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Q: How can I scale my bitmap in Android based on the width and height pixels? I want to be able to scale my image based on the screen size. In a normal java applet I would do something like the following....
int windowWidth = 1280;
int windowHeight = 720;
Image image;
public void paint(Graphics g)
{
g.drawImage(image, x, y, windowWidth / 4, windowHeight / 16, null);
}
I've been searching for an answer for a while and everything I find seems to turn up some weird result. From what I read I might need to do something with Resolution Independent Pixels but I'm not %100 sure.
The thing I am trying to avoid is having to create a whole new set of images and icons just for different screen densities. The method I showed above works for resizing desktop apps without a problem.
Edit:
This is what I have been using to draw an image in android.
Matrix matrix = new Matrix();
Bitmap image;
Constuctor....()
{
image = BitmapFactory.decodeResource(context.getResources(), R.drawable.play);
}
public void render(Canvas c)
{
c.drawBitmap(image, matrix, null);
}
A: Hi see thsi question I have posted scale bitmap
If you are using canvas get the width and height of the canvas. or if you want to have it formal normal layouts then get the width and height by using
DisplayMetrics metrics = new DisplayMetrics();
getWindowManager().getDefaultDisplay().getMetrics(metrics);
dispWidth=metrics.widthPixels;
dispheight=metrics.heightPixels;
and then scale our bitmap according to your requirement like this. In this I Have to have 8 bricks so I have taken the width by dividing with the Number of columns
String strwidth=String.valueOf(((float)(bmp.getWidth())/NO_COLUMNS));
if(strwidth.contains("."))
{
scalebit=Bitmap.createScaledBitmap(bmp, (int)(Math.ceil(((float)bmp.getWidth())/NO_COLUMNS))*NO_COLUMNS, bmp.getHeight(), true);
}
else
{
scalebit=bmp;
}
| {
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Eparchie Goroděc je eparchie ruské pravoslavné církve nacházející se v Rusku.
Území a titul biskupa
Zahrnuje správní hranice území Varnavinského, Vetlužského, Voskresenského, Goroděckého, Krasnobakovského, Koverninského, Sokolského, Semjonovského, Tonkinského, Tonšajevského, Ureňského, Šarangského a Šachuňského rajónu Nižněnovgorodské oblasti.
Eparchiálnímu biskupovi náleží titul biskup goroděcký a vetlužský.
Historie
Roku 1927 byl založen goroděcký vikariát nižněnovgorodské eparchie. Dne 8. května stejného roku byl archimandrita Neofit (Korobov) jmenován prvním biskupem vikariátu. Během krátké doby řízení vikariátu vysvětil biskup Neofit několik kněží a podpořil rodiny těch, kteří byli utlačováni pro svou víru. Dne 1. srpna 1929 byl přeložen do vetlužského vikariátu. Poté nebyl vikariát obsazen.
Dne 15. března 2012 byla rozhodnutím Svatého synodu zřízena goroděcká eparchie oddělením území z nižněnovgorodské eparchie. Stala se součástí nově vzniklé nižněnovgorodské metropole.
Seznam biskupů
Goroděcký vikariát
1927–1929 Neofit (Korobov)
Goroděcká eparchie
od 2012 Avgustin (Anisimov)
Reference
Externí odkazy
Oficiální stránky eparchie
Moskevský patriarchát
Goroděc | {
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CBM7026 is low-cost single chip solution for capacitive touch sensor controller. The chip is mainly used for mechanical buttons replacement in home appliances, consumer electronics, industrial areas. With robust sensing technology, it has high performance across a variety materials and thickness, high noise immunity, waterproof and dustproof.
CBM7026 is 8-bit RISC architecture microcontroller devices with I2C Host/Slave, UART interface. For function application, CBM7026 support Button information for customers. In operation mode, it can support protocol and I/O mode for customer. Developer can use I/O mode to get a valid button message and develop their system very easily, and no longer need to decode the communication package.
The capacitive touch sensor can be designed by placing a copper pad on the PCB directly, covered with a plastic or glass case. It provides auto-calibrate the parameter for a wide range of capacitance on the touch sensor(1pF ~ 40pF). The system controller converts finger data to button presses, depending on finger location and human interface context.
CBM7026 robust sense solutions leverage our flexible programmable system-on-chip architecture, which accelerate time-to-market, integrates critical system functions and reduced BOM costs. CBM7026 supports multi-package for various application. | {
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Malinowski () ist eine Siedlung (ländlichen Typs) in der Oblast Kursk in Russland. Sie gehört zum Rajon Chomutowka und zur Landgemeinde (selskoje posselenije) Skoworodnewski selsowjet.
Geographie
Der Ort liegt gut 89 km Luftlinie westlich des Oblastverwaltungszentrums Kursk im südwestlichen Teil der Mittelrussischen Platte, 25,5 km südöstlich des Rajonverwaltungszentrums Chomutowka, 2,5 km vom Sitz des Dorfsowjet – Skoworodnewo, 34 km von der Grenze zwischen Russland und der Ukraine, am Fluss Schichowka (Nebenfluss der Swapa).
Klima
Das Klima im Ort ist wie im Rest des Rajons kalt und gemäßigt. Es gibt während des Jahres eine erhebliche Niederschlagsmenge. Dfb lautet die Klassifikation des Klimas nach Köppen und Geiger.
Bevölkerungsentwicklung
Anmerkung: Volkszählungsdaten
Verkehr
Malinowskij liegt 24 km von der Fernstraße föderaler Bedeutung A 142 (Trosna – Kalinowka), 19,5 km von der Straße regionaler Bedeutung 38K-040 (Chomutowka – Rylsk – Gluschkowo – Tjotkino – Grenze zur Ukraine), 6 km von der Straße 38K-003 (Dmitrijew – Berjosa – Menschikowo – Chomutowka), 3,5 km von der Straße interkommunaler Bedeutung 38N-024 (Bogomolow – Kapystitschi – Grenze zum Rajon Rylsk), 3 km von der Straße 38N-023 (38N-024 – Skoworodnewo), 4,5 km von der Straße 38N-708 (38-040 – Pody – Petrowskoje) und 23,5 km von der nächsten Eisenbahnhaltestelle 536 km (Eisenbahnstrecke Nawlja – Lgow-Kijewskij) entfernt.
Der Ort liegt 180 km vom internationalen Flughafen von Belgorod entfernt.
Einzelnachweise
Malinowskij | {
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} | 3,635 |
Угорський національний музей () — національний музей у столиці Угорщини місті Будапешті, один з головних музеїв міста та країни; найважливіше та найбільше зібрання пам'яток з угорської історії, національного матеріального та духовного надбання, культури і мистецтва Угорщини, видатних їх представників.
Загальні дані
Угорський національний музей розташований у монументальному комплексі споруд, спеціально зведених для цього закладу, в середмісті Будапешта за адресою:
Múzeum krt. 14-16, м. Будапешт—1088 (Угорщина).
Велична споруда Угорського національного музею в стилі класицизму — з великою колонадою, була зведена за проектом угорського архітектора Міхая Поллака. У оформленні інтер'єрів музею брали участь національні митці Карой Лотц і Мор Тан.
Музей працює щоденно з 10:00 до 18:00. Щопонеділка вихідний.
Директор музею — доктор Ласло Чорба (László Csorba).
З історії музею
Угорський національний музей був заснований 1802 року, коли граф Ференц Сечені (Ferenc Széchényi) створив Національну бібліотеку Сечені. Спершу музейне зібрання формувалось на базі колекції монет, книжок і рукописів, мінералів тощо самого́ графа Сечені та його родини, що вивело заклад суто за збірку письмових матеріалів, зробивши справжнім природничо-історичним музеєм.
У 1807 році угорський національний парламент ухвалив закон, підтримавши створення цієї інституції та закликав угорську націю робити пожертви на її розбудову.
Розкішну неокласицистичну споруду Національного музею було зведено у 1837—47 роки.
Вже наступного (1848) року Угорський національний музей відіграв важливу роль у подіях Угорської національної революції. Са́ме на ґанку музею національний культурний лідер Шандор Петефі прочитав свій знаменитий вірш Nemzeti, який був справжнім закликом до початку революції. Відтоді музей став головним осередком ідентифікації угорської нації. На згадку про революцію дві статуї були додані до будівлі музею. Першою з них є статуя Яноша Араня (її встановили 1883 року). Згодом (1890) було розміщено барельєфну меморіальну дошку поруч зі сходами музею на честь Петефі. У цей же час у церемоніальній залі музею засідала верхня палата угорського парламенту, причому це тривало аж до будівництва споруди угорського парламенту. Сучасним нагадуванням про це є те, що заходи національного Дня поминання 1848 відбуваються са́ме перед будівлею музею.
У 1949 році згідно із законом колекції етнографічна та природнича Угорського національного музею були передані у новостворені профільні музейні заклади. Ці процеси — відгалуження спеціалізованих музеїв, афільованих з головним музейним осередком Угорщини, тривали і надалі, в тому числі вони прислужилися і до створення сучасної Національної бібліотеки Сечені. Востаннє філією Угорського національного музею в 1985 році став Замковий музей у Естергомі.
З експозиції
Фонди Угорського національного музею в теперішній час (2000-ні) налічують понад 1 млн одиниць зберігання. Постійні музейні виставки охоплюють історію Угорщини від моменту заснування держави і до сьогодення:
у лапідарії музею представлено багату колекцію артефактів римського періоду;
добу Середньовіччя та османської навали в Угорщині ілюструють королівські регалії, зібрання предметів середньовічного побуту, ювелірні вироби, нумізматична та зброярська колекції;
багатою є експозиція зали, присвяченої подіям революції 1848—49 років;
зала історії XX століття має велике зібрання документів, предметів, фотографій та свідчень 2 Світових воєн, подій в Угорській народній республіці, становлення сьогоденної Угорщини.
Окрема зала музею присвячена королівській мантії, в якій коронувався засновник угорської держави король Іштван — її виконано з візантійського шовку із золотим гаптуванням та оздобленням з коштовного каміння.
У портретній галереї музею репрезентовано велике цінне зібрання зображень королів з династії Арпадів.
Експозиція, присвячена визначним діячам культури та музичній культурі, надає можливість побачити клавікорд Моцарта, арфу Марії-Антуанетти, роялі Бетховена та Ліста.
Виноски
Джерела, посилання та літеартура
Офіційна вебсторінка музею
Угорський національний музей серед інших музеїв Будапешта на Будапештський офіс туризму
Венгерский национальный музей (к 175-летию создания)., Будапешт: «Корвина», 1978
Будапештские музеи., Будапешт: «Корвина», 1985
Венгрия., М.: «Вокруг света», 2009 ISBN 978-5-98652-226-5
Михаэль Херл. Будапешт., Polyglott, 1996 ISBN 5-88395-021-3
Музеї Угорщини
Культура Будапешта
Історичні музеї
Музеї, засновані 1802
1802 у Європі
Національні музеї | {
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When you need to replace the headlight, turning signal or parking light bulb on your 2007 Nissan Murano with HID, finding out what the replacement size you need can be a time consuming task. Whether you want to replace your headlamp bulb with a HID (High Intensity Discharge) Headlight Kit, Xenon Headlamp Kit or replace your interior lighting bulbs with LED (Light Emitting Diode) bulbs, we're here to help you with your 2007 Nissan Murano with HID.
The Modified Life staff has taken all its Nissan Murano with HID headlight bulb replacement guide, Nissan Murano with HID light bulb size guides, Nissan Murano with HID light bulb replacement guides and cataloged them online for use by our visitors for free. Our Nissan Murano with HID light bulb guides allow you to easily replace light bulbs, replace headlight bulb, change a broken lightbulb, install a hid headlight conversion or install led light bulbs instead of spending countless hours trying to figure out which light bulb sizes in your 2007 Nissan Murano with HID.
Feel free to use any replacement light bulb sizes that are listed on Modified Life but keep in mind that all information here is provided as is without any warranty of any kind. Use of the replacement bulb size information is at your own risk. Always verify all light bulb sizes, bulb voltage usage and bulb wattage before applying any information found here to your 2007 Nissan Murano with HID.
I need to replace the HID-D2S bulb in the driver side of my 2007 Nissan Murano. I have the bulb but in looking at the task, I am not sure how to gain access to change. Does anyone have instructions on how to change this bulb or do I have to take to dealer? | {
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I worked with Francis on a weekly basis for a few months in 2012/13 and then on an ad hoc basis after that.
Francis helped me work through many issues related to my childhood, family, bereavement and self confidence. His kind and empathetic manner helped a great deal as well as his questions that were thought provoking and challenging and allowed me to confront and work through my issues. By working with Francis I was really able to move forward in my life to a place where I am very happy and have a set of coping tools for difficult situations. As someone who was completely new to counselling, I cannot recommend Francis enough.
I had therapy with Francis when I was experiencing some strong anxiety due to a stressful period of my life. It was the first time I had therapy. The induction session was about defining what I wanted to achieve and sharing my personal background. Then Francis helped me deal with anxiety and some grief by working on techniques such as behavioural and cognitive therapies and some relaxation techniques such as mindfulness. Very quickly I felt we built a space where I could focus on what I wanted to work on. I had a very positive experience with Francis and I would recommend him for his methodological approach, his knowledge of psychotherapy and his strong interest in his patients. After less than one year, I manage to cope with periods of anxiety better and I now feel more confident in my ability to take care of myself.
I've seen a number of therapists over the years, but it was only after spending time with Francis that I recognised how I hadn't been able to express myself honestly with the other therapists I'd worked with.
This was particularly important as I came to Francis at a time of crisis. I was battling with some aspects of my upbringing that, for most of my life, I hadn't paid enough attention to and they had begun to impact my relationship.
Francis' created an environment where I felt comfortable enough to explore these issues and relieve the pressure they were placing on me. I found him to be professional, but also able to empathise and draw on his own experiences in order to help me. Looking back this was a crucial factor in us forming a good working relationship.
I would recommend Francis to anyone seeking professional help at times of difficulty.
My self belief & conviction of my thoughts & emotions were very low, which is what brought me to Francis. He was recommended to me by a friend & I would not hesitate to do the same for others. I had found myself feeling emotionally isolated in London with no family or long term friends as I am not originally from the UK. I was in a marriage which I knew deep down was not fulfilling my needs but I lacked the support & confidence to be able to feel strong enough to rely on & trust my feelings. Through 14 months of counselling with Francis, he made me believe in myself again. He became the support that I needed to take myself on to the next stage of my life as we worked together to help me regain my trust in my own intuition. I feel confident that I made the right decision to come out of my marriage, the right decision to have Francis to guide me through the process & I now feel I have the strength to make better decisions for myself in the future as I can now trust myself again.
'I was recommended Francis and I would recommend him to others. Being a psychotherapist myself finding a therapist with a certain level of experience was very important to me.
Having the free initial meeting gave me a chance to meet Francis and decide in person whether I would like to continue. I saw Francis short term (6 months) after experiencing some difficulties in my personal life that where impacting my work and my general well being.
I found Francis to be professional, flexible and I felt his consulting room was comfortable and confidential. The location and building were really accessible, but at the same time offered privacy and a bit of sanctuary away from the hustle and bustle of the city.
Finding the right Therapist or a good Therapist is a daunting task particularly when faced with lengthy counselling and psychotherapy directories whilst also being in a difficult or challenging time in life. I was very fortunate in choosing Francis from the BACP directory whom I was in therapy with for several years. Having various deeply seated issues which were having a serious impact upon my life, I was able to work through these issues in a safe and trusting space which has had a highly reparative influence upon my life. Francis's non-judgmental, emphatic and containing presence has enabled me to lead a more authentic, happier and fulfilling life and I will be forever grateful to him. He has a wide range of experience and takes an integrative approach with his clients. I have recommended him to various friends and colleagues who have been contemplating having counselling or psychotherapy and will continue to do so.
This has greatly improved my self belief and my ability to articulate myself. Upon finalising my sessions with Francis I feel more able to communicate - both with myself and others. I also feel more confident in my choices and trust myself more.
Francis' guidance and demeanor was invaluable in helping me to come my own conclusions about what arose from the therapy and I would have no hesitation in recommending Francis to anyone who feels they would benefit. | {
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\section{Introduction}
Complex, or dusty plasmas exist in different environments and take various forms \cite{Ivlev:2012book,VladimirovBook}. One interesting variety is a two-dimensional (2D) plasma crystal \cite{Thomas:1996,Nosenko:2009,Melzer:2013,Nosenko:2007, Nosenko:2011PRL,Nosenko:2012, Nunomura:2002,Piel:02,Couedel:2010,Nosenko:2004,Gavrikov:2005,Hartmann:2011,Nunomura:2005a,Nosenko:2008,Nunomura:2006,Liu:2008}. It is a single-layer suspension of micron-size solid particles in a weakly ionized gas. Due to the high thermal speed of plasma electrons,
microparticles become negatively charged and interact with each other via a screened Coulomb (Yukawa) pair potential. Under certain conditions, these charged particles can self-organize in a triangular lattice with hexagonal symmetry, forming a 2D plasma crystal. Individual particle motion can be easily recorded in real time using video microscopy. This makes 2D complex plasmas excellent model systems where dynamics can be studied at the level of individual particles which can be regarded as proxy ``atoms.''
Various aspects of 2D complex plasmas that were recently studied in experiments include phase transitions \cite{Thomas:1996,Nosenko:2009,Melzer:2013}, dynamics of defects \cite{Nosenko:2007,Nosenko:2011PRL}, microstructure \cite{Nosenko:2012}, waves \cite{Nunomura:2002,Piel:02}, mode coupling instability \cite{Couedel:2010}, and transport phenomena: momentum \cite{Nosenko:2004,Gavrikov:2005,Hartmann:2011} and heat \cite{Nunomura:2005a,Nosenko:2008} transport, self-diffusion \cite{Nunomura:2006}, superdiffusion \cite{Liu:2008}. Systems other than complex plasmas were also reported to allow superdiffusion: supercooled liquids \cite{Furrukawa:2012}, solid surfaces \cite{Sancho:2004}, granular media \cite{Ganti:2010}.
It was recently discovered that single-layer plasma crystals sometimes include extra particles levitating slightly above the main layer \cite{Du:2012}. Such particles were called upstream particles since they are upstream of the flow of ions in the plasma sheath, with respect to the lattice particles. These extra particles move about and create disturbances in the lattice layer by both Coulomb repulsion and the effective attraction due to the ion wake effect. If a particle moves faster than the sound speed of the plasma crystal, it creates a Mach cone behind it. Compared to other disturbance sources such as a moving extra particle beneath the lattice layer \cite{Samsonov:1999PRL} or a laser beam \cite{Nosenko:2003}, the disturbance is rather weak and thus hard to notice. In certain conditions, the upstream particles tend to travel between the rows of lattice particles, analogous to channeling of energetic particles in regular crystals \cite{Fliller:2006}.
\begin{figure}[!bh]
\includegraphics[width=0.9\columnwidth]{figure1}
\caption{(Color) Channeling of an upstream extra particle in a 2D plasma crystal (a movie is also available \cite{movie001}). The figure is an overlay of a sequence of thresholded images recorded by a top-view video camera. The upstream particle trajectory is the curve across the figure starting at the red cross mark.
The inset is a zoom-in of the area where the upstream particle encounters a chain of dislocations. Black dots represent the lattice particles in a snapshot taken $1.5$~s before the upstream particle came inside the field of view. Colored dots represent the ``wall'' particles and the upstream particle positions in a $0.7$-s time sequence coded from blue to red. The chain of dislocations is marked by a gray stripe, the 7-fold and 5-fold cells are indicated by squares and triangles, respectively.
\label{figure1}}
\end{figure}
In this paper, we study channeling of upstream extra particles in a 2D plasma crystal in more detail and show its implications for experiments on transport phenomena.
The experiments were performed in a modified Gaseous Electronics Conference (GEC) rf reference cell \cite{Nosenko:2012}. Argon plasma was sustained using a capacitively coupled rf discharge at $13.56$~MHz. The input power was set at $20$~W. We used monodisperse polystyrene (PS) particles to create a 2D plasma crystal suspended above the bottom rf electrode. The particles have a diameter of $11.36\pm0.12$~$\mu$m and mass density of $1.05$~g/cm$^{3}$. The gas pressure was maintained at $0.65$~Pa, corresponding to the gas friction rate $\gamma\simeq0.91$~s$^{-1}$ \cite{Liu:2003}. The lattice layer was illuminated by a horizontal laser sheet shining through a side window of the chamber. A high-resolution video camera (Photron FASTCAM~1024~PCI) was mounted above the chamber, capturing a top view with a size of $42.7\times42.7$~mm$^2$. The recording rate was set at $250$~frames per second.
We prepared 2D plasma crystals using a standard technique \cite{Nosenko:2011PRL}. After the particles were injected into plasma, they formed a single-layer suspension; in addition, some heavier particles always levitated beneath the main layer \cite{Samsonov:1999PRL,Schweigert:2002}. These were mainly agglomerations of two \cite{Chaudhuri:2012} or more particles \cite{Du:2012PoP}. We removed these particles from the suspension by dropping them on the lower rf electrode while reducing the discharge power. However, even after all heavier particles were removed, we sometimes observed moving disturbances in the lattice. These were caused by the upstream extra particles that levitated {\it above} the main layer as was recently discovered in Ref.~\cite{Du:2012}. In the present paper, we study in more detail the dynamics of upstream extra particles and their effect on the transport phenomena in the main layer. To maximize the rate of occurrence of upstream extra particles, we used polystyrene (PS) particles in this study. The reason why plasma crystals composed of PS particles contained upstream extra particles more often than those composed of melamine formaldehyde (MF) particles is not clear.
An upstream extra particle usually moved about in a horizontal plane approximately $0.25\Delta$ higher than the main layer, where $\Delta$ is the in-plane interparticle distance. Since a moving particle must overcome the ambient gas friction, there must be a driving force acting on it; the origin of this force is not clear. Possible candidates are the wake-field interaction \cite{Schweigert:2002} and photophoretic force from the illuminating laser \cite{Nosenko:2010PoP}. From a different perspective, upstream particles can be regarded as self-propelled (active) particles \cite{Buttinoni:2013}. They may be smaller particles from within the natural size distribution or particles with some defects on their surface. The rather small vertical separation between the main layer and the upstream particles allowed us to illuminate and observe them simultaneously by properly adjusting the height of the illuminating laser sheet.
\begin{figure}
\includegraphics[width=\columnwidth]{figure2}
\caption{(Color) Analysis of the channeling event shown in Fig.~\ref{figure1}. The $S$-axis represents a natural coordinate system along the upstream particle trajectory, where $S=0$ is marked by a red cross in Fig.~\ref{figure1}. Panel (a) shows the instantaneous direction of motion $\theta$ (black) and the global bending angle (red) of the upstream particle. The inset shows the particle transverse acceleration $a$ as a function of its deviation $d$ from the central line of the channel. Color-coding from blue to orange represents timing during $1.85$~s. Panel (b) presents the local bond-orientational order parameter $\langle|\psi_6|\rangle$ (see text) of the lattice at different delay times ${\Delta}t$. Panel (c) shows the local interparticle separation. The gray stripes correspond to the chain of dislocations marked in Fig.~\ref{figure1}.
\label{figure2}}
\end{figure}
A particularly long trajectory of an upstream particle is shown in Fig.~\ref{figure1}. By tracking its motion and analyzing its interaction with the lattice particles, we observed several new features of the channeling process. First, the channeling path is not necessarily straight. As the upstream particle follows the channel formed by the lattice particles, it is constantly redirected and the resulting trajectory may be curved as in Fig.~\ref{figure1}. Here, the total angle of deflection is about $50^{\circ}$, see Fig.~\ref{figure2}(a). The rows of lattice particles are curved due to the presence of defects in the lattice, e.g. a chain of dislocations marked by a grey stripe in Fig.~\ref{figure1} \footnote{At the periphery of the crystal, shell structure \cite{Dubin:1989} may also account for the curved channels.}. This situation is analogous to the channeling effect in regular crystals. It is well known that bent crystals are used to redirect energetic particle beams \cite{Fliller:2006}. Despite the totally different energy, space, and time scales involved, our experiment presents the first direct observation of the dynamics of such process.
Second, we found that the upstream particle scattering angle is determined by two competing factors: number density and regularity of the lattice. As an upstream particle travels in the channel, it bounces on the channel walls and its trajectory has a zig-zag pattern. The scattering angle is related to the lattice particle number density. As shown in Figs.~\ref{figure2}(a),(c), the magnitude of scattering is relatively large at two ends where the local lattice density is relatively low. The scattering angle decreases as particle moves into the central region of the particle suspension, where the particle number density is higher. However, when the upstream particle encounters lattice defects (e.g., a chain of dislocations), the scattering angle increases, see Fig.~\ref{figure2}(a). The channeling process may continue, as in Fig.~\ref{figure2}(a), or it may be interrupted, if the lattice distortion is too large. In Ref.~\cite{Du:2012}, the upstream particle ``jumped'' out of the lattice layer upon encountering a chain of dislocation. Depending on the local structure, this particle either found a new channel and started channeling again or kept jumping until it left the particle suspension.
Third, we gained insight into the channel internal structure by analyzing the particle transverse acceleration $a$
as a function of its deviation $d$ from the central line of the channel, see inset in Fig.~\ref{figure2}(a). The slope here is proportional to the square of the transverse oscillation frequency of channeling according to Hooke's law. The dashed line corresponds to the oscillation frequency of $11$~Hz obtained from the Fourier-transformation analysis of the transverse velocity. The double-strand shape of the $a(d)$ plot suggests that the particle actually moves along one of two lines in the channel closer to its walls and not along the central line.
\begin{figure}
\includegraphics[bb=60 20 800 350, width=\columnwidth]{figure3}
\caption{Velocity and trajectory length distributions for the observed channeling events. Panel (a) is for the ratio of instantaneous velocity of upstream particle to the local sound speed $C_s$ for the event shown in Fig.~\ref{figure1}. Panels (b) and (c) are respectively for the mean velocity $\langle v\rangle$ and trajectory length $L_{\rm ch}$ of the channeling events from a series of experiment realizations.
\label{figure3}}
\end{figure}
We observed channeling upstream particles in several experiments. In all cases, their speed was comparable to the sound speed of compressional waves $C_s$. In Fig.~\ref{figure3}, we summarize the velocity and trajectory length distributions for such channeling events from a series of experiments with the same conditions. The local sound speed is given by the relation $C_s=2.44\sqrt{{Q^2\lambda_D}/m{\Delta}^2}$, where $\Delta$ is the interparticle separation in the respective lattice location, $Q$ is the particle charge, $\lambda_D$ is screening length, and $m$ is the particle mass \cite{Zhdanov:2002}.
The average channeling length in different experiments was $\langle L_{\rm ch} \rangle \approx 12$~mm, see Fig.~\ref{figure3}(c). Very long trajectories like that in Fig.~\ref{figure1} were rare. Note, however, that $\langle L_{\rm ch} \rangle \gg\Delta$ (the latter was $0.65-0.75$~mm). This may present a problem for studying transport phenomena in this system. For example, if one intends to study diffusion, the existence of such long particle trajectories apparently contradicts the basic assumption that diffusion results from the random walk of particles.
The diffusion coefficient $D$ is usually calculated by measuring the mean squared displacement (MSD) of particles in an ensemble \cite{Ohta:2000}:
\begin{equation}
\label{diffusion}
D=\lim_{t \to \infty} (2nt)^{-1} \langle |{\bf r}_i(t)-{\bf r}_i(0)|^2 \rangle,
\end{equation}
where ${\bf r}_i(t)$ is the position of the $i$th particle and $n$ is the space dimensionality ($n=2$ in our case). If upstream particles are accidentally recorded in an experiment and enter the calculation of MSD in Eq.~(\ref{diffusion}) along with the regular particles, their high velocity (on the order of $C_s$) will lead to a spurious rise in the diffusion coefficient. In fact, in this case MSD will not even have a $\propto t$ asymptote at large times, since the displacement of an upstream particle is (roughly) proportional to time. Therefore, an accurate measurement of diffusion coefficient under these circumstances is problematic.
Furthermore, a moving upstream particle interacts with the lattice particles, particularly those in the channel walls, and transfers some of its momentum and energy to the surrounding particles. The momentum and energy will then spread further in the particle suspension. This may affect any measurements of the transport properties.
To quantify the disturbance created by an upstream particle in the lattice, we use the bond-orientational order parameter defined at each site as $\psi_6=n^{-1}\Sigma_{j=1}^{n}\exp(6i\Theta_j)$, where $\Theta_j$ are bond orientation angles for $n$ nearest neighbors \cite{Grier:1994}. In a perfect hexagonal structure, $|\psi_6|=1$. The smaller $|\psi_6|<1$, the more disordered the structure is, e.g. $|\psi_6|\approx 0.35$ in a dislocation core \cite{Nosenko:2007}. Here, we average $|\psi_6|$ within a small circular area with a radius of $1.2$~mm to quantify the local degree of order in the lattice along the trajectory of upstream particle, see Fig.~\ref{figure2}(b). ${\Delta}t$ is the delay time of the measurement, where negative and positive values correspond respectively to the time before and after the upstream particle reaches the corresponding site along the $S$-axis.
As can be seen in Fig.~\ref{figure2}(b), a dip in $\langle|\psi_6|\rangle$ at $S\approx15$~mm corresponds to a chain of dislocations that was present in the crystal even before the arrival of the upstream particle. As the upstream particle travels in the channel between $S\approx3$~mm and $\approx13$~mm, the local lattice structure is slightly disturbed with a decrease of $\langle|\psi_6|\rangle$ from $0.99$ to $0.93$. However, when the upstream particle encounters the chain of defects, it starts to move more irregularly and the scattering angle increases. Though it is still confined in the channel, the disturbance in the lattice becomes significant and $\langle|\psi_6|\rangle$ drops from $0.93$ to $0.74$ (black and blue curves). It takes about $80$~ms for the system to restore initial order; the chain of defects is preserved thereafter, as shown by the yellow curve.
As an example of the effect of upstream particles on transport phenomena, we consider heat transport in the lattice. As discussed in the previous paragraph, a moving upstream particle disturbs the lattice and generates local disorder. During this process, its kinetic energy is transferred to the lattice particles and therefore it acts as a moving heat source. The transferred heat (in-plane particle kinetic energy) is concentrated on the channel walls for about $5$ interparticle spacings behind the upstream particle, see Fig.~\ref{figure4}. Then the heat starts to spread out transversely. The peak value is located slightly behind the upstream particle and can on average reach a few tens of eV.
\begin{figure}
\includegraphics[width=\columnwidth]{figure4}
\caption{Upstream particle as a moving heat source. Panel (a) shows a map of in-plane kinetic energy of the lattice particles averaged from data for 30 consecutive video frames. The upstream particle is marked by a circle located at (0,0) and is moving from right to left. Here $x^*$ and $y^*$ represent the longitudinal and the transverse axes along the trajectory. Panel (b) shows the in-plane kinetic energy $E_k$ of the wall particles. The black plus signs are the means of a gauss-fit of the data along the evolution and error bars represent the standard deviation.
\label{figure4}}
\end{figure}
The usual analysis of heat transport (based on the idea of heat diffusion) is clearly not applicable in this situation. The moving heat source can be properly taken into account in the following way. If we only concentrate on the kinetic energy of the particles in the walls, namely in the longitudinal direction, we can simplify the heat transport model to a quasi-one-dimensional model \cite{Williams:2012} with a thermal diffusivity $\chi = 2 \gamma L^2 + v L$, where $L$ is the inhomogeneity (gradient) length. The RHS of the equation contains two terms, the first one is associated with conduction and the second with convection. Solving the equation, we obtain $L_{1,2} = \sqrt{(\frac{v}{4\gamma})^2+\frac{\chi}{2\gamma}} \mp \frac{v}{4\gamma}$, corresponding to the inhomogeneity length in front of ($L_1$) and behind ($L_2$) the moving heat source, respectively. The experimental kinetic energy profile can be well fitted by an exponential function $E_k \propto \exp(-|x|/L_{f,b})$. For the inhomogeneity length in front of the upstream particle, $L_f=0.75\pm0.04$~mm. Since the upstream particle travels with a high velocity ($v\approx22$~mm/s), we can simplify $L_1$ to $\chi/v$ and thus from $L_1=L_f$ the thermal diffusivity $\chi=16$~mm$^2$/s. This value is between two previous measurements of $30$~mm$^2$/s \cite{Nunomura:2005a} and $9$~mm$^2$/s \cite{Nosenko:2008} reported in solid and melted 2D complex plasmas, respectively. As to the inhomogeneity length behind the upstream particle, we simplify $L_2$ to $v/2\gamma$, which turns out to be $12$~mm and is associated with the relaxation due to gas friction. However, by fitting the kinetic energy profile behind the upstream particle, we measured $L_b$ to be $6.4\pm0.2$~mm, which is about half of the value $L_2$ predicted by the model. This can be explained by the fact that the heat transport behind the upstream particle has both longitudinal and transverse component. The latter is mainly associated with conduction and thus can be estimated as $L\approx\sqrt{\chi/2\gamma}\approx3$~mm, using the thermal diffusivity obtained earlier. Apart from the in-plane motion of particles, their vertical motion also has a certain contribution to the heat transport. Therefore, our simple one-dimensional model has limited applicability in this case. For other experimental runs we observed that $L_f$ ranges from $0.4$ to $0.8$~mm and $L_b$ ranges from $6$ to $7.1$~mm for the same experimental conditions.
To summarize, we reported new features of the recently discovered effect of upstream extra particle channeling in 2D complex plasma crystals. The implications of particle channeling for transport phenomena were discussed using heat transfer as an example.
So far, we did not find any reliable way of removing the upstream extra particles from the suspension. Since they may pose problems in delicate experiments, it is recommended that the particle suspension be crystalized and carefully checked for the presence of upstream particles. Depending on the particle type and size, the gap between the upstream particles and the main layer may be large and they might not be illuminated by the laser sheet. Therefore, scanning the illumination laser above the lattice layer is recommended.
An important open question is whether particle channeling is also possible in liquid complex plasmas. New dedicated experiments will be necessary to answer this question.
This work was supported by the European Research Council under the European Union's Seventh Framework Programme (FP7/2007-2013) / ERC Grant agreement.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 1,923 |
Balancing Humility and Ambition for the Inner Ring
K. E. Colombini
In his new book How to Think, Alan Jacobs brings up a 1944 lecture by the British writer C.S. Lewis that summarizes well the state of politics three-quarters of a century later and an ocean away—not surprising perhaps, given that his observation is one that humanity has experienced for millennia.
In his lecture at King's College at the University of London, Lewis cites the phenomenon of "The Inner Ring," the unofficial group of people that really run things—whether it's a grade-school playground, a men's club lodge or even a national government. "I can assure you that in whatever hospital, inn of court, diocese, school, business, or college you arrive after going down, you will find the Rings—what Tolstoy calls the second or unwritten systems."
"The pastor is not always the most influential person in a church, not the boss in the workplace," Jacobs explains. "Sometimes groups of people with no formal titles or authority are the ones who determine how the organization works. They form its Inner Ring." To Lewis, all men have not only a desire to be part of this Inner Ring, but a "terror"—his word—of being left outside.
Politically, we can see this Inner Ring at work in what has been called "the deep state," the idea that career bureaucrats are really the ones running things in Washington and, one assumes, other levels of government. There is some truth to this; whether it's true to the degree the accusers claim is another matter. Just about every conspiracy theory, whether it's about the assassination of JFK or the terror attack of 9/11, hinges on the idea of an Inner Ring.
It's the underlying moral issue that needs to be considered robustly. We all want to be part of the Inner Ring, Lewis asserts. It's the perennial drive to be "popular" and esteemed by all. The question is: What are we willing to do to reach that point, to join the Inner Ring?
For C.S. Lewis in 1944 and Alan Jacobs in 2017, the answer is the same. Lewis notes that the desire to be part of the Inner Ring is what can corrupt someone—"Of all the passions, the passion for the Inner Ring is most skillful in making a man who is not yet a very bad man do very bad things." It can be as simple as bullying someone at the lunchtable who does not fit in, or taking part in a whisper campaign against a coworker. Jacobs sees this also in How to Think: "The draw of the Inner Ring has such a corrupting power because it never announces itself as evil—indeed, it never announces itself at all."
When it comes to the Inner Ring, Jacobs argues that this is the opposite of a true community of people. One finds this in his "Thinking Person's Checklist" at the end of the book, after reminding people that they don't have to respond to every opposing comment: "If you do have to respond to what everyone else is responding to in order to signal your virtue and right-mindedness, or else lose your status in your community, then you should realize that it's not a community but an Inner Ring."
Rather, Jacobs asserts, it's true community we should be striving for, one where differences are respected and make the whole stronger, not the Inner Ring, where everyone must think and do the same.
Among Lewis's voluminous writings one also finds a series of letters to an Italian priest, Don Giovanni Calabria. Calabria had sent Lewis a copy of the Catholic devotional Litany of Humility in 1948, to which the Anglican writer responded, in part: "You did not know, did you, that all the temptations against which he pours forth these prayers I have long been exceedingly conscious of?"
The Litany of Humility is the sort of prayer a person can certainly find difficult to pray, for all the right reasons—most of all, because it challenges our modern assumptions about career and ambition. It asks us, for example, to pray to be delivered from the desires of being praised, preferred to others, consulted and approved. Rather, we would pray for the desire that others may be esteemed more than us, that others may be chosen and we set aside, that others may be praised and we unnoticed, and that others may be preferred to us in everything. It's a prayer that makes us a little uncomfortable, as all such prayers should.
Does this mean we give up ambition and advancement? No; rather, it means we strive to do it on our own terms. Recently in the New York Times, Adam Grant of the Wharton School wrote about the trend toward career networking, and that what people often forget is the other part of it, that we can build better and stronger networks by standing out in our work first. And it's an idea with Christian roots that Lewis would find appealing.
In a curious detail lurking in a local gender discrimination lawsuit and related complaints about a major business group in St. Louis, a woman alleges that during a performance review, her supervisor encouraged her to pray the Litany of Humility and stop seeking a promotion. Perhaps needless to say, the employee was not receptive to the idea and rightly found it wrong-headed in a secular professional setting.
In his Times article, Grant cites the work of sociologist Robert Merton, whose research indicated what he referred to as the "Matthew Effect," based on the biblical parable of the talents. "If you establish a track record of achievement, advantages tend to accumulate," Grant writes. "Who you'll know tomorrow depends on what you contributed yesterday."
This may not help the poor woman in the lawsuit, but it shows a path forward. If we work hard at what we are supposed to be doing, a network arises naturally and a way forward—even at another employer if one is in a bad spot with a bad boss—can more easily present itself.
This is exactly what Lewis is striving for at the end of his lecture: "If in your working hours you make the work your end, you will presently find yourself all unawares inside the only circle in your profession that really matters. You will be one of the sound craftsmen, and other sound craftsmen will know it."
Lewis concludes: "To a young person, just entering on adult life, the world seems full of 'insides,' full of delightful intimacies and confidentialities, and he desires to enter them. But if he follows that desire he will reach no 'inside' that is worth reaching. The true road lies in quite another direction."
Tagged as C.S. Lewis, cultural / political elite, Humility
By K. E. Colombini
K. E. Colombini is a former journalist who served as a political speechwriter before a career in corporate communications. A Thomas Aquinas College alumnus, he also studied English literature at Sonoma State University in Northern California. In addition to Crisis, Colombini has been published in First Things, Inside the Vatican, The American Conservative and the Homiletic and Pastoral Review. He and his wife live in suburban St. Louis, and have five children and four grandchildren. | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 7,496 |
Q: There was no /etc/default/grub file, so how come my system was able to boot? I'm running Ubuntu 13.10. I wanted to change the GRUB timeout by editing the /etc/default/grub file. But there was no such file in my system.
On startup the GRUB menu is displayed and I'm not experiencing any boot problems. If this file was so important for booting, why am I nit experiencing any boot problems?
A: just copy example from /usr/share/grub/default/grub:
sudo cp /usr/share/grub/default/grub /etc/default/
A: Reinstall the package and hope for the best:
sudo apt-get --reinstall install grub-pc
Actually the file is not critical for the boot, just to configure the boot when you execute sudo update-grub. If the file doesn't exist, then all values are default ones, like you didn't set them up.
A: If you really only want to know "why this works", the answer is that LiveCD Ubuntu that you used for installation had this unimportant for boot file \etc\default\grub.cfg, so created yours important for boot /boot/grub/grub.cfg using it.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 7,844 |
package pajama.js;
public class JSNum extends JSAtom<Integer>{
public JSNum(int value){
super(value);
}
} | {
"redpajama_set_name": "RedPajamaGithub"
} | 3,021 |
Open Class: ARTS1631 Introductory Japanese B with Professor Chihiro Thomson
by Open Class Initiative
Mon., 14 October 2019
11:00 am – 1:00 pm AEDT
Ritchie Theatre, UNSW Kensington Campus
Library Road
Kensington, NSW 2033
Interactive introductory Japanese language lecture with the support of senior students.
This lecture will be an interactive introduction to the Japanese language, with the support of senior students. Techniques and approaches used will include:
Student peer learning
Active student participation
This course is part of the School of Humanities and Languages within the Faculty of Arts and Social Sciences.
About Professor Chihiro Thomson
Chihiro is a Professor of Japanese Studies in the School of Humanities and Languages in Arts & Social Sciences. She has been a Scientia Education Academy Fellow since 2016, when the Academy was first established.
Chihiro is an internationally recognised educator and reseacher of Japanese language, having served as President of the Japanese Studies Association of Australia (2009-2011) and the Chair of the Board of the Global Network of Japanese Language Education (2007 - 2009 & 2012 - 2016), an alliance of Japanese language education associations in 11 countries and regions.
To read more, visit Chihiro's bio page on the Teaching Gateway
Open Class: ARTS1631 Introductory Japanese B with Professor Chihiro Thomson at Ritchie Theatre, UNSW Kensington Campus
Library Road, Kensington, NSW 2033, Australia
Browse Kensington Events | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 8,215 |
Razborca – wieś w Słowenii, w gminie Mislinja. W 2018 roku liczyła 114 mieszkańców.
Przypisy
Miejscowości w gminie Mislinja | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 5,034 |
Kosekans (znanstv. lat. cosecans, skaćeno od lat. complementi secans, 'sekans komplementarnoga kuta') jest trigonometrijska funkcija. Jednak je omjeru hipotenuze pravokutnog trokuta i katete koja je kutu nasuprot, to jest jednak je recipročnoj vrijednosti sinusa. Graf ove funkcije nalikuje onomu sekansa, ali pomaknut je udesno za π/2. Oznaka je cosec.
Izvori
Trigonometrijske funkcije | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 1,071 |
Andrew McGee is the American former guitarist for God Lives Underwater. He currently plays guitar in Wired All Wrong.
References
Living people
American rock guitarists
Year of birth missing (living people)
God Lives Underwater members
Place of birth missing (living people) | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 9,686 |
It's time for another round of MSG! We are looking for plays to be developed for our staged reading series in January 2020.
Through the MSG Lab, VACT showcases Asian Canadian playwrights and their stories and seeks to develop more Asian Canadian plays to a professional level for potential production by VACT or with other theatre companies in the Greater Vancouver area.
The selected emerging or established playwright(s) will be offered 3 months of dramaturgy through PTC, two workshop sessions and a staged reading. One play may be selected for further development with the potential to be produced in the future.
A cover letter (2 pages maximum) discussing the rationale behind your proposed project, for example: What is your project about? What is your vision for this project? Why this particular project now? Why the MSG Lab? What are your needs and expectations of this process?
Submissions are read by a panel of curatorial staff and guest artists. Shortlisted candidates may be contacted for a follow-up interview. Decisions will be made by end of May.
If you have any questions regarding the application process or your eligibility, please contact playsubmission@vact.ca. We look forward to receiving your submissions! | {
"redpajama_set_name": "RedPajamaC4"
} | 794 |
Brian Jolly (born 1 March 1946) is a former British cyclist. He competed in the individual road race at the 1968 Summer Olympics.
References
External links
1946 births
Living people
British male cyclists
Olympic cyclists of Great Britain
Cyclists at the 1968 Summer Olympics
Sportspeople from Sheffield
20th-century British people | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 1,175 |
Q: Passing this pointer to inner class as a weak_ptr I am trying the figure out a way to use shared_ptr and weak_ptr in my code. I have two classes - Class One and Class Two. Two in an inner class of One. Class two's constructor takes in a weak_ptr of Class One as shown below and stores it for later use.
Class One
{
Class Two
{
private:
std::weak_ptr<One> m_wptrOne;
public:
Two(std::weak_ptr<One> wptrOne)
{
m_wptrOne = wptrOne;
/* m_wptr is used later by class Two if not expired and valid ofcourse */
}
}; // End Class Two
.....
void foo()
{
std::shared_ptr sptrOne(this);
Two obj(sptrOne);
.... /* do my work */
} // Program crashes when foo terminates
}; //End Class One
I get a crash when my function foo returns because I think "sptr" is trying to free "this" pointer thinking that it is the only owner of it.
How can I solve this problem? Or is my program architecturally incorrect? Any suggestion would be greatly appreciated.
Thanks,
Tushar
A:
I don't want to make it compulsory for my library users to create the object on the heap as a shared pointer.
Then your inner class cannot require a weak_ptr. The use of a weak_ptr requires the use of a shared_ptr; it relies on the machinery that shared_ptr creates to know when the pointer has been destroyed. So if you don't want users to have to use shared_ptr, you cannot do anything that expects the class to be wrapped in a shared_ptr. Like create a weak_ptr from it.
Therefore, you need to make your inner class independent of weak_ptr if you want users to be able to not create these objects on the heap.
You could try something where you force users to wrap their stack objects in a shared_ptr that uses a special deleter. But that'd be far more annoying than just heap allocating it.
A: The below is an example of using enable_shared_from_this to pass weak ownership semantics from this.
Note that it is not possible to express weak ownership semantics to an object with automatic storage duration.
The crash you mention regarding shared_from_this is probably an exception of type std::bad_weak_ptr caused by the attempt to gain a shared_ptr from an object that is not shared.
#include <memory>
class One : public std::enable_shared_from_this<One>
{
public:
class Two
{
private:
std::weak_ptr<One> m_wptrOne;
public:
Two(std::weak_ptr<One> wptrOne)
{
m_wptrOne = wptrOne;
/* m_wptr is used later by class Two if not expired and valid ofcourse */
}
}; // End Class Two
//.....
void foo()
{
std::shared_ptr<One> sptrOne = shared_from_this();
Two obj(sptrOne);
//.... /* do my work */
} // Program crashes when foo terminates
}; //End Class One
int main()
{
auto one = std::make_shared<One>();
one->foo();
}
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 7,338 |
Q: PDFStamper not working for newer versions of iTextSharp The latest version of iTextSharp is 5.5.11.
When I run the exact same code against 5.5.0 it works.
It's a pretty simple request - use iTextSharp to set a watermark on an existing PDF in-memory.
Here's the offending code:
private static Stream Watermark(Stream inStream, WatermarkRequest request)
{
using (var outStream = new MemoryStream((int)(inStream.Length + 4096)))
{
using (var reader = new PdfReader(inStream))
using (var stamper = new PdfStamper(reader, outStream))
{
var opacity = request.Opacity.HasValue ? request.Opacity.Value : 0.75f;
var angle = request.Angle.HasValue ? request.Angle.Value : 45;
var size = request.Size.HasValue ? request.Size.Value : 72f;
var color = new BaseColor(1f, 0f, 0f, 0f);
if (!String.IsNullOrEmpty(request.Color))
{
var rgba = PDF.Lib.RGBA.convert(request.Color);
color = new BaseColor((float)rgba.R, (float)rgba.G, (float)rgba.B, 0f);
}
for (var n = 1; n <= reader.NumberOfPages; n++)
{
var pcb = stamper.GetOverContent(n);
var gstate = new PdfGState() { FillOpacity = opacity, StrokeOpacity = opacity };
pcb.SaveState();
pcb.SetGState(gstate);
pcb.SetColorFill(color);
pcb.BeginText();
pcb.SetFontAndSize(BaseFont.CreateFont(BaseFont.HELVETICA, Encoding.Default.WebName, true), size);
var ps = reader.GetPageSize(n);
var x = (ps.Right + ps.Left) / 2;
var y = (ps.Bottom + ps.Top) / 2;
pcb.ShowTextAligned(Element.ALIGN_CENTER, request.Text, x, y, angle);
pcb.EndText();
pcb.RestoreState();
}
}
return new MemoryStream(outStream.ToArray());
}
}
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 6,978 |
{"url":"https:\/\/stats.stackexchange.com\/questions\/321153\/two-questions-about-explanatory-variables-and-transformations","text":"# Two questions about explanatory variables and transformations\n\nI'm using Excel and R to do this. I'm new to R.\n\nI have a dataset of energy savings arising from capital investments supported by government grants. I've done a multiple linear regression of energy savings against grant size, total investment cost, and several dummies about investment type. The data is far from normal and looks like it needs transformation.\n\nI did a Box-Cox test on y (ie energy savings), resulting in lambda of -0.14, so I transformed y using a natural log. Then, thinking the explanatory variables also needed transforming, I followed this (where it speaks about transformations of predictors), and the lambdas of cost and grant are 2.25 and 1.75 respectively (both have significant p-values), suggesting a X^2 transformation. Additionally (ie as an alternative), I did log transformations on both grant and cost as I had a hunch that would normalise them.\n\nMy questions are as follows.\n\nQuestion 1: Using transformed explanatory variables\n\nSo, looking solely at the adjusted R^2 and not removing any non-significant predictors, the adj-R^2 values are as follows.\n\n1. model 1 (no transformation): 0.5596\n2. model 2 (y = ln(y)): 0.6070\n3. model 3 (y=ln(y), x=x^2 where not dummy): 0.5478\n4. model 4 (y=ln(y), x=ln(x) where not dummy): 0.6809\n\nI think I should use the ln transformation (ie model 4), but the box.tidwell lambdas recommended a x^2 transformation.\n\nAm I justified using a ln transformation on the explanatory variables despite what the box.tidwell test says? Why? This page suggests I can compare models in R with anova(fit1, fit2), but I don't really know how to evaluate the output (the smaller RSS is better?).\n\nQuestion 2: Addressing multicollinearity\n\nWhile doing all this, I realised I had forgotten about multicollinearity. Grant size and investment cost are highly correlated. I want to use the model to estimate energy savings for various grant sizes, so I think it's okay that they are correlated. But I was wondering, can I transform the grant column to be something like 'percentage granted'? I suspect that would remove the correlation, but might do something weird because it would be a constrained value (ie always < 1). Or maybe it could make it a dummy or something?\n\nHow can I deal with the multicollinearity of grant size and project cost? Is it just a matter of trying different transformations and seeing how they affect the fit?\n\nThanks so much.\n\n\u2022 The main reason for transforming here should be that way you get closer to linearity. Normality of any marginal distribution is not a problem if you get it, but it is not an assumption for regression. Every good text on regression explains this point. That aside, you are asking for advice on how to model the data, but you never show the data either in listings or in graphs. \u2013\u00a0Nick Cox Jan 2 '18 at 12:10\n\u2022 (1) Adjusted $R^2$ may be useful for comparing transformations of predictors--at least if you adjust it for the additional parameters implied by the freedom to transform the predictors--but it is meaningless for comparing transformations of the responses. (2) Don't use Box-Cox parameters like $2.25$ or $1.75$. Indeed, be very suspicious of any estimate of $\\lambda$ that lies beyond the interval $[-1,1]$ unless theory suggests it should. (3) Maximum likelihood makes strong assumptions; its Box-Cox estimates can be poor. (4) For a robust approach, see stats.stackexchange.com\/a\/35717\/919. \u2013\u00a0whuber Jan 2 '18 at 13:39\n\n## 1 Answer\n\nFor question 1, I would suggest not transforming variables based on statistical considerations but on substantive ones. Presumably, energy savings is in dollar terms (or some other currency). It often makes sense to take the log of this sort of variable since we often think of money in multiplicative terms rather than additive ones. For instance, the difference between a \\$10,000 house and a \\$12,000 one is large; the difference between a \\$1,000,000 house and a \\$1,002,000 house is rounding error. Whether this is the case in your situation isn't completely clear to me, but it seems possible.\n\nIf you decide to not transform, based on the above, note that OLS regression does not assume that the variables are normally distributed; it assumes that the errors (estimated by the residuals) are normally distributed. If you find that this assumption is violated, then you can use methods that don't assume linearity of the residuals (e.g. robust regression, quantile regression).\n\n\u2022 Peter: Linearity of the residuals is just a typo here for normality of the errors, but not quite trivial errors should be corrected by the OP! \u2013\u00a0Nick Cox Jan 2 '18 at 12:17","date":"2021-06-24 18:25:24","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\": 0.7622259855270386, \"perplexity\": 1080.5392753402787}, \"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-2021-25\/segments\/1623488556482.89\/warc\/CC-MAIN-20210624171713-20210624201713-00567.warc.gz\"}"} | null | null |
Brachycephalus fuscolineatus é uma espécie de anfíbio da família Brachycephalidae. Endêmica do Brasil, pode ser encontrada no Morro do Baú no município de Ilhota, estado de Santa Catarina.
fuscolineatus
Anfíbios descritos em 2015
Anfíbios do Brasil | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 8,947 |
It's like a community garden, planted with solar panels!
Community solar is ideal for members who want to harness the power of the sun's energy, but lack a suitable site or funds for a solar array of their own.
Phase I Generation Monitor
Phase II Generation Monitor
Phase II Q & A
Crow Wing Power has now built and commissioned our second Community Solar Array or Phase II. It's located at the Swanburg Outpost east of Pine River. Would you be interested in purchasing and being a co-owner with other Cooperative members at $1,300/share? Members have the option for the following:
1/2 share
3 or more shares
or finance the $1,300 over twenty years at $7.50/month.
Phase II Interest Form
Array Size:
50.47 KW DC – 39.9 KW AC
Average Predicted Annual Energy Production:
55,000 kWhs
Enough to supply five homes
Solar Panel Manufacturer:
Solar World, made in the US
Solar Panel Installer:
Viking Electric In. out of Spring Grove MN
Panel Size:
Number of Panels:
Average Annual kWh projected per share:
500/kWhs/share/annually
Q. Why would Crow Wing Power be involved in building and selling shares in a community solar array, when renewable energy costs more than traditional electricity?
A. We did extensive surveys of our members in 2016 and found a substantial amount of support from a select group of members that were very committed to solar energy and said they would pay additional money to be a part of a green project such as this.
Q. How much energy is the solar array anticipated to produce?
A. The anticipated output is 450 to 500 kWhs/year per share. There will be 111 shares available. As a point of reference, an average household uses 10,080 kWhs /year of electricity.
Q. What monthly credit can a member anticipate receiving for the output produced by a community solar share?
A. Participants will receive a bill credit for the amount of solar energy their subscription produces (Current calculations show that to be approximately $4.00/month to $4.50/month.)
Q. What is the cost of purchasing a share and what does that cover?
A. Subscription cost of one share is $1300. Crow Wing Power will take care of maintenance and insurance. The contract is for 20 years. Payback depends on the future price of electricity and solar energy produced.
Q. Can I buy a one-half share or more than one share of the community solar array?
A. You can purchase from one-half share up to, but not to exceed, your annual kWh use. Maximum shares will be determined by comparing the estimated annual output of the solar share(s) the the annual General Service use historically consumed at the service address. (An average member uses 10,080 kWhs/year in electricity.)
Q. Can I make a down payment and pay the balance later?
A. Members can elect to pay in full when signing the contract, or they can elect to pay half as a down payment with the balance due and payable by June 1, 2018. Each full share will require a $500 down payment and 1/2 share will require a $250 down payment.Members will receive a monthly invoice after making their down payment, showing the amount yet due allowing multiple payment opportunities prior to June1, 2018.
Q. What do I do next to subscribe to a share or shares in the solar array project?
A. Members who wish to participate must call our customer service department. Our representatives can help members decide how much to purchase and send members a contract to review before enrolling.
Q. When will the solar array be built and commence operations?
A. We will begin construction once 80% of the array is subscribed, or when 88 shares have been subscribed.
Q. Why wait until 80 shares are sold to commence building?
A. Crow Wing Power members that do not wish to participate or pay additional money to be a part of this renewable energy project will not be burdened by the added cost of a community solar array. Only members that wish to be a part of the project will financially support it.
Q. Can I buy shares and gift it to my favorite charitable organization or another family?
A. If that organization or family lives in Crow Wing Power's service territory and is a member, you can purchase shares on their behalf. The total amount you can purchase can be no more than that service location's annual electric use allows.
Signing up is easy. After calling our Member Service Department at 218-829-2827 or 1-800-648-9401 to determine your electric needs and matching that to the amount of shares you desire to purchase, we will send you a twenty-year contract. You will be able to pay for your share(s) by credit card or check. You can review the contract online, but you must call us to purchase your share(s).
Crow Wing Power Community Solar Contract
The Fall of 2017 we began selling shares in a new Community Solar Project and within a few months we were 100% sold out.
On March 22, 2017 the panels were installed and hooked up to the grid and solar production began. Subscribers began seeing credits on their May bills for the month of April and March.
Our community solar project features a solar photovoltaic (PV) system manufactured and assembled in Minnesota. The panels are installed in tandem, facing back-to-back to maximize installed watts and energy yield per square foot. Because this arrangement offers complete PV coverage of a space, the performance of the system is virtually unaffected by its system azimuth. This is ideal for applications where maximum energy generation is required in a limited space.
We celebrated our new Community Solar Project with a ribbon cutting on Thursday April 20, 2017 with our member supporters/subscribers and Board of Directors. | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 2,638 |
gallery | What Is An Expat? - Part 3What Is An Expat?
I didn't even know they still made these pants. Seen under the Eiffel Tower in Paris.
I have finally updated the Gallery with pictures from our June trip to the US and the July Mexico trip.
2007 06 USA Just a few really. Mainly at the river running through Golden, Colorado.
2007 07 Mexico Lots of good stuff in here. Dolphins, cats, other animals, Mayan ruins, underground caverns, jungle, the beach and sea…you name it. Unfortunately I didn't have the underwater housing for my camera so there are no cave diving pictures, though there are a couple lower quality ones from a cave snorkeling trip.
I've updated the gallery with pictures from our recent trip to France and Italy. It's broken down into subgalleries of Paris, Versailles, Rome, Sorrento, and Pompeii for easy viewing, though the Paris and Rome galleries are pretty good sized. These were actually uploaded several days ago, but I've been kind of busy preparing to leave again. Enjoy!
Does anything about this picture look difficult to you? Does any of it look pointless? This was the most amusing of several "we're in a ruined city with broken, cobbled pave stones and we are toting a kid in a stroller" type moments we happened across in Pompeii. Traveling with kids seems like it would be nightmare enough to avoid whenever possible. I mean sure, trips to see the folks are a different issue and point to point travel is not the same as vacation, but I am non-plussed to think that people would travel with a child for non-essential reasons.
For the first 5-6 years, the kid certainly won't get anything out of it — won't understand what's going on nor remember it. They can't really move under their own power, and if they can it's probably the wrong direction and you'd want to avoid that. You're looking at mobile, and therefore more difficult, babysitting. It costs the parents extra money and would dilute their enjoyment of whatever they are doing since they wouldn't be able to focus on it. It seems like there's only cost and no benefit. Leave the kids with relatives or stay home.
For the next 5-6 years, it'd be a little better. The kids would be generally self-motile at least, and would probably remember the vacation, though if you're going to travel places as opposed to beach or camping trips, they may have trouble to "get" what they're seeing. Stick to natural wonders and getting to know nature type journeys.
Once you're dealing with teens, I think things open up more. Supervision is less necessary as they are now less dependent and not so much a danger to themselves. You can go places that take more appreciation and will broaden the kid's world. You might be able to learn and teach something that they can grow on later on, making them a better person by seeing places and things in new towns, states, or countries. Give 'em something to have as stories back at school. Show them there is a whole world out there, full of people that are kind of like us, even! Keep them open minded.
I like to call this one "We're Gonna Need Another Maid".
Where labor is so cheap, many Arab families actually have 1 maid/nanny per kid. I know a couple who pay their live-in about $150 a month. They got her before they even had a kid or she was even pregnant, no less. While that's on the low end of the pay scale, $250-300 is going to be the top end. A round-trip ticket home once a year will be included as well.
Cheap labor is everything here. The city is built on it. Because of it, anything you could go buy in a store or restaurant is available for delivery. It costs almost nothing to have a couple of guys that do nothing but drive stuff back and forth on a moped. The easy availability of delivery has a tendency to make certain sorts of people lazy. I'm lazy, but we've only ever had food delivered a couple-three times in these almost 2 years in Dubai. It's important to get out and about every so often, even if it's only to go downstairs and over 2-3 buildings to get lunch instead of someone bringing it to you so you never have to leave your desk. That said, come summer time you certainly won't want to go any further than 2-3 buildings away as you will stink at your co-workers the rest of the day. | {
"redpajama_set_name": "RedPajamaC4"
} | 8,444 |
{"url":"https:\/\/math.stackexchange.com\/questions\/1337355\/what-is-the-difference-between-functions-and-operations","text":"# What is the difference between functions and operations?\n\nWikipedia says that\n\nan operation $\\omega$ is a function of the form $\\omega: V \\to Y$, where $V \\subset X_1 \\times\\cdots\\times X_k$.\n\nBut as far as I know, every function's domain is a set, so every function can be seen as an operation where $V \\subset X_1$. Thus, all functions are operations. Since all operations are also functions by the definition given, the terms \"function\" and \"operation\" are equivalent. What am I missing? Or are they truly equivalent?\n\n\u2022 Any function can be seen as a unary operation over its domain. Jun 24, 2015 at 8:56\n\nWhile your reasoning is correct that \"every function is an operation\" under that extremely general definition of \"operation\", I would say that a more common definition of an \"operation\" on a set $S$ would be a function $$\\alpha: S^n\\to S\\quad\\text{ for some }n\\geq 0$$ or, to allow \"partial\" operations, $$\\alpha: X\\to S\\quad\\text{ where }X\\subset S^n\\text{ for some }n\\geq 0$$ (and we would say $\\alpha$ is an $n$-ary operation). Under this definition, there are clearly many functions that are not operations.","date":"2022-05-16 22:54:45","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\": 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\": 0.8872423768043518, \"perplexity\": 240.62676884872874}, \"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\/1652662512249.16\/warc\/CC-MAIN-20220516204516-20220516234516-00616.warc.gz\"}"} | null | null |
<?php
namespace J2t\Rewardpoints\Block\Adminhtml\Customer\Edit\Tab\Points\Grid\Renderer;
/**
* Adminhtml newsletter queue grid block status item renderer
*/
class Pointstype extends \Magento\Backend\Block\Widget\Grid\Column\Renderer\AbstractRenderer {
protected $_pointModel = null;
protected $_pointData = null;
protected $_customerModel = null;
protected $_orderModel = null;
public function __construct(
\Magento\Backend\Block\Context $context, \J2t\Rewardpoints\Model\Point $point, \J2t\Rewardpoints\Helper\Data $pointHelper, \Magento\Customer\Model\Customer $customer, \Magento\Sales\Model\Order $order, array $data = []
) {
$this->_pointModel = $point;
$this->_customerModel = $customer;
$this->_orderModel = $order;
$this->_pointData = $pointHelper;
parent::__construct($context, $data);
}
/**
* Constructor for Grid Renderer Status
*
* @return void
*/
/* protected function _construct()
{
self::$_statuses = [
\Magento\Newsletter\Model\Queue::STATUS_SENT => __('Sent'),
\Magento\Newsletter\Model\Queue::STATUS_CANCEL => __('Cancel'),
\Magento\Newsletter\Model\Queue::STATUS_NEVER => __('Not Sent'),
\Magento\Newsletter\Model\Queue::STATUS_SENDING => __('Sending'),
\Magento\Newsletter\Model\Queue::STATUS_PAUSE => __('Paused'),
];
parent::_construct();
} */
/**
* @param \Magento\Framework\Object $row
* @return string
*/
public function render(\Magento\Framework\DataObject $row) {
$statusField = $this->_pointData->getStatusField();
$orderId = $row->getData($this->getColumn()->getIndex());
$model = $this->_pointModel;
$pointsType = $model->getPointsDefaultTypeToArray();
$pointsType[\J2t\Rewardpoints\Model\Point::TYPE_POINTS_ADMIN] = __('Store input %1', ($row->getRewardpointsDescription()) ? ' - ' . $row->getRewardpointsDescription() : '');
if ($orderId == \J2t\Rewardpoints\Model\Point::TYPE_POINTS_REFERRAL_REGISTRATION) {
$currentModel = $model->load($row->getRewardpointsAccountId());
$model = $this->_customerModel->load($currentModel->getRewardpointsLinker());
if ($model->getName()) {
return __('Referral registration points (%1)', $model->getName());
}
}
if (\J2t\Rewardpoints\Model\Point::TYPE_POINTS_REQUIRED == $orderId) {
$currentModel = $model->load($row->getRewardpointsAccountId());
if ($currentModel->getQuoteId()) {
$orderModel = $this->_orderModel->loadByAttribute('quote_id', $currentModel->getQuoteId());
if ($orderModel->getIncrementId()) {
$pointsType[\J2t\Rewardpoints\Model\Point::TYPE_POINTS_REQUIRED] = __('Required point usage for order #%1 (%2)', $orderModel->getIncrementId(), __($orderModel->getData($statusField)));
}
}
}
if (($orderId > 0) || ($orderId != "" && !is_numeric($orderId))) {
$order = $this->_orderModel->loadByIncrementId($orderId);
return __('Points related to order #%1 (%2)', $orderId, __($order->getData($statusField)));
} elseif (isset($pointsType[$orderId])) {
return $pointsType[$orderId];
}
return null;
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 713 |
Drastically Reduced Commercial real estate. Motivated Seller!!! Bring offer. This location also sits south of large storage facility and shopping center. This location is ideal for store front, convenience store, gas station, etc... Prime location just minutes from 1-10 and recently zoned B-2. All utilities are within 50 feet. | {
"redpajama_set_name": "RedPajamaC4"
} | 92 |
Q: Is there a way to read a csv column when its name contains spaces in Angular? I use d3.js library to read a CSV in Angular. Here's how I read it.
d3.csv('../assets/datasets/tests-per-million.csv').then((data: any) => {
this.testsData = data;
for (let i = 0; i < this.testsData.length; i++) {
this.testCountries.push(this.testsData[i].Entity.split(' ')[0]);
this.testDates.push(this.testsData[i].Date);
this.cumulativeTests.push(this.testsData[i].Cumulative);
}
});
The problem is that the last column of the CSV contains spaces on its name. Here is a sample of a CSV row.
Entity: "Australia - units unclear"
Code: ""
Date: "Mar 22, 2020"
Cumulative total tests per million: "5632.53024026095"
I have successfully read the three first columns but I cannot catch the last one which contains space on its name.
A: Since the key (Cumulative total tests per million) has spaces, you need to access the JSON using bracket notation:
this.cumulativeTests.push(this.testsData[i]['Cumulative total tests per million']);
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 8,541 |
Kent captain Darren Stevens is hoping his side can finish off Somerset's second innings early on the last day of their County Championship Division One opener on Monday.
Having scored 209 in reply to the home side's first innings 171 at Taunton – following a day one washout – Kent reduced their rivals to 171-7 on Sunday.
Mitch Claydon followed his 5-46 haul with another 3-30, as Stevens (2-34) also had success, to leave Somerset just 133 runs ahead with three wickets in hand.
And Stevens is looking for Kent to complete their task in the field quickly, to keep the target down as much as possible, saying: "It's hard to say what a par score will be in the final innings, but we have talked about taking quick wickets in the morning. We would be happy to chase anything up to 180, but even if the target ended up being 230 we would have a decent chance. | {
"redpajama_set_name": "RedPajamaC4"
} | 2,922 |
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Effective Date: 13.06.2017
© 2017 TravelTradeDaily.com. All Rights Reserved. | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 4,371 |
Q: Custom NativeActivity to show Arguments on a rehosted designer I've a custom NativeActivity with a bunch of InArguments. It will always be the root activity when the client is editing it's workflow on a rehosted designer. So far so good.
The problem is, although InArguments are showed on PropertyInspector, no 'Arguments' tab is showed. I understand why it happens but I really want to make this NativeActivity to be the root activity for a variety of reasons: I can customize it on my way, I can give it a custom Designer, etc.
How can I had argument properties to a NativeActivity in a way that they're showed on 'Arguments' tab within the designer making them available and allowing the client to do whatever he/she wants with the InArguments without the "'argumentName' is not declared. It may be inaccessible due to its protection level" think?
What I've tried:
*
*Just like variables, where we can use a Collection{Variable} and the designer recognizes it as the place where the activity variables are, I've tried Collection{Argument}. No luck!
*Simply write InArgument within NativeActivity. They're showed on Property Inspector but can't be used within the workflow through the designer. Again they cannot be resolved and"'argumentName' is not declared. It may be inaccessible due to its protection level" happens.
*Bind previous InArguments to RuntimeArguments through CacheMetadata() using metadata.AddArgument() and metadata.Bind(). No luck whatsoever.
I known that ActivityBuilder is probably the best way to solve this. I successfully tried something like:
ActivityBuilder builder = new ActivityBuilder();
builder.Implementation = new MyCustomNativeActivity();
DynamicActivityProperty property = new DynamicActivityProperty()
{
Name = "TestArg",
Type = typeof(InArgument<string>),
};
builder.Properties.Add(property);
Designer.Load(builder);
But this leads to bad code because I've to had the arguments when I'm loading the designer. This prevents me from having a library with my custom activities separated from the forms code itself. Mainly because ActivityBuilder is a sealed class, much like DynamicActivity which, I assume, would work too.
Is there any way to simulate the ActivityBuilder behaviour, that allow to add properties at codetime, but using a NativeActivity which can be easily inherited and customized?
I hope I've made myself clear.
P.S.: I also took a look at IExecutionProperty but I didn't understand well how it works, what's its purpose and if it can be applied here.
Thanks
A: Not really sure bout your problem here. That said only yesterday I solved another problem by deriving from Activity directly and not using NativeActivity for a root of the workflow. Not sure if it will help in your case but it might just do it.
So something like this:
public class RootActivity : Activity
{
public InArgument<string> TestArg {get; set;}
public RootActivity(Activity child)
{
Implementation = () => child;
}
}
Note: Typed in notepad so not tested and be careful with typos.
A: It seems that my problem is known and there's no plan to support it on next WF releases.
Links with explanations and a few workarounds: here and here.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 2,487 |
Happiness Blooms Designer series coordinates so well with other sets I wanted to show off another example. In this card I paired the paper with the Amazing Life Stamp Set for an entirely different look than yesterday's card .
Using the Big Shot cut out a Stitched Rectangle Frame 3 x 2 1/4″ in Call Me Clover and rectangle in Whisper White.
Stamp sentiment and the clovers in Call Me Clover Ink on the Whisper White rectangle.
Color clovers and luck with Wink of Stella.
Adhere the rectangle frame to the white rectangle with snail.
Adhere the framed rectangle to the DSP with a dimensional.
Stamp a clover on scrap Whisper White and color with Wink of Stella.
Cut out the clover and adhere to the rectangle with snail.
Previous postHope your day is amazing card. | {
"redpajama_set_name": "RedPajamaC4"
} | 2,392 |
{"url":"http:\/\/mamclain.com\/?page=Research_Engineering_UNCC_RF_1GHz_HW_Dipole_Antenna","text":"Posted:4\/08\/2015 2:10PM\n\n### The Creation of a 1 GHz Half-Wave Dipole Antenna\n\nMike Mclain discusses the creation of a 1 GHz half-wave dipole antenna and presents the necessary calculations associated with that antenna's development.\n\n## Preface:\n\nThe purpose of this article is to discuss the development and creation of a 1 GHz half-wave dipole antenna from its conception to its physical hardware implementation. This article will accomplish this task by providing design calculations, experimental results, and diagrams of the materials utilized in the antennas physical implementation.\n\n## General Design Calculations:\n\nThe first step needed to create a 1 GHz half-wave dipole antenna is to calculate the fundamental wavelength of the desired signal (in this particular case 1 GHz) by using the wavelength equation, as shown by:\n\n$\\lambda = \\frac{v_p}{f}$\n\nin which ($\\lambda$) is the wavelength, ($v_p$) is the phase velocity, and ($f$) is the frequency.1 [p.15]\n\nLikewise, the phase velocity ($v_p$) of the signal can then be calculated by using the phase velocity equation, as shown by:\n\n$v_p = \\frac{1}{\\sqrt{\\mu_r\\mu_0\\varepsilon_r\\varepsilon_0}}$\n\nin which ($\\mu_r$) is the relative permeability, ($\\mu_0$) is the permeability of free-space, ($\\varepsilon_r$) is the relative permittivity, and ($\\varepsilon_0$) is the permittivity of free-space.1 [p.15]\n\nConversely, substitution of these parameters into Equation: 2 results in the creation of:\n\n$v_p = \\frac{1}{\\sqrt{\\mu_r\\mu_0\\varepsilon_r\\varepsilon_0}}\\\\\\\\ = \\frac{1}{\\sqrt{(1)(4\\pi\\times 10^{-7})(1)(8.854\\times 10^{-12})}}\\\\\\\\ = 2.99796\\times10^{8}$\n\n, and substitution of these parameters into Equation: 1 results in the creation of:\n\n$\\lambda = \\frac{v_p}{f}\\\\\\\\ = \\frac{2.99796\\times10^{8}}{1\\times 10^{9}}\\\\\\\\ = .299796m$\n\nNow, since most building materials are specified in inches rather than in metric values, conversion from meters to inches can be done by:\n\n$1m \\approx 39.3700787in$\n\nand substitution of Equation: 5 into Equation: 4 results in the creation of:\n\n$\\lambda = .299796m\\\\\\\\ = \\left(.299796m\\right)\\left(\\frac{39.3700787in}{1m}\\right)\\\\\\\\ = 11.803in$\n\n.2\n\nLikewise, (because a half-wave dipole antenna is being designed) the half-wavelength of the desired signal (1 GHz) can be calculated from Equation: 6, as shown by:\n\n$\\lambda_{\\frac{1}{2}} = \\frac{\\lambda}{2}\\\\\\\\ = \\frac{11.803in}{2}\\\\\\\\ = 5.90149in$\n\nand similarly, the fundamental structure of a half-wave dipole antenna, as shown by:\n\n, can be conceptually summarized as being an antenna with two conductive elements that are separated by a small gap (in the middle) to aid in defining the antenna's impedance. Furthermore, the half-wave dipole antenna gets its name from the fact that the total length of the antenna (including both radiating elements and the gap) is half the wavelength of the desired (in this case 1 GHz) transmitting\/receiving frequency.\n\nConversely, because the total length of the antenna being created is half the wavelength of the desired 1 GHz signal, the half-wavelength value found in Equation: 7 was divided (once again) by one half in order to find the length of each dipole element as shown by:\n\n$\\lambda_{\\frac{1}{4}} = \\frac{\\lambda_{\\frac{1}{2}}}{2}\\\\\\\\ = \\frac{5.90149}{2}\\\\\\\\ = 2.950745in$\n\n. Likewise, because a finite gap must exist between the two antenna elements (and this gap is not accounted for within Equation: 8) some modifications to the length equation must be made to account for this gap. Conversely, since the gap shown (within Figure: 1) extends outwards from the center of the antenna symmetrically, the gap must also be divided by two in order to obtain the size (or half gap) that each antenna element needs to have subtracted from its element length in order to maintain the antenna's overall half-wavelength, as shown by:\n\n$Gap_{\\frac{1}{2}} = \\frac{Gap}{2}$\n\n.\n\nSimilarly, upon accounting for the half gap (observed within Equation: 9) in the element's quarter wavelength equation (defined within Equation: 8) results in the creation of the total element length equation, shown by:\n\n$\\lambda_{Gap} = \\lambda_{\\frac{1}{4}}-Gap_{\\frac{1}{2}}\\\\\\\\ = 2.950745in-Gap_{\\frac{1}{2}}$\n\n## Performance Calculations:\n\n. Now, because the gaps overall (or total) size will have real world design consequences, along with the fact that the mathematical equations needed to consider the size of the gap are very complicated (as observed upon examining :\n\n$I_e(x^{'},y^{'},z^{'}) = \\left\\{\\begin{array}{rc} \\hat{a}_z\\left\\{I_0\\sin\\left[k\\left(\\frac{\\ell}{2}-z^{'}\\right)\\right]+jpI_0\\left[\\cos\\left(kz^{'}\\right)-\\cos\\left(\\frac{k}{2}\\ell\\right)\\right]\\right\\} & 0\\leq z^{'}\\leq\\frac{\\ell}{2} \\\\ \\hat{a}_z\\left\\{I_0\\sin\\left[k\\left(\\frac{\\ell}{2}+z^{'}\\right)\\right]+jpI_0\\left[\\cos\\left(kz^{'}\\right)-\\cos\\left(\\frac{k}{2}\\ell\\right)\\right]\\right\\} & \\frac{-\\ell}{2}\\leq z^{'}\\leq 0 \\end{array}\\right.$\n\n, which is the current distribution equation for a antenna with a gap where $p$ is an unspecified coefficient, likely empirical, that relates the gap to the antenna length), it is best to neglect the size of the gap when performing back of the envelop calculations and once the general design is known then a RF computational tool (like NEC or HFSS) can be utilized to analyze the effects the gap has on the overall design.3 [p.181]\n\nLikewise, because of the overall mathematical complexity created by including the gap into the antenna's theoretical design, it is best (for the moment) to neglect the length of the gap within the fundamental design equations; however, it is also important to remember the existence of the gap and possible effects that the gap will have when fine tuning the design within software simulations.\n\nConversely, upon neglecting the gap from the design equations, the ideal current distribution (across the antenna) can be modeled mathematically, by:\n\n$I_e(0,0,z^{'}) = \\left\\{ \\begin{array}{rc}\\hat{a}_z I_0\\sin\\left[k\\left(\\frac{\\ell}{2}-z^{'}\\right)\\right] & 0\\leq z^{'}\\leq\\frac{\\ell}{2} \\\\ \\hat{a}_zI_0\\sin\\left[k\\left(\\frac{\\ell}{2}+z^{'}\\right)\\right] & \\frac{-\\ell}{2}\\leq z^{'}\\leq 0\\end{array} \\right.$\n\nin which ($I_0$) is the current maximum, ($k$) is the wave number which is defined by:\n\n$k = \\frac{2\\pi}{\\lambda}$\n\n, ($\\ell$) is the length of the antenna, and ($z^{'}$) is a position on the antenna with respect to the z axis (or element axis).3 [pp.170,179]\n\nNow, assuming a wavelength equal to the one calculated for 1 GHz (along with a unity maximum current) results in a current distribution plot which (when superimposed onto a figure of the dipole antenna) produces a graph as shown by:\n\n, that conceptually illustrates the current distribution assumed for all mathematical calculations while designing the antenna.\n\nNow that the current distribution has been assumed, the next thing that can be found is the magnetic vector potential which is described by:\n\n$A_r = \\frac{\\mu_0I_0\\cos(\\theta)}{2\\pi rk\\sin^2(\\theta)}e^{-jkr}\\left[\\cos\\left(\\frac{k\\ell}{2}\\cos(\\theta)\\right)-\\cos\\left(\\frac{k\\ell}{2}\\right)\\right]$\n\nand :\n\n$A_{\\theta} = -\\frac{\\mu_0I_0}{2\\pi rk}e^{-jkr}\\left[\\frac{\\cos\\left(\\frac{k\\ell}{2}\\cos(\\theta)\\right)-\\cos\\left(\\frac{k\\ell}{2}\\right)}{\\sin(\\theta)}\\right]$\n\n, in which ($r$) represents the distance away from the antenna.4\n\nLikewise, graphing $|A_r|$ produces:\n\n, while graphing $|A_\\theta|$ produces:\n\n; however, neither of these values are particularly useful (in their current context) because the magnetic vector potential is an intermediary step before obtaining values of electric and magnetic intensity.\n\nConversely, after performing a few mathematical operations that involve substitution with the magnetic vector potential equations, the electric intensity equation, as shown by:\n\n$\\bar{E} \\approx j\\eta\\frac{I_0}{2\\pi r}e^{-jkr}\\left[\\frac{\\cos\\left(\\frac{k\\ell}{2}\\cos(\\theta)\\right)-\\cos\\left(\\frac{k\\ell}{2}\\right)}{\\sin(\\theta)}\\right]\\hat{a}_{\\theta}$\n\n, and the magnetic intensity equation, as shown by:\n\n$\\bar{H} \\approx \\frac{jI_0}{2\\pi r}e^{-jkr}\\left[\\frac{\\cos\\left(\\frac{k\\ell}{2}\\cos(\\theta)\\right)-\\cos\\left(\\frac{k\\ell}{2}\\right)}{\\sin(\\theta)}\\right]\\hat{a}_{\\phi}$\n\n, can be obtained, In which ($\\eta$) is defined by:\n\n$\\eta = \\sqrt{\\frac{\\mu_r\\mu_0}{\\varepsilon_r\\varepsilon_0}}$\n\n.4 Likewise, Graphing $|\\bar{H}|$ produces:\n\n, while graphing $|\\bar{E}|$ produces:\n\n, noting that both graphs have a similar shape (but $\\bar{H}$ varies with $\\hat{a}_{\\phi}$ while $\\bar{E}$ varies with $\\hat{a}_{\\theta}$) and such similarities should be expected given the direct connection between the magnetic and electric fields.\n\nSimilarly, with values of magnetic and electric intensity known, the time-average power density can be calculated by:\n\n$\\mathscr{\\bar{P}}_{av} = \\frac{\\eta\\left|I_0\\right|^2}{8\\pi^2r^2}\\left[\\frac{\\cos\\left(\\frac{k\\ell}{2}\\cos(\\theta)\\right)-\\cos\\left(\\frac{k\\ell}{2}\\right)}{\\sin(\\theta)}\\right]^2\\hat{a}_r$\n\n.4\n\nLikewise, graphing $|\\mathscr{\\bar{P}}_{av}|$ produces:\n\n, and the time-average power density can now be used to calculate the radiation intensity, as shown by:\n\n$\\bar{U} = r^2|\\mathscr{\\bar{P}}_{av}|\\\\\\\\ = \\frac{\\eta\\left|I_0\\right|^2}{8\\pi^2}\\left[\\frac{\\cos\\left(\\frac{k\\ell}{2}\\cos(\\theta)\\right)-\\cos\\left(\\frac{k\\ell}{2}\\right)}{sin(\\theta)}\\right]^2$\n\n.4\n\nConversely, graphing $|\\bar{U}|$ produces:\n\nwhich, because of the unity value selected for ($r$), the time-average power density is identical to the radiation intensity. Likewise, more mathematics can be applied to the time-average power density in order to obtain the total radiation power, as shown by:\n\n$\\mathscr{\\bar{P}}_{rad} = \\frac{\\eta\\left|I_0\\right|^2}{4\\pi}\\left\\{C+\\ln(k\\ell)+\\frac{1}{2}\\sin(k\\ell)\\left[S_i\\left(2k\\ell\\right)-2S_i\\left(k\\ell\\right)\\right]\\right. \\\\ +\\left.\\frac{1}{2}\\cos\\left(k\\ell\\right)\\left[C+\\ln\\left(\\frac{k\\ell}{2}\\right)+C_i\\left(2k\\ell\\right)-2C_i\\left(k\\ell\\right)\\right]-C_i(k\\ell)\\right\\}$\n\n, in which ($C_i(x)$) is defined by:\n\n$C_i(x) = \\int_x^{\\infty}\\frac{\\cos(y)}{y}dy$\n\n, ($S_i(x)$) is defined by:\n\n$S_i(x) = \\int_0^x\\frac{\\sin(y)}{y}dy$\n\n, and ($C$) is defined by:\n\n$C = 0.5772$\n\n.4\n\nSimilarly, substitution into Equation: 21 results in the creation of:\n\n$\\mathscr{\\bar{P}}_{rad} = 36.5393$\n\n, and now that the value for radiation intensity and total radiated power has been found, directivity can be calculated by:\n\n$D = \\frac{4\\pi\\bar{U}}{\\mathscr{\\bar{P}}_{rad}}$\n\n.4 Likewise, Graphing $|D|$ produces:\n\n, which is the same plot observed within Figure: 8 but this plot is scaled by a numerical factor. Conversely, (at this point) the radiation resistance can be calculated by:\n\n$R_r = \\frac{2\\mathscr{\\bar{P}}_{rad}}{|I_0|^2}$\n\n, and further substitution results in obtaining the value of:\n\n$R_r \\approx 73.0785$\n\n.4\n\nNow, assuming a half-wavelength current density (as defined within Equation: 12 the input resistance for the antenna can be found by application of:\n\n$R_{in} = \\frac{|I_0|^2R_r}{|I_0|^2\\sin^2\\left(\\frac{k\\ell}{2}\\right)}$\n\nand further substitution results in obtaining the value of:\n\n$R_{in} \\approx 73.0785$\n\n.4\n\n## Experimental Results:\n\nAfter all of the mathematical calculations were simulations can be performed in both HFSS and NEC. The first set of simulations i performed were done in NEC, as shown by:\n\n, in which the values needed to create a half-wave dipole antenna (from the math above) were entered into the NEC software. Likewise, upon running a NEC simulation, a graph of both the two dimensional total gain plot, as shown by:\n\n, and a three dimensional total gain plot, as shown by\n\nwere obtained.\n\nSimilarly, a second set of simulations were performed in HFSS and a dipole structure was created, as shown by:\n\n, that was separated by a small gap, as shown by:\n\nand simulations were performed on the model.\n\nLikewise, a three dimensional total gain plot for the HFSS simulation was found, as shown by:\n\n, which (upon comparison with Figure: 12) appears to match the overall shape of the NEC 3D simulation. While HFSS is a more rigorous simulation software than NEC; however, one issue that arose (upon comparing results between the two) was a discrepancy between the input impedance calculated by HFSS and NEC, although the mathematics (provided above) seemed to agree more with the NEC simulation (vs. the HFSS simulation) yet HFSS is designed to account for more complex EM effects and this fact must also be considered while evaluating results.\n\nNote: during this part of the design process, I considered utilizing a single stub tuner to match the impedance of the antenna to a 50$\\Omega$ coaxial cable; however, the occurrence of such discrepancies (between the two simulations) made such notions a very bad idea, especially since a exact input impedance is required for single stub tuning to work correctly. Likewise, because the only option available (to obtain a exact input impedance given such discrepancies) was to physically build the antenna and measure the input impedance (prior to stubing), thus such options were not very viable (because of time and material constraints) so i decided not to utilize a single stub tuner for this particular antenna design.\n\n## Hardware Implementation:\n\nConversely, one of the first hardware challenges that needed to be resolved (when building the antenna) was deciding upon the type of coaxial cable that was going to be used since both $50\\Omega$ and $75\\Omega$ coaxial cable was available for this particular antenna design.\n\nLikewise, originally $50\\Omega$ coaxial cable was selected (in conjunction with a single stub tuner in order to match the antenna impedance to the coaxial cable impedance); however, (as discussed earlier) problems with knowing the physical impedance of the antenna made this selection a poor choice overall and the $50\\Omega$ coaxial cable was then changed to $75\\Omega$ coaxial cable at the cost of having a higher reflection coefficient between the RF 1 GHz power supply (that had a $50\\Omega$ input impedance) and the $75\\Omega$ coaxial cable.\n\nSimilarly, after deciding upon the $75\\Omega$ coaxial cable, additional information about the dielectric of the cable needed to be found and some modifications made to the phase velocity calculation and to the wavelength calculation in order for a RF balun to be created correctly using the selected $75\\Omega$ coaxial cable, as shown by:\n\n$\\varepsilon_{r_{Polyethylene}} = 2.25$\n\n,:\n\n$v_p = \\frac{1}{\\sqrt{\\mu_r\\mu_0\\varepsilon_r\\varepsilon_0}}\\\\\\\\ = \\frac{1}{\\sqrt{(1)(4\\pi\\times 10^{-7})(2.25)(8.854\\times 10^{-12})}}\\\\\\\\ = 1.9986\\times10^8$\n\n,:\n\n$\\lambda = \\frac{v_p}{f}\\\\\\\\ = \\frac{1.9986\\times10^{8}}{1\\times 10^{9}}\\\\\\\\ = .1999m\\\\\\\\ = \\left(.1999m\\right)\\left(\\frac{39.3700787in}{1m}\\right)\\\\\\\\ = 7.8687in$\n\n, and by:\n\n$\\lambda_{\\frac{1}{2}} = \\frac{\\lambda}{2}\\\\\\\\ = \\frac{7.8687in}{2}\\\\\\\\ = 3.9343in$\n\nLikewise, because the half-wave dipole must be driven (by the 1 GHz signal) differentially, i decided that a half-wavelength coaxial balun would be utilized, as shown by:\n\n, because constructing this type of balun is relatively simplistic (when compared to the bazooka balun) and is easier to understand at a conceptual level.3 [p.541]\n\nConversely, upon building the half-wavelength balun and assembling the half-wave dipole (using the numbers calculated from above) resulted in the creation of:\n\n, in which the balun assembly (within the photo) is observed to be a looped coaxial wire (noting that the electrical attachment from the balun to the antenna assembly was done experimentally via a twisted pair (at $\\approx 70\\Omega$ impedance) transmission structure to ensure that only a TEM mode of RF operation was in effect.\n\nLikewise, the overall antenna was mounted on a PVC pipe stand to allow for easy deployment in an outside test environment, as shown by:\n\n.\n\n## Experimental Measurements:\n\nConversely, this antenna was tested in a open outdoor area (to avoid RF reflections) at 1 GHz and a 3D radiation pattern was observed from a series of time average power density measurements obtained via a RF signal strength analyzer.\n\nLikewise, in order to obtain this information, the antenna was first mounted with the end-fire elements pointed along the z axis (towards the sky and ground) and a RF signal strength analyzer was moved along a fixed radius (outside the transmission near field) while the time average power density (in dB) was recorded at fixed intervals, as shown by:\n\n. Similarly, after the $\\phi$ axis time-average power density measurements were taken, the antenna was remounted with the end-fire elements pointed along the y\/x axis (towards the horizon) and a RF signal strength analyzer was moved along a fixed radius (outside the transmission near field) while the time average power density (in dB) was recorded at fixed intervals, as shown by:\n\n.\n\n## Conclusion:\n\nBased upon the measurements obtained within Figure: 19 it becomes apparent that the created dipole antenna is functioning in a manner similar to the HFSS\/NEC predicted results (an attribute that should be expected) given the consistent time average power observed across the $\\phi$ axis (noting that the minor 5 dB deviations observed are likely the combined result of hardware defects, mathematical approximations, impedance imbalances, and testing apparatus errors. Likewise, the $\\theta$ axis measurements (obtained within Figure: 20) are also consistent with the HFSS\/NEC predicted results (as dipole antennas do not transmit from an end-fire configuration, thus the transmission power falls off rapidly at the end of each antenna element).\n\n## Acknowledgments\n\nSpecial thanks to my friend Aaron Hatley for his help in the development, design, construction, and testing of this antenna!\n\n## References:\n\n[1] D. M. Pozar, Microwave Engineering. John Wiley & Sons, Inc, third ed., 2005","date":"2020-04-02 11:28:16","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\": 0.8508533835411072, \"perplexity\": 1040.08790352047}, \"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-2020-16\/segments\/1585370506959.34\/warc\/CC-MAIN-20200402111815-20200402141815-00105.warc.gz\"}"} | null | null |
Ferdinand Hodler (14. března 1853 – 19. května 1918) byl švýcarský malíř 19. století.
Biografie
Hodler se narodil v Bernu jako nejstarší ze šesti dětí v rodině truhláře Jeana Hodlera a jeho manželky Marguerite. V osmi letech mu zemřel otec a dva bratři na tuberkulózu. Matka se pak provdala za malíře Gottlieba Schüpacha, který měl z prvního manželství 5 dětí a do nově rodiny se narodily další.
Ferdinand Hodler v roce 1871 odešel do Ženevy, aby zde zahájil svou malířskou kariéru. Jeho dílo se skládá především z krajinomaleb, portrétů a figurálních skladeb. V poslední části svého života přešel k symbolismu a secesi. Vytvořil také vlastní styl, kterému říkal paralelizmus, jehož rysem je symetrické uspořádání figur jakoby v tanci.
V roce 1889 se Hodler oženil s Berthe Jacques. Jedním z jejich dětí byl Hector Hodler, zakladatel Světového esperantského svazu. Často stál otci modelem v nepřirozených polohách, oblečený nebo nahý. Když ve 33 letech nečekaně zemřel, přičítalo se to tomu, že stání modelem podlomilo jeho zdraví, není to však lékařsky ověřeno.
V roce 1914 Ferdinand Hodler odsoudil německý útok na Remeš, za což si vysloužil odstranění svých obrazů z německých obrazáren.
V následujícím roce jeho milenka Valentine Godé-Darel zemřela na rakovinu. Hodler strávil hodiny u jejího lože a maloval její utrpení.
Odkazy
Reference
Externí odkazy
Život a dílo F. Hodlera v němčině
Publikace o Ferdinandu Hodlerovi v Národní knihovně
Film o Ferdinandu Hodlerovi
Román o Hodlerovi v esperantu
Švýcarští malíři
Symbolističtí malíři
Secesní malíři
Narození v roce 1853
Narození 14. března
Narození v Bernu
Úmrtí v roce 1918
Úmrtí 19. května
Úmrtí v Ženevě
Muži | {
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} | 5,173 |
Residents Of Tamale Lauds Zoomlion
By Justice Dzido On Mar 8, 2019
Zoomlion Ghana Limited the waste management giant has given Tamale, the regional capital of the Northern Region a sanitation boost and urge to continue to keep their environment clean during and after the 62nd Independence Day Anniversary celebration.
The company started the celebration preparation by thoroughly cleaning the city of Tamale and it's environs to ensure that the city befits the President H. E. Nana Addo Dankwa Akufo-Addo's vision of a clean Ghana.
All major roads were cleaned and washed with mechanized street sweepers and men, all curbs painted, branding of the principal streets with the national colours and the introduction of astro turf technology at all round abouts making them so green and colourful among others.
Hundreds of men were deployed by the company through the week long preparation to ensure a cleaner Tamale.
Also, the company provided sweeping and cleaning services at the venue of the programme, the Aliu Mahama Sports Stadium, water to service all the washrooms, positioned waste bins on the streets and at vantage points in and around the stadium to further enhance the cleanliness of the city and the stadium.
In an interview, the Group Head of Communication of the Jospong Group of Companies, Mrs. Sophia Lissah, appealed to the media to intensify public education to augment Zoomlion's routine public education programmes to educate the populace on proper and acceptable waste management practices. She urged citizens of Tamale and visitors to continue to keep the environment clean to attract tourists to boost the Ghanaian economy.
Mrs. Lissah gave the assurance that Zoomlion will continue to do the daily cleaning and create the needed awareness to ensure a clean environment saying that is "the job we have chosen for ourselves but the people owe it a duty to ensure the waste is properly kept for collection and onward management."
The Regional Minster Alhaji Salifu Saed and other government officials were full of praise of the excellent work done by Zoomlion in making the Tamale City clean and colourful for the historic independence day celebrations. He reiterated the need for sustaining the clean environment and beautify the city.
The Northern Regional General Manager, Mr. Peter Dawuni, appealed to residents to take advantage of the nationwide one million nationwide bin distribution exercise being undertaken by Zoomlion to adopt a waste bin in their households in order to keep their refuse for appropriate management.
Mr. John Emmanuel Kwose a Ph.D student of the University for Development Studies and a development worker in an interview described the neatness of Tamale during the one week into the anniversary as the cleanest city in the world taking the President H.E. Nana Addo Dankwa Akufo-Addo for the innovation of regionalizing the independence anniversary and making the city of Tamale this clean. He was however quick to express his doubts if the neatness of the city was anything that could be sustained.
"My only fear is that the exercise could be political, because the president was coming to the region the power players did this to camouflage him and the president and after his visit we go back to the old days".
Mr. Kwose who is also the Chief Executive Officer of Universal Developers Consultancy (UniDeC) commended Zoomlion Ghana Limited for the show of competence and capability to the Ghanaian society that they can really keep Ghana clean. He said the city is highly amazed at the work that Zoomlion had put in and urged the authorities to ensure that their activities are sustained and not occasional.
He said the forest reserves in the Tamale metropolis should be developed into forest resorts instead of leaving it for people to engage in open defaecation and amorous meetings.
A businessman resident in Tamale, Mr. Jones Pedavoah on his part called for the Metropolitan and Municipal authorities to allow Zoomlion to manage their roundabouts for a fee saying the beauty of the city lies in the beauty of our roundabouts and entry points.
He said due to the independence celebration I noticed some streets that were not swept by Zoomlion have been added for cleaning and urged the writer to ensure that all such streets are covered during their routine cleaning.
62nd Independence CelebrationGhanaZoomlion
Soldier In Prison – Over 'Drop That Chamber' Video
AMIDU Opens Fire …My Request For Information To Work Are Disregarded
Don't torture suspects – CHRAJ Advices Police | {
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Q: Leaflet map reload onclick I have a gallery of image thumbnails and a image viewer. Each image has different #id has lat and long attached to it. I am triggering series of onclick events on clicking the thumbnail. The imagename, lat and long info is stored in a mysql table. The following are the functions I have written sofar.
function clickedImage(clicked_id) {
//series of functions
$.post("getLatLong.php",{"clickedimag":clickedImg},function(data){
var ll = JSON.parse(data);
var lat = parseFloat(ll[0]);
var long = parseFloat(ll[1]);
var pname = ll[2];
localStorage.setItem('lat',lat);
localStorage.setItem('long',long);
localStorage.setItem('cpname',pname);
//alert([lat,long]);
});
//Series of functions
}
All my other functions are working including I am able to save lat, long info into localStorage. The alert in the above function, clearly pops up with the lat and long attached to the image. To render the map, I am using the template given in the leaflet page.
$(function(){
var la = localStorage.getItem('lat');
var lon = localStorage.getItem('long');
var cpname = localStorage.getItem('cpname');
var mymap = L.map('mapid').setView([la,lon], 13);
L.tileLayer('https://api.mapbox.com/styles/v1/{id}/tiles/{z}/{x}/{y}?access_token={accessToken}', {
attribution: 'Map data © <a href="https://www.openstreetmap.org/">OpenStreetMap</a> contributors, <a href="https://creativecommons.org/licenses/by-sa/2.0/">CC-BY-SA</a>, Ima$
maxZoom: 18,
id: 'mapbox/streets-v11',
tileSize: 512,
zoomOffset: -1,
accessToken: 'my.access.token'
}).addTo(mymap);
var marker = L.marker([la, lon]).addTo(mymap);
marker.bindPopup("<b>" + cpname +"</b>").openPopup();
});
The above code works and renders the map correctly, however, only when I refresh the page. It is not changing the map asynchronously. I have tried this function too:
function getMyMap(la,lon) {
var la = localStorage.getItem('lat');
var lon = localStorage.getItem('long');
var mymap = L.map('mapid').setView([la,lon], 13);
L.tileLayer('https://api.mapbox.com/styles/v1/{id}/tiles/{z}/{x}/{y}?access_token={accessToken}', {
attribution: 'Map data © <a href="https://www.openstreetmap.org/">OpenStreetMap</a> contributors, <a href="https://creativecommons.org/licenses/by-sa/2.0/">CC-BY-SA</a>, Ima$
maxZoom: 18,
id: 'mapbox/streets-v11',
tileSize: 512,
zoomOffset: -1,
accessToken: 'myaccesstoken'
}).addTo(mymap);
}
and called this function inside my onClick event. I am getting the same result.
HTML
<div class="rhsbar" id="rhsbar" style="overflow:hidden;display:flex;flex-direction:column;">
<div id="mapid" >Loading map</div>
</div>
var h = $("#rhsbar").height();
var w = $("#rhsbar").width()+100;
document.getElementById("mapid").style.height=h+"px";
document.getElementById("mapid").style.width=w+"px";
How do I update the map with new coordinates without refreshing the page.
A: What you need is:
your_mapid.fitBounds(your_mapid.getBounds());
It will do a soft refresh of the map whenever something changes.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 1,372 |
Q: Azure Hyperscale (Citus) couldn't create database with the citus account After creating the Azure Hyperscale (Citus) on the Azure portal, I use DbBeaver to access to the created database. I can connect to that database (succeed login to with the citus account and its password on the Azure portal), but when I tried to create the new database, then I couldn't create it.
I have got the exception as below
After investigation a while, I found out that the citus account didn't have permission for create database as following
But I couldn't check that checkbox and save it. It has always thrown exception that I don't have permission with the citus account (the default account created after I created the Citus Postgres Db)
Just noticed that I can create the new database successfully with the same approach above in the last week. But now I couldn't.
Have anyone got the same issue as me? And how can we solve it? I tried to search around Google but have got no luck.
Thank you very much.
A: Hyperscale (Citus) requires that users work on the default database (citus) and does not let users to create new databases.
The behavior has been always like this.
Could you tell about why you need a new database ?
If you need separation of the tables, you can use a namespace, if you need security you can set user based access levels on tables.
Thanks
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 7,465 |
Agononida sabatesae is a species of squat lobster in the family Munididae. The species name is dedicated to A. Sabates. The males measure from and the females from . A. sabates is found off of New Caledonia and Vanuatu, at depths between . It is also found off of Tonga, where it resides between depths of about . There are no common names for Agononida sabatesae.
References
Squat lobsters
Crustaceans described in 1994 | {
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} | 3,078 |
<!doctype html>
<meta charset=utf-8>
<title></title>
<script src=/resources/testharness.js></script>
<script src=/resources/testharnessreport.js></script>
<iframe src="Element-getElementsByTagName-change-document-HTMLNess-iframe.xml"></iframe>
<script>
setup({ single_test: true });
onload = function() {
var parent = document.createElement("div");
var child1 = document.createElementNS("http://www.w3.org/1999/xhtml", "a");
child1.textContent = "xhtml:a";
var child2 = document.createElementNS("http://www.w3.org/1999/xhtml", "A");
child2.textContent = "xhtml:A";
var child3 = document.createElementNS("", "a");
child3.textContent = "a";
var child4 = document.createElementNS("", "A");
child4.textContent = "A";
parent.appendChild(child1);
parent.appendChild(child2);
parent.appendChild(child3);
parent.appendChild(child4);
var list = parent.getElementsByTagName("A");
assert_array_equals(list, [child1, child4],
"In an HTML document, should lowercase the tagname passed in for HTML " +
"elements only");
frames[0].document.documentElement.appendChild(parent);
assert_array_equals(list, [child1, child4],
"After changing document, should still be lowercasing for HTML");
assert_array_equals(parent.getElementsByTagName("A"),
[child2, child4],
"New list with same root and argument should not be lowercasing now");
// Now reinsert all those nodes into the parent, to blow away caches.
parent.appendChild(child1);
parent.appendChild(child2);
parent.appendChild(child3);
parent.appendChild(child4);
assert_array_equals(list, [child1, child4],
"After blowing away caches, should still have the same list");
assert_array_equals(parent.getElementsByTagName("A"),
[child2, child4],
"New list with same root and argument should still not be lowercasing");
done();
}
</script>
| {
"redpajama_set_name": "RedPajamaGithub"
} | 6,258 |
{"url":"https:\/\/ci.nii.ac.jp\/naid\/10017178462\/en\/","text":"Martin boundary points of a John domain and unions of convex sets\n\nAbstract\n\nWe show that a John domain has finitely many minimal Martin boundary points at each Euclidean boundary point. The number of minimal Martin boundary points is estimated in terms of the John constant. In particular, if the John constant is bigger than $\\sqrt3$\/2, then there are at most two minimal Martin boundary points at each Euclidean boundary point. For a class of John domains represented as the union of convex sets we give a sufficient condition for the Martin boundary and the Euclidean boundary to coincide.\n\nJournal\n\n\u2022 Journal of the Mathematical Society of Japan\n\nJournal of the Mathematical Society of Japan 58(1), 247-274, 2006-01-01\n\nThe Mathematical Society of Japan\n\nCodes\n\n\u2022 NII Article ID (NAID)\n10017178462\n\u2022 NII NACSIS-CAT ID (NCID)\nAA0070177X\n\u2022 Text Lang\nENG\n\u2022 Article Type\nJournal Article\n\u2022 ISSN\n00255645\n\u2022 NDL Article ID\n7783405\n\u2022 NDL Source Classification\nZM31(\u79d1\u5b66\u6280\u8853--\u6570\u5b66)\n\u2022 NDL Call No.\nZ53-A209\n\u2022 Data Source\nCJP\u00a0 CJPref\u00a0 NDL\u00a0 J-STAGE\n\nPage Top","date":"2020-06-04 02:31:53","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\": 0.852594792842865, \"perplexity\": 1819.8354940330805}, \"config\": {\"markdown_headings\": false, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.3, \"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-2020-24\/segments\/1590347436828.65\/warc\/CC-MAIN-20200604001115-20200604031115-00040.warc.gz\"}"} | null | null |
A priceless sixth century mosaic depicting St. Mark which was looted from an Orthodox Christian church in Cyprus after the Turkish invasion in 1974 was returned to the island on Sunday.
According to a Cyprus Mail report, the precious mosaic was handed over to the Cypriot embassy in the Hague by a Dutch detective who is specializes in stolen art, and was then returned to Cyprus by a delegation from the Church of Cyprus and the Antiquities Department.
The mosaic depicting a young St. Mark was one of several which went missing from the Church of Panayia Kanakaria after Cyprus was invaded by Turkish troops in 1974. A mosaic of St. Andrew from the same church was repatriated in April, Cyprus Mail says.
Dutch detective Arthur Brand found the rare artifact after a nearly two-year hunt for it across Europe. Mosaics and other treasures from the church were stolen around the year 1976 and ended up in the hands of art dealers and collectors.
"This is a very special piece that is more than 1,600 years old. It's one of the last and most beautiful examples of art from the early Byzantine era," Brand told AFP.
The St. Mark mosaic is among a handful of such works which survived a period of Iconoclasm during the 8th and 9th centuries, which saw many Orthodox icons destroyed by religious fanatics, who considered religious art to be idolatrous.
Twelve major Kanakaria pieces have been returned so far; however, some are yet to be located. | {
"redpajama_set_name": "RedPajamaC4"
} | 1,731 |
Q: Prove $\bigcap\limits_{i\in I} (A\cap U_i) =A\cap \left(\bigcap\limits_{i\in I} U_i\right)$ Is it true that
$$\bigcap\limits_{i\in I} (A\cap U_i) =A\cap \left(\bigcap\limits_{i\in I} U_i\right)?$$
I guess it follows from the properties $A\cap A=A$ and $A\cap B=B\cap A$ as for the finite case, but if $I$ is infinite does it still hold?
A: The proof is exactly the same for a finite intersection or an infinite one. If $x$ belongs to the LHS it belongs to $A\cap U_i$ for all $i \in I$ and therefore to $A$ and all U$_i$. Hence $x$ belongs to the RHS. And conversely.
In the proof, no hypothesis on the fact that $I$ is finite is required.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 4,884 |
Netflix: Full Trailer For Marvel's 'Luke Cage' Released Today
Dustin Murrell
Netflix and Marvel have teamed up to produce some of the best television series of the last few years. Fans of the Netflix-exclusive Daredevil and Jessica Jones can't wait for the next season of either show. Jon Bernthal's portrayal of the Punisher in the second season of Daredevil has fans anxiously awaiting his own show, while long-time Marvel Comics fans have been waiting for The Defenders cross-over series the entire time. The Marvel Cinematic Universe faithful will get their next dose on September 30, when Luke Cage debuts on Netflix. Just today, the full trailer was finally released.
The new trailer shows the titular character's reluctance to embrace his role in Harlem as a superhero. In the Marvel comic book series, Luke Cage's character took on the alias Power Man as he teamed up with Iron Fist to form the Heroes for Hire. It's unclear whether Iron Fist (played by Finn Jones) will make a cameo in Luke Cage or whether their first interaction on Netflix will be on The Defenders in 2017. Almost immediately, however, the trailer shows Claire Temple (played by Rosario Dawson), who appeared in both seasons of Daredevil as well as Jessica Jones.
Also visible in the full trailer is Misty Knight (played by Simone Missick), a Harlem police detective with romantic ties to both Luke Cage/Power Man and Danny Rand/Iron Fist in the comics. As noted by Complex, Misty Knight will appear in Luke Cage, but not in Iron Fist. It was believed that she played a significant role in both series when Head of Marvel Television Jeph Loeb made that comment in error.
Simone Missick will play Misty Knight in Marvel's Luke Cage [image by Dave Mangels/Getty Images for Netflix]
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Netflix: Tired Of Waiting For Marvel's Daredevil Season 2? Check Out The Cast In These Other Netflix Options
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Can X-Men: Apocalypse Compete With Marvel's Civil War And Dawn Of Justice At The Box Office?
Netflix Unveils Luke Cage Key Art And Promises Trailer Tomorrow
Just yesterday, Netflix officially gave fans the new Luke Cage poster, as seen in the Instagram post below. As the trailer and poster both suggest, the show will be darker and grittier than the blockbuster Marvel movies, just like Daredevil and Jessica Jones before it.
Sweet Christmas! Watch the brand new #LukeCage trailer on Marvel.com now.
A photo posted by Marvel Entertainment (@marvel) on Aug 9, 2016 at 7:01am PDT
As actor Mike Colter recently told Time, Cage will be dealing with relationship issues as well as criminal issues in Harlem.
"Jessica had brought on quite a bit of baggage, things he didn't want to deal with. By the time we get to the beginning of Luke Cage, he's had time to get himself back on his feet a little bit. But he's weary of women. He's weary of trusting people."
ICYMI: #JessicaJones star @KrystenRitter submerges herself underwater to take a stand for orcas trapped at #SeaWorld https://t.co/r5JGcYuK7s
— PETA (@peta) August 7, 2016
The full series will be released all at once on September 30, so fans of the MCU can binge the entire season of Luke Cage in one sitting. Iron Fist is scheduled to be released some time in 2017, and The Defenders is scheduled to be released later in the year. Premiere dates for The Punisher and new seasons of Daredevil and Jessica Jones have yet to be confirmed by Netflix or Marvel.
[Image via Netflix] | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 1,864 |
We since become the slaves to one man's lust. --B.
before this or now ago.
How many ages since has Virgil writ? --Roscommon.
brought to a great lady's house. --Sir P.
windmill in St George's field? --Shak.
The Lord hath blessed thee, since my coming. --Gen.
extant since the ancients. --Dryden.
Then let example be obeyed. --Granville. | {
"redpajama_set_name": "RedPajamaC4"
} | 713 |
package poker.table;
import java.util.ArrayList;
import poker.card.Card;
import poker.card.CardList;
import poker.handrank.HandRank;
import poker.handrank.HandRankType;
/* @author: Hammad Quddus
* Constructor allows for playerCards and playerName to be initialized. Setter method for boardCards
* also calls the init method which initialized the HandRank for board and player card combinations.
*/
public class Player implements Comparable<Player> {
public String playerName;
private CardList playerCards;
private HandRank rank;
public Player(String playerName) {
this.playerName = playerName;
}
public Player(String playerName, ArrayList<String> arr) {
this.playerName = playerName;
playerCards = new CardList(arr);
}
public CardList getPlayerCards() {
return playerCards;
}
public void setPlayerCards(CardList playerCards) {
this.playerCards = playerCards;
}
public HandRank getRank() {
return rank;
}
/*
* Finds HandRank and assigns it to rank variable.
*/
public void initRank(CardList boardCards) {
ArrayList<Card> playerNBoardCards = new ArrayList<>();
playerNBoardCards.addAll(playerCards.cards);
playerNBoardCards.addAll(boardCards.cards);
rank = HandRankType.getHandRank(playerNBoardCards);
}
@Override
public String toString() {
return "Player [playerName=" + playerName + ", playerCards=" + playerCards + ", rank=" + rank + "]";
}
@Override
public int compareTo(Player o) {
return rank.compareTo(o.rank);
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 3,455 |
Home ⌂→Trimdon History→When I Was Young In A Pit Village→Scarlet Fever
I had just started school when I got scarlet fever, very prevalent then. It left me with a swollen gland and a swelling that used to burst and look awful because I had to wear a dressing round my neck. Some children went to Durham Hospital to have their necks 'treated' and an awful mess it made. My mother would not have me go and had faith in Scott the chemist's salve. It took several years, but it healed and left the scar I still have. In the middle of all this, I fell off a high see-saw and broke my left elbow badly. Again my mother would not let me go to hospital, so my father took me to Byers Green to have it 'stretched' by a bone setter he knew who had worked at Kelloe pit. He was a nice man and gave my father big tins of jelly to put on my arm and showed him how to get the elbow to bend. I still cannot touch my shoulder or neck, but I can manage everything else.
Our house was in Green Street at the end of one row. It had three bedrooms, a kitchen, a sitting room and a passage. At the other side of the yard, which was not big, was the W.C., the coal house and the wash house where all the cooking was done. We had an oven in the fireplace in the kitchen, but the whole place had been electro-plated and my mother would not have it used. On the mantelpiece there was much brass and all these candlesticks and other pieces were cleaned every week. In the sitting room the brass round the fireplace was elegant and so were the long pokers and shovels. We had a big pantry where there was always a big bowl of water, kept for the sink in the wash house. There was a big bowl for bread which my mother baked, and baked well, every Thursday. She made big loaves and little ones and fadges. Always a piece of bacon and ham hung from the top and my father cut it when needed.
There was plenty of furniture in the bedrooms. There were many drawers especially in one big elegant piece of mahogany, a wood greatly used and lovely to look at. In my parents' room there was a grandfather clock which never stopped ticking and also a cupboard over the stairs where we hung coats and where my father had a safe. The sitting room was quite something with a good piano, a real suite of chairs and settee, a big piece of furniture in one corner with drawers and a china cabinet, and had we some china! At the other side were dozens of small ornaments and pairs of ornaments big and small, some of which, now, I can see were really nice. A gold clock with a glass cover, stuffed birds in a cage were there too. There was a what-not where the music was kept and there was a table of rose wood, heavily carved with three monkeys and round leaves. Helen has this now in her hall. Over the fireplace was a big mirror. It made three after my mother died and I have one now. My father collected rents for Mr. Robinson who had a big shop in Stockton, and we went there twice a year and this was where my mother saw all these goods that were then coming from the Far East, Japan possibly, and bought. When Tom went to college his trunk came from Robinson's and I had it afterwards.
When I Was Young In A Pit Village
Young In A Pit Village
World War 1 at Trimdon
Garmondsway Anniversary Mornings
Quilting Clubs and Allotments
Men With Carts And Horses
New Row Houses
Rabbit Pie and Cakes
East Hetton Colliery | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 1,190 |
{"url":"https:\/\/stuartl.longlandclan.id.au\/blog\/2016\/04\/16\/","text":"# April 16, 2016\n\n## Solar cluster: Trying out the MIC29712s\n\nI figured, rather than letting these loose directly on the nodes themselves, I\u2019d give them a try with a throw-away dummy load. For this, I grabbed an old Philips car cassette player that\u2019s probably older than I am and hooked that up. I shoved some old cassette in.\n\nThe datasheet for the regulators defines the output voltage as: $V_{OUT}=1.240 \\big({R_1 \\over R_2} + 1\\big)$\n\nPlaying with some numbers, I came out with R1 being a 2.7k\u03a9 and 560\u03a9 resistors in series, and R2 being a 330\u03a9. So I scratched around for those resistors, grabbed one of the MIC29172s and hooked it all up for a test.\n\nThe battery here is not the one I\u2019ll use in the cluster ultimately, I have a 100Ah AGM cell battery for that. The charger seen there is what will be used, initially as a sole power source, then in combination with solar when I get the panels. It\u2019s capable of producing 20A of current, and runs off mains power.\n\nThis is the power drain from the battery, with the charger turned on.\n\nNot as much as I thought it\u2019d be, but still a moderate amount.\n\nThis is what the output side of the regulator looked like:\n\nSo from 14.8V down to 13.1V. It also showed 13.1V when I had the charger unplugged, so it\u2019s doing its job pretty well I think. That\u2019s a drop of 1.7V, so dissipating about 600mW. Efficiency is therefore about 93%, not bad for linear regulators.","date":"2023-03-28 11:06:03","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\": 1, \"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\": 0.38306093215942383, \"perplexity\": 3336.4165391849187}, \"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-2023-14\/segments\/1679296948858.7\/warc\/CC-MAIN-20230328104523-20230328134523-00182.warc.gz\"}"} | null | null |
Q: How to animate child prop changes in react? I'm trying to create a component that takes a child prop and when this child prop changes I'd like the old component that was the child prop to animate out of screen and the new component to animate in. I attempted to do this using react-spring using the code below, you can run it here https://codesandbox.io/s/react-spring-issue-898-forked-ril7mc?file=/src/App.js.
What am I missing here? Shouldn't the component animate left when it unmounts and animate in from the right when it mounts?
/** @jsx jsx */
import { useState } from "react";
import { useTransition, animated } from "react-spring";
import { jsx } from "@emotion/core";
function FadeTransition({ children, onClick }) {
const [toggle, setToggle] = useState(true);
const transition = useTransition(toggle, {
from: {
opacity: 0,
transform: "translateX(100%)"
},
enter: {
opacity: 1,
transform: "translateX(0)"
},
leave: { transform: "translateX(-100%)" }
});
return (
<div>
{transition(
(props, item) =>
item && <animated.div style={props}>{children}</animated.div>
)}
<button
onClick={() => {
setToggle(!toggle);
onClick();
}}
style={{ marginTop: "2rem" }}
>
Change
</button>
</div>
);
}
export default function App() {
const [currentComponent, setCurrentComponent] = useState(0);
let components = [<div>Thing 1</div>, <div>Thing 2</div>];
const onClick = () => {
setCurrentComponent((currentComponent + 1) % 2);
};
return (
<div style={{ paddingTop: "2rem" }}>
<FadeTransition onClick={onClick}>
{components[currentComponent]}
</FadeTransition>
</div>
);
}
A: I didn't use rect-sprint as much,There is issue with toggle state so when thing 1 is rendered animation component (FadeTransition) is rendered with setting up toggle state to true.
Now when you click on change button we are setting up the toggle button to opposite of current state which will be false this time and without unmounting animation component (FadeTransition) it will update dom and so the item which is thing 2.
As we are passing toggle in useTransition and if it's values will be false then animation will not be performed.
So I have updated snippet of yours, Here is the working example codesandbox
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 7,847 |
Family uses foot power to help Myanmar
When Sydneysider Matthew Clark returns to school after the summer holidays, he will have an incredible story to tell his classmates. Source: Catholic Mission.
The 15-year-old will head to Myanmar next week with his parents, Vera and Stuart, to take part in the Trek to Reach Out, a 12-day adventure exploring the rugged landscape of a country steeped in history.
Together, the trio will celebrate Matthew's 16th birthday in the trip's final days, making the occasion extra special.
"I think it'll be cool to celebrate [my birthday] in another country," Matthew said, adding that he won't be expecting gifts this year. "Giving to others is a good gift in itself."
The Clarks are taking up the challenge – which is run by Catholic Mission and Inspired Adventures – to raise vital funds for children's and community projects, some of which they will visit during their trip.
Mr Clark said the trek will be an eye-opening experience. "Meeting and talking to children in need will be a challenge for me," he said. "I want to help people immediately, but I have to look at the bigger picture; it has to be done in a sustainable way. The funds we raise will sustain these programs."
The idea to get involved came from Mrs Clark, a long-time supporter of Catholic Mission who was inspired by a friend working in international aid in Myanmar. "He highly recommended going to the country soon, before tourism really takes off and while it is still unspoiled," Mrs Clark said.
Mr Clark was quick to sign up with his wife, relishing the opportunity to see another part of the world and to give back to the community, but Matthew says his decision took some time. "I needed a little convincing, but soon realised it would be fun and a great experience, not just for me, but for those I'm raising money for."
The family has set up a fundraising page, working towards a target of nearly $10,000, funds that will support projects in education, health and community development.
