web2.0calc.com\/questions\/challenge-problem","text":"+0\n\nChallenge Problem\n\n0\n425\n3\n+445\n\nIn how many ways can you spell the word COOL in the grid below? You can start on any letter, then on each step, you can step one letter in any direction (up, down, left, right, or diagonal).\n\n$$\\begin{array}{ccccc} C&C&C&C&C\\\\ L&O&O&O&L\\\\ L&O&O&O&L\\\\ L&O&O&O&L\\\\ C&C&C&C&C\\\\ \\end{array}$$\n\nOct 23, 2017\n\n#1\n+7612\n0\n\nI count \u00a04\n\nOct 24, 2017\nedited by hectictar \u00a0Oct 24, 2017\n#2\n+445\n0\n\n4 was incorrect... Do you have any other answers?\n\nI counted 64 ways, that was wrong too.\n\nMr.Owl \u00a0Oct 24, 2017\n#3\n+638\n0\n\nhttp:\/\/web2.0calc.com\/questions\/in-how-many-ways-can-you-spell-the-word-cool-in-the-grid-below-you-can-start-on-any-letter-then-on-each-step-you-can-step-one-letter-in-an\n\nOct 24, 2017","date":"2019-05-26 12:33:38","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 1, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.6231075525283813, \"perplexity\": 4405.506886875632}, \"config\": {\"markdown_headings\": false, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2019-22\/segments\/1558232259126.83\/warc\/CC-MAIN-20190526105248-20190526131248-00380.warc.gz\"}"} | null | null |
<?php
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$query1 = "INSERT INTO user_msg (name,email,affiliation,subject,msg) VALUES ";
$query2 = "(".$name.",".$email.",".$aff.",".$sbj.",".$msg.");";
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$mail->isSMTP(); // set PHPMailer to use SMTP
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$mail->Host = 'smtp.gmail.com'; // GMail's SMTP hostname
$mail->SMTPAuth = true;
$mail->Username = $ijahMail;
$mail->Password = $ijahMailPass;
$mail->SMTPSecure = 'ssl'; // setting TLS encryption (GMail needs this)
$mail->Port = 465; // TCP port to connect, 465 for SSL, 587 for TLS
$mail->From = 'ijahweb@gmail.com';
$mail->FromName = 'Ijah Webserver';
$mail->addAddress('vektor.dewanto@gmail.com');
$mail->addCC('hzbarkan@gmail.com');
$mail->addCC('w.ananta.kusuma@gmail.com');
$mail->isHTML(false);
$mail->Subject = "ijahws: user-message: {$email}";
$mail->Body = "Name: {$name} \nE-mail: {$email} \nAffiliation: {$aff} \nFeedback Type: {$sbj} \n\nMessage: \n {$msg}";
if(!$mail->send()) {
echo 'Message could not be sent.';
echo 'Mailer Error: ' . $mail->ErrorInfo;
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echo 'Message has been sent';
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## PRAISE FOR THE WRITING OF THEODORE STURGEON
"One of the greatest... I can't recommend his work too highly!"—Stephen King
"I look upon Sturgeon with a secret and growing jealousy."—Ray Bradbury
"A master storyteller certain to fascinate."—Kurt Vonnegut Jr.
"One of the masters of modern science fiction."— _The Washington Post Book World_
"The Sturgeon magic does not diminish with the years. His stories have a timeless quality and a universality which is beyond fantasy and science fiction."—Madeleine L'Engle
"The corpus of science fiction produced by Theodore Sturgeon is the single most important body of science fiction by an American."—Samuel R. Delany
### _More Than Human_
"A quantum leap in the development of science fiction as an art."— _The Washington Post_
"One of the best science fiction novels of the year."— _The New York Times_
### _Godbody_
"Embodies the very best of Theodore Sturgeon... a master."— _San Francisco Chronicle_
"The capstone of Sturgeon's art... Read it, enjoy it, reread it, give it to somebody you love."—Robert A. Heinlein
"You will do more than enjoy; you will be increased."—Stephen King
### _To Marry Medusa_
"Dazzling... Sturgeon swerves around cliché and dull language like a maniac. At times, it seems like he's working in his own personal version of the English language. It's like taking a road trip with an incredibly eccentric dude: You may know the most logical or efficient route, but the offbeat guy will know the way past the most stunning vistas. Read a little of _Medusa_ , and you'll see what I mean."—SF Site
"A fine example of what science fiction is supposed to be: simultaneously plot- and character-driven and completely devoid of fluff... . A fantastic classic."—SF Signal
### _The Dreaming Jewels_
"An intensely written and very moving novel of love and retribution." _—Washington Star_
### _Venus Plus X_
"It's interesting to read _Venus_ ' sexual commentary in the wake of a second wave of feminism, the gay liberation, and the sexual revolution of the '60s. Obviously, in 1960 the novel was way ahead of its time. It has lost some of that power, but its critique of American prudence still holds."— _City Paper_ (Baltimore)
# The Dreaming Jewels
### Theodore Sturgeon
###
# Contents
A Biography of Theodore Sturgeon
# 1
THEY CAUGHT THE KID DOING SOMETHING disgusting out under the bleachers at the high-school stadium, and he was sent home from the grammar school across the street. He was eight years old then. He'd been doing it for years.
In a way it was a pity. He was a nice kid, a nice-looking kid too, though not particularly outstanding. There were other kids, and teachers, who liked him a little bit, and some who disliked him a little bit; but everyone jumped on him when it got around. His name was Horty—Horton, that is—Bluett. Naturally he caught blazes when he got home.
He opened the door as quietly as he could, but they heard him, and hauled him front and center into the living room where he stood flushing, with his head down, one sock around his ankle, and his arms full of books and a catcher's mitt. He was a good catcher, for an eight-year-old. He said, "I was—"
"We know," said Armand Bluett. Armand was a bony individual with a small mustache and cold wet eyes. He clapped his hands to his forehead and then threw up his arms. "My God, boy, what in Heaven's name made you do a filthy thing like that?" Armand Bluett was not a religious man, but he always talked like that when he clapped his hands to his head, which he did quite often.
Horty did not answer. Mrs. Bluett, whose name was Tonta, sighed and asked for a highball. She did not smoke, and needed a substitute for the smoker's thoughtful match-lit pause when she was at a loss for words. She was so seldom at a loss for words that a fifth of rye lasted her six weeks. She and Armand were not Horton's parents. Horton's parents were upstairs, but the Bluetts did not know it. Horton was allowed to call Armand and Tonta by their first names.
"Might I ask," said Armand icily, "how long you have had this nauseating habit? Or was it an experiment?"
Horty knew they weren't going to make it easy on him. There was the same puckered expression on Armand's face as when he tasted wine and found it unexpectedly good.
"I don't do it much," Horty said, and waited.
"May the Lord have mercy on us for our generosity in taking in this little swine," said Armand, clapping his hands to his head again. Horty let his breath out. Now that was over with. Armand said it every time he was angry. He marched out to mix Tonta a highball.
"Why did you do it, Horty?" Tonta's voice was more gentle only because her vocal cords were more gently shaped than her husband's. Her face showed the same implacable cold.
"Well, I—just felt like it, I guess." Horty put his books and catcher's mitt down on the footstool.
Tonta turned her face away from him and made an unspellable, retching syllable. Armand strode back in, bearing a tinkling glass.
"Never heard anything like it in my life," he said scornfully. "I suppose it's all over the school?" I guess so.
"The children? The teachers too, no doubt. But of course. Anyone say anything to you?"
"Just Dr. Pell." He was the principal. "He said—said they could..."
"Speak up!"
Horty had been through it once. Why, why go through it all again? "He said the school could get along without f-filthy savages."
"I can understand how he felt," Tonta put in, smugly.
"And what about the other kids? They say anything?"
"Hecky brought me some worms. And Jimmy called me Sticky-tongue." And Kay Hallowell had laughed, but he didn't mention that.
"Sticky-tongue. Not bad, that, for a kid. Ant-eater." Again the hand clapped against the brow. "My God, what am I going to do if Mr. Anderson greets me with 'Hi Sticky-tongue!' Monday morning? This will be all over town, sure as God made little apples." He fixed Horty with the sharp wet points of his gaze. "And do you plan to take up bug-eating as a profession?"
"They weren't bugs," Horty said diffidently and with accuracy. "They were ants. The little brown kind."
Tonta choked on her highball. "Spare us the details."
"My God," Armand said again, "what'll he grow up as?" He mentioned two possibilities. Horty understood one of them. The other made even the knowledgeable Tonta jump. "Get out of here."
Horty went to the stairs while Armand thumped down exasperatedly beside Tonta. "I've had mine," he said. "I'm full up to here. That brat's been the symbol of failure to me ever since I laid eyes on his dirty face. This place isn't big enough— _Horton!"_
"Huh."
"Come back here and take your garbage with you. I don't want to be reminded that you're in the house."
Horty came back slowly, staying out of Armand Bluett's reach, picked up his books and the catcher's mitt, dropped a pencil-box—at which Armand my-Godded again—picked it up, almost dropped the mitt, and finally fled up the stairs.
"The sins of the stepfathers," said Armand, "are visited on the stepfathers, even unto the thirty-fourth irritation. What have I done to deserve this?"
Tonta swirled her drink, keeping her eyes on it and her lips pursed appreciatively as she did so. There had been a time when she disagreed with Armand. Later, there was a time when she disagreed and said nothing. All that had been too wearing. Now she kept an appreciative exterior and let it soak in as deeply as it would. Life was so much less trouble that way.
Once in his room, Horty sank down on the edge of the bed with his arms still full of his books. He did not close the door because there was none, due to Armand's conviction that privacy was harmful for youngsters. He did not turn on the light because he knew everything in the room, knew it with his eyes closed. There was little enough. Bed, dresser, closet with a cracked cheval glass. A child's desk, practically a toy, that he had long outgrown. In the closet were three oiled-silk dress-covers stuffed full of Tonta's unused clothes, which left almost no space for his.
His...
None of this was really his. If there had been a smaller room, he would have been shoved into it. There were two guest bedrooms on this floor, and another above, and they almost never had guests. The clothes he wore weren't his; they were concessions to something Armand called "my position in this town"; rags would have done if it weren't for that.
He rose, the act making him conscious of the clutter he still clutched in his arms. He put it down on the bed. The mitt was his, though. He'd bought it for seventy-five cents from the Salvation Army store. He got the money by hanging around Dempledorff's market and carrying packages for people, a dime a trip. He had thought Armand would be pleased; he was always talking about resourcefulness and earning ability. But he had forbidden Horty ever to do that again. "My God! People will think we are paupers!" So the mitt was all he had to show for the episode.
All he had in the world—except, of course, Junky.
He looked, through the half-open closet door, at the top shelf and its clutter of Christmas-tree lights (the Christmas tree was outside the house, where the neighbors could see—never inside), old ribbons, a lampshade, and—Junky.
He pulled the oversized chair away from the undersized desk and carried it—if he had dragged it, Armand would have been up the stairs two at a time to see what he was up to, and if it was fun, would have forbidden it—and set it down carefully in the closet doorway. Standing on it, he felt behind the leftovers on the shelf until he found the hard square bulk of Junky. He drew it out, a cube of wood, gaudily painted and badly chipped, and carried it to the desk.
Junky was the kind of toy so well-known, so well-worn, that it was not necessary to see it frequently, or touch it often, to know that it was there. Horty was a foundling—found in a park one late fall evening, with only a receiving blanket tucked about him. He had acquired Junky while he was at the Home, and when he had been chosen by Armand as an adoptee (during Armand's campaign for City Counsellor, which he lost, but which he thought would be helped along if it were known he had adopted a "poor little homeless waif") Junky was part of the bargain.
Horty put Junky softly on the desk and touched a worn stud at the side. Violently at first, then with rusted-spring hesitancy, and at last defiantly, Junky emerged, a jack-in-the-box, a refugee from a more gentle generation. He was a Punch, with a chipped hooked nose which all but met his upturned, pointed chin. In the gulch between these stretched a knowing smile.
But all Junky's personality—and all his value to Horty—was in his eyes. They seemed to have been cut, or molded, blunt-faceted, from some leaded glass which gave them a strange, complex glitter, even in the dimmest room. Time and again Horty had been certain that those eyes had a radiance of their own, though he could never quite be sure.
He murmured, "Hi, Junky."
The jack-in-the-box nodded with dignity, and Horty reached and caught its smooth chin. "Junky, let's get away from here. Nobody wants us. Maybe we wouldn't get anything to eat, and maybe we'd be cold, but gee... Think of it, Junky. Not being scared when we hear _his_ key in the lock, and never sitting at dinner while he asks questions until we have to lie, and—and all like that." He did not have to explain himself to Junky.
He let the chin go, and the grinning head bobbed up and down, and then nodded slowly, thoughtfully.
"They shouldn't 'a been like that about the ants," Horty confided. "I didn't _drag_ nobuddy to see. Went off by myself. But that stinky Hecky, he's been watching me. An' then he sneaked off and got Mr. Carter. That was no way to do, now was it, Junky?" He tapped the head on the side of its hooked nose, and it shook its head agreeably. "I hate a sneak."
"You mean me, no doubt," said Armand Bluett from the doorway.
Horty didn't move, and for a long instant his heart didn't either. He half crouched, half cowered behind the desk, not turning toward the doorway.
"What are you doing?"
"Nothin'."
Armand belted him across the cheek and ear. Horty whimpered, once, and bit his lip. Armand said, "Don't lie. You are obviously doing something. You were talking to yourself, a sure sign of a degenerating mind. What's this—oh. Oh yes, the baby toy that came with you. Your estate. It's as repulsive as you are." He took it from the desk, dropped it on the floor, wiped his hand on the side of his trousers, and carefully stepped on Junky's head.
Horty shrieked as if it were his own head which was being crushed, and leapt at Armand. So unexpected was the attack that the man was bowled right off his feet. He fell heavily and painfully against the bedpost, grabbed at it and missed, and went to the floor. He sat there for a moment grunting and blinking, and then his little eyes narrowed and fixed themselves on the trembling Horty. "Mmm _—hm!"_ said Armand in a tone of great satisfaction, and rose. "You should be exterminated." He grasped the slack of Horty's shirt and struck him. As he spoke, he hit the boy's face, back and forth, back and forth, by way of punctuation. "Homicidal, that's what you are. I was going to. Send you away. To a school. But it isn't safe. The police will. Take care of you. They have a place. For juvenile delinquents. Filthy little. Pervert."
He rushed the sodden child across the room and jammed him into the closet. "This will keep you safe until the police get here," he panted, and slammed the door. The hinge side of it caught three fingers of Horty's left hand.
At the boy's shriek of very real agony Armand snapped the door open again. "No use in your yelling. You—My God! What a mess. Now I suppose I'll have to get a doctor. There's no end—absolutely no end to the trouble you cause. Tonta!" He ran out and down the stairs. "Tonta!"
"Yes, Peaches."
"That young devil stuck his hand in the door. Did it on purpose, to excite sympathy. Bleeding like a stuck pig. You know what he did? He struck me. He attacked me, Tonta! It's not safe to have him in the house!"
"You poor darling! Did he hurt you?"
"A wonder he didn't kill me. I'm going to call the police."
"I'd better go up while you're phoning," said Tonta. She wet her lips.
But when she reached the room, Horty was gone. There was a lot of excitement for a while after that. At first Armand wanted to get his hands on Horty for his own purposes, and then he began to be afraid of what people might say if the boy gave his own garbled version of the incident. Then a day went by, and a week, and a month, and it was safe to look to heaven and say mysteriously, "He's in safe hands now, the poor little tyke," and people could answer, "I understand..." Everyone knew he was not Armand's child, anyway.
But Armand Bluett tucked one idea snugly away in the corner of his mind. That was to look out, in the future, for any young man with three fingers missing from his left hand.
# 2
THE HALLOWELLS LIVED AT THE EDGE of town, in a house that had only one thing wrong with it; it was at the intersection where the State Highway angled into the end of Main Street, so that the traffic roared night and day past both the front and back gates.
The Hallowell's taffy-headed daughter, Kay, was as full of social consciousness as only a seven-year-old can be. She had been asked to empty the trash, and as usual she opened the back gate a crack and peeped out at the highway, to see if anyone she knew would catch her at the menial task.
_"Horty!"_
He shrank into the fog-swirled shadows of the traffic-light standard.
"Horton Bluett, I see you."
"Kay..." He came to her, staying close to the fence. "Listen, don't tell nobody you saw me, huh?"
"But wh—oh. You're running away!" she blurted, noticing the parcel tucked under his arm. "Horty—are you sick?" He was white, strained. "Did you hurt your hand?"
"Some." He held his left wrist with his right hand, tightly. His left hand was wrapped in two or three handkerchiefs. "They was going to get the police. I got out the window onto the shed roof and hid there all afternoon. They was lookin' all over the street and everywhere. You won't tell?"
"I won't tell. What's in the package?"
"Nothin'."
If she had demanded it, grabbed at it, he would probably never have seen her again. Instead she said, "Please, Horty."
"You can look." Without releasing his wrist, he turned so she could pull the package out from under his arm. She opened it—it was a paper bag—and took out the hideous broken face of Junky. Junky's eyes glittered at her, and she squeaked. "What is it?"
"It's Junky. I had him since before I was born. Armand, he stepped on it."
"Is that why you're running away?"
_"Kay! What are you doing out there?"_
"Coming, Mother! Horty, I got to go. Horty, are you coming back?"
"Not _ever."_
"Gee"... that mister Bluett, he's so _mean..._ "
_"Kay Hallowell! Come in this instant. It's raining!"_
"Yes, Mother! Horty, I wannit to tell you. I shouldn'ta laughed at you today. Hecky brought you the worms, and I thought it was a joke, thass all. I didn't know you really did eat ants. Gee... I et some shoe-polish once. That's nothin'."
Horty held out his elbow and she carefully put the package under it. He said, as if he had just thought of it—and indeed he had—"I _will_ come back, Kay. Someday."
_"Kay!"_
"'Bye, Horty." And she was gone, a flash of taffy hair, yellow dress, a bit of lace, changed before his eyes to a closed gate in a board fence and the sound of dwindling quick footsteps.
Horton Bluett stood in the dark drizzle, cold, but with heat in his ruined hand and another heat in his throat. This he swallowed, with difficulty, and, looking up, saw the broad inviting tailgate of a truck which was stopped for the traffic light. He ran to it, tossed his small bundle on it, and squirmed up, clawing with his right hand, trying to keep his left out of trouble. The truck lurched forward; Horty scrabbled wildly to stay on. The package with Junky in it began to slide back toward him, past him; he caught at it, losing his own grip, and began to slip.
Suddenly there was a blur of movement from inside the truck, and a flare of terrible pain as his smashed hand was caught in a powerful grip. He came very close to fainting; when he could see again he was lying on his back on the jolting floor of the truck, holding his wrist again, expressing his anguish in squeezed-out tears and little, difficult grunts.
"Gee, kid, you don't care how long you live, do you?" It was a fat boy, apparently his own age, bending over him, his bowed head resting on three chins. "What's the matter with your hand?"
Horty said nothing. He was quite beyond speech for the moment. The fat boy, with surprising gentleness, pressed Horty's good hand away from the handkerchiefs and began laying back the cloth. When he got to the inner layer, he saw the blood by the wash of light from a street-light they passed, and he said "Man."
When they stopped for another traffic signal at a lighted intersection, he looked carefully and said, "Oh, man," with all the emphasis inside him somewhere, and his eyes contracted into two pitying little knots of wrinkles. Horty knew the fat boy was sorry for him, and only then did he begin to cry openly. He wished he could stop, but he couldn't, and didn't while the boy bound up his hand again and for quite a while afterward.
The fat boy sat back on a roll of new canvas to wait for Horty to calm down. Once Horty subsided a little and the boy winked at him, and Horty, profoundly susceptible to the least kindness, began to wail again. The boy picked up the paper bag, looked into it, grunted, closed it carefully and put it out of the way on the canvas. Then to Horty's astonishment, he removed from his inside coat pocket a large silver cigar case, the kind with five metal cylinders built together, took out a cigar, put it all in his mouth and turned it to wet it down, and lit up, surrounding himself with sweet-acrid blue smoke. He did not try to talk, and after a while Horty must have dozed off, because he opened his eyes to find the fat boy's jacket folded as a pillow under his head, and he could not remember its being put there. It was dark then; he sat up, and immediately the fat boy's voice came from the blackness.
"Take it easy, kid." A small pudgy hand steadied Horty's back. "How do you feel?"
Horty tried to talk, choked, swallowed and tried again. "All right, I guess. Hungry... gee! We're out in the country!"
He became conscious of the fat boy squatting beside him. The hand left his back; in a moment the flame of a match startled him, and for an etched moment the boy's face floated before him in the wavering light, moonlike, with delicate pink lips acrawl on the black cigar. Then with a practiced flick of his fingers, he sent the match and its brilliance flying out into the night. "Smoke?"
"I never did smoke," said Horty. "Some corn-silk, once." He looked admiringly at the red jewel at the end of the cigar. "You smoke a lot, huh."
"Stunts m'growth," said the other, and burst into a peal of shrill laughter. "How's the hand?"
"It hurts some. Not so bad."
"You got a lot of grit, kid. I'd be screamin' for morphine if I was you. What happened to it?"
Horty told him. The story came out in snatches, out of sequence, but the fat boy got it all. He questioned briefly, and to the point, and did not comment at all. The conversation died after he had asked as many questions as he apparently wanted to, and for a while Horty thought the other had dozed off. The cigar dimmed and dimmed, occasionally sputtering around the edges, once in a while brightening in a wavery fashion as vagrant air touched it from the back of the truck.
Abruptly, and in a perfectly wide-awake voice, the fat boy asked him, "You lookin' fer work?"
"Work? Well—I guess maybe."
"What made you eat them ants?" came next.
"Well, I—I don't know. I guess I just—well, I wanted to."
"Do you do that a lot?"
"Not too much." This was a different kind of questioning than he had had from Armand. The boy asked him about it without revulsion, without any more curiosity, really, than he had asked him how old he was, what grade he was in.
"Can you sing?"
"Well—I guess so. Some."
"Sing something. I mean, if you feel like it. Don't strain y'self. Uh—know _Stardust?"_
Horty looked out at the starlit highway racing away beneath the rumbling wheels, the blaze of yellow-white which turned to dwindling red tail-light eyes as a car whisked by on the other side of the road. The fog was gone, and a lot of the pain was gone from his hand, and most of all he was gone from Armand and Tonta. Kay had given him a feather-touch of kindness, and this odd boy, who talked in a way he had never heard a boy talk before, had given him another sort of kindness. There were the beginnings of a wonderful warm glow inside him, a feeling he had had only once or twice before in his whole life—the time he had won the sack-race and they gave him a khaki handkerchief, and the time four kids had whistled to a mongrel dog, and the dog had come straight to him, ignoring the others. He began to sing, and because the truck rumbled so, he had to sing out to be heard; and because he had to sing out, he leaned on the song, giving something of himself to it as a high-steel worker gives part of his weight to the wind.
He finished. The fat boy said "Hey." The unaccented syllable was warm praise. Without any further comment he went to the front of the truck body and thumped on the square pane of glass there. The truck immediately slowed, pulled over and stopped by the roadside. The fat boy went to the tailgate, sat down, and slid off to the road.
"You stay right there," he told Horty. "I'm gonna ride up front a while. You hear me now—don't go 'way."
"I won't," said Horty.
"How the hell can you sing like that with your hand mashed?"
"I don't know. It doesn't hurt so much now."
"Do you eat grasshoppers too? Worms?"
"No!" cried Horty, horrified.
"Okay," said the boy. He went to the cab of the truck; the door slammed, and the truck ground off again.
Horty worked his way carefully forward until, squatting by the front wall of the truck-body, he could see through the square pane.
The driver was a tall man with a curious skin, lumpy and grey-green. He had a nose like Junky's, but almost no chin, so that he looked like an aged parrot. He was so tall that he had to curve over the wheel like a fern-frond.
Next to him were two little girls. One had a round bush of white hair—no; it was platinum—and the other had two thick ropes of pigtails, bangs, and beautiful teeth. The fat boy was next to her, talking animatedly. The driver seemed not to pay any attention to the conversation at all.
Horty's head was not clear, but he did not feel sick either. Everything had an exciting, dreamlike quality. He moved back in the truck body and lay down with his head on the fat boy's jacket. Immediately he sat up, and crawled among the goods stacked in the truck until his hand found the long roll of canvas, moved along it until he found his paper bag. Then he lay down again, his left hand resting easily on his stomach, his right inside the bag, with his index and little fingers resting between Junky's nose and chin. He went to sleep.
# 3
WHEN HE WOKE AGAIN THE TRUCK HAD stopped, and he opened unfocussed eyes to a writhing glare of light—red and orange, green and blue, with an underlying sheet of dazzling gold.
He raised his head, blinking, and resolved the lights into a massive post bearing neon signs: ICE TWENTY FLAVORS CREAM and CABINS and BAR—EAT. The wash of gold came from floodlights over the service area of a gas station. Three tractortrailer trucks were drawn up behind the fat boy's truck; one of them had its trailer built of heavily-ribbed stainless steel and was very lovely under the lights.
"You awake, kid?"
"Uh-Hi! Yes."
"We're going to grab a bite. Come on."
Horty rose stiffly to his knees. He said, "I haven't got any money."
"Hell with that," said the fat boy. "Come on."
He put a firm hand under Horty's armpit as he climbed down. A jukebox throbbed behind the grinding sound of a gasoline pump, and their feet crunched pleasantly on cinders. "What's your name?" Horty asked.
"They call me Havana," said the fat boy. "I never been there. It's the cigars."
"My name's Horty Bluett."
"We'll change that."
The driver and the two girls were waiting for them by the door of a diner. Horty hardly had a chance to look at them before they all crowded through and lined up at the counter. Horty sat between the driver and the silver-haired girl. The other one, the one with dark ropes of braided hair took the next stool, and Havana, the fat-boy, sat at the end.
Horty looked first at the driver—looked, stared, and dragged his eyes away in the same tense moment. The driver's sagging skin was indeed a grey-green, dry, loose, leather-rough. He had pouches under his eyes, which were red and inflamed-looking, and his underlip drooped to show long white lower incisors.
The backs of his hands showed the same loose sage-green skin, though his fingers were normal. They were long and the nails were exquisitely manicured.
"That's Solum," said Havana, leaning forward over the counter and talking across the two girls. "He's the Alligator-Skinned Man, an' the ugliest human in captivity." He must have sensed Horty's thought that Solum might resent this designation, for he added, "He's deef. He don't know what goes on."
"I'm Bunny," said the girl next to him. She was plump—not fat like Havana, but round—butter-ball round, skin-tight round. Her flesh was flesh colored and blood-colored—all pink with no yellow about it. Her hair was as white as cotton, but glossy, and her eyes were the extraordinary ruby of a white rabbit's. She had a little midge of a voice and an all but ultrasonic giggle, which she used now. She stood barely as high as his shoulder, though they sat at the same height. She was out of proportion only in this one fact of the long torso and the short legs. "An' this is Zena."
Horty turned his gaze full on her and gulped. She was the most beautiful little work of art he had ever seen in his life. Her dark hair shone, and her eyes shone too, and her head planed from temple to cheek, curved from cheek to chin, softly and smoothly. Her skin was tanned over a deep, fresh glow like the pink shadows between the petals of a rose. The lipstick she chose was dark, nearly a brown red; that and the dark skin made the whites of her eyes like beacons. She wore a dress with a wide collar that lay back on her shoulders, and a neckline that dropped almost to her waist. That neckline told Horty for the very first time that these kids, Havana and Bunny and Zena, weren't kids at all. Bunny was girl-curved, puppy-fat curved, the way even a four-year-old girl—or boy—might be. But Zena had breasts, real, taut, firm, separate breasts. He looked at them and then at the three small faces, as if the faces he had seen before had disappeared and were replaced by new ones. Havana's studied, self-assured speech and his cigars were his badges of maturity, and albino Bunny would certainly show some such emblem in a minute.
"I won't tell you his name," said Havana. "He's fixin' to get a new one, as of now. Right, kid?"
"Well," said Horty, still struggling with the strange shifting of estimated place these people had made within him, "Well, I guess so."
"He's cute," said Bunny. "You know that, kid?" She uttered her almost inaudible giggle. "You're cute."
Horty found himself looking at Zena's breasts again and his cheeks flamed. "Don't rib him," said Zena.
It was the first time she had spoken... One of the earliest things Horty could remember was a cattail stalk he had seen lying on the bank of a tidal creek. He was only a toddler then, and the dark-brown sausage of the cat-tail fastened to its dry yellow stem had seemed a hard and brittle thing. He had, without picking it up, run his fingers down its length, and the fact that it was not dried wood, but velvet, was a thrilling shock. He had such a shock now, hearing Zena's voice for the first time.
The short-order man, a pasty-faced youth with a tired mouth and laugh-wrinkles around his eyes and nostrils, lounged up to them. He apparently felt no surprise at seeing the midgets or the hideous green-skinned Solum. "Hi, Havana. You folks setting up around here?"
"Not fer six weeks or so. We're down Eltonville way. We'll milk the State Fair and work back. Comin' in with a load o' props. Cheeseburger fer the glamor-puss there. What's yer pleasure, ladies?"
"Scrambled on rye toast," said Bunny.
Zena said, "Fry some bacon until it's almost burned—"
"—an' crumble it over some peanut-butter on whole wheat. I remember, princess," grinned the cook. "What say, Havana?"
"Steak. You too, huh?" he asked Horty. "Nup—he can't cut it. Ground sirloin, an' I'll shoot you if you bread it. Peas an' mashed."
The cook made a circle of his thumb and forefinger and went to get the order.
Horty asked, timidly, "Are you with a circus?"
"Carny," said Havana.
Zena smiled at his expression. It made his head swim. "That's a carnival. You know. Does your hand hurt?"
"Not much."
"That kills me," Havana exploded. "Y'oughta see it." He drew his right hand across his left fingers and made a motion like crumbling crackers. "Man."
"We'll get that fixed up. What are we going to call you?" asked Bunny.
"Let's figure out what he's going to do first," said Havana. "We got to make the Maneater happy."
"About those ants," said Bunny, "would you eat slugs and grasshoppers, and that?" She asked him straight out, and this time she did not giggle.
"No!" said Horty, simultaneously with Havana's "I already asked him that. That's out, Bunny. The Maneater don't like to use a geek anyway."
Regretfully, Bunny said, "No carny ever had a midge that would geek. It would be a card."
"What's a geek?" asked Horty.
"He wants to know what's a geek."
"Nothing very nice," said Zena. "It's a man who eats all sorts of nasty things, and bites the heads off live chickens and rabbits."
Horty said, "I don't think I'd like doing that," so soberly that the three midgets burst into a shrill explosion of laughter. Horty looked at them all, one by one, and sensed that they laughed with, not at him, and so he laughed too. Again he felt that inward surge of warmth. These folk made everything so easy. They seemed to understand that he could be a little different from other folks, and it was all right. Havana had apparently told them all about him, and they were eager to help.
"I told you," said Havana, "he sings like an angel. Never heard anything like it. Wait'll you hear."
"You play anything?" asked Bunny. "Zena, could you teach him guitar?"
"Not with that left hand," said Havana.
_"Stop_ it!" Zena cried. "Just when did you people decide he was going to work with us?"
Havana opened his mouth helplessly. Bunny said, "Oh—I thought..." and Horty stared at Zena. Were they trying to give and take away all at the same time?
"Oh, kiddo, don't look at me like that," said Zena. "You'll tear me apart..." Again, in spite of his distress, he could all but feel her voice with fingertips. She said, "I'd do anything in the world for you, child. But—it would have to be something good. I don't know that this would be good."
"Sure it'd be good," scoffed Havana. "Where's he gonna eat? Who's gonna take him in? Listen, after what he's been through he deserves a break. What's the matter with it, Zee? The Maneater?"
"I can handle the Maneater," she said. Somehow, Horty sensed that in that casual remark was the thing about Zena that made the others await her decision. "Look, Havana," she said, "what happens to a kid his age makes him what he will be when he grows up. Carny's all right for us. It's home to us. It's the one place where we can be what we are and like it. What would it be for him, growing up in it? That's no life for a kid."
"You talk as if there was nothing in a carnival but midges and freaks."
"In a way that's so," she murmured. "I'm sorry," she added. "I shouldn't have said that. I can't think straight tonight. There's something..." She shook herself. "I don't know. But I don't think it's a good idea."
Bunny and Havana looked at each other. Havana shrugged helplessly. And Horty couldn't help himself. His eyes felt hot, and he said "Gee."
"Oh, Kid, don't."
"Hey!" barked Havana. "Grab him! He's fainting!"
Horty's face was suddenly pale and twisted with pain. Zena slid off her stool and put her arm around him. "Sick, honey? Your hand?"
Gasping, Horty shook his head. "Junky," he whispered, and grunted as if his windpipe were being squeezed. He pointed with his bandaged hand toward the door. "Truck," he rasped. "In—Junky—oh, truck!"
The midgets looked at one another, and then Havana leaped from his stool and, running to Solum, punched his arm. He made quick motions, pointing outside, turning an imaginary steering wheel, beckoning toward the door.
Moving with astonishing speed, the big man slipped to the door and was gone, the others following. Solum was at the truck almost before the midgets and Horty were outside. He bounded catlike past the cab, throwing a quick glance into it, and in two more jumps was at the tail gate and inside. There were a couple of thumps and Solum emerged, the tattered figure of a man dangling from his parti-colored hands. The tramp was struggling, but when the brilliant golden light fell on Solum's face, he uttered a scratchy ululation which must have been clearly audible a quarter of a mile away. Solum dropped him on to the cinders; he landed heavily on his back and lay there writhing and terrified, fighting to get wind back into his shocked lungs.
Havana threw away his cigar stub and pounced on the prone figure, roughly going through the pockets. He said something unprintable and then, "Look here—our new soupspoons and four compacts and a lipstick and—why, you little sneak," he snarled at the man, who was not large but was nearly three times his size. The man twitched as if he would throw Havana off him; Solum immediately leaned down and raked a large hand across his face. The man screamed again, and this time did surge up and send Havana flying; not, however, to attack, but to run sobbing and slobbering with fear from the gaunt Solum. He disappeared into the darkness across the highway with Solum at his heels.
Horty went to the tailgate. He said, timidly, to Havana, "Would you look for my package?"
"That ol' paper bag? Sure." Havana swung up on the tailgate, reappeared a moment later with the bag, and handed it to Horty.
Armand had broken Junky very thoroughly, breaking the jack-in-the-box's head away from the rest of the toy, flattening it until all that Horty could salvage was the face. But now the ruin was complete.
"Gee," said Horty. "Junky. He's all busted." He drew out the two pieces of the hideous face. The nose was crushed to a coarse powder of papier-mâché, and the face was cracked in two, a large piece and a small piece. There was an eye in each, glittering. "Gee," Horty said again, trying to fit them together with one hand.
Havana, busy gathering up the loot, said over his shoulder, "'Sa damn shame, kid. The guy must've put his knee on it while he was goin' through our stuff." He tossed the odd collection of purchases into the cab of the truck while Horty wrapped Junky up again. "Let's go back inside. Our order'll be up."
"What about Solum?" asked Horty.
"He'll be along."
Horty was conscious, abruptly, that Zena's deep eyes were fixed on him. He almost spoke to her, didn't know what to say, flushed in embarrassment, and led the way into the restaurant. Zena sat beside him this time. She leaned across him for the salt, and whispered, "How did you know someone was in the truck?"
Horty settled his paper bag in his lap, and saw her eyes on it as he did so. "Oh," she said; and then in quite a different tone, slowly, "Oh-h." He had no answer to her question, but he knew, suddenly, that he would not need one. Not now.
"How'd you know there was someone out there?" demanded Havana, busy with a catsup bottle.
Horty began to speak, but Zena interrupted. "I've changed my mind," she said suddenly. "I think carny can do the kid more good than harm. It's better than making his way on the outside."
"Well now." Havana put down the bottle and beamed. Bunny clapped her hands. _"Good,_ Zee! I knew you'd see it."
Havana added, "So did I. I... see somp'n else, too." He pointed.
"Coffee urn?" said Bunny stupidly. "Toaster?"
"The mirror, stoopid. Will you look?" He leaned close to Horty and put an arm around his head, drawing his and Zena's faces together. The reflections looked back at them—small faces, both brown, both deep-eyed, oval, dark-haired. If Horty were wearing lipstick and braids, his face would have been different from hers—but very little.
"Your long-lost brother!" breathed Bunny.
"My cousin—and I mean a _girl_ cousin," said Zena. "Look—there are two bunks in my end of the wagon... stop that cackling, Bunny; I'm old enough to be his mother and besides—oh, shut up. No; this is the perfect way to do it. The Maneater never has to know who he is. It's up to you two."
"We won't say anything," said Havana.
Solum kept on eating.
Horty asked, "Who's the Maneater?"
"The boss," said Bunny. "He used to be a doctor. He'll fix up your hand."
Zena's eyes looked at something that was not in the room. "He hates people," she said. "All people."
Horty was startled. This was the first indication among these odd folk that there might be something to be afraid of. Zena, understanding, touched his arm. "Don't be afraid. His hating won't hurt you."
# 4
THEY REACHED THE CARNIVAL in the dark part of the morning, when the distant hills had just begun to separate themselves from the paling sky.
To Horty it was all thrilling and mysterious. Not only had he met these people, but there was also the excitement and mystery ahead, and the way of starting it, the game he must play, the lines he must never forget. And now, at dawn, the carnival itself. The wide dim street, paved with wood shavings, seemed faintly luminous between the rows of stands and bally-platforms. Here a dark neon tube made ghosts of random light rays from the growing dawn; there one of the rides stretched hungry arms upward in bony silhouette. There were sounds, sleepy, restless, alien sounds; and the place smelled of damp earth, popcorn, perspiration, and sweet exotic manures.
The truck threaded its way behind the western row of midway stands and came to a stop by a long house-trailer with doors at each end.