Sydney family trekking trio keen to make a difference in Myanmar(Catholic Mission) | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 2,922 |
\section{Introduction}
\label{sec:intro}
The study of Borel equivalence relations and their reducibility springs from
the interest in classification problems in mathematics. The classical theory
studies Borel and analytic equivalence relations on Polish spaces and the partial
order formed by these equivalence relation with respect to Borel reducibility $\la\mathcal{E},\le_B\rangle$.
The generalised descriptive theory, initiated in the 1990's by the work of Halko, Mekler, Väänänen
and Shelah \cite{Hal,HalShe,MekVaa}, and recently developed further \cite{FHK,Kul,Lucke} studies
the classification problems on generalised Baire and Cantor spaces, $\k^{\k}$ and $2^\k$ for
uncountable regular $\k$. As is already a custom we concentrate on cardinals with $\k^{<\k}=\k$.
We show that the classical result, known as the Silver dichotomy, fails in the generalised
setting in the following two ways.
It was shown in \cite{Kul}, in particular, that the power set of $\k$ ordered by inclusion,
$\la\Po(\k),\subset\rangle$ can be embedded into $\la\mathcal{E},\le_B\rangle$.
In this paper we show that if $\k$ is inaccessible and $V=L$, then
$\la\la\Po(\k),\subset\rangle$ can be embedded into $\la\mathcal{E},\le_B\rangle$ \emph{below}
the identity relation (Theorem~\ref{thm:Main1}). Then we show that if $V=L$ and $\k$ is uncountable
and regular, then there is an antichain with respect to $\le_B$ of length $2^{\k}$ of Borel equivalence relations
and each of these relations is also incomparable with the identity relation.
In this paper we always work in ${\operatorname{ZFC}}+V=L$ unless stated otherwise.
\paragraph{Acknowledgement}
The authors wish to thank the FWF for its support through Project P 24654 N25.
\section{Preliminaries}
\subsection{Fat Diamond}
\label{ssec:FatDiamond}
\begin{Def}[Fat diamond]
A \emph{fat diamond} on $\k$, denoted $\text{\rotatebox[origin=c]{90}{$\diamondsuit$}}_\k$, is a sequence $\la S_\a\mid \a<\k\rangle$ such that
for all $\a<\k$, $S_\a\subset \a$ and for every set $S\subset\k$, cub set $C\subset \k$ and
$\gamma<\k$ there is a continuous increasing sequence of order type $\gamma$ inside the set
$$C\cap \{\a<\k\mid S\cap \a=S_\a\}.$$
\end{Def}
\begin{Thm}[$V=L$]
A $\text{\rotatebox[origin=c]{90}{$\diamondsuit$}}_\k$-sequence exists for uncountable regular $\k$.
\end{Thm}
\begin{proof}
Suppose $\b\le\k$, $(S_\a)_{\a<\b}$ is defined and
$(C^*,S^*,\gamma^*)$ is a triple such that $C^*,S^*\subset \b$, $\gamma^*<\b$,
$C^*$ is cub in $\b$ and there
exists no continuous increasing sequence of order type $\gamma^*$ in $C^*\cap \{\a\mid S^*\cap\a=S_\a\}$.
Then we abbreviate this by $R(C^*,S^*,\gamma^*)$.
Let $(C_0,S_0)=(\varnothing,\varnothing)$. By induction, suppose that $(C_\a,S_\a)$ is defined for all $\a<\b$.
Let $(C_\b,S_\b)$ be the $L$-least pair such that for some $\gamma<\b$ we have $R(C_\b,S_\b,\gamma)$,
if such exists and set $(C_\b,S_\b)=(\varnothing,\varnothing)$ otherwise.
Let us show that the sequence $\la S_\a\rangle_{\a<\k}$ obtained in this way is a $\text{\rotatebox[origin=c]{90}{$\diamondsuit$}}_\k$-sequence.
Note that this sequence is definable in~$L$.
Suppose on the contrary that it is not a $\text{\rotatebox[origin=c]{90}{$\diamondsuit$}}_\k$-sequence. Then
there exists a cub $C$ and $S\subset \k$ such that there exists an ordinal $\gamma<\k$
with~$R(C,S,\gamma)$.
Suppose that $(C,S)$ is the $L$-least such pair and $\gamma$ the least ordinal
witnessing this. Note that then $(C,S)$ and $\gamma$ are definable in~$L$.
Then build a continuous increasing sequence $(M_\b)_{\b<\gamma}$ of elementary
submodels of $L_{\k^+}$
such that $(M_\b\cap\k)_{\b<\gamma}$ is a continuous increasing
sequence of ordinals in~$C$.
Let us show that each $M_\b\cap\k$ is also in $\{\a\mid S\cap\a=S_\a\}$ which is a contradiction.
So let $\b<\gamma$, denote $\b'=M_\b\cap \k$ and let $\pi$ be the transitive
collapse of $M_\b$ onto some $L_{\b''}$.
Then $\pi(C)=C\cap\b'$ and $\pi(S)=S\cap\b'$. Moreover by the elementarity of $M_\b$ in $L$
$$M\models (C,S)\text{ is the }L\text{-least pair s.t. }\exists\d<\k R(C,S,\d).$$
Applying $\pi$ we get
$$L_{\b''}\models (C\cap\b',S\cap \b')\text{ is the }L
\text{-least pair s.t. }\exists\d<\b'R(C\cap\b',S\cap \b',\d)$$
But then by absoluteness of the $L$-ordering, this holds also in $L$, so
in fact $S\cap\b'=S_{\b'}$, so $\b'$ is in $\{\a\mid S\cap\a=S_\a\}$ as intended.
\end{proof}
\begin{Def}
A stationary set $S\subset \k$ is \emph{fat}, if for all cub sets $C\subset\k$
and all $\gamma<\k$ there is a continuous increasing sequence of length $\gamma$ in $S\cap C$.
\end{Def}
\begin{Thm}[$V=L$]\label{thm:fatsetInL}
If $\k>\o$ is regular, then there exists a fat stationary set $S\subset\k$ such that $\k\setminus S$
is also fat stationary.
\end{Thm}
\begin{proof}
Let $S=\{\a\mid S_\a=\a\}$ where $(S_\a)_{\a<\k}$ is the $\text{\rotatebox[origin=c]{90}{$\diamondsuit$}}_\k$-sequence
defined above. Then $S$ is fat by definition.
But also $S'=\{\a\mid S_\a=\varnothing\}$ is fat stationary and is disjoint from~$S$.
\end{proof}
\subsection{Trees}
\label{sec:Trees}
\begin{Def}
$\k^{<\k}$ is the tree consisting of all functions $p\colon\a\to\k$ for $\a<\k$ ordered by end-extension:
$p<q\iff p\subset q$.
By a \emph{tree} we mean a downward closed suborder of $\k^{<\k}$.
A subtree is a subset of a tree which is itself a tree.
Let $T\subset \k^{<\k}$ be a tree. A \emph{branch} through
$T$ is a set $b$ which is a maximal linear suborder of $T$. The set of all branches of $T$ is denoted by
$[T]$. The set of all branches of length $\k$ is denoted by $[T]_{\k}$.
The \emph{height} of an element $p\in T$, denoted ${\operatorname{ht}}(p)$, is the order type of $\{q\in t\mid q<p\}$.
Let $\a<\k$ be an ordinal. Denote by $T^{\a}$ the subtree of $T$ formed
by all the elements with ${\operatorname{ht}}(p)<\a$.
\end{Def}
\section{For an Inaccessible $\k$.}
\label{sec:EmbeddingOfPk}
In this section we show that, if $V=L$ and $\k$ is a strongly inaccessible cardinal,
then there exists an embedding $F$ of $\la \Po(\k),\subset\rangle$ into $\la\mathcal{E},\le_B\rangle$ where
$\mathcal{E}$ is the set of Borel equivalence relations on $2^\k$ such that
for all $A\in \Po(\k)$, $F(A)\lneq_B \operatorname{id}_{2^\k}$.
\begin{Def}\label{def:Trees}
Let ${\operatorname{sing}_\omega}(\k)$ be the set of singular $\omega$-cofinal cardinals below $\k$.
We will construct for each set $S\subset {\operatorname{sing}_\omega}(\k)$ a \emph{weak $S$-Kurepa tree} $T_S$ as follows.
For each $\a\in{\operatorname{sing}_\omega}(\k)$ let $f(\a)$ be the least limit ordinal $\b$ such that
$L_{\b}\models(\a\text{ is singular})$.
Then let
$$T_S=\{s\in 2^{<\k}\mid \forall \a\le\operatorname{dom}(s)(\a\in S\rightarrow s\!\restriction\! \a\in L_{f(\a)})\}.$$
\end{Def}
\begin{Lemma}\label{lem:SmallLevels}
Let $S\subset{\operatorname{sing}_\omega}(\k)$. Then for every $\gamma\in S$ we have $|T_S^{\gamma+1}|=|\gamma|$.
\end{Lemma}
\begin{proof}
When $\gamma\in S$, then $T_S^{\gamma+1}$ is a subset of $L_{f(\gamma)}$ whose cardinality is
$|f(\gamma)|$. But $|f(\gamma)|<|\gamma|^+$, so we are done.
\end{proof}
\begin{Remark}
For the converse,
if $S$ is as in Lemma \ref{lem:SmallLevels} and $\gamma\in S^\k_\o\setminus S$, then
we have $|T^{\gamma+1}_S|=2^{\gamma}$: by GCH and $\cf\gamma=\o$, we have $|2^{\gamma}|=|\gamma^\o|$ and
it is enough to consider increasing cofinal sequences in $\gamma$. But if $s$ is an increasing
cofinal sequence in $\gamma$ of length $\o$, then it is an element of $T_S$.
\end{Remark}
\begin{Lemma}\label{lem:KurepaCo-meager}
Let $T=T_S$ be a tree as above for some $S\subset{\operatorname{sing}_\omega}(\k)$.
Let $(D_i)_{i<\k}$ be a sequence of dense open subsets of $T$, i.e.
such that for all $i<\k$ we have $\forall p\in T\exists q\in D_i(q>p)$ (density)
and $\forall p\in D_i(N_p\cap T\subset D_i)$ (openness).
Then there is a branch of length $\k$ in $\Cap_{i<\k}D_i$ through $T$.
\end{Lemma}
\begin{proof}
Suppose there is a sequence $(D_i)_{i<\k}$ of dense open subsets of $[T_S]_\k$ such that
$\Cap_{i<\k}D_i$ is empty. Suppose $(D_i)_{i<\k}$ is the $L$-least such sequence.
For a contradiction it is enough to find a branch through $T_{{\operatorname{sing}_\omega}(\k)}$ in $\Cap_{i<\k} D_i$.
Let $(M_{\gamma})_{\gamma<\k}$ be
a definable
continuous increasing sequence of sufficiently
elementary submodels of $(L_{\k^+},\in)$ of size $<\k$
such that $M_\gamma\cap\k=\gamma'$
for some $\gamma'<\k$ and $M_\gamma$ contains a Borel code for $D_\gamma$.
Let $L_{\gamma''}$ be the result of the transitive collapse of $M_\gamma$. Now pick the $L$-least
$p_0\in L_{0''}$ such that
$L_{0''}\models (N_{p_0}\subset D_0\land p_0\in T_{{\operatorname{sing}_\omega}(\k)})$.
Note that this implies that $p_0\in T_{{\operatorname{sing}_\omega}(\k)}$.
If $p_\gamma$ is defined to be an element of $L_{\gamma''}$, let $p_{\gamma+1}$
be the $L$-least element of $L_{(\gamma+1)''}\cap T_S$ extending $p_\gamma$ such that
$L_{(\gamma+1)''}\models (N_{p_{\gamma+1}}\subset D_{\gamma+1}\land p_{\gamma+1}\in T_{{\operatorname{sing}_\omega}(\k)})$
and $\operatorname{dom} p_{\gamma+1}>\gamma'$. If $\gamma$ is a limit and $p_\b$
are defined for all $\b<\gamma$, then let $p_{\gamma}=\Cup_{\b<\gamma}p_\b$. The sequence $(M_{\b})_{\b<\gamma}$ is definable
in $L_{\gamma''}$, and so is $p_\gamma$. On the other hand $\operatorname{dom} p=\gamma'$ is regular from the viewpoint of $L_{\gamma''}$,
so $p_\gamma\in T_{{\operatorname{sing}_\omega}(\k)}$. In this way we obtain a branch through $T_{{\operatorname{sing}_\omega}\k}\subset T_S$
in $\Cap_{i<\k}D_i$.
\end{proof}
\begin{Lemma}
For every $S\subset {\operatorname{sing}_\omega}(\k)$, $T_S$ has $\k^+$ branches of length $\k$, i.e. \mbox{$|[T_S]_{\k}|=\k^+$}.
\end{Lemma}
\begin{proof}
As remarked above, $T_{{\operatorname{sing}_\omega}(\k)}\subset T_S$, so
it is sufficient to show that $T_{{\operatorname{sing}_\omega}(\k)}$ has $\k^+$ branches.
For each $\b<\k^+$ let $C(\b)=\{\gamma<\k\mid {\operatorname{SH}}^{L_{\b}}(\gamma \cup\{\k\})\cap\k=\gamma\}$.
We want to show that there is an unbounded set $G\subset\k^+$ such that
for all $\b,\b'\in G$ the sets $C(\b)$ and $C(\b')$ are all different if $\b\ne\b'$
and that the characteristic function of each $C(\b)$ is a branch through $T_{{\operatorname{sing}_\omega}(\k)}$.
We claim that $G=\{\b<\k^+\mid {\operatorname{SH}}^{L_\b}(\k\cup\{\k\})=L_\b\}$ is such a set.
To show that $G$ is unbounded, let $\b<\k^+$ and let $X\subset \k$ be a set
such that $X\notin L_\b$. Let $\b'<\k^+$ be the least ordinal
such that $X$ is definable in $L_{\b'}$ with parameters, so $\b'\ge \b$.
Let $\varphi(p)$ be a formula with parameters $p$,
which defines $X$ and let $p_0$ be the $L$-least sequence of parameters such that
$\varphi(p_0)$ defines a subset of $L_{\b'}$ which is not an element of $L_{\b'}$. Now $p_0$ is in
${\operatorname{SH}}^{L_{\b'}}(\k\cup\{\k\})$. Let $\bar\b$ be such that
${\operatorname{SH}}^{L_{\b'}}(\k\cup\{\k\})\cong L_{\bar\b}$. We want to show that $\bar\b=\b'$.
But since $p_0\in{\operatorname{SH}}^{L_{\b'}}(\k\cup\{\k\})$, the set defined by $\varphi(p_0)$ in $L_{\bar\b}$
is in $L_{\bar\b+1}$. But by the definition of $p_0$, this set cannot be in $L_{\b'}$, so
$\bar\b=\b'$.
Suppose $\b,\b'\in G$ and $\b<\b'$. We claim
that $C(\b')\subset^* \lim C(\b)$, where $\subset^*$ means inclusion modulo
a bounded set and $\lim$ denotes the limit points of a set. This clearly implies that $C(\b)\ne C(\b')$.
Suppose $\gamma\in C(\b')$. Then since $\b'\in G$, we have $\b\in {\operatorname{SH}}^{L_{\b'}}(\gamma\cup\{\k\})$ for
any $\gamma$ greater than some~$\gamma^*<\k$. Now every Skolem function of $L_{\b}$ is definable in $L_{\b'}$
with parameters from $\gamma\cup \{\k\}$, so $\b'\in C(\b)$. But in fact, also $C(\b)$ is definable
in $L_{\b'}$ with these parameters, so in fact $\b'\in \lim C(\b)$.
Thus $C(\b')\setminus\gamma^*\subset \lim C(\b)$.
Let $f_\b$ be the characteristic function of $C(\b)$ and let us show that $(f_\b\!\restriction\!\a)_{\a<\k}$
is a branch of $T=T_{{\operatorname{sing}_\omega}(\k)}$. By the definition of $T$ it is sufficient to show that
$f_\b\!\restriction\!\a$ is in $L_{f(\a)}$ for all singular $\a\in \k$; this is of course equivalent
to $C(\b)\cap\a$ being in $L_{f(\a)}$. There are two cases: either $\a\in C(\b)$ or $\a\notin C(\b)$.
If $\a$ is in $C(\b)$, then by the definition
of $C(\b)$, ${\operatorname{SH}}^{L_\b}(\a\cup\{\k\})\cap\k=\a$. Let $\bar\b$ be such that
$L_{\bar\b}$ is the transitive collapse of ${\operatorname{SH}}^{L_\b}(\a\cup\{\k\})$. Then
$C(\b)\cap\a\in L_{\bar\b+2}$ and since $\k$ becomes $\a$ in the collapse, $\a$ is
regular in $L_{\bar\b}$ and so $\bar\b<f(\a)$. But $f(\a)$ was chosen to be a limit ordinal,
$\bar\b+2<f(\a)$ as well. Thus $C(\b)\cap\a\in L_{f(\a)}$.
Suppose that $\a\notin C(\b)$. But then $C(\b)\cap \a$ is bounded in $\a$ and since $\a$
is a cardinal, $C(\b)\cap\a\in L_{\a}\subset L_{f(\a)}$.
\end{proof}
\begin{Thm}\label{thm:Main1}
Suppose $V=L$ and $\k$ is inaccessible. Then the order $\la\Po(\k),\subset\rangle$ can be embedded
into $\la \mathcal{E},\le_B\rangle$ (Borel equivalence relations) strictly below the identity on~$2^\k$.
More precisely, there exists $F\colon \Po(\k)\to \mathcal{E}$ such that
for all $A_0,A_1\subset \Po(\k)$ we have
$A_0\subset A_1\iff F(A_0)\le_B F(A_1)$ and $F(A_0)\lneq_B\operatorname{id}_{2^\k}$.
\end{Thm}
\begin{proof}
For a tree $T\subset 2^{<\k}$ let $E(T)$ be the equivalence relation on $2^{\k}$
such that two elements are equivalent if and only if both of them are not branches of $T$
\emph{or} they are identical.
\begin{claim}\label{claim:Something}
Suppose $S_0\subset \k$ is a fat stationary set such that $S^\k_\o\setminus S_0$ is stationary.
(Such sets $S_0$ exist by Theorem~\ref{thm:fatsetInL}.)
Then if $S'$ and $S$ are stationary subsets of $S_\o^\k\setminus S_0$
such that $S'\setminus S$ is stationary, we have $E(T_S)\not\le_B E(T_{S'})$.
\end{claim}
\begin{proof}
Suppose to the contrary that $f\colon 2^{\k}\to 2^\k$ is a Borel reduction from
$E(T_S)$ to $E(T_{S'})$.
The space $[T_S]_\k$ is equipped with the subspace topology inherited from $2^\k$
and we can define Borel, meager and co-meager subsets of $[T_S]_\k$.
Note that the meager and co-meager subsets of $[T_S]_\k$ do not coincide with
those in $2^\k$, for example
$[T_S]_\k$ is not meager in $[T_S]_\k$ by Lemma \ref{lem:KurepaCo-meager}
but meager in~$2^\k$. Now we can define the Baire property
relativised to $[T_S]_\k$: a set $A\subset [T_S]_\k$ has the Baire property, if
there exists open $U\subset [T_S]_\k$ such that $U\sd [T_S]_\k$ is meager in $[T_S]_\k$.
A standard proof gives that all Borel sets of $[T_S]_\k$ have the Baire property.
For every $p\in T_{S'}$, the inverse image of $N_{p}$ under $f$ is Borel and so there is
open $U_p$ such that $U_p\sd f^{-1}N_p$ is meager.
Now let $D=[T_S]_\k\setminus \Cup_{p\in 2^{<\k}} U_p\sd f^{-1}N_p$. By
Lemma~\ref{lem:KurepaCo-meager} an intersection of $\k$ many dense
open sets is non-empty in $[T_S]_\k$ whereas it follows
that the space is co-meager in itself and $D$ is co-meager.
So $D\subset [T_S]_\k$ is dense and $f$ is continuous on~$D$.
By removing one point from $D$, we may assume without loss of generality
that $f\colon [T_S]_\k\to [T_{S'}]_\k$.
Let $c(S_0)$ be the set of increasing continuous sequences
$(\a_\b)_{\b<\gamma}$ in $S_0$ with the property that also
$\sup_{\b<\gamma}\a_\b\in S_0$.
We will now define a function
$$\tau\colon c(S_0)\to\k\times \Po(T_S)\times\Po(T_{S'})$$
by induction on the length of the sequence
$(\a_\b)_{\b<\gamma}\in c(S_0)$. The projection of $\tau$ to the
first coordinate, $\operatorname{pr}_1\circ\tau$ can be thought as a strategy of a player in a climbing game
(where the players pick ordinals below $\k$ in an increasing way).
Let $\tau(\varnothing)=(0,\{\eta\!\restriction\!\d\mid\d<\k\},\{f(\eta)\!\restriction\!\d\mid \d<\k\})$
where $\eta$ is any element of $[T_S]_\k\cap D$
and suppose $\tau((\a_\d)_{\d<\gamma+1})$ is defined to be
$(\a,A,A')$ such that $A$ and $A'$ are subtrees of $T_S$ and $T_{S'}$ respectively
such that
\begin{enumerate}
\item $\a>\a_\d$ for all $\d<\gamma+1$,
\item all the branches of $A$ and $A'$ have length $\k$,
\item for each branch $\eta$ of $A$, $N_{\eta\restl\a}\cap [A]_\k=\{\eta\}$, i.e. there are no splitting nodes above $\a$, and the same for $A'$,
\item for each branch $\eta$ of $A$, we have $f[D\cap N_{\eta\restl\a}]\subset N_{\xi\restl\a_\gamma}$
for some unique branch $\xi$ of $A'$.\label{condition_4_definition_of_str}
\end{enumerate}
Note that the last condition defines an embedding from $[A]_\k$ to $[A']_\k$.
Now we want to define $\tau((\a_\d)_{\d\le \gamma+1})=(\b,B,B')$ where
$\a_{\gamma+1}\in S_0 \setminus (\a_\gamma+1)$.
For each branch $\eta$ of $A$, there is a branch $\xi_\eta$ in
$[T_S]_\k\cap D\cap N_{\eta\restl\a_{\gamma+1}}$ such that $\xi_\eta(\a_{\gamma+1}+1)\ne\eta(\a_{\gamma+1}+1)$
(for example find $\xi_\eta$ as follows: first note that the function $\xi_\eta'$
such that $\xi_\eta'(\d)=\eta(\d)$ for $\d\le \a_{\gamma+1}$
and $\xi_\eta'(\d)=1-\eta(\d)$ for $\d>\a_{\gamma+1}$ is a branch of $T_S$ (because $\eta$ is a branch
and $\xi'_{\eta}\!\restriction\!\b$ is definable from $\eta\!\restriction\!\b$ for all~$\b$), so by the density of $D$, there
is $\xi_\eta\in D\cap N_{\xi_{\eta}'\restl\a_{\gamma+1}+1}$). By condition (3) this branch is
new (i.e. not in~$A$).
Let $B$ be the downward closed subtree of $T_S$ such that
$[B]_\k=\Cup_{\eta\in A}\{\eta,\xi_\eta\}$ and $B'$ the same for $T_{S'}$ such that
$[B']_\k=\{f(\eta)\mid \eta\in B\}$.
Then pick $\b$ high enough so that condition~\eqref{condition_4_definition_of_str}
is satisfied for $\a$, $A$ and $A'$ replaced by $\b$, $B$ and $B'$ which is possible by the
continuity of $f$ on~$D$.
Suppose $\gamma$ is a limit and $(\a_{\d})_{\d<\gamma}$ is in $c(S_0)$ and
$\tau((\a_{\d})_{\d<\varepsilon})=(\b_\varepsilon,B_\varepsilon,B_\varepsilon')$ is defined for all $\varepsilon<\gamma$. Let us define
$\tau((\a_\varepsilon)_{\varepsilon<\gamma})=(\b,B,B')$. Let $\b$ be
the supremum of $\{\b_\varepsilon\mid \varepsilon<\gamma\}$.
Note that $\Cup_{\varepsilon<\gamma}B_\varepsilon$ is a downward closed subset of $T_S$.
Let $p$ be any branch of length~$\b$ through $\Cup_{\varepsilon<\gamma}B_\varepsilon \cap 2^{<\b}$.
If there is a branch in $\Cup_{\varepsilon<\gamma}B_\varepsilon$ that continues $p$, let $\eta(p)$ be that branch.
Otherwise, since $\b\notin S$
and $p\!\restriction\!\gamma\in T_S$ for all $\gamma<\b$, $p$ can be continued
to some branch $\eta$ in $D\cap T_S$ and we define $\eta(p)$ to be that $\eta$.
Let
$$B=\{\eta(p)\mid p\text{ is a branch through }\Cup_{\varepsilon<\gamma}B_\varepsilon\}$$
and
$$B'=\{f(\eta)\mid \eta\in B\}.$$
Let $C$ be the cub set of ordinals $\a$ that are closed under $\tau$, in the sense that
$C$ is the set of those $\a$ such that
for all sequences $s\in c(S_0)$ that are bounded in $\a$, we have
$(\operatorname{pr}_1\circ \tau)(s)<\a$, $|(\operatorname{pr}_2\circ \tau)(s)|<\a$ and $|(\operatorname{pr}_3\circ\tau)(s)|<\a$.
For each pair of ordinals $(\a_1,\a_2)\in\k$, let $\pi(\a_1,\a_2)$ be the least ordinal such that
there is an increasing continuous sequence of order type $\a_1$ starting
above $\a_2$ with supremum at most $\pi(\a_1,\a_2)$ and let $C_1$ be the cub set of
ordinals closed under $\pi$.
Now by the stationarity of $S'\setminus S$, pick $\a\in C\cap C_1\cap S'\setminus S$.
Now it is easy to construct a continuous increasing sequence $s$ in $c(S_0)$ of
order type $\a$, cofinal in $\a$, and a cofinal sequence $(\gamma_n)_{n<\o}$ in $\a$ such that
$s\!\restriction\! \gamma_n$ is in $c(S_0)$ for all $n$ and $\tau(s\!\restriction\!\gamma_n)=(\d_n,A_n,A_n')$
has the following properties:
\begin{itemize}
\item $\gamma_n\le \d_n<\a$,
\item $A_n$ has at least $2^{\gamma_n}$ many branches and
\item $f$ defines a bijection between the branches of $A_n$ and the branches of $A'_n$
\end{itemize}
Let $A_\o$ be the tree which consists
of those branches $\eta$ of $T_S$ in $D$
that for every $\d<\a$ there is a branch $\xi$ in $\Cup_{n<\o}[A_n]_\k$ such that
the common initial segment of $\eta$ and $\xi$ has height at least $\d$. Since $\a\notin S$,
the number of branches of $A_\o$ is $2^{\a}$. So it means
that the set $f[[A_\o]_\k]$ must have $2^{\a}$ branches too. The contradiction will follow
once we show that this implies that $T_{S'}^{\a+1}$ must have $2^{\a}$ elements,
contradicting Lemma~\ref{lem:SmallLevels}, because $\a\in S'$.
But if $\eta$ and $\xi$ are any two branches in $A_\o$, their images must disagree below $\a$
by the construction, hence $T_{S'}^{\a+1}$ should have at least the same cardinality as~$|A_\o|$.
\end{proof}
\begin{claim}
If $S\subset S'\subset\k$, then $E(T_{S'})\le_B E(T_S)$.
\end{claim}
\begin{proof}
By the assumption we have $T_{S'}\subset T_{S}$.
Let $i\colon [T_{S'}]_\k\to [T_S]_\k$ be the inclusion map and let $\xi$
be a fixed element of $2^\k\setminus[T_S]_\k$.
For $\eta\in 2^{\k}$, let $f(\eta)=i(\eta)$, if $\eta\in [T_{S'}]_\k$
and $f(\eta)=\xi$ otherwise.
\end{proof}
To prove the Theorem, let $S_0$ be a fat stationary set such that $S^\k_\o\setminus S_0$ is stationary.
Let $\{S_i\mid i<\k\}$ be a partition of $S^\k_\o\setminus S_0$ into $\k$ many disjoint
stationary sets. Then by the claims above, the function defined by
$$A\mapsto E(T_{\Cup_{i\notin A}S_i})$$
is an embedding $F$ of $\la \Po(\k),\subset\rangle$ into $\la \mathcal{E},\le_B\rangle$
such that for all $A\in \Po(\k)$, we have $F(A)\le_B\operatorname{id}_{2^\k}$ and by the same
argument as in the proof of Claim~\ref{claim:Something}, we have $\operatorname{id}_{2^\k}\not\le_B F(A)$
\end{proof}
\section{An Antichain Containing the Identity}
\label{sec:Antichain}
In this section $\k$ is regular and uncountable, but not necessarily inaccessible.
We now redefine the meaning of ${\operatorname{sing}_\omega}(\k)$ to be the
set of all $\o$-cofinal ordinals below $\k$ (instead of just cardinals as in the previous section).
Let $T=T_{{\operatorname{sing}_\omega}(\k)}$ (see Definition~\ref{def:Trees}).
As in Lemma \ref{lem:KurepaCo-meager}, $T$ is not meager in itself and we can define the ideal of meager
sets relativised to $T$. In this way, the Borel subsets of $T$ will have the Baire property in~$T$.
Note that $T$ is a meager subset of $2^\k$, so the meager ideal on subsets of $T$ is not a straightforward
restriction of the meager ideal on the subsets of $2^\k$.
\begin{Lemma}\label{lem:ContOnCoM}
Suppose $f\colon T\to 2^{\k}$ is a Borel function. Then there is a co-meager set $D\subset T$
such that $f$ is continuous on $D$.
\end{Lemma}
\begin{proof}
Using Lemma \ref{lem:KurepaCo-meager} as in the beginning of the proof of Claim \ref{claim:Something}
\end{proof}
\begin{Thm}\label{thm:Main2}
Suppose $V=L$. Then there is an antichain of Borel equivalence relations
with respect to $\le_B$ of size $2^{\k}$ such that one of the relations is the identity.
\end{Thm}
\begin{proof}
Let $[T]_\k$ be the set of branches of length $\k$ of $T$. Let $S\subset\k$ be stationary.
Then let $\eta$ and $\xi$ be $F_S$-equivalent, either if both $\eta$ and $\xi$
are not in $[T]_\k$, or if both $\eta$ and $\xi$ are in $[T]_\k$ \emph{and} are $E_S$-equivalent,
where $E_S$ is as in~\cite{Kul}:
$\eta$ and $\xi$ are $E_S$ equivalent if they are $E_0$-equivalent \emph{and} for every $\a\in S$
there exists $\b<\a$ such that $\forall \gamma\in \left[\b,\a\right[$,
$|\eta(\gamma)-\xi(\gamma)|=|\eta(\b)-\xi(\b)|$.
For a tree $T\subset 2^{<\k}$ and a stationary $S\subset\k$
define the following game $G(T,S)$ of length $\o$ for two players $\,{\textrm{\bf I}}$ and $\textrm{\bf I\hspace{-1pt}I}$:
At move $n<\o$, player $\,{\textrm{\bf I}}$ picks a pair $(p^0_{n},p^1_{n})$ of
elements of $T$ with $\operatorname{dom} p_n^0=\operatorname{dom} p_n^1$
and then player $\textrm{\bf I\hspace{-1pt}I}$ picks an ordinal $\a_n$ above $\operatorname{dom} p^0_n$.
Additionally
the following conditions should be satisfied by the moves of player~$\,{\textrm{\bf I}}$:
\begin{enumerate}
\item $p_{n-1}^0\subset p_{n}^0$ and $p_{n-1}^1\subset p_{n}^1$,
\item $\operatorname{dom} p_n^0=\operatorname{dom} p_n^1>\a_{n-1}$.
\end{enumerate}
Suppose that $(p_n^i)_{n<\o}$ for $i\in\{0,1\}$ are the sequences obtained in this way by player $\,{\textrm{\bf I}}$.
Player $\textrm{\bf I\hspace{-1pt}I}$ wins, if player $\,{\textrm{\bf I}}$ didn't follow the rules, or else
$\Cup_{n<\o}p_n^0$ and $\Cup_{n<\o}p_n^1$ are both in $T$ and $\sup_{n<\o}\operatorname{dom} p_n^0\in S$.
\begin{claim}
Suppose $S\subset {\operatorname{sing}_\omega}(\k)$ is stationary and $T$
is the weak Kurepa tree defined above.
Then Player $\,{\textrm{\bf I}}$ has no winning strategy in $G(T,S)$.
\end{claim}
\begin{proof}
Suppose $\tau$ is a strategy of Player $\,{\textrm{\bf I}}$.
Let $M$ be an elementary submodel of $(L_{\k^+},\tau,S,\in)$ of size $\k$ such that
$M\cap L_{\k}$ is transitive and
$M\cap \k=\a$ for some $\a\in S$. Let $f(\a)$ be the least ordinal such that $\a$ is
singular in $L_{f(\a)}$ and let $r=(r_i)_{i<\o}$ be some cofinal sequence in $\a$ in $L_{f(\a)}$.
Now player $\textrm{\bf I\hspace{-1pt}I}$ can play against $\tau$ in $L_{f(\a)}$
towards $\a$ using $r$. The replies of $\,{\textrm{\bf I}}$ will be in fact in $M$ and the
eventual sequences $(p^k_n)_{n<\o}$,
$k\in\{0,1\}$, constructed by $\,{\textrm{\bf I}}$ will be in $L_{f(\a)}$ and so by definition
$\Cup_{i<\o}p^k_n$ will be in $T$ and so player $\textrm{\bf I\hspace{-1pt}I}$ wins this game.
\end{proof}
\begin{claim}
If $S'\setminus S$ is $\o$-stationary, then $F_{S}$ is not Borel-reducible to $F_{S'}$.
\end{claim}
\begin{proof}
The argument is as in~\cite{Kul}. Suppose $f$ is a Borel
function from $[T]_\k$ to $[T]_\k$ which reduces $F_{S}$ to $F_{S'}$ for
some stationary $S$ and $S'$ such that $S'\setminus S$ is
stationary. We will derive a contradiction. By Lemma~\ref{lem:ContOnCoM}
there exists a sequence $(D_{i})_{i<\k}$ of dense open sets
such that $f$ is continuous on the co-meager set $D=\Cap_{i<\k}D_i$.
We will now define a strategy of
player $\,{\textrm{\bf I}}$ in $G(T,S'\setminus S)$ such that if it is not a winning
strategy, then the contradiction is achieved, so we are done by the
claim above.
The strategy is as follows. At the first move, player $\,{\textrm{\bf I}}$ picks
a function $\eta\in 2^{\k}$ with the property that both $\eta$ and $1-\eta$
are branches of $T$ and in $D$. Since $\eta$ and $1-\eta$ are non-equivalent in $F_S$,
$f(\eta)$ and $f(1-\eta)$ are non-equivalent in $F_{S'}$.
So there is a point $\a$ such that $f(\eta)(\a)\ne f(1-\eta)(\a)$.
Player $\,{\textrm{\bf I}}$ then finds $\a_0$ such that $f[D\cap N_{\eta\restl\a_0}]\subset N_{f(\eta)\!\restriction\!(\a+1)}$
and $f[D\cap N_{(1-\eta)\restl\a_0}]\subset N_{f(1-\eta)\restl (\a+1)}$.
The first move is the pair $(p_0^0,p_0^1)$ where
$p_0^0=\eta\restl\a_0$ and $p_0^1=(1-\eta)\restl\a_0$.
Additionally player $\,{\textrm{\bf I}}$ keeps in mind the elements
$q_0^0=f(\eta)\!\restriction\!(\a+1)$ and $q_0^1=(f(1-\eta)\restl (\a+1))$. Suppose the players have played
$n$ moves and $(\b_0,\dots,\b_n)$ are the ordinals picked by player $\textrm{\bf I\hspace{-1pt}I}$ and
$((p_0^0,p_0^1),\dots,(p_n^0,p_n^1))$
the pairs picked by player $\,{\textrm{\bf I}}$. Player $\,{\textrm{\bf I}}$ has also constructed a sequence
$(q_i^0,q_i^1)_{i\le n}$. If $n$ is even, then player $\,{\textrm{\bf I}}$ extends
$p_n^0$ and $p_n^1$ into branches $\eta$ and $\xi$ of $T$
such that $\eta(\a)=\xi(\a)$ implies $\a<\operatorname{dom} p_n^0=\operatorname{dom} p_n^1=\a_n$
and such that $\eta$ and $\xi$ are both in $D$. By the induction
hypothesis $f(\eta)$ extends $q_n^0$ and $f(\xi)$
extends $q_n^1$, so we can find $\b_n'>\b_n$ such that the
continuations $q_{n+1}^0=f(\eta)\!\restriction\!\b_n'$ and $q_{n+1}^1=f(\xi)\!\restriction\!\b_{n'}$ are
of equal length and for some $\b\in \operatorname{dom} q_{n+1}^0\setminus \b_n$ with $q_{n+1}^0(\b)\ne q_{n+1}^1(\b)$
(if such $\b'_n$ does not exist,
then it implies that $q_{n}^0$ and $q_{n}^{1}$ cannot be extended
to $F_{S'}$-equivalent branches whereas $p_{n}^0$ and $p_{n}^1$ can be
extended to $F_S$-equivalent branches, which would be a contradiction).
Then player $\,{\textrm{\bf I}}$ finds
an $\a_{n+1}>\b_n'$ such that, denoting $p_{n+1}^0=\eta\!\restriction\!\a_{n+1}$
and $p_{n+1}^{1}=\xi\!\restriction\!\a_{n+1}$, we have
$$f[D\cap N_{p_{n+1}^0}]\subset N_{q_{n+1}^0}$$
and
$$f[D\cap N_{p_{n+1}^1}]\subset N_{q_{n+1}^1}.$$
The pair $(p^0_{n+1},p^1_{n+1})$ is the next move.
If $n$ is odd, then player $\,{\textrm{\bf I}}$ proceeds in the same way,
but with the only differences that now
he picks $\eta$ and $\xi$ such that $\eta(\a)=\xi(\a)$ for all $\a>\operatorname{dom} \a_n$
and finds $q_{n+1}^0$ and $q_{n+1}^1$ such that $q_{n+1}^0(\b)=q_{n+1}^{1}(\b)$ for some
$\b\in \operatorname{dom} q_{n+1}^0\setminus \b_n$.
This describes the strategy.
If player $\textrm{\bf I\hspace{-1pt}I}$ beats this strategy in $G(T,S'\setminus S)$, it means
that the limit of her moves, which is the same as the limit of the sequence
$(\operatorname{dom} p_n^i)_{n<\o}$, $i\in\{0,1\}$, is in $S'$ and not in $S$. So by looking at
the things that player $\,{\textrm{\bf I}}$ has constructed, we note that $p_\o^0=\Cup_{n<\o}p_n^0$
and $p_\o^1=\Cup_{n<\o}p_n^1$ can be extended
to equivalent branches on the side of $F_S$, but
$q_\o^0=\Cup_{n<\o}q_n^0$ and $q_\o^1=\Cup_{n<\o}q_n^1$
cannot be extended (in $D$) to equivalent branches
on the range side $F_{S'}$ which is a contradiction, because
$f[D\cap N_{p_\o^i}]\subset N_{q_\o^i}$, $i\in\{0,1\}$.
\end{proof}
To prove the Theorem, let $(S_i)_{i<\k}$ be a partition of $S^\k_\o$ into disjoint stationary pieces.
Then let $\mathcal{A}$ be a maximal antichain in $\Po(\k)$ (a set of size $2^\k$ of subsets of $\k$ incomparable
under inclusion)
and define $G\colon\mathcal{A}\to \mathcal{E}$
by $G(A)=F_{\Cup_{i\notin A}S_i}$. Then $G[\mathcal{A}]$ is an antichain by the claims above.
Every element of this antichain
is incomparable with identity: identity is not reducible to any of them, because
of the small levels guaranteed by the weak Kurepa tree $T$. On the other hand any of the relations
is not reducible to $\operatorname{id}$ because of the $E_0$-component: the equivalence classes are dense in $T$ which
violates the continuity of any reduction even on an (arbitrary) co-meager set.
\end{proof}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 8,143 |
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Pages (25) [ 1 2 3 4 5 ... ] | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 3,202 |
Q: What's the best classical physics (mechanics, electromagnetism, etc) textbook to read as a regular book? Looking for recommendations for someone who loves physics but is not in college/university. I'm a 39-year old engineer.
A: I highly recommend Six ideas that shaped Physics by Thomas Moore. Every volume in the series just oozes physics, emphasizing symmetries and how the physics flows from some core concept.
It's written at the first year University level, but I think it's too rich in physics for the typical 1st year student mostly interested in just passing a required course.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 2,691 |
Duke stars Smith, Singler to play in Crossfire Game
By Brittany JacksonTimes-News Staff Writer
Duke stars Nolan Smith and Kyle Singler, and UNC Asheville's John Williams will be suiting up for the ACC All-Stars in the annual ACC/Crossfire All-Star Game on Saturday, April 16, at A.C. Reynolds High School.
It's time again for the annual ACC/Crossfire All-Star Game, and this year Duke fans will get a treat in seeing Associated Press first-team All-America and ACC Player of the Year Nolan Smith up close locally, as the game will be held for the second straight year at A.C. Reynolds High School in Asheville on Saturday, April 16.
Duke fans will also get to see first-team All-ACC pick Kyle Singler in action at the 18th annual showdown, as well as Duke senior Casey Peters. Smith led the ACC with 21.6 points per game, as well as 4.6 rebounds and 5.2 assists per game. He finished his senior season with an 81 percent free throw shooting percentage.
With no seniors starting from UNC this season, the Tar Heels will be represented by graduate student Justin Knox and seniors Daniel Bolick, D.J. Johnston and Van Hatchell, the son of Carolina womens' head basketball coach Sylvia Hatchell.
"We love basketball and love to play basketball but our purpose as former collegiate players is to have an opportunity to share the Gospel of our Lord Jesus Christ," said Crossfire Ministries co-founder Randy Shepherd. "Fans can come see the guys they've seen on TV but more importantly to listen to Christ's message."
The Crossfire team will be made up of co-founders Shepherd and Jamie Johnson, both former collegiate basketball players who will give a sermon at halftime, as well as former Brevard College standout Chuck Peterson, who led the Crossfire to a 104-95 win over ACC with 20 second-half points.
Another big name in the Crossfire lineup will be John Williams, who led the UNC Asheville Bulldogs to the second round of the NCAA Tournament, just the second time in the program's history.
But before playing in the Crossfire game and dunk contest on April 16, Williams will partake in the State Farm Slam Dunk & 3-Point Championship contest tonight in Houston, TX, at the site of the NCAA Final Four. He is the first player in the history of the Big South Conference to be invited to the competition, and just one of 24 senior standouts from around the country.
"Thank the Lord that we can still get up and down the court, but we are wise enough to recruit younger and taller players as well. John Williams from UNC Asheville will play with us and he had a great year," Shepherd said.
"I've seen him do some dunks that other people haven't done. I've also heard that he's got a 45-inch vertical. Eddie Biedenbach recruited David Thompson, who was on the 1974 NCAA championship team and is one of Michael Jordan's idols, and said he's never seen anyone jump like David could until John came along."
At just 6-4, Williams is the No. 1 all-time in UNCA history with 255 blocks in 120 games. He completed his career as a Bulldog with 1,171 points and 684 rebounds.
Clemson, one of the four ACC teams to make it into the NCAA Tournament, will be represented by third-team all-ACC Demontez Stitt, along with Jonah Baize, Jerai Grant (honorable mention all-ACC) and Zavier Anderson. Rounding up the ACC team are Javier Gonzalez and Tracy Smith (honorable mention all-ACC) from N.C. State, and from Wake Forest, Gary Clark.
Joining the Crossfire cast will be former collegiate players Clark Camp (Southern Wesleyan), Tim Lewis (Montreat College), Zachary Chandler (Newberry College), Jazz Cathcart (Montreat), Willie Battle (Western Carolina University) and Joel Flemming (WCU).
"They play hard and we play hard," Shepherd said of the ACC and Crossfire teams. "In 18 years, they've won about nine times and we've won eight. It's a good competitive game, 3-point and dunk competition; it's a chance to get autographs, shake hands with and meet some guys that you see play on TV, but the halftime message is the clutch for Crossfire."
Doors will open at 5 p.m. and the game will start at 6 p.m. To help fund the event, tickets are $10 and can be purchased at Showtime Sports & Trophies (298-4808); Arsenal Athletics at Asheville Mall (298-3303); Biltmore Baptist Church bookstore (650-6500); Leicester Carpet in Asheville (254-8937) and Leicester Carpet in Hendersonville (233-0500). | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 2,478 |
Q: Do electromagnetic waves travel in the same speed only in vacuum? Why so? I've read that all electromagnetic waves travel in the same speed in vacuum. But what about air, water and other materials, is their speed different in those materials, if yes then why?
A: Your question can be answered in a variety of ways that are quite different from each other depending on whether a classical or quantum type answer is most meaningful to you. However, I will take a very simple Classical argument to answer your question.
Wave propagation in Free space is the Speed of Light usually denoted by the letter $c$. Maxwell's equations tell us that this value of $c$ can be computed from the following relationship with the permeability of free space $\mu_0$ and the permittivity of free space $\epsilon_0$:
$$
c = \frac{1}{\sqrt{\mu_0 \epsilon_0}}
$$
However, in material mediums, both of these values of permeability and permittivity can change in such a manner that the resultant speed computed is less than the standard speed of light in a vacuum.
These values of permeability and permittivity are sort of like an impedance to the flow of the electromagnetic waves. As a result, the medium is sometimes denoted as having a velocity factor for electromagnetic waves which is a percentage of the speed of light for that medium.
A: The phase and group velocity of light in vacuum is identical and constant $$c_{ph}=\frac {\omega}{k}=c_{gr}=\frac {d\omega}{dk}=c$$ due to the linear dispersion relationship $$k(\omega) =\frac {2\pi}{\lambda}=\frac {\omega}{c}$$ Therefore , in vacuum all electromagnetic waves travel at the same speed. In usual dielectric materials (with $\mu_r=1$) the electromagnetic field interacts with the molecules producing a dielectric polarization which leads to (frequency dependent) a relative dielectric permittivity $\epsilon_r \gt 1$. This, in general, changes the phase velocity to a value smaller than the vacuum light velocity $$c_{ph}=\frac {c}{n} \lt c$$ where the (frequency dependent) refractive index is $n =\sqrt \epsilon_r$. Thus the phase velocity of light in a dielectric medium is (with exceptions) smaller than the speed of light in vacuum. It usually also depends on the frequency of the light, which is called dispersion.
A: Yes, all EM waves travel with same speed $c$ only in vacuum. In other media, their velocity varies w.r.t their frequency.
In an EM wave the teo characteristic properties are frequency an wavelength. Here, frequency can be considered a constant as is decided by the source producing the EM wave. On the other hand, the wavelength does npt remain constant and is susceptible to change when travelling between media. So, on the whole the frequency is constant whereas the wavelength changes.
Now, the speed of an EM wave = frequency $\times$ wavelength.
From the above facts, we understand that when travelling between media, the wavelength changes and therefore the speed of EM wave also changes. Also this change is different for different frequency. So, different EM waves possess different velocity of propagation in different medium.
| {
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Савез радио-аматера Србије ( СРС ) (на енглеском, Amateur Radio Union of Serbia ) је национална непрофитна организација за радио-аматере у Србији . Кључне бенефиције чланства СРС-а укључују спонзорство награда за рад радио-аматера и радио такмичења, као и КСЛ биро за оне чланове који редовно комуницирају са радио-аматерима у другим земљама. СРС заступа интересе српских радио-аматера пред националним, европским и међународним регулаторним органима за телекомуникације. СРС је национално друштво које представља Србију у Међународној унији радио-аматера .
Списак чланова СРС-а, са њиховим објављеним подацима као што су КТХ локатор, фотографија и сл., можете погледати у YU Callbook- у.
Види још
Међународни радио-аматерски савез
Референце
Организације из Београда
Клуб радио аматера
Радио у Србији
Интернационално удружење радио аматера | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 9,148 |
{"url":"https:\/\/mcqmojo.com\/quiz\/percentage-bank-clerk-mcq-questions-with-answers\/","text":"\u2022 25%\n\u2022 20%\n\u2022 33.33%\n\u2022 15%\n\u2022 66.66%\n\u2022 #### 2.\u00a0 The population of a town 2 years ago was 245000. It increased by 15% in the first year and then increased by 20% in the second year. What is the current population of the town?\n\n\u2022 338100\n\u2022 274400\n\u2022 351560\n\u2022 294400\n\u2022 None of these\n\u2022 #### 3.\u00a0 Raman\u2019s height is $$33\\frac{1}{3}\\%$$ more than that of Dhirendra. If Dhirendra\u2019s height is 60 inches, then how much percent Dhirendra\u2019s height is less than that of Raman?\n\n\u2022 10%\n\u2022 50%\n\u2022 25%\n\u2022 $$33\\frac{1}{3}\\%$$\n\u2022 None of these\n\u2022 #### 4.\u00a0 In a medical certificate, by mistake a candidate gave his height as 23% more than normal. In the interview panel, he clarified that his height was 5 feet 7 inches. Find the percentage correction made by the candidate from his stated height to his actual height.\n\n\u2022 23%\n\u2022 18.7%\n\u2022 25%\n\u2022 15%\n\u2022 None of the above\n\n\u2022 12.5\n\u2022 15.7625\n\u2022 17.625\n\u2022 22.3625\n\u2022 25\n\n\u2022 Rs. 2000\n\u2022 Rs. 2040\n\u2022 Rs. 2100\n\u2022 Rs. 2200\n\u2022 Rs.\u00a02400\n\n\u2022 81\n\u2022 100\n\u2022 64\n\u2022 121\n\u2022 196\n\n\u2022 44.44%\n\u2022 45%\n\u2022 50%\n\u2022 55.55%\n\u2022 35%\n\u2022 #### 9.\u00a0 Sophia invests 37% of her monthly salary in insurance policies. She spends 17% of her monthly salary in shopping and 21% of her salary on household expenses. She saves the remaining amount of Rs. 9,025. What is Sophia\u2019s annual income?\n\n\u2022 Rs. 3,61,000\n\u2022 Rs. 4,33,200\n\u2022 Rs. 4,23,200\n\u2022 Rs. 4,34,400\n\u2022 None of these\n\u2022 #### 10.\u00a0 For annual income in the slab of Rs. 1200000 to 2000000, a person pays tax at 20% over the surplus on Rs. 1200000. Sunil\u2019s annual salary is Rs. 1500000. How much tax does he pays per annum?\n\n\u2022 Rs. 300000\n\u2022 Rs. 240000\n\u2022 Rs. 120000\n\u2022 Rs. 60000\n\u2022 None of these\n\n\u2022 12%\n\u2022 5.75%\n\u2022 8.50%\n\u2022 13.25%\n\u2022 15%\n\u2022 #### 12.\u00a0 Mrs. Sheela spends 18% of her monthly income for the children\u2019s education. She spends 32% of her monthly income in household expenses and 12% in travelling. She spends 55% of the remaining amount in gambling and manages to save only Rs. 7315 at the end of the month. What is Mrs. Sheela\u2019s monthly income?\n\n\u2022 Rs. 36,000\n\u2022 Rs. 50,000\n\u2022 Rs. 42,777.76\n\u2022 Rs. Cannot be determined\n\u2022 None of these\n\u2022 #### 13.\u00a0 A vessel of 80 liters is filled with milk and water. 70% of milk and 30% of water is taken out of the vessel. It is found that the vessel is vacated by 55%, find the initial quantity of water.\n\n\u2022 50 L\n\u2022 60L\n\u2022 40 L\n\u2022 30 L\n\u2022 None of these\n\u2022 #### 14.\u00a0 The C.P. of an article is 40% of the S.P. The percent that the S.P. is of C.P. is\n\n\u2022 250%\n\u2022 240%\n\u2022 60%\n\u2022 40%\n\u2022 None of these\n\u2022 #### 15.\u00a0 A sum of Rs. 825 is divided among A, B and C such that \u2018A\u2019 receives 50% more than \u2018B\u2019 and \u2018B\u2019 receives 30% less than \u2018C\u2019. What is the \u2018A\u2019s share in the amount?\n\n\u2022 Rs. 328\n\u2022 Rs. 347\n\u2022 Rs. 315\n\u2022 Rs. 304\n\u2022 None of these\n\n\u2022 40%\n\u2022 80%\n\u2022 48.8%\n\u2022 51.2%\n\u2022 52.3%\n\u2022 #### 17.\u00a0 A student has to secure a minimum 35% marks to pass an examination. If he gets 200 marks and fails by 10 marks, then the maximum marks in the examination are:\n\n\u2022 600\n\u2022 450\n\u2022 780\n\u2022 650\n\u2022 None of these\n\u2022 #### 18.\u00a0 When 30% of a number is added to another number, the second number increases by 60%. What is the respective ratio between the first and the second number?\n\n\u2022 3 : 1\n\u2022 2 : 1\n\u2022 5 : 2\n\u2022 Cannot be determined\n\u2022 None of these\n\u2022 #### 19.\u00a0 63% of $$3\\frac{4}{7}$$ is:\n\n\u2022 2.25\n\u2022 2.40\n\u2022 2.50\n\u2022 2.75\n\u2022 None of these\n\n\u2022 Rs. 8571\n\u2022 Rs. 7862\n\u2022 Rs. 8230\n\u2022 Rs. 8200\n\u2022 Rs. 7986\n\u2022 #### 21.\u00a0 A number is increased by 40% then again by 40%. Then it is decreased by 40%. By what percent the original number has changed?\n\n\u2022 40%\n\u2022 56%\n\u2022 17.6%\n\u2022 16%\n\u2022 None of these\n\u2022 #### 22.\u00a0 A salesman is appointed on the basic salary of Rs. 1200 per month and the condition that for every sales of Rs. 10,000 and above Rs. 10,000, he will get 50% of basic salary and 10% of the sales as a reward. This incentive scheme does not operate for the first Rs. 10000 of sales. What should be the value of sales if he wants to earn Rs. 7600 in a particular month?\n\n\u2022 Rs. 60,000\n\u2022 Rs. 50,000\n\u2022 Rs. 40,000\n\u2022 Rs. 70000\n\u2022 None of these\n\u2022 #### 23.\u00a0 What value should come in place of the question mark (?) in the following question? ?% of 280 + 18% of (2200 \u00f7 4) = 143.8\n\n\u2022 11\n\u2022 18\n\u2022 21\n\u2022 16\n\u2022 None of these\n\u2022 #### 24.\u00a0 An insurance collector receives 15% commissions on the premium collected per week. How much must he collect per week in order that his annual income may be Rs.6500\/-?\n\n\u2022 Rs.1250\n\u2022 Rs.\u00a0$$833\\frac{1}{3}$$\n\u2022 Rs.720\n\u2022 Rs.\u00a0$$650\\frac{2}{3}$$\n\u2022 Rs.750\n\u2022 #### 25.\u00a0 Shreyas bought an article and sold for 150% per cent of its cost price. What was the cost price of the article, if Shreyas sold it for Rs. 750?\n\n\u2022 Rs. 500\n\u2022 Rs. 480\n\u2022 Rs. 520\n\u2022 Rs. 560\n\u2022 None of these\n\n\u2022 18%\n\u2022 20%\n\u2022 22.5%\n\u2022 25%\n\u2022 30%\n\u2022 #### 27.\u00a0 Robin decided to donate 16% of his monthly salary to an NGO. On the day of donation, he changed his mind and donated Rs. 7,705 which was 67% of what he had decided earlier. How much is Robin\u2019s monthly salary?\n\n\u2022 Rs. 80,756\n\u2022 Rs. 71,875\n\u2022 Rs. 56,700\n\u2022 Rs. 45,696\n\u2022 None of these\n\n\u2022 99.22%\n\u2022 95%\n\u2022 98.22%\n\u2022 100%\n\u2022 96.28%\n\u2022 #### 29.\u00a0 Find the original fraction if in a fraction, numerator in increased by 25% and the denominator is decreased by 10%. The new fraction then obtained is 5\/9.\n\n\u2022 2\/5\n\u2022 5\/9\n\u2022 3\/5\n\u2022 1\/5\n\u2022 None of these\n\n\u2022 90\n\u2022 78.5\n\u2022 72.5\n\u2022 85\n\u2022 12.5\nReport Question\naccess_time","date":"2023-03-20 09:41:47","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\": 0.9253100156784058, \"perplexity\": 12810.568772404526}, \"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-2023-14\/segments\/1679296943471.24\/warc\/CC-MAIN-20230320083513-20230320113513-00361.warc.gz\"}"} | null | null |
Laeops is a genus of small lefteye flounders from the Indo-Pacific. They are mainly found in deep water, although a few species have been recorded shallower than .
Species
There are currently 13 recognized species in this genus:
Laeops clarus Fowler, 1934 (Clear fin-base flounder)
Laeops cypho Fowler, 1934
Laeops gracilis Fowler, 1934 (Philippine slender flounder)
Laeops guentheri Alcock, 1890 (Günther's flounder)
Laeops kitaharae (H. M. Smith & T. E. B. Pope, 1906)
Laeops macrophthalmus (Alcock, 1889)
Laeops natalensis Norman, 1931 (Khaki flounder)
Laeops nigrescens Lloyd, 1907
Laeops nigromaculatus von Bonde, 1922 (Blackspotted flounder)
Laeops parviceps Günther, 1880 (Small headed flounder)
Laeops pectoralis (von Bonde, 1922) (Longarm flounder)
Laeops sinusarabici Chabanaud, 1968
Laeops tungkongensis J. S. T. F. Chen & H. T. C. Weng, 1965
References
Bothidae
Marine fish genera
Taxa named by Albert Günther | {
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set -ex
# change to grpc repo root
cd $(dirname $0)/../../..
source tools/internal_ci/helper_scripts/prepare_build_linux_rc
# Submodule name is passed as the RUN_TESTS_FLAGS variable
SUBMODULE_NAME="${RUN_TESTS_FLAGS}"
# Update submodule to be tested at HEAD
(cd "third_party/${SUBMODULE_NAME}" && git fetch origin && git checkout origin/master)
echo "This suite tests whether gRPC HEAD builds with HEAD of submodule '${SUBMODULE_NAME}'"
echo "If a test breaks, either"
echo "1) some change in the grpc repository has caused the failure"
echo "2) some change that was just merged in the submodule head has caused the failure."
echo ""
echo "submodule '${SUBMODULE_NAME}' is at commit: $(cd third_party/${SUBMODULE_NAME}; git rev-parse --verify HEAD)"
tools/buildgen/generate_projects.sh
if [ "${SUBMODULE_NAME}" == "protobuf" ]
then
tools/distrib/python/make_grpcio_tools.py
fi
# commit so that changes are passed to Docker
git -c user.name='foo' -c user.email='foo@google.com' commit -a -m 'Update submodule'
tools/run_tests/run_tests_matrix.py -f linux --inner_jobs 4 -j 4 --internal_ci --build_only
| {
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Happy National Pizza Month! In 1984, October was designated as National Pizza Month by Gerry Durnell, the founder of Pizza Today magazine. Durnall conveniently chose that month because the first issue of his magazine debuted in October of 1984. As the owner of a local pizzeria, Durnall's goal for the pizza-oriented publication was to support not only his growing pizza business but also the pizza industry as a whole.
At its core, pizza consists of three basic ingredients: dough, sauce and toppings. However, countless recipe variations have emerged. To celebrate National Pizza Month, we've rounded up our favorite Moore's pizza recipes. These recipes are especially unique because each uses a different ingredient to form the pizza crust and a different Moore's sauce!
Made with Moore's Buffalo Wing Sauce and tortillas, this first recipe comes to us from Lambert's Lately.
Chop the chicken breast into 1/2" pieces. Place chicken and ½-cup of Moore's Buffalo Wing Sauce in a plastic bag and marinate for 2-3 hours.
Once marinated, cook chicken on a hot grill pan until just cooked through and browned on each side. Remove chicken and place to the side. Cover grill pan with a little bit of oil (there will be sauce left on the pan...that's ok) and cook onions and celery for 1-2 minutes, just to take some of the crunch off. Remove from pan and reserve.
Mix together remaining ½-cup of Moore's Buffalo Wing Sauce and ranch dressing. Spread a small amount on each tortilla, acting as the pizza sauce. Spread most of the mozzarella cheese over the sauce on each pizza. Layer with chicken, onion/celery, and blue cheese. Sprinkle remaining mozzarella cheese on top.
Cook in a 400-degree oven for 12-15 minutes, or until edges of tortilla and cheese on top are starting to brown. Remove from oven and garnish with cilantro. Cut each tortilla into quarters.
Get the full recipe from Small Town Women here.
Brynn Hunter from the Jefferson State Culinary and Hospitality Institute created this quick and easy recipe for Buffalo Ranch Chicken Pizza.
Season chicken breasts with salt and pepper and grill or bake until cooked throughout (165 degrees).
Remove chicken from grill, cut in half, and set aside.
Toss the tomatoes and arugula with salt and pepper.
Slice the Chicken Breasts into small strips.
Build the pizzas by topping each pita with sliced chicken, bacon, and cheese.
Place on baking sheet and cook in 350-degree oven just until the cheese is melted.
Remove pizzas from oven and add arugula, tomatoes and avocado.
Drizzle each pizza with the Creamy Ranch Buffalo Sauce and serve. | {
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\section{Introduction}
\indent
The transition of the electrical energy system towards sustainability is paralleled by grid decentralization and increasing percentage
of renewables. This development requires novel grid operation and design concepts.
Electrical storage will be a key component of future energy systems to balance feed-in variations and mitigate
power quality problems induced by stochastic
renewables \cite{albadi2009electricalPowerSystemsResearch,ibrahim2011energyProcedia,ren2017overviewWindPowerIntermittency}.
Therefore, new research issues emerge concerning
optimal grid embedding and sizing of storage facilities as well as
smart storage control strategies, which are customized to the specific application purpose and feed-in properties.\\
\indent
Wind and solar have characteristic non-Gaussian statistics over a broad range of time
scales from seasonal and diurnal imbalances down to sub-second fluctuations \cite{anvari2016njp,ren2017overviewWindPowerIntermittency}.
Short-term fluctuations on the second and sub-second scale
are a particular challenge for power system operation, since standard load balancing such as primary control\cite{entsoeHandbookPolicy1} does not operate yet on these time scales.
As a consequence, frequency quality is significantly
reduced \cite{albadi2009electricalPowerSystemsResearch,ibrahim2011energyProcedia,schmietendorf2017TurbulentRenewableEnergyProductionGridStabilityQuality}.
This problem is exacerbated by a side effect: as conventional power plants are progressively substituted by renewables, system inertia is decreased and
the grid becomes more sensitive to sudden perturbations in terms of feed-in
fluctuations \cite{entsoeHPoPEIPSl2017,tielens2012gridInertiaFrequencyControlRenewables}.\\
\indent
Against this backdrop, the focus of power grid research shifts towards short-term dynamics and
multidisciplinary approaches including self-organization and collective phenomena. This requires a profound mathematical modeling framework
mediating between the simple conceptual models from nonlinear dynamics and the detailed models used for case studies in electrical
engineering. Over the past decade, the Kuramoto-like modeling framework has been established as a suitable instrument for this purpose. It is derived from the original Kuramoto model, which describes the phase dynamics of coupled
oscillators, in particular the phase transition from incoherence to self-organized
synchronization \cite{kuramoto1975IntSympMathProblems,strogatz2000physica_d}.
Kuramoto-like models have been used to address various issues of power system dynamics and
topology-stability interplay \cite{filatrella2008epjb,menck2014natcomm,motter2013nature,rohden2012prl,rohden2016CascadingFailuresACgrids,rohden2017CuringCriticalLinks,witthaut2016criticalLinksNonlocalRerouting,wolff2018powerGridStabilityHeterogeneityInternalNodes}.
In a previous work \cite{schmietendorf2017TurbulentRenewableEnergyProductionGridStabilityQuality}, it was shown how the turbulent-like
character of wind feed-in, in particular its intermittency, is directly transferred into frequency and voltage fluctuations. This was confirmed by
real-world frequency measurements \cite{haehne2018footprintTurbulenceGridFreq}.
Other recent studies on Kuramoto-like grids with stochastic feed-in investigated the propagation of frequency quality
deterioration \cite{auer2017stabilitySynchronyIntermittentFluctTreeLikePowerGrids,zhang2016} and potential routes to system instability
\cite{schaefer2017EscapeRoutesFluctDrivenNetworks}. However, the current Kuramoto-like framework does not allow to implement grid components, which are not modelled
by oscillator equations. This shortcoming affects the integration of storage units with arbitrary
control strategies from scratch and hence prevents from fundamental investigations of storage-related issues.\\
\indent
With our study, we fill this gap. The primary target was to integrate a flexible storage
model, which does not imply any restrictive assumptions on storage features or control strategies
beforehand.
For this purpose, we introduce a novel approach by bringing together Kuramoto-like
differential and algebraic load-flow equations, which are a standard tool in power-flow analysis.
The general idea of embedding grid components by means of load-flow equations has a broader range of application: it can serve as a starting
point to implement arbitrary grid components into Kuramoto-like power grids, e.\,g. power inverters with various types of control or nodes connecting
different grid levels. This broadens the scope of KM-like models significantly. At the same time,
the modeling framework is still a reduced
approach compared to the detailed models used in electrical engineering and yet simple enough to address power grid dynamics
from the viewpoint of self-organization and collective dynamics, i.\,e. methods beyond the standard engineering practice.
\\
\indent
For demonstration purpose, we consider frequency quality improvement by means of a storage facility with
limited capacity in a simplified power system subjected to realistic wind feed-in. This application example has been identified as
one of the key issues on the road to power systems with high percentage of wind and solar by electrical engineering
communities \cite{albadi2009electricalPowerSystemsResearch,jabir2017IntermittentSmoothingWindPowerReview,Li2013BESSsmoothingControlWindPV,ibrahim2011energyProcedia,zhao2015reviewStorageWindPowerIntegrationSupport}.
In order to provide a guide to the implementation of storage units as a starting point for follow-up research, we demonstrate
basic control strategies adopted from electrical engineering and give insights into their potential and limitations with respect to
different aspects of frequency quality improvement.\\
\indent
The paper is organized as follows: First, we briefly address electrical storage in wind and solar applications and list the
features of real storage units, which should be implementable into the model.
Then we outline Kuramoto-like power grid modeling and describe how to integrate storage units by means of
load-flow equations. After that, we specify the simplified power system with realistic wind power input, which we use in this study.
We close the subsection with an explanation of the frequency quality assessment we use, and how this is related to established electrical
engineering practice. Then we turn to the application example:
We start with a preliminary performance assessment by considering the ability to ensure stationary operation as function of maximum capacity.
Subsequently, we demonstrate how to implement three basic control strategies, namely: \textit{state-of-charge feedback} reinterpreted as storage
resource management, \textit{droop control} and \textit{ramp-rate control}. We
investigate their potential with respect to frequency quality improvement.
It shows that these control concepts have different advantages
according to their underlying main target.
It is pointed out that the ambition in terms of control strength or tolerance range has to be carefully
adjusted to the storage dimension in order to perform optimally.
Finally, we demonstrate that these strategies are sensitive against finite-time response and confirm that
short-term frequency quality applications require storage and control systems with rapid response.
We conclude with a summary of the main results
and give an outlook to storage-related problems, which can now be addressed within the context of Kuramoto-like power grid models.
\section{Model and Methods}
\subsection{Electrical storage}\label{subsubsec:storageEquipment}
Electrical energy storage \cite{luo2015overviewStorageTechApplications} denotes the process of converting surplus electrical power into a
storable form and reserving it, until it is
converted back when required. It is commonly categorized
by the form of energy stored, but also in terms of their technical features like response time or capacity, or their
function. Electrical storage is already or considered as a promising candidate for various wind and solar power applications.
The type of storage follows its function, or to be more precise, the underlying time scale of power variability.
For long-term storage applications like time-shifting, peak-shaving, seasonal storage and mid-term
frequency control, storage types with large energy dimensions are used, which do not necessarily feature fast response,
e.\,g. pumped hydro, hydrogen-based or compressed air storage. Short-term frequency quality
improvement and power output smoothing require rapid response (ranging from few seconds to
milliseconds), which is usually paralleled by smaller energy capacity.
Candidates for this application are flywheels, batteries, superconducting magnetic energy storage and (super) capacitors.
\cite{diazgonzales2012ReviewStorageTechWindPowerApplications,jabir2017IntermittentSmoothingWindPowerReview}\\
\indent
The storage model to be developed has to meet two requirements:
On the one hand, simplifications are essential in order to fit the model into the Kuramoto-like framework.
On the other hand, all relevant characteristics of real storage operation
have to be implementable,
namely\cite{diazgonzales2012ReviewStorageTechWindPowerApplications,jabir2017IntermittentSmoothingWindPowerReview,luo2015overviewStorageTechApplications}:
\begin{itemize}
\item[\textbullet] \textit{Efficiency}\\
In practice, the energy conversion processes can not be realized without losses. The efficiency factor $\eta$ gives the ratio of input to
output energy. It depends on the type of storage.
\item[\textbullet] \textit{Maximum energy capacity} and \textit{power rating}\\
These define the main dimensions of the storage facility. The energy capacity is the maximum energy the storage unit is able to deliver and hence
serves as an upper limit for the amount of energy stored. The power rating corresponds to the maximum instantaneous supply.
\item[\textbullet] \textit{Control strategies}\\
The storage control strategy determines the storage output at time $t$ as a function of one or more feedback variables.
It can be used to manage the storage resources or to provide system services like frequency control and power output smoothing.
\item[\textbullet] \textit{Response time}\\
The storage unit has a finite response time effecting a time delay between the feedback signal and its reaction.
The response time depends on the type of storage and the underlying control mechanism.
\end{itemize}
\subsection{Kuramoto-like power grid models}\label{subsubsec:KMlike_models}
Kuramoto-like grid models are based on networks of synchronous machines with producers (generators) and consumers (motors)
converting mechanical power into electrical power and vice versa. Real and reactive power is transferred among these
nodes via transmission lines. The topology of the underlying network is condensed in the nodal admittance matrix,
$\{Y_{ij}\}_{i,j=1,..,N}$, with $N$ being the number of nodes.
The common assumption of lossless transmission yields $Y_{ij}\approx \mathrm i \mathfrak{Im}(Y_{ij})=\mathrm i B_{ij}$ with susceptance $B_{ij}$.
Each node $i\,\in\mathcal{M}_\mathrm{grid}$ of the grid is associated with
a complex nodal voltage $\bm E_i=E_i\mathrm e^{\mathrm i \delta_i}$ with $E_i$ being the voltage magnitude and $\delta_i$ the phase with respect to
a reference frame rotating with nominal frequency. (Hence, $\dot\delta_i=\omega_i=0$ means that node $i$ is at nominal
frequency.)\\
\indent
The coupled frequency-voltage dynamics
of the synchronous machines are given by \cite{machowski2008book,schmietendorf2014epjsti}:
\begin{subequations}
\begin{align}
m_i\ddot\delta_i=&\,\gamma_i\dot\delta_i+P_i-\sum_{j=1}^N B_{ij}E_iE_j\sin\delta_{ij}, \label{eq:KM1}\\
\alpha_i\dot E_i=&\,C_i+\beta_i(\bar E_i-E_i)-E_i\hspace{2cm}\phantom{x}\notag\\
&+\chi_i\sum_{j=1}^N B_{ij}E_j\cos\delta_{ij}\label{eq:KM2}.
\end{align}
\end{subequations}
The parameters $m_i$ and $\gamma_i$ denote the total inertia and effective damping, $P_i$ is the mechanical power feed-in or consumption.
$P_{ij}=B_{ij}E_iE_j\sin\delta_{ij}$ is the real power transfer between nodes $i$ and $j$, and the interaction term in
Eq.\,(\ref{eq:KM2}) is related to reactive power flows.
$\alpha_i$, $C_i$, and $\chi$ can be calculated from
machine parameters. The term $\beta_i(\bar E_i-E_i)$ mimics a proportional voltage controller, which pulls the voltage
towards its nominal value \cite{schmietendorf_toBePublished}.
\subsection{Integration of storage units by means of load-flow equations}\label{subsubsec:implementingStorage}
We now assume a power network consisting of a set of conventional synchronous machines $\mathcal{M}_\mathrm{syn}$
(modeled acc. to Eqs.\.(\ref{eq:KM1}) and (\ref{eq:KM2})) and a set of storage units
with control equipment $\mathcal{M}_\mathrm{SCU}$.
The storage units are also associated with nodal voltages $\bm E_i=E_i\mathrm e^{\mathrm i \delta_i}$.
However, their dynamics differ from synchronous machines in that they lack inertia and have no inherent physical relationship
between frequency and electrical power output. The most direct approach, which does not include any restrictive assumptions
on storage features or control, is to calculate
the phase $\delta_i$, $i\in\mathcal{M}_\mathrm{SCU}$, by solving the algebraic load flow equation \cite{kundur1994book,machowski2008book}
\begin{equation}\label{eq:storageModel}
P_{\mathrm{SCU},i}^\mathrm{out}=\sum\limits_{j=1}^N B_{ij}E_iE_j\sin\delta_{ij},\label{eq:loadflow_p}
\end{equation}
with the nodal voltage $E_i$ assumed to be constant.
$P_{\mathrm{SCU},i}^\mathrm{out}$ is the power being injected into the grid by storage-control unit.\\
\indent
Load-flow analysis is a standard tool in electrical engineering for power-flow calculations on power networks.
It is can be derived from basic physical relationships given by Kirchhoff's and Ohm's laws.
This approach is particularly qualified as a starting point for the investigation of smart control since the power fed into the
grid $P_{\mathrm{SCU},i}^\mathrm{out}$ can be determined by arbitrary control strategies and the model can flexibly be complemented
with the other realistic storage characteristics listed above.
The combination with load-flow equations provides a general method for the straight-forward implementation
of arbitrary grid components into Kuramoto-like networks. This includes, for example, for power inverters and nodes linking
different voltage levels or microgrid-macrogrid connections.
\subsection{Simplified power system with wind feed-in and storage facility}\label{subsec:theSystem}
In this study, we demonstrate the operation of a storage unit with control equipment by means of a simplified system consisting of a
generating unit with wind power feed-in and a synchronous machine mimicking the response of the grid
in terms of frequency $\omega_\mathrm{sys}$ and voltage $E_\mathrm{sys}$ in a coarse-grained view (see Fig.\,\ref{fig:simpleSystem}).
We consider its application with respect to frequency quality under
fluctuating wind power feed-in.\\
\indent
The storage unit is assumed to have a finite maximum capacity $K_\mathrm{max}$.
The actual capacity $K(t)$, or in more casual terms: the \enquote{filling level} at time $t$, corresponds to the
\textit{state of charge} with respect to batteries.
$K_\mathrm{max}$ serves as an upper bound: if $K(t)=K_\mathrm{max}$, surplus
energy cannot be stored and has to be discarded. On the other hand, the storage unit can only provide balancing power, if $K(t)$ is
sufficient. For $K(t)=0$, only positive power mismatch can be mitigated, whereas the system is exposed to negative power deficiencies.
\\
\indent
For the sake of simplification, we neglect \textit{efficiency} and limitations due to \textit{power rating} here. This means we assume
lossless conversion and that the power to be delivered according to the specific control strategy is provided completely if permitted
by the storage filling $K(t)$.
The storage unit can be equipped with different control strategies. Three standard strategies adopted from engineering practice
(\textit{state-of-charge dependent resource management}, \textit{droop control}, \textit{ramp rate control}) are specified and investigated
below.
Our intention is to demonstrate the general impact of different basic storage strategies and their limitations due to
maximum capacity and time-delay on system behaviour rather than to model a detailed situation or derive concrete
guidelines. The approach can be applied to more concrete situations in the course of follow-up research.
Complex optimization problems may evolve depending on multiple factors such as operation conditions and technical system requirements,
cost concerns, legal and economic framework etc.
\begin{figure*}[t!]
\begin{center}
\includegraphics[scale=0.66]{Figure_powerSystem.pdf}
\end{center}
\caption[Simplified model of a power system subjected to wind feed-in with local storage and control.]
{Simplified model of a power system subjected to wind feed-in with local storage and control.
(a) Exemplary part of the feed-in time series $P_\mathrm{WPP}(t)=|P_\mathrm{sys}^\mathrm{cons}|+x(t)$ delivered by a wind power plant or park
(with $|P_\mathrm{sys}^\mathrm{cons}|=0.25$ (magenta line), mean value $\langle x\rangle=0$ and standard deviation
$\sigma_x=0.084\cdot|P_\mathrm{sys}^\mathrm{cons}|$).
(b) On the basis of the actual wind power feed-in $P_\mathrm{WPP}(t)$, the \underline{s}torage and \underline{c}ontrol
\underline{u}nit (SCU) first calculates the desired
power output value $P_\mathrm{SCU}(t)$ according to the specific control strategy and the mismatch $\Delta_P(t)=P_\mathrm{WPP}(t)-P_\mathrm{SCU}(t)$.
For $\Delta_P(t)<0$, the mismatch is delivered by the storage, if the filling level $K(t)$ is sufficient. Conversely, surplus power $\Delta_P(t)>0$
can only be stored with $K_\mathrm{max}$ as an upper bound. The power actually fed into the grid by the storage unit is
denoted as $P^\mathrm{out}_\mathrm{SCU}(t)\leq P_\mathrm{SCU}(t)$.
The \enquote{grid node} is modeled as a synchronous machine with parameters
(following \cite{schmietendorf2017TurbulentRenewableEnergyProductionGridStabilityQuality}) read: $m=1.0, \gamma=0.2,
P_\mathrm{sys}^\mathrm{cons}=-0.25$, $B_{12}=B_{21}=1.0$, $B_{11}=B_{22}=-0.95$, $\alpha=2.0$, $C=0.9101$, $\beta=1.0$, $\chi=0.5$.}
\label{fig:simpleSystem}
\end{figure*}
\subsection{Wind power feed-in}\label{subsec:windPowerGeneration}
Due to atmospheric turbulence, wind power has specific turbulent-like characteristics\cite{milan2013prl,anvari2016njp}:
extreme events, correlations, Kolmogorov
power spectrum, and intermittent increment statistics.
We implement realistic wind feed-in time series taking these basic properties into account
\begin{equation}
P_\mathrm{WPP}(t)=|P_\mathrm{sys}^\mathrm{cons}|+x(t)\label{eq:feed_in}
\end{equation}
with the constant part $|P_\mathrm{sys}^\mathrm{cons}|$ meeting the consumption of the system.
The fluctuating time series $x(t)$ is generated as follows \cite{schmietendorf2017TurbulentRenewableEnergyProductionGridStabilityQuality}:
first, a time series $\tilde{x}(t)$
is generated by means of the Langevin-type system of equations
\begin{eqnarray}
\dot y&=&-\gamma y+\Gamma(t),\\
\dot{\tilde{x}}&=&\tilde x\left( g-\frac{\tilde x}{x_0}\right)+\sqrt{D\tilde x^2}y,
\end{eqnarray}
with $\gamma=1.0$, $g=0.5$, $x_0=2.0$, $D=2.0$ and $\delta$-correlated Gaussian white noise $\Gamma$.
Then the corresponding Fourier spectrum is modified so that the final
power spectrum $S(f)=|F(f)|^2$ roughly reproduces real data sets, in particular the Kolmogorov $\frac 53$-decay. Transforming back to real space
yields $x(t)$ (see Fig.\,\ref{fig:simpleSystem}(a)
for an exemplary part of the feed-in time series $P_\mathrm{WPP}(t)$). Since
$\langle x\rangle=0$, power balance is given over time: $\langle P_\mathrm{WPP}\rangle=|P_\mathrm{sys}^\mathrm{cons}|$.
Due to the generating process, $P_\mathrm{WWP}$ features a smallest frequency mode. Lower frequencies corresponding to power variations on longer time scales
are assumed to be handled by other mechanisms like standard load balancing.
In practice, different time scales can actually be divided up
and assigned to different control mechanisms by low-pass filtering \cite{zhao2015reviewStorageWindPowerIntegrationSupport}.
\subsection{Power quality assessment}\label{subsec:performanceAssessment}
\textit{Power quality} is a wide ranging notion, which includes different aspects of voltage and frequency stability and supply reliability.
In this study, we focus on short-term frequency quality.
We use different criteria for performance assessment, which in combination give a more detailed picture of frequency
quality\footnote{Electrical engineering literature indicates \textit{power output smoothing} as
another major issue with respect to stochastic
feed-in\cite{jabir2017IntermittentSmoothingWindPowerReview,Li2013BESSsmoothingControlWindPV}.
This is related to frequency quality are related, as system frequency and power balance are coupled.}.\\
\indent
\textit{Frequency quality} \cite{entsoeHandbookPolicy1,entsoeSupportLoadFreqControl2013,entsoeHPoPEIPSl2017} refers to the systems ability to
maintain nominal frequency $\omega_\mathrm{sys}^\mathrm{nom}$, or keep the frequency within a pre-defined range
(the \textit{standard frequency range}) for a large percentage of operation time.
It can be evaluated on different time scales. Short-term frequency quality
is referred to \textit{instantaneous frequency deviations} in electrical engineering. It is commonly evaluated by means of
the percentage of time the system frequency is outside the standard frequency range \cite{entsoeSupportLoadFreqControl2013}.\\
\indent
Against this practical backdrop, we define
\begin{equation}
q_{\bar\omega}=
\begin{cases}
\int^\infty_{\bar\omega}p(\omega)\mathrm d\omega \qquad\mathrm{for} \,\,\bar\omega>0,\\
\phantom{\tiny x}\\
\int^{-\bar\omega}_{-\infty}p(\omega)\mathrm d\omega \qquad\mathrm{for} \,\,\bar\omega<0,
\end{cases}
\end{equation}
with $\bar\omega$ denoting the bound given by the standard frequency range $\omega_\mathrm{sys}^\mathrm{nom}\pm\bar\omega$ and $p(\omega)$
being the
probability distribution of frequency deviations $\omega(t)=\omega_\mathrm{sys}(t)-\omega_\mathrm{sys}^\mathrm{nom}$.
In real power grids, the
nominal frequency is 50\,Hz or 60\,Hz. Kuramoto-like power grid models are usually transformed into a reference frame rotating with nominal frequency
so that here $\omega_\mathrm{sys}^\mathrm{nom}=0$.
For sufficiently long simulation, $q_{\bar\omega}$ corresponds to the percentage of time the system is expected to operate outside
of the frequency range defined by $\bar\omega$ on average.
With a specified standard frequency range $\bar\omega$, $q_{|{\bar\omega}|}=q_{-\bar\omega}+q_{\bar\omega}$ was introduced
as the \textit{exceedance} measure\cite{auer2017stabilitySynchronyIntermittentFluctTreeLikePowerGrids}.
In this study, we evaluate $q_{\bar\omega}$ as a function
of $\bar\omega$ rather than for one specified $\bar\omega$ for two reasons: firstly, this gives a more informative picture of system dynamics; secondly, the value of standard frequency range
is not unequivocally defined\footnote{In \cite{entsoeSupportLoadFreqControl2013}, this fact is mentioned with view to the different characteristics of transmission grid areas.
This applies even more for the heterogeneous operation conditions on the distribution grid level and in islanded microgrids.}.
\\
\indent
Frequency quality not only involves deviations from nominal frequency, but also the time derivative $\mathrm{d}\omega/\mathrm{d}t$, commonly
referred as the \textit{rate of change of frequency} \cite{entsoeSupportLoadFreqControl2013,entsoeHPoPEIPSl2017}.
The rate of change of frequency reflects the sensitivity against sudden perturbations and is
inversely proportional to system inertia \cite{tielens2012gridInertiaFrequencyControlRenewables}.
In former times, it was of interest mainly during transient periods after significant imbalances
\cite{entsoeHPoPEIPSl2017}. Nowadays, due to the loss of system inertia and the increasing percentage of stochastic renewables
inducing continuous perturbations, the rate of frequency change becomes
relevant also during \enquote{normal operation}.\\
\indent
The frequency changes can be related to the increments
$\Delta\omega_\mathrm{sys}=\omega_\mathrm{sys}(t+\Delta t)-\omega_\mathrm{sys}(t)$.
It was shown that intermittency of wind power
in terms of heavy-tailed probability density functions is directly transferred into frequency fluctuations and significantly
contribute to frequency quality decrease \cite{schmietendorf2017TurbulentRenewableEnergyProductionGridStabilityQuality}.
Therefore, we here capture increments statistics not only by their mean value $\mu_{|\Delta\omega|}$ and
standard deviation $\sigma_{\Delta\omega}$ (as it is standard in electrical engineering) but also their kurtosis
$\kappa_{\Delta\omega}=\mu_{4,\Delta\omega}/\sigma_{\Delta\omega}^4$
(with $\mu_4$ denoting the fourth moment of the distribution)\footnote{The statistical measures $\mu_{|\Delta\omega|}$,
$\sigma_{\Delta\omega}$
and $\kappa_{\Delta\omega}$ refer to a given
time lag $\Delta t$, i.\,e. strictly speaking we have $\mu_{|\Delta\omega|}^{\Delta t}$ etc. We drop the upper index, but specify $\Delta t$
in the following analysis}.
The kurtosis serves as a measure for the tailed-ness of the distribution. With $\kappa=3$ being the value of the Gaussian distribution,
$\kappa>3$ means that there are more extreme events or outliers than in the Gaussian case, and vice versa for $\kappa<3$. Note that $\kappa$
entails information about the shape of the distribution, not about the magnitude of the outliers.\\
\indent
As we will see, the extreme events observed in the increment statistics in this study have two reasons: on the one hand, the system is exposed to the
feed-in fluctuations $P_\mathrm{WPP}(t)$ during time intervals, in which the storage facility is not able or supposed to fully compensate
power imbalances. As stated above, these fluctuations are known to transfer intermittency into the frequency statistics.
On the other hand, new extreme events can be induced when the storage steps in. For example, if the system runs out of storage in the course of a longer
time period with power deficiency or discontinues balancing quite suddenly due to its control specifications,
the frequency may face an instantaneous drop. The following analysis will show that the kurtosis serves as a good indicator for an
inaccurate adjustment of control strength.
\section{Results}
\subsection{Storage control limited by maximum capacity}\label{subsec:simplestStorage}
We start with a simple storage strategy, which provides a first insight into the performance of the storage facility as a function of its
maximum capacity.
We assume the storage facility to have a maximum energy capacity $K_\mathrm{max}$.
The actual power mismatch is $\Delta_P(t)=P_\mathrm{WPP}(t)-P_\mathrm{SCU}(t)$ (see Fig.\,\ref{fig:simpleSystem}).
In this simplified operation mode,
the storage unit is intended to ensure power balance between feed-in and consumption
whenever possible:
\begin{itemize}
\item[\textbullet] For positive power mismatch $\Delta_P(t)\geq 0$,
$P_\mathrm{SCU}^\mathrm{out}(t)=|P_\mathrm{sys}^\mathrm{cons}|$ is fed into the system. The corresponding energy surplus
$\Delta_K$ is stored with the maximum capacity $K_\mathrm{max}$ being an upper bound.
\item[\textbullet] For $\Delta_P(t)<0$, the power mismatch is
\begin{itemize}
\item[\textbullet] either fully compensated, i.\,e. $P_\mathrm{SCU}^\mathrm{out}(t)=|P_\mathrm{sys}^\mathrm{cons}|$, if the storage is sufficiently
filled,
\item[\textbullet] or the rest capacity is used to provide $P_\mathrm{SCU}^\mathrm{out}(t)$ with
$P_\mathrm{WPP}(t)< P_\mathrm{SCU}^\mathrm{out}(t)<|P_\mathrm{sys}^\mathrm{cons}|$.
\item[\textbullet] While $K(t)=0$, no balancing power can be provided.
\end{itemize}
\end{itemize}
\begin{figure*}[t!]
\begin{center}
\includegraphics[scale=0.98]{Figure_maxStorageCapacity.pdf}
\caption[Storage limited by maximum capacity $K_\mathrm{max}$.]
{Storage limited by maximum capacity $K_\mathrm{max}$. (a) Exemplary time series of system frequency
$\omega_\mathrm{sys}(t)$ and storage filling level $K(t)$ for maximum storage capacity
$K_\mathrm{max}=1.0$. Empty or insufficient storage filling is paralleled by frequency fluctuations. (b)
$q_\mathrm{non-stat}=q_\mathrm{|10^{-3}|}$ gives the percentage of non-stationary operation.}
\label{fig:maxCapacitiy}
\end{center}
\end{figure*}
In this mode of operation, the system dynamics alternate between stationary operation and time intervals with $\omega_\mathrm{sys}<0$,
in which the system either fluctuates in reaction to the stochastic feed-in, or is on its way to return to stationary operation
(see Figure\,\ref{fig:maxCapacitiy}\,(a)).
Figure \ref{fig:maxCapacitiy}\,(b) shows how the percentage of non-stationary operation time
$q_\mathrm{non-stat}=q_{|\mathrm{10^{-3}}|}$\footnote{We define $|\omega_\mathrm{sys}(t)|<|10^{-3}|$ as stationary
operation. This is in accordance with real
power grids in the sense that, strictly speaking, these are constantly subjected to disturbances and hence, never in a stationary state
with $\omega_\mathrm{sys}=\omega_\mathrm{sys}^\mathrm{nom}$ and
$\frac{\mathrm{d}}{\mathrm dt}\omega_\mathrm{sys}=0$.} decreases to zero with
maximum storage capacity $K_\mathrm{max}$. The behaviour for the $K\rightarrow\infty$ limit is trivial in qualitative respects,
as it implies that
a sufficiently large storage capacity is able to continually balance power differences and guarantee stationary operation.
It was to be expected as we restricted our analysis to a
limited time scale, i.\,e. we assumed the feed-in fluctuations to be balanced by other load control mechanisms on longer time scales.\\
\indent
However, our analysis so far was only to serve as a first storage capacity assessment.
In reality, the equipment of wind and PV plants with large storage capacity is cost expensive.
In the following, we therefore consider the more
realistic and less trivial situation of a storage facility with \enquote{insufficient capacity}.\\
\subsection{Storage control strategies}\label{sec:controlStrategies}
We now investigate different basic storage control strategies with
regard to their potential to improve frequency quality.
We set the maximum storage capacity $K_\mathrm{max}=2.0$.
This would allow for stationary operation in about 90 \% of time assuming the simplified
strategy considered above.
The control strategies refer to accepted methods in engineering practice, and rely on different control
feedback signals:
\begin{itemize}
\item[\textbullet] the actual storage level $K(t)$, or the \textit{state-of-charge} (here used for the purpose of \textit{storage resource management}),
\item[\textbullet] system frequency as an indicator for power imbalances (\textit{droop control}), and
\item[\textbullet] power differences between certain time steps (\textit{ramp rate control}).
\end{itemize}
These methods cover the elementary strategies for power quality improvement and therefore provide a basic structure for
the development of smart control techniques by refining the conventional strategies or composing hybrid systems.
\subsubsection{Storage resource management}\label{subsec:resourceManagement}
In the current technical application, the \textit{state-of-charge} is applied as a feed-back signal in battery storage systems
mainly to guarantee operation within proper state-of-charge
range and to prevent shut-down due to over-charge \cite{Li2013BESSsmoothingControlWindPV}.
Here, we shift the scope of application to the grid side and reinterpret the basic idea as a form of intelligent storage management.\\
\indent
We complement the simple storage strategy presented above and make the balance power at time $t$ dependent
on the current capacity $K(t)$.
To be specific, the power to be delivered by the storage is the power deficiency $\Delta_P$
multiplied by a factor $f=f(K(t))\in[0,1]$ with $f(0)=0$ and $f(K_\mathrm{max})=1$.
Here, we consider the different realizations for $f(K)$ depicted in Fig.\,\ref{fig:resourceManagement}\,(a):
$f(K)=-(1-K/K_\mathrm{max})^n+1$ for $n=12,\,4,\,2$ (denoted as scenarios I,\,II and III) and $f(K)=(K/K_\mathrm{max})^n$ for
$n=1,\,2,\,4,\,12$ (scenarios IV-VII).\\
\indent
Fig.\,\ref{fig:resourceManagement}\,(b) shows $q_{\bar\omega}(\bar\omega)$ for the different realization of storage resource management
in comparison to the simplified storage strategy described in the previous subsection.
With proper a choice of $f(K)$, the proposed storage resource management can in fact serve to prevent large frequency deviations.
Of course, a higher percentage of small deviations has to be tolerated in exchange.
The best option can only be chosen in knowledge of the operational circumstances and the specific guidelines for $\bar\omega$.
Furthermore, storage resource
management can be applied as part of a multi-pronged control strategy in combination with other storage control mechanisms.
\begin{figure*}[t!]
\begin{center}
\includegraphics[scale=0.9]{Figure_storageResourceManagement.pdf}
\caption[Storage resource management for $K_\mathrm{max}=2.0$.]
{Storage resource management for $K_\mathrm{max}=2.0$.
(a) Different realizations of storage resource management $f(K)$ denoted as strategies I-VII. \enquote{0} corresponds to the simplified storage mechanism
with $f(K)=1$.
(b) $q_{\bar\omega}(\bar\omega)$ giving
the percentage of time the system frequency is outside the $\bar\omega$ boundaries for realizations 0-VII.
(As in the case described above, $\omega_\mathrm{sys}(t)\leq 0$ in this mode of operation.)}
\label{fig:resourceManagement}
\end{center}
\end{figure*}
\subsubsection{Droop control}\label{subsec:droopControl}
\textit{Droop control} is based on the relationship between power imbalances and system frequency:
a positive power mismatch is paralleled by frequency increase,
whereas negative mismatch leads to frequency decrease.
This fact can also be observed in Kuramoto-like grids.\\
\indent
Droop control has a broad range of application, which
includes frequency control services provided by wind power plants \cite{zhao2015reviewStorageWindPowerIntegrationSupport}.
The standard practice is to use a linear droop control mechanism, whose slope is given by the control strength $k_\mathrm{DC}$.
The balancing droop power is
\begin{equation}\label{eq:droopControl}
P_\mathrm{droop}(t)=k_\mathrm{DC}(\omega_{\mathrm{sys}}^\mathrm{nom}-\omega_\mathrm{sys}(t))=-k_\mathrm{DC}\omega(t),
\end{equation}
as $\omega_{\mathrm{sys}}^\mathrm{nom}=0$ here.
If $\omega_\mathrm{sys}(t)<0$, this is interpreted as an indicator of negative power balance, and consequently more
power is injected into the system in order to keep system frequency close to its nominal value.
For $\omega_\mathrm{sys}(t)>0$, power feed-in is reduced accordingly.\\
\indent
This adjustment of power input to the actual system frequency obviously requires storage capacity in the background.
A specific type of inverter with linear droop control was shown to behave analogue to a synchronous machine \cite{schiffer2013SyncDroopControlledMicroGrids}, and
was already implemented into Kuramoto-like grids with stochastic feed-in \cite{auer2017stabilitySynchronyIntermittentFluctTreeLikePowerGrids}.
Our approach here is different
in the sense that we explicitely
take into account the limits of the installed background storage capacity $K_\mathrm{max}$ but
do not assume any further specifications on the grid feed-in process.
As explained above, in case of $\omega_\mathrm{sys}(t)<0$, the balancing power $P_\mathrm{droop}>0$ can only be provided
to the extent that the storage level $K(t)$ is
sufficient, and for $\omega_\mathrm{sys}(t)>0$, surplus power can only be stored with $K_\mathrm{max}$ as an upper bound.\\
\indent
We first investigate system performance under standard droop control according to Eq.\,(\ref{eq:droopControl}) for fixed maximum storage capacity
$K_\mathrm{max}=2.0$ and varying control strength $k_\mathrm{DC}$.
Fig.\,\ref{fig:droopControl}(a) shows system frequency in response to the same power feed-in $P_\mathrm{WPP}(t)$ for different $k_\mathrm{DC}$.
With increasing control strength, the positive frequency
deviations are more and more eliminated, as it is always possible to feed in less power than available.
In contrast, balancing negative frequency deviations requires sufficient storage level.
This asymmetry can also be seen in
Fig.\,\ref{fig:droopControl}(b1), which reveals the dilemma of standard droop control under limited storage capacity: On the one hand, for
sufficiently large control strength $k_\mathrm{DC}$, the positive frequency deviations can be more or less eliminated; but in this case, the storage unit
runs out of capacity quickly at the beginning of longer timer periods with negative power mismatch.
Fig.\,\ref{fig:droopControl}(a) highlights an concrete example of a frequency dip not being prevented due to overambitious control strength.
On the other hand, for small control strength, the storage facility performs better in the sense that it
reduces the probability of large negative frequency deviations.
But at the same time, the droop control mechanism remains sub-optimal with respect to positive frequency deviations.\\
\indent
To overcome this problem, we propose a non-symmetric droop control strategy, which treats positive an negative frequency
deviations differently:
\begin{equation}\label{eq:droopControl_mod}
P_\mathrm{droop}(t)=\begin{cases}
-k^\mathrm{DC}_1\omega_{\mathrm{sys}}(t)\qquad \forall\,\omega_{\mathrm{sys}}\geq0,\\
k^\mathrm{DC}_2(\omega_{\mathrm{sys}}(t))^n\qquad \forall\,\omega_{\mathrm{sys}}<0.
\end{cases}
\end{equation}
We choose large control strength $k^\mathrm{DC}_1$ in order to counteract the positive frequency deviations
and tested two control schemes for the negative frequency range:
(i) quartic droop control\footnote{Nonlinear droop control is technically feasible \cite{nonLinDroopControl2018ieee}.}
; and (ii) linear droop control with small control strength $k^\mathrm{DC}_2$.
Fig.\,\ref{fig:droopControl}(b2) shows that these alternative strategies combine the best of both small and large
control strength in standard droop control: they effectively mitigate positive deviations and prevent
large frequency dips.
A nonlinear control-term, inter alia, gives the opportunity to focus the onset of control to a specific frequency bound.
For example, the drop of $q_{\bar\omega}(\bar\omega)$ indicates that the quartic control term actually starts acting around
$\omega_{\mathrm{sys}}\approx 0.05$.\\
\indent
With view to the frequency increments statistics (see Fig.\,\ref{fig:droopControl}(c)),
the mean value $\mu_{|\Delta\omega|}$ and standard deviation $\sigma_{\Delta\omega}$ decrease with increasing control strength, finally
converging to a minimum value. However, the non-Gaussianity in terms of kurtosis $\kappa_{\Delta\omega}$ grows, even when $\mu_{|\Delta\omega|}$ and
$\sigma_{\Delta\omega}$ have nearly approached their minima and barely change\footnote{The corresponding values of the statistical measures for the alternative strategies (i) and (ii)
are: (i) $\mu_{|\Delta\omega|}=2.25\cdot 10^{-5}$, $\sigma_{\Delta\omega}=3.21\cdot 10^{-5}$, $\kappa_{\Delta\omega}=9.00$, and
(ii) $\mu_{|\Delta\omega|}=2.26\cdot 10^{-5}$, $\sigma_{\Delta\omega}=3.16\cdot 10^{-5}$, $\kappa_{\Delta\omega}=8.50$.}.
This is an indicator that the control strength $k_\mathrm{DC}$ is getting too ambitious and the storage facility runs out of capacity
more frequently. It therefore becomes evident that increment statistics are an essential part of a comprehensive
picture of frequency quality.
\begin{figure*}[t!]
\begin{center}
\includegraphics[scale=0.86]{Figure_droopControl_1.pdf}\\
\includegraphics[scale=0.76]{Figure_droopControl_2.pdf}
\caption[Droop control with maximum storage capacity $K_\mathrm{max}=2.0$.]
{Droop control with maximum storage capacity $K_\mathrm{max}=2.0$. (a) System frequency response to the same feed-in time series
with standard droop control acc. to Eq.\,(\ref{eq:droopControl})
for different control strengths $k_\mathrm{DC}=0.0$ (no control), $k_\mathrm{DC}=0.5$, and $k_\mathrm{DC}=10.0$. The time interval indicated by
the red dotted lines illustrates the drawback of too ambitious control strength: the frequency dip is prevented for $k_\mathrm{DC}=0.5$,
but no longer for $k_\mathrm{DC}=10.0$.
(b1) $q_{\bar\omega}(\bar\omega)$ for standard droop control with different control strengths $k_{DC}$. For
$k_\mathrm{DC}=10.0$ the curve for positive deviations is not displayed due to its rapid decay ($q_{0.01}$ has already dropped
to $\mathcal{O}(10^{-5})$).
(b2) $q_{\bar\omega}(\bar\omega)$ for the alternative non-symmetric droop control
strategies acc. to Eq.\,(\ref{eq:droopControl_mod}): (i) $n=4$, $k^\mathrm{DC}_2=10.0$, $k^\mathrm{DC}_2=200.0$, and (ii)
$n=1$, $k^\mathrm{DC}_2=10.0$, $k^\mathrm{DC}_2=0.1$. For negative $\bar\omega$, (ii) resembles the standard droop control case.
(c) Increment statistics during non-stationary operation (according to the definition given above). For increasing $k_\mathrm{DC}$ in standard droop control,
the mean value $\mu_{|\Delta\omega|}$ and standard deviation $\sigma_\omega$
decrease, while the non-Gaussianity of the distribution in terms of the kurtosis $\kappa_{\Delta\omega}$ grows.}
\label{fig:droopControl}
\end{center}
\end{figure*}
\subsubsection{Ramp rate control}\label{subsec:rampRateControl}
A \textit{power ramp} is defined as a normalized power change or power increment:
\begin{equation}
\Delta P_\mathrm{ramp}(t)=\frac{P_\mathrm{in}(t)-P_\mathrm{ref}}{P_\mathrm{norm}}
\end{equation}
with input power $P_\mathrm{in}(t)$ and reference power $P_\mathrm{ref}$.
\textit{Ramp rate control} \cite{marcos2014storageRequirementsPVrampRate,schnabel2016storageRequirementsPVrampRateNorthernEurope}
aims at keeping power ramps within specified tolerance bounds:
\begin{equation}\label{eq:rampCondition}
|\Delta P_\mathrm{ramp}|\leq r_\mathrm{tol}.
\end{equation}
It is utilized in wind and solar power applications. In the latter case, power ramps play a even major role due to passing clouds.\\
\indent
The basic idea opens up numerous opportunities for concrete realization depending of the choice of $P_\mathrm{ref}$. For example, it can
be given by prior values $P(t-\Delta t)$ defined by a sampling time
$\Delta t$ or be calculated as a function of the actual demand.
We here demonstrate a version of ramp rate control, which mainly targets on short-term ramps:
First, we set $P_\mathrm{ref}=P_\mathrm{SCU}^\mathrm{out}(t-\Delta t)$ with $\Delta t=0.005$.
As long as the ramp condition Eq.\,(\ref{eq:rampCondition}) is satisfied, no balancing is necessary and
$P_\mathrm{SCU}^\mathrm{out}(t)=P_\mathrm{WPP}(t)$. If the condition is violated, the storage facility steps in: For $\Delta P_\mathrm{ramp}>0$
(upward ramps),
$P_\mathrm{SCU}^\mathrm{out}$ is decreased so that $|\Delta P_\mathrm{ramp}|=r_\mathrm{tol}$ and surplus power is stored.
For $\Delta P_\mathrm{ramp}<0$ (downward ramps), the storage is supposed provide balance power in order to fulfill
$|\Delta P_\mathrm{ramp}|=r_\mathrm{tol}$.
Again, the storage of surplus power is limited by the maximum capacity $K_\mathrm{max}$ and balancing power can only be delivered if
the actual storage level $K(t)$ is sufficient.\\
\indent
The performance of ramp rate control is usually assessed with respect to the power feed-in statistics. Here,
we consider frequency statistics instead, for two reasons: first, this is the scope of our study and consistent with the previous analysis.
Secondly, we investigate a consequential phenomenon, as power fluctuations are directly transferred into frequency variations.
Fig.\,\ref{fig:rampRateControl} (a) and (b) show how the likelihood of tolerance bound violations in terms of $q_{\bar\omega}(\bar\omega)$ and
the frequency increment statistics evolve as functions of the tolerance ramp rate $r_\mathrm{tol}$.
It shows that the ramp rate control strategy fulfills its
main purpose with view to the increment statistics: by suppressing power ramps, frequency increments can be mitigated significantly.
In parallel, the percentage of operation time
beyond certain tolerance bounds can be decreased.\\
\begin{figure*}[t!]
\begin{center}
\includegraphics[scale=0.92]{Figure_rampRateControl.pdf}
\caption[Ramp rate control with $K_\mathrm{max}=2.0$.]
{Ramp rate control with $K_\mathrm{max}=2.0$. (a) $q_{\bar\omega}(\bar\omega)$ for different tolerance ramp rates
$r_{tol}$ given in percent of the standard deviation of the wind power feed-in $\sigma$.
(b) Increment statistics during non-stationary operation: with decreasing ramp tolerance $r_\mathrm{tol}$, frequency increments are mitigated in terms of their
mean value $\mu_{\Delta\omega}$, standard deviation and kurtosis $\kappa_{\Delta\omega}$. For large tolerance
ramps, the influence of control diminishes and $\mu_{\Delta\omega}$, $\sigma_{\Delta\omega}$ and $\kappa_{\Delta\omega}$ approach their values
of the no-control case (horizontal lines).}
\label{fig:rampRateControl}
\end{center}
\end{figure*}
\indent
Again, the ambition of control, here in terms of $r_\mathrm{tol}$, has to be chosen carefully.
On the one hand, if $r_\mathrm{tol}$
is too large, the control does not achieve its potential. It has no influence on $q_{\bar\omega}(\bar\omega)$ and barely improves the
increment statistics. On the other hand, if $r_{tol}$ is too
small, the storage tends to run out of capacity. This is indicated by a steep rise of $\kappa_{\Delta\omega}$ and
increasing likelihood of large negative frequency deviations.
Compounding the problem in this specific version of \textit{ramp-rate control} is the fact that
if the power input $P_\mathrm{SCU}^\mathrm{out}$ drops to a low $P_\mathrm{WPP}$ during a feed-in deficit period with empty storage,
this value serves as the new reference $P_\mathrm{ref}$. In the following, the input power and system frequency can return to their nominal values
only slowly due to the tight tolerance range, even if storage capacity is available. As explained before, running out of
storage is paralleled by sudden frequency drops, which is indicated by the drastic increase of the non-Gaussianity of the increment distribution.\\
\indent
Note that the dissymmetry between positive and negative frequency deviations
(which can be seen in Fig.\,\ref{fig:rampRateControl}\,(a)) is not completely analogue to the droop control case. First, ramp rate
control responds to power input fluctuations (which cause of frequency fluctuations) and not to the deviation from nominal frequency
directly.
Secondly, upward ramps
can always be balanced, irrespective whether the actual system frequency is below or above its nominal value. In contrast,
balancing downward ramps requires sufficient storage. This particularly affects
power deficit periods accompanied by $\omega_\mathrm{sys}<0$, during which the storage is depleted.\\
\indent
Comparing droop control and the applied version of ramp rate control, the latter has the advantage to be
able to mitigate frequency increments to a certain extent without being paralleled by increasing non-Gaussianity. For example, droop control
with $k_\mathrm{DC}=6.0$ and ramp control with $r_{tol}=0.03\sigma\%$ both reduce the mean value to
$\mu_{\Delta\omega}\approx 1.6\cdot 10^{-5}$. At
the same time, the statistics for the droop control case contain considerably more extreme events ($\kappa_{\Delta\omega}=13.7$) than the
system with ramp rate mechanism ($\kappa_{\Delta\omega}=1.8$).
On the other hand, the ramp rate control does not take into account the absolute deviation from nominal frequency, and hence is
not designed to prevent large frequency excursion as efficiently as droop control.
\
\subsection{Finite response time}
Real control equipment does not react instantaneously but in response
to the feedback signal at time $t-\tau$. In the following, we investigate and compare the sensitivity of the three control strategies
introduced in the previous section.
We implemented finite time response as follows\footnote{Alternatively, the time delay could be modelled by
an ordinary differential equation with an appropriate time constant.}:
\begin{itemize}
\item[\textbullet] In case of \textit{storage resource management}, the balance power to be delivered by the storage facility is
$\Delta_P\cdot f(K(t-\tau))$\footnote{We here assumed finite-time response solely for the filling-level feedback, while the calculation
of $\Delta_P$ and hence the decision
whether the storage facility has to step in, happens instantaneously in response to the actual mismatch $\Delta_P(t)$. If this was not
the case (this process would perhaps be associated with another
time delay $\tau^\prime$), the situation would of course be exacerbated and also positive frequency deviations could occur.}.
We picked the linear storage resource management scenario VI as
example.
\item[\textbullet] For \textit{droop control}, we instance the asymmetric control strategy (ii)
with $k^\mathrm{DC}_1=10.0$ and $k^\mathrm{DC}_2=0.5$.
The balancing power $P_\mathrm{droop}(t)$ is calculated on the basis of
$\omega_\mathrm{sys}(t-\tau)$.
\item[\textbullet] The \textit{ramp-rate control} realization we presented above is very sensitive due to the
short sampling rate. In view of finite time response, we consider another variant of ramp rate control and define the reference power
$P_\mathrm{ref}=|P_\mathrm{sys}^\mathrm{cons}|$ $\forall t$ and the tolerance range $r_\mathrm{tol}=0.5\sigma$ here.
The balancing power at time $t$ is calculated as the response to $\Delta P_\mathrm{ramp}(t-\tau)$.
\end{itemize}
Fig.\,\ref{fig:timeDelay} shows how time delay limits the possibilities for frequency quality improvement
with focus on the main target of each control strategy.\\
\indent
Storage resource management was introduced in order to prevent overspending and save capacity to mitigate large frequency deviations.
From Figure\,\ref{fig:timeDelay}\,(a), one can see that finite response time has negligible impact up to $\tau=10.0$.
Then the deviations from the instantaneous-response case become more and more apparent. In particular,
the control strategy increasingly misses its main objective as the probability of large deviations from nominal frequency grows.\\
\indent
Droop control is intended to mitigate deviations from nominal frequency.
Fig.\,\ref{fig:timeDelay}\,(b) shows that in this respect the system is able to handle a finite response time up to $\tau=0.1$ quite well.
Then $q_{\bar\omega}$ starts to increase for small $\bar\omega$ as
the feedback delay causes trouble when the system fluctuates close around
nominal frequency. As the control switches between positive and negative balancing too late, oscillations around nominal frequency are
induced.
A nonlinear droop scheme could mitigate these oscillations as it interfers less for small deviations.\\
\indent
Ramp rate control was shown to be a promising candidate for frequency quality improvement with respect to increment statistics.
The version of ramp-rate control considered here is very sensitive towards time delay.
Fig.\,\ref{fig:timeDelay}\,(c) shows that the introduction of finite response time leads to a reduction of frequency
quality as $\mu_{|\Delta\omega|}$, $\sigma_{\Delta\omega}$ and $\kappa_{\Delta\omega}$ immediately increase,
even beyond the no-control case.\\
\begin{figure*}[t!]
\begin{center}
\includegraphics[scale=0.78]{Figure_timeDelay_1.pdf}\\
\includegraphics[scale=0.78]{Figure_timeDelay_2.pdf}
\caption[How finite response time undermines frequency quality improvement.]
{How finite response time undermines frequency quality improvement.
(a) Linear storage resource management: $q_{\bar\omega}(\bar\omega)$ for different
response times $\tau$. The curves for $\tau<5.0$ almost resemble the instantaneous-response ($\tau=0$) case. (b) Droop control for the
asymmetric control strategy (ii) . Again, $q_{\bar\omega}(\bar\omega)$ is shown
for different response times. For $\tau<0.1$, the impact of finite-time response on $q_{\bar\omega}(\bar\omega)$ is negligible. (c)
Increment statistics for ramp rate control as a function of $\tau$. The horizontal lines indicate the values for the no-control case.}
\label{fig:timeDelay}
\end{center}
\end{figure*}
\indent
These results indicate that short-term frequency quality applications require rapid response of the underlying control
mechanism (on sub-second scale\footnote{As we use a
greatly simplified power system in dimensionless units in order to point out general relationships, we are careful with
specifying concrete values. However, at this point, we want to give a rough idea about the time scale. For
$\mathcal{O}(m)\sim10^4\,\mathrm{kg\,m}^2$,
$\omega^\mathrm{nom}_\mathrm{sys}=2\pi\cdot 50\,$Hz, $\mathcal{O}(BE_\mathrm{sys}E_\mathrm{SCU})\sim 1\,\mathrm{GW}$ and
$\mathcal{O}(1/\gamma)\sim 0.1\,\mathrm{s}-1\,\mathrm{s}$
(cf.\,\cite{menck2014natcomm}), $t=\mathcal{O}(t)\sim 0.1\,\mathrm{s}-1\,\mathrm{s}$ and hence the response time of the storage facility has to be
in the sub-second range. This coarse evaluation is in accordance with the technical features of electrical
storage with the fastest response times being in the range of milliseconds \cite{zhao2015reviewStorageWindPowerIntegrationSupport}.}).
As this is usually cost-expensive, it may be advisable to use hybrid systems and treat the
high-frequency and lower frequency fluctuations separately with different storage and control systems.
\section{Discussion and Outlook}
We extended the current Kuramoto-like modeling framework with flexible storage units.
With that, the scope of Kuramoto-like models opens up to one of the most important research topics in power grid engineering.
On the way to this goal, we brought together
Kuramoto-like equations and load-flow analysis. This is a substantial extension, which
can serve as a starting point for the straight-forward
implementation of arbitrary grid components. \\
\indent
For demonstration purposes, we considered short-term frequency quality
improvement by means of storage facility with maximum capacity in a power system subjected to realistic wind feed-in.
Motivated by recent findings, we assessed system performance not only with respect to frequency range violations, but also
took into account frequency increment statistics.\\
\indent
We demonstrated how to implement three basic control methods, which cover the elementary strategies
for power quality improvement in engineering practice.
First, we adopted \textit{state-of-charge feedback control} and reinterpreted it as a form of storage resource management.
It has been proven that this concept can actually serve to save capacity in order to prevent large frequency deviations.
Secondly, it was shown that \textit{droop control} can improve frequency quality not only with view to deviations
from nominal frequency but also with respect to frequency increment statistics. We pointed out that, particularly in case of limited capacity,
it is
favorable to handle positive and negative frequency deviations with different droop schemes and consider non-linear mechanisms.
Thirdly, we implemented a version of \textit{ramp rate control}. Originally designed for power-output-smoothing
applications, we demonstrated that this strategy entails frequency quality improvement.\\
\indent
For both droop and ramp-rate control, it became apparent that
the corresponding control strength or ramp tolerance range may not be too ambitious and have
to be carefully proportioned to the dimensions of the storage facility. Furthermore, it was shown that the finite response time of the control
mechanism limits the potential of the storage facility. Short-term frequency quality applications in particular require a rapid response.
\\
\indent
With this study, we created a sound starting point for follow-up research on various aspects of storage implementation
from the viewpoint of self-organized synchronization and collective phenomena.
This includes stability-topology issues like
\textit{optimal siting} of storage units as well as comparative studies on global vs. local storage location or \textit{optimal sizing} and
rough cost-benefit assessment.
Another current topic is the development and refinement of smart control strategies,
which are customized to the realistic features of wind and solar power and, at the same time, take into account the impact of
collective network dynamics. This study has already shown that
the presented basic control strategies have different advantages and disadvantages. Against this backdrop, and with view to the impact of
finite response times, systems with combined
control techniques are conceivable solutions and novel smart control strategies should be developed.
\section*{Acknowledgements}
Financial support from the Deutsche Forschungsgemeinschaft (PE 478/16-1 and MA 1636/9-1) is gratefully acknowledged.