"Home," yawned Bunny. Horty was riding in front with the girls now, and Havana had curled up in the back. "Out you get. Scoot, now; right into that doorway. The Maneater'll be asleep, and no one will see you. When you come out you'll be somebody different, and then we'll go fix your hand up."
Horty stood on the truck step, glanced around, and then arrowed to the door of the trailer and skinned inside. It was dark there. He stood clear of the door and waited for Zena to come in, close it, and draw the curtains on the small windows before turning on the lights.
The light seemed very bright. Horty found himself in a small square room. There was a tiny bunk on each side, a compact kitchenette in one corner, and what appeared to be a closet in the other.
"All right," said Zena, "take off your clothes."
" _All_ of 'em?"
"Of course, all of them." She saw his startled face, and laughed. "Listen, Kiddo. I'll tell you something about us little people. Uh—how old did you say you were?"
"I'm almost nine."
"Well, I'll try. Ordinary grown-up people are very careful about seeing each other without clothes. Whether or not it makes any sense, they are that way because there's a big difference between men and women when they're grown up. More than between boys and girls. Well, a midget stays like a child, in most ways, all his life except for maybe a couple of years. So a lot of us don't let such things bother us. As for us, you and me, we might as well make up our minds right now that it's not going to make any difference. In the first place, no one but Bunny and Havana and me know you're a boy. In the second place, this little room is just too small for two people to live in if they're going to be stooping and cringing and hiding from each other because of something that doesn't matter. See?"
"I—I guess so."
She helped him out of his clothes, and he began his careful education on how to be a woman from the skin outward.
"Tell me something, Horty," she said, as she turned out a neat drawer, looking for clothes for him. "What's in the paper bag?"
"That's Junky. It's a jack-in-the-box. It was, I mean. Armand busted it—I told you. Then the man in the truck busted it more."
"Could I see?"
Worrying into a pair of her socks, he nodded toward one of the bunks. "Go ahead."
She lifted out the tattered bits of papier-mâché. _"Two_ of them!" she exploded. She turned and looked at Horty as if he had turned bright purple, or sprouted rabbit's ears.
"Two!" she said again. "I thought I saw only one, there at the diner. Are they really yours? Both of them?"
"They're Junky's eyes," he explained.
"Where did Junky come from?"
"I had him before I was adopted. A policeman found me when I was a baby. I was put in a Home. I got Junky there. I guess I never had any folks."
"And Junky stayed with you—here, let me help you into that—Junky stayed with you from then on?"
"Yes. He had to."
"Why had to?"
"How do you hook this?"
Zena checked what seemed to be an impulse to push him into a corner and hold him still until she extracted the information from him. "About Junky," she said patiently.
"Oh. Well, I just had to have him near me. No, not near me. I could go a long way away as long as Junky was all right. As long as he was mine, I mean. I mean, if I didn't even see him for a year it was all right, but if somebody moved him, I knew it, and if somebody hurt him, I hurt too. See?"
"Indeed I do," said Zena surprisingly. Again Horty felt that sweet shock of delight; these people seemed to understand everything so well.
Horty said, "I used to think everybody had something like that. Something they'd be sick if they lost it, like. I never thought to ask anyone about it, even. And then Armand, he picked on me about Junky. He used to hide Junky to get me excited. Once he put him on a garbage truck. I got so sick I had to have a doctor. I kept yelling for Junky, until the doctor told Armand to get this Junky back to me or I would die. Said it was a fix something. Ation."
"A fixation. I know the routine," Zena smiled.
"Armand, he was mad, but he had to do it. So anyway he got tired of fooling with Junky, and put him in the top of the closet and forgot about him pretty much."
"You look like a regular dream-girl," said Zena admiringly. She put her hands on his shoulders and looked gravely into his eyes. "Listen to me, Horty. This is _very_ important. It's about the Maneater. You're going to see him in a few minutes, and I'm going to have to tell a story—a whopper of a story. And you've got to help me. He just _has_ to believe it, or you won't be able to stay with us."
"I can remember real good," said Horty anxiously. "I can remember anything I want to. Just tell me."
"All right." She closed her eyes for a moment, thinking hard. "I was an orphan," she said presently. "I went to live with my Auntie Jo. After I found out I was going to be a midget I ran away with a carnival. I was with it for a few years before the Maneater met me and I came to work for him. Now..." She wet her lips. "Auntie Jo married again and had two children. The first one died and you were the second. When she found out you were a midget too she began to be very mean to you. So you ran away. You worked a while in summer stock. One of the stagehands—the carpenter—took a shine to you. He caught you last night and took you into the wood shop and did a terrible thing to you—so terrible that you can't even talk about it. Understand? If he asks you about it, just cry. Have you got all that?"
"Sure," said Horty casually. "Which one is going to be my bed?"
Zena frowned. "Honey—this is _terribly_ important. You've got to remember every single word I say."
"Oh, I do," said Horty. And to her obvious astonishment he reeled off everything she had said, word for word.
"My!" she said, and kissed him. He blushed. "You _are_ a quick study! That's wonderful. All right then. You're nineteen years old and your name's—uh—Hortense. (That's in case you hear someone say 'Horty' some day and the Maneater sees you look around.) But everybody calls you Kiddo. All right?"
"Nineteen and Hortense and Kiddo. Uh-huh."
"Good. Gosh, honey, I'm sorry to give you so many things to think of at once! Now, this is something just between us. First of all, you must never, _never_ let the Maneater know about Junky. We'll find a place for him here, and I don't want you to ever talk about him again, except to me. Promise?"
Wide eyed, Horty nodded. "Uh-huh."
"Good. And one more thing, just as important. The Maneater's going to fix your hand. Don't worry; he's a good doctor. But I want you to push every bit of old bandage, every little scrap of cotton he uses, over toward me if you can, without letting him notice it. I don't want you to leave a drop of your blood in his trailer, understand? Not a drop. I'm going to offer to clean up for him—he'll be glad; he hates to do it—and you help me as much as you can. All right?"
Horty promised. Bunny and Havana pounded just then. Horty went out first, holding his bad hand behind him, and they called him Zena, and Zena pirouetted out, laughing, while they goggled at Horty. Havana dropped his cigar and said "Hey."
"Zee, he's _beautiful!"_ cried Bunny.
Zena help up a tiny forefinger. _"She's_ beautiful, and don't you forget it."
"I feel awful funny," said Horty, twitching his skirt.
"Where on earth did you get that hair?"
"A couple of false braids. Like 'em?"
"And the dress?"
"Bought it and never wore it," said Zena. "It won't fit my chest expansion... Come on, kids. Let's go wake the Maneater."
They made their way among the wagons. "Take smaller steps," said Zena. "That's better. You remember everything?"
"Oh, sure."
"That's a good—a good girl, Kiddo. And if he should ask you a question and you don't know, just smile. Or cry. I'll be right beside you."
A long silver trailer was parked next to a tent bearing a brilliantly colored poster of a man in a top-hat. He had long pointed mustachios and zig-zags of lightning came from his eyes. Below it, in flaming letters, was the legend
WHAT DO YOU THINK?
Mephisto Knows.
"His name isn't Mephisto," said Bunny. "It's Monetre. He used to be a doctor before he was a carny. Everyone calls him Maneater. He don't mind."
Havana pounded on the door. "Hey, Maneater! Y'going to sleep all afternoon?"
"You're fired," growled the silver trailer.
"Okay," said Havana casually. "Come on out and see what we got."
"Not if you want to put it on the payroll," said the sleepy voice. There were movements inside. Bunny pushed Horty over near the door and waved to Zena to hide. Zena flattened against the trailer wall.
The door opened. The man who stood there was tall, cadaverous, with hollows in his cheeks and a long bluish jaw. His eyes seemed, in the early morning light, to be just inch-deep black sockets in his head. "What is it?"
Bunny pointed at Horty. "Maneater, who's that?"
"Who's that?" He peered. "Zena, of course. Good morning, Zena," he said, his tone suddenly courtly.
"Good morning," laughed Zena, dancing out from behind the door.
The Maneater stared from Zena to Horty and back. "Oh, my aching bankroll," he said. "A sister act. And if I don't hire her you'll quit. And Bunny and Havana will quit."
"A mind-reader," said Havana, nudging Horty.
"What's your name, kid sister?"
"My pa named me Hortense," recited Horty, "but everyone calls me Kiddo."
"I don't blame them," said the Maneater in a kindly voice. "I'll tell you what I'm going to do, Kiddo. I'm going to call your bluff. Get off the lot, and if the rest of you don't like it, you can go along with her. If I don't see any of you on the midway at eleven o'clock this morning, I'll know what you decided." He closed the door softly and with great firmness.
"Oh _—gee!"_ said Horty.
"It's all right," grinned Havana. "He don't mean it. He fires everybody 'most every day. When he means it he pays 'em. Go get 'im, Zee."
Zena rippled her knuckles on the aluminum door. "Mister Maneater!" she sang.
"I'm counting your pay," said the voice from inside.
"Oh-oh," said Havana.
"Please. Just a minute," cried Zena.
The door opened up again. The Maneater had one hand full of money. "Well?"
Horty heard Bunny mutter, "Do good, Zee. Do good!"
Zena beckoned to Horty. He stepped forward hesitantly. "Kiddo, show him your hand."
Horty extended his ruined hand. Zena peeled off the soiled, bloody handkerchiefs one by one. The inner one was stuck fast; Horty whimpered as she disturbed it. Enough could be seen, however, to show the Maneater's trained eye that three fingers were gone completely and the rest of the hand in a bad way.
"How in creation did you do a thing like this, girl?" he barked. Horty fell back, frightened.
"Kiddo, go over there with Havana, hm?"
Horty retreated, gratefully. Zena began talking rapidly in a low voice. He could only hear part of it. "Terrible shock, Maneater. Don't remind her of it, ever... carpenter... and took her to his shop... when she... and her hand in the vise."
"No wonder I hate people," the Maneater snarled. He asked her a question.
"No," said Zena. "She got away, but her hand..."
"Come here, Kiddo," said the Maneater. His face was something to see. His whip of a voice seemed to issue from his nostrils which, suddenly, were not carven slits but distended, circular holes. Horty turned pale.
Havana pushed him gently. "Go on, Kiddo. He's not mad. He's sorry for you. Go _on!"_
Horty inched forward and timidly climbed the step. "Come in here."
"We'll see you," called Havana. He and Bunny turned away. As the door closed behind him and Zena, Horty looked back and saw Bunny and Havana gravely shaking hands.
"Sit down there," said the Maneater.
The inside of his trailer was surprisingly spacious. There was a bed across the front end, partially curtained. There was a neat galley, a shower, and a safe; a large table, cabinets, and more books than one would ever expect to fit into such space.
"Does it hurt?" murmured Zena.
"Not much."
"Don't you worry about that," growled the Maneater. He put alcohol, cotton, and a hypodermic case on the table. "Tell you what I'm going to do. (Just to be different from other doctors.) I'm going to block the nerve on your whole arm. When I poke the needle into you it'll hurt, like a bee-sting. Then your arm will feel very funny, as if it were a balloon being blown up. Then I'll clean up that hand. It won't hurt."
Horty smiled up at him. There was something in this man, with his frightening changes of voice and his treacherous humor, his kindness and his cruel aura, which the boy found deeply appealing. There was a kindness like Kay's, little Kay who hadn't cared if he ate ants. And there was a cruelty like Armand Bluett's. If nothing else, the Maneater would serve as a link with the past for Horty—for a while at least. "Go ahead," said Horty.
"That's a good girl."
The Maneater bent to his work, with Zena, fascinated, looking on, deftly moving things out of his way, making things more convenient for him. So absorbed he became that if he had any further questions to ask about "Kiddo" he forgot them.
Zena cleaned up afterward.
# 5
PIERRE MONETRE HAD GRADUATED from college three days before he was sixteen, and from medical school when he was twenty-one. A man died under his hands during a simple appendectomy, which was not Pierre Monetre's fault.
But someone—a hospital trustee—made a slighting reference to it. Monetre went to him to protest and stayed to break the man's jaw. He was immediately banned from the surgical theater, and rumor blamed it on the appendectomy alone. Instead of proving to the world matters which he felt needed no proof, he resigned from the hospital. He then began to drink. He took his drunkenness before the world as he had taken his brilliance and his skill—front and center, and damn the comments. The comments on his brilliance and his skill had helped him. The comments on his drunkenness shut him out.
He got over the drunkenness; alcoholism is not a disease, but a symptom. There are two ways of disposing of alcoholism. One is to cure the disorder which causes it. The other is to substitute some other symptom for it. That was Pierre Monetre's way.
He chose to despise the men who had shut him out, and let himself despise the rest of humanity because it was kin to those men.
He enjoyed his disgust. He built himself a pinnacle of hatred and stood on it to sneer at the world. This gave him all the altitude he needed at the time. He starved while he did it; but since riches were of value to the world at which he sneered, he enjoyed his poverty too. For a while.
But a man with such an attitude is like a child with a whip—or a nation with battleships. For a while it is sufficient to stand in the sun, with one's power in plain sight for all to see. Soon, however, the whip must whistle and crack, the rifles must thunder, the man must take more than a stand; he must take action.
Pierre Monetre worked for a while with subversive groups. It was of no importance to him which group, or what it stood for, as long as its aim was to tear down the current structure of the majority. He did not confine this to politics, but also did what he could to introduce modern non-objective art into traditional galleries, agitated for atonal music in string quartets, poured beef-extract on the serving tables of a vegetarian restaurant, and made a score of other stupid, petty rebellions—rebellions for their own sake always, having nothing to do with the worth of any art or music or food-taboos.
His disgust, meanwhile, fed on itself, until it was neither stupid nor petty. Again he found himself at a loss for a means of expressing it. He grew increasingly bitter as his clothes wore out, as he was forced out of one sordid garret after another. He never blamed himself, but felt victimized by humanity—a humanity that was, part and parcel, inferior to him. And suddenly he was given what he wanted.
He had to eat. All his corrosive hatreds focused there. There was no escaping it, and for a while there was no means of eating except doing work which would be of some value to some part of humanity. This galled him, but there was no other way of inducing humanity to pay him for his work. So he turned to a phase of his medical training and got a job in a biological laboratory doing cellular analyses. His hatred of mankind could not change the characteristics of his interested, inquiring, brilliant mind; he loved the work, hating only the fact that it benefited people-employers and their clients, who were mostly doctors and their patients.
He lived in a house—an ex-stable—near the edge of a small town, where he could take long walks by himself in the woods and think his strange thoughts. Only a man who had consciously turned away, for years, from everything human would have noticed what he had noticed one fall afternoon, or would have had the curiosity to examine it. Only a man with his unusual combination of training and ability would have had the equipment to explain it. And certainly, only such a social monster could have used it as he did.
He saw two trees.
Each was a tree like any other tree—an oak sapling, twisted from some early accident, young and alive. Never in a thousand years would he have noticed either of them, particularly, had he seen it alone. But he saw them together; his eye swept over them, he raised his eyebrows in slight surprise and walked on. Then he stopped and went back and stood staring at them. And suddenly he grunted as if he had been kicked, and went between the trees—they were twenty feet apart—and gaped from one to the other.
The trees were the same size. Each had a knotted primary limb snaking off to the north. Each had a curling scar on the first shoot from it. The first cluster on the primary on each tree had five leaves on it.
Monetre went and stood closer, running his gaze from tree to tree, up and down, one, then the other.
What he saw was impossible. The law of averages permits of such a thing as two absolutely identical trees, but at astronomical odds. Impossible was the working word for such a statistic.
Monetre reached and pulled down a leaf from one tree, and from the other took down its opposite number.
They were identical—veining, shape, size, texture.
That was enough for Monetre. He grunted again, looked searchingly around to fix the location in his mind, and headed back to his shack at a dead run.
Far into the night he labored over the oak leaves. He stared through a magnifying glass until his eyes ached. He made solutions of what he had in the house—vinegar, sugar, salt, a little phenol—and marinated parts of the leaves. He dyed corresponding parts of them with diluted ink.
What he found out about them checked and double-checked when he took them to the laboratory in the morning. Qualitative and quantitative analysis, volumetric and kindling temperature and specific gravity tests, spectrographics and pH ratings—all said the same thing; these two leaves were incredibly and absolutely identical.
Feverishly, in the months that followed, Monetre worked on parts of the trees. His working microscopes told the same story; he talked his employer into letting him use the 300-power mike which the lab kept in a bell-jar, and it said the same thing. The trees were identical, not leaf for leaf, but cell for cell. Bark and cambium and heartwood, they were the same.
It was his own incessant sampling which gave him his next lead. He took his specimens from the trees after the most meticulous measurements. A core-drill "take" from Tree A was duplicated on Tree B, to the fractional millimeter. And one day Monetre positioned his drill on both trees, got his sample from Tree A, and, in removing it, broke the drill before he could obtain his specimen from the second tree.
He blamed it, of course, on the drill, and therefore on the men who made it, and therefore on all men; and he fumed home, happily in his own ground.
But when he came back the next day he found a hole in Tree B, exactly on the corresponding spot to his tap on Tree A.
He stood with his fingers on the inexplicable hole, and for a long moment his active mind was at a complete stop. Then, carefully, he took out his knife and cut a cross in Tree A, and, in the same place on Tree B, a triangle. He cut them deep and clear, and went home again to read more esoteric books on cell structure.
When he returned to the forest, he found both trees bearing a cross.
He made many more tests. He cut odd shapes in each tree. He painted swatches of color on them.
He found that overlays, like paint and nailed-on pieces of board, remained as he applied them. But anything effecting the structure of the tree—a cut or scrape or laceration or puncture—was repeated, from Tree A to Tree B.
Tree A was the original. Tree B was some sort of a... copy.
Pierre Monetre worked on Tree B for two years before he found out, with the aid of an electron microscope, that aside from the function of exact duplication, Tree B was different. In the nucleus of each cell of Tree B was a single giant molecule, akin to the hydrocarbon enzymes, which could transmute elements. Three cells removed from a piece of bark or leaf-tissue meant three cells replaced within an hour. The freak enzyme, depleted, would then rest for an hour or two, and slowly begin to restore itself, atom by captured atom, from the surrounding tissue.
The control of restoration in damaged tissue is a subtle business at its simplest. Any biologist can give a lucid description of what happens when cells begin to rebuild—what metabolistic factors are present, what oxygen exchange occurs, how fast and how large and for what purpose new cells are developed. But they cannot tell you _why._ They cannot say what gives the signal, "Start!" to a half-ruined cell, and what says "stop." They know that cancer is a malfunction of this control mechanism, but what the mechanism is they do not say. This is true of normal tissue.
But what of Pierre Monetre's Tree B? It never restored itself normally. It restored itself only to duplicate Tree A. Notch a twig of Tree A. Break off the corresponding twig of Tree B and take it home. For twelve to fourteen hours, that twig would work on the laborious process of reforming itself to be notched. After that it would stop, and be an ordinary piece of wood. Return then to Tree B, and you would find another restored twig, and this one with its notch perfectly duplicated.
Here even Pierre Monetre's skill bogged down. Cell regeneration is a mystery. Cell duplication is a step beyond an unfathomable enigma. But somewhere, somehow, this fantastic duplication was controlled, and Monetre doggedly set about finding what did it.
He was a savage, hearing a radio and searching for the signal source. He was a dog, hearing his master cry out in pain because a girl wrote that she did not love him. He saw the result, and he tried, without adequate tools, without the capacity to understand it if it were thrust under his nose, to determine the cause.
A fire did it for him.
The few people who knew him by sight—none knew him any other way—were astonished that he joined the volunteer fire-fighters that autumn, when the smoke blasted through the hills driven by a flame-whipped wind. And for years there was a legend about the skinny feller who fought the fire like a soul promised release from hell. They told about cutting the new fire-trail, and how the skinny feller threatened to kill the forest ranger if he did not move his fire line a hundred yards north of where it had been planned. The skinny feller made history with his battle of the back-blaze, watering it with his very sweat to keep it out of a certain patch of wood. And when the fire advanced to the edge of the back-blaze, and the men broke and fled before it, the skinny feller was not with them, but stayed, crouched in the smoking moss between two oak saplings, with a spade and an axe in his bleeding hands and a fire in his eyes hotter than any that ever touched a tree. They saw all of that—
They did not see Tree B begin to tremble. Their eyes were not with Monetre's, to peer through heat and smoke and the agonized cloud of exhaustion which hovered around him, and see the scientist's mind reaching out to seize on the fact that the shuddering of Tree B was timed exactly with the rolling flames over a clearing fifty feet away.
He watched it, red-eyed. Flame touched the rocky clearing, and the tree shivered. Flame tugged the earth like hair in a hurricane pulling a scalp, and when the fire wavered and streamed upward, Tree B stood firm. But when a tortured gust of cold air rushed in to fill the heat-born vacuum, and was pursued along the ground by fingers of fire, the tree shook and tensed, wavered and trembled.
Monetre dragged his half-flayed body to the clearing and watched the flames. A spear of red-orange there; the tree stood firm. A lick of a fiery tongue here, and the tree moved.
So he found it, in the middle of a basalt outcropping. He turned over a rock with fingers which sizzled when they touched it, and under it he found a muddy crystal. He thrust it under his armpit and staggered, tottered, back to his trees, which were now in a small island built of earth and sweat and fire by his own demonaic energy, and he collapsed between the oak saplings while the fire roared past him.
Just before dawn he staggered through a nightmare, a spitting, dying inferno, to his house, and hid the crystal. He dragged himself a quarter of a mile further toward the town before he collapsed. He regained consciousness in the hospital and immediately began demanding to be released. First they refused, next they tied him to his bed, and finally he left, at night, through the window, to be with his jewel.
Perhaps it was because he was at the ragged edge of insanity, or because the fusion between his conscious and unconscious minds was almost complete. More likely it was because he was peculiarly equipped, with that driving, searching mind of his. Certainly few, if any, men had ever done it before, but he did it. He established a contact with the jewel.
He did it with the bludgeon of his hatred. The jewel winked passively at him through all his tests—all that he dared give it. He had to be careful, once he found out that it was alive. His microscope told him that; it was not a crystal, but a supercooled liquid. It was a single cell, with a faceted wall. The solidified fluid inside was a colloid, with an index of refraction like that of polystyrene, and there was a complex nucleus which he did not understand.
His eagerness quarreled with his caution; he dared not run excessive heat, corrosion, and bombardment tests on it. Wildly frustrated, he sent to it a blast of the refined hatred which he had developed over the years, and the thing—screamed.
There was no sound. It was a pressure in his mind. There was no word, but the pressure was an agonized negation, a "no"-flavored impulse.
Pierre Monetre sat stunned at his battered table, staring out of the dark of his room at the jewel, which he had placed in the pool of light under a gooseneck lamp. He leaned forward and narrowed his eyes, and with complete honesty—for he had a ravening dislike of anything which bid to defy his understanding—he sent out the impulse again.
_"No!"_
The thing reacted, by that soundless cry, as if he had prodded it with a hot pin.
He was, of course, quite familiar with the phenomena of piezoelectricity, wherein a crystal of quartz or Rochelle salts would yield a small potential when squeezed, or would slightly change its dimensions when voltage was applied across it. Here was something analagous, for all the jewel was not a true crystal. His thought-impulse apparently brought a reaction from the jewel in thought "frequencies."
He pondered.
There was an unnatural tree, and it had been connected, in some way, with this buried jewel, fifty feet away; for when flame came near the jewel, the tree trembled. When he flicked the jewel with the flame of his hatred, it reacted.
Could the jewel have _built_ that tree, with the other as a model? But how? _How?_
"Never mind how," he muttered. He'd find that out in good time. He could hurt the thing. Laws and punishment hurt; oppression hurts; power is the ability to inflict pain. This fantastic object would do what he wanted it to do or he would flog it to death.
He caught up a knife and ran outside. By the light of a waning moon he dug up a sprig of basil which grew near the old stable and planted it in a coffee can. In a similar can he put earth. Bringing them inside, he planted the jewel in the second can.
He composed himself at the table, gathering a particular strength. He had known that he had an extraordinary power over his own mind; in a way he was like a contortionist, who can make a shoulder muscle, or a thigh or part of an arm, jump and twitch individually. He did a thing like tuning an electronic instrument, with his brain. He channeled his mental energy into the specific "wave-length" which hurt the jewel, and suddenly, shockingly, spewed it out.
Again and again he struck out at the jewel. Then he let it rest while he tried to bring into the cruel psychic blows some directive command. He visualized the drooping basil shrub, picturing it in the second can.
> _Grow one._
>
> _Copy that._
>
> _Make another._
>
> _Grow one._
Repeatedly he slashed and slugged the jewel with the order. He could all but hear it whimper. Once he detected, deep in his mind, a kaleidoscopic flicker of impressions—the oak tree, the fire, a black, star-studded emptiness, a triangle cut into bark. It was brief, and nothing like it was repeated for a long time, but Monetre was sure that the impressions had come from the jewel; that it was protesting something.
It gave in; he could feel it surrender. He bludgeoned it twice more for good measure, and went to bed.
In the morning he had two basil plants. But one was a freak.
# 6
CARNIVAL LIFE PLODDED STEADILY along, season holding the tail of the season before. The years held three things for Horty. They were—belonging; Zena; and a light with a shadow.
After the Maneater fixed up his—"her"—hand, and the pink scar-tissue came in, the new midget was accepted. Perhaps it was the radiation of willingness, the delighted, earnest desire to fit in and to be of real value that did it, and perhaps it was a quirk or a carelessness on the Maneater's part, but Horty stayed.
In the carnival the pinheads and the roustabouts, the barkers and their shills, the dancers and fireaters and snake-men and ride mechanics, the layout and advance men, had something in common which transcended color and sex and racial and age differences. They were carny, all of them, interested in gathering their tips and turning them—which is carnivalese for collecting a crowd and persuading it to file past the ticket-taker—for this, and for this alone, they worked. And Horty was a part of it.
Horty's voice was a part of Zena's in their act, which followed Bets and Bertha, another sister team with a total poundage in the seven hundreds. Billed as The Little Sisters, Zena and Kiddo came on with a hilarious burlesque of the preceding act, and then faded to one of their own, a clever song-and-dance routine which ended in a bewildering vocal—a harmonizing yodel. Kiddo's voice was clear and true, and blended like keys on an organ with Zena's full contralto. They also worked in the Kiddie's Village, a miniature town with its own fire station, city hall, and restaurants, all child-size; adults not admitted. Kiddo served weak tea and cookies to the round-eyed, freckle-faced moppets at the country fairs, and felt part of their wonder and part of their belief in this magic town. Part of... part of... it was a deep-down, thrilling theme to everything that Kiddo did; Kiddo was part of Horty, and Horty was part of the world, for the first time in his life.
Their forty trucks wound among the Rockies and filed out along the Pennsylvania Turnpike, snorted into the Ottawa Fairgrounds and blended themselves into the Fort Worth Exposition. Once, when he was ten, Horty helped the giant Bets bring her child into the world, and thought nothing of it, since it was so much a part of the expected-unexpected of being a carny. Once a pinhead, a happy, brainless dwarf who sat gurgling and chuckling with joy in a corner of the freak show, died in Horty's arms after drinking lye, and the scar in Horty's memory of that frightening scarlet mouth and the pained and puzzled eyes—that scar was a part of Kiddo, who was Horty, who was part of the world.
And the second thing was Zena, who was hands for him, eyes for him, a brain for him until he got into the swing of things, until he learned to be, with utter naturalness, a girl midget. It was Zena who made him belong, and his starved ego soaked it up. She read to him, dozens of books, dozens of kinds of books, in that deep, expressive voice which quite automatically took the parts of all the characters in a story. She led him, with her guitar and her phonograph records, into music. Nothing he learned changed him; but nothing he learned was forgotten. For Horty-Kiddo had eidetic memory.
Havana used to say it was a pity about that hand. Zena and Kiddo wore black gloves in their act, which seemed a little odd; and besides, it would have been nice if they both played guitar. But of course that was out of the question. Sometimes Havana used to remark to Bunny, at night, that Zena was going to wear her fingers plumb off if she played all day on the bally-platform and all night to amuse Horty; for the guitar would cry and ring for hours after they bedded down. Bunny would say sleepily that Zena knew what she was doing—which was, of course, perfectly true.
She knew what she was doing when she had Huddie thrown out of the carnival. That was bad, for a while. She violated the carny's code to do it, and she was carny through and through. It wasn't easy, especially because there was no harm in Huddie. He was a roustabout, with a broad back and a wide, tender mouth. He idolized Zena, and was happy to include Kiddo in his inarticulate devotion. He brought them cookies and cheap little scatter-pins from the towns, and squatted out of sight against the base of their bally-platform to listen raptly while they rehearsed.
He came to the trailer to say goodbye when he was fired. He had shaved, and his store suit didn't fit very well. He stood on the step holding a battered straw "keyster" and chewed hard on some half-formed words that he couldn't quite force out. "I got fired," he said finally.
Zena touched his face. "Did—did the Maneater tell you why?"
Huddie shook his head. "He jus' called me in and handed me my time. I ain't done nothin', Zee. I—I didn' say nothin' t' him, though. Way he looked, he like to kill me. I—I jus' wish..." He blinked, set down his suitcase, and wiped his eyes on his sleeve. "Here," he said. He reached into his breast pocket, thrust a small package at Zena, turned and ran.
Horty, sitting on his bunk and listening wide-eyed, said, "Aw... Zee, what's he done? He's such a _nice_ feller!"
Zena closed the door. She looked at the package. It was wrapped in gilt gift paper and had a red ribbon with a multiple, stringy bow. Huddie's big hands must have spent an hour over it. Zena slipped the ribbon off. Inside was a chiffon kerchief, gaudy and cheap and just the bright present that Huddie would choose after hours of careful searching.
Horty suddenly realized that Zena was crying. "What's the matter?"
She sat beside him and took his hands. "I went and told the Maneater that Huddie was—was bothering me. That's why he was fired."
"But—Huddie never did anything to you! Nothing bad."
"I know," Zena whispered. "Oh, I know. I lied. Huddie had to go—right away."
Horty stared at her. "I don't understand about that, Zee."
"I'm going to explain it to you," she said carefully. "It's going to hurt, Horty, but maybe that'll prevent something else happening that will hurt much more. Listen. You always remember everything. You were talking to Huddie yesterday, remember?"
"Oh, yes. I was watching him and Jemmy and Ole and Stinker drive stakes. I love to watch 'em. They stand around in a circle with their big heavy sledgehammers and each one taps easy—plip, plip, plip, plip—and then each one swings the hammer right over their head and hits with all their might—blap, blap, blap, blap!—so _fast!_ An' that ol' stake, it jus' _melts_ into the ground!" He stopped, his eyes shining, hearing and seeing the machine-gun rhythm of the sledge crew with all the detail of his sound-camera mind.
"Yes, dear," said Zena patiently. "And what did you say to Huddie?"
"I went to feel the top of the stake inside the iron band, where it was all splintery. I said, 'my, it's all mashed!' And Huddie, he said, 'Jus' think how mashed your hand'd be iff'n you lef' it there while we-uns drove it.' And I laughed at him an' said, 'It wouldn't bother me for long, Huddie. It would grow back again.' That's all, Zee."
"None of the others heard?"
"No. They were starting the next stake."
"All right, Horty. Huddie had to go because you said that to him."
"But—but he thought it was a joke! He just laughed... what did I _do,_ Zee?"
"Horty sweetheart, I told you that you must never say the slightest, tiniest word to anyone about your hand, or about anything growing back after it gets cut off, or anything at all like that. You've got to wear a glove on your left hand day and night, and never do a thing with—"
"—with my three new fingers?"
She clapped a hand over his mouth. "Never talk about it," she hissed, "to anyone but me. _No one_ must know. Here." She rose and tossed the dazzling kerchief on his lap. "Keep this. Look at it and think about it and—and leave me alone for a while. Huddie was—I... I can't like you very much for a little while, Horty. I'm sorry."
She turned away from him and went out, leaving him shocked and hurt and deeply ashamed. And when, very late that night, she came to his bed and slid her warm, small arms around him and told him it was all right now, he needn't cry any more, he was so happy he could not speak. He burrowed his face into her shoulder and trembled, and he made a promise—a deep promise, to himself, not to her, that he would always, always do as she said. They never spoke of Huddie again.
Sights and smells were treasures; he treasured the books they read together—fantasies like _The Worm Ouroborus_ and _The Sword in the Stone_ and _The Wind in the Willows;_ strange, quizzical, deeply human books, each the only one of its kind, like _Green Mansions,_ Bradbury's _Martian Chronicles,_ Capek's _War with the Newts,_ and _The Innocent Voyage._
Music was a treasure—laughing music like the Polka from the "Isle of Gold" and the cacaphonous ingenuities of Spike Jones and Red Ingalls; the rich romanticism of Crosby, singing "Adeste Fideles" or "Skylark" as if each were his only favorite, and Tchaikovsky's azure sonorities; and the architects, Franck building with feathers, flowers and faith, Bach with agate and chrome.
But the things Horty treasured most were the drowsy conversations in the dark, sometimes on a silent fairgrounds after hours, sometimes bumping along a moonwashed road.
"Horty—" (She was the only one who called him Horty. No one else heard her do it. It was like a private pet-name.)
"Mmm?"
"Can't you sleep?"
"Thinkin'..."
"Thinking about your childhood sweetheart?"
"How'd you know? Uh—don't kid me, Zee."
"Oh, I'm sorry, honey."
Horty said into the darkness, "Kay was the only one who ever said anything nice to me, Zee. The only one. It wasn't only that night I ran away. Sometimes in school she'd just smile, that's all. I—I used to wait for it. You're laughing at me."
"No, Kiddo, I'm not. You're so sweet."
"Well," he said defensively, "I like to think about her sometimes."
He did think about Kay Hallowell, and often; for this was the third thing, the light with a shadow. The shadow was Armand Bluett. He could not think of Kay without thinking of Armand, though he tried not to. But sometimes the cold wet eyes of a tattered mongrel in some farmyard, or the precise, heralding sound of a key in a Yale lock, would bring Armand and Armand's flat sarcasm and Armand's hard and ready hands right into the room with him. Zena knew of this, which is why she always laughed at him when he mentioned Kay...
He learned so much in those somnolent talks. About the Maneater, for example. "How'd he ever get to be a carny, Zee?"
"I can't say exactly. Sometimes I think he hates carny. He seems to despise the people who come in, and I guess he's in the business mostly because it's the only way he can keep his—" She fell silent.
"What, Zee?"
She was quiet until he spoke again. "He has some people he—thinks a lot of," she explained at length. "Solum. Gogol, the Fish Boy. Little Pennie was one of them." Little Pennie was the pinhead who had drunk lye. "A few others. And some of the animals. The two-legged cat, and the Cyclops. He—likes to be near them. He kept some of them before he got into show business. But it must have cost a lot. This way, he can make money out of them."
"Why does he like them, 'specially?"
She turned restlessly. "He's the same kind they are," she breathed. Then, "Horty, don't _ever_ show him your hand!"
One night in Wisconsin something woke Horty.
_Come here._
It wasn't a sound. It wasn't in words. It was a call. There was a cruel quality to it. Horty lay still.
_Come here, come here. Come! Come!_
Horty sat up. He heard the prairie wind, and the crickets.
_Come!_ This time it was different. There was a coruscating blaze of anger in it. It was controlled and directive, and had in it a twinge of the pleasure of an Armand Bluett in catching a boy in an inarguable wrong. Horty swung out of bed and stood up, gasping.
"Horty? Horty—what is it?" Zena, naked, came sliding out of the dim whiteness of her sheets like the dream of a seal in surf.
"I'm supposed to—go," he said with difficulty.
"What is it?" she whispered tensely. "Like a voice inside you?"
He nodded. The furious command struck him again, and he twisted his face.
"Don't go," Zena whispered. "You hear me, Horty? Don't you move." She spun into a robe. "You get back into bed. Hold on tight; whatever you do, don't leave this trailer. The—it will stop. I promise you it will stop quickly." She pressed him back to his bunk. "Don't you go, now, no matter what happens."
Blinded, stunned by this urgent, painful pressure, he sank back on the bunk. The call flared again within him; he started up. "Zee—" But she was gone. He stood up, his head in his hands, and then remembered the furious urgency of her orders, and sat down again.
It came again and was—incomplete. Interrupted.
He sat quite still and felt for it with his mind, timidly, as if he were tonguing a sensitive tooth. It was gone. Exhausted, he fell back and went to sleep.
In the morning Zena was back. He had not heard her come in. When he asked her where she had been, she gave him a curious look and said, "Out." So he did not ask her anything more. But at breakfast with Bunny and Havana, she suddenly gripped his arm, taking advantage of a moment when the others had left the table to stove and toaster. "Horty! If you ever get a call like that again, wake me. Wake me right away, you hear?" She was so fierce he was frightened; he had only time to nod before the others came back. He never forgot it. And after that, there were not many times when he woke her and she slipped out, wordlessly, to come back hours later; for when he realized the calls were not for him, he no longer felt them.
The seasons passed and the carnival grew. The Maneater was still everywhere in it, flogging the roustabouts and the animal men, the daredevils and the drivers, with his weapon—his contempt, which he carried about openly like a naked sword.
The carnival grew—larger. Bunny and Havana grew—older, and so did Zena, in subtle ways. But Horty did not grow at all.
He—she—was a fixture now, with a clear soprano voice and black gloves. He passed with the Maneater, who withheld his contempt in saying "Good Morning"—a high favor—and who had little else to say. But Horty-Kiddo was loved by the rest, in the earnest, slap-dash way peculiar to carnies.
The show was a flat-car rig now, with press-agents and sky-sweeping searchlights, a dance pavilion and complicated, epicyclic rides. A national magazine had run a long picture story on the outfit, with emphasis on its "Strange People" ("Freak Show" being an unpopular phrase.) There was a press office now, and there were managers, and annual re-bookings from big organizations. There were public-address systems for the bally-platforms, and newer—not new, but newer—trailers for the personnel.