\nocite{*}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 4,484 |
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"redpajama_set_name": "RedPajamaC4"
} | 3,772 |
{"url":"https:\/\/www.orbiter-forum.com\/threads\/failing-to-spawn-a-vessel-at-the-desired-position-solved-use-clbkprestep.40101\/","text":"# ProblemFailing to spawn a vessel at the desired position [solved, use clbkPreStep]\n\n#### N_Molson\n\n##### Addon Developer\nAddon Developer\nDonator\nHello,\n\nStill on the Tianwen Mars project, I'm trying to spawn the parts jettisoned by the lander during the descent (shell and parachute, heatshield...). So I searched for a code sample doing this and I came with the below piece of code. It sort of works, but it seems it isn't accurate enough. The discarded vessel is created like 10 meters above my lander, and it seems tilted at a 90\u00b0 angle. I'd expect it to spawn exactly where my vessel is, and oriented the same way. Right now, it looks crappy. So, what am I missing there ? ? My lander is oriented like a capsule, -Z axis is towards the heatshield +Z axis is towards the parachute. It is a rather small vessel so I don't see where that 10 meters (or more) offset is coming from...\n\nC++:\nvoid TIANWEN_LANDER::SpawnObject(void)\n{\nVESSELSTATUS2 vs;\nvs.flag = 0;\nvs.version = 2;\n\nchar name[256];\nVECTOR3 sofs = { 0, 0, 0 }; \/\/ Seperation offset\nVECTOR3 sdir = { 0, 0, 1 }; \/\/ Seperation direction\ndouble svel = 0; \/\/ Separation velocity[\/COLOR]\n\n\/\/ Get vessel status structure\nVECTOR3 rofs;\nGetStatusEx(&vs);\nLocal2Rel(sofs, vs.rpos);\nGlobalRot(sdir, rofs);\nvs.rvel += rofs * svel;\n\n\/\/ Create descent stage as seperate vessel\nstrcpy(name, GetName());\nstrcat(name, \"-Shell\");\noapiCreateVesselEx(name, \"Tianwen_shell\", &vs);\n}\n\n#### jedidia\n\n##### shoemaker without legs\nAddon Developer\nThe discarded vessel is created like 10 meters above my lander, and it seems tilted at a 90\u00b0 angle. I'd expect it to spawn exactly where my vessel is, and oriented the same way.\nRotation and position of a vessel when spawning in landed state depend on the first three touchdown points!\n\n#### asbjos\n\n##### tuanibrO\nAddon Developer\nAre you spawning the vessel from PostStep? I've experienced the same offset in position when spawning from there. Moving it to PreStep fixed it for me.\n\nAs for the rotation, is it precisely 90\u00b0? If so, it could be you have forgotten that Orbiter uses a left-handed coordinate system, so you may have to swap y and z.\n\n#### N_Molson\n\n##### Addon Developer\nAddon Developer\nDonator\nAre you spawning the vessel from PostStep? I've experienced the same offset in position when spawning from there. Moving it to PreStep fixed it for me.\n\nYou're the best ! That was it, now it works perfectly. The rotation issue was related.\n\nRotation and position of a vessel when spawning in landed state depend on the first three touchdown points!\n\nIn this particular case you have \"vs.flag = 0;\" which means \"in flight\". Landed state is 1 (or the now famous \"special\" -10 value).\n\n### Similar threads\n\nReplies\n2\nViews\n632\nReplies\n10\nViews\n1K\nAdvanced Question Spawning a landed vessel\nReplies\n7\nViews\n2K\nReplies\n8\nViews\n1K\nReplies\n0\nViews\n946","date":"2023-01-27 21:18:04","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\": 0.7157869338989258, \"perplexity\": 5403.989407342596}, \"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-2023-06\/segments\/1674764495012.84\/warc\/CC-MAIN-20230127195946-20230127225946-00303.warc.gz\"}"} | null | null |
Background Antenatal care (ANC) has been recognised as a way to improve health outcomes for pregnant women and their babies. However, only 29% of pregnant women receive the recommended four antenatal visits in Nepal but reasons for such low utilisation are poorly understood. As in many countries of South Asia, mothers-in-law play a crucial role in the decisions around accessing health care facilities and providers. This paper aims to explore the mother-in-law's role in (a) her daughter-in-law's ANC uptake; and (b) the decision-making process about using ANC services in Nepal. Methods In-depth interviews were conducted with 30 purposively selected antenatal or postnatal mothers (half users, half non-users of ANC), 10 husbands and 10 mothers-in-law in two different (urban and rural) communities. Results Our findings suggest that mothers-in-law sometime have a positive influence, for example when encouraging women to seek ANC, but more often it is negative. Like many rural women of their generation, all mothers-in-law in this study were illiterate and most had not used ANC themselves. The main factors leading mothers-in-law not to support/ encourage ANC check ups were expectations regarding pregnant women fulfilling their household duties, perceptions that ANC was not beneficial based largely on their own past experiences, the scarcity of resources under their control and power relations between mothers-in-law and daughters-in-law. Individual knowledge and social class of the mothers-in-law of users and non-users differed significantly, which is likely to have had an effect on their perceptions of the benefits of ANC. Conclusion Mothers-in-law have a strong influence on the uptake of ANC in Nepal. Understanding their role is important if we are to design and target effective community-based health promotion interventions. Health promotion and educational interventions to improve the use of ANC should target women, husbands and family members, particularly mothers-in-law where they control access to family resources. | {
"redpajama_set_name": "RedPajamaC4"
} | 131 |
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| {
"redpajama_set_name": "RedPajamaGithub"
} | 2,288 |
{"url":"https:\/\/mathematica.stackexchange.com\/tags\/index\/hot","text":"# Tag Info\n\n45\n\nI think Leonid's answer deserves to be expanded upon. Most other languages are not symbolic, and thus the \"variable name\" is not something one needs to keep track of --- ultimately the interpreted or compiled code is keeping track of pointers or something. In contrast, in Mathematica the Head of an expression is arbitrary. This is somewhat along the lines ...\n\n34\n\nGeneral usage Here is what I think Using strings and subsequently ToString - ToExpression just to generate variable names is pretty much unacceptable, or at the very least should be the last thing you try. I don't know of a single case where this couldn't be replaced with a better solution Using subscripts is also pretty bad and should be avoided, except ...\n\n21\n\nMy answer is based on a modification of a binary heap. Basically the construction looks something like this. We start with a binary tree: Notice that if we label the nodes breadth-first, the labels have an interesting property. Each parent node $n$ has two children, $2n$ and $2n+1$. This also works in reverse: the parent of node $n$ is node $\\left\\... 17 Preface Below, you will find two different solutions. For understanding the problem itself, the first, iterative solution is better suited since it gives insight in how the solution can be found without directly executing the instructions given as input. Iterative Solution Detailed explanation To explain the idea behind this approach let us work with a ... 16 Many index-specific operations can be implemented via MapIndexed with a level specificaton. Your Power example can be written as: MapIndexed[#1^(#2[[1]]*#2[[2]]) &, test2D, {2}] If you want better readability of indices you can define an auxiliary function: myPower[x_, {n1_, n2_}] := x^(n1 n2); MapIndexed[myPower, test2D, {2}] Some index-specific ... 14 I think the best way to understand this behavior is with this example, In[99]:= x = {a, b, c, d, e}; In[101]:= Length@x Out[101]= 5 In[102]:= x[[3]] = Nothing; In[103]:= Length[x] Out[103]= 4 In[104]:= Block[{Nothing}, Length[x]] Out[104]= 5 When you say x[[3]] = Nothing, you are not deleting an element from x. x is still a 5-element list. But x is ... 13 I have a tree-based method that has the right asymptotics but a very high coefficient. The upshot being, it will not compete with other methods until we get past 10^6 or so in list size. With considerable work that tree structure could be flattened so that Compile might be brought into play. The basic tree layout is {left subtree, node, right subtree} where ... 12 Some years ago, a friend of mine was in the supermarket with his son, small kid, who asked him to buy some candy. After some resistance, my friend agreed, but told him that should be just one. In the cashier, his son had two candy, and my friend said: What is this? haven't we agreed One? And the answer (very smart) was: Yes! Here it is. Zero and One... (... 12 I don't know if this is useful to you but it seems a little cleaner than your own code: asc = <|\"z\" -> 11, \"x\" -> 22, \"b\" -> 33, \"a\" -> 44|>; keySpan[k_Span][asc_Association] := asc[[k \/. First \/@ PositionIndex@Keys@asc]] asc \/\/ keySpan[\"x\" ;; \"a\"] asc \/\/ keySpan[\"z\" ;; \"a\" ;; 2] asc \/\/ keySpan[\"b\" ;;] <|\"x\" -> 22, \"b\" -> 33,... 11 MapIndexed: MapIndexed[#2[[1]] + # &, {a, b, c, d}] {1 + a, 2 + b, 3 + c, 4 + d} Also Range[Length @ #] + # & @ {a,b,c,d} {1 + a, 2 + b, 3 + c, 4 + d} 11 The variable i is a dummy one. The evaluated expression: Sum[f[i], {i, 1, 10}] f[1] + f[2] + f[3] + f[4] + f[5] + f[6] + f[7] + f[8] + f[9] + f[10] contains no explicit variable f[i], hence, the result is 0. Try to first Inactivate the sum, and only then to calculate the derivative: expr1 = D[Inactivate[Sum[f[i], {i, 1, 100}], Sum], f[i]] The result is ... 10 The example JSON string: json = ExportString[<| \"Names\" -> <| \"Sister\" -> \"Nina\", \"Brothers\" -> {<|\"Older\" -> \"John\", \"Younger\" -> \"Jake\"|>}, \"somethingElse\" -> \"answer\" |>, \"DOB\" -> { <|\"Nina\" -> 2001, \"location\" -> \"Miami\"|>, <|\"John\" -> 2017, \"location\" ->... 10 If you put your cursor on the Position command and press F1 for help, you will see the following under Properties and Relations: \"Use Extract to extract parts based on results from Position.\" There is also an example. For your case: p = Position[{a, b, c, d, e, f}, c] Extract[list,p] where list is the list you want to extract from. 10 My goal is to add an index to all elements of a list in the form {\"a\", \"b\", \"c\", ... }, so it becomes {\"N1 a\", \"N2 b\", \"N3 c\" ... } may be seq={\"a\",\"b\",\"c\",\"d\"}; MapIndexed[\"N\"<>ToString[First@#2]<>\" \"<>#1&,seq] gives {\"N1 a\", \"N2 b\", \"N3 c\", \"N4 d\"} 10 We don't need to avoid Table in my view. In cases that Table is more straightforward, just use Table. If speed is concerned, Compile it. Here is an example: Can I generate a \"piecewise\" list from a list in a fast and elegant way? Nevertheless, your 2 examples (especially 2nd one) don't belong to the cases that Table is more straightforward, at ... 9 strs = {\"first block of text with random content\", \"different block of text\", \"1 2 3 4\"}; Nearest[(StringPadRight[#, 50] & \/@ strs), \"content random with\"] This is a deep and complex question apparently: Mathematica has a menagerie of built in goodies to assemble your own variant. EditDistance DamerauLevenshteinDistance ... 9 As of Mathematica 11: filenames = Table[CreateFile[], 3]; content = {\"first block of text with random content\", \"different block of text\", \"1 2 3 4\"}; MapThread[Put, {content, filenames}]; index = CreateSearchIndex[filenames]; Perform searches using TextSearch: Snippet \/@ Normal@TextSearch[index, \"block\"] In order to rank search results, score them ... 9 Following the comments I am encouraging the use of brackets rather than subscripts or superscripts. Here is an example where a function may take a variable with a subscript or a variable without a subscript. The function will use pattern recognition to sort out how to behave. First we define the function. Note I start with a ClearAll[f] so that previous ... 7 Prompted by comments conversation with Mr. Wizard, a routine I use: findMultiPosXX[list_, find_, allowBits_: False, skipCands_: True] := Module[{f = DeleteDuplicates[find], o, l, oo, bitmax = 20, cands, dims}, If[allowBits && Length@f <= bitmax, With[{r = If[Length@(dims = Dimensions@list) == 1, Range@Length@list, Array[... 7 Using DownValues enables you to format the display in the subscripted form without using Notation and Symbolize (Format[#[n_]] := Subscript[#, n]) & \/@ {x, \u03c3, a}; kvar[k_] := Through[{x, \u03c3, a}[k]] kvar[3] kvar[n] If you will never use a symbolic index then you can restrict the argument of kvar to Integer as you did originally. 7 What are the requirements for well behaved variables? Functions are not variables, although in most cases, the kernel treats undefined variables and functions identically. Sometimes it doesn't. After all, there are places in mathematics where the difference between a number and a function is important. One extreme and undocumented example is Dt[], the ... 7 func1[a_, b_] := a + b; func2[a_, b_] := 1 + a + b; func3[a_, b_] := a + 2 b; n = 10; sa = SparseArray[{Band[{1, 1}] -> (func1[0, #] & \/@ Range[n]), Band[{1, 2}] -> (func2[0, #] & \/@ Range[n - 1]), Band[{2, 1}] -> (func3[0, #] & \/@ Range[n - 1])}, {n, n}]; sa \/\/ MatrixForm or sa = Quiet@SparseArray[{Band[{1, 1}] -> (... 7 Try Pick[foo, mask] (* {a, c}*) 6 Module[{p = Range[Length@#]}, Reap@Fold[(Sow[#[[#2]]]; Drop[#, {#2}]) &, p, #]] &@{1, 1, 2, 1, 1} (* {{}, {{1, 2, 4, 3, 5}}} *) 6 This seems rather be a error of determining the type of i, j and k than Sum itself. When you introduce your iterator variables in a Module and make clear that they are of type integer, it compiles fine for me: fc = Compile[{{x, _Real, 2}}, Module[{i = 0, j = 0, k = 0}, Sum[x[[i, k]]*x[[j, k]], {k, 5}, {i, 12}, {j, i + 1, 12}]] ] Btw, I want to note ... 6 You could use Evaluate[list] = ConstantArray[0, Length[list]] or MapThread[Set, {list, ConstantArray[0, Length[list]]}] to Set each indexed variable inside of list to 0. If the indexes for the variables inside list follow a known condition, one can use for example p[i_ \/; i < 3, j_ \/; j < 5] = 0 or memorization p[i_ \/; i < 3, j_ \/; j < 5] ... 6 An alternative to subscripts as indices... Instead of: {Subscript[x, 1], Subscript[x, 2]} Let's use: x[1], x[2] And this can be generalized to: Subscript[x, i, j] --> x[i,j] This will uniquely identify any number of variables to any dimension. 6 From version 10.4 onward, we can define keySpan like this: keySpan[k1_, k2_] := Replace[<|___, s:PatternSequence[k1 -> _, ___, k2 -> _], ___|> :> <|s|>] so that:$a = <| \"z\" -> 1, \"x\" -> 2, \"b\" -> 3, \"a\" -> 4 |>; \\$a \/\/ keySpan[\"x\", \"a\"] (* <|\"x\" -> 2, \"b\" -> 3, \"a\" -> 4|> *) We can make this more ...\n\n5\n\nYou can use this (where t is your dataset): Ordering[#, -1] & \/@ Transpose[t] which produces {{1}, {1}, {2}, {4}, {2}, {2}} Incidentally, the list of positions you gave in your question is wrong (the 4th element should be {4,4}, and the 6th element should be {2,6}). The above method omits the first coordinates in your expected output, since they ...\n\n5\n\nThis is indeed somewhat confusing when you are new to Mathematica. In Mathematica, == stands for mathematical equality. Thus a == 0 does not evaluate to either True or to False until a is replaced by a numerical value. a is considered to be a variable that may or may not be zero. A pattern like x_ \/; condition will only match if condition is explicitly ...\n\nOnly top voted, non community-wiki answers of a minimum length are eligible","date":"2021-09-16 18:45:38","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\": 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\": 0.44076836109161377, \"perplexity\": 714.8281945515729}, \"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-2021-39\/segments\/1631780053717.37\/warc\/CC-MAIN-20210916174455-20210916204455-00501.warc.gz\"}"} | null | null |
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"redpajama_set_name": "RedPajamaCommonCrawl"
} | 1,068 |
I Loved the Half Time Show, But I'm a Hedonist
Posted by Kevin Lynn | Feb 9, 2020 | Current Events | 0 |
With the Eagles going down in the playoffs, I was less than enthused about Superbowl 54 and last Sunday chose to skip it altogether. I awoke the next morning to several texts and DMs raving about the halftime show. What was the source of all the brouhaha? It certainly appeared to be much more than a wardrobe malfunction this time around.
I went to YouTube and watched it. Then I watched it again, and again, and again. Then I told two people to watch it; and they told two people, and they told two people … By now I am sure everyone in the world has watched it.
By any measure, it was a heck of a show that J. Lo and Shakira put on. The scantily clad, suggestive, brazen stars and cast danced, strutted and gyrated their way across the stage in what I gathered to be a cross between an old time Vegas stage show and a bacchanalian festival that invited the world to leave their seats and join in.
Great music and great choreography. What was not to like? I liked it! But then, I'm a hedonist. As I will explain, it was hedonistic in the literal sense. Whether it was intentional or unintentional, through a very well planned and executed usage of symbols, it wrapped decadence in a red, black, white, silver and gold shroud of virtue in a sophisticated attempt to rewire traditional virtues and values.
The show opens with Shakira singing her song "She Wolf." She and her troupe of dancers are clad in red, and there is reason for this, as anyone who graced the rooms a brothel can tell you. The symbolism of red is important and shouldn't be missed. "Red is the color of fire and blood, so it is associated with energy, war, danger, strength, power and determination, as well as passion, desire and love. Red is a very emotionally intense color. It enhances human metabolism, increases respiration rate and raises blood pressure." Well, the routine raised my blood pressure. The lyrics shouldn't be overlooked either:
There's a she wolf in the closet
Open up and set her free (ahoo)
Let it out so it can breathe
S-O-S she is in disguise
There's a she wolf in disguise
Coming out, coming out, coming out
Later in the routine Shakira performs a belly dance routine only this time with her hands bound. Bondage anyone? Well, one should be fit, to be tied, I suppose.
A bit later during her rendition of "I Like It Like That," she lays herself on the stage in sexually suggestive poses, while rapper Bad Bunny, clad in silver from the shoes on his feet to his earring loomed large over her.
It is interesting that he was in silver, as there is symbology there too and silver will make its appearance again. The color silver is associated with meanings of industrial, sleek, high-tech and modern, as well as ornate, glamourous, graceful, sophisticated and elegant. Silver is a precious metal and, like gold, often symbolizes riches and wealth. So, who is in the power position there? The women in red, or the guy in silver? Sex for money anyone?
Then enter J. Lo looking amazing as she took over center stage grasping a pole belting out "Jenny from the Block." If J. Lo's shear physicality weren't astounding enough, her performance on the pole would have put the best dancer I'd ever seen at a strip club to shame. And she is how old?
So, we have left the brothel, and are now at the strip club. Still with me, America?
At this point J. Lo and her troupe of dancers are wearing mostly black and leather to boot. Black is associated with power, fear, mystery, strength, authority, elegance, formality, death, evil, aggression, authority, rebellion and sophistication. Black is required for all other colors to have depth and variation of hue. The black color is the absence of color. The leather is of course emblematic of sadomasochism or appropriate attire when watching a band doing Village People covers. Well, they did spare us the hoods and dildos. And if there were any question to who is on top, I think J. Lo answered that for us.
Later in her routine J. Lo remounts the pole, only this time clad in silver. Yep, the message here being the stripping paid off and now throngs of dancers clad in white are massing at the base of her pole and at times holding her up while a few have poles of their own.
Let's talk about the color white. White plays a pivotal role in the show. White is associated with light, goodness, innocence and virginity. It is considered to be the color of perfection. White means safety, purity and cleanliness. As opposed to black, white usually has a positive connotation.
Then enter Emme, J. Lo's 11-year-old daughter, and things start to get really strange. Emme opens to the camera clad in white, singing from what appears to be a cage, possibly a bird cage, and she is accompanied by other children, also clad in white and in cages.
A camera angle from way above shows the entire stage. At one end are the children illuminated in their cages, and on the other end of the stage is J Lo and the other adult dancers. What appears to be separating them is an illuminated yellow cross on the stage floor. I won't go into the symbology of the color yellow, but there is a reason buses are often painted yellow with black trim.
The scene then cuts back to Emme singing her song which is entitled "Let's Get Loud":
If you want to live your life
Live it all the way and don't waste it
Every feelin' every beat
Can be so very sweet you gotta taste it
You gotta do it, you gotta do it your way
You gotta prove it
You gotta mean what you say
Emme is joined in the song by a host of other children like her, clad in white, only on their tops are American flags in sequins, silver sequins.
The children make their way to the main stage where Emme is joined by her mother who approaches her draped in the colors of the American flag. The routine picks up as J. Lo is joined by Shakira now clad in a gold outfit and all the other adult dancers who are now clad in gold.
Gold is significant as it is a precious metal that is associated with wealth, grandeur and prosperity, as well as sparkle, glitz and glamour. So, we have moved from red to black, to silver, to white and then to gold.
There are those who say there was a political message to the halftime show. That it was not so much about showcasing Miami's Latino culture as it was about promoting diversity and other woke culture themes. And if its intent was to promote diversity, why was Shakira the only Caucasian on stage? Surely there are Caucasians living in Miami! Isn't it pretty much Cuban culture that makes Miami interesting?
If there were some kind of political subtext, its intent was to mask the didactic message of this exhibition. What was the didactic meaning you ask? What were the creators of the show (either wittingly or unwittingly) attempting to teach us?
Again, I do not believe the subtext was political. Rather it was cultural, ideological in a classic neoliberal context. You see neoliberalism is the unfettered movement of people and wealth across borders so as to maximize profits.
So, the message from America's elites as portrayed by the halftime show is pretty simple; people of color from across the globe, come to America, surrender your virtues and family values to our special brand of consumerism and you too will achieve the success of your fantasies. Just look at J. Lo and Emme. Don't you want to be like them?
Tragically, unlike the song, you won't get to "do it your way." The machine that keeps our stock markets propped up and the debt and death paradigm going demands fresh meat. And you are here to be milked, until you no longer can be. Then you get to experience the U.S. healthcare system which leads to debt and then the death of a thousand vicissitudes. Well, I suppose it beats seeing your village bombed by Team America. That is unless you are rich like J. Lo!
Am I being some old fuddy duddy who is reading way too much into the whole thing? Quite possibly. For certain, I doubt it was intentional, as creatives like Shakira and J. Lo better than most have a talent for tapping into the zeitgeist and crafting metaphors to describe it. And you can bet the deciders weren't going to stop this show from rolling right along.
In closing, if this were really to be exhibition of Latin Miami culture where was all that is great about Miami's Latin culture. The children of Cuban immigrants to America have much higher levels of tertiary education than the native born. They also pretty much run everything in Miami and pretty darn competently too. Why couldn't that have been on display?
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Kevin Lynn
Kevin is a long-time student of history and world affairs. He considers himself a lapsed Republican, an unrepentant Perotist, a failed Green and frustrated Democrat. He is an astute observer of the hijinks being played not only on both sides of the political aisle, but also with numerous NGOs and activist groups across the U.S.
Amazon Must be Destroyed
We Are In A Fourth Turning
Hooray for Hollywood Part 2
Policy Watch: Protect and Serve? | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 5,875 |
Q: why is my flutter and dart plug in not installed? I already did installed it but when I use flutter doctor it says that dart in not installed/flutter is not installed. how to fix this?
A: Try to follow official step by step procedure to install flutter
Install flutter
A: check your flutter path and setup.
Go to Setting and check your path.and after this apply this.
| {
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{"url":"https:\/\/www.cpsyjournal.org\/articles\/10.1162\/CPSY_a_00022\/","text":"A- A+\nAlt. Display\n\n# Active Inference and Auditory Hallucinations\n\n## Abstract\n\nAuditory verbal hallucinations (AVH) are often distressing symptoms of several neuropsychiatric conditions, including schizophrenia. Using a Markov decision process\u00a0formulation of active inference, we develop a novel model of AVH as false (positive)\u00a0inference. Active inference treats perception as a process of hypothesis testing, in which\u00a0sensory data are used to disambiguate between alternative hypotheses about the world.\u00a0Crucially, this depends upon a delicate balance between prior beliefs about unobserved (hidden) variables and the sensations they cause. A false inference that a voice is present,\u00a0even in the absence of auditory sensations, suggests that prior beliefs dominate perceptual\u00a0inference. Here we consider the computational mechanisms that could cause this imbalance\u00a0in perception. Through simulation, we show that the content of (and confidence in) prior\u00a0beliefs depends on beliefs about policies (here sequences of listening and talking) and\u00a0on beliefs about the reliability of sensory data. We demonstrate several ways in which\u00a0hallucinatory percepts could occur when an agent expects to hear a voice in the presence\u00a0of imprecise sensory data. This model expresses, in formal terms, alternative computational\u00a0mechanisms that underwrite AVH and, speculatively, can be mapped onto neurobiological changes associated with schizophrenia. The interaction of action and perception is\u00a0important in modeling AVH, given that speech is a fundamentally enactive and interactive\u00a0process\u2014and that hallucinators often actively engage with their voices.\n\nKeywords:\nHow to Cite: Benrimoh, D. A., Parr, T., Vincent, P., Adams, R. A., & Friston, K. (2018). Active Inference and Auditory Hallucinations. Computational Psychiatry, 2, 183\u2013204. DOI: http:\/\/doi.org\/10.1162\/CPSY_a_00022\nPublished on 01 Dec 2018\nAccepted\u00a0on 03\u00a0Oct\u00a02018 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0Submitted\u00a0on 02\u00a0May\u00a02018\n\n## INTRODUCTION\n\nThe phenomenology of auditory verbal hallucinations (AVH) is rich and heterogeneous (McCarthy-Jones et al., 2014), but at its simplest, it involves the perception of a voice in the absence of (verbal) auditory data. Bayesian theories of brain function (Friston, 2010) provide a way to operationalize this notion. Hallucinations can be conceptualized, in Bayesian terms, as representing false (positive) inference (Adams, Stephan, Brown, Frith, & Friston, 2013; Brown, Adams, Parees, Edwards, & Friston, 2013; Corlett & Fletcher, 2009; Powers, Mathys, & Corlett, 2017; Teufel, Fletcher, & Davis, 2010; Teufel et al., 2015). The inference that a voice is present in the absence of auditory input implies that internally generated \u201cprior\u201d beliefs about the presence of stimuli dominate perception, even in the absence of supportive sensory evidence. This suggests that overly precise (confident) prior beliefs (Friston, 2005) could be important in the genesis of schizophrenic hallucinations. A complementary perspective (Adams et al., 2013) is that false inference may be due to a down-weighting of the precision of sensations (i.e., silence) that contradict the expected percept (i.e., a voice). Both forms of imbalance lead to excessive confidence in prior beliefs relative to sensory evidence. A Bayesian approach to understanding hallucinations affords an opportunity for the construction of computational models of the phenomenon, which could shed light on information processing deficits in schizophrenia and suggest potential neural mechanisms for further investigation.\n\nThese formal models serve as a foundation for more complete computational accounts of auditory hallucinations. However, neither takes the active nature of perception into account. In this article, we argue for the importance of action, noting that the experience of auditory hallucinations is often in the form of a dialog with a voice, in which the hallucinating person may take an active role (i.e., speaking to or attempting to ignore the voice). So, in addition to the degraded perceptual processing of sensory input, our model\u2019s perceptual inferences also depend upon its beliefs about how it interacts with another (speaking) agent. In addition, we will use a discrete state formulation that is more consistent with the discrete nature of language (words, sentences, etc.) than the continuous formulations used in previous work. These two agendas (the prominence of action and the discrete formulation) are naturally modeled using a Markov decision process (MDP). This serves as the generative model for a synthetic subject who engages in active (Bayesian) inference (Mirza, Adams, Mathys, & Friston, 2016; Parr & Friston, 2017). We begin with a brief overview of active inference before specifying the generative model we have used and show how altering this model can lead to hallucinations. Note that here we explore only one possible computational alteration that could underpin AVH; we discuss some alternative mechanisms in the discussion. We conclude with a (speculative) discussion of the neurobiological plausibility of this model\u2014and its implications for understanding the link between pathophysiology and psychopathology.\n\n## MATERIALS AND METHODS\n\n### Active Inference\n\nUnder active inference, agents use a generative model to infer the causes of their sensory experiences. Crucially, agents are equipped with the ability to act (e.g., sample their environment) to gather evidence for their beliefs about those causes. Formally, this means agents act to minimize their variational free energy (Friston, Kilner, & Harrison, 2006). Technically, free energy is a variational approximation to the surprise, surprisal, self-information, or the negative log (marginal) likelihood of an observation under an internal model of the world (Friston, 2012). Crucially, surprise or self-information is negative Bayesian model evidence. This means that self-evidencing is the same as minimizing variational free energy. The free energy can be written as (see Table 1 for a list of variables)\n\n((1))\n$F=-{E}_{Q\\left(\\stackrel{~}{s}\\right)}\\left[lnP\\left(\u00f5,\\stackrel{~}{s}\\right)-lnQ\\left(\\stackrel{~}{s}\\right)\\right].$\nHere F is the free energy, \u00f5 is the sequence of observations through time, s are unobserved or hidden states, and Q is an approximate probability distribution over s. P is the generative model that expresses beliefs about how sensory data are generated, that is, the co-occurrence of observations and hidden states. It is this that takes the form of a MDP.\n\nTable 1.\n\nVariables used\n\nVariable Description\nAa, Ap Likelihood matrix (superscript denotes auditory or proprioceptive)\nBa, Bp Transition matrix (superscript denotes auditory or proprioceptive)\no\u03c4a, o\u03c4p Outcomes (agent observations; superscript denotes auditory or proprioceptive)\n\u03c0 Policies\n\u03b6 Likelihood precision\n\u03b3 Prior precision over policies\ns Hidden state (superscripts can be used to denote modality and subscripts to denote parameterization)\nG Expected free energy\nF Free energy\nC Prior preferences matrix\n\n#### Markov Decision Process and Generative Model\n\nA MDP is a framework for modeling the beliefs of agents who, like us, navigate environments in which they have control over some variables but not over others. MDPs have two important types of hidden variables that need to be inferred by the agent: hidden states and hidden policies. By hidden, we mean those variables that cannot be directly observed. The hidden states s\u03c4 represent the beliefs of an agent about the causes of her sensations; for our purposes, these are \u201cspeaking\u201d (or not) and \u201clistening to a voice present in the environment\u201d (or not). The hidden states are inferred from the sensory outcomes o\u03c4 (where \u03c4 indexes time): auditory input being present (or not) and speech movements (i.e., proprioception) being present (or not). Put simply, the hidden state of \u201clistening to a voice\u201d implies outcomes in the auditory domain only; the hidden state of \u201cspeaking\u201d (note that the agent does not automatically \u201cknow\u201d it is speaking: It must infer it) implies outcomes in the proprioceptive domain only. Given that our agent employs sensory attenuation, when speaking, it reduces the precision of the auditory modality, which may affect its inference about the auditory state of the world. This means that speaking involves attending away from the auditory domain (to attenuate any auditory evidence that one is in fact not speaking). Instead of influencing outcomes directly, speaking modulates the precision of outcomes, given the listening state.\n\nThe probability of a sensory observation, given a hidden state, can be expressed as a likelihood matrix with elements P(o\u03c4 = i|s\u03c4 = j) =Aij. For the proprioceptive modality in our model, this was simply an identity matrix (mapping speaking to proprioception). For the auditory modality, the likelihood matrix is\n\n((2))\n${\\stackrel{-}{A}}_{a}=\\frac{1}{Z}{\\left({I}_{2}+exp\\left(-30\\right)\\right)}^{\\zeta }.$\nHere Z is a normalizing constant (i.e., partition function) that ensures that each column in this matrix sums to 1. \u03b6 is the likelihood precision.1I indicates the identity matrix; exp(\u221230) denotes a small amount of imprecision (added to each element of the matrix to avoid numerical overflow). The bar notation means that Aa has been normalized (subscript denotes the auditory modality). As \u03b6 increases, this mapping comes to resemble the identity matrix. As it decreases, the probabilities become close to uniform, and the mapping becomes more uncertain. In other words, even if one knows the hidden state of the world, all outcomes are equally likely.\n\nCrucially, this formulation allows the fidelity of the mapping between states and outcomes to be modulated by \u03b6 (see Figure 1 for a graphical illustration of this). In continuous state space formulations of active inference, optimizing the equivalent quantity is the process of attending, for example, attention to a sensory channel increases its precision (Feldman & Friston, 2010). Decreased likelihood precision is analogous to a reduction in signal to noise. Previously, we have shown that synthetic subjects tend to \u201cignore\u201d low-precision mappings, as these contain relatively imprecise information (Parr & Friston, 2017; cf. the streetlight effect, Demirdjian et al., 2005).\n\nFigure 1.\n\nEffect of likelihood precision on state\u2013outcome mapping. Top: A relatively precise likelihood matrix (high likelihood precision \u03b6) leads to a high-fidelity mapping between the state, s, and the outcome, o (in this case, if the state is equal to 1, then the probability of the outcome being equal to 1 is 0.9). Black = precise belief; gray = uncertainty regarding belief; white =4 very imprecise belief. Blue box highlights the probabilities of the outcomes associated with an arbitrary state. Bottom: In this case, the likelihood matrix has been made much less precise (all of its entries are now equal probabilities), corresponding to a low likelihood precision \u03b6. This leads to an uncertain state\u2013outcome mapping.\n\nWhen \u201cI am speaking,\u201d we set the auditory likelihood matrix to have equal probabilities for both listening and not listening to simulate \u201csensory attenuation.\u201d This is the reduction of the precision afforded to self-generated sensory stimuli (i.e., the inability to tickle oneself; Blakemore, Wolpert, & Frith, 2000; Shergill, Bays, Frith, & Wolpert, 2003). Sensory attenuation is a fundamental aspect of intentional behavior, because it protects prior (intentional) beliefs about acting from sensory evidence that the act is not being executed (Brown, Adams, Parees, Edwards, & Friston, 2013). In effect, this means that while speaking, the presence and absence of sounds are deemed equally likely. This means I can maintain the belief that \u201cI am speaking\u201d in the absence of any auditory evidence to the contrary. Hence the agent\u2019s inference about the current state depends only on prior beliefs. The complement of this is that, if an agent is listening and not speaking, it must deploy a higher level of likelihood (i.e., sensory) precision to \u201cattend\u201d to its conversational partner. In short, only when the agent is listening can the likelihood precision affect the mapping between sensations (outcomes) and percepts (states). See Equation 2 and Figures 2 and3.\n\nFigure 2.\n\nNeuronal message passing. This schematic illustrates the form of the (variational) message passing implied by active inference. Here sensory observations o\u03c4 inform beliefs about states under each policy s\u03c0\u03c4 (and this depends on the likelihood precision, \u03b6). These reciprocally influence beliefs about states in the past and future. Beliefs about states under each policy are used to compute the expected free energy for each policy. This informs the beliefs about policies, \u03c0, and is modulated by precision over policies, \u03b3. Beliefs about policies are combined with beliefs about the states under each policy to compute the marginal beliefs about states (averaged under all policies), s\u03c4. By manipulating \u03b3, \u03b6, and \u03c0, we sought to induce changes in s\u03c4. Bold terms represent vector quantities; italics are model parameters. A is the likelihood matrix; B is the state transition matrix; C is the prior preferences matrix. G is the expected free energy. H is the entropy of A. The filled circle containing o\u03c4 is sensory data (outcomes).\n\nFigure 3.\n\nGenerative model. Here the generative model is presented more explicitly. In blue are the hidden states for listening or not listening and their associated outcomes\u2014hearing or not hearing a voice. Mapping between states and outcomes occurs via the likelihood matrix A. This is either the mapping from Equation 2 or (when the hidden state for speaking has been inferred) a matrix with equal entries to simulate sensory attenuation (the reduction of auditory likelihood precision during speaking, to prevent inference that one is not speaking but listening). In pink are the hidden states for speaking or not speaking, mapped via identity matrices to the speaking or not speaking proprioceptive outcomes. Transitions between hidden states are accomplished via the transition matrices Ba and Bp.\n\nThe second type of hidden variable is the policy. Each policy is a sequence of actions that an agent can pursue. Crucially, the policy that the agent is currently pursuing must be inferred (Botvinick & Toussaint, 2012). Policies are simply various combinations of actions (e.g., listening or speaking) that, in our case, mimic the flow of a conversation. For example, the agent could take turns listening and speaking, could engage in a monologue, or could listen for the whole trial. This is closely related to formulations of active inference for birdsong, in which beliefs about the narrative of a given song are shared by two birds (Friston & Frith, 2015). In this article, the implicit \u201cturn taking\u201d is determined by the sequence of choices our agent pursues, as in real conversations.\n\nThe subject\u2019s states change over time according to a probability transition matrix, conditioned upon the previous state and the policy, \u03c0, currently being pursued. This matrix is defined as B(u)ij = P(s\u03c4 +1 = i|s\u03c4 = j, u = \u03c0(\u03c4)). In other words, the policy influences, via its effect on probability transitions, how a state at a given time step changes to become the state at the next. We constrained these policies so that deciding to listen requires one to stop talking, and vice versa. Heuristically, we consider listening as an action to be a composite of mental and physical actions: all the things one might do when expecting to hear someone speak (i.e., pay attention, turn your head to hear better, etc.; Holzman, 1972).\n\nThe subject was paired with a generative process that determined the sensory input she experienced: an environment that produces alternating sounds and silences. Whenever she chose to speak, this generated sound at the next time step (and attenuated the likelihood precision). Whenever the subject chose to listen, the likelihood precision was determined by Equation 2 at the next time step. In this way, our subject interacted with the (simple) generative process in the environment, generating sequences of sounds and silences dependent on both the environment and her actions.\n\nThe synthetic subject began the simulation with a probability distribution over possible initial hidden states, Di = P(s1 = i). In our simulations, all initial states were equally likely. The subject is also equipped with a probability distribution over possible outcomes, which sets its prior preferences, C\u03c4i = P(o\u03c4 = i). These prior preferences influence policy choice by making some outcomes\u2014and therefore the policies that tend to lead to those outcomes\u2014more likely than others. In general, priors can be learned (empirical priors) or be \u201chardwired\u201d into a phenotype by the pressures of natural selection (Friston, 2010). In this article, prior preferences did not differ between outcomes (i.e., there were flat priors over outcomes). This means that the imperatives for action (i.e., talking and listening) were driven purely by epistemic affordances, namely, the imperative to resolve uncertainty about states of affairs in the world (see later). Note that making the probability distributions over either initial states or preferred outcomes unequal did not affect the nature of the model\u2019s hallucinations.\n\nImportantly, there is also a prior probability distribution over policies. In this scheme, policies are treated as alternative models and are chosen via Bayesian model selection, where the policy selected leads to the lowest expected free energy, G(\u03c0). This is equivalent to saying that agents choose the policies that are most likely to resolve uncertainty (such as choosing to make a saccade to an informative location). Formally, one can express a prior belief over policies as\n\n((3))\n$P\\left(\\pi \\right)=\\sigma \\left(-\\gamma \\cdot G\\left(\\pi \\right)\\right).$\nHere the expected free energy under alternative policies is multiplied by a scalar \u03b3 and passed through a softmax function (i.e., normalized exponential) to return a prior distribution over policies. In this setting, \u03b3 is a sensitivity or inverse temperature parameter that signifies the precision, or confidence, the agent has about its beliefs about policies. How confident an agent is about its policies a priori will have an impact on the relative weighting of sensory information when the agent tries to infer its policy. The balance of likelihood and policy precisions will, in turn, determine whether sensory data\u2014or the agent\u2019s inferred policy\u2014contribute most to the agent\u2019s beliefs about states (i.e., listening to a voice or speaking). This policy precision is important in the current context, as midbrain dopamine has been suggested to be its in vivo homolog (Schwartenbeck, FitzGerald, Mathys, Dolan, & Friston, 2015). Before we can describe the form of expected free energy under each policy, we need to consider the form of the posterior beliefs.\n\nTo simulate active inference in a tractable manner, we adopt a mean-field approximation (Friston & Buzsaki, 2016) to update approximate posterior beliefs Q about hidden variables:\n\n((4))\n$Q\\left(\\stackrel{~}{s},\\pi \\right)=Q\\left(\\pi \\right)\\prod _{\\tau }Q\\left({s}_{\\tau }|\\pi \\right).$\nThis formulation allows for independent optimization of each factor on the right-hand side of Equation 4 and for the expression of the free energy under a given policy as\n((5))\n$F\\left(\\pi \\right)={E}_{Q}\\left[lnQ\\left(\\stackrel{~}{s}|\\pi \\right)-lnP\\left(\u00f5,\\stackrel{~}{s}|\\pi \\right)\\right].$\nFree energy scores the information gained via observation. This poses a problem for policy selection: Policies should be chosen to reduce free energy in the future, but one can only define the free energy with respect to the present or the past (i.e., the times for which the agent has access to observations). To remedy this, we use the free energy expected under the policy to guide policy selection:\n((6))\n$\\begin{array}{ccc}G\\left(\\pi \\right)& =& {E}_{\\stackrel{~}{Q}}\\left[lnQ\\left(\\stackrel{~}{s}|\\pi \\right)-lnP\\left(\u00f5,\\stackrel{~}{s}|\\pi \\right)\\right],\\\\ \\stackrel{~}{Q}\\left(\u00f5,\\stackrel{~}{s}|\\pi \\right)& =& Q\\left(\\stackrel{~}{s}|\\pi \\right)P\\left(\u00f5|\\stackrel{~}{s}\\right),\\\\ P\\left(\\pi \\right)& =& \\sigma \\left(-G\\left(\\pi \\right)\\right).\\hfill \\end{array}$\n\nHaving specified the generative model (see Figure 4)\u2014and in particular the prior beliefs about policies\u2014in terms of expected free energy, it is relatively straightforward to derive belief update equations that underwrite perception and action. These equations of up-to-date posterior beliefs Q($\\stackrel{~}{s}$, \u03c0) in response to new observations provide a minimization of free energy. Crucially, the posterior belief computed at one time step becomes the (empirical) prior for the next. We have argued previously (e.g., de Vries & Friston, 2017) that these update equations (aka variational message passing) can be interpreted in terms of neuronal message passing. The architecture of this message passing implies a connectivity scheme that closely resembles the functional architecture of cortico-subcortical loops (Friston, Rosch, Parr, Price, & Bowman, 2017).\n\nFigure 4.\n\nAn imprecise likelihood matrix can cause false positive inference. The plots shown in this figure illustrate posterior beliefs about hidden states. True outcomes are noted above each trial. Darker colors mean greater probabilities of each state. Here we show that decreasing likelihood precision \u03b6 (from left to right, \u03b6 = 0.7; 0.525; 0.3) can lead to a false inference about the state of the world. Note that all other parameters were unchanged across these three simulations. Here we are looking at the beliefs of the agent about whether or not it is listening. In this example, it should listen (black box) and not listen (white box) in an alternating pattern to match the true outcomes of sound and no sound, respectively. Note that as there is an identity likelihood mapping between states and outcomes for speaking, the proprioceptive outcomes also indicate what the agent has inferred (i.e., whether it believes it is or is not speaking). In the leftmost figure, the agent infers the state of the world correctly, as reflected in its very certain beliefs (black squares) about states that correspond to external reality. In the middle figure, we have decreased the precision of the likelihood matrix, leading to uncertainty about whether or not the agent is in a listening state at the third and fifth time points; this decreased certainty is represented by the gray coloring over both possible states. In the third figure, a further decrease in \u03b6 has led the agent to believe firmly that it is in the \u201clistening to a voice\u201d state (dark boxes at the third and fifth time points). These are the hallucinations (red circles). For Figures 46, the lower part of each panel, labeled \u201cpolicies,\u201d represents the inferences (posterior probabilities), over time, by the agent about which policy she is pursuing. Darker shading represents increasing probabilities. Each row represents an alternative policy. Plotted between the columns are the actions (listening or speaking) that would be selected during the transition between the preceding and the next time step, if the policy in that row were to be followed (i.e., in this figure, if Policy 1 is selected at the end of Time Step 3, the agent would choose to listen during Time Step 4). Note that the legend of icons is conserved for all figures.\n\nIn the foregoing, posterior beliefs about states are conditioned upon the policy pursued (consistent with the interpretation of planning-as-model-selection). We can marginalize out this dependence on policies (through Bayesian model averaging) to obtain a belief about hidden states:\n\n((7))\n$Q\\left({s}_{\\tau }\\right)={\\sum }_{\\pi }Q\\left({s}_{\\tau }|\\pi \\right)Q\\left(\\pi \\right).$\nEquation 7 is crucial because it means that beliefs about states (to the left of the equation) depend upon beliefs about the policy being pursued (the terms on the right, which correspond to the approximate probability distribution over policies, and over states, given policies). This is a fundamental observation that further ties perception to action. This speaks to a quintessentially enactive aspect of hallucinations\u2014which we wanted to understand through simulations.\n\n### In Silico Hallucinations\n\nGiven the definition of hallucinations outlined earlier, how might we induce hallucinations in our model? Our aim is to dissociate inferred states or beliefs about the world from sensory constraints. Given that the mapping between states and outcomes depends on the likelihood matrix and its likelihood precision parameter \u03b6, we can hypothesize the following:\n\n\u2022 1.\u2003\u00a0 Decreasing likelihood precision \u03b6 will affect the mapping between states and outcomes, with reduced \u03b6 leading directly to a disconnect between the inferred state and the outcome (Equation 2). This follows because decreasing \u03b6 is equivalent to reducing the confidence in sensory information, impairing the use of sensory evidence to inform perceptual synthesis.\n\u2022 2.\u2003\u00a0 Increasing the prior precision over policies \u03b3 will change the prior distribution over policies (Equation 3) and hence the posterior beliefs about policies Q(\u03c0) and, therefore, inferred states through model averaging (Equation 7). Put simply, an increased policy precision will bias perceptual inference away from sensory evidence and toward its current action.\n\u2022 3.\u2003\u00a0 Changing the policy space will also change Q(\u03c0) and therefore affect the state inferred through model averaging (Equation 7). Removing one or more policies may change which policy is inferred to be most likely, which in turn will affect state estimation.\n\n## RESULTS: SIMULATIONS\n\n### Reducing Likelihood Precision Leads to False Positive Inferences\n\nInference about hidden states (e.g., \u201clistening to a voice\u201d or not) depends on both outcomes (whether sound is present) and policies (whether one is speaking or listening). We hypothesized that decreasing \u03b6 could cause false inferences through a state\u2013outcome dissociation. We additionally noted that, as inferences about states depend upon policies via Bayesian Model Averaging (see earlier), the inferred state will be influenced by the policy set. In Figure 5, we present the finding that reducing \u03b6 moderately, from baseline, leads to perceptual \u201cconfusion\u201d (i.e., posterior beliefs are ambiguous about which state is in play); further reducing the likelihood precision leads to a false positive inference, namely, the belief that the subject was listening in the absence of sound. The reason for this reversal is that the subject has inferred that it is following a policy consistent with the presence of heard speech. This induces an empirical prior belief that she is listening to something, despite the fact that there is nothing to hear, leading the agent to infer that she is actually hearing a sound in the environment. In the high-precision condition, this belief is corrected by precise sensory evidence\u2014even if the agent chooses to listen, she should be able to infer when silence is present and conclude that she is not actually listening to anything in the environment. Low precision allows the (false) prior belief to dominate, producing a false inference. This illustrates that low precision has a permissive effect on hallucinations. As we will see in the third result, this effect of reduced likelihood (i.e., sensory) precision is highly dependent on the policy space; reduction of likelihood precision does not lead to hallucinations in all policy spaces.\n\nFigure 5.\n\nIncreasing prior precision over policies given a noisy likelihood matrix can induce hallucinations. Darker boxes represent increasing levels of confidence. True outcomes are represented above each trial. On the left, an agent with a somewhat imprecise sensory mapping (likelihood precision \u03b6 = 0.525) but a low prior precision over policies [\u03b3 = exp(64)] is uncertain about the state of the world and makes incorrect inferences (the inferred state does not concord with the outcome) at the second, third, and fourth time steps; at the third time step, the agent has hallucinated (infers it is listening to a voice when there is no voice present). Effects of low likelihood precision are denoted by the blue box. If \u03b3 is increased [\u03b3 = exp(\u221264)], the agent begins to hallucinate at the fifth time step as well (red rectangle). Note that this does not occur with baseline likelihood precision (\u03b3 = 0.7).\n\n### Increasing Prior Precision over Policies Can Elicit Hallucinations in the Presence of Reduced Likelihood Precision\n\nWe next demonstrate that increasing policy precision induces hallucinations in the presence of a permissive imprecise likelihood mapping between states and outcomes \u03b6 and a suitable policy space. This effect is shown in Figure 5. Given normal \u03b3, a lower \u03b6 leads to some confusion at the last time step, but there is a weak (unconfident) belief in the not listening state, and as such, the agent concludes, accurately, that it has not heard a voice. Increasing \u03b3 then causes a switch to the (inaccurate) belief that a voice was heard.\n\nOnly a few policy spaces showed this effect of changing \u03b3. This demonstrates an important finding: The generation of hallucinations is highly context dependent. More formally, the different policy options determine which policy is ultimately inferred. The most probable policy under the resulting approximate posterior distribution Q(\u03c0) has sensory consequences that might be consistent with, or may conflict with, the sensory evidence. False inference is induced by the latter. As such, hallucinations can only occur when the agent is equipped with a policy space likely to conflict with the sensory evidence (hence the policy space dependence of the effect of increasing \u03b3). The degree to which the winning policy influences state expectations depends upon \u03b3, which gives rise to more precise posteriors over policies. It is important to note again the permissive effect of unattenuated likelihood precision; at baseline \u03b6, increasing the prior precision over policies does not induce hallucinations. This is because conflict between policy and sensory evidence is more likely to be resolved in the winning policy\u2019s favor if the influence of sensory evidence is down-weighted by a low precision.\n\n### Hallucinations Are Context Dependent\n\nFigure 6 demonstrates the importance of the policy space in determining whether an agent will hallucinate. We equipped an agent with a set of six policies and decreased \u03b6 in a stepwise manner, producing high-, moderately decreased, and low-precision conditions. With the full policy space (Figure 6, top row), the agent inferred a policy that induced no hallucinations. It did begin to stop hearing its own voice at one time at the lowest likelihood precision; this was enabled by the sensory attenuation associated with self-generated speech and by the way in which our policies are set up to be oppositional (which contributed to the attenuation). To illustrate the mechanisms by which certain policy spaces predispose to hallucinations, we \u201clesioned\u201d the policy space by deleting the policy originally inferred by the agent (Figure 6, bottom row). This left policies that are worse explanations for active exchange with the same environment. The subject with the remaining five policies did not hallucinate at a high \u03b6 but did hallucinate when \u03b6 was lowered. This followed a dose\u2013response relationship\u2014at a moderately decreased \u03b6, the subject was confused about the state she was in but did not experience frank hallucinations. The lesioned policy space left the agent unable to appeal to the policy that better accounted for evidence in the auditory environment. When she was not able to use sensory information to correct her beliefs\u2014about the actions to pursue (the low likelihood precision condition)\u2014she was forced to infer a policy that led to a hallucination (Figure 6, bottom left). The subject hallucinated at the third time step because of the following sequence of events. At the second time step, she used the relatively precise information from the proprioceptive domain (which is not affected by the reduced likelihood precision in the auditory domain) to infer that she was speaking. Given that she was not speaking at the third time step, this left two plausible policies that allowed for speaking at the second and not speaking at the third time steps. Both of these mandated listening to a voice at the third time step, even though none was present. The subject was unable to use the imprecise auditory information to correct the prior engendered by the policy. The key point of these results is that different policy spaces are prone to different kinds of false inference under low likelihood precision\u2014while false inference may occur in any policy space when the likelihood precision has been reduced, given the same environment and initial conditions, different policy spaces produce different kinds of false inference.\n\nFigure 6.\n\nContext sensitivity of synthetic hallucinations. At high likelihood precision, outcomes dominate state inference; at low likelihood precision, priors derived from policies dominate and hallucinations can occur in a policy-dependent manner. This figure demonstrates that only certain policy spaces can lead to hallucinations, and only in the presence of permissively low likelihood precision. When the agent has a relatively large policy space (top row) and likelihood precision is reduced (left to right likelihood precision \u03b6 = 0.7, 0.525, 0.3), the agent correctly infers the presence of externally generated speech. When likelihood precision is high (left), it believes that it is able to listen (third time step) while generating speech. This is inconsistent with the policies and indicates that precise sensory evidence dominates the empirical priors derived from policy selection. At a sufficiently low likelihood precision (right), beliefs about the policy dominate sensory evidence about states, and this causes our agent to believe that she was not listening, having selected the speak policy. This is permitted by the sensory attenuation we have associated with the speaking state and the oppositional nature of our policies (which contributed to the attenuation). On the plots of posterior probabilities of policies over time, we see that the agent has inferred that she is pursuing the second policy by the fourth time step. As such, the agent has concluded that this policy is the most likely given the data observed and the actions taken. If this policy is removed (bottom row), this agent now no longer has access to Policy 2 and has selected alternative policies. This poses no problem when likelihood precision is high (left), as sensory evidence again dominates the inference. When precision is low, however, this policy does not explain the data well, leading to a hallucination at the third time step (highlighted in red).\n\n## DISCUSSION\n\nWe simulated a free energy\u2013minimizing agent using a MDP formalism that engaged in a simple turn-taking conversation. We performed three in silico experiments on this synthetic subject to induce hallucinations (false positive inferences). We found that decreasing auditory precision, given a vulnerable policy space, could induce hallucinations, as could increasing the prior precision over policies in the presence of low likelihood precision. We found that the effect of both decreased likelihood precision and increased prior precision\u2014over policies\u2014was highly dependent on the policies available to the agent, with only a subset of policy spaces producing specifically false positive inference. Here we discuss our findings and relate them to possible underlying neurobiology.\n\n### Precision and Prior Beliefs\n\nOur first finding was that decreasing likelihood precision \u03b6 can lead to hallucinations\u2014in fact, no hallucinations in any of our simulations could occur without this deficit. \u03b6 affects the (likelihood) mapping between states and outcomes. By attenuating the precision of this mapping, sensory evidence has less influence on belief updating; in other words, perception is dissociated from sensation. This is consistent with attractor and neural network models that produce hallucinations via similar means (Adams et al., 2013; Hoffman & McGlashan, 2006). As we will see shortly, the effects of a reduced \u03b6 depend on the policy space; simply decreasing \u03b6 does not always produce false positive inference.\n\nOur second finding was that increasing the prior precision over policies also led to hallucinations, given reduced likelihood precision. This is because beliefs about policies (e.g., \u201cnow is my turn to listen\u201d) are a source of prior beliefs about states (e.g., \u201cI am hearing a voice\u201d), which may be imposed on perception. This emphasizes the conflict between likelihood (i.e., sensory) and policy (i.e., prior) precision. An agent with strong (precise) prior beliefs about the actions it will take hallucinates because it is unable to correct for this prior belief using imprecise sensory evidence. An interesting consequence of this is that the form false inferences take becomes highly dependent on the action plans entertained.\n\nOur third finding was that deleting the policy originally inferred by the agent to explain her actions (given the environment) leads her to hallucinate, given low likelihood precision; that is, the agent hallucinates when she does not have access to a model of the world that does not bias toward false positive inference and cannot use sensory information to correct false positive inferences under this poor model (note that the deleted \u201cbest\u201d model of the world shown here does not necessarily result in perfect inference; it is only a relatively better model). Our model differs from previous models by incorporating actions specified by the policies employed by our simulated subject. As such, reduced likelihood precision is not sufficient to generate a hallucination (see Figure 6), and we were able to generate policy spaces that did not support false positive inference at low likelihood precision. Crucially, it is the interaction between likelihood precision and beliefs about the policy that then impacts the state the agent infers. This can be stated succinctly as follows: Agents will respond to reduced likelihood precision in a policy space\u2013dependent manner.\n\nThe nature of this interaction is important. As in existing Bayesian accounts of hallucinations, our approach requires the imposition of (empirical) prior beliefs on perception. We propose a source for these empirical prior beliefs\u2014beliefs about actions. By bringing in policies, we show how pathological empirical priors (over states) can develop from beliefs about \u201cwhat would happen if I were to do that.\u201d In short, we suggest that the balance between prior beliefs and sensory evidence can be framed as a balance between the confidence in plans of action and in the sensory consequences of these actions. An interesting consequence of this is that the form false inferences take becomes highly dependent on the plans selected.\n\nThus we arrive at a conceptualization of auditory hallucinations as requiring some defect in policy space (a policy space vulnerable to hallucinations, potentially with an increased prior precision) and decreased likelihood precision, which renders it impossible for sensory evidence to prevent the emergence of a false positive percept. This fits comfortably with Bayesian accounts of auditory hallucinations as resulting from prior beliefs about hearing a sound dominating over sensory evidence that a sound is not present.\n\nPrevious empirical work supports the hypothesis that both clinical and nonclinical positive symptoms are associated with greater reliance on priors relative to sensory evidence (Cassidy et al., 2018; Powers et al., 2017; Teufel et al., 2015), in particular, priors about voices in those who hear voices (Alderson-Day et al., 2017).\n\nConversely, many other phenomena in schizophrenia have been related to an increased weighting of sensory evidence relative to prior beliefs\u2014the opposite imbalance (Adams et al., 2013). Reduced vulnerability to certain perceptual illusions, such as the hollow-mask illusion, is another key element of schizophrenia phenomenology and has been found in patients with positive symptoms (Notredame, Pins, Den\u00e8ve, & Jardri, 2014). Classically, perceptual illusions have been explained as obtaining when a prior belief (e.g., a bias toward detecting convex faces) dominates over sensory evidence (e.g., a concave face). This would seem to imply a weakening of priors in schizophrenia, allowing for a resistance to illusions. Indeed, Jardri and Den\u00e8ve (2013) used a belief propagation model (another Bayesian message-passing scheme) to show one way in which sensory evidence could come to dominate prior beliefs in the disorder. They proposed that a failure of interneurons to inhibit messages that have been passed up the hierarchy could lead to the content of those same messages being passed back down the hierarchy, in effect, mistaking sensory evidence for a prior belief and thus overweighting it. Likewise, an overweighting of sensory precision in a predictive coding hierarchy (as in Adams et al., 2013) would have the same effect. In addition to these purely perceptual tasks, an increased relative weighting of sensory evidence also accounts for sensorimotor (e.g., loss of smooth pursuit gain and failure to attenuate self-produced sensations; Shergill, Samson, Bays, Frith, & Wolpert, 2005), electrophysiological (e.g., diminished oddball responses), and belief-updating changes (Jardri & Den\u00e8ve, 2013) in schizophrenia. The apparent discrepancy between abnormally precise prior beliefs and a failure of sensory attenuation is resolved by noting that a failure to attenuate sensory precision does not preclude\u2014and may even lead to\u2014a relatively high prior precision (Adams et al., 2013). We now consider this in more detail.\n\nHow can these apparently conflicting findings be resolved? One potential explanation\u2014developed in Sterzer et al. (2018)\u2014notes that the cortical hierarchy is many levels deep and that prior beliefs in sensorimotor hierarchies are themselves contextualized by priors higher in the hierarchy. For example, the perceptual (predictive coding) hierarchies modeled by Adams et al. (2013), in which prior beliefs are relatively imprecise, essentially correspond to the likelihood matrix in the current model (i.e., perceiving a voice). Thus auditory sensations may be more vivid (increased sensory precision), but their content may be harder to resolve (decreased intermediate precision), and therefore they may be more easily dominated by beliefs about the speech narratives at a higher level. Indeed, increased sensory precision may give resulting hallucinations their realistic, \u201cout-loud\u201d quality.\n\nThe key element of both this and our previous modeling work is that for one to hallucinate, one\u2019s prior beliefs must be unconstrained by incoming sensory evidence at some level of perceptual synthesis; that is, (empirical prior) precision must be reduced somewhere in the perceptual hierarchy. Our previous (predictive coding) model of birdsong could hallucinate only when sensory precision was reduced. While some auditory hallucinations are vague and ill formed, many have clear speech content, which in Bayesian terms must reflect priors of some sort. The model in this article is an initial attempt to show how beliefs about dialog with other agents are good candidates for such priors. Indeed, a propensity for dialog was the only attribute of inner speech associated with AVH proneness in a large population sample (McCarthy-Jones & Fernyhough, 2011).\n\nOf course, there are numerous possible sources of priors for auditory hallucinations. Cluster analysis of phenomenological surveys indicates four different kinds (McCarthy-Jones et al., 2014): nonverbal, memories of a voice, one\u2019s own voice, and another\u2019s voice. So aside from prior beliefs about other agents, one\u2019s memories and inner speech are also likely sources of priors in AVH, and in those with psychosis, delusional beliefs are likely to interact with these mechanisms. The idea that one\u2019s own inner speech could contribute to AVH is long established and stems from the influential idea of a corollary discharge (i.e., descending predictions) failure in schizophrenia (Feinberg, 1978; Frith & Done, 1989). One difference between these accounts and current formulations\u2014in terms of precision\u2014is that in the case of motor passivity symptoms, an imprecise prediction is dominated by (precise) sensory evidence; in our AVH model, an imprecise likelihood is dominated by a prior belief. Clearly these failures of precision weighting may coexist in a deep hierarchy.\n\nIt is also realistic to suppose that delusional ideas may arise in circumstances of higher sensory precision (i.e., overly salient prediction errors causing unwarranted belief updates), but it is hard to think of fully formed delusions as anything but (high-level) prior beliefs held with undue precision. Delusions may also therefore arise from a similar pattern of altered precisions at different hierarchical levels. To address this empirically, it may be necessary to design tasks that can probe the precision beliefs at sensory, intermediate, and high hierarchical levels (Karvelis, Seitz, Lawrie, & Seri\u00e8s, 2018).\n\nOf course, other accounts could also be addressed. It is possible that precision imbalances are specific to different sensorimotor modalities\u2014although work in the visual domain seems to argue against this (Schmack et al., 2013; Teufel et al., 2015). It is also possible that it is easier for subjects with schizophrenia to learn \u201cempirical\u201d perceptual priors over short timescales (e.g., if greater variance permits greater belief updating) but then harder to maintain their precision over time.\n\n### Action and Planning\n\nThe current formulation distinguishes itself from previous models of AVH by incorporating action. In our model\u2014and in active inference more generally\u2014perception depends upon action, and what one perceives will depend on the balance between priors and the outcomes solicited by actions. As such, for a false positive inference to ensue, it is necessary for the policy that contributes to that inference to have a strong posterior probability, which is more likely to be the case if it has a greater prior probability. Both of our policy-related manipulations would have the effect of increasing this prior probability\u2014this is caused directly by increasing prior precision over policies and indirectly by reducing the number of competing policies.\n\nOur model\u2019s use of action differentiates it somewhat from the \u201cinner speech\u201d models described earlier, which posit that hallucinations occur via the misattribution of self-generated internal speech as being generated by others (Allen, Aleman, & McGuire, 2007). In our model, hallucinations are generated to satisfy expectations of external speech derived from active perception. This generation is perhaps more in line with the phenomenology of AVH, which do not always take the form of one\u2019s own thoughts that can be misattributed to another: They can have rich content that can take the form of conversations or that can have the grammatical structure of another\u2019s speech (i.e., heard as \u201cyou\u201d or \u201cshe\/he\u201d instead of \u201cI\u201d), and they can have their own personas (making them seem less like thoughts whose source was simply misattributed). That being said, as noted earlier, dialogic inner speech may certainly be one of the priors that drive AVH (McCarthy-Jones & Fernyhough, 2011). This may have implications for understanding the perceived loss of agency that often (but not always; McCarthy-Jones et al., 2014) accompanies AVH. We hope to explore this aspect in further work.\n\nThe inclusion of action, and the context of a dialog as opposed to the recognition of sequences of previously learned words, sets our model apart from Hoffman and McGlashan (2006). This is a key difference, because hallucinations can take the form of a conversational partner in schizophrenia. This indicates that the genesis and maintenance of hallucinations is unlikely to be solely sensory-perceptual in nature. The conversational aspect is naturally modeled using our approach, as it is a discrete time model, consistent with the sequential form of a conversation. A further attraction is that, as Dietrich and Markman (2003) argued, discrete representations are needed to allow for complex cognitive processes, such as categorization, which are likely relevant to psychotic phenomena.\n\n### False Positives and False Negatives\n\nInterestingly, Hoffman and McGlashan (2006) predicted\u2014based on their simulations\u2014that patients with hallucinations would fail to detect words at a higher rate than nonhallucinators. They confirmed this effect in a psychophysics study of hallucinating and nonhallucinating patients with schizophrenia-spectrum disorders as well as healthy controls. Furthermore, this effect was stronger when sensory stimuli were degraded (which can be regarded as an external modulation of the precision of the stimuli). Our model reproduces this latter effect\u2014lower likelihood precision led to speech detection errors (i.e., inferring \u201cnot listening\u201d when the true state was that sound was present) in several of our simulations.\n\n### Neurobiological Plausibility\n\nActive inference can be formulated in terms of neurobiologically plausible processes (Friston, FitzGerald, Rigoli, Schwartenbeck, & Pezzulo, 2016). We can therefore draw some tentative parallels between the requisite neuronal computations and the pathophysiology that may underlie hallucinations. Figure 7 shows the MDP model mapped onto a putative functional architecture. Here we have represented auditory outcomes, oa, in Wernicke\u2019s area. Wernicke\u2019s area is a key region for the recognition of sequences of phonemes as constituting words (Ardila, Bernal, & Rosselli, 2016) and, as such, is a good candidate for one of the earliest processing centers in the brain capable of representing the presence or absence of meaningful speech. Wernicke\u2019s area is connected via the arcuate fasciculus (which represents the likelihood mapping, Aa, for the auditory modality) to the inferior frontal gyrus (IFG), where auditory states sa may be inferred. IFG is an important part of the language network and has been suggested as a source of priors during speech recognition (Sohoglu, Peelle, Carlyon, & Davis, 2012) and the selection of semantic information (Grindrod, Bilenko, Myers, & Blumstein, 2008) and has been implicated in AVH in fMRI studies (Raij et al., 2009). In addition, a dynamic causal modeling study by \u0106ur\u010di\u0107-Blake et al. (2013) found reduced connectivity from Wernicke\u2019s to Broca\u2019s areas that correlated with patient AVH status, further supporting our model. Proprioceptive outcomes, op, are assigned to primary somatosensory cortex and map via Ap to proprioceptive states, sp, in laryngeal motor cortex (LMC; Simonyan & Horwitz, 2011). Expected states in IFG and LMC inform the selection of policies, as performed by the striatum, which is well recognized to be involved in action planning. Under a neuronal process theory associated with active inference, probability distributions over policies are usually represented in the striatum, connected to expected states in the cortex via cortico-striato-thalamo-cortical loops (Friston, Rosch et al., 2017). Policy precision is often considered to be the computational homolog of dopamine (Schwartenbeck et al., 2015) and is represented here by the ventral tegmental area (VTA). Likelihood or sensory precision \u03b6, which may represent the computational homolog of acetylcholine (Dayan & Yu, 2001; Vossel et al., 2014), is located in the nucleus basalis (Liu, Chang, Pearce, & Gentleman, 2015).\n\nFigure 7.\n\nPutative mapping of Markov decision process (MDP) model to neurobiology. This figure shows the MDP model mapped onto putative neurobiology. This mapping should not be taken too seriously but serves as an illustration of how our model may relate to underlying functional anatomy. Here we have placed auditory outcomes, oa, in Wernicke\u2019s area. Wernicke\u2019s is connected via the arcuate fasciculus to the inferior frontal gyrus (IFG). The arcuate fasciculus represents the likelihood mapping between outcomes and auditory states, sa, which are located in IFG. Proprioceptive outcomes, op, are located in primary somatosensory cortex and map to proprioceptive states (representation of whether or not one is speaking), sp, in laryngeal motor cortex (LMC). States in IFG and LMC inform the selection of policies assigned to the striatum; the blue lines represent corticostriatal connections between IFG and striatum and LMC and striatum. The nucleus basalis represents cholinergic signaling, and the likelihood precision \u03b6 modulates (blue arrow) the state\u2013outcome mapping in the auditory modality. The ventral tegmental area\/substantia nigra (VTA) represents dopaminergic signaling, encodes the prior precision over policies \u03b3, and modulates the striatum (light blue arrow).\n\nLet us consider the potential neurobiological correlates of our modulation of \u03b6, \u03b3, and \u03c0. Anticholinergic drugs like scopolamine can induce auditory hallucinations (Perry & Perry, 1995); muscarinic agonists have been shown, in small studies, to improve psychotic symptoms; and dysfunction of the muscarinic system is hypothesized to play a role in schizophrenia (for a review, see Raedler, Bymaster, Tandon, Copolov, & Dean, 2006). Thus there is some merit to the idea that the reduced \u03b6 in our model may represent a cholinergic defect. However, the integrity of gray (Ohi et al., 2016) and white matter (including the arcuate fasciculus) is compromised in schizophrenia (\u0106ur\u010di\u0107-Blake et al., 2015; Gavrilescu et al., 2010), as is synaptic efficacy and NMDA receptor function (Coyle & Tsai, 2004). These abnormalities are also candidates for reduced likelihood precision, as they may represent a failure to propagate ascending sensory information.\n\nIn psychosis, there is increased presynaptic synthesis and release of striatal dopamine (see Howes & Kapur, 2009). This increase in striatal dopamine release may correspond to an increase in precision of policies \u03b3, though this requires much more empirical validation.\n\nFinally, what is the potential significance in neurobiological terms of our manipulation of the policy space? Policies could be encoded in the cortex, selected in the striatum through cortico-striatal loops, and accessed by other regions of cortex via cortico-cortical connections. Synaptic loss within these areas or functional dysconnections between state- and policy-representing regions could effectively reduce the policy space (see Friston, Brown, Siemerkus, & Stephan, 2016, for a discussion of the disconnection hypothesis of schizophrenia). It is therefore interesting to note that reductions in frontal and temporal gray matter and reduced synaptic density are features of schizophrenic illness (for a review, see Faludi & Mirnics, 2011).\n\n### Limitations\n\nOne limitation of our model is that it assumes a dialogic structure. While this allowed us to produce an agent that can act instead of only perceiving prelearned stimuli, it does not necessarily reflect all cases of AVH\u2014as these do not always have a dialogic component. Real patients can hallucinate in many contexts, but our model produces hallucinations strictly in the context of an ongoing conversation. We acknowledge that our model is not a comprehensive model of communication or language, in that our agent chooses to speak or listen only as a function of the policy she infers and in reaction to her conversational partner\u2019s actions. She does not speak or listen to minimize uncertainty beyond the proprioceptive and auditory domains; that is, the agent does not use language to impart meaning or to satisfy prior preferences (i.e., asking questions to resolve uncertainty). Our agent therefore does not properly employ language but rather a simple form of turn-taking behavior that reflects aspects of language in a rudimentary way. Another limitation is that, for simplicity, we restricted our policy spaces to be oppositional\u2014an agent could only choose to listen or to speak, but not both. However, the purpose of the simulation was to demonstrate the importance of active perception, not to comment on specific sequences of inferred actions that may exist in vivo. In addition, we only employed two states, and it is unclear if our findings would generalize to agents with higher dimensional state and policy spaces. Our model simulates the emergence of hallucinations given overly strong priors and a permissively low likelihood precision. While this may be attractive for disorders like schizophrenia, it may be less appropriate for the description of hallucinations in patients losing their sight (Charles Bonnet syndrome) or hearing, where decreased likelihood precision caused by a dysfunctional sensory apparatus allows priors to dominate (without there being any abnormality of the priors), leading to hallucinations (Friston, 2005). As such, this model represents only one way in which false positive inferences can be generated.\n\n### Future Directions\n\nIn the future, we hope to produce an agent that can modulate its own policy space, perhaps under the influence of affective or memory-related cues from a simulated medial temporal lobe. This would help to explain how an agent might sculpt a hallucinogenic policy space, perhaps when constrained to reduce that space over time to simulate overpruning. We hope to construct models with more complex state spaces to simulate language or conversations. A more complete model may also allow us to investigate the content of hallucinations and how they respond to context. We hope to design tasks that test model predictions that could be completed during fMRI to probe neurobiological correlates.\n\n## CONCLUSION\n\nWe have simulated hallucinations using a Markov decision process, under an active inference framework. Hallucinations, defined as false positive inferences, emerged with decreased likelihood precision, combined with a high prior precision over policies or a hallucinogenic policy space. In other words, hallucinations occurred when aberrant but strongly held priors over policies entailed predictions about sensory states that could not be corrected because of imprecise sensory information. This leads to a \u201cprecise (prior) belief, imprecise (sensory) evidence\u201d view of AVH. Agents that hallucinate do so because they believe that only certain sequences of events are likely, and they are unable to use sensory information to update these beliefs.\n\n## ACKNOWLEDGMENTS\n\nAll simulations were run using the DEM toolbox included in the SPM12 package. This is open-access software provided and maintained by the Wellcome Trust Center for Neuroimaging and can be accessed here: http:\/\/www.fil.ion.ucl.ac.uk\/spm\/software\/spm12\/.\n\n## AUTHOR CONTRIBUTIONS\n\nDavid A. Benrimoh: Conceptualization: Lead; Software: Equal; Visualization: Equal; Writing\u2014original draft: Lead; Writing\u2014review & editing: Equal. Thomas Parr: Conceptualization: Equal; Software: Equal; Supervision: Equal; Visualization: Equal; Writing\u2014review & editing: Supporting. Peter Vincent: Conceptualization: Supporting; Writing\u2014review & editing: Supporting. Rick A. Adams: Conceptualization: Supporting; Supervision: Equal; Writing\u2014review & editing: Equal. Karl Friston: Methodology: Lead; Software: Supporting; Supervision: Equal; Writing\u2014review & editing: Supporting.\n\n## FUNDING INFORMATION\n\nDB is supported by a Richard and Edith Strauss Fellowship (McGill University) and the Fonds de Recherche due Quebec\u2013Sant\u00e9 (FRQS). TP is supported by the Rosetrees Trust (Award Number 173346). RAA is supported by the Academy of Medical Sciences (AMS-SGCL13-Adams) and the National Institute of Health Research (CL-2013\u201318\u2013003). 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\section{Introduction}
Bosonic degrees of freedom arise generically and naturally in theories of fundamental physics, both in the Standard Model and beyond. The Higgs boson is of paramount importance, being the only fundamental scalar in the Standard Model \cite{Higgs:1964pj,Chatrchyan:2012xdj}, but many other scalar degrees of freedom have been proposed to extend particle physics to high energy scales. These include (among many others) the axion of QCD \cite{Peccei:1977hh} or the scalar which drives the expansion of the universe in quintessence models \cite{Ratra:1987rm}.
These bosonic particles often make good Dark Matter (DM) candidates as well. One reason for this is that unlike the Higgs, many of these new scalars would be stable or long-lived enough that they could coalesce into DM halos which constitute the seeds of galaxy formation. Unlike the usual collisionless cold DM picture, however, we are interested in the scenario where large collections of these bosons form bound states of macroscopic size due to their self-gravitation (and self-interaction generically). For this picture to be consistent, the scalars are taken to be sufficiently cold so that they may coalesce into a Bose-Einstein Condensate (BEC) state, and can thus be described by a single condensate wavefunction. These wavefunctions can indeed encompass an astrophysically large volume of space and have thus been termed ``boson stars'' \cite{Colpi:1986ye}.
It was shown many years ago that objects of this type are allowed by the equations of motion, first by Kaup \cite{Kaup:1968zz} and subsequently by Ruffini and Bonazzola \cite{Ruffini:1969qy} in non-interacting systems. They found a maximum mass for boson stars of the form $M_\text{max} \approx 0.633 M_\text{P}^2/m$, where $M_\text{P}=1.22\times10^{19}$ GeV is the Planck mass and $m$ is the mass of the individual bosons. (This is very different from the analogous limit for fermionic stars, termed the Chandrasekhar limit, which scales as $M_\text{P}^3/m^2$). Later, it was shown by Colpi et al. \cite{Colpi:1986ye} that self interactions in these systems can cause significant phenomenological changes. In particular, they examined systems with repulsive self-interactions, and show that the upper limit on the mass is $M_\text{max}\approx 0.02\sqrt{\lambda}M_\text{P}^3/m^2$, where $\lambda$ is a dimensionless $\phi^4$ coupling.\footnote{Note that the Colpi et al. result does not reduce to the Kaup bound as $\lambda\rightarrow 0$ because the former is derived by rescaling the equations of motion and dropping higher-order terms in the strong coupling limit, as we see in Section \ref{BosonSec}.}
This extra factor of $M_\text{P}/m$ as compared to the noninteracting case makes it more plausible that boson stars can have masses even larger than a solar mass. A different method of constraining the boson star parameter space, which fits the coupling strength using data from galaxy and galaxy cluster sizes, has been considered in \cite{Souza1,Souza2}.
The situation for attractive self-interactions is slightly more complex. The simplest case involves a self-interaction of the form $\lambda \phi^4$, where $\lambda<0$ for attractive interactions. If this were the highest-order term in the potential, then it would not be bounded below, and so one typically stabilizes it by the addition of a positive $\phi^6$ term. We will assume that the contribution of such higher-order terms is negligible phenomenologically (we address the validity of that assumption in Section \ref{3.2}). Furthermore, in this scenario the typical sizes of gravitationally bound BEC states is significantly smaller than the repulsive or non-interacting cases. This is because the only force supporting the condensate against collapse comes from the uncertainty principle. Gravity and attractive self-interactions tend to shrink the condensate. We will see in Section \ref{BosonSec} that the maximum mass for an attractive condensate scales as $M_\text{max} \sim M_\text{P}/\sqrt{|\lambda|}$. This result was originally derived using an approximate analytical method \cite{Chavanis:2011zi}, and was later confirmed by a precise numerical calculation \cite{Chavanis:2011-2}.
DM self-interactions have already been proposed and studied in different contexts~\cite{Spergel:1999mh,Wandelt:2000ad,Faraggi:2000pv,Mohapatra:2001sx,Kusenko:2001vu,Loeb:2010gj,Kouvaris:2011gb,
Rocha:2012jg,Peter:2012jh,Vogelsberger:2012sa,Zavala:2012us,Tulin:2013teo,Kaplinghat:2013xca,Kaplinghat:2013yxa,Cline:2013pca,
Cline:2013zca,Petraki:2014uza,Buckley:2014hja,Boddy:2014yra,Schutz:2014nka}. One of the main reasons why DM self-interactions can play an important role is due to the increasing tension between numerical simulations of collisionless cold DM and astrophysical observations, the resolution of which (for the moment) is unknown. The first discrepancy, known as the ``cusp-core problem'', is related to the fact that dwarf galaxies are observed to have flat density profiles in their central regions \cite{Moore:1994yx,Flores:1994gz}, while N-body simulations predict cuspy profiles for collisionless DM \cite{Navarro:1996gj}. Second, the number of satellite galaxies in the Milkly Way is far fewer than the number predicted in simulations \cite{Klypin:1999uc,Moore:1999nt,Kauffmann:1993gv,Liu:2010tn,Tollerud:2011wt,Strigari:2011ps}. Last is the so-called ``too big to fail'' problem: simulations predict dwarf galaxies in a mass range that we have not observed, but which are too large to have not yet produced stars \cite{BoylanKolchin:2011de}.
The solution of these problems is currently unknown, but a particularly well-motivated idea involves self-interacting DM (SIDM). Simulations including such interactions suggest that they have the effect of smoothing out cuspy density profiles, and could solve the other problems of collisionless DM as well \cite{Dave:2000ar, Vogelsberger:2013eka, Rocha:2012jg}. These simulations prefer a self-interaction cross section of $0.1$ cm$^2$/g $\lesssim \sigma/m \lesssim 10$ cm$^2$/g. There are, however, upper bounds on $\sigma/m$ from a number of sources, including the preservation of ellipticity of spiral galaxies \cite{Feng:2009mn,Feng:2009hw}. The allowed parameter space from these constraints nonetheless intersects the range of cross sections which can resolve the small-scale issues of collisionless DM, in the range $0.1$ cm$^2$/g $\lesssim \sigma/m \lesssim 1$ cm$^2$/g.
Self-gravitation and additionally extra self-interactions among DM particles can lead in some cases to the collapse of part of the DM population into formation of dark stars. The idea of DM forming star-like compact objects is not new. Dark stars that consist of annihilating DM might have existed in the early universe~\cite{Spolyar:2007qv,Freese:2008hb,Freese:2008wh}. Dark stars have been also studied in the context of hybrid compact stars made of baryonic and DM~\cite{Leung:2011zz,Leung:2013pra,Tolos:2015qra,Mukhopadhyay:2015xhs} as well as in the context of mirror DM~\cite{Khlopov:1989fj,Silagadze:1995tr,Foot:1999ex,Foot:2000iu}. Additionally some of the authors of the current paper studied the possibility of dark star formation from asymmetric fermionic DM that exhibits Yukawa type self-interactions that can alleviate the problems of the collisionless cold DM paradigm~\cite{Kouvaris:2015rea}. Unlike the dark stars of annihilating DM, asymmetric dark stars can be stable and observable today. \cite{Kouvaris:2015rea} displays the parameter space where it is possible to observe such dark stars, providing mass radius relations, corresponding Chandrasekhar mass limits and density profiles. Self-interactions in dark stars have also been considered in \cite{Schaffner-Bielich_Fermions} for fermionic particles, as well as in \cite{Schaffner-Bielich_Bosons} for bosonic ones.
In this paper we examine the dark stars composed of asymmetric self-interacting bosonic DM. The study is fundamentally different from that of ~\cite{Kouvaris:2015rea} because unlike the case of fermionic DM where the stability of the star is achieved by equilibrium between the Fermi pressure and gravitation, bosonic DM does not have a Fermi surface. They form a BEC in the ground state and it is the uncertainty principle that keeps the star from collapsing. We are going to demonstrate how DM self-interactions affect the mass radius relation, the density profile and the maximum mass of these DM bosonic stars in the context of the self-interactions that reconcile cold DM with the observational findings.
Note that we set $\hbar=c=1$ in what follows.
\section{SIDM parameter space} \label{AstroSec}
As we mentioned above, galactic scale $N$-body simulations of cold, non-interacting DM indicate that the central regions of galaxies should have a ``cuspy'' density profile, contrary to the cored profiles one observes. This, along with the ``missing satellites'' and ``too big to fail'' problems, has led some to question the non-interacting DM paradigm. While some believe that the inclusion of baryonic physics could alleviate these issues \cite{Oh:2010mc,Brook:2011nz,Pontzen:2011ty,Governato:2012fa}, it remains an open question. On the other hand, the inclusion of self-interactions in the DM sector could resolve these issues without creating tension with other astrophysical constraints. These two conditions can be simultaneously satisfied if the cross section per unit mass for DM satisfies
\begin{equation} \label{csConstraint}
0.1 \frac{\text{cm}^2}{\text{g}} \lesssim \frac{\sigma}{m} \lesssim 1 \frac{\text{cm}^2}{\text{g}}.
\end{equation}
Assuming a velocity independent cross section, \cite{Rocha:2012jg} found that $\sigma/m = 1$ cm$^2$/g tends to over-flatten dwarf galaxy cores and that it is marginally consistent with ellipticity constraints of the Milky Way. On the other hand a value of $0.1$ cm$^2$/g satisfies all constraints and flattens dwarf galaxy cores sufficiently.
Let us consider a potential of the form
\begin{equation}
V(\phi) = \frac{m^2}{2}\phi^2 + \frac{\lambda}{4!} \phi^4.
\end{equation}
Note that $\lambda>0$ ($\lambda<0$) signifies a repulsive (attractive) interaction. The resulting DM-DM scattering cross-section is
\begin{equation}
\sigma = \frac{\lambda^2}{64\pi m^2}
\label{eq:sigmaSC}
\end{equation}
at tree level. Plugging this into Eq. (\ref{csConstraint}), we get the constraint
\begin{equation}\label{bounds}
\Big(\frac{m}{1\text{ MeV}}\Big)^{3/2} < \frac{|\lambda|}{10^{-3}}
< 3 \Big(\frac{m}{1\text{ MeV}}\Big)^{3/2}.
\end{equation}
This matches the results of \cite{AmaroSeoane:2010qx}. For perturbativity, we should restrict $\lambda \lesssim 4\pi$, which would imply that our results are valid only for $m\lesssim 100$ MeV. In this mass range, it is plausible that these DM particles coalese into boson stars at some point in early cosmology.
If a large fraction of DM is contained inside boson stars, the derived parameter space may be significantly altered \cite{Enqvist:2001jd}, since boson star-DM interactions and boson star self-interactions may become significant. We will however assume that boson stars are rather scarce and the DM self-interactions are dominated by DM-DM scattering.
\subsection{DM scattering with boson stars}
To quantify how scarce boson stars have to be within this approximation, we assume that boson stars have a characteristic radius $R$, mass $M$ and number density $n_\text{BS}$. The mass, number density and self-interaction cross section of free DM is taken to be $m$, $n$ and $\sigma$. The mean free path a DM particle travels before hitting another DM particle or a boson star will be $\lambda_{DM}= (n \sigma)^{-1} $ and $\lambda_\text{BS} \sim (n_\text{BS} \pi R^2)^{-1} $, respectively. Scattering with boson stars has to be much rarer than with other free DM in our approximation. Therefore we require $\lambda_\text{DM} \ll \lambda_\text{BS}$. For the DM density we use the typical value of the solar system, i.e. $\rho_\text{DM} = M n_\text{BS} + m n \approx 0.3 \text{GeV/cm}^3$. These requirements lead to the following condition
\begin{equation}
n_\text{BS} \ll \frac{\sigma \rho_\text{DM}}{m\pi R^2+M\sigma}.
\label{eq:BSnumberdensity}
\end{equation}
Taking self-interactions to be that of Eq.~(\ref{eq:sigmaSC}), and the boson star radius to be comparable to the minimum radius (which scales the same for both signs of interaction)
$R \sim \sqrt{|\lambda|} M_\text{P}/m^2$ (see Eq.~(\ref{R_Att})),
Eq.~(\ref{eq:BSnumberdensity}) becomes
\begin{equation}
n_\text{BS} \ll \frac{\rho_\text{DM}}{64 \pi^2\tfrac{M_\text{P}^2}{|\lambda| m} + M}.
\label{eq:BSnumberdensity2}
\end{equation}
The maximum mass of a boson star with non-negligible attractive interactions is $\sim M_\text{P}/\sqrt{|\lambda|}$. Since this scaling is only proportional to a single power of $M_\text{P}$, the first term in the denominator of Eq.~(\ref{eq:BSnumberdensity2}) tends to dominate. We obtain in the attractive scenario
\begin{equation}
n_\text{BS}^\text{att} \ll \frac{|\lambda| m\rho_\text{DM}}{64 \pi^2M_\text{P}^2 } \approx 2\times 10^{-5} |\lambda| \frac{m}{\text{MeV}} \text{AU}^{-3},
\end{equation}
where AU is an astronomical unit. The minimum mean distance between attractive boson stars can therefore within this approximation be $(n_\text{BS}^\text{att})^{-1/3} \approx 40 (|\lambda| m/\text{MeV})^{-1/3} \text{AU}$.
In the scenario with repulsive interactions the maximum mass scales as $\sqrt{\lambda} M_\text{P}^3/m^2$. Therefore the second term in the denominator of Eq.~(\ref{eq:BSnumberdensity2}) dominates. The number density must satisfy
\begin{equation}
n_\text{BS}^\text{rep} \ll \frac{m^2\rho_\text{DM}}{\sqrt{\lambda} M_\text{P}^3} \approx 9\times 10^{-9} \lambda^{-1/2}\left( \frac{m}{\text{MeV}}\right)^2 \text{pc}
^{-3}.
\end{equation}
The minimum mean distance between repulsive boson stars which leaves our approximation valid can at most be $(n_\text{BS}^\text{rep})^{-1/3} \approx 5 \times 10^2 \lambda^{2/3} (m/\text{MeV})^{-2/3} \text{pc}$.
\section{Bosonic Dark Matter} \label{BosonSec}
An important property of light scalar particles that has been examined extensively in the literature \cite{Matos:2008ag,Suarez:2011yf} is that large collections (particle number $N\gg1$) can transition to a BEC phase at relatively high temperature, as compared to terrestrial experiments with cold atoms. The critical temperature for condensate occurs when the de Broglie wavelength is equal to the average interparticle distance, $\lambda_\text{dB}=[\zeta(3/2)/n]^{1/3}$, where $n$ is the average number density of the particles and $\zeta(x)$ is the Riemann Zeta function. This implies a critical temperature for transition to the BEC phase of the form
\begin{equation}
k T_\text{c} = \frac{2\pi}{m}\Big(\frac{n}{\zeta(3/2)}\Big)^{2/3}.
\end{equation}
In this paper, we will assume that all relevant scalar field particles are condensed, i.e. that the system is in its ground state, a perfect BEC. The effect of thermal excitations is examined in \cite{Harko:2011dz} and they are expected to be negligible as long as $T<T_\text{c}$ is satisfied.
\subsection{Non-Interacting Case}
It is instructive to begin with the case of boson stars bound only by gravity, first analyzed in \cite{Kaup:1968zz}. In this seminal work, Kaup considers the free field theory of a complex scalar in a spacetime background curved by self-gravity. The equations of motion\footnote{The non-interacting equations of motion are equivalent to Eq.s (\ref{EKG_Colpi}) and (\ref{Metric_Colpi}) in the limit $\Lambda\rightarrow0$.} were solved numerically. The maximum mass of these solutions was found to be $M_{max}\approx 0.633 M_P^2/m$, the oft-quoted Kaup limit for non-interacting boson stars. This value was later confirmed by Ruffini and Bonazzola \cite{Ruffini:1969qy}, who used a slightly different method by taking expectation values of the equations of motion in an $N$-particle quantum state.
Interacting field theories are more complex. In particular, for cross sections satisfying Eq. (\ref{csConstraint}), the phenomenology of repulsive and attractive interactions are very different, and accordingly, the methods required to analyze them are different as well. We outline the relevant methods in the sections below.
\subsection{Repulsive Interactions}
If the self-interaction is repulsive, we can make use of the result of Colpi et al. \cite{Colpi:1986ye}. Like Kaup, their method begins with the relativistic equations of motion for a boson star, the coupled Einstein and Klein-Gordon equations, but including a self-interaction term represented by $\Lambda$:
\begin{align} \label{EKG_Colpi}
\frac{A'}{A^2 x} + \frac{1}{x^2}\left(1-\frac{1}{A}\right)
&= \left(\frac{\Omega^2}{B}+1\right)\sigma^2 + \frac{\Lambda}{2}\sigma^4
+ \frac{(\sigma')^2}{A} \nonumber \\
\frac{B'}{B^2 x} + \frac{1}{x^2}\left(1-\frac{1}{A}\right)
&= \left(\frac{\Omega^2}{B}+1\right)\sigma^2 - \frac{\Lambda}{2}\sigma^4
+ \frac{(\sigma')^2}{A} \nonumber \\
\sigma'' + \left(\frac{2}{x} + \frac{B'}{2B} - \frac{A'}{2A}\right)&\sigma'
+ A \left[\left(\frac{\Omega^2}{B}-1\right)\sigma
- \Lambda\sigma^3\right] = 0,
\end{align}
where the rescaled variables are $x=mr$, $\sigma=\sqrt{4\pi G}\Phi$ ($\Phi$ the scalar field), $\Omega = \omega/m$ ($\omega$ the particle energy), and $\Lambda=\lambda M_\text{P}^2/(4\pi m^2)$. In addition to the scalar field itself, $A(r)$ and $B(r)$ must be solved for; these represent the deviations from the flat metric due to the self-gravity of the condensate,
\begin{equation} \label{Metric_Colpi}
ds^2 = -B(r) dt^2 + A(r) dr^2 + r^2 d\Omega^2.
\end{equation}
In practice, one can trade the metric function $A(r)$ for the mass $\mathcal{M}(x)$ by the relation $A(x)=[1-2\mathcal{M}(x)/x]^{-1}$. In the limit that the interactions are strong (precisely, $\Lambda \gg 1$), the system can be simplified significantly, as one can perform a further rescaling of the equations: $\sigma_* = \sigma\Lambda^{1/2}$, $x_*=x\Lambda^{-1/2}$, and $\mathcal{M}_*=\mathcal{M}\Lambda^{-1/2}$. The relevant parameters of Section \ref{AstroSec} suggest a value of $\Lambda = \mathcal{O}(10^{40})$ or higher, so it is completely safe to neglect terms proportional to $\Lambda^{-1}$. In this limit the equations simplify to
\begin{align}
\sigma_* &= \sqrt{\frac{\Omega^2}{B}-1} \nonumber \\
\mathcal{M}_*' &= 4\pi x_*^2\rho_* \nonumber \\
\frac{B'}{Bx_*}\left(1-\frac{2\mathcal{M}_*}{x_*}\right)
&-\frac{2\mathcal{M}_*}{x_*^3}=8\pi p_*,
\end{align}
where the pressure $p_*$ and density $\rho_*$ are given by
\begin{align}
\rho_* &= \frac{1}{16\pi}\left(\frac{3\Omega^2}{B}+1 \right)
\left(\frac{\Omega^2}{B}-1 \right) \nonumber \\
p_* &= \frac{1}{16\pi}\left(\frac{\Omega^2}{B}-1 \right)^2 .
\label{Eq:EoS}
\end{align}
In this limit, the equations do not depend on $\Lambda$, and one finds numerically that there is a maximum (dimensionless) mass $\mathcal{M}_{*\text{max}}\approx 0.22$. Restoring the appropriate dimensions, one finds
\begin{equation} \label{mc_att}
M < M_\text{max}^\text{rep} = 0.22\sqrt{\frac{\lambda}{4\pi}}\frac{M_\text{P}^3}{m^2}.
\end{equation}
This bound on the mass of repulsive boson stars was confirmed very precisely using a hydrodynamic approach as well \cite{Chavanis-GR}.
Figures \ref{Colpi_MassRad_plot} and \ref{Colpi_Density_plot} show the mass-radius relation and selected density profiles, respectively. The branch to the left of the peak in Figure \ref{Colpi_MassRad_plot} represents unstable equilibria, where the ground state energy is higher than the equilibrium on the right branch with the same number of particles (and thus the same quantum numbers).
\begin{figure}[htc]
\centering
\begin{minipage}{0.46\textwidth}
\centering
\includegraphics[width=.9\textwidth]{Repulsive_Mass_Radius.pdf}
\caption{The mass-radius relation for a boson star with strong repulsive coupling. The 3 circles correspond to the density profiles in Figure \ref{Colpi_Density_plot}. The dimensionless variables in the plot are defined in terms of the dimensionful ones as $M_*= m M^2\Lambda^{-1/2}/M_\text{P}$ and $X_*= mR\Lambda^{-1/2}$.}
\label{Colpi_MassRad_plot}
\end{minipage}\hfill
\begin{minipage}{0.46\textwidth}
\centering
\includegraphics[width=.9\textwidth]{Repulsive_Density_Profiles.pdf}
\caption{Three examples of density profiles in the case of repulsive interactions. The red profile corresponds to the profile of the maximum mass equilibrium, while the blue and green are taken on the stable branch of equilibria. The dimensionless variables in the plot are defined in terms of the dimensionful ones as $\rho_*$ defined in Eq. (\ref{Eq:EoS}) and $x_*=mr\Lambda^{-1/2}$.}
\label{Colpi_Density_plot}
\end{minipage}
\end{figure}
If we take the allowed range of $\lambda$ to be given by Eq. (\ref{bounds}), then we find the following range for $M_\text{max}^\text{rep}$:
\begin{equation}
\Big(\frac{1 \text{ MeV}}{m}\Big)^{5/4} 3.42\times10^{4} M_{\odot}
\lesssim M_\text{max}^\text{rep}
\lesssim \Big(\frac{1 \text{ MeV}}{m}\Big)^{5/4} 6.09\times10^{4} M_{\odot},
\end{equation}
where $M_{\odot}=1.99\times10^{30}$ kg is the solar mass. The range of masses allowed by these inequalities are represented in Figure \ref{M_Rep_plot}. Because of the strength of the repulsive interactions, these solutions can have masses several orders of magnitude above $M_\odot$. If there is a significant number of such objects in the Milky Way, it could have important observational signatures. However, a detailed analysis of the formation of these objects is required, in order to give some indication of whether DM boson stars in galaxies have masses close to the maximum value or lower.
\begin{figure}[htc]
\begin{center}
\includegraphics[width=6in]{repulsiveMaxM_blue.pdf}
\caption{The maximum mass of a boson star with \emph{repulsive} self-interactions satisfying Eq. (\ref{bounds}), as a function of DM particle mass $m$. The green band is the region consistent with solving the small scale problems of collisionless cold DM. The blue region represents generic allowed interaction strengths (smaller than $0.1$ cm$^2$/g) extending down to the Kaup limit which is shown in black. The red shaded region corresponds to $\lambda\gtrsim4\pi$. Note that the horizontal axis is measured in solar masses $M_\odot$.}
\label{M_Rep_plot}
\end{center}
\end{figure}
\subsection{Attractive Interactions}\label{3.2}
If DM self-interactions are attractive, then the method of \cite{Colpi:1986ye} does not apply. However, assuming relativistic corrections are negligible, one can instead solve the nonrelativistic equations of motion numerically and analyze the solutions. To be precise, the dynamics of a dilute, nonrotating BEC are governed by the Gross-Pit\"aevskii equation for a single condensate wavefunction $\phi(r,t)=\psi(r)e^{-iEt}$ \cite{Boehmer:2007um}
\begin{equation}\label{GP}
E \psi(r) = \Big(-\frac{\vec{\nabla}^2}{2m} + V(r) + \frac{4\pi a}{m}|\psi(r)|^2 \Big)\psi(r)
\end{equation}
where $V$ is the trapping potential, which in our case is the gravitational potential of the BEC and satisfies the Poisson equation
\begin{equation} \label{Poisson}
\vec{\nabla}^2 V(r) = 4 \pi G m \rho(r).
\end{equation}
The s-wave scattering length $a$ is related to a dimensionless $\phi^4$ coupling $\lambda$ by $a=\lambda/(32\pi m)$.
Here, $\rho(r) = m\cdot n(r) = m\cdot |\psi(r)|^2$ is the mass density of the condensate, which is normalized such that $\int{d^3 r \rho(r)} = M$, the total mass. The three terms on the right-hand side of Eq. (\ref{GP}) correspond to the kinetic, gravitational, and self-interaction potentials, respectively. As our notation signifies, we will assume that the density function is spherically symmetric, i.e. $\rho(\vec{r}) = \rho(r)$, which should be correct for a ground state solution.
Because the Gross-Pit\"aevskii + Poisson system (hereafter GP, defined by Eqs. (\ref{GP}) + (\ref{Poisson})) cannot be solved analytically in general, we use a shooting method to integrate the system numerically over a large range of parameters. As boundary conditions, we choose the values of $\psi(0)$ and $V(0)$ so that both functions are regular as $r\rightarrow 0$, and so that asymptotically $\psi(r)\rightarrow 0$ and $r V(r)\rightarrow 0$ exponentially as $r\rightarrow\infty$. Some examples of integrated density functions are given in Figure \ref{Att_Density_plot}. Our numerical procedure requires the following rescaling of the dimensionful quantities:
\begin{align}
\psi &= \sqrt{\frac{m}{4\pi G}}\frac{1}{|\tilde a|} \tilde\psi \qquad
V - E = \frac{m}{|\tilde a|} \tilde V \nonumber \\
a &= m G |\tilde a| \qquad \qquad \quad
r = \frac{\sqrt{|\tilde a|}}{m} \tilde r,
\end{align}
where the dimensionless quantities on the RHS are denoted with a tilde. The equations take the form
\begin{align}
\left[-\frac{1}{2}\tilde{\nabla}^2 + \tilde{V}
- |\tilde{\psi}^2|\right]\tilde{\psi} = 0 \nonumber \\
\tilde{\nabla}^2\tilde{V} = |\tilde{\psi}^2|,
\end{align}
where $\tilde{\nabla}$ denotes a gradient with respect to $\tilde{r}$, and we have explicitly taken $a<0$. These are the equations we solve. Similar rescaled equations were used in \cite{Guzman1}, but for repulsive interactions, and unlike \cite{Guzman1}, we also scale away the scattering length $a$. This makes our solutions valid for any generic $a<0$.
In Figure \ref{Att_MassRad_plot} we show the mass-radius relation for the bosonic stars, which agrees well with the results obtained in \cite{Chavanis:2011-2}. As in the repulsive case, there is a maximum mass for these condensates, but this mass is significantly smaller for attractive interactions. For parameters satisfying Eq. (\ref{bounds}), our analysis shows that condensates of this type would be light and very dilute, having masses $<1$ kg and radii $R\sim\mathcal{O}$(km). (Our assumption that the General Relativistic effects could be neglected in this case is therefore well supported a posteriori.)
\begin{figure}[htc]
\centering
\begin{minipage}{0.46\textwidth}
\centering
\includegraphics[width=0.9\textwidth]{Attractive_Mass-Radius.pdf}
\caption{The mass-radius relation for a boson star with attractive interactions. The three circles correspond to the density profiles in Figure \ref{Att_Density_plot}. The dimensionless variables in the plot are defined in terms of the dimensionful ones as $\displaystyle{\tilde{M}=\sqrt{\frac{\lambda}{32\pi}}\frac{M}{M_P}}$ and $\displaystyle{\tilde{R}_{99}=\sqrt{\frac{32\pi}{\lambda}}\frac{m^2}{M_P}R_{99}}$.}
\label{Att_MassRad_plot}
\end{minipage}\hfill
\begin{minipage}{0.46\textwidth}
\centering
\includegraphics[width=.9\textwidth]{Attractive_Density_Profiles.pdf}
\caption{Three examples of density profiles in the case of attractive interactions. The red profile corresponds to the profile of the maximum mass equilibrium, while the blue and green are taken on the stable branch of equilibria. The dimensionless variables in the plot are defined in terms of the dimensionful ones as $\displaystyle{\tilde{\rho}=\frac{\lambda}{m^4}\rho}$ and $\displaystyle{\tilde{r}=\sqrt{\frac{32\pi}{\lambda}}\frac{m^2}{M_P}r}$.}
\label{Att_Density_plot}
\end{minipage}
\end{figure}
One can arrive at a good, order of magnitude analytic estimate on the size and mass of condensates by a variational method which minimizes the total energy. To this end, we follow the approach of \cite{Chavanis:2011zi} by using the GP energy functional,
\begin{equation}\label{Energy}
E[\psi] = \int{d^3r\Bigg[\frac{|\vec{\nabla}\psi|^2}{2m} + V |\psi|^2} + \frac{2\pi a}{m}|\psi|^4\Bigg].
\end{equation}
As input, we choose an ansatz for the wavefunction $\psi(r)$, and subsequently compute the energy of the condensate by integrating Eq.~(\ref{Energy}) up to some maximum size $R$. Minimizing the energy with respect to $R$ should give a good estimate for the size of stable structures. Note that the gravitational potential $V(r)$ must be chosen self-consistently to satisfy Eq. (\ref{Poisson}) for a given choice of $\psi(r)$.
In order to illustrate the salient features of the method, we will choose a simple ansatz for the wavefunction:
\begin{equation} \label{Wavefunction}
\psi(r) =
\begin{cases}\sqrt{\frac{3N}{4\pi R^3}}e^{i r/R} &\text{if $r \leq R$,}\\
0 &\text{if $r>R$,}
\end{cases}
\end{equation}
which is normalized as above. Performing the energy integral gives the result
\begin{equation}
E = N \Big[\frac{A}{R^2} - \frac{B N}{R} + \frac{3A N a}{R^3}\Big],
\end{equation}
where $A\equiv 1/(2m)$ and $B\equiv 6 G m^2/5$. Minimizing $E(R)$ with respect to $R$ gives two critical points
\begin{equation}
R_\text{c} = \frac{A}{B N}\Big(1\pm \sqrt{1+\frac{9a}{A/B}N^2}\Big).
\end{equation}
In this calculation, a natural length scale $X\equiv A/B$ emerges. For any $a\neq 0$ (repulsive or attractive), the minimum of the energy lies at the solution with the ``+'' sign, i.e.
\begin{equation} \label{Radius}
R_0 = \frac{X}{N}\Big(1 + \sqrt{1+\frac{9a}{X}N^2}\Big).
\end{equation}
In the case of attractive interactions, there is a critical number of particles $N_\text{max} \equiv \sqrt{X/(9|a|)}$, above which the real energy minimum disappears and no stable condensate exists. Using $M_\text{max} = m N_\text{max}$, this analysis sets a value for the maximum mass for stable condensates with attractive interactions:
\begin{equation} \label{M_Att}
M < M_\text{max}^\text{att} = m \sqrt{\frac{X}{9|a|}} = \sqrt{\frac{320}{27}}\frac{M_\text{P}}{\sqrt{|\lambda|}}.
\end{equation}
The corresponding limit on the radius is a lower bound, attractive boson stars being stable only for
\begin{equation} \label{R_Att}
R > R_\text{min}^\text{att} = \sqrt{\frac{15}{16}|\lambda|}\frac{M_\text{P}}{m^2}.
\end{equation}
Note that while the coefficient depends on the details of the wavefunction ansatz, the scaling relations $M_\text{max}^\text{att}\sim M_\text{P}/\sqrt{|\lambda|}$ and $R_\text{min}^\text{att}\sim \sqrt{\lambda}M_\text{P}/m^2$ are completely generic.
Using Eq. (\ref{bounds}), we find
\begin{equation}
\Big(\frac{1 \text{ MeV}}{m}\Big)^{3/4} 7.37\times10^{-9} \text{ kg}
\lesssim M_\text{max}^\text{att}
\lesssim \Big(\frac{1 \text{ MeV}}{m}\Big)^{3/4} 1.31\times10^{-8} \text{ kg}
\end{equation}
The range of masses allowed by these inequalities is given by the green band in Figure \ref{MaxMass_Att}. We plot the maximum masses over many orders of magnitude, between $1$ eV and $1$ GeV, but the maximum mass of boson stars with such strong attractive self-interactions is still $<1$ kg.
\begin{figure}[htc]
\begin{center}
\includegraphics[width=6in]{attractiveMaxM_blue.pdf}
\caption{The maximum mass of a boson star with \emph{attractive} self-interactions satisfying Eq.~(\ref{bounds}), as a function of DM particle mass $m$. The green band is the region consistent with solving the small scale problems of collisionless cold DM. The blue region represents generic allowed interaction strengths (smaller than $0.1$ cm$^2$/g) extending up to the Kaup limit which is shown in black. The red shaded region corresponds to $\lambda\gtrsim4\pi$. Note that the horizontal axis is measured in grams.}
\label{MaxMass_Att}
\end{center}
\end{figure}
Note that the numerical results agree well with the predictions of the variational method to within an order of magnitude, even for the na\"ive constant density ansatz in Eq. (\ref{Wavefunction}). These estimates can be improved further by a more robust ansatz for the wavefunction.
As an example of a physical model, field theories describing axions exhibit an attractive self-coupling through the expansion of the axion potential $V(A)=m^2 f^2\Big(1-\cos(A/f)\Big)$, where $A$ is the axion field, $m$ is the axion mass, and $f$ is the axion decay constant. Gravitationally bound states, particularly in the context of QCD axions, have become the topic of much recent interest \cite{Barranco_2011,Eby_2015,Guth_2015}. These states typically have maximum masses of roughly $10^{-11} M_\odot$, far below the bounds set in this section, because the couplings are typically many orders of magnitude smaller.
As we pointed out in the introduction, in the case of attractive interactions the potential is unbounded from below since $\lambda < 0$. Therefore there must exist higher dimensional operators suppressed by some cutoff. The first irrelevant operator with a $\mathbb{Z}_2$ symmetry is $\phi^6/\mu_\text{c}^2$ where $\mu_\text{c}$ is the cutoff scale. We will now set a lower limit for $\mu_\text{c}$ by requiring that the $\phi^6$ term is negligible with respect to the $\phi^4$ term for typical boson star field values. Assuming that the kinetic energy of the field is negligible, the energy density is roughly equal to the potential. The maximum mass and minimum radius in Eqs. (\ref{M_Att}) and (\ref{R_Att}) can also be used to estimate the energy density as $\rho \approx M_\text{max}/R_\text{min}^3 \approx m^6/|\lambda|^2 M_\text{P}^2$. Now we can estimate the field value $\tilde{\phi}$ inside the boson star with attractive interactions to be
\begin{equation}
|\tilde{\phi}| \approx \frac{m}{\sqrt{2|\lambda|}} \left(1+ \left(1-\frac{4m^2}{ |\lambda| M_\text{P}^2}\right)^{\tfrac{1}{2}}\right)^{\tfrac{1}{2}} \approx \frac{m}{\sqrt{|\lambda|}}.
\end{equation}
Requiring $|\lambda|\tilde{\phi}^4 \gg \tilde{\phi}^6/\mu_\text{c}^2$ we obtain the inequality $\mu_\text{c} \gg m/ |\lambda|$.
\section{Conclusions}
In this paper we studied the possibility that self-interacting bosonic DM forms stars. We assumed that self-interactions are mediated by a $\lambda \phi^4$ interaction and we investigated what type of stars can be formed in the case of both attractive and repulsive self-interactions, giving particular emphasis to the parameter phase space of masses and couplings where the DM bosons alleviate the problems of collisionless DM. We have considered DM particles that populate the BEC ground state. We estimated the maximum mass where these dark stars are stable, the mass-radius relation and the density profile for generic values of DM mass and self-interacting coupling $\lambda$.
We leave several things for future work. The first and most important is the mechanism of formation for these bosonic dark stars. Sufficiently strong self-interactions can lead to the gravothermal collapse of part or the whole amount of DM to dark stars~\cite{Balberg:2002ue}. In this case, DM self-interactions can facilitate the formation of bosonic stars because DM particles get confined to deeper self-gravitating wells simply by expelling high energetic DM particles out of the core. As the core loses energy, the virial theorem dictates that the core shrinks and heats up the same time. This leads to further energy loss and thus to the gravothermal collapse. Such a scenario could also explain why the black hole at the center of the Galaxy is so heavy, since DM bosonic stars could provide the initial seed required for the further growth of the supermassive black hole~\cite{Pollack:2014rja}. It is interesting to note that boson stars can coexist in equilibrium with black holes, as shown in \cite{Herdeiro1,Herdeiro2}. One should also notice that if the whole density of DM collapses to dark stars, one does not have to be within the narrow band of parameter space depicted in Figures \ref{M_Rep_plot} and \ref{MaxMass_Att}. Another possibility is the creation of high DM density regions due to adiabatic contraction, caused by baryons~\cite{Blumenthal:1985qy,Gustafsson:2006gr}. Moreover, bosonic DM particles can get trapped inside regular stars via DM-nucleon collisions. The DM population is inherited by subsequent white dwarfs that, in case of supernovae 1a explosions, can leave the bosonic matter intact, either alone or with some baryonic matter~\cite{Kouvaris:2010vv}.
Asymmetric bosonic dark stars where no substantial number of annihilations take place will not be very visible in the sky, although present. Gravitational lensing could be one way to deduce the presence of such stars in the universe. Additionally, if the DM boson interacts with the Standard Model particles via some portal (e.g. kinetic mixing between a photon and a dark photon), thermal Bremmstrahlung could potentially produce an observable amount of luminosity. This is particularly interesting since such a photon spectrum would probe directly the density profile of the boson star. Bosonic stars could also disguise themselves as ``odd" neutron stars. For example, it is hard to explain sub-millisecond pulsars with typical neutron stars. XTE J1739-285 could possibly be such a case, since it allegedly rotates with a frequency of 1122Hz \cite{Kaaret:2006gr}. Compact enough bosonic stars would have no problem to explain such high rotational frequencies. Another possibility is the observation of compact stars with masses higher than the maximum mass a neutron star can support. Such might be the case of the so-called ``black widow" PSR B1957+20, with a mass of 2.4 solar masses~\cite{vanKerkwijk:2010mt}.
Therefore, abnormal neutron stars can well be the smoking gun for the existence of asymmetric dark stars either with fermionic constituents like~\cite{Kouvaris:2015rea}, or with the bosonic ones studied here.
{\bf Acknowledgements}\\
The research of C.K. and N.G.N. is supported by the Danish National Research Foundation, Grant No. DNRF90. J.E. is supported by a Mary J. Hanna Fellowship. L.C.R.W.'s research is partially supported by a faculty development award at UC. C.K. and L.C.R.W. acknowledge support by the Aspen Center for Physics through the NSF Grant PHY-1066293. J.E. and L.C.R.W. also acknowledge M. Ma, C. Prescod-Weinstein, and P. Suranyi for valuable discussions about Bose-Einstein Condensation.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 1,076 |
\section{Introduction}
In the study of quantum gravity one encounters many technical complications,
and it is often desirable to test one's ideas and tools in a simpler setting.
The NonLinear Sigma Models (NLSMs) have striking similarities to gravity:
they are nonpolynomially interacting theories, and from the point of view
of power counting, they have exactly the same structure as gravity.
On the other hand, they lack the complications due to gauge invariance.
They are therefore a good theoretical laboratory where one can study various
technical aspects of the renormalization of gravity without having to
consider the complications due to gauge fixing, and with the certainty
that one's results are not gauge artifacts.
Recent work on the beta functions of gravity suggests that there
might exist a nontrivial fixed point with finitely many UV attractive directions,
making this theory ``asymptotically safe''.
This means that if one considers all the terms in the derivative expansion
of the effective action, the corresponding (renormalized) couplings would
all run towards a Fixed Point (FP) in the UV limit, and that only finitely many
combinations of couplings would be relevant (attracted to the FP in the UV).
Then, the requirement of tending to the FP in the UV would constrain the theory
to lie on a finite dimensional surface, and the theory would then be predictive.
See the original work \cite{weinberg} for the definition of asymptotic safety,
\cite{reviews} for reviews and \cite{cpr,bms} for more recent results.
Understanding the UV behaviour of the NLSM may shed some light on
the analogous issue for gravity.
Aside from this,
the NLSMs also play an important role in particle physics phenomenology:
they are used as low energy effective field theories both for strong
and weak interactions. In the former case the scalar fields are identified with the
light mesons \cite{gl}, in the latter with the three Goldstone degrees of freedom
of the complex Higgs doublet \cite{ab}.
These effective field theories are usually thought to break down at some cutoff scale,
of the order of the GeV in the strong case and of the TeV in the weak case.
It is an interesting question in itself, and one that may have some relevance
also for particle physics, whether some of these NLSM's
might actually be asymptotically safe.
Old work on the epsilon expansion and $1/N$ expansion suggests that a fixed
point with the right properties may exist \cite{polyakov,zinnjustin,bardeen,arefeva}.
More recently, the beta functions of the NLSM were recalculated using
a two derivative truncation of an exact RG equation,
and it was found in the case of the $O(N)$ models
that they have a nontrivial UV FP \cite{codello2}.
In the present work we begin addressing the issue of asymptotic safety in the NLSM
taking into account also four derivative interactions.
The beta functions of four derivative NLSM were considered before in
\cite{hasenfratz} and \cite{bk}.
The former reference uses a formalism that applies only to group-valued models;
the latter uses dimensional regularization and therefore cannot properly compute the
running of the two derivative terms, which is necessary to establish asymptotic
safety. In this paper we extend and partly correct the results of these earlier
works.
This paper is organized as follows: in section 2 we discuss
the models and the techniques we use; in section 3 we evaluate the beta functions,
first for arbitrary manifolds and then for the $O(N)$ models and the chiral models;
in section 4 we list their fixed points;
in section 5 we close with a discussion.
The comparison with \cite{hasenfratz} is given in appendix A.
\section{The theory}
\subsection{Geometry and action}
In general the NLSM is a field theory whose configurations are maps
from $\varphi:X\to Y$, where $X$ is a $d$-dimensional manifold
interpreted as spacetime and $Y$ is some $n$-dimensional internal manifold.
We will always take $X$ to be four dimensional and to have a fixed flat Euclidean metric,
and we will call $h$ a riemannian metric on $Y$.
Given a map $\varphi$, one calls ``vectorfield along $\varphi$''
a rule that assigns to each point $x$ of $X$ a vector tangent to $Y$
at $\varphi(x)$.
\footnote{The vectorfields along $\varphi$ should be thought of, in geometrical terms,
as sections of the pullback bundle $\varphi^*TY$.}
For example, given a fixed vector $v$ tangent to $X$ at $x$,
the image of $v$ under the tangent map $T\varphi$ is a vectorfield along $\varphi$.
Its components are $v^\mu\partial_\mu\varphi^\alpha$.
Thus we can view the matrix $\partial_\mu\varphi^\alpha$ as the components of four
vectorfields along $\varphi$.
The Levi-Civita connection of the metric $h$ in $TY$
can be used to define the covariant derivative of vectorfields
along $\varphi$.
Let $\mit\Gamma_\alpha{}^\beta{}_\gamma$ be the Christoffel symbols of $h$ and
$R_{\alpha\beta}{}^\gamma{}_\delta=\partial_\alpha{\mit\Gamma}_\beta{}^\gamma{}_\delta
-\partial_\beta{\mit\Gamma}_\alpha{}^\gamma{}_\delta
+{\mit\Gamma}_\alpha{}^\gamma{}_\epsilon{\mit\Gamma}_\beta{}^\epsilon{}_\delta
-{\mit\Gamma}_\beta{}^\gamma{}_\epsilon{\mit\Gamma}_\alpha{}^\epsilon{}_\delta$
its Riemann tensor.
The covariant derivative of a vectorfields along $\varphi$ is
\begin{equation}
\label{covariant}
\nabla_\mu\xi^\alpha=
\partial_\mu\xi^\alpha+\partial_\mu\varphi^\gamma{\mit\Gamma}_\gamma{}^\alpha{}_\beta\xi^\beta
\end{equation}
A diffeomorphism $f$ of $Y$ can be represented in coordinates by $y'=f(y)$.
It maps vectorfields along $\varphi$ to vectorfields along $\varphi'=f\circ\varphi$.
One can check explicitly using the transformation properties
\begin{equation}
\xi'^\alpha={\partial\varphi^{\prime\alpha}\over\partial\varphi^\beta}\xi^\beta\ ;\qquad
{\mit\Gamma}'{}_\gamma{}^\alpha{}_\beta
={\partial \varphi^\eta\over\partial \varphi^{\prime\gamma}}
{\partial \varphi^{\prime\alpha}\over\partial \varphi^\delta}
{\partial \varphi^\epsilon\over\partial \varphi^{\prime\beta}}
{\mit\Gamma}_\eta{}^\delta{}_\epsilon
+{\partial \varphi^{\prime\alpha}\over\partial \varphi^\delta}
{\partial^2 \varphi^\delta\over\partial \varphi^{\prime\gamma}\partial \varphi^{\prime\beta}}
\end{equation}
that the covariant derivative transforms in the same way as $\xi$ under diffeomorphisms of $Y$.
We also note for future reference that the curvature of the pullback connection
is the pullback of the curvature of the Levi-Civita connection:
\begin{equation}
\label{curvature}
[\nabla_\mu,\nabla_\nu]\xi^\gamma
\equiv\Omega_{\mu\nu}{}^\gamma{}_\delta\xi^\delta
=\partial_\mu\varphi^\alpha\partial_\nu\varphi^\beta
R_{\alpha\beta}{}^\gamma{}_\delta \xi^\delta
\end{equation}
We can now discuss the dynamics of the NLSM.
Since the ordinary derivatives of $\varphi^\alpha$ are the components of vectorfields
along $\varphi$, the second covariant derivatives of the scalars are given by
\begin{equation}
\nabla_\mu\partial_\nu\varphi^\alpha=
\partial_\mu\partial_\nu\varphi^\alpha
+\partial_\mu\varphi^\beta{\mit\Gamma}_\beta{}^\alpha{}_\gamma
\partial_\nu\varphi^\gamma\ .
\end{equation}
Note that due to the symmetry of the Christoffel symbols
$\nabla_\mu\partial_\nu\varphi^\alpha=\nabla_\nu\partial_\mu\varphi^\alpha$.
We also define $\Box\varphi^\alpha=D^\mu\partial_\mu\varphi^\alpha$.
After these preliminaries, the most general Lorentz-- and parity--invariant
NLSM with up to four derivatives has an action of the form:
\begin{eqnarray}
\label{action}
&\frac{1}{2}\int d^4x\,\Bigl[&\!\!\!
\partial_{\mu}\varphi^\alpha\partial^\mu\varphi^\beta \g2_{\alpha\beta}(\varphi)
+\Box\varphi^\alpha \Box\varphi^\beta \g4_{\alpha\beta}(\varphi)
\nonumber
\\
&&+\nabla_\mu\partial_{\nu}\varphi^\alpha\partial^\mu\varphi^\beta\partial^\nu\varphi^\gamma
A_{\alpha\beta\gamma}(\varphi)
+\partial_{\mu}\varphi^\alpha\partial^\mu\varphi^\beta
\partial_{\nu}\varphi^\gamma\partial^\nu\varphi^\delta T_{\alpha\beta\gamma\delta}(\varphi)
\Bigr]
\ .
\end{eqnarray}
Here we defined parity to correspond to the reflection
$\varphi^\alpha(x_1,x_2,x_3,x_4)\mapsto \varphi^\alpha(-x_1,x_2,x_3,x_4)$.
This is the only parity operation one can define in full generality.
We will discuss below other ``parities'' that can be defined
on special manifolds.
At the classical level, $\g2$, $\g4$ $A$, and $T$ are fixed tensorfields on $Y$.
They represent, in general, an infinite number of interaction terms.
In the quantum theory these tensors will be subject to RG flow.
The tensors $\g2$, $\g4$ are assumed to be positive definite metrics.
In the present work we will always use $\g4$ to raise and lower indices,
while $\g2$ is treated as any tensor. Of course nothing ultimately can depend
on this convention.
The tensor $A$ can be assumed to be totally symmetric without loss of generality.
The tensor $T$ must have the following symmetry properties:
$$
T_{\alpha\beta\gamma\delta}=
T_{\beta\alpha\gamma\delta}=
T_{\alpha\beta\delta\gamma}=
T_{\gamma\delta\alpha\beta}\,.
$$
In \eqref{action} we have not considered (parity violating) terms that involve
the $\epsilon$ tensor, of the form
\begin{equation}
c\int d^4x\,\epsilon^{\mu\nu\rho\sigma}
\partial_\mu\varphi^\alpha\partial_\nu\varphi^\beta
\partial_\rho\varphi^\gamma\partial_\sigma\varphi^\delta
B_{\alpha\beta\gamma\delta}(\varphi)\,,
\end{equation}
where $B$ is some four-form on $Y$. These could be called ``Wess-Zumino-Witten terms''
in a generalized sense. A proper Wess-Zumino-Witten term is one for which the
four form $B$ is not defined everywhere on $Y$, but the five-form $H=dB$ is.
Then $H$ defines a nontrivial fifth-cohomology class and the coefficient $c$
has to obey a quantization condition.
The original Wess-Zumino term corresponds to the case $Y=SU(N)$
and $H=\mathrm{tr}(g^{-1}dg)^5$.
We will briefly return to these terms in the discussion.
We observe that since the field $\varphi$ appears nonpolynomially
in the action, it must be dimensionless.
Then, $\g2$ must have dimension of mass squared, whereas
the other tensors are dimensionless.
Later on we will find it convenient to split off a dimensionful coupling from
the dimensionful tensors, so that all the tensors are dimensionless.
We will be especially interested in cases in which the theory has some global
symmetries. Let $\Phi$ be a diffeomorphism of $Y$ that leaves the tensors
$\g2$, $\g4$, $A$, $T$ invariant, for example
$$
T_{\alpha\beta\gamma\delta}(y)=
\frac{\partial\Phi^{\alpha'}}{\partial y^\alpha}
\frac{\partial\Phi^{\beta'}}{\partial y^\beta}
\frac{\partial\Phi^{\gamma'}}{\partial y^\gamma}
\frac{\partial\Phi^{\delta'}}{\partial y^\delta}
T_{\alpha'\beta'\gamma'\delta'}(\Phi(y))\ .
$$
In particular, $\Phi$ is an isometry of $\g4$.
Then the action is invariant under the transformation $\varphi\mapsto \Phi\circ\varphi$.
Such global symmetries may be discrete, or they may form a continuous group $G$.