The Maneater had long since abandoned his mind-reading act, and, increasingly, was a presence only to those working on the lot. In the magazine stories, he was a "partner," if mentioned at all. He was seldom interviewed and never photographed. He spent his working hours with his staff, and stalking about the grounds, and his free time with his books and his rolling laboratory and his "Strange People." There were stories of his being found in the dark hours of the morning, standing in the breathing blackness with his hands behind him and his gaunt shoulders stooped, staring at Gogol in his tank, or peering over the two-headed snake or the hairless rabbit. Watchmen and animal men had learned to keep away from him at such times; they withdrew silently, shaking their heads, and left him alone.
"Thank you, Zena." The Maneater's tone was courtly, mellow.
Zena smiled tiredly, closed the door of the trailer against the blackness outside. She crossed to the chrome and plastic-web chair by his desk and curled up with her robe tucked over her toes. "I've had enough sleep," she said.
He poured wine—shimmering Moselle. "An odd hour for it," he offered, "but I know you like it."
She took the glass and set it on the corner of the desk. She waited. She had learned to wait.
"I found some new ones today," said the Maneater. He opened a heavy mahogany box and lifted a velvet tray out of it. "Mostly young ones."
"That's good," said Zena.
"It is and it isn't," said Monetre irascibly. "They're easier to handle—but they can't do as much. Sometimes I wonder why I bother."
"So do I," said Zena.
She thought his eyes moved to her and away in their deep sockets, but she couldn't be sure. He said, "Look at these."
She took the tray on her lap. There were eight crystals lying on the velvet, gleaming dully. They had been freshly cleaned of the layer of dust, like dried mud, that always covered them when they were found—the layer that made them look like clods, like stones. They were not quite translucent, yet the nucleus could be seen by one who knew just what internal hovering shadow to look for.
Zena picked one up and held it to the light. Monetae grunted, and she met his gaze.
"I was wondering which one you would pick up first," he said. "That one's very alive." He took it from her and stared at it, narrowing his eyes. The bolt of hatred he aimed at it made Zena whimper. "Please don't..."
"Sorry... but it screams so," he said softly, and put it back with the others. "If I could only understand how they think," he said. "I can hurt them. I can direct them. But I can't talk to them. But some day I'll find out..."
"Of course," said Zena, watching his face. Was he going to have another of his furies? He was due for one...
He slumped into his chair, put his clasped hands between his knees and stretched. She could hear his shoulders crackle. "They dream," he said, his organ voice dwindling to an intense whisper. "That's as close to describing them as I've come yet. They dream."
Zena waited.
"But their dreams live in our world—in our kind of reality. Their dreams are not thoughts and shadows, pictures and sounds like ours. They dream in flesh and sap, wood and bone and blood. And sometimes their dreams aren't finished, and so I have a cat with two legs, and a hairless squirrel, and Gogol, who should be a man, but who has no arms, no sweat glands, no brain. They're not finished... they all lack formic acid and niacin, among other things. But—they're alive."
"And you don't know—yet—how the crystals do it."
He looked up at her without moving his head, so that she saw his eyes glint through his heavy brows. "I hate you," he said, and grinned. "I hate you because I have to depend on you—because I have to talk to you. But sometimes I like what you do. I like what you said— _yet._ I don't know how the crystals do their dreaming—yet."
He leaped to his feet, the chair crashing against the wall as he moved. "Who understands a dream fulfilled?" he yelled. Then, quietly, as if there were no excitement in him, he continued evenly, "Talk to a bird and ask it to understand that a thousand-foot tower is a man's finished dream, or that an artist's sketch is part of one. Explain to a caterpillar the structure of a symphony—and the dream that based it. Damn structure! Damn ways and means!" His fist crashed down on the desk. Zena quietly picked up her wine glass. "How this thing happens isn't important. Why it happens isn't important. But it _does_ happen, and I can control it." He sat down and said to Zena, courteously, "More wine?"
"Thank you, no. I still—"
"The crystals are alive," Monetre said conversationally. "They think. They think in ways which are utterly alien to ours. They've been on this earth for hundreds, thousands of years... clods, pebbles, shards of stone... thinking their thoughts in their own way... striving for nothing mankind wants, taking nothing mankind needs... intruding nowhere, communing only with their own kind. But they have a power that no man has ever dreamed of before. And I want it. I want it. I want it, and I mean to have it."
He sipped his wine and stared into it. "They breed," he said. "They die. And they do a thing I don't understand. They die in pairs, and I throw them away. But some day I'll force them to give me what I want. I'll make a perfect thing—a man, or a woman... one who can communicate with the crystals... one who will do what I want done."
"How do—how can you be sure?" Zena asked carefully.
"Little things I get from them when I hurt them. Flashes, splinters of thought. For years I've been prodding them, and for every thousand blows I give them, I get a fragment. I can't put it into words; it's a thing I _know._ Not in detail, not quite clearly... but there's something special about the dream that gets _finished._ It doesn't turn out like Gogol, or like Solum—incomplete or wrongly made. It's more like that tree I found. And that finished thing will probably be human, or near it... and if it is, I can control it."
"I wrote an article about the crystals once," he said after a time. He began to unlock the deep lower desk drawer. "I sold it to a magazine—one of those veddy lit'ry quarterly reviews. The article was pure conjecture, to all intents and purposes. I described these crystals in every way except to say what they look like. I demonstrated the possibility of other, alien life-forms on earth, and how they could live and grow all around us without our knowledge— _provided they didn't compete._ Ants compete with humans, and weeds do, and amoebae. These crystals do not—they simply live out their own lives. They may have a group consciousness like humans—but if they do, they don't use it for survival. And the only evidence mankind has of them is their dreams—their meaningless, unfinished attempts to copy living things around them. And what do you suppose was the learned refutation stimulated by my article?"
Zena waited.
"One," said Monetre with a frightening softness, "countered with a flat statement that in the asteroid belt between Mars and Jupiter there is a body the size of a basketball which is made of chocolate cake. That, he said, is a statement which must stand as a truth because it cannot be scientifically disproved. _Damn_ him!" he roared, and then went on, as softly as before, "Another explained away every evidence of malformed creatures by talking eclectic twaddle about fruit-flies, x-rays, and mutation. It's that blind, stubborn, damnable attitude that brought such masses of evidence to prove that planes wouldn't fly (for if ships needed power to keep them afloat as well as to drive them, we'd have no ships) or that trains were impractical (because the weight of the cars on the tracks would overcome the friction of the locomotives' wheels, and the train would never start.) Volumes of logical, observer's proof showed the world was flat. Mutations? Of course there are natural mutations. But why must one answer be the only answer? Hard radiation mutations—demonstrable. Purely biochemical mutations—very probable. And the crystals' dreams..."
From the deep drawer he drew a labelled crystal. He took his silver cigarette lighter from the desk, thumbed it alight, and stroked the yellow flame across the crystal.
Out of the blackness came a faint, agonized scream.
"Please don't," said Zena.
He looked sharply at her drawn face. "That's Moppet," he said. "Have you now bestowed your affections on a two-legged cat, Zena?"
"You didn't have to hurt her."
"Have to?" He brushed the crystal with the flame again, and again the scream drifted to them from the animal tent. "I had to develop my point." He snapped the lighter out, and Zena visibly relaxed. Monetre dropped lighter and crystal on the desk and went on calmly, "Evidence. I could bring that fool with his celestial chocolate cake here to this trailer, and show him what I just showed you, and he'd tell me the cat was having a stomach ache. I could show him electron photomicrographs of a giant molecule inside that cat's red corpuscles actually transmuting elements—and he'd accuse me of doctoring the films. Humanity has been accursed for all its history by its insistence that what it already knows must be right, and all that differs from that must be wrong. I add my curse to the curse of history, with all my heart. Zena..."
"Yes, Maneater." His abrupt change in tone startled her; she had never gotten used to it.
"The complex things—mammals, birds, plants—the crystals only duplicate them if they want to—or if I flog them half to death. But some things are easy."
He rose, and drew drapes aside from the shelves behind and above him. He lifted down a rack on which was a row of chemist's watch-glasses. Setting it under the light, he touched the glass covers fondly. "Cultures," he said, in a lover's voice. "Simple, harmless ones, now. Rod bacilli in this one, and spirilla here. The _cocci_ are coming along slowly, but coming for all that. I'll plant glanders, Zena, if I like, or the plague. I'll carry nuisance-value epidemics up and down this country—or wipe out whole cities. All I need to be sure of it is that middle-man—that fulfilled dream of the crystals that can teach me how they think. I'll find that middle-man, Zee, or make one. And when I do, I'll do what I like with mankind, in my own time, in my own way."
She looked up at his dark face and said nothing.
"Why do you come here and listen to me, Zena?"
"Because you call. Because you'll hurt me if I don't," she said candidly. Then, "Why do you talk to me?"
Suddenly, he laughed. "You never asked me that before, in all these years. Zena, thoughts are formless, coded... impulses without shape or substance or direction—until you convey them to someone else. Then they precipitate, and become ideas that you can put out on the table and examine. You don't know what you think until you tell someone else about it. That's why I talk to you. That's what you're _for._ You didn't drink your wine."
"I'm sorry." Dutifully, she drank it, looking at him wide-eyed over the rim of the glass that was too big to be her glass.
After that he let her go.
The seasons passed and there were other changes. Zena very seldom read aloud any more. She heard music or played her guitar, or busied herself with costumes and continuities, quietly, while Horty sprawled on his bunk, one hand cupping his chin, the other flipping pages. His eyes moved perhaps four times to scan each page, and their turning was a rhythmic susurrus. The books were Zena's choice, and now they were almost all quite beyond her. Horty swept the books of knowledge, breathed it in, stored it, filed it. She used to look at him, sometimes, in deep astonishment, amazed that he was Horty... he was Kiddo, a girl-child, who, in a few minutes would be on the bally-platform singing the "Yodelin' Jive" with her. He was Kiddo, who giggled at Cajun Jack's horseplay in the cook-tent and helped Lorelei with her brief equestrienne costumes. Yet, still giggling, or still chattering about bras and sequins, Kiddo was Horty, who would pick up a romantic novel with a bosomy dust-jacket, and immerse himself in the esoteric matter it concealed—texts disguised under the false covers—books on microbiology, genetics, cancer, dietetics, morphology, endocrinology. He never discussed what he read, never; apparently, evaluated it. He simply stored it—every page, every diagram, every word of every book she brought him. He helped her put the false covers on them, and he helped her secretly dispose of the books when he had read them—he never needed them for reference—and he never questioned her once about why he was doing it.
Human affairs refuse to be simple... human goals refuse to be clear. Zena's task was a dedication, yet her aims were speckled and splotched with surmise and ignorance, and the burden was heavy...
The rain drove viciously against the trailer in one morning's dark hours, and there was an October chill in the August air. The rain spattered and hissed like the churning turmoil she sensed so often in the Maneater's mind. Around her was the carnival. It was around her memories too, for more years than she liked to count. The carnival was a world, a good world, but it exacted a bitter payment for giving her a place to belong. The very fact that she belonged meant a stream of goggling eyes and pointing fingers: _You're different. You're different._
_Freak!_
She turned restlessly. Movies and love-songs, novels and plays... here was a woman—they called her dainty, too—who could cross a room in five strides instead of fifteen, who could envelop a doorknob in one _small_ hand. She stepped up into trains instead of clambering like a little animal, and used restaurant forks without having to distort her mouth.
And they were loved, these women. They were loved, and they had choice. Their problems of choice were subtle ones, easy ones—differences between men which were so insignificant they really couldn't matter. They didn't have to look at a man and think first, first of all before anything else, _What will it mean to him that I'm a freak?_
She was little, little in so many ways. Little and stupid. The one thing she had been able to love, she had put into deadly jeopardy. She had done what she could, but there was no way of knowing if it was right.
She began to cry, silently.
Horty couldn't have heard her, but he was there. He slid into bed beside her. She gasped, and for a moment could not release her breath from her pounding throat. Then she took his shoulders, turned him away from her. She pressed her breasts against his warm back, crossed her arms over his chest. She drew him close, close, until she heard breath hissing from his nostrils. They lay still, curled, nested together like two spoons.
"Don't move, Horty. Don't say anything."
They were quiet for a long time.
She wanted to talk. She wanted to tell him of her loneliness, her hunger. Four times she pursed her lips to speak, and could not, and tears wet his shoulder instead. He lay quiet, warm and with her—just a child, but so much _with_ her.
She dried his shoulder with the sheet, and put her arms around him again. And gradually, the violence of her feeling left her, and the all but cruel pressure of her arms relaxed.
At last she said two things that seemed to mean the pressures she felt. For her swollen breasts, her aching loins, she said, "I love you, Horty. I love you."
And later, for her hunger, she said, "I wish I was big, Horty. I want to be big..."
Then she was free to release him, to turn over, to sleep. When she awoke in the dripping half-light, she was alone.
He had not spoken, he had not moved. But he had given her more than any human being had ever given her in her whole life.
# 7
"ZEE..."
"Mmm?"
"Had a talk with the Maneater today while they were setting up our tent."
"What'd he say?"
"Just small-talk. He said the rubes like our act. Guess that's as near as he can get to saying he likes it himself."
"He doesn't," said Zena with certainty. "Anything else?"
"Well—no, Zee. Nothing."
"Horty, darling. You just don't know how to lie."
He laughed. "Well, it'll be all right, Zee."
There was a silence. Then, "I think you'd better tell me, Horty."
"Don't you think I can handle it?"
She turned over to face him across the trailer. "No."
She waited. Although it was pitch black, she knew Horty was biting his lower lip, tossing his head.
"He asked to see my hand."
She sat bolt upright in her bunk. "He didn't!"
"I told him it didn't give me any trouble. Gosh—when was it that he fixed it? Nine years ago? Ten?"
"Did you show it to him?"
"Cool down, Zee! No, I didn't. I said I had to fix some costumes, and got away. But he called after me and said to come to his lab before ten tomorrow. I'm just trying to think of some way to duck it."
"I was afraid of this," she said, her voice shaking. She put her arms around her knees, resting her chin on them.
"It'll be all right, Zee," said Horty sleepily. "I'll think of something. Maybe he'll forget."
"He won't forget. He has a mind like an adding machine. He won't attach any importance to it until you don't show up; then, look out!"
"Well, s'pose I do show it to him."
"I've told you and told you, Horty, you must _never_ do that!"
"All right, all right.—Why?"
"Don't you trust me?"
"You know I do."
She did not answer, but sat rigidly, in thought. Horty dozed off.
Later—probably two hours later—he was awakened by Zena's hand on his shoulder. She was crouched on the floor by his bunk. "Wake up, Horty. Wake up!"
"Wuh?"
"Listen to me, Horty. You remember all you've told _me—please_ wake up!—remember, about Kay, and all?"
"Oh, sure."
"What was it you were going to do, some day?"
"You mean about going back there and seeing Kay again, and getting even with that old Armand?"
"That's right. Well, that's exactly what you're going to do."
"Well, sure." He yawned and closed his eyes. She shook him again. "I mean now, Horty. Tonight. Right now."
"Tonight? Right now?"
"Get up, Horty. Get dressed. I mean it."
He sat up blearily. "Zee... it's night time!"
"Get dressed," she said between her teeth. "Hop to it, Kiddo. You can't be a baby all your life."
He sat on the edge of the bed and shivered away the last smoky edges of sleep. "Zee!" he cried. "Go away? You mean, leave here? Leave the carnival and Havana and—and you?"
"That's right. Get dressed, Horty."
"But—where will I go?" He reached for his clothes. "What will I do? I don't know anybody out there!"
"You know where we are? It's only fifty miles to the town you came from. That's as near as we'll get this year. Anyway, you've been here too long," she added, her voice suddenly gentle. "You should have left before—a year ago, two years, maybe." She handed him a clean blouse.
"But why do I have to?" he asked pitiably.
"Call it a hunch, though it isn't really. You wouldn't get through that appointment with the Maneater tomorrow. You've got to get out of here and stay out."
"I can't go!" he said, childishly protesting even as he obeyed her. "What are you going to tell the Maneater?"
"You had a telegram from your cousin, or some such thing. Leave it to me. You won't ever have to worry about it."
"Not ever—can't I ever come back?"
"If you ever see the Maneater again, you turn and run. Hide. Do anything, but never let him near you as long as you live."
"What about you, Zee? I'll never see you again!" He zipped up the side of a grey pleated skirt and held still for Zee's deft application of eyebrow pencil.
"Yes you will," she said softly. "Some day. Some way. Write to me and tell me where you are."
"Write to you? Suppose the Maneater should get my letter? Would that be all right?"
"It would not." She sat down, casting a woman's absent, accurate appraisal over Horty. "Write to Havana. A penny postcard. Don't sign it. Pick it out on a typewriter. Advertise something—hats or haircuts, or some such. Put your return address on it but transpose each pair of numbers. Will you remember that?"
"I'll remember," said Horty vaguely.
"I know you will. You never forget anything. You know what you're going to learn now, Horty?"
"What?"
"You're going to learn to _use_ what you know. You're just a child now. If you were anyone else, I'd say you were a case of arrested development. But all the books we've read and studied... you remember your anatomy, Horty? And the physiology?"
"Sure, and the science and history and music and all that. Zee, what am I going to do out there? I got nobody to tell me anything!"
"You'll have to tell yourself now."
"I don't know what to do _first!"_ he wailed.
"Honey, honey..." She came to him and kissed his forehead and the tip of his nose. "You walk out to the highway, see? And stay out of sight. Go down the road about a quarter of a mile and flag a bus. Don't ride in anything else but a bus. When you get to town wait at the station until about nine o'clock in the morning and then find yourself a room in a rooming house. A quiet one on a small street. Don't spend too much money. Get yourself a job as soon as you can. You better be a boy, so the Maneater won't know where to look."
"Am I going to grow?" he asked, voicing the professional fear of all midgets.
"Maybe. That depends. Don't go looking for Kay and that Armand creature until you're ready for it."
"How will I know when I'm ready?"
"You'll know. Got your bankbook? Keep on banking by mail, the way you always have. Got enough money? Good. You'll be all right, Horty. Don't ask anyone for anything. Don't tell anyone anything. Do things for yourself, or do without."
"I don't—belong out there," he muttered.
"I know. You will, though; just the way you came to belong here. You'll see."
Moving gracefully and easily on high heels, Horty went to the door. "Well, good-by, Zee. I—I wish I—Couldn't you come with me?"
She shook her glossy dark head. "I wouldn't dare, Kiddo. I'm the only human being the Maneater talks to—really talks to. And I've—got to watch what he's doing."
"Oh." He never asked what he should not ask. Childish, helpless, implicitly obedient, the exact, functional product of his environment, he gave her a frightened smile and turned to the door. "Good bye, honey," she whispered, smiling.
When he had gone she sank down on his bunk and cried. She cried all night. It was not until the next morning that she remembered Junky's jeweled eyes.
# 8
A DOZEN YEARS HAD PASSED since Kay Hallowell had seen, from the back window, Horty Bluett climb into a brilliantly painted truck, one misty night. Those years had not treated the Hallowells kindly. They had moved into a smaller house, and then into an apartment, where her mother died. Her father had hung on for a while longer, and then had joined his wife, and Kay, at nineteen, left college in her junior year and went to work to help her brother through pre-medical school.
She was a cool blonde, careful and steady, with eyes like twilight. She carried a great deal on her shoulders, and she kept them squared. Inwardly she was afraid to be frightened, afraid to be impressionable, to be swayed, to be moved, so that outwardly she wore carefully constructed poise. She had a job to do; she had to get ahead herself so that she could help Bobby through the arduous process of becoming a doctor. She had to keep her self-respect, which meant decent housing and decent clothes. Maybe some day she could relax and have fun, but not now. Not tomorrow or next week. Just some day. Now, when she went out to dance, or to a show, she could only enjoy herself cautiously, up to the point where late hours, or a strong new interest, or even enjoyment itself, might interfere with her job. And this was a great pity, for she had a deep and brimming reservoir of laughter.
"Good morning, Judge." How she hated that man, with his twitching nostrils and his limp white hands. Her boss, T. Spinney Hartford, of Benson, Hartford and Hartford, was a nice enough man but he certainly hobnobbed with some specimens. Oh well; that's the law business. "Mr. Hartford will be with you in a moment. Please sit down, Judge."
Not there, Wet-Eyes! Oh dear, right next to her desk. Well, he always did.
She flashed him a meaningless smile and went to the filing cabinets across the room before he could start that part weak, part bewildering line of his. She hated the waste of time; there was nothing she needed from the files. But she couldn't sit there and ignore him, and she knew he wouldn't shout across the office at her; he preferred the technique described by Thorne Smith as "a voice as low as his intentions."
She felt his moist gaze on her back, on her hips, rolling up and down the seams of her stockings, and she had an attack of gooseflesh that all but itched. This wouldn't do. Maybe short range would be better; perhaps she could parry what she couldn't screen. She returned to her desk, gave him the same lipped smile, and pulled out her typewriter, swinging it up on its smooth countersprung swivels. She ran in some letterhead and began to type busily.
"Miss Hallowell."
She typed.
"Miss Hallowell." He reached and took her wrist. "Please don't be so very busy. We have such a brief moment together."
She let her hands fall into her lap—one of them, at least. She let the other hang unresisting in the Judge's limp white clasp until he let it go. She folded her hands and looked at them. That voice! If she looked up she was sure she would see a trickle of drool on his chin. "Yes, Judge?"
"Do you enjoy it here?"
"Yes. Mr. Hartford is very kind."
"A most agreeable man. Most agreeable." He waited until Kay felt so stupid, sitting there staring at her hands, that she had to raise her face. Then he said, "You plan to stay here for quite a while, then."
"I don't see why—that is, I'd like to."
"The best-laid plans..." he murmured. Now, what was that? A threat to her job? What did this slavering stuffed-shirt have to do with her job? _"Mr. Hartford is a most agreeable man."_ Oh. Oh dear. Mr. Hartford was a lawyer, and frequently had cases in Surrogate. Some of those were hairline decisions on which a lot depended. _"Most agreeable."_ Of course Mr. Hartford was an agreeable man. He had a living to make.
Kay waited for the next gambit. It came.
"You really won't have to work here more than two more years, as I understand it."
"Wh—why? Oh. How did you know about that?"
"My dear girl," he said, with an insipid modesty. "I naturally know the contents of my own files. Your father was most provident, and very wise. When you are twenty-one, you'll be in for a comfortable bit of money, eh?"
It's none of your business, you old lynx. "Why, I'll hardly notice that, Judge. That's earmarked for Bobby, my brother. It will put him through his last two years and a year of specialization too, if he wants it. And we won't have to lose a wink of sleep over anything from then on. We're just keeping above water until then. But I'll go on working."
"Admirable." He twitched his nostrils at her, and she bit her lip and looked down at her hands again. "Very lovely," he added appreciatively. Again she waited. Move Three took place. He sighed. "Did you know there was a lien on your father's estate, for an old partnership matter?"
"I—had heard that. The old agreements were torn up when the partnership was dissolved in Daddy's trucking business."
"One set of papers were not torn up. I still have them. Your father was a trusting man."
"That account was squared twice over, Judge!" Kay's eyes could, sometimes, take on the slate color of thunderclouds. They did now.
The Judge leaned back and put his fingertips together. "It is a matter which could get to court. To Surrogate, by the way."
He could get her job. Maybe he could get the money and with it, Bobby's career. The alternative... well, she could expect that now.
She was so right.
"Since my dear wife departed—" (She remembered his dear wife. A cruel, empty-headed creature with wit enough to cater to his ego in the days before he became a judge, and nothing else) "—I am a very lonely man, Miss Hallowell. I have never met anyone quite like you. You have beauty, and you could be clever. You can go far. I would like to know you better," he simpered.
Over my dead body. "You would?" she said inanely, stiff with disgust and fear.
He underlined it. "A lovely girl like you, with such a nice job, and with that little nest-egg coming to you—if nothing happens." He leaned forward. "I'm going to call you Kay from now on. I'm sure we understand each other."
"No!" She said it because she did understand, not because she didn't.
He took it his way. "Then I'd be happy to explain further," he chuckled. "Say tonight. Quite late tonight. A man in my position can't—haw!—trip the light fantastic where the lights are bright."
Kay said nothing.
"There's a little place," sniggered the Judge, "called Club Nemo, on Oak Street. Know it?"
"I think I have—noticed it," she said with difficulty.
"One o'clock," he said cheerfully. He stood up and leaned over her. He smelled like soured after-shave. "I do not like to stay up late for nothing. I'm sure you'll be there."
Her thoughts raced. She was furious, and she was frightened, two emotions which she had avoided for years. She wanted to do several things. She wanted primarily to scream, and to get rid of her breakfast then and there. She wanted to tell him some things about himself. She wanted to storm into Mr. Hartford's office and demand to know if this, this, and that were included in her duties as a stenographer.
But then, there was Bobby, so close to a career. She knew what it was to have to quit on the homestretch. And poor, fretting, worried Mr. Hartford; he meant no harm, but he wouldn't know how to handle a thing like this. And one more thing, a thing the Judge apparently did not suspect—her proven ability to land on her feet.
So instead of doing any of the things she wanted to do, she smiled timidly and said, "We'll see..."
"We'll see each other," he amended. "We'll see a great deal of each other." She felt that moist gaze again on the nape of her neck as he moved off, felt it on her armpits.
A light on her switchboard glowed. "Mr. Hartford will see you now, Judge Bluett," she said.
He pinched her cheek. "You can call me Armand," he whispered. "When we're alone, of course."
# 9
HE WAS THERE WHEN she arrived. She was late—only a few minutes, but they cost a great deal. They were minutes added to the hours of fuming hatred, of disgust, and of fear which she had gone through after the Judge's simpering departure from the Hartford offices that morning.
She stood for a moment just inside the club. It was quiet—quiet lights, quiet colors, quiet music from a three-piece orchestra. There were very few customers, and one she knew. She caught a glimpse of silver hair in the corner back of the jutting corner of the bandstand at a shadowed table. She went to it more because she knew he would choose such a spot than because she recognized him.
He stood up and pulled out a chair for her. "I knew you'd come."
How could I get out of it, you toad? "Of course I came," she said. "I'm sorry you had to wait."
"I'm glad you're sorry. I'd have to make you sorry, if you weren't." He laughed when he said it, and only served to stress the pleasure he felt at the thought. He ran the back of his hand over her forearm, leaving a new spoor of gooseflesh. "Kay. Pretty little Kay," he moaned. "I've got to tell you something. I really put some pressure on you this morning."
You don't say! "You did?" she asked.
"You must have realized it. Well, I want you to know right away, right now, that I didn't mean any of that—except about how lonely I am. People don't realize that as well as being a judge, I'm a man."
That makes me one of the people. She smiled at him. This was a rather complicated process. It involved the fact that in this persuasive, self-pitying speech his voice had acquired a whine, and his features the down-drawn character of a spaniel's face. She half-closed her eyes to blur his image, and got such a startling facsimile of a mournful hound's head over his wing collar that she was reminded of an overheard remark: "He's that way through having been annoyed, at an early age, by the constant barking of his mother." Hence the smile. He misunderstood it and the look that went with it and stroked her arm again. Her smile vanished, though she still showed her teeth.
"What I mean is," he crooned, "I just want you to like me for myself. I'm sorry I had to use any pressure. It's just that I didn't want to fail. Anyway, all's fair... you know."
"—in love and war," she said dutifully. And this means war. Love me for myself alone, or else.
"I won't ask much of you," he said out of wet lips. "It's only that a man wants to feel cherished."
She closed her eyes so he could not see them roll heavenward. He wouldn't ask much. Just sneaking and skulking to protect his "position" in the town. Just that face, that voice, those hands... the swine, the blackmailer, the doddering, slimy-fingered old _wolf! Bobby, Bobby,_ she thought in anguish, _be a_ good _doctor..._
There was more of it, much more. A drink arrived. His choice for a sweet young girl. A sherry flip. It was too sweet and the foam on it grabbed unpleasantly at her lipstick. She sipped and let the Judge's sentimental slop wash over her, nodded and smiled, and, as often as she could, tuned out the sound of his voice and listened to the music. It was competent and clean—Hammond Solovox, string bass, and guitar—and for a while it was the only thing in the whole foul world she could hold on to.
Judge Bluett had, it seemed, a little place tucked away over a store in the slums. "The Judge works in the court and his chambers," he intoned, "and has a fine residence on The Hill. But Bluett the Man has a place too, a comfortable spot, a diamond in a rough setting, a place where he can cast aside the black robes, his dignities and his honors, and learn again that he has red blood in his veins."
"It must be very nice," she said.
"One can hide there," he said expansively. "I should say, _two_ can hide there. All the conveniences. A cellar at your elbow, a larder at your beck and call. A civilized wilderness for a loaf of bread, a jug of wine, and—th-h-owoo." He ended with a hoarse whisper, and Kay had the insane feeling that if his eyes protruded another inch, a man could sit on one and saw the other off.
She closed her eyes again and explored her resources. She felt that she had possibly twenty seconds of endurance left. Eighteen. Sixteen. Oh, this is fine. Here goes Bobby's career up in smoke—in a mushroom-shaped cloud at a table for two.
He gathered his feet under him and rose. "You'll excuse me for a moment," he said, not quite clicking his heels. He made a little joke about powder rooms, and obviously being human. He turned away and turned back and pointed out that this was only the first of the little intimacies they would come to learn of each other. He turned away and turned back and said "Think it over. Perhaps we can slip away to our little dreamland this very night!" He turned away and if he had turned back again he would have gotten a French heel in the area of his watchpocket.
Kay sat alone at the table and visibly wilted. Anger and scorn had sustained her; now, for a moment, fear and weariness took their places. Her shoulders sagged and turned forward and her chin went down, and a tear slid out onto her cheek. This was three degrees worse than awful. This was too much to pay for a Mayo Clinic full of doctors. She wanted out. Something had to happen, right now.
Something did. A pair of hands appeared on the tablecloth in front of her.
She looked up and met the eyes of the young man who stood there. He had a broad, unremarkable face. He was nearly as blond as she, though his eyes were dark. He had a good mouth. He said, "A lot of people don't know the difference between a musician and a potted palm when they go to pour their hearts out. You're in a spot, Ma'am."
Some of her anger returned, but it subsided, engulfed in a flood of embarrassment. She could say only, "Please leave me alone."
"I can't. I heard that routine." He tossed his head toward the rest rooms. "There's a way out, if you'll trust me."
"I'll keep the devil I know," she said coldly.
"You listen to me. I mean listen, until I'm finished. Then you can do as you like. When he comes back, stall him off for tonight. Promise to meet him here tomorrow night. Make it a real good act. Then tell him you shouldn't leave here together; you might be seen. He'll think of that anyway."
"And he leaves, and I'm at your tender mercies?"
"Don't be a goon! Sorry. No, you leave first. Go straight to the station and catch the first train out. There's a northbound at three o'clock and a south-bound at three-twelve. Take either one. Go somewhere else, hole up, find yourself another job, and stay out of sight."
"On what? Three dollars mad-money?"
He flipped a long wallet out of his inside jacket pocket. "Here's three hundred. You're smart enough to make out all right on that."
"You're crazy! You don't know me, and I don't know you. Besides, I haven't anything up for sale."
He made an exasperated gesture. "Who said anything about that? I said take a train—any train. No one's going to follow you."
"You _are_ crazy. How could I get it back to you?"
"You worry about that. I work here. Drop by some time—during the day when I'm not here, if you like, and leave it for me."
"What on earth makes you want to do a thing like that?"
His voice was very gentle. "Say it's the same thing makes me bring raw fish to alley-cats. Oh, stop arguing. You need an out and this is it."
"I can't do a thing like that!"
"You got a good imagination? The kind that makes pictures?"
"I—suppose so."
"Then, forgive me, but you need a kick in the teeth. If you don't do what I just told you, that crumb is going to—" and in a half-dozen simple, terse words, he told her exactly what that crumb was going to do. Then, with a single deft motion, he slipped the bills into her handbag and got back on the bandstand.
She sat, sick and shaken, until Bluett returned from the men's room. She had an unusually vivid pictorial imagination.
"While I was gone" he said, settling into his chair and beckoning to the waiter for the check, "know what I was doing?"
That, she thought, is just the kind of question I need right now. Limpidly, she asked, "What?"
"I was thinking about that little place, and how wonderful it would be if I could slip away after a hard day at court, and find you there waiting for me." He smiled fatuously. "And no one would ever know."
Kay sent up a "Lord-forgive-me, I-know-not-what-I-do," and said distinctly, "I think that's a charming idea. Just charming."
"And it wouldn't— _what?"_
For a moment she almost pitied him. Here he had his lines flaked out, his hooks sharpened and greased, and his casting arm worked up to a fine snap, and she'd robbed him of his sport. She'd driven up behind him with a wagon-load of fish. She'd surrendered.
"Well," he said. "Well, I, hm. Hm-m-m! Waiter!"
"But," she said archly, "Not tonight, Ar-mand."
"Now, Kay. Just come up and look at it. It's not far."
She figuratively spit on her hands, took a deep breath and plunged—wondering vaguely at just what instant she had decided to take this fantastic course. She batted her eyelashes only a delicate twice, and said softly, "Ar-mand, I'm not an experienced person like you, and I—" she hesitated and dropped her eyes—"I want it to be perfect. And tonight, it's all so sudden, and I haven't been able to look forward to anything, and it's terribly late and we're both tired, and I have to work tomorrow but I won't the day after, and besides—" and here she capped it. Here she generated, on the spot, the most diffuse and colorful statement of her entire life—"Besides," she said, fluttering her hands prettily, "I'm not _ready."_
She peeped at him from the sides of her eyes and saw his bony face undergo four distinct expressions, one after the other. Again there was that within her which was capable of astonishment; she had been able to think of only three possible reactions to a statement like that. At the same moment the guitarist behind her, in the middle of a fluid _glissando,_ got his little finger trapped underneath his A string.
Before Armand Bluett could get his breath back, she said, "Tomorrow, Ar-mand. But—" She blushed. When she was a child, reading "Ivanhoe" and "The Deerslayer," she used to practice blushing before the mirror. She never could do it. Yet she did it now. "But earlier," she finished.
Her astonishment factor clicked again, this time with the thought, why haven't I ever tried this before?
"Tomorrow night? You'll come?" he said. "You really will?"
"What time, Ar-mand?" she asked submissively.
"Well now. Hmp. Ah—say eleven?"
"Oh, it would be crowded here then. Ten, before the shows are over."
"I knew you were clever," he said admiringly.
She grasped the point firmly and pressed it. "There are always too many people," she said, looking around. "You know, we shouldn't leave together. Just in case."
He shook his head in wonder, and beamed.
"I'll just—" she paused, looking at his eyes, his mouth. "I'll just go, like that." She snapped her fingers. "No goodbyes..."
She skipped to her feet and ran out, clutching her purse. And as she passed the end of the bandstand, the guitarist, speaking in a voice just loud enough to reach her, and barely moving his lips, said,
"Lady, you ought to have your mouth washed out with bourbon."
# 10
HIS HONOR, THE SURROGATE Armand Bluett, left his chambers early the next afternoon. Dressed in a dark brown business suit and seeing alternately from the corners of his eyes, he taxied across town, paid off the driver, and skulked down a narrow street. He strolled past a certain doorway twice to be sure he was not followed, and then dodged inside, key in hand.
Upstairs, he went through the compact two-and-kitchenette with a fine-toothed comb. He opened all the windows and aired the place out. Stuffed between the cushions on the couch he found a rainbow-hued silk scarf redolent with cheap, dying scent. He dropped it in the incinerator with a snort. "Won't need _that_ any more."
He checked the refrigerator, the kitchen shelves, the bathroom cabinet. He ran the water and tested the gas and the lights. He tried the end-table lamps, the torchère, the radio. He ran a small vacuum cleaner over the rugs and the heavy drapes. Finally, grunting with satisfaction, he went into the bathroom and shaved and showered. There followed clouds of talc and a haze of cologne. He pared his toenails, after which he stood before the cheval glass in various abnormal chest-out poses, admiring his reflection through a rose-colored ego.
He dressed carefully in a subdued hound's-tooth check and a tie designed strictly for the contracting pupil, returned to the mirror for a heady fifteen minutes, sat down and painted his nails with colorless polish, and wandered dreamily around fluttering his flabby hands and thinking detailed thoughts, reciting, half-aloud, little lines of witty, sophisticated dialogue. "Who polished your eyes?" he muttered, and "My dear, dear child, that was nothing, really nothing. A study in harmony, before the complex instrumentation of the flesh... no, she's not old enough for that one. Hm. You're the cream in my coffee. No! _I'm_ not old enough for that."
So he passed the evening, very pleasantly indeed. At 8:30 he left, to dine sumptuously at a seafood restaurant. At 9:50 he was ensconced at the corner table at Club Nemo, buffing his glittering nails on his lapel and alternately wetting his lips and dabbing at them with a napkin.
At ten o'clock she arrived.
Last night he had risen to his feet as she crossed the dance floor. Tonight he was up out of his chair and at her side before she reached it.
This was Kay transformed. This was the concretion of his wildest dreams of her.
Her hair was turned back from her face in soft small billows which framed her face. Her eyes were skillfully shadowed, and seemed to have taken on a violet tinge with their blue. She wore a long cloak of some heavy material, and under it, a demure but skin-tight jacket of black ciré satin and a black hem-slashed skirt.
"Armand..." she whispered, holding out both hands.
He took them. His lips opened and closed twice before he could say anything at all, and then she was past him, walking with a long, easy stride to the table. Walking behind her, he saw her pause as the orchestra started up, and throw a glance of disdain at the guitarist. At the table she unclasped the cloak at her throat and let it fall away confidently. Armand Bluett was there to receive it as she slid into her chair. He stood there goggling at her for so long that she laughed at him. "Aren't you going to say anything at all?"
"I'm speechless," he said, and thought, my word, that came out effectively.
A waiter came, and he ordered for her. Daiquiri, this time. No woman he had ever seen reminded him less of a sherry flip.
"I am a very lucky man," he said. That was twice in a row he had said something unrehearsed.
"Not as lucky as I am," she said, and she seemed quite sincere as she said it. She put out just the tip of a pink tongue; her eyes sparkled, and she laughed. For Bluett, the room began to gyrate. He looked down at her hands, toying with the clasp of a tiny cosmetic case.
"I don't think I ever noticed your hands before," he said.
"Please do," she twinkled. "I love the things you say, Ar-mand," and she put her hands in his. They were long, strong hands with square palms and tapered fingers and what certainly must be the smoothest skin in the world.
The drinks came. He let go reluctantly and they both leaned back, looking at each other. She said, "Glad we waited?"
"Oh, yes. Hm. Yes indeed." Suddenly, waiting was intolerable. Almost inadvertently he snatched up his drink and drained it.
The guitarist fluffed a note. She looked pained. Armand said, "It's not too nice here tonight, is it?"
Her eyes glistened. "You know a better place?" she asked softly.
His heart rose up and thumped the lower side of his Adam's apple. "I certainly do," he said when he could.
She inclined her head with an extraordinary, controlled acquiescence that was almost like a deep pain to him. He threw a bill on the table, put her cloak over her shoulders, and led her out.
In the cab he lunged for her almost before the machine was away from the curb. She hardly seemed to move at all, but her body twisted away from him inside the cloak; he found himself with a double handful of cloth while Kay's profile smiled slightly, shaking its head. It was unspoken, but it was a flat "no." It was also a credit to the low frictional index of ciré satin.
"I never knew you were like this," he said.
"Like what?"
"You weren't this way last night," he floundered.
"What way, Ar-mand?" she teased.
"You weren't so—I mean, you didn't seem to be sure of yourself at all."
She looked at him. "I wasn't—ready."
"Oh, I see," he lied.
Conversation lapsed after that, until he paid off the cab at the street intersection near his hideout. He was beginning to feel that the situation was out of his control. If she controlled it, however, as she had so far, he was more than willing to go along.
Walking down the dirty, narrow street, he said, "Don't look at any of this, Kay. It's quite different upstairs."
"It's all the same, when I'm with you," she said, stepping over some garbage. He was very pleased.
They climbed the stairs, and he flung open the door with a wide gesture. "Enter, fair lady, the land of the lotus-eaters."
She pirouetted in and cooed over the drapes, the lamps, the pictures. He closed the door and shot the bolt, dropped his hat on the couch and stalked toward her. He was about to put his arms around her from behind when she darted away. "What a way to begin!" she sang. "Putting your hat there. Don't you know it's bad luck to put a hat on a bed?"
"This is my lucky day," he pronounced.
"Mine too," she said. "So let's not spoil it. Let's pretend we've been here forever, and we'll be here forever."
He smiled. "I like that."
"I'm glad. That way," she said, stepping away from a corner as he approached, "there's no hurry. Could we have a drink?"
"You may have the moon," he chanted. He opened the kitchenette. "What would you like?"
"Oh, how wonderful. Let me, let me. You go into the other room and sit down, Mister Man. This is woman's work." She shunted him out, and began to mix, busily.
Armand lounged back on the couch with his feet on the rock-maple coffee table, and listened to the pleasant clinking and swizzling noises from the other room. He wondered idly if he could get her to bring his slippers every evening.
She glided in, balancing two tall highballs on a small tray. She kept one hand behind her back as she knelt and put the tray down on the coffee table and slipped into an easy-chair.
"What are you hiding?" he asked.
"It's a secret."
"Come over here."
"Let's talk a little while first. Please."
"A little while." He sniggered. "It's your fault, Kay. You're so beautiful. Hm. You make me feel mad—impetuous." He began rubbing his hands together. She closed her eyes. "Armand..."
"Yes, my little one," he answered, patronizingly.
"Did you ever hurt anyone?"
He sat up. "I? Kay, are you afraid?" He puffed his chest out a bit. "Afraid of me? Why, I won't hurt you, baby."
"I'm not talking about me," she said, a little impatiently. "I just asked you—did you ever hurt anyone?"
"Why, of course not. Not intentionally, that is. You must remember—my business is justice."
"Justice." She said it as if it tasted good. "There are two ways of hurting people, Armand—outside, where it shows, and inside, in the mind, where it scars and festers."
"I don't follow you," he said, his pomposity returning as his confusion grew. "Whom have I ever hurt?"
"Kay Hallowell, for one," she said detachedly, "with the kind of pressure you've been putting on. Not because she's a minor; you are only a criminal on paper for that, and even that wouldn't apply in some states."
"Now, look here, young lady—"
"—but because," she went on calmly, "you have been systematically wrecking what faith she has in humanity. If there is a basic justice, then for that you are a criminal by its standards."
"Kay—what's come over you? What are you talking about? I won't have any more of this!" He leaned back and folded his arms. She sat quietly.
"I know," he said, half to himself, "you're joking. Is that it, baby?"
In the same level, detached tone, she went on speaking. "You are guilty of hurting others in both the ways I mentioned. Physically, where it shows, and psychically. You will be punished in both those ways, _Justice_ Bluett."
He blew air from his nostrils. "That is quite enough. I did not bring you here for anything like this. Perhaps I shall have to remind you, after all, that I am not a man to be trifled with. Hm. The matter of your estate—"
"I am not trifling, Armand." She leaned across the low table to him. He put up his hands. "What do you want?" he breathed, before he could stop himself.
"Your handkerchief."
"My h _-what?"_
She plucked it out of his breast pocket. "Thank you." As she spoke she shook it out, brought up two corners and knotted them together. She slipped her left hand through the loop and settled the handkerchief high on her forearm. "I am going to punish you first in the way it doesn't show," she said informatively, "by reminding you, in a way you can't forget, of how you once hurt someone else."
"What kind of nonsense—"
She reached behind her with her right hand and brought out what she had been hiding—a new, sharp, heavy cleaver.
Armand Bluett cowered away, back into the couch cushions. "Kay—no! No!" he panted. His face turned green. "I haven't touched you, Kay! I only wanted to talk. I wanted to help you and—and your brother. Put that thing down, Kay!" He was drooling with terror. "Can't we be friends, Kay?" he whimpered.
"Stop it!" she hissed. She lifted the cleaver high, resting her left hand on the table and leaning toward him. Her face made, line upon plane upon carven curve, a mask of utter contempt. "I told you that your physical punishment comes later. Think about this while you wait for it."
The cleaver arced over and came down, with every ounce of a lithe body behind it. Armand Bluett screamed—a ridiculous, hoarse, thin sound. He closed his eyes. The cleaver crashed into the heavy top of the coffee-table. Armand twisted and scrabbled back into the cushions, crabbed sidewise and backward along the wall until he could go no farther. He stopped ludicrously, on all fours, on the couch, backed into the corner, sweat and spittle running off his chin. He opened his eyes.
It had apparently taken him only a split second to make the hysterical move, for she still stood over the table; she still held the handle of the cleaver. Its edge had buried itself in the thick wood, after passing through the flesh and bone of her hand.
She snatched up the bronze letter-opener and thrust it under the handkerchief on her forearm. As she straightened, bright arterial blood spouted from the stumps of three severed fingers. Her face was pale under the cosmetics, but not one whit changed otherwise; it still wore its proud, unadulterated contempt. She stood straight and tall, twisting the handkerchief with the handle of the letter-opener, making a tourniquet, and she stared him down. As his eyes fell, she spat, "Isn't this better than what you planned? Now you've got a part of me to keep for your very own. That's much better than using something and giving it back."
The spurting blood had slowed to a dribble as she twisted. Now she went to the chair on which she had left her cosmetic case. Out of it she worried a rubber glove. Holding the tourniquet against her side, she pulled the glove over her hand and snugged it around the wrist.
Armand Bluett began to vomit.
She shouldered into her cloak and went to the door.
When she had drawn back the bolt and opened it, she called back in a seductive voice, "It's been so wonderful, Ar-mand darling. Let's do it again soon..."
It took Armand's mind nearly an hour to claw its way up out of the pit of panic into which it had fallen. During the hour he hunkered there on the couch in his own filth, staring at the cleaver and the three still white fingers.
Three fingers.
Three _left_ fingers.
Somewhere, deep in his mind, that meant something to him. At the moment he refused to let it surface. He feared it would. He knew it would. He knew that when it did, he would know consuming terror.
# 11
BOBBY DEAR, SHE WROTE, I can't bear to think of you getting letters back with "address unknown" on them. I'm all right. That's first and foremost. I'm all right, monkey-face, and you're not to worry. Your big sister is _all right._
I'm also all mixed up. Maybe in that nice orderly hospital this will make more sense to you. I'll try to make it short and simple.
I was working one morning at the office when that awful Judge Bluett came in. He had to wait for a few minutes before he could see old Wattles Hartford, and he used it to make his usual wet soggy string of verbal passes. My brush worked fine until the seamy old weasel got on the subject of Daddy's money. You know that we'll get it when I'm 21—unless that old partnership deal comes up again. It would have to go to court. Bluett not only was the partner—he's the Surrogate. Even if we could get him disqualified from hearing the case, you know how he could fix anyone else who might take the bench. Well, the idea was that if I would be nice and sweet to Hizzoner, in any nasty way he wanted, the will wouldn't be contested. I was terribly frightened, Bobby; you know the rest of your training has to come out of that money. I didn't know what to do. I needed time to think. I promised to meet him that night, real late, in a nightclub.
Bobby, it was awful. I was just at the point of blowing up, there at the table, when the old drooler left the room for a minute. I didn't know whether to fight or run away. I was scared, believe me. All of a sudden there was somebody standing there talking to me. I think he must be my guardian angel. Seems he had overheard the Judge talking to me. He wanted me to cut and run. I was afraid of him, too, at first, and then I saw his face. Oh, Bobby, it was such a _nice_ face! He wanted to give me some money, and before I could say no he told me I could return it whenever I wanted to. He told me to get out of town right now—take a train, any train; he didn't even want to know which one. And before I could stop him he shoved $300 into my bag and walked off. The last thing he said was to accept a date for the next evening with the Judge. I couldn't do a thing—he'd only been there two minutes and he was talking practically every second of it. And then the Judge came back. I flapped my eyelids at the old fool like a lost woman, and cut out. I got a train to Eltonville twenty minutes later and didn't even register in a hotel when I got here. I waited around until the stores opened and bought an overnight case and a tooth brush and got myself a room. I slept a few hours and the very same afternoon I had a job in the only record shop in the place. It's $26 a week but I can make it fine.
Meanwhile I don't know what's happening back home. I'm sort of holding my breath until I hear something. I'm going to wait, though. We have time, and in the meantime, I'm all right. I'm not going to give you my address, honey, though I'll write often. Judge Bluett just might be able to get his hands on mail, some way. I think it pays to be careful. He's dangerous.
So, honey, that's the situation as far as it's gone. What next? I'll watch the home town papers for any item about His Dishonor the Surrogate, and hope for the best. As for you, don't worry your little square head about me, darling. I'm doing fine. I'm only making a few dollars a week less than I was at home and I'm a lot safer here. And the work isn't hard; some of the nicest people like music. I'm sorry I can't give you my exact address, but I do think it's better not to just now. We can let this thing ride for a year if we have to, and small loss. Work hard, baby; I'm behind you a thousand percent. I'll write often.
XXX
Your loving
Big Sis Kay.
(This is the letter that Armand Bluett's hired second-story man found in Undergraduate Robert Hallowell's room at the State Medical School.)
# 12
"YES—I AM PIERRE Monetre. Come in." He stood aside and the girl entered.
"This is good of you, Mr. Monetre. I know you must be terribly busy. And probably you won't be able to help me at all."
"I might not if I were able," he said. "Sit down."
She took a molded plywood chair which stood at the end of the half desk, half lab bench which took up almost an end wall of the trailer. He looked at her coldly. Soft yellow hair, eyes sometimes slate-blue, sometimes a shade darker than sky-blue; a studied coolness through which he, with his schooled perceptions, could readily see. She is disturbed, he thought; frightened and ashamed of it. He waited.
She said, "There's something I've got to find out. It happened years ago. I'd almost forgotten about it, and then saw your posters, and I remembered... I could be wrong, but if only—" She kneaded her hands together. Monetre watched them, and then returned his cold stare to her face.
"I'm sorry, Mr. Monetre. I can't seem to get to the point. It's all so vague and so—terribly important. The thing is, when I was a little girl, seven or eight years old, there was a boy in my class in school who ran away. He was about my age, and had some sort of horrible run-in with his stepfather. I think he was hurt. His hand. I don't know how badly. I was probably the last one in town to see him. No one ever saw him again."
Monetre picked up some papers, shifted them, put them down again. "I really don't know what I can do about that, Miss—"
"Hallowell. Kay Hallowell. Please hear me out, Mr. Monetre. I've come thirty miles just to see you, because I can't afford to pass up the slightest chance—"
"If you cry, you'll have to get out," he rasped. His voice was so rough that she started. Then he said, with gentleness, "Please go on."
"Th-thank you. I'll be quick... it was just after dark, a rainy, misty night. We lived by the highway, and I went out back for something... I forget... anyway, he was there, by the traffic light. I spoke to him. He asked me not to tell anyone that I had seen him, and I never have, till now. Then—" she closed her eyes, obviously trying to bring back every detail of the memory—"—I think someone called me. I turned to the gate and left him. But I peeped out again, and saw him climbing on the back of a truck that was stopped for the light. It was one of your trucks. I'm sure it was. The way it was painted... and yesterday, when I saw your posters, I thought of it."
Monetre waited, his deep-set eyes expressionless. He seemed to realize, suddenly, that she had finished. "That happened twelve years ago? And, I suppose, you want to know if that boy reached the carnival."
"Yes."
"He did not. If he had, I should certainly have known of it."
"Oh..." It was a faint sound, stricken, yet resigned; apparently she had not expected anything else. She pulled herself together visibly, and said, "He was small for his age. He had very dark hair and eyes and a pointed face. His name was Horty—Horton."
"Horty..." Monetre searched his memory. There was a familiar ring to those two syllables, somehow. Now, where... He shook his head. "I don't remember any boy called Horty."
"Please try. _Please!_ You see—" She looked at him searchingly, her eyes asking a question. He answered it, saying, "You can trust me."
She smiled. "Thank you. Well, there's a man, a horrible person. He was once responsible for that boy. He's doing a terrible thing to me; it's something to do with an old law case, and he might be able to keep me from getting some money that is due me when I come of age. I need it. Not for myself; it's for my brother. He's going to be a doctor, and—"
"I don't like doctors," said Monetre. If there is a great bell for hatred as there is one for freedom, it rang in his voice as he said that. He stood up. "I know nothing about any boy named Horty, who disappeared twelve years ago. I am not interested in finding him in any case, particularly if doing so would help a man make a parasite of himself and fools of his patients. I am not a kidnapper, and will have nothing to do with a search which reeks of that and blackmail to boot. Good-by."
She had risen with him. Her eyes were round. "I—I'm sorry. Really, I—"
"Good-by." It was the velvet this time, used with care, used to show her that his gentleness was a virtuosity, an overlay. She turned to the door, opened it. She stopped and looked back over her shoulder. "May I leave you my address, just in case, some day, you—"
"You may not," he said. He turned his back on her and sat down. He heard the door close.
He closed his eyes, and his arched, slit nostrils expanded until they were round holes. Humans, humans, and their complex, useless, unimportant machinations. There was no mystery about humans; no puzzle. Everything human could be brought to light by asking simply, "What does it gain you?"... What could humans know of a life-form to which the idea of gain was alien? What could a human say of his crystal-kin, the living jewels which could communicate with each other and did not dare to, which could co-operate with each other and scorned to?
And what—he let himself smile—what would humans do when they had to fight the alien? When they were up against an enemy which would make an advance and then scorn to consolidate it—and then make a different _kind_ of advance, in a different way, in another place?
He sank into an esoteric reverie, marshaling his crystallines against teeming, stupid mankind; losing, in his thoughts, the pointless perturbations of a girl in a search for a child long missing, for some petty gainful reason of her own.
"Hey—Maneater."
_"Damn_ it! What now?"
The door opened diffidently. "Maneater, there's—"
"Come in, Havana, and speak up. I don't like mumblers."
Havana edged in, after setting his cigar down on the step. "There's a man outside wants to see you."
Monetre glowered over his shoulder. "Your hair's getting gray. What's left of it. Dye it."
"Okay, okay. Right away, this afternoon. I'm sorry." He shifted his feet miserably. "About this man—"
"I've had my quota for today," said Monetre. "Useless people wanting impossible things of no importance. Did you see that girl go out of here?"
"Yes. That's what I'm trying to tell you. So did this guy. See, he was waiting to see you. He asked Johnward where he could find you, and—"
"I think I'll fire Johnward. He's an advance man, not an usher. What business has he, bringing people to annoy me?"
"I guess he thought you ought to see this one. A big-shot," said Havana timidly. "So when he got your trailer, he asked me were you busy. I told him yes, you were talking to someone. He said he'd wait. About then the door opens, and that girl comes out. She puts a hand on the side and turns back to say something to you, and this guy, this big-shot, he blows a fuse. No kidding, Maneater, I never seen anything like it. He grabs my shoulder. I'll have a bruise there for a week. He says, 'It's her! It's her!' and I says 'Who?' and he says, 'She mustn't see me! She's a devil! She cut those fingers off, and they've grown back again!'"
Monetre sat bolt upright and turned in his swivel chair to face the midget. "Go on, Havana," he said in his gentle voice.
"Well, that's all. 'Cept he ducked back behind Gogol's bally-platform and hunkered down out of sight, and peeped out at that girl as she walked past him. She never saw him."
"Where is he now?"
Havana glanced through the door. "Still right there. Looks pretty bad. I think he's having some kind of a fit."
Monetre left his chair and shot through the door, leaving it completely up to Havana whether he got out of the way or not. The midget leaped to the side, out of Monetre's direct path, but not far enough to avoid the bony edge of Monetre's pelvis, which glanced stunningly off Havana's pudgy cheekbone.
Monetre bounded to the side of the man who cowered down behind the bally platform. He knelt and placed a sure hand on the man's forehead, which was clammy and cold.
"It's all right now, sir," he said in a deep, soothing voice. "You'll be perfectly safe with me." He urged the idea "safe," because, whatever the cause might be, the man was sodden, trembling, all but ecstatic with fear. Monetre asked no questions, but kept crooning, "You're in good hands now, sir. Quite safe. Nothing can happen now. Come along; we'll have a drink. You'll be all right."
The man's watery eyes fixed themselves on him, slowly. Awareness crept into them, and a certain embarrassment. He said, "Hm. Uh—slight attack of—hm... vertigo, you know. Sorry to be... hm."
Monetre courteously helped him up, picked up a brown homburg and dusted it off. "My office is just there. Do come in and sit down."
Monetre kept a firm hand on the man's elbow, led him to the trailer, handed him up the two steps, reached past him and opened the door. "Would you like to lie down for a few minutes?"
"No, no. Thank you; you're very kind."
"Sit here, then. I think you'll find it comfortable. I'll get you something that will make you feel better." He fingered a simple combination latch, chose a bottle of tawny port. From a desk drawer he took a small phial and put two drops of liquid into a glass, filling it with the wine. "Drink this. It will make you feel better. A little sodium amytal—just enough to quiet your nerves."
"Thank you, thank—" He drank it greedily, "—you. Are you Mr. Monetre?"
"At your service."
"I am Judge Bluett. Surrogate, you know. Hm."
"I am honored."
"Not at all, not at all. I am the one who... I drove fifty miles to see you, sir, and would gladly have done twice that. You have a wide reputation."
"I hadn't realized it," said Monetre, and thought, this deflated creature is as insincere as I am. "What can I do for you?"
"Hm. Well, now. Matter of—ah—scientific interest. I read about you in a magazine, you know. Said you know more about fr—ah, strange people, and things like that, than anyone alive."
"I wouldn't say that," said Monetre. "I have worked with them for a great many years, of course. What was it you wanted to know?"
"Oh... the kind of thing you can't get out of reference books. Or ask any so-called scientist, for that matter; they just laugh at things that aren't in some book, somewhere."
"I have experienced that, Judge. I do not laugh readily."
"Splendid. Then I shall ask you. Namely, do you know anything about—ah—regeneration?"
Monetre cloaked his eyes. Would the fool ever get to the point? "What kind of regeneration? The girdle of the nematodes? Cellular healing? Or are you talking about old-time radio receivers?"
"Please," said the judge, and made a flabby gesture. "I'm quite the layman, Mr. Monetre. You'll have to use simple language. What I want to know is—how much of a restoration is possible after a serious cut?"
"How serious a cut?"
"Hm. Call it an amputation."
"Well, now. That depends, Judge. A fingertip, possibly. A chipped bone can grow surprisingly. You—you know of a case where a regeneration has been, shall we say, a bit more than normal?"
There was a long pause. Monetre noticed that the Judge was paling. He poured him more port, and filled a glass for himself. Excitement mounted within him.
"I do know of such a case. At least, I mean... hm. Well, it seemed so to me. That is, I saw the amputation."
"An arm? A leg, perhaps, or a foot?"
"Three fingers. Three whole fingers," said the Judge. "It would seem that they grew back. And in forty-eight hours. A well-known osteologist treated the whole thing as a great joke when I asked him about it. Refused to believe I was serious." Suddenly he leaned forward so abruptly that the loose skin of his jaw quivered. "Who was the girl who just left here?"
"An autograph hound," said Monetre in a bored tone. "A person of no importance. Do proceed."
The Judge swallowed with difficulty. "Her name is—Kay Hallowell."
"Perhaps so, perhaps so. Have you changed the subject?" asked Monetre impatiently.
"I have not, sir," the Judge answered hotly. "That girl, that monster—in good light, and right before my eyes, _chopped off three fingers of her left hand!"_ He nodded, pushing his lower lip out, and sat back.
If he expected a sharp reaction, he was not disappointed. Monetre leaped to his feet and bellowed, "Havana!" He strode to the door and yelled again. "Where is that little fat—oh; there you are, Havana. Go and find that girl who just left here. Understand? Find her and bring her back. I don't care what you tell her; find her and bring her back here." He clapped his hands explosively. _"Run!"_
He returned to his chair, his face working. He looked at his hands, then at the judge. "You're quite sure of this."
"I am."
"Which hand?"
"The left." The Judge ran a finger around his collar. "Ah—Mr. Monetre. If that boy should bring her back here, why, ah—I, that is—"
"I gather you are afraid of her."
"Now, ah—I wouldn't say that," said the Judge. "Startled, yes. Hm. Wouldn't you be?"
"No," said Monetre. "You are lying, sir."
"I? Lying?" Bluett puffed up his chest and glowered at the carny boss.
Monetre half-closed his eyes and began ticking off items on his fingers. "It would seem that what frightened you a few minutes ago was the sight of that girl's left hand. You told the midget that the fingers had grown back. It was obviously the first time you had seen the hand regenerated. And yet you tell me that you have already consulted an osteologist about it."
"There are no lies involved," said Bluett stiffly. "True, I did see the restored hand when she stood in this doorway, and it was the first time. But I also saw her cut those fingers off!"
"Then why," asked Monetre, "come to me to ask questions about regeneration?" Watching the Judge flounder about for an answer, he added, "Come now, Judge Bluett. Either you have not stated your original purpose in coming here, or—you have seen a case of this regeneration before. Ah. I see that's it." His eyes began to burn. "I think you'd better tell me the whole story."
"That _isn't_ it!" the Judge protested. "Really, sir, I am not enjoying this cross-questioning. I fail to see—"
Shrewdly, Monetre reached out to touch the fear which hovered so close to this wet-eyed man. "You are in greater danger than you suspect," he interrupted. "I know what that danger is, and I am probably the only man in the world who can help you. You will co-operate with me, sir, or you will leave this instant—and take the consequences." He said this with his flexible voice toned down to a soft, resonating diapason, which apparently frightened the Judge half out of his wits. The chain of imaginary horrors which mirrored themselves on Bluett's paling face must have been colorful, to say the least. Smiling slightly, Monetre leaned back in his chair and waited.
"M-may I..." The Judge poured himself more wine. "Ah. Now, sir. I must tell you at the outset that this whole matter has been one of—ah—conjecture on my part. That is, up until I saw the girl just now. By the way—I do not want to have her see me. Could you—"
"When Havana brings her back, I'll get you out of sight. Go on."
"Good. Thank you, sir. Well, some years ago I brought a child into my house. Ugly little monster. When he was seven or eight years old, he ran away from home. I have not heard of him since. I imagine he would be nineteen or so by this time—if he's alive. And—and there seems to be some connection between him and this girl."
"What connection?" Monetre prompted.
"Well, sh-she seemed to know something about him." As Monetre shifted his feet impatiently, he blurted, "Fact is, there was a little trouble. The boy was downright rebellious. I thrashed him and shoved him into a closet. His hand—quite accidentally, you realize—his hand was crushed in the hinge of the door. Hm. Yes—very unpleasant."
"Go on."
"I've been—ah—looking, you know—that is, if that boy has grown up, he might be resentful, you understand... besides, he was a most unbalanced child, and one never knows how these things might affect a weak mind—"
"You mean you feel guilty as hell and scared to boot, and you've been watching for a young man with some fingers missing. Fingers—get to the point! What has this to do with the girl?" Monetre's voice was a whip.
"I can't—say exactly," mumbled the Judge. "She seemed to know something about the boy. I mean, she hinted something about him—said that she was going to remind me of a way I had—hurt someone once. And then she took a cleaver and cut off her fingers. She disappeared. I had a man locate her. He found out she was due here—my man sent for me. That's all."
Monetre closed his eyes and thought hard. "There was nothing wrong with her fingers when she was in here."
"Damn it, I know that! But I tell you, I saw, with my own eyes—"
"All right, all right. She cut them off. Now, exactly why did you come here?"
"I—that's all. When something like that happens it makes you forget everything you know and start right from scratch. What I saw was impossible, and I began thinking in a way that let anything be possible... anyth—"
"Come to the point!" roared the Maneater.
"There is none!" Bluett roared back. They glared at each other for a crackling moment. "That's what I'm trying to tell you; I don't know. I remembered that child and his crushed fingers, and there was this girl and what she did. I began wondering if she and the boy were the same... I told you 'impossible' didn't matter any more. Well, the girl had a perfectly good hand before she chopped into it. If, somehow, she was that boy, he must have grown the fingers back. If he could do it once, he could do it again. If he knew he could do it again, he wouldn't be afraid to cut them off." The judge threw up his hands and shrugged, and let his arms fall limply. "So I began to wonder what manner of creature could grow fingers at will. That's all."
Monetre made wide eaves of his lids, his burning dark eyes studying the Judge. "This—boy who might be a girl," he murmured. "What was his name?"
"Horton. Horty, we called him. Vicious little scut."
"Think, now. Was there anything strange about him as a child?"
"I should say so! I don't think he was sane. Clinging to baby-toys—that sort of thing. And he had filthy habits."
"What filthy habits?"
"He was expelled from school for eating insects."
"Ah! Ants?"
"How did you know?"
Monetre rose, paced to the door and back. Excitement began to thump in his chest. "What baby-toys did he cling to?"
"Oh, I don't remember. It isn't important."
"I'll decide that," snapped Monetre. "Think, man! If you value your life—"
"I can't think! I can't!" Bluett looked up at the Maneater, and quailed before those blazing eyes. "It was some sort of a jack-in-the-box. A hideous thing."
"What did it look like? Speak up, damn it!"
"What does it—oh, all right. It was this big, and it had a head on it like a Punch—you know, Punch and Judy. Big nose and chin. The boy hardly ever looked at it. But he had to have it near him. I threw it away one time and the doctor made me find it and bring it back. Horton almost died."
"He did, eh?" grunted Monetre tautly, triumphantly. "Now tell me—that toy had been with him since he was born, hadn't it? And there was something about it—some sort of jeweled button, or something glittery?"
"How did you know—" Bluett began again, and again quailed under the radiation of furious, excited impatience from the carny boss. "Yes. The eyes."
Monetre flung himself on the Judge. He grasped his shoulders, shook him. "You said 'eye,' didn't you? There was only one jewel?" he panted.
"Don't—don't—" wheezed Bluett, pushing weakly at Monetre's taloned hands. "I said 'eyes.' Two eyes. They were both the same. Nasty looking things. Seemed to have a light of their own."
Monetre straightened slowly, backed off. "Two of them," he breathed. _"Two..."_
He closed his eyes, his brain humming. Disappearing boy, fingers... fingers crushed. Girl... the right age, too... Horton. Horton... Horty. His mind looped and wheeled back over the years. A small brown face, peaked with pain, saying, "My folks called me Hortense, but everyone calls me Kiddo." Kiddo, who had arrived with a crushed hand, and had left the carnival two years ago. What had happened when she left? He had wanted something, wanted to examine her hand, and she left during the night.
That hand. When she first arrived, he had cleaned it up, trimmed away the ruined flesh, sewed it up. He had treated it every day for weeks, until the scartissue was fused over, and there was no further danger of infection; and then, somehow or other he had never looked at it again. Why not? Oh—Zena. Zena had always told him how Kiddo's hand was getting along.
He opened his eyes—slits, now. "I'll find him," he snarled.
There was a knock at the door, and a voice. "Maneater—"
"It's the midget," babbled Bluett, leaping up. "With the girl. What shall I—where shall—"
Monetre sent him a look which wilted him, tumbled him back in his chair. The carny boss rose and stilted to the door, opening it a crack. "Get her?"
"Gosh, Maneater, I—"
"I don't want to hear it," said Monetre in a terrible whisper. "You didn't bring her back. I sent you to get her and you didn't do it." He closed the door with great care and turned to the Judge. "Go away."
"Eh? Hm. But what about the—"
"Go away!" It was a scream. As his glare had made Bluett limp, his voice stiffened him. The Judge was on his feet and moving doorward before the scream had ceased to be a sound. He tried to speak, and succeeded only in moving his wet mouth.
"I'm the only one in the world who can help you," said Monetre; and the Judge's face showed that this easy, quiet, conversational tone was the most shocking thing of all. He went to the door and paused. Monetre said, "I will do what I can, Judge. You'll hear from me very soon, you may be sure of that."
"Ah," said the Judge. "Mm. Anything I can do, Mr. Monetre. Call on me. Anything at all."
"Thank you. I shall certainly need your help." Monetre's bony features froze the instant he stopped speaking. Bluett fled.
Pierre Monetre stood staring at the space where the Judge's bloated face had just been. Suddenly he balled his fist and smashed it into his palm. "Zena!" said only his lips. He went pale with fury, weak with it, and went to his desk. He sat down, put his elbows on the blotter and his chin in his hand, and began to send out waves of feral hatred and demand.
_Zena!_
_Zena!_
_Here! Come Here!_
# 13
HORTY LAUGHED. HE LOOKED at his left hand, at the three stubs of fingers which rose, like unspread mushrooms, from his knuckles, touched the scar-tissue around them with his other hand, and he laughed.
He rose from the studio couch and crossed the wide room to the cheval glass, to stare at his face, to stand back and look critically at his shoulders, his profile. He grunted in satisfaction and went to the telephone in the bedroom.
"Three four four," he said. His voice was resonant, well suited to the cast of his solid chin and his wide mouth. "Nick? This is Sam Horton. Oh, fine. Sure, I'll be able to play again. The doc says I was lucky. A broken wrist usually heals pretty stiff, but this one won't. No—don't worry. Hm? About six weeks. Positively... Gold? Thanks Nick, but I'll get along. No, don't worry—I'll yell if I need any. Thanks, though. Yeah, I'll drop by every once in a while. I was in there a couple days ago. Where did you find that three-chord bubblehead you have on guitar? He does by accident what Spike Jones does on purpose. No, I didn't want to hit him. I wanted to husk him." He laughed. "I'm kidding. He's okay. Well, thanks, Nick. 'Bye."
Going to the studio couch, he flung himself down with the confident relaxation of a well-fed feline. He pressed his shoulders luxuriously into the foam mattress, rolled and reached for one of the four books on the end table.
They were the only books in the apartment. Long ago he had learned of the physical encroachment of books, and the difficulties of overflowing book-cases. His solution was to get rid of them all, and make an arrangement with his dealer to send him four books a day—new books, on a rental basis. He read them all, and always returned them on the next day. It was a satisfactory solution, for him. He had total recall. What use, then, were book-cases?
He owned two pictures—a Markell, meticulously unmatched irregular shapes, varying in their apparent transparency, superimposed one on the other so that the tone of each affected the others, and so that the color of the background affected everything. The other was a Mondrian, precise and balanced, and conveying an almost-impression of something which could never quite be anything.
He owned, however, miles of magnetic tape on which was recorded a magnificent collection of music. Horty's fabulous mind could retain the whole mood of a book, and recall any part of it. It could do the same with music; but to recall music is to generate it to a certain degree, and there is a decided difference in the coloration of a mind which hears music and one which makes it. Horty could do both, and his music library made it possible for him to do either.
He had the classics and the romantics which had been Zena's favorites, the symphonies, concerti, ballads and virtuosic showpieces which had been his introduction to music. But his tastes had widened and deepened, and now included Honnegger and Copland, Shostakovitch and Walton. In the popular field he had discovered Tatum's somber chordings and the incredible Thelonius Monk. He had the occasionally inspired trumpet of Dizzy Gillespie, the bewildering cadenzas of Ella Fitzgerald, the faultless production of Pearl Bailey's voice. His criterion in all of it was humanity and the extensions of humanity. He lived with books that led to books, art that led him to conjecture, music that led him to worlds beyond worlds of experience.
Yet for all these riches, Horty's rooms were simply furnished. The only unconventional article of furniture was the tape recorder and reproducer—a massive incorporation of high-fidelity components which Horty had been led to assemble because of an ear that demanded every nuance, every overtone, of every instrumental voice. Otherwise his rooms were like anyone's comfortably appointed, tastefully decorated apartment. It occurred to him, fleetingly and at long intervals, that with his resources he could surround himself with automatic luxury-machines like back-kneading chairs and air-conditioned drying chambers for after his shower. But he was never moved in such directions. His mind was simply and steadily acquisitive. His analytical abilities were phenomenal, but he was seldom moved to use them extensively. Therefore to acquire knowledge was sufficient; its use could wait for demand, and there was little demand coexistent with his utter and demonstrable confidence in his own powers.
Halfway through his book he stopped, a puzzled expression in his eyes. It was as if a special sound had reached him—yet none had.
He closed the book and racked it, rose to stand listening, turning his head slightly as if he were trying to fix the source of the sensation.
The doorbell rang.
Horty stopped moving. It was not a freeze, the startled immobilization of a frightened animal. It was more a controlled, relaxed split second for thought. Then he moved again, balanced and easily.
At the door he paused, staring at the lower panel. His face tightened, and a swift frown rippled on his brow. He flung the door open.
She stood crookedly in the hallway, looking up at him with her eyes. Her head was turned sidewise and a little downward. She had to strain her eyes painfully to meet his; she was only four feet tall.
She said, faintly, "Horty?"
He made a hoarse sound and knelt, pulling her into his arms, holding her with power and gentleness. "Zee... Zee, what happened? Your face, your—" He picked her up and kicked the door shut and carried her over to the studio couch, to sit with her across his knees, cradled in his arms, her head resting in the warm strong hollow of his right hand. She smiled at him. Only one side of her mouth moved. Then she began to cry, and Horty's own tears curtained from him the sight of her ravaged face.
Her sobs stopped soon, as if she were simply too tired to continue. She looked at his face, all of it, part by part. She brought her hand up and touched his hair. "Horty..." she whispered. "I loved you so much the way you were..."
"I haven't changed," he said. "I'm a big grown-up man now. I have an apartment and a job. I have this voice and these shoulders and I weigh a hundred pounds more than I did three years ago." He bent and kissed her quickly. "But I haven't changed, Zee. I haven't changed." He touched her face, a careful, feathery contact. "Do you hurt?"
"Some." She closed her eyes and wet her lips. Her tongue seemed unable to reach one corner of her mouth. "I've changed."
"You've _been_ changed," he said, his voice shaking. "The Maneater?"
"Of course. You knew, didn't you?"
"Not really. I thought once you were calling me. Or he was... it was far away. But anyway, no one else would have—would... what happened? Do you want to tell me?"
"Oh yes. He—found out about you. I don't understand how. Your—that Armand Bluett—he's a judge or something now. He came to see the Maneater. He thought you were a girl. A big girl, I mean."
"I was, for a while." He smiled tensely.
"Oh. Oh, I see. Were you really at the carnival that day?"
"At the carnival? No. What day, Zee? You mean when he found out?"
"Yes. Four—no; five days ago. You weren't there. I don't under—" She shrugged. "Anyway, a girl came to see the Maneater and the Judge followed her and thought she was you. The Maneater thought so too. He sent Havana looking for her. Havana couldn't find her."
"And then the Maneater got hold of you."
"Mm. I didn't mean to tell him, Horty. I didn't. Not for a long time, anyway. I—forget." She closed her eyes again. Horty trembled suddenly, and then could breathe.
"I don't... remember," she said with difficulty.
"Don't try. Don't talk any more," he murmured.
"I want to. I've got to. He mustn't find you!" she said. "He's hunting for you right this minute!"
Horty's eyes narrowed and he said, "Good."
Her eyes were still closed. She said, "It was a long time. He talked very quietly. He gave me cushions and some wine that tasted like autumn. He talked about the carnival and Solum and Gogol. He mentioned 'Kiddo' and then talked about the new flat cars and the commissary tent and the trouble with the roustabouts' union. He said something about the musicians' union and something about music and something about the guitar and then about the act we used to have. Then he was off again about the menageries and the shills and the advance men, and back again. You see? Just barely mentioning you and going away and coming back and back. All night, Horty, all, all _night!"_
"Sh-h-h."
"He wouldn't ask me! He talked with his head turned away watching me out of the corners of his eyes. I sat and tried to sip the wine, and tried to eat when Cooky brought dinner and midnight lunch and breakfast, and tried to smile when he stopped for a minute. He didn't touch me, he didn't hit me, he didn't _ask_ me!"
"He did later," breathed Horty.