In the latter case there exist vector fields $K_a$ on $Y$ (with $a=1\ldots\mathrm{dim}G$)
whose Lie brackets form an algebra isomorphic to the Lie algebra of $G$,
and such that $\g2$, $\g4$, $A$, $T$ are invariant under $G$:
\begin{equation}
\nonumber
{\cal L}_{K_a}\g2=0\ ;\quad
{\cal L}_{K_a}\g4=0\ ;\quad
{\cal L}_{K_a}A=0\ ;\quad
{\cal L}_{K_a}T=0\ .\quad
\end{equation}
In particular, $K_a$ are Killing vectors for the metric $\g4$:
$\nabla_\alpha K_{a\beta}+\nabla_\beta K_{a\alpha}=0$.
Then, the action \eqref{action} is invariant under the
infinitesimal transformation $\delta_\epsilon\varphi^\alpha=\epsilon^a K_a^\alpha(\varphi)$.
Discrete isometries may appear in the definition of parity or time reversal.
In linear scalar theories one can define the operation $\phi\mapsto -\phi$.
For example the pions transform as
$(P\pi)^a(x_1,x_2,x_3,x_4)=-\pi^a(-x_1,x_2,x_3,x_4)$ under parity.
In a general NLSM the transformation
$\varphi^\alpha\mapsto -\varphi^\alpha$ has no intrinsic meaning.
However, suppose that every point $y\in Y$ is the fixed point
of an involutive isometry $\Phi_y$.
Such a manifold is said to be a symmetric space \cite{helgason}.
We can then define a new parity operation, let us call it ``Parity''
with capital P, by
$(P\varphi)^\alpha(x_1,x_2,x_3,x_4)=\Phi_0\circ\varphi(-x_1,x_2,x_3,x_4)$,
where $\Phi_0$ is the involutive isometry of the vacuum.
The transformation properties of the action under this new definition
of parity are different than under the previous definition.
In particular, if
$A_{\alpha\beta\gamma}(\Phi_0(y))=A_{\alpha\beta\gamma}(y)$,
then the $A$-term will not be Parity--invariant.
On the other hand if
$B_{\alpha\beta\gamma\delta}(\Phi_0(y))=-B_{\alpha\beta\gamma\delta}(y)$,
then the Wess-Zumino-Witten term is Parity--invariant
\cite{witten}.
\subsection{Background field expansion}
We use the background field techniques developed in \cite{honerkamp,afm,bb,hps}.
We review here some of the main points.
Having chosen a (not necessarily constant) background $\bar\varphi$,
any other field $\varphi$ in an open neighborhood of $\bar\varphi$ can be written
$\varphi^\alpha=\bar\varphi^\alpha+\eta^\alpha$.
In principle one could work with the quantum fields $\eta^\alpha$,
but this is not convenient because, as differences of coordinates, they do not have
nice transformation properties.
It is therefore convenient to proceed as follows.
For each $x$ one can find a unique vector $\xi(x)$ tangent to $\bar\varphi(x)$
such that $\varphi(x)$ is the point on the geodesic passing through
$\bar\varphi(x)$ and tangent to $\xi(x)$, the distance between $\varphi(x)$ and $\bar\varphi(x)$
being equal to $|\xi(x)|$.
We can thus write $\varphi(x)=Exp_{\bar\varphi(x)}\xi(x)$, where $Exp$ is the exponential map.
The field, $\xi^\alpha(x)$ is a vectorfield along $\bar\varphi$,
and its covariant derivative is defined as in \eqref{covariant}.
In principle, then, the action $\varphi$ can be rewritten as $S(\varphi)=\bar S(\bar\varphi,\xi)$.
In practice one can compute the first few terms in an expansion
$
\bar S(\bar\varphi,\xi)=\bar S^{(0)}(\bar\varphi,\xi)+\bar S^{(1)}(\bar\varphi,\xi)
+\bar S^{(2)}(\bar\varphi,\xi)+\ldots\ ,
$
where $\bar S^{(n)}$ contains $n$ powers of $\xi$.
The first term is clearly $\bar S^{(0)}(\bar\varphi,\xi)=\bar S(\bar\varphi,0)=S(\bar\varphi)$.
To compute the next terms we use the following formulae (whose derivation can be found in \cite{honerkamp}):
\begin{eqnarray}
\partial_\mu\varphi^\alpha&=&
\partial_\mu\bar\varphi^\alpha
+\bar\nabla_\mu\xi^\alpha
-\frac{1}{3}\partial_\mu\bar\varphi^\gamma\bar R_{\gamma\epsilon}{}^\alpha{}_\eta\xi^\epsilon\xi^\eta+\ldots
\nonumber\\
t_{\alpha\beta\ldots}(\varphi)&=&
t_{\alpha\beta\ldots}(\bar\varphi)
+\xi^\epsilon \bar\nabla_\epsilon t_{\alpha\beta\ldots}(\bar\varphi)
+\frac{1}{2}\xi^\epsilon\xi^\eta\bar\nabla_\epsilon\bar\nabla_\eta t_{\alpha\beta\ldots}(\bar\varphi)
-\frac{1}{6}\xi^\epsilon\xi^\eta \bar R^\gamma{}_{\epsilon\alpha\eta}t_{\gamma\beta\ldots}
-\frac{1}{6}\xi^\epsilon\xi^\eta \bar R^\gamma{}_{\epsilon\beta\eta}t_{\alpha\gamma\ldots}
+\ldots\nonumber
\end{eqnarray}
A bar over the derivatives and the curvatures indicates that they have to be computed
with the background field $\bar\varphi$.
In particular for the metric $g$ we have
$$
g_{\alpha\beta}(\varphi)=
g_{\alpha\beta}(\bar\varphi)
-\frac{1}{3}\bar R_{\alpha\epsilon\beta\eta}\xi^\epsilon\xi^\eta +\ldots
$$
Inserting in \eqref{action}, with $A=0$, and keeping terms of second order in $\xi$ we obtain
\begin{eqnarray}
\label{secondvariation}
&&\!\!\!\!\!\!\! \frac{1}{2}\int\! d^4x \Bigl[
\g2_{\alpha\beta} \nabla_\mu\xi^\alpha \nabla^\mu\xi^\beta
-\xi^\alpha\xi^\beta R_{\alpha\gamma\beta}{}^\epsilon
\g2_{\epsilon\delta}\partial_\mu\varphi^\gamma\partial_\mu\varphi^\delta
+2 \xi^\alpha\nabla_\mu\xi^\beta \nabla_\alpha\g2_{\beta\gamma} \partial_\mu\varphi^\gamma
+\frac{1}{2}\xi^\alpha\xi^\beta \nabla_\alpha \nabla_\beta\g2_{\gamma\delta}
\partial_\mu\varphi^\gamma\partial_\mu\varphi^\delta
\nonumber\\
&&
+\g4_{\alpha\beta}\Box\xi^\alpha\Box\xi^\beta
+2\xi^\alpha\Box\xi^\beta R_{\alpha\gamma\beta\delta}
\partial_\mu\varphi^\gamma\partial^\mu\varphi^\delta
-4\xi^\alpha\nabla_\mu\xi^\beta R_{\alpha\gamma\beta\delta}
\partial^\mu\varphi^\gamma \Box\varphi^\delta
-\xi^\alpha\xi^\beta R_{\alpha\gamma\beta\delta}\Box\varphi^\gamma\Box\varphi^\delta
\nonumber\\
&&
+\xi^\alpha\xi^\beta
\left(\nabla_\alpha R_{\epsilon\gamma\beta\delta}+\nabla_\gamma R_{\epsilon\alpha\beta\delta}\right)
\partial_\mu\varphi^\gamma\partial^\mu\varphi^\delta \Box\varphi^\epsilon
+\xi^\alpha\xi^\beta R_{\phi\gamma\delta\alpha}R^\phi\,\!_{\epsilon\eta\beta}
\partial_\mu\varphi^\gamma\partial^\mu\varphi^\delta
\partial_\nu\varphi^\epsilon\partial^\nu\varphi^\eta
\nonumber\\
&&
+
2\nabla_\mu\xi^\alpha\nabla^\mu\xi^\beta \partial_\nu\varphi^\gamma\partial^\nu\varphi^\delta
T_{\alpha\beta\gamma\delta}
+4\nabla_\mu\xi^\alpha\nabla_\nu\xi^\beta \partial^\mu\varphi^\gamma\partial^\nu\varphi^\delta
T_{\alpha\gamma\beta\delta}
-2 \xi^\alpha\xi^\beta R^\phi{}_{\alpha\gamma\beta}T_{\phi\delta\epsilon\eta}
\partial_\mu\varphi^\gamma\partial^\mu\varphi^\delta
\partial_\nu\varphi^\epsilon\partial^\nu\varphi^\eta
\nonumber\\
&&
+4\xi^\alpha\nabla_\mu\xi^\beta\nabla_\alpha T_{\beta\gamma\delta\epsilon}
\partial^\mu\varphi^\gamma\partial_\nu\varphi^\delta\partial^\nu\varphi^\epsilon
+\frac{1}{2}\xi^\alpha\xi^\beta\nabla_\alpha\nabla_\beta T_{\gamma\delta\epsilon\eta}
\partial_\nu\varphi^\gamma\partial^\nu\varphi^\delta
\partial_\mu\varphi^\epsilon\partial^\mu\varphi^\eta
\Bigr]\ .
\end{eqnarray}
For notational simplicity here and in the following
we drop the bars over $\varphi$, $\nabla$ and $R$,
but it is always understood that they are computed at the background field.
The terms have been kept in the order in which they appear in \eqref{action}, namely the
first line comes from the variation of the two derivative term,
the second and third lines come from the variation of the term containing $\g4$
the last two lines come from the variation of the term containing $T$.
\subsection{The running effective action}
Our procedure for calculating the beta functions is a particular implementation of
Wilson's prescription that physics at the scale $k$ is described by an
effective action $\Gamma_k$ where all modes with momenta $q>k$ have been integrated out.
We define formally an ``effective average action'' $\Gamma_k$
by implementing an {\it infrared} cutoff $k$
in the functional integral over the quantum field $\xi$.
If $\bar S(\varphi,\xi)$ is the bare action of the theory,
the IR cutoff can be implemented by adding
to $\bar S$ a term $\Delta S_k(\varphi,\xi)$, quadratic in $\xi^\alpha$,
which in Fourier space would have the general structure:
\begin{equation}
\label{cutoff1}
\Delta S_k(\varphi,\xi)=\int d^4q\, \xi^\alpha(-q) R_{k\alpha\beta}(q^2)\xi^\beta(q)\ .
\end{equation}
The kernel $R_{k\alpha\beta}(q^2)$, sometimes also called the cutoff,
is chosen in such a way that the propagation of
field modes $\xi^\alpha(q)$ with $|q|<k$ is suppressed, while
field modes with $|q|>k$ are unaffected.
There is a vast freedom in the choice of the cutoff $\Delta S_k$, and in principle
physical predictions should turn out to be independent of this choice.
One can use this freedom to simplify calculations to some extent.
One possibility would be to write the cutoff exactly as in \eqref{cutoff1},
with $R_{k\alpha\beta}(q^2)=\g4_{\alpha\beta}R_k(q^2)$,
where $R_k(q^2)$ is a scalar function of the modulus of the momentum.
Note that $q^2$ is just the eigenvalue of the operator $-\partial^2$
acting on the quantum field.
It is more conveniente to write the cutoff in terms of the eigenvalues of some covariant
operator, such as the Laplacian constructed with the background field $-\nabla^2$.
This is the choice that was used in \cite{codello2}.
In this paper we will find it expedient to use instead of $-\nabla^2$
the full covariant fourth order operator
$\Delta=\frac{\delta^2 S}{\delta\varphi\delta\varphi}$:
\begin{equation}
\label{cutoff2}
\Delta S_k(\varphi,\xi)=\frac{1}{2}\int d^4x\,
\xi^\alpha \g4_{\alpha\beta}(\varphi)R_k(\Delta)\xi^\beta\ .
\end{equation}
Because $\Delta$ depends only on the background field, and not on the quantum fields,
this cutoff is still quadratic in the quantum fields, as required.
Having modified the propagator of the theory, we define a generating functional $W_k(\varphi,j)$,
depending on the background $\varphi$ and on a source field $j$ coupled linearly to the quantum field $\xi$, by
\begin{equation}
\label{pathintegral}
W_k(\varphi,j)=-\log \int (d\xi^\alpha) \exp\left(-\bar S(\varphi,\xi)
-\Delta S_k(\varphi,\xi)-\int j_\alpha\xi^\alpha\right)
\end{equation}
Then we define a modified $k$--dependent Legendre transform
$$
\Gamma_k(\varphi,\xi)=W_k(\varphi,j)-\int j_\alpha\xi^\alpha -\Delta S_k(\varphi,\xi)\ ,
$$
where $\Delta S_k$ has been subtracted.
The ``classical fields'' $\frac{\delta W_k}{\delta j_\alpha}$
are denoted again $\xi^\alpha$ for notational simplicity.
The functional $\Gamma_k$ reduces for $k\to 0$ to the usual
background field effective action $\Gamma(\varphi,\xi)$,
the generating functional of one--particle irreducible Green functions of $\xi$.
\subsection{The one loop beta functional}
At one loop one can evaluate the functional $\Gamma_k$:
\begin{equation}
\Gamma_k^{(1)}=S+\frac{1}{2}\mathrm{Tr}\log\left(\frac{\delta^2 S}{\delta\varphi\delta\varphi}+R_k\right)\ .
\end{equation}
Note that $\Delta S_k$ has cancelled out.
The only remaining dependence on $k$ is in $R_k$, so
\begin{equation}
\label{onelooperge}
k\frac{d\Gamma_k^{(1)}}{dk}=
\frac{1}{2}\mathrm{Tr}\left(\frac{\delta^2 S}{\delta\varphi\delta\varphi}+R_k\right)^{-1}
k\frac{dR_k}{dk}\ .
\end{equation}
The r.h.s. can be regarded as the one loop beta functional of the theory.
The individual beta functions can be read off by isolating the coefficients
of various operators.
Of course one could derive the one loop beta functions in other, more traditional ways.
We prefer this route because it has a few advantages.
First, due to the rapid fall off
of the function $k\partial_k R_k$, the beta functional is itself finite
and one does not actually need to introduce any ultraviolet regularization.
So, even though the derivation of the equation from a functional integral
was formal, because the functional integral is itself ill defined,
the functional RG equation is itself perfectly well defined.
A second important point is that the RG improvement of this equation,
where one replaces the bare action $S$ by $\Gamma_k$ in the r.h.s.,
is actually an exact equation \cite{wetterich}.
So although in the present work we shall restrict ourselves to the one loop approximation,
the formalism is ready for the calculation of the beta functions based on a truncation
of the exact RG equation, which amount to resumming infinitely many orders of perturbation theory.
A final, important point is that experience with other systems shows that this
procedure gives exactly the same results as any other procedure for the
universal (scheme-independent) one loop beta functions.
We will see in section IIID that, to the extent that a comparison is possible, this expectation
will be confirmed also in this case.
\subsection{Global symmetries}
If there are any symmetries, one can define the RG flow so as to preserve them.
To see this, let $\Phi$ be an internal symmetry, as in section IIA.
Since it is an isometry of $\g4$, it also leaves the connection invariant,
so it maps the geodesic through $y$ tangent to $\xi$
to the geodesic through $\Phi(y)$ tangent to $T\Phi(\xi)$ \cite{kn}:
$$
\Phi(Exp_y(\xi))=Exp_{\Phi(y)}(T\Phi(\xi))\ .
$$
We call $\varphi'=\Phi\circ\varphi$
and $\xi'=T\Phi(\xi)$ the transform of $\varphi$ and $\xi$ under $\Phi$.
Then
$\varphi'=\Phi(Exp_{\varphi}\xi)=Exp_{\Phi(\varphi)}(T\Phi(\xi))
=Exp_{\varphi'}\xi'$.
There follows that
\begin{equation}
\bar S(\varphi',\xi')=S(\varphi')=S(\varphi)=\bar S(\varphi,\xi)\,,
\end{equation}
{\it i.e.} the background field action $\bar S$ is $G$-invariant provided both
background and quantum field are transformed.
The operator $\Delta$ is covariant, so
$ \Delta'(\xi')=T\Phi(\Delta(\xi))$
or abstractly $\Delta'=T\Phi\circ(\Delta)\circ T\Phi^{-1}$,
so also the cutoff term \eqref{cutoff2} is invariant:
\begin{equation}
\Delta S_k(\varphi',\xi')=\Delta S_k(\varphi,\xi)\,,
\end{equation}
One can formally choose the measure in the functional
integral \eqref{pathintegral} to be invariant under $\Phi$.
Since both measure and integrand are invariant, the effective action $\Gamma_k$ will
also be invariant, for all $k$.
Somewhat less formally, one can arrive at the same conclusion as follows:
observe that the cutoff as defined in \eqref{cutoff2} is a suppression term that depends
on the eigenvalue of the operator $\Delta$ on the normal modes of the field.
From the transformation properties of $\Delta$ one sees that
if $\xi$ is an eigenvector of $\Delta$ with eigenvalue $\lambda$,
then $\xi'$ is an eigenvector of $\Delta'$ with the same eigenvalue.
Therefore the spectrum of $\Delta$ is invariant.
Equation \eqref{onelooperge} gives the (one loop)
scale variation of $\Gamma_k(\varphi)$ as a sum of terms,
each term being a fixed function evaluated on an eigenvalues of $\Delta$.
Since the eigenvalues are invariant, the sum is also invariant,
so it follows that $\partial_t\Gamma_k(\varphi)$ is invariant.
This implies that if the starting action $\Gamma_{k_0}(\varphi)$ is invariant,
also the action at any other $k$ is.
This argument is mathematically more meaningful, because unlike the one
based on the path integral, it involves only statements about finite expressions.
The previous argument can be applied both to discrete and continuous symmetries.
For example in the case of discrete symmetries, it implies that the flow preserves Parity.
If the $A$ term violates Parity, it must be set to zero in order to have a Parity invariant theory.
The flow will preserve this property, so the beta function of $A$ will be zero.
In other words the condition $A=0$ will be ``protected by Parity''.
We will see this in an explicit calculation in section IIIb.
\section{Evaluation of beta functions}
The one loop RG flow equation \eqref{onelooperge} can be approximated by resorting to a truncation,
which means keeping only a finite number of terms in $\Gamma_k$,
inserting this ansatz in the flow equation and deriving from it
the beta functions of the couplings that enter in the ansatz.
The best way of truncating $\Gamma_k$ is to do so consistently with a derivative expansion,
{\it i.e.} to keep all the terms with a given number of derivatives.
In this paper we will approximate $\Gamma_k$ by a functional of the form \eqref{action},
where the tensors $\g2$, $\g4$ and $T$ are $k$-dependent, and $A=0$.
In general this is still a functional flow, because the tensors
actually contain infinitely many couplings.
We will be able to say more in the case when a global symmetry restricts
the possible form of these tensors, so that only finitely many couplings remain.
In this paper we will explicitly compute the beta functions in the case when $Y$ is a
sphere or a special unitary group. Since these are symmetric spaces,
it will be consistent to neglect the $A$ terms altogether.
\subsection{The inverse propagator}
Integrating by parts one can rewrite \eqref{secondvariation} in the form
$\bar S^{(2)}(\varphi,\xi)=\frac{1}{2}(\xi,\Delta\xi)$,
where the inner product of vectorfields along $\varphi$ is
$(\xi,\zeta)=\int d^4x\, \g4_{\alpha\beta}\xi^\alpha\zeta^\beta$
and $\Delta$ is a self-adjoint operator of the form:
\begin{equation}
\label{operator}
\Delta_{\alpha\beta} =
\g4_{\alpha\beta}\Box^2 + {\cal B}^{\mu\nu}_{\alpha\beta}\nabla_\mu \nabla_\nu
+ {\cal C}^\mu_{\alpha\beta}\nabla_\mu+{\cal D}_{\alpha\beta}\,.
\end{equation}
Self-adjointness means that $(\xi,\Delta\zeta)=(\Delta\xi,\zeta)$ and implies the properties:
\begin{eqnarray}
{\cal B}^{\mu\nu}_{\alpha\beta}&=&{\cal B}^{\nu\mu}_{\beta\alpha}\ ,
\\
{\cal C}^\mu_{\alpha\beta}&=&-{\cal C}^\mu_{\beta\alpha}
+\nabla_\nu{\cal B}^{\mu\nu}_{\beta\alpha}
+\nabla_\nu{\cal B}^{\nu\mu}_{\beta\alpha}\ ,
\\
{\cal D}_{\alpha\beta}&=&{\cal D}_{\beta\alpha}
+\nabla_\nu{\cal C}^\mu_{\beta\alpha}
+\nabla_\nu\nabla_\nu{\cal B}^{\nu\mu}_{\beta\alpha}\ .
\end{eqnarray}
In addition by commuting derivatives we can arrange the operator so that
${\cal B}^{\mu\nu}_{\alpha\beta}={\cal B}^{\nu\mu}_{\alpha\beta}$.
In order to arrive at the operator $\Delta$ we proceed in two steps.
First we put all the derivatives of \eqref{secondvariation} on one of the $\xi$'s, so that
$\bar S^{(2)}(\varphi,\xi)=\frac{1}{2}(\xi,\tilde\Delta\xi)$,
where $\tilde\Delta$ is of the form \eqref{operator}, with
\begin{eqnarray}
\label{bi}
\tilde{\cal B}^{\mu\nu}_{\alpha\beta}&=&\delta^{\mu\nu}
(-h^{(2)}_{\alpha\beta}
+2\partial_\rho\varphi^\gamma\partial^\rho\varphi^\delta
(R_{\alpha\gamma\beta\delta}
- T_{\alpha\beta\gamma\delta}))
-4\partial_\mu\varphi^\gamma\partial_\nu\varphi^\delta
T_{\alpha\gamma\beta\delta}\ ,
\\
\tilde{\cal D}_{\alpha\beta}&=&
\partial_\mu\varphi^\gamma\partial^\mu\varphi^\delta
\left(\frac{1}{2}\nabla_\alpha\nabla_\beta\g2_{\gamma\delta}
-\g2_{\gamma\epsilon}R^\epsilon{}_{\beta\delta\alpha}\right)
-\Box\varphi^\gamma\Box\varphi^\delta R_{\alpha\gamma\beta\delta}
-2 \partial_\rho\varphi^\gamma\partial^\rho\varphi^\delta\Box\varphi^\epsilon
\nabla_{(\delta}R_{\alpha)\epsilon\beta\gamma}
\nonumber\\
&&
+\partial_\rho\varphi^\gamma\partial^\rho\varphi^\delta
\partial_\sigma\varphi^\epsilon\partial^\sigma\varphi^\eta
\left(
R_{\alpha\gamma\delta\phi}R_{\beta\epsilon\eta}{}^\phi
+\frac{1}{2}\nabla_\alpha\nabla_\beta T_{\gamma\delta\epsilon\eta}
+2R_{\phi\alpha\beta\epsilon}T^\phi{}_{\eta\gamma\delta}\right)
\ .
\end{eqnarray}
We do not display the form of $\tilde{\cal C}^\mu_{\alpha\beta}$,
since it does not contribute to the expressions we want to calculate,
as will become clear in due course.
This operator $\tilde\Delta$ is not self-adjoint, and we define
$\Delta={\tiny\frac{1}{2}}(\tilde\Delta+\tilde\Delta^\dagger)$.
Its coefficients are
\begin{eqnarray}
{\cal B}^{\mu\nu}_{\alpha\beta}&=&
\frac{1}{2}\left(\tilde{\cal B}^{\mu\nu}_{\alpha\beta}+\tilde{\cal B}^{\nu\mu}_{\beta\alpha}\right)\ ,
\nonumber\\
{\cal C}^\mu_{\alpha\beta}&=&\frac{1}{2}\left(\tilde{\cal C}^\mu_{\alpha\beta}-\tilde{\cal C}^\mu_{\beta\alpha}
+\nabla_\nu\tilde{\cal B}^{\mu\nu}_{\beta\alpha}
+\nabla_\nu\tilde{\cal B}^{\nu\mu}_{\beta\alpha}\right)\ ,
\nonumber\\
{\cal D}_{\alpha\beta}&=&\frac{1}{2}\left(\tilde{\cal D}_{\alpha\beta}
+\tilde{\cal D}_{\beta\alpha}
-\nabla_\mu\tilde{\cal C}^\mu_{\beta\alpha}
+\nabla_\mu\nabla_\nu\tilde{\cal B}^{\nu\mu}_{\beta\alpha}
\right)\ .
\end{eqnarray}
Note that the last two terms in ${\cal C}^\mu_{\alpha\beta}$
and ${\cal D}^\mu_{\alpha\beta}$ are total derivatives,
and will not contribute to our final formulae.
Finally we can symmetrize ${\cal B}^{\mu\nu}_{\alpha\beta}$
in $\mu$, $\nu$ at the cost of generating a commutator term that
contributes to ${\cal D}_{\alpha\beta}$.
The final form of the operator $\Delta$ is \eqref{operator}, with
\begin{eqnarray}
{\cal B}^{\mu\nu}_{\alpha\beta}&=&\delta^{\mu\nu}
(-h^{(2)}_{\alpha\beta}
+2\partial_\rho\varphi^\gamma\partial^\rho\varphi^\delta
(R_{\alpha\gamma\beta\delta}
- T_{\alpha\beta\gamma\delta}))
-2\partial_\mu\varphi^\gamma\partial_\nu\varphi^\delta
(T_{\alpha\gamma\beta\delta}+T_{\alpha\delta\beta\gamma})\ ,
\nonumber\\
{\cal D}_{\alpha\beta}&=&\frac{1}{2}\left(\tilde{\cal D}_{\alpha\beta}
+\tilde{\cal D}_{\beta\alpha}\right)
-\partial_\rho\varphi^\gamma\partial^\rho\varphi^\delta
\partial_\sigma\varphi^\epsilon\partial^\sigma\varphi^\eta
(T_{\alpha\gamma\epsilon\phi}R_{\delta\eta\beta}{}^\phi
+T_{\beta\gamma\epsilon\phi}R_{\delta\eta\alpha}{}^\phi)
+\mathrm{TD}\ ,
\end{eqnarray}
where TD stands for ``total derivatives''.
Again we omit to give ${\cal C}^\mu_{\alpha\beta}$, because it
does not contribute to the beta functions.
These formulae agree with (3.17-21) in \cite{bk}, except for a factor 2 in the coefficient
of the first term containing $T_{\alpha\beta\gamma\delta}$ in equation \eqref{bi}.
\subsection{Beta functionals}
We begin by discussing the general case of the action \eqref{action}
with arbitrary $h^{(2)}$, $h^{(4)}$ and $T$, and $A=0$.
We evaluate the trace in \eqref{onelooperge} by heat kernel methods.
The advantage of this procedure is that pieces of the
calculation are readily available in the literature.
Given a differential operator $\Delta$ of order $p$,
and some function $W$, we have
\begin{equation}
\label{HKE}
Tr W(\Delta) = \frac{1}{(4\pi)^{2}}\Bigl[Q_{\frac{4}{p}}(W)B_0(\Delta)+
Q_{\frac{2}{p}}(W)B_2(\Delta)+ Q_{0}(W)B_4(\Delta)+\ldots\Bigr]\,.
\end{equation}
The heat kernel coefficients are defined by the asymptotic expansion
\begin{equation}
\label{heatkernel}
{\rm Tr} (e^{-s\Delta})=
\frac{1}{(4\pi)^2}
\left[B_0s^{-4/p}+B_2s^{-2/p}+B_0+\ldots\right]\ ,
\end{equation}
with $B_n=\int d^4x \mathrm{tr} b_n$; $b_n$ are matrices with indices
$\alpha$, $\beta$ and tr denotes the trace over such indices.
For a fourth order operator of the form
\eqref{operator}, they can be found in \cite{barth}.
The quantities $Q_n(W)$ in \eqref{HKE} are given by
$Q_n(W) = \frac{1}{\Gamma(n)} \int_0^\infty dz z^{n-1} W(z)$
for $n>0$ and $Q_0(W)=W(0)$.
We do not need any higher coefficients.
In order to be able to evaluate the integrals in closed form we choose the
``optimized'' cutoff function $R_k(z)=(k^4-z)\theta(k^4-z)$ \cite{optimized}.
The scale derivative of the cutoff is
${\tiny k\frac{dR_k}{dk}}=4k^4\theta(k^4-z)$, and the
modified inverse propagator $P_k(z)=z+R_k(z)$ is equal to $k^4$ for $z<k^4$.
Then the function to be traced in the ERGE is just a step function:
$W(z)=
\frac{1}{2}\frac{1}{P_k}k\frac{dR_k}{dk}=2\theta(1-z/k^4)$,
and the integrals are very simple:
\begin{equation}
Q_1 = 2 k^4\,,\qquad
Q_{\frac{1}{2}}=\frac{4}{\sqrt{\pi}}k^2\,,\qquad
Q_0 = 2\,.
\end{equation}
The first term in \eqref{HKE} is field independent and will be omitted.
Putting together the remaining pieces:
\begin{equation}
k\frac{d\Gamma_k}{dk}=
\frac{1}{(4\pi)^{2}}\int d^4x
\left(
\frac{1}{4}k^2{\cal B}^{\alpha}_{\alpha}
+\frac{1}{6}\Omega_{\mu\nu}^{\alpha\beta}\Omega^{\mu\nu}_{\beta\alpha}
+\frac{1}{24}{\cal B}^{\alpha\beta}_{\mu\nu}{\cal B}^{\mu\nu}_{\beta\alpha}
+\frac{1}{48}{\cal B}_{\alpha\beta}{\cal B}^{\beta\alpha}
-{\cal D}^\alpha_\alpha
\right)
\end{equation}
where $\Omega$ is defined as in \eqref{curvature} and ${\cal B}={\cal B}^{\mu}_{\mu}$.
The first term comes from $B_2$, the others from $B_4$.
One finds
\begin{eqnarray}
\label{traceb}
\frac{1}{4}{\cal B}^{\alpha}_{\alpha}&=&
\partial_\mu\varphi^\gamma\partial^\mu\varphi^\delta
\left(2R_{\gamma\delta}-2T^\alpha{}_{\alpha\gamma\delta}
-T^\alpha{}_{\gamma\alpha\delta}\right)
\\
\label{omegasquare}
\frac{1}{6}\Omega_{\mu\nu}^{\alpha\beta}\Omega^{\mu\nu}_{\beta\alpha}&=&
-\frac{1}{6}\partial_\mu\varphi^\alpha\partial^\mu\varphi^\beta
\partial_\nu\varphi^\gamma\partial^\nu\varphi^\delta
R_{\alpha\gamma\epsilon\eta}R_{\beta\delta}{}^{\epsilon\eta}
\\
\label{tracebb}
\frac{1}{24}{\cal B}^{\alpha\beta}_{\mu\nu}{\cal B}^{\mu\nu}_{\beta\alpha}
+\frac{1}{48}{\cal B}_{\alpha\beta}{\cal B}^{\beta\alpha}&=&
\frac{1}{2}\g2_{\alpha\beta}{\g2}^{\alpha\beta}
+\partial_\mu\varphi^\alpha\partial^\mu\varphi^\beta
\left(T_\alpha{}^\gamma{}_\beta{}^\delta
+2T_{\alpha\beta}{}^{\gamma\delta}
-2 R_\alpha{}^\gamma{}_\beta{}^\delta\right)
\g2_{\gamma\delta}
\nonumber\\
&&
+\partial_\mu\varphi^\alpha\partial^\mu\varphi^\beta
\partial_\nu\varphi^\gamma\partial^\nu\varphi^\delta
\Bigl[
\frac{2}{3}T_{\alpha\epsilon\gamma\eta}T_\beta{}^{(\epsilon}{}_\delta{}^{\eta)}
+\frac{1}{3}T_{\alpha\epsilon\beta\eta}T_\gamma{}^{(\eta}{}_\delta{}^{\epsilon)}
+4T_{\alpha(\epsilon\gamma)\eta}T_\beta{}^{(\epsilon}{}_\delta{}^{\eta)}
\nonumber\\
&&
-4R_{\alpha\epsilon\gamma\eta}T_\beta{}^{(\epsilon}{}_\delta{}^{\eta)}
-2R_{\alpha\epsilon\beta\eta}T_\gamma{}^{(\eta}{}_\delta{}^{\epsilon)}
+2R_{\alpha\epsilon\beta\eta}R_\gamma{}^{(\eta}{}_\delta{}^{\epsilon)}
\Bigr]
\\
\label{traced}
-{\cal D}^\alpha_\alpha&=&
\Box\varphi^\alpha\Box\varphi^\beta R_{\alpha\beta}
+\Box\varphi^\alpha\partial^\mu\varphi^\beta
\partial_\mu\varphi^\gamma
(2\nabla_\gamma R_{\alpha\beta}-\nabla_\alpha R_{\beta\gamma})
\nonumber\\
&&
+\partial_\mu\varphi^\alpha\partial^\mu\varphi^\beta
\left(\g2_{\alpha\gamma}R^\gamma{}_\beta
-\frac{1}{2}\nabla_\gamma\nabla^\gamma\g2_{\alpha\beta}\right)
+\partial_\mu\varphi^\alpha\partial^\mu\varphi^\beta
\partial_\nu\varphi^\gamma\partial^\nu\varphi^\delta\times
\nonumber\\
&&
\Bigl(
2R_\alpha{}^\epsilon T_{\epsilon\beta\gamma\delta}
+2R_{\beta\delta}{}^{\epsilon\eta}T_{\alpha\eta\gamma\epsilon}
-\frac{1}{2}\nabla_\epsilon\nabla^\epsilon T_{\alpha\beta\gamma\delta}
-R_{\alpha\epsilon\beta\eta}R_\gamma{}^\epsilon{}_\delta{}^\eta
\Bigr)
\end{eqnarray}
From here one can read off the beta functionals of $\g2$, $A$, $T$
as the coefficients of terms containing two, three and four
powers of $\partial_\mu\varphi^\alpha$, respectively.
We do not give these general formulae, but just make some observations.
The only term proportional to $\Box\varphi^\alpha\Box\varphi^\beta$ is contained
in $-{\cal D}^\alpha_\alpha$, so the beta functional of $h^{(4)}$
is easily obtained:
\begin{equation}
k\frac{d}{dk}h^{(4)}_{\alpha\beta}=\frac{1}{8\pi^2}R_{\alpha\beta}\,.
\end{equation}
This is very similar to the result for the two derivative truncation.
In order to compare results obtained with the same type of cutoff, we should
repeat the calculation of \cite{codello2} using a cutoff constructed with the
full inverse propagator
$\Delta_{\alpha\beta}=-\g2_{\alpha\beta}\nabla^2-
\partial_\mu\varphi^\gamma\partial_\mu\varphi^\delta R_{\alpha\gamma\beta\delta}$.
This is a cutoff of type III in the terminology used in \cite{cpr}.
In this case the general beta function of the metric is
\begin{equation}
\label{betatrunc2III}
k\frac{d}{dk}\g2_{\alpha\beta}=
\frac{1}{(4\pi)^2} Q_1\left(\frac{\dot R_k}{P_k}\right)R_{\alpha\beta}
=\frac{1}{8\pi^2}k^2 R_{\alpha\beta}\ ,
\end{equation}
where $R$ denotes now the curvature of $\g2_{\alpha\beta}$.
As a side remark, this little calculation is also useful to test the scheme dependence
of the results: with the type I cutoff used in \cite{codello2} the result was
\begin{equation}
\label{betatrunc2I}
k\frac{d}{dk}\g2_{\alpha\beta}
=\frac{1}{(4\pi)^2} Q_2\left(\frac{\dot R_k}{P_k^2}\right)R_{\alpha\beta}
=\frac{1}{16\pi^2} k^2 R_{\alpha\beta}\ ,
\end{equation}
which differs by a factor 2.
Another fact that follows from \eqref{traced} is that the beta function
of $A$ (coming from the coefficient of
$\Box\varphi^\alpha\partial^\mu\varphi^\beta\partial_\mu\varphi^\gamma$)
is proportional to covariant derivatives of the Ricci tensor.
For symmetric spaces the covariant derivative of the curvature vanishes
and therefore on such spaces it is consistent to set $A=0$.
This confirms the general statement made in section IIE.
The particular models that we shall consider in the following are
symmetric spaces.
\subsection{The spherical models}
We now consider the class of models for which the target space $Y$ is the sphere $S^n$.
Such models are often called the $O(N)$ models, with $N=n+1$, because they have global
symmetry $O(N)$.
There is only one $O(n+1)$-invariant nonvanishing rank two tensor on the sphere,
there is no invariant rank three tensor and
there are only two invariant rank four tensors with the desired index symmetries,
up to overall constant factors.
If we regard $S^n$ as embedded in $\mathbf{R}^{n+1}$, we call $h_{\alpha\beta}$ the metric
of the sphere of unit radius.
Its Riemann and Ricci tensors are given by
$$
R_{\alpha\beta\gamma\delta}=h_{\alpha\gamma}h_{\beta\delta}-h_{\alpha\delta}h_{\beta\gamma}\ ;\quad
R_{\alpha\beta}=(n-1)h_{\alpha\beta}\ ;\quad
R=n(n-1)\ .
$$
Therefore both $\g2$ and $\g4$ must be proportional to $h$, and $T$ is a combination of $h$'s:
$$
\g2_{\alpha\beta}=\frac{1}{g^2}h_{\alpha\beta}\ ;\qquad
\g4_{\alpha\beta}=\frac{1}{\lambda}h_{\alpha\beta}\ ;\qquad
T_{\alpha\beta\gamma\delta}=\frac{\ell_1}{2}\left(h_{\alpha\gamma}h_{\beta\delta}+h_{\alpha\delta}h_{\beta\gamma}\right)
+\ell_2 h_{\alpha\beta}h_{\gamma\delta}\ .
$$
Here $g^2$ has mass dimension $2$, while $\lambda$, $\ell_1$, $\ell_2$ are dimensionless
\footnote{the names $\ell_1$ and $\ell_2$ are used commonly in chiral perturbation theory \cite{gl}.}.
It is convenient to regard $\tiny\frac{1}{\lambda}$ as the overall factor of the fourth order terms;
then we define the ratios between the three coefficients of the four-derivative terms as
$f_1=\lambda\ell_1$ and $f_2=\lambda\ell_2$.
For the reader's convenience we rewrite the action of the $S^n$ models:
\begin{equation}
\label{sphericalaction}
\int d^4x\Biggl[
\frac{1}{2g^2}h_{\alpha\beta}\partial_\mu\varphi^\alpha\partial^\mu\varphi^\beta
+\frac{1}{2\lambda}\left(h_{\alpha\beta}\Box\varphi^\alpha\Box\varphi^\beta
+\partial_\mu\varphi^\alpha\partial^\mu\varphi^\beta\partial_\nu\varphi^\gamma\partial^\nu\varphi^\delta
(f_1h_{\alpha\gamma}h_{\beta\delta}+f_2h_{\alpha\beta}h_{\gamma\delta})\right)
\Biggr]
\end{equation}
One then finds the following beta functions:
\begin{eqnarray}
\label{bls}
\beta_\lambda & = & -\frac{n-1}{8\pi^2}\lambda^2
\\
\label{b1s}
\beta_{f_1} & = &\frac{\lambda}{48\pi^2}
\left((n+21)f_1^2+20f_2f_1
+4f_2^2+6(n+3)f_1+24f_2+8\right)
\\
\label{b2s}
\beta_{f_2} & = &\frac{\lambda}{8\pi^2}
\left(\frac{n+15}{12}f_1^2+\frac{3n+17}{3}f_1f_2
+\frac{6n+7}{3}f_2^2-(n+3)f_1-(3n+1)f_2+n-\frac{7}{3}\right)
\\
\label{bgs}
\beta_{\tilde g^2} & = & 2\tilde g^2
+\frac{\tilde g^4}{16\pi^2}\left((5+n)f_1+(2+4n)f_2+4(1-n)\right)
-\frac{\lambda\tilde g^2}{16\pi^2}\left((5+n)f_1+(2+4n)f_2+2(1-n)\right)\phantom{aaa}
\end{eqnarray}
Equations \eqref{b1s} and \eqref{b2s} differ in a significant way from equation
(5.11) in \cite{bk}. This is due to the already mentioned factor 2 in a term in \eqref{bi}.
Unfortunately, this changes completely the picture of the fixed points.
It will be instructive to compare the results of this four derivative truncation
with those of the simpler two derivative truncation discussed in \cite{codello2}.
If we specialize \eqref{betatrunc2III} to $Y=S^n$, it gives
\begin{equation}
\label{betatrunc2sphere}
k\frac{d\tilde g^2}{dk}=2\tilde g^2-\frac{n-1}{8\pi^2}\tilde g^4\,,
\end{equation}
whereas from \eqref{bgs}, setting for simplicity
$\ell_1=\ell_2=0$ and in the limit $\lambda\to 0$ one gets
\begin{equation}
\label{betatrunc4sphere}
k\frac{d\tilde g^2}{dk}=2\tilde g^2-\frac{n-1}{4\pi^2}\tilde g^4\,.
\end{equation}
The difference is just a factor $2$, which is within the
range of variation due to the scheme dependence.
It is quite remarkable that the beta function
is so similar in spite of the very different dynamics.
We shall see in section IVA that this fact is quite general.
\subsection{The chiral models}
Next we consider the case where $Y$ is the group $SU(N)$.
In this case it is customary to denote $U(x)$ the matrix (in the fundamental representation)
that corresponds to the coordinates $\varphi^\alpha$.
We demand that the theory be invariant under left and right multiplications
$U(x)\mapsto g_L^{-1}U(x)g_R$, forming the group $SU(N)_L\times SU(N)_R$
(``chiral symmetry'').
Further we demand that the theory be invariant under the discrete symmetries
$U(x)\mapsto U^T(x)$, which corresponds physically to charge conjugation,
to the simple parity $x_1\mapsto-x_1$, to the involutive isometry
$\Phi_0:U\to U^{-1}$
and hence to Parity $U(x_1,x_2,x_3,x_4)\mapsto U^{-1}(-x_1,x_2,x_3,x_4)$.
More details on the translation between the tensor and the matrix formalism
are given in Appendix A.
Let $e_a$ be a basis of the Lie algebra, with $a=1\ldots n^2-1$.
We denote $T_a$ the corresponding matrices in the fundamental representation;
they are a set of hermitian, traceless $N\times N$ matrices.
We fix the normalization of the basis by the equation
\begin{equation}
\label{normalization}
T_aT_b=\frac{1}{2N}\delta_{ab}+\frac{1}{2}\left(d_{abc}+if_{abc}\right)T_c\,.
\end{equation}
(In the case of $SU(3)$ these matrices are one half the Gell-Mann $\lambda$ matrices.)
A tensor on $SU(N)$ which is invariant under
$SU(N)_L\times SU(N)_R$ is said to be ``biinvariant''.
There is a one to one correspondence between
biinvariant tensors on $SU(N)$ and $Ad$-invariant tensors
in the Lie algebra of $SU(N)$, where $Ad$ is the adjoint representation.
Given an $Ad$-invariant tensor $t_{ab\ldots}{}^{cd\ldots}$ on the algebra, the
corresponding biinvariant tensorfield on the group is
$$
t_{\alpha\beta\ldots}{}^{\gamma\delta\ldots}=
t_{ab\ldots}{}^{cd\ldots}L^a_\alpha L^b_\beta\ldots L_c^\gamma L_d^\delta\ldots
$$
where $L^a_\alpha$ are the components of the left-invariant Maurer Cartan form
$L=U^{-1}dU=L^a_\alpha dy^\alpha (-iT_a)$ and $L_a^\alpha$ are the components of the left-invariant
vectorfields on $SU(N)$. The matrix $L_a^\alpha$ is the inverse of $L^a_\alpha$.
(In this construction we could use equivalently right-invariant objects.)
Up to rescalings, there is a unique $Ad$-invariant inner product in the Lie algebra,
which we choose as $h_{ab}=2\mathrm{Tr}T_aT_b=\delta_{ab}$
\footnote{Here the matrices are in the fundamental representation.
The Cartan-Killing form just differs by a constant: $B_{ab}=\mathrm{Tr}(Ad(T_a)Ad(T_b))=N\delta_{ab}$.}.
Then the corresponding biinvariant metric is
\begin{equation}
\label{biinvariantmetric}
h_{\alpha\beta}=L^a_\alpha L^b_\beta\delta_{ab}\ ,
\end{equation}
so that the left-invariant vectorfields $L_a$ can also be regarded as a vierbein.
The Riemann and Ricci tensors and the Ricci scalar of $h$ are given by
\begin{equation}
\label{curvaturesun}
R_{\alpha\beta\gamma\delta}=\frac{1}{4}L^a_\alpha L^b_\beta L^c_\gamma L^d_\delta f_{ab}{}^e f_{ecd}\ ;\qquad
R_{\alpha\beta}=\frac{1}{4}N h_{\alpha\beta}\ ;\qquad
R=\frac{1}{4}N(N^2-1)\ .
\end{equation}
As with the sphere, we define
$\g2_{\alpha\beta}=\frac{1}{g^2}h_{\alpha\beta}$,
$\g4_{\alpha\beta}=\frac{1}{\lambda}h_{\alpha\beta}$.
The tensors $d_{abc}$ and $f_{abc}$ are a totally symmetric and a
totally antisymmetric $Ad$-invariant three tensor in the algebra.
In principle chiral invariance would permit a term in the action with
$A_{\alpha\beta\gamma}=L^a_\alpha L^b_\beta L^c_\gamma d_{abc}$;
however using $L^a_\alpha(\Phi_0(y))=R^a_\alpha(y)$,
$L_a^\alpha(y)R^b_\alpha(y)=Ad(g(y))^b{}_a$
and the $Ad$-invariance of $d_{abc}$, one sees that
$A_{\alpha\beta\gamma}(\Phi_0(y))=A_{\alpha\beta\gamma}(y)$,
so this term violates Parity.
For $T$ we have the following $Ad$-invariant
four-tensors in the algebra with the correct symmetries:
\begin{eqnarray}
T^{(1)}_{abcd}&=&\frac{1}{2}\left(\delta_{ac}\delta_{bd}+\delta_{ad}\delta_{bc}\right)\ ;\qquad
T^{(2)}_{abcd}=\delta_{ab}\delta_{cd}\ ;\qquad
T^{(3)}_{abcd}=\frac{1}{2}\left(f_{ace}f_{bd}{}^e+f_{ade}f_{bc}{}^e\right)\ ;
\nonumber
\\
T^{(4)}_{abcd}&=&\frac{1}{2}\left(d_{ace}d_{bd}{}^e+d_{ade}d_{bc}{}^e\right)\ ;\qquad
T^{(5)}_{abcd}=d_{abe}d_{cd}{}^e\ .
\end{eqnarray}
They are not all independent, however.
The identity (2.10) of \cite{macfarlane} implies that
\begin{equation}
\label{firstrelation}
\frac{2}{N}T^{(1)}-\frac{2}{N}T^{(2)}+T^{(3)}+T^{(4)}-T^{(5)}=0\ ,
\end{equation}
so that $T^{(5)}$ can be eliminated.
In the case $N=3$ the identity (2.23) of \cite{macfarlane},
together with the preceding relation, further implies
\begin{equation}
\label{secondrelation}
T^{(2)}-T^{(3)}-3 T^{(4)}=0\ ,
\end{equation}
so that we can also eliminate $T^{(4)}$.
Finally in the case $N=2$ the tensor $d_{abc}$ is identically zero,
so we can keep only $T^{(1)}$ and $T^{(2)}$ as independent combinations,
and use $T^{(3)}=T^{(2)}-T^{(1)}$.
The action of the generic $SU(N)$ models can then be written in the form:
\begin{equation}
\label{actionsun}
\int d^4x\Biggl[
\frac{1}{2g^2}h_{\alpha\beta}\partial_\mu\varphi^\alpha\partial^\mu\varphi^\beta
+\frac{1}{2\lambda}h_{\alpha\beta}\Box\varphi^\alpha\Box\varphi^\beta
+\frac{1}{2}\partial_\mu\varphi^\alpha\partial^\mu\varphi^\beta
\partial_\nu\varphi^\gamma\partial^\nu\varphi^\delta
\sum_{i=1}^4 \ell_i T^{(i)}_{\alpha\beta\gamma\delta}
\Biggr]
\end{equation}
and the sum stops at $i=3$ and $i=2$ for $N=3$ and $N=2$ respectively.
As in \eqref{sphericalaction}, it will be convenient to use instead of the couplings $\ell_i$
the combinations $f_i=\lambda\ell_i$.
Making repeated use of traces given in \cite{macfarlane2} one finds the following beta functions:
\begin{eqnarray}
\label{bsunlambda}
\beta_\lambda & = & -\frac{N}{32\pi^2}\lambda^2
\\
\label{bsun1}
\beta_{f_1} & = &
\frac{\lambda}{768\pi^2 N^2}
\Bigl[16N^2(N^2+20)f_1^2+64N^2f_2^2+180N^2f_3^2+4(149N^2-1280)f_4^2
\nonumber\\
&&
\qquad\qquad
+320N^2f_1f_2-32N^3f_1f_3+32N(N^2+4)f_1f_4+128Nf_2f_4-120N^2f_3f_4
\nonumber\\
&&
\qquad\qquad
+24N^3f_1-108N^2f_3+36N^2f_4+9N^2
\Bigr]
\\
\label{bsun2}
\beta_{f_2} & = &
\frac{\lambda}{768\pi^2 N^2}
\Bigl[
8N^2(N^2+14)f_1^2+32N^2(6N^2+1)f_2^2+60N^2f_3^2+4(7N^2+656)f_4^2
\nonumber\\
&&
\qquad\qquad
+32N^2(3N^2+14)f_1f_2+80N^3f_1f_3+16N(7N^2-44)f_1f_4+288N^3f_2f_3
\\
&&
\qquad\qquad
+32N(15N^2-64)f_2f_4+120N^2f_3f_4-24N^3(f_1+3f_2)-36N^2(f_3+f_4)+3N^2
\Bigr]
\nonumber\\
\label{bsun3}
\beta_{f_3} & = &
\frac{\lambda}{1536\pi^2 N}
\Bigl[52N^2f_3^2+12(23N^2-320)f_4^2+768Nf_1f_3+256Nf_1f_4
+384Nf_2f_3+128Nf_2f_4
\nonumber\\
&&
\qquad\qquad
+24(11N^2-64)f_3f_4-192N(f_1+f_2)-60N^2(f_3+f_4)+384f_4+N^2
\Bigr]
\\
\label{bsun4}
\beta_{f_4} & = &
\frac{\lambda}{1536\pi^2 N}
\Bigl[60N^2f_3^2+4(87N^2-1728)f_4^2+1536Nf_1f_4
+768Nf_2f_4+216N^2f_3f_4
\nonumber\\
&&
\qquad\qquad
-36N^2(f_3+f_4)+3N^2
\Bigr]
\\
\label{bsung}
\beta_{\tilde g^2} & = & 2\tilde g^2
+\frac{\tilde g^4}{16 N\pi^2}
\left(N(N^2+4)f_1+2N(2N^2-1)f_2+3N^2f_3+5(N^2-4)f_4-N^2\right)
\nonumber\\
&&
-\frac{\lambda\tilde g^2}{16 N\pi^2}
\left(N(N^2+4)f_1+2N(2N^2-1)f_2+3N^2f_3+5(N^2-4)f_4-N^2/2\right)
\qquad
\end{eqnarray}
In appendix A we establish the dictionary between our notation and that
used in \cite{hasenfratz}. When the beta functions are compared, we find
perfect agreement, except for one small difference:
the very last term in the first line of $\beta_{\tilde g^2}$ would be $N^2/2$ according to
\cite{hasenfratz}, {\it i.e.} $\tilde g^4$ and $\lambda\tilde g^2$
would have the same coefficients.
This is the same difference that we observed between \eqref{betatrunc2III}
(type III cutoff) and \eqref{betatrunc2I} (type I cutoff),
so, effectively the calculation in \cite{hasenfratz} is equivalent to a type I cutoff.
Given that the calculation in \cite{hasenfratz} was done using completely
different techniques, this agreement confirms that the one loop beta functions
of the dimensionless couplings (which in a calculation of the effective action
would correspond to logarithmic divergences) is scheme independent.
The cases $N=3$ and $N=2$ have to be treated separately, because in these
cases only three, respectively two, of the couplings $f_i$ are independent.
In the case $N=3$ one can eliminate $f_4$ in favor of the other three couplings.
Then using \eqref{secondrelation}
one can obtain the beta functions of $f_1$, $f_2$ and $f_3$
from the ones given above by
\begin{eqnarray*}
\label{betasuthree}
\beta_{f_1}\Big|_{N=3} & = &\beta_{f_1}\Big|_{N=3,f_4=0}
=\frac{\lambda}{768\pi^2}
\left[464f_1^2+64f_2^2+180f_3^2+320f_1f_2-96f_1f_3+72f_1-108f_3+9\right]
\\
\beta_{f_2}\Big|_{N=3} & = &\beta_{f_2}+\frac{1}{3}\beta_{f_4}\Big|_{N=3,f_4=0}
\\
= \frac{\lambda}{1536\pi^2}&&\!\!\!\!\!\!\!\!
\Bigl[368f_1^2+3520f_2^2+180f_3^2+2624f_1f_2+480f_1f_3+1728f_2f_3-144f_1-432f_2-108f_3+9\Bigr]
\\
\beta_{f_3}\Big|_{N=3} &=&\beta_{f_3}-\frac{1}{3}\beta_{f_4}\Big|_{N=3,f_4=0}
=\frac{\lambda}{32\pi^2}
\left[2f_3^2+16f_1f_3+8f_2f_3-4f_1-4f_2-3f_3\right]
\end{eqnarray*}
In the case $N=2$ we can set $f_4=0$, because $T^{(4)}=0$ identically,
and we can eliminate $f_3$.
One can obtain the beta functions of $f_1$, $f_2$ from the ones given above by
\begin{eqnarray*}
\label{betasutwo}
\beta_{f_1}\Big|_{N=2} & = &\beta_{f_1}-\beta_{f_3}\Big|_{N=2,f_3=0,f_4=0}
=\frac{\lambda}{96\pi^2}
\left[48f_1^2+8f_2^2+40f_1f_2+18f_1+12f_2+1\right]
\\
\beta_{f_2}\Big|_{N=2} & = &\beta_{f_2}+\beta_{f_3}\Big|_{N=2,f_3=0,f_4=0}
=\frac{\lambda}{192\pi^2}
\left[36f_1^2+200f_2^2+208f_1f_2-36f_1-60f_2+1\right]
\end{eqnarray*}
The latter result can be used to check also our beta functions for the spherical sigma model.
In fact there is exactly one manifold which is simultaneously a sphere
and a special unitary group: it is $SU(2)=S^3$.
Thus the beta functions should agree in this case.
Before comparing, a little point needs
to be addressed. In section IIIC we chose the metric $h_{\alpha\beta}$
to be that of a sphere of unit radius.
In this section we have fixed the metric by the conditions \eqref{normalization},
\eqref{biinvariantmetric}.
It turns out that in the case $N=2$ this normalization corresponds to a sphere of radius two.
This can be seen for example from equation \eqref{curvaturesun}, specialized
to $N=2$, with $f_{abc}=\varepsilon_{abc}$.
In order to compare the beta functions of $S^3$
with those for $SU(2)$ we therefore have to redefine $\lambda\to\lambda/4$,
$f_1\to 4f_1$, $f_2\to 4f_2$, $g^2\to g^2/4$.
With these redefinitions, the beta functions do indeed agree.
\section{Fixed points}
\subsection{The spherical models}
We now discuss solutions of the RG flow equations.
The beta function of $\lambda$ depends only on $\lambda$ and the solution is
\begin{equation}
\label{lambdasol}
\lambda(t)=\frac{\lambda_0}{1+\lambda_0\frac{n-1}{8\pi^2}(t-t_0)}\ ,
\end{equation}
where $\lambda_0=\lambda(t_0)$. We assume $\lambda_0>0$, thus $\lambda$ is asymptotically free.
The beta functions of $f_1$ and $f_2$ do not depend on $g$,
so their flow can be studied independently.
Here we do not discuss general solutions but merely look for fixed points.
The overall factor $\lambda$ in these beta functions can be eliminated by
a simple redefinition $t=t(\tilde t)$ of the parameter along the RG trajectories:
\begin{equation}
\label{redef}
\frac{d}{d\tilde t}=\frac{1}{\lambda}\frac{d}{dt}\,.
\end{equation}
Since $\tilde t$ is a monotonic function of $t$,
the FPs for $f_1$ and $f_2$ are the zeroes of the modified beta functions
$$
\tilde\beta_{f_i}=\frac{df_i}{d\tilde t}=\frac{1}{\lambda}\beta_{f_i}\,.
$$
They are just polynomials in $f_1$ and $f_2$.
The model has no real FP for $n=2$, but there are FPs for all $n>2$.
For $n=3,\ldots 8$ they are given in the fifth and sixth column in Table I.
One can then insert the FP values of $f_1$ and $f_2$ in $\beta_{\tilde g^2}$
and look for FP of $\tilde g^2$.
In each case there are two solutions, one at $\tilde g^2=0$, the other at
some nonzero value. These solutions are reported in the fourth column in Table I,
for $n=3,\ldots 8$.
The first solution describes the Gaussian FP (GFP),
where all the couplings $\tilde g^2$, $\lambda$, $1/\ell_1$, $1/\ell_2$ are zero,
the others non Gaussian FP's (NFP) where $\tilde g^2$ has finite limits instead.
Each FP can be approached only from specific directions in the space parametrized by
$\lambda$, $\ell_1$, $\ell_2$, {\it i.e.} the ratios $f_1$ and $f_2$ take specific values.
For each NFP these values are unique, while for the GFP there may be several possible values:
two if $n=3,4,5$ and four if $n=6,7,8$.
When one considers the linearized flow around any of the GFPs,
one finds as expected that the critical exponents, defined as minus the eigenvalues
of the matrix $\frac{\partial\beta_i}{\partial g_j}$,
are -2,0,0,0, corresponding to the canonical dimensions of the couplings.
The critical exponents at the NGP are instead 2,0,0,0.
Thus the dimensionless couplings are marginal, and of the two FPs,
the trivial one is IR attractive and the nontrivial one UV attractive for $\tilde g$.
For $\lambda$ it is clear that the FP is UV attractive
(if we had chosen $\lambda<0$ it would be IR attractive).
In order to establish the attractive or repulsive character of $f_1$ and $f_2$,
one can look at the linearized flow in the variable $\tilde t$,
which is described by the $2\times2$ matrix
$$
\frac{\partial\tilde\beta_{f_i}}{\partial f_j}\,.
$$
We define the ``critical exponents'' $\theta_{1,2}$ to be minus the eigenvalues
of this matrix. They are reported in the last two columns of table I, for $n=3,\ldots 8$.
It is important to realize that even for the GFP the eigenvectors of the stability matrix are
not the operators that appear in the action but mixings thereof.
We do not report the eigenvectors here.
\begin{table}
\begin{center}
\begin{tabular}{|l|l|l|l|l|l|l|l|l|} \hline
$n$ & $\tilde g_*^{(III)}$\ & FP &\ $\tilde g_*$\ & \ $f_{1*}$\ &\ $f_{2*}$\ &\ $\theta_1$\ &\ $\theta_2$\ \\
\hline
3& 8.886 & NFP1 & 6.626 & -0.693 & 0.453 &\ 0.094 & -0.0121 \\
3& & NFP2 & 6.390 & -1.042 & 0.615 &\ 0.103 &\ 0.0119 \\
3& & GFP1 & 0 & -0.693 & 0.453 &\ 0.094 & -0.0121 \\
3& & GFP2 & 0 & -1.042 & 0.615 &\ 0.103 &\ 0.0119 \\
\hline
4& 7.255 & NFP1 & 5.877 & -0.479 & 0.398 &\ 0.105 & -0.0412 \\
4& & NFP2 & 5.442 & -1.555 & 0.852 &\ 0.132 &\ 0.0392 \\
4& & GFP1 & 0 & -0.479 & 0.398 &\ 0.105 & -0.0412 \\
4& & GFP2 & 0 & -1.555 & 0.852 &\ 0.132 &\ 0.0392 \\
\hline
5& 6.283 & NFP1 & 5.310 & -0.400 & 0.400 &\ 0.118 & -0.0608 \\
5& & NFP2 & 4.924 & -1.875 & 0.988 &\ 0.154 &\ 0.0567 \\
5& & GFP1 & 0 & -0.400 & 0.400 &\ 0.118 & -0.0608 \\
5& & GFP2 & 0 & -1.875 & 0.988 &\ 0.154 &\ 0.0567 \\
\hline
6& 5.620 & NFP1 & 4.883 & -0.350 & 0.408 &\ 0.131 & -0.0780 \\
6& & NFP2 & 4.577 & -2.131 & 1.091 &\ 0.171 & -0.0717 \\
6& & GFP1 & 0 & -0.350 & 0.408 &\ 0.131 & -0.0780 \\
6& & GFP2 & 0 & -2.131 & 1.091 &\ 0.171 &\ 0.0717 \\
6& & GFP3 & 0 & -0.814 & 1.369 & -0.161 & -0.0539 \\
6& & GFP4 & 0 & -2.363 & 2.091 & -0.164 & -0.0617 \\
\hline
7& 5.130 & NFP1 & 4.548 & -0.314 & 0.417 &\ 0.143 & -0.0939 \\
7& & NFP2 & 4.322 & -2.347 & 1.175 &\ 0.185 &\ 0.0851 \\
7& & GFP1 & 0 & -0.314 & 0.417 &\ 0.143 & -0.0939 \\
7& & GFP2 & 0 & -2.347 & 1.175 &\ 0.185 &\ 0.0851 \\
7& & GFP3 & 0 & -2.790 & 2.130 & -0.181 & -0.0647 \\
7& & GFP4 & 0 & -0.598 & 1.241 & -0.174 & -0.0716 \\
\hline
8& 4.750 & NFP1 & 4.274 & -0.286 & 0.424 &\ 0.156 & -0.1092 \\
8& & NFP2 & 4.125 & -2.535 & 1.247 &\ 0.197 &\ 0.0976 \\
8& & GFP1 & 0 & -0.286 & 0.424 &\ 0.156 & -0.1092 \\
8& & GFP2 & 0 & -2.535 & 1.247 &\ 0.197 &\ 0.0976 \\
8& & GFP3 & 0 & -2.790 & 2.131 & -0.180 &\ 0.1023 \\
8& & GFP4 & 0 & -0.598 & 1.247 & -0.187 & -0.0872 \\
\hline
\end{tabular}
\end{center}
\caption{Gaussian and non-Gaussian fixed points of the $S^n$ model at one loop.
The first column gives the dimension $n$. The second column gives the
position of the NGFP in the two-derivative truncation, using a type III cutoff.
The rest of the table refers to the four-derivative truncation, also using a type III cutoff.
The third column gives the name of the FP. Columns 4,5,6 give the
position of the NGFP, columns 7,8 the critical exponents, as defined in the text.
The coupling $\lambda$, not listed, goes to zero and is marginal in this approximation.}
\label{table1}
\end{table}
Beyond the values given in Table I, we have checked numerically the existence of the FP up to $n=200$.
For large $n$ one can study the theory analytically, to some extent.
There are four FPs for the system of the $f_i$'s, which are:
$f_1=0,\,f_2=1$ with critical exponents $\theta_1=6,\,\theta_2=12$,
$f_1=0,\,f_2=1/2$ with critical exponents $\theta_1=6,\,\theta_2=-12$,
$f_1=-6,\,f_2=5/2$ with critical exponents $\theta_1=-6,\,\theta_2=12$,
$f_1=-6,\,f_2=2$ with critical exponents $\theta_1=-6,\,\theta_2=-12$.
The numerical values at finite $n$ do indeed tend towards these limits for growing $n$.
\subsection{The chiral models}
The chiral model with $N=2$ is equivalent to the spherical model with $n=3$
(up to the redefinition of the couplings mentioned in the end of section IIID)
so we need not discuss this case further.
For ease of comparison we just report the properties of its nontrivial FPs
in the parametrization we used for the chiral models:
\begin{eqnarray}
NFP1: &&f_{1*}=-0.173\ ;\qquad f_{2*}=0.113\ ;\qquad \tilde g=13.25\nonumber \\
NFP2: &&f_{1*}=-0.261\ ;\qquad f_{2*}=0.154\ ;\qquad \tilde g=12.78\nonumber
\end{eqnarray}
The critical exponents don't depend on the definition of the couplings and therefore
are the same as in Table I;
they do however depend on the choice of RG parameter and they
differ from those given in \cite{hasenfratz} by a factor
$4\pi^2$, which is due to the definition of the parameter $x$ there.
In the case $N=3$ the system of the $f_i$'s has two FPs at
\begin{eqnarray}
FP1:&&\ \ f_{1*}=-0.154\ ;\qquad f_{2*}=0.050\ ;\qquad f_{3*}=0.085\ ;\nonumber\\
FP2:&&\ \ f_{1*}=-0.108\ ;\qquad f_{2*}=0.043\ ;\qquad f_{3*}=0.061\ .\nonumber
\end{eqnarray}
The attractivity properties in the space spanned by the $f_i$'s is given,
as in the spherical case, by studying the modified flow with parameter $\tilde t$.
The critical exponents at FP1 are:
0.0303 with eigenvector (0.411, 0.630, 0.658);
0.0123 with eigenvector (0.515, -0.570, 0.640);
0.00289 with eigenvector (0.869, -0.148, -0.473),
whereas at FP2 they are:
0.0280 with eigenvector (0.366, 0.618, 0.695);
0.0108 with eigenvector (0.513, -0.575, 0.638) and
-0.00293 with eigenvector (0.887, -0.125, -0.445).
Therefore FP1 is attractive in all three directions,
while FP2 is attractive in two directions.
For each of these two FP's, the beta function of $\tilde g$ has two FP's:
the trivial FP, which has always critical exponents -2,
and a nontrivial FP, which is located at $\tilde g=11.17$ for NFP1
or $11.50$ for NFP2, and having critical exponent 2 in both cases.
We have found no FP's for $N>3$:
the system of equations $\tilde\beta_{f_i}=0$ for $i=1,2,3,4$ only has complex solutions.
To cover all of theory space we have checked this statement also
in the parametrization of the $\ell_i$ and in the parametrization of $u_i=1/\ell_i$.
This is true also in the large $N$ limit.
If we keep only the leading terms (of order $N^2$ for $f_1$ and $f_2$
and of order $N$ for $f_3$ and $f_4$), again the resulting polynomials
do not have any real zero.
\section{Discussion}
We have calculated the one loop beta functionals of the NLSM with values in any manifold,
in the presence of a very general class of four derivative terms.
We have then specialized our results to two infinite families of models:
the $O(N)$ models, with values in spheres, and the chiral models
with values in the groups $SU(N)$.
Such calculations had been done before, but since the results are rather
complicated, it is useful to have independent verifications.
Our approach is calculationally very similar to \cite{bk},
but after correcting some small errors at the general level,
we find that the FP structure of the $O(N)$ models is completely different
from their findings.
On the other hand our results for the chiral models agree completely
with \cite{hasenfratz} for what concerns the dimensionless couplings,
even though the calculation was done using very different techniques.
Since $SU(2)=S^3$, this provides a check also for our results for the spheres.
Since our aim is to establish asymptotic safety, or lack thereof,
it is important for us to have also the beta functions of the dimensionful
coupling $g$, which in the chiral models is the inverse of the pion decay constant.
This had not been considered at all in \cite{bk}, but it had been calculated
in \cite{hasenfratz} for the chiral models. Again we have agreement with the
result of \cite{hasenfratz}, up to a single factor 2 in one term; as discussed before,
since this beta function is scheme dependent, we believe
that this is not an error on either side, but the result of the different way
in which the calculation was done.
This difference results in a shift of the FP value of $\tilde g$;
for example in the case of $SU(2)$ one would find $\tilde g=19.88$ instead of 13.25 for NFP1
and 18.39 instead of 12.78 for NFP2.
Such variations by a factor of order 2 are to be expected.
One of the motivations of this work was to use the NLSM as a toy model for gravity.
From this point of view we have a perfect correspondence of results.
If we use the $1/p^2$ propagator that comes from the two derivative term,
both theories are perturbatively unitary but nonrenormalizable;
if on the other hand we use the $1/p^4$ propagator that comes from
the four derivative terms both theories are renormalizable
(see \cite{stelle} for gravity and \cite{slavnov} for the NLSM) but contain ghosts.
In the latter case it had also been established (see \cite{julve,fradkin,avramidi,shapiro}
for gravity and \cite{hasenfratz, bk} for the NLSM) that the four derivative terms,
whose couplings are dimensionless, are asymptotically free.
Actually the analogy works even in greater detail.
The coefficient of the square of the Weyl tensor (for gravity)
and the square of $\Box\varphi^\alpha$ (for the NLSM)
have at one loop a beta function that is constant.
These coefficients diverge logarithmically in the UV,
so their inverses, which are the perturbative couplings,
are asymptotically free.
The coefficients of the other four derivative terms have more complicated
beta functions, but overall there is asymptotic freedom, provided the
Gaussian FP is approached from some special direction.
There have been many attempts to avoid the effects of the ghosts,
see \cite{julve,salam} for gravity and \cite{hasenfratz} for the NLSM.
In any case, the existence of the ghosts is only established at tree level.
Whether they exist in the full quantum theory is a deep dynamical question
whose answer is not known.
All these ``old'' works on higher derivatives theories concentrated on the
behavior of the couplings that multiply the four derivative terms;
much less attention, if any, was payed to the coefficient of the two derivative term,
which has dimension of square of a mass:
the inverse of Newton's constant in gravity and
the square of the pion decay constant in the chiral NLSM.
In several papers this issue was ignored, or incorrect results were given,
because of the use of dimensional regularization.
The correct RG flow of these couplings is quadratic
in $k$, and is best seen when a momentum cutoff is used.
In \cite{codello1} this point was made for gravity and it was shown
that when this quadratic running is taken into account
the beta function for Newton's constant (and for the cosmological constant)
has, in addition to the Gaussian one, also a nontrivial FP.
In this paper we have found that the same is true in a large class of NLSM.
This is crucial for asymptotic safety.
It is somewhat gratifying to see that the FP does not always exist for all NLSM:
in particular we have seen that within the one loop approximation,
adding the higher derivative terms destroys the FP that is present in the
two derivative truncation for the sphere $S^2$ and for the chiral models with $N>3$.
If there was any doubt,
this shows that the existence of the FP is not ``built into the formalism''
but is a genuine property of the theory.
This is somewhat analogous to the situation when one adds minimally
coupled matter fields to gravity \cite{perini1}.
The next step will be to replace the one loop functional RG equation \eqref{onelooperge}
by its {\it exact} counterpart, which only differs in the replacement of
the bare action $S$ by $\Gamma_k$ in the r.h.s. \cite{wetterich}.
There are at least two good reasons to do this calculation.
One of the points of \cite{codello2} that needed further clarification was the value
of the lowest critical exponent. In the two derivative truncation at one loop it was
always 2 at the nontrivial FP.
Thus the critical exponent $\nu$ that governs the rate at which the correlation length
diverges was given by
$$
\nu=-\frac{1}{\frac{d\beta}{dt}_*}=\frac{1}{\theta}=\frac{1}{2}\ ,
$$
which is the value of mean field theory.
Using the ``exact'' RG truncated at two derivatives gave $\nu=3/8$ for the $O(N)$ models,
independent of $N$.
One would like to understand what effect the higher derivative terms have on this exponent.
Since here we restricted ourselves to one loop, we found again $\nu=1/2$,
so the calculations of this paper are of no use in this respect.
Another motivation comes from recent calculations in higher derivative gravity
\cite{bms} that go beyond one loop and find that the theory is not asymptotically free,
but rather all couplings reach nonzero values at the UV FP.
It would be interesting to see similar behaviour in (some) NLSM.
Concerning possible direct phenomenological applications of the NLSM,
regarded as an effective field theory,
it is interesting to ask what relation, if any, the UV properties
of the NLSM may have to the properties of the underlying fundamental theory.
Regarding the chiral NLSM as the low energy approximations of a QCD-like theory,
one may note that there is rough agreement between the range of existence
of the NLSM FP and the ``conformal window'' for the existence of an IR FP
in the case when the quarks are in the adjoint or in the symmetric tensor
representation \cite{sannino}.
One could get a better understanding of this issue if the beta functions of
the NLSM depended on the number of ``colors'' of the underlying theory,
which in the effective theory are reflected in the coefficient of the
Wess-Zumino-Witten term \cite{witten}.
The one loop beta function of the Wess-Zumino-Witten term is zero
\cite{braaten,bk};
this is consistent with the quantization of the coefficient $c$.
Unfortunately the beta functions of the remaining couplings are completely
independent of this coefficient, so the low energy theory seems to be
insensitive to this parameter.
Another possible application is to electroweak chiral perturbation theory \cite{ab}.
If the NLSM turned out to be asymptotically safe in the presence of gauge fields and
fermions, then one may envisage a higgsless standard model up to very high energies.
This will also require a separate investigation.
A related application of asymptotic safety to the standard model has been
discussed recently in \cite{gies}.
To summarize we believe that the NLSM are interesting theoretical laboratories
in which one may test various theoretical ideas, and they have also important
phenomenological applications.
The question whether some NLSM could be asymptotically safe seems to us
to be a particularly important one, and to deserve more attention.
\medskip
\centerline{Acknowledgements}
\noindent
We would like to thank J. Ambj\o rn and F. Sannino for
discussions and S. Ketov for correspondence.
\goodbreak
\section{Appendix A}
In \cite{hasenfratz} the action for the chiral $SU(N)$ model is written in the form:
\begin{eqnarray}
\frac{1}{f^2}&&\!\!\!\!\!\!\int d^4x\,\Bigl(
c_0 \mathrm{Tr} L_\mu L^\mu
+\frac{1}{2}\mathrm{Tr}(\partial_\mu L^\mu\partial_\nu L^\nu+\partial_\mu L_\nu\partial^\mu L^\nu)
-\frac{1}{2}c_2\mathrm{Tr}(\partial_\mu L^\mu\partial_\nu L^\nu-\partial_\mu L_\nu\partial^\mu L^\nu)
\nonumber\\
&&\!\!\!\!\!\!
-\frac{1}{2}c_3\mathrm{Tr}(L_\mu L^\mu L_\nu L^\nu+L_\mu L_\nu L^\mu L^\nu)
-c_4\mathrm{Tr}(L_\mu L^\mu) \mathrm{Tr}(L_\mu L^\mu)
-c_5\mathrm{Tr}(L_\mu L^\mu) \mathrm{Tr}(L_\mu L^\mu)
\Bigr]\ .
\label{hasenfratzaction}
\end{eqnarray}
where $L_\mu=U^{-1}\partial_\mu U$.
We want to translate this action into the form \eqref{actionsun}.
Deriving the equation $L_\mu=\partial_\mu\varphi^\alpha L^a_\alpha (-iT_a)$
we obtain
$$
\partial_\mu L_\nu=-i T_a(\nabla_\mu\partial_\nu\varphi^\alpha L_\alpha^a
-\partial_\mu\varphi^\alpha\partial_\nu\varphi^\beta \nabla_\alpha L^a_\beta)
$$
The antisymmetric part of this equation is
$$
\partial_\mu L_\nu-\partial_\nu L_\mu=-[L_\mu,L_\nu]
$$
whereas using Killing's equation, the symmetric part is
$$
\partial_{(\mu} L_{\nu)}=-i T_a\nabla_\mu\partial_\nu\varphi^\alpha L_\alpha^a
$$
The terms of \eqref{hasenfratzaction} have the following translation into
our
tensorial language:
\begin{eqnarray*}
\int d^4x\,\mathrm{Tr} L_\mu L^\mu&=&-\frac{1}{2}
\int d^4x\,\partial_\mu\varphi^\alpha\partial^\mu\varphi^\beta h_{\alpha\beta}
\\
\int d^4x\,\mathrm{Tr}\partial_\mu L^\mu\partial_\nu L^\nu&=&
-\frac{1}{2}\int d^4x\,
\Box\varphi^\alpha \Box\varphi^\beta h_{\alpha\beta}
\\
\int d^4x\,\mathrm{Tr}\partial_\mu L_\nu\partial^\mu L^\nu&=&
\!-\frac{1}{2}\!\int d^4x\!\left(
\nabla^\mu\partial^\nu\varphi^\alpha \nabla_\mu\partial_\nu\varphi^\beta h_{\alpha\beta}
+\frac{1}{4}
\partial_\mu\varphi^\alpha\partial^\mu\varphi^\beta
\partial_\nu\varphi^\gamma\partial^\nu\varphi^\delta
T^{(3)}_{\alpha\beta\gamma\delta}\right)
\\
\int d^4x\,\mathrm{Tr}L_\mu L^\mu L_\nu L^\nu&=&
\int d^4x\,
\partial_\mu\varphi^\alpha\partial^\mu\varphi^\beta
\partial_\nu\varphi^\gamma\partial^\nu\varphi^\delta
\left(\frac{1}{4N}T^{(2)}_{\alpha\beta\gamma\delta}
+\frac{1}{8}T^{(5)}_{\alpha\beta\gamma\delta}\right)
\\
\int d^4x\,\mathrm{Tr}L_\mu L_\nu L^\mu L^\nu&=&
\int d^4x\,
\partial_\mu\varphi^\alpha\partial^\mu\varphi^\beta
\partial_\nu\varphi^\gamma\partial^\nu\varphi^\delta
\left(\frac{1}{4N}T^{(1)}_{\alpha\beta\gamma\delta}
-\frac{1}{8}T^{(3)}_{\alpha\beta\gamma\delta}
+\frac{1}{8}T^{(4)}_{\alpha\beta\gamma\delta}\right)
\\
\int d^4x\,\mathrm{Tr}(L_\mu L^\mu) \mathrm{Tr}(L_\nu L^\nu)&=&
\frac{1}{4}\int d^4x\,
\partial_\mu\varphi^\alpha\partial^\mu\varphi^\beta
\partial_\nu\varphi^\gamma\partial^\nu\varphi^\delta
T^{(2)}_{\alpha\beta\gamma\delta}
\\
\int d^4x\,\mathrm{Tr}(L_\mu L_\nu) \mathrm{Tr}(L^\mu L^\nu)&=&
\frac{1}{4}\int d^4x\,
\partial_\mu\varphi^\alpha\partial^\mu\varphi^\beta
\partial_\nu\varphi^\gamma\partial^\nu\varphi^\delta
T^{(1)}_{\alpha\beta\gamma\delta}
\end{eqnarray*}
where $\Box\varphi^\alpha=\nabla^\mu\partial_\mu\varphi^\alpha$.
One can further manipulate the third term integrating by parts and
commuting covariant derivatives. One finds
$$
\int d^4x\,
\nabla^\mu\partial^\nu\varphi^\alpha \nabla_\mu\partial_\nu\varphi^\beta h_{\alpha\beta}
=\int d^4x\,\left(
\Box\varphi^\alpha \Box\varphi^\beta h_{\alpha\beta}
+\partial_\mu\varphi^\alpha\partial^\mu\varphi^\beta
\partial_\nu\varphi^\gamma\partial^\nu\varphi^\delta
R_{\alpha\beta\gamma\delta}\right)
$$
and using \eqref{curvaturesun} one can further substitute the Riemann tensor by $T^{(3)}$.
In the fourth term one can eliminate $T^{(5)}$.
One has to note that Hasenfratz's action has to be compared to {\it minus}
our action. This is because it appears with the positive sign in the
exponent of the functional integral (this is consistent with the fact
that the $(\Box\varphi)^2$ term has a negative coefficient in \eqref{hasenfratzaction}).
It is then straightforward to calculate the following relations between the
couplings used in \cite{hasenfratz} and our couplings:
$$
g^2=\frac{f^2}{c_0}\ ;\qquad
\lambda=f^2\ ;\qquad
f_1=\frac{c_3}{2N}+\frac{c_5}{2}\ ;\qquad
f_2=\frac{c_4}{2}\ ;\qquad
f_3=\frac{1+c_2}{4}\ ;\qquad
f_4=\frac{c_3}{4}\ ;\qquad
$$
With these relations, one can translate his beta functions and one finds that
they agree with those given in section IIID, with a single exception:
the term proportional to $\tilde g^4$ and containing no $f_i$ in $\beta_{\tilde g^2}$.
We observe that the two polynomials in the $c$'s
in equation (39) in \cite{hasenfratz} are the same, up to an overall factor 2.
As a consequence, when one extracts the beta function of $c_0/f^2=1/g^2$ and
rewrites it in terms of the $f_i$'s, the coefficients of $\tilde g^4$ and $\tilde g^2\lambda$
are exactly the same. This differs from the beta function given in (\ref{bsung}),
where the two coefficients differ in the last term.
We believe that this difference can be attributed to the different cutoff scheme.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 8,682 |
Gridiron Software has released a public beta of their revolutionary new workflow software called Flow. Their website contains lots of information about the product, some videos, and the like, but most importantly, you can now download a beta version and see the thing firsthand for yourself.
Today, only the Mac version is available, but the company claims a Windows version is on the way shortly (probably within a week or so).
Don't we already have a Flow tool from the mensa who brought us Enfocus PitStop - Peter Camps of Gradual ? I guess I will have to take a peek at this, but I can't imagine what it might do better or faster (maybe easier?) that Peter's stuff. | {
"redpajama_set_name": "RedPajamaC4"
} | 3,279 |
\section{Introduction}
We consider two closely-related derivative-free black-box global optimization problems with expensive-to-evaluate objective functions,
\begin{equation}
\max_{x\in A\subset\mathbb{R}^{d}} G(x) := \max_{x\in A\subset\mathbb{R}^{d}}\sum_{w=1}^{n}F\left(x,w\right)p(w)
\label{eq:goal1}
\end{equation}
and
\begin{equation}
\max_{x\in A\subset\mathbb{R}^{d}} G(x) := \max_{x\in A\subset\mathbb{R}^{d}}\int_{}F\left(x,w\right)p(w)dw,
\label{eq:goal2}
\end{equation}
where $A$ is a simple compact set (e.g., a hyperrectangle, simplex, or finite collection of points); $w$ is a vector belonging to a set $W$; $p$ is finite and inexpensive to evaluate with a known analytic form; and $F$ is expensive to evaluate, does not provide derivatives with its evaluations, and may be observable either directly or with independent normally distributed noise. We also assume in \eqref{eq:goal2} that $F(x,\cdot)p(\cdot)$ is integrable for each $x$.
Here, ``expensive-to-evaluate'' functions are ones that consume a great deal of time per evaluation, e.g., minutes or hours each, or whose number is otherwise severely restricted \citep[see, e.g.,][]{sacks1989, booker1998}.
We treat $F$ as a black box and assume that it is continuous in $x$ and also in $w$ in problem \eqref{eq:goal2} and well-represented by a Gaussian process prior \citep{RaWi06} as described below, but make no other assumptions on its structure.
This pair of closely related problems arises in three settings:
\begin{enumerate}
\item First, both \eqref{eq:goal1} and \eqref{eq:goal2} arise when optimizing average-case performance of an engineering system or business process across environmental conditions,
where $F\left(x,w\right)$ is the performance of system design $x$ under environmental condition $w$, and $p(w)$ represents the fraction of time that condition $w$ occurs.
This arises, for example, when choosing
the shape of an aircraft's wing \citep{MLA},
the configuration of a cardiovascular bypass graft \citep{marsden2010},
or the parameters of an algorithm that dispatches cars in a ride-sharing service.
\item Second, both \eqref{eq:goal1} and \eqref{eq:goal2} arise when we wish to optimize the expected value of a system modeled by a discrete-event simulation $f(x,\omega)$, where $\omega$ is random. In this setting, we may choose a random variable $w$ whose distribution $p(w)$ we know, and for which we can simulate $\omega$ given $w$. We may then define $F\left(x,w\right) = E[f(x,\omega) | w]$. Our objective $E[f(x,\omega)]$ becomes either $\sum_{w=1}^n F(x,w) p(w)$ if $w$ is discrete or $\int F(x,w) p(w) dw$ if $w$ is continuous, and we can obtain noisy observations of $F(x,w)$ by simulating $f(x,\omega)$ with $\omega$ drawn from its conditional distribution given $w$. This arises, for example, when building a transportation system to maximize expected service quality subject to stochastic patterns of arrivals through the day $\omega$, which we can simulate given the total number of arrivals in a day $w$. When used for variance reduction in simulation rather than optimization, this technique is known as stratification \citep{glasserman2003monte}.
\item Third, \eqref{eq:goal1} arises when tuning
a machine learning algorithm's hyperparameters by using
k-fold cross-validation. In this application,
$F\left(x,w\right)$ is the error on fold $w$ using hyperparameters $x$ and our goal is to minimize $\sum_w F\left(x,w\right)$. This problem arises more generally when optimizing average performance across multiple prediction tasks \citep{bardenet2013collaborative,hutter2011sequential,swersky2013multi} and is called multi-task Bayesian optimization \citep{swersky2013multi}.
\end{enumerate}
Although \eqref{eq:goal1} and \eqref{eq:goal2} can be considered jointly as maximization of an objective
$\int F\left(x,w\right)d\mu\left(w\right)$ where $\mu$ is a measure,
and our theoretical analysis will at times take this view, \eqref{eq:goal1} and \eqref{eq:goal2} have very different properties computationally and have been considered separately in the literature so we refer to them separately here.
\paragraph{Potential Solution Approaches:}
Problems \eqref{eq:goal1} and \eqref{eq:goal2} may be solved by optimizing $G(x)$ directly with a method designed for derivative-free black box global optimization of expensive and possibly noisy functions. These methods include Bayesian optimization methods \citep{jones1998efficient, forrester2008engineering,brochu2010tutorial} and other surrogate-based optimization methods \citep{barthelemy1993,torczon1997, shoemaker2014}.
Indeed, $G(x)$ can be evaluated using multiple evaluations of $F(x,w)$ by summing in \eqref{eq:goal1} or with numerical quadrature in \eqref{eq:goal2}. However, the expense of evaluating $G(x)$ is many times larger than for $F(x,w)$, especially if $n$ in \eqref{eq:goal1} is large or numerical quadrature in \eqref{eq:goal2} is performed accurately.
This approach is inefficient because it is unable to adjust the computational effort spent on evaluating $G(x)$: it either evaluates it fully or not at all. This inefficiency is most apparent when the first evaluations of $F(x,w)$ at $x$ indicate $G(x)$ is substantially sub-optimal: the extra expense of fully evaluating $G(x)$ is wasted.
These problems may also be solved by applying a black box global optimization method to noisy observations of $G(x)$ obtained via Monte Carlo sampling. One may sample $w_{1},\ldots,w_{m}$ from $p$ and use $\frac{1}{m}\sum_{i=1}^{m}F(x,w_{i})$ as a noisy estimate of $G(x)$. This approach is inefficient because it ignores information about $w$ when building its surrogate for $G$. This inefficiency is most apparent when the first two evaluations of $F(x,w)$ are at the same or very similar $w$. If $F$ is free from noise and varies slowly with $w$, the second such evaluation provides little information beyond the first.
This inefficiency could perhaps be mitigated by using Quasi Monte Carlo \citep[QMC; see, e.g.,][]{glasserman2003monte} together with an optimization method (ideally a multifidelity one, e.g., \citealt{forrester2007multi}) that is tolerant to bias in its noisy observations, but even this would become inefficient when most of $G(x)$'s variability is driven by values of $w$ not sampled until later in the QMC sequence.
These inefficiencies in optimization based on surrogate models of $G$ suggest one may create a more efficient method through surrogate models of $F$, coupled with intelligent selection of points $x$ and $w$ at which to evaluate $F$.
\citet{williams2000sequential} and \citet{swersky2013multi}
developed methods of this type for solving \eqref{eq:goal1},
and \citet{groot2010bayesian,Xie:2012} for solving \eqref{eq:goal2}.
While these approaches can improve performance over modeling $G$ alone, we show in this article that they leave substantial room for improvement.
Indeed, all of these previous approaches except \citet{Xie:2012} use heuristic two-step rules that choose $x$ first without considering $w$, and then choose $w$ with $x$ fixed. As we show below, not considering $x$ and $w$ together causes these methods to perform poorly in certain settings, even sometimes failing to be asymptotically consistent.
Moreover, these previous methods are insufficiently general: \cite{Xie:2012} requires $p$ and the kernel of the covariance of the Gaussian process to be Gaussian, and all previous methods require evaluations of $F$ to be free from noise, significantly restricting their applicability.
\paragraph{Contributions:}
In this paper, we significantly generalize and improve over this previous work by developing a novel method, Bayesian Quadrature Optimization (BQO), that uses a one-step value of information analysis to select the pair of points $x,w$ at which to evaluate $F$.
This method is general and supports solving
either \eqref{eq:goal1} or \eqref{eq:goal2} with noisy or noise-free evaluations of $F$, and this support for noisy observations significantly expands the applicability of our approach within optimization via simulation.
This algorithm is Bayes-optimal by construction when only a single evaluation of $F$ may be made. We also prove that it provides a consistent estimator of the global optimum of $G$ as the number of samples allowed extends to infinity in both the finite and continuum domain settings for the finite sum problem \eqref{eq:goal1}.
Performing the one-step value of information analysis at the heart of BQO requires solving a challenging optimization problem, and we present novel computational methods that solve this problem efficiently,
including a novel discretization-free method for estimating the gradient of the value of information, a new convergence analysis of a different and less efficient discretized scheme more closely related to past work, and a novel transformation that provides a more computationally convenient form of $F$.
We demonstrate that our algorithm substantially outperforms state-of-the-art Bayesian optimization methods when observations are noisy or the integrand varies smoothly in the integrated variables, and performs as well as state-of-the-art methods in the remaining settings. Our demonstrations use a variety of problems from optimization via simulation and hyperparameter tuning in machine learning.
We also provide a robust implementation of our method at \url{https://github.com/toscanosaul/bayesian_quadrature_optimization}.
Our method improves over the previous literature in three ways:
First, it is more \textit{general}, as it is the first to allow noise in the evaluation of $F$, the first to simultaneously support solving both \eqref{eq:goal1} and \eqref{eq:goal2}, and allows general $p$ in contrast with \citet{groot2010bayesian} and \citet{Xie:2012}'s requirement that $p$ be a normal density.
Second, it is more \textit{well-supported theoretically}, as its one-step optimality justification contrasts with the heuristic justification offered in \citet{williams2000sequential,groot2010bayesian} and \citet{swersky2013multi}. (\citealt{Xie:2012} is one-step Bayes-optimal for the special case of \eqref{eq:goal2} that it considers.) Also none of these previous methods come with a proof of consistency, and \citet{williams2000sequential} may fail to be consistent if a poor tie-breaking rule is chosen as we note below. Third, it provides \textit{better empirical performance} in problems with noisy evaluations or when the integrand varies smoothly in the integrated variables, and performs comparably in other problems. We discuss this previous literature in more detail below.
This paper significantly extends the conference paper \cite{MCQMC}, where an early version of the BQO method was referred to as Stratified Bayesian Optimization (SBO). We have re-named our method to reflect its more general ability to solve problems beyond the second use-case based on stratification described at the start of this section. Beyond that conference paper, the current paper includes proofs of consistency for both finite and continuum domains for the finite sum problem \eqref{eq:goal1}, a proof of convergence of the discretized computational method used in that paper, a new discretization-free computational method that is substantially more efficient in higher dimensions, and additional numerical experiments on new problems with new benchmark algorithms.
\paragraph{Detailed Discussion of Related Work:}
\citet{williams2000sequential} considers the problem \eqref{eq:goal1} when $F$ is noiseless, and uses a small modification of the well-known expected improvement acquisition function \citep{Mockus1989,jones1998efficient}. Their acquisition function is a two step procedure which first uses expected improvement to choose $x\in A$ by maximizing the conditional expectation of $\mbox{max}\left\{ 0,G\left(x\right)-\mbox{max}_{1\leq i\leq n}G\left(x_{i}\right)\right\} $ given the past $n$ observations, and then chooses $w\in W$ by minimizing the posterior mean squared prediction error. This algorithm is not consistent for finite $A$ for the following reason: After $F(x,w)$ has been evaluated for all $x\in A$, (but not necessarily all $w \in W$),
$G\left(x\right)-\mbox{max}_{x\in A}G\left(x\right)\leq0$ almost surely. This implies that the conditional expectation of $\mbox{max}\left\{ 0,G\left(x\right)-\mbox{max}_{x\in A}G\left(x\right)\right\} $ is $0$ for all $x$. If the tie-breaking rule used chooses the same $x$ on each iteration, then this method will fail to evaluate each $x$ infinitely often.
\citet{lehman2004} considers minor modifications of the previous algorithm, and their M-robust algorithm can also fail to be consistent with a poor tie-breaking rule.
\citet{groot2010bayesian} considers problem \eqref{eq:goal2} when $F$ is noiseless, $p(w)$ is Gaussian, and the kernel of the Gaussian process on $F$ is the squared exponential kernel. Its acquisition function is a minor modification of the active learning method ALC \citep{cohn1996}. Numerical experiments in that paper do not demonstrate an improvement over evaluating $G$ directly. While the method proposed for choosing $x,w$ is motivated by minimizing the expected variance of the objective $G$ after one evaluation, it does not do so optimally. Instead, like \citet{williams2000sequential}, it chooses $x$ ignoring what $w$ will be chosen, and then chooses $w$ with $x$ fixed. This is in contrast with our approach, which chooses $x$ and $w$ jointly in a one-step optimal way.
\citet{swersky2013multi} considers problem \eqref{eq:goal1} when $F$ is noiseless, and uses a small modification of the expected improvement acquisition function \citep{jones1998efficient}. Like \citet{williams2000sequential} and \citet{groot2010bayesian}, and in contrast with our joint optimization approach, it first chooses $x$ ignoring $w$, and then in a second step it chooses $w$ with $x$ fixed.
Although one should typically choose $w$ to reduce uncertainty about $G(x)$, \citet{swersky2013multi} instead chooses $w$ using an expected improvement criterion over $F(x,w)$ even though we are not maximizing over $w$. This can select points whose posterior mean of $F(x,w)$ is high but posterior variance is extremely low, essentially wasting a measurement.
This leads in turn to examples where the policy repeatedly samples the same $x$ and under-explores, as we discuss in the appendix ($\mathsection$\ref{bad_example_mt}).
Our numerical experiments show this method can perform well in problems with a small number of homogeneous tasks, but tends to underperform significantly as the number of tasks increase.
\citet{Xie:2012} considers problem \eqref{eq:goal2} when $F$ is noiseless, $p$ is Gaussian and independent, and the Gaussian process on $F$ has a squared exponential kernel. BQO generalizes the method in that paper to the significantly more applicable setting where $F$ can be noisy (required for application to optimization via simulation), with any $p$ (required for applications to cross-validation in machine learning) and any kernel (required for good performance on a wider variety of problems). These generalizations significantly increase the difficulty of the problem, because they preclude closed-form expressions used in \citet{Xie:2012}.
We also provide significantly improved computational methods:
the discretized method used in \citet{Xie:2012} to optimize the acquisition function requires computation that scales exponentially in the dimension, preventing its use for more than $3$ dimensions, while our discretization-free method has sub-exponential scaling and numerical experiments demonstrate excellent performance on problems in up to $7$ dimensions. Although \citet{Xie:2012} does not provide theoretical analysis, one can see our convergence proof for the discretized method as addressing theoretical questions left unanswered by that previous work. We also go beyond \citet{Xie:2012} in extending our methodology to be fully Bayesian by sampling Gaussian process hyperparameters from their posterior distribution using slice sampling.
Other related work includes \citet{marzat2013}, which considers a related but different formulation of \eqref{eq:goal1} based on maximizing worst-case performance over a discrete set of environmental conditions. \citet{lam2008} considers a modification of \citet{williams2000sequential} where the criterion used is for response surface model fit instead of global optimization.
Our consistency proof for the finite sum problem \eqref{eq:goal1} is the first for any algorithm that evaluates $F$ instead of $G$. Consistency of some Bayesian optimization algorithms that evaluate $G$ have, however, been shown in the literature. \citet{frazier2009knowledge} proved consistency of the knowledge gradient algorithm for any Gaussian process for finite domains. Later \citet{bull2011} proved consistency of expected improvement for functions that belong to the reproducing kernel Hilbert space (RKHS) of the covariance function. The recent working paper \citet{bect2016} also contains consistency results for knowledge gradient and expected improvement over any Gaussian process with continuous paths.
Our BQO algorithm can be considered to be within the class of knowledge gradient policies \citep{PowellFrazier2008}, because it selects the ($x,w$) to sample that maximizes the expected utility of the final solution, under the assumption, made for tractability, that we may take only one additional sample. Work on knowledge gradient algorithms in other settings includes \citet{frazier2009knowledge,poloczek2017multi,wugradientsbo}. Our discretization-free approach leverages ideas in particular from \cite{wugradientsbo}.
Our algorithm also leverages Bayesian quadrature techniques \citep{o1991bayes}, which build a Gaussian process model of the function $F(x,w)$, and then use the relationships given by the sum or integral to imply a second Gaussian process model on the objective $G$.
The rest of this paper is organized as follows: $\mathsection$\ref{model} presents our statistical model. $\mathsection$\ref{SBO} presents the conceptual value of information analysis underlying the BQO algorithm. $\mathsection$\ref{sec:VOI} describes computation of the value of information and its derivative, and presents the BQO algorithm in a practically implementable form. $\mathsection$\ref{sec:asymptotic} presents theoretical results on consistency of BQO. $\mathsection$\ref{experiments} presents simulation experiments. $\mathsection$\ref{conclusion} concludes.
\section{Statistical Model}
\label{model}
Our BQO algorithm relies on a Gaussian process (GP) model of the underlying function $F$, which then implies
a Gaussian process model over $G$. Before presenting BQO in $\mathsection$\ref{SBO}, we present this statistical model to provide notation used through the rest of the paper. The first part of our development is standard in Bayesian optimization \citep{jones1998efficient} and Bayesian quadrature \citep{o1991bayes}, while the second part, in which a Gaussian process on the function's integral or sum is obtained, is only standard in Bayesian quadrature.
We suppose that observing $F$ at $x,w$ provides an observation
$y\left(x,w\right)$ equal to $F\left(x,w\right)$ optionally perturbed by
additive independent normally distributed noise with mean $0$ and variance
$\lambda_{\left(x,w\right)}$.
To permit estimation, we require one of two additional assumptions on this noise:
either that $\lambda_{\left(x,w\right)}$ is constant across the domain;
or that observing at $x,w$ also provides an observation of $\lambda_{\left(x,w\right)}$.
The first assumption has been shown to be effective in a wide range of applications in the Bayesian optimization literature \citep{snoek2012practical}. The second is reasonable in discrete-event simulation applications in which $y(x,w)$ is the average of a large batch of independent replications. In such applications, the difference between $y(x,w)$ and its mean $F(x,w)$ converges to a normal distribution by the central limit theorem as the batch size grows large, and $\lambda_{\left(x,w\right)}$ can be estimated by dividing the sample variance of these samples by their number \citep{kim:2007}.
We assume that the function $F$ follows a Gaussian process prior distribution:
\[
F\left(\cdot,\cdot\right)\mid \theta\sim GP\left(\mu_{0}\left(\cdot,\cdot;\theta\right),\Sigma_{0}\left(\cdot,\cdot,\cdot,\cdot; \theta\right)\right),
\]
where $\mu_{0}$ is a real-valued function taking arguments $x,w$ (the {\it mean function}), $\Sigma_{0}$ is a positive semi-definite function taking arguments $x,w,x',w'$ (the {\it kernel}), and $\theta$ are the hyperparameters of the mean function and kernel. $\theta$ contains $\lambda_{(x,w)}$ when the variance of the observational noise is assumed to be unknown and constant. Common choices for $\mu_0$ and $\Sigma_0$ from the Gaussian process regression literature \citep{RaWi06,murphy2012machine,goovaerts1997geostatistics,seeger2005semiparametric,bonilla2007multi}
appropriate for problem \eqref{eq:goal2}
include setting $\mu_0$ to a constant and letting $\Sigma_0$ be the squared exponential or Mat\'{e}rn $5/2$ kernel. In the case of the finite sum \eqref{eq:goal1}, kernels from the intrinsic model of coregionalization are appropriate \citep{seeger2005semiparametric, goovaerts1997geostatistics, bonilla2007multi} and will be discussed in $\mathsection$\ref{experiments}.
Following work on fully Bayesian inference in GP regression \citep{Neal:GPBayesian}, we additionally place a Bayesian prior distribution $\pi$ on $\theta$. This prior can regularize values of $\theta$ used in inference, pushing them toward regions of the space of hyperparameters believed to best correspond to the data. The prior can also be set constant if there is enough data to obviate such regularization.
We now discuss inference supposing that we have $n$ points in the historical data $H_{n}:= \left(y_{1:n},w_{1:n},x_{1:n}\right)$, where $y_{i} = y(x_i,w_i)$ is a (possibly noisy) observation of $F(x_{i},w_{i})$ with the conditional distribution given $x_i$,$w_i$ described above.
Within our inference procedure we sample $\theta$ from its posterior distribution given $H_n$ via slice sampling \citep{radford2003Slice}. One may also replace this sampling-based fully Bayesian treatment of $\theta$ by using the maximum {\it a posteriori} estimate (MAP), which sets $\theta$ to its posterior mode \citep{murphy2012machine}. This is less computationally intensive, but tends to be less accurate. The maximum likelihood estimate (MLE) of $\theta$ is a particular case of the MAP when the prior distribution on $\theta$ is flat.
Using this procedure to sample $\theta$,
we now describe computation of the posterior distribution on both $F$ and $G$ given $\theta$. The posterior distribution on $F$ given $\theta$ at time $n$ is
\[
F\left(\cdot,\cdot;\theta\right)\mid H_{n}, \theta\sim GP\left(\mu_{n}\left(\cdot,\cdot; \theta\right),\Sigma_{n}\left(\cdot,\cdot,\cdot,\cdot;\theta\right)\right),
\]
where the parameters $\mu_{n}$, $\Sigma_{n}$ can be computed using standard
results from Gaussian process regression \citep{RaWi06}.
To support later analysis, we provide these expressions here, suppressing dependence on $\theta$ in our notation:
\begin{eqnarray}
\mu_{n}\left(x,w\right) & = & \mu_{0}\left(x,w\right)+\left[\Sigma_{0}\left(x,w,x_{1},w_{1}\right)\mbox{ }\cdots\mbox{ }\Sigma_{0}\left(x,w,x_{n},w_{n}\right)\right]A_{n}^{-1}\left(\begin{array}{c}
y_{1}-\mu_{0}\left(x_{1},w_{1}\right)\\
\vdots\\
y_{n}-\mu_{0}\left(x_{n},w_{n}\right)
\end{array}\right)\label{post_mean_F}\\
\Sigma_{n}\left(x,w,x',w'\right) & = & \Sigma_{0}\left(x,w,x',w'\right)-\left[\Sigma_{0}\left(x,w,x_{1},w_{1}\right)\mbox{ }\cdots\mbox{ }\Sigma_{0}\left(x,w,x_{n},w_{n}\right)\right]A_{n}^{-1}\left(\begin{array}{c}
\Sigma_{0}\left(x',w',x_{1},w_{1}\right)\\
\vdots\\
\Sigma_{0}\left(x',w',x_{n},w_{n}\right)
\end{array}\right) \label{post_cov_F}
\end{eqnarray}
where
\[
A_{n}=\left[\begin{array}{ccc}
\Sigma_{0}\left(x_{1},w_{1},x_{1},w_{1}\right) & \cdots & \Sigma_{0}\left(x_{1},w_{1},x_{n},w_{n}\right)\\
\vdots & \ddots & \vdots\\
\Sigma_{0}\left(x_{n},w_{n},x_{1},w_{n}\right) & \cdots & \Sigma_{0}\left(x_{n},w_{n},x_{n},w_{n}\right)
\end{array}\right]+\mbox{diag}\left(\lambda_{(x_{1},w_{1})},\ldots,\lambda_{(x_{n},w_{n})}\right).
\]
We now describe the posterior distribution on the objective function $G$ given $\theta$.
We assume that $G$ is written in its integral form \eqref{eq:goal2}. Results for \eqref{eq:goal1} are similar, where the resulting expressions are obtained by replacing integration over $w$ by a sum over $w$ (or equivalently Lebesgue integration with respect to a counting measure).
We denote by $E_{n}$, $\mathrm{Cov}_{n}$, and $\mathrm{Var}_n$ the conditional expectation, conditional covariance, and conditional variance with respect to the Gaussian process posterior given $H_{n}$ and $\theta$.
By results from Bayesian quadrature \citep{o1991bayes}, for
$G(x) := \int F(x,w) p(w)\,dw$, we have that
\begin{align}
E_{n}\left[G(x) \right]
&= \int\mu_{n}(x,w)p\left(w\right)dw := a_n(x;\theta), \label{eq:a_n} \\
\mbox{Cov}_{n}\left(G(x),G(x')\right)
& = \int\int\Sigma_{n}\left(x,w,x',w'\right)p\left(w\right)p\left(w'\right)dw\,dw'. \label{eq:cov_n_g}
\end{align}
Ignoring some technical details, the first line is derived using interchange of integral and expectation, as in
$E_{n}\left[G(x) \right]
= E_{n}\left[\int F(x,w) p(w)\,dw \right]
= \int E_{n}\left[F(x,w) p(w) \right] \,dw
= \int\mu_{n}(x,w)p\left(w\right)dw$.
The second line is derived similarly, though with more effort, by writing the covariance in terms of the expectation and interchanging expectation and integration.
The posterior distributions of $F$ and $G$ given $H_n$ and marginalizing over $\theta$ are infinite mixtures of Gaussian processes. Means of these posterior distributions can be obtained by averaging \eqref{post_mean_F} or \eqref{eq:a_n} with respect to the posterior on $\theta$.
\section{Conceptual Description of the BQO Algorithm}
\label{SBO}
Our BQO algorithm uses the statistical model described in $\mathsection$\ref{model} and samples $F$ sequentially.
It chooses where to sample $F$ using a value of information analysis \citep{Ho66}.
This analysis measures the expected quality of the best solution we can provide to \eqref{eq:goal1} or \eqref{eq:goal2} after $n$ samples, and how this quality improves with an additional sample of $F(x,w)$.
In this section we describe this value of information analysis from a conceptual perspective in preparation for describing in $\mathsection$\ref{sec:VOI} the novel computational methodology we create to make its implementation possible.
This conceptual value of information analysis echos the use of value of information in Bayesian optimization in works on knowledge-gradient methods \citep{frazier2009knowledge} and in problem (2) by \citet{Xie:2012}.
The value of information analysis we describe here generalizes \citet{Xie:2012} to support noisy observations, integrals with dependent normally distributed densities, non-normally distributed densities, and sums, and general kernels.
While the generalization of the {\it conceptual} form of the value of information analysis from \citet{Xie:2012} to handle this richer class of problems is straightforward, it presents a host of new computational challenges that requires new methodology, as fully described in $\mathsection$\ref{sec:VOI}.
We conduct our value of information analysis assuming the hyperparameters $\theta$ are given, as is common practice in Bayesian optimization \citep{swersky2013multi,reviewBO}. Then, in the implementation of the BQO algorithm, because $\theta$ is unknown, we average this $\theta$-dependent value of information over the posterior on $\theta$. While in principle one could instead conduct the full value of information analysis acknowledging that $\theta$ is unknown, proceeding as we do provides substantial computational benefits.
To conduct this analysis,
we first study the expected quality of the best solution we can provide.
Given $n$ samples, $\theta$, and a risk-neutral utility function, we would choose as our solution to \eqref{eq:goal1} or \eqref{eq:goal2},
\begin{equation*}
x_{n,\theta}^{*} \in \mbox{argmax}_{x \in A} E_{n}\left[G(x)\mid \theta\right]=\mbox{argmax}_{x\in A} a_n(x;\theta),
\end{equation*}
where $a_n(x;\theta):=E_{n}[G(x)\mid \theta]$.
This solution has expected value (again, with respect to the posterior after $n$ samples given $\theta$),
\begin{equation*}
a_{n,\theta}^* := E_{n}\left[G\left(x_{n,\theta}^{*}\right)\mid \theta\right] = \max_{x}E_{n}\left[G(x)\mid \theta\right]=\max_{x} a_n(x; \theta).
\end{equation*}
Consequently, the improvement in expected solution quality resulting from a sample at $(x,w)$ at time $n$ is
\begin{equation}
V_n(x,w; \theta) = E_n\left[ a_{n+1,\theta}^* - a_{n,\theta}^* \mid x_{n+1}=x, w_{n+1}=w\right],
\label{eq:VOI}
\end{equation}
and we refer to this quantity as the {\it value of information}. Our Bayesian Quadrature Optimization (BQO) algorithm
then is defined as the algorithm that samples where this value of information (marginalized over $\theta$) is maximized,
\begin{equation}
\left(x_{n+1},w_{n+1}\right)\in\mbox{argmax}_{x,w} E\left[V_{n}\left(x,w; \theta\right) \mid H_n\right]. \label{eq:max_VOI}
\end{equation}
Here, the expectation is over the posterior on $\theta$, as indicated by the subscript.
This policy is one-step Bayes optimal in the known-hyperparameter case (i.e., the prior on $\theta$ is concentrated on a single value), in the sense that if we can take one more sample before reporting a final solution then its sampling decision maximizes the expected value of $G$ at this reported final solution.
It is not necessarily Bayes-optimal if we can take more than one sample, but we argue that it remains a reasonable heuristic, and our numerical experiments in $\mathsection$\ref{experiments} support this. It is also Bayes-optimal for the problem \eqref{eq:goal1} in the known-hyperparameter case when the number of iterations converge to infinity, as we show in $\mathsection$\ref{sec:asymptotic}.
Figure~\ref{fig:tahi10-IBO} illustrates BQO, showing one step in the algorithm applied to a simple analytic test problem
\begin{equation}
\label{eq:test}
\max_{x\in\left[-\frac12,\frac12\right]}E\left[z x^{2}+w\right],
\end{equation}
where $w\sim N\left(0,1\right)$, $z\sim N\left(-1,1\right)$, and $F(x,w)=E[zx^2+w\,|\,w] = -x^2+w$. Direct computation shows $G(x)=-x^{2}$. In the figure, we fix $\theta$ to a maximum likelihood estimate obtained using 15 training points.
\begin{figure}[!htbp]
\centering
\subcaptionbox{The contours of $F\left(x,w\right)$. $G$ is determined \\ from $F$ by $G(x) = \int F(x,w) p(w)\, dw$.}[0.45\linewidth]{
\includegraphics[width=0.45\linewidth,height=2.1in]{F.pdf}}
\subcaptionbox{The contours of BQO's estimate $\mu_n(x,w; \theta)$ of $F\left(x,w\right)$ after $n=9$ evaluations of $F$ by BQO.}[0.45\linewidth]{
\includegraphics[width=0.45\linewidth,height=2.1in]{9muN.pdf}}
\subcaptionbox{
The contours of BQO's value of information $V_n(x,w; \theta)$ versus $x$ and $w$. $F$ was evaluated previously at the red points, chosen according to a random uniform design in an initial phase of training, and at $n=9$ black points, chosen by BQO.}[0.45\linewidth]{
\includegraphics[width=0.45\linewidth,height=2.1in]{9VOI_3.pdf}
}
\quad
\subcaptionbox{
The objective $G(x)$, BQO's estimate $a_n(x;\theta)$ of $G(x)$, and BQO's $95\%$ credible interval for $G(x)$ after $n=9$ evaluations of $F$ by BQO.
The estimate of $G$ is extremely close to its true value, especially near its maximum.
}[0.45\linewidth]{
\includegraphics[width=0.37\linewidth,height=2.0in]{9a_n_crop.pdf}\hfill{}
}
\smallskip
\caption{
Illustration of the BQO algorithm on an analytic test problem after evaluating $F$ at points chosen uniformly at random in an initial phase of training and $n=9$ points chosen by BQO.
\label{fig:tahi10-IBO}}
\end{figure}
The figure's first row shows the contours of $F(x,w)$ (left panel) and BQO's estimate (right panel) after evaluating $F$ at points chosen uniformly at random in an initial training phase, and at an additional $n=9$ points chosen by BQO.
The second row's left panel shows the value of information $V_n(x,w;\theta)$.
The value of information is small near where BQO has already sampled, because it has less uncertainty about $F(x,w)$ in this region.
BQO's value of information is also small for extreme values of $x$, because its posterior on $G$ suggests that these $x$ are far from its maximum, and small for extreme values of $w$ because $p(x,w)$ is small there. BQO's value of information is thus largest for points that are far from previous samples, relatively close to the maximizer of $G$'s posterior mean, and have moderate values of $w$.
BQO samples next at the point with the largest value of information, near $x=-0.2$ and $w=1.8$.
The second row's right panel shows the posterior on $G$.
This posterior is accurate and almost perfectly estimates $G$'s maximizer.
Figure~\ref{fig:tahi10-KG} shows equivalent quantities for the knowledge-gradient (KG) method \citep{frazier2009knowledge},
after noisy evaluations of $G$
at points chosen uniformly at random in an initial training phase, and at an additional $n=9$ points chosen by KG. Like other traditional Bayesian optimization methods, KG models $G(x)$ directly, ignoring valuable information from $w$, and computes a value of information as a function of $x$ only while leaving the choice of $w$ to chance. As a consequence, KG's estimates of $G$ and its maximizer have significantly more error than BQO's estimates.
\begin{figure}[!htbp]
\centering
\subcaptionbox{
The value of information versus $x$ under a traditional Bayesian optimization method (KG). The value pictured is after noisy evaluations of $G$ at the red points, chosen in an initial phase of training, and the $n=9$ black points chosen by KG.}[0.45\linewidth]{
\includegraphics[width=0.45\linewidth,height=2.1in]{9Voi_n.pdf}
}
\quad
\subcaptionbox{
The objective $G(x)$, KG's estimate $\mu_n(x; \theta)$ of $G(x)$; and KG's $95\%$ credible interval for $G(x)$, after $n=9$ noisy evaluations of $G$. This estimate is of lower quality than BQO's because they do not use the observed values of $w$.
}[0.45\linewidth]{
\includegraphics[width=0.34\linewidth,height=2.0in]{mu_image.pdf}
}
\smallskip
\caption{
Illustration of a traditional Bayesian optimization algorithm in the same problem setting as Figure~\ref{fig:tahi10-IBO}. The algorithm pictured is the knowledge gradient (KG) method \citep{frazier2009knowledge}. This algorithm evaluates $G$, unlike BQO's evaluations of $F$. As a consequence, it tends to provide lower-quality estimates of $G$ within a given sampling budget.
\label{fig:tahi10-KG}}
\end{figure}
\section{Computation of the BQO Algorithm}
\label{sec:VOI}
In this section we develop methods to compute the value of information \eqref{eq:VOI} and its gradient, to support implementation of the BQO algorithm.
We introduce a new and powerful method in $\mathsection$\ref{sec:monte_carlo} for computing
unbiased stochastic estimators of the gradient of the value of information, which we refer to more briefly as stochastic gradients.
These stochastic gradients are used within a stochastic gradient ascent method to optimize the value of information.
We also show in $\mathsection$\ref{sec:discretization} how a deterministic discretized method for approximating the value of information and its gradient, first developed in \citet{Xie:2012} for the setting without noise and independent Gaussian $p(w)$ and kernel, can be extended to our more general setting. When it was first proposed in \citet{Xie:2012} it lacked a theoretical analysis of its discretization error. To address this shortcoming, we demonstrate that this discretization error vanishes asymptotically when the discretizations are sufficiently well-designed.
We refer to this method as the ``discretized method,'' and refer to the first method (which does not rely on discretization) as the ``discretization-free method.''
We provide an analysis of the computational complexity of each method, showing that the time and space complexity of the discretization-free method scale better in the dimension of $x$. In concert with this theoretical observation, empirical observations show that the discretized method is fastest when $x$ has one or two dimensions but is too slow to be practical in higher dimensions. In contrast, our numerical experiments ($\mathsection$\ref{experiments}) show that our novel discretization-free method is practical in dimensions as large as $7$.
To simplify proofs, we assume that $G$ has the integral form defined in \eqref{eq:goal2}. As we mentioned in $\mathsection$\ref{model}, results for \eqref{eq:goal1} are similar, where the resulting expressions are obtained by replacing integration over $w$ by a sum.
We also assume in our computation of the value of information that $\theta$ is given, as discussed in $\mathsection$\ref{SBO}, and drop the dependence on $\theta$ from our notation (except in $\mathsection$\ref{sec:IBO-alg} where we write it explicitly to support a high-level summary of the BQO algorithm).
Table~\ref{table:notation} summarizes notation used in this section, including both notation introduced in previous sections and new notation defined later in this section.
\begin{table}[htbp]
\caption{Table of Notation.}
\label{table:notation}
\centering
\begin{center}
\begin{tabular}{rclcc}
\hline
$G(x)$ & $\triangleq$ & $\sum_{w=1}^{m}F\left(x,w\right)p(w)$ or $\int_{}F\left(x,w\right)p(w)dw$ \\
$V_{n}$ & $\triangleq$ & Value of information at time $n$\\
$a_{n}\left(x\right)$ & $\triangleq$ & $E_{n}\left[G(x) \right]$\\
$H_{n}$ & $\triangleq$ & History observed by time $n$\\
$\Sigma_{0}$ & $\triangleq$ & Kernel of the Gaussian process prior distribution over the function $F$\\
$B\left(x,i\right)$ & $\triangleq$ & $\int\Sigma_{0}\left(x,w,x_{i},w_{i}\right)p(w)dw$ if $G(x)=\int_{}F\left(x,w\right)p(w)dw$, or \\
& & $\sum_{w=1}^{m}\Sigma_{0}\left(x,w,x_{i},w_{i}\right)p(w)$ if $G(x)=\sum_{w=1}^{m}F\left(x,w\right)p(w)$ for $i=1,\ldots,n+1$\\
$\gamma$ & $\triangleq$ & $(\Sigma_{0}(x_{n+1},w_{n+1},x_{1},w_{1}),\ldots,\Sigma_{0}(x_{n+1},w_{n+1},x_{n},w_{n}))^T$ \\
$\lambda_{\left(x,w\right)}$ & $\triangleq$ & Variance of the noise in evaluations of $F\left(x,w\right)$
\\
$A_{n}$ & $\triangleq$ & $\left(\Sigma_{0}\left(x_{i},w_{i},x_{j},w_{j}\right)\right)_{i,j=1}^{n} + \mbox{diag}\left(\left(\lambda_{\left(x_{i},w_{i}\right)}\right)_{i=1}^{n}\right) $ \\
$\tilde{\sigma}^2_n(x,x_{n+1},w_{n+1})$ & $\triangleq$ & $\mbox{Var}_{n}\left[G\left(x\right)\right]-E_{n}\left[\mbox{Var}_{n+1}\left[G\left(x\right)\mid x_{n+1},w_{n+1}\right]\right]$\\
$\mbox{Var}_{n}$ & $\triangleq$ & conditional variance given $H_{n}$\\
\hline
\end{tabular}
\end{center}
\vskip -0.1in
\end{table}
\subsection{Preliminary Representation of the Value of Information}
\label{sec:BO1}
In this section, we find a useful representation of the value of information \eqref{eq:VOI} that will allow us to develop the discretization-free ($\mathsection$\ref{sec:monte_carlo}) and discretized ($\mathsection$\ref{sec:discretization}) methods to approximate it and its gradient.
We first observe that we can rewrite the value of information \eqref{eq:VOI} as
\begin{align*}
V_{n}\left(x_{n+1},w_{n+1}\right) =&E_{n}\left[\mbox{max}_{x'\in A}a_{n+1}\left(x'\right) \mid x_{n+1},w_{n+1}\right] -\mbox{max}_{x' \in A} a_{n}\left(x'\right). \numberthis
\label{eq:VOI_a}
\end{align*}
This expression is not directly useful from a computational perspective, so we take one step further and find the joint distribution of $a_{n+1}\left(x\right)$ across all $x$ conditioned on $x_{n+1},w_{n+1}$ and $H_{n}$ for any $x$. This is provided by the following lemma.
The lemma is a generalization of Section 2.1 in \citet{frazier2009knowledge}, and we include the proof in the appendix $\mathsection$\ref{proofs_monte_carlo_sect}.
\begin{lemma}
\label{postdist}
There exists a random variable $Z_{n+1}$, whose conditional distribution given $H_{n}$ is standard normal, such that
$a_{n+1}\left(x\right) = a_{n}\left(x\right)+\tilde{\sigma}_n(x,x_{n+1},w_{n+1})Z_{n+1}$
for all $x$,
with
\begin{align*}
\tilde{\sigma}^2_n(x,x_{n+1},w_{n+1}) :=& \mbox{Var}_{n}\left[G\left(x\right)\right]-E_{n}\left[\mbox{Var}_{n+1}\left[G\left(x\right)\right]\mid x_{n+1},w_{n+1}\right].
\end{align*}
The posterior mean $a_{n}(x)$ of $G(x)$ can be represented by
\begin{eqnarray}
a_{n}\left(x\right)&=&\int\mu_{0}\left(x,w\right)p(w)dw+\left[B\left(x,1\right)\mbox{ }\cdots\mbox{ }B\left(x,n\right)\right]A_{n+1}^{-1}\left(\begin{array}{c}
y_{1}-\mu_{0}\left(x_{1},w_{1}\right)\\
\vdots\\
y_{n}-\mu_{0}\left(x_{n},w_{n}\right)\end{array}\right),\label{eq:post_mean_formula}
\end{eqnarray}
where $B(x,i):=\int\Sigma_{0}\left(x,w,x_{i},w_{i}\right)p\left(w\right)dw$ for
$1\leq i\leq n$.
We also have that
\begin{equation}
\begin{split}
&\tilde{\sigma}_{n}\left(x,x_{n+1},w_{n+1}\right) = \\
&\frac{B\left(x,n+1\right)-\left[B\left(x,1\right)\mbox{ }\cdots\mbox{ }B\left(x,n\right)\right]A_{n}^{-1}\gamma}{\sqrt{\Sigma_{0}\left(\!x_{n+1}\!,\!w_{n+1}\!,\!x_{n+1}\!,\!w_{n+1}\!\right)-\gamma^{T}A_{n}^{-1}\gamma+\lambda_{(x_{n+1},w_{n+1})}}}
\times 1{\left\{ \lambda_{\left(x_{n+1},w_{n+1}\right)}\!>\!0 \text{ or } \left(x_{n+1},w_{n+1}\right)\!\notin\!\{\left(x_{i},w_{i}) : i\!\leq\!n\right\} \right\}},
\end{split}
\label{eq:post_var_formula}
\end{equation}
where $\gamma^{T}:=(\Sigma_{0}(x_{n+1},w_{n+1},x_{1},w_{1}),\ldots,\Sigma_{0}(x_{n+1},w_{n+1},x_{n},w_{n}))$.
\end{lemma}
The expressions in this lemma require that $\lambda_{(x_{n+1},w_{n+1})}$ be known to compute the value of information.
This is seldom true in practice, but this quantity can be estimated and the estimate used in its place.
If the noise is homogeneous then it can be estimated by including it as a hyperparameter in our Gaussian-process-based inference.
If each observation is an average of many i.i.d. replications, allowing the variance of the noise in each observation to be estimated with high accuracy,
and we believe that the noise does not change abruptly in the domain,
then we can use the mean of the variance estimates from previous observations as our estimator of $\lambda_{(x_{n+1},w_{n+1})}$.
Finally, if we are in neither of these situations, then we can use the approach developed in \citet{Kersting2007} in which a Gaussian process is used to estimate the variance of heteroscedastic noise across the domain.
We now use \Cref{postdist} to estimate the value of information and its gradient in the next subsection.
\subsection{Discretization-Free Computation of the Value of Information and its Gradient}
\label{sec:monte_carlo}
In this subsection, we provide unbiased and strongly consistent Monte Carlo estimators of the value of information and its gradient. Our techniques use the envelope theorem \citep{milgrom} and were inspired by \citet{wugradientsbo}, which uses this theorem to build an unbiased estimator of the gradient of the knowledge-gradient in a different setting.
First, \Cref{postdist}, equation (\ref{eq:VOI_a}), and the strong law of large numbers show that if $\left\{ Z_{i}\right\} _{i=1}^{\infty}$ are independent standard
normal random variables, then
\[
\widehat{V}_{n,m}\left(x,w\right):=\frac{1}{m}\sum_{i=1}^{m}\left[\mbox{max}_{x'\in A}\left(a_{n}\left(x'\right)+\tilde{\sigma}_{n}\left(x',x,w\right)Z_{i}\right)-\mbox{max}_{x'\in A}a_{n}\left(x'\right)\right]
\]
is an unbiased and strongly consistent estimator of the value of information $V_{n}(x,w)$.
The inner optimization problems $\mbox{max}_{x'\in A}\left(a_{n}\left(x'\right)+\tilde{\sigma}_{n}\left(x',x,w\right)Z_{i}\right)$ can be solved using standard optimization methods such as LBFGS \citep{byrd:1997} or
Newton methods (if the Hessian of the kernel exists). Gradients of the inner optimization problem's objective can
be computed using (\ref{eq:200}).
We now build an unbiased and strongly consistent estimator of $\frac{\partial}{\partial r}V_{n}\left(x,w\right)$ where $x=\left(x_{1},\ldots,x_{d}\right),w=\left(w_{1},\ldots,w_{p}\right)$, $r\in\left\{ x_{i}:1\leq i\leq d\right\} \bigcup\left\{ w_{i}:1\leq i\leq p\right\} $, and give sufficient conditions for existence of $\frac{\partial}{\partial r}V_{n}\left(x,w\right)$. We use the envelope theorem \citep{milgrom}, along with the following lemma, which shows some smoothness properties of $a_{n}$ and $\tilde{\sigma}_{n}$. The proof of this lemma may be found in $\mathsection$\ref{proofs_monte_carlo_sect}.
\begin{lemma}
\label{smooth_ibo}
We assume $\mu_{0}$ is constant, and the kernel $\Sigma_{0}$ of the prior distribution on $F$ is continuously differentiable and bounded. We also suppose there is a non-negative function $h$ such that $\int h(x,w',x')p(w')dw'$ is finite for all $x,x'\in A$, and $\left|\frac{\partial\Sigma_{0}\left(x,w',x',w\right)}{\partial w}\right|<h(x,w',x')$ for all $x,x'\in A$ and $w,w'\in W$. Then:
\begin{enumerate}
\item $a_{n}$ and $\tilde{\sigma}_{n}\left(\cdot,x,w\right)$ are both continuously differentiable for any $x,w$ if $\Sigma_{n}(x,w,x,w) > 0$.
\item For any $x'$, $\tilde{\sigma}_{n}\left(x',x,w\right)$ is continuously differentiable with respect to $x,w$ if $\Sigma_{n}(x,w,x,w) > 0$ and $\lambda_{(x,w)}$ is continuously differentiable.
\end{enumerate}
\end{lemma}
The condition $\Sigma_{n}(x,w,x,w) > 0$ in the previous lemma is always true in the noisy case, as shown by \eqref{post_cov_F}. In the noiseless case, $\Sigma_{n}(x,w,x,w)$ can be zero only if $x,w$ is a previously measured point.
The following lemma shows how to compute stochastic gradients of $V_{n}\left(x,w\right)$ and allows us to optimize $V_{n}$ with a stochastic gradient ascent method.
\begin{lemma}
\label{monte_carlo_voi}
Suppose that the hypotheses of \cref{smooth_ibo} on $\Sigma_0$ and $p$ are satisfied. Also assume that for a given $\left(x_{n+1},w_{n+1}\right)$, $\mbox{argmax}_{x'\in A}\left(a_{n}\left(x'\right)+\tilde{\sigma}_{n}\left(x',x_{n+1},w_{n+1}\right)Z\right)$
is almost surely a singleton, where $Z$ is a standard normal random variable.
Also assume
$\Sigma_{n}(x_{n+1},w_{n+1},x_{n+1},w_{n+1}) > 0$ and
$\lambda_{(x,w)}$ is continuously differentiable at $x=x_{n+1}$ and $w=w_{n+1}$.
Let $\left\{ Z_{i}\right\} _{i=1}^{\infty}$ be independent standard normal random variables. Then
\[
\nabla V_{n}\left(x_{n+1},w_{n+1}\right)=\lim_{m\rightarrow\infty}\frac{1}{m}\sum_{i=1}^{m}\left(\nabla_{\left(x_{n+1},w_{n+1}\right)}\tilde{\sigma}_{n}\left(y_{i},x_{n+1},w_{n+1}\right)Z_{i}\right) \mbox{ a.s.,}
\]
where $y_{i}=\mbox{argmax}_{x'\in A}\left(a_{n}\left(x'\right)+\tilde{\sigma}_{n}\left(x',x_{n+1},w_{n+1}\right)Z_{i}\right)$. Furthermore, $E\left[\left(\nabla_{\left(x_{n+1},w_{n+1}\right)}\tilde{\sigma}_{n}\left(y_{i},x_{n+1},w_{n+1}\right)Z_{i}\right)\right]=V_{n}\left(x_{n+1},w_{n+1}\right)$ for all $i$.
\end{lemma}
\proof{}
Let $Z$ be a standard normal random variable, and $f\left(x',\left(x,w\right)\right):=a_{n}\left(x'\right)+\tilde{\sigma}_{n}\left(x',x,w\right)Z$ where $x',x\in A$ and $w\in W$. By \cref{smooth_ibo}, $f$ is continuously differentiable, and so
by the envelope theorem \citep[Corollary 4]{milgrom},
\begin{equation*}
\nabla_{\left(x_{n+1},w_{n+1}\right)}f\left(y,\left(x_{n+1},w_{n+1}\right)\right)=\nabla_{\left(x_{n+1},w_{n+1}\right)}\left(a_{n}\left(y\right)+\tilde{\sigma}_{n}\left(y,x_{n+1},w_{n+1}\right)Z\right)=\nabla_{\left(x_{n+1},w_{n+1}\right)}\tilde{\sigma}_{n}\left(y,x_{n+1},w_{n+1}\right)Z\mbox{ a.s.}
\end{equation*}
where $y=\mbox{argmax}_{x'\in A}\left(a_{n}\left(x'\right)+\tilde{\sigma}_{n}\left(x',x_{n+1},w_{n+1}\right)Z\right)$ and the dependence of $y$ on $x_{n+1}$,$w_{n+1}$ is ignored when taking the gradient.
We now show that $\nabla V_{n}\left(x_{n+1},w_{n+1}\right)=E_{n}\left[\nabla_{\left(x_{n+1},w_{n+1}\right)}\mbox{max}_{x'\in A}\left(a_{n}\left(x'\right)+\tilde{\sigma}_{n}\left(x',x_{n+1},w_{n+1}\right)Z\right)\right]$. First observe that if $z$ is a real number such that $\mbox{argmax}_{x'\in A}\left(a_{n}\left(x'\right)+\tilde{\sigma}_{n}\left(x',x_{n+1},w_{n+1}\right)z\right)=\left\{y_{z}\right\}$, by the envelope theorem, we then have that,
\[
\nabla_{\left(x_{n+1},w_{n+1}\right)}\mbox{max}_{x'\in A}\left(a_{n}\left(x'\right)+\tilde{\sigma}_{n}\left(x',x_{n+1},w_{n+1}\right)z\right)=\nabla_{\left(x_{n+1},w_{n+1}\right)}\left(a_{n}\left(y_{z}\right)+\tilde{\sigma}_{n}\left(y_{z},x_{n+1},w_{n+1}\right)z\right).
\]
Furthermore, we have that
\begin{eqnarray*}
\left|\nabla_{\left(x_{n+1},w_{n+1}\right)}\left(a_{n}\left(y_{z}\right)+\tilde{\sigma}_{n}\left(y_{z},x_{n+1},w_{n+1}\right)z\right)\right| & = & \left|\nabla_{\left(x_{n+1},w_{n+1}\right)}\left(\tilde{\sigma}_{n}\left(y_{z},x_{n+1},w_{n+1}\right)z\right)\right| \leq L_{\left(x_{n+1},w_{n+1}\right)}\left|z\right|
\end{eqnarray*}
where $L_{\left(x_{n+1},w_{n+1}\right)}=\mbox{sup}_{y\in A}\left|\nabla_{\left(x_{n+1},w_{n+1}\right)}\tilde{\sigma}_{n}\left(y,x_{n+1},w_{n+1}\right)\right|$, which is finite because $\tilde{\sigma}_{n}\left(\cdot,x_{n+1},w_{n+1}\right)$ is continuously differentiable
and $A$ is a compact set. Consequently, $E_{n}\left[L_{\left(x_{n+1},w_{n+1}\right)}\left|Z\right|\right]<\infty$.
By Corollary 5.9 of \citet{bartle}, $\nabla V_{n}\left(x,w\right)=E_{n}\left[\nabla_{\left(x_{n+1},w_{n+1}\right)}\mbox{max}_{x'\in A}\left(a_{n}\left(x'\right)+\tilde{\sigma}_{n}\left(x',x_{n+1},w_{n+1}\right)Z\right)\right]$. Finally, the first claim of the lemma follows from the strong law of large numbers.
\endproof
As a reminder, we assumed that the objective function $G$ has the integrated form at the beginning of the section. In the case of the finite sum \eqref{eq:goal1}, the assumptions we have made in \cref{monte_carlo_voi} and \cref{smooth_ibo} may no longer hold.
In particular, $\frac{\partial}{\partial w_{j}}\Sigma_{0}\left(x,w,x',w\right)$
will typically not exist for $1\leq j\leq p$ where $w=(w_{1},\ldots,w_{p})$, and so $\frac{\partial}{\partial w_{j}}V_{n}(x,w)$ may not exist either (see \cref{postdist}).
However, our approach remains applicable in this setting:
we use $\nabla_{x} V_{n}(x,w)$ to maximize $V_{n,x}(x,w)$ for each $w\in W$, and then easily solve $\mbox{max}_{w\in W}V_{n}^{*}\left(w\right)=\mbox{max}_{w\in W}\left[\mbox{max}_{x\in A}V_{n}\left(x,w\right)\right]$ observing that $W$ is a finite set.
Using $\nabla_{x} V_{n}(x,w)$ in this way requires showing similar results to the ones presented in this section to compute stochastic gradients of $V_{n}(x,w)$ with respect to $x$ for any fixed $w$, under the assumption that $\Sigma_{0}(\cdot, w, \cdot, w')$ and $\lambda_{(\cdot,w)}$ are sufficiently smooth for any $w,w'\in W$. We do not include these results here, because their proofs follow the same ideas already presented.
\subsection{Computation and Complexity of the BQO Algorithm}
\label{sec:IBO-alg}
In this section, we summarize computation of the BQO algorithm, which combines the tools developed in previous sections, and discuss its complexity.
First, recall we previously described methods for obtaining unbiased samples of $V_n$ and $\nabla V_n$ using a fixed value of $\theta$.
Because $\theta$ was fixed, we suppressed it in our notation, but here we indicate it explicitly, writing these values as $V_n(\theta)$ and $\nabla V_n(\theta)$ and their estimators as $\widehat{V}_n(\theta)$ and $\widehat{\nabla V}_n(\theta)$ respectively. We will use these within a stochastic gradient
algorithm, ADAM \citep{adam}, for solving problem \eqref{eq:max_VOI}, i.e., for maximizing $E[V_n(\theta) \mid H_n]$.
Within this stochastic gradient algorithm,
each stochastic gradient is obtained by first taking $J$ independent samples $\widehat{\theta}_j : j=1,\ldots,J$ from the posterior distribution on $\theta$ given $H_n$ using slice sampling, and then computing $\frac{1}{J}\sum_{j=1}^J \widehat{\nabla V}_n(\widehat{\theta}_j)$, where each $\widehat{\nabla V}_n(\theta)$ uses a single independent standard normal random variable (so $m=1$ as defined \Cref{monte_carlo_voi}).
We use a similar approach in the final step of our algorithm to select a point with maximal posterior mean $E[a_N(x;\theta) | H_N]$, except that the only source of randomness in our stochastic gradient estimator is $\theta$.
The BQO algorithm using this approach is summarized in \cref{alg:SBO}.
\begin{algorithm}[!htb]
\caption{IBO Algorithm}
\label{alg:SBO}
\begin{algorithmic}[1]
\STATE Evaluate $F$ at $n_0$ points, chosen uniformly at random from $A\times W$.
\FOR{$n=0$ {\bfseries to} $N-1$}
\STATE Use the multi-start ADAM algorithm to maximize $E[V_n(\theta) \mid H_n]$,
using the stochastic gradient estimator $\frac{1}{J} \sum_{j=1}^J \widehat{\nabla V}_n(\widehat{\theta}_j)$ on each iteration using independent samples $\widehat{\theta}_j$ from the posterior on $\theta$.
Include all $n_0 + n$ samples of $F$ when computing the posterior.
Let $\left(x_{n+1},w_{n+1}\right)$ be the resulting maximizer.
\STATE Sample $F\left(x_{n+1},w_{n+1}\right)$ to obtain $y_{n+1}$.
\ENDFOR
\STATE
Use the multi-start ADAM algorithm to maximize $E[a_N(x;\theta) | H_N]$ using the stochastic gradient estimator
$\frac{1}{J'}\sum_{j=1}^{J'} \nabla a_N(x;\widehat{\theta}_j)$, using independent samples
$\hat{\theta}_j$ from the posterior on $\theta$. Return $x^{*}\in\mbox{argmax}_{x\in A} E[a_{N}\left(x;\theta\right) \mid H_N]$.
\end{algorithmic}
\end{algorithm}
Observe that in the noise-free case, $V_{n}\left(x,w\right)$ is not differentiable at any previously evaluated point $x,w$, as shown by the last equation of \cref{postdist}.
The set of previously evaluated points, however, is finite and so we can still use ADAM by perturbing the algorithm's current iterate whenever it resides at a non-differentiable point. A similar idea can be found in \citealt{jordan:saddle}.
Finally we discuss BQO's time and space complexity, assuming we use it to select $N$ points $(x,w)$ to sample. To select each point $(x,w)$ to sample, we use \Cref{alg:SBO}, which runs the ADAM algorithm for $T$ iterations. Each iteration requires a stochastic gradient computed using $J$ independent standard Gaussian random variables, $J$ independent samples from the posterior on $\theta$ (let $Q$ be the number of iterations of slice sampling used for each), and $J$ runs of LBFGS to maximize
$\left(a_{n}\left(x'\right)+\tilde{\sigma}_{n}\left(x',x_{n+1},w_{n+1}\right)Z_{i}\right)$.
Let $K$ be the number of steps in a single run of LBFGS, where each step requires an evaluation of $a_n$, $\tilde{\sigma}_n$, and their gradients. Let $O(L)$ be the complexity of computing the kernel and its gradient, and let $O(S)$ be the complexity of computing (\ref{eq:200}),$\nabla_{n+1}B\left(x,n+1\right)$, and $B(x,i)$ for all $i\leq n$.
With this notation, we show in the appendix ($\mathsection$\ref{bqo_complexity}) that the BQO algorithm has time complexity $O(JQN^4 + JQLN^3 + JTK(SN^2 + N^3)+JTLN^2)$ and space complexity $O(N^2)$.
The integrals in (\ref{eq:200}), $\nabla_{n+1}B\left(x,n+1\right)$, and $B(x,i)$ do not necessarily have closed-form expressions.
While we might estimate them via Monte Carlo or numerical integration, this can be inconvenient and increase computational cost.
Consequently it may be better to first perform a change of variables from $w$ to another $w'$ for which integrals may be evaluated in closed form.
One such transformation, to the Gaussian distribution, is discussed in the appendix $\mathsection$\ref{gaussian_case}.
In addition, a change of variables from $w$ to $w'$ induces a change from $F(x,w)$ to some other $F'(x,w')$, which might change more slowly with $w'$ (requiring fewer samples to model it) or be better modeled by a Gaussian process. We illustrate this change of variable technique in our numerical experiments, in the inventory ($\mathsection$\ref{sec:IPexample}) and Citi Bike ($\mathsection$\ref{sec:citibike}) problems.
\subsection{Discretized Computation of the Value of Information and its Gradient}
\label{sec:discretization}
In this section we describe an alternate approach to that in $\mathsection$\ref{sec:monte_carlo} for estimating the value of information and its gradient. It uses a discretization of $A$.
Although this approach was already considered in \citet{Xie:2012}, which presents a particular case of our method for the integral objective (\ref{eq:goal2}) in the noise-free setting with an independent Gaussian $p$ and Gaussian kernel, we extend this analysis by generalizing it to our setting and showing that a sequence of increasingly fine discretizations produces a sequence of estimators whose estimates converge to the value of information. Thus, these estimators, while biased, are strongly consistent. In practice, computational intractability limits this approach when $A$ has more than $3$ dimensions. This lack of scalability is also demonstrated by a complexity analysis we present at the end of this subsection.
\begin{lemma}
\label{discretization_converge_past}
We assume that $\Sigma_{0}(\cdot,w,x',w
')$ is continuous for all $w,w'\in W$, $x'\in A$, $\Sigma_{0}$ is bounded, and $\mu_{0}$ is a constant. Suppose that we have an increasing sequence of finite discretizations $\left\{ A'_{L}\right\} _{L=1}^{\infty}$ of $A$, such that $\bigcup_{L=1}^{\infty}A'_{L}$ is dense in $A$. Then
\begin{align*}
V_n(x_{n+1},w_{n+1})=\lim_{L\rightarrow\infty}\left(E_{n}\left[\max_{x\in A'_{L}} \left(a_{n}\left(x\right) + \tilde{\sigma}(x, x_{n+1},w_{n+1})Z_{n+1}\right)\right]-\max_{x \in A'_{L}} a_{n}\left(x\right)\right).
\end{align*}
\end{lemma}
The proof of this lemma may be found in $\mathsection$\ref{proofs_monte_carlo_sect}. A
sequence of discretizations that satisfy the properties of the lemma can be built by considering the rationals, as we do in the proof of \cref{continuum_th}.
Using the previous lemma, we have that $V_n(x_{n+1},w_{n+1})=\mbox{lim}_{L\rightarrow\infty}h(a_n(A'_{L}),\tilde{\sigma}_n(A'_{L},x_{n+1},w_{n+1}))$,
where
$a_{n}(A'_{L})=\left(a_{n}\left(x_{i}\right)\right)_{i=1}^{L}$,
$\tilde{\sigma}_{n}\left(A'_{L},x,w\right)=\left(\tilde{\sigma}_{n}\left(x_{i},x,w\right)\right)_{i=1}^{L}$, and
$h:\mathbb{R}^{L}\times\mathbb{R}^{L}\rightarrow\mathbb{R}$ is defined
by
\begin{equation*}
h\left(a,b\right)=E\left[\mbox{max}_{i}a_{i}+b_{i}Z\right]-\mbox{max}_{i}a_{i},
\end{equation*}
where $a$ and $b$ are any deterministic vectors, and $Z$ is a one-dimensional
standard normal random variable. We can then approximate the value of information by $h(a_n(A'_{L}),\tilde{\sigma}_n(A'_{L},x_{n+1},w_{n+1}))$ for some $L$. By convenience, we denote $a_{n}\left(x_{i}\right)$ by $q_{i}$ and
$\tilde{\sigma}_{n}\left(x_{i},x,w\right)$ by $r_{i}$ for each $i$ in $\{1,\ldots,L\}$. If $A = A'_{L}$, which is possible if $A$ is a finite set, then the approximation is exact.
Algorithm 1 of \citet{frazier2009knowledge} applied to $h$, gives a subset of indexes
$\left\{ j_{1},\ldots,j_{\ell}\right\}$ from $\left\{ 1,\ldots,L\right\}$, such that
$V_n(x_{n+1},w_{n+1})=h(a_n(A'_{L}),\tilde{\sigma}_n(A'_{L},x_{n+1},w_{n+1}))=\sum_{i=1}^{\ell-1}\left(r_{j_{i+1}}-r_{j_{i}}\right)f\left(-\left|c_{i}\right|\right)$, where $f\left(z\right):=\varphi\left(z\right)+z\Phi\left(z\right)$, $c_{i} := \frac{q_{j_{i+1}}-q_{j_{i}}}{r_{j_{i+1}}-r_{j_{i}}}$ for $1\leq i \leq \ell -1$, and $\varphi,\Phi$ are the standard normal cdf and pdf, respectively. This shows how to approximate the value of information $V_n$ using the discretization $A'_{L}$ of $A$.
We now show how to approximate the gradient of the value of information $V_n$ using the discretization $A'_{L}$ of $A$. Observe that if $\ell=1$, $V_{n}\left(x,w\right)=0$ and so $\nabla V_{n}\left(x,w\right)=0$. On
the other hand, if $\ell>1$, one can show via direct computation that $\nabla V_{n}\left(x,w\right) = \sum_{i=1}^{\ell-1}\left(-\nabla r_{j_{i+1}}+\nabla r_{j_{i}}\right)\varphi\left(\left|c_{i}\right|\right)$.
Consequently, we only need to compute $\nabla r_{j_{i}}$ for each $i$ in $\{1,\ldots,\ell\}$ . Another direct computation shows that $\nabla_{(x_{n+1},w_{n+1})}\tilde{\sigma}_{n}\left(x,x_{n+1},w_{n+1}\right) =
\beta_{1} \beta_{3}-\frac{1}{2}\beta_{1}^{3}\beta_{2}\left[\beta_{5}-\beta_{4}\right]$, where
\begin{eqnarray*}
\beta_{1} & =& \left[\Sigma_{0}\left(x_{n+1},w_{n+1},x_{n+1},w_{n+1}\right)-\gamma^{T}A_{n}^{-1}\gamma\right]^{-1/2},\\
\beta_{2} & = & B\left(x,n+1\right)-[B\left(x,1\right)\mbox{ }\cdots\mbox{ }B\left(x,n\right)]A_{n}^{-1}\gamma,\\
\beta_{3} & = & \left(\nabla B\left(x,n+1\right)-\nabla\left(\gamma^{T}\right)A_{n}^{-1}\left[B\left(x,1\right),\cdots,B\left(x,n\right)\right]^{T}\right),\\
\beta_{4} & = & 2\nabla\left(\gamma^{T}\right)A_{n}^{-1}\gamma,\\
\beta_{5} & = & \nabla\Sigma_{0}\left(x_{n+1},w_{n+1},x_{n+1},w_{n+1}\right).\\
\end{eqnarray*}
\paragraph*{Complexity of the Discretized Version of the BQO Algorithm.}
Here we discuss the time and space complexity of a version of BQO based on discretized computation of the value of information and its gradient.
We use the same notation and a similar analysis to that in the previous section. As a reminder, $O(L)$ is the complexity of the computation of the kernel and its gradient, and $O(S)$ is the complexity of computing $\nabla_{n+1}B\left(x,n+1\right)$, and $B(x,i)$ for all $i\leq n$. We sample $J$ parameters $(\widehat{\theta}_{1},\cdots,\widehat{\theta}_{J})$ from the posterior on $\theta$, running slice sampling for at most $Q$ iterations for each, and we optimize the value of information with the ADAM algorithm for at most $T$ iterations. If we use a uniform discretization of size $E^d$ where $d$ is the dimension of $A$, the time complexity of the BQO algorithm run for $N$ iterations is $O(E^d TJ(SN^2 + N^3) + JTLN^2 + JQLN^3 + JQN^4)$ , and the space complexity is $O(N^2+E^d)$. Thus if $E$ increases, the time complexity and space complexity increase exponentially.
This makes this method impractical when $d>3$.
\section{Asymptotic Analysis for BQO}
\label{sec:asymptotic}
In this section, we show consistency of BQO. We show that if $p$ and $W$ are finite, $A$ is finite or a closed box in $\mathbb{R}^{d}$, the integrand function $F$ follows a Gaussian process prior with continuous paths for a fixed $w$, and the prior on the hyperparameters of the kernel is concentrated on a single value, then as the number of iterations of the algorithm tends to infinity, the optimal solution given by the BQO algorithm converges in expectation to an optimal solution $\mbox{argmax}_{x\in A}G\left(x\right)$. We omit the explicit dependence of the expressions on $\theta$ since the prior on $\theta$ is assumed concentrated on a single value.
We state two consistency results, one for continuum $A$ (Theorem~\ref{continuum_th_intuitive}) and the other for finite $A$ (Theorem~\ref{main_theorem_simplified}). The proofs for both results may be found in the appendix.
The proof for finite $A$ has a similar structure to the proof of consistency for the knowledge-gradient method for finite domains in problems without integrated objectives from \citet{frazier2009knowledge}. We present a proof for the finite case partly because finite $A$ arises in practice, and partly because it is substantially simpler than the continuum case and provides a starting point for understanding the continuum proof.
Our proof for the continuum goes substantially beyond the techniques required for the finite case, and develops techniques that may also be useful for proving consistency of other Bayesian optimization methods in continuum settings. Consistency of Bayesian optimization in continuum settings has been largely unexplored, with the authors being aware of only two other papers on this topic: The working paper \citet{bect2016} contains consistency results for Bayesian optimization algorithms over Gaussian processes with continuous paths in the continuum setting; and \citet{bull2011} proved consistency of expected improvement for functions that belong to the reproducing kernel Hilbert space (RKHS) of the covariance function in the continuum setting, though Driscoll's Theorem \citep{beder2001} shows that, under some regularity conditions, sample paths of the Gaussian process almost surely do not belong to the RKHS.
We first introduce notation needed for the theorems. Define $A'=A\times W$. Define the set $\mathcal{H}:=D\left(A'\right)\times D_{\mbox{kernel}}\left(A'\times A'\right)$,
where $D\left(A'\right)$ is the set of functions defined on $A'$, and $D_{\mbox{kernel}}\left(A'\times A'\right)$ is the set of positive semidefinite functions defined on $A'\times A'$. We define the set $\mathcal{H_{0}} \subset \mathcal{H}$ as
\[
\left\{ \left(\mu,\Sigma\right):\mu\equiv0, \Sigma_{w,w'}\left(x,y\right):=\Sigma\left(x,w,y,w'\right)\mbox{ is in }C^{1}\left(\mathbb{R}^{d}\times\mathbb{R}^{d}\right) \mbox{ and is isotropic for
all } w,w'\in W\right\}.
\]
We first state our result for continuum $A$ and prove it in the appendix $\mathsection$\ref{proof_continuum_case},
\begin{theorem}
\label{continuum_th_intuitive}
Suppose that $A=\left[a_{1},b_{1}\right]\times\cdots\times\left[a_{d},b_{d}\right]\subset\mathbb{R}^{d}$,
$a_{i}<b_{i}$ for all $i$, $W$ is a finite set, and the probability space is complete.
Assume $(\mu_0,\Sigma_0) \in \mathcal{H_{0}}$.
We assume that the function
$g_{w}\left(x\right):=\lambda_{\left(w,x\right)}$ is continuous in
$A$ for all $w\in W$, and there exists $k_{\lambda},K_{\lambda}>0$
such that $k_{\lambda}<\lambda_{\left(x,w\right)}<K_{\lambda}$ for
all $w\in W$ and $x\in A$. Then
\[
\lim_{N}\mbox{ }E^{BQO}\left[\max_{x\in A}a_{N}\left(x\right)\right] =
E\left[\max_{x\in A}G\left(x\right) \right],
\]
where $E^{BQO}$ indicates expectation with respect to the distribution over sampling decisions induced by BQO.
\end{theorem}
We also state our result for finite $A$ and prove it in the appendix $\mathsection$\ref{proof_finite_case},
\begin{theorem}
\label{main_theorem_simplified}
Suppose that $A$ and $W$ are finite.
Assume $(\mu_0,\Sigma_0) \in \mathcal{H}$.
We have that
\[
\lim_{N}\mbox{ }E^{BQO}\left[\max_{x\in A}a_{N}\left(x\right)\right] =
E\left[\max_{x\in A}G\left(x\right)\right].
\]
\end{theorem}
\section{Numerical Experiments}
\label{experiments}
In this section we present numerical experiments motivated by applications in operations research and machine learning. We compare the BQO algorithm against baseline Bayesian optimization algorithms and algorithms from the literature designed for the specific problems considered. These experiments illustrate how the BQO algorithm can be applied in practice, and demonstrate it performs at least as well as these benchmarks on all problems considered, and often much better.
We compare on seven test problems: a test problem with a simple analytic form ($\mathsection$\ref{sec:test}); a composition of Branin functions ($\mathsection$\ref{sec:branin}) used in \citet{williams2000sequential}; a realistic problem arising in the design of the New York City's Citi Bike system ($\mathsection$\ref{sec:citibike}); cross-validation of convolutional neural networks ($\mathsection$\ref{sec:CVexample_cnn}) and recommendation engines ($\mathsection$\ref{sec:CVexample_filtering}); an inventory problem with substitution ($\mathsection$\ref{sec:IPexample}); and a collection of problems simulated from Gaussian process priors ($\mathsection$\ref{sec:GPexample}) that provide insight into how the benefit provided by BQO is determined by problem characteristics, and that identify problems where BQO is most helpful.
As benchmark algorithms we consider the multi-task algorithm in Section 3.2 of \citet{swersky2013multi} and the algorithm of \citet{williams2000sequential}, which both place a Gaussian process prior on $F(x,w)$, as BQO does. In addition, we consider two baseline Bayesian optimization algorithms: the knowledge-gradient (KG) policy of \citet{frazier2009knowledge} and the Expected Improvement criterion of \citet{jones1998efficient}, which both place the Gaussian process prior directly on $G(x)$. The KG policy is equivalent to BQO in problems where all components of $w$ are moved into $x$. We also solved the problems from $\mathsection$\ref{sec:test} and $\mathsection$\ref{sec:citibike} with Probability of Improvement (PI) \citep{brochu2010tutorial}, but do not include these results because both KG and EI significantly outperform PI.
We now discuss the kernels used in these experiments.
When implementing BQO and the benchmark algorithms in the Branin ($\mathsection$\ref{sec:branin}) and inventory ($\mathsection$\ref{sec:IPexample}) problems, we use the $5/2$-Mat\'ern kernel
$\Sigma_{0}\left(x,w,x',w'\right)=\sigma^{2}\left(1+\sqrt{5}r+\frac{5}{3}r^{2}\right)\mbox{exp}\left(-\sqrt{5}r\right)$,
where $r=\sqrt{\sum_{i=1}^{n}\alpha_{1}^{\left(i\right)}\left(x_{i}-x'_{i}\right)^{2}+\sum_{i=1}^{d_{1}}\alpha_{2}^{\left(i\right)}\left(w_{i}-w'_{i}\right)^{2}}$.
In the cross-validation problems ($\mathsection$\ref{sec:CVexample_filtering}, $\mathsection$\ref{sec:CVexample_cnn}), we use the expected improvement algorithm with the $5/2$-Mat\'ern kernel, and the BQO algorithm (\cref{alg:SBO} in $\mathsection$\ref{SBO}) and multi-task Bayesian algorithm with the task kernel \citep{swersky2013multi}, which is the Kronecker product of a $5/2$-Mat\'ern kernel and a kernel defined only over the finite set $W$. Specifically this kernel is defined by
$\Sigma_{0}\left(x,t,x',t'\right)=\sigma_{t,,t'}\left(1+\sqrt{5}r+\frac{5}{3}r^{2}\right)\mbox{exp}\left(-\sqrt{5}r\right)$
where $r=\sqrt{\sum_{k=1}^{n}\alpha^{\left(k\right)}\left[x_{k}-x'_{k}\right]^{2}}$, $n$ is the number of tasks, and $\left\{ \sigma_{t,t'}\right\} _{t,t'\in\left\{ 1,\ldots,n\right\} }$ are real numbers, such that $\sigma_{t,t'} = \sigma_{t_{1},t'_{1}}$ whenever $t_{1} \neq t'_{1}$ and $t\neq t'$, and
the matrix $(\sigma_{t,t'} : t,t' \in \{1,\ldots,n\})$
is symmetric and positive definite.
In the analytic test problem ($\mathsection$\ref{sec:test}), Citi Bike problem ($\mathsection$\ref{sec:citibike}), and the problems simulated from Gaussian process priors ($\mathsection$\ref{sec:GPexample}),
we implemented BGO
and the benchmark algorithms with the squared exponential kernel
$\Sigma_{0}\left(x,w,x',w'\right)=
\sigma_{0}^{2}\mbox{exp}\left(-\sum_{k=1}^{n}\alpha_{1}^{\left(k\right)}\left[x_{k}-x'_{k}\right]^{2}-\sum_{k=1}^{d_{1}}\alpha_{2}^{\left(k\right)}\left[w_{k}-w'_{k}\right]^{2}\right)$,
where $\sigma_{0}^{2}$ is the common prior variance and $\alpha_{1}^{\left(1\right)},\ldots,\alpha_{1}^{\left(n\right)},\alpha_{2}^{\left(1\right)},\ldots,\alpha_{2}^{\left(d_{1}\right)}\in\mathbb{R}_{+}$
are the length scales parameters.
In the majority of our experiments
($\mathsection$\ref{sec:branin} and
$\mathsection$\ref{sec:CVexample_filtering}-$\mathsection$\ref{sec:IPexample})
we implement BGQO using the discretization-free approach with fully Bayesian inference over hyperparameters (\cref{alg:SBO} in $\mathsection$\ref{SBO}).
In the analytic test problem ($\mathsection$\ref{sec:test}), the Citi Bike problem ($\mathsection$\ref{sec:citibike}), and the problems simulated from Gaussian process priors ($\mathsection$\ref{sec:GPexample}),
we use the discretized version ($\mathsection$\ref{sec:discretization}) of the BQO algorithm. In these problems we also calculate the hyperparameters of the kernels, $\sigma^{2}$ and $\mu_{0}$, using maximum likelihood estimation following the first stage of samples. We do not use the discretization-free version of BQO in these problems because they were performed as part of an initial conference paper version of this work \citep{MCQMC}, and only the discretized version of BQO existed at that time.
Benchmark algorithms in each problem were implemented using the same approach to hyperparameter estimation as used by BQO.
\subsection{An Analytic Test Problem}
\label{sec:test}
In our first example, we consider the problem \eqref{eq:test} stated in $\mathsection$\ref{SBO}. BQO is well-suited to this problem because evaluations of $F(x,w)$ have much lower noise than those of $G(x)$. We do not compare against the multi-task algorithm \citep{swersky2013multi} and SDE algorithm \citep{williams2000sequential} because they can only be applied when the objective function is a finite sum.
We do not compare against \citet{Xie:2012} because this problem has noisy evaluations.
Figure~\ref{fig:analytic} compares the performance of BQO, KG and EI on this problem, plotting the number of samples beyond the first stage on the $x$ axis, and the average true quality of the solutions provided, $G(\mathrm{argmax}_x E_n[G(x)])$, averaging over 3000 independent runs of the three algorithms. We see that BQO substantially outperforms both benchmark methods. This is because BQO reduces the noise in its observations by conditioning on $w$, allowing it to more swiftly localize the objective's maximum.
\begin{figure}[!htb]
\centering
\includegraphics[width=0.50\linewidth]{comparisonSameConfigurationanal.pdf}
\caption{Performance comparison between BQO and two Bayesian optimization benchmark, the KG and EI methods, on the analytic test problem \eqref{eq:test} from $\mathsection$\ref{SBO}, as described in $\mathsection$\ref{sec:test}.
\label{fig:analytic}}
\end{figure}
\subsection{Branin Function}
\label{sec:branin}
In this example problem we compare BQO against the SDE \citep{williams2000sequential} and multi-task \citep[Section~3.2]{swersky2013multi} algorithms. We consider the Branin problem proposed in \citet{williams2000sequential} where $F\left(x_{1},x_{2},x_{3},x_{4}\right)=y_{b}\left(15x_{1}-5,15x_{2}\right)y_{b}\left(15x_{3}-5,15x_{4}\right)$,
\[
y_{b}\left(u,v\right)=\left(v-\frac{5.1}{4\pi^{2}}u^{2}+\frac{5}{\pi}u-6\right)^{2}+10\left(1-\frac{1}{8\pi}\right)\cos\left(u\right)+10
\]
is the Branin function and $x_{1},x_{2},x_{3},x_{4}\in\left[0,1\right]$. We define $x:=\left(x_{1},x_{4}\right)$ and $w:=\left(x_{2},x_{3}\right)$. The joint distribution $p$ of $w$ is defined in \Cref{probability_dist_w}. We maximize the function $G\left(x_{1},x_{4}\right):=\sum_{\left(x_{2},x_{3}\right)\in \left\{ 0.25,0.5,0.75\right\} \times\left\{ 0.2,0.4,0.6,0.8\right\} }p\left(x_{2},x_{3}\right)F\left(x_{1},x_{2},x_{3},x_{4}\right)$.
\begin{table}[htb]
\centering
\begin{tabular}{|l||r|r|r|c|}
\hline
$x_{2} / x_{3}$ & 0.2 & 0.4 & 0.6 & 0.8\\
\hline\hline
0.25 & 0.0375 & 0.0875 & 0.0875 & 0.0375\\
\hline
0.5 & 0.0750 & 0.1750 & 0.1750 & 0.0750\\
\hline
0.75 & 0.0375 & 0.0875 & 0.0875 & 0.0375\\
\hline
\end{tabular}
\caption{Probability distribution of $w=(x_{2},x_{3})$ for the Branin problem from $\mathsection$\ref{sec:branin}.}
\label{probability_dist_w}
\end{table}
Figure~\ref{fig:branin} compares the performance of BQO, SDE and the multi-task algorithm on this problem, plotting the number of samples beyond the first stage on the $x$ axis, and the average true quality of the solutions provided, $G(\mathrm{argmax}_x E_n[G(x)])$. We average over 100 independent runs of the BQO algorithm, 126 independent runs of the multi-task algorithm, and 230 independent runs of the SDE algorithm. We see that BQO substantially outperforms both the SDE and multi-task optimization benchmarks, despite the fact that these competing methods also model $F$. We believe this is because SDE and the multi-task optimization algorithm both choose points using a heuristic rule that performs poorly in certain settings, as explained in the introduction, rather than using a one-step optimality analysis like BQO.
\begin{figure}[!htb]
\centering
\includegraphics[width=0.50\linewidth]{plot_branin_mt_None_12.pdf}
\caption{Performance comparison between BQO, the SDE algorithm \citep{williams2000sequential}, and the multi-task algorithm \citep{swersky2013multi} on the Branin problem from $\mathsection$\ref{sec:branin}.\label{fig:branin}}
\end{figure}
\subsection{New York City's Citi Bike System}
\label{sec:citibike}
We now consider a queuing simulation based on New York City's Citi Bike system in which system users may remove an available bike from a station at one location within the city and ride it to a station with an available dock in some other location. The optimization problem that we consider is the allocation of a constrained number of bikes (6000) to available docks within the city at the start of rush hour, so as to minimize, in simulation, the expected number of potential trips in which the rider could not find an available bike at their preferred origination station, or could not find an available dock at their preferred destination station. We call such trips ``negatively affected trips.''
We simulate the demand for bike trips on days from January 1st to December 31st between 7:00am and 11:00am. We use 329 actual bike stations, locations, and numbers of docks from the Citi Bike system. In our simulator, we choose a day at random from the 365 days of the year and then simulate the demand for trips between each pair of bike stations on that day using an independent Poisson process whose rate is given by historical data from that day in 2014 available from Citi Bike's website \citep{citibike}.
Travel times between pairs of stations are modeled using an exponential distribution with parameters estimated from this same dataset. If a potential trip's origination station has no available bikes, then that trip does not occur, and we increment our count of negatively affected trips. If a trip does occur, and its preferred destination station does not have an available dock, then we also increment our count of negatively affected trips, and the bike is returned to the closest bike station with available docks.
We divide the bike stations into $4$ groups using k-nearest neighbors, and let $x$ be the number of bikes in each group at 7:00 AM. We suppose that bikes are allocated uniformly among stations within a single group.
The random variable $w$ is the total demand for bike trips during the period of our simulation, summed over all pairs of bike stations. The distribution of $w$ is a mixture of Poisson distributions. Evaluations of $F(x,w)$ for $w$ fixed are noisy due to additional sources of randomness beyond $w$ within our simulation.
We solve this problem with BQO, KG, EI and the multi-task algorithm. The multi-task algorithm cannot solve problems where the objective function is an infinite sum, as it is in this problem, so we modify the objective function it uses to a truncated expectation over finitely many values of $w$. Because implementing the multi-task algorithm become computationally intractable when there are thousands of tasks, we restrict this truncated expectation to 181 values of $w$.
Figure~\ref{fig:citibike} compares the performance of BQO, KG, EI and the multi-task algorithm, plotting the number of samples beyond the first stage on the $x$ axis, and the average true quality of the solutions provided, $G(\mathrm{argmax}_x E_n[G(x)])$, averaging over 300 independent runs of BQO, EI and KG, and 100 independent runs of the multi-task algorithm. We see that BQO quickly finds an allocation of bikes to groups that attains a small expected number of negatively affected trips. We believe that multi-task does poorly because of the large number of tasks, and its inability to leverage information across related tasks.
\begin{figure}[!htb]
\centering
\subcaptionbox{Performance comparison between BQO and two Bayesian optimization benchmark, the KG and EI methods, on the Citi Bike Problem from $\mathsection$\ref{sec:citibike} \label{fig:citibike}}[0.45\linewidth]{
\includegraphics[width=0.45\linewidth]{plot_citi_bike_mt_None_50.pdf}}
\quad
\subcaptionbox{Location of bike stations (circles) in New York City, where size and color represent the ratio of available bikes to available docks.}[0.45\linewidth]{
\includegraphics[width=0.45\linewidth]{testStationMapPng0-full.png}}
\caption{Performance results for the Citi Bike problem (plot a), and a screenshot from our simulation of the Citi Bike problem (plot b), as described in $\mathsection$\ref{sec:citibike}.
\label{fig:stuff}}
\end{figure}
\subsection{Hyperparameter Tuning in Recommender Systems}
\label{sec:CVexample_filtering}
In this subsection and the following we consider optimization of a machine learning model's hyperparameters where error is evaluated using cross-validation. Cross-validation is a method for estimating a machine learning model's error. In more detail, $n$-fold cross-validation randomly splits the training data into $n$ datasets of roughly equal size.
Then, for each dataset (or ``fold''), it trains the machine learning model holding out that data, and evaluates the error of the resulting estimates on the held out data.
The average of these errors is called the cross-validation error, and is used as an objective in optimization of a machine learning model's hyperparameters. In this approach, we minimize $\frac{1}{n}\sum_{i=1}^{n}L\left(x;D_{i}\right)$ over $x$, where $L\left(x;D_{i}\right)$ is the error of the model with hyperparameters $x$ evaluated on the $i$-th dataset $D_{i}$.
In this subsection we consider the problem of optimizing hyperparameters for probabilistic matrix factorization (PMF) models used in recommender systems \citep{mnih2008}. We apply this PMF model to a dataset from arxiv.org \citep{arxiv}, with information about downloads from $2752$ users on $2018$ papers.
We treat a user as providing a positive binary rating for a paper if that user downloaded the paper, which creates $263,238$ positive binary ratings.
We use $5$-fold cross-validation to provide an estimate of the test error as a function of four PMF model hyperparameters: the learning rate, the $\ell_{2}$ regularizer, the number of epochs, and the matrix rank. We then use EI, BQO and the multi-task algorithm to choose these hyperparameters to minimize this cross-validation error. EI simply selects a set of hyperparameters $x$ at each step and evaluates all 5 folds, while BQO and the multi-task algorithm select an $x$ and a fold $w$.
Figure~\ref{fig:pmf} compares the cross-validation error of these algorithms, plotting the number of folds queried beyond the first stage on the $x$ axis, and the best error obtained, averaging over $35$ independent runs of BQO and multi-task, and $65$ of EI. We see that BQO and multi-task perform similarly, and both outperform EI. We conjecture that multi-task's competitive performance in this problem is due to the small number of tasks and the homogeneity of the folds.
\begin{figure}[!htb]
\centering
\subcaptionbox{Performance comparison between BQO and two Bayesian optimization benchmark, the multi-task and EI methods, on the recommender system problem $\mathsection$\ref{sec:CVexample_filtering}\label{fig:pmf}}[0.40\linewidth]{
\includegraphics[width=0.40\linewidth]{plot_arxiv_None_1.pdf}}
\quad
\subcaptionbox{Performance comparison between BQO and two Bayesian optimization benchmark, the multi-task and EI methods, on the convolutional neural network problem $\mathsection$\ref{sec:CVexample_cnn}\label{fig:cnn}}[0.40\linewidth]{
\includegraphics[width=0.40\linewidth]{plot_cnn_cifar10_ei_None_10.pdf}}
\caption{Performance results for the recommender system (plot a) and convolutional neural network (plot b) problems.
\label{fig:stuff}}
\end{figure}
\subsection{Hyperparameter Tuning in Convolutional Neural Networks}
\label{sec:CVexample_cnn}
We consider the problem of training convolutional neural networks (CNNs) to classify images \citep{cnncifar}. We use $5$-fold cross validation on the CIFAR10 dataset \citep{cifar}, which consists of 10 classes and 50,000 training images. We choose the network architecture described in the pytorch tutorial \citep{pytorch}, which consists of two convolutional layers, two fully connected layers, and on top of them a softmax layer for final classification. We tune the following hyperparameters: the number of epochs, the batch size, the learning rate, the number of channels in the first convolutional layer, the size of the kernel in the convolutional layers, and the number of hidden units in the first fully connected layer. The number of channels in the second convolutional layer is the number of channels in the first convolutional layer plus 10, and the number of hidden units in the second fully connected layer is 84.
Figure~\ref{fig:cnn} compares the performance of EI, BQO and the multi-task algorithm, plotting the number of folds queried beyond the first stage on the $x$ axis, and the best error obtained, averaging over $90$ independent runs of BQO and the multi-task algorithm, and $75$ of EI. We see that BQO and multi-task perform similarly, but better than EI. As in the recommender system problem, we conjecture that multi-task is competitive because of the small number of tasks and their homogeneity.
\subsection{Newsvendor Problem under Dynamic Consumer Substitution}
\label{sec:IPexample}
The newsvendor problem under dynamic consumer substitution is adapted from \citet{ryzin2001Stocking}, and was considered in \citet{simopt}. In this problem, we choose the initial inventory levels of the products sold, each with given cost $c_{j}$ and price $p_{j}$. Our goal is to optimize profit.
\newcommand{\Psi}{\Psi}
\newcommand{\Upsilon}{\Upsilon}
A sequence of $T$ customers indexed by $t$ arrive in order and either buy an in-stock product, or decide to not buy anything. Here, $T$ is known. Customer $t$ assigns a utility $U^{j}_{t}$ to each product $j$, and to the no-purchase option (indexed by $j=0$). Customer $t$ decides which product to buy, if any, by choosing the $j$ with the largest $U^j_t$ among the in-stock $j$ and the no-purchase option.
Utilities for products ($j>0$) are modeled with the multinomial logit model, where $U_{t}^{j}=u^{j}+\xi_{t}^{j}$, $u^{j}$ is constant, and
$\left\{ \xi_{t}^{j}\right\}$ are mutually independent Gumbel random
variables with distribution function $P\left(\xi_{t}^{j}\leq z\right)=\mbox{exp}\left(-e^{-\left(z/\mu+\gamma\right)}\right) =: \Psi^j_t(z)$
where $\gamma$ is Euler's constant.
The utility for the no-purchase option is $U_t^0 = 0$.
In this problem, the objective function $G(x)$ is defined as the expected overall profit considering the $T$ customers starting from a vector $x$ of initial inventory positions for each product. This profit is computed as sum of the prices of the products sold minus the cost of the initial inventory.
We consider the setting where there are $1000$ customers and $2$ products whose costs are $5$ and $10$ dollars respectively and prices are $8$ and $18$ dollars respectively. We assume that $u^{j}$ is equal to $1$ for all $j>0$.
We now describe how we use BQO in this problem. Fix a product $j$. Observe that since $\xi_{t}^{j}$ for $j>0$ follows a Gumbel distribution with cdf $\Psi_{t}^{j}$, then $\Psi_{t}^{j}\left(Z\right)$ is uniformly
distributed on $\left[0,1\right]$. Consequently, $\Upsilon^{-1}\left(\Psi_{t}^{j}\left(\xi_{t}^{j}\right)\right)$
follows a gamma distribution where $\Upsilon$ is the gamma cumulative distribution function.
Now define the vector $W:=\left(W_{1},W_{2}\right)$ where $W_{j}=\sum_{t=1}^{T}\Upsilon^{-1}\left(\Psi_{t}^{j}\left(\xi_{t}^{j}\right)\right)$.
It is straightforward to simulate $\xi = (\xi_t^j : t, j)$ given $W$, for example by noting that the distribution of $(\Upsilon^{-1}(\Psi^j_t(\xi^j_t)) / W_j : t \ge 1)$ is Dirichlet and independent of $W_j$ (see Theorem 2.1 of Section 2.1.2 of \citealt{dirichletbook}). Thus, simulating from this Dirichlet and multiplying by the given value of $W_j$ provides a sample of $\xi^j$ given $W_j$. Alternatively, we can use a simple modification of Example 10e in Section 10.2 of \citet{rossSimulation}.
We can also simulate $\xi^j$ given that $W^j$ resides in an interval by acceptance-rejection sampling,
or by simulating $W^j$ from a truncated gamma distribution, and then simulating $\xi^j$ conditioned on $W^j$.
We then apply BQO with $F(x,w)$ equal to the conditional expectation of the profit given $W=w$ and the initial inventory levels $x$ for each product.
To observe this conditional expectation we average results from $25$ independent simulations, where the collection of values for $\xi_t^j$ are simulated conditioned on $W$.
\newcommand{q_{1/2}}{q_{1/2}}
We similarly apply the multi-task algorithm with $F(x,i)$ equal to the conditional expectation of the profit given the initial inventory level $x$ and that $W \in R_i$.
Here, each $R_{i}$ is a rectangular region of values for $W$, given by
$R_1 = [0,q_{1/2}]^2$, $R_2 = (q_{1/2},\infty)^2$,
$R_3 = [0,q_{1/2}]\times(q_{1/2},\infty)$, and
$R_4 = (q_{1/2},\infty)\times[0,q_{1/2}]$,
where $q_{1/2}$ is the median of $W_j$ (this is the same for $j=1$ and $j=2$).
For each observation
of this conditional expectation we average $25$ independent simulations. The EI algorithm observes the profit without conditioning, averaging $25$ independent simulations.
In Figure~\ref{fig:vendor} we compare the performance of EI, BQO and the multi-task algorithm, plotting the number of samples beyond the first stage on the $x$ axis, and the best profit obtained, averaging over $100$ independent runs of BQO, $80$ of multi-task, and $230$ of EI. We see that BQO outperforms the benchmark algorithms, and the multi-task algorithm underperforms the other algorithms considered.
\begin{figure}[!htb]
\centering
\includegraphics[width=0.50\linewidth]{plot_vendor_problem_None_4.pdf}
\caption{Performance results for the vendor problem $\mathsection$\ref{sec:IPexample}\label{fig:vendor}}
\end{figure}
\subsection{Problems Simulated from Gaussian Process Priors}
\label{sec:GPexample}
We now compare the performance of BQO against a benchmark Bayesian optimization algorithm on synthetic problems drawn at random from Gaussian process priors. By varying the parameters of the Gaussian process prior, we study how BQO's performance relative to a benchmark (the KG algorithm) varies with problem characteristics, offering insight into the types of real-world problems on which BQO is likely to provide the most substantial benefit in comparison with using a traditional Bayesian optimization method.
These experiments show that the most important factor influencing BQO's relative performance is the speed with which $F(x,w)$ varies with $w$.
BQO provides the most value when this variation is large enough to influence performance, and small enough to allow $F(x,w)$ to be modeled with a Gaussian process.
Thus, users of BQO should choose a $w$ that plays a big role in overall performance, and whose influence on performance is smooth enough to support predictive modeling.
These experiments also show that when settings are favorable, BQO provides substantial benefit, in some cases offering an improvement of almost $1000\%$. On those few problems in which BQO underperforms the benchmark, it underperforms by a much smaller margin of less than $50\%$.
We now construct these problems in detail.
Let $f(x,w,z)=h(x,w)+r(z)$ on $\left[0,1\right]^{2}\times\mathbb{R}$, where:
$r(z)$ is drawn, for each $z$ in a fine discretization of $[0,1]$, independently from a normal distribution with mean $0$ and variance $\alpha_{d}$
(we could have set $r$ to be an Orstein-Uhlenbeck process with large volatility, and obtained an essentially identical result); and
$h$ is drawn from a Gaussian Process with mean $0$ and Gaussian covariance function $\Sigma\left(\left(x,w\right),\left(x',w'\right)\right)=\alpha_{h}\exp\left(-\beta\left\Vert \left(x,w\right)-\left(x',w'\right)\right\Vert _{2}^{2}\right)$.
We then define $F$ by $F(x,w)=E[f(x,W,Z)\mid W=w]$ where the expectation is over $Z$, and $G$ by $G(x)=E[f(x,W,Z)]$, where the expectation is over both $W$ and $Z$,
$W$ is drawn uniformly from $\left\{ 0,1/49,2/49,\ldots,1\right\}$ and $Z$ is drawn uniformly from the discretization of $[0,1]$.
To observed $F$, we draw 1 sample of $W$ and $Z$ and average $f(x,W,Z)$.
(We also performed experiments, not shown here, that observed $F$ by averaging multiple samples, and found the same qualitative behavior.)
We thus have a class of problems parametrized by $\alpha_h$, $\alpha_d$, $\beta$, and an outcome measure determined by the overall number of samples.
Before displaying results,
we reparametrize the dependence on $\alpha_h$ and $\alpha_d$ in what will be a more interpretable way. We first set $\mathrm{Var}[f(x,W,Z)|W,Z] = \alpha_h + \alpha_d$ to 1, as multiplying both $\alpha_h$ and $\alpha_d$ by a scalar simply scales the problem.
Then, the variance reduction ratio
$\mathrm{Var}[f(x,W,Z)|W] / \mathrm{Var}[f(x,W,Z)]$
achieved by BQO in conditioning on $W$ is approximately
$\alpha_h / (\alpha_d + \alpha_h)$, with this estimate becoming exact as $\beta$ grows large and the values of $h(x,w)$ become uncorrelated across $w$.
We define $A = \alpha_h / (\alpha_d + \alpha_h)$ equal to this approximate variance reduction ratio.
Thus, our problems are parametrized by the approximate variance reduction ratio $A$,
the overall number of samples, and by $\beta$, which measures the speed with which $F(x,w)$ varies with $w$.
Given this parametrization, we sampled problems from Gaussian process priors using all combinations of
$A\in\left\{ \frac{1}{2}, \frac14, \frac18, \frac1{16} \right\}$ and
$\beta \in \left\{ 2^{-4}, 2^{-3},\ldots, 2^{9}, 2^{10} \right\}$.
We also performed additional simulations at $A=\frac12$ for $\beta \in \left\{ 2^{11},\ldots,2^{15}\right\}$.
Figure~\ref{fig:simulated} shows Monte Carlo estimates of the normalized performance difference between BQO and KG for these problems, as a function of $\log(\beta)$ ($\log$ is the natural logarithm), $A$, and the number of samples taken overall.
The normalized performance difference is estimated for each set of problem parameters by taking a randomly sampled problem generated using those problem parameters, discretizing the domain into 2500 points, running each algorithm independently 500 times on that problem, and averaging $(G(x^*_{\mathrm{BQO}}) - G(x^*_{\mathrm{KG}})) / |G(x^*_{\mathrm{KG}})|$ across these 500 samples, where $x^*_{\mathrm{BQO}}$ is the final solution calculated by BQO, and similarly for $x^*_{\mathrm{KG}}$.
\begin{figure}[!htbp]
\centering
\subcaptionbox{Normalized performance difference as a function of $\beta$ and $A$, when the overall number of samples is 50.
\label{fig:sim3}}[0.35\linewidth]{
\includegraphics[width=0.35\linewidth]{contourPlotbetahN1iter50ver3.pdf}}
\quad
\subcaptionbox{Normalized performance difference as a function of $\beta$ and the overall number of iterations, when $A=1/2$.
\label{fig:sim1}}[0.35\linewidth]{
\includegraphics[width=0.35\linewidth]{contourPlotbetahN1A2ver3.pdf}}
\caption{ Normalized performance difference between BQO and KG in problems simulated from a Gaussian process, as a function of $\beta$, which measures how quickly $F(x,w)$ varies with $w$, the approximate variance reduction ratio $A$, and the overall number of samples. BQO outperforms KG over most of the parameter space, and is approximately 10 times better when $\beta$ is near $\exp(4)$.
\label{fig:simulated}}
\end{figure}
The normalized performance difference is robust to $A$ and the overall number of samples, but is strongly influenced by $\beta$. BQO is always better than KG whenever $\beta \ge 1$. Moreover, it is substantially better than KG when $\mbox{log}(\beta)\in(3,5)$, with BQO outperforming KG by as much as a factor of $10$. For larger $\beta$, BQO remains better than KG, but by a smaller margin. This unimodal dependence of the normalized performance difference on $\beta$ can be understood as follows: BQO provides value by modeling the dependence of $F(x,w)$ on $w$. Modeling this dependence is most useful when $\beta$ takes moderate values because it is here where observations of $F(x,w)$ at one value of $w$ are most useful in predicting the value of $F(x,w)$ at other values of $w$. When $F$ varies very quickly with $w$ (large $\beta$), it is more difficult to generalize, and when $F$ varies very slowly with $w$ ($\beta$ close to $0$), then modeling dependence on $w$ is comparable with modeling $F$ as constant.
\section{Conclusions}
\label{conclusion}
We have presented a new Bayesian optimization algorithm, Bayesian Quadrature Optimization, designed for objectives that are sums or integrals of expensive-to-evaluate integrands. This method is derived from a conceptual one-step optimality analysis for which we provide novel computational techniques that support efficient implementation. We demonstrated that this method is consistent when the objective is a finite sum, and showed via extensive numerical experiments that it performs as well or better than the state of the art, providing substantial value when evaluations are noisy or the integrand varies smoothly in the integrated variables.
\section*{Acknowledgments}
The authors were partially supported by NSF CAREER CMMI-1254298, NSF CMMI-1536895, AFOSR FA9550-15-1-0038, AFOSR FA9550-16-1-0046, and DMR-1120296.
\putbib
\end{bibunit}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 5,924 |
Raj Kundra Porn Case: Bombay High Court dismisses Raj Kundra's plea for immediate release
The troubles of Raj Kundra, imprisoned for making porn movies, usually are not taking the title of lessening. On Saturday, August 7, the classes courtroom deferred the listening to on Raj Kundra's bail plea. Thereafter () has dismissed the petition for immediate release of Raj Kundra. Raj Kundra and his affiliate Ryan Thorpe had given this petition towards the order of the Magistrate Court. The Magistrate Court had already rejected Raj Kundra's bail plea. Let us inform that on July 19 final month, the Crime Branch of Mumbai Police arrested Raj Kundra for making porn movies. During the listening to within the Bombay High Court, public prosecutor Aruna Pai had instructed that the police had arrested Raj as a result of he was destroying proof by deleting some vital WhatsApp chats. During this listening to, it was additionally claimed that the police have recovered 51 porn movies from 2 apps of Raj Kundra. Earlier on August 4, the Metropolitan Magistrate's courtroom had rejected the bail plea of Raj Kundra and his aide Ryan Thorpe. In February, the Mumbai Police had registered a case of forcibly making porn movies. Several arrests have been additionally made on this case and the title of Raj Kundra got here up within the investigation. Police declare that the Hotshots app serving porn movies was run by Raj Kundra's firm Armspine. The app was later bought to Kenrin Pvt Ltd, London, owned by Raj's brother-in-law Pradeep Bakshi. | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 2,232 |
Il Rio Batalha è un affluente del fiume Tietê nello Stato di San Paolo in Brasile.
Percorso
Il fiume nasce in comune di Agudos, passa per Piratininga e Bauru dirigendosi verso nord-ovest. Passa poi ancora per Avaí dove devia verso nord e attraversa Reginópolis prima di sfociare nel fiume Tietê nei pressi di Uru. Il suo corso è lungo 167 km.
Note
Altri progetti
Batalha
Batalha | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 7,798 |
package edu.ucsf.lava.core.file;
import edu.ucsf.lava.core.file.exception.DeleteReadOnlyFileAccessException;
import edu.ucsf.lava.core.file.exception.SaveReadOnlyFileAccessException;
import edu.ucsf.lava.core.file.model.LavaFile;
public class ReadOnlyLocalFileSystemRepositoryStrategy extends LocalFileSystemRepositoryStrategy {
public void deleteFile(LavaFile file) throws DeleteReadOnlyFileAccessException {
this.logRepositoryError("Attempt to delete read-only resource from repository:", file);
throw new DeleteReadOnlyFileAccessException("Attempt to delete read-only resource from repository",file);
}
public LavaFile saveFile(LavaFile file) throws SaveReadOnlyFileAccessException {
this.logRepositoryError("Attempt to save to a read-only resource from repository:", file);
throw new SaveReadOnlyFileAccessException("Attempt to save to a read-only resource from repository.",file);
}
public LavaFile saveOrUpdateFile(LavaFile file) throws SaveReadOnlyFileAccessException {
this.logRepositoryError("Attempt to save to a read-only resource from repository:", file);
throw new SaveReadOnlyFileAccessException("Attempt to save to a read-only resource from repository.", file);
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 6,878 |
OTHER BOOKS BY CHESTER HIMES
AVAILABLE IN VINTAGE
_Cotton Comes to Harlem
The Heat's On_
FIRST VINTAGE BOOKS EDITION, DECEMBER 1988
Copyright © 1959 by Chester Himes
Copyright renewed 1987 by Lesley Himes
All rights reserved under International and Pan-American Copyright Conventions. Published in the United States by Random House, Inc., New York. Originally published in France as _Il pleut des coups dur_ and in the United States in 1958.
Library of Congress Cataloging-in-Publication Data
Himes, Chester B., 1909–1984
The real cool killers.
I. Title.
PS3515.1713R44 1988 813′.54 88-40121
eISBN: 978-0-307-80327-6
DISPLAY TYPOGRAPHY BY BARBARA M. BACHMAN
v3.1
# Contents
_Cover_
_Other Books by This Author_
_Title Page_
_Copyright_
Chapter 1
Chapter 2
Chapter 3
Chapter 4
Chapter 5
Chapter 6
Chapter 7
Chapter 8
Chapter 9
Chapter 10
Chapter 11
Chapter 12
Chapter 13
Chapter 14
Chapter 15
Chapter 16
Chapter 17
Chapter 18
_About the Author_
# 1
_"I'm gwine down to de river,
Set down on de ground.
If de blues overtake me,
I'll jump overboard and drown..."_
Big Joe Turner was singing a rock-and-roll adaptation of _Dink's Blues_. The loud licking rhythm blasted from the jukebox with enough heat to melt bones.
A woman leapt from her seat in a booth as though the music had struck her full of tacks. She was a lean black woman clad in a pink jersey dress and red silk stockings. She pulled up her skirt and began doing a shake dance as though trying to throw off the tacks one by one.
Her mood was contagious. Other women jumped down from their high stools and shook themselves into the act. The customers laughed and shouted and began shaking too. The aisle between the bar and the booths became stormy with shaking bodies.
Big Smiley, the giant-sized bartender, began doing a flatfooted locomotive shuffle up and down behind the bar.
The colored patrons of Harlem's Dew Drop Inn on 129th Street and Lenox Avenue were having the time of their lives that crisp October night.
A white man standing near the middle of the bar watched them with cynical amusement. He was the only white person present.
He was a big man, over six feet tall, dressed in a dark gray flannel suit, white shirt and blood-red tie. He had a big-featured, sallow face with the blotched skin of dissipation. His thick black hair was shot with gray. He held a dead cigar butt between the first two fingers of his left hand. On the third finger was a signet ring. He looked about forty.
The colored women seemed to be dancing for his exclusive entertainment. A slight flush spread over his sallow face.
The music stopped.
A loud grating voice said dangerously above the panting laughter: "Ah feels like cutting me some white mother-raper's throat."
The laughter stopped. The room became suddenly silent.
The man who had spoken was a scrawny little chicken-necked bantamweight, twenty years past his fist-fighting days, with gray stubble tinging his rough black skin. He wore a battered black derby green with age, a ragged plaid mackinaw and blue denim overalls.
His small enraged eyes were as red as live coals. He stalked stiff-legged toward the big white man, holding an open spring-blade knife in his right hand, the blade pressed flat against his overalled leg.
The big white man turned to face him, looking as though he didn't know whether to laugh or get angry. His hand strayed casually to the heavy glass ashtray on the bar.
"Take it easy, little man, and no one will get hurt," he said.
The little knifeman stopped two paces in front of him and said, "Efn' Ah finds me some white mother-raper up here on my side of town trying to diddle my little gals Ah'm gonna cut his throat."
"What an idea," the white man said. "I'm a salesman. I sell that fine King Cola you folks like so much up here. I just dropped in here to patronize my customers."
Big Smiley came down and leaned his ham-sized fists on the bar.
"Looka here, big, bad, and burly," he said to the little knifeman. "Don't try to scare my customers just 'cause you're bigger than they is."
"He doesn't want to hurt anyone," the big white man said. "He just wants some King Cola to soothe his mind. Give him a bottle of King Cola."
The little knifeman slashed at his throat and severed his red tie neatly just below the knot.
The big white man jumped back. His elbow struck the edge of the bar and the ashtray he'd been gripping flew from his hand and crashed into the shelf of ornamental wine glasses behind the bar.
The crashing sound caused him to jump back again. His second reflex action followed so closely on the first that he avoided the second slashing of the knife blade without even seeing it. The knot of his tie that had remained was split through the middle and blossomed like a bloody wound over his white collar.
"... throat cut!" a voice shouted excitedly as though yelling Home Run!
Big Smiley leaned across the bar and grabbed the red-eyed knifeman by the lapels of his mackinaw and lifted him from the floor.
"Gimme that chiv, shorty, 'fore I makes you eat it," he said lazily, smiling as though it were a joke.
The knifeman twisted in his grip and slashed him across the arm. The white fabric of his jacket sleeve parted like a burst balloon and his black-skinned muscles opened like the Red Sea.
Blood spurted.
Big Smiley looked at his cut arm. He was still holding the knifeman off the floor by the mackinaw collar. His eyes had a surprised look. His nostrils flared.
"You cut me, didn't you?" he said. His voice sounding unbelieving.
"Ah'll cut you again," the little knifeman said, wriggling in his grip.
Big Smiley dropped him as though he'd turned hot.
The little knifeman bounced on his feet and slashed at Big Smiley's face.
Big Smiley drew back and reached beneath the bar counter with his right hand. He came up with a short-handled fireman's axe. It had a red handle and a honed, razor-sharp blade.
The little knifeman jumped into the air and slashed at Big Smiley again, matching his knife against Big Smiley's axe.
Big Smiley countered with a right cross with the red-handled axe. The blade met the knifeman's arm in the middle of its stroke and cut it off just below the elbow as though it had been guillotined.
The severed arm in its coat sleeve, still clutching the knife, sailed through the air, sprinkling the nearby spectators with drops of blood, landed on the linoleum tile floor, and skidded beneath the table of a booth.
The little knifeman landed on his feet, still making cutting motions with his half arm. He was too drunk to realize the full impact. He saw that the lower part of his arm had been chopped off; he saw Big Smiley drawing back the red-handled axe. He thought Big Smiley was going to chop at him again.
"Wait a minute, you big mother-raper, till Ah finds my arm!" he yelled. "It got my knife in his hand."
He dropped to his knees and began scrambling about the floor with his one hand, searching for his severed arm. Blood spouted from his jerking stub as though from the nozzle of a hose.
Then he lost consciousness and flopped on his face.
Two customers turned him over; one tied a necktie as a tourniquet about the bleeding arm, the other inserted a chair leg to tighten it.
A waitress and another customer were twisting a knotted towel about Big Smiley's arm. He was still holding the fireman's axe in his right hand, a look of surprise on his face.
The white manager stood on top of the bar and shouted, "Please remain seated, folks. Everybody go back to his seat and pay his bill. The police have been called and everything will be taken care of."
As though he'd fired a starting gun, there was a race for the door.
When Sonny Pickens came out on the sidewalk he saw the big white man looking inside through one of the small front windows.
Sonny had been smoking marijuana cigarettes and he was tree-top high. Seen from his drugged eyes, the dark night sky looked bright purple and the dingy smoke-blackened tenements looked like brand new skyscrapers made of strawberry-colored bricks. The neon signs of the bars and pool rooms and greasy spoons burned like phosphorescent fires.
He drew a blue steel revolver from his inside coat pocket, spun the cylinder and aimed it at the big white man.
His two friends, Rubberlips Wilson and Lowtop Brown, looked at him in pop-eyed amazement. But before either could restrain him, Sonny advanced on the white man, walking on the balls of his feet.
"You there!" he shouted. "You the man what's been messing around with my wife."
The big white man jerked his head about and saw a pistol. His eyes stretched and the blood drained from his sallow face.
"My God, wait a minute!" he cried. "You're making a mistake. All of you folks are confusing me with someone else."
"Ain't going to be no waiting now," Sonny said and pulled the trigger.
Orange flame lanced toward the big white man's chest. Sound shattered the night.
Sonny and the white man leapt simultaneously straight up into the air. Both began running before their feet touched the ground. Both ran straight ahead. They ran head on into one another at full speed. The white man's superior weight knocked Sonny down and he ran over him.
He plowed through the crowd of colored spectators, scattering them like ninepins, and cut across the street through the traffic, running in front of cars as though he didn't see them.
Sonny jumped up to his feet and took out after him. He ran over the people the big white man had knocked down. Muscles rolled on bones beneath his feet. He staggered drunkenly. Screams followed him and car lights came down on him like shooting stars.
The big white man was moving between parked cars across the street when Sonny shot at him again. He gained the sidewalk safely and began running south along the inner edge.
Sonny followed between the cars and kept after him.
People in the line of fire did acrobatic dives for safety. People up ahead crowded into the doorways to see what was happening. They saw a big white man with wild blue eyes and a stubble of red tie which made him look as though his throat were cut, being chased by a slim black man with a big blue pistol. They drew back out of range.
But the people behind, who were safely out of range, joined the chase.
The white man was in front. Sonny was next. Rubberlips and Lowtop were running at Sonny's heels. Behind them the spectators stretched out in a ragged line.
The white man ran past a group of eight Arabs at the corner of 127th Street. All of the Arabs had heavy, grizzly black beards. All wore bright green turbans, smoke-colored glasses, and ankle-length white robes. Their complexions ranged from stovepipe black to mustard. They were jabbering and gesticulating like a frenzied group of caged monkeys. The air was redolent with the pungent scent of marijuana.
"An infidel!" one yelled.
The jabbering stopped abruptly. They wheeled in a group after the white man.
The white man heard the shout. He saw the sudden movement through the corners of his eyes. He leaped forward from the curb.
A car coming fast down 127th Street burnt rubber in an ear-splitting shriek to keep from running him down.
Seen in the car's headlights, his sweating face was bright red and muscle-ridged; his blue eyes black with panic; his gray-shot hair in wild disorder.
Instinctively he leaped high and sideways, away from the oncoming car. His arms and legs flew out in grotesque silhouette.
At that instant Sonny came abreast of the Arabs and shot at the leaping white man while he was still in the air.
The orange blast lit up Sonny's distorted face and the roar of the gunshot sounded like a fusillade.
The big white man shuddered and came down limp. He landed face down and in a spread-eagled posture. He didn't get up.
Sonny ran up to him with the smoking pistol dangling from his hand. He was starkly spotlighted by the car's headlights. He looked at the white man lying face down in the middle of the street and started laughing. He doubled over laughing, his arms jerking and his body rocking.
Lowtop and Rubberlips caught up. The eight Arabs joined them in the beams of light.
"Man, what happened?" Lowtop asked.
The Arabs looked at him and began to laugh.
Rubberlips began to laugh too, then Lowtop.
All of them stood in the stark white light, swaying and rocking and doubling up with laughter.
Sonny was trying to say something but he was laughing so hard he couldn't get it out.
A police siren sounded nearby.
# 2
The telephone rang in the captain's office at the 126th Street precinct station. The uniformed officer behind the desk reached for the outside phone without looking up from behind the record sheet he was filling out.
"Harlem precinct, Lieutenant Anderson," he said.
A high-pitched correct voice said, "Are you the man in charge?"
"Yes, lady," Lieutenant Anderson said patiently and went on writing with his free hand.
"I want to report that a white man is being chased down Lenox Avenue by a colored man with a gun," the voice said with the smug sanctimoniousness of a saved sister.
Lieutenant Anderson pushed aside the record sheet and pulled forward a report pad.
When he'd finished taking down the essential details of her incoherent account, he said, "Thank you, Mrs. Collins," hung up and reached for the closed line to central police on Centre Street.
"Give me the radio dispatcher," he said.
Two colored men were driving east on 135th Street in the wake of a crosstown bus. Shapeless dark hats sat squarely on their clipped kinky hair and their big frames filled up the front seat of a small, battered black sedan.
Static crackled from the shortwave radio and a metallic voice said: "Calling all cars. Riot threatens in Harlem. White man running south on Lenox Avenue at 128th Street. Chased by drunken Negro with gun. Danger of murder."
"Better goose it," the one on the inside said in a grating voice.
"I reckon so," the driver replied laconically.
He gave a short sharp blast on the siren and gunned the small sedan in a crying U-turn in the middle of the block, cutting in front of a taxi coming fast from the direction of The Bronx.
The taxi tore its brakes to keep from ramming into the sedan. Seeing the private license plates, the taxi driver thought they were two small-time hustlers trying to play big shots with the siren on their car. He was an Italian from The Bronx who had grown up with bigtime-gangsters and Harlem hoodlums didn't scare him.
He leaned out of his window and yelled, "You ain't plowing cotton in Mississippi, you black son of a bitch. This is New York City, the Big Apple, where people drive–"
The colored man riding with his girl friend in the back seat leaned quickly forward and yanked at his sleeve. "Man, come back in here and shut yo' mouth," he warned anxiously. "Them is Grave Digger Jones and Coffin Ed Johnson you is talking to. Can't you see that police antenna stuck up from their tail."
"Oh, that's them," the driver said, cooling off as quickly as a showgirl on a broke stud. "I didn't recognize 'em."
Grave Digger had heard him but he mashed the gas without looking around.
Coffin Ed drew his pistol from its shoulder sling and spun the cylinder. Passing street light glinted from the long nickel-plated barrel of the special .38 revolver, and the five brass-jacketed bullets looked deadly in the six chambers. The one beneath the trigger was empty. But he kept an extra box of shells along with his report book and handcuffs in his greased-leather-lined right coat pocket.
"Lieutenant Anderson asked me last night why we stick to these old-fashioned rods when the new ones are so much better. He was trying to sell me on the idea of one of those new hydraulic automatics that shoot fifteen times; said they were faster, lighter and just as accurate. But I told him we'd stick to these."
"Did you tell him how fast you could reload?" Grave Digger carried its mate beneath his left arm.
"Naw, I told him he didn't know how hard these Harlem Negroes' heads are," Coffin Ed said.
His acid-scarred face looked sinister in the dim panel light.
Grave Digger chuckled. "You should have told him that these people don't have any respect for a gun that doesn't have a shiny barrel half a mile long. They want to see what they're being shot with."
"Or else hear it, otherwise they figure it can't do any more damage than their knives."
When they came onto Lenox, Grave Digger wheeled south through the red light with the siren open, passing in front of an eastbound trailer truck, and slowed down behind a sky blue Cadillac Coupe de Ville trimmed in yellow metal, hogging the southbound lane between a bus and a fleet of northbound refrigerator trucks. It had a New York State license plate numbered B-H-21. It belonged to Big Henry who ran the "21" numbers house. Big Henry was driving. His bodyguard, Cousin Cuts, was sitting beside him on the front seat. Two other rugged-looking men occupied the back seat.
Big Henry took the cigar from his thick-lipped mouth with his right hand, tapped ash in the tray sticking out of the instrument panel, and kept on talking to Cuts as though he hadn't heard the siren. The flash of a diamond in his cigar hand lit up the rear window.
"Get him over," Grave Digger said in a flat voice.
Coffin Ed leaned out of the right side window and shot the rear-view mirror off the door hinge of the big Cadillac.
The cigar hand of Big Henry became rigid and the back of his fat neck began to swell as he looked at his shattered mirror. Cuts rose up in his seat, twisting about threateningly, and reached for his pistol. But when he saw Coffin Ed's sinister face staring at him from behind the long nickel-plated barrel of the .38 he ducked like an artful dodger from a hard thrown ball.
Coffin Ed planted a hole in the Cadillac's front fender.
Grave Digger chuckled. "That'll hurt Big Henry more than a hole in Cousin Cut's head."
Big Henry turned about with a look of pop-eyed indignation on his puffed black face, but it sank in like a burst balloon when he recognized the detectives. He wheeled the car frantically toward the curb and crumpled his right front fender into the side of the bus.
Grave Digger had space enough to squeeze through. As they passed, Coffin Ed lowered his aim and shot Big Henry's gold lettered initials from the Cadillac's door.
"And stay over!" he yelled in a grating voice.
They left Big Henry giving them a how-could-you-do-this-to-me-look with tears in his eyes.
When they came abreast the Dew Drop Inn they saw the deserted ambulance and the crowd running on ahead. Without slowing down, they wormed between the cars parked haphazardly in the street and pushed through the dense jam of people, the sirens shrieking. They dragged to a stop when their headlights focused on the macabre scene.
"Split!" one of the Arabs hissed. "Here's the things."
"The monsters," another chimed.
"Keep cool, fool," the third admonished. "They got nothing on us."
The two tall, lanky, loose-jointed detectives hit the pavement in unison, their nickel-plated .38 specials gripped in their hands. They looked like big-shouldered plowhands in Sunday suits at a Saturday night jamboree.
"Straighten up!" Grave Digger yelled at the top of his voice.
"Count off!" Coffin Ed echoed.
There was movement in the crowd. The morbid and the innocent moved in closer. Suspicious characters began to blow.
Sonny and his two friends turned startled, pop-eyed faces.
"Where they come from?" Sonny mumbled in a daze.
"I'll take him," Grave Digger said.
"Covered," Coffin Ed replied.
Their big flat feet made slapping sounds as they converged on Sonny and the Arabs. Coffin Ed halted at an angle that put them all in line of fire.
Without a break in motion, Grave Digger closed on Sonny and slapped him on the elbow with the barrel of his pistol. With his free hand he caught Sonny's pistol when it flew from his nerveless fingers.
"Got it," he said as Sonny yelped in pain and grabbed his numb arm.
"I ain't–" Sonny tried to finish but Grave Digger shouted, "Shut up!"
"Line up and reach!" Coffin Ed ordered in a threatening voice, menacing them with his pistol. He sounded as though his teeth were on edge.
"Tell the man, Sonny," Lowtop urged in a trembling voice, but it was drowned by Grave Diggers's thundering at the crowd: "Back up!" He lined a shot overhead.
They backed up.
Sonny's good arm shot up and his two friends reached. He was still trying to say something. His Adam's apple bobbed helplessly in his dry wordless throat.
But the Arabs were defiant. They dangled their arms and shuffled about.
"Reach where, man?" one of them said in a husky voice.
Coffin Ed grabbed him by the neck, lifted him off his feet.
"Easy, Ed," Grave Digger cautioned in a strangely anxious voice. "Easy does it."
Coffin Ed halted, his pistol ready to shatter the Arab's teeth, and shook his head like a dog coming out of water. Releasing the Arab's neck, he backed up one step and said in his grating voice: "One for the money... and two for the show..."
It was the first line of a jingle chanted in the game of hide-and-seek as a warning from the "seeker" to the "hiders" that he was going after them.
Grave Digger took the next line, "Three to get ready..."
But before he could finish it with "And here we go," the Arabs had fallen into line with Sonny and had raised their hands high into the air.
"Now keep them up," Coffin Ed said.
"Or you'll be the next ones lying on the ground," Grave Digger added.
Sonny finally got out the words, "He ain't dead. He's just fainted."
"That's right," Rubberlips confirmed. "He ain't been hit. It just scared him so he fell unconscious."
"Just shake him and he'll come to," Sonny added.
The Arabs started to laugh again, but Coffin Ed's sinister face silenced them.
Grave Digger stuck Sonny's revolver into his own belt, holstered his own revolver, and bent down and lifted the white man's face. Blue eyes stared fixedly at nothing. He lowered the head gently and picked up a limp, warm hand, feeling for a pulse.
"He ain't dead," Sonny repeated. But his voice had grown weaker. "He's just fainted, that's all."
He and his two friends watched Grave Digger as though he were Jesus Christ bending over the body of Lazarus.
Grave Digger's eyes explored the white man's back. Coffin Ed stood without moving, his scarred face like a bronze mask cast with trembling hands. Grave Digger saw a black wet spot in the white man's thick gray-shot black hair, low down at the base of the skull. He put his fingertips to it and they came off stained. He straightened up slowly, held his wet fingertips in the white headlights; they showed red. He said nothing.
The spectators crowded nearer. Coffin Ed didn't notice; he was looking at Grave Digger's bloody fingertips.
"Is that blood?" Sonny asked in a breaking whisper. His body began to tremble, coming slowly upward from his grasshopper legs.
Grave Digger and Coffin Ed stared at him, saying nothing.
"Is he dead?" Sonny asked in a terror-stricken whisper. His trembling lips were dust dry and his eyes were turning white in a black face gone gray.
"Dead as he'll ever be," Grave Digger said in a flat toneless voice.
"I didn't do it," Sonny whispered. "I swear 'fore God in heaven."
"He didn't do it," Rubberlips and Lowtop echoed in unison.
"How does it figure?" Coffin Ed asked.
"It figures for itself," Grave Digger said.
"So help me God, boss. I couldn't have done it," Sonny said in a terrified whisper.
Grave Digger stared at him from agate hard eyes and said nothing.
"You gotta believe him, boss, he couldn't have done it," Rubberlips vouched.
"Naw, suh," Lowtops echoed.
"I wasn't trying to hurt him, I just wanted to scare him," Sonny said. Tears were trickling from his eyes.
"It were that crazy drunk man with the knife that started it," Rubberlips said. "Back there in the Dew Drop Inn."
"Then afterwards the big white man kept looking in the window," Lowtop said. "That made Sonny mad."
The detectives stared at him with blank eyes. The Arabs were motionless.
"He's a comedian," Coffin Ed said finally.
"How could I be mad about my old lady," Sonny argued. "I ain't even got any old lady."
"Don't tell me," Grave Digger said in an unrelenting voice, and handcuffed Sonny. "Save it for the judge."
"Boss, listen, I beg you, I swear 'fore God–"
"Shut up, you're under arrest," Coffin Ed said.
# 3
A police car siren sounded from the distance. It was coming from the east; it started like the wail of an anguished banshee and grew into a scream. Another sounded from the west; it was joined by other from the north and south, one sounding after another like jets taking off from an aircraft carrier.
"Let's see what these real cool Moslems are carrying," Grave Digger said.
"Count off, you sheiks," Coffin Ed said.
They had the case wrapped up before the prowl cars arrived. The pressure was off. They felt cocky.
"Praise Allah," the tallest of the Arabs said.
As though performing a ritual, the others said, "Mecca," and all bowed low with outstretched arms.
"Cut the comedy and straighten up," Grave Digger said. "We're holding you as witnesses."
"Who's got the prayer?" the leader asked with bowed head.
"I've got the prayer," another replied.
"Pray to the great monster," the leader commanded.
The one who had the prayer turned slowly and presented his white-robed backside to Coffin Ed. A sound like a hound dog baying issued from his rear end.
"Allah be praised," the leader said, and the loose white sleeves of their robes fluttered in response.
Coffin Ed didn't get it until Sonny and his friends laughed in amazement. Then his face contorted in black rage.
"Punks!" he grated harshly, somersaulted the bowed Arab with one kick, and leveled on him with his pistol as if to shoot him.
"Easy man, easy," Grave Digger said, trying to keep a straight face. "You can't shoot a man for aiming a fart at you."
"Hold it, monster," a third Arab cried, and flung liquid from a glass bottle toward Coffin Ed's face. "Sweeten thyself."
Coffin Ed saw the flash of the bottle and the liquid flying and ducked as he swung his pistol barrel.
"It's just perfume," the Arab cried in alarm.
But Coffin Ed didn't hear him through the roar of blood in his head. All he could think of was a con man called Hank throwing a glass of acid into his face. And this looked like another acid thrower. Quick scalding rage turned his acid-burnt face into a hideous mask and his scarred lips drew back from his clenched teeth.
He fired two shots together and the Arab holding the half-filled perfume bottle said, "Oh," softly and folded slowly to the pavement. Behind, in the crowd, a woman screamed as her leg gave beneath her.
The other Arabs broke into wild flight. Sonny broke with them. A split second later his friends took off in his wake.
"God damn it, Ed!" Grave Digger shouted and lunged for the gun.
He made a grab for the barrel, deflecting the aim as it went off again. The bullet cut a telephone cable in two overhead. It fell into the crowd, setting off a cacophony of screams.
Everybody ran.
The panic-stricken crowd stampeded for the nearest doorways, trampling the woman who was shot and two others who fell.
Grave Digger grappled with Coffin Ed and they crashed down on top of the dead white man. Grave Digger had Coffin Ed's pistol by the barrel and was trying to wrest it from his grip.
"It's me, Digger, Ed," he kept saying. "Let go the gun."
"Turn me loose, Digger, turn me loose. Let me kill 'im," Coffin Ed mouthed insanely, tears streaming from his hideous face. "They tried it again, Digger."
They rolled over the corpse and rolled back.
"That wasn't acid, that was perfume," Grave Digger said, gasping for breath.
"Turn me loose, Digger, I'm warning you," Coffin Ed mumbled.
While they threshed back and forth over the corpse, two of the Arabs followed Sonny into the doorway of a tenement. The other people crowding into the doorway stepped aside and let them pass. Sonny saw the stairs were crowded and kept on going through, looking for a back exit. He came out onto a small back courtyard, enclosed with stone walls. The Arabs followed him. One put a noose over his head, knocking off his hat, and drew it tight. The other pulled a switch-blade knife and pressed the point against his side.
"If you holler you're dead," the first one said.
The Arab leader joined them.
"Let's get him away from here," he said.
At that moment the patrol cars began to unload. Two harness cops and Detective Haggerty hit the deck and were the first on the murder scene.
"Holy mother!" Haggerty exclaimed.
The cops stared aghast.
It looked to them as though the two colored detectives had the big white man locked in a death struggle.
"Don't just stand there," Grave Digger panted. "Give me a hand."
"They'll kill him," Haggerty said, wrapping his arms about Grave Digger and trying to pull him away. "You grab the other one," he said to the cops.
"To hell with that," the cop said, swinging his black-jack across Coffin Ed's head, knocking him unconscious.
The other cop drew his pistol and took aim at the corpse. "One move out of you and I'll shoot," he said.
"He won't move; he's dead," Grave Digger said to Haggerty.
"Well, Hell," Haggerty said indignantly, releasing him. "You asked me to help. How in hell do I know what's going on?"
Grave Digger shook himself and looked at the third cop. "You didn't have to slug him," he said.
"I wasn't taking no chances," the cop said.
"Shut up and watch the Arab," Haggerty said.
The cop moved over and looked at the Arab. "He's dead, too."
"Holy Mary, the plague," Haggerty said. "Look after that woman then."
Four more cops came running. At Haggerty's order, two turned toward the woman who'd been shot. She was lying in the street, deserted.
"She's alive, just unconscious," the cop said.
"Leave her for the ambulance," Haggerty said.
"Who're you ordering about?" the cop said. "We know our business."
"To hell with you," Haggerty said.
Grave Digger bent over Coffin Ed, lifted his head and put an open bottle of ammonia to his nose. Coffin Ed groaned.
A red-faced uniformed sergeant built like a General Sherman tank loomed above him.
"What happened here?" he asked.
Grave Digger looked up. "A rumpus broke and we lost our prisoner."
"Who shot your partner?"
"He's not shot, he's just knocked out."
"That's all right then. What's your prisoner look like?"
"Black man, about five eleven, twenty-five to thirty years, one-seventy to one-eighty pounds, narrow face sloping down to chin, wearing light gray hat, dark gray hickory-striped suit, white tab collar, red striped tie, beige chukker boots. He's handcuffed."
The sergeant's small china-blue eyes went from the big white corpse to the bearded Arab corpse.
"Which one did he kill?" he asked.
"The white man," Grave Digger said.
"That's all right, we'll get him," he said. Raising his voice, he called, "Professor!"
The corporal who'd stopped to light a cigarette said, "Yeah."
"Rope off this whole goddamned area," the sergeant said. "Don't let anybody out. We want a Harlem-dressed Zulu. Killed a white man. Can't have gotten far 'cause he's handcuffed."
"We'll get 'im," the corporal said.
"Pick up all suspicious persons," the sergeant said.
"Right," the corporal said, hurrying off towards the cops that were just arriving.
"Who shot the Arab?" the sergeant asked.
"Ed shot him," Grave Digger said.
"That's all right then," the sergeant said. "We'll get your prisoner. I'm sending for the lieutenant and the medical examiner. Save the rest for them."
He turned and followed the corporal.
Coffin Ed stood up shakily. "You should have let me killed that son of a bitch, Digger," he said.
"Look at him," Grave Digger said, nodding toward the Arab's corpse.
Coffin Ed stared.
"I didn't even know I hit him," he said as though coming out of a daze. After a moment he added, "I can't feel sorry for him. I tell you, Digger, death is on any son of a bitch who tries to throw acid into my eyes again."
"Smell yourself, man," Grave Digger said.
Coffin Ed bent his head. The front of his dark wrinkled suit reeked with the scent of dime-store perfume.
"That's what he threw. Just perfume," Grave Digger said. "I tried to warn you."
"I must not have heard you."
Grave Digger took a deep breath. "God damn it, man, you got to control yourself."
"Well, Digger, a burnt child fears fire. Anybody who tries to throw anything at me when they're under arrest is apt to get shot."
Grave Digger said nothing.
"What happened to our prisoner?" Coffin Ed asked.
"He got away," Grave Digger said.
They turned in unison and surveyed the scene.
Patrol cars were arriving by the minute, erupting cops as though for an invasion. Others had formed blockades across Lenox Avenue at 128th and 126th Streets, and had blocked off 127th Street on both sides.
Most of the people had gotten off the street. Those that stayed were being arrested as suspicious persons. Several drivers trying to move their cars were protesting their innocence loudly.
The packed bars in the area were being rapidly sealed by the police. The windows of tenements were jammed with black faces and the exits blocked by police.
"They'll have to go through this jungle with a fine-toothed comb," Grave Digger said. "With all these white cops about, any colored family might hide him."
"I'll want those gangster punks too," Coffin Ed said.
"Well, we'll just have to wait now for the men from homicide."
But Lieutenant Anderson arrived first, with the harness sergeant and Detective Haggerty latched on to him. The five of them stood in a circle in the car's headlights between the two corpses.
"All right, just give me the essential points first," Anderson said. "I put out the flash so I know the start. The man hadn't been killed when I got the first report."
"He was dead when we got here," Grave Digger said in a flat, toneless voice. "We were the first here. The suspect was standing over the victim with the pistol in his hand–"
"Hold it," a new voice said.
A plain-clothes lieutenant and a sergeant from downtown homicide bureau came into the circle.
"These are the arresting officers," Anderson said.
"Where's the prisoner?" the homicide lieutenant asked.
"He got away," Grave Digger said.
"Okay, start over," the homicide lieutenant said.
Grave Digger gave him the first part then, went on:
"There were two friends with him and a group of teenage gangsters around the corpse. We disarmed the suspect and handcuffed him. When we started to frisk the gangster punks we had a rumble. Coffin Ed shot one. In the rumble the suspect got away."
"Now let's get this straight," the homicide lieutenant said.
"Were the teenagers implicated too?"
"No, we just wanted them as witnesses," Grave Digger said. "There's no doubt about the suspect."
"Right."
"When I got here Jones and Johnson were fighting, rolling all over the corpse," Haggerty said. "Jones was trying to disarm Johnson."
Lieutenant Anderson and the men from homicide looked at him, then turned to look at Grave Digger and Coffin Ed in turn.
"It was like this," Coffin Ed said. "One of the punks turned up his ass and farted toward me and–"
Anderson said, "Huh!" and the homicide lieutenant said incredulously, "You killed a man for farting?"
"No, it was another punk he shot," Grave Digger said in his toneless voice. "One who threw perfume on him from a bottle. He thought it was acid the punk was throwing."
They looked at Coffin Ed's acid-burnt face and looked away embarrassedly.
"The fellow who was killed is an Arab," the sergeant said.
"That's just a disguise," Grave Digger said. "They belong to a group of teenage gangsters who call themselves Real Cool Moslems."
"Hah!" the homicide lieutenant said.
"Mostly they fight a teenage gang of Jews from The Bronx," Grave Digger elaborated. "We leave that to the welfare people."
The homicide sergeant stepped over to the Arab corpse and removed the turban and peeled off the artificial beard. The face of a colored youth with slick conked hair and beardless cheeks stared up. He dropped the disguises beside the corpse and sighed.
"Just a baby," he said.
For a moment no one spoke.
Then the homicide lieutenant asked, "You have the homicide gun?"
Grave Digger took it from his pocket, holding the barrel by the thumb and first finger, and gave it to him.
The lieutenant examined it curiously for some moments. Then he wrapped it in his handkerchief and slipped it into his coat pocket.
"Had you questioned the suspect?" he asked.
"We hadn't gotten to it," Grave Digger said. "All we know is the homicide grew out of a rumpus at the Dew Drop Inn."
"That's a bistro a couple of blocks up the street," Anderson said. "They had a cutting there a short time earlier."
"It's been a hot time in the old town tonight," Haggerty said.
The homicide lieutenant raised his brows enquiringly at Lieutenant Anderson.
"Suppose you go to work on that angle, Haggerty," Anderson said. "Look into that cutting. Find out how it ties in."
"We figure on doing that ourselves," Grave Digger said.
"Let him go on and get started," Anderson said.
"Right-o," Haggerty said. "I'm the man for the cutting."
Everybody looked at him. He left.
The homicide lieutenant said, "Well, let's take a look at the stiffs."
He gave each a cursory examination. The teenager had been shot once, in the heart.
"Nothing to do but wait for the coroner," he said.
They looked at the unconscious woman.
"Shot in the thigh, high up," the homicide sergeant said. "Loss of blood but not fatal – I don't think."
"The ambulance will be here any minute," Anderson said.
"Ed shot at the gangster twice," Grave Digger said. "It must have been then."
"Right."
No one looked at Coffin Ed. Instead, they made a pretense of examining the area.
Anderson shook his head. "It's going to be a hell of a job finding your prisoner in this dense slum," he said.
"There isn't any need," the homicide lieutenant said. "If this was the pistol he had, he's as innocent as you and me. This pistol won't kill anyone." He took the pistol from his pocket and unwrapped it. "This is a thirty-seven caliber blank pistol. The only bullets made to fit it are blanks and they can't be tampered with enough to kill a man. And it hasn't been made over into a zip gun."
"Well," Lieutenant Anderson said at last. "That tears it."
# 4
There was a rusty sheet-iron gate in the concrete wall between the small back courts. The gang leader unlocked it with his own key. The gate opened silently on oiled hinges.
He went ahead.
"March!" the henchman with the knife ordered, prodding Sonny.
Sonny marched.
The other henchman kept the noose around his neck like a dog chain.
When they'd passed through, the leader closed and locked the gate.
One of the henchman said, "You reckon Caleb is bad hurt?"
"Shut up talking in front of the captive," the leader said. "Ain't you got no better sense than that."
The broken concrete paving was strewn with broken glass bottles, rags and diverse objects thrown from the back windows: a rusty bed spring, a cotton mattress with a big hole burnt in the middle, several worn-out automobile tires, the half-dried carcass of a black cat with its left foot missing and its eyes eaten out by rats.
They picked their way through the debris carefully.
Sonny bumped into a loose stack of garbage cans. One fell with a loud clatter. A sudden putrid stink arose.
"God damn it, look out!" the leader said. "Watch where you're going."
"Aw, man, ain't nobody thinking about us back here," Choo-Choo said.
"Don't call me man," the leader said.
"Sheik, then."
"What you jokers gonna do with me?" Sonny asked.
His weed jag was gone; he felt weak-kneed and hungry; his mouth tasted brackish and his stomach was knotted with fear.
"We're going to sell you to the Jews," Choo-Choo said.
"You ain't fooling me, I know you ain't no Arabs," Sonny said.
"We're going to hide you from the police," Sheik said.
"I ain't done nothing," Sonny said.
Sheik halted and they all turned and looked at Sonny. His eyes were white half moons in the dark.
"All right then, if you ain't done nothing we'll turn you back to the cops," Sheik said.
"Naw, wait a minute, I just want to know where you're taking me."
"We're taking you home with us."
"Well, that's all right then."
There was no back door to the hall as in the other tenement. Decayed concrete stairs led down to a basement door. Sheik produced a key on his ring for that one also. They entered a dark passage. Foul water stood on the broken pavement. The air smelled like molded rags and stale sewer pipes. They had to remove their smoked glasses in order to see.
Halfway along, feeble yellow light slanted from an open door. They entered a small, filthy room.
A sick man clad in long cotton drawers lay beneath a ragged horse blanket on a filthy pallet of burlap sacks.
"You got anything for old Bad-eye," he said in a whining voice.
"We got you a fine black gal," Choo-Choo said.
The old man raised up on his elbows. "Whar she at?"
"Don't tease him," Inky said.
"Lie down and shut," Sheik said. "I told you before we wouldn't have nothing for you tonight." Then to his henchmen, "Come on, you jokers, hurry up."
They began stripping off their disguises. Beneath their white robes they wore sweat shirts and black slacks. The beards were put on with make-up gum.
Without their disguises they looked like three high-school students.
Sheik was a tall yellow boy with strange yellow eyes and reddish kinky hair. He had the broad-shouldered, trim-waisted figure of an athlete. His face was broad, his nose flat with wide, flaring nostrils, and his skin freckled. He looked disagreeable.
Choo-Choo was shorter, thicker and darker, with the egg-shaped head and flat, mobile face of the born joker. He was bowlegged and pigeon-toed but fast on his feet.
Inky was an inconspicuous boy of medium size, with a mild, submissive manner, and black as the ace of spades.
"Where's the gun?" Choo-Choo asked when he didn't see it stuck in Sheik's belt.
"I slipped it to Bones."
"What's he going to do with it?"
"Shut up and quit questioning what I do."
"Where you reckon they all went to, Sheik?" Inky asked, trying to be peacemaker.
"They went home if they got sense," Sheik said.
The old man on the pallet watched them fold their disguises into small packages.
"Not even a little taste of King Kong," he whined.
"Naw, nothing!" Sheik said.
The old man raised up on his elbows. "What do you mean, naw? I'll throw you out of here. I'se the janitor. I'll take my keys away from you. I'll–"
"Shut your mouth before I shut it and if any cops come messing around down here you'd better keep it shut too. I'll have something for you tomorrow."
"Tomorrow? A bottle?"
The old man lay back mollified.
"Come on," Sheik said to the others.
As they were leaving he snatched a ragged army overcoat from a nail on the door without the janitor noticing. He stopped Sonny in the passage and took the noose from about his neck, then looped the overcoat over the handcuffs. It looked as though Sonny were merely carrying an overcoat with both hands.
"Now nobody'll see those cuffs," Sheik said. Turning to Inky, he said, "You go up first and see how it looks. If you think we can get by the cops without being stopped, give us the high sign."
Inky went up the rotten wooden stairs and through the doorway to the ground-floor hall. After a minute he opened the door and beckoned.
They went up in single file.
Strangers who'd ducked into the building to escape the shooting were held there by two uniformed cops blocking the outside doorway. No one paid any attention to Sonny and the three gangsters. They kept on going to the top floor.
Sheik unlocked a door with another key on his ring, and led the way into a kitchen.
An old colored woman clad in a faded blue Mother Hubbard with darker blue patches sat in a rocking chair by a coal-burning kitchen stove, darning a threadbare man's woolen sock on a wooden egg, and smoking a corncob pipe.
"Is that you, Caleb?" she asked, looking over a pair of ancient steel-rimmed spectacles.
"It's just me and Choo-Choo and Inky," Sheik said.
"Oh, it's you, Samson." The very note of expectancy in her voice died in disappointment. "Whar's Caleb?"
"He went to work downtown in a bowling alley, Granny. Setting up pins," Sheik said.
"Lord, that chile is always out working at night," she said with a sigh. "I sho hope God he ain't getting into no trouble with all this night work, 'cause his old Granny is too old to watch over him as a mammy would."
She was so old the color had faded in spots from her dark brown skin so that it looked like the skin of a dried speckled pea, and once-brown eyes had turned milky blue. Her bony cranium was bald at the front and the speckled skin was taut against the skull. What remained of her short gray hair was gathered into a small tight ball at the back of her head. The outline of each finger bone plying the darning needle was plainly visible through the transparent parchment-like skin.
"He ain't getting into no trouble," Sheik said.
Inky and Choo-Choo pushed Sonny into the kitchen and closed the door.
Granny peered over her spectacles at Sonny. "I don't know this boy. Is he a friend of Caleb's too?"
"He's the fellow Caleb is taking his place," Sheik said. "He hurt his hands."
She pursed her lips. "There's so many of you boys coming and going in here all the time I sho hope you ain't getting into no mischief. And this new boy looks older than you others is."
"You worry too much," Sheik said harshly.
"Hannh?"
"We're going on to our room," Sheik said. "Don't wait up for Caleb. He's going to be late."
"Hannh?"
"Come on," Sheik said. "She ain't hearing no more."
It was a shotgun flat, one room opening into the other. The next room contained two small white enameled iron beds where Caleb and his grandmother slept, and a small potbellied stove on a tin mat in one corner. A table held a pitcher and washbowl; there was a small dime-store mirror on top of a chest of drawers. As in the kitchen, everything was spotlessly clean.
"Give me your things and watch out for Granny," Sheik said, taking their bundled-up disguises.
Choo-Choo bent his head to the keyhole.
Sheik unlocked a large old cedar chest with another key from his ring and stored their bundles beneath layers of old blankets and house furnishings. It was Granny's hope chest; there she stored things given her by the white folks she worked for to give Caleb when he got married. Sheik locked the chest and unlocked the door to the next room. They followed him and he locked the door behind them.
It was the room he and Choo-Choo rented. There was a double bed where he and Choo-Choo slept, chest of drawers and mirror, pitcher and bowl on the table, as in the other room. The corner was curtained off with calico for a closet. But a lot of junk lay around and it wasn't as clean.
A narrow window opened to the platform of the red-painted iron fire escape that ran down the front of the building. It was protected by an iron grille closed by a padlock.
Sheik unlocked the grille and stepped out onto the fire escape.
"Look at this," he said.
Choo-Choo joined him; Inky and Sonny squeezed into the window.
"Watch the captive, Inky," Sheik said.
"I ain't no captive," Sonny said.
"Just look," Sheik said, pointing toward the street.
Below, on the broad avenue, red-eyed prowl cars were scattered thickly, like monster ants about an ant-hill. Three ambulances were threading through the maze, two police hearses, and cars from the police commissioner's office and the medical examiner's office. Uniformed cops and men in plain clothes were coming and going in every direction.
"The men from Mars," Sheik said. "The big dragnet. What you think about that, Choo-Choo?"
Choo-Choo was busy counting.
The lower landings and stairs of the fire escape were packed with other people watching the show. Every front window as far as the eye could see on both sides of the street was jammed with black heads.
"I counted thirty-one prowl cars," Choo-Choo said. "That's more than was up on Eighth Avenue when Coffin Ed got that acid throwed in his eyes."
"They're shaking down the buildings one by one," Sheik said.
"What we're going to do with our captive?" Choo-Choo asked.
"We got to get the cuffs off first. Maybe we can hide him up in the pigeon's roost."
"Leave the cuffs on him."
"Can't do that. We got to get ready for the shakedown."
He and Choo-Choo stepped back into the room. He took Sonny by the arm, and pointed toward the street.
"They're looking for you, man."
Sonny's black face began graying again.
"I ain't done nothing. That wasn't a real pistol I had. That was a blank gun."
The three of them stared at him disbelievingly.
"Yeah, that ain't what they think," Choo-Choo said.
Sheik was staring at Sonny with a strange expression. "You sure, man?" he asked tensely.
"Sure I'm sure. It wouldn't shoot nothing but thirty-seven caliber blanks."
"Then it wasn't you who shot the big white stud?"
"That's what I been telling you. I couldn't have shot him."
A change came over Sheik. His flat, freckled yellow face took on a brutal look. He hunched his shoulders, trying to look dangerous and important.
"The cops are trying to frame you, man," he said. "We got to hide you now for sure."
"What you doing with a gun that don't shoot bullets?" Choo-Choo asked.
"I keep it in my shine parlor as a gag, is all," Sonny said.
Choo-Choo snapped his fingers. "I know you. You're the joker what works in that shoe shine parlor beside the Savoy."
"It's my own shoe shine parlor."
"How much marijuana you got stashed there?"
"I don't handle it."
"Sheik, this joker's a square."
"Cut the gab," Sheik said. "Let's get these handcuffs off this captive."
He tried keys and lockpicks but he couldn't get them open. So he gave Inky a triangle file and said, "Try filing the chain in two. You and him set on the bed." Then to Sonny, "What's your name, man?"
"Aesop Pickens, but people mostly call me Sonny."
"All right then, Sonny."
They heard a girl's voice talking to Granny and listened silently to rubber-soled shoes crossing the other room.
A single rap, then three quick ones, then another single rap sounded on the door.
"Gaza," Sheik said with his mouth against the panel.
"Suez," a girl's voice replied.
Sheik unlocked the door.
A girl entered and he locked the door behind her.
She was a tall sepia-colored girl with short black curls, wearing a turtle-necked sweater, plaid skirt, bobby socks, and white buckskin shoes. She had a snub nose, wide mouth, full lips, even white teeth, and wide-set brown eyes fringed with long black lashes.
She looked about sixteen years old, and was breathless with excitement.
Sonny stared at her and muttered to himself, "If this ain't it, it'll have to do."
"Hell, it's just Sissie. I thought it was Bones with the gun," Choo-Choo said.
"Stop beefing about the gun. It's safe with Bones. The cops ain't going to shake down no garbage collector's house. His old man works for the city same as they do."
"What's this about Bones and the gun?" Sissie asked.
"Sheik's got–"
"It's none of Sissie's business," Sheik cut him off.
"Somebody said an Arab had been shot and at first I thought it was you," Sissie said.
"You hoped it was me," Sheik said.
She turned away, blushing.
"Don't look at me," Choo-Choo said to Sheik. "You tell her. She's your girl."
"It was Caleb," Sheik said.
"Caleb! Jesus!" Sissie dropped onto the bed beside Sonny. She looked stunned. "Jesus! Poor little Caleb. What will Granny do?"
"What the hell can she do?" Sheik said brutally. "Raise him from the dead?"
"Does she know?"
"Does it look like she knows?"
"Jesus! Poor little Caleb. What did he do?"
"I gave old Coffin Ed the stink gun and–" Choo-Choo began.
"You didn't!" she exclaimed.
"The hell I didn't."
"What did Caleb do?"
"He threw perfume over the monster. It's the Moslem salute for cops. I told you about it before. But the monster must have thought Cal was throwing some more acid into his eyes. He blasted so fast we couldn't tell him any better."
"Jesus!"
"Where's Sugartit?" Sheik asked.
"At home. She didn't come into town tonight. I phoned her and she said she was sick."
"Yeah. Did you have any trouble getting in here?"
"No. I told the cops at the door that I live here."
They heard the signal rapped on the door.
Sissie gasped.
Sheik looked at her suspiciously. "What the hell's the matter with you?" he asked.
"Nothing."
He hesitated before opening the door. "You ain't expecting nobody?"
"Me? No. Who could I expect?"
"You're acting mighty funny."
"I'm just nervous."
The signal was rapped again.
Sheik stepped to the door and said, "Gaza."
"Suez," a girl's lilting voice replied.
Sheik gave Sissie a threatening look as he unlocked the door.
A small-boned chocolate-brown girl dressed like Sissie slipped hurriedly into the room.
At sight of Sissie she stopped and said, "Oh!" in a guilty tone of voice.
Sheik looked from one to the other. "I thought you said she was at home," he accused Sissie.
"I thought she was," Sissie said.
He turned his gaze on Sugartit. "What the hell's the matter with you? What the hell's going on here?"
"A Moslem's been killed and I thought it was you," she said.
"All you little bitches were hoping it was me," he said.
She had sloe eyes with long black lashes that looked secretive. She threw a quick defiant look at Sissie and said, "Don't include me in that."
"Did you tell Granny?" Sheik asked.
"Of course not."
"It was your lover, Caleb," Sheik said brutally.
She gave a shriek and charged at Sheik, clawing and kicking.
"You dirty bastard!" she cried. "You're always picking at me."
Sissie pulled her off. "Shut up and keep your mouth shut," she said tightly.
"You tell her," Sheik said.
"It was Caleb, all right," Sissie said.
"Caleb!" Sugartit screamed and flung herself face down across the bed. She was up in a flash, hurling accusations at Sheik. "You did it. You got him killed. On account of me. 'Cause he had the best go and you couldn't get me to do what you made Sissie do."
"That's a lie," Sissie said.
"Caleb!" Sugartit screamed at the top of her voice.
"Shut up, Granny will hear you," Choo-Choo said.
"Granny! Caleb's dead! Sheik killed him!" she screamed again.
"Stop her," Sheik commanded Sissie. "She's getting hysterical and I don't want to have to hurt her."
Sissie clutched her from behind, put one hand over her mouth and twisted her arm behind her back with the other.
Sugartit looked furiously at Sheik over the top of Sissie's hand.
"Granny can't hear," Inky said.
"The hell she can't," Choo-Choo said. "She can hear when she wants to."
"Let me go!" Sugartit mumbled and bit Sissie's hand.
"Stop that!" Sissie said.
"I'm going to him," Sugartit mumbled. "I love him. You can't stop me. I'm going to find out who shot him."
"Your old man shot him," Sheik said brutally. "The monster, Coffin Ed."
"Did I hear someone calling Caleb?" Granny asked from the other side of the door.
Sheik closed his hands quickly about Sugartit's throat and choked her into silence.
"Naw, Granny," he called. "It's just these silly girls arguing about their cubebs."
"Hannh?"
"Cubebs!" Sheik shouted.
"You chillen make so much racket a body can't hear herself think," she muttered.
They heard her shuffling back to the kitchen.
"Jesus, she's sitting up waiting for him," Sissie said.
Sheik and Choo-Choo exchanged glances.
"She don't even know what's happening in the street," Choo-Choo said.
Sheik took his hands away from Sugartit's throat.
# 5
"How soon can you find out what he was killed with?" the chief of police asked.
"He was killed with a bullet, naturally," the assistant medical examiner said.
"You're not funny," the chief said. "I mean what caliber bullet."
His brogue had begun thickening and the cops who knew him best began getting nervous.
The deputy coroner snapped his bag shut with a gesture of coyness and peered at the chief through magnified eyeballs encircled by black gutta-percha.
"That can't be known until after the autopsy. The bullet will have to be removed from the corpse's brain and subjected to tests–"
The chief listened in red-faced silence.
"I don't perform the autopsy. I'm the night man. I just pass on whether they're dead. I marked this one as D.O.A. That means dead on arrival – my arrival, not his. You know more about whether he was dead on his arrival than I do, and more about how he was killed, too."
"I asked you a civil question."
"I'm giving you a civil answer. Or, I should say, a civil service answer. The men who do the autopsy come on duty at nine o'clock. You ought to get your report by ten."
"That's all I asked you. Thanks. And damn little good that'll do me tonight. And by ten o'clock tomorrow morning the killer ought to be hell and gone to another part of the United States if he's got any sense."
"That's your affair, not mine. You can send the stiffs to the morgue when you've finished with them. I'm finished with them now. Good night, everyone."
No one answered. He left.
"I never knew why we needed a goddamned doctor to tell us whether a stiff was dead or not," the chief grumbled.
He was a big weather-beaten man dressed in a lot of gold braid. He'd come up from the ranks. Everything about him from the armful of gold hash stripes to the box-toed custom-made shoes said "flatfoot." Behind his back the cops on Centre Street called him Spark Plug, after the tender-footed nag in the comic strip "Barney Google."
The group near the white man's corpse, of which he was the hub, had grown by then, to include, in addition to the principals, two deputy police commissioners, an inspector from homicide, and nameless uniformed lieutenants from adjoining precincts.
The deputy commissioners kept quiet. Only the commissioner himself had any authority over the chief, and he was at home in bed.
"This thing's hot as hell," the chief said at large. "Have we got our stories synchronized?"
Heads nodded.
"Come on then, Anderson, we'll meet the press," he said to the lieutenant in charge of the 126th Street precinct station.
They walked across the street to join a group of newsmen who were being held in leash.
"Okay, men, you can get your pictures," he said.
Flash bulbs exploded in his face. Then the photographers converged on the corpses and left him facing the reporters.
"Here it is, men. The dead man has been identified by his paper as Ulysses Galen of New York City. He lives alone in a two-room suite at Hotel Lexington. We've checked that. They think his wife is dead. He's a sales manager for the King Cola Company. We've contacted their main office in Jersey City and learned that Harlem is in his district."
His thick brogue dripped like milk and honey through the noisy night. Stylos scratched on pads. Flash bulbs went off around the corpses like an anti-aircraft barrage.
"A letter in his pocket from a Mrs. Helen Kruger, Wading River, Long Island, begins with Dear Dad. There's an unposted letter addressed to Homer Galen in the sixteen hundred block on Michigan Avenue in Chicago. That's a business district. We don't know whether Homer Galen is his son or another relation–"
"What about how he was killed?" a reporter interrupted.
"We know that he was shot in the back of the head by a Negro man named Sonny Pickens who operates a shoe shine parlor at 134th Street and Lenox Avenue. Several Negroes resented the victim drinking in a bar at 129th Street and Lenox–"
"What was he doing at a crummy bar up here in Harlem?"
"We haven't found that out yet. Probably just slumming. We know that the barman was cut trying to protect him from another colored assailant–"
"How did the shine assail him?"
"This is not funny, men. The first Negro attacked him with a knife – tried to attack him; the bartender saved him. After he left the bar Pickens followed him down the street and shot him in the back."
"You expect him to shoot a white man in the front."
"Two colored detectives from the 126th Street precinct station arrived on the scene in time to arrest Pickens virtually in the act of homicide. He still had the gun in his hand," the chief continued. "They handcuffed the prisoner and were in the act of bringing him in when he was snatched by a teenage Harlem gang that calls itself Real Cool Moslems."
Laughter burst from the reporters.
"What, no Mau-Maus?"
"It's not funny, men," the chief said again. "One of them tried to throw acid in one of the detective's eyes."
The reporters were silenced.
"Another gangster threw acid in an officer's face up here about a year ago, wasn't it?" a reporter said. "He was a colored cop, too. Johnson, Coffin Ed Johnson, they called him."
"It's the same officer," Anderson said, speaking for the first time.
"He must be a magnet," the reporters said.
"He's just tough and they're scared of him," Anderson said. "You've got to be tough to be a colored cop in Harlem. Unfortunately, colored people don't respect colored cops unless they're tough."
"He shot and killed the acid thrower," the chief said.
"You mean the first one or this one?" the reporter asked.
"This one, the Moslem," Anderson said.
"During the excitement, Pickens and the others escaped into the crowd," the chief said.
He turned and pointed toward a tenement building across the street. It looked indescribably ugly in the glare of a dozen powerful spotlights. Uniformed police stood on the roof, others were coming and going through the entrance; still others stuck their heads out of front windows to shout to other cops in the street. The other front windows were jammed with colored faces, looking like clusters of strange purple fruit in the stark white light.
"You can see for yourselves we're looking for the killer," the chief said. "We're going through those buildings with a fine-toothed comb, one by one, flat by flat, room by room. We have the killer's description. He's wearing toolproof handcuffs. We should have him in custody before morning. He'll never get out of that dragnet."