"Much later. I don't remember... his face over me like a moon, once. I hurt all over. He shouted. Who is Horty, where is Horty, who is Kiddo, why did I hide Kiddo.... I woke up and woke up. I don't remember the times I slept, or fainted, or whatever it was. I woke up with my blood in my eyes, drying, and he was talking about the ride mechanics and the power for the floodlights. I woke up in his arms, he was whispering in my ear about Bunny and Havana, they must have known what Horty was. I woke up on the floor. My knee hurt. There was a terrible light. I jumped up with the pain of it. I ran out the door and fell down, my knee wouldn't work, it was in the afternoon and he caught me and dragged me back again and threw me on the floor and made the light again. He had a burning glass and he gave me vinegar to drink. My tongue swelled, I—"
"Sh-h-h. Zena, honey, hush. Don't say any more."
The flat, uninflected voice went on. "I lay still when Bunny looked in and the Maneater didn't know she saw what he was doing and Bunny ran away and Havana came and hit the Maneater with a piece of pipe and the Maneater broke his neck he's going to die and I—"
Horty's eyelids felt dry. He raised a careful hand and slapped her smartly across her undamaged cheek. "Zena. _Stop it!"_
At the impact she uttered a great shriek, and screamed, "I don't _know_ any more, _truly_ I don't!" and burst into painful, writhing sobs. Horty tried to speak to her but could not be heard through her weeping. He stood, turned, put her down gently on the couch, ran and wrung out a cloth in cold water and bathed her face and wrists. She stopped crying abruptly and fell asleep.
Horty watched her until her breathing assured him that she was at peace. He put his head slowly down beside hers as he knelt on the floor beside the couch. Her hair was on his forehead. Half-crossing his arms, he grasped his elbows and began to pull them. He kept the tension until his shoulders and chest throbbed with pain. He needed to be near her, would not move, yet must relieve the black tension of fury which built in him, and the work his muscles did against each other saved his sanity without the slightest movement to disturb the sleeping girl. He knelt there for a long time.
At breakfast the next morning she could laugh again. Horty had not moved her or touched her except to remove her shoes and cover her with a down quilt. In the small hours of the morning he had taken a pillow from the bedroom and put it on the floor between the studio couch and the door, and had stretched out to listen to her breathing and, with feline attention, to each sound from the stairway and hall outside.
He was standing, bent over her, when she opened her eyes. He said immediately, "I'm Horty and you're safe, Zee." The spiraling panic in her eyes died unborn, and she smiled.
While she bathed, he took her clothes to a neighborhood machine laundry and in half an hour was back with them washed and dried. The food he had picked up on the way was not needed; she had breakfast well on the way when he returned—"gas-house" eggs (fried in the center of slices of bread punched out with a water glass) and crisp bacon. She took the groceries from him and scolded him. "Kippers—papaya juice—Danish ring. Horty, that's _company_ eatments!"
He smiled, more at her courage and her resilience than at her protests. He leaned against the wall with his arms folded, watching her hobbling about the kitchen, draped from neck to heels in what was, for him, a snug-fitting bathrobe, and tried not to think of the fact that she had used it at all. He understood, though, seeing the limp, seeing what had happened to her face...
It was a gay breakfast, during which they happily played "Remember when—" which is, in the final analysis, the most entrancing game in the world. Then there was a silent time, when to each, the sight of the other was enough communication. At last Horty asked, "How did you get away?"
Her face darkened. The effort for control was evident—and successful. Horty said, "You'll have to tell me everything, Zee. You'll have to tell me about—me, too."
"You've found out a lot about yourself." It was not a question.
Horty waved this aside. "How did you get away?"
The mobile side of her face twitched. She looked down at her hands, slowly lifted one, put it on and around the other, and as she talked, squeezed. "I was in a coma for days, I guess. Yesterday I woke up on my bunk, in the trailer. I knew I had told him everything—except that I knew where you were. He still thinks you are that girl.
"I heard his voice. He was at the other end of the trailer, in Bunny's room. Bunny was there. She was crying. I heard the Maneater taking her away. I waited and then dragged myself outside and over to Bunny's door. I got in. Havana was there on the bed with a stiff thing around his neck. It hurt him to talk. He said the Maneater was taking care of him, fixing his neck. He said the Maneater is going to make Bunny do a job for him." She looked up swiftly at Horty. "He can, you know. He's a hypnotist. He can make Bunny do anything."
"I know." He considered her. "Why the hell didn't he use it on you?" he flared.
She fingered her face. "He can't. He—it doesn't work like that on me. He can reach me, but he can't make me do anything. I'm too—"
"Too what?"
"Human," she said.
He stroked her arm and smiled at her. "That you are... Go on."
"I went back to my part of the trailer and got some money and a few other things and left. I don't know what the Maneater will do when he learns I'm gone. I was very careful, Horty. I hitch-hiked fifty miles and then took a bus to Eltonville—that's three hundred miles from here—and a train from there. But I know he'll find me somehow, sooner or later. He doesn't give up."
"You're safe here," he said, and there was blued steel in his soft voice.
"It isn't me! Oh, Horty—don't you understand? It's you he's after!"
"What does he want with me? I left the carnival three years ago and it didn't seem to bother him much." He caught her eye; she was looking at him in amazement. "What is it?"
"Aren't you curious about yourself at all, Horty?"
"About myself? Well, sure. Everybody is, I guess. But about what, especially?"
She was silent a moment, thinking. Abruptly she asked, "What have you done since you left the carnival?"
"I've told you in my letters."
"The bare outlines, yes. You got a furnished room and lived there for a while, reading a lot and feeling your way. Then you decided to grow. How long did that take?"
"About eight months. I got this by mail and moved in at night so no one saw me, and changed. Well, I had to. I'd be able to get a job as a grown man. I buskined a while—you know, playing the clubs for whatever the customers would throw to me—and bought a really good guitar and went to work at the Happy Hours. When that closed I went to Club Nemo. Been there ever since, biding my time. You told me I'd know when it was time... that's always been true."
"It would be," she nodded. "Time to stop being a midget, time to go to work, time to start on Armand Bluett—you'd know."
"Well, sure," he said, as if the fact deserved no further comment. "And when I needed money, I wrote things... some songs and arrangements, articles and even a story or two. The stories weren't so good. It's easy to put things together, but awful hard to make them up. Hey—you don't know what I did to Armand, do you?"
"No." She looked at his hand. "It has something to do with that, hasn't it?"
"It has." He inspected it and smiled. "Last time you saw my hand like this was about a year after I came to the carnival. Want to know something? I lost these fingers just three weeks ago."
"And they've grown that much?"
"It doesn't take as long as it did," he said.
"It did start slowly," she said.
He looked at her, seemed about to ask a question, and then went on. "One night at Club Nemo he walked in with her. I'd never dreamed that I'd seen them together—I know what you're thinking! I always thought of them at the same time! Ah, but that was check and balance. Good and evil. Well..." He drank coffee. "They sat right where I could hear them talk. He was the oily wolf and she was the distressed maiden. It was pretty disgusting. So, he got up to powder his nose, and I made like Lochinvar. I mixed right in. I gave her some succinct language and some carfare, and she got away, after promising him a date for the next night."
"You mean she got away from him for the moment."
"Oh no. She got clear away, by train. I don't know where she went. Well, I sat there chording that guitar and thinking hard. You said that I'd always know when it was time. I knew that night that it was time to get Armand Bluett. Time to start, that is. He gave me a treatment once that lasted for six years. The least I could do was to give him a long stretch too. So I made my plans. I put in a tough night and day." He stopped, smiling without humor.
"Horty—"
"I'll tell it, Zee. It's simple enough. He got his date. Took the gal to a sybaritic little pest-hole he had hidden away in the slums. He was very easy to lead along the primrose paving. At the critical point his 'conquest' said a few well-chosen words about cruelty to children and left him to mull them over while staring at the three fingers she had chopped off as souvenirs."
Zena glanced at his left hand again. "Uh! What a treatment! But Horty—you got ready in one night and day?"
"You don't know the things I can do," he said. He rolled back his sleeve. "Look."
She stared at the brown, slightly hairy right forearm. Horty's face showed deep concentration. There was no tension; his eyes were quiet and his brow un-furrowed.
For a moment the arm remained unchanged. Suddenly the hair on it moved— _writhed._ One hair fell off; another; a little shower of them, finding their way down among the small checks of the tablecloth. The arm remained steady and, like his brow, showed no tension beyond its complete immobility. It was naked now, and the creamy brown color that was typical of both him and Zena. But—was it? Was it the effect of staring with such concentration? No; it was actually paler, paler and more slender as well. The flesh on the back of the hand and between the fingers contracted until the hand was slim and tapered rather than square and thick as it had been.
"That's enough," said Horty conversationally, and smiled. "I can restore it in the same length of time. Except for the hair, of course. That will take two or three days."
"I knew about this," she breathed. "I did know, but I don't think I ever really believed... your control is quite complete?"
"Quite. Oh, there are things I can't do. You can't create or destroy matter. I could shrink to your size, I suppose. But I'd weigh the same as I do now, pretty much. And I couldn't become a twelve-foot giant overnight; there's no way to assimilate enough mass quickly enough. But that job with Armand Bluett was simple. Hard work, but simple. I compacted my shoulders and arms and the lower part of my face. Do you know I had twenty-eight toothaches the whole time? I whitened my skin. The hair was a wig, of course, and as for the female form deevine, that was taken care of by what Elliot Springs calls the 'bust-bucket and torso-twister trade.'"
"How can you joke about it?"
His voice went flat as he said, "What should I do; grind my teeth _every_ minute? This kind of wine needs a shot of bubbles every now and then, honey, or you can't swallow much. No; what I did to Armand Bluett was just a starter. I'm making him do it himself. I didn't tell him who I am. Kay's out of the picture; he doesn't know who she is or who I am or, for that matter, who he is himself." He laughed; an unpleasant sound. "All I gave him was a powerful association with three ruined fingers from 'way back. They'll work in his sleep. The next thing I do to him will be as good—and nothing like that at all."
"You'll have to change your plans some."
"Why?"
"Kay isn't out of the picture. I'm beginning to understand now. She came out to the carnival to see the Maneater."
_"Kay_ did? But why?"
"I don't know. Anyway, the Judge followed her there. She left, but Bluett and the Maneater got together. I know one thing, though. Havana told me—the Judge is terrified of Kay Hallowell."
Horty slapped the table. "With her hand intact! Oh, how wonderful! Can you imagine what that must have been?"
"Horty, darling—it isn't all fun. Don't you see that that's what started all this—that's what made the Maneater suspect that 'Kiddo' was something else besides a girl midget? Don't you realize that the Maneater thinks you and Kay are the same one, no matter what the Judge thinks?"
"Oh, my God."
"You remember everything you hear," said Zena. "But you just don't figure things out very fast, sweetheart."
"But—but—getting smashed up like this... Zena, it's my fault! It's as if I'd done it to you!"
She came around the table and put her arms around him, pulling his head to her breast. "No, darling. That was coming to me, from years back. If you want to blame someone—besides the Maneater—blame me. It was my fault for taking you in twelve years ago."
"What did you do it for? I never really knew."
"To keep you away from the Maneater."
"Away fr—but you kept me right next to him!"
"The last place in the world he'd think of looking."
"You're saying he was looking for me then."
"He's been looking for you ever since you were one year old. And he'll find you. He'll find you, Horty."
"I hope he does," grated Horty. The doorbell rang.
There was a frozen silence. It rang again.
"I'll go," said Zena, rising.
"You will like hell," said Horty roughly. "Sit down."
"It's the Maneater," she whimpered. She sat down.
Horty stood where he could look through the living room at the front door. Studying it, he said, "It isn't. It's—it's—well, what do you know! Old Home Week!"
He strode out and flung the door open. "Bunny!"
"Wh-Excuse m—is this where..." Bunny hadn't changed much. She was a shade more roly-poly, and perhaps a little more timid.
"Oh, Bunny..." Zena came running unevenly out, tripped on the hem of the bathrobe. Horty caught her before she could fall. The girls hugged each other frantically, shouting tearful endearments over the rich sound of Horty's relieved laughter. "But darling, how did you find—" "It's so good to—" "I thought you were—" "You doll! I never thought I'd—"
_"Cut!"_ roared Horty. "Bunny, come in and have some breakfast."
Startled, she looked at him, her albino eyes round. Gently he asked, "How's Havana?"
Without taking her eyes off his face, Bunny fumbled for Zena and held on. "Does he know Havana?"
"Honey," said Zena, "That's _Horty!"_
Bunny shot Zena a rabbit-like glance, craned to peer behind Horty, and suddenly seemed to realize just what Zena had said. "That?" she demanded, pointing. "Him?" She stared. "He's—Kiddo, too?"
Horty grinned. "That's right."
"He grew," said Bunny inanely. Zena and Horty bellowed with laughter, and, as Horty had done once so long ago, so Bunny gaped from one to the other, sensed that they were laughing with and not at her, and joined her tinkling giggle to the noise. Still laughing, Horty went into the kitchen and called out, "You still take canned milk and half a teaspoon of sugar, Bunny?" and Bunny began to cry. Into Zena's shoulder she sobbed happily. "It is Kiddo, it is, it is..."
Horty put the steaming cup on the end table and settled down beside the girls. "Bunny, how in time did you find me?"
"I didn't find you. I found Zee. Zee, maybe Havana's goin' to die."
"I—remember," Zee whispered. "Are you sure?"
"The Maneater did what he could. He even called in another doctor."
"He _did?_ Since when has he taken to doctors?"
Bunny sipped her coffee. "You just can't know how he's changed, Zee. I couldn't believe it myself until he did that, called a doctor in, I mean. You know about m-me and Havana. You know how I feel about what the Maneater did to him. But—it's as if he had come up from under a cloud that he's lived with for years. He's really changed. Zee, he wants you to come back. He's _so_ sorry about what happened. He's really broken up."
"Not enough," muttered Horty.
"Does he want Horty to come back too?"
"Horty—oh. Kiddo." Bunny looked at him. "He couldn't do an act now. I don't know, Zee. He didn't say."
Horty noticed the swift, puzzled frown on Zena's brow. She took Bunny's upper arm and seemed to squeeze it impatiently. "Honey—start from the beginning. Did the Maneater send you?"
"Oh no. Well, not exactly. He's changed so, Zee. You don't believe me... Well, you'll see for yourself. He needs you and I came to get you back, all by myself."
"Why?"
"Because of Havana!" Bunny cried. "The Maneater might be able to save him, don't you see? But not when he's all torn apart by what he did to you."
Zena turned a troubled face to Horty. He rose. "I'll fix you a bite to eat, Bunny," he said. A slight side-wise movement of his head beckoned to Zena; she acknowledged it with an eyelid and turned back to Bunny. "But how did you know where I was, honey?"
The albino leaned forward and touched Zena's cheek. "You poor darling. Does it hurt much?"
Horty, in the kitchen, called, "Zee! What did you do with the tabasco?"
"Be right back, Bun," said Zee. She hobbled across to the kitchen. "It should be right there on the... yes. Oh—you haven't started the toast! I'll do it, Horty."
They stood side by side at the stove, busily. Under his breath Horty said, "I don't like it, Zee."
She nodded. "There's something... we've asked her twice, three times, how she found this place, and she hasn't said." She added clearly, "See? _That's_ the way to make toast. Only you have to watch it."
A moment later, "Horty. How did you know who it was at the door?"
"I didn't. Not really. I knew who it _wasn't._ I know hundreds of people, and I knew it wasn't any of them." He shrugged. "That left Bunny. You see?"
"I can't do that. Nobody I know can do that. 'Cept maybe the Maneater." She went to the sink and clattered briskly. "Can't you tell what people are thinking?" she whispered when she came close to him again.
"Sometimes, a little. I never tried, much."
"Try now," she said, nodding toward the living room.
His face took on that unruffled, deeply occupied expression. At the same moment there was a flash of movement past the open kitchen door. Horty, who had his back to it, turned and sprang through into the living room. "Bunny!"
Bunny's pink lips curled back from her teeth like an animal's and she scuttled to the front door, whipped it open and was gone. Zena screamed. "My purse! She's got my purse!"
In two huge bounds Horty was in the hall. He pounced on Bunny at the head of the stairs. She squealed and sank her teeth into his hand. Horty clamped her head under his arm, jamming her chin against his chest. Having taken a bite, she was forced to keep it—and meanwhile was efficiently gagged.
Inside, he kicked the door closed and pitched Bunny to the couch like a sack of sawdust. Her jaws did not relax; he had to lean over her and pry them apart. She lay with her eyes red and glittering, and blood on her mouth.
"Now, what do you suppose made her go off like that?" he asked, almost casually.
Zena knelt by Bunny and touched her forehead. "Bunny. Bunny, are you all right?"
No answer. She seemed conscious. She kept her mad ruby eyes fixed on Horty. Her breath came in regular, powerful pulses like those of a slow freight. Her mouth was rigidly agape. "I didn't do anything to her," said Horty. "Just picked her up."
Zena rescued her handbag from the floor and fumbled through it. Seemingly satisfied, she set it down on the coffee table. "Horty, what did you do in the kitchen just now?"
"I—sort of..." He frowned. "I thought of her face, and I made it kind of open like a door, or—well, blow away like fog, so I could see inside. I didn't see anything."
"Nothing at all?"
"She moved," he said simply.
Zena began to knead her hands together. "Try again."
Horty went to the couch. Bunny's eyes followed him. Horty folded his arms. His face relaxed. Bunny's eyes closed immediately. Her jaw slackened. Zena barked, "Horty—be careful!"
Without moving otherwise, Horty nodded briefly.
For a moment nothing happened. Then Bunny trembled. She threw out an arm, clenched her small hand. Tears appeared between her lids, and she relaxed. In a few seconds she began to move vaguely, purposelessly, as if unfamiliar hands tested her motor centers. Twice she opened her eyes; once she half sat up, and then lay back. At last she released a long, shuddering sigh, pitched almost as low as Zena's voice, and lay still, breathing deeply.
"She's asleep," said Horty. "She fought me, but now she's asleep." He fell into a chair and covered his face for a moment. Zena watched him restore himself as he had restored his whitened arm earlier. He sat up briskly and said, his voice strong again, "It was more than her strength, Zee. She was full to the brim with something that wasn't hers."
"Is it all gone now?"
"Sure. Wake her and see."
"You've never done anything like this before, Horty? You seem as sure of yourself as old Iwazian." Iwazian was the carnival's photo-gallery operator. He had only to take a picture to know how good it was; he never looked at a proof.
"You keep saying things like that," said Horty with a trace of impatience. "There are things a man can do and things he can't. When he does something, what's the point of wondering whether or not he's actually done it? Don't you think he knows?"
"I'm sorry, Horty. I keep underestimating you." She sat beside the albino midget. "Bunny," she cooed. Bunny...
Bunny turned her head, turned it back, opened her eyes. They seemed vague, unfocussed. She turned them on Zena, and recognition crept into them. She looked around the room, cried out in fear. Zena held her close. "It's all right, darling," she said. "That's Kiddo, and I'm here, and you're all right now."
"But how—where—"
"Sh-h. Tell us what's happened. You remember the carnival? Havana?"
"Havana's goin' to die."
"We'll try to help, Bunny. Do you remember coming here?"
"Here." She looked around, as if one part of her mind were trying to catch up with the rest. "The Maneater told me to. He was nothing but eyes. After a while I couldn't even see his eyes. His voice was inside my head. I don't remember," she said piteously. "Havana's going to die." She said this as if it were the first time.
"We'd better not ask her questions now," said Zena.
"Wrong," said Horty. "We'd better, and fast." He bent over Bunny. "How did you find this place?"
"I don't remember."
"After the Maneater talked inside your head, what did you do?"
"I was on a train." Her answers were almost vague; she did not seem to be withholding information—rather, she seemed unable to extend it. It had to be lifted out.
"Where did you go when you got off the train?"
"A bar. Uh—Club... Nemo. I asked the man where I could find the fellow who hurt his hand."
Zena and Horty exchanged a look. "The Maneater said Zena would be with this fellow."
"Did he say the man was Kiddo? Or Horty?"
"No. He didn't say. I'm hungry."
"All right, Bunny. We'll get you a big breakfast in a minute. What were you supposed to do when you found Zena? Bring her back?"
"No. The jewels. She had the jewels. There had to be two of them. He'd give me twice what he gave Zena if I came back without them. But he'd kill me if I came with only one."
"How he's changed," Zena said, scornful horror in her voice.
"How did he know where I was?" Horty demanded.
"I don't know. Oh; that girl."
"What girl?"
"She's a blonde girl. She wrote a letter to someone. Her brother. A man got the letter."
"What man?"
"Blue. Judge Blue."
"Bluett?"
"Yes, Judge Bluett. He got the letter and it said the girl was working in a record shop in town. There was only one record shop. They found her easily."
_"They found her?_ Who?"
"The Maneater. And that Blue. Bluett."
Horty brought his fists together. "Where is she?"
"The Maneater's got her at the carnival. Can I have my breakfast now?"
# 14
HORTY LEFT.
He slipped into a light coat and found his wallet and keys, and he left. Zena screamed at him. Intensity injected raucousness into her velvet voice. She caught his arm; he did not shake her off, but simply kept moving, dragging her as if she were smoke in the suction of his movement. She turned to the table, snatched up her bag, found two glittering jewels. "Horty, wait, wait!" She held out the jewels. "Don't you remember, Horty? Junky's eyes, the jewels—they're _you,_ Horty!"
He said, "If you need anything at all, no matter what, call Nick at Club Nemo. He's all right," and opened the door.
She hobbled after him, caught at his coat, missed her hold, staggered against the wall. "Wait, wait. I have to tell you, you're not ready, you just don't _know!"_ She sobbed. "Horty, the Maneater—"
Halfway down the stairs he turned. "Take care of Bunny, Zee. Don't go out, not for anything. I'll be back soon."
And he left.
Holding the wall, Zena crept down the hall and into the apartment. Bunny sat on the couch, sobbing with fright. But she stopped when she saw Zena's twisted face, and ran to her. She helped her to the easy-chair and crouched on the floor at her feet, hugging her legs, her round chin against Zena's knees. The vibrant color was gone from Zena; she stared dryly down, black eyes in a grey face.
The jewels fell from her hand and glittered on the rug. Bunny picked them up. They were warm, probably from Zena's hand. But the little hand was so cold... They were hard, but Bunny felt that if she squeezed them they would be soft. She put them on Zena's lap. She said nothing. She knew, somehow, that this was not the time to say anything.
Zena said something. It was unintelligible; her voice was a hoarseness, nothing more. Bunny made a small interrogative sound, and Zena cleared her throat and said, "Fifteen years."
Bunny waited quietly after that, for minutes, wondering why Zena did not blink her eyes. Surely that must hurt her... she reached up presently and touched the lids. Zena blinked and stirred uneasily. "Fifteen years I've been trying to stop this from happening. I knew what he was the instant I saw those jewels. Maybe even before... but I was sure when I saw the jewels." She closed her eyes; it seemed to give more vitality to her voice, as if her intense gaze had been draining her. "I was the only one who knew. The Maneater only hoped. Even Horty didn't know. Only me. Only me. Fifteen years—"
Bunny stroked her knee. A long time passed. She became certain that Zena was asleep, and had begun to think thoughts of her own when the deep, tired voice came again.
"They're alive." Bunny looked up; Zena's hand was over the jewels. "They think and they speak. They mate. They're alive. These two are Horty."
She sat up and pushed her hair back. "That's how I knew. We were in that diner, the night we found Horty. A man was robbing our truck, remember? The man put his knee on these crystals, and Horty got sick. He was indoors and a long way from the truck but he knew. Bunny, do you remember?"
"Mm-hm. Havana, he used to talk about it. Not to you, though. We always knew when you didn't want to talk, Zee."
"I do now," said Zena wearily. She wet her lips. "How long have you been with the show, Bun?"
"I guess eighteen years."
"Twenty for me. Almost that, anyway. I was with Kwell Brothers when the Maneater bought into it. He had a menagerie. He had Gogol and a pinhead and a two-headed snake and a bald squirrel. He used to do a mind-reading act. Kwell sold out for nothing. Two late springs and a tornado taught Kwell all the carny he ever wanted to know. Lean years. I stuck with the show because I was there, mostly. Just as tough there as anywhere else." She sighed, scanning over twenty years. "The Maneater was obsessed by what he called a hobby. Strange people aren't his hobby. Carny isn't his hobby. Those things are because of his hobby." She lifted the jewels and clicked them together like dice. "These are his hobby. These things sometimes make strange people. When he got a new freak—" (The word jolted both of them as she said it)—"he kept it by him. He got into show business so he could keep them and make money too. That's all. He kept them and studied them and made more of them."
"Is that really what makes strange people?"
"No! Not all of them. You know about glands and mutations, and all that. These crystals make them too, that's all. They do it—I _think_ they do it—on purpose."
"I don't understand, Zee."
"Bless your heart! Neither do I. Neither does the Maneater, although he knows an awful lot about them. He can talk to them, sort of."
"How?"
"It's like his mind-reading. He puts his mind on them. He—hurts them with his mind until they do what he wants."
"What does he want them to do?"
"Lots of things. They all amount to one thing, though. He wants a—a middle-man. He wants them to make something that he can maybe talk to, give orders to. Then the middle-man would turn around and make the crystals do what he wanted."
"I guess I'm sort of stupid, Zee."
"No you're not, honey... oh. Bunny, Bunny, I'm so _glad_ you're here!" She pulled the albino up into the chair and hugged her fervently. "Let me talk, Bun. I've got to talk! Years and years, and I haven't said a word..."
"I won't understand one word in ten, I bet."
"Yes you will, lamb. Comfy? Well... you see, these crystals are a sort of animal, kind of. They're not like any other animal that ever lived on earth. I don't think they came from anywhere on earth. The Maneater told me he sees a picture sometimes of white and yellow stars in a black sky, the way space would look away outside the earth. He thinks they drifted here."
"He told you? You mean he talked to you about them?"
"By the hour. I guess everybody has to talk to someone. He talked to me. He threatened to kill me, time and time again, if I ever said a word. But that's not why I kept it a secret. See, he was good to me, Bunny. He's mean and crazy, but he was always good to me."
"I know. We used to wonder."
"I didn't think it made any difference to anyone. Not at first, not for years. When I did learn what he was really trying to do, I _couldn't_ tell anyone; no one would've believed me. All I could do was to learn as much as I could and hope I could stop him when the time came."
"Stop him from what, Zee?"
"Well—look; let me tell you a little more about the crystals. Then you'll see. These crystals used to _copy_ things. I mean, one would be near a flower, and it would make another flower almost like it. Or a dog, or a bird. But mostly they didn't come out right. Like Gogol. Like the two-headed snake."
"Gogol is one of those?"
Zena nodded. "The Fish-Boy. I think he was supposed to be a human being. No arms, no legs, no teeth, and he can't sweat so he has to be kept in a tank or he'll die."
"But what do the crystals do that for?"
She shook her head. "That's one of the things the Maneater was trying to find out. There isn't anything regular about the things the crystals make, Bunny. I mean, one will look like the real thing and another will come up all strange, and another won't live at all, it's such a botch. That's why he wanted a middle-man—someone who could communicate with the crystals. He couldn't except in flashes. He could no more understand them than you or I could understand advanced chemistry or radar or something. But one thing did not come clear. There are different kinds of crystals; some are more complicated than others, and can do more. Maybe they're all the same kind, but some are older. They never helped each other; didn't seem to have anything to do with each other.
"But they bred. The Maneater didn't know that. He knew that sometimes a pair of crystals would sometimes stop responding when he hurt them. At first he thought they were dead. He dissected one pair. And once he gave a couple to old Worble."
"I remember him! He used to be a strong man, but he was too old. He used to help the cook, and all. He died."
"Died—that's one way to say it. Remember the things he used to whittle?"
"Oh, yes—dolls and toys and all like that."
"That's right. He made a jack-in-the-box and used these for eyes." She tossed the crystals and caught them. "He was always giving things away to kids. He was a good old man. I know what happened to that jack-in-the-box. The Maneater never found out, but Horty told me. Somehow or other it passed from hand to hand and got into an orphanage. That's where Horty was, when he was a tiny baby. Inside of six months they were a part of Horty—or he was a part of them."
"But what about Worble?"
"Oh, maybe a year later the Maneater began wondering if the crystals bred, and what happened when they did. He was afraid that he had given away two big, well-developed crystals that weren't dead after all. When Worble told him he had put them in toys he made and some kid had them, he didn't know where, why, the Maneater hit him. Knocked him down. Old Worble never woke up again though it was two weeks before he died. No one knew about it but me. It was out behind the cook-tent. I saw."
"I never knew," breathed Bunny, her ruby eyes wide.
"No one did," Zena repeated. "Let's have some coffee—why, _honey!_ You never did get your breakfast, you poor baby!"
"Oh gosh," said Bunny, "that's all right. Go on talking."
"Come into the kitchen," she said as she rose stiffly. "No, don't be surprised when the Maneater seems to be inhuman. He— _isn't_ human."
"What is he, then?"
"I'll get to it. About the crystals; the Maneater says that the closest you can come to the way they make things—plants and animals, and so on, is to say they _dream_ them. You dream sometimes. You know how the things in your dreams are sometimes sharp and clear, and sometimes fuzzy or crooked or out of proportion?"
"Yup. Where's the eggs?"
"Here, dear. Well, the crystals dream sometimes. When they dream sharp and clear they make pretty good plants, and real rats and spiders and birds. They usually don't, though. The Maneater says they're erotic dreams."
"What d'ye mean?"
"They dream when they're ready to mate. But some are too—young, or undeveloped, and maybe some just don't find the right mate at that time. But when they dream that way, they change molecules in a plant and make it like another plant, or change a pile of mold into a bird... no one can say what they'll choose to make, or why."
"But—why should they make things so they can mate?"
"The Maneater doesn't think they do it so they _can_ mate, exactly," said Zena, her voice patient. She skillfully flipped an egg in the pan. "He calls it a byproduct. It's as if you were in love and you were thinking of nothing but the one you love, and you made a song. Maybe the song wouldn't be about your lover at all. Maybe it'd be about a brook, or a flower, or something. The wind. Maybe it wouldn't be a whole song, even. That song would be a by-product. See?"
"Oh. And the crystals make things—even complete things—like Tin Pan Alley makes songs."
"Something like it." Zena smiled. It was the first smile in a long while. "Sit down, honey; I'll bring the toast. Now—this is my guess—when two crystals mate, something different happens. They make a whole thing. But they don't make it from just anything the way the single crystals do. First they seem to die together. For weeks they lie like that. After that they begin a together-dream. They find something near them that's alive, and they make it over. They replace it, cell by cell. You can't see the change going on in the thing they're replacing. It might be a dog; the dog will keep on eating and running around; it will howl at the moon and chase cats. But one day—I don't know how long it takes—it will be completely replaced, every bit of it."
"Then what?"
"Then it can change itself—if it ever thinks of changing itself. It can be almost anything if it wants to be."
Bunny stopped chewing, thought, swallowed, and asked, "Change how?"
"Oh, it could get bigger or smaller. Grow more limbs. Go into a funny shape—thin and flat, or round like a ball. If it's hurt it can grow new limbs. And it could do things with thought that we can't even imagine. Bunny, did you ever read about werewolves?"
"Those nasty things that change from wolves to men and back again?"
Zena sipped coffee. "Mmm. Well, those are mostly legends, but they could have started when someone saw a change like that."
"You mean these crystal-things aren't new on earth?"
"Oh, heavens no! The Maneater says they're arriving and living and breeding and dying here all the time."
"Just to make strange people and werewolves," breathed Bunny in wonder.
"No, darling! Making those things is nothing to them! They live a life of their own. Even the Maneater doesn't know what they do, what they think about. The things they make are absent-minded things, like doodles on a piece of paper that you throw away. But the Maneater thinks he could understand them if he could get that middle-man."
"What's he want to understand a crazy thing like that for?"
Zena's small face darkened. "When I found that out, I began listening carefully—and hoping that some day I could stop him. Bunny, the Maneater hates people. He hates and despises all people."
"Oh, yes," said Bunny.
"Even now, with the poor control he has over the crystals, he's managed to make some of them do what he wants. Bunny, he's planted crystals in swampland with malaria mosquito eggs all around them. He's picked up poisonous coral snakes in Florida and planted them in Southern California. Things like that. It's one of the reasons he keeps the carnival. It covers the country, the same route year after year. He goes back and back, finding the crystals he's planted, seeing how much harm they've done to people. He keeps finding more. He finds them all over. He walks in the woods and out on the prairies, and every once in a while he sends out a—a kind of thought he knows how to do. It hurts the crystals. When they feel pain, he knows it. He hunts around, hurting the crystals until their pain leads him right to them. But anyway, there are plenty around. They look like pebbles or clods until they're cleaned."
"Oh, how—how awful!" Tears brightened Bunny's eyes. "He ought to be—killed!"
"I don't know if he can be killed."
"You mean he really is one of those things from the crystals?"
"Do you think a human being could do what he does?"
"But—what would he do if he got that middleman?"
"He'd train him up. Those creatures that are made by two crystals, they're whatever they think they are. The Maneater would tell the middle-man that he was a servant; he was under orders. The middle-man would believe him, and think that of himself. Through him the Maneater would have real power over the crystals. He could probably even make them mate, and dream-together any horrible thing he wanted. He could spread disease and plant-blight and poison until there wouldn't be a human being left on earth! And the worst thing about it is that the crystals don't even seem to want that! They're satisfied to go on as they are, making a flower or a cat once in a while, and thinking their own thoughts, and living whatever strange sort of life they live. They aren't after people! They just don't _care."_
"Oh, Zee! And you've been carrying all this around with you for years!" Bunny ran around the table and kissed her. "Oh, baby, why didn't you tell someone?"
"I didn't dare, sweetheart. They would think I was out of my mind. And besides—there's Horty."
"What about Horty?"
"Horty was a baby in an orphanage when, somehow, that toy with the crystal eyes was brought in. The crystals picked on him. It all fits. He told me that when the jack-in-the-box—he called it Junky—was taken away from him; he almost died. The doctors there thought it was some kind of psychosis. It wasn't, of course; the child was in some strange bondage to the married crystals and could not exist away from them. It seems that it was far simpler to leave the toy with the child—it was an ugly toy, Horty tells me—than to try to cure the psychosis. In any case, Junky went along with Horty when he was adopted—by that Armand Bluett, incidentally; that judge."
"He's awful! He looks all soft and— _wet."_
"The Maneater has been looking for one of those twin-crystal creatures for twenty years or more, only he didn't know it. Why, the very first crystal he found was probably one of a pair, and he didn't realize it. Not ever—not until he found out about Horty. He guessed it, but he never _knew_ until now. I knew that night we picked up Horty. The Maneater would give everything he owns in the world for Horty—a human. Not a human; Horty isn't human and hasn't been since he was a year old. But you know what I mean."
"And that would be his middle-man?"
"That's right. So when I saw what Horty was, I jumped at the chance to hide him in the last place in the world Pierre Monetre would think of looking—right under his nose."
"Oh, Zee! What a terrible chance to take! He was bound to find out!"
"It wasn't too much of a chance. The Maneater can't read my mind. He can prod it; he can call me in a strange way; but he can't find out what's in it. Not the way Horty did on you before. The Maneater hypnotized you to make you steal the jewels and bring them back. Horty went right into your mind and cleared all that away."
"I—I remember. It was crazy."
"I kept Horty by me and worked on him constantly. I read everything I could get my hands on and fed it to him. Everything, Bunny—comparative anatomy and history and music and mathematics and chemistry—everything I could think of that would help him to a knowledge of human things. There's an old Latin saying, Bunny: _Cogito ergo sum_ —'I think, therefore I am.' Horty is the essence of that saying. When he was a midget he believed he was a midget. He didn't grow. He never thought of his voice changing. He never thought of applying what he learned to himself; he let me make all his decisions for him. He digested everything he learned in a reservoir with no outlet, and it never touched him until he decided himself that it was time to use it. He has eidetic memory, you know."
"What's that?"
"Camera memory. He remembers perfectly everything he has seen or read or heard. When his fingers began to grow back—they were smashed hopelessly, you know—I kept it a secret. That was the one thing that would have told the Maneater what Horty was. Humans can't regenerate fingers. Single-crystal creatures can't either. The Maneater used to spend hours in the dark, in the menagerie tent, trying to force the bald squirrel to grow hair, or trying to put gills on Gogol the Fish Boy, by prodding at them with his mind. If any of them had been twin-crystal creatures, they would have repaired themselves."
"I think I see. And what you were doing was to convince Horty that he was human?"
"That's right. He had to identify himself first and foremost with humanity. I taught him guitar for that reason, after his fingers grew back, so that he could learn music quickly and thoroughly. You can learn more music theory in a year on guitar than you can in three on a piano, and music is one of the most human of human things... He trusted me completely because I never let him think for himself."
"I—never heard you talk like this before, Zee. Like out of books."
"I've been playing a part too, sweet," said Zena gently. "First, I had to keep Horty hidden until he had learned everything I could teach him. Then I had to plan some way to make him stop the Maneater, without danger of the Maneater's making a servant of him."
"How could he do that?"
"I think the Maneater is a single-crystal thing. I think if Horty could only learn to use that mental whip that the Maneater has, he could destroy him with it. If I should kill the Maneater with a bullet, it won't kill his crystal. Maybe that crystal will mate, later, and produce him all over again—with all the power that a twin-crystal creature has."
"Zee, how do you know the Maneater isn't a twin-crystal thing?"
"I don't," Zee said bleakly. "If that's the case, then I can only pray that Horty's estimate of himself as a human being is strong enough to fight what the Maneater wants to make of him. Hating Armand Bluett is a human thing. Loving Kay Hallowell is another. Those are two things that I needled him with, drilled into him, teased him about, until they became part of his blood and bone."
Bunny was silent before this bitter flood of words. She knew that Zena loved Horty; that she was enough of a woman to feel Kay Hallowell's advent as a deep menace to her; that she had fought and won against the temptation to steer Horty away from Kay; and that, more than anything else, she was face to face with terror and remorse now that her long campaign had come to a head.