"If he isn't already out," a reporter said.
"He's not out. We got here too fast for that."
The reporters then began to question him.
"Is Pickens one of the Real Cool Moslems?"
"We know he was rescued by seven of them. The eighth was killed."
"Was there any indication of robbery?' "
"Not unless the victim had valuables we don't know about. His wallet, watch and rings are intact."
"Then what was the motive? A woman?"
"Well, hardly. He was an important man, well off financially. He didn't have to chase up here."
"It's been done before."
The chief spread his hands. "That's right. But in this case both Negroes who attacked him did so because they resented his presence in a colored bar. They expressed their resentment in so many words. We have colored witnesses who heard them. Both Negroes were intoxicated. The first had been drinking all evening. And Pickens had been smoking marijuana also."
"Okay, chief, it's your story," the dean of the police reporters said, calling a halt.
The chief and Anderson recrossed the street to the silent group.
"Did you get away with it?" one of the deputy commissioners asked.
"God damn it, I had to tell them something," the chief said defensively. "Did you want me to tell them that a fifteen-thousand-dollar-a-year white executive was shot to death on a Harlem street by a weedhead Negro with a blank pistol who was immediately rescued by a gang of Harlem juvenile delinquents while all we got to show for the efforts of the whole god-damned police force is a dead adolescent who's called a Real Cool Moslem?"
"Sho' 'nuff cool now," Haggerty slipped in _sotto voce_.
"You want us to become the laughing stock of the whole goddamned world," the chief continued, warming up to the subject. "You want it said the New York City police stood by helpless while a white man got himself killed in the middle of a crowded nigger street?"
"Well, didn't he?" the homicide lieutenant said.
"I wasn't accusing you," the deputy commissioner said apologetically.
"Pickens is the one it's rough on," Anderson said. "We've got him branded as a killer when we know he didn't do it."
"We don't know any such goddamned thing," the chief said, turning purple with rage. "He might have rigged the blanks with bullets. It's been done, God damn it. And even if he didn't kill him, he hadn't ought to've been chasing him with a goddamned pistol that sounded as if it was firing bullets. We haven't got anybody to work on but him and it's just his black ass."
"Somebody shot him, and it wasn't with any blank gun," the homicide lieutenant said.
"Well, God damn it, go ahead and find out who did it!" the chief roared. "You're on homicide; that's your job."
"Why not one of the Moslems," the deputy commissioner offered helpfully. "They were on the scene, and these teenage gangsters always carry guns."
There was a moment of silence while they considered this.
"What do you think, Jones?" the chief asked Grave Digger. "Do you think there was any connection between Pickens and the Moslems?"
"It's like I said before," Grave Digger said. "It didn't look to me like it. The way I figure it, those teenagers gathered around the corpse directly after the shooting, like everybody else was doing. And when Ed began shooting, they all ran together, like everybody else. I see no reason to believe that Pickens even knows them."
"That's what I gathered too," the chief said disappointedly.
"But this is Harlem," Grave Digger amended. "Nobody knows all the connections here."
"Furthermore, we don't have but one of them and that one isn't carrying a gun," Anderson said. "And you've heard Haggerty's report on the statement he took from the bartender and the manager of the Dew Drop Inn. Both Pickens and the other man resented Galen making passes at the colored women. And none of the Moslem gang were even there at the time."
"It could have been some other man feeling the same way," Grave Digger said. "He might have seen Pickens shooting at Galen and thought he'd get in a shot, too."
"These people!" the chief said. "Okay, Jones, you begin to work on that angle and see what you can dig up. But keep it from the press."
As Grave Digger started to walk away, Coffin Ed fell in beside him.
"Not you, Johnson," the chief said. "You go home."
Both Grave Digger and Coffin Ed turned and faced the silence.
"Am I under suspension?" Coffin Ed asked in a grating voice.
"For the rest of the night," the chief said. "I want you both to report to the commissioner's office at nine o'clock tomorrow morning. Jones, you go ahead with your investigation. You know Harlem, you know where you have to go, who to see." He turned to Anderson. "Have you got a man to work with him?"
"Haggerty," Anderson offered.
"I'll work alone," Grave Digger said.
"Don't take any chances," the chief said. "If you need help, just holler. Bear down hard. I don't give a goddamn how many heads you crack; I'll back you up. Just don't kill any more juveniles."
Grave Digger turned and walked with Coffin Ed to their car.
"Drop me at the Independent Subway," Coffin Ed said.
Both of them lived in Jamaica and rode the E train when they didn't use the car.
"I saw it coming," Grave Digger said.
"If it had happened earlier I could have taken my daughter to a movie," Coffin Ed said. "I see so little of her it's getting so I hardly know her."
# 6
"Let her loose now," Sheik said.
Sissie let her go.
"I'll kill him!" Sugartit raved in a choked voice. "I'll kill him for that!"
"Kill who?" Sheik asked, scowling at her.
"My father. I hate him. The ugly bastard. I'll steal his pistol and shoot him."
"Don't talk like that," Sissie said. "That's no way to talk about your father."
"I hate him, the dirty cop!"
Inky looked up from the handcuffs he was filing. Sonny stared at her.
"Shut up," Sissie said.
"Let her go ahead and croak him," Sheik said.
"Stop picking on her," Sissie said.
Choo-Choo said, "They won't do nothing to her for it. All she got to say is her old man beat her all the time and they'll start crying and talking 'bout what a poor mistreated girl she is. They'll take one look at Coffin Ed and believe her."
"They'll give her a medal," Sheik said.
"Those old welfare biddies will find her a fine family to live with. She'll have everything she wants. She won't have to do nothing but eat and sleep and go to the movies and ride around in a big car," Choo-Choo elaborated.
Sugartit flung herself across the foot of the bed and burst into loud sobs.
"It'll save us the trouble," Sheik said.
Sissie's eyes widened. "You wouldn't!" she said.
"You want to bet we wouldn't?"
"If you keep talking like that I'm going to quit."
Sheik gave her a threatening look. "Quit what?"
"Quit the Moslems."
"The only way you can quit the Moslems is like Caleb quit," Sheik said.
"If I'd ever thought that poor little Caleb–"
Sheik cut her off. "I'll kill you myself."
"Aw, Sheik, she don't mean nothing," Choo-Choo said nervously. "Why don't you light up a couple of sticks and let us Islamites fly to Mecca."
"And let the cops smell it when they shake us down and take us all in. Where are your brains at?"
"We can go up on the roof."
"There're cops on the roof, too."
"On the fire escape then. We can close the window."
Sheik gave it grave consideration. "Okay, on the fire escape. I ain't got but two left and we got to get rid of them anyway."
"I'm going to look and see where the cops is at by now," Choo-Choo said, putting on his smoked glasses.
"Take those cheaters off," Sheik said. "You want the cops to identify you?"
"Aw hell, Sheik, they couldn't tell me from nobody else. Half the cats in Harlem wear their smoke cheaters all night long."
"Go 'head and take a gander at the avenue. We ain't got all night," Sheik said.
Choo-Choo started climbing out the window.
At that moment the links joining the handcuffs separated with a small clinking sound beneath Inky's file.
"Sheik, I've got 'em filed in two," Inky said triumphantly.
"Let's see."
Sonny stood up and stretched his arms.
"Who's he?" Sissie asked as though she'd noticed him for the first time.
"He's our captive," Sheik said.
"I ain't no captive," Sonny said. "I just come with you 'cause you said you was gonna hide me."
Sissie looked round-eyed at the severed handcuffs dangling from the wrists. "What did he do?" she asked.
"He's the gangster who killed the syndicate boss," Sheik said.
Sugartit stopped sobbing abruptly and rolled over and looked up at Sonny through wide wet eyes.
"Was that who he is?" Sissie asked in an awed tone. "The man who was killed, I mean."
"Sure. Didn't you know?" Sheik said.
"I done told you I didn't kill him," Sonny said.
"He claims he had a blank gun," Sheik said. "He's just trying to build up his defense. But the cops know better."
"It was a blank gun," Sonny said.
"What did he kill him for?" Sissie asked.
"They're having a gang war and he got assigned by the Brooklyn mob to make the hit."
"Oh, go to hell," Sissie said.
"I ain't killed nobody," Sonny said.
"Shut up," Sheik said. "Captives ain't allowed to talk."
"I'm getting tired of that stuff," Sonny said.
Sheik looked at him threateningly. "You want us to turn you over to the cops?"
Sonny backtracked quickly. "Naw, Sheik, but hell, ain't no need of taking advantage of me–"
Choo-Choo stuck his head in the window and cut him off: "Cops is out here like white on rice. Ain't nothing but cops."
"Where they at now?" Sheik asked.
"They're everywhere, but right now they's taking the house two doors down. They got all kinds of spotlights turned on the front of the house and cops is walking around down the street with machine guns. We better hurry if we're going to move the prisoner."
"Keep cool, fool," Sheik said. "Take a look at the roof."
"Praise Allah," Choo-Choo said, backing away on his hands and knees.
"Get out of that coat and shirt," Sheik ordered Sonny.
When Sonny had stripped to his underwear shirt, Sheik looked at him and said, "Nigger, you sure are black. When you was a baby your mama must'a had to chalk your mouth to tell where to stick it."
"I ain't no blacker than Inky," Sonny said defensively.
"I ain't in that," Inky said.
Sheik grinned at him derisively. "You didn't have no trouble, did you, Inky? Your mama used luminous paint on you."
"Come on, man, I'm getting cold," Sonny said.
"Keep your pants on," Sheik said. "Ladies present."
He hung Sonny's coat with his own clothes on the wire line behind the curtain and threw the shirt in the corner. Then he tossed Sonny an old faded red turtle-necked sweater.
"Pull the sleeves down over the irons and put on that there overcoat," he directed, indicating the old army coat he'd taken from the janitor.
"It's too hot," Sonny protested.
"You gonna do what I say, or do I have to slug you?"
Sonny put on the coat.
Sheik then took a pair of leather driving gauntlets from his pasteboard suitcase beneath the bed and handed them to Sonny, too.
"What am I gonna do with these?" Sonny asked.
"Just put them on and shut up, fool," Sheik said.
He then took a long bamboo pole from behind the bed and began passing it through the window. On one end was attached a frayed felt New York Giants pennant.
Choo-Choo came down the fire escape in time to take the pole and lean it against the ladder.
"Ain't no cops on this roof yet but the roof down where they's shaking down is lousy with 'em," he reported.
His face was shiny with sweat and the whites of his eyes had begun to glow.
"Don't chicken out on me now," Sheik said.
"I just needs some pot to steady my nerves."
"Okay, we're going to blow two now." Sheik turned to Sonny and said, "Outside, boy."
Sonny gave him a look, hesitated, then climbed out on the fire-escape landing.
"Let me come, too," Sissie said.
Sugartit sat up with sudden interest.
"I want both you little jailbaits to stay right here in this room and don't move," Sheik ordered in a hard voice, then turned to Inky, "You come on, Inky, I'm gonna need you."
Inky joined the others on the fire escape. Sheik came last and closed the window. They squatted in a circle. The landing was crowded.
Sheik took two limp cigarettes from the roll of his sweatshirt and stuck them into his mouth.
"Bombers!" Choo-Choo exclaimed. "You've been holding out on us."
"Give me some fire and less of your lip," Sheik said.
Choo-Choo flipped a dollar lighter and lit both cigarettes. Sheik sucked the smoke deep into his lungs, then passed one of the sticks to Inky.
"You and Choo-Choo take halvers and me and the captive will split this one."
Sonny raised both gloved hands in a pushing gesture. "Pass me. That gage done got me into more trouble now than I can get out of."
"You're chicken," Sheik said contemptuously, sucking another puff. He swallowed back the smoke each time it started up from his lungs. His face swelled and began darkening with blood as the drug took hold. His eyes became dilated and his nostrils flared.
"Man, if I had my heater I bet I could shoot that sergeant down there dead between the eyes," he said. The cigarette was stuck to his bottom lip and dangled up and down when he talked.
"What I'd rather have me is one of those hard-shooting long-barreled thirty-eights like Grave Digger and Coffin Ed have got," Choo-Choo said. "Them heaters can kill a rock. Only I'd want me a silencer on it and I could sit here and pick off any mother-raper I wanted. But I wouldn't shoot nobody unless he was a big shot or the chief of police or somebody like that."
"You're talking about rathers, what you'd rather have; me, I'm talking about facts," Sheik said, the cigarette bobbing up and down.
"What you're talking about will get you burnt up in Sing-Sing if you don't watch out," Choo-Choo said.
"What you mean!" Sheik said, jumping to his feet threateningly. "You're going to make me throw your ass off this fire escape."
Choo-Choo jumped to his feet, too, and backed against the rail. "Throw whose ass off where? This ain't Inky you're talking to. My ass ain't made of chicken feathers."
Inky scrambled to his feet and stepped between them. "What about the captive, Sheik?" he asked in alarm.
"Damn the captive!" Sheik raved and whipped out a bone-handled knife, shaking open the six-inch blade with the same motion.
"Don't cut 'em!" Inky cried.
He knocked Inky into the iron steps with a back-handed slap and grabbed a handful of Choo-Choo's sweat shirt collar.
"You blab and I'll cut your mother-raping throat," he said.
Violence surged through him like runaway blood.
Choo-Choo's eyes turned three-quarters white and a feverish sweat popped out on his dark brown skin.
"I didn't mean nothing, Sheik," he whined desperately, talking low. "You know I didn't mean nothing. A man can talk 'bout his rathers, can't he?"
The violence receded but Sheik was still gripped in a murderous compulsion.
"If I thought you'd pigeon I'd kill you."
"You know I ain't gonna pigeon, Sheik. You know me better than that."
Sheik let go of his collar. Choo-Choo took a deep sighing breath.
Inky straightened up and rubbed his bruised shin. "You done made me lose the stick," he complained.
"Hell with the stick," Sheik said.
"That's what I mean," Sonny said. "This here gage they sells now will make you cut your own mamma's throat. They must be mixing it with loco weed or somethin'."
"Shut up!" Sheik said, still holding the open knife in his hand. "I ain't gonna tell you no more."
Sonny cast a look at the knife and said, "I ain't saying nothing."
"You better not," Sheik said. Then he turned to Inky. "Inky, you take the captive up on the roof and you and him start flying Caleb's pigeons. You, Sonny, when the cops come you tell them your name is Caleb Bowee and you're just trying to teach your pigeons how to fly at night. You got that?"
"Yeah," Sonny said skeptically.
"You know how to make pigeons fly?"
Sonny hesitated. "Chunk rocks at 'em?"
"Hell, nigger, your brain ain't big as a mustard seed. You can't chunk no rocks up there with all those cops about. What you got to do is take this pole and wave the end with the flag at 'em every time they try to light."
Sonny looked at the bamboo pole skeptically. "S'posin' they fly away and don't come back."
"They ain't going nowhere. They just fly in circles trying all the time to get back into the coop." Sheik doubled over suddenly and started laughing. "Pigeons ain't got no sense, man."
The rest of them just looked at him.
Finally Inky asked, "What you want me to do?"
Sheik straightened up quickly and stopped laughing. "You guard the captive and see that he don't escape."
"Oh!" Inky said. After a moment he asked, "What I'm gonna tell the cops when they ask me what I'm doin'?"
"Hell, you tell the cops Caleb is teaching you how to train pigeons."
Inky bent over and started rubbing his shins again. Without looking up he said, "You reckon the cops gonna fall for that, Sheik? You reckon they gonna be crazy enough to believe anybody's gonna be flying pigeons with all this going on all around here?"
"Hell, these is white cops," Sheik said contemptuously. "They believe spooks are crazy anyway. You and Sonny just act kind of simpleminded. They gonna to swallow it like it's chocolate ice cream. They ain't going to do nothing but kick you in the ass and laugh like hell about how crazy spooks are. They gonna go home and tell their old ladies and everybody they see about two simpleminded spooks up on the roof teaching pigeons how to fly at night all during the biggest dragnet they ever had in Harlem. You see if they don't."
Inky kept on rubbing his shin. "It ain't that I doubt you, Sheik, but s'posin' they don't believe it."
"God damn it, go ahead and do what I told you and don't stand there arguing with me," Sheik said, hit by another squall of fury. "I'd take me one look at you and this nigger here and I'd believe it myself, and I ain't even no gray cop."
Inky turned reluctantly and started up the stairs toward the roof. Sonny gave another sidelong look at Sheik's open knife and started to follow.
"Wait a minute, simple, don't forget the pole," Sheik said. "I've told you not to try chunking rocks at those pigeons. You might kill one and then you'd have to eat it." He doubled over laughing at his joke.
Sonny picked up the pole with a sober face and climbed slowly after Inky.
"Come on," Sheik said to Choo-Choo, "open the window and let's get back inside."
Before turning his back and bending to open the window, Choo-Choo said, "Listen, Sheik, I didn't mean nothing by that."
"Forget it," Sheik said.
Sissie and Sugartit were sitting silently side by side on the bed, looking frightened and dejected. Sugartit had stopped crying but her eyes were red and her cheeks stained.
"Jesus Christ, you'd think this is a funeral," Sheik said.
No one replied. Choo-Choo fidgeted from one foot to the other.
"I want you chicks to wipe those sad looks off your faces," Sheik said. "We got to look like we're balling and ain't got a thing to worry about when the cops get here."
" _You_ go ahead and ball by yourself," Sissie said.
Sheik lunged forward and slapped her over on her side.
She got up without a word and walked to the window.
"If you go out that window I'll throw you down on the street," Sheik threatened.
She stood looking out the window with her back turned and didn't answer.
Sugartit sat quietly on the edge of the bed and trembled.
"Hell," Sheik said disgustedly and flopped lengthwise behind Sugartit on the bed.
She got up and went to stand in the window beside Sissie.
"Come on, Choo-Choo, to hell with those bitches," Sheik said. "Let's decide what to do with the captive."
"Now you're getting down to the gritty," Choo-Choo said enthusiastically, straddling a chair. "You got any plans?"
"Sure. Give me a butt."
Choo-Choo fished two Camels from a squashed package in his sweat shirt roll and lit them, passing one to Sheik.
"This square weed on top of gage makes you crazy," he said.
"Man, my head already feels like it's going to pop open, it's so full of ideas," Sheik said. "If I had me a real mob like Dutch Schultz's I could take over Harlem with the ideas I got. All I need is just the mob."
"Hell, you and me could do it alone," Choo-Choo said.
"We'd need some arms and stuff, some real factory-made heaters and a couple of machine guns and maybe some pineapples."
"If we croaked Grave Digger and the Monster we'd have two real cool heaters to start off with," Choo-Choo suggested.
"We ain't going to mess with those studs until after we're organized," Sheik said. "Then maybe we can import some talent to make the hit. But we'd need some dough."
"Hell, we can hold the prisoner for ransom," Choo-Choo said.
"Who'd ransom that nigger," Sheik said. "I bet even his own mamma wouldn't pay to get him back."
"He can ransom hisself," Choo-Choo said. "He got a shine parlor, ain't he? Shine parlors make good dough. Maybe he's got a chariot too."
"Hell, I knew all along he was valuable," Sheik said. "That's why I had us snatch him."
"We can take over his shine parlor," Choo-Choo said.
"I got some other plans too," Sheik said. "Maybe we can sell him to the Stars of David for some zip guns. They got lots of zip guns and they're scared to use them."
"We could do that or we could swap him to the Puerto Rican Bandits for Burrhead. We promised Burrhead we'd pay his ransom and they been saying if we don't hurry up and get 'im they're gonna cut his throat."
"Let 'em cut the black mother-raper's throat," Sheik said. "That chicken-hearted bastard ain't no good to us."
"I tell you what, Sheik," Choo-Choo said exuberantly. "We could put him in a sack like them ancient cats like the Dutchman and them used to do and throw him into the Harlem river. I've always wanted to put some bastard into a sack."
"You know how to put a mother-raper into a sack?" Sheik asked.
"Sure, you–"
"Shut up, I'm gonna to tell you how. You knock the mother-raper unconscious first; that's to keep him from jumping about. Then you put a noose with a slip-knot 'round his neck. Then you double him up into a Z and tie the other end of the wire around his knees. Then when you put him in the gunny sack you got to be sure it's big enough to give him some space to move around in. When the mother-raper wakes up and tries to straighten out he chokes hisself to death. Ain't nobody killed 'im. The mother-raper has just committed suicide." Sheik rolled with laughter.
"You got to tie his hands behind his back first," Choo-Choo said.
Sheik stopped laughing and his face became livid with fury. "Who don't know that, fool!" he shouted. " 'Course you got to tie his hands behind his back. You trying to tell me I don't know how to put a mother-raper into a sack. I'll put _you_ into a sack."
"I know you know how, Sheik," Choo-Choo said hastily. "I just didn't want you to forget nothing when we put the captive in a sack."
"I ain't going to forget nothing," Sheik said.
"When we gonna put him in a sack?" Choo-Choo asked. "I know where to find a sack."
"Okay, we'll put him in a sack just soon as the police finish here; then we take him down and leave him in the basement," Sheik said.
# 7
Grave Digger flashed his badge at the two harness bulls guarding the door and pushed inside the Dew Drop Inn.
The joint was jammed with colored people who'd seen the big white man die, but nobody seemed to be worrying about it.
The jukebox was giving out with a stomp version of "Big-Legged Woman." Saxophones were pleading; the horns were teasing; the bass was patting; the drums were chatting; the piano was catting, laying and playing the jive, and a husky female voice was shouting:
_"... you can feel my thigh
But don't you feel up high."_
Happy-tail women were bouncing out of their dresses on the high bar stools.
Grave Digger trod on the sawdust sprinkled over the bloodstains that wouldn't wash off and parked on the stool at the end of the bar.
Big Smiley was serving drinks with his left arm in a sling.
The white manager, the sleeves of his tan silk shirt rolled up, was helping.
Big Smiley shuffled down the wet footing and showed Grave Digger most of his big yellow teeth.
"Is you drinking, Chief, or just sitting and thinking?"
"How's the wing?" Grave Digger asked.
"Favorable. It wasn't cut deep enough to do no real damage."
The manager came down and said, "If I'd thought there was going to be any trouble I'd have called the police right away."
"What do you calculate as trouble in this joint?" Grave Digger asked.
The manager reddened. "I meant about the white man getting killed."
"Just what started all the trouble in here?"
"It wasn't exactly what you'd call trouble, Chief," Big Smiley said. "It was only a drunk attacked one of my white customers with his shiv and naturally I had to protect my customer."
"What did he have against the white man?"
"Nothing, Chief. Not a single thing. He was sitting over there drinking one shot of rye after another and looking at the white man standing here tending to his own business. Then he gets red-eyed drunk and his evil tells him to get up and cut the man. That's all. And naturally I couldn't let him do that."
"He must have had some reason. You're not trying to tell me he got up and attacked the man without any reason whatever."
"Naw suh, Chief, I'll bet my life he ain't had no reason at all to wanta cut the man. You know how our folks is, Chief; he was just one of those evil niggers that when they get drunk they start hating white folks and get to remembering all the bad things white folks ever done to them. That's all. More than likely he was mad at some white man that done something bad to him twenty years ago down South and he just wanted to take it out on this white man in here. It's like I told that white detective who was in here, this white man was standing here at the bar by hisself and that nigger just figgered with all those colored folk in here he could cut him and get away with it."
"Maybe. What's his name?"
"I ain't ever seen that nigger before tonight, Chief; I don't know what is his name."
A customer called from up the bar, "Hey, boss, how about a little service up here?"
"If you want me, Jones, just holler," the manager said, moving off to serve the customer.
"Yeah," Grave Digger said, then asked Big Smiley, "Who was the woman?"
"There she is," Big Smiley said, nodding toward a booth.
Grave Digger turned his head and scanned her.
The black lady in the pink jersey dress and red silk stockings was back in her original seat in a booth surrounded by three workers.
"It wasn't on account of her," Big Smiley added.
Grave Digger slid from his stool, went over to her booth and flashed his badge. "I want to talk to you."
She looked at the gold badge and complained, "Why don't you folks leave me alone? I done already told a white cop everything I know about that shooting, which ain't nothing."
"Come on, I'll buy you a drink," Grave Digger said.
"Well, in that case..." she said and went with him to the bar.
At Grave Digger's order Big Smiley grudgingly poured her a shot of gin and Grave Digger said, "Fill it up."
Big Smiley filled the glass and stayed there to listen.
"How well did you know the white man?" Grave Digger asked the lady.
"I didn't know him at all. I'd just seen him around here once or twice."
"Doing what–"
"Just chasing."
"Alone?"
"Yeah."
"Did you see him pick up anyone?"
"Naw, he was one of those particular kind. He never saw nothing he liked."
"Who was the colored man who tried to cut him?"
"How the hell should I know?"
"He wasn't a relative of yours?"
"A relation of mine. I should hope not."
"Just exactly what did he say to the white man when he started to attack him?"
"I don't remember exactly; he just said something 'bout him messing about with his gal."
"That's the same thing the other man, Sonny Pickens, accused him of."
"I don't know nothing about that."
He thanked her and wrote down her name and address.
She went back to her seat.
He returned back to Big Smiley. "What did Pickens and the man argue about?"
"They ain't had no argument, Chief. Not in here. It wasn't on account of nothing that happened in here that he was shot."
"It was on account of something," Grave Digger said. "Robbery doesn't figure, and people in Harlem don't kill for revenge."
"Naw suh, leastwise they don't shoot."
"More than likely they'll throw acid or hot lye," Grave Digger said.
"Naw, suh, not on no white gennelman."
"So what else is there left but a woman," Grave Digger said.
"Naw suh," Big Smiley contradicted flatly. "You know better'n that, Chief. A colored woman don't consider diddling with a white man as being unfaithful. They don't consider it no more than just working in service, only they is getting better paid and the work is less straining. 'Sides which, the hours is shorter. And they old men don't neither. Both she and her old man figger it's like finding money in the street. And I don't mean no cruisers neither; I means church people and Christians and all the rest."
"How old are you, Smiley?" Grave Digger asked.
"I be forty-nine come December seventh."
"You're talking about old times, son. These young colored men don't go in for that slavery-time deal anymore."
"Shucks, Chief, you just kidding. This is old Smiley. I got dirt on these women in Harlem ain't never been plowed. Shucks, you and me both can put our finger on high society colored ladies here who got their whole rep just by going with some big important white man. And their old men is cashing in on it, too; makes them important, too, to have their old ladies going with some big-shot gray. Shucks, even a hard-working nigger wouldn't shoot a white man if he come home and found him in bed with his old lady with his pants down. He might whup his old lady just to show her who was boss, after he done took the money 'way from her, but he wouldn't sure 'nough hurt her like he'd do if he caught her screwing some other nigger."
"I wouldn't bet on it," Grave Digger said.
"Have it your own way, Chief, but I still think you're barking up the wrong tree. Lissen, the only way I figger a colored man in Harlem gonna kill a white man is in a fight. He'll draw his shiv if he getting his ass whupped and maybe stab him to death. But I'll bet my life ain't no nigger up here gonna shoot down no white man in cold blood – no important white gennelman like him."
"Would the killer have to know he was important?"
"He'd know it," Big Smiley said positively.
"You knew him?" Grave Digger said.
"Naw suh, not to say knew him. He come in here two, three times before but I didn't know his name."
"You expect me to believe he came in here two or three times and you didn't find out who he was?"
"I didn't mean exactly I didn't know his name," Big Smiley hemmed. "But I'se telling you, Chief, ain't no leads 'round here, that's for sure."
"You're going to have to tell me more than that, son," Grave Digger said in a flat, toneless voice.
Big Smiley looked at him; then suddenly he leaned across the bar and said in a low voice, "Try at Bucky's, Chief."
"Why Bucky's?"
"I seen him come in here once with a pimp what hangs 'round in Bucky's."
"What's his name?"
"I don't recollect his name, Chief. They driv up in his car and just stopped for a minute like they was looking for somebody and went out and drive away."
"Don't play with me," Grave Digger said with a sudden show of anger. "This ain't the movies; this is real. A white man has been killed in Harlem and Harlem is my beat. I'll take you down to the station and turn a dozen white cops loose on you and they'll work you over until the black comes off."
"Name's Ready Belcher, Chief, but I don't want nobody to know I told you," Big Smiley said in a whisper. "I don't want no trouble with that starker."
"Ready," Grave Digger said and got down from his stool.
He didn't know much about Ready; just that he operated up-town on the swank side of Harlem, above 145th Street in Washington Heights.
He drove up to the 154th Street precinct station at the corner of Amsterdam Avenue and asked for his friend, Bill Cresus. Bill was a colored detective on the vice squad. No one knew where Bill was at the time. He left word for Bill to contact him at Bucky's if he called within the hour. Then he got into his car and coasted down the sharp incline to St. Nicholas Avenue and turned south down the lesser incline past 149th Street.
Outwardly it was a quiet neighbourhood of private houses and five- and six-story apartment buildings flanking the wide black-paved street. But the houses had been split up into bed-sized one-room kitchenettes, renting for $25 weekly, at the disposal of frantic couples who wished to shack up for a season. And behind the respectable-looking facades of the apartment buildings were the plush flesh cribs and poppy pads and circus tents of Harlem.
The excitement of the dragnet hadn't reached this far and the street was comparatively empty.
He coasted to a stop before a sedate basement entrance. Four steps below street level was a black door with a shiny brass knocker in the shape of three musical notes. Above it red neon lights spelled out the word _BUCKY'S_.
It felt strange to be alone. The last time had been when Coffin Ed was in the hospital after the acid throwing. The memory of it made his head tight with anger and it took a special effort to keep his temper under wraps.
He pushed and the door opened.
People sat at white-clothed tables beneath pink-shaded wall lights in a long narrow room, eating fried chicken daintily with their fingers. There was a white party of six, several colored couples, and two colored men with white women. They looked well-dressed and reasonably clean.
The walls behind them were covered with innumerable small pink-stained pencil portraits of all the great and the near-great who had ever lived in Harlem. Musicians led nine to one.
The hat-check girl stationed in a cubicle beside the entrance stuck out her hand with a supercilious look.
Grave Digger kept his hat on and strode down the narrow aisle between the tables.
A chubby pianist with shining black skin and a golden smile who was dressed in a tan tweed sport jacket and white silk sport shirt open at the throat sat at a baby grand piano wedged between the last table and the circular bar. Soft white light spilled on his partly bald head while he played nocturnes with a bedroom touch.
He gave Grave Digger an apprehensive look, got up and followed him to the semi-darkness of the bar.
"I hope you're not on business, Digger. I pay to keep this place off-limits for cops," he said in a fluttery voice.
Grave Digger's gaze circled the bar. Its high stools were inhabited by a varied crew: a big dark-haired white man, two slim young colored men, a short heavy-set white man with blond crew-cut hair, two dark women dressed in white silk evening gowns, a chocolate dandy in a box-backed double-breasted tuxedo sporting a shoestring dubonet bow. A high-yellow waitress waited nearby with a serving tray. Another tall, slim ebony young man presided over the bar.
"I'm just looking around, Bucky," Grave Digger said. "Just looking for a break."
"Many folks have found a break in here," Bucky said suggestively.
"I don't doubt it."
"But that's not the kind of break you're looking for."
"I'm looking for a break on a case. An important white man was shot to death over on Lenox Avenue a short time ago."
Bucky gestured with lotioned hands. His manicured nails flashed in the dim light. "What has that to do with us here? Nobody ever gets hurt in here. Everything is smooth and quiet. You can see for yourself. Genteel people dining in leisure. Fine food. Soft music. Low lights and laughter. Doesn't look like business for the police in this respectable atmosphere."
In the pause that followed, one of the marcelled ebonies was heard saying in a lilting voice, "I positively did not even look at her man, and she upped and knocked me over the head with a whisky bottle."
"These black bitches are so violent," his companion said.
"And strong, honey."
Grave Digger smiled sourly.
"The man who was killed was a patron of yours," he said. "Name of Ulysses Galen."
"My God, Digger, I don't know the names of all the ofays who come into my place," Bucky said. "I just play for them and try to make them happy."
"I believe you," Grave Digger said. "Galen was seen about town with Ready. Does that stir your memory?"
"Ready?" Bucky exclaimed innocently. "He hardly ever comes in here. Who gave you that notion?"
"The hell he doesn't," Grave Digger said. "He panders out of here."
"You hear that!" Bucky appealed to the barman in a shrill horrified voice, then caught himself as the silence from the diners reached his sensitive ears. With hushed indignation he added, "This flatfoot comes in here and accuses me of harboring panderers."
"A little bit of that goes a long way, son," Grave Digger said in his flat voice.
"Oh, that man's an ogre, Bucky," the barman said. "You go back to your entertaining and I'll see what he wants." He switched over to the bar, put his hands on his hips and looked down at Grave Digger with a haughty air. "And just what can we do for you, you mean rude grumpy man?"
The white men at the bar laughed.
Bucky turned and started off.
Grave Digger caught him by the arm and pulled him back. "Don't make me get rough, son," he muttered.
"Don't you dare manhandle me," Bucky said in a low tense whisper, his whole chubby body quivering with indignation. "I don't have to take that from you. I'm covered."
The bartender backed away, shaking himself. "Don't let him hurt Bucky," he appealed to the white men in a frightened voice.
"Maybe I can help you," the white man with the blond crew cut said to Grave Digger. "You're a detective, aren't you?"
"Yeah," Grave Digger said, holding on to Bucky. "A white man was killed in Harlem tonight and I'm looking for the killer."
The white man's eyebrows went up an inch.
"Do you expect to find him here?"
"I'm following a lead, is all. The man has been seen with a pimp called Ready Belcher who hangs out here."
The white man's eyebrows subsided.
"Oh, Ready; I know him. But he's merely–"
Bucky cut him off: "You don't have to tell him anything; you're protected in here."
"Sure," the white man said. "That's what the officer is trying to do, protect us all."
"He's right," one of the evening-gowned colored women said. "If Ready has killed some trick he was steering to Reba's the chair's too good for him."
"Shut your mouth, woman," the barman whispered fiercely.
The muscles in Grave Digger's face began to jump as he let go of Bucky. He stood up with his heels hooked into the rungs of the barstool and leaned over the bar. He caught the barman by the front of his red silk shirt as he was trying to dance away. The shirt ripped down the seam with a ragged sound but enough held for him to jerk the barman close to the bar.
"You got too goddamned much to say, Tarbelle," he said in a thick cottony voice, and slapped the barman spinning across the circular enclosure with the palm of his open hand.
"He didn't have to do that," the first woman said.
Grave Digger turned on her and said thickly, "And you, little sister, you and me are going to see Reba."
"Reba!" her companion replied. "Do I know anybody named Reba. Lord no!"
Grave Digger stepped down from his high stool.
"Cut that Aunt Jemima routine and get up off your ass," he said thickly, "or I'll take my pistol and break off your teeth."
The two white men stared at him as though at a dangerous animal escaped from the zoo.
"You mean that?" the woman said.
"I mean it," he said.
She scrunched out of the stool and said, "Gimme my coat, Jule."
The chocolate dandy took a coat from the top of the jukebox behind them.
"That's putting it on rather thick," the blond white man protested in a reasonable voice.
"I'm just a cop," Grave Digger said thickly. "If you white people insist on coming up to Harlem where you force colored people to live in vice-and-crime-ridden slums, it's my job to see that you are safe."
The white man turned bright red.
# 8
The sergeant knocked at the door. He was flanked by two uniformed cops and a corporal.
Another search party led by another sergeant was at the door across the hall.
Other cops were working all the corridors starting at the bottom and sealing off the area they'd covered.
"Come in," Granny called in a querulous voice. "The door ain't locked." She bit the stem of her corn-cob pipe with toothless gums.
The sergeant and his party entered the small kitchen. It was crowded.
At the sight of the very old woman working innocently at her darning, the sergeant started to remove his cap, then remembered he was on duty and kept it on.
"You don't lock your door, Grandma?" he observed.
Granny looked at the cops over the rims of her ancient spectacles and her old fingers went lax on the darning egg.
"Naw suh, Ah ain't got nuthin' for nobody to steal and ain't nobody want nuthin' else from an old 'oman like me."
The sergeant's beady blue eyes scanned the kitchen. "You keep this place mighty clean, Grandma," he remarked in surprise.
"Yes suh, it don't kill a body to keep clean and my old missy used to always say de cleaness is next to the goddess."
Her old milky eyes held a terrified question she couldn't ask and her thin old body began to tremble.
"You mean goodness," the sergeant said.
"Naw suh, Ah means goddess; Ah knows what she said."
"She means cleanliness is next to godliness," the corporal interposed.
"The professor," one of the cops said.
Granny pursed her lips. "Ah know what my missy said; goddess, she said."
"Were you in slavery?" the sergeant asked as though struck suddenly by the thought.
The others stared at her with sudden interest.
"Ah don't rightly know, suh. Ah 'spect so though."
"How old are you?"
Her lips moved soundlessly; she seemed to be trying to remember.
"She must be all of a hundred," the professor said.
She couldn't stop her body from trembling and slowly it got worse.
"What for you white 'licemen wants with me, suh?" she finally asked.
The sergeant noticed that she was trembling and said reassuringly, "We ain't after you, Grandma; we're looking for an escaped prisoner and some teenage gangsters."
"Gangsters!"
Her spectacles slipped down on her nose and her hands shook as though she had the palsy.
"They belong to a neighbourhood gang that calls itself Real Cool Moslems."
She went from terrified to scandalized. "We ain't no heathen in here, suh," she said indignantly. "We be God-fearing Christians."
The cops laughed.
"They're not real Moslems," the sergeant said. "They just call themselves that. One of them, named Sonny Pickens, is older than the rest. He killed a white man outside on the street."
The darning dropped unnoticed from Granny's nerveless fingers. The corncob pipe wobbled in her puckered mouth; the professor looked at it with morbid fascination.
"A white man! Merciful hebens!" she exclaimed in a quavering voice. "What's this wicked world coming to?"
"Nobody knows," the sergeant said, then changed his manner abruptly. "Well, let's get down to business, Grandma. What's your name?"
"Bowee, suh, but e'body calls me Granny."
"Bowee. How do you spell that, Grandma?"
"Ah don't rightly know, suh. Hit's just short for boll weevil. My old missy name me that. They say the boll weevil was mighty bad the year Ah was born."
"What about your husband, didn't he have a name?"
"Ah neber had no regular 'usban', suh. Just whosoever was thar."
"You got any children?"
"Jesus Christ, sarge," the professor said. "Her youngest child would be sixty years old."
The two cops laughed; the sergeant reddened sheepishly.
"Who lives here with you, Granny?" the sergeant continued.
Her bony frame stiffened beneath her faded Mother Hubbard. The corncob pipe fell into her lap and rolled unnoticed to the floor.
"Just me and mah grandchile, Caleb, suh," she said in a forced voice. "And Ah rents a room to two workin' boys; but they be good boys and don't neber bother nobody."
The cops grew suddenly speculative.
"Now this grandchild, Caleb, Grandma–" the sergeant began cunningly.
"He might be mah great-grandchile, suh," she interrupted.
He frowned, "Great, then. Where is he now?"
"You mean right now, suh?"
"Yeah, Grandma, right this minute."
"He at work in a bowling alley downtown, suh."
"How long has he been at work?"
"He left right after supper, suh. We gennally eats supper at six o'clock."
"And he has a regular job in this bowling alley?"
"Naw suh, hit's just for t'night, suh. He goes to school – Ah don't rightly 'member the number of his new P.S."
"Where is this bowling alley he's working at tonight?"
"Ah don't know, suh. Ah guess you all'll have to ast Samson. He is one of mah roomers."
"Samson, yeah." The sergeant stored it in his memory. "And you haven't seen Caleb since supper – about seven o'clock, say?"
"Ah don't know what time it was but it war right after supper."
"And when he left here he went directly to work?"
"Yas suh, you find him right dar on de job. He a good boy and always mind me what Ah say."
"And your roomers, where are they?"
"They is in they room, suh. Hit's in the front. They got visitors with 'em."
"Visitors?"
"Gals."
"Oh!" Then to his assistants he said, "Come on."
They went through the middle room like hounds on a hot scent. The sergeant tried the handle to the front-room door without knocking, found it locked and hammered angrily.
"Who's that?" Sheik asked.
"The police."
Sheik unlocked the door. The cops rushed in. Sheik's eyes glittered.
"What the hell do you keep your door locked for?" the sergeant asked.
"We didn't want to be disturbed."
Four pairs of eyes quickly scanned the room.
Two teenaged colored girls sat side by side on the bed, leafing through a colored picture magazine. Another youth stood looking out the open window at the excitement on the street.
"Who the hell you think you're kidding with this phony stage setting?" the sergeant roared.
"Not you, ace," Sheik said flippantly.
The sergeant's hand flicked out like a whip, passing inches in front of Sheik's eyes.
Sheik jumped back as though he'd been scalded.
"Jagged to the gills," the sergeant said, looking minutely about the room. His eyes lit on Choo-Choo's half-smoked package of Camels on the table. "Dump out those fags," he ordered a cop, watching Sheik's reaction. "Never mind," he added. "The bastard's got rid of them."
He closed in on Sheik like a prizefighter and shoved his red sweaty face within a few inches of Sheik's. His veined blue eyes bored into Sheik's pale yellow eyes.
"Where's that A-rab costume?" he asked in a browbeating voice.
"What Arab costume? Do I look like an A-rab to you?"
"You look like a two-bit punk to me. You got the eyes of a yellow cur."
"You ain't got no prize-winning eyes yourself."
"Don't give me none of your lip, punk; I'll knock out your teeth."
"I could knock out your teeth too if I had on a sergeant's uniform and three big flatfeet backing me up."
The cops stared at him from blank shuttered faces.
"What do they call you, Mo-hammed or Nasser?" the sergeant hammered.
"They call me by my name, Samson."
"Samson what?"
"Samson Hyers."
"Don't give me that crap; we know you're one of those Moslems."
"I ain't no Moslem; I'm a cannibal."
"Oh, so you think you're a comedian."
"You the one asking the funny questions."
"What's that other punk's name?"
"Ask him."
The sergeant slapped him with such force it sounded like a .22-caliber shot.
Sheik reeled back from the impact of the slap but kept his feet. Blood darkened his face to the color of beef liver; the imprint of the sergeant's hand glowed purple red. His pale yellow eyes looked wildcat crazy. But he kept his lip buttoned.
"When I ask you a question I want you to answer it," the sergeant said.
He didn't answer.
"You hear me?"
He still didn't answer.
The sergeant loomed in front of him with both fists cocked like red meat axes.
"I want an answer."
"Yeah, I hear you," Sheik muttered sullenly.
"Frisk him," the sergeant ordered the professor, then to the other two cops; said, "You and Price start shaking down this room."
The professor set to work on Sheik methodically, as though searching for lice, while the other cops started dumping dresser drawers onto the table.
The sergeant left them and turned his attention to Choo-Choo.
"What kind of Moslem are you?"
Choo-Choo started grinning and fawning like the original Uncle Tom.
"I ain't no Moslem, boss, I'se just a plain old unholy roller."
"I guess your name is Delilah."
"He-he, naw suh boss, but you're warm. It's Justice Broome."
All three cops looked about and grinned, and the sergeant had to clamp his jaws to keep from grinning too.
"You know these Moslems?"
"What Moslems, boss?"
"The Harlem Moslems in this neighbourhood."
"Naw suh, boss, I don't know no Moslems in Harlem."
"You think I was born yesterday? They a neighbourhood gang. Every black son of a bitch in this neighbourhood knows who they are."
"Everybody 'cept me, boss."
The sergeant's palm flew out and caught Choo-Choo unexpectedly on the mouth while it was still open in a grin. It didn't rock his short thick body, but his eyes rolled back in their sockets. He spit blood on the floor.
"Boss, suh, please be careful with my chops – they're tender."
"I'm getting damn tired of your lying."
"Boss, I swear 'fore God, if I knowed anything 'bout them Moslems you'd be the first one I'd tell it to."
"What do you do?"
"I works, boss, yes suh."
"Doing what?"
"I helps out."
"Helps out with what? You want to lose some of your pearly teeth?"
"I helps out a man who writes numbers."
"What's his name?"
"His name?"
The sergeant cocked his fists.
"Oh, you mean his name, boss: Hit's Four-Four Row."
"You call that a name?"
"Yas suh, that's what they calls him."
"What does your buddy do?"
"The same thing," Sheik said.
The sergeant wheeled on him. "You keep quiet; when I want you I'll call you." Then he said to the professor, "Can't you keep that punk quiet?"
The professor unhooked his sap. "I'll quiet him."
"I don't want you to quiet him; just keep him quiet. I got some more questions for him." Then he turned back to Choo-Choo. "When do you punks work?"
"In the morning, boss. We got to get the numbers in by noon."
"What do you do the rest of the day?"
"Go 'round and pay off."
"What if there isn't any payoff?"
"Just go 'round."
"Where's your beat?"
" 'Round here."
"God damn it, you mean to tell me you write numbers in this neighbourhood and you don't know anything about the Moslems?"
"I swear on my mother's grave, boss, I ain't never heard of no Moslems 'round here. They must not be in this neighbourhood, boss."
"What time did you leave the house tonight?"
"I ain't never left it, boss. We come here right after we et supper and ain't been out since."
"Stop lying; I saw you both when you slipped back in here a half-hour ago."
"Naw suh, boss, you musta seen somebody what looks like us 'cause we been here all the time."
The sergeant crossed to the door and flung it open. "Hey, Grandma!" he called.
"Hannh?" she answered querulously from the kitchen.
"How long have these boys been in their room?"
"Hannh?"
"You have to talk louder; she can't hear you," Sissie volunteered.
Sheik and Choo-Choo gave her threatening looks.
The sergeant crossed the middle room to the kitchen door. "How long have your roomers been back from supper?" he roared.
She looked at him from uncomprehending eyes.
"Hannh?"
"She can't hear no more," Sissie called. "She gets that way sometime."
"Hell," the sergeant said disgustedly and stormed back to Choo-Choo. "Where'd you pick up these girls?"
"We didn't pick 'em up, boss; they come here by themselves."
"You're too goddam innocent to be alive." The sergeant was frustrated. He turned to the professor: "What did you find on that punk?"
"This knife."
"Hell," the sergeant said. He took it and dropped it into his pocket without a glance. "Okay, fan this other punk – Justice."
"I'll do Justice," the professor punned.
The two cops crossed glances suggestively.
They had dumped out all the drawers and turned out all the boxes and pasteboard suitcases and now they were ready for the bed.
"You gals rise and shine," one said.
The girls got up and stood uncomfortably in the center of the room.
"Find anything?" the sergeant asked.
"Nothing that I'd even care to have in my dog house," the cop said.
The sergeant began on the girls. "What's your name?" he asked Sissie.
"Sissieratta Hamilton."
"Sissie what?"
"Sissieratta."
"Where do you live, Sissie?"
"At 2702 Seventh Avenue with my aunt and uncle, Mr. and Mrs. Coolie Dunbar."
"Ummm," he said, "And yours?" he asked Sugartit.
"Evelyn Johnson."
"Where do you live, Eve?"
"In Jamaica with my parents, Mr. and Mrs. Edward Johnson."
"It's mighty late for you to be so far from home."
"I'm going to spend the night with Sissieratta."
"How long have you girls been here?" he asked of both.
"About half an hour, more or less," Sissie replied.
"Then you saw the shooting down on the street?"
"It was over when we got here."
"Where did you come from?"
"From my house."
"You don't know if these punks have been in all evening or not."
"They were here when we got here and they said they'd been waiting here since supper. We promised to come at eight but we had to stay help my aunty and we got here late."
"Sounds too good to be true," the sergeant commented.
The girls didn't reply.
The cops finished with the bed and the talkative one said, "Nothing but stink."
"Can that talk," the sergeant said. "Grandma's clean."
"These punks aren't."
The sergeant turned to the professor. "What's on Justice besides the blindfold?"
His joke laid an egg.
"Nothing but his black," the professor said.
His joke drew a laugh.
"What do you say, shall we run 'em in?" the sergeant asked.
"Why not," the professor said. "If we haven't got space in the bullpen for everybody we can put up tents."
The sergeant wheeled suddenly on Sheik as though he'd forgotten something.
"Where's Caleb?"
"Up on the roof tending his pigeons."
All four cops froze. They stared at Sheik with those blank shuttered looks.
Finally the sergeant said carefully, "His grandma said you told her he was working in a bowling alley downtown."
"We just told her that to keep her from worrying. She don't like for him to go up on the roof at night."
"If I find you punks are holding out on me, God help you," the sergeant said in a slow sincere voice.
"Go look then," Sheik said.
The sergeant nodded to the professor. The professor climbed out of the window into the bright glare of the spotlights and began ascending the fire escape.
"What's he doing with them at night?" the sergeant asked Sheik.
"I don't know. Trying to make them lay black eggs, I suppose."
"I'm going to take you down to the station and have a private talk with you, punk," the sergeant said. "You're one punk who needs talking to privately."
The professor came down from the roof and called through the window, "They're holding two coons up here beside a pigeon loft. They're waiting on you."
"Okay, I'm coming. You and Price hold these punks on ice," he directed the other cops and climbed out of the window behind the professor.
# 9
"Get in," Grave Digger said.
She pulled up the skirt of her evening gown, drew the black coat tight, and eased her jumbo hams into the seat usually occupied by Coffin Ed.
Grave Digger went around on the other side and climbed beneath the wheel and waited.
"Does I just have to go along, honey," the woman said in a wheedling voice. "I can just as well tell you where she's at."
"That's what I'm waiting for."
"Well, why didn't you say so? She's in the Knickerbocker Apartments on 45th Street – the old Knickerbocker, I mean. She on the six story, 669."
"Who is she?" Grave Digger asked, probing a little.
"Who she is? Just a landprop is all."
"That ain't what I mean."
"Oh, I know what you means. You means who is she. You means you don't know who Reba is, Digger?" She tried to sound jocular but wasn't successful. "She the landprop what used to be old cap Murphy's go-between 'fore he got sent up for taking all them bribes. It was in all the papers."
"That was ten years ago and they called her Sheba then," he said.
"Yare, that's right, but she changed her name after she got into that last shooting scrape. You musta 'member that. She caught the nigger with some chippie or 'nother and made him jump buck naked out the third-story window. That wouldn't 'ave been so bad but she shot 'im through the head as he was going down. That was when she lived in the valley. Since then she done come up here on the hill. 'Course it warn't nobody but her husband and she didn't get a day. But Reba always has been lucky that way."
He took a shot in the dark. "What would anybody shoot Galen for?"
She grew stiff with caution, "Who he?"
"You know damn well who he was. He's the man who was shot tonight."
"Naw suh, I didn't know nothing 'bout that gennelman. I don't know why nobody would want to shoot him."
"You people give me a pain in the seat with all that ducking and dodging every time someone asks you a question. You act like you belong to a race of artful dodgers."
"You is asking me something I don't know nothing 'bout."
"Okay, get out."
She got out faster than she got in.
He drove down the hill of St. Nicholas Avenue and turned up the hill of 145th Street toward Convent Avenue.
On the left-hand corner, next to a new fourteen-story apartment building erected by a white insurance company, was the Brown Bomber Bar; across from it Big Crip's Bar; on the right-hand corner Cohen's Drug Store with its iron-grilled windows crammed with electric hair straightening irons, Hi-Life hair cream, Black and White bleaching cream, SSS and 666 blood tonics, Dr. Scholl's corn pads, men's and women's nylon head caps with chin straps to press hair while sleeping, a bowl of blue stone good for body lice, tins of Sterno canned heat good for burning or drinking, Halloween postcards and all the latest in enamelware hygiene utensils; across from it Zazully's Delicatessen with a white-lettered announcement on the plate-glass window: _We Have Frozen Chitterlings and Other Hard-to-find Delicacies_.
Grave Digger parked in front of a big frame house with peeling yellow paint which had been converted into offices, got out and walked next door to a six-story rotten-brick tenement long overdue at the wreckers.
Three cars were parked at the curb in front; two with upstate New York plates and the other from mid-Manhattan.
He pushed open a scaly door beneath the arch of a concrete block on which the word _KNICKER-BOCKER_ was embossed.
An old gray-haired man with a splotched brown face sat in a chair just inside the doorway to the semi-dark corridor. He cautiously drew back gnarled feet in felt bedroom slippers and looked Grave Digger over with dull, satiated eyes.
"Evenin'," he said.
Grave Digger glanced at him. "Evenin'."
"Fourth story on de right. Number 421," the old man informed him.
Grave Digger stopped. "That Reba's?"
"You don't want Reba's. You want Topsy's. Dat's 421."
"What's happening at Topsy's?"
"What always happen. Dat's where the trouble is."
"What kind of trouble?"
"Just general trouble. Fightin' and cuttin'."
"I'm not looking for trouble. I'm looking for Reba."
"You're the man, ain't you?"
"Yeah, I'm the man."
"Then you wants 421. I'se de janitor."
"If you're the janitor then you know Mr. Galen."
A veil fell over the old man's face. "Who he?"
"He's the big Greek man who goes up to Reba's."
"I don't know no Greeks, boss. Don't no white folks come in here. Nothin' but cullud folks. You'll find 'em all at Topsy's."
"He was killed over on Lenox tonight."
"Sho nuff?"
Grave Digger started off.
The old man called to him, "I guess you wonderin' why we got them big numbers on de doors."
Grave Digger paused. "All right, why?"
"They sounds good." The old man cackled.
Grave Digger walked up five flights of shaky wooden stairs and knocked on a red-painted door with a round glass peephole in the upper panel.
After an interval a heavy woman's voice asked, "Who's you?"
"I'm the Digger."
Bolts clicked and the door cracked a few inches on the chain. A big dark silhouette loomed in the crack, outlined by blue light from behind.
"I didn't recognize you, Digger," a pleasant bass voice said. "Your hat shades your face. Long time no see."
"Unchain the door, Reba, before I shoot it off."
A deep bass laugh accompanied chain rattling and the door swung inward.
"Same old Digger, shoot first and talk later. Come on in; we're all colored folks here."
He stepped into a blue-lit carpeted hall reeking of incense.
"You're sure?"
She laughed again as she closed and bolted the door. "Those are not folks, those are clients." Then she turned casually to face him. "What's on your mind, honey?"
She was as tall as his six feet two, with snow-white hair cut short as a man's and brushed straight back from her forehead. Her lips were painted carnation red and her eyelids silver but her smooth unlined jet black skin was untouched. She wore a black sequined evening gown with a red rose in the V of her mammoth bosom, which was a lighter brown than her face. She looked like the last of the Amazons blackened by time.
"Where can we talk?" Grave Digger said. "I don't want to strain you."
"You don't strain me, honey," she said, opening the first door to the right. "Come into the kitchen."
She put a bottle of bourbon and a siphon beside two tall glasses on the table and sat in a kitchen chair.
"Say when," she said as she started to pour.
"By me," Grave Digger said, pushing his hat to the back of his head and planting a foot on the adjoining chair.
She stopped pouring and put down the bottle.
"You go ahead," he said.
"I don't drink no more," she said. "I quit after I killed Sam."
He crossed his arms on his raised knee and leaned forward on them, looking at her.
"You used to wear a rosary," he said.
She smiled, showing gold crowns on her outside incisors.
"When I got real religion I quit that too," she said.
"What religion did you get?"
"Just the faith, Digger, just the spirit."
"It lets you run this joint?"
"Why not. It's nature, just like eating. Nothing in my faith 'gainst eating. I just make it convenient and charge 'em for it."
"You'd better get a new steerer; the one downstairs is simple-minded."
Her big bass laugh rang out again. "He don't work for us; he does that on his own."
"Don't make it hard on yourself," he said. "This can be easy for us both."
She looked at him calmly. "I ain't got nothing to fear."
"When was the last time you saw Galen?"
"The big Greek? Been some time now, Digger. Three or four months. He don't come here no more."
"Why?"
"I don't let him."
"How come?"
"Be your age, Digger. This is a sporting house. If I don't let a white john with money come here, I must have good reasons. And if I want to keep my other white clients I'd better not say what they are. You can't close me up and you can't make me talk, so why don't you let it go at that?"
"The Greek was shot to death tonight over on Lenox."
"I just heard it over the radio," she said.
"I'm trying to find out who did it."
She looked at him in surprise. "It said on the radio the killer was known. A Sonny Pickens. Said a teenage gang called the something-or-'nother Moslems snatched him."
"He didn't do it. That's why I'm here."
"Well, if he didn't do it, you got your job cut out," she said. "I wish I could help you but I can't."
"Maybe," he said. "Maybe not."
She raised her eyebrows slightly. "By the way, where's your sidekick, Coffin Ed? The radio said he shot one of the gang."
"Yeah, he got suspended."
She became still, like an animal alert to danger. "Don't take it out on me, Digger."
"I just want to know why you stopped the Greek from coming here."
She stared into his eyes. She had dark brown eyes with clear whites and long black lashes.
"I'll let you talk to Ready. He knows."
"Is he here now?"
"He got a little chippie here he can't stay 'way from for five minutes. I'm going to throw 'em both out soon. Would have before now but my clients like her."
"Was the Greek her client?"
She got up slowly, sighing slightly from the effort.
"I'll send him out here."
"Bring him out."
"All right. But take him away, Digger. I don't want him talking in here. I don't want no more trouble. I've had trouble all my days."
"I'll take him away," he said.
She went out and Grave Digger heard doors being discreetly opened and shut and then her controlled bass voice saying, "How do I know? He said he was a friend."
A tall man with pockmarked skin a dirty shade of black stepped into the kitchen. An old razor scar cut a purple ridge from the lobe of his ear to the tip of his chin. There was a cast in one eye, the other was reddish brown. Thin corked hair stuck to a double-jointed head shaped like a peanut. He was flashily dressed in a light tan suit. Glass glittered from two gold-plated rings. His pointed tan shoes were shined to mirror brilliance.
At sight of Grave Digger he drew up short and turned a murderous look on Reba.
"You tole me hit was a friend," he accused in a rough voice.
She didn't let it bother her. She pushed him into the kitchen and closed the door.
"Well, ain't he?" she asked.
"What's this, some kind of frame-up?" he shouted.
Grave Digger chuckled at the look of outrage on his face. "How can a buck as ugly as you be a pimp?" he asked.
"You're gonna make me talk about you mamma," Ready said, digging his right hand into his pants pocket.
With nothing moving but his arm, Grave Digger back-handed him in the solar plexus, knocking out his wind, then pivoted on his left foot and followed with a right cross to the same spot, and with the same motion raised his knee and sunk it into Ready's belly as the pimp's slim frame jack-knifed forward. Spit showered from Ready's fishlike mouth, and the sense was already gone from his eyes when Grave Digger grabbed him by the back of the coat collar, jerked him erect, and started to slap him in the face with his open palm.
Reba grabbed his arm, saying. "Not in here, Digger, I beg you; don't make him bleed. You said you'd take him out."
"I'm taking him out now," he said in a cottony voice, shaking off her hold.
"Then finish him without bleeding him; I don't want nobody coming in here finding blood on the floor."
Grave Digger grunted and eased off. He propped Ready against the wall, holding him up on his rubbery legs with one hand while he took the knife and frisked him quickly with the other.
The sense came back into Ready's good eye and Grave Digger stepped back and said, "All right, let's go quietly, son."
Ready fussed about without looking at him, straightening his coat and tie, then fished a greasy comb from his pocket and combed his rumpled conk. He was bent over in the middle from pain and breathing in gasps. A white froth had collected in both corners of his mouth.
Finally he mumbled, "You can't take me outa here without no warrant."
"Go ahead with the man and shut up," Reba said quickly.
He gave her a pleading look. "You gonna let him take me outa here?"
"If he don't I'm going to throw you out myself," she said. "I don't want any hollering and screaming in here scaring my white clients."
"That's gonna cost you," Ready threatened.
"Don't threaten me, nigger," she said dangerously. "And don't set your foot in my door again."
"Okay, Reba, that's the lick that killed Dick," Ready said slowly. "You and him got me outnumbered." He gave her a last sullen look and turned to go.
Reba walked to the door and let them out.
"I hope I get what I want," Grave Digger said. "If I don't I'll be back."
"If you don't it's your own fault," she said.
He marched Ready ahead of him down the shaky stairs.
The old man in the ragged red chair looked up in surprise.
"You got the wrong nigger," he said. "Hit ain't him what's makin' all the trouble."
"Who is it?" Grave Digger asked.
"Hit's Cocky. He the one what's always pulling his shiv."
Grave Digger filed the information for future reference.
"I'll keep this one since he's the one I've got," he said.
"Balls," the old man said disgustedly. "He's just a halfass pimp."
# 10
White light coming from the street slanted upward past the edge of the roof and made a milky wall in the dark.
Beyond the wall of light the flat tar roof was shrouded in semi-darkness.
The sergeant emerged from the edge of light like a hammerhead turtle rising from the deep. In one glance he saw Sonny frantically beating a flock of panic-stricken pigeons with a long bamboo pole, and Inky standing motionless as though he'd sprouted from the tar.
"By God, now I know why they're called tarbabies!" he exclaimed.
Gripping the pole for dear life with both gauntleted hands, Sonny speared desperately at the pigeons. His eyes were white as they rolled toward the red-faced sergeant. His ragged overcoat flapped in the wind. The pigeons ducked and dodged and flew in lopsided circles. Their heads were cocked on one side as they observed Sonny's gymnastics with beady apprehension.
Inky stood like a silhouette cut from black paper, looking at nothing. The whites of his eyes gleamed in the dark.
The pigeon loft was a rickety coop about six feet high, made of scraps of chicken wire, discarded screen windows and assorted rags tacked to a frame of rotten boards propped against the low brick wall separating the roofs. It had a tarpaulin top and was equipped with precarious roosts, tin cans of rusty water, and a rusty tin feeding pan.
Blue-uniformed white cops formed a jagged semi-circle in front of it, staring at Sonny in silent and bemused amazement.
The sergeant climbed onto the roof, puffing, and paused for a moment to mop his brow.
"What's he doing, voodoo?" he asked.
"It's only Don Quixote in blackface dueling a windmill," the professor said.
"That ain't funny," the sergeant said. "I like Don Quixote."
The professor let it go.
"Is he a halfwit?" the sergeant said.
"If he's got that much," the professor said.
The sergeant pushed to the center of the stage, but once there hesitated as though he didn't know how to begin.
Sonny looked at him through the corners of his eyes and kept working the pole. Inky stared at nothing with silent intensity.
"All right, all right, so your feet don't stink," the sergeant said. "Which one of you is Caleb?"
"Dass me," Sonny said, without an instant neglecting the pigeons.
"What the hell you call yourself doing?"
"I'se teaching my pigeons how to fly."
The sergeant's jowls began to swell. "You trying to be funny?"
"Naw suh, I didn' mean they didn' know how to fly. They can fly all right at day but they don't know how to night fly."
The sergeant looked at the professor. "Don't pigeons fly at night?"
"Search me," the professor said.
"Naw suh, not unless you makes 'em," Inky said.
Everybody looked at him.
"Hell, he can talk," the professor said.
"They sleeps," Sonny added.
"Roosts," Inky corrected.
"We're going to make some pigeons fly, too," the sergeant said. "Stool pigeons."
"If they don't fly, they'll fry," the professor said.
The sergeant turned to Inky. "What do they call you, boy?"
"Inky," Inky said. "But my name's Rufus Tree."
"So you're Inky," the sergeant said.
"They're both Inky," the professor said.
The cops laughed.
The sergeant smiled into his hand. Then he wheeled abruptly on Sonny and shouted, "Sonny! Drop that pole!"
Sonny gave a violent start and speared a pigeon in the craw, but he hung on to the pole. The pigeon flew crazily into the light and kept on going. Sonny watched it until he got control of himself, then he turned slowly and looked at the sergeant with big innocent white eyes.
"You talking to me, boss?" His black face shone with sweat.
"Yeah, I'm talking to you, Sonny."
"They don't calls me, Sonny, boss; they calls me Cal."
"You look like a boy called Sonny."
"Lots of folks is called Sonny, boss."
"What did you jump for if your name isn't Sonny? You jumped halfway out of your skin."
"Most anybody'd jump with you hollerin' at 'em like that, boss."
The sergeant wiped off another smile. "You told your grandma you were going downtown to work."
"She don't want me messin' 'round these pigeons at night. She thinks I might fall off'n the roof."
"Where have you been since supper?"
"Right up here, boss."
"He's just been up here about a half an hour," one of the cops volunteered.
"Naw suh, I been here all the time," Sonny contradicted. "I been inside the coop."
"Ain't nobody in heah but us pigeons, boss," the professor cracked.
"Did you look in the coop?" the sergeant asked the cop.
The cop reddened. "No, I didn't; I wasn't looking for a screwball."
The sergeant glanced at the coop. "By God, boy, your pigeons lead a hard life," he said. Then turning suddenly to the other cops, he asked, "Have these punks been frisked?"
"We were waiting for you," another cop replied.
The sergeant sighed theatrically. "Well, who are you waiting for now?"
Two cops converged on Inky with alacrity; the professor and a third cop took on Sonny.
"Put that damn pole down!" the sergeant shouted at Sonny.
"No, let him hold it," the professor said. "It keeps his hands up."
"What the hell are you wearing that heavy overcoat for?" The sergeant kept on picking at Sonny. He was frustrated.
"I'se cold," Sonny said. Sweat was running down his face in rivers.
"You look it," the sergeant said.
"Jesus Christ, this coat stinks," the professor complained, working Sonny over fast to get away from it.
"Nothing?" the sergeant asked when he'd finished.
"Nothing," the professor said. In his haste he hadn't thought to make Sonny put down the pole and take off his gauntlets.
The sergeant looked at the cops frisking Inky. They shook their heads.
"What's Harlem coming to?" the sergeant complained. "All right, you punks, get downstairs," the sergeant ordered.
"I got to get my pigeons in," Sonny said.
The sergeant looked at him.
Sonny leaned the pole against the coop and began moving. Inky opened the door of the coop and began moving too. The pigeons took one look at the open door and began rushing to get inside.
"IRT subway at Times Square," the professor remarked.
The cops laughed and moved on to the next roof.
The sergeant and the professor followed Inky and Sonny through the window and into the room below.
Sissie and Sugartit sat side by side on the bed again. Choo-Choo sat in the straight-backed chair. Sheik stood in the center of the floor with his feet wide apart, looking defiant. The two cops stood with their buttocks propped against the edge of the table, looking bored.
With the addition of the four others, the room was crowded.
Everybody looked at the sergeant, waiting his next move.
"Get Grandma in here," he said.
The professor went after her.
They heard him saying, "Grandma, you're needed."
There was no reply.
"Grandma!" they heard him shout.
"She's asleep," Sissie called to him. "She's hard to wake once she gets to sleep."
"She's not asleep," the professor called back in an angry tone of voice.
"All right, let her alone," the sergeant said.
The professor returned, red-faced with vexation. "She sat there looking at me without saying a word," he said.
"She gets like that," Sissie said. "She just sort of shuts out the world and quits seeing and hearing anything."
"No wonder her grandson's a halfwit," the professor said, giving Sonny a malicious look.
"Well, what the hell are we going to do with them?" the sergeant said in a frustrated tone of voice.
The cops had no suggestions.
"Let's run them all in," the professor said.
The sergeant looked at him reflectively. "If we take in all the punks who look like them in this block, we'll have a thousand prisoners," he said.
"So what," the professor said. "We can't afford to risk losing Pickens because of a few hundred shines."
"Well, maybe we'd better," the sergeant said.
"Are you going to take her in too?" Sheik said, nodding toward Sugartit on the bed. "She's Coffin Ed's daughter."
The sergeant wheeled on him. "What! What's that about Coffin Ed?"
"Evelyn Johnson there is his daughter," Sheik said evenly.
The cops turned as though their heads were synchronized and stared at her. No one spoke.
"Ask her," Sheik said.
The sergeant's face turned bright red.
It was the professor who spoke. "Well, girl? Are you Detective Johnson's daughter?"
Sugartit hesitated.
"Go on and tell 'em," Sheik said.
The red started crawling up the back of the sergeant's neck and engulfed his ears. "I don't like you," he said to Sheik, his voice constricted.
Sheik threw him a careless look, started to say something, then bit it off.
"Yes, I am," Sugartit said finally.
"We can soon check on that," the professor said, moving toward the window. "He and his partner must be in the vicinity."
"No, Jones might be, but Johnson was sent home," the sergeant said.
"What! Suspended?" the professor asked in surprise.
Sugartit looked startled; Sheik grinned smugly; the others remained impassive.
"Yeah, for killing the Moslem punk."
"For that?" the professor exclaimed indignantly. "Since when did they start penalizing policemen for shooting in self-defense?"
"I don't blame the chief," the sergeant said. "He's protecting himself. The punk was under-age and the newpapers are sure to put up a squawk."
"Anyway, Jones ought to know her," the professor said, going out on the fire escape and shouting to the cops below.
He couldn't make himself understood so he started down.
The sergeant asked Sugartit, "Have you got any identification?"
She drew a red leather card case from her skirt pocket and handed it to him without speaking.
It held a black, white-lettered identification card with her photograph and thumbprint, similar to the one issued to policemen. It had been given to her as a souvenir for her sixteenth birthday and was signed by the chief of police.
The sergeant studied it for a moment and handed it back. He had seen others like it, his own daughter had one.
"Does your father know you're here visiting these hoodlums?" he asked.
"Certainly," Sugartit said. "They're friends of mine."
"You're lying," the sergeant said wearily.
"He doesn't know she's over here," Sissie put in.
"I know damn well he doesn't," the sergeant said.
"She's supposed to be visiting me."
"Well, do your folks know you're here?"
She dropped her gaze. "No."
"Eve and I are engaged," Sheik said with a smirk.
The sergeant wheeled toward him with his right cocked high. Sheik ducked automatically, his guard coming up. The sergeant hooked a left to his stomach underneath his guard, and when Sheik's guard dropped, he crossed his right to the side of Sheik's head, knocking him into a spinning stagger. Then he kicked him in the side of the stomach as he spun and, when he doubled over, the sergeant chopped him across the back of the neck with the meaty edge of his right hand. Sheik shuddered as though poleaxed and crashed to the floor. The sergeant took dead aim and kicked him in the valley of the buttocks with all his force.
The professor returned just in time to see the sergeant spit on him.
"Hey, what's happened to him?" he asked, climbing hastily through the window.
The sergeant took off his hat and wiped his perspiring forehead with a soiled white handkerchief. "His mouth did it," he said.
Sheik was groaning feebly, although unconscious.
The professor chuckled. "He's still trying to talk." Then he said, "They couldn't find Jones. Lieutenant Anderson says he's working on another angle."
"It's okay, she's got an ID card," the sergeant said. Then asked, "Is the chief still there?"
"Yeah, he's still hanging around."
"Well, that's his job."
The professor looked about at the silent group. "What's the verdict?"
"Let's get on to the next house," the sergeant said. "If I'm here when this punk comes to I'll probably be the next one to get suspended."
"Can we leave the building now?" Sissie asked.
"You two girls can come with us," the sergeant offered.
Sheik groaned and rolled over.
"We can't leave him like that," she said.
The sergeant shrugged. The cops passed into the next room. The sergeant started to follow, then hesitated.
"All right, I'll fix it," he said.
He took the girls out on the fire escape and got the attention of the cops guarding the entrance below.
"Let these two girls pass!" he shouted.
The cops looked at the girls standing in the spotlight glare.
"Okay."
The sergeant followed them back into the room.
"If I were you I'd get the hell away from this punk fast," he advised, prodding Sheik with his toe. "He's headed straight for trouble, big trouble."
Neither replied.
He followed the professor out of the flat.
Granny sat unmoving in the rocking chair where they'd left her, tightly gripping the arms. She stared at them with an expression of fierce disapproval on her puckered old face and in her dim milky eyes.
"It's our job, Grandma," the sergeant said apologetically.
She didn't reply.
They passed on sheepishly.
Back in the front room, Sheik groaned and sat up.
Everyone moved at once. The girls moved away from him. Sonny began taking off the heavy overcoat. Inky and Choo-Choo bent over Sheik and, each taking an arm, began helping him to his feet.
"How you feel, Sheik?" Choo-Choo asked.
Sheik looked dazed. "Can't no copper hurt me," he muttered thickly, wobbling on his legs.
"Does it hurt?"
"Naw, it don't hurt," he said with a grimace of pain. Then he looked about stupidly. "They gone?"
"Yeah," Choo-Choo said jubilantly and cut a jig step. "We done beat 'em, Sheik. We done fooled 'em two ways sides and flat."
Sheik's confidence came back in a rush. "I told you we was going to do it."
Sonny grinned and raised his clasped hands in the prize-fight salute. "They had me sweating in the crotch," he confessed.
A look of crazed triumph distorted Sheik's flat, freckled face. "I'm the Sheik, Jack," he said. His yellow eyes were getting wild again.
Sissie looked at him and said apprehensively, "Me and Sugartit got to go. We were just waiting to see if you were all right."
"You can't go now – we got to celebrate," Sheik said.
"We ain't got nothing to celebrate with," Choo-Choo said.
"The hell we ain't," Sheik said. "Cops ain't so smart. You go up on the roof and get the pole."
"Who, me, Sheik?"
"Sonny then."
"Me!" Sonny said. "I done got enough of that roof."
"Go on," Sheik said. "You're a Moslem now and I command you in the name of Allah."
"Praise Allah," Choo-Choo said.
"I don't want to be no Moslem," Sonny said.
"All right, you're still our captive then," Sheik decreed. "You go get the pole, Inky. I got five sticks stashed in the end."
"Hell, I'll go," Choo-Choo said.
"No, let Inky go, he's been up there before and they won't think it's funny."
When Inky left for the pole, Sheik said to Choo-Choo, "Our captive's getting biggety since we saved him from the cops."
"I ain't gettin' biggety," Sonny declared. "I just want to get the hell outen here and get these cuffs off'n me without havin' to become no Moslem."
"You know too much for us to let you go now," Sheik said, exchanging a look with Choo-Choo.
Inky returned with the pole and, pulling the plug out of the end joint, he shook five cigarettes onto the table top.
"A feast!" Choo-Choo exclaimed. He grabbed one, opened the end with his thumb, and lit up.
Sheik lit another.
"Take one, Inky," he said.
Inky took one.
Everybody put on smoked glasses.
"Granny will smell it if you smoke in here," Sissie said.
"She thinks they're cubebs." Choo-Choo mimicked Granny: "Ah wish you chillens would stop smokin' them coo-bebs 'cause they make a body feel moughty funny in de head."
He and Sheik doubled over with laughter.
The room stank with the pungent smoke.
Sugartit picked up a stick, sat on the bed and lit it.
"Come on, baby, strip," Sheik urged her. "Celebrate your old man's flop by getting up off of some of it."
Sugartit stood up and undid her skirt zipper and began going into a slow striptease routine.
Sissie clutched her by the arms. "You stop that," she said. "You'd better go on home before your old man gets there first and comes out looking for you."
In a sudden rage, Sheik snatched Sissie's hands away from Sugartit and flung her across the bed.
"Leave her alone," he raved. "She's going to entertain the Sheik."
"If her old man's really Coffin Ed you oughta let her go on home," Sonny said soberly. "You just beggin' for trouble messin' round with his kinfolks."
"Choo-Choo, go to the kitchen and get Granny's wire clothesline," Sheik ordered.
Choo-Choo went out grinning.
When he saw Granny staring at him with such fierce disapproval, he said guiltily, "Pay no 'tention to me, Granny," and began clowning.
She didn't answer.
He tiptoed with elaborate pantomime to the closet and took out her coil of clothesline.
"Just wanna hang out the wash," he said.
Still she didn't answer.
He tiptoed close to the chair and passed his hand slowly in front of her face. She didn't bat an eyelash. His grin widened. Returning to the front room, he said, "Granny's dead asleep with her eyes wide open."
"Leave her to Gabriel," Sheik said, taking the line and beginning to uncoil it.
"What you gonna do with that?" Sonny asked apprehensively.
Sheik made a running loop in one end. "We going to play cowboy," he said. "Look."
Suddenly he threw the loop over Sonny's head and pulled on the line with all his strength. The loop tightened about Sonny's neck and jerked him off his feet.
Sissie ran toward Sheik and tried to pull the wire from his hands. "You're choking him," she said.
Sheik knocked her down with a backhanded blow.
"You can let up on him now," Choo-Choo said. "We got 'im."
"Now I'm gonna show you how to tie up a mother-raper to put him in a sack," Sheik said.
# 11
Grave Digger halted on the sidewalk in front of the yellow frame house next door to the Knickerbocker. It had been partitioned into offices and all of the front windows were lettered with business announcements.
"Can you read that writing on those windows?" Grave Digger asked Ready Belcher.
Ready glanced at him suspiciously. "Course I can read that writing."
"Read it then," Grave Digger said.
Ready stole another look. "Read what one?"
"Take your choice."
Ready squinted his good eye against the dark and read aloud, _"Joseph_ C. _Clapp, Real Estate and Notary Public."_ He looked at Grave Digger like a dog who has retrieved a stick. "That one?"
"Try another."
He hesitated. Passing car lights played on his pockmarked black face, brought out the white cast in his bad eye and lit up his flashy tan suit.
"I haven't got much time," Grave Digger warned.
He read, " _Amazing 100-year-old Gypsy Bait Oil – Makes Catfish Go Crazy_." He looked at Grave Digger again like the same dog with another stick.
"Not that one," Grave Digger said.
"What the hell is this, a gag?" he muttered.
"Just read!"
_"JOSEPH, The Only and Original Skin Lightener. I guarantee to lighten the darkest skin by twelve shades in six_ _months."_
"You don't want your skin lightened?"
"My skin suit me," he said sullenly.
"Then read on."
_"Magic Formula For Successful PRAYER_... That it?"
"Yeah, that's it. Read what it says underneath."
_"Here are some of the amazing things it tells you about: When to pray; Where to pray; How to pray; The Magic Formulas for Health and Success through prayer; for conquering fear through prayer; for obtaining work through prayer; for money through prayer; for influencing others through prayer; and–"_
"That's enough." Grave Digger took a deep breath and said in a voice gone thick and cottony again, "Ready, if you don't tell me what I want to know, you'd better get yourself one of those prayers. Because I'm going to take you over to 129th Street near the Harlem River. You know where that is? It's a deserted jungle of warehouses and junk yards beneath the New York Central bridge."
"Yare, I know where it's at."
"And I'm going to pistol whip you until your own whore won't recognize you again. And if you try to run, I'm going to let you run fifty feet and then shoot you through the head for attempting to escape. You understand me?"
"Yare, I understand you."
"You believe me?"
Ready took a quick look at Grave Digger's rage-swollen face and said quickly, "Yare, I believes you."
"My partner got suspended tonight for killing a criminal rat like you and I'd just as soon they suspended me too."
"You ain't asted me yet what you want to know."
"Get into the car."
The car was parked at the curb. Ready got into Coffin Ed's seat. Grave Digger went around and climbed beneath the wheel.
"This is as good a spot as any," he said. "Start talking."
" 'Bout what?"
"About the Big Greek. I want to know who killed him."
Ready jumped as though he'd been stung. "Digger, I swear 'fore God–"
"Don't call me Digger, you lousy pimp."
"Mista Jones, lissen–"
"I'm listening."
"Lots of folks mighta killed him if they'd knowed–" He broke off. The pockmarks in his skin began filling with sweat.
"Known what? I haven't got all night."
Ready gulped and said, "He was a whipper."
"What?"
"He liked to whip 'em."
"Whores?"
"Not 'specially. If they was regular whores he wanted them to be big black mannish-looking bitches like what might cut a mother-raper's throat. But what he liked most was little colored school gals."
"That's it? That's why Reba barred him?"
"Yas suh. He proposition her once. She got so mad she drew her pistol on him."
"Did she shoot him?"
"Naw, suh, she just scared him."
"I mean tonight. Was she the one?"
Ready's eyes started rolling in their sockets and the sweat began to trickle down his mean black face.
"You mean the one what killed him? Naw suh, she was home all evening."
"Where were you?"
"I was there, too."
"Do you live there?"
"Naw, suh, I just drops by for a visit now and then."
"Where did he find the girls?"
"You mean the school girls?"
"What other girls would I mean?"
"He picked 'em up in his car. He had a little Mexican bull whip with nine tails he kept in his car. He whipped 'em with that."
"Where did he take them?"
"He brung 'em to Reba's till she got suspicious 'bout all the screaming and carrying on. She didn't think nothing of it at first; these little chippies likes to make lots of noise for a white man. But they was making more noise than seemed natural and she went in and caught 'im. That's when he proposition her."
"How did he get 'em to take it?"
"Get 'em to take what?"
"The whipping."
"Oh, he paid 'em a hundred bucks. They was glad to take it for that."
"You're certain of that, that he paid them a hundred dollars?"
"Yas suh. Not only me but lots of chippies all over Harlem knew about him. A hundred bucks didn't mean nothing to him. They boy friends knew too. Lots of times they boy friends made 'em. There was chippies all over town on the lookout for him. 'Course one time was enough for most of 'em."
"He hurt them?"
"He got his money's worth. Sometime he whale hell out of 'em. I s'pect he hurt more'n one of 'em bad. 'Member that kid they picked up in Broadhurst Park. It were all in the paper. She was in the hospital three, four days. She said she'd been attacked but the police thought she was beat up by a gang. I believes she was one of 'em."
"What was her name?"
"I don't recollect."
"Where'd he take them after Reba barred him from her place?"
"I don't know."
"Do you know the names of any of them?"
"Naw suh, he brung 'em and took 'em away by hisself. I never even seen any of 'em."
"You're lying."
"Naw suh, I swear 'fore God."
"How did you know they were school girls if you never saw any of them?"
"He tole me."
"What else he tell you?"
"Nuthin' else. He just talk to me 'bout gals."
"How old is your girl?"
"My gal?"
"The one you have at Reba's?"
"Oh, she twenty-five or more."
"One more lie and off we go."
"She sixteen, boss."
"She had him, too?"
"Yas suh. Once."
The sweat was streaming down Ready's face.
"Once. Why only once?"
"She got scared."
"You tried to fix it up for another time?"
"Naw suh, boss, she didn't need to. Hit cost her more'n it was worth."
"What were you doing with him in the Dew Drop Inn?"
"He was looking for a little gal he knew and he ast me to come 'long, that's all, boss."
"When was that?"
" 'Bout a month ago."
"You said you didn't know where he took them after he was barred from Reba's."
"I don't, boss, I swear 'fore–"
"Can that Uncle Tom crap. Reba said she barred him three or four months ago."
"Yas suh, but I didn't say I hadn't seed him since."
"Did Reba know you were seeing him?"
"I only seed him that once, boss. I was in the Alabama-Georgia bar and he just happen in."
Grave Digger nodded towards the three alien cars parked ahead, in front of the Knickerbocker.
"One of those cars his?"
"Them struggle buggies!" Scorn pushed the fear from Ready's voice. "Naw suh, he had a dream boat, a big green Caddy Coupe de Ville."
"Who was the girl he and you were looking for?"
"I wasn't looking for her; I just went 'long with him to look for her."
"Who was she, I asked."
"I didn't know her. Some little chippie what hung 'round in that section."
"How did he come to know her?"
"He said he'd done whipped her girl friend once. That's how come he knew her. Said Sissie's boy friend brought her to 'im."
"Sissie! You said you didn't know the name of any of them."
"I'd forgotten her, boss. He didn't bring her to Reba's. I didn't know nuthin' 'bout her but just what he said."
"What did he say exactly?"
"He just say Sissie's boy friend, some boy they call Sheik, arrange it for him and he pay Sheik. Then he wanted Sheik to arrange for the other one but Sheik couldn't do it."
"What was the other one called? The one he and you were looking for?"
"He call her Sugartit. She was Sissie's girl friend. He'd seen 'em walking together down Seventh Avenue one time after he'd whipped Sissie."
"Where did you find her?"
"We didn't find her, I swear 'fore–"
"Does your girl know them?"
"I didn' hear you."
"Your girl, does she know them?"
"Know who, boss?"
"Either Sissie or Sugartit."
"Naw suh. My gal's a pro and them is just chippies. I recollect him saying one time they all belonged to a kid gang over in that section. I means them two chippies and Sheik. He say Sheik was the chief."
"What's the name of the gang?"
"He say they call themselves the Real Cool Moslems. He thought it were funny."
"Did you listen to the news on the radio tonight?"
"You mean what it say 'bout him getting croaked? Naw suh, I was lissening to the Twelve-Eighty Club. Reba tole me 'bout it. She were lissening. That were just 'fore you come. She were telling me when the doorbell rang. She say the big Greek's croaked over on Lenox Avenue and I say so what."
"You said before that lots of people might have killed him if they'd known about him. Who?"
"All I meant was some of those gal's pas. Like Sissie's or some of 'em. He might have been hanging 'round over there looking for Sugartit again and her pa might have got hep to it some kind of way and been layin' for him and when he seed him coming down the street might have lowered the boom on 'im."
"You mean slipped up behind him?"
"He were in his car, warn't he?"
"How about the Moslems – the kid gang?"
"Them! What they'd wanta do it for? He was money in the street for them."
"Who's Sugartit's father?"
"You mean her old man?"
"I mean her father."
"How am I gonna know that, boss? I ain't never heard of her 'fore he talk 'bout her."
"What did he say about her?"
"Just say she was the gal for him."
"Did he say where she lived?"
"Naw, suh, he just say what I say he say, boss, I swear 'fore God."
"You stink. What are you sweating so much for?"
"I'se just nervous, that's all."
"You stink with fear. What are you scared of?"
"Just naturally scared, boss. You got that big pistol and you mad at everybody and talkin' 'bout killin' me and all that. Enough to make anybody scared."
"You're scared of something else, something in particular. What are you holding out?"
"I ain't holding nothing out. I done tole you everything I know, I swear boss, I swears on everything that's holy in this whole green world."
"I know you're lying. I can hear it in your voice. What are you lying about?"
"I ain't lying, boss. If I'm lying I hope God'll strike me dead on the spot."
"You know who her father is, don't you?"
"Naw suh, boss. I swear. I done tole you everything I know. You could whup me till my head is soft as clabber but I couldn't tell you no more than I'se already tole you."
"You know who her father is and you're scared to tell me."
"Naw suh, I swear–"
"Is he a politician?"
"Boss, I–"
"A numbers banker?"
"I swear, boss–"
"Shut up before I knock out your goddamned teeth."
He mashed the starter as though tromping on Ready's head. The motor purred into life. But he didn't slip in the clutch. He sat there listening to the softly purring motor in the small black nondescript car, trying to get his temper under control.
Finally he said, "If I find out that you're lying I'm going to kill you like a dog. I'm not going to shoot you, I'm going to break all your bones. I'm going to try to find out who killed Galen because that's what I'm paid for and that was my oath when I took this job. But if I had my way I'd pin a medal on him and I'd string up every goddamned one of you who were up with Galen. You've turned my stomach and it's all I can do right now to keep from beating out your brains."
# 12
The reception room of the Harlem Hospital, on Lenox Avenue ten blocks south from the scene of the murder, was wrapped in a midnight hush.
It was called an interracial hospital; more than half of its staff of doctors and nurses were colored people.
A graduate nurse sat behind the reception desk. A bronze-shaded desk lamp spilled light on the hospital register before her while her brown-skinned face remained in shadow. She looked up inquiringly as Grave Digger and Ready Belcher approached, walking side by side.
"May I help you," she said in a trained courteous voice.
"I'm Detective Jones," Grave Digger said, exhibiting his badge.
She looked at it but didn't touch it.
"You received an emergency patient here about two hours ago; a man with his right arm cut off."
"Yes?"
"I would like to question him."
"I will call Dr. Banks. You may talk to him. Please be seated."
Grave Digger prodded Ready in the direction of chairs surrounding a table with magazines. They sat silently, like relatives of a critical case.
Dr. Banks came in silently, crossing the linoleum-tiled floor on rubber-soled shoes. He was a tall, athletic-looking young colored man dressed in white.
"I'm sorry to have kept you waiting, Mr. Jones," he said to Grave Digger whom he knew by sight. "You want to know about the case with the severed arm." He had a quick smile and a pleasant voice.
"I want to talk to him," Grave Digger said.
Dr. Banks pulled up a chair and sat down. "He's dead. I've just come from him. He had a rare type of blood – Type O – which we don't have in our blood bank. You realize transfusions were imperative. We had to contact the Red Cross blood bank. They located the type in Brooklyn, but it arrived too late. Is there anything I can tell you?"
"I want to know who he was."
"So do we. He died without revealing his identity."
"Didn't he make a statement of any kind before he died?"
"There was another detective here earlier, but the patient was unconscious at the time. The patient regained consciousness later, but the detective had left. Before leaving, he examined the patient's effects, however, but found nothing to establish his identity."
"He didn't talk at all, didn't say anything?"
"Oh yes. He cried a great deal. One moment he was cursing and the next he was praying. Most of what he said was incoherent. I gathered he regretted not killing the man whom he had attacked – the white man who was killed later."
"He didn't mention any names?"
"No. Once he said 'the little one' but mostly he used the word _mother-raper_ which Harlemites apply to everybody, enemies, friends and strangers."
"Well, that's that," Grave Digger said. "Whatever he knew he took with him. Still I'd like to examine his effects too, whatever they are."
"Certainly; they're just the clothes he wore and the contents of his pockets when he arrived here." He stood up. "Come this way."
Grave Digger got to his feet and motioned his head for Ready to walk ahead of him.
"Are you an officer too?" Dr. Banks asked Ready.
"No, he's my prisoner," Grave Digger said. "We're not that hard up for cops as yet."
Dr. Banks smiled. He led them down a corridor smelling strongly of ether to a room at the far end where the clothes and personal effects of the emergency and ward patients were stored in neatly wrapped bundles on shelves against the walls. He took down a bundle bearing a metal tag and placed it on the bare wooden table.
"Here you are."
From the adjoining room an anguished male voice was heard reciting the Lord's Prayer.
Ready stared as though fascinated at the number 219 on the metal tag fastened to the bundle of clothes and whispered, "Death row."
Dr. Banks flicked a glance at him and said to Grave Digger, "Most of the attendants play the numbers. When an emergency patient arrives they put this tag with the death number on his bundle and if he dies they play it."
Grave Digger grunted and began untying the bundle.
"If you discover anything leading to his identity, let us know," Dr. Banks said. "We'd like to notify his relatives." He left them.
Grave Digger spread the blood-caked mackinaw and overalls on the table. It contained two incredibly filthy one-dollar bills, some loose change, a small brown paper sack of dried roots, two Yale keys and a skeleton key on a rusty key ring, a dried rabbit's foot, a dirty piece of resin, a cheese cloth rag that had served as a handkerchief, a putty knife, a small piece of pumice stone, and a scrap of dirty writing paper folded into a small square. The putty knife and pumice stone indicated that the man had worked somewhere as a porter, using the putty to scrape chewing gum from the floor and the pumice stone for cleaning his hands. That didn't help much.
He unfolded the square of paper and found a note on cheap school paper written in a childish hand.
GB, you want to know something. The Big John hangs out in the Inn. How about that. Just like those old Romans.
Bee.
Grave Digger folded it again and slipped it into his pocket.
"Is your girl called Bee?" he asked Ready.
"Naw, suh, she called Doe."
"Do you know any girl called Bee – a school girl?"
"Naw suh."
"GB?"
"Naw suh."
Grave Digger turned out the pockets of the clothes but found nothing more. He wrapped the bundle and attached the tag. He noticed Ready staring at the number on the tag again.
"Don't let that number catch up with you," he said. "Don't you end up with that tag on your fine clothes."
Ready licked his dry lips.
They didn't see Dr. Banks on their way out. Grave Digger stopped at the reception desk to tell the nurse he hadn't found anything to identify the corpse.
"Now we're going to look for the Greek's car," he said to Ready.
They found the big green Cadillac beneath a street lamp in the middle of the block on 130th Street between Lenox and Seventh Avenues. It had an Empire State license number–UG-16 – and it was parked beside a fire hydrant. It was as conspicuous as a fire truck.
He pulled up behind it and parked.
"Who covered for him in Harlem?" he asked Ready.
"I don't know, Mista Jones."
"Was it the precinct captain?"
"Mista Jones, I–"
"One of our councilmen?"
"Honest to God, Mista Jones–"
Grave Digger got out and walked toward the big car.
The doors were locked. He broke the glass of the left-side wind screen with the butt of his pistol, reached inside past the wheel and unlocked the door. The interior lights came on.
A quick search revealed the usual paraphernalia of a motorist: gloves, handkerchiefs, Kleenex, half-used packages of different brands of cigarettes, insurance papers, a woman's plastic overshoes and compact. A felt monkey dangled from the rear view mirror and two medium-sized dolls, a black-faced Topsy and a blonde Little Eva, sat in opposite corners on the back seat.
He found the miniature bull whip and a Manila envelope of postcard-sized photos in the right-hand glove compartment. He studied the photos in the light. They were pictures of nude colored girls in various postures, each photo revealing another developed technique of the sadist. On most of the pictures the faces of the girls were distinct although distorted by pain and shame.
He put the whip in his leather-lined coat pocket, kept the photos in his hand, slammed the door, walked back to his own car and climbed beneath the wheel.
"Was he a photographer?" he asked Ready.
"Yas suh, sometime he carry a camera."
"Did he show you the pictures he took?"
"Naw suh, he never said nothing 'bout any pictures. I just seen him with the camera."
Grave Digger snapped on the top light and showed Ready the photos.
"Do you recognize any of them?"
Ready whistled softly and his eyes popped as he turned over one photo after another.
"Naw suh, I don't know none of them," he said, handing them back.
"Your girl's not one of them?"
"Naw suh."
Grave Digger pocketed the envelope and mashed the starter.
"Ready, don't let me catch you in a lie," he said again, letting out the clutch.
# 13
He parked directly in front of the Dew Drop Inn and pushed Ready through the door. On first sight it looked just as he had left it; the two white cops guarding the door and the colored patrons celebrating noisily. He ushered Ready between the bar and the booths, toward the rear. The varicolored faces turned toward them curiously as they passed.
But in the last booth he noticed an addition. It was crowded with teenagers, three school boys and four school girls, who hadn't been there before. They stopped talking and looked at him intently as he and Ready approached. Then at sight of the bull whip all four girls gave a start and their young dark faces tightened with sudden fear. He wondered how they'd got past the white cops on the door.
All the places at the bar were taken.
Big Smiley came down and asked two men to move.
One of them began to complain. "What for I got to give up my seat for some other niggers."
Big Smiley thumbed toward Grave Digger. "He's the man."
"Oh, one of them two."
Both rose with alacrity, picked up their glasses and vacated the stools, grinning at Grave Digger obsequiously.
"Don't show me your teeth," Grave Digger snarled. "I'm no dentist. I don't fix teeth. I'm a cop. I'll knock your teeth out."
The men doused their grins and slunk away.
Grave Digger threw the bull whip on top of the bar and sat on the high bar stool.
"Sit down," he ordered Ready, who stood by hesitantly. "Sit down, Goddamn it."
Ready sat down as though the stool were covered with cake icing.
Big Smiley looked from one to another, smiling warily.
"You held out on me," Grave Digger said in his thick cottony voice of smoldering rage. "And I don't like that."
Big Smiley's smile got a sudden case of constipation. He threw a quick look at Ready's impassive face, found nothing there to reassure him, then fell back on his cut arm which he carried in a sling.
"Guess I must be runnin' some fever, Chief, 'cause I don't remember what I told you."
"You told me you didn't know who Galen was looking for in here," Grave Digger said thickly.
Big Smiley stole another look at Ready, but all he got was a blank. He sighed heavily.
"Who he were looking for? Is dat what you ast me?" he stalled, trying to meet Grave Digger's smoldering hot gaze. "I dunno who he were looking for, Chief."
Grave Digger rose up on the bar stool rungs as though his feet were in stirrups, snatched the bull whip from the bar and slashed Big Smiley across one cheek after another before Big Smiley could get his good hand moving.
Big Smiley stopped smiling. Talk stopped suddenly along the length of the bar, petered out in the booths. In the vacuum that followed, Lil Green's voice whined from the jukebox:
_"Why don't you do right
Like other mens do..."_
Grave Digger sat back on the stool, breathing hard, struggling to control his rage. Veins stood out in his temples, growing out of his short-cropped kinky hair like strange roots climbing toward the brim of his misshapen hat. His brown eyes laced with red veins generated a steady white heat.
The white manager, who'd been working the front end of the bar, hastened down toward them with a face full of outrage.
"Get back," Grave Digger said thickly.
The manager got back.
Grave Digger stabbed at Big Smiley with his left forefinger and said in a voice so thick it was hard to understand, "Smiley, all I want from you is the truth. And I ain't got long to get it."
Big Smiley didn't look at Ready any more. He didn't smile. He didn't whine.
He said, "Just ask the questions, Chief, and I'll answer 'em the best of my knowledge."
Grave Digger looked around at the teenagers in the booth. They were listening with open mouths, staring at him with popping eyes. His breath burned from his flaring nostrils. He turned back to Big Smiley. But he sat quietly for a moment to give the blood time to recede from his head.
"Who killed him?" he finally asked.
"I don't know, Chief."
"He was killed on your street."
"Yas suh, but I don't know who done it."
"Do Sissie and Sugartit come in here?"
"Yas suh, sometimes."
Out of the corners of his eyes Grave Digger noticed Ready's shoulders begin to sag as though his spine were melting.
"Sit up straight, God damn it," he said. "You'll have plenty of time to lie down if I find out you've been lying."
Ready sat up straight.
Grave Digger addressed Big Smiley. "Galen met them in here?"
"Naw suh, he met Sissie in here once but I never seen him with Sugartit."
"What was she doing in here then?"
"She come in here twice with Sissie."
"How'd you know her name?"
"I heard Sissie call her that."
"Was Sheik with her when Galen met her?"
"You mean with Sissie, when she met the big man? Yas suh."
"He paid Sheik the money?"
"I couldn't be sure, Chief, but I seen money being passed. I don't know who got it."
"He got it. Did they both leave with him?"
"You mean both Sheik and Sissie?"
"That's what I mean."
Big Smiley took out a blue bandana handkerchief and mopped his sweating black face.
The four school girls in the booth began going through the motions of leaving. Grave Digger wheeled toward them.
"Sit down! I want to talk to you later," he ordered.
They began a shrill protest: "We got to get home... Got to be at school tomorrow at nine o'clock... Haven't finished homework... Can't stay out this late... Get into trouble..."
He got up and went over to show them his gold badge. "You're already in trouble. Now I want you to sit down and keep quiet."
He took hold of the two girls who were standing and forced them back into their seats.
"He can't hold you 'less he's got a warrant," the boy in the aisle seat said.
Grave Digger slapped him out of his seat, reached down and lifted him from the floor by his coat lapels and slammed him back into his seat.
"Now say that again," he suggested.
The boy didn't speak.
Grave Digger waited for a moment until they had settled down and were quiet, then he returned to his bar stool.
Neither Big Smiley nor Ready had moved; neither had looked at the other.
"You didn't answer my question," Grave Digger said.
"When he took Sissie off Sheik stayed in his seat," Big Smiley said.
"What kind of a goddamned answer is that?"
"That's the way it was, Chief."
"Where did he take her?"
Rivers of sweat poured from Big Smiley's face. He sighed.
"Downstairs," he said.
"Downstairs! In here?"
"Yas suh. They's stairs in the back room."
"What's downstairs?"
"Just a cellar like any other bar's got. It's full of bottles an' old bar fixtures and beer barrels. The compression unit for the draught beer is down there and the refrigeration unit for the ice boxes. That's all. Some rats and we keeps a cat."
"No bed or bedroom?"
"Naw suh."
"He whipped them down there in that kind of place?"
"I don't know what he done."
"Couldn't you hear them?"
"Naw suh. You can't hear nothin' through this floor. You could shoot off your pistol down there and you couldn't hear it up here."
Grave Digger looked at Ready. "Did you know that?"
Ready began to wilt again. "Naw suh, I swear 'fore–"
"Sit up straight, God damn it! I don't want to have to tell you again."
He turned back to Big Smiley. "Did he know it?"
"Not so far as I know, unless he told him."
"Is Sissie or Sugartit among those girls over there?"
"Naw suh," Big Smiley said without looking.
Grave Digger showed him the pornographic photos.
"Know any of them?"
Big Smiley leafed through them slowly without a change of expression. He pulled out three photos. "I've seen them," he said.
"What're their names?"
"I don't know only two of 'em." He separated them gingerly with his fingertips as though they were coated with external poison. "Them two. This here one is called Good Booty, t'other one is called Honey Bee. This one here, I never heard her name called."
"What are their family names?"
"I don't know none of 'em's square monicker's."
"He took these downstairs?"
"Just them two."
"Who came here with them?"
"They came by theyself, most of 'em did."
"Did he have appointments with them?"
"Naw suh, not with most of 'em, anyway. They just come in here and laid for him."
"Did they come together?"
"Sometime, sometime not."
"You just said they came by themselves."
"I meant they didn't bring no boy friends."
"Had he known them before?"
"I couldn't say. When he come in if he seed any of 'em he just made his choice."
"He knew they hung around here looking for him?"
"Yas suh. When he started comin' here he was already known."
"When was that?"
"Three or four months ago. I don't remember 'zactly."
"When did he start taking them downstairs?"
" 'Bout two months ago."
"Did you suggest it?"
"Naw suh, he propositioned me."
"How much did he pay you?"
"Twenty-five bucks."
"You're talking yourself into Sing-Sing."
"Maybe."
Grave Digger examined the note addressed to GB and signed Bee that he'd taken from the dead man's effects, then passed it over to Big Smiley.
"That came from the pocket of the man you cut," he said. Big Smiley read the note carefully, his lips spelling out each word. His breath came out in a sighing sound.
"Then he must be a relation of her," he said.
"You didn't know that?"
"Naw suh, I swear 'fore God. If I knowed that I wouldn't 'ave chopped him with the axe."
"What exactly did he say to Galen when he started toward him with the knife?"
Big Smiley wrinkled his forehead. "I don't 'member 'zactly. Something 'bout if he found a white mother-raper trying to diddle his little gals he'd cut his throat. But I just took that to mean colored women in general. You know how our folks talk. I didn't figure he meant his own kin."
"Maybe some other girl's father had the same idea with a pistol," Grave Digger suggested.
"Could be," Big Smiley said cautiously.
"So evidently he's the father and he's got more girls than one."
"Looks like it."
"He's dead."
Big Smiley's expression didn't change. "I'm sorry to hear it."
"You look like it. Who went your bail?"
"My boss."
Grave Digger looked at him soberly. "Who's covering for you?" he asked.
"Nobody."
"I know that's a lie but I'm going to pass it. Who was covering for Galen?"
"I don't know."
"I'm going to pass that lie too. What was he doing here tonight?"
"He was looking for Sugartit."
"Did he have a date with her?"
"I don't know. He said she was coming by with Sissie."
"Did they come by after he'd left?"
"Naw suh."
"Okay, Smiley, this one is for keeps. Who is Sugartit's father?"
"I don't know none of 'em's kinfolks nor neither where they lives, Chief, like I told you before. It didn't make no difference."
"You must have some idea."
"Naw suh, it's just like I say, I never thought about it. You don't never think 'bout where a gal lives in Harlem, 'less you goin' home with her. What do anybody's address mean up here?"
"Don't let me catch you in a lie, Smiley."
"I ain't lying, Chief. I went with a woman for a whole year once and never did know where she lived. Didn't care neither."
"Who are the Real Cool Moslems?"
"Them punks! Just a kid gang around here."
"Where do they hang out?"
"I don't know 'zactly. Somewhere down the street."
"Do they come in here?"
"Only three of 'em sometime. Sheik – I think he's they leader – and a boy called Choo-Choo and the one they call Bones."
"Where do they live?"
"Somewhere near here, but I don't know 'zactly. The boy what keeps the pigeons oughtta know. He lives a coupla blocks down the street on t'other side. I don't know his name but he got a pigeon coop on the roof."
"Is he one of 'em?"
"I don't know for sure but you can see a gang of boys on the roof when he's flying his pigeons."
"I'll find him. Do you know the ages of those girls in the booth?"
"Naw suh, when I ask 'em they say they're eighteen."
"You know they're under age."
"I s'pect so but all I can do is ast 'em."
"Did he have any of them?"
"Only one I knows of."
Grave Digger turned and looked at the girls again.
"Which one?" he asked.
"The one in the green tam." Big Smiley pushed forward one of the three photos. "She's this one here, the one called Good Booty."
"Okay, son, that's all for the moment," Grave Digger said.
He got down from the stool and walked forward to talk to the manager.
As soon as he left, without saying a word or giving a warning Big Smiley leaned forward and hit Ready in the face with his big ham-sized fist. Ready sailed off the stool, crashed into the wall and crumpled to the floor.
Grave Digger looked down in time to see his head disappearing beneath the edge of the bar, then turned his attention to the white manager across from him.
"Collect your tabs and shut the bar; I'm closing up this joint and you're under arrest," he said.
"For what?" the manager challenged hotly.
"For contributing to the delinquency of minors."
The manager sputtered, "I'll be open again by tomorrow night."
"Don't say another God damned word," Grave Digger said and kept looking at him until the manager closed his mouth and turned away.
Then he beckoned to one of the white cops on the door and told him, "I'm putting the manager and the bartender under arrest and closing the joint. I want you to hold the manager and some teenagers I'll turn over to you. I'm going to leave in a minute and I'll send back the wagon. I'll take the bartender with me."
"Right, Jones," the cop said, as happy as a kid with a new toy.
Grave Digger walked back to the rear.
Ready was down on the floor on his hands and knees, spitting out blood and teeth.
Grave Digger looked at him and smiled grimly. Then he looked up at Big Smiley who was licking his bruised knuckles with a big red tongue.
"You're under arrest, Smiley," he said. "If you try to escape, I'm going to shoot you through the back of the head."
"Yas suh," Big Smiley said.
Grave Digger shook a customer loose from a plastic-covered chair and sat astride it at the end of the table in the booth, facing the scared, silent teenagers. He took out his notebook and stylo and wrote down their names, addresses, numbers of the public schools they attended, and their ages. The oldest was a boy of seventeen.
None of them admitted knowing either Sissie, Sugartit, the big white man Galen, or anyone connected with the Real Cool Moslems.
He called the second cop away from the door and said, "Hold these kids for the wagon."
Then he said to the girl in the green tam who'd given her name as Gertrude B. Richardson. "Gertrude, I want you to come with me."
One of the girls tittered. "You might have known he'd take Good Booty," she said.
"My name is Beauty," Good Booty said, tossing her head disdainfully.
On sudden impulse Grave Digger stopped her as she was about to get up.
"What's your father's name, Gertrude?"
"Charlie."
"What does he do?"
"He's a porter."
"Is that so? Do you have any sisters?"
"One. She's a year younger than me."
"What does your mother do?"
"I don't know. She don't live with us."
"I see. You two girls live with your father."
"Where else we going to live?"
"That's a good question, Gertrude, but I can't answer it. Did you know a man got his arm cut off in here earlier tonight?"
"I heard about it. So what? People are always getting cut around here."
"This man tried to knife the white man because of his daughters."
"He did?" She giggled. "He was a square."
"No doubt. The bartender chopped off his arm with an axe to protect the white man. What do you think about that?"
She giggled again, nervously. "Maybe he figured the white man was more important than some colored drunk."
"He must have. The man died in Harlem Hospital less than an hour ago."
Her eyes got big and frightened. "What are you trying to say, mister?"
"I'm trying to tell you that he was your father."
Grave Digger hadn't anticipated her reaction. She came up out of her seat so fast that she was past him before he could grab her.
"Stop her!" he shouted.
A customer wheeled from his bar stool into her path and she stuck her fingers into his eye. The man yelped and tried to hold her. She wrenched from his grip and sprang towards the door. The white cop headed her off and wrapped his arms about her. She twisted in his grip like a panic-stricken cat and clawed at his pistol. She had gotten it out the holster when a colored man rushed in and wrenched it from her grip. The white cop threw her onto the floor on her back and straddled her, pinning down her arms. The colored man grabbed her by the feet. She writhed on her back and spat into the cop's face.
Grave Digger came up and looked down at her from sad brown eyes. "It's too late now, Gertrude," he said. "They're both dead."
Suddenly she began to cry. "What did he have to mess in it for?" she sobbed. "Oh, Pa, what did you have to mess in it for?"
# 14
Two uniformed white cops standing guard on a dark roof-top were talking.
"Do you think we'll find him?"
"Do I think we'll find him? Do you know who we're looking for? Have you stopped to think for a moment that we're looking for one colored man who supposedly is handcuffed and seven other colored men who were wearing green turbans and false beards when last seen. Have you turned that over in your mind? By this time they've got rid of those phony disguises and maybe Pickens has got rid of his handcuffs too. And then what does that make them, I ask you? That makes them just like eighteen thousand or one hundred and eighty thousand other colored men, all looking alike. Have you ever stopped to think there are five hundred thousand colored people in Harlem – one half of a million people with black skin. All looking alike. And we're trying to pick eight out of them. It's like trying to find a cinder in a coal bin. It ain't possible."
"Do you think all these colored people in this neighbourhood know who Pickens and the Moslems are?"
"Sure they know. Every last one of them. Unless some other colored person turns Pickens in he'll never be found. They're laughing at us."
"As much as the chief wants that coon, whoever finds him is sure to get a promotion," the first cop said.
"Yeah, I know, but it ain't possible," the second cop said. "If that coon's got any sense at all he would have filed those cuffs in two a long time ago."
"What good would that do him if he couldn't get them off?"
"Hell, he could wear heavy gloves with gauntlets like—Hey! Didn't we see some coon wearing driving gauntlets?"
"Yeah, that halfwit coon with the pigeons."
"Wearing gauntlets and a ragged old overcoat. And a coal black coon at that. He certainly fits the description."
"That halfwitted coon. You think it's possible he's the one?"
"Come on! What are we waiting for?"
Sheik said, "Now all we've got to do is get this mother-raper past the police lines and throw him into the river."
"Doan do that to me, please, Sheik," Sonny's muffled voice pleaded from inside the sack.
"Shhhh," Choo-Choo cautioned. "Chalk the walking Jeffs."
The two cops leaned over and peered in through the open window.
"Where's that boy who was wearing gloves?" the first cop asked.
"Gloves!" Choo-Choo echoed, going into his clowning act like a chameleon changing color. "You means boxing gloves?"
The second cop sniffed. "A weed pad!" he exclaimed.
They climbed inside. Their gazes swept quickly over the room.
The roof reeked of marijuana smoke. Everyone was high. The ones who hadn't smoked were high from inhaling the smoke and watching the eccentric motions of the ones who had smoked.
"Who's got the sticks?" the first cop demanded.
"Come on, come on, who's got the sticks?" the second cop echoed, looking from one to the other. He passed over Sheik who stood in the center of the floor where he'd been arrested in motion by Choo-Choo's warning and stared at them as though trying to make out what they were; then over Inky who was caught in the act of ducking behind the curtains in the corner and stood there half in and half out, like a billboard advertisement for a movie about bad girls; and landed on Choo-Choo who seemed the most vulnerable because he was grinning like an idiot. "You got the sticks, boy?"
"Sticks! You mean that there pigeon stick," Choo-Choo said, pointing at the bamboo pole on the floor beside the bed.
"Don't get funny with me, boy!"
"I just don't know what you means, boss."
"Forget the sticks," the first cop said. "Let's find the boy with the gloves."
He looked about. His gaze lit on Sugartit who was sitting in the straight-backed chair and staring with a fixed expression at what appeared to be a gunny sack filled with huge lumps of coal lying in the middle of the bed.
"What's in that sack?" he asked suspiciously.
For an instant no one replied.
Then Choo-Choo said, "Just some coal."
"On the bed?"
"It's clean coal."
The cop pinned a threatening look on him.
"It's my bed," Sheik said. "I can put what I want on it."
Both cops turned to stare at him.
"You're a kind of lippy bastard," the first cop said. "What's your name?"
"Samson."
"You live here?"
"Right here."
"Then you're the boy we're looking for. That's your pigeon loft on the roof."
"No, that's not him," the second cop said. "The boy we want is blacker than he is and has another name."
"What's a name to these coons?" the first cop said. "They're always changing about."
"No, the one we want is called Inky. He was the one wearing the gloves."
"Now I remember. He was called Caleb. He was the one wearing the gloves. The other one was Inky, the one who couldn't talk."
The second cop wheeled on Sheik. "Where's Caleb?"
"I don't know anybody named Caleb."
"The hell you don't! He lives here with you."
"Naw suh, you means that boy what lives down on the first floor," Choo-Choo said.
"Don't tell me what I mean. I mean the boy who lives here on this floor. He's the boy who's got the pigeon loft."
"Naw suh, boss, if you means the Caleb what's got the pigeon roost, he lives on the first floor."
"Don't lie to me, boy. I saw the sergeant bring him down the fire escape to this floor."
"Naw suh, boss, the sergeant taken him on by this floor and carried him down on the fire escape to the first floor. We seen 'em when they come by the window. Didn't we, Amos?" he called to Inky.
"That's right, suh," Inky said. "They went right past that window there."
"What other window could they go by?"
"None other window, suh."
"They had another boy with 'em called Inky," Choo-Choo said. "It looked like they had 'em both arrested."
The second cop was staring at Inky. "This boy here looks like Inky to me," he said. "Aren't you Inky, boy?"
"Naw suh–" Inky began, but Choo-Choo quickly cut him off: "They calls him Smokey. Inky is the other one."
"Let him talk for himself," the first said.
The second cop pinned another threatening look on Choo-Choo. "Are you trying to make a fool out of me, boy!"
"Naw suh, boss, I'se just tryin' a help."
"Let up on him," the first cop said. "These coons are jagged on weed; they're not strictly responsible."
"Responsible or not, they'd better be careful before they get some lumps on their heads."
The first cop noticed Sissie standing quietly in the corner, holding her hand to her bruised cheek.
"You know them, Caleb and Inky, don't you girl?" he asked her.
"No sir, I just know Smokey," she said.
Suddenly Sonny sneezed.
Sugartit giggled.
The cop wheeled toward the bed, looked at the sack and then looked at her.
"Who was that sneezed?"
She put her hand to her mouth and tried to stop laughing.
The cop turned slightly pinkish and drew his pistol.
"Someone's underneath the bed," he said. "Keep the other covered while I look."
The second cop drew his pistol.
"Just relax and no one will get hurt," he said calmly.
The first cop got down on his hands and knees, holding his cocked pistol ready to shoot, and looked underneath the bed.
Sugartit put both hands over her mouth and bit into her palm. Her face swelled with suppressed laughter and tears flowed down her cheeks.
The cop straightened to his knees and braced himself on the edge of the bed. There was a perplexed look on his red face.
"There's something funny going on here," he said. "There's someone else in this room."
"Ain't nobody here but us ghosts, boss," Choo-Choo said.
The cop threw him a look of frustrated fury, and started to his feet.
"By God, I'll–" His voice dried up when he heard the choking sounds issuing from inside the sack.
He jumped upward and backward as though one of the ghosts had sure enough groaned. Leveling his pistol, he said in a quaking voice, "What's in that sack?"
Sugartit burst into hysterical laughter.
For an instant no one spoke.
Then Choo-Choo said hastily, "Hit's just Joe."
"What!"
"Hit's just Joe in the sack."
"Joe!"
Gingerly, the cop leaned over, holding his cocked pistol in his right hand, and with his left untied the cord closing the sack. He drew the top of the sack open.
Popping eyes in a gray-black face stared up at him.
The cop drew back in horror. His face turned white and a shudder passed over his big solid frame.
"It's a body," he said in a choked voice. "All trussed up."
"Hit ain't no body, hit's just Joe," Choo-Choo said, not intending to play the comic.
The second cop hastened over to look. "It's still alive," he said.
"He's choking!" Sissie cried and ran over and began loosening the noose about Sonny's neck.
Sonny sucked in breath with a gasp.
"My God, what's he doing in there?" the first cop asked in amazement.
"He's just studying magic," Choo-Choo said. He was beginning to sweat from the strain.
"Magic!"
The second cop noticed Sheik inching toward the window and aimed his pistol at him.
"Oh no, you don't," he said. "You come over here."
Sheik turned and came closer.
"Studying magic!" the first cop said. "In a sack?"
"Yas suh, he's trying to learn how to get out, like Houdini."
Color flooded back into the cop's face. "I ought to take him in for indecent exposure," he said.
"Hell, he's wearing a sack, ain't he," the second cop said, amused by his own wit.
Both of them grinned at Sonny as though he were a harmless halfwit.
Then the second cop said suddenly, "It ain't possible! There can't be two such halfwits in the whole world."
The first cop looked closely at Sonny and said slowly, "I believe you're right." Then to the others at large, "Get that boy out of that sack."
Sheik didn't move, but Choo-Choo and Inky hastened over and pulled Sonny out while Sissie held the bottom of the sack.
The cops stared at Sonny in awe.
"Looks like barbecued coon, don't he?" the first cop said.
Sugartit burst into laughter again.
Sonny's black skin had a gray pallor as though he'd been dusted over lightly with wood ash. He was shaking like a leaf.
The second cop reached out and turned him around.
Everyone stared at the handcuff bracelets clamped about each wrist.
"That's our boy," the first cop said.
"Lawd, suh, I wish I'd gone home and gone to bed," Sonny said in a moaning voice.
"I'll bet you do," the cop said.
Sugartit couldn't stop laughing.
# 15
The bodies had been taken to the morgue. All that remained were chalk outlines on the pavement where they had lain.
The street had been cleared of private cars. Police tow trucks had carried away those that had been abandoned in the middle of the street. Most of the patrol cars had returned to duty; those remaining blocked the area.
The chief of police's car occupied the center of the stage. It was parked in the middle of the intersection of 127th Street and Lenox Avenue.
To one side of it, the chief, Lieutenant Anderson, the lieutenant from homicide and the precinct sergeant who'd led one of the search parties were grouped about the boy called Bones.
The lieutenant from homicide had a zip gun in his hand.
"All right then, it isn't yours," he said to Bones in a voice of tried patience. "Whose is it then? Who were you hiding it for?"
Bones stole a glance at the lieutenant's face and his gaze dropped quickly to the street. It crawled over the four pairs of big black copper's boots. They looked like the Sixth Fleet at anchor. He didn't answer.
He was a slim black boy of medium height with girlish features and short hair almost straight at the roots and parted on one side. He wore a natty topcoat over his sweat shirt and tight-fitting black pants above shiny tan pointed-toed shoes.
An elderly man, a head taller, with a face grizzled from hard outdoor work, stood beside him. Kinky hair grew like burdock weeds on his shiny black dome, and worried brown eyes looked down at Bones from behind steel-rimmed spectacles.
"Go 'head, tell 'em, so, don't be no fool," he said; then he looked up and saw Grave Digger approaching with his prisoners. "Here comes Digger Jones," he said. "You can tell him, cain't you?"
Everybody looked about.
Grave Digger held Good Booty by the arm and Big Smiley and Ready Belcher, handcuffed together, were walking in front of him.
He looked at Anderson and said, "I closed up the Dew Drop Inn. The manager and some juvenile delinquents are being held by the officers on duty. You'd better send a wagon up there."
Anderson whistled for a patrol car team and gave them the order.
"What did you find out on Galen?" the chief asked.
"I found out he was a pervert," Grave Digger said.
"It figures," the homicide lieutenant said.
The chief turned red. "I don't give a goddamn what he was," he said. "Have you found out who killed him?"
"No, right now I'm still guessing at it," Grave Digger said.
"Well, guess fast then. I'm getting goddamned tired of standing up here watching this comedy of errors."
"I'll give you a quick fill-in and let you guess too," Grave Digger said.
"Well, make it short and sweet and I damn sure ain't going to guess," the chief said.
"Listen, Digger," the colored civilian interposed. "You and me is both city workers. Tell 'em my boy ain't done no harm."
"He's broken the Sullivan law concerning concealed weapons by having this gun in his possession," the homicide lieutenant said.
"That little thing," Bones's father said scornfully. "I don't b'lieve that'll even shoot."
"Get these people away from here and let Jones report," the chief said testily.
"Well, do something with them, Sergeant," Lieutenant Anderson said.
"Come on, both of you," the sergeant said, taking the man by the arm.
"Digger–" the man appealed.
"It'll keep," Grave Digger said harshly. "Your boy belonged to the Moslem gang."
"Naw-naw, Digger–"
"Do I have to slug you," the sergeant said.
The man allowed himself to be taken along with his son across the street.
The sergeant turned them over to a corporal and hurried back. Before he'd gone three steps the corporal was summoning two cops to take charge of them.
"What kind of city work does he do?" the chief asked.
"He's in the sanitation department," the sergeant said. "He's a garbage collector."
"All right, get on Jones," the chief ordered.
"Galen picked up colored school girls, teenagers, and took them to a crib on 145th Street," Grave Digger said in a flat toneless voice.
"Did you close it?" the Chief asked.
"It'll keep; I'm looking for a murderer now," Grave Digger said. Taking the miniature bull whip from his pocket, he went on, "He whipped them with this."
The chief reached out silently and took it from his hand.
"Have you got a list of the girls, Jones?" he asked.
"What for?"
"There might be a connection."
"I'm coming to that–"
"Well, get to it then."
"The landprop, a woman named Reba – used to call herself Sheba – the one who testified against Captain Murphy–"
"Ah, that one," the chief said softly. "She won't slip out of this."
"She'll take somebody with her," Grave Digger warned. "She's covered and Galen was, too."
The chief looked at Lieutenant Anderson reflectively.
The silence ran on until the sergeant blurted, "That's not in this precinct."
Anderson looked at the sergeant. "No one's charging you with it."
"Get on, Jones," the chief said.
"Reba got scared of the deal and barred him. Her story will be that she barred him when she found out what he was doing. But that's neither here nor there. After she barred him Galen started meeting them in the Dew Drop Inn. He arranged with the bartender so he could whip them in the cellar."
Everyone except Grave Digger seemed embarrassed.
"He ran into a girl named Sissie," Grave Digger said. "How doesn't matter at the moment. She's the girl friend of a boy called Sheik, who is the leader of the Real Cool Moslems."
Sudden tension took hold of the group.
"Sheik sold Sissie to him. Then Galen wanted Sissie's girl friend Sugartit. Sheik couldn't get Sugartit, but Galen kept looking for her in the neighbourhood. I have the bartender here and a two-bit pimp who has a girl at Reba's. He steered for Galen. I got this much from them."
The officers stared appraisingly at the two handcuffed prisoners.
"If they know that much, they know who killed him," the chief said.
"It's going to be their asses if they do," Grave Digger said. "But I think they're leveling. The way I figure it, the whole thing hinges on Sugartit. I think he was killed because of her."
"By who?"
"That's the jackpot question."
The chief looked at Good Booty. "Is this girl Sugartit?"
The others stared at her, too.
"No, she's another one."
"Who is Sugartit then?"
"I haven't found out yet. This girl knows but she doesn't want to tell."
"Make her tell."
"How?"
The chief appeared to be embarrassed by the question. "Well, what the hell do you want with her if you can't make her talk?" he growled.
"I think she'll talk when we get close enough. The Moslem gang hangs out somewhere near here. The bartender here thinks it might be in the flat of a boy who has a pigeon loft."
"I know where that is!" the sergeant exclaimed. "I searched there."
Everyone, including the prisoners, stared at him. His face reddened. "Now I remember," he said. "There were several boys in the flat. The boy who kept pigeons, Caleb Bowee is his name, lives there with his Grandma; and two of the others roomed there."
"Why the hell didn't you bring them in?" the chief asked.
"I didn't find anything on them to connect them with the Moslem gang or the escaped prisoner," the sergeant said, defending himself. "The boy with the pigeons is a halfwit–he's harmless, and I'm sure the grandma wouldn't put up with a gang in there."
"How in the hell do you know he's harmless?" the chief stormed. "Half the murderers in Sing-Sing look like you and me."
The homicide lieutenant and Anderson exchanged smiles.
"They had two girls with them and–" the sergeant began to explain but the chief wouldn't let him.
"Why in the hell didn't you bring them in, too?"
"What were the girls' names?" Grave Digger asked.
"One was called Sissieratta and–"
"That must be Sissie," Grave Digger said. "It fits. One was Sissie and the other was Sugartit. And one of the boys was Sheik." Turning to Big Smiley, he asked, "What does Sheik look like?"
"Freckle-faced boy the color of a bay horse, with yellow cat eyes," Big Smiley said impassively.
"You're right," the sergeant admitted sheepishly. "He was one of them. I should have trusted my instinct; I started to haul that punk in."
"Well, for God's sake, get the lead out of your ass now," the chief roared. "If you still want to work for the police department."
"Well, Jesus Christ, the other girl, the one Jones calls Sugartit, was Ed Johnson's daughter," the sergeant exploded. "She had one of those souvenir police ID cards signed by yourself and I thought–"
He was interrupted by the flat whacking sound of metal striking against a human skull.
No one had seen Grave Digger move.
What they saw now was Ready Belcher sagging forward with his eyes rolled back into his head and a white cut – not yet beginning to bleed – two inches wide in the black pockmarked skin of his forehead. Big Smiley reared back on the other end of the handcuffs like a dray horse shying from a rattlesnake.
Grave Digger gripped his nickel-plated thirty-eight by the long barrel, making a club out of the butt. The muscles were corded in his rage-swollen neck and his face was distorted with violence. Looking at him, the others were suspended in motion as though turned to stone.
"Stop him, God damn it!" the chief roared. "He'll kill them."
The sculptured figures of the police officers came to life. The sergeant grabbed Grave Digger from behind in a bear hug. Grave Digger doubled over and sent the sergeant flying over his head toward the chief, who ducked in turn and let the sergeant sail on by.
Lieutenant Anderson and the homicide lieutenant converged on Grave Digger from opposite directions. Each grabbed an arm while he was still in a crouch and lifted upward and backward.
Ready was lying prone on the pavement, blood trickling from the dent in his skull, a slack arm drawn tight by the handcuffs attached to Big Smiley's wrist. He looked dead already.
Big Smiley gave the appearance of a terrified blind beggar caught in a bombing raid; his giant frame trembled from head to foot.
Grave Digger had just time enough to kick Ready in the face before the officers jerked him out of range.
"Get him to the hospital, quick!" the chief shouted; and in the next breath added, "Rap him on the head!"
Grave Digger had carried the lieutenants to the ground and it was more than either could to do to follow the chief's command.
The sergeant had already picked himself up and at the chief's order set off at a gallop.
"God damn it, phone for it, don't run after it!" the chief yelled. "Where the hell is my chauffeur, anyway?"
Cops came running from all directions.
"Give the lieutenants a hand," the chief said. "They've got a wild man."
Four cops jumped into the fray. Finally they pinned Grave Digger to the ground.
The sergeant climbed into the chief's car and began talking into the telephone.
Coffin Ed appeared suddenly. No one had noticed him approaching from his parked car down the street.
"Great God, what's happening, Digger?" he exclaimed.
Everybody was quiet, their embarrassment noticeable.
"What the hell!" he said, looking from one to the other. "What the hell's going on."
Grave Digger's muscles relaxed as though he'd lost consciousness.
"It's just me, Ed," he said, looking up from the ground at his friend. "I just lost my head, is all."
"Let him go," Anderson ordered his helpers. "He's back to normal now."
The cops released Grave Digger and he got to his feet.
"Cooled off now?" the homicide lieutenant asked.
"Yeah. Give me my gun," Grave Digger said.
Coffin Ed looked down at Ready Belcher's bloody head.
"You too, eh, partner," he said. "What did this rebel do?"
"I told him if I caught him holding out on me I'd kill him."
"You told him no lie," Coffin Ed said. Then asked, "Is it that bad?"
"It's dirty, Ed. Galen was a rotten son of a bitch."
"That doesn't surprise me. Have you got anything on it so far?"
"A little, not much."
"What the hell do you want here?" the chief said testily. "I suppose you want to help your buddy beat up some more of your folks."
Grave Digger knew the chief was trying to steer the conversation away from Coffin Ed's daughter, but he didn't know how to help him.
"You two men act as if you want to kill off the whole population of Harlem," the chief kept on.
"You told me to crack down," Grave Digger reminded him.
"Yeah, but I didn't mean in front of my eyes where I would have to be a witness to it."
"It's our beat," Coffin Ed spoke up for his friend. "If you don't like the way we handle it why don't you take us off."
"You're already off," the chief said. "What in the hell did you come back for, anyway?"
"Strictly on private business."
The chief snorted.
"My little daughter hasn't come home and I'm worried about her," Coffin Ed explained. "It's not like her to stay out this late and not let us know where she is."
The chief looked away to hide his embarrassment.
Grave Digger swallowed audibly.
"Hell, Ed, you don't have to worry about Eve," he said in what he hoped was a reassuring tone of voice. "She'll be home soon. You know nothing can happen to her. She's got that police ID card you got for her on her last birthday, hasn't she?"
"I know, but she always phones her mother if she's going to stay out."
"While you're out here looking for her she's probably gone home. Why don't you go back home and go to bed? She'll be all right."
"Jones is telling you right, Ed," the chief said brusquely. "Go home and relax. You're off duty and you're in our way here. Nothing is going to happen to your daughter. You're just having nightmares."
A siren sounded in the distance.
"Here comes the ambulance," Lieutenant Anderson said.
"I'll go and phone home again," Coffin Ed said. "Take it easy, Digger. Don't get yourself docked, too."
As he turned and started off a fusillade of shots sounded from the upper floor of some nearby tenement. Ten shots from regulation .38 police specials were fired so fast that by the time the sounds had reached the street they were chained together.
Every cop within earshot froze to alert attention. They strained their ears in almost superhuman effort to place the direction from which the shots had come. Their eyes scanned the fronts of the tenements until not a spot escaped their observation.
But no more shots were fired.
The only signs of life left were the lights going out. With the rapidity of gun shots, one light after another went out until only one lighted window remained in the whole block of darkened dingy buildings. It was behind a fire-escape landing on the top floor of the tenement half a block up the street.
All eyes focused on that spot.
The grotesque silhouette of something crawling over the window sill appeared in the glare of light. Slowly it straightened and took the shape of a short, husky man. It staggered slowly along the three feet of grilled iron footing and leaned against the low outer rail. For a moment it swayed back and forth in a macabre pantomime and then, slowly, like a roulette ball climbing the last hurdle before the final slot, it fell over the railing, turned in the air, missed the second landing by a breath. The body turned again and struck the third railing and started to spin faster. It landed with a resounding thud on top of a parked car and lay there with one hand hanging down beside the driver's window as though signaling for a stop.
"Well, God damn it, get going!" the chief shouted in stentorian tone. Then, on second thought, he added, "Not you, Jones. Not you!" and ran toward his car to get his megaphone.
Already motion had broken out. Cops were heading toward the tenement like the Marines landing.
The two cops guarding the entrance ran out into the street to locate the scene of the disturbance.
The chief grabbed his megaphone and shouted, "Get the lights on that building."
Two spotlights that had been extinguished were turned back on immediately and beamed on the tenement's top floor.
A patrolman stepped from the window onto the fire-escape landing and raised his hands in the light.
"Hold it, everybody!" he shouted. "I want the chief! Is the chief there?"
"Lower the lights," the chief megaphoned. "I'm here. What is it?"
"Send for an ambulance. Petersen is shot–"
"An ambulance is coming."
"Yes sir, but don't let anybody in here yet–"
Grave Digger took hold of Coffin Ed's arm.
"Hang on tight, Ed," he said. "Your daughter's up there."
He felt Coffin Ed's muscles tighten beneath his grip as the cop went on, "We found Pickens but one of the Moslem gangsters grabbed Pete's pistol and shot him. He used his buddy as a shield and I got his buddy but he snatched one of the girls here and escaped into the back room. He's locked himself in there and there's no other way out of this shotgun shack. He says the girl is Detective Ed Johnson's daughter. He threatened to cut her throat if he can't talk to you and Grave Digger Jones. Whatcha want me to do?"
The ambulance approached and the chief had to wait until the siren had died away to make himself heard.
"Has he still got Petersen's pistol?"
"Yes, sir, but he emptied it."
"All right, Officer, sit pat," the chief megaphoned. "We'll get Petersen down the fire escape and I'll go up and see what it's all about."
Coffin Ed's acid-burnt face was hideous with fear.
# 16
"You stay down here, Johnson," the chief ordered. "I'll take Anderson and Jones."
"Not unless you shoot me," Coffin Ed said.
The chief looked at him.
"Let him come," Grave Digger said.
"I ought to come too; I know the flat," the sergeant said.
"It's my job to come," the lieutenant from homicide said.
"Who the hell's running this police department," the chief said.
"We haven't got any time," Grave Digger replied.
All of them went quickly and quietly as possible. No one spoke again until the chief said through the kitchen door, "All right, I'm the chief. Come out and give yourself up and you won't get hurt."
"How do I know you're the chief?" asked a fuzzy voice from within.
"If you open the door and come out you'll see."
"Don't get so mother-raping smart. You're the chief, but I'm the Sheik."
"Well, all right, you're a big-shot gang boss. What do you want?"
"Keep him talking," Coffin Ed whispered. "I'm going up on the roof."
"Who's that with you?" Sheik asked sharply.
Grave Digger pointed to the sergeant and Lieutenant Anderson.
"The precinct lieutenant and a sergeant," the chief said.
"Where's Grave Digger?"
"He's not here yet. I had to send for him."
"Send those other mother-rapers away. Let's you and me settle this, the Sheik and the Chief."
"How will you know if they're gone if you're scared to come out and look?"
"Let 'em stay then. I don't give a good goddam. And don't think I'm scared. I don't need to take any chances. I got Coffin Ed's daughter by the hair with my left hand and I'm holding a razor-edged butcher knife against her throat with my right hand. If you try to take me I'll cut her mother-raping head off before you can get through the door."
"All right, Sheik, you got us by the short hair, but you know you can't get away. Why don't you come out peaceably and give yourself up like a man. I give you my word that no one will abuse you. The officer you shot ain't seriously hurt. There's no other charge against you. You ought to get off with five years. With time off for good behavior, you'll be back in the big town in three years. Why risk sudden death or the hot seat just for a moment of playing the big shot?"
"Don't hand me that mother-raping crap. You'll hang a kidnapping charge on me for snatching your prisoner."
"What the hell! You can keep him. We don't want him anymore. We found out he didn't kill the man. All he had was a blank pistol."
"So he didn't kill the man?"
"No."
"Who killed him?"
"We don't know yet."
"So you don't know who killed the big Greek, do you?"
"All right, all right, what's that to you? What do you want to get mixed up in something that doesn't concern you?"
"You're one of those smart mother-rapers, ain't you? You're going to be so smart you're going to make me cut her mother-raping throat just to show you."
"Please don't argue with him, Mr. Chief, please," said a small scared voice from within. "He'll kill me. I know he will."
"Shut up!" Sheik said roughly. "I don't need you to tell 'im I'm going to kill you."
Beads of sweat formed on the ridge of the chief's red nose and about the blue bags beneath his eyes.
"Why don't you be a man," he urged, filling his voice with contempt. "Don't be a mad dog like Vincent Coll. Be a man like Dillinger was. You won't get much. Three years and no more. Don't hide behind an innocent little girl."
"Who the hell do you think you're kidding with that stale crap. This is the Sheik. Can't no dumb cop like you make a fool out of the Sheik. You got the chair waiting for me and you think you're going to kid me into walking out there and sitting in it."
"Don't play yourself too big, punk," the chief said, losing his temper for a moment. "You shot an officer but you didn't kill him. You snatched a prisoner but we don't want him. Now you want to take it out on a little girl who can't defend herself. And you call yourself the Sheik, the big gang leader. You're just a cheap tinhorn punk, yellow to the core."
"Keep on, just keep on. You ain't kidding me with that mother-raping sucker bait. You know it was me who killed him. You've had me tabbed ever since you found out that nigger was shooting blanks."
"What!" The chief was startled. Forgetting himself, he asked Grave Digger, "What the hell's he talking about?"
"Galen." Grave Digger formed the word with his lips.
"Galen!" the chief exclaimed. "You're trying to tell me you killed the white man, you chicken-livered punk?" he roared.
"Keep on, just keep on. You know damn well it was me lowered the boom on the big Greek." He sounded as though he bitterly resented an oversight. "Who do you think you're kidding? You're talking to the Sheik. You think 'cause I'm colored I'm dumb enough to fall for that rock-a-bye-baby crap you're putting down."
The chief had to readjust his train of thought.
"So it was you who killed Galen?"
"He was just the Greek to me," Sheik said scornfully. "Just another gray sucker up here trying to get his kicks. Yeah, I killed him." There was pride in his voice.
"Yeah, it figures," the chief said thoughtfully. "You saw him running down the street and you took advantage of that and shot him in the back. Just what a yellow son of a bitch like you would do. You were probably laying for him and were scared to go out and face him like a man."
"I wasn't laying for the mother-raper no such goddam thing," Sheik said. "I didn't even know he was anywhere about."
"You were nursing a grudge against him."
"I didn't have nothing against the mother-raper. You must be having pipedreams. He was just another gray sucker to me."
"Then why the hell did you shoot him?"
"I was just trying out my new zip gun. I saw the mother-raper running by where I was standing so I just blasted at him to see how good my gun would shoot."
"You God damned little rat," the chief said, but there was more sorrow in his voice than anger. "You sick little bastard. What the God damned hell can be done with somebody like you?"
"I just want you to quit trying to kid me, 'cause I'd just as soon cut this girl's throat right now as not."
"All right, _Mister_ Sheik," the chief said in a cold, quiet voice. "What do you want me to do?"
"Is Grave Digger come yet?"
Grave Digger nodded.
"Yeah, he's here, _Mister_ Sheik."
"Let him say something then, and you better can that mister crap."
"Eve, this is me, Digger Jones," Grave Digger said, spurning Sheik.
"Answer him," Sheik said.
"Yes, Mr. Jones," she said in a voice so weightless it floated out to the tense group listening like quivering eiderdown.
"Is Sissie in there with you?"
"No, sir, just Granny Bowee and she's sitting in her chair asleep."
"Where's Sissie?"
"She and Inky are in the front room."
"Has he hurt you?"
"Quit stalling," Sheik said dangerously. "I'm going to give you until I count to three."
"Please, Mr. Jones, do what he says. He's going to kill me if you don't."
"Don't worry child, we're going to do what he says," he reassured her and then said, "What do you want, boy?"
"These are my terms: I want the street cleared of cops; all the police blockades moved–"
"What the hell!" the chief exploded.
"We'll do it," Grave Digger said.
"I want to hear the chief say it," Sheik demanded.
"I'll be damned if I will," the chief said.
"Please," came a tiny voice no bigger than a prayer.
"What if she was your daughter," Grave Digger said.
"I'm going to give you until I count three," Sheik said.
"All right, I'll do it," the chief said, sweating blood.
"On your word of honor as a great white man," Sheik persisted.
The chief's red sweating face drained of color.
"All right, all right, on my word of honor," he said.
"Then I want an ambulance driven up to the door downstairs. I want all its doors left open so I can see inside, the back doors and both the side doors, and I want the motor left running."
"All right, all right, what else? The Statue of Liberty?"
"I want this house cleared—"
"All right, all right, I said I'd do that."
"I don't want any mother-raping alarm put out. I don't want anybody to try to stop me. If anybody messes with me before I get away you're going to have a dead girl to bury. I'll put her out somewhere safe when I get clear away, clear out of the state."
"Don't cross him," Grave Digger whispered tensely. "He's teaed to the eyes."
"All right, all right," the chief said. "We'll give you safe passage. If you don't hurt the girl. If you hurt her we won't kill you, but you'll beg us to. Now take five minutes and come out and we'll let you drive away."
"Who do you think you're kidding?" Sheik said. "I ain't that big a fool. I want Grave Digger to come inside of here and put his pistol down on the table, then I'm going to come out."
"You're crazy if you think we're going to give you a pistol," the chief roared.
"Then I'm going to kill her now."
"I'll give it to you," Grave Digger said.
"You're under suspension as of now," the chief said.
"All right," Grave Digger said: then to Sheik, "What do you want me to do?"
"I want you to stand outside the door with the pistol held by the barrel. When I open the door I want you to stick it forward and walk into the room so's the first thing I see is the butt. Then I want you to walk straight ahead and put it on the kitchen table. You got that?"
"Yeah, I got it."
"The rest of you mother-rapers get downstairs," Sheik said.
The two lieutenants and the sergeant looked at the chief for orders.
"All right, Jones, it's your show," the chief said, adding on second thought, "I wish you luck."
He turned and started down the stairs.
The others hesitated. Grave Digger motioned violently for them to leave too. Reluctantly they followed the chief.
It was silent in the kitchen until the sound of the officers' receding footsteps diminished into silence below.
Grave Digger stood facing the kitchen door, holding the pistol as instructed. Sweat poured down his lumpy cordovan-colored face and collected in the collar about his neck.
Finally the sound of movement came from the kitchen. The bolt of the Yale lock clicked open, a hand bolt was pulled back with a grating snap, a chain was unfastened. The door swung slowly inward.
Only Granny was visible from the doorway. She sat bolt upright in the immobile rocking chair with her hands gripping the arms and her old milky eyes wide open and staring at Grave Digger with a fixed look of fierce disapproval.
Sheik spoke from behind the door, "Turn the butt this way so I can see if it's loaded."
Without looking around, Grave Digger turned the pistol so that Sheik could see the shells in the chambers of the cylinder.
"Go ahead, keep walking," Sheik ordered.
Still without looking around, Grave Digger moved slowly across the room. When he came to the table he looked swiftly toward the small window at the far end of the back wall. It was on the other side of an old-fashioned homemade cupboard which partially blocked the view of the kitchen from the outside, so that only the section between the table and the side wall was visible.
He saw what he was looking for. He leaned slowly forward and placed the pistol on the far side of the table.
"There," he said.
Raising his hands high above his head, he turned slowly away from the table and faced the back wall. He stood so that Sheik had to either pass in front of him to reach the pistol or go around on the other side of the table.
Sheik kicked the door shut, revealing himself and Sugartit, but Grave Digger didn't turn his head or even move his eyes to look at them.
Sheik gripped Sugartit's pony tail tightly in his left hand, pulling her head back hard to make her slender brown throat taut beneath the blade of the butcher knife. They began a slow shuffling walk, like a weird Apache dance in a Montmartre night club.
Sugartit's eyes had the huge liquid look of a dying doe's, and her small brown face looked as fragile as toasted meringue. Her upper lip was sweating copiously.
Sheik kept his gaze pinned on Grave Digger's back while slowly skirting the opposite walls of the room and approaching the table from the far side. When he came within reach of the pistol he released his hold on Sugartit's pony tail, pressed the knife blade tighter against her throat and reached out with his left hand for the pistol.
Coffin Ed was hanging head downward from the roof, only his head and shoulders visible below the top edge of the kitchen window. He had been hanging there for twenty minutes waiting for Sheik to come into view. He took careful aim at a spot just above Sheik's left ear.
Some sixth sense caused Sheik to jerk his head around at the exact instant Coffin Ed fired.
A third eye, small and black and sightless, appeared suddenly in the exact center of Sheik's forehead between his two startled yellow cat's eyes.
The high-powered bullet had cut only a small round hole in the window glass, but the sound of the shot shattered the whole pane and blasted a shower of glass into the room.
Grave Digger wheeled to catch the fainting girl as the knife clattered harmlessly onto the table top.
Sheik was dead when he started going down. He landed crumpled up beside Granny's immobile rocking chair.
The room was full of cops.
"That was too much of a risk, too much of a risk," Lieutenant Anderson said, shaking his head, a dazed expression of his face.
"What isn't risky on this job?" the chief said authoritatively. "We cops got to take risks."
No one disputed him.
"This is a violent city," he added belligerently.
"There wasn't that much risk," Coffin Ed said. He had his arm about his daughter's trembling shoulders. "They don't have any reflexes when you shoot them in the head."
Sugartit winced.
"Take Eve and go home," Grave Digger said harshly.
"I guess I'd better," Coffin Ed said, limping painfully as he guided Sugartit gently toward the door.
"Geez," a young patrol-car rookie was saying. "Geez. He hung there all that time on just some wire tied around his ankles. I don't know how he stood the pain."
"You'd've stood it too if she was your daughter," Grave Digger said.
"Forget what I said to you about being under suspension, Jones," the chief said.
"I didn't hear you," Grave Digger said.
"Jesus Christ, look at that!" the sergeant exclaimed in amazement. "All that noise and Grandma's still sleeping."
Everybody turned and looked at him. They were solemn for a moment.
"Nothing's ever going to wake her up again," the lieutenant from homicide said. "She must have been dead for hours."
"All right, all right, all right," the chief shouted. "Let's clean up here and get away. We've got this case tied up tighter than Dick's hatband." Then he added in a pleased tone of voice, "That wasn't too difficult, was it?"
# 17
It was eleven o'clock the next morning.
Inky and Bones had spilled their guts.
It had gone hard for them and when the cops got through with them they were as knotty as fat pine.
The remaining members of the Real Cool Moslems–Camel Mouth, Beau Baby, Punkin Head and Slow Motion – had been rounded up, questioned and were now being held along with Inky and Bones.
Their statements had been practically identical:
They had been standing on the corner of 127th Street and Lenox Avenue.
_Q_. What for?
_A_. Just having a dress rehearsal.
_Q_. What? Dress rehearsal?
_A_. Yas suh. Like they do on Broadway. We was practicing wearing our new A-rab costumes.
_Q_. _And_ you saw Mr. Galen when he ran past?
_A_. Yas suh, that's when we seed him.
_Q_. Did you recognize him?
_A_. Naw suh, we didn't know him.
_Q_. Sheik knew him.
_A_. Yas suh, but he didn't say he knew 'im and we'd never seen him before.
_Q_. Choo-Choo must have known him, too.
_A_. Yas suh, must'ave. Him and Sheik usta room together.
_Q_. But you saw Sheik shoot him?
_A_. Yas suh. He said, "Watch this," and pulled out his new zip gun and shot at him.
_Q_. How many times did he shoot?
_A_. Just once. That's all a zip gun will shoot.
_Q_. Yes, these zip guns are single shots. But you knew he had the gun?
_A_. Yas suh. He'd been working on it for 'most a week.
_Q_. He made it himself?
_A_. Yas suh.
_Q_. Had you ever seen him shoot it previously?
_A_. Naw suh. It were just finished. He hadn't tried it out.
_Q_. But you knew he had it on his person?
_A_. Yas suh. He were going to try it out that night.
_Q_. And after he shot the white man, what did you do?
_A_. The man fell down and we went up to see if he'd hit him.
_Q_. Were you acquainted with the first suspect, Sonny Pickens?
_A_. Naw suh, we seed him for the first time too when he come past there shootin'.
_Q_. When you saw the white man had been killed, did you know Sheik had shot him?
_A_. Naw suh, we thought the other fellow had did it.
_Q_. Which one of you, er, passed the wind?
_A_. Suh?
_Q_. Which one of you broke wind?
_A_. Oh, that were Choo-Choo, suh, he the one farted.
_Q_. Was there any special significance in that?
_A_. Suh?
_Q_. Why did he do it?
_A_. That were just a salute we give to the cops.
_Q_. Oh! Was the perfume throwing part of it?
_A_. Yas suh, when they got mad Caleb thew the perfume on them.
_Q_. To allay their anger, er, ah, make them jolly?
_A_. Naw suh, to make them madder.
_Q_. Oh! Well, why did Sheik kidnap Pickens, the other suspect?
_A_. Just to put something over on the cops. He hated cops.
_Q_. Why?
_A_. Suh?
_Q_. Why did he hate cops? Did he have any special reason to hate cops?
_A_. Special reason? To hate cops? Naw suh. He didn't need none. Just they was cops, is all.
_Q_. Ah, yes, just they was cops. Is this the zip gun Sheik had?
_A_. Yas suh. Leastwise it looks like it.
_Q_. How did Bones come to be in possession of it?
_A_. He gave it to Bones when he was running off. Bones's old man work for the city and he figgered it was safe with Bones.
_Q_. That's all for you, boy. You had better be scared.
_A_. Ah is.
That was the case. Open and shut.
Sonny Pickens could not be implicated in the murder. He was being held temporarily on a charge of disturbing the peace while a district attorney's assistant was studying the New York State criminal code to see what other charge could be lodged against him for shooting a citizen with a blank gun.
His friends, Lowtop Brown and Rubberlips Wilson had been hauled in as suspicious persons.
The cases of the two girls had been referred to the probation officers, but as yet nothing had been done. Both were supposedly at their respective homes, suffering from shock.
The bullet had been removed from the victim's brain and given to the ballistics bureau. No further autopsy was required. Mr. Galen's daughter, Mrs. Helen Kruger of Wading River, Long Island, had claimed the body for burial.
The bodies of the others, Granny and Caleb, Choo-Choo and Sheik, lay unclaimed in the morgue. Perhaps the Baptist church in Harlem, of which Granny was a member, would give her a decent Christian burial. She had no life insurance and it would be financially inconvenient for the church, unless the members contributed to defray the costs.
Caleb would be buried along with Sheik and Choo-Choo in potters field, unless the medical college of one of the universities obtained their bodies for dissection. No college would want Choo-Choo's, however, because it had been too badly damaged.
Ready Belcher was in Harlem Hospital, in the same ward where Charlie Richardson, whose arm had been chopped off, had died earlier. His condition was serious, but he would live. He would never look the same, however, and should his teenage whore ever see him again she wouldn't recognize him.
Big Smiley and Reba were being held for contributing to the delinquency of minors, manslaughter, operating a house of prostitution, and sundry other charges.
The woman who was shot in the leg by Coffin Ed was in Knickerbocker Hospital. Two ambulance-chasing shysters were vying with each other for her consent to sue Coffin Ed and the New York police department on a fifty-fifty split of the judgement, but her husband was holding out for a sixty-percent cut.
That was the story; the second and corrected story. The late editions of the morning newspapers had gone hog wild with it:
The prominent New York Citizen hadn't been shot, as first reported, by a drunken Negro who had resented his presence in a Harlem bar. No, not at all. He had been shot to death by a teenage Harlem gangster called Sheik, who was the leader of a teenage gang called the Real Cool Moslems. Why? Well, Sheik had wanted to find out if his zip gun would actually shoot.
The copy writers used a book of adjectives to describe the bizarre aspects of the three-ring Harlem murder; meanwhile they tossed a bone of commendation to the brave policemen who had worked through the small hours of the morning, tracking down the killer in the Harlem jungle and shooting him to death in his lair less than six hours after the fatal shot had been fired.
The headlines read:
POLICE PUT HEAT ON REAL COOL MOSLEMS
DEATH IS THE KISS-OFF FOR THRILL KILL
HARLEM MANIAC RUNS AMUCK
But already the story was a thing of the past, as dead as the four main characters.
"Kill it," ordered the city editor of an afternoon paper. "Someone else has already been murdered somewhere else."
Uptown in Harlem, the sun was shining on the same drab scene it illuminated every other morning at eleven o'clock. No one missed the few expendable colored people being held on various charges in the big new granite skycraper jail on Centre Street that had replaced the old New York City tombs.
In the same building, in a room high up on the southwest corner, with a fine clear view of the Battery and North River, all that remained of the case was being polished off.
Earlier the police commissioner and the chief of police had had a heart-to-heart talk about possible corruption in the Harlem branch of the police department.
"There are strong indications that Galen was protected by some influential person up there, either the police department or in the city government," the police commissioner said.
"Not in the department," the chief maintained. "In the first place, that low license number of his – UG-Sixteen – tells me he had friends higher up than a precinct captain, because that kind of license number is issued only to the specially privileged, and that don't even include me."
"Did you find any connections with politicians in that area?"
"Not connecting Galen; but the woman, Reba, telephoned a colored councilman this morning and ordered him to get down here and get her out on bail."
The commissioner sighed. "Perhaps we'll never know the extent of Galen's activities up there."
"Maybe not, but one thing we do know," the chief said. "The son of a bitch is dead, and his money won't corrupt anybody else."
Afterward the police commissioner reviewed the suspension of Coffin Ed. Grave Digger and Lieutenant Anderson were present along with the chief at this conference. Coffin Ed had exercised his privilege to be absent.
"In the light of subsequent developments in this case, I am inclined to be lenient toward Detective Johnson," the commissioner said. "His compulsion to fire at the youth is understandable, if not justifiable, in view of his previous unfortunate experience with an acid thrower." The commissioner had come into office by way of a law practice and could handle those jaw-breaking words with much greater ease than the cops, who'd learned their trade pounding beats.
"What's your opinion, Jones?" he asked.
Grave Digger turned from his customary seat, one ham propped on the window ledge and one foot planted on the floor, and said, "Yes sir, he's been touchy and on edge ever since that con-man threw the acid in his eyes, but he was never rough on anybody in the right."
"Hell, I wasn't disciplining Johnson so much as I was just taking the weight off the whole God damned police department," the chief said in defense of his action. "We'd have caught holy hell from all the sob sisters, male and female, in this town if those punks had turned out to be innocent pranksters."
"So you are in favor of his reinstatement?" the commissioner asked.
"Why not?" the chief said. "If he's got the jumps let him work them off on those hoodlums up in Harlem who gave them to him."
"Right ho," the commissioner said, then turned to Grave Digger again: "Perhaps you can tell me, Jones; one aspect of the case has me puzzled. All of the reports state that there was a huge crowd of people present at the victim's death, and witnessed the actual shooting. One report states–" he fumbled among the papers on his desk until he found the page he wanted. " 'The street was packed with people for a distance of two blocks when deceased met death by gunfire.' Why is this? Why do the people up in Harlem congregate at the scene of a killing as though it were a three-ring circus?"
"It is," Grave Digger said tersely. "It's the greatest show on earth."
"That happens everywhere," Anderson said. "People will congregate at a killing wherever it takes place."
"Yes, of course, out of morbid curiosity. But I don't mean that exactly. According to reports, not only the reports on this case, but all reports that have come into my office, this, er, phenomenon, let us say, is more evident in Harlem that any place else. What do you think, Jones?"
"Well, it's like this, Commissioner," Grave Digger said. "Every day in Harlem, two and three times a day, the colored people see some colored man being chased by another colored man with a knife or an axe or a club. Or else being chased by a white cop with a gun, or by a white man with his fists. But it's only once in a blue moon they get to see a white man being chased by one of them. A big white man at that. That was an event. A chance to see some white blood spilled for a change, and spilled by a black man, at that. That was greater than Emancipation Day. As they say up in Harlem, that was the greatest. That's what Ed and I are always up against when we try to make Harlem safe for white people."
"Perhaps I can explain it," the commissioner said.
"Not to me," the chief said drily. "I ain't got the time to listen. If the folks up there want to see blood, they're going to see all the blood they want if they kill another white man."
"Jones is right," Anderson said. "But it makes for trouble."
"Trouble!" Grave Digger echoed. "All they know up there is trouble. If trouble was money, everybody in Harlem would be a millionaire."
The telephone rang. The commissioner picked up the receiver.
"Yes...? Yes, send him up." He replaced the receiver and said, "It's the ballistics report. It's coming up."
"Fine," the chief said. "Let's write it in the record and close this case up. It was a dirty business from start to finish and I'm sick and God damned tired of it."
"Right ho," the commissioner said.
Someone knocked.
"Come in," he said.
The lieutenant from homicide who had worked on the case came in and placed the zip gun and the battered lead pellet taken from the murdered man's brain on the commissioner's desk.
The commissioner picked up the gun and examined it curiously.
"So this is a zip gun?"
"Yes sir. It's made from an ordinary toy cap pistol. The barrel of the toy pistol is sawed off and this four-inch section of heavy brass pipe is fitted in in its place. See, it's soldered to the frame, then for greater stability it's bound with adjustable cables in place. The shell goes directly into the barrel, then this clip is inserted to prevent it from backfiring. The firing pin is soldered to the original hammer. On this one it's made from the head and a quarter-inch section of an ordinary Number Six nail, filed down to a point."
"It is more primitive than I had imagined, but it is certainly ingenious."
The others looked at it with bored indifference; they had seen zip guns before.
"And this will project a bullet with sufficient force to kill a man, to penetrate his skull?"
"Yes sir."
"Well, well, so this is the gun which killed Galen and led the boy who made it to be killed in turn."
"No sir, not this gun."
"What!"
Everybody sat bolt upright, eyes popping and mouths open. Had the lieutenant said the Empire State Building had been stolen and smuggled out of town, he couldn't have caused a greater sensation.
"What do you mean, not that gun!" the chief roared.
"That's what I came to tell you," the lieutenant said. "This gun fires a twenty-two caliber bullet. It contained the case of a twenty-two shell when the sergeant found it. Galen was killed with a thirty-two caliber fired from a more powerful pistol."
"This is where we came in," Anderson said.
"I'll be God damned if it is!" the chief bellowed like an enraged bull. "The papers have already gotten the story that he was killed with this gun and they've gone crazy with it. We'll be the laughingstock of the world."
"No," the commissioner said quietly but firmly. "We have made a mistake, that is all."
"I'll be God damned if we have," the chief said, his face turning blood red with passion. "I say the son of a bitch was killed with that gun and that punk lying in the morgue killed him, and I don't give a God damn what ballistics show."
The commissioner looked solemnly from face to face. There was no question in his eyes, but he waited for someone else to speak.
"I don't think it's worth re-opening the case," Lieutenant Anderson said. "Galen wasn't a particularly lovable character."
"Lovable or not, we got the killer and that's the gun and that's that," the chief said.
"Can we afford to let a murderer go free?" the chief said.
The commissioner looked again from face to face.
"This one," Grave Digger said harshly. "He did a public service."
"That's not for us to determine, is it?" the commissioner said.
"You'll have to decide that, sir," Grave Digger said. "But if you assign me to look for the killer, I resign."
"Er, what? Resign from the force?"
"Yes sir. I say the killer will never kill again and I'm not going to track him down to pay for this killing even if it costs me my job."
"Who killed him, Jones?"
"I couldn't say, sir."
The commissioner looked grave. "Was he as bad as that?"
"Yes sir."
The commissioner looked at the lieutenant from homicide.
"But this zip gun was fired, wasn't it?"
"Yes sir. But I've checked with all the hospitals and the precinct station in Harlem and there has been no gunshot injury reported."
"Someone could have been injured who was afraid to report it."
"Yes sir. Or the bullet might have landed harmlessly against a building or an automobile."
"Yes. But there are the other boys who are involved. They might be indicted for complicity. If it is proved that they were his accomplices, they face the maximum penalty for murder."
"Yes sir," Anderson said. "But it's been pretty well established that the murder – or rather the action of the boy firing the zip gun – was not premeditated. And the others knew nothing of his intention to fire at Galen until it was too late to prevent him."
"According to their statements."
"Well, yes sir. But it's up to us to accept their statements or have them bound over to the grand jury for indictment. If we don't charge complicity when they go up for arraignment the court will only fine them for disturbing the peace."
The commissioner looked back at the lieutenant from homicide. "Who else knows about this?"
"No one outside of this office, sir. They never had the gun in ballistics; they only had the bullet."
"Shall we put it to a vote?" the commissioner asked.
No one said anything.
"The ayes have it," the commissioner said. He picked up the small lead pellet that had murdered a man. "Jones, there is a flat roof on a building across the park. Do you think you can throw this so it will land there?"
"If I can't sir, my name ain't Don Newcombe," Grave Digger said.
# 18
The old stone apartment house at 2702 Seventh Avenue was heavy with pseudo-Greek trimmings left over from the days when Harlem was a fashionable white neighbourhood and the Negro slums were centred around San Juan Hill on West 42nd Street.
Grave Digger pushed open the cracked glass door and searched for the name of Coolie Dunbar among the row of mail boxes nailed to the front hall wall. He found the name on a fly-specked card, followed by the apartment number 3-B.
The automatic elevator, one of the first made, was out of order.
He climbed the dark ancient stairs to the third floor and knocked on the left-hand door at the front.
A middle-aged brown-skinned woman with a worried expression opened the door and said, "Coolie's at work and we've told the people already we'll come in and pay our rent in the office when—"
"I'm not the rent collector, I'm a detective," Grave Digger said, flashing his badge.
"Oh!" The worried expression turned to one of apprehension. "You're Mr. Johnson's partner. I thought you were finished with her."
"Almost. May I talk to her?"
"I don't see why you got to keep on bothering her if you ain't got nothing on Mr. Johnson's daughter," she complained as she guarded the entrance. "They were both in it together."
"I'm not going to arrest her. I would just like to ask her a few questions to clear up the last details."
"She's in bed now."
"I don't mind."
"All right," she consented grudgingly. "Come on in. But if you've got to arrest her, then keep her. Me and Coolie have been disgraced enough by that girl. We're respectable church people–"
"I'm sure of it," he cut her off. "But she's your niece, isn't she?"
"She's Coolie's niece. I haven't got any wild ones in my family."
"You're lucky," he said.
She pursed her lips and opened a door next to the kitchen.
"Here's a policeman to see you, Sissie," she said.
Grave Digger entered the small bedroom and closed the door behind him.
Sissie lay on a narrow single bed with the covers pulled up to her chin. At sight of Grave Digger her red, tear-swollen eyes grew wide with terror.
He drew up the single hard-backed chair and sat down.
"You're a very lucky little girl," he said. "You have just missed being a murderer."
"I don't know what you mean," she said in a terrified whisper.
"Listen," he said. "Don't lie to me. I'm dog-tired and you children have already made me as depressed as I've ever been. You don't know what kind of hell it is sometimes to be a cop."
She watched him like a half-wild kitten poised for flight.
"I didn't kill him. Sheik killed him," she whispered.
"We know Sheik killed him," he said in a flat voice. He looked weary beyond words. "Listen, I'm not here as a cop. I'm here as a friend. Ed Johnson is my closest friend and his daughter is your closest friend. That ought to make us friends too. As a friend I tell you we've got to get rid of the gun."
She hesitated, debating with herself, then said quickly before she could change her mind, "I threw it down a water drain on 128th Street near Fifth Avenue."
He sighed. "That's good enough. What kind of gun was it?"
"It was a thirty-two. It had the picture of an owl's head on the handle and Uncle Coolie called it an Owl's Head."
"Has he missed it?"
"He missed it out of the drawer this morning when he started for work and asked Aunt Cora if she'd moved it. But he ain't said nothing to me yet. He was late for work and I think he wanted to give me all day to put it back."
"Does he need it in his work?"
"Oh no, he works for a garage in the Bronx."
"Good. Does he have a permit for it?"
"No, sir. That's what he's so worried about."
"Okay. Now listen. When he asks you about it tonight, you tell him you took it to protect yourself against Mr. Galen and that during the excitement you left it in Sheik's room. Tell him that I found it there but I don't know to whom it belongs. He won't say any more about it."
"Yes sir. But he's going to be awfully mad."
"Well, Sissie, you can't escape all punishment."
"No, sir."
"Why did you shoot at Mr. Galen anyway? You can tell me now since it doesn't matter."
"It wasn't account of myself," she said. "It was on account of Sugartit – Evelyn Johnson. He was after her all the time and I was afraid he was going to get her. She tries to be wild and does crazy things sometimes and I was afraid he was going to get her and do to her what he did to me. That would ruin her. She ain't an orphan like me with nobody to really care what happens to her; she's from a good family with a father and a mother and a good home and I wasn't going to let him ruin her."
He sat there listening to her, a big, tough lumpy-faced cop, looking as though he might cry.
"How'd you plan to do it?" he asked.
"Oh, I was just going to shoot him. I'd made a date with him at the Inn for me and Sugartit, but I wasn't going to take her. I was going to make him drive me out somewhere in his car by telling him we were going to pick her up; and then I was going to shoot him and run away. I took Uncle Coolie's pistol and hid it downstairs in the hall in a hole in the plaster so I could get it when I went out. But before time came for me to go, Sugartit came by here. I wasn't expecting her and I couldn't tell her I wanted to go out, so it was late before I could get rid of her. I left her at the subway at 125th Street, thinking she was going home, then I ran all the way over to Lenox to meet Mr. Galen; but when I got over on Lenox I saw all the commotion going on. Then I saw him come running down the street and Sonny chasing him and shooting at him with a gun. It looked like half the people in Harlem were running after him. I got in the crowd and followed and when I caught up with him at 127th Street I saw that Sonny was going to shoot at him again, so I shot at him, too. I don't think anyone even saw me shoot; everybody was looking at Sonny. But when I saw him fall and all the Moslems in their costumes run up and ganged up around him I was scared one of them was going to see me, so I ran around the block and threw the gun in a drain, then came back to Caleb's from the other way and made out like I didn't know what had happened. I didn't know then that Caleb had been shot."
"Have you told anyone else about this?"
"No sir. When I saw Sugartit come sneaking into Caleb's, I was going to tell her I'd shot him because I knew she'd come back looking for him. But Choo-Choo had let it slip out that Sheik was carrying his zip gun, and then after Sonny said his gun wouldn't shoot anything but blanks I knew right away it was Sheik who'd shot him; and I was scared to say anything."
"Good. Now listen to me. Don't tell anybody else. I won't tell anybody either. We'll just keep it to ourselves, our own private secret. Okay?"
"Yes sir. You can bet I won't tell anybody else. I just want to forget it – if I ever can."
"Good. I don't suppose there's any need to tell you to keep away from bad company; you ought to have learned your lesson by now."
"I'm going to do that, I promise."
"Good. Well, Sissie," Grave Digger said, getting slowly to his feet, "you made your bed hard; if it hurts lying on it, don't complain."
It was visiting hour next day in the Centre Street jail.
Sissie said, "I brought you some cigarettes, Sonny. I didn't know whether you had a girl to bring you any."
"Thanks," Sonny said. "I ain't got no girl."
"How long do you think they'll give you?"
"Six months, I suppose."
"That much. Just for what you did."
"They don't like for people to shoot at anybody, even if you don't hit them, or even if they ain't shooting nothing but blanks like what I did."
"I know," she said sympathetically. "Maybe you're getting off easy at that."
"I ain't complaining," Sonny said.
"What are you going to do when you get out?"
"Go back to shining shoes, I suppose."
"What's going to happen to your shine parlor?"
"Oh, I'll lose that one, but I'll get me another one."
"You got a car?"
"I had one but I couldn't keep up the payments and the man took it back."
"You need a girl to look after you."
"Yeah, who don't? What you going to do yourself, now that your boy friend's dead?"
"I don't know. I just want to get married."
"That shouldn't be hard for you."
"I don't know anybody who'll have me."
"Why not?"
"I've done a lot of bad things."
"Like what?"
"I'd be ashamed to tell you everything I've done."
"Listen, to show you I ain't scared of nothing you might have done, I want you to be my girl."
"I don't want to play around any more."
"Who's talking about playing around. I'm talking about for keeps."
"I don't mind. But there's something I've got to tell you first. It's about me and Sheik."
"What about you and Sheik?"
"I'm going to have a baby by the time you get out of jail."
"Well, that makes it different," he said. "We'd better get married right away. I'll talk to the man and ask him to see if he can't arrange it."
# ABOUT THE AUTHOR
CHESTER HIMES was born in Missouri in 1909. He began writing while serving a prison sentence for a jewel theft and published just short of twenty novels before his death in 1984. Among his best-known thrillers are _Cotton Comes to Harlem, The Real Cool Killers_ , and _The Heat's On_ , all available from Vintage.
| {
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\section{Introduction}
\label{sec-2}
\subsection{Overview and motivation}\label{sec:intro-overview}
Michel--Venkatesh
(see
\cite[\S4.5.3]{michel-2009}, \cite{MichelVenkateshICM})
suggested a strategy
for establishing spectral
identities between moments of $L$-functions, generalizing those
introduced by Motohashi \cite{MR1226527}. They emphasized the
further problem \cite[\S4.5.4]{michel-2009} of implementing that
strategy in a sufficiently flexible form, and suggested that
doing so might lead to strong subconvex bounds.
We address that
problem and apply their strategy to establish a general formula
for the cubic moment of central values of automorphic
$L$-functions on $\operatorname{PGL}_2$.
As a first application of that formula,
we generalize a theorem of Conrey--Iwaniec
\cite{MR1779567}
(see also \cite{MR3394377, MR3968874})
from the rational numbers
to general number fields:
\begin{theorem}\label{thm:CI}
Let $F$ be a number field with adele ring $\mathbb{A}$,
let $\chi$ be a quadratic character of
$\mathbb{A}^\times/F^\times$,
and let $\sigma$ be
either a cuspidal automorphic representation
of $\operatorname{PGL}_2(\mathbb{A})$
or a unitary Eisenstein series.
Then the Weyl-type subconvex bound
\begin{equation}\label{eq:weyl-bound-sigma-chi}
L(\sigma \otimes \chi, 1/2)
\ll_{\sigma}
C(\chi)^{1/3+\varepsilon}
\end{equation}
holds,
with the implied constant depending polynomially
upon $C(\sigma)$.
In particular,
\begin{equation}\label{eqn:weyl-bound-chi}
L(\chi,1/2) \ll C(\chi)^{1/6+\varepsilon}.
\end{equation}
\end{theorem}
Here and henceforth $\varepsilon$ denotes a fixed sufficiently small
positive quantity, whose precise value may change from line to
line, and the asymptotic notation $A \ll B$
or $A = \operatorname{O}(B)$ denotes an estimate of the form
$|A| \leq C |B|$, where the \emph{implied constant}
$C \geq 0$ depends only upon $\varepsilon$ and
the number field $F$.
The refined notation
$A \ll_{x,y,z} B$
or $A = \operatorname{O}_{x,y,z}(B)$
signifies
that $C$ may depend also upon
$x,y,z$.
We write $C(\chi)$ for the analytic conductor,
given by the product over all places $\mathfrak{p}$
of the local analytic conductor $C(\chi_\mathfrak{p})$
as defined in \cite[\S3.1.8]{michel-2009} or \S\ref{sec:intro-l-factors}.
The above ``subconvex'' estimates improve upon the
respective ``trivial''
or ``convexity'' bounds of $C(\chi)^{1/2+\varepsilon}$ and
$C(\chi)^{1/4+\varepsilon}$, and also upon earlier nontrivial subconvex
bounds (see \cite{2009arXiv0904.2429B, MR3977317, MR3213837, MR3594414} and
references). Via period formulas as in \cite{MR3885172, MR4001088,MR2322488} (see also \cite{MR646366,
MR783554, MR1233447,MR1404335,MR3112415, MR3112415,MR3649366}) these
estimates lead to improved bounds for the Fourier coefficients
of half-integral weight modular forms over number fields
(cf. \cite[Cor 1]{2009arXiv0904.2429B}), hence to improved
estimates for representation numbers of ternary quadratic forms
over number fields.
For instance,
we obtain the following
numerical improvement upon \cite[Cor 2]{2009arXiv0904.2429B},
reducing the exponent
$7/16 + \vartheta/8$
to $5/12$:
\begin{corollary}
Let $Q$ be a positive integral ternary quadratic form over
a
totally real number field $F$.
For an element $n$ of the ring of integers of $F$,
let $r_Q(n)$ denote the number of integral representations
of $n$ by $Q$.
For squarefree $n$ locally represented by $Q$,
\begin{equation}
r_Q(n)
= r(n)
+ \operatorname{O}_Q(\norm(n)^{5/12+\varepsilon}),
\end{equation}
where
$r(n) = \norm(n)^{1/2+o(1)}$
is the
product of local densities as in the Siegel mass formula.
\end{corollary}
As a further application, we may combine Theorem \ref{thm:CI}
with
recent work of Balog, Bir{\'o}, Cherubini and Laaksonen
(see
\cite[Cor 1.4, Rmk 2]{2019arXiv191101800B})
to sharpen the error term in the prime
geodesic theorem for $\mathbb{Q}(i)$ \cite[Thm 5.1]{MR723012},
reducing the exponent $\approx 1.60023$ of \cite[Cor
1.2]{2019arXiv191101800B} to $67/42 \approx 1.59524$:
\begin{corollary}
Let $\Psi(X)$ denote the Chebyshev-type counting function for
primitive geodesics on
$\PSL_2(\mathbb{Z}[i]) \backslash \mathbb{H}^3$,
as defined in
\cite[p1]{2019arXiv191101800B}.
We have
\begin{equation}
\Psi(X) = (1/2) X^2
+ \operatorname{O}(X^{67/42+ \varepsilon}).
\end{equation}
\end{corollary}
\begin{proof}
Theorem \ref{thm:CI} implies the bound
$L(\chi,1/2 + it) \ll (1 + |t|)^{\operatorname{O}(1)} C(\chi)^{1/6+\varepsilon}$
for quadratic characters $\chi$ of
$\mathbb{A}^\times /F^\times$,
$F = \mathbb{Q}(i)$. Inserting this bound into
\cite[Cor 1.4]{2019arXiv191101800B}
(and computing
that $3/2 + (4/7) \cdot (1/6) = 67/42$)
gives the required
estimate.
\end{proof}
\subsection{The proposed strategy}\label{sec:intro-motivation}
We summarize the strategy proposed by Michel--Venkatesh for
establishing spectral identities
generalizing those of Motohashi.
Let $F$ be a number field with adele ring $\mathbb{A}$. Set
$G := \operatorname{PGL}_2(F)$, let $A \leq G$ denote the diagonal subgroup,
and write
$[G] := G \backslash G_{\mathbb{A}}$ and $[A] := A \backslash
A_{\mathbb{A}}$ for the corresponding adelic quotients. Let
$\mathcal{I}(0)$ denote the representation of $G_\mathbb{A}$
defined by normalized induction of the trivial character
of the standard Borel
(see \S\ref{sec:intro-representations}). For
$f \in \mathcal{I}(0)$, write $\Eis^*(f)$ for the associated
normalized Eisenstein series (\S\ref{sec-5-2}).
For instance, if $F = \mathbb{Q}$
and $f$ is normalized spherical,
then $\Eis^*(f)$
corresponds
to the derivative $\frac{d }{d s} E(s,z)|_{s=1/2}$
at the central point
of the classical
$\SL_2(\mathbb{Z})$-invariant
Eisenstein series
$E(s,z) = y^s + \dotsb$.
Working formally for the moment
(ignoring important issues of convergence and regularization),
consider the
(divergent) diagonal integral of the product of two such Eisenstein series,
attached to $f_1, f_2 \in \mathcal{I}(0)$, over
the adelic quotient $[A]$ of the diagonal subgroup:
\begin{equation}\label{eq:integral-product-eis}
\int_{[A]} \Eis^*(f_1) \Eis^*(f_2).
\end{equation}
We may expand \eqref{eq:integral-product-eis} in two ways.
On the one hand, expanding the product of Eisenstein series
over the spectrum of $[G]$
yields
\begin{equation}\label{eq:intro-spectral-expn}
\int_{\sigma:\text{generic}}
\sum _{\varphi \in \mathcal{B}(\sigma)}
\int_{[G]} \Eis^*(f_1) \Eis^*(f_2) \overline{\varphi }
\int_{[A]} \varphi
+ (\dotsb),
\end{equation}
where the integral is over generic standard automorphic
representations $\sigma$ of $\operatorname{PGL}_2(\mathbb{A})$
and $(\dotsb)$ denotes the ``contribution of the
one-dimensional representations.''
(We note that the integral
over $\sigma$ is typically
written as a sum over the cuspidal representations
plus an integral over Eisenstein series,
see \S\ref{sec-9-3} for details
and precise
normalizations.)
On the other
hand, the Parseval relation on the group $[A]$ yields
\begin{equation}\label{eq:intro-parseval-A}
\int_{\omega\text{:unitary}}
\left(
\int_{[A]} \Eis^*(f_1)
\omega
\right)
\left(
\int_{[A]} \Eis^*(f_2) \omega^{-1}
\right),
\end{equation}
where the integral is taken over unitary characters $\omega$ of
$[A]$.
By unfolding the global Hecke and
Rankin--Selberg integrals (\S\ref{sec-5-4}, \S\ref{sec-5-6}, \S\ref{sec:global-inv-func})
in each of these expansions,
we ``deduce'' that for factorizable vectors
$f_i = \otimes f_{i \mathfrak{p}}$ and some large enough finite
collection $S$ of places of $F$,
\begin{align}
\label{eqn:basic-moment-identity}
&\int_{\substack{
\sigma:\text{generic}, \\
\text{unram. outside $S$}
}}
\frac{L^{(S)}(\sigma,1/2)^3}{L^{(S),*}(\sigma \times \sigma,
1)}
h(\sigma)
\\ \nonumber
&\quad =
(\dotsb) +
\int_{\substack{
\omega:\text{unitary}, \\
\text{unram. outside $S$}
}}
\frac{|L^{(S)}(\omega,1/2)|^4}{\zeta_F^{(S),*}(1)^2}
\tilde{h}(\omega),
\end{align}
where $L^{(S)}(\dotsb)$ denotes a partial $L$-function,
an asterisk signifies taking the first nonvanishing
Laurent coefficient,
and the weights
\[
h(\sigma) =
\prod_{\mathfrak{p} \in S}
h_\mathfrak{p}(\sigma_\mathfrak{p}),
\quad
\tilde{h}(\omega) = \prod_{\mathfrak{p} \in S}
\tilde{h}_\mathfrak{p}(\omega_\mathfrak{p})
\]
are products of local weights given
in terms of local Hecke and Rankin--Selberg integrals
(\S\ref{sec-4-3}, \S\ref{sec-4-5}):
\begin{equation}\label{eq:intro-ell-sigma}
h_\mathfrak{p}(\sigma_\mathfrak{p})
:=
\sum _{W_\mathfrak{p} \in \mathcal{B}(\sigma_{\mathfrak{p}})}
\int_{N_\mathfrak{p} \backslash G_\mathfrak{p}}
\overline{W_{\mathfrak{p}}}
W_{f_{1 \mathfrak{p}}}
f_{2 \mathfrak{p}}
\int_{A_{\mathfrak{p}}} W_\mathfrak{p},
\end{equation}
\begin{equation}\label{eq:intro-ell-omega}
\tilde{h}_\mathfrak{p}(\omega_\mathfrak{p})
:=
\int_{A_\mathfrak{p}}
W_{f_{1 \mathfrak{p}}} \omega
\int_{A_\mathfrak{p}}
W_{f_{2 \mathfrak{p}}} \omega^{-1}.
\end{equation}
\subsection{The basic spectral identity}
While the argument just recorded was highly non-rigorous, we
prove that the conclusion is nevertheless valid
(see Corollary \ref{cor:basic-identity-summary}
for the precise statement):
\begin{theorem}\label{thm:basic-spectr-ident}
\label{thm:intro-basic-decomp-mv}
The identity \eqref{eqn:basic-moment-identity} holds
with
$(\dotsb)$ the limit of a sum of fifteen ``degenerate
functionals''
$\mathcal{I}(0) \otimes \mathcal{I}(0) \rightarrow \mathbb{C}$
defined in \S\ref{sec-12}.
\end{theorem}
The proof begins by deforming the Eisenstein series
generically and ends by taking a limit.
Following Zagier \cite{MR656029} and Michel--Venkatesh
\cite[\S4.3]{michel-2009}, the
product of deformed Eisenstein series differs by a
finite linear combination of
Eisenstein series from an
$L^2$-function, which in turn admits a spectral expansion. On
the other hand, the diagonal integral
\eqref{eq:integral-product-eis} of that product admits a
canonical regularization and enjoys a modified form of
Parseval's identity \eqref{eq:intro-parseval-A}.
Interchanging the regularized diagonal
integral with the spectral expansion introduces
additional terms. Some of these arguments were sketched or
suggested by Michel--Venkatesh \cite[\S4.5]{michel-2009} in the
special case that $F = \mathbb{Q}$ and the $f_i$ are spherical.
\subsection{Achieving arbitrarily weighted cubic moments}\label{sec:achi-arbitr-weight}
We turn to the main problem addressed by this paper,
which was
raised in \cite[\S4.5.4]{michel-2009}.
In seeking to apply
Theorem \ref{thm:intro-basic-decomp-mv}, several fundamental
questions arise. Which weights $h$ (or $\tilde{h}$) arise from
the above scheme applied to some choice of
$f_1, f_2 \in \mathcal{I}(0)$, or more generally of some tensor
$f \in \mathcal{I}(0) \otimes \mathcal{I}(0)$? Let us call such
weights \emph{admissible}. It is not \emph{a priori} obvious
that the collection of admissible weights is sufficiently rich
for the sake of applications. Can one find a nonnegative-valued
admissible $h$ (or $\tilde{h}$) that localizes to a given subset
of its domain? Can one explicitly relate $h$ to $\tilde{h}$?
Can one efficiently estimate the degenerate terms $(\dotsb)$?
The main point of this paper is to initiate the systematic study
of such questions.
We focus here on studying the cubic moment
via \eqref{eqn:basic-moment-identity}.
To do so effectively,
we need to know that the characteristic functions of
interesting spectral families of automorphic representations
$\sigma$ may be approximated by nonnegative admissible weights
$h(\sigma)$. This possibility is confirmed by one of our main local
results
(Theorem \ref{thm:constr-admiss-weight}),
summarized here informally:
\begin{theorem}\label{thm:summarize-kuznetsov-weights-adm}
Let $h(\sigma)$ be a weight function that shows up on the
spectral side of the pre-Kuznetsov formula (i.e., the relative
trace formula for twisted unipotent periods with a
compactly-supported test function). Then $h$ is admissible.
The corresponding dual weight $\tilde{h}(\omega)$ may be
evaluated via explicit integral transforms.
\end{theorem}
We show by
example in \S\ref{sec:appl-cubic-moment}
that the ``pre-Kuznetsov weights''
$h(\sigma)$ alluded to in
Theorem \ref{thm:summarize-kuznetsov-weights-adm}
are adequate for applications.
We show also that the degenerate
terms may be either discarded altogether
(\S\ref{sec:handl-degen-terms-1})
or evaluated explicitly (\S\ref{sec:handl-degen-terms-2}).
The idea behind the proof
of Theorem \ref{thm:summarize-kuznetsov-weights-adm}
is to consider $f$ belonging to a
specific class of generalized vectors (``Whittaker vectors'')
for which the weights $h(\sigma)$ on the cubic moment side
resemble those appearing in the pre-Kuznetsov formula. We hope
that this technique will be useful more broadly in
period-based approaches to proving spectral identities
between families of $L$-functions.
Theorems \ref{thm:intro-basic-decomp-mv} and
\ref{thm:summarize-kuznetsov-weights-adm} combine to yield
spectral identities for the cubic moment with nonnegative
weights that localize to any given subset of the spectrum.
One can likely adapt the basic proof technique of Theorem
\ref{thm:summarize-kuznetsov-weights-adm} to produce a rich
class of admissible $\tilde{h}$ and study the inverse transform
from $\tilde{h}$ to $h$,
but we leave such pursuits and
their applications to future work.
\subsection{Applications to subconvexity}
The $L$-values $L(\sigma,1/2)$ are known to be nonnegative, so
taking $h$ nonnegative and estimating the RHS of
\eqref{eqn:basic-moment-identity} yields an upper bound for such
$L$-values. This feature has been exploited over
$F = \mathbb{Q}$ in the work of Conrey--Iwaniec
\cite{MR1779567}, Ivic \cite{MR1879668} and Petrow and Young
\cite{MR3394377, MR3635360, MR3968874,
2018arXiv181102452P, 2019arXiv190810346P}.
We hope the methods of this paper may
be useful in generalizing such results to
the setting of number fields.
We have already recorded in \S\ref{sec:intro-overview} a
generalization of the theorem of Conrey--Iwaniec
\cite{MR1779567}.
As a further application,
we present a partial generalization of
a recent result of Petrow--Young \cite{2018arXiv181102452P}:
\begin{theorem}\label{thm:PY1}
Let $F$ be a number field. Let $\chi$ be a
unitary character of $\mathbb{A}^\times/F^\times$
whose infinite component $\chi_\infty$ is
trivial and whose finite conductor is cubefree.
Then the Weyl-type subconvex bound \eqref{eqn:weyl-bound-chi}
holds.
\end{theorem}
The point of departure for our proof of Theorems \ref{thm:CI}
and \ref{thm:PY1} is similar to that of earlier work over
$\mathbb{Q}$ in that we seek to bound a cubic moment. The
conclusion of our proof is likewise similar: we eventually
reduce to essentially the same character sum estimates (relying
in turn upon Deligne's results) and fourth moment bounds for
$\GL_1$ $L$-functions as in those works. The remaining parts of
our arguments differ somewhat. For instance, a major difficulty
encountered in \cite{MR3968874} (see especially
\cite[\S1.3]{MR3968874}) is that the relevant cubic moment
involves newforms of different conductors, so the approximate
functional equations for their $L$-values have different
lengths, which causes problems when one seeks to estimate an odd
power moment of nonnegative $L$-values via the Petersson
formula. The authors of \cite{MR3968874} overcome this
difficulty by developing general Petersson formulas for
newforms, which are of independent interest. Such difficulties
are avoided in the approach pursued here. We do not use
approximate functional equations. The cubic moment formula
implied by Theorems \ref{thm:basic-spectr-ident} and
\ref{thm:summarize-kuznetsov-weights-adm} reduces our task to
the local problem of producing weights $h(\sigma)$ majorizing
the family of interest and estimating the integral transforms
defining the dual weights $\tilde{h}(\omega)$ and degenerate
terms $(\dotsb)$. The same formula can likely be supplemented
with additional local analysis to prove many variations on
Theorems \ref{thm:CI} and \ref{thm:PY1}. We note also that the
delicate archimedean stationary phase calculations required
already in \cite{MR1779567} are not needed in our approach.
The restriction in Theorem \ref{thm:PY1} to cubefree conductor
arises here due to local phenomena as in
\cite{2018arXiv181102452P}. Recently Petrow--Young
\cite{2019arXiv190810346P} have removed this restriction from
their earlier result (over $F = \mathbb{Q}$) and established the
bound \eqref{eqn:weyl-bound-chi} over $F = \mathbb{Q}$ for all
unitary $\chi$, i.e., without the quadraticity assumption. We
establish many of the local estimates relevant for adapting
their strategy to our setting (compare Proposition
\ref{prop:non-arch-estimates-key} with
\cite[\S3]{2019arXiv190810346P}).
To complete this adaptation
requires verifying the following generalization of
\cite[Thm 1.4]{2019arXiv190810346P}:
\begin{conjecture}\label{conj:fourth-moment-cosets}
Let $F$ be a number field.
Let $\chi$ be a unitary character of $\mathbb{A}^\times$. Let
$B = (B_\mathfrak{p})_{\mathfrak{p}}$ be a collection of real
numbers $B_\mathfrak{p} \geq 1$, indexed by the places of $F$,
with $B_\mathfrak{p}$ belonging to the value group of
$F_\mathfrak{p}$ for all $\mathfrak{p}$ and
satisfying
$B_\mathfrak{p} = 1$ for all but finitely many $\mathfrak{p}$.
Let $\mathcal{F}(\chi,B)$ denote the set of all characters
$\omega$ of $\mathbb{A}^\times/F^\times$ such that for each
place $\mathfrak{p}$ of $F$, we have
\[C(\omega_\mathfrak{p}/\chi_\mathfrak{p}) \leq
B_\mathfrak{p}.
\]
Assume that for each $\mathfrak{p}$,
we have
\[
B_\mathfrak{p} \geq C(\chi_\mathfrak{p})^{2/3}.
\]
Then
\begin{equation}\label{eq:required-fourth-moment-bound-for-generalization}
\int_{\omega \in \mathcal{F}(\chi,B)}
|L(\omega,1/2)|^4
\ll
(\prod_\mathfrak{p} B_\mathfrak{p})^{1+\varepsilon}.
\end{equation}
\end{conjecture}
Conjecture \ref{conj:fourth-moment-cosets} is also relevant for
removing the restriction on $\chi_\infty$ in Theorem
\ref{thm:PY1}: the local phenomena responsible for the
restriction to cubefree conductors show up over any local field
in which $-1$ is a square, and thus arise whenever the number
field $F$ has a complex embedding.
One can study the LHS of
\eqref{eq:required-fourth-moment-bound-for-generalization}
by applying
the basic formula \eqref{eqn:basic-moment-identity}
in the direction opposite to the primary one considered in this
paper (i.e., as in the original work of Motohashi),
but we leave this for future work.
\subsection{Further remarks}
It might be interesting to compare the formulas obtained here
with those in \cite{MR1226527} and \cite{MR1879668}, or to
replace the degenerate Eisenstein series occurring in
\eqref{eq:integral-product-eis} with other Eisenstein or with
cusp forms; the case treated here is in some sense the most
degenerate and (apparently) the most relevant for applications.
In another direction, with some refined local analysis it should
be possible to improve the estimate
\eqref{eq:weyl-bound-sigma-chi} to be simultaneously subconvex
in $\sigma$ and $\chi$ (compare with \cite{MR3635360}).
We mention the works \cite{MR2124019, MR2279942,
2019arXiv190207042B}, which offer other perspectives on
Motohashi's formula and its generalizations.
\subsection{Organization of this
paper}\label{sec:organ-this-paper}
In Part \ref{part:main-ideas}, we aim to introduce the main
ideas of this paper with minimal technical overhead. In
\S\ref{sec-6}, we give the precise statement and proof of
Theorem \ref{thm:summarize-kuznetsov-weights-adm}, which allows
us to study arbitrarily weighted cubic moments in terms of
periods of Eisenstein series. In
\S\ref{sec:appl-cubic-moment}, we define and study a class of
weights concentrating on the ``short families'' of primary
interest in applications of the cubic moment to subconvexity.
We explain in particular how the two-variable exponential sums
that featured in the work of Conrey--Iwaniec and Petrow--Young
arise naturally from local representation-theoretic
considerations. In \S\ref{sec-8}, we take for granted the basic
spectral identity, Theorem \ref{thm:basic-spectr-ident}, and
give the proofs of our main applications,
Theorems \ref{thm:CI} and \ref{thm:PY1}, modulo some
technicalities concerning the degenerate terms $(\dotsb)$ and
the required polynomial dependence of
\eqref{eq:weyl-bound-sigma-chi} upon $\sigma$. The remainder of
the paper is then devoted to addressing those technicalities
and proving
Theorem \ref{thm:basic-spectr-ident}.
Part \ref{part:preliminaries} contains preliminaries of a
general nature, concerning regularized integration
(\S\ref{sec-9-1}), Sobolev norms on representations of reductive
groups (\S\ref{sec:norms-repr}), the local (\S\ref{sec-4}) and
global (\S\ref{sec-5}) theory of integral representations of
$L$-functions, and (regularized) spectral decompositions of the
space of automorphic forms (\S\ref{sec-5}). Our Sobolev norm
discussion is similar to that of \cite[\S2]{michel-2009}, but
applies also to non-unitary representations, as is convenient
when studying degenerate integrals like
\eqref{eq:integral-product-eis} via deformation.
Part \ref{part:basic-identity} gives the precise statement and
proof of Theorem \ref{thm:basic-spectr-ident}. We begin by
studying in detail the functionals \eqref{eq:intro-ell-sigma}
and \eqref{eq:intro-ell-omega}, in local
(\S\ref{sec:local-inv-func}) and global
(\S\ref{sec:global-inv-func}) settings. In \S\ref{sec-12}, we
relate these two families of functionals by decomposing in two
ways a deformation of the integral \eqref{sec-12}.
Part \ref{part:analys-local-weights} contains several results
needed for the proofs of our main applications.
In particular, we estimate the degenerate terms
in cases relevant to our applications.
\subsection{General notation and conventions}\label{sec:gener-notat-conv}
\subsubsection{Asymptotic notation and terminology}
Let us recall and elaborate upon the conventions introdued earlier.
The notation $A = \operatorname{O}(B)$ or $A \ll B$ signifies that
$|A| \leq C |B|$ for some \emph{fixed} quantity $C$,
while $A \asymp B$ is shorthand for $A \ll B \ll A$.
Here and henceforth, we consider
a quantity to be \emph{fixed} if we explicitly label it as such,
or if it depends only upon some previously defined fixed
quantities. The small positive parameter $\varepsilon > 0$ is always
regarded as fixed.
For instance, we have $x^n \ll \exp(\varepsilon x)$ for any fixed
natural number $n$ and all positive reals $x$.
When considering a number field $F$ and a nontrivial unitary
character $\psi$ of its adele class group $\mathbb{A}/F$, we
regard the pair $(F,\psi)$ as fixed. When working over a local
field $F$ equipped with a nontrivial unitary character $\psi$,
we consider the pair $(F,\psi)$ as fixed \emph{except} when $F$
is non-archimedean and $\psi$ is unramified, in which case we
only regard the absolute degree of $F$ as fixed. This
convention ensures that implied constants remain uniform as the
pair $(F,\psi)$ traverses the local components of corresponding
global data.
\subsubsection{Local fields}
Let $F$ be a local field,
thus $F$ is either $\mathbb{R}$, $\mathbb{Q}_p, \mathbb{F}_p(t)$
or a finite extension of one of these fields.
When $F$ is non-archimedean, we denote
by $\mathfrak{o}$ the ring of integers and by $\mathfrak{p}$ the
maximal ideal, and set $q := \# \mathfrak{o}/\mathfrak{p}$.
When $F$ is archimedean,
it will be convenient to set $q := 1$.
We will often consider local fields $F$ equipped with nontrivial
unitary characters $\psi : F \rightarrow \U(1)$. Recall that if
$F$ is non-archimedean, then $\psi$ is \emph{unramified} if it
is trivial on $\mathfrak{o}$ but not on $\mathfrak{p}^{-1}$. We
say that the pair $(F,\psi)$ is \emph{unramified} if $F$ is
non-archimedean and $\psi$ is unramified.
\subsubsection{Characters}
A \emph{character} of a topological group $G$ is a continuous
homomorphism $\chi : G \rightarrow \mathbb{C}^\times$; a
\emph{unitary character} is one with image in the unit circle
$\U(1)$.
When $G$ is the multiplicative group $F^\times$ of a local field
$F$ or the idele class group $\mathbb{A}^\times/F^\times$ of a
number field $F$, the normalized absolute value $|.|$
defines a character of $G$, and every positive-valued character
is of the form $|.|^c$ for some real number $\chi$. The
\emph{real part} $\Re(\chi)$ of any character
$\chi : G \rightarrow \mathbb{C}^\times$ is then characterized
by the identity $|\chi| = |.|^{\Re(\chi)}$.
\subsubsection{Local zeta functions}
For a local field $F$,
we write $\zeta_F(s)$ for the local zeta function, given
respectively by $\pi^{-s/2} \Gamma(s/2)$ or
$2 (2 \pi)^{-s} \Gamma(s)$ or $(1 - q^{-s})^{-1}$ according as
$F$ is real or complex or non-archimedean.
\subsubsection{Matrix notation}
When working with $\operatorname{PGL}_2$ over a ring,
we use the notation
\[
n(x) := \begin{pmatrix}
1 & x \\
0 & 1
\end{pmatrix},
\quad a(y) :=
\begin{pmatrix}
y & 0 \\
0 & 1
\end{pmatrix},
\quad
n'(z) :=
\begin{pmatrix}
1 & 0 \\
z & 1
\end{pmatrix}
\quad
w := \begin{pmatrix}
& -1 \\
1 &
\end{pmatrix}.
\]
We will use that $w n(x) = n(-1/x) a(1/x^2) n'(1/x)$
for invertible $x$.
\subsubsection{Representations}\label{sec:intro-representations}
Let $F$ be a local field,
and let $G$ be a reductive group over $F$. By a
``representation'' $\sigma$ of $G$, we always mean a smooth
representation in the non-archimedean case (see
\cite[\S4.2]{MR1431508}) and a smooth moderate growth
Fr{\'e}chet representation in the archimedean case (see
\cite{MR1013462}, \cite[\S11]{MR1170566}, \cite{MR3219530}). In
particular, any vector $v \in \sigma$ is assumed smooth.
Given natural numbers $r_1,\dotsc,r_k$ with sum
$r := \sum_j r_j$, we may form the standard parabolic subgroup
$P = M U$ of $\GL_{r}(F)$, with
$M \cong \GL_{r_1}(F) \times \dotsb \times \GL_{r_k}(F)$. Given
representations $\chi_j$ of $\GL_{r_j}(F)$ ($j=1..k$), we write
\begin{equation}\label{eqn:parabolic-induction-chi-1-through-chi-k}
\Ind(\chi_1 \boxtimes \dotsb \boxtimes \chi_k)
\end{equation}
for the
normalized induction from $P$ to $G$ of the corresponding
representation of $M$.
Given a character $\chi$ of $F^\times = \GL_1(F)$,
we abbreviate
\begin{equation}
\mathcal{I}(\chi) := \Ind(\chi \boxtimes
\chi^{-1});
\end{equation}
it defines a representation of $\GL_2(F)$ with trivial central
character, hence a representation of $\operatorname{PGL}_2(F)$
consisting of smooth functions
$f : \operatorname{PGL}_2(F) \rightarrow \mathbb{C}$
satisfying
\begin{equation}\label{eq:f-of-n-x-a-y-g-equals-blah}
f(n(x) a(y) g)
= |y|^{1/2} \chi(y) f(g).
\end{equation}
We write
simply $\mathcal{I}(s)$ for $\mathcal{I}(|.|^s)$. We will never
use the potentially ambiguous notation $\mathcal{I}(1)$.
Following \cite[\S1.5]{MR3753910}, by a \emph{Whittaker type}
representation $\sigma$ of $\GL_r(F)$ we mean one of the form
\eqref{eqn:parabolic-induction-chi-1-through-chi-k} with the
$\chi_j$ essentially square-integrable, and in particular
generic.
Thus
$\chi_j = \chi_j^0 \otimes |.|^{c_j}$ with $\chi_j^0$
square-integrable and $c \in \mathbb{R}$,
and
\begin{equation}\label{eqn:whittaker-type-induction}
\sigma =
\Ind(
\chi_1^0 \otimes |.|^{c_1}
\boxtimes
\dotsb
\boxtimes
\chi_k^0 \otimes |.|^{c_k}
).
\end{equation}
(In \cite[\S2.1]{MR701565}, ``Whittaker type'' has a more
general meaning.)
We distinguish between a Whittaker type representation
and its isomorphism class.
Every generic irreducible representation is
isomorphic to at least one representation of Whittaker type.
Writing
$\GL_r(F)^\wedge_{\gen}$ for the set of isomorphism classes of
generic irreducible representations of $\GL_r(F)$, we have a
surjective map with finite fibers
\begin{equation}\label{eqn:from-whittaker-type-to-generic-0}
\{\text{Whittaker type representations of } \GL_r(F)\}
\rightarrow \GL_r(F)^\wedge_{\gen}
\end{equation}
given by taking the unique generic subquotient.
A Whittaker type representation of $\operatorname{PGL}_2(F)$ is either
square-integrable or of the form $\mathcal{I}(\chi)$ for some
character $\chi$ of $A$.
\subsubsection{Bounds toward
Ramanujan}\label{sec:bounds-towards-raman}
Let $F$ be a local field. We say that a Whittaker type
representation $\sigma$ of $\GL_r(F)$ is
\emph{$\vartheta$-tempered} if in
\eqref{eqn:whittaker-type-induction}, each $c_j$ is bounded in
magnitude by $\vartheta$. Then $\sigma$ is $0$-tempered iff it
is tempered in the customary sense.
A Whittaker type representation $\sigma$ of $\GL_2(F)$ is
$\vartheta$-tempered precisely when
\begin{itemize}
\item $\sigma = \sigma_0 \otimes |.|^c$
with $\sigma_0$
square-integrable and $|c| \leq \vartheta$, or
\item $\sigma = \Ind(\mu_1 \boxtimes \mu_2)$
of some characters $\mu_1, \mu_2$ of $F^\times$
with $|\Re(\mu_i)| \leq \vartheta$.
\end{itemize}
It is known \cite{MR2811610} that for $\GL_2$ over a number
field, the local component of any cuspidal automorphic
representation with unitary central character is
$7/64$-tempered.
\subsubsection{$L$-factors}\label{sec:intro-l-factors}
The local $L$-factors considered in this paper are defined and
studied in work of Jacquet--Piatetski-Shapiro--Shalika
\cite{MR701565} and Jacquet \cite{MR0401654}.
Over any local field $F$,
those works attach
to any pair of Whittaker type representations
$\sigma_1, \sigma_2$ of $\GL_{r_1}(F), \GL_{r_2}(F)$
an $L$-factor $L(\sigma_1 \otimes \sigma_2, s)$,
$\varepsilon$-factor $\varepsilon(\psi,\sigma_1 \otimes \sigma_2, s)$
and $\gamma$-factor
\begin{equation}\label{eqn:defn-local-rs-gamma-factor}
\gamma(\psi,\sigma_1 \otimes \sigma_2, s) = \varepsilon(\psi,\sigma_1
\otimes \sigma_2, s) \frac{L(\tilde{\sigma}_1 \otimes
\tilde{\sigma}_2, 1 - s)}{L(\sigma_1 \otimes \sigma_2, s)}.
\end{equation}
When $r_2 = 1$ and $\sigma_2$ is trivial
one writes simply $L(\sigma_1,s)$ for $L(\sigma_1 \otimes
\sigma_2, s)$, and similarly for the $\varepsilon$- and
$\gamma$-factors.
The local $L$-factors
$L(\sigma_1 \otimes \sigma_2, s)$
may be written
\begin{equation}\label{eqn:factorization-of-local-L-factor-into}
\prod_{j=1}^n \zeta_F(s -
u_j),
\end{equation}
where $\{u_j\}_{j=1}^n$ is a collection of complex
numbers with $n \leq r_1 r_2$.
If $\sigma_i$
is $\vartheta_i$-tempered,
then
$\Re(u_j) \leq \vartheta_1 + \vartheta_2$.
If is a character of $F^\times$, then $L(\chi,s)$
has the form \eqref{eqn:factorization-of-local-L-factor-into}
with $n \leq 1$ and $\Re(u_j) \leq \Re(\chi)$.
The $L$-factors are multiplicative with respect to induction
in the sense that
\begin{equation}\label{eqn:multiplicativity-L-factors-wrt-induction}
L(
\Ind(\chi_1 \boxtimes \dotsb \boxtimes \chi_k)
\otimes
\Ind(\omega_1 \boxtimes \dotsb \boxtimes \omega_{\ell})
,s)
= \prod_{i,j}
L(\chi_i \otimes \omega_j, s),
\end{equation}
and similarly for the $\varepsilon$- and $\gamma$-factors.
The local $\gamma$-factors may be characterized by the local
functional equation of \emph{loc. cit.}, which we recall below
in special cases as needed. The local $\varepsilon$-factors will not
play an important role for us.
We define the analytic conductor
$C(\sigma_1 \otimes \sigma_2,s) = C(\sigma_1 \otimes \sigma_2
\otimes |.|^s) \in \mathbb{R}_{\geq 1}$ as in
\cite[\S3.1.8]{michel-2009}: in
the non-archimedean case it is defined by the relation
$\varepsilon(\psi_0, \sigma_1 \otimes \sigma_2, s) =
C(\sigma_1
\otimes \sigma_2, s)^{1/2-s}
\varepsilon(\psi_0, \sigma_1 \otimes \sigma_2, 1/2)$
for an unramified character
$\psi_0$ of $F$, while in the archimedean case it is defined as
$C(\sigma_1 \otimes \sigma_2, s) := \prod_{j} (2 + |\mu_j + s|)$
if
$L(\sigma_1 \otimes \sigma_2,s) = \prod_j \zeta_{\mathbb{R}}(s +
\mu_j)$.
We abbreviate
$C(\sigma_1 \otimes \sigma_2) := C(\sigma_1 \otimes \sigma_2,0)$.
For our purposes, the key property of the analytic conductor is
that
if $\sigma_1$ and $\sigma_2$ are unitary,
then
\begin{equation}\label{eq:stirling-for-general-RS-gamma}
\gamma(\psi,\sigma_1 \otimes \sigma_2,1/2 + s) \asymp
C(\sigma_1 \otimes \sigma_2,s)^{-\Re(s)} q^{\operatorname{O}(1)}
\end{equation}
provided that $s$ is at least some fixed positive distance away
from any poles or zeros of the LHS of
\eqref{eq:stirling-for-general-RS-gamma}. In the archimedean
case, the estimate \eqref{eq:stirling-for-general-RS-gamma} is a
consequence of Stirling's formula. In the non-archimedean case,
it follows from the definition of the analytic conductor
in terms of the $\varepsilon$-factor and the fact that
$|\varepsilon(\psi_0,\sigma_1 \otimes \sigma_2,1/2)| = 1$ in the
unitary case; the factor $q^{\operatorname{O}(1)}$ crudely bounds the ratio of
$L$-factors in \eqref{eqn:defn-local-rs-gamma-factor} and may be
omitted when those $L$-factors are identically $1$ or
when $s$ is chosen so that their ratio is $\asymp 1$.
The
estimate \eqref{eq:stirling-for-general-RS-gamma} may be usefully applied in conjunction with the
multiplicativity property of $\gamma$-factors with respect to
induction.
\subsubsection{$L$-functions}
When $F$ is a number field,
we denote completed $L$-functions (including archimedean
factors) by $\Lambda(\dotsb)$ and their finite parts by
$L(\dotsb)$. For a finite set of places $S$ that contains every
archimedean place, we write $L^{(S)}(\dotsb)$ for the Euler
product taken over $\mathfrak{p} \notin S$.
We write
$\xi_F(s) = \prod_{\mathfrak{p}} \zeta_{F_\mathfrak{p}}(s)$ for
the Dedekind zeta function and $\zeta_F^{(S)}(s)$ for the
corresponding product over $\mathfrak{p} \notin S$.
We use a superscripted asterisk,
as in $\xi_F^*(1)$ or $\zeta_F^{(S)*}(1)$
or $\Lambda^*(\sigma \times \sigma, 1)$ (for a generic
automorphic representation $\sigma$ of $\operatorname{PGL}_2$),
to denote the first nonvanishing Laurent coefficient (typically
the residue).
\subsubsection{Groups and measures}
\label{sec-4-1}
When working over a local field $F$, we
generally use the notation
\[
G := \operatorname{PGL}_2(F),
\]
$N := \{n(x) : x \in F\},A :=\{a(y) : y \in F^\times \}$,
$N' := \{n'(z) : z \in F\}$, and $B := N A$.
(The exception to this convention is
\S\ref{sec:norms-repr},
in which we work more generally). We let
$K \leq G$ denote the standard maximal compact subgroup. We
identify $N \cong F$, $A \cong F^\times$, and $N' \cong F$ in
the evident way; in particular, we identify their character
groups.
We denote by $A^\wedge$ the group of characters
$\chi : A \rightarrow \mathbb{C}^\times$, or equivalently,
$\chi : F^\times \rightarrow \mathbb{C}^\times$.
It has the
natural structure of a Riemann surface with at most countably
connected components, with charts given by
$\chi |.|^s \mapsto s$.
We denote by $G^\wedge_{\gen}$ the set of isomorphism classes of
generic irreducible representations $\sigma$ of $G$. Via the
map \eqref{eqn:from-whittaker-type-to-generic-0}, we may regard
$G^\wedge_{\gen}$ as a quotient of the set of Whittaker type
representations of $G$. The latter set is naturally a complex
manifold.
We equip $G^\wedge_{\gen}$ with the quotient topology,
and say that a function
$h : G^\wedge_{\gen} \rightarrow \mathbb{C}$ is
\emph{holomorphic} if it pulls back to a holomorphic function on
the set of Whittaker type representations.
Given a nontrivial unitary character $\psi$ of $F$, we equip the
additive group $F$ with the $\psi$-self-dual Haar measure $d x$,
so that the Fourier inversion formula
$\int_{x \in F} (\int_{y \in F} f(y) \psi(x y) \, d y) \, d x =
f(0)$ holds for $f \in C_c^\infty(F)$. We equip the
multiplicative group $F^\times$ with the measure
$\frac{d y}{|y|}$. We note that if $(F,\psi)$ is unramified,
then
\begin{equation}\label{eq:volume-of-unit-group}
\vol(\mathfrak{o}^\times, \tfrac{d y}{|y|})
= 1/\zeta_F(1).
\end{equation}
By our identifications, we obtain Haar measures on $N, A, N'$.
We equip $G$ with the Haar given by the pushforward of
$d x \, \frac{d y}{|y|} \, d z$ under the map
$F \times F^\times \times F \rightarrow G$,
$(x,y,z) \mapsto n(x) a(y) w n(z)$, or equivalently, under
$(x,y,z) \mapsto n(x) a(y) n'(z)$. If $(F,\psi)$ is unramified,
then the chosen Haar measure on $G$ assigns volume
$1/\zeta_F(2)$ to $K$ (see \cite[\S3.1.6]{michel-2009} and
\eqref{eq:volume-of-unit-group}).
It will be convenient
for us to equip $K$ with the Haar measure $d k$
for which the Haar measure on $G$
is given by the pushforward
of $d x \, \frac{d y}{|y|} \, d k$
under $(x,y,k) \mapsto n(x) a(y) k$;
the volume of $d k$ is then $\asymp 1$.
We will use freely that any irreducible representation of $G$
is self-dual, i.e., isomorphic to its own contragredient
(see \cite[Exercise 2.5.8, Thm 4.2.2]{MR1431508}).
\part{Main ideas}\label{part:main-ideas}
In this part, we precisely formulate and prove Theorem
\ref{thm:summarize-kuznetsov-weights-adm}. We then record most
of the arguments required to deduce our main applications
(Theorems \ref{thm:CI} and \ref{thm:PY1}). Along the way, we
explain how the character sums appearing in the works of
Conrey--Iwaniec and Petrow--Young arise naturally from the
perspective of representation theory.
The reader who is not familiar with integral representations of
$L$-functions and the related local representation theory might
wish to peruse \S\ref{sec-4} and \S\ref{sec-5} before
proceeding.
\section{Basic study of local weights}
\label{sec-6}
Let $F$ be a local field, and let $\psi$ be a nontrivial unitary
character of $F$.
\subsection{Some induced representations of $G$}
\label{sec-6-2}
Recall that for a character $\chi$ of $F^\times \cong A$,
we write $\mathcal{I}(\chi)$ for the corresponding normalized
induced representation of $G$,
consisting of
smooth functions $f : G \rightarrow \mathbb{C}$
satisfying
\eqref{eq:f-of-n-x-a-y-g-equals-blah}.
We are interested primarily in the
representation
$\mathcal{I}(0) = \mathcal{I}(|.|^0)$ induced by the trivial character,
consisting of smooth functions $f : G \rightarrow \mathbb{C}$
satisfying
\begin{equation}\label{eqn:transformation-law-I-of-0}
f(n(x) a(y) g)
= |y|^{1/2} f(g).
\end{equation}
It is a generic irreducible unitary representation. It arises
naturally as a local component of the Eisenstein series central
to this paper.
For $f \in \mathcal{I}(0)$,
we write $W_f$
for the Whittaker function,
given for $g \in G$ by
\begin{equation}\label{eqn:W-f-via-f-0-defn}
W_f(g) :=
\int_{x \in F}
f(w n(x)g) \psi(-x) \, d x.
\end{equation}
This and similar integrals must be understood in general via
meromorphic continuation or regularization. For instance, in
the non-archimedean case,
a standard regularization
\cite[\S1.9]{MR3889963}
is
given by summing the integrals over cosets of the
unit group. One may similarly regularize in the archimedean
case via a smooth dyadic partition of unity
or convolution (see \eqref{eq:whittaker-function-explicated-defn} for details).
We also write $W_f : F^\times \rightarrow \mathbb{C}$
for the corresponding Kirillov model element,
given for $t \in F^\times$ by
\begin{equation}
W_f(t) :=
W_f(a(t))
= |t|^{1/2}
\int_{x \in F}
f(w n(x)) \psi(-t x) \, d x,
\end{equation}
which satisfies the (convergent) Fourier inversion formula
\begin{equation}\label{eqn:f-from-Wf-Fourier-inv}
\int_{t \in F}
|t|^{-1/2} W_f(t) \psi(t x) \, d t
= f (w n(x))
\end{equation}
(see around \eqref{eq:fourier-inversion-for-whittaker-intertwiner}
for details).
\subsection{A representation $\pi$ of $G \times G$}\label{sec:representation-pi-g}
We denote by $\mathcal{I}(0) \otimes \mathcal{I}(0)$ the space
of smooth functions $f : G \times G \rightarrow \mathbb{C}$
satisfying
\[
f(n(x_1) a(y_1) g_1, n(x_2) a(y_2) g_2)
= |y_1 y_2|^{1/2} f(g_1,g_2).
\]
It defines a representation of $G \times G$.
We will refer often to this representation,
so we abbreviate
\begin{equation}\label{eqn:defn-pi-I0-I0}
\pi := \mathcal{I}(0) \otimes \mathcal{I}(0)
\end{equation}
A pair of elements $f_1, f_2 \in \mathcal{I}(0)$
defines an element $f_1 \otimes f_2 \in \pi$
by the rule $(f_1 \otimes f_2)(g_1,g_2)
= f_1(g_1) f_2(g_2)$.
In the non-archimedean case, $\pi$ identifies with the usual tensor product;
in the archimedean case, it may be interpreted
as the completed tensor product,
regarding $\mathcal{I}(0)$
as a
nuclear Fr{\'e}chet space.
In all cases, $\pi$ satisfies the universal property: any
continuous bilinear form on $\mathcal{I}(0)$ extends uniquely to
a continuous linear functional on $\pi$,
as follows (for instance) from
\S\ref{sec:reduct-pure-tens} below.
In our global analysis, an element of $\pi$
gives local data describing a linear combination of products of
pairs of Eisenstein series, as in
\eqref{eq:integral-product-eis}.
\subsection{Some $A$-invariant functionals on $\pi$}
\label{sec-6-3}
We consider two families
of $A$-invariant functionals
$\pi \rightarrow \mathbb{C}$,
the motivation for which
should be clear from \S\ref{sec:intro-motivation}.
(Here we identify $G$ with a subgroup of $G \times G$
via the diagonal embedding;
in particular, we regard $A$
as a subgroup of $G \times G$.)
\subsubsection{$G \times G \geq G \geq A$}
One family is indexed by generic irreducible
representations
$\sigma$ of $G$:
for each such representation,
we define
\begin{equation}\label{eq:defn-ell-sigma-basic}
\ell_\sigma (f_1 \otimes f_2)
:=
\sum_{W \in \mathcal{B}(\sigma)}
(\int_{N \backslash G}
\tilde{W} W_{f_1} f_2)
\int_A W
\end{equation}
Here and henceforth $W$ traverses a basis
for the Whittaker model $\mathcal{W}(\sigma,\psi)$
and
$\tilde{W}$
the corresponding element
of a dual basis for the dual Whittaker model
$\mathcal{W}(\sigma,\bar{\psi})$,
with the duality given by regularized integration over $A$
(\S\ref{sec:whitt-intertw-dual}).
We note in passing that $\sigma$,
like any irreducible representation of $G$, is self-dual,
so it is unnecessary
to pass to the contragredient.
The integral over $N \backslash G$ is to
be understood as a local Rankin--Selberg integral (see
\S\ref{sec-4-5} for details); it converges if $\sigma$ is
unitary, and may be meromorphically continued to all $\sigma$
for which $L(\sigma,1/2)$ is finite. The integral over $A$ is
a local Hecke integral to which similar remarks apply (see
\S\ref{sec-4-3} for details).
The precise choice of basis $\mathcal{B}(\sigma)$
is described in \S\ref{sec:norms-whit-type-reps}.
In particular,
we take $W, \tilde{W}$
to be $K$-isotypic.
The sum \eqref{eq:defn-ell-sigma-basic}
then
converges as written;
see \S\ref{sec:defn-ell-sigma-s-makes-sense} for details.
The definition may be understood
without reference to a basis:
$f_1 \otimes f_2 \mapsto \sum_{W \in \mathcal{B}(\sigma)} (\int_{N
\backslash G} \tilde{W} W_{f_1} f_2) W$ is the
linear map $\pi \rightarrow \mathcal{W}(\sigma,\psi)$ adjoint to
the Rankin--Selberg functional
$\pi \otimes \mathcal{W}(\sigma,\overline{\psi }) \rightarrow
\mathbb{C}$.
The existence of the required adjoint
is closely related to the convergence
of the indicated sum over suitable bases.
We will be primarily (although not exclusively) concerned with
the case that $\sigma$ is unitary, in which case one may take
for $\tilde{W}$ the complex conjugate of $W$ and for
$\mathcal{B}(\sigma)$ an orthonormal basis of
$\mathcal{W}(\sigma,\psi)$, with the norm defined by the
absolutely-convergent integral over $A$.
\subsubsection{$G \times G \geq A \times A \geq A$}
The other family of functionals is indexed
by characters $\omega$ of $F^\times$:
for each such character,
we define
\[
\ell_{\omega}(f_1 \otimes f_2)
:=
(\int_{A}
W_{f_1} \omega )
(\int_{A}
W_{f_2} \omega^{-1} ).
\]
The integrals are understood as local Hecke integrals (\S\ref{sec-4-3});
they converge for unitary $\omega$,
and extend meromorphically to all $\omega$
for which $L(\omega, 1/2)$ is finite.
\subsubsection{Further remarks}
These definitions extend continuously
from pure tensors to general elements of $\pi$
(see \S\ref{sec:defn-ell-sigma-s-makes-sense} for
proof of continuity).
They may be
understood as compositions
\begin{equation}\label{eqn:ell-sigma-arrows}
\ell_{\sigma} : \pi \rightarrow \sigma \rightarrow \mathbb{C},
\end{equation}
\begin{equation}\label{eqn:ell-omega-arrows}
\ell_{\omega} : \pi \rightarrow
\omega^{-1} \otimes \omega \rightarrow \mathbb{C}.
\end{equation}
The first arrow in \eqref{eqn:ell-sigma-arrows}
is $G$-invariant,
and described by the Rankin--Selberg integral;
the second is $A$-invariant,
and described by the Hecke integral.
The first arrow in \eqref{eqn:ell-omega-arrows}
is $A \times A$-invariant,
and given by a pair of Hecke integrals;
the second is the $A$-invariant canonical pairing.
\subsection{Variation with respect to the additive character}\label{sec:vari-with-resp-1}
The definitions of the functionals $\ell_{\sigma}$ and
$\ell_{\sigma}$ depend upon the choice of additive character
$\psi$. The dependence is mild: changing $\psi$ to
$\psi^b(x) := \psi(b x)$ has the effect of multiplying these
functionals by an explicit power of $|b|$. For the purpose of
proving estimates, there is thus no harm in assuming $\psi$ to
be of a convenient form, e.g., unramified when $F$ is
non-archimedean. We record explicit formulas describing this
variation, in a more general setting, in
\S\ref{sec:vari-with-resp}.
\subsection{Definition of admissible weights}
\label{sec-6-4}
Recall the general notation of \S\ref{sec-4-1}.
We say that
a function
\[
h : G^\wedge_{\gen} \rightarrow \mathbb{C},
\]
respectively,
\[
\tilde{h} : A^{\wedge} \rightarrow \mathbb{C},
\]
is
\emph{admissible} if for some $f \in \pi$
we have
\[\text{$h(\sigma) = \ell_\sigma(f)$ for all $\sigma$,}\]
respectively,
\[\text{$\tilde{h}(\omega) = \ell_{\omega}(f)$
for all $\omega$.}\]
We say that
$h$ and $\tilde{h}$
are dual
if they may be defined using
the same $f$.
\subsection{Crude estimates}\label{sec:crude-estimates}
\begin{lemma}\label{lem:crude-estimates}
Any admissible weight $h$ or $\tilde{h}$ as above is
meromorphic on its domain, with poles controlled by local
$L$-factors in the sense
described in \S\ref{sec:local-inv-func}.
There
are no poles on the unitary subsets of
$A^\wedge, G^\wedge_{\gen}$.
Any admissible weight is bounded and rapidly-decreasing
on the unitary subset
$\{\text{unitary } \sigma \in G^\wedge_{\gen}\}$.
More precisely:
\begin{itemize}
\item If $F$ is non-archimedean, then $h$ and $\tilde{h}$ are
uniformly bounded on the unitary subsets, and vanish on
arguments of sufficiently large conductor.
\item If $F$ is archimedean, then the restrictions of
$h$ and
$\tilde{h}$ to the unitary subset are bounded by constant
multiples of $C(\cdot)^{-d}$ for each $d \geq 0$.
\end{itemize}
\end{lemma}
The conclusion follows from the smoothness of $f$ and
``integration by parts''
(see around
\eqref{eq:crude-estimate-for-ell-sigma-s}
and
\eqref{eq:crude-ell-omega-s-estimate}
for details).
\subsection{Models}
\label{sec-6-5}
Here we consider a pair of models of the representation
$\pi$ that arise
naturally when studying the functionals
$\ell_{\sigma}$ and $\ell_{\omega}$.
\subsubsection{}
We write $C^\infty(N \backslash G, \psi)$
for the space of smooth functions $V$ on $G$
satisfying $V(n(x) g) = \psi(x) V(g)$.
For $f = f_1 \otimes f_2 \in \pi$,
we define
\[V_f \in C^\infty(N \backslash G, \psi),
\quad
W_f \in C^\infty(F^\times \times F^\times)
\]
by
the formulas
\begin{equation}
V_f(g) := W_{f_1}(g) f_2(g),
\end{equation}
\begin{equation}
W_f(t_1,t_2) := W_{f_1}(t_1) W_{f_2}(t_2).
\end{equation}
The map $f \mapsto V_f$
arose also in \cite[\S3.2.7]{michel-2009}.
These definitions extend continuously to all $f \in
\pi$,
and may be defined directly
by the formulas
\begin{equation}\label{eqn:V_f-formula}
V_f(g)
= \int_{x \in F} f(w n(x) g, g) \psi(-x) \, d x,
\end{equation}
\begin{equation}\label{eqn:W-f-formula}
W_f(t_1,t_2)
=
|t_1 t_2|^{1/2}
\int_{x_1,x_2 \in F}
f(w n(x_1), w n(x_2)) \psi(- t_1 x_1 - t_2 x_2) \, d x_1 \, d x_2.
\end{equation}
The functionals defined previously
are then given by
\begin{equation}\label{eqn:ell-sigma-via-V-f-sigma}
\ell_\sigma(f)
=
\int_{A} (V_f)_{\sigma}
\text{ with }
(V_f)_\sigma :=
\sum_{W \in \mathcal{B}(\sigma)}
(\int_{N \backslash G}
\tilde{W} V_f)
W,
\end{equation}
\begin{equation}
\ell_\omega(f)
=
\int_{A \times A}
W_f \cdot (\omega \otimes \omega^{-1} ).
\end{equation}
The maps $f \mapsto V_f$ and $f \mapsto W_f$ define models of
$\pi$.
Our aim in this section is to verify that these models
contain the compactly-supported elements and to establish
integral transforms relating them.
We will see later that such transforms
are at the heart
of the spectral identities
discussed in \S\ref{sec-2}.
\subsubsection{}
We define for any $V \in C_c^\infty(N \backslash G,\psi)$
the integral transform
$V^\wedge : F^2 \rightarrow \mathbb{C}$ by
\begin{equation}\label{eqn:V-wedge-defn}
V^\wedge(\xi,z)
:=
\int_{y \in F^\times}
|y|^{-1} V(a(y) n'(z)) \psi(\xi y) \, d y.
\end{equation}
For $f \in \pi$, we may define
$V_f^\wedge : F^2 \rightarrow \mathbb{C}$ by the same formula.
We note that
if $V$ is supported on
the dense open
neighborhood $N A N'$ of the identity in $G$,
then the map $F^\times \times F \ni (y,z) \mapsto V(a(y) n'(z))$
is compactly-supported,
and so $V^\wedge$ belongs to the Schwartz space
$\mathcal{S}(F^2)$.
\begin{lemma}
For $f \in \pi$
and almost all $(\xi,z) \in F^2$,
we have
\begin{align}\label{eqn:formula-Vf-wedge-via-f-1}
V_f^\wedge(\xi,z)
&=
f(w n(\xi) n'(z), n'(z))
\\
\label{eqn:formula-Vf-wedge-via-f-2}
&=
|x_2 - x_1|
f(w n(x_1), w n(x_2))
\end{align}
where
\begin{equation}
x_1 =
\frac{\xi }{1 + \xi z},
\quad
x_2 = \frac{1}{z},
\end{equation}
so that
\begin{equation}\label{eqn:xi-z-via-x2-x1}
\xi = \frac{x_2 x_1}{x_2 - x_1},
\quad
z = \frac{1}{x_2}.
\end{equation}
\end{lemma}
\begin{proof}
The first identity follows (for $f = f_1 \otimes f_2$,
hence in general)
from
\eqref{eqn:f-from-Wf-Fourier-inv}, the second from the readily
verified identities
\[
w n(\xi) n'(z)
=
n(\dotsb)
a (\frac{1}{(1 + \xi z)^2})
w n(x_1),
\]
\[
n'(z) = n(1/z) a(1/z^2) w n(x_2),
\]
\[
x_2 - x_1
= \frac{1}{z(1+\xi z)}
\]
and the transformation law \eqref{eqn:transformation-law-I-of-0}.
\end{proof}
\subsubsection{}
We verify here that our models contain
the compactly-supported elements.
\begin{lemma}\label{lem:many-W-f}
The set $\{W_f : f \in \pi \}$
contains $C_c^\infty(F^\times \times F^\times)$.
\end{lemma}
\begin{proof}
This may be deduced from the corresponding property of the
Kirillov model of $\mathcal{I}(0)$, or proved in the same way.
\end{proof}
We have recorded lemma \ref{lem:many-W-f} for motivational
purposes; we do not use it directly in the present work.
More relevant for our purposes is the following analogue:
\begin{lemma}\label{lem:many-V-f}
The set $\{V_f : f \in \pi \}$ contains
$C_c^\infty(N \backslash G, \psi)$.
\end{lemma}
Before giving the proof, we sketch informally why this should be
true. Take $f = f_1 \otimes f_2$, where
$y \mapsto W_{f_1}(a(y))$ and $z \mapsto f_2(n'(z))$ approximate
$\delta$-masses
at
$y = 1$ and $z=0$, respectively. Then
$V_f(a(y) n'(z))$ approximates a $\delta$-mass at $y=1, z=0$.
By taking linear combinations of translates of $f$, we can
approximate any element of $C_c^\infty(N \backslash G, \psi)$,
giving something like the required conclusion.
\begin{proof
Both sets in question are invariant under right translation by
$G$, so we reduce to verifying that $\{V_f : f \in \pi \}$
contains every element $V$ of
$C_c^\infty(N \backslash G, \psi)$ whose support is contained
in some small neighborhood of the identity. In particular, we
may assume that $V$ is supported on $N A N'$,
so that $V^\wedge$ is Schwartz.
We aim to construct $f \in \pi$ with $V_f = V$. It is enough
to check that $V_f^\wedge = V^\wedge$. It seems simplest to
us to use the following Schwartz space parametrization: for
each $\Phi \in \mathcal{S}(F^3)$, there is a unique $f \in \pi$
for which
\[
f(g, n'(z))
= |\det g|^{1/2}
\int_{r \in F}
\Phi((0,r) g, z) \, d r.
\]
Let us compute $V_f^\wedge$ for such an $f$.
First, we specialize \eqref{eqn:V_f-formula}
to see that
\begin{align*}
V_f(a(y) n'(z))
=
|y|
\int_{\xi \in F}
f(w n(\xi) n'(z), n'(z))
\psi(-\xi y) \, d \xi.
\end{align*}
Since $(0,r) w n(\xi) n'(z)
= (r (1 + \xi z), r \xi)$,
the definition of $f$
then gives
\[
V_f(a(y) n'(z))
=
|y|
\int_{\xi,r \in F}
\Phi(r ( 1 + \xi z), r \xi, z)
\psi(-\xi y) \, d \xi \, d r,
\]
hence
\[
V_f^\wedge(\xi,z) =
\int_{r \in F}
\Phi(r (1 + \xi z), r \xi, z)
\, d r.
\]
This formula suggests the choice
\begin{equation}\label{eqn:Phi-via-phi-0-V-wedge}
\Phi(x,y,z)
:=
\phi_0(x - y z)
V^\wedge (\frac{y}{x - y z}, z)
\end{equation}
for some
$\phi_0 \in C_c^\infty(F^\times)$
with $\int_{r \in F} \phi_0(r) \, d r = 1$.
Then $V_f^\wedge = V^\wedge$.
\end{proof}
\subsubsection{}
For $V \in C_c^\infty(N \backslash G, \psi)$,
we define the integral
transform $V^\sharp : F^2 \rightarrow \mathbb{C}$ by
the convergent integral
\begin{equation}\label{eqn:V-sharp-defn}
V^\sharp(x,y)
:=
\int _{\xi \in F}
V^\wedge(\xi,-x/\xi) \psi(-\xi y) \, d \xi.
\end{equation}
\begin{lemma}\label{lem:W_f-via-V_f-0}
For $f \in \pi$,
we have
\begin{equation}\label{eqn:W-f-via-V-f-sharp}
W_f(t_1,t_2)
=
|t_1 t_2|^{1/2}
\int_{x \in F}
V_f^\sharp (x, \frac{(t_1 + t_2) x - t_2}{x(1-x)})
\, \frac{d x}{|x(1-x)|}.
\end{equation}
\end{lemma}
The RHS of \eqref{eqn:W-f-via-V-f-sharp} does not in general
converge absolutely, but may be regularized as in the definition
\eqref{eqn:W-f-via-f-0-defn} of the Whittaker intertwiner,
either by analytic continuation with respect to the parameters
of the induced representations
$\mathcal{I}(s_1) \otimes \mathcal{I}(s_2)$ deforming
$\pi = \mathcal{I}(0) \otimes \mathcal{I}(0)$, or using smooth
dyadic partitions of unity near the singular points $x =0$ and
$x = 1$. Indeed, the proof below will show that the $x$ and
$\xi$ integrals in \eqref{eqn:V-sharp-defn} and
\eqref{eqn:W-f-via-V-f-sharp} arise via a change of variables
from a pair of Whittaker intertwiners
\eqref{eqn:W-f-via-f-0-defn}.
\begin{proof}
By the formulas \eqref{eqn:W-f-formula} and
\eqref{eqn:formula-Vf-wedge-via-f-2} relating $W_f$ and
$V_f^\wedge$ to $f$,
we have
\begin{equation}
W_f(t_1,t_2)
=
|t_1 t_2|^{1/2}
\int_{x_1, x_2 \in F}
\psi(-t _1 x_1 - t_2 x_2)
V_f^\wedge(\xi,z) \, \frac{d x_1 \, d x_2}{|x_2-x_1|},
\end{equation}
where $\xi, z$ are given by \eqref{eqn:xi-z-via-x2-x1}.
We set $x := - \xi/x_2$, so that
$x_1 = \xi / (1-x), x_2 = - \xi/x$
and
\[
t_1 x_1 + t_2 x_2 = \xi \frac{(t_1 + t_2) x - t_2}{x(1-x)}.
\]
The Jacobian calculation
$|\partial(x_1,x_2)/\partial(\xi,z)|
= |x_2 - x_1|/|x(1-x)|$
then yields \eqref{eqn:W-f-via-V-f-sharp}.
\end{proof}
\begin{remark}
Lemmas \ref{lem:many-V-f} and \ref{lem:W_f-via-V_f-0}
should hold (in modified form) for
any representation $\pi$ of $G \times G$ of the form
$\pi_1 \otimes \pi_2$, with $\pi_1$ generic irreducible and
$\pi_2$ belonging to the principal series. The proof
technique indicated above applies when $\pi_1$ also belongs to
the principal series (see \S\ref{sec:holom-famil-local}); in
general, one can argue using the local functional equation for
$\pi_1$. The case indicated above is the most relevant one
for our applications.
\end{remark}
\subsection{Pre-Kuznetsov weights}\label{sec:pre-kuzn-weights}
\begin{definition}
By a \emph{pre-Kuznetsov weight}
$h : G_{\gen}^\wedge \rightarrow \mathbb{C}$,
we mean
a function
for which there exists
$\phi \in C_c^\infty(G)$
so that
\begin{equation}\label{eqn:defn-h-via-phi}
h(\sigma) =
\sum_{W \in \mathcal{B}(\sigma)}
(\int_G \tilde{W} \phi)
W(1)
\end{equation}
for all $\sigma$.
In that case, we refer to $\phi$ as a \emph{kernel} for $h$.
\end{definition}
We give several examples of such weights in
\S\ref{sec:constr-suit-weight}. We note the sum
\eqref{eqn:defn-h-via-phi} converges absolutely and that any
such $h$ is holomorphic (\S\ref{sec:holomorphy-of-pre-K-wts}).
Moreover, if $\phi = \phi_1 \ast \phi_2$ is the convolution
of $\phi_1, \phi_2 \in C_c^\infty(G)$,
then
\begin{equation}
h(\sigma) = \sum_{W \in \mathcal{B}(\sigma)}
(\int_G \tilde{W} \phi_1)
(\int_G W \phi_2^\iota),
\quad
\phi_2^{\iota}(g) := \phi_2(g^{-1}).
\end{equation}
\begin{remark}
In the archimedean case, one could generalize the above
definition to allow rapidly decaying $\phi$ in the sense of
\cite[p392]{MR1013462}, but doing so is not necessary for our
purposes.
\end{remark}
We record here a cheap construction that is often useful at
``bad primes.''
\begin{lemma}\label{lem:crude-lower-bound-individual}
For each compact subset $\Sigma \subseteq G^\wedge_{\gen}$
consisting of unitary representations,
there is a pre-Kuznetsov weight
$h : G^\wedge_{\gen} \rightarrow \mathbb{C}$ for which
\begin{itemize}
\item $h(\sigma) \geq 0$ for all
unitary $\sigma \in G^\wedge_{\gen}$, and
\item $h(\sigma) \geq 1$ for all $\sigma \in \Sigma$.
\end{itemize}
\end{lemma}
\begin{proof}
By continuity and compactness, we may assume that
$\Sigma = \{\sigma _0\}$ is a singleton. Let
$W_0 \in \sigma_0$ with $W_0(1) = 1$. By continuity, we may
find a small neighborhood $U$ of the identity in $G$ so that
$\Re(W_0(u)) \geq 1/2$ for all $u \in U$. Choose a
nonnegative-valued $\phi_0 \in C_c^\infty(U)$ with
$\int \phi_0 = 1$. Take for $\phi$ the convolution
$\phi_0^* \ast \phi_0$, where
$\phi_0^*(g) := \overline{\phi_0(g^{-1})}$. Take for $h$ the
pre-Kuznetsov weight with kernel $\phi$. Then
\[
h(\sigma)
= \sum_{W \in \mathcal{B}(\sigma)}
|\phi_0 \ast W|^2(1) \geq 0.
\]
Moreover, $\Re(\phi_0 \ast W_0) \geq 1/2$, so likewise
$h(\sigma_0) \geq (1/4) \|W_0\|^{-2} > 0$. We conclude by
replacing $h$ with a suitable positive multiple.
\end{proof}
Lemma \ref{lem:crude-lower-bound-individual} has the
disadvantage of being ineffective. For instance, invoking it in
the proof of Theorem \ref{thm:CI} would lead to a weaker form of
the estimate \eqref{eq:weyl-bound-sigma-chi} in which the
implied constant depends in an unspecified manner (rather than
polynomially) upon $\sigma$. A more complicated but effective
variant will be given below in \S\ref{sec:crude-local-estim}.
\subsection{Whittaker vectors}\label{sec:whittaker-vectors}
\subsubsection{}
Let $\sigma$ be a smooth representation of $G$ (smooth moderate
growth Fr{\'e}chet
in the archimedean case).
The main example here is when $\sigma$
is the restriction to $G \hookrightarrow G \times G$
of $\pi$.
By a \emph{generalized vector} in $\sigma$, we mean an element
of the algebraic dual of the contragredient. Any closely
related definition (e.g., via negative index Sobolev spaces as
in \cite[\S2]{michel-2009} or \cite[\S3.2]{nelson-venkatesh-1}
or \S\ref{sec:norms-repr}) would also work for us. We identify
$\sigma$ with a subspace of the space of generalized vectors.
The $G$-action extends naturally to this larger space.
By a \emph{$\psi$-Whittaker vector} (or
simply \emph{Whittaker vector} when $\psi$ is understood) we
mean a generalized vector $v$ for which $n(x) v = \psi(x) v$ for
all $x \in F$.
\subsubsection{}
Given any (smooth) vector $v$,
the formula
\begin{equation}
N_\psi v := \int_{x \in F}
\psi(-x) n(x) v \, d x
\end{equation}
defines a Whittaker vector in $\sigma$.
We note that $N_\psi$ commutes with $G$-equivariant homomorphisms.
\subsubsection{}
Given any $\phi_0 \in C_c^\infty(A)$,
the convolution
\begin{equation}\label{eqn:phi0-compose-N-psi-v}
\phi_0 \ast N_\psi v :=
\int_{a \in A}
\phi_0(a) a N_\psi v
\end{equation}
defines an actual vector, i.e., an element of $\sigma$.
We may give a direct definition of this vector
via the formula
\begin{equation}\label{eqn:direct-defn-phi0-N-psi}
\phi_0 \ast N_\psi v
=
\int_{x \in F}
(\int_{u \in F^\times}
\phi_0(1/u)
\psi(-x u)
n(x) a(1/u) v
\, \frac{d u }{|u|})
\, d x
\end{equation}
obtained from \eqref{eqn:phi0-compose-N-psi-v}
by writing $a = a(1/u)$ substituting
$x \mapsto xu$.
The iterated integral
\eqref{eqn:direct-defn-phi0-N-psi}
converges in the indicated order.
(To see this,
integrate by parts in the $u$-integral with
respect to the phase $\psi(-x u)$
in
the archimedean case,
and
use that $v$ is invariant by an open subgroup of the unit group
in the non-archimedean case.)
It follows that for any
functional $\ell$ on $\pi$ that transforms under $A$ with
respect to a character $\nu$, we may canonically define
$\ell(N_\psi v)$ to be $\ell(\phi_0 \ast N_\psi V)$ for any
$\phi_0$ whose Mellin transform takes the value $1$ at $\nu$.
\subsubsection{}
We record how $N_\psi$ acts on the $\psi$-Kirillov model of a
generic irreducible representation $\sigma$ of $G$. Let
$W \in \sigma$, realized as
$W : F^\times \rightarrow \mathbb{C}$ with
the actions
$n(x) W(t) = \psi(x t ) W(t)$, $a(y) W(t) = W(t y)$. Then
$N_\psi W = W(1) \delta_1$, where $\delta_1$ denotes the
$\delta$-mass at $1 \in F^\times$. It follows that
\begin{equation}\label{eqn:action-phi0-N-psi-on-kirillov-model}
\phi_0 \ast N_\psi W(t) =
\phi_0(1/t) W(1).
\end{equation}
\subsubsection{}
For the sake of concreteness, we will work throughout the paper
primarily with the smooth vectors defined by
\eqref{eqn:phi0-compose-N-psi-v} rather than the generalized
vectors $N_\psi v$.
\subsection{Admissibility of
pre-Kuznetsov weights}\label{sec:constr-admiss-weight}
\label{sec-6-6}
We are now prepared to state the precise form of Theorem
\ref{thm:summarize-kuznetsov-weights-adm}.
\begin{theorem}\label{thm:constr-admiss-weight}
Let $h : G_{\gen}^\wedge \rightarrow \mathbb{C}$
be a pre-Kuznetsov
weight
with kernel $\phi \in C_c^\infty(G)$.
Then $h$ is admissible (\S\ref{sec-6-4}).
An admissible dual
$\tilde{h} : A^\wedge \rightarrow \mathbb{C}$
is
given by the convergent formulas
\begin{equation}\label{eqn:h-tilde-of-omega-via-h-sharp}
\tilde{h}(\omega)
=
\int_{t \in F}
h^\sharp(t)
\omega (\frac{1-t}{t})
\, \frac{d t}{|t(1-t)|^{1/2}}
\end{equation}
where
\begin{equation}
h^\sharp(t)
= \int_{x \in F}
V^\sharp(x, \frac{x-t}{x(1-x)})
\, \frac{d x}{|x(1-x)|}
\end{equation}
where $V^\sharp$ is defined
as above in terms of the element
$V \in C_c^\infty(N \backslash G, \psi)$
defined by
\begin{equation}\label{eqn:defn-V-via-phi}
V(g) := \int_{x \in F} \phi(n(x) g) \psi(-x) \, d x.
\end{equation}
\end{theorem}
\begin{proof}
Note that $\int_G \tilde{W} \phi
= \int_{N \backslash G} \tilde{W} V$.