She watched Zena's proud, battered face, the lips which drooped slightly on one side, the painfully canted head, the shoulders squared under the voluminous robe, and she knew that here was a picture she would never forget. Humanity is a concept close to the abnormals, who are wistfully near it, who state their membership with aberrated breath, who never cease to stretch their stunted arms toward it. Bunny's mind struck a medallion of this torn and courageous figure—a token and a tribute.
Their eyes met, and slowly Zena smiled. "Hi, Bunny..."
Bunny opened her mouth and coughed, or sobbed. She put her arms around Zena and snuggled her chin into the cool hollow of the dark-skinned neck. She closed her eyes tight to squeeze away tears. When she opened them she could see again. And then she couldn't speak.
She saw, over Zena's shoulder through the kitchen door, out in the living room, a huge, gaunt figure. Its lower lip swung loosely as it bent over the coffee table. Its exquisite hands plucked up one, two jewels. It straightened, sent her a look of dull pity from its sage-green face, and went silently out.
"Bunny, darling, you're hurting me."
_Those jewels are Horty,_ Bunny thought. _Now I'll tell her Solum has taken them back to the Maneater._ Her face and her voice were as dry and as white as chalk as she said, "You haven't been hurt yet..."
# 15
HORTY POUNDED UP THE stairs and burst into his apartment. "I'm walking under water," he gasped. "Every damn thing I reached for is snatched away from me. Everything I do, everywhere I go, it's too early or too late or—" Then he saw Zena on the easy-chair, her eyes open and staring, and Bunny crouched at her feet. "What's the matter here?"
Bunny said, "Solum came in when we were in the kitchen and took the jewels and we couldn't do anything and Zena hasn't said a word since and I'm scared and I don't know what to do—hoo..." and she began to cry.
"Oh Lord." He was across the room in two strides. He lifted Bunny up and hugged her briefly and set her down. He knelt beside Zena. "Zee—"
She did not move. Her eyes were all pupil, windows to a too-dark night. He tilted her chin up and fixed his gaze on her. She trembled and then cried out as if he had burned her, twisted into his arms. "Don't, don't..."
"Oh, I'm sorry, Zee. I didn't know it would hurt you."
She leaned back and looked up at him, seeing him at last. "Horty, you're all right..."
"Well, sure. What's this about Solum?"
"He got the crystals. Junky's eyes."
Bunny whispered, "For twelve years she's been keeping them away from the Maneater, Horty; and now—"
"You think the Maneater sent him for them?"
"Must have. I guess he must have followed me, and waited until he saw you leave. He was in here and out again before we could do so much as turn and look."
"Junky's eyes..." There was the time he had almost died, as a child, when Armand threw the toy away. And the time when the tramp had crushed them under his knee, and Horty, in the lunch room two hundred feet away, had felt it. Now the Maneater might... oh, no. This was too much.
Bunny suddenly clapped her hand to her mouth. "Horty—I just thought—the Maneater wouldn't've sent Solum by himself. He wanted those jewels... you know how he gets when he wants something. He can't bear to wait. He must be in town right now."
"No." Zena rose stiffly. "No, Bun. Unless I'm quite wrong, he was here and is on his way back to the carnival. If he thinks Kay Hallowell is Horty, he'll want to have the jewels where he can work on them and watch her at the same time. I'll bet he's burning up the road back to the carnival this minute."
Horty moaned. "If only I hadn't gone out! I might've been able to stop Solum, maybe even get to the Maneater and—Damn it! Nick's car was in the garage; first I had to find Nick and borrow it, and then I had to get a parked truck out from in front of the garage, and then there was no water in the radiator, and—oh, you know. Anyway, I have the car now. It's downstairs. I'm going to take off right now. In three hundred miles I ought to be able to catch up with... how long ago was Solum here?"
"An hour or so. You just can't, Horty. And what will happen to you when he goes to work on those jewels, I hate to think."
Horty took out keys, tossed and caught them. "Maybe," he said suddenly, "Just maybe we can—" He dove for the phone.
Listening to him talk rapidly into the instrument, Zena turned to Bunny. "A plane. But of course!"
Horty put the phone down, looking at his watch. "If I can get out to the airport in twelve minutes I can get a feeder flight."
"You mean 'we.'"
"You're not coming. This is my party, from here on out. You kids have been through enough."
Bunny was pulling on her light coat. "I'm going back to Havana," she said grimly, and for all her baby features, her face showed case-hardened purpose.
"You're not going to leave me here," said Zena flatly. She went for her coat. "Don't argue with me, Horty. I have a lot to tell you, and maybe a lot to do."
"But—"
"I think she's right," said Bunny. "She has a lot to tell you."
The plane was wobbling out to the runway when they arrived. Horty drove right out onto the tarmac, horn blasting, and it waited. And after they were settled in their seats, Zena talked steadily. They were ten minutes away from their destination when she was finished.
After a long, thoughtful pause, Horty said, "So that's what I am."
"It's a big thing to be," said Zena.
"Why didn't you tell me all this years ago?"
"Because there were too many things I didn't know. There still are... I didn't know how much the Maneater might be able to dig out of your mind if he tried; I didn't know how deep your convictions on yourself had to go before they settled. All I tried to do was to have you accept, without question, that you were a human being, a part of humanity, and grow up according to that idea."
He turned on her suddenly. "Why did I eat ants?"
She shrugged. "I don't know. Perhaps even two crystals can't do a perfect job. Anyway your formic acid balance was out of adjustment. (Did you know the French word for 'ant' is _fourmi?_ They're full of the stuff.) Some kids eat plaster because they need calcium. Some like burned cake for the carbon. If you had an imbalance, you can bet it would be an important one."
The flaps went down; they felt the braking effect. "We're coming in. How far is the carnival from here?"
"About four miles. We can get a cab."
"Zee, I'm going to leave you outside the grounds somewhere. You've been through too much."
"I'm going in with you," said Bunny firmly. "But Zee—I think he's right. Please stay outside until—until it's over."
"What are you going to do?"
He spread his hands. "Whatever I can. Get Kay out of there. Stop Armand Bluett from whatever filthy thing he plans to do with her and her inheritance. And the Maneater... I don't know, Zee. I'll just have to play it as it comes. But I have to do it. You've done all you can. Let's face it; you're not fast on your feet just now. I'd have to keep looking out for you."
"He's right, Zee. Please—" said Bunny.
"Oh, be careful, Horty— _please_ be careful!"
No bad dream can top this, Kay thought. Locked in a trailer with a frightened wolf and a dying midget, with a madman and a freak due back any minute. Wild talk about missing fingers, about living jewels, and about—wildest of all—Kay not being Kay, but someone or something else.
Havana moaned. She wrung out a cloth and sponged his head again. Again she saw his lips tremble and move, but words stuck in his throat, gurgled and fainted there. "He wants something," she said. "Oh, I wish I knew what he wanted! I wish I knew, and could get it quickly..."
Armand Bluett leaned against the wall by the window, one sack-suited elbow thrust through it. Kay knew he was uncomfortable there and that, probably, his feet hurt. But he wouldn't sit down. He wouldn't get away from the window. Oh no. He might want to yell for help. Old Crawly-Fingers was suddenly afraid of her. He still looked at her wet-eyed and drooling, but he was terrified. Well, let it go. No one likes having his identity denied, but in this case it was all right with her. Anything to keep a room's-breadth between her and Armand Bluett.
"I wish you'd leave that little monster alone," he snapped. "He's going to die anyway."
She turned a baleful glance on him and said nothing. The silence stretched, punctuated only by the Judge's painful foot-shifting. Finally he said, "When Mr. Monetre gets back with those crystals, we'll soon find out who you are. And don't tell me again that you don't know what all this is about," he snapped.
She sighed. "I don't know. I wish you'd stop shouting like that. You can't jolt information out of me that I haven't got. And besides, this little fellow's sick."
The Judge snorted, and moved even closer to the window. She had an impulse to go over there and growl at him. He'd probably go right through the wall. But Havana moaned again. "What is it, fellow? What is it?"
Then she stiffened. Deep within her mind she sensed a presence, a concept connected somehow with delicate, sliding music, with a broad pleasant face and a good smile. It was as if a question had been asked of her, to which she answered silently, _I'm here. I'm all right—so far._
She turned to look at the Judge, to see if he shared the strange experience. He seemed tense. He stood with his elbow on the sill, nervously buffing his nails on his lapel.
And a hand came through the window.
It was a mutilated hand. It rose into the trailer like the seeking head and neck of a waterfowl, passed in over Armand's shoulder and spread itself in front of has face. The thumb and index fingers were intact. The middle finger was clubbed; the other two were mere buttons of scar-tissue.
Armand Bluett's eyebrows were two stretched semi-circles, bristling over bulging eyes. The eyes were as round as the open mouth. His upper lip turned back and upward, almost covering his nostrils. He made a faint sound, a retch, a screech, and dropped.
The hand disappeared through the window. There were quick footsteps outside, around to the door. A knock. A voice. "Kay. Kay Hallowell. Open up."
Inanely, she quavered, "Wh-who is it?"
"Horty." The doorknob rattled. "Hurry. The Maneater's due back, but quick."
"Horty. I—the door's locked."
"The key must be in the Judge's pocket. Hurry."
She went with reluctant speed to the prone figure. It lay on its back, the head propped against the wall, the eyes screwed shut in a violent psychic effort to shut out the world. In the left jacket pocket were keys on a ring—and one single. This she took. It worked.
Kay stood blinking at sunlight. "Horty."
"That's right." He came in, touched her arm, grinned. "You shouldn't write letters. Come on, Bunny."
Kay said, "They thought I knew where you were."
"You do." He turned away from her and studied the supine form of Armand Bluett. "What a sight. Something the matter with his stomach?"
Bunny had arrowed to the bunk, knelt beside it. "Havana... Oh Havana..."
Havana lay stiffly on his back. His eyes were glazed and his lips pouted and dry. Kay said, "Is—is he... I've done what I could. He wants something. I'm afraid he—" She went to the bedside.
Horty followed. Havana's pale chubby lips slowly relaxed, then pursed themselves. A faint sound escaped. Kay said, "I _wish_ I knew what he wants!" Bunny said nothing. She put her hands on the hot cheeks, gently, but as if she would wrest something up out of him by brute force.
Horty frowned. "Maybe I can find out," he said.
Kay saw his face relax, smoothed over by a deep placidity. He bent close to Havana. The silence was so profound, suddenly, that the carnival noises outside seemed to wash in on them, roaring.
The face Horty turned to Kay a moment later was twisted with grief. "I know what he wants. There may not be time before the Maneater gets here... but—There's got to be time," he said decisively. He turned to Kay. "I've got to go to the other end of the trailer. If he moves—" indicating the Judge—"hit him with your shoe. Preferably with a foot in it." He went out, his hand, oddly, on his throat, kneading.
"What's he going to do?"
Bunny, her eyes fixed on Havana's comatose face, answered, "I don't know. Something for Havana. Did you see his face when he went out? I don't think Havana's going to—to—"
From the partition came the sound of a guitar, the six open strings brushed lightly. The A was dropped, raised a fraction. The E was flatted a bit. Then a chord...
Somewhere a girl began to sing to the guitar. _Stardust._ The voice was full and clear, a lyric soprano, pure as a boy's voice. Perhaps it was a boy's voice. There was a trace of vibrato at the ends of the phrases. The voice sang to the lyric, just barely trailing the beat, not quite ad lib, not quite stylized, and as free as breathing. The guitar was not played in complicated chords, but mostly in swift and delicate runs in and about the melody.
Havana's eyes were still open, and still he did not move. But his eyes were wet now, and not glazed, and gradually he smiled. Kay knelt beside Bunny. Perhaps she knelt only to be nearer... Havana whispered, through his smile, "Kiddo."
When the song was done, his face relaxed. Quite clearly he said "Hey." There was a world of compliment in the single syllable. After that, and before Horty came back, he died.
Entering, Horty did not even glance at the cot. He seemed to be having trouble with his throat. "Come on," he said hoarsely. "We've got to get out of here."
They called Bunny and went to the door. But Bunny stayed by the bunk, her hands on Havana's cheeks, her soft round face set.
"Bunny, come on. If the Maneater comes back—"
There was a step outside, a thump against the wall of the trailer. Kay wheeled and looked at the suddenly darkened window. Solum's great sad face filled it. Just then Horty screamed shrilly and dropped writhing to the floor. Kay turned to face the opening door.
"Good of you to wait," said Pierre Monetre, looking about.
# 16
ZENA HUDDLED ON THE edge of the lumpy motel bed and whimpered. Horty and Bunny had been gone for nearly two hours; for the past hour, depression had grown over her until it was like bitter incense in the air, like clothes of lead sheeting on her battered limbs. Twice she had leapt up and paced impatiently, but her knee hurt her and drove her back to the bed, to punch the pillow impotently, to lie passive and watch the doubts circling endlessly about her. Should she have told Horty about himself? Should she not have given him more cruelty, more ruthlessness, about more things than revenging himself on Armand Bluett? How deep had her training gone in the malleable entity which was Horty? Could not Monetre, with his fierce, directive power undo her twelve years' work in an instant? She knew so little; she was, she felt, so small a thing to have undertaken the manufacture of a—a human being.
She wished, fiercely, that she could burrow her mind into the strange living crystals, as the Maneater tried to do, but completely, so that she could find the rules of the game, the facts about a form of life so alien that logic seemed not to work on it at all. The crystals had a rich vitality; they created, they bred, they felt pain; but to what end did they live? Crush one, and the others seemed not to mind. And why, why did they make these "dream-things" of theirs, laboriously, cell by cell—sometimes to create only a horror, a freak, an unfinished, unfunctional monstrosity, sometimes to copy a natural object so perfectly that there was no real distinction between the copy and its original; and sometimes, as in Horty's case, to create something new, something that was not a copy of anything but, perhaps, a mean, a living norm on the surface, and a completely fluid, polymorphic being at its core? What was their connection with these creations? How long did a crystal retain control of its product—and how, having built it, could it abruptly leave it to go its own way? And when the rare syzygy occurred by which two crystals made something like Horty—when would they release him to be his own creature... and what would become of him then?
Perhaps the Maneater had been right when he had described the creatures of the crystals as their dreams—solid figments of their alien imaginations, built any way they might occur, patterned on partial suggestions pictured by faulty memories of real objects. She knew—the Maneater had happily demonstrated—that there were thousands, perhaps millions of the crystals on earth, living their strange lives, as oblivious to humanity as humanity was to them, for the life-cycles, the purposes and aims of the two species were completely separate. Yet—how many men walked the earth who were not men at all; how many trees, how many rabbits, flowers, amoebae, sea-worms, redwoods, eels and eagles grew and flowered, swam and hunted and stood among their prototypes with none knowing that they were an alien dream, having, apart from the dream, no history?
"Books," Zena snorted. The books she had read! She had snatched everything she could get her hands on that would give her the slightest lead on the nature of the dreaming crystals. And for every drop of information she had gained (and passed on to Horty) about physiology, biology, comparative anatomy, philosophy, history, theosophy and psychology, she had taken in a gallon of smug certitude, of bland assumptions that humanity was the peak of creation. The answers... the books had answers for everything. A new variety of manglewort appears, and some learned pundit places his finger alongside his nose and pronounces, "Mutation!" Sometimes, certainly. But—always? What of the hidden crystal-creatare dreaming in a ditch, absently performing, by some strange telekinesis, a miracle of creation?
She loved, she worshipped Charles Fort, who refused to believe that any answer was the only answer.
She looked at her watch yet again, and whimpered. If she only knew; if she could only guide him... if she could get guidance herself, somewhere, somewhere...
The doorknob turned. Zena froze, staring at it. Something heavy pressed against the door. There was no knock. The crack between door and frame, high up, widened. Then the bolt let go, and Solum burst into the room.
His loose-skinned, grey-green face and dangling lower lip seemed to pull more than usual at the small, inflamed eyes. He took a half-step back to swing the door closed behind him, and crossed the room to her, his great arms away from his body as if to check any move she might make.
His presence told her some terrible news. No one knew where she was but Horty and Bunny, who had left her in this tourist cabin before they crossed the highway to the carnival. And when last heard of, Solum had been on the road with the Maneater.
So—the Maneater was back, and he had contacted Bunny or Horty, or both, and, worst of all, he had been able to extract information that neither would give willingly.
She looked up at him out of a tearing flurry of deadening resignation and mounting terror. "Solum—"
His lips moved. His tongue passed over his brilliant pointed teeth. He reached for her, and she shrank back.
And then he dropped to his knees. Moving slowly, he took her tiny foot in one of his hands, bent over it with an air that was, unmistakably, reverence.
He kissed her instep, ever so gently, and he wept. He released her foot and crouched there, immersed in great noiseless shuddering sobs.
"But, _Solum—"_ she said, stupidly. She put out a hand and touched his wet cheek. He pressed it closer. She watched him in utter astonishment. Long ago she used to wonder at what went on in the mind behind this hideous face, a mind locked in a silent, speechless universe, with all the world pouring in through the observant eyes and never an expression, never a conclusion or an emotion coming out.
"What is it, Solum?" she whispered. "Horty—"
He looked up and nodded rapidly. She stared at him. "Solum—can you hear?"
He seemed to hesitate; then he pointed to his ear, and shook his head. Immediately he pointed to his brow, and nodded.
"Oh-h-h..." Zena breathed. For years there had been idle arguments in the carnival as to whether the Alligator-skinned Man was really deaf. There was instance after instance to prove both that he was, and that he was not. The Maneater knew, but had never told her. He was—telepathic! She flushed as she thought of it, the times that carnies, half-kidding, had hurled insults at him; worse, the horrified reactions of the customers.
"But—What's happened? Have you seen Horty? Bunny?"
His head bobbed twice.
"Where are they? Are they safe?"
He thumbed toward the carnival, and shook his head gravely.
"Th-the Maneater's got them?"
Yes.
"And the girl?"
Yes.
She hopped off the bed, strode away and back, ignoring the pain. "He sent you here to get me?"
Yes.
"But why don't you scoop me up and take me back, then?"
No answer. He motioned feebly. She said, "Let's see. You took the jewels when he asked you to..."
Solum tapped his forehead, spread his hands. Suddenly she understood. "He hypnotized you then."
Solum shook his head slowly.
She understood that it had been a matter of indifference to him. But this time it was different. Something had happened to change his mind, and drastically.
"Oh, I wish you could talk!"
He made anxious, lateral circular motions with his right hand. "Oh, of _course!"_ she exploded. She limped to the splintery bureau and her purse. She found her pen; she had no paper but her checkbook. "Here, Solum. Hurry. Tell me!"
His huge hands enveloped the pen, completely hid the narrow paper. He wrote rapidly while Zena wrung her hands in impatience. At last he handed it to her. His script was delicate, almost microscopic, and as neat as engraving.
He had written, tersely, "M. hates people. Me too. Not so much. M. wants help, I helped him. M. wanted Horty so he could hurt more people. I didn't care. Still helped. People never liked me.
"I am human, a little. Horty is not human at all. But when Havana was dying, he wanted Kiddo to sing. Horty read his mind. He knew. There was no time. There was danger. Horty knew. Horty didn't save himself. He made Kiddo's voice. He sang for Havana. Too late then. M. came. Caught him. Horty did this so Havana could die happy. It didn't help Horty. Horty knew; did it anyway. Horty is love. M. is hate. Horty more human than I am. I am ashamed. You made Horty. Now I help you."
Zena read it, her eyes growing very bright. "Havana's dead, then."
Solum made a significant gesture, twisting his head in his hands, pointing to his neck, snapping his fingers loudly. He shook his fist at the carnival.
"Yes. The Maneater killed him.... How did you know about the song?"
Solum tapped his forehead.
"Oh. You got it from Bunny, and the girl Kay; from their minds."
Zena sat on the bed, pressing her knuckles hard against her cheekbones. Think, think... oh, for guidance; for a word of advice about these alien things! The Maneater, crazed, inhuman; surely a warped crystalline product; there must be some way of stopping him. If only she could contact one of the jewels and ask it what to do... surely it would know. If only she had the "middle-man," the interpreter, that the Maneater had been seeking all these years...
_The middle-man!_ "I'm blind, I'm stone blind and stupid!" she gasped. All these years her single purpose had been to keep Horty away from the crystals; he must have nothing to do with them, lest the Maneater use him against humanity. But Horty was what he was; he was the very thing the Maneater wanted; he was the one who could contact the crystals. There must be a way in which the crystals could destroy what they created!
But would the crystals tell him of such a thing?
They wouldn't have to, she decided instantly. All Horty would have to do would be to understand the strange mental mechanism of the crystals, and the method would be clear to him.
If only she could tell him! Horty learned quickly, thought slowly; for eidetic memory is the enemy of methodical thought. Ultimately he would think of this himself—but by then he might be the Maneater's crippled slave. What could she do? Write him a note? He might not even be conscious to read it! If only she were a telepath... Telepath!
"Solum," she said urgently, "Can you— _speak,_ up here" (she touched her forehead) as well as hear?"
He shook his head. But at the same time he picked up the check on which he had written and pointed to a word.
"Horty. You can speak to Horty?"
He shook his head, and then made outgoing motions from his brow. "Oh," she said. "You can't project it, but he can read it if he tries." He nodded eagerly.
"Good!" she said. She drew a deep breath; she knew, at last, exactly what she must do. But the cost... it didn't matter. It couldn't matter.
"Take me back there, Solum. You've caught me. I'm frightened, I'm angry. Get to Horty. You can think of a way. Get to him and think _hard._ Think: _Ask the crystals how to kill one of their dream-things. Find out from the crystals._ Got that, Solum?"
The wall had gone up years ago, when Horty came to the very simple conclusions that the peremptory summonses which awakened him at night in his bunk were for Zena, and not for him. _Cogito, ergo sum;_ the wall, once erected, stood untended for years, until Zena suggested that he try reaching into the hypnotized Bunny's mind. The wall had come down for that; it was still down when he used his new sense to locate the trailer in which Kay was a prisoner, and when he sought the nature of Havana's dying wish. His sensitive mind was therefore open and unguarded when the Maneater arrived and hurled at him his schooled and vicious lance of hatred. Horty went down in flames of agony.
In ordinary terms, he was completely unconscious. He did not see Solum catch the fainting Kay Hallowell and tuck her under his long arm while his other hand darted out to snatch up soft-faced, tenderhearted Bunny, who fought and spit as she dangled there. He had no memory of being carried to Monetre's big trailer, of the tottering advent, a few minutes later, of a shaken and murderous Armand Bluett. He was not aware of Monetre's quick hypnotic control of hysterical Bunny, nor of her calm flat voice revealing Zena's whereabouts, nor of Monetre's crackling command to Solum to go to the motor court and bring Zena back. He did not hear Monetre's blunt order to Armand Bluett: "I don't think I need you and the girl for anything any more. Stand back there out of the way." He did not see Kay's sudden dash for the door, nor the cruel blow of Armand Bluett's fist which sent her sliding back into the corner as he snarled, "I need _you_ for something, sweetheart, and you're not getting out of my sight again."
But the blacking out of the ordinary world revealed another. It was not strange; it had coexisted with the other. Horty saw it now only because the other was taken away.
There was nothing about it to relieve the utter lightlessness of oblivion. In it, Horty was immune to astonishment and quite without curiosity. It was a place of flickering impressions and sensations; of pleasure in an integration of abstract thought, of excitement at the approach of one complexity to another, of engrossing concentration in distant and exoteric constructions. He felt the presence of individuals, very strongly indeed; the liaison between them was non-existent, except for the rare approach of one to another and, somewhere far off, a fused pair which he knew were exceptional. But for these, it was a world of self-developing entities, each evolving richly according to its taste. There was a sense of permanence, of life so long that death was not a factor, save as an aesthetic termination. Here there was no hunger, no hunting, no co-operation, and no fear; these things had nothing to do with the bases of a life like this. Basically trained to accept and to believe in that which surrounded him, Horty delved not at all, made no comparisons, and was neither intrigued nor puzzled.
Presently he sensed the tentative approach of the force which had blasted him, used now as a goad rather than as a spear. He rebuffed it easily, but moved to regain consciousness so that he might deal with the annoyance.
He opened his eyes and found them caught and held by those of Pierre Monetre, who sat at his desk facing him. Horty was sprawled back in an easy-chair, his head propped in the angle of the back and a small rounded wing. The Maneater was radiating nothing. He simply watched, and waited.
Horty closed his eyes, sighed, moved his jaws as a man does on awakening.
"Horty." The Maneater's voice was mellow, friendly. "My dear boy. I have looked forward so long to this moment. This is the beginning of great things for us two."
Horty opened his eyes again and looked about. Bluett stood glowering at him, a shuddering mixture of fear and fury. Kay Hallowell huddled in the corner opposite the entrance, on the floor. Bunny squatted next to her, holding limply to Kay's forearm, looking out into the room with vacant eyes.
"Horty," said the Maneater insistently. Horty met his gaze again. Effortlessly he blocked the hypnotic force which the Maneater was exerting. The mellow voice went on, soothingly, "You're home at last, Horty—really home. I am here to help you. You belong here. I understand you. I know the things you want. I will make you happy. I will teach you greatness, Horty. I will protect you, Horty. And you will help me." He smiled. "Won't you, Horty?"
"You can drop dead," said Horty succinctly.
The reaction was instant—a shaft of brutal hatred whetted to a razor-edge, a needle-point. Horty rebuffed it, and waited.
The Maneater's eyes narrowed and his eyebrows went up. "Stronger than I thought. Good. I'd rather have you strong. You _are_ going to work with me, you know."
Horty blankly shook his head. Again, and twice more, the Maneater struck at him, timing the psychic blows irregularly. Had Horty's defense been a counter-act, like that of a rapier or a boxing-glove, the Maneater would have gotten through. But it was a wall.
The Maneater leaned back, consciously relaxing. His weapon apparently took quantities of energy. "Very well," he purred. "We'll dull you down a bit." He drummed his fingers idly.
Long moments passed. For the first time Horty realized that he was paralyzed. He could breathe fairly easily, and, with difficulty, move his head. But his arms and legs were leaden, numb. A vague ache in the nape of his neck—and his profound knowledge of anatomy—informed him of a skillfully administered spinal injection.
Kay stirred and was quiet. Bunny looked at her and away, still with that vacant gaping look on her sweet round face. Bluett shifted uncomfortably on his feet.
The door was elbowed aside. Solum came in with Zena in his arms. She was limp. Horty tried frantically and uselessly to move. The Maneater smiled engagingly and motioned with his head. "Into the corner with the rest of the trash," he said. "We might be able to use her. Think our friend would be more co-operative if we cut her down a bit?"
Solum grinned wolfishly.
"Of course," said the Maneater thoughtfully. "She isn't very big to begin with. We'd have to be careful. A little at a time." Belying his offhand tone, his eyes watched every move of Horty's face. "Solum, old fellow, our boy Horty is a little too alert. Suppose you jolt him a bit. The edge of your hand at the side of his neck, right at the base of the skull. The way I showed you. You know."
Solum stalked over to Horty. He put one hand on Horty's shoulder, and took careful aim with the other. The hand which rested on his shoulder squeezed slightly, over and over again. Solum's eyes burned down to Horty's. Horty watched the Maneater. He knew the major blow would come from there.
Solum's other hand came down. A fraction of a second after it hit his neck, Monetre's psychic bolt smashed against Horty's barrier. Horty felt a faint surprise; Solum had pulled the punch. He looked up quickly. Solum, his back turned to the Maneater, touched his forehead, worked his lips anxiously. Horty shrugged this off. He had no time for idle wonderments... he heard Zena whimper.
"You're in my way, Solum!" Solum moved reluctantly. "You'll have another chance at him," said the Maneater, He opened the drawer in front of him and took out two objects. "Horty, d'ye know what these are?"
Horty grunted and nodded. They were Junky's eyes. The Maneater chuckled. "If I smash these, you die. You know that, don't you?"
"Wouldn't be much help to you then, would I?"
"That's right. But I just wanted to let you know I have them handy." Ceremoniously he lighted a small alcohol blow-lamp. "I don't have to destroy them. Single-crystal creatures react beautifully to fire. You should do twice as well." His voice changed abruptly. "Oh, Horty, my boy, my dear boy—don't force me to play with you like this."
"Play away," gritted Horty.
"Hit him again, Solum." Now the voice crackled.
Solum swept down on him. Horty caught a glimpse of Armand's avid face, the flick of a tongue across his wet lips. The blow was heavier this time, though still surprisingly less powerful than he expected—less powerful, for that matter, than it looked. Horty rolled his head with the impact, and slumped down with his eyes closed. The Maneater hurled no bolts this time, apparently in an attempt to force Horty to use up counter-ammunition while saving his own.
"Too hard, you idiot!"
Kay's voice moaned out of the corner, "Oh, stop it, stop it..."
"Ah." The Maneater's chair scraped as he turned. "Miss Hallowell! How much would the young man do for you? Drag her out here, Bluett."
The Judge did. He said, with a leer, "Save some for me, Pierre."
"I'll do as I like!" snapped the Maneater.
"All right, all right," said the Judge, cowed. He went back to his corner.
Kay stood erect but trembling before the desk. "You'll have the police to answer to," she flared.
"The Judge will take care of the police. Sit down, my dear." When she did not move, he roared at her. _"Sit down!"_ She gulped and sat in the chair at the end of the long desk. He reached out and trapped her wrist, pulled it toward him. "The Judge tells me you like having your fingers cut off."
"I don't know what you m-mean. Let me g—"
Meanwhile Solum was on his knees beside Horty, rolling his head, slapping his cheeks. Horty submitted patiently, quite conscious. Kay screamed.
"Nice noisy carnival we have here," smiled the Maneater. "That's quite useless, Miss Hallowell." He pulled a heavy pair of shears out of the drawer. She screamed again. He put them down and took up the blow-lamp, passing the flame lightly over the crystals which lay winking before him. By some fantastic stroke of luck—or perhaps some subtler thing than luck, Horty flashed a quick look through his lashes at that precise second. As the pale flame touched the jewels, he threw his head back, twisted his features—
But he did it on purpose. He felt nothing.
He looked at Zena. Her face was strained, her whole soul streaming through it, trying to tell him something...
He opened his mind to it. The Maneater saw his eyes open and hurled another of those frightful psychic impulses. Horty slammed his mind shut barely in time; part of the impulse got in and jolted him to the core.
For the first time he fully recognized his lack, his repeated failure to figure things clearly out for himself. He made a grim effort. Zena trying to tell him something. If he had just a second to receive her... but he was lost if he submitted to another such blow as the first one. There was something else, something about— _Solum!_ The signaling hand on his shoulder, the hot eyes, bursting with something unsaid...
"Hit him again, Solum." The Maneater picked up the shears. Kay screamed again.
Again Solum bent over him; again the hand pressed his shoulder secretly, urgently. Horty looked the green man full in the eyes and opened up to the message which rolled there.
_ASK THE CRYSTALS. Ask the crystals how to kill one of their dream-things. Find out from the crystals._
"What are you waiting for, Solum?"
Kay screamed and screamed. Horty closed his eyes and his mind. Crystals... not the ones on the table. The—the— _all_ the crystals, which lived in—in—
Solum's hard hand landed on his neck. He let it drive him under, down and down into that lightless place full of structural, shimmering sensations. Resting in it, he drove his mind furiously about, questing. He was ignored completely, majestically. But there was no guard against him, either. What he wanted was there; he had only to understand it. He would not be helped or hindered.
He recognized now that the crystal-world was not loftier than the ordinary one. It was just—different. These self-sufficient abstracts of ego were the crystals, following their tastes, living their utterly alien existences, thinking with logic and with scales of values impossible to a human being.
He could understand some of it, untrammeled as he was with fixed ideas, though he was hammered into human mold too solidly to be able to merge himself completely with these unthinkable beings. He understood almost immediately that Monetre's theory of the crystal-dreams was true and not-true, like the convenient theory that an atom-nucleus had planetary particles rotating about it. The theory worked in simple practice. The manufacture of living things was a function with a purpose, but that purpose could never be explained in human terms. The one thing that was borne in on Horty was the almost total unimportance, to the crystals, of this function. They did it, but it served them about as much as a man is served by his appendix. And the fate of the creatures they created mattered as little to them as does the fate of a particular molecule of CO2 exhaled by a man.
Nevertheless, the machinery by which the creation was done was there before Horty. Its purpose was beyond him, but he could grasp its operation. Studying it with his gulping, eidetic mind, he learned... things. Two things. One had to do with Junky's eyes, and the other—It was a thing to do. It was a thing like stopping a rolling boulder by blocking it with another rolled in its path. It was a thing like lifting the brush-holder on a DC motor, like cutting the tendons at the back of the hind legs of a running horse. It was a thing done with the mind, with a tremendous effort, which said a particular _stop!_ to a particular kind of life.
Understanding, he withdrew, not noticed—or ignored—by the strange egos about him. He let in the light. He emerged, and felt his first real astonishment. His neck stung from the blow of Solum's hand, which was still rebounding. The same scream which had begun when he went under came to its gasping conclusion as he came up. Bunny still stared between the slow blink of her drugged-looking lids; Zena still crouched with the same tortured expression of concentration in her pointed face.
The Maneater hurled his bolt. Horty turned it aside, and now he laughed.
Pierre Monetre rose, his face blackening with rage. Kay's wrist slipped out of his hand. Kay bounded for the door; Armand Bluett blocked her. She cowered away, across to Zena's corner, and slumped down, sobbing.
Horty knew what to do, now; he had learned a thing. He tested it with his mind, and knew immediately that it was not a thing which could be done casually. It meant a gathering of mental powers, a shaping of the mass of them, an aiming, a triggering. He turned his mind in on itself and began to work.
"You shouldn't have laughed at me," said the Maneater hoarsely. He raked in the two jewels and dropped them into a metal ash tray. He picked up the blow-lamp, meticulously adjusting the flame.
Horty worked. And still, a part of his mind was not occupied with the task. You can kill crystal-creatures, it said. The Maneater, yes, but—this is a big thing you are going to do. It may kill others... what others? Moppet? The two-headed snake? Gogol? _Solum?_
Solum, ugly, mute, imprisoned Solum, who had, at the last moment, turned against the Maneater and had helped him. He had carried Zena's message, and it was his own death warrant.
He looked up at the green man, who was backing away, his flaring eyes still anxiously filled with the message, not knowing that Horty had read it and acted upon it seconds before. Poor, trapped, injured creature...
But it was Zena's message. Zena had always been his arbiter and guide. The fact that it was hers meant that she had considered the cost and had decided accordingly. Perhaps it was better this way. Perhaps Solum could, in some unfathomable way, enjoy a peace that life had never yielded him.
The strange force mounted within him, his polymorphic metabolism draining itself into the arsenal of his mind. He felt the drugged strength drain out of his hands, out of the calves of his legs.
"Does this tickle?" snarled the Maneater. He swept the flame over the winking jewels. Horty sat rigidly, waiting, knowing that now this mounting pressure was out of his control, and that it would release itself when it reached its critical pressure. He kept his gaze fixed on the purpling, furious face.
"I wonder," said the Maneater, "which crystal builds which part, when two of them go at it." He lowered the flame like a scalpel, stroking it back and across one of the crystals. "Does that—"
Then it came. Even Horty was unprepared for it. It burst from him, the thing he had understood from the crystals. There was no sound. There was a monstrous flare of blue light, but it was inside his head; when it had passed he was quite blind. He heard a throttled cry, the fall of a body. Slowly, then, knees, hip, head, another body. Then he gave himself up to pain, for his mind, inside, was like a field after a wind-driven brush fire, raw and burnt and smoking, speckled with hot and dying flames.
Blackness crept over it slowly, with here and there a stubborn luminous pain. His vision began to clear. He lay back, drained.
Solum had tumbled to the floor by his side. Kay Hallowell sat against the wall with her hands over her face. Zena leaned against her, her eyes closed. Bunny still sat on the floor, staring, weaving very slightly. Near the door, Armand Bluett was stretched out. Horty thought, the fool passes out like a corseted Victorian. He looked at the desk.
Pale and shaken, but erect, the Maneater stood. He said, "You seem to have made a mistake."
Horty simply stared at him dully. The Maneater said, "I would think that, with your talents, you would know the difference between a crystalline and a human being."
_I never thought to look,_ he cried silently. _Will I ever learn to doubt? Zena always did my doubting for me!_
"You disappoint me. I always have the same trouble. My average is pretty high, though. I can spot 'em about eight times out of ten. I will admit, though, that _that_ was a surprise to me." He tossed a casual thumb at Armand Bluett. "Oh well. Another heart case on the Fair Grounds. A dead crystalline looks just the same as a dead human. Unless you know what to look for." With one of those alarming changes of voice, he said, _"You tried to kill me..._ " He wandered over to Horty's chair and looked down at Solum. "I'll have to learn to get along without old Solum. Nuisance. He was very useful." He kicked the long body idly, and suddenly swung around and landed a stinging slap on Horty's mouth. "You'll do twice what he did, and like it!" he shouted. "You'll jump when I so much as whisper!" He rubbed his hands.
"Oh-h-h..."
It was Kay. She had moved slightly. Zena's head had thumped down into her lap. She was chafing the little wrists.
"Don't waste your time," said the Maneater, casually. "She's dead."
Horty's fingertips, especially the growing stubs on his left hand, began to tingle. _She's dead. She's dead._
At his desk, the Maneater picked up one of the crystals and tossed it, glancing at Zena. "Lovely little thing. Treacherous snake, of course, but pretty. I'd like to know where the crystal that made her got its model. As nice a job as you'll find anywhere." He rubbed his hands together. "Not a patch on what we'll have from now on, hey, Horty?" He sat down, fondling the crystal. "Relax, boy, relax. That was one hell of a blast. I'd like to learn a trick like that. Think I could?... Maybe I'll leave it to you, at that. Seems to be quite a drain on you."
Horty tensed muscles without moving. Strength was seeping back into his exhausted frame. Not that it would do him much good. The drug would hold him if he were twice his normal strength.
_She's dead. She's dead._ When he said that, he meant Zena. Zena had wanted to be a real live normal human being... well, all strange people do, but Zena especially, because she wasn't human, not at all. That was why she'd never let him read her mind. She didn't want anyone to know. She wanted so _much_ to be human. But she'd known. She must have known when she sent him the message through Solum. She knew it would kill her too. She was—more of a human being than any woman born.
_I'll move now,_ he thought.
"You'll sit there without food or water until you rot," the Maneater said pleasantly, "or at least until you weaken enough to let me into that stubborn head of yours so I can blast out any silly ideas you may have about being your own master. You belong to _me_ —three times over." He handled the two crystals lovingly. "Stay where you are!" he snarled, whirling on Kay Hallowell, who had begun to rise. Startled and broken, she sank down again. Monetre rose, went and stood over her. "Now, what to do with you. Hm."
Horty closed his eyes, and with all his mounting energy, he thought. What was the drug Monetre had used? One of the 'ocaines, surely—benzocaine, monocaine... He was conscious of approaching vertigo, the first hint of nausea. Which drug would yield just this effect, then demonstrate just this much toxicity? In the back of his mind, he saw the riffling pages of a drug dictionary.
_Think!_
A dozen drugs could have this effect. But Monetre would certainly choose one that would do all he wanted—and he wanted more than immobility. He wanted psychic stimulation with it.
_Got it!_ The old standby—cocaine hydrochloride. Antidote... epinephrine.
Now I've got to be a pharmacy, he thought grimly. Epinephrine...
Adrenalin! Close enough—and very easy to supply under the circumstances. He had only to open his eyes and look at the Maneater. His lips curled. The vertigo faded. His heart began to thump. He controlled it. He could feel his body going into a forced-draft condition. His feet began to tingle almost unbearably.
"You could be a heart failure case too," the Maneater was saying pensively to Kay. "A little _curare..._ no. The Judge is enough for one day."
Watching Monetre's back, Horty flexed his hands, pressed his elbows against his sides until his pectoral muscles crackled. He tried to rise, tried again. He all but collapsed, and then freedom and hate combined to accelerate the return of strength to his body. He rose, clenching his hands, trying not to breathe noisily.
"Well, we'll dispose of you in some way," said the Maneater, returning to his desk, talking over his shoulder at the frightened girl. "And soon— _uh!"_ He found himself face to face with Horty.
The Maneater's hand crept out and closed around the jewels. "Don't come one small half-inch nearer," he rasped, "or I'll smash these. You'll slump together like a bag of rotten potatoes. Don't move, now."
"Is Zena really dead?"
"As a doornail, son. I'm sorry. I'm sorry that it was so quick, I mean. She deserved a more artistic treatment. _Don't move!"_ He held the crystals together in one hand, like walnuts about to be cracked. "Better go back and sit down where it's comfortable." Their eyes met, held. Once, twice, the Maneater sent Horty his barbed hate. Horty did not flinch. "Wonderful defense," said the Maneater admiringly. "Now go and _sit_ down!" His fingers tightened on the crystals.
Horty said, "I know a way to kill humans too." He came forward.
The Maneater scuttled back. Horty rounded the desk and came on. "You asked for it!" panted the Maneater. He closed his bony hand. There was a faint, tinkling crackle.
"I call it Havana's way," said Horty thickly, "after a friend of mine."
The Maneater's back was against the wall, round-eyed, pasty-faced. He goggled at the single intact crystal in his hand—like walnuts, only one broke when the two were crushed together—uttered a birdlike squeak, dropped the crystal, and ground it under his feet. Then Horty had both hands on his head. He twisted. They fell together. Horty wrapped his legs around the Maneater's chest, got another grip on the head, and twisted again with all his strength. There was a sound like a pound of dry spaghetti being broken in two, and the Maneater slumped.
Blackness showered in descending streamers around Horty. He crawled off the inert figure, pushing his face almost into Bunny's. Bunny's face was looking down and past him, and was no longer vacant and staring. Her lips were curled back from her teeth. Her neck was arched, the cords showing starkly. Gentle Bunny... she was looking at the dead Maneater, and she was laughing.
Horty lay still. Tired, tired... it was almost too much effort to breathe. He raised his chin to make it easier for air to pass his throat. This pillow was so soft, so warm... Feather-touches of hair lay on his upturned face, delicately stroked his closed eyelids. Not a pillow; a round arm curved behind his head. Scented breath at his lips. She was big, now; a regular human girl, the way she always wanted to be. He kissed the lips. "Zee. Big Zee," he murmured.
"Kay. It's Kay, darling, you poor brave darling..."
He opened his eyes and looked up at her, his eyes a child's eyes for the moment, full of weariness and wonder. "Zee?"
"It's all right. Everything's all right now," she said soothingly. "I'm Kay Hallowell. Everything's all right."
"Kay." He sat up. There was Armand Bluett, dead. There was the Maneater, dead. There was—was—He uttered a hoarse sound and scrambled uncertainly to his feet. He ran to the wall and picked Zena up and put her gently on the table. She had plenty of room... Horty kissed her hair. He gathered her hands together and called her quietly, twice, as if she were hiding somewhere near and was teasing him.
"Horty—"
He did not move. With his back to her, he said thickly, "Kay—where'd Bunny go?"
"She went to sit with Havana. Horty—"
"Go stay with her a little. Go on. Go on..."
She hesitated, and when she left, she ran.
Horty heard a mourning sound, but he did not hear it with his ears. It was inside his head. He looked up. Solum stood there, silent. The mourning sound appeared again in Horty's head.
"I thought you were dead," Horty gasped.
_I thought you were dead,_ the silent, startled response came. _The Maneater smashed your jewels._
"They were through with me. They've been through with me for years. I'm grown... complete.... finished, and I have been since I was eleven. I just found out, when you sent me to—to speak to the crystals. I didn't know. Zena didn't know. All these years she's been... oh, Zee, Zee!" Horty raised his eyes after a bit and looked at the green man. "What about you?"
_I'm not a crystalline, Horty. I'm human. I happen to be a receptive telepath. You gave me a nasty jolt right where I felt it most. I don't blame you and the Maneater for thinking I was dead. I did myself for a while. But Zena—_
Together they stood over the tiny, twisted body, and their thoughts were their own.
After a time they talked.
"What'll we do with the Judge?"
_It's dark now. I'll leave him near the midway. It will be heart failure._
"And the Maneater?"
_The swamp. I'll take care of it after midnight._
"You're a big help, Solum. I feel sort of—lost. I would be, too, if it hadn't been for you."
_Don't thank me. I haven't the brains for a thing like that. She did it. Zena. She told me exactly what to do. She knew what was going to happen. She knew I was human, too. She knew everything. She did everything._
"Yeah. Yeah, Solum... What about the girl? Kay?"
_Oh. I don't know._
"I think she better go back where she was working. Eltonville. I wish she could forget the whole thing."
_She can._
"She—oh, of course. I can do that. Solum, she—"
_I know. She loves you, just as if you were human. She thinks you are. She doesn't understand any of this._
"Yes. I—wish... Never mind. No I don't. She's not my—my kind. Solum—Zena... loved me."
_Yes. Oh, yes... and what are you going to do?_
"Me? I don't know. Cut out, I guess. Play guitar somewhere."
_What would she want you to do?_
"I—"
_The Maneater did a lot of harm. She wanted to stop him. Well, he's stopped. But I think perhaps she would like you to right some of the wrongs he's done. All over our carnival route, Horty—anthrax in Kentucky, deadly nightshade in the pasture lands up and down Wisconsin, puff adders in Arizona, polio and Rocky Mountain spotted fever in the Alleghenies; why, he even planted tsetse flies in Florida with his infernal crystals! I know where some of them are, but you could find the rest better even than he could._
"My God... and they mutate, the diseases, the snakes..."
_Well?_
"Who would I be working for? Who's going to run the—Solum! Why are you staring at the Maneater like that? What's your idea? You—you think I—"
_Well?_
"He was three inches taller... long hands... narrow face... I don't really see why not, Solum. I could play it that way for a while—at least until 'Pierre Monetre' wound up the arrangements to have 'Sam Horton' run the carnival so he can retire. Solum, you have a brain."
_No. She told me to suggest it to you if you didn't think of it yourself._
"She—Oh Zee, Zee... Solum, if it's all the same to you, I've got to be by myself a while."
_Yes. I'll get this carrion out of here. Bluett first, I'll just tote him to the First Aid tent. No one ever asks old Solum any questions._
Horty stroked Zena's hair, once. His eyes strayed around the trailer and fixed on the Maneater's body. He walked abruptly over to it and turned it over on its face. "I don't like to be stared at..." he muttered.
He sat down at the desk on which Zena's body lay. He pulled the chair up close, crossed his forearms and rested his cheek on them. He didn't touch Zena, and his face was turned away from her. But he was _with_ her, close, close. Softly, he talked to her, using their old idioms, just as if she were alive.
"Zee...?
"Does it hurt you, Zee? You look as if you hurt. 'Member about the kitten on the carpet, Zee? We used to tell each other. It's a soft carpet, see, and the kitten digs its claws in and str-r-etches. It goes down in front and up behind, and it yawns, _yeeowarrgh!_ And then it tips one shoulder under and jus' _pours_ out flat. And if you lift a paw with your finger it's as limp as a tassel and drops back _phup!_ on the deep soft rug. And if you think about that until you see it, all of it, the place where the fur's tousled a bit, and the little line of pink that shows on the side because the kitten's just too relaxed to close his mouth all the way—why then, you just _can't_ hurt any more.
"There, now...
"It hurt you to be different from—from folks, didn't it, Zee? I wonder if you know how much there is of that in everybody. The strange people, the little people—they have more than most. And you had more than any of them. Now I know, _now_ I know why you wished and wished you were big. You pretended you were human, and had a human sorrow that you weren't big; and that way you hid from yourself that you weren't human at all. And that's why you tried so hard to make me the best kind of human you could think of; because you'd have to be pretty human yourself to do all that for humanity. I think you believed, really believed you were human—until today, when you had to face it.
"So you faced it, and you died.
"You're full of music and laughter and tears and passion like a real woman. You share, and you know about _with_ ness _._
"Zena, Zena, a jewel dreamed a truly beautiful dream when it made you!
_"Why didn't it finish the dream?_
"Why don't they finish what they start? Why these sketches and no paintings, these chords with no key signatures, these plays cut off at the second-act climax?
_"Wait!_ Shh—Zee! Don't say anything...
"Must there be a painting for every sketch? Do you have to compose a symphony for every theme? Wait, Zee... I've got a big think in my head...
"It comes straight from you. Remember all you taught me—the books, the music, the pictures? When I left the carnival I had Tchaikowsky and Django Rheinhart; I had _Tom Jones, a Foundling_ and _1984._ And when I went away I built on these things. I found new beauties. I have Bartok and Gian-Carlo Menotti now, _Science and Sanity_ and _The Garden of the Plynck._ Do you see what I mean, honey? New beauties... things I'd never dreamed of before.
"Zena, I don't know whether it's a large or a small part of the crystals' life, but they have an art. When they're young—as they develop—they try their skills at copying. And when they mate (if it is mating) they make a new something. Instead of copying, they take over a living thing, cell by cell, and build it to a beauty of their own invention.
"I'm going to show them a new beauty. I'm going to point a new direction for them—something they've never dreamed before."
Horty rose and went to the door. He pulled down the louvres and locked them, and shot the inside bolt. Returning to the desk, he sat down and went through the drawers. From the deep one at the left he lifted a heavy mahogany box, opened it with the Maneater's keys, and took out the trays of crystals. He glanced at them curiously under the desk light. Ignoring the labels, he piled all the crystals in a heap beside Zena's body, and put his head in his hands among them. It was quite dark except for the desklamp; very little light filtered into the draped oval windows of the trailer.
Horty leaned forward and kissed the smooth, cool elbow. "Now stay here," he whispered. "I'll be right back, honey."
He bowed his head and closed his eyes, and let his mind go dark. His sense of presence in the trailer slipped away, and he became detached, a wanderer in lightlessness.
Again another sense replaced his sight, and once again he found himself aware of Presences. Profoundly, this time, all "group" atmosphere was lacking, but for one—no, three quite distant pairs. But all the rest were single, isolated, sharing nothing, each pursuing esoteric, complicated lines of thought... not thought, but something like it. Horty felt the differences between the creatures sharply. One was concentrated grandeur, dignity and peace. Another's aura was dynamic, haughty, and another closely hid a strange, pulsating, secret idea-series that entranced him, though he knew he'd never understand it.
The strangest thing of all was this: that he, a stranger, was not strange among them. Strangers anywhere on earth, on entering a club, or auditorium, or swimming pool, are, to some extent, made conscious of their lack of membership. But Horty felt no trace of such a thing. And neither did he feel included. Or ignored. He knew they noticed him. They knew he watched them. He could feel it. No one here, however long he stayed, would try communication—he was sure of that. And no one would avoid it.
And in a flash he understood. All earthborn life proceeds and operates from one command: Survive! A human mind cannot conceive of any other base.
The crystals had one—and a very different one.
Horty almost grasped it, but not quite. As simple as "survive!", it was a concept so remote from anything he'd ever heard or read that it escaped him. By that token, he was sure that they would find his message complex and intriguing.
So—he spoke to them. There are no words for what he said. He used no words; the thing he had to say came out in one great surge of rich description. Holding every thought that had been sleeping in his mind for twenty years, his books and music, all his fears and joys and puzzlements, and all his motives, this single flash of message coursed among the crystals.
It told of her perfect white teeth and her musical diction. It told of the time she had sent Huddie off, and the turn of her cheek, and the depth of expression which lay in her eyes. It told of her body, and cited a thousand and one human standards by which she was beautiful. It told of the eloquent rustling chords of her half-size guitar, and her generous voice, and the danger she faced in defense of the species denied her by one of the crystals. It pictured her artlessly naked; it brought back the difficult, half-concealed weeping; outbalanced her tears with a peal of arpeggio laughter; and told of her pain, and her death.
Implicit in this was humanity. With it, the base of Survival emerged, a magnificent ethic: _the highest_ c _ommand is in terms of the species, the next is survival of group. The lowest of three is survival of self._ All good and all evil, all morals, all progress, depend on this order of basic commands. To survive for the self at the price of the group is to jeopardize species. For a group to survive at the price of the species is manifest suicide. Here is the essence of good and of greed, and the wellspring of justice for all of mankind.
And back to the girl, the excluded. She has given her life for an alien caste, and has done it in terms of its noblest ethic. It might be that "justice" and "mercy" are relative terms; but nothing can alter the fact that her death, upon earning her right to survive, is bad art.
And that, in brief, all weighted down with clumsy, partial words, describes his single phrase of message. Horty waited. Nothing. No response, no greeting... nothing.
He came back. He felt the desk under his forearms, his forearm on his cheek. He raised his head and blinked at the desk light. He moved his legs. No stiffness. Some day he must investigate the anomaly in time-perception in that atmosphere of alien thought.
It hit him then—his failure.
He cried out, hoarsely, and put his arms out to Zena. She lay quite still, quite dead. He touched her. She was rigid. Rigor had accented the crooked smile resulting from the damage the Maneater had done to her motor centers. She looked brave, rueful, and full of regret. Horty's eyes burned. "You dig a hole, see," he growled, "and you drop this in it, and you cover it up. And then what the hell do you do with the rest of your life?"
He sensed someone at the door. He took out his handkerchief and wiped his eyes. They still burned. He turned out the desk lamp and went to the door. Solum.
Horty went out, closed the door behind him, and sat down on the mounting step.
_As bad as that?_
"I guess it is," said Horty. "I—didn't really think she was going to stay dead until just now." He waited a moment, then said harshly, "Make conversation, Solum."
_We lost about a third of our strange people. Every one of them within two hundred feet of that blast of yours._
"May they rest in peace." He looked up at the looming green man. "I meant that, Solum. It wasn't just a line."
_I know._
A silence. "I haven't felt like this since I was kicked out of school for eating ants."
_What did you do that for?_
"Ask my crystals. While they operate they cause a hell of a formic acid deficiency. I don't know why. I couldn't keep away from 'em." He sniffed. "I can smell 'em now." He bent, sniffed again. "Got a light?"
Solum handed him a lighter, flaming. "Thought so," said Horty. "Stepped smack on an anthill." He took up a pinch of the hill and sifted it on his palm. "Black ants. The little brown ones are much better." Slowly, almost reluctantly, he turned his hand over and dropped the rubble. He dusted his hands.
_Come on over to the mess tent, Horty._
"Yeah." He rose. On his face was a dawning perplexity. "No, Solum. You go ahead. I got something to do."
Solum shook his head sadly and strode off. Horty went back into the trailer, felt his way to the back wall where the Maneater had kept his laboratory racks. "Ought to have some here," he muttered, switching on the light. Muriatic, sulphuric, nitric, acetic—ah, here we go." He took down the bottle of formic acid and opened it. He found a swab, wet it in the acid and touched it to his tongue. "That goes good," he muttered. "Now, what is this? A relapse?" He lifted the swab again.
"That smells _so_ good! What is it? Could I have some?"
Horty bit his tongue violently, and whirled.
She came into the light, yawning. "Of all the crazy places for me to go to sleep... Horty! What's the matter? You're—are you crying?" Zena asked.
"Me? Never," he said. He took her into his arms and sobbed. She cradled his head and sniffed at the acid.
After a time, when he had quieted, and when she had a swab of her own, she asked, "What is it, Horty?"
"I have a lot to tell you," he said softly. "Mostly it's about a little girl who was an undesirable alien until she saved a country. Then there was a sort of international citizen's committee that saw to it that she got her first papers, and her husband as well. It's quite a story. Real artistic..."
# 17
PART OF A LETTER:
...in the hospital just resting up, Bobby Baby. I guess I just cracked under the strain. I don't remember a thing. They tell me I walked out of the store one evening and was found wandering four days later. Nothing had happened to me, really nothing, Bob. It's a weird thing to look back on—a hole in your life. But I'm none the worse for wear.
But here's some good news. Old Crawly-Fingers Bluett died of a heart attack at the carnival.
My job at Hartford's is waiting for me whenever I get back. And listen—remember the wild tale about the young guitarist that lent me $300 that awful night? He sent a note around to Hartford's for me. It said he had just inherited a business worth two million and I was to keep the money. I just don't know what to do. No one knows where he is or anything about him. He's left town permanently. One of the neighbors told me he had two little daughters. Anyway he had two little girls with him when he left. So the money's in the bank and Daddy's legacy in the bag.
So don't worry. Specially about me. As for those four days, they didn't leave a mark on me; well, a little bruise on one cheek, but that's nothing. They were probably good days. Sometimes when I'm waking up, I have a feeling—I can almost put my finger on it—it's sort of a half memory about loving somebody who was very, very good. But maybe I made that up. Now you're laughing at me...
# A Biography of Theodore Sturgeon
Theodore Hamilton Sturgeon (1918–1985) is the acclaimed author of eleven novels and more than two hundred short stories. Considered to be among the most influential writers of science fiction's "Golden Age," he won the International Fantasy Award for his novel _More Than Human_ , and the Hugo and Nebula Awards for his short story "Slow Sculpture."
Born Edward Hamilton Waldo in Staten Island, New York, Sturgeon was the son of Edward Molineaux Waldo, a paint and dye manufacturer, and Christine Hamilton Waldo, a teacher. At the age of eleven, following his mother's remarriage, his name was legally changed to Theodore Sturgeon.
Sturgeon began writing stories and poems during the three years he spent working as an engine room laborer on a freighter. Beginning in 1938, he published short stories for genre and general market publications including _Astounding_ (now _Analog Science Fiction and Fact_ ), _Unknown Worlds of Science Fiction_ , and _Argosy_. His groundbreaking short story "The World Well Lost" (1953), which was among the first science fiction stories to include positive themes of homosexuality, went on to win the Gaylactic Spectrum Award in 2000.
Sturgeon's 1953 novel _More Than Human_ was considered groundbreaking for science fiction in its stylistic daring, fine characterization, and visionary impact. Offering the idea that the next step in human evolution was a gestalt organism composed of people with different and strange talents who "bleshed," _More Than Human_ was an inspiration to many in the 1960s counterculture, including artists and musicians such as the Grateful Dead and Crosby, Stills and Nash.
In the 1960s, Sturgeon ventured into television writing, penning the screenplays for two of the most popular _Star Trek_ episodes: "Shore Leave" (1966) and "Amok Time" (1967). He is credited with inventing the story of Spock's sex life, as well as the famous Vulcan greeting, "Live long and prosper," and (with Leonard Nimoy) its accompanying hand signal. Two of Sturgeon's stories were adapted for _The New Twilight Zone_ , and his novella _Killdozer!_ (1944) became a television movie in 1974. He is also the creator of Sturgeon's Law—90 percent of everything is crap—which he developed to counter the common denigration of science fiction as a genre.
Beloved by critics and readers alike, Sturgeon inspired a generation of authors across genres, such as Samuel R. Delany, Michael Chabon, Jonathan Lethem, Octavia E. Butler, Karen Joy Fowler, and Rad Bradbury. Kurt Vonnegut considered Sturgeon to be one of the best writers in America, and Sturgeon served as inspiration for Vonnegut's recurring character, Kilgore Trout.
Survived by his seven children, Sturgeon died in Eugene, Oregon, on May 8, 1985. In 2000, he was posthumously inducted into the Science Fiction Hall of Fame.
The decree wherein Sturgeon is officially adopted by his stepfather (William "Argyll" D. Sturgeon) and his mother, and his last name is changed accordingly, from "Waldo" to "Sturgeon." (Photo courtesy of Special Collections, Kenneth Spencer Research Library, University of Kansas.)
Sturgeon's report card from the Pennsylvania State Nautical Schoolship "Annapolis" postmarked April 10, 1937, showing his rank as last in his class of cadets. (Photo courtesy of Special Collections, Kenneth Spencer Research Library, University of Kansas.)
Sturgeon with his third wife, Marion McGahan, and (left to right) daughter Tandy (b. 1954), son Robin (b. 1952), and daughter Noël (b. 1956).
A typescript page from _More Than Human_ with handwritten edits. (Photo courtesy of Special Collections, Kenneth Spencer Research Library, University of Kansas.)
A page from Sturgeon's "inspiration file." (Photo courtesy of Special Collections, Kenneth Spencer Research Library, University of Kansas.)
Notes, dated 6/12/66 and entitled "Spock Blows Top," for an episode of _Star Trek_ that Sturgeon wrote, ultimately titled "Amok Time." (Photo courtesy of Special Collections, Kenneth Spencer Research Library, University of Kansas.)
Sturgeon in his library.
The Theodore Sturgeon Memorial Award trophy, designed by Elden Tefft. The Gunn Center for the Study of Science Fiction at the University of Kansas presents the award annually for the genre's best short stories of the year. (Photo courtesy of the Gunn Center for the Study of Science Fiction.)
From left to right, author N. K. Jemisin; Noël Sturgeon, Theodore's daughter and trustee of the Theodore Sturgeon Literary Trust; and author Samuel R. Delany at a May 4, 2011, tribute to Sturgeon, sponsored by the _New York Review of Science Fiction._ (Photo courtesy of Marc Blackman.)
All rights reserved under International and Pan-American Copyright Conventions. By payment of the required fees, you have been granted the non-exclusive, non-transferable right to access and read the text of this ebook onscreen. No part of this text may be reproduced, transmitted, downloaded, decompiled, reverse engineered, or stored in or introduced into any information storage and retrieval system, in any form or by any means, whether electronic or mechanical, now known or hereinafter invented, without the express written permission of the publisher.
This is a work of fiction. Names, characters, places, and incidents either are the product of the author's imagination or are used fictitiously. Any resemblance to actual persons, living or dead, businesses, companies, events, or locales is entirely coincidental.
Copyright © 1950 by Theodore Sturgeon
Copyright renewed 1978 by the Theodore Sturgeon Literary Trust
Cover design by Mauricio Diaz
978-1-4532-9542-7
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**THE COMPLETE STORIES SERIES**
I. The Ultimate Egoist (1994)
II. Microcosmic God (1995)
III. Killdozer! (1996)
IV. Thunder and Roses (1997)
V. The Perfect Host (1998)
VI. Baby Is Three (1999)
VII. A Saucer of Loneliness (2000)
VIII. Bright Segment (2002)
IX. And Now the News... (2003)
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XI. The Nail and the Oracle (2007)
XII. Slow Sculpture (2009)
XIII. Case and the Dreamer (2010)
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| {
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\section{Introduction}
Regime switching models have been used extensively in econometric time series analysis. In most of these models, two regimes are introduced with a state process determining one of the regimes to take place in each period. The bivalued state process is typically modeled as a Markov chain. The autoregressive model with this type of Markov switching was first considered by Hamilton (1989), and later analyzed by Kim (1994).
Markov-switching models with endogenous explanatory variables have
been considered by Kim (2004, 2009). The most authors assume that the Markov chain, which determine regimes, is completely independent from all other parts of the model. Diebold et al. (1994) and Kim (2009) considers a Markov-switching driven by a set of observed variables.
Chang et al. (2016) introduces a new approach to model regime switching using an autoregressive latent factor, which determines regimes depending upon whether it
takes a value above or below some threshold level.
Despite numerous generalizations of this type of models, there is still little known about their theoretical properties. For example, one of the problems is the likelihood ratio test and other tests for comparing two regime switching models, the second is the indication of regularity conditions for the efficiency of maximum likelihood estimator of unknown model parameters.
Various statistical properties of the model have been studied by Hansen (1992), Hamilton (1996), Garcia (1998), Timmermann (2000), and Cho and White (2007), among others. The overview of the literature is in monograph by Kim and Nelson (1999).
In order to solve many problems related to testing hypothesis or some estimator efficiency, it is enough to prove Central Limit Theorem.
In regime switching models the random variables, which distributions are controled by hidden Markov chains we have dependent variables.
The central limit theorem has been extended to the case of
dependent random variables by several authors. The conditions under which these theorems are stated either are very restrictive or involve conditional distributions, which makes them difficult to apply. Hoeffding and Robbins (1994) prove central limit theorems for sequences of dependent random variables of a certain special type which occurs frequently in mathematical statistics.
In this paper we prove Central Limit Theorem for random variables, which distributions are controled by hidden Markov chains. We prove that when a Markov chain is ergodic and random variables fullfiled Lindeberg's condition then the Central Limit Theorem is true.
\section{Asymptotic independence}
Let consider a process:
\begin{equation}
\left(S_{t},X_{t}\right)_{t=0}^{\infty}
\end{equation}
where
\begin{itemize}
\item $S_{t}$ is an unobservable hidden Markow chain with $N$ states;
\item realizations of process $X_{t}$ are observed;
\item the distribution of $X_{t}$ conditional on history $\boldsymbol{{\cal R}}_{t-1}=\left(x_0,\ldots, x_{t-1}\right) $ has a form:
\begin{equation}
f\left (\boldsymbol{x}_{t}\mid\boldsymbol{{\cal R}}_{t-1};\boldsymbol{\theta}\right)
= \sum_{j=1}^{l}f\left (\boldsymbol{x}_{t}\mid S_{t}=j,\boldsymbol{{\cal R}}_{t-1};\boldsymbol{\theta}\right)P\left (S_{t}=j\mid\boldsymbol{{\cal R}}_{t-1};\boldsymbol{\theta}\right). \label{distr}\end{equation}
\end{itemize}
We prove the following lemma:
\begin{lemma}
Let consider a proces $\left(S_t, X_t\right)$ where the conditional distribution of $X_t\mid \boldsymbol{{\cal R}}_{t-1}$ is defined as \eqref{distr}. Let assume that $S_t$ is an ergodic process. Then the random variables$\{X_t\}$ are asymptotically independent, i.e.
\begin{equation}\label{as-ind}
\lim_{s_1,\ldots,s_k \rightarrow \infty} \left|P(\bigcap_{\nu=1}^k\{X_{t+\sum_{\rho=0}^{\nu}s_{\rho}}\in A_{\rho}\})-\prod_{\nu=1}^{k}P(X_{t+\sum_{\rho=0}^{\nu}s_{\rho}}\in A_{\rho})\right|=0
\end{equation}
\end{lemma}
{\bf Proof:}
At first we estimate the difference
\begin{multline*}\left|P(\bigcap_{\nu=1}^k\{X_{t+\sum_{\rho=0}^{\nu}s_{\rho}}\in A_{\rho}\})-\prod_{\nu=1}^{k}P(X_{t+\sum_{\rho=0}^{\nu}s_{\rho}}\in A_{\rho})\right|\\ \leq \left|P(\bigcap_{\nu=1}^k\{X_{t+\sum_{\rho=0}^{\nu}s_{\rho}}\in A_{\rho}\})-P(\{X_{t+\sum_{\rho=0}^{k}s_{\rho}}\in A_{\rho}\})P(\bigcap_{\nu=1}^{k-1}\{X_{t+\sum_{\rho=0}^{\nu}s_{\rho}}\in A_{\rho}\})\right|\\+ \left|P(\{X_{t+\sum_{\rho=0}^{k}s_{\rho}}\in A_{\rho}\}P(\bigcap_{\nu=1}^{k-1}\{X_{t+\sum_{\rho=0}^{\nu}s_{\rho}}\in A_{\rho}\})-\prod_{\nu=1}^{k}P(X_{t+\sum_{\rho=0}^{\nu}s_{\rho}}\in A_{\rho})\right|=I_1+I_2\end{multline*}
\begin{multline*}I_1=\left|P(\bigcap_{\nu=1}^k\{X_{t+\sum_{\rho=0}^{\nu}s_{\rho}}\in A_{\rho}\})-P(\{X_{t+\sum_{\rho=0}^{k}s_{\rho}}\in A_{\rho}\})P(\bigcap_{\nu=1}^{k-1}\{X_{t+\sum_{\rho=0}^{\nu}s_{\rho}}\in A_{\rho}\})\right|\\=
\left|P(\bigcap_{\nu=1}^k\{X_{t+\sum_{\rho=0}^{\nu}s_{\rho}}\in A_{\rho}\}|\bigcap_{\nu=1}^{k-1}\{X_{t+\sum_{\rho=0}^{\nu}s_{\rho}}\in A_{\rho}\})P(\bigcap_{\nu=1}^{k-1}\{X_{t+\sum_{\rho=0}^{\nu}s_{\rho}}\in A_{\rho}\})\right.\\-\left.P(\{X_{t+\sum_{\rho=0}^{k}s_{\rho}}\in A_{\rho}\})P(\bigcap_{\nu=1}^{k-1}\{X_{t+\sum_{\rho=0}^{\nu}s_{\rho}}\in A_{\rho}\})\right|\\ \leq \left|P(\bigcap_{\nu=1}^k\{X_{t+\sum_{\rho=0}^{\nu}s_{\rho}}\in A_{\rho}\}\mid\{X_{t+\sum_{\rho=0}^{k-1}s_{\rho}}\in A_{\rho}\})-P(\{X_{t+\sum_{\rho=0}^{k}s_{\rho}}\in A_{\rho}\})\right|
\end{multline*}
Denote\begin{itemize}
\item$ T=t+\sum_{\rho=0}^{k-1}\tau{\rho}$
\item $\tau=\tau_k$
\item $A=A_k$
\item $B=A_{k-1}$
\end{itemize}
We notice, that \[P(X_{T+\tau}\in A, X_T\in B)=P(X_{T+\tau}\in A|X_T\in B)P(X_T\in B)\]
Moreover for $A=\{s_1,\ldots, s_p\}\times \mathfrak{A}, B=\{s_0\}\times\mathfrak{B}$ \begin{equation}\label{war}
P(X_{T+\tau}\in A |X_T\in B)=\sum_{\iota=1}^{p}p_{s_0,s_{\iota}}(\tau)\int_{\mathfrak{A}}f_{\iota}(x)dx
\end{equation}
and
\begin{multline} \label{bezwar}
P(X_{T+\tau}\in A )=\sum_{\sigma \in\boldsymbol{{\cal S}}}P(X_{T+\tau}\in A \mid X_T\in \{\sigma\}\times \mathbb{R}^N)P(X_T\in \{\sigma\}\times \mathbb{R}^N) \\= \sum_{\sigma\in \boldsymbol{{\cal S}}}\sum_{\iota=1}^{p}p_{\sigma,s_{\iota}}(\tau)\int_{\mathfrak{A}}f_{\iota}(x)dxP(X_T\in \{\sigma\}\times \mathbb{R}^N)
\end{multline}
From the ergodic theorem \cite{} follows, that \[\lim_{s\rightarrow \infty}p_{ij}(s)=p_j^*(s)\] and\[ |p_{ij}(s)-p_j^*(s)|\leq \alpha^s \] for some $\alpha <1$.
From \eqref{bezwar}, \eqref{war} and the obvious equality \[\sum_{\sigma\in \boldsymbol{{\cal S}}}P(X_T\in \{\sigma\}\times \mathbb{R}^N)=1\] follows, that
\begin{multline*}
P(X_{T+\tau}\in A |X_T\in B)- P(X_{T+\tau}\in A ) \\ =\sum_{\iota=1}^{p}p_{s_0,s_{\iota}}(\tau)\int_{\mathfrak{A}}f_{\iota}(x)dx-\sum_{\sigma\in \boldsymbol{{\cal S}}}\sum_{\iota=1}^{p}p_{\sigma,s_{\iota}}(\tau)\int_{\mathfrak{A}}f_{\iota}(x)dxP(X_T\in \{\sigma\}\times \mathbb{R}^N) \\=\sum_{\iota=1}^{p}[p_{s_0,s_{\iota}}(\tau)-p_{s_{\iota}}^*(\tau)]\int_{\mathfrak{A}}f_{\iota}(x)dx +\sum_{\sigma\in \boldsymbol{{\cal S}}}\sum_{\iota=1}^{p}[p_{s_{\iota}}^*(\tau)-p_{\sigma,s_{\iota}}(\tau)]\int_{\mathfrak{A}}f_{\iota}(x)dxP(X_T\in \{\sigma\}\times \mathbb{R}^N)
\end{multline*} and in consequence \[| P(X_{T+\tau}\in A |X_T\in B)- P(X_{T+\tau}\in A )|\leq 2 \alpha^{\tau}\]
Hence $I_1\leq 2 \alpha^{\tau}$. Now. notice, that
\begin{multline*}
I_2=\left|P(\{X_{t+\sum_{\rho=0}^{k}\tau_{\rho}}\in A_{\rho}\})P(\bigcap_{\nu=1}^{k-1}\{X_{t+\sum_{\rho=0}^{\nu}\tau_{\rho}}\in A_{\rho}\})-\prod_{\nu=1}^{k}P(X_{t+\sum_{\rho=0}^{\nu}\tau_{\rho}}\in A_{\rho})\right|\\= P(\{X_{t+\sum_{\rho=0}^{k}\tau_{\rho}}\in A_{\rho}\})\left|P(\bigcap_{\nu=1}^{k-1}\{X_{t+\sum_{\rho=0}^{\nu}\tau_{\rho}}\in A_{\rho}\})-\prod_{\nu=1}^{k-1}P(X_{t+\sum_{\rho=0}^{\nu}\tau_{\rho}}\in A_{\rho})\right|\\ \leq \left|P(\bigcap_{\nu=1}^{k-1}\{X_{t+\sum_{\rho=0}^{\nu}\tau_{\rho}}\in A_{\rho}\})-\prod_{\nu=1}^{k-1}P(X_{t+\sum_{\rho=0}^{\nu}\tau_{\rho}}\in A_{\rho})\right|
\end{multline*}
By simple induction we can conclude, that \[\left|P(\bigcap_{\nu=1}^k\{X_{t+\sum_{\rho=0}^{\nu}\tau_{\rho}}\in A_{\rho}\})-\prod_{\nu=1}^{k}P(X_{t+\sum_{\rho=0}^{\nu}\tau_{\rho}}\in A_{\rho})\right| \leq k\alpha^{\sum_{\rho=0}^{k}\tau_{\rho}}\] which completes the proof.
\section{Property of $\varepsilon$-independence}
Next, let define the notion of $\varepsilon$-independence. This concept will be useful to prove central limit theorem.
\begin{defn}
The random variables sequence $\{X_k\}_{k\in \mathbb{N}}$ are $\varepsilon$-independent, when for any $n\in\mathbb{N}$ and any sets $A_1,\ldots,A_n$ we have the following inequality:
\begin{equation} \label{eps-indep}
|P(X_1\in A_1, \ldots, X_n\in A_n)- P(x_1\in A_1)\ldots P(X_n\in A_n)|\leq\varepsilon.
\end{equation}
\end{defn}
We prove, that for $\varepsilon$-independent variables the following lemma is true.
\begin{lemma}
When random variables $X_1, \ldots, X_n$ are $\varepsilon$-independent, then \[ |\varphi_{X_1+\ldots+X_n}(t)-\varphi_{X_1}(t)\ldots\varphi{X_n}(t)|\leq 2\varepsilon. \]
\end{lemma}
{\bf Proof:}
From the formula \eqref{eps-indep} follows, that
\[|E(\mathbbm{1}_{A_1}(X_1)\ldots\mathbbm{1}_{A_n}(x_n))-E\mathbbm{1}_{A_1}(X_1)\ldots E\mathbbm{1}_{A_n}(x_n)|\leq \varepsilon.\]
Since every coninuous function can be approximatad by simple functions consider at first the real function \begin{equation} \label{prosta}
f=\sum_{j=1}^{m}c_{j}\mathbbm{1}_{A_{j}},
\end{equation} where the sets $A_j$ are pairwise disjoint and $|c_j|\leq 1$.
In the first step, we estimate the real part of $\left(Ef(X_1)\cdots f(X_n)-Ef(X_1)\cdots Ef(X_n)\right)$. Since $|c_{j}|\leq 1$ we get that:
\begin{multline*}
\Re(Ef(X_1)\cdots f(X_n)-Ef(X_1)\cdots Ef(X_n)) \\ = \sum_{j_1,j_2,\ldots, j_n=1}^n\Re(c_{j_1}c_{j_2}\cdots c_{j_n})\left[P(X_1\in A_{j_1}, \ldots , X_n\in A_{j_n})-P(X_1\in A_{j_1})\cdots P(X_n\in A_{j_n}) \right] \\ \leq \sum_{j_1,j_2,\ldots, j_n=1}^n\left[P(X_1\in A_{j_1}, \ldots , X_n\in A_{j_n})-P(X_1\in A_{j_1})\cdots P(X_n\in A_{j_n}) \right].
\end{multline*}
Since the sets $A_j$ are pairwise disjoint then:
\[\sum_{j_1,j_2,\ldots, j_n=1}^nP(X_1\in A_{j_1}, \ldots , X_n\in A_{j_n})=P(X_1\in A, \ldots, x_n\in A),\]
\[\sum_{j=1}^{m}P(X_k\in A_j)=P(X_k\in A),\]
where \[ A=\bigcup_{j=1}^m A_j.\]
From it follows, that
\[\Re(Ef(X_1)\cdots f(X_n)-Ef(X_1)\cdots Ef(X_n)) \leq P(X_1\in A, \ldots, x_n\in A)- P(X_1\in A) \cdots P(X_n\in A)\leq \varepsilon. \]
Analogously
\[\Re(Ef(X_1)\cdots f(X_n)-Ef(X_1)\cdots Ef(X_n)) \geq -P(X_1\in A, \ldots, x_n\in A)+ P(X_1\in A) \cdots P(X_n\in A)\geq -\varepsilon \]
In the second step, we estimate the imaginary part of $\left(Ef(X_1)\cdots f(X_n)-Ef(X_1)\cdots Ef(X_n)\right)$. So, we have:
\[\Im(Ef(X_1)\cdots f(X_n)-Ef(X_1)\cdots Ef(X_n)) \leq P(X_1\in A, \ldots, x_n\in A)- P(X_1\in A) \cdots P(X_n\in A)\leq \varepsilon \]
\[\Im(Ef(X_1)\cdots f(X_n)-Ef(X_1)\cdots Ef(X_n)) \geq -P(X_1\in A, \ldots, x_n\in A)+ P(X_1\in A) \cdots P(X_n\in A)\geq -\varepsilon. \]
In consequence we obtain:
\[|Ef(X_1)\cdots f(X_n)-Ef(X_1)\cdots Ef(X_n)| \leq \sqrt{2}\varepsilon. \]
The last inequality is true for any $X_1, \ldots, X_n$ for any $n$ and for any $f$ Now, let fix $t\in \mathbb{R}$, $n$ and $\eta>0$. Let define
the function $f_{\eta}$ which can be presented as \eqref{prosta} and satisfying the inequality \[|f_{\eta}(x)-e^{itx}|\leq \eta\] for every $x\in \mathbb{R}$.
Let estimate the difference
\begin{equation}\varphi_{X_1+\ldots+X_n}(t)- Ef_{\eta}(X_1)\cdots f_{\eta}(X_n)
\label{sacowanie}
\end{equation}
This difference is equal to \begin{multline}
Ee^{itX_1}\cdots e^{itX_n}-Ef_{\eta}(X_1)\cdots f_{\eta}(X_n) \\ = E(e^{itX_1}-f_{\eta}(X_1))e^{itX_2}\cdots e^{itX_n}+ Ef_{\eta}(X_1)(e^{itX_2}-f_{\eta}(X_2))e^{itX_3}\cdots e^{itX_n}+ \ldots \label{rownosc}
\end{multline}
Each component of the sum \eqref{rownosc} has the form $ EZ(e^{itX_k}- f_{\eta}(X_k)) $ where $|Z|\leq 1$. From this fact implies that \[|Ee^{itX_1}\cdots e^{itX_n}-Ef_{\eta}(X_1)\cdots f_{\eta}(X_n)|\leq n\eta.\]
By conducting a similar reasoning, we obtain that:
\[|Ee^{itX_1 +\cdots +itX_n}-E(f_{\eta}(X_1)\cdots f_{\eta}(X_n))|\leq n\eta.\]
From it follows, that \begin{multline*}
|Ee^{itX_1 +\cdots +itX_n}-Ee^{itX_1}\cdots e^{itX_n}| \\ \leq |Ee^{itX_1 +\cdots +itX_n}-E(f_{\eta}(X_1)\cdots f_{\eta}(X_n))|\\+Ef(X_1)\cdots f(X_n)-Ef(X_1)\cdots Ef(X_n)|+Ee^{itX_1}\cdots e^{itX_n}-Ef_{\eta}(X_1)\cdots f_{\eta}(X_n)| \\ \leq \sqrt{2}\varepsilon+ 2\eta n
\end{multline*}
Choosing such $\eta$, that $2n\eta\leq 2\varepsilon - \sqrt{2}\varepsilon$ we complete the proof.
\section{Central Limit Theorem }
Using the thesis of the above lemma, one can prove a central limit theorem for $\varepsilon$-independent variables.
\begin{theorem}
Let assume that for each $ n $, $ X_{{1n}}, X_{{2n}}, \ldots, X_{{r_{n}n}} $ are $\varepsilon$-independent random variables with expected value equal $ 0 $ and:
\begin{equation}
\sum _{{k=1}}^{{r_{n}}}\mathbb{E}X_{{kn}}^{2}\xrightarrow{n\to\infty}1.\label{porownanie} \end{equation}
Additionally, let us assume that Lindeberg's condition is fulfilled:
\[\sum _{{k=1}}^{{r_{n}}}\mathbb{E}X_{{kn}}^{2}1_{{\{|X_{{kn}}|>\eta\}}}\xrightarrow{n\to\infty}0\quad\mbox{for each }\eta>0.\]
Let $Y_n$ be a sum:
\[= X_{{1n}}+X_{{2n}}+\ldots+X_{{r_{n}n}}.\]
Then
\[\limsup|\varphi_{Y_n}(t)-e^{-\frac{1}{2}t^2}|<\varepsilon.\]
\end{theorem}
{\bf Proof:}
For Lindeberg Theorem (Loeve, 1977) implies that:
\[\lim_{n\rightarrow \infty}\varphi_{X_{1n}}(t)\ldots\varphi_{X_{r_nn}}(t)=e^{-\frac{1}{2}t^2}\]
From last Lemma implies that: \[|\varphi_{Y_n}(t)-\varphi_{X_{1n}}(t)\ldots\varphi_{X_{r_nn}}(t)|<\varepsilon.\]
Now, our goal is to prove the fact that asymptotically independent variables also satisfy the Central Limit Theorem.
\begin{theorem}
Assume, that
\begin{enumerate}
\item Random Variables $X_1, \ldots,X_n$ are asymptoticaly independent i.e., they satisfy \eqref{as-ind}.
\item $EX_i=0$ for $i=1,\ldots,n.$
\item \begin{equation}
E\left(\frac{1}{\sqrt{n}}\sum_{k=1}^n X_{i+k}\right)^2 \rightarrow 1 \text{ uniformly in }i. \label{wariancja}\end{equation}
\end{enumerate}
Then, the sequence of the distributions of the random variables \[ \frac{1}{\sqrt{v}}(X_1+\cdots+X_n)\]
tend to the standard normal distribution.
\end{theorem}
{\bf Proof:}
At first we fix some arbitrary $ \varepsilon >0 $. From \eqref{as-ind} follows, that there exists such $m$, that \[|P(X_{k_1}\in A_1,\ldots, X_{k_r}\in A_r)=P(X_{k_1}\in A_1)\ldots P(x_{k_r}\in A_r)|\leq \varepsilon,\] where $k_{j+1}>k_j+m.$
\\
Fix $n$ and let $0<\alpha<\frac{1}{4}$. Let define $ k=[n^{\alpha}] $ and $ \nu=\left[\frac{k}{n}\right] $, so clearly $k\leq n^{\alpha}$ and $n=k\nu+r$. \\
Let define:
\[U_i=X_{ik-k+1}+\ldots + X_{ik-m}.\]
Because $(i+1)k-k+1-(ik-m)=m+1$ so $U_i$ are $\varepsilon$-independent. Let us consider a sum:
\[X_1+\ldots+X_n. \]
This sum we can seperate into two parts: a sum of $U_1+\ldots, U_{\nu}$ and the rest. We can notice that each component $U_i$ includes $k+1-m$ elements, so the sum $U_1+\ldots +U_{\nu}$ consists of $(k+1-m)\nu= n-r-(m-1)\nu$ elements. So the rest includes $r+(m-1)\nu$ ingredients, which we denote as $Z_1^{(n)},\ldots,Z_p^{(n)}$, for fixed $n$. So:
\[X_1+\ldots +X_n=U_1+\ldots+ U_{\nu} +Z_1^{(n)}+\ldots Z_p^{(n)}=\sqrt{n}\mathfrak{U}_n+\sqrt{n}\mathfrak{Z}_n.\]
From Schwarz and H\"older inequality (Vuong, 1989) implies that:
\[E(Z_1^{(n)}+\ldots +Z_p^{(n)})^2\leq p^2R^2,\]
where $R=E|X_i|^3$, so:
\[E(\frac{1}{\sqrt{n}}(Z_1^{(n)}+\ldots+Z_p^{(n)}))^2\leq \frac{p^2}{n}R^2.\]
From a fact that $p=r+(m-1)\nu\leq m\nu \leq m\frac{k}{n}\leq m\frac{n^{\alpha}}{n}$ implies that
$\frac{p^2}{n}\leq m^2\frac{n^{2\alpha}}{n^3}$.
So, the sum \[\sum_{n=1}^{\infty}\frac{p^2}{n}R^2\]
is consistent, so with probability $1$, we have:
\begin{equation} \label{z}
\lim_{n\rightarrow \infty}\frac{1}{\sqrt{n}}(Z_1^{(n)}+\ldots+Z_p^{(n)})=0.
\end{equation}
Now, we need estimate the
\[\frac{1}{\sqrt{\nu}}(U_1^{(n)}+\ldots+U_{\nu}^{(n)})\] \[\frac{1}{\sqrt{n}}(U_1^{(n)}+\ldots+U_{\nu}^{(n)})=\frac{\sqrt{k\nu}}{\sqrt{n}}\frac{1}{\sqrt{k\nu}}(U_1^{(n)}+\ldots+U_{\nu}^{(n)})=\frac{\sqrt{k\nu}}{\sqrt{n}}\frac{1}{\sqrt{\nu}}(\frac{1}{\sqrt{k}}U_1^{(n)}+\ldots+\frac{1}{\sqrt{k}}U_{\nu}^{(n)}).
\] Moreover
\[U_i^{(n)}=X_{ik-k+1}+\ldots + X_{ik-m},\] from which follows, that
\[E\left(\frac{1}{\sqrt{k}}U_i^{(n)}\right)^2=\frac{k-m}{k}E\left(\frac{1}{\sqrt{k-m}}U_i^{(n)}\right)^2.\]
From the assumption \eqref{wariancja}follows, that
\[\lim_{k\rightarrow \infty}E\left(\frac{1}{\sqrt{k}}U_i^{(n)}\right)^2=1.\] In consequencewe obtain that:
\[\lim_{n\rightarrow \infty}\frac{1}{\nu}\left(\sum_{i=1}^{\nu}E\left(\frac{1}{\sqrt{k}}U_i^{(n)}\right)^2\right)=1.\]
Having above estimates we go to the next step of the proof. We have to estimate the difference:
\begin{equation}\left| \varphi_{\frac{1}{\sqrt{n}}(X_1+\ldots+X_n)}(t)- e^{-\frac{t^2}{2}}\right|.
\end{equation}
So,
\begin{multline*}\left| \varphi_{\frac{1}{\sqrt{n}}(X_1+\ldots+X_n)}(t)- e^{-\frac{t^2}{2}}\right|=
\left| \varphi_{\mathfrak{U}_n+\mathfrak{Z}_n}(t)- e^{-\frac{t^2}{2}}\right|=\\
= \left| \varphi_{\mathfrak{U}_n+\mathfrak{Z}_n}(t) -\varphi_{\mathfrak{U}_n}(t)+\varphi_{\mathfrak{U}_n}(t) - e^{-\frac{t^2}{2}}\right| \leq
\left|\varphi_{\mathfrak{U}_n+\mathfrak{Z}_n}(t) -\varphi_{\mathfrak{U}_n}(t)\right|+\left|\varphi_{\mathfrak{U}_n}(t) - e^{-\frac{t^2}{2}}\right|\\
\leq E\left|e^{it(\mathfrak{U}_n+\mathfrak{Z}_n)} -e^{it\mathfrak{U}_n}\right|+\left|\varphi_{\mathfrak{U}_n}(t) - e^{-\frac{t^2}{2}}\right|=E\left|e^{it(\mathfrak{U}_n}(e^{it\mathfrak{Z}_n)} -1)\right|+\left|\varphi_{\mathfrak{U}_n}(t) - e^{-\frac{t^2}{2}}\right|\\ \leq E\left|e^{it\mathfrak{Z}_n} -1\right|+\left|\varphi_{\mathfrak{U}_n}(t) - e^{-\frac{t^2}{2}}\right|
\end{multline*}
From \eqref{z} follows, that \[\lim_{n \rightarrow \infty }E\left|e^{it\mathfrak{Z}_n} -1\right|=0\] and from lemma follows, that \[\limsup_{n \rightarrow \infty }\left|\varphi_{\mathfrak{U}_n}(t) - e^{-\frac{t^2}{2}}\right|\leq 2 \varepsilon\]
Then
\[\limsup_{n \rightarrow \infty }\left| \varphi_{\frac{1}{\sqrt{n}}(X_1+\ldots+X_n)}(t)- e^{-\frac{t^2}{2}}\right|\leq 2 \varepsilon\]
Since the last estimation can be proved for any $\varepsilon$
\[\lim_{n \rightarrow \infty }\left| \varphi_{\frac{1}{\sqrt{n}}(X_1+\ldots+X_n)}(t)- e^{-\frac{t^2}{2}}\right|=0\]
which completes the proof.
\section*{Conclusion}
I this paper we prove a Central Limit Theorem. The assumptions to this thorem is not restrictive and they are not difficult to apply in the practical. We proved that when a Markov chain is ergodic and random variables fullfiled Lindeberg's condition then the Central Limit Theorem is true.
\section*{References}
\[\]
Garcia, R., (1998), Asymptotic null distribution of the likelihood ratio test in
Markov switching
models. International Economic Review 39, 763-88. \\
Hamilton, J., (1989), A new approach to the economic analysis of nonstationary
time series
and the business cycle. Econometrica 57, 357-384.\\
Hamilton, J., (1996), Specification testing in Markov-switching time-series models.
Journal
of Econometrics 70, 127-157. \\
Hansen, B. E., (1992), The likelihood ratio test under non-standard conditions.
Journal of Applied Econometrics 7, S61-82.\\
Hoeffding W., Robbins H. ,(1994), The Central Limit Theorem for Dependent
Random Variables. In: Fisher N.I., Sen P.K. (eds) The Collected Works
of Wassily Hoeffding. Springer Series in Statistics. Springer, New York. \\
Kim, C.-J., (2009), Markov-switching models with endogenous explanatory
variables II: A two-step MLE procedure. Journal of Econo
metrics 148, 46-55. \\
Kim, C.-J., Nelson, C., (1999), State-Space Models with Regime Switching.
MIT Press, Cambridge, MA.\\
Loeve, M.,(1977), Probability theory, Springer. \\
Timmermann, A., (2000), Moments of Markov switching models. Journal of Econo
metrics 96, 75-111. \\
Vuong Q. H., (1989),
Likelihood ratio tests for model selection and non-nested
hypotheses, Econometrica 57, 307 - 333.
\end{document}
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Sir Robert Hamilton
Former MP for Orkney and Shetland
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Oral Answers to Questions — Post Office.: Wireless Service (Shetland Islands). 31 Jul 1935
69. asked the Postmaster-General the position regarding the application by the Zetland County Council for the installation of wireless communication between the islands of Papa Stour and Foula and the mainland of Shetland?
Oral Answers to Questions — Scotland.: Herring Industry (Advertisement). 30 Jul 1935
5. asked the Secretary of State for Scotland what steps have been taken by the Herring Board to carry out a scheme of advertisement in foreign countries?
Orders of the Day — Pensions (Governors of Dominions, etc.) [Money]. 26 Jul 1935
I do not think the Committee need hesitate to allow this financial resolution to go through. The Bill that will be founded on it is merely to fill up certain gaps in legislation already passed.
More of Sir Robert Hamilton's recent appearances
Entered the House of Commons on 15 November 1922 — General election
Left the House of Commons on 25 October 1935 — General election
Early Day Motions signed by this MP
Previous MPs in this constituency
John Wason
Leonard Lyell
Samuel Laing
Frederick Dundas
Arthur Anderson
Thomas Balfour
George Traill
George Dundas
John Balfour
Richard Honyman
Malcolm Laing
Robert Honyman
Future MPs in this constituency
Basil Neven-Spence
Jo Grimond
Jim Wallace
Alistair Carmichael
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This data was produced by TheyWorkForYou from a variety of sources. Voting information from Public Whip. | {
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|}
Die Überlandstraßenbahn Puerto Supe–Alpas war eine eingleisige Bahnstrecke in der Spurweite von 600 mm im mittleren Peru in der heutigen Provinz Barranca.
Geografische Lage
Die Strecke verband den Hafen von Supe (Puerto Supe) an der Pazifikküste mit den Ortschaften Barranca und Alpas im Hinterland. Nördlich von Barranca verlief die Strecke parallel zur Trasse der – allerdings in Kapspur (1067 mm) ausgeführten – Bahnstrecke Puerto Supe–Paramonga, teils begleitet, teils bestand ein Dreischienengleis für Fahrzeuge beider Spurweiten. Die Überlandstraßenbahn Puerto Supe–Alpas stellte ihren Betrieb in den 1950er Jahren ein. Zwischen Barranca und Supe gab es in gleicher Verbindung ab 1928 eine dritte Trasse, eine Zweigstrecke der Bahnstrecke Ancón–Sayan, die in 914 mm-Spur ausgeführt und bis 1960 betrieben wurde.
Geschichte
Die Strecke zwischen Supe und Barranca ging 1903 in Betrieb, die Verlängerung 1909. Als Bezeichnung der Betreibergesellschaft werden genannt: Empresa del Ferrocarril Supe á Barranca und Ferrocarril de Supe á Barranca y Alpas. Von der 16 km langen Stammstrecke aus entwickelte sich bis 1927 ein Netz im Umfang von 54,5 km, das hauptsächlich angrenzenden landwirtschaftlichen Großbetrieben diente, die überwiegend Zuckerrohr anbauten. Der Betrieb des Netzes endete in den 1950er Jahren.
Zweigstrecken
An diese Strecke schlossen sich Dreischienengleise an, die von anderen Bahnen verwaltet wurden, zur Zuckerfabrik Paramonga und zur Hacienda San Nicólas.
Einzelnachweise
Bahnstrecke in Peru
Bahnstrecke Puerto Supe-Alpas | {
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{"url":"https:\/\/gmatclub.com\/forum\/gmat-practice-test-quant-3-question-94076.html?kudos=1","text":"It is currently 24 Jun 2017, 08:49\n\n### GMAT Club Daily Prep\n\n#### Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.\n\nCustomized\nfor You\n\nwe will pick new questions that match your level based on your Timer History\n\nTrack\n\nevery week, we\u2019ll send you an estimated GMAT score based on your performance\n\nPractice\nPays\n\nwe will pick new questions that match your level based on your Timer History\n\n# Events & Promotions\n\n###### Events & Promotions in June\nOpen Detailed Calendar\n\n# GMAT Practice Test Quant #3 - question 3\n\nAuthor Message\nIntern\nJoined: 05 Apr 2010\nPosts: 6\nGMAT Practice Test Quant #3 - question 3\u00a0[#permalink]\n\n### Show Tags\n\n11 May 2010, 15:15\nHello,\n\nI hope someone can help me on this, because I really can't get the correct answer. On GMAT practice test Quant #3, the third question is the following data sufficiency question:\n\nIn a demographic study, the population and total income of a certain region were estimated from other data, and both estimates had lower and upper limits. At the time of the estimates, was the per capita income for the region greater than $16,500? (1) the lower limit for the estimate of the population was 330,000 people. (2) the lower limit for the estimate of the total income was$5,500,000,000.\n\nMy answer choice was C (Both statements), but when I reviewed all my questions, it seems that the correct answer is E (both are not sufficient). If you calculate the per capita income, using both statements, it gives that the lower possible income for this area is $16,666...., which is higher than$16,500. If this is not right, could someone explain me why?\n\nThank you!\nVP\nJoined: 05 Mar 2008\nPosts: 1469\nRe: GMAT Practice Test Quant #3 - question 3\u00a0[#permalink]\n\n### Show Tags\n\n11 May 2010, 15:38\npepemelo wrote:\nHello,\n\nI hope someone can help me on this, because I really can't get the correct answer. On GMAT practice test Quant #3, the third question is the following data sufficiency question:\n\nIn a demographic study, the population and total income of a certain region were estimated from other data, and both estimates had lower and upper limits. At the time of the estimates, was the per capita income for the region greater than $16,500? (1) the lower limit for the estimate of the population was 330,000 people. (2) the lower limit for the estimate of the total income was$5,500,000,000.\n\nMy answer choice was C (Both statements), but when I reviewed all my questions, it seems that the correct answer is E (both are not sufficient). If you calculate the per capita income, using both statements, it gives that the lower possible income for this area is $16,666...., which is higher than$16,500. If this is not right, could someone explain me why?\n\nThank you!\n\nBecause both have upper and lower limits. The limit can be 330,000 < x < z\nAs z increases the per capita income decreases.\n\nIf the upper limit is 550,000 then the per capita income is 11,000\nIntern\nJoined: 05 Apr 2010\nPosts: 6\nRe: GMAT Practice Test Quant #3 - question 3\u00a0[#permalink]\n\n### Show Tags\n\n11 May 2010, 15:46\nOf course it is! I feel really stupid now... I just assumed that only the lower population number had the lower total income, which cannot be assumed by the question stem...\n\nThank you very much!\nRe: GMAT Practice Test Quant #3 - question 3 \u00a0 [#permalink] 11 May 2010, 15:46\nSimilar topics Replies Last post\nSimilar\nTopics:\nGMAT Practice Test Quant #4 - question 14 3 13 May 2010, 03:52\nHelp - Probabilty question 3 on test 25 3 06 Apr 2010, 12:04\ntest m02\/question 3 7 01 Dec 2009, 20:14\n1 Test 3, Question 14 2 27 Oct 2009, 04:56\n25 GMAT Diagnostic Test Question 3 33 22 Oct 2013, 23:26\nDisplay posts from previous: Sort by\n\n# GMAT Practice Test Quant #3 - question 3\n\nModerator: Bunuel\n\n Powered by phpBB \u00a9 phpBB Group and phpBB SEO Kindly note that the GMAT\u00ae test is a registered trademark of the Graduate Management Admission Council\u00ae, and this site has neither been reviewed nor endorsed by GMAC\u00ae.","date":"2017-06-24 15:49:58","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.5649750828742981, \"perplexity\": 2363.2137243843204}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2017-26\/segments\/1498128320264.42\/warc\/CC-MAIN-20170624152159-20170624172159-00601.warc.gz\"}"} | null | null |
La gare de Kropyvnytskyï (ukrainien : Кропивницький (станція)) est une gare ferroviaire située dans la ville de Kropyvnytskyï en Ukraine.
Situation ferroviaire
Histoire
La gare est ouverte en 1868, elle est sur la ligne Znamianka-Pomitchna-Odessa. Le bâtiment fut détruit lors de la seconde guerre mondiale pour être relevé en 1954. La ligne est électrifiée en 1971. Le nom actuel a été adopté en 2017.
Service des voyageurs
Accueil
Desserte
Intermodalité
Notes et références
Voir aussi
Articles connexes
Transport ferroviaire en Ukraine,
Liste de gares en Ukraine.
Liens externes
Gare dans l'oblast de Kirovohrad
Gare mise en service en 1868
Gare | {
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Saint Nicholas and the AI MYTH : About the Creation of Myths !
AInarrative
Katja Rausch December 6, 2021 9 min read
Saint Nicholas and the AI MYTH : About the Creation of Myths.
December 6, Saint Nicholas Day, is the perfect moment to ponder on the essence, the creation and the "sustainability" of myths.
What makes a myth? How is a myth created? What are its main characteristics?
And paradoxically, why demythifying is part of the development of a myth?
On the one hand, we have ancient sacred myths – like the myth of Saint Nicholas.
On the other hand, there are modern techno-myths. The most impactful at the moment: the AI MYTH. And a potentially emerging new augmented myth, the METAVERSE MYTH.
Ever since John McCarthy, Minsky, Shannon and Rochester coined the term "Artificial Intelligence" in their "pitch" for subvention for a Summer Workshop in Dartmouth in 1956, the narrative, rhetorical and semantic universes of AI have been crafted and defined. (The article "What if in 1956 Minsky, McCarthy, Shannon and Rochester had named it "Machine Intelligence"?" by the House of Ethics deals with this particular aspect.)
What is a myth?
In 1957 the French semiologist Roland Barthes (1915-1980) introduced his groundbreaking book "Mythologies" about the importance of signs and symbols through words and objects leading to ideological and cultural meanings and identities.
As a side reflection, he evokes the "Myth of Santa Claus", pointing at strange cognitive process leading us to believe in something even though we know it does not exist.
He calls this mechanism "Le mythe du Père Noël"- "The Myth of Santa Claus".
This cognitive process juggles between imagination, reality, beliefs, culture and personal values.
Notwithstanding facts that deny the existence of a fact, we still can believe in a constructed system with approximate values.
"We believe in Santa Claus even though we know he does not exist." An oscillation between signs and meanings.
In literature, analyzing the concept of the word myth and the process of creating a myth have a long tradition.
Fundamental questions being "what is a myth" and "how to create a myth"?
Professor Pierre Brunel, professor emeritus in comparative literature at the Sorbonne in Paris, leading specialist of myths, and author of the "Dictionary of literary myths", whose extraordinary courses I was lucky enough to attend, points at the "terminological vagueness" of the word myth.
The myth is polysemous and therefore multi-conceptual. A myth always conveys meanings. It nevertheless needs interpretation. It talks and explains. It is a narrative.
"A myth is words," says Roland Barthes. It is a narrative. A special kind of narrative.
According to the Romanian author and philosopher Mircea Eliade (1907-1986), a myth is foremost a Narrative of Creation. It relates the beginning, the genesis of something fabuluous. A phenomenon being narrated. For Eliade, it holds a sacred dimension.
Rhetorical figures to feed the mythical Narrative
Three rhetorical figures are particularly effective in creating and maintaining a myth.
1) The Hyperbola meaning Exaggeration, with an overuse of superlatives, extraordinary projections and forecasts, and fabulous product descriptions to create a sensational and cutting-edge character. This level of narrative can be found in numerous press releases, mainstream press articles and titles for conferences.
Mostly the impressive future-oriented discourse with exceptional outcome boils down to a deceiving and misleading state of the art. "AI will be more intelligent than humans, Singularity will outperform the human brain" even though we now know that ML and DL models are super-effective but not super-intelligent.
2) Repetition by redundancy, pleonasms and paraphrasing. To tell the same story over and over again, always differently is an integral part of a myth. Generating multiple variations/versions with a core message is also the business models of softwares providers.
3) Excessive use of comparisons and illustrations through analogies, allegories and technological metaphors.
The use of metaphors and analogies is an important factor within a mythical narrative.
In the field of AI, specific metaphors hold a leading role : the "brain-as-a-computer", "global village (McLuhan)", "cortical CPU", "black box" or "neural net" metaphors, to name a few.
Analogies, on the other hand are very present in robotics. The creative universe of humanoids like Sophia, Ai-da, Erica and others, only add to their singularity. By creating human features, describing them as "companions", "mothers", "citizens", "artists", soon "musicians", "doctors", "priests", they paradoxically are set apart in their category. By making them "normal", "human", they reversely become "singular". And not just in an "uncanny valley"- sense.
Leading scientists agree that such uses of metaphors are misleading and contribute to a false narrative, a misguiding perception.
"I think the brain-as-a-computer metaphor has led us astray a little bit," says Floris de Lange, a cognitive neuroscientist at the Donders Institute in the Netherlands.
And yet contrary, misleading versions are part of the development of a myth. It even strengthens it.
Three Characteristics of a Myth
Most specialists agree that a the concept of myth is intrinsicly linked to a specific form of narrative.
Beyond their transmissive nature, myths are meaning-laden. They act as open questions, they are both objects to be explained and principles of explanation.
Three essential characteristics to myths
1) The myth falls under the category of discourse, it is a Narrative.
It is in large part thanks to this dynamic of telling a story over and over, in various forms and over a long period of time, that myths are able to survive. It's their "power of irradiation" according to Pierre Brunel.
2) A myth relies on a collective transmission.
For the anthropologist Lévi-Strauss (1908-2009) each myth is linked to a collective transmission. It has a social, and cultural dimension.
3) A myth is not "true or false" but merely "alive or dead".
"Because a myth is made up of all of its variations. […] There is no "True" version of which all others would be copies or distorted echoes. All versions belong to the myth "(Lévi-Strauss 1974)
These characteristics still hold true for the modern techno-myth. Especially the AI Myth and more importantly the emerging Metaverse Myth.
Saint Nicholas and Santa Claus : from a religious myth to a media myth
Saint Nicholas (15 March 270 – 6 December 343) was born into an affluent Greek family in Myra, what is now part of Turkey. He is best known for secretly leaving gifts out for the poor, but also being a devout Christian who became a bishop.
Dutch immigrants helped take the tradition to America, so that by 1822 the poem, "A Visit From St. Nicholas" immortalized him as a jolly sled driver pulled by reindeers, with rosy cheeks and a round belly. But the bishop of Myra and ancestor of Saint Nicolas, will be significantly relifted and pimped.
First in 1860, by cartoonist Thomas Nest from the New York's Harper's Illustrated Weekly. He chooses the famous red color and dresses the "merry old" Santa Claus with a red gown and white fur, wearing a wide leather belt.
Later it is the artist Haddon Sundblom who stressed his human character by giving him a normal height, a well rounded belly, a sympathetic and jovial face, and drinking Coca Cola. This current image of Santa Claus was created in 1931 for Coca Cola in Atlanta, Giorgia. It was the first time for a product, which is to be consumed ice-cold, to come through the chimney and entering the sacro-saint intimate family space : the living room.
Myth and advertising
Using the newly created jolly Santa Claus as messenger and salesperson for Coca Cola's flagship product was a coup de génie that impacted the entire ad industry, as well as our religious subconscious.
It was a successful business operation of
product mythification and simultanuous religious demythification. The first TV spots in 2002 accelerated this phenomenon.
Thus any myth, be it ancient or techno modern, calls for demythification!
Any myth tells a story, just like advertising
And the myth itself morphs into a vector of feelings and values.
The most powerful feature of the myth for the advertising industry was its capacity to constantly reinvent itself without changing its essence. It has a high power of adherence. Later we even saw a Harley Davidson Santa Claus.
But the real stroke of genius by Coca
Cola was to have been able to capitalize on the values and the meanings attached to the initial image while adding new values. The religious myth had been transformed into a media myth.
Advanced AI technology used to update image of the real Saint Nick
The image of Saint Nicholas on the left was created in 2006 after the Vatican granted scientists access to X-rays and measurements taken from the saint's actual remains.
The updated image on the right was created using advanced AI technology. (Photo courtesy of Image Foundry). Source
Anand Kapoor, the co-founder and director of Image Foundry, the CGI company that produced the photo.
"So we've revisited the image we created years ago, using cutting-edge facial reconstruction software. We're like a virtual plastic surgeon, breathing new life into Saint Nick!"
The shock sits deep. According to the findings, Saint Nicholas "was a short swarthy man with dark brown eyes and a broken nose." Quite a galaxy away from the Coca Cola Santa Claus.
The emergence of techno-myths
"Nothing is lost, nothing is created: everything is transformed". Antoine Laurent de Lavoisier (1743-1794).
Besides ancient myths, our modern techno- and data-driven world will create its own new myths.
Having a look at techno-myths such as the AI Myth or the hype Metaverse can be extremely elucidating. Triggering the question: Has the Metaverse the potential to morph into a myth?
In "Imagining the thinking machine: Technological myths and the rise of artificial intelligence", the authors Simone Natale and Andrea Ballatore identified three major characteristics of technological myths.
1 ) Discursive shifts
Alan Turing introduced the original sin of discursive shifts with his now famous question « Can Machines think ? » merging computational technologies with cognitive sciences.
The migration of concepts from different contexts, carrying with them their own cultural associations and meanings, are often resulting in misleading cross-domain translations (Ekbia, 2008).
Combining concepts, shared meanings, blurred bounderies are fertile grounds to to misleading narratives.
2 ) Future-oriented discourse forecasting exceptional success
Predictions and visions of AGI, Singularity (2045), TransHumanism, Humanoid Robotics with Emotional AI, self-replicating robotics, auto-generating xenobots, …
3) The role of controversies
Controversies based on factual grounds are normal in the scientific world. They even are necessary to prove the validity of a theory or a model.
These three identified characterists echo the general myths' caracteristics without stressing on the social and cultural impact and the collective dissemination emphazised by Lévi-Strauss.
The Creation of a New techo-myth : the Metaverse Myth!
With the emerging Metaverse, we perhaps assist live and collectively, for the very first time, in the creation of a new form of a techno-myth.
Does the Metaverse have the potential to be the next techno-myth?
The Metaverse goes way beyond the AI MYTH whose mythical character is predominently, like all existing myths, attached to verbal narrative and transmission.
With virtual reality, augmented reality, and XR new layers of transmission will be added.
It not only shapes and adds a new form narrative but impacts the very meaning of the concept of myth. It might be transformed, even augmented.
The power of metamorphosis of mythical data
The power of metamorphosis constitutes the richness of the myth and allows a dialogue/multilogue to be woven in between the imaginary, the reality or the possible. The virtual myth transforms into a hybrid and augmented creation.
From Ovid's Metamorphoses to Kafka's The Metamorphosis, contradiction and conflict, imaginary and reality, all is needed to fuel the survival of a myth.
Metamorphosis is part of the myth. It nourishes its singularity and sustainability.
Myths are in perpetual gestation or transformation. "The interesting thing, for a literary man, is that this myth of today [the oneof the plane] joins the ancient myth of Icarus […] "(Brunel).
Profoundly modern, contemporary techno-myths draw their substance from archetypes that refer to pre-existing models before morphing into new futuristic forms.
All the ingredients are present for the genesis of a new techno-myth, the Metaverse. The beginning of a new story, the opposing dynamic of imagination and reality, and collective/social participation and dissemination. It propably will be the first real universal myth blurring cultural and historical differences.
Utopia and distopia, apotheosis and apocalypse are substrats of a mythical fundation. Criticism and disbelieve leading to rejection are part of its character and growth.
Can a myth be ethical?
Before closing the article, we need to ask the question : Can a myth be ethical? From our standpoint the answer is no. The question simply does not apply.
Why? Because the levels of abstraction are totally different. Whereas, for example, science deals with "true or false", ethics concentrates on "right or wrong" whereas a myth can just be "dead or alive" in our belief.
Like Saint Nicholas or Santa Claus or Klees'chen, Sinterklaas, San Nicolò di Bari, Agios Nikolaos, Saint Nicolas, Sankt Nikolaus, Santa Claus, Helgen Nicholas, São Nicolau, Aziz Nikolas!
Let's believe, for a while, even though we know it does not exist.
Happy Saint Nicholas Day for all Children and the Child within Us.
Tags: #ai #ainarrative #myth
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ЕуроНатур - Европска природна баштина је фондација која подржава пројекте заштите природе у Европи. Фондација је смештена у немачком граду Радолфцелу.
Фондација је основана 1987. године, а покренуле су је Немачка федерација за заштиту животне средине и природе (Bund für Umwelt und Naturschutz Deutschland - BUND), Немачка организација за заштиту животне средине (Naturschutzbund Deutschland - DUH) и немачка асоцијација за заштиту природе (Naturschutzbund Deutschland - NABU).
Циљеви
Циљ фондације је очување европске природне баштине у свим њеним разноликостима.
Фокус пројеката су дивље животиње као што су вук, мрки медвед, рис и угрожене врсте птица селица, као и вредни природни и културни пејзажи. Важан приступ у раду фондације је повезивање људи и природе како би се осигурао дугорочни успех пројеката заштите угрожених дивљих животиња и њихових станишта у Европи.
Фондација је снажно укључена у иницијативу Европски зелени појас. Циљ ове иницијативе је очување природних вредности дуж трасе којом се некада пружала Гвоздена завеса, путем међународне сарадње, као коридора станишта са изузетном биолошком разноликошћу. Поред тога што је укључена у организовање ове иницијативе, ЕуроНатур фондација од 2004. године координира активности на очувању природе на Балканском зеленом појасу, најјужнијем делу Европског зеленог појаса.
Начин рада
Фондација окупља научнике, конзерваторе, пољопривреднике, политичаре и предузетнике из прекограничних подручја, а посебан фокус је на просторима Балкана. У својим пројектима Фондација не оснива сопствене структуре, већ сарађује са локалним и конзерваторима и волонтерима. Укључена је у планирање и спровођење великих заштићених подручја и на научној основи развија пројекте заштите врста вукова, медведа, рисова и птица селица. Радионице, семинари и програми обуке нуде се за промоцију размене корисног искуства и знања о заштити дивљих животиња у Европи. Политичко лобирање се спроводи и унутар Европске уније.
Финансирање
Фондација се финансира донацијама, средствима треће стране и приходима од капитала фондације.
Остало
Фондација од 1993. године додељује награду ЕуроНатур, а од 1994. и титулу Село европских рода.
Еуронатур, заједно са филмским фестивалом Натур вижн и часописима Zeitschrift natur и Gelsenwasser, организује фото конкурс Природно благо Европе. Резултати се објављују сваке године на годишњој изложби "Фасцинација природом", која се може видети у више градова. Најбоље фотографије одабирају се за мотиве ЕуроНатур календара.
Еуронатур је члан Немачког прстена за заштиту природе (нем. Deutschen Naturschutzring).
Види још
Мартин Шнајдер-Јакоби
Референце
Заштита природе
Екологија
Организације
Организације за конзервацију
Међународне организације за заштиту животне средине | {
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