By lemma \ref{lem:many-V-f},
we may find $f_0 \in \pi$ so that
$V_{f_0} = V$.
Define
\begin{equation}\label{eqn:V-sig-proj-defn}
V_\sigma(g) := \sum_{W \in
\mathcal{B}(\sigma)}
(\int_{N \backslash G} V \tilde{W}) W(g).
\end{equation}
(See \S\ref{sec:holomorphy-of-pre-K-wts}
for details regarding convergence.)
Then $V_\sigma(1) = h(\sigma)$.
Choose $\phi_0 \in C_c^\infty(F^\times) \cong C_c^\infty(A)$
with $\int_{A} \phi_0 = 1$,
set $\phi_0^{\iota}(y) := \phi_0(1/y)$,
and define
(with notation as in \S\ref{sec:whittaker-vectors})
\begin{equation}\label{eqn:defn-f-via-f0-basic-case}
f
:=
\phi_0^{\iota} \ast N_\psi f_0
\in \pi.
\end{equation}
In the Kirillov model,
we
may expand
\[
W_f(t_1,t_2)
=
\int_{x \in F}
\int_{u \in F^\times}
\phi_0(u)
\psi(((t_1 + t_2)/ u - 1) x)
W_{f_0}(t_1 /u, t_2 /u)
\, d x
\, \frac{d u}{|u|}
\]
and apply Fourier inversion
to see that
\begin{equation}\label{eqn:W-f-via-W-f0}
W_f(t_1,t_2)
=
\phi_0(t_1 + t_2)
W_{f_0} (\frac{t_1}{t_1 + t_2}, \frac{t_2}{t_1 + t_2}).
\end{equation}
For each $\sigma$, the projection
$(V_f)_{\sigma}$
defined by analogy to \eqref{eqn:V-sig-proj-defn}
is given by
\begin{equation}
(V_f)_\sigma(a(y)) = \phi_0(y) (V_{f_0})_\sigma(1),
\end{equation}
by \eqref{eqn:action-phi0-N-psi-on-kirillov-model}.
Thus
\begin{equation}
\ell_{\sigma}(f) = (V_{f_0})_\sigma(1) = h(\sigma).
\end{equation}
It remains only to verify the required formula
for $\ell_{\omega}(f)$.
By definition,
\[
\ell_{\omega}(f)
=
\int_{t_1,t_2}
\omega(t_1/t_2)
W_f(t_1,t_2)
\, \frac{d t_1 \, d t_2}{|t_1 t_2|}
=
\int_{s,t}
\omega(s)
W_f(s t,t)
\, \frac{d s \, d t}{|s t|}.
\]
Inserting \eqref{eqn:W-f-via-W-f0}
gives
\begin{equation}
\ell_{\omega}(f)
=
\int_{s,t}
\omega(s)
\phi_0((s+1)t)
W_{f_0}
(\frac{s}{1+s},
\frac{1}{s + 1})
\, \frac{d s \, d t}{|s t|}.
\end{equation}
Invoking the normalization of $\phi_0$,
the above integral simplifies to
\[
\int_{s}
\omega(s)
W_{f_0}
(\frac{s}{s + 1},
\frac{1}{s + 1})
\, \frac{d s}{|s|}.
\]
The substitution $s := 1/(1-t)$
then yields
\[
\int_{t}
\omega(\frac{1-t}{t})
W_{f_0}
(1 - t, t)
\, \frac{d t}{|t(1-t)|}.
\]
We evaluate
$W_{f_0}$ using \eqref{eqn:W-f-via-V-f-sharp}
to arrive at
\eqref{eqn:h-tilde-of-omega-via-h-sharp}.
The convergence in each step above follows
from that of the Hecke integrals $\ell_{\omega}(f)$.
\end{proof}
\begin{remark}
One can verify, using the Whittaker--Plancherel
theorem for $G$, that
\begin{equation}
\int_{\text{unitary }\sigma \in G^\wedge_{\gen}}
|h(\sigma)|^2
= \int_{\text{unitary }\omega \in A^\wedge}
|\tilde{h}(\omega)|^2,
\end{equation}
where the integrals are taken with respect
to the Plancherel measures dual to the measures
defined on $G$ and $A$, respectively.
\end{remark}
\begin{remark}
A pre-Kuznetsov weight $h$ typically admits many kernels
$\phi$, so the reader may find it unnatural that $\tilde{h}$
is expressed in terms of $\phi$ rather than $h$.
One can express $\tilde{h}$ directly in
terms of the Bessel transform
$J_h(g) := \int_\sigma h(\sigma) J_\sigma(g)$, where
$J_\sigma(g)$ denotes the Bessel distribution
$\sum_{W \in \mathcal{B}(\sigma)}\tilde{W}(1) W(g)$,
but our experience suggests that it is more efficient
to pass first through $\phi$.
\end{remark}
\section{Local estimates for short families}\label{sec:appl-cubic-moment}
\subsection{Families}\label{sec:short-families}
Here we record some notation for referring in a unified manner
to families of representations corresponding to ``short''
families of automorphic forms, such as
\begin{itemize}
\item for a large positive real $T$,
the set of Maass forms of eigenvalue $1/4 + t^2$
for some $t \in [T-1, T + 1]$
(corresponding below to the family
$\Sigma_{\mathbb{R}}(\omega)$
with $\omega = |.|^{i T}$), or
\item
for a large prime $p$,
the set of twists by the quadratic character $(\cdot|p)$
of a newform on $\SL_2(\mathbb{Z})$ or $\Gamma_0(p)$
(corresponding below to the family $\Sigma_{\mathbb{Q}_p}(\omega)$,
with $\omega$ a ramified quadratic character of $\mathbb{Q}_p^\times$).
\end{itemize}
Let $F$ be a local field. We call a character $\chi$
of
$F^\times$ \emph{analytically unramified} if
\begin{itemize}
\item $F$ is non-archimedean and $\chi$ is unramified
in the usual sense (trivial restriction
to the unit group), or
\item $F$ is archimedean and $\chi = |.|^{s}$
for some $s \in \mathbb{C}$ with $|\Im(s)| \leq 1$.
\end{itemize}
For a character $\chi$ of $F^\times$, let $\Omega_F(\chi)$
denote the set of characters $\omega$ of $F^\times$ for which
the ratio $\omega/\chi$ is analytically unramified,
and let
$\Sigma_F(\chi)$ denote the set of irreducible representations
$\sigma$ of $\operatorname{PGL}_2(F)$ for which
there exists $\omega \in \Omega_F(\chi)$
so that either
\begin{itemize}
\item $\sigma$ is the principal series representation
$\mathcal{I}(\omega)$ induced by
$\omega$ (\S\ref{sec:intro-representations}), or
\item
$F$ is non-archimedean, $\omega$ is quadratic and $\sigma$ is the twist by
$\omega$ of the Steinberg representation of $G$
(thus $\sigma$ is the unique irreducible subrepresentation
of $\mathcal{I}(\omega |.|^{1/2})$).
\end{itemize}
\subsection{Construction of suitable weights}\label{sec:constr-suit-weight}
Let $F$ be a local field,
let $\psi$ be a nontrivial unitary character of $F$,
and let $\chi$ be a unitary character
of $F^\times$.
We denote by $Q := C(\chi)$ its analytic conductor.
We aim to construct a pre-Kuznetsov weight
$h$ that is nonnegative on unitary representations
and uniformly bounded from below on the family $\Sigma_F(\chi)$.
We give the construction of $h$ separately in the
non-archimedean and archimedean cases, but the reader will
notice very close parallels.
\subsubsection{Non-archimedean case}\label{sec:constr-wt-non-archimedean-case}
Suppose that $F$ is non-archimedean.
We assume in this case that $\chi$ is ramified,
so that $Q \in \{q, q^2, q^3, \dotsc \}$.
We assume also that $\psi$ is unramified;
this last assumption simplifies slightly the construction,
but has no effect on the subsequent estimates,
as explained in \S\ref{sec:vari-with-resp-1}.
Let $J \leq G$ denote the compact open subgroup consisting of
elements of the form
\[
\text{$g = n(x) a(y) n'(z)$
with $|x| \leq 1, |y| = 1, |z| \leq 1/Q$.}
\]
We note that $\vol(J) = \zeta_F(1)/Q$.
Let $\chi_J$ denote the
character of $J$ given by $n(x) a(y) n'(z) \mapsto \chi(y)$.
Define
$\phi \in C_c^\infty(G)$ to be supported on $J$ and given there
by $\chi_J^{-1}$.
Let $h : G^\wedge_{\gen} \rightarrow \mathbb{C}$ denote the
pre-Kuznetsov weight with kernel $\phi$.
\subsubsection{Archimedean case}\label{sec:constr-wt-archimedean-case}
Suppose now that $F$ is archimedean, thus $F = \mathbb{R}$ or
$F = \mathbb{C}$.
We take $\alpha_0 \in (0,1)$ sufficiently small but fixed, and
then take $\alpha_1 \in (0,1)$ fixed but small enough in terms
of $\alpha_0$. We set
\begin{equation}
J := \left\{ n(x) a(y) n'(z) :
|x|, |y-1|, Q |z| \leq \alpha_1
\right\},
\end{equation}
and note that $\vol(J) \asymp 1/Q$.
We take for $\phi_0 \in C_c^\infty(G)$ a ``smoothened
characteristic function of $J$,'' by which we mean more
precisely an element with the following properties:
\begin{itemize}
\item $\phi_0$ is supported in $J$, and nonnegative.
\item $\int_G \phi_0 \asymp \vol(J)$.
\item
For any fixed multi-indices $\alpha,\beta,\gamma
\in \mathbb{Z}_{\geq 0}^{[F:\mathbb{R}]}$,
\[
\partial_x^{\alpha}
\partial_y^{\beta}
\partial_z^{\gamma}
\phi_0 (n(x) a(y) n'(z))
\ll
Q^{|\gamma|}.
\]
Here the partial derivatives are the usual ones when
$F = \mathbb{R}$ and are given by differentiating with respect
to the real and imaginary coordinates when $F = \mathbb{C}$.
Here
$|\gamma| = \sum_{1 \leq j \leq [F:\mathbb{R}]} |\gamma_j|$,
as usual.
\end{itemize}
Such an element exists (e.g., take a product of suitably rescaled bump
functions of the coordinates $x,y,z$).
Let $\chi_J$ denote the function on $J$ given by
$n(x) a(y) n'(z) \mapsto \chi(y)$.
We may define
the multiple $\phi_1 := \chi_J^{-1} \phi_0 \in C_c^\infty(G)$ of
$\phi_0$. Let $\phi_1^*(g) := \overline{\phi_1(g^{-1})}$ denote
its adjoint, and define the convolution product
$\phi := \vol(J)^{-1} \phi_1^* \ast \phi_1 \in C_c^\infty(G)$.
Let $h$ denote the
pre-Kuznetsov weight with kernel $\phi$.
\subsection{Lower bounds for weights}\label{sec:lower-bounds-for-wts}
\begin{theorem}\label{thm:lower-bounds-weights}
Fix $\vartheta \in [0,1/2)$.
Let $h$ be as in
\S\ref{sec:constr-wt-non-archimedean-case}
($F$ non-archimedean, $\chi$ ramified)
or
\S\ref{sec:constr-wt-archimedean-case}
($F$ archimedean, $\chi$ general).
Let $\sigma \in G_{\gen}^\wedge$ be unitary.
\begin{enumerate}[(i)]
\item We have $h(\sigma) \geq 0$.
\item If $\sigma$ is $\vartheta$-tempered (\S\ref{sec:bounds-towards-raman})
and belongs to $\Sigma_F(\chi)$,
then
$h(\sigma) \gg_{\vartheta} 1/Q$.
\item\label{item:assertion-about-unramified-vanishing-of-h}
If $F$ is non-archimedean
and $\sigma$ is unramified,
then
$h(\sigma) = 0$.
\end{enumerate}
\end{theorem}
\begin{proof}[Proof in the non-archimedean case]
Since $\chi_J$ is a character of $J$, the normalized element
$\vol(J)^{-1} \phi$ defines an idempotent in the convolution
algebra $C_c^\infty(G)$.
Thus $h(\sigma) \geq 0$.
More precisely, by \eqref{eqn:defn-h-via-phi},
we see that $h(\sigma)$ is the sum of $\vol(J) |W(1)|^2$ taken
over $W$ in an orthonormal basis for the space
\[
\sigma^{\chi_J} := \{W \in \sigma : g W = \chi_J(g) W
\text{ for all } g \in J\}.
\]
We now determine a
criterion for when $\sigma^{\chi_J}$
is nontrivial
(see \cite[Lem 22]{nelson-padic-que}
for a related argument).
We verify readily that the image of $\sigma^{\chi_J}$
under the twisting map
$\sigma \rightarrow \sigma \otimes \chi^{-1}$
consists of all vectors
transforming under the group
\[
K_0(Q) := \{\begin{pmatrix}
a & b \\
c & d
\end{pmatrix} : |a| = |d| = 1, |b| \leq 1, |d| \leq 1/Q\}
\leq \GL_2(F)
\]
via the character
\[
\begin{pmatrix}
a & b \\
c & d
\end{pmatrix} \mapsto \chi^{-2}(d).
\]
By newvector theory,
it follows that
\begin{equation}\label{eq:nontrivial-chi-J-invariants-criterion}
\sigma^{\chi_J} \neq \{0\}
\iff
C(\sigma \otimes \chi^{-1}) \leq Q = C(\chi).
\end{equation}
Assertion \eqref{item:assertion-about-unramified-vanishing-of-h}
follows:
under its hypotheses,
we have $C(\sigma \otimes \chi^{-1}) = C(\chi^{-1})^2 = Q^2 >
Q$,
so $\sigma^{\chi_J} = \{0\}$ and thus $h(\sigma) = 0$.
Since $\vol(J) \asymp 1/Q$,
the proof of the remaining
assertions will be complete
once we show that for each
$\vartheta$-tempered $\sigma \in \Sigma_F(\chi)$
that there is a nonzero
$W \in \sigma^{\chi_J}$
with $|W(1)| \gg_{\vartheta} \|W\|$.
If $\sigma$ belongs to the principal
series, then
$\sigma = \mathcal{I}(\chi \eta)$
with $\eta$ unramified,
so
$C(\sigma \otimes \chi^{-1}) = C(\eta) C(\chi^{-2} \eta^{-1})
= C(\chi^{-2}) \leq C(\chi)$
(see \cite[p8]{Sch02}). If
$\sigma$ is a twist of Steinberg, then
$\sigma$ is an irreducible subrepresentation
of $\mathcal{I}(|.|^{1/2} \chi \eta)$
with $\eta$ unramified,
so
$C(\sigma \otimes \chi^{-1}) = q \leq C(\chi)$
(see \emph{loc. cit.}).
In either case, the
inequality in \eqref{eq:nontrivial-chi-J-invariants-criterion}
holds,
and so $\sigma^{\chi_J} \neq 0$.
The proof of the criterion
\eqref{eq:nontrivial-chi-J-invariants-criterion}
shows moreover that if
$\sigma \in \Sigma_F(\chi)$, then $\sigma^{\chi_J}$
contains the inverse image of a newvector $W$ under
the
map $\sigma \rightarrow \sigma '$
for some unitary
twist $\sigma '$ of $\sigma$. As recalled in
\S\ref{sec:whitt-intertw-dual},
we have $\|W\|^2 = |W(1)|^2 L(\ad \sigma ', 1) / \zeta_F(2)$
for any such $W$.
Since $\sigma$ and hence also $\sigma '$
is $\vartheta$-tempered, we have $L(\ad \sigma ', 1)
\asymp_{\vartheta} 1$,
thus $W(1) \gg_{\vartheta} \|W\|$.
(The parameter $\vartheta$
is relevant only when $\sigma$ is a quadratic twist of a non-tempered
unramified
representation,
since otherwise both representations are tempered.)
The twisting map
$\sigma \rightarrow \sigma '$ is unitary, so the inverse image
of $W$ has the same norm as $W$, and the required lower bound
$W(1) \gg_{\vartheta} \|W\|$ follows.
\end{proof}
\begin{proof}[Proof in the archimedean case]
(We note that the results proved
here are not used
in the present paper, but should be of
direct use in future work.)
We have
\begin{equation}
h(\sigma) =
\vol(J)^{-1} \sum_{W \in \mathcal{B}(\sigma)}
| \phi_1 \ast W (1) |^2,
\end{equation}
so $h(\sigma) \geq 0$. Suppose now that
$\sigma \in \Sigma_F(\chi)$.
Since $\vol(J) \asymp 1/Q$,
it suffices to show that there
is a unit vector $W \in \sigma$ with
$|\phi_1^* \ast W(1)| \gg 1/Q$. The proof will be very
similar to that given above in the non-archimedean case, but
making use of the ``analytic newvector theory''
given in \S\ref{sec:analyt-newv-oper} rather than the ``usual''
newvector theory.
The reader might wish to skim \S\ref{sec:analyt-newv-oper}
before proceeding.
Set $\sigma ' := \sigma \otimes \chi^{-1}$. We may assume
$\alpha_0$ taken small enough that Theorem
\ref{lem:produce-whittaker-against-random-central-character-wannabe}
holds with $\delta = 1/2$ and $\varepsilon = \alpha_0$. Let
$W' \in \sigma'$ denote the unit vector produced by that
result, and let $W \in \sigma$ denote the inverse image of
$W'$.
With notation as in \S\ref{sec:analyt-newv-oper},
set \[J ' := K_1(C(\sigma'), C(\omega_{\sigma '} \omega^{-1}),
\alpha_0) \subseteq \GL_2(F).\]
The conclusion of Theorem
\ref{lem:produce-whittaker-against-random-central-character-wannabe}
then reads
\begin{equation}
|W'(g) - \eta_{\chi^{-2}}(g)| \leq 1/2
\text{ for all } g \in J'.
\end{equation}
The central
character $\omega_{\sigma '}$ is $\chi^{-2}$, so with
$\omega := \chi^{- 2}$ we have
$C(\omega_{\sigma '} \omega^{-1}) \ll 1$ (perhaps equal to
$1$, depending upon one's convention).
We may write $\sigma$
as the normalized induction of some character $\eta$ of
$F^\times$, with $\eta/\chi$ analytically unramified. From
this it follows that
$C(\sigma ') = C(\eta/\chi) C(\eta^{-1}/\chi) \asymp
C(\chi^{-2}) \asymp Q$.
Since $\alpha_1$ is small in terms of
$\alpha_0$, we see that
$J$ is contained
in the image of $J'$ under the quotient map $\GL_2(F)
\twoheadrightarrow G$.
In other words, each $g \in J$ admits
a lift $\tilde{g} \in J'$.
We note also that the product $\chi(\det g)
\eta_{\chi^{-2}}(g)$,
defined initially for $g \in J'$,
descends to a well-defined function
of $g \in J$.
More precisely,
by lifting $g = n(x) a(y) n'(z)$
to the matrix
\[
\tilde{g}
= \begin{pmatrix}
1 & x \\
& 1
\end{pmatrix}
\begin{pmatrix}
y & \\
& 1
\end{pmatrix}
\begin{pmatrix}
1 & \\
z & 1
\end{pmatrix}
= \begin{pmatrix}
y + x z & x \\
z & 1
\end{pmatrix}
\in \GL_2(F)
\]
we see that
\begin{equation}
\chi(\det g)
\eta_{\chi^{-2}} (g) =
\chi_J(g)
\end{equation}
Since $W(g) = \chi(\det g) W'(g)$,
we deduce that
\begin{equation}
|W(g) - \chi_J(g)| \leq 1/2
\text{ for all } g \in J.
\end{equation}
Expanding out
\begin{equation}
\phi_1 \ast W(1) = \int_{g \in G} \phi_0(g) \chi_J^{-1}(g)
W(g),
\end{equation}
it follows that
\begin{equation}
|\phi_1 \ast W(1)
-
\int_G \phi_0
|
\leq (1/2) \int_G \phi_0,
\end{equation}
hence that $|\phi_1 \ast W(1)| \geq (1/2) \int_G \phi_0 \gg 1/Q$, as required.
\end{proof}
\subsection{Upper bounds for dual weights}\label{sec:upper-bounds-dual-wts}
Let $\tilde{h}$
denote
the dual admissible weight
furnished by Theorem \ref{thm:constr-admiss-weight},
and let $\omega$ be a unitary character
of $F^\times \cong A$.
We aim to estimate $\tilde{h}(\omega)$.
We do this in the case that $F$ is non-archimedean and, as
before, $\chi$ is ramified and $\psi$ is unramified. This case
is the relevant one for our immediate applications. It should
be possible to carry out analogous archimedean calculations, but
we leave those for future work.
We assume moreover that \emph{$F$ is of characteristic zero},
since it will be convenient in some proofs to refer to the
exponential map without fuss. This case is anyway the relevant
one in applications to the subconvexity problem.
\begin{definition}
We say that the pair $(\chi,\omega)$ is
\emph{atypical}
if
\begin{itemize}
\item $q$ is odd and sufficiently large
in terms of the degree of $F$,
\item $-1$ is a square in $\mathfrak{o}/\mathfrak{p}$,
\item $C(\chi) \geq q^3$,
say
$C(\chi) = q^{\alpha + \alpha '}$
with $1 \leq \alpha \leq \alpha ' \leq \alpha + 1$,
\item $C(\omega) = C(\chi)$, and
\item the class
$\xi \in \mathfrak{o}^\times / (1 + \mathfrak{p}^{\alpha'})$
characterized by $\omega(\exp(a)) = \chi(\exp(\xi a))$ for
$a \in \mathfrak{p}^{\alpha}$ satisfies
$1 + 4 \xi^2 \in \mathfrak{p}$.
\end{itemize}
\end{definition}
We note that $(\chi,\omega)$ is atypical
if and only if $(\chi,\omega^{-1})$ is atypical.
\begin{proposition}\label{prop:non-arch-estimates-key}
Let $\omega$ be a unitary character of $F^\times \cong A$.
\begin{enumerate}[(i)]
\item If $C(\omega) = 1$,
then $\tilde{h}(\omega) \ll 1$.
\item If $C(\omega) > Q$,
then $\tilde{h}(\omega) = 0$.
\item If $1 < C(\omega) \leq Q$,
then
$\tilde{h}(\omega) \ll 1/Q$
unless $(\chi,\omega)$ is atypical.
\item Suppose that $(\chi,\omega)$
is atypical.
Let $\alpha$ and $\xi$ be as above.
Then
\begin{equation}
\tilde{h}(\omega) \ll
\frac{N_\alpha(\xi)}{Q}
\cdot
\begin{cases}
q^{1/2} & \text{ if } C(\chi) = q^{2 \alpha + 1}, \\
1& \text{ if } C(\chi) = q^{2 \alpha},
\end{cases}
\end{equation}
where
$N_\alpha(\xi) := \# \{\tau \in \mathfrak{o}/\mathfrak{p}^\alpha : \xi^2
\tau^2 - \tau - 1 \equiv 0 \}$.
\end{enumerate}
\end{proposition}
Note the similarity between these estimates and
those of \cite[\S3]{2019arXiv190810346P}.
\begin{proof}
Define $V$ in terms of $\phi$ as in \S\ref{sec-6-6}.
Then
\begin{equation}
V(a(y) n'(z)) = 1_{|y| = 1}
1_{|z| \leq 1/Q} \chi(y),
\end{equation}
and so
\begin{equation}
V^\wedge(\xi,z)
=
1_{|z| \leq 1/Q}
\int_y
1_{|y| = 1}
\chi(y) \psi(\xi y) \, d y.
\end{equation}
Standard properties of Gauss sums
imply that this last integral
vanishes unless $|\xi| = Q$.
Thus
\begin{equation}
V^\wedge(\xi,-x/\xi)
=
1_{|x| \leq 1}
\int_y
1_{|y| = 1}
\chi(y)
\psi(\xi y) \, d y
\end{equation}
and so by Fourier inversion,
\begin{equation}\label{eqn:V-sharp-non-arch-evald-0-0}
V^\sharp(x,y)
= 1_{|x| \leq 1} 1_{|y| = 1} \chi(y).
\end{equation}
Theorem \ref{thm:constr-admiss-weight}
now gives
\begin{equation}
\tilde{h}(\omega)
=
\int_{\substack{
x, t \in F : \\
|x| \leq 1, \\
|t - x| = |x(1-x)|
}
}
\omega \left( \frac{1-t}{t} \right)
\chi \left( \frac{x - t}{x(1-x)} \right)
\, \frac{d t \, d x}{|t(1-t)|^{1/2} |x(1-x)|}.
\end{equation}
Observe that the support conditions
imply that at least one of the following
pair of conditions holds:
\begin{itemize}
\item $|x| = |t| = 1$
\item $|1 - x| = |1 - t| = 1$
\end{itemize}
(For instance, if $|x| < 1$ and $|1-t| < 1$,
then $|t| = 1 = |1-x|$,
thus $1 = |t - x| = |x| < 1$,
contradiction.)
The two pairs of conditions are swapped
by the substitution
$x \mapsto 1-x, t \mapsto 1-t$,
which also swaps
$\omega$ with $\omega^{-1}$.
Thus
\[
|\tilde{h}(\omega)|
\leq
|\rho(\omega)|
+ |\rho(\omega^{-1})|
+ |\rho'(\omega)|
\]
where
\begin{equation}
\rho(\omega)
:=
\int_{\substack{
x, t \in F : \\
|x| \leq 1, \\
|t - x| = |x(1-x)|, \\
|1 - x| = |1-t| = 1
}
}
\omega \left( \frac{1-t}{t} \right)
\chi \left( \frac{x-t}{x(1-x)} \right)
\, \frac{d t \, d x}{|t(1-t)|^{1/2} |x(1-x)|}.
\end{equation}
and $\rho '$ is defined similarly but with the
additional conditions
$|x| = |t| = 1$.
We thereby reduce
to bounding $\rho(\omega)$ and $\rho ' (\omega)$.
We introduce the new variables
$u := x/t, v := 1/x$,
so that $x = 1/v, t = 1/u v$.
We compute that $|\partial (x, t) / \partial (u, v)|
= 1/|u^2 v^3| = |t^2 x|$,
from which it follows
that
\begin{equation}
\frac{d t \, d x}{|t(1-t)|^{1/2} |x(1-x)|}
=
\left\lvert \frac{u v}{u v - 1} \right\rvert^{1/2}
\left\lvert \frac{v}{v-1} \right\rvert
\, \frac{d u \, d v}{|u v|^{3/2}},
\end{equation}
thus
\begin{equation}
\rho(\omega)
=
\int_{\substack{
u,v \in F: \\
|u - 1| = |u|, \\
|v - 1| = |v|, \\
|u v - 1| = |u v|
}
}
\omega \left( u v - 1 \right)
\chi \left( \frac{1 - 1/u}{1 - 1/v} \right)
\, \frac{d u \, d v}{ |u v|^{3/2} }.
\end{equation}
(Compare with the exponential
sums occurring in \cite{MR1779567, 2018arXiv181102452P,
2019arXiv190810346P}.)
The integration conditions
force $|u|, |v| \geq 1$,
so we may split the integral
dyadically as
\[
\rho(\omega)
= \sum_{U,V \in \{1, q, q^2, \dotsc \} }
(U V)^{-1/2} \rho_{U, V},
\]
where
\begin{equation}
\rho_{U,V}
:=
\int_{\substack{
u,v \in F: \\
|u - 1| = |u| = U, \\
|v - 1| = |v| = V, \\
|u v - 1| = U V
}
}
\omega \left( u v - 1 \right)
\chi \left( \frac{1 - 1/u}{1 - 1/v} \right)
\, \frac{d u \, d v}{ |u v| }.
\end{equation}
We note that $\rho '(\omega) = \rho_{1,1}$.
Our task thereby reduces to estimating
the dyadic integrals $\rho_{U,V}$
suitably.
We record proofs of adequate estimates
in Appendix \ref{lem:char-sum-CI}.
\end{proof}
\section{Reduction of the proofs of the main applications}
\label{sec-8}
Here we record the essential
parts of the proofs of our main applications, Theorems
\ref{thm:CI} and \ref{thm:PY1}.
More precisely,
-- that is to say, assuming
that the basic identity \eqref{eqn:basic-moment-identity} holds
and ignoring the degenerate terms $(\dotsb)$,
whose definition and treatment we postpone.
\subsection{The case of Theorem \ref{thm:CI}}\label{sec:CI}
Let $F$ be a number field
and
$\psi$
a nontrivial unitary character of $\mathbb{A}/F$. Let $\sigma_0$ be either a cuspidal
automorphic representation of $\operatorname{PGL}_2(\mathbb{A})$ or a unitary
Eisenstein series. Let $\chi$ be a quadratic character of
$\mathbb{A}^\times/F^\times$,
with analytic conductor $Q := C(\chi)$.
We regard $\sigma_0$ as fixed.
We explain first how to prove the estimate
\begin{equation}\label{eqn:desired-bound-for-L-sigma-0-chi-one-half}
L(\sigma_0 \otimes \chi, 1/2) \ll Q^{1/3+\varepsilon},
\end{equation}
with polynomial
dependence of the implied constant upon $C(\sigma_0)$.
We will do so using \eqref{eqn:basic-moment-identity}, the lower
bounds established for various $h_\mathfrak{p}$, and the upper
bounds established for various $\tilde{h}_\mathfrak{p}$.
It suffices to consider the following cases:
\begin{itemize}
\item $\sigma_0$ is cuspidal.
\item $\sigma_0 = \Eis^*(\mathcal{I}(0))$,
i.e., $\sigma_0$ is the unitary Eisenstein series
with degenerate parameter
for which
$L(\sigma_0, s) = \zeta_F(s)^2$.
In that case,
we use the standard estimate
\begin{equation}\label{eqn:eis-upper-bound-integral-trick}
L(\chi,1/2)^6
\ll
Q^{\varepsilon}
\int_{t \in \mathbb{R} : |t| \leq 1}
t^2
|L(\chi, 1/2 + i t)|^6
\, d t,
\end{equation}
which may be deduced (in sharper forms) via the convexity
bound for the derivatives of $t \mapsto L(\chi,1/2+it)$.
(The exponents $2$ and $6$ are not special.)
\end{itemize}
Let $S_0$ denote the set consisting of all infinite places of
$F$ together with all finite places at which $\sigma_0$
ramifies. Let $S_1$ denote the set of finite places not in
$S_0$ at which $\chi$ ramifies. Set $S := S_0 \cup S_1$.
The consequence $\exp(\# S) \ll Q^{\varepsilon}$ of the divisor bound
may be used to control products of implied constants indexed by
$S$. We note also that, thanks to any fixed bound
$\vartheta < 1/2$ towards Ramanujan, any local $L$-factor
appearing on either side of \eqref{eqn:basic-moment-identity} is
$\exp(\operatorname{O}(1))$. These observations imply that any product of
such factors taken over $\mathfrak{p} \in S$ is of size
$Q^{o(1)}$ as $Q \rightarrow \infty$.
To each $\mathfrak{p} \in S$
we attach an admissible weight $h_\mathfrak{p}$,
as follows:
\begin{itemize}
\item Let $\mathfrak{p} \in S_0$.
By lemma
\ref{lem:crude-lower-bound-individual},
we may find
$h_\mathfrak{p}$ nonnegative on unitary representations and
bounded from below by $1$ on the local component
$\sigma_{0, \mathfrak{p}}$ together with each of its quadratic
twists.
If $\sigma = \Eis^*(\mathcal{I}(0))$,
we may assume moreover
that $h_\mathfrak{p}$
is $\geq 1$ on $\{\mathcal{I}_\mathfrak{p} (i t) : |t| \leq
1\}$,
where
$\mathcal{I}_\mathfrak{p}(i t)$
denotes the corresponding induced
representation of $\operatorname{PGL}_2(F_\mathfrak{p})$.
We then bound the dual
$\tilde{h}_\mathfrak{p}$ crudely via lemma
\ref{lem:crude-estimates}.
The arguments just recorded do not quite suffice for our purposes:
they lead to estimates
\eqref{eqn:desired-bound-for-L-sigma-0-chi-one-half}
with an \emph{unspecified}
dependence upon $\sigma_0$. The required polynomial
dependence follows from the slightly finer arguments given below
in \S\ref{sec:crude-local-estim}.
\item For $\mathfrak{p} \in S_1$,
our assumptions imply that
$\chi_\mathfrak{p}$ is ramified. We construct $h_\mathfrak{p}$
as in \S\ref{sec:constr-suit-weight} to majorize the family
$\Sigma_{F_\mathfrak{p}}(\chi_\mathfrak{p})$, and estimate the
dual $\tilde{h}_\mathfrak{p}$ via
\S\ref{sec:upper-bounds-dual-wts}.
\end{itemize}
By \eqref{eqn:eis-upper-bound-integral-trick},
the lower bounds for $h_\mathfrak{p}$
proved in \S\ref{sec:lower-bounds-for-wts},
and
standard upper bounds
for $L^{(S),*}(\sigma \times \sigma, 1)$,
we have
\begin{equation}\label{eqn:bound-cube-via-integral}
L(\sigma_0 \otimes \chi,1/2)^3
\ll_{\sigma_0}
Q^{1+\varepsilon}
\int_{\substack{
\sigma:\text{generic}, \\
\text{unram. outside $S$}
}
}
\frac{L^{(S)}(\sigma,1/2)^3}{L^{(S),*}(\sigma \times \sigma,
1)}
\prod_{\mathfrak{p} \in S} h_\mathfrak{p}(\sigma_\mathfrak{p}).
\end{equation}
By the precise form of Theorem \ref{thm:constr-admiss-weight}
stated below in \S\ref{sec:summary-main-results}, the integral
on the RHS of \eqref{eqn:bound-cube-via-integral}
is equal to a degenerate term $(\dotsb)$ plus
\begin{equation}\label{eqn:dual-fourth-moment-in-pf-sketch}
\int_{\substack{
\omega:\text{unitary}, \\
\text{unram. outside $S$}
}}
\frac{|L^{(S)}(\omega,1/2)|^4}{\zeta_F^{(S),*}(1)^2}
\prod_{\mathfrak{p} \in S}
\tilde{h}_\mathfrak{p}(\omega_\mathfrak{p}),
\end{equation}
at least if $\psi_\mathfrak{p}$ is unramified for all
$\mathfrak{p} \in S$; in general, the indicated relation holds
up to a scalar $\asymp 1$, by the remarks of
\S\ref{sec:vari-with-resp-1}.
By the upper bounds for $\tilde{h}_\mathfrak{p}$
noted above,
we may majorize \eqref{eqn:dual-fourth-moment-in-pf-sketch} by
\begin{equation}\label{eqn:fourth-moment-after-1}
Q^{-1+\varepsilon}
\int_{\substack{
\omega:\text{unitary}, \\
\text{unram. outside $S$}, \\
C(\omega_\mathfrak{p})
\ll 1 \text{ for finite }
\mathfrak{p} \in S_0, \\
C(\omega_\mathfrak{p})
\leq C(\chi_\mathfrak{p}) \text{ for }
\mathfrak{p} \in S_1
}
}
|L(\omega,1/2)|^4
\prod_{\text{infinite } \mathfrak{p}}
C(\omega_\mathfrak{p})^{-d}
\end{equation}
for any fixed $d$,
which we take sufficiently large.
We claim that \eqref{eqn:fourth-moment-after-1} is $\ll Q^{2 \varepsilon}$.
To see this, it suffices to show
for any collection $(X_\mathfrak{p})_{\mathfrak{p}}$
of parameters $X_\mathfrak{p} \geq 1$,
with $X_\mathfrak{p} = 1$ for almost all $\mathfrak{p}$ and
$X_\mathfrak{p}$
belonging to the value group of $F_\mathfrak{p}$
for all $\mathfrak{p}$,
that
\begin{equation}\label{eqn:fourth-moment-after-2}
\int_{\substack{
\omega:\text{unitary}, \\
C(\omega_\mathfrak{p})
\leq X_\mathfrak{p} \text{ for all } \mathfrak{p}
}
}
|L(\omega,1/2)|^4
\ll (\prod_\mathfrak{p} X_\mathfrak{p} )^{1+\varepsilon}.
\end{equation}
Such an estimate is implicit in work of Han Wu \cite{MR3977317}
on the subconvexity problem for the $L$-functions
$L(\omega,1/2)$. Wu gives a geometric proof, using unipotent
translates of Eisenstein series as in Sarnak \cite{MR780071}
and Michel--Venkatesh \cite{michel-2009}, of bounds for amplified fourth moments of
$L(\omega,1/2)$ over families as in
\eqref{eqn:fourth-moment-after-2}. Omitting the amplifier, his
arguments give the required estimate. In fact, such arguments
may be understood as special cases of
\eqref{eqn:basic-moment-identity} in which the cubic moment side
is particularly simple, so it should be instructive in future work
to revisit those cases from this perspective.
We verify below in \S\ref{sec:handl-degen-terms-1},
\S\ref{sec:handl-degen-terms-2} that the degenerate term
$(\dotsb)$ in \eqref{eqn:basic-moment-identity} satisfies the
estimate $(\dotsb) \ll Q^{\varepsilon}$.
(Morally, this corresponds ``up to logarithms''
to the fact that the kernels
$\phi_\mathfrak{p}$
used to define $h_\mathfrak{p}$ for $\mathfrak{p} \in S_1$
satisfy $\phi_\mathfrak{p}(1) \ll 1$.)
Assuming this, the required bound for $L(\sigma_0 \otimes
\omega,1/2)$
follows.
\subsection{The case of Theorem \ref{thm:PY1}}\label{sec:PY1}
Recall that $\chi$ is a character of
$\mathbb{A}^\times/F^\times$ with $\chi_\infty$ trivial and
finite conductor cubefree. We define $S = S_0 \cup S_1$, with
$S_0$ the set of archimedean places and $S_1$ the set of finite
places at which $\chi$ ramifies. For $\mathfrak{p} \in S_0$, we
choose $h_\mathfrak{p}$ to be nonnegative on unitary
representations and bounded from below by $1$ on
$\{\mathcal{I}_p(i t) : |t| \leq 1\}$. For
$\mathfrak{p} \in S_1$, we choose $h_\mathfrak{p}$ as in
\S\ref{sec:constr-suit-weight} to majorize
$\Sigma_{F_\mathfrak{p}}(\chi_\mathfrak{p})$. We then argue
exactly as in \S\ref{sec:CI}.
The cubefree hypothesis
ensures that we avoid the atypical case
of Proposition \ref{prop:non-arch-estimates-key}
when we estimate $\tilde{h}_\mathfrak{p}$ for
$\mathfrak{p} \in S_1$.
\part{Preliminaries}\label{part:preliminaries}
We recall here the basic local and global theory relevant for studying $L$-functions on $\operatorname{PGL}_2$ via their (regularized) integral
representations, together with some basics concerning
automorphic forms and their (regularized) spectral expansions.
As general references, we mention \cite{MR2508768, michel-2009, MR0379375,
MR1431508, MR0401654,Ja72, MR546600}.
\section{Regularized integration}
\label{sec-9-1}
We
record a notion of regularized integration, adapted from
\cite[\S4.3]{michel-2009} (see also \cite{zagier-mellin,
MR656029}), that is adequate for our purposes.
\subsection{Regularizable functions}
Let $A$ be a
locally compact abelian group,
but not a finite group. Let $X$ be an
$A$-space, equipped with an invariant measure $\mu$. We say that a
measurable function $f : X \rightarrow \mathbb{C}$ is
\emph{regularizable} (with respect to $\mu$ and $A$)
if there is a bounded variation complex
Borel measure $r$ on $A$, with $\int_A r \neq 0$, so that the
convolution $r \ast f$ lies in $L^1(X,\mu)$; the
\emph{regularized integral}
is then
defined to be the ratio
\[
\int_X^{\reg} f \, d \mu
:=
\frac{\int_X r \ast f}{\int_A r }.
\]
The definition is independent of the choice of $r$, and defines
an $A$-invariant functional on the space of regularizable
functions on $X$. This notion of regularized
integration generalizes
those
cited above, but applies a
bit more generally, e.g., to the integrals used to define
Whittaker functions of principal series representations.
The definition extends with the evident modifications to the
case that $\mu$ is quasi-invariant, i.e., transforms under $A$
with respect to a character.
\subsection{Finite functions}
A \emph{finite}
function $\phi : X \rightarrow \mathbb{C}$ is one whose
$A$-translates span a finite-dimensional space $\langle A \phi \rangle$; if that space
does not contain the trivial representation of $A$, then $\phi$
is called \emph{admissible}.
An \emph{exponent} $\chi$ of a finite function $\phi$ is a
character
of $A$ that arises as a generalized eigenvalue
for the action of $A$ on $\langle A \phi \rangle$.
We say that $f$ is \emph{strongly regularizable}
if there is a finite cover
$X = \cup U_i$ and admissible finite functions $\varphi_i$ on
$X$ so that $f - \varphi_i$ is $\mu$-integrable on $U_i$;
in that case,
$f$ is regularizable.
\subsection{Holomorphy criteria}
Given a family of integrable functions $f_s$ depending pointwise
holomorphically upon a complex parameter $s$,
a standard criterion for their integrals
$\int f_s$ to vary holomorphically is that $|f_s| \leq h$ for
some integrable function $h$.
This criterion may be applied locally in $s$.
We have a similar criterion for regularized integrals,
which may also be applied locally:
\begin{lemma}\label{lem:reg-int-holom-var}
Suppose given a complex manifold $M$
and a family of measurable functions
$f_s : X \rightarrow \mathbb{C}$
indexed by $s \in M$
that vary pointwise holomorphically
(i.e., for each $x \in X$,
the map $s \mapsto f_s(x)$ is holomorphic).
Suppose that we may find
\begin{itemize}
\item a finite cover $X = \cup_{i=1}^n U_i$,
\item admissible finite functions $\varphi_{i,s}$ on $X$
that vary pointwise holomorphically, and
\item integrable functions $h_i$ on $U_i$
so that $|f_s - \varphi_{i,s}| \leq h_i$.
\end{itemize}
Assume that
there exists $r \geq 0$ so that for each
$i$ and each $s$, the dimension of the span of the
$A$-translates of $\varphi_{i,s}$ is bounded by $r$.
Then each $f_s$ is strongly regularizable
and the map
$M \ni s \mapsto \int_X^{\reg} f_s \, d \mu$ is holomorphic.
\end{lemma}
\begin{proof}
Since $A$ is infinite, we may find distinct elements
$a_1,\dotsc,a_{r+2} \in A$. For each $i \in \{1..n\}$ and
$s \in M$, consider the system of linear equations in the
variables $c^i_{1}(s),\dotsc,c^i_{r+2}(s) \in \mathbb{C}$
given by
\[
\sum_j c^i_j(s) = 1,
\]
\[
\sum_j c^i_j(s) \varphi_{i,s}(a_j x) = 0
\text{ for all } x \in X.
\]
The assumption on the dimension of the span of the translates
of the $\varphi_{i,s}$ implies that there are more variables
than independent equations. The assumption that $\varphi_{i,s}$ is
admissible implies then that the system is solvable for each
$s$. By passing to an open subset of $M$, we may find a
specific invertible minor in the matrix describing this system
and hence a family of solutions $c^i_j(s)$ that vary
holomorphically with $s$. Having chosen one such family, let
$\kappa_s^i$ denote the measure on $A$ given by the linear
combination of point masses $\sum_j c^i_j(s) \delta_{a_j}$.
Take for $\kappa_s := \kappa_s^1 \ast \dotsb \ast \kappa_s^n$
their convolution product. Then $\int \kappa_s = 1$, while
each convolution $\kappa_s \ast \varphi_{i,s}$ vanishes.
Thus $\kappa_s \ast f_s$ is integrable
and
\begin{equation}\label{eq:integrate-regularized-via-convolution-holomorphy}
\int_X^{\reg} f_s \, d \mu
=
\int_X (\kappa_s \ast f_s) \, d \mu.
\end{equation}
Moreover, $|\kappa_s \ast f_s|$ is bounded on $U_i$,
locally uniformly in $s$,
by a linear combination of $h_i$ and its translates.
Thus the standard criterion implies
that the RHS of
\eqref{eq:integrate-regularized-via-convolution-holomorphy} is
holomorphic in $s$.
\end{proof}
\subsection{Main examples}\label{sec:regularization-specialized-to-A}
We will apply these notions primarily when either
\begin{itemize}
\item $A$ is the multiplicative group of a local field $F$
and $X = A$, or
\item $A$ is the multiplicative group $\mathbb{A}^\times$ of the
adele ring $\mathbb{A}$
of a global field $F$
and $X = \mathbb{A}^\times / F^\times$,
\end{itemize}
equipped with suitable Haar measures.
In either case, a convenient cover is given by $X = U_{\infty} \cup U_0$
with
$U_\infty := \{x \in X : |x| \geq 1\}$
and
$U_0 := \{x \in X : |x| < 1\}$.
The finite
functions are the linear combinations of functions of the form
$y \mapsto \chi(y) \log^{m-1} |y|$ for some character $\chi$ of
$X$ and some positive integer $m$.
If $f$ is strongly regularizable
(with respect to this cover),
then its regularized integral may be defined
as above using convolution,
or (as noted in \cite[\S4.3]{michel-2009}) in other equivalent ways:
\begin{itemize}
\item By truncation:
We may write
$\int_{x \in X : 1/T \leq |x| \leq T}
f(x) \, d \mu(x)$
as the sum $g(T) + h(T)$,
where $g$ is an admissible finite
function on the value group $\{ |x| : x \in X \}$
and $h(T)$ has a limit as $T \rightarrow \infty$;
then $\int_X^{\reg} f = \lim_{T \rightarrow \infty} h(T)$.
\item By meromorphic continuation:
The integrals
$\int_{x \in U_\infty} f(x) |x|^{s} \, d \mu(x)$ and
$\int_{x \in U_0} f(x) |x|^s \, d \mu(x)$ converge absolutely for $\Re(s)$
sufficiently negative and positive, respectively.
They extend meromorphically to
functions $F_\infty, F_0$ on the complex plane
for which
$\int_X^{\reg} f = F_\infty(0) + F_0(0)$.
Alternatively, we may find an admissible finite function $\phi_\infty$
so that the integral
$F(s) := \int_{x \in X} (f - \phi_\infty)(x) |x|^s \, d
\mu(x)$
converges
absolutely for $\Re(s)$ sufficiently large.
Then $F$
continues meromorphically to the complex plane, and
\begin{equation}\label{eqn:regularized-integral-via-mellin-analysis}
\int_X^{\reg} f = F(0).
\end{equation}
\end{itemize}
\section{Norms on representations}\label{sec:norms-repr}
Let $F$ be a local field. Let $G$ be a reductive group over
$F$.
(The groups $\GL_1(F)$, $\operatorname{PGL}_2(F)$ and products thereof
are the relevant ones for this paper.)
Our aim in this section is to define a system of Sobolev
norms $\mathcal{S}_d$ ($d \in \mathbb{R}$) on certain
representations $\sigma$ of $G$. Michel--Venkatesh
\cite[\S2]{michel-2009} constructed a suitable system when
the representation $\sigma$ is unitary. It will be important for
us to work with \emph{non-unitary} representations (possibly far
from the ``tempered axis''), so we give a more general
construction sufficient for our aims.
We will often use these norms to estimate linear functionals
$\ell : \sigma \rightarrow \mathbb{C}$ by asserting that
$\ell(v) \ll \mathcal{S}_d(v)$ for some fixed $d$. In the
archimedean case, the existence of such an estimate is
equivalent to the continuity of $\ell$. In the non-archimedean
case (where our conventions imply that any linear functional is
continuous), such estimates should be understood informally as
asserting that $\ell(v)$ is bounded polynomially with respect to
the ramification of $v$.
The purpose of these norms is to give a
convenient way to formulate such estimates \emph{uniformly} as
the representation $\sigma$ and the underlying local field $F$
vary. We use them in this paper primarily to verify that
certain estimates (e.g., \eqref{eq:weyl-bound-sigma-chi}) depend
polynomially upon auxiliary parameters.
\subsection{Setting}\label{sec:norms-reps-setting}
We assume that $G$ comes equipped with a faithful linear
representation $G \hookrightarrow \SL_r(F)$ and a maximal
compact subgroup $K$. In the non-archimedean case, we assume
that $K$ is special and contains $G \cap \GL_r(\mathfrak{o})$.
We assume given
a Haar measure $d k$ on $K$ of volume $\asymp 1$
(e.g., the probability Haar).
Let $P = M U$ be a parabolic subgroup of $G$, with Levi $M$ and
unipotent radical $U$. Let $\chi_0$ be an irreducible
unitary representation
of $M$, with inner product $\langle , \rangle$ and norm $\|.\|$.
We may then define the normalized induction
$\sigma_0 := \Ind_P^G(\chi_0)$, consisting of smooth
$f : G \rightarrow \chi_0$ satisfying
$f(n m g) = \delta_P^{1/2}(m) \chi_0(m) f(g)$. It is a unitary
representation with respect to the inner product
$\langle f_1, f_2 \rangle := \int_{k \in K} \langle f_1(k),
f_2(k) \rangle \, d k$. By the decomposition $G = P K$, we see that restriction to
$K$ identifies the restricted representation $\sigma_0|_K$ with
$\Ind_{M \cap K}^K (\chi_0|_{M \cap K})$.
Let $\theta : M \rightarrow \mathbb{R}^\times_+$ be a
positive-valued character of $M$. We may then form the twisted
representation $\chi := \chi_0 \otimes \theta$ of $M$ and its
induction $\sigma := \Ind_P^G(\chi)$.
We regard $\chi$ as the same underlying space
as $\chi_0$, but with a modified action. Since $\theta$ is
positive-valued and $M \cap K$ is compact,
the representations $\chi$ and $\chi_0$ have
the same restrictions to $M \cap K$, hence their inductions $\sigma$
and $\sigma_0$ have the same restrictions to $K$.
We equip $\sigma$ with the inner product $\langle , \rangle$
and norm $\|.\|$ transferred from $\sigma_0$.
The inner product on $\sigma$ is thus $K$-invariant,
but not in general $G$-invariant.
We note that if $G$ is a quasi-split classical group (e.g., if $G =
\GL_r(F)$),
then
each generic irreducible representation $\sigma$
is known to
arise (essentially uniquely) from the above construction
with $\chi_0$ tempered and
$\theta$ strictly dominant
(see the second paragraph of
\cite{MR2046512}, or \cite{MR507800,GKdim}).
\subsection{Construction of the Sobolev norms}
\label{sec:constr-sobol-norms}
We now define Sobolev norms $\mathcal{S}_d$ on $\sigma$. In
summary, we first apply (a slight modification of) the
construction of \cite[\S2]{michel-2009} to obtain such norms on
$\sigma_0$. We then transfer those norms to $\sigma$ via the
$K$-equivariant identification $\sigma \cong \sigma_0$.
We formulate the construction in
terms of $K$-types. Let $K^\wedge$ denote the set of
isomorphism classes of
irreducible representations of $K$. For $\nu \in K^\wedge$, we
denote by $\sigma^{\nu}$ the $\nu$-isotypic component of
$\sigma$.
The notation applies also to $\sigma_0$,
and we have $\sigma_0^{\nu} = \sigma^\nu$ as representations
of $K$.
For each $\nu \in K^\wedge$
such that $\sigma^{\nu} \neq \{0\}$,
we define a scalar $N_{\sigma \nu} \geq 1$
as follows.
In the non-archimedean case, we define for $n \geq 0$ the
principal congruence subgroup
\[
K[n] := G \cap \{g \in \GL_r(\mathfrak{o}) : g \equiv 1
(\mathfrak{p}^n)
\}
\]
of $K$. We set
\[
N_{\sigma \nu} := q^n
\]
if $n \geq 0$ is the
smallest nonnegative integer such that $K[n]$ acts trivially
on $\nu$.
In the archimedean case, we fix an orthonormal basis
$\{x_i\} \cup \{y_j\}$ for $\mathfrak{g} := \Lie(G)$ with
respect to the trace pairing derived from the given linear
embedding, where the $x_i \in \mathfrak{k}$ and
$y_j \in \mathfrak{p}$ belong to summands of the Cartan
decomposition
$\mathfrak{g} = \mathfrak{k} \oplus \mathfrak{p}$. The
corresponding Casimir elements for $G$ and $K$ are then given
by $\mathcal{C}_G = -\sum_i x_i^2 + \sum_j y_j^2$ and
$\mathcal{C}_K = -\sum_i x_i^2$. The subspaces $\sigma^{\nu}$
and $\sigma_0^{\nu}$ are eigenspaces for the actions of these
Casimir elements. We write $c_{\sigma}, c_{\sigma_0}$ for the
corresponding eigenvalues of $\mathcal{C}_G$ and $c_\nu$ for
that of $\mathcal{C}_K$. Set
\begin{equation}\label{eqn:Delta-G-defn}
\Delta_G := - \sum_i x_i^2 - \sum_j y_j^2 = - \mathcal{C}_G +
2 \mathcal{C}_K.
\end{equation}
Since $\sigma_0$ is unitary, the element
$\Delta_G$ acts on $\sigma_0$ by a positive-definite operator,
as do its summands $- \sum_i x_i^2$ and $- \sum_j y_j^2$. It
follows that the quantities $c_\nu, c_\nu - c_{\sigma_0}$ and
$- c_{\sigma_0} + 2 c_\nu$ are nonnegative. We set
\begin{equation}
N_{\sigma \nu} :=
(1 - c_{\sigma_0} + 2 c_{\nu})^{1/2} \in \mathbb{R}_{\geq 1}.
\end{equation}
In other words,
$N_{\sigma \nu}$ is the eigenvalue
for $(1 + \Delta_G)^{1/2}$ on $\sigma_0^{\nu}$.
In either case, we define for $d \in \mathbb{R}$ the Sobolev
norm $\mathcal{S}_d$ on $\sigma$ by the rule
\begin{equation}\label{eq:defn-S-d-of-v-squared}
\mathcal{S}_d(v)^2 :=
\sum_{\nu}
N_{\sigma \nu}^{2 d}
\|v_\nu \|^2,
\end{equation}
where $v = \sum_{\nu} v_\nu$ is the decomposition of the
(smooth) vector $v \in \sigma$ into $K$-isotypic components. In
the non-archimedean case, this is really a finite sum, while in
the archimedean case the sum converges in the natural
Fr{\'e}chet topology on $\sigma$.
\begin{remark}
The norms $\mathcal{S}_d$ coincide with those defined in
\cite[\S2]{michel-2009} when $\sigma = \sigma_0$, except for a
slight difference in normalization
(our $\mathcal{S}_d$ is the
``$\mathcal{S}_{d/2}$'' of \cite[\S2]{michel-2009}
when $F$ is archimedean).
In the non-archimedean case, the construction of
\cite[\S2]{michel-2009} refers only to the restriction of
$\sigma$ to $K$, hence applies directly even when $\sigma$ is
non-unitary. The subtlety in the archimedean case responsible
for the roundabout definition given above is that the operator
$\sigma(\Delta_G)$ need not be positive-definite, or even
self-adjoint. For that definition to be useful in practice, we
still need to control the operator $\sigma(\Delta_G)$ and the
norms $\mathcal{S}_d$ in terms of one another, or in other
words, to compare the eigenvalues of $\sigma(\Delta_G)$ and
$\sigma_0(\Delta_G)$ on their common eigenspaces
$\sigma_0^\nu \cong \sigma_0^{\nu}$. The required comparison is
given below in lemma \ref{lem:comp-betw-lapl}.
\end{remark}
\begin{remark}
When $\sigma$ is a non-tempered unitary representation, the
norms defined here typically differ from those defined in
\cite[\S2]{michel-2009}.
\end{remark}
\begin{remark}
We note that in the archimedean case, the seminorms
$\mathcal{S}_d$ define the natural Fr{\'e}chet topology on
$\sigma$. Thus the continuous functionals
$\ell : \sigma \rightarrow \mathbb{C}$ are precisely those for
which $|\ell(v)| \leq C \mathcal{S}_d$ for some $C$ and $d$.
\end{remark}
\subsection{Norms on Schwartz spaces}\label{sec:norms-schw-spac}
Here we
define Sobolev norms $\mathcal{S}_d$ ($d \in \mathbb{R}$) on
the Schwartz spaces $\mathcal{S}(F^r)$ ($r \geq 1$). These may
be obtained by adapting the above construction to standard
representations of Heisenberg groups, but it seems simpler for
our purposes to give the definition more directly.
We assume given a nontrivial unitary character $\psi$ of $F$,
hence a Haar measure on $F$ and on $F^r$.
In the archimedean case, we choose a basis for
$F$ over $\mathbb{R}$ to identify $F^r$ with
$\mathbb{R}^{[F:\mathbb{R}] r}$.
We write $x \in F^r$
in coordinates as $x = (x_1,\dotsc,x_{[F:\mathbb{Q}]})$
For multi-indices $\alpha, \beta \in \mathbb{Z}_{\geq
0}^{[F:\mathbb{R}]}$,
we denote by $M^\alpha$ the function
$\mathcal{S}(F^r) \ni x \mapsto \prod_{j=1}^{[F:\mathbb{R}]}
x_j^{\alpha_j}$
and by $\partial^\beta$ the differential
operator
$\prod_{j=1}^{[F:\mathbb{R}]} \frac{\partial }{\partial x_j}$.
For $\phi \in \mathcal{S}(F^r)$,
we then define $\mathcal{S}_d(\phi)$
to be the sum, over all multi-indices
$\alpha, \beta \in \mathbb{Z}_{\geq 0}^{[F:\mathbb{R}]}$ with
$\sum \alpha_i + \sum \beta_j \leq d$, of
$\|M^\alpha \partial^{\beta} \phi \|_{L^2(F^r)}$.
We turn to the non-archimedean case. Choose an unramified
character $\psi_0$ of $F$. For $x,y \in \mathfrak{o}^{r}$ and
$\phi \in \mathcal{S}(F^r)$, define
$(x,y) \cdot \phi \in \mathcal{S}(F^r)$ by the formula
$(x,y) \cdot f(z) = \phi(z + x) \psi_0(y z)$. This defines an
action of the compact group $\mathfrak{o}^{2 r}$ on
$\mathcal{S}(F^r)$. Each $\phi \in \mathcal{S}(F^r)$ may be
decomposed accordingly as a finite sum $\phi = \sum \phi_\nu$,
where $\nu$ runs over the characters of $\mathfrak{o}^{2 r}$.
We define $N_\nu := q^n$ if $n \geq 0$ is the smallest
nonnegative integer for which $\nu$ has trivial restriction to
$(\mathfrak{p}^n)^{\oplus 2 r}$, and set
$\mathcal{S}_d(\phi)^2 := \sum_{\nu} N_{\nu}^{2 d}
\|\phi_\nu\|^2_{L^2(F^r)}$. Explicitly, we may write each
$\phi \in \mathcal{S}(F^r)$ uniquely as a linear combination
$\phi = \sum_{x,\xi} c(x,\xi) \phi_{x,\xi}$ of the functions
$\phi_{x,\xi} \in \mathcal{S}(F^r)$ defined for
$x,\xi \in F^r / \mathfrak{o}^r$ by
$\phi_{x,\xi}(y) := 1_{x + \mathfrak{o}}(y) \psi_0(\xi y)$, and
we have
$\mathcal{S}_d(\phi)^2 = \sum_{x,\xi} \max(1,|x_1|,\dotsc,|\xi_r|)^2
|c(x,\xi)|^2$.
Many of the results stated below for the norms $\mathcal{S}_d$
attached above to representations of reductive groups hold with
minor modifications for the norms $\mathcal{S}_d$ defined here
on Schwartz spaces.
\subsection{Uniformity}\label{sec:uniformity}
For the estimates stated below,
we assume given a \emph{fixed} number field $\mathbf{F}$, a
\emph{fixed} reductive group $\mathbf{G}$ over $\mathbf{F}$ with
a \emph{fixed} linear embedding
$\mathbf{G} \hookrightarrow \mathbf{G} \mathbf{L}_r$ from which
the local field $F$, the group $G$ and its linear embedding
$G \hookrightarrow \GL_r(F)$ arise as local components at some
place $\mathfrak{p}$ of $\mathbf{F}$.
Implied constants
are thus allowed to depend upon $\mathbf{F}$
and $\mathbf{G}$,
but not upon $F$ or $G$.
In the case that $F$ is archimedean, we assume that
$\theta$ belongs to a \emph{fixed} bounded collection $\Theta$
of positive-valued characters of $M$. For instance, if
$G = \operatorname{PGL}_2(F)$ and $M = A \cong F^\times$, then
$\theta = |.|^c$ for some $c \in \mathbb{R}$, and we may define
$\Theta$ by requiring that $|c| \leq \vartheta$ for some fixed
$\vartheta > 0$.
As in \cite[\S2]{michel-2009}, we adopt the ``implied index''
convention that $\mathcal{S}(v)$ denotes a Sobolev norm of the form
$\mathcal{S}_d(v)$ for some fixed index $d$.
\subsection{Comparison between Laplace eigenvalues}
\label{sec:comp-betw-lapl}
\begin{lemma}\label{lem:comp-betw-lapl}
Suppose that $F$ is archimedean.
Let $\nu \in K^\wedge$ be such that
the isotypic component $\sigma^{\nu}$
(equivalently, $\sigma_0^{\nu}$) is nontrivial.
Let $\delta = - c_{\sigma} + 2 c_\nu$
and $\delta_0 = - c_{\sigma_0} + 2 c_\nu$
denote the respective eigenvalues
for $\Delta_G$.
Then
\begin{equation}
\delta
=
\delta_0 + \operatorname{O}(\delta_0^{1/2} + 1).
\end{equation}
In particular, if $C \geq 1$ is fixed
but large enough
(in terms of $\Theta$),
then
\begin{equation}\label{eqn:C-plus-delta-comparison}
C + \delta \asymp C + \delta_0.
\end{equation}
\end{lemma}
\begin{proof}
We denote by $\langle , \rangle$ the norm on
$\Ind_{M \cap K}^{K}(\chi_0) = \sigma_0|_K \cong \sigma|_K$
defined as in \S\ref{sec:norms-reps-setting} by
$\langle f_1, f_2 \rangle = \int_{k \in K} \langle f_1(k), f_2(k)
\rangle\, d k$. Let $f$ be a unit vector in $\sigma_0^{\nu}$,
so that $\delta_0 = \langle \Delta f, f \rangle$ with $\Delta := \Delta_G$. We may
extend $\theta$ from $M$ to $P$ via pullback and then to $G$
via the decomposition $G = P K$. Since $\theta|_K \equiv 1$,
the product $\theta f$ is a unit vector in $\sigma^{\nu}$, and
so $\delta = \langle \Delta (\theta f), \theta f \rangle$. On
the other hand, we have
$\Delta(\theta f) = \theta (\Delta f) + u_1 + u_0$, where
\begin{equation}
u_1 :=
-2\sum_{x \in \mathcal{B}(\mathfrak{g})}
(x \theta) (x f),
\quad
u_0
:=
(\Delta \theta ) f.
\end{equation}
Since $\langle \theta (\Delta f), f \rangle = \langle \Delta
f, f \rangle = \delta_0$,
it follows that
$\delta = \delta_0 + \langle u_1 + u_0, f \rangle$,
hence by Cauchy--Schwarz that
$|\delta - \delta_0| \leq \|u_1\| + \|u_0\|$.
Since $\theta$ lies in the fixed bounded collection $\Theta$,
the $L^\infty(K)$-norms of $\theta$ and $x \theta$
are $\operatorname{O}(1)$.
Thus $\|u_0\| \ll 1$.
It follows from (e.g.)
\cite[\S3.5]{nelson-venkatesh-1}
that
\begin{equation}\label{eq:estimate-for-first-order-diff-op}
\|x f\| \ll 1 + \delta_0^{1/2},
\end{equation}
thus $\|u_1\| \ll 1 + \delta_0^{1/2}$.
The required estimate
follows.
\end{proof}
In the non-archimedean case, we \emph{define} the operators
$\Delta^d := \Delta_G^d$ ($d \in \mathbb{R}$) by
\begin{equation}\label{eqn:defn-Delta-nonarch}
\Delta^d \sum v_\nu = \sum N_{\sigma \nu}^{d/2} v_\nu,
\end{equation}
so that
$\mathcal{S}_d(v)^2 = \langle \Delta^d v, v \rangle$.
In either case, we see that if $C$ is large enough but fixed,
then
\begin{equation}\label{eqn:sobolev-norms-via-C-plus-delta}
\mathcal{S}_d(v)^2 \asymp
\langle (C + \Delta)^d v, v \rangle.
\end{equation}
\subsection{Uniform trace class property}\label{sec:uniform-trace-class}
\begin{lemma}\label{lem:uniform-bounds-N-sigma-nu}
There is a fixed $d_0 \geq 0$
so that
for all all irreducible
representations $\sigma$ of $G$,
\begin{equation}\label{eq:sum-N-sigma-nu-neg-d-0}
\sum_{\nu}
\dim(\sigma^\nu)
N_{\sigma \nu}^{-d_0}
\ll 1.
\end{equation}
\end{lemma}
\begin{proof}
This estimate is the same for $\sigma$ as for $\sigma_0$, so
the arguments of \cite[\S2.6.3]{michel-2009} apply directly.
\end{proof}
\subsection{Sobolev conductor}\label{sec:sobolev-conductor}
We define, as in \cite[\S2.6.5]{michel-2009},
the ``Sobolev conductor''
\begin{equation}
C'(\sigma) := \min \{N_{\sigma \nu} :
\nu \in K^\wedge \text{ with } \sigma^\nu \neq \{0\}\}
\end{equation}
The following result is (the local analogue of) \cite[Lem 2.6.6]{michel-2009}.
\begin{lemma}\label{lem:compare-conductors}
Suppose that $G = \GL_r(F)$ and that $\sigma$ is irreducible
and generic. Then
\begin{equation}\label{eqn:log-conductor-comparison}
\log(1 + C(\sigma)) \asymp \log(1 + C'(\sigma)).
\end{equation}
\end{lemma}
Note that the assumptions of \S\ref{sec:uniformity} imply that
$\sigma$ is $\vartheta$-tempered for some fixed $\vartheta > 0$.
The proof of lemma \ref{lem:compare-conductors} is attributed in
\emph{loc. cit.}, to a personal communication from W.T. Gan, but
no proof is recorded.
The case $r=1$ is straightforward
(see Lemma
\ref{lem:gl1-character-analytic-conductor-invariance}
and \cite[\S1.1]{JN19a} for related discussion).
We explain
the case $r = 2$ relevant to this paper (and also to
\cite{michel-2009}).
The archimedean
case can be verified
by comparing the Casimir eigenvalue and analytic
conductor of each irreducible representation.
In the non-archimedean
case, the main input is the following:
\begin{lemma}
Assume $F$ non-archimedean.
Let $\sigma$ be an irreducible representation of $\GL_2(F)$
and $\omega$ a character of $F^\times$.
Then
\begin{equation}\label{eqn:conductor-inequality-twists}
C(\sigma \otimes \omega)
\leq
\max
( C(\sigma) C(\omega), C(\omega)^2 )
\end{equation}
\end{lemma}
\begin{proof}
This is a special case of \cite[Thm 1]{MR1462836};
see also \cite[Prop
3.4]{xoMR0476703} and \cite{MR1606410}
for sharper bounds in the supercuspidal case.
\end{proof}
We now deduce \eqref{eqn:log-conductor-comparison} from \eqref{eqn:conductor-inequality-twists}.
Set $K := \GL_2(\mathfrak{o})$ and
\[
K_1[n] :=
\left\{\begin{pmatrix}
a & b \\
c & d
\end{pmatrix} \in K : c, d- 1 \in \mathfrak{p}^n \right\},
\quad K[n] := \left\{ k \in K : k \equiv 1 (\mathfrak{p}^n)
\right\}.
\]
If $\sigma$ contains a nonzero vector invariant by $K_1[n]$,
then the same vector is invariant by $K[n]$, and so
$C'(\sigma) \leq C(\sigma)$. Conversely, write
$C'(\sigma) = q^n$, so that $\sigma$ contains a nonzero vector
invariant by $K[n]$. The translate of that vector by a suitable
diagonal element of $G$
is then invariant by the following conjugate of
$K[n]$:
\[
K \cap
(1 + \begin{pmatrix}
\mathfrak{p}^n & \mathfrak{o} \\
\mathfrak{p}^{2 n} & \mathfrak{p}^n
\end{pmatrix}).
\]
Let $\omega_\sigma$ denote the central character of $\sigma$.
By Fourier decomposition with respect to the diagonal subgroup
of $K$, we may find a character $\chi$ of $F^\times$ with
$C(\chi) \leq q^n$ and a nonzero vector in $\sigma$ transforming
under $J_{1}[2n]$ by the character $\left(
\begin{smallmatrix}
a&b\\
c&d
\end{smallmatrix}
\right) \mapsto \omega_{\sigma}(d) \chi(a/d)$. Then the twisted
representation $\sigma \otimes \chi^{-1}$ contains a nonzero
vector invariant by $J_1[2 n]$. By newvector theory
\cite{MR0337789}, it follows that
$C(\sigma \otimes \chi^{-1}) \leq q^{2 n}$.
By \eqref{eqn:conductor-inequality-twists}
we deduce the sufficient estimate
\begin{equation}
C(\sigma) = C( (\sigma \otimes \chi^{-1}) \otimes \chi)
\leq \max(q^{2 n} C(\chi), C(\chi)^2)
\leq q^{3 n} = C'(\sigma)^3.
\end{equation}
\subsection{Reduction to isotypic vectors}\label{sec:reduct-isotyp-vect}
Let
$\sigma^{\flat}$ denote the subspace of $K$-finite vectors in
$\sigma$.
\begin{lemma}
Let
$\ell$ be a linear functional on $\sigma^{\flat}$ such that the estimate
$\ell(v) \ll \mathcal{S}_d(v)$ holds for some fixed $d$ and all
$K$-isotypic vectors $v \in \sigma$. Then $\ell$ extends to a
continuous linear functional on $\sigma$ for which the estimate
$\ell(v) \ll \mathcal{S}_{d'}(v)$ holds for some fixed $d'$ and
all $v \in \sigma$.
\end{lemma}
\begin{proof}
For $K$-finite $v$,
our hypothesis gives
\[
|\ell(v)|
\leq \sum_{\nu} |\ell(v_\nu)|
\ll \sum_{\nu} \mathcal{S}_d(v_\nu).
\]
On the other hand, for any $d_0 \geq 0$, we have
\[
\mathcal{S}_d(v_\nu)
=
\mathcal{S}_{d+d_0}(v_\nu)
N_{\sigma \nu}^{-d_0}
\leq
\mathcal{S}_{d+d_0}(v)
N_{\sigma \nu}^{-d_0}.
\]
Choosing $d_0$ as in lemma
\ref{lem:uniform-bounds-N-sigma-nu},
the required estimate follows
(with $d' := d + d_0$) from
\eqref{eq:sum-N-sigma-nu-neg-d-0}.
\end{proof}
\subsection{Integration by parts}\label{sec:integration-parts}
Suppose given a pair of groups $G$ and $H$ as above, with
$H$
contained in $G$.
Let $\pi$ and $\sigma$ be representations of $G$ and $H$,
respectively, of the sort considered in
\S\ref{sec:norms-reps-setting}. We assume that the conditions
of \S\ref{sec:uniformity} are satisfied for both $(G,\pi)$ and
$(H,\sigma)$, with $\mathbf{H}$ an algebraic subgroup of
$\mathbf{G}$.
We will refer subsequently to the following lemma
as \emph{integration
by parts with respect to $\sigma$}.
\begin{lemma}
Suppose given a linear functional
$\ell : \pi \otimes \sigma \rightarrow \mathbb{C}$
that is invariant by the diagonal action of $H$
and satisfies
\begin{equation}\label{eq:int-by-parts-hyp}
\ell(u \otimes v)
\ll \mathcal{S}_{d_0}(u) \mathcal{S}_{d_0}(v)
\end{equation}
for some fixed $d_0$.
Then for fixed $d, d'$ with $d'$ large in terms of $d$,
\begin{equation}\label{eq:int-by-parts-conc-better}
\ell(u \otimes v)
\ll \mathcal{S}_{d'}(u) \mathcal{S}_{-d}(v) C'(\sigma)^{-d}.
\end{equation}
\end{lemma}
\begin{proof}
By the definition of $C'(\sigma)$,
we have $\mathcal{S}_{-2 d}(v) \ll \mathcal{S}_{-d}(v)
C'(\sigma)^{-d}$,
so by replacing $d$ with $2 d$
it will suffice to show that
\begin{equation}\label{eq:int-by-parts-conc}
\ell(u \otimes v)
\ll \mathcal{S}_{d'}(u) \mathcal{S}_{-d}(v).
\end{equation}
We choose $C \geq 1$ fixed but large enough.
For any $b \in \mathbb{Z}_{\geq 0}$,
the $H$-invariance of $\ell$
and the initial estimate
\eqref{eq:int-by-parts-hyp}
give
\begin{align*}
\ell(u \otimes v)
&= \ell( (C + \Delta_H)^b u \otimes (C + \Delta_H)^{-b} v)
\\
&\ll
\mathcal{S}_{d_0}( (C + \Delta_H )^b u)
\mathcal{S}_{d_0}( ( C + \Delta_H)^{-b} v).
\end{align*}
By \eqref{eqn:sobolev-norms-via-C-plus-delta},
we have
\begin{equation}
\mathcal{S}_{d_0}( ( C + \Delta_H)^{-b} v)
\asymp
\mathcal{S}_{d_0 - 2 b}(v).
\end{equation}
On the other hand,
we claim that
\begin{equation}\label{eq:estimate-S-d0-C-plus-delta-H-b-u}
\mathcal{S}_{d_0}( ( C + \Delta_H)^{b} u)
\ll
\mathcal{S}_{d_0 + 2 b}(v).
\end{equation}
In the archimedean case, this last estimate follows from
\eqref{eq:estimate-for-first-order-diff-op}. In the
non-archimedean case, denote by $K_G$ and $K_H$ the
corresponding maximal compact subgroups. Then
$K_H[m + \operatorname{O}(1)] \subseteq K_G[m]$ in general, and
$K_H[m] \subseteq K_G[m]$ if $q$ is sufficiently large. Thus
the eigenvalues of $\Delta_H$ on the $K_G$-isotypic subspaces
$\pi^\nu$ of $\pi$ are $\operatorname{O}(N_{\pi \nu}^2)$. The claim
\eqref{eq:estimate-S-d0-C-plus-delta-H-b-u} follows. We
conclude that \eqref{eq:int-by-parts-conc} holds with
$d' := d_0 + 2 b$ for any $b \geq (d + d_0)/2$.
\end{proof}
\subsection{Reduction to pure
tensors}\label{sec:reduct-pure-tens}
Let $\sigma_1$ and $\sigma_2$ be representations of a
pair of groups $G_1$ and $G_2$ as above. Let
$\sigma_i^{\flat} \subseteq \sigma_i$ denote the subspace of
$K_i$-finite vectors.
\begin{lemma}
Suppose given a bilinear form $\ell$ on
$\sigma_1^{\flat} \times \sigma_2^{\flat}$ satisfying the
estimate
\begin{equation}\label{eqn:hypothesis-ell-v1-otimes-v2-bound}
\ell(v_1, v_2)
\ll
\mathcal{S}_d(v_1) \mathcal{S}_d(v_2)
\end{equation}
for some $d$.
Then $\ell$ extends to a continuous linear functional
on the completed tensor product $\sigma_1 \otimes \sigma_2$
satisfying the estimate
\begin{equation}\label{eqn:ell-v-bounded-by-S-d-prime-v}
\ell(v) \ll \mathcal{S}_{d'}(v).
\end{equation}
\end{lemma}
\begin{proof}
By \S\ref{sec:reduct-isotyp-vect},
we may assume
that
$v$ is $K_1 \times K_2$-isotypic,
say transforming under
$\nu_1 \otimes \nu_2$
for some $\nu_j \in K_j$.
We
may decompose $v$ with respect
to an orthonormal basis of its isotypic component as $v = \sum_i v_i$,
where the number of summands is
$\dim(\sigma_1^{\nu_1}) \dim(\sigma_2^{\nu_2})$
and $\mathcal{S}_d(v)^2 = \sum_i \mathcal{S}_d(v_i)^2$.
By \eqref{eqn:hypothesis-ell-v1-otimes-v2-bound}
and Cauchy--Schwarz,
we obtain
\[
\ell(v)
\ll
\sum_i \mathcal{S}_d(v_i)
\leq
\sqrt{\dim(\sigma_1^{\nu_1}) \dim(\sigma_2^{\nu_2})}
\mathcal{S}_d(v).
\]
Choosing $d_0$ large enough that
$\dim(\sigma_j^{\nu_j}) \leq N_{\sigma_j \nu_j}^{2 d_0}$ and
taking $d' := d + d_0$ then gives
\eqref{eqn:ell-v-bounded-by-S-d-prime-v}.
\end{proof}
In particular, a functional $\ell$ on $\sigma_1 \otimes \sigma_2$
satisfies
$\ell(v_1 \otimes v_2) \ll \mathcal{S}(v_1) \mathcal{S}(v_2)$
if and only if it satisfies $\ell(v) \ll \mathcal{S}(v)$.
\subsection{Adelic setting}\label{sec:repr-adel-groups}
Let $F$ be a number field and
$\mathbf{G} \hookrightarrow \mathbf{G} \mathbf{L} _r$ a linear
reductive $F$-group, as in \S\ref{sec:uniformity}. Take now for
$G$ the set of points of $\mathbf{G}$ over the adele ring
$\mathbb{A}$ of $\mathbf{F}$. Let $\sigma$ be a representation
of $G$ arising as a restricted tensor product
$\sigma = \otimes_{\mathfrak{p}} \sigma_\mathfrak{p}$, where
each $\sigma_\mathfrak{p}$ arises as in
\S\ref{sec:norms-reps-setting}. Let
$K = \prod_{\mathfrak{p}} K_\mathfrak{p}$ be a maximal compact
subgroup of $G$, with $K_\mathfrak{p}$ satisfying the
assumptions of \S\ref{sec:norms-reps-setting}. Each irreducible
representation $\nu$ of $K$ is a tensor product
$\otimes \nu_\mathfrak{p}$ of irreducible representations
$\nu_\mathfrak{p}$ of $K_\mathfrak{p}$. We define
$N_{\sigma \nu}$ to be the product of the local quantities
$N_{\sigma_\mathfrak{p} \nu_\mathfrak{p}}$ defined in
\S\ref{sec:constr-sobol-norms}, and then define Sobolev norms
$\mathcal{S}_d$ on $\sigma$ by the same formula
\eqref{eq:defn-S-d-of-v-squared} as in the local case. We
define the operator $\Delta_G$ on $\sigma$ to be the tensor
product $\otimes \Delta_{G_\mathfrak{p}}$ of the operators
$\Delta_{G_\mathfrak{p}}$ on $\sigma_\mathfrak{p}$ defined in
\eqref{eqn:Delta-G-defn} and \eqref{eqn:defn-Delta-nonarch}.
We may similarly adapt the definitions of
\S\ref{sec:norms-schw-spac}
to adelic Schwartz spaces.
As in \cite[\S2]{michel-2009}, many of the local results given
above extend readily to the adelic setting. Assume that for
each archimedean place $\mathfrak{p}$, $\sigma_\mathfrak{p}$
satisfies the uniformity conditions specified in
\S\ref{sec:uniformity}. Then the estimate
\eqref{eqn:sobolev-norms-via-C-plus-delta} and the results of
\S\ref{sec:uniform-trace-class}, \S\ref{sec:sobolev-conductor},
\S\ref{sec:reduct-isotyp-vect}, \S\ref{sec:integration-parts}
and \S\ref{sec:reduct-pure-tens} hold. The reduction to the
local setting is explained in
\cite[\S2.6.3, Proof of (S1d)]{michel-2009}, and follows
ultimately from the divisor bound.
\section{Local preliminaries}
\label{sec-4}
Let $F$ be a local field, and let $\psi$ be a nontrivial unitary
character of $F$.
\subsection{Norms}\label{sec:norms-whit-type-reps}
Let $\sigma$ be a Whittaker type representation of
$G$.
Let $\sigma$ be a Whittaker type representation of $G$.
We obtain from \S\ref{sec:norms-repr} a system of Sobolev norms
$\mathcal{S}_d$ ($d \in \mathbb{R}$), and in particular a norm
$\|.\| = \mathcal{S}_0$ and inner product $\langle , \rangle$.
More precisely:
\begin{itemize}
\item If $\sigma$ is square-integrable, then we regard it as
coming equipped with an invariant inner product and apply the
construction of \S\ref{sec:norms-repr} with $M = G$.
\item If $\sigma = \mathcal{I}(\chi)$, then we write
$\chi = |.|^c \chi_0$ with $\chi_0$ unitary and apply the
construction of \S\ref{sec:norms-repr} with $M = A$. Thus for
$f \in \mathcal{I}(\chi)$ we have
$\|f\|^2 = \int_K |f|^2$.
This norm is invariant if $\chi$ is unitary.
\end{itemize}
We choose an orthonormal basis $\mathcal{B}(\sigma)$
for $\sigma$
consisting of $K$-isotypic elements.
\subsection{Whittaker intertwiners and duality}\label{sec:whitt-intertw-dual}
Let $\sigma$ be a Whittaker type
representation of $G$.
We denote by $\mathcal{W}(\sigma,\psi)$ the $\psi$-Whittaker
model, which consists of smooth functions
$W : G \rightarrow \mathbb{C}$ satisfying
$W(n(x) g) = \psi(x) W(g)$. We similarly define
$\mathcal{W}(\sigma, \bar{\psi})$. We normalize intertwiners
\[
\sigma \rightarrow
\mathcal{W}(\sigma,\psi),
\quad
\sigma \rightarrow
\mathcal{W}(\sigma,\bar{\psi})
\]
\[
f \mapsto W_f,
\quad
f \mapsto \tilde{W}_f
\]
as follows.
If $\sigma$ is square-integrable,
then we require that
\[
\|f\|^2 =
\int_A |W_f|^2
=
\int_A |\tilde{W}_f|^2
\]
(see \cite[\S6.4]{MR748505}
and \cite[\S10.2]{MR1999922}).
The remainder of this section
treats the
case $\sigma = \mathcal{I}(\chi)$.
We define
\begin{equation}\label{eq:W-f-of-g-definition}
W_f(g) :=
\int_{x \in F}^{\reg} f(w n(x) g) \psi(-x) \, d x,
\quad
\tilde{W}_f(g) :=
\int_{x \in F}^{\reg} f(w n(x) g) \psi(x) \, d x.
\end{equation}
The integrals \eqref{eq:W-f-of-g-definition}
converge absolutely for $\Re(\chi)$ sufficiently
large and may be defined in general by
analytic
continuation \cite[Thm 15.4.1]{MR1170566},
or equivalently, by regularization
with respect to the action
of $F^\times$.
For instance, for any
$\kappa \in C_c^\infty(F^\times)$
with $\int_u \kappa(u) \, \frac{d u}{|u|} = 1$,
we have
\begin{equation}\label{eq:whittaker-function-explicated-defn}
W_f(g)
=
\int_{x \in F}
\left(
\int_u
\kappa(u)
f(w n(x u) g)
\psi(- x u)
\, d u
\right)
\, d x,
\end{equation}
first for $\Re(\chi)$ sufficiently large, then in general by
analytic continuation with respect to a holomorphic family of
vectors. The point is that the integral
\eqref{eq:whittaker-function-explicated-defn},
taken in the indicated order, converges
absolutely for all $\chi$.
We note the formula
\begin{equation}\label{eq:W-f-viz-f}
W_f(y)
:=
W_f(a(y))
= |y|^{1/2}
\chi^{-1}(y)
\int_{x \in F}^{\reg}
f(w n(x)) \psi(- y x) \, d x,
\end{equation}
which follows by a simple rearrangement of
\eqref{eq:W-f-of-g-definition}.
If $\Re(\chi) > -1/2$,
then we have moreover the absolutely convergent
inversion formula
\begin{equation}\label{eq:fourier-inversion-for-whittaker-intertwiner}
f(w n(x))
= \int_{y \in F^\times}
W_f(y)
|y|^{-1/2} \chi(y)
\psi(y x)
\, d y,
\end{equation}
as follows from Fourier inversion on $C_c^\infty(F)$ together
with the absolute convergence of the double integral
$\int_{y \in F^\times} \int_{x \in F} |\int_u \kappa(u) f(w n(x
u)) \psi(- x y u) \, d u| \, d x \, d y$.
We note that if $\chi$ is unitary (i.e., $c = 0$), then
by
\eqref{eqn:local-parseval-f-vs-W-f} and Parseval,
\begin{equation}
\|f\|^2 = \int_K |f|^2 = \int_{B \backslash G} |f|^2 =
\int_A |W_f|^2
=
\int_A |\tilde{W}_f|^2.
\end{equation}
On the other hand, the unitary structures on complementary
series representations
play no role in this paper.
For general $\chi$,
there is a natural duality between $\mathcal{I}(\chi)$ and
$\mathcal{I}(\chi^{-1})$, given by
$(f,\tilde{f}) \mapsto \int_{B \backslash G} f \tilde{f}$. Here
we equip $B \backslash G$ with the quotient measure
corresponding to the left Haar on $B$ described by the map
$A \times N \rightarrow B$, $(a,n) \mapsto a n$. In view of our
choice (\S\ref{sec-4-1}) of Haar measures on $G$ and on $K$, we
may write $\int_{B \backslash G} f \tilde{f}$ as
$\int_{n \in N} f (w n) \tilde{f}(w n)$ or as
$\int_K f \tilde{f}$.
We then have
\begin{equation}\label{eqn:local-parseval-f-vs-W-f}
\int_K \tilde{f} f =
\int_{B \backslash G} \tilde{f}f = \int_{A}^{\reg} \tilde{W}_{\tilde{f}} W_f,
\end{equation}
first when
$-1/2 < \Re(\chi) < 1/2$
(so that the latter integral converges absolutely, see \cite[\S1.9, \S1.10]{MR3889963}),
then in general by analytic continuation along a flat family.
\subsection{Holomorphic families of vectors}\label{sec:families-vectors}
By a holomorphic family of vectors
$f[\chi] \in \mathcal{I}(\chi)$
defined for $\chi$ in some open subset $X$ of $A^\wedge$,
we mean a continuous map
\[
X \times G \rightarrow \mathbb{C}
\]
\[
(\chi,g) \mapsto f[\chi](g)
\]
with the following properties.
\begin{itemize}
\item $f[\chi] \in \mathcal{I}(\chi)$ for each $\chi \in X$.
\item The map $X \ni \chi \mapsto f[\chi](g) \in \mathbb{C}$ is
holomorphic for each $g \in G$.
\item If $F$ is non-archimedean,
then for each $\chi_0 \in X$
there is a compact open subgroup $U$ of $G$ so
that the vector $f[\chi]$ is $U$-invariant
for all $\chi$ in some neighborhood of $\chi_0$.
\item If $F$ is archimedean, then for every continuous seminorm
$\mathcal{N}$ on $C^\infty(K)$, the composition
$X \ni \chi \mapsto \mathcal{N}(f[\chi])$ is locally bounded;
equivalently,
$\chi \mapsto \mathcal{S}_d(f[\chi])$
is locally bounded for each $d \in \mathbb{R}$
(\S\ref{sec:norms-repr}).
\end{itemize}
By a flat family of vectors $f[\chi] \in \mathcal{I}(\chi)$ we
mean one for which $f[\chi](g)$ is locally constant in $\chi$
for each $g \in K$.
We note that
any holomorphic family $f[\chi]$ may be written as a sum
$\sum_v c_v(\chi) v[\chi]$ of flat families $v[\chi]$ consisting
of $K$-isotypic vectors with holomorphic coefficients
$c_v(\chi)$. For $F$ non-archimedean, the sum over $v$ is
finite. For $F$ archimedean, it converges rapidly in the sense
that if $v[\chi]$ has $K$-type $\nu$, then
$\|v[\chi]\| \ll_{d,\chi} N_{\mathcal{I}(\chi),\nu}^{-d}$ for
any fixed $d$ (see \S\ref{sec:constr-sobol-norms}), locally
uniformly in $\chi$.
\subsection{Standard intertwining operator}
The standard intertwining operator
is the $G$-equivariant map
$M : \mathcal{I}(\chi) \rightarrow \mathcal{I}(\chi^{-1})$ is
given for $\chi$ of large real part by
$M f(g) := \int_{n \in N} f(w n g)$ and in general
by meromorphic continuation.
One may check
(see e.g. \cite[Prop 4.5.9]{MR1431508})
that
\begin{equation}\label{eqn:W-Mf-vs-W-f}
W_{M f}
=
\gamma(\psi, \chi^{-2},1)
W_f.
\end{equation}
\subsection{Normalized elements
and unramified calculations}\label{sec:normalized-spherical-induced-rep}
Suppose that $F$ is non-archimedean and $\chi$ is unramified.
The \emph{normalized
spherical element} $f \in \mathcal{I}(\chi)$ is defined to be
the unique $K$-invariant vector with $f(1) = 1$, thus
$f (n(x) a(y) k) = |y|^{1/2} \chi(y)$ for
$x \in F, y \in F^\times, k \in K$.
Suppose $(F,\psi)$ is unramified.
Let $\sigma$ be an unramified
generic irreducible representation.
Then the nonzero $K$-invariant vectors
$W \in \mathcal{W}(\sigma,\psi)$
satisfy $W(1) \neq 0$
\cite[Thm 4.6.5]{MR1431508}.
The \emph{normalized spherical Whittaker function} $W \in
\mathcal{W}(\sigma,\psi)$
is defined to be the unique $K$-invariant vector
with $W(1) = 1$.
Such an element exists
(see \emph{loc. cit.}).
If $W \in \mathcal{W}(\sigma,\psi)$ and
$\tilde{W} \in \mathcal{W}(\sigma,\bar{\psi})$ are normalized
spherical, then
$\int_{A}^{\reg}\tilde{W} W = L(\ad \sigma, 1) / \zeta_F(2)$
(see \cite[Proof of Prop 3.8.1]{MR1431508}).
For any local field $F$
and character $\chi$ of $F^\times$ for which $L(\chi^2,1)$
is finite, we attach to each
$f \in \mathcal{I}(\chi)$ its multiple
\begin{equation}\label{eq:defn-f-asterisk}
f^* := L(\chi^2,1) f.
\end{equation}
Suppose now that $F$ is non-archimedean
and $\chi$ is unramified,
so that $\mathcal{I}(\chi)$ is unramified.
Let $f \in \mathcal{I}(\chi)$
and $\tilde{f} \in \mathcal{I}(\chi^{-1})$
denote the normalized spherical elements.
Then:
\begin{itemize}
\item For unramified $\psi$, the Whittaker function $W_{{f}^*}$
and $\tilde{W}_{\tilde{f}^*}$ are normalized spherical (i.e.,
$f(1) = 1 \implies W_{f^*}(1) = 1$). (When
$\mathcal{I}(\chi)$ is reducible, we define $W_{f^*}$ and
$\tilde{W}_{\tilde{f}^*}$ by analytic continuation.)
\item $M ( f^*) = (\tilde{f} ) ^*$;
\item
$\int_{B \backslash G} \tilde{f} f = \zeta_F(1)/\zeta_F(2)$.
\end{itemize}
For proofs of these facts we refer
respectively to \cite[Thm 4.6.5]{MR1431508},
\cite[Prop 4.6.7]{MR1431508}
and \cite[\S3.1.6]{michel-2009}.
\subsection{Hecke integrals}
\label{sec-4-3}
\label{sec:local-hecke-basic}
We recall
some special cases
of the results of \cite{MR701565,MR0401654}.
Let $\sigma$ be a Whittaker type representation of $G$, and let
$W \in \mathcal{W}(\sigma,\psi)$. For almost all $\omega \in A^\wedge$, the
\emph{local Hecke integral} $\int_{A}^{\reg} W \omega$ is
defined, and meromorphic in $\chi$. The integral converges for
$\Re(\omega)$ large enough, and the ratio
\begin{equation}\label{eqn:local-hecke-ratio}
\frac{\int_A^{\reg} W \omega }{ L(\sigma \otimes \omega,1/2)/\zeta_F(1)}
\end{equation}
extends to a holomorphic function of $\omega$.
For a character $\omega$ of $A$, we denote by
$\mathbb{C}(\omega)$ the corresponding one-dimensional
representation. For any representation $V$ of $A$, we use the
map $v \otimes 1 \mapsto v$ to identify
$V \otimes \mathbb{C}(\omega)$ with $V$, but equipped with the
twisted $A$-action. When $\omega = |.|^s$, we write simply
$\mathbb{C}(s)$. (We will never use potentially ambiguous
notation such as $\mathbb{C}(1)$.)
Thus
after choosing an identification
$\sigma \cong \mathcal{W}(\sigma,\psi)$,
the (normalized) Hecke integral
defines an $A$-invariant functional
\[
\sigma \otimes \mathbb{C}(\omega) \rightarrow \mathbb{C}.
\]
If $(F,\psi)$ and $\sigma$ are unramified and
$W \in \mathcal{W}(\sigma,\psi)$ is the normalized spherical
vector, then the ratio
\eqref{eqn:local-hecke-ratio}
vanishes unless $\omega$ is
unramified, in which case
it evaluates to $1$
(see \cite[Prop 3.5.3]{MR1431508} and
\eqref{eq:volume-of-unit-group}).
\subsection{Rankin--Selberg integrals}
\label{sec-4-5}
\label{sec:local-rs-basic}
Let $\sigma_1, \sigma_2$ be Whittaker type representations
of $G$. Let $\chi$ be a character of $A$. Let
$W_1 \in \mathcal{W}(\sigma_1, \psi), W_2 \in
\mathcal{W}(\sigma_2, \bar{\psi})$ and
$f \in \mathcal{I}(\chi)$. If $\Re(\chi)$ is large enough, then
the \emph{local Rankin--Selberg integral}
$\int_{N \backslash G} W_1 W_2 f$ converges absolutely. It
extends meromorphically. We denote that extension by
$\int_{N \backslash G}^{\reg} W_1 W_2 f$.
The notation reflects that the extension may be defined
equivalently by regularized integration on the quotient
$N \backslash G$, regarded as a left $A$-space,
as follows from lemma \ref{lem:unif-polyn-type-hecke}
and \eqref{eqn:regularized-integral-via-mellin-analysis}.
Letting $f \in \mathcal{I}(\chi)$ vary over a
holomorphic family, the ratio
\begin{equation}\label{eqn:normalized-local-RS}
\frac{\int_{N \backslash G}^{\reg} W_1 W_2 f^* }{ L(\sigma_1 \otimes
\sigma_2 \otimes \chi,1/2)/\zeta_F(2)}
\end{equation}
extends to a holomorphic function of $\chi$.
To deduce this last assertion from the cited references,
we use
(see the proof of \cite[Lemma, p6]{MR871663})
that any holomorphic
family $f[\chi] \in \mathcal{I}(\chi)$
is locally of the form
$f[\chi] = f_{\chi,\Phi}$ for some
$\Phi \in \mathcal{S}(F^2)$,
where $f_{\chi,\Phi}$
is defined by the local Tate integral
\begin{equation}\label{eq:godement-parametrization-induced-rep}
f_{\chi,\Phi}(g)
:=
\frac{|\det g|^{1/2} \chi(\det g)}{L(\chi^2,1)}
\int_{r \in F^\times}^{\reg}
\Phi(r e_2 g)
|r| \chi^2(r)
\, \frac{d r}{|r|}.
\end{equation}
For each $\chi$, we may find $f$ so that the ratio
\eqref{eqn:normalized-local-RS} (defined in general by analytic
continuation) is nonzero. Thus
when $\sigma_1, \sigma_2$ and $\mathcal{I}(\chi)$ are irreducible,
\eqref{eqn:normalized-local-RS} defines a basis element for the
one-dimensional space of $G$-invariant trilinear functionals on
$\sigma_1 \otimes \sigma_2 \otimes \mathcal{I}(\chi)$ (see
\cite{MR1198305, MR1810211}).
If $(F,\psi),\sigma_1, \sigma_2$ and $\chi$ are unramified and
$W_1, W_2,f$ are normalized spherical, then the ratio
\eqref{eqn:normalized-local-RS} evaluates to $1$ (see \cite[Prop
3.8.1]{MR1431508}, \cite[\S3.1.6]{michel-2009} and
\eqref{eq:volume-of-unit-group}).
\subsection{Uniform polynomial-type
estimates}\label{sec:local-hecke-rs-continuity}
We record here some crude polynomial estimates for local Tate,
Hecke and Rankin--Selberg integrals, as well as some related
estimates for Whittaker functions, that will be used later to
verify some technical assertions (concerning convergence,
continuity, etc.) in the archimedean case and to verify the
polynomial dependence of our estimates upon auxiliary parameters
(e.g., upon $\sigma$ in Theorem \ref{thm:CI}).
Recall
(\S\ref{sec:norms-schw-spac})
that we have equipped
the Schwartz space $\mathcal{S}(F)$
with Sobolev norms $\mathcal{S}_d$
indexed by real numbers $d$,
and that $\mathcal{S}$ denotes a Sobolev norm of some fixed
index.
\begin{lemma}\label{lem:crude-tate-integral-estimate}
Let $\chi$ be a character
of $F^\times$.
Suppose that the real part of $\chi$ is
$\operatorname{O}(1)$
and that $\chi$ is at least some fixed positive
distance away from any pole of $L(\chi,0)$.
Then for each fixed real number $d$
and any $\phi \in \mathcal{S}(F)$,
we have the
local Tate integral
estimate
\begin{equation}
\int_{x \in F^\times}^{\reg} \phi(x) \chi(x) \, \frac{d x}{|x|}
\ll \mathcal{S}(\phi) C(\chi)^{-d}.
\end{equation}
\end{lemma}
\begin{proof
Suppose first that $\Re(\chi) \geq 2$.
Then the indicated integral converges
absolutely, and the required estimate follows readily: In the
archimedean case, we appeal to the Sobolev lemma and partial
integration. In the non-archimedean case, we may assume
as in \S\ref{sec:reduct-isotyp-vect}
that
$\phi(y) = 1_{x + \mathfrak{o}}(y) \psi_0(\xi y)$ for some
unramified character $\psi_0$ of $F$
and
$x,\xi \in F$.
Then $\mathcal{S}_{d'}(\phi) \asymp \max(1,|x|,|\xi|)^{d'}$.
The integral $\int_{x \in F^\times} \phi(x) \chi(x) \,
\frac{d x}{|x|}$
vanishes unless one of the following
conditions holds,
in which case its magnitude is $\operatorname{O}(1)$:
\begin{itemize}
\item $x \in \mathfrak{o}, \xi \in \mathfrak{o}$,
$C(\chi) = 1$.
\item $x \in \mathfrak{o}, \xi \notin \mathfrak{o}$,
$C(\chi) = 1, |\xi| \leq q$.
\item $x \in \mathfrak{o}, \xi \notin \mathfrak{o},
C(\chi) > 1, |\xi| = C(\chi)$.
\item
$x \notin \mathfrak{o}$,
$|x| \max(1,|\xi|) \geq C(\chi)$.
\end{itemize}
In each case the required estimate follows.
The required estimate follows then for $\Re(s) \leq -1$ by the
Tate local functional equation
\begin{equation}\label{eqn:Tate-local-func-eqn}
\int_{x \in F^\times}^{\reg}
\phi(x)
\chi(x) |x|^s \, \frac{d x}{|x|}
=
\frac{\int_{x \in F^\times}^{\reg}
(\int_{y \in F} \phi(y) \psi(-y x) \, d y)
\chi^{-1}(x) |x|^{1-s} \, \frac{d x}{|x|}}{ \gamma(\psi,\chi,s)}
\end{equation}
and the
Stirling consequence \eqref{eq:stirling-for-general-RS-gamma}, and
finally for general $s$ by Phragmen--Lindel{\"o}f.
\end{proof}
\begin{lemma}\label{lem:unif-polyn-type-hecke}
Let $\sigma$ be a Whittaker type representation of $G$.
Assume $\sigma$ is $\vartheta$-tempered
for some $\vartheta = \operatorname{O}(1)$.
For each
$f \in \sigma$,
we have the following estimates:
\begin{enumerate}[(i)]
\item Let $\omega \in A^\wedge$
be
such that $\Re(\omega) = \operatorname{O}(1)$
and the distance between $\omega$
and each pole of $L(\sigma \otimes \omega,1/2)$
is $\gg 1$. Then for each fixed $d$,
\begin{equation}\label{eqn:upper-bound-Mellin-transform-W}
\int_A^{\reg} W_f \omega \ll
\mathcal{S}(f)
C(\omega)^{-d} q^{\operatorname{O}(1)}.
\end{equation}
\item
For $|y| \geq 1$ and fixed $d$,
\begin{equation}\label{eqn:estimate-W-near-infinity}
W_f(y) \ll \mathcal{S}(f) |y|^{-d} q^{\operatorname{O}(1)}.
\end{equation}
\item
For $|y| \leq 1$,
\begin{equation}\label{eq:estimate-W-near-0-with-eps}
W_f(y) \ll \mathcal{S}(f)
|y|^{1/2-\vartheta-\varepsilon} q^{\operatorname{O}(1)}.
\end{equation}
\item
There is a finite function
$\phi$ on $F^\times$
so that for $|y| \leq 1$ and fixed $d$,
\begin{equation}\label{eqn:estimate-W-near-zero}
W_f(y) - \phi(y) \ll \mathcal{S}(W_f) |y|^{d} q^{\operatorname{O}(1)}.
\end{equation}
Moreover, we may choose $\phi$ so that:
\begin{enumerate}[(a)]
\item The dimension of the span of the $A$-translates
of $\phi$ is $\operatorname{O}(1)$.
\item We may write
$\phi(y) = \sum c_i \chi_i(y) (\log |y|)^{m_i}$,
corresponding to the principal part of
$\int_A^{\reg} W_f \omega$ for $\Re(\omega) > -d$. Thus the
$\chi_i \in A^\wedge$ are poles of
$L(\sigma \otimes \chi_i^{-1}, 1/2)$, the $m_i$ are
nonnegative integers with $m_i + 1$ bounded by the
multiplicity of the corresponding pole, and the
coefficients $c_i$ depend linearly upon $W$.
\item If $\sigma = \mathcal{I}(\chi)$ and we take
$f = f[\chi]$ for some holomorphic family
$f[\chi] \in \mathcal{I}(\chi)$ defined for $\chi$ in a
small open subset of $A^\wedge$, then
$\phi = \phi_{f[\chi]}$ varies pointwise holomorphically
with $\chi$.
\end{enumerate}
\end{enumerate}
\end{lemma}
\begin{proof
We loosely follow an argument of Jacquet (see \cite[Thm 2.3,
\S12.2]{MR2533003} and also \cite[\S3.2.3]{michel-2009}). The
estimates \eqref{eqn:estimate-W-near-infinity},
\eqref{eq:estimate-W-near-0-with-eps} and
\eqref{eqn:estimate-W-near-zero}, together with the noted
refinement of \eqref{eqn:upper-bound-Mellin-transform-W} and
the noted properties of $\phi$, follow readily from
\eqref{eqn:upper-bound-Mellin-transform-W}, Mellin expansion
and Cauchy's theorem. By Phragmen--Lindel{\"o}f, it suffices
to establish \eqref{eqn:upper-bound-Mellin-transform-W} when
$\Re(\omega)$ is sufficiently positive or negative in terms of
$\vartheta$. Recall the local functional equation (see
\S\ref{sec:local-gamma-factors}):
\[
\gamma(\psi,\sigma
\otimes \omega,1/2)
\int^{\reg}_A W \omega
=
\int^{\reg}_A w W |.| \omega^{-1}.
\]
For $\Re(\omega)$ sufficiently positive in terms of
$\vartheta$, the the Stirling asymptotics
\eqref{eq:stirling-for-general-RS-gamma} imply
that $\gamma(\psi,\sigma \otimes \omega,1/2) \ll q^{\operatorname{O}(1)}$. We
thereby reduce to establishing
\eqref{eqn:upper-bound-Mellin-transform-W} (both for $W$ and
its Weyl translate $w W$)
when $\Re(\omega)$ is sufficiently positive.
In that case, $\int_A W \omega$
converges absolutely (as follows either from standard
estimates for individual $W$ or from the uniform estimates
verified below). Using the $A$-equivariance of the Hecke
integral
and integration by parts with respect to $\omega$
(\S\ref{sec:integration-parts}), we may reduce further to establishing that
\begin{equation}\label{eqn:very-weak-bound-for-Hecke-integral-uniform-0}
\int_A W \omega \ll \mathcal{S}(W)
C(\omega)^{\operatorname{O}(1)} q^{\operatorname{O}(1)}.
\end{equation}
Since $\Re(\omega)$ is large enough in terms
of $\vartheta$ but of size $\operatorname{O}(1)$,
we reduce further to verifying for each fixed $d$ that
\begin{equation}\label{eqn:very-weak-bound-Whittaker-upper-bound-all-over}
W(y)
\ll
\mathcal{S}(W)
|y|^{-d} q^{\operatorname{O}(1)}.
\end{equation}
If $\sigma$ is square-integrable, the required
estimate
\eqref{eqn:very-weak-bound-Whittaker-upper-bound-all-over}
follows (in stronger form) from \cite[\S3.2.3]{michel-2009}.
We may thus suppose that $\sigma = \mathcal{I}(\chi)$,
so that $W = W_f$ for some $f \in \mathcal{I}(\chi)$.
Define $\kappa \in C_c^\infty(F^\times)$
in the non-archimedean case as the normalized characteristic
function of $\mathfrak{o}^\times$
and in the archimedean case as a fixed bump function,
with the normalization in either case so that
$\int \kappa(u) \, \frac{d u }{|u|} = 1$.
We may then express $W_f$
as the absolutely-convergent integral
\begin{equation}\label{eqn:W-f-actual-defn-via-convolution}
W_f(y)
=
|y|^{1/2} \chi^{-1}(y)
\int_{x \in F}
I_f(x,y)
\, d x,
\end{equation}
where
\begin{equation}
I_f(x,y) :=
\int_{u \in F^\times}
\kappa(u)
f(w n(x u)) \psi(- y x u)
\, \frac{d u}{|u|}.
\end{equation}
We thereby reduce to verifying that
\begin{equation}\label{eqn:estimate-I-f-small-x}
\int_{x : |x| \leq 1}
I_f(x,y) \, d x
\ll |y|^{-d}
\mathcal{S}(f) q^{\operatorname{O}(1)},
\end{equation}
and that
\begin{equation}\label{eqn:estimate-I-f-large-x}
I_f(x,y)
\ll |x y|^{-d}
\mathcal{S}(f) q^{\operatorname{O}(1)}
\text{ for } |x| \geq 1.
\end{equation}
To that end, we first verify some pointwise estimates for $f$.
We may assume that $f$ is a unit vector in $\sigma^{\nu}$ for
some $\nu \in K^\wedge$. Let us say that a quantity is
\emph{bounded polynomially} if it is of the form
$N_{\sigma \nu}^{\operatorname{O}(1)} q^{\operatorname{O}(1)}$. By the Sobolev lemma on
$K$, we see that the $L^\infty$-norms on $K$ of $f$ and any
archimedean derivatives of fixed degree are bounded polynomially.
(In the non-archimedean case,
we use that $f$ is invariant by $K[n]$
with $N_{\sigma \nu} = q^n$
to pass from an $L^2$ bound to an $L^\infty$ bound.)
By lemma
\ref{lem:compare-conductors}, $C(\chi)$ is
bounded polynomially. By the definition of
$\mathcal{I}(\chi)$, it follows that the $L^\infty$-norms on
$\{x : |x| \leq 10\}$ of the functions
$x \mapsto f(w n(x)), x \mapsto f(n'(x))$ and any archimedean
derivatives of fixed degree are bounded polynomially.
The estimate \eqref{eqn:estimate-I-f-small-x}
follows by partial integration.
(In the archimedean case,
this carries the usual meaning.
In the non-archimedean case, we use that $x \mapsto f(w n(x))$ is invariant
under translation by $\mathfrak{p}^n$ with $N_{\sigma \nu} =
q^n$
to see that
$\int_{x : |x| \leq 1}
I_f(x,y) \, d x$ vanishes unless $|y| \ll N_{\sigma \nu}$,
in which case it is bounded polynomially.)
The estimate \eqref{eqn:estimate-I-f-large-x}
follows similarly,
by partial integration
and the identity
\begin{equation}\label{eqn:identity-f-w-n-x-large-x-useful}
f(w n(x)) = |x|^{-1} \chi^{-2}(x) f(n'(x)).
\end{equation}
\end{proof}
\begin{lemma}\label{lem:polyn-bound-rs}
Let $\sigma_1, \sigma_2$
be Whittaker type representations
of $G$.
Let $\chi$ be a character
of $A$.
Assume that $\sigma_1, \sigma_2$ are $\vartheta$-tempered
for some $\vartheta = \operatorname{O}(1)$.
Assume that $\chi$ has real part $\operatorname{O}(1)$ and is some fixed
positive distance away from the poles of
$L(\sigma_1 \otimes \sigma_2 \otimes \chi, 1/2)$.
Then for $f_1 \in \sigma_1, f_2 \in \sigma_2, f_3 \in \mathcal{I}(\chi)$,
\begin{equation}\label{eq:sobolev-bound-RS}
\int_{N \backslash G}^{\reg} W_{f_1} \tilde{W}_{f_2} f_3 \ll
\mathcal{S}(f_1) \mathcal{S}(f_2) \mathcal{S}(f_3) q^{\operatorname{O}(1)}.
\end{equation}
\end{lemma}
\begin{proof
Suppose first that $\Re(\chi)$ is sufficiently large
in terms of $\vartheta$.
Then the integral converges
absolutely,
and the Whittaker function estimates
\eqref{eqn:estimate-W-near-infinity},
\eqref{eq:estimate-W-near-0-with-eps}
and the Sobolev lemma consequence
$\|f\|_{L^\infty(K)} \ll \mathcal{S}(f)$
(see the paragraph after \eqref{eqn:estimate-I-f-large-x})
imply the required estimate \eqref{eq:sobolev-bound-RS}.
Integrating by parts with respect
to $\mathcal{I}(\chi)$
(\S\ref{sec:integration-parts})
yields for any fixed $d \geq 0$ the refined estimate
\begin{equation}\label{eqn:refined-RS-sobolev-large-real-part-chi}
\int_{N \backslash G} W_1 W_2 f \ll
\mathcal{S}(W_1) \mathcal{S}(W_2) \|f\| C(\chi)^{-d} q^{\operatorname{O}(1)}
\end{equation}
(We have used here
that
$C(\mathcal{I}(\chi)) \asymp C(\chi)^2$.)
The following local functional equation may be
derived from \cite[Thm 2.7]{MR701565} and \cite[Thm
2.1]{MR2533003}, using
\eqref{eq:godement-parametrization-induced-rep}:
\begin{equation}\label{eq:RS-local-func-eqn}
\int_{N \backslash G}^{\reg} W_1 \cdot W_2
\cdot f
=
\frac{
\gamma(\psi,\chi^2,0)
}
{
\gamma(\psi, \sigma_1 \otimes \sigma_2 \otimes \chi, 1/2)
}
\int_{N \backslash G}^{\reg} W_1 \cdot W_2
\cdot M f.
\end{equation}
(We have used also that $\sigma_1, \sigma_2$ are
representations of $G = \operatorname{PGL}_2(F)$, hence have trivial central
character when regarded as representations of $\GL_2(F)$.) If
$\Re(\chi)$ is sufficiently \emph{negative} in terms of
$\vartheta$ (but chosen, as indicated, to avoid the poles of
$L(\sigma_1 \otimes \sigma_2 \otimes \chi,1/2)$), then
we have by \eqref{eq:stirling-for-general-RS-gamma}
the crude Stirling-type estimates
$1/\gamma(\psi, \sigma_1 \otimes \sigma_2 \otimes \chi, 1/2)
\ll q^{\operatorname{O}(1)}$ and $\gamma(\psi,\chi^2,0) \ll (q C(\chi))^{\operatorname{O}(1)}$.
We claim that $\|M f\| \ll \mathcal{S}(f)$. To see this, it
suffices (by the $K$-invariance of the Sobolev norms) to
verify that
$M f(1) = \int_{x \in F} f(w n(x)) \, d x \ll \mathcal{S}(f)$.
To that end, we smoothly decompose the $x$-integral according
as $|x| \leq 2$ and $|x| \geq 1$. The contribution from the
former range is estimated as in the proof of lemma
\ref{lem:unif-polyn-type-hecke}. After the change of
variables $x \mapsto 1/x$ and the identity
\eqref{eqn:identity-f-w-n-x-large-x-useful}, the contribution
from the latter range is a Tate integral for which adequate
bounds follow from lemma
\ref{lem:crude-tate-integral-estimate}.
We now embed $f$ in a flat family $f[\chi]$.
Note that $\|f[\chi]\|$ is locally constant in such a family.
By the local functional equation
\eqref{eq:RS-local-func-eqn}
and the noted estimates for the $\gamma$-factors and $M f[\chi]$,
we see that the estimate
\eqref{eqn:refined-RS-sobolev-large-real-part-chi}
holds also when $\chi$ has sufficiently \emph{negative} real part.
By Phragmen--Lindel{\"o}f,
we deduce
\eqref{eqn:refined-RS-sobolev-large-real-part-chi}
for all indicated $\chi$,
and in particular its consequence \eqref{eq:sobolev-bound-RS}.
\end{proof}
\begin{remark}
The stated estimates
that require separation from some pole
may be generalized
via Cauchy's
theorem.
For illustration,
we record the general form of
\eqref{eqn:upper-bound-Mellin-transform-W}.
For any $\omega \in A^\wedge$, let
$P(s)$ be the smallest monic polynomial such that
$P(s) L(\sigma \otimes \omega, 1/2 + s)$ is holomorphic for
$|s| < 2$. Then $P(s) \int_A^{\reg} W_f \omega |.|^s$ is
holomorphic for $|s| < 2$ and satisfies the analogue of
\eqref{eqn:upper-bound-Mellin-transform-W} for $|s| \leq 1$.
\end{remark}
\section{Global preliminaries}
\label{sec-5}
\subsection{Groups, measures, etc}
\label{sec-5-1}
Let $F$ be a number field with adele ring $\mathbb{A}$. We fix
a nontrivial unitary character
$\psi$ of $\mathbb{A}/F$.
We write $\mathfrak{p}$ for a place of $F$,
finite or infinite.
We write $\mathbf{G}$
for the $F$-algebraic groups given by $\operatorname{PGL}_2$. We define
subgroups $\mathbf{B}, \mathbf{N}, \mathbf{A}$ of $\mathbf{G}$
by analogy to the local case. We set $G := \mathbf{G}(F)$,
$G_\mathbb{A} := \mathbf{G}(\mathbb{A}), G_\mathfrak{p} :=
\mathbf{G}(F_\mathfrak{p})$ and
$[G] := G \backslash G_\mathbb{A}$. We similarly define $A$,
$A_{\mathbb{A}}, A_\mathfrak{p}$ and $[A]$, and likewise for
$\mathbf{N}$ and $\mathbf{B}$. Note that
$[A] \cong \mathbb{A}^\times/F^\times$ is an abelian group. We
define $K = \prod_{\mathfrak{p}} K_\mathfrak{p}$ with
$K_\mathfrak{p} \leq G_\mathfrak{p}$ as in the local case. We
identify $N$ and $A$ with $F$ and $F^\times$ as in the local
case, and similarly for $N_\mathbb{A}, A_\mathbb{A}$, etc.
We equip $N_\mathbb{A}$ with the product of the local measures
and $[N]$ with the quotient measure
(of volume $1$).
The
product of the local measures on $A_\mathfrak{p}$ does not
literally converge to a measure on
$A_{\mathbb{A}} \cong \mathbb{A}^\times$, but may be regularized
as follows: for $f \in C_c^\infty(A_\mathbb{A})$ of the form
$f(a) = \prod_{\mathfrak{p}} f_\mathfrak{p}(a_\mathfrak{p})$,
with $f_\mathfrak{p}$ the characteristic function of the unit
group for almost all finite $\mathfrak{p}$, we set
$\int_{A_\mathbb{A}} f := \xi_F^*(1)^{-1} \prod_\mathfrak{p}
\zeta_{F_\mathfrak{p}}(1) \int_{A_\mathfrak{p}} f_\mathfrak{p}$.
We equip $[A] \cong \mathbb{A}^\times/F^\times$ with the
quotient measure.
The pushforward of this measure under the
idelic norm
$\mathbb{A}^\times/F^\times \xrightarrow{|.|}
\mathbb{R}^\times_+$ is then $\tfrac{d t}{t}$, with $d t$
Lebesgue measure.
As in the local case,
the character group $[A]^\wedge$ of $[A]$
is a complex manifold
with charts $\chi |.|^s \mapsto s$.
We equip the group of unitary characters of
$[A]$, hence also its cosets consisting of characters of given
real part, with the measure dual to the chosen measure
on $[A]$.
Then for a real number $c$ and $f : [A]^\wedge \rightarrow \mathbb{C}$,
\begin{equation}\label{eq:measure-on-dual-omega}
\int_{\chi : \Re(\chi) = c}
f(\chi)
=
\sum_{\chi_0}
\int_{s : \Re(s) = c}
f(\chi_0 |.|^s) \, \frac{d s}{2 \pi i },
\end{equation}
where $\chi_0$ runs over a set of representatives for the group
of unitary characters of $[A]$ modulo the subgroup
$\{|.|^{i t} : t \in \mathbb{R} \}$. The normalization
\eqref{eq:measure-on-dual-omega}
is well-suited for the applications of Cauchy's
theorem and contour shift arguments
given in \S\ref{sec-12}.
We define the measure on $G_\mathbb{A}$
using the measures on $N_\mathbb{A}$ and $A_\mathbb{A}$
as in the local case,
and equip $[G]$ with the quotient measure,
which is then the Tamagawa measure of volume $2$.
\subsection{Induced representations and Eisenstein series}
\label{sec-5-2}
For a character $\chi$ of $[A]$, we denote by
$\mathcal{I}(\chi)$ the corresponding smooth induction, defined
by analogy to the local case and denoted $\mathcal{I}(s)$
when $\chi = |.|^s$.
The factorizable vectors $f \in \mathcal{I}(\chi)$
have the form
$f = \otimes_\mathfrak{p} f_\mathfrak{p} :
g
\mapsto \prod_\mathfrak{p}
f_\mathfrak{p}(g_\mathfrak{p})$,
where $f_\mathfrak{p}$ belongs to
$\mathcal{I}(\chi_\mathfrak{p})$
for all $\mathfrak{p}$ and is the normalized
spherical element for almost all finite $\mathfrak{p}$.
As in the local case, we may speak of holomorphic families of vectors.
For $\chi^2 \neq |.|^{\pm 1}$,
the standard intertwining operator
$M : \mathcal{I}(\chi) \rightarrow \mathcal{I}(\chi^{-1})$
is defined on factorizable vectors by
\[
M f(g)
=
\int_{x \in \mathbb{A}}
f(w n(x) g)
\, d x
:=
\frac{\Lambda(\chi^2,0)}{\Lambda(\chi^2,1)}
\prod_{\mathfrak{p}}
\frac{L(\chi_\mathfrak{p}^2,1)}{L(\chi_\mathfrak{p}^2,0)}
M f_{\mathfrak{p}}(g_\mathfrak{p}),
\]
with the product really a finite product.
The duality between
$\mathcal{I}(\chi)$ and $\mathcal{I}(\chi^{-1})$
is given on factorizable vectors by
the (finite) product
\[
(f,\tilde{f})
\mapsto
\int_{B_\mathbb{A} \backslash G_\mathbb{A}}
\tilde{f} f
:=
\frac{\xi_F^*(1)}{\xi_F(2)
}
\prod_{\mathfrak{p}}
\frac{\zeta_{F_\mathfrak{p}}(2)}{\zeta_{F_\mathfrak{p}}(1)}
\int_{B_\mathfrak{p} \backslash G_\mathfrak{p} }
\tilde{f}_\mathfrak{p} f_\mathfrak{p}.
\]
Suppose now that $\Re(\chi) > -1/2$ and $\chi^2 \neq |.|$. We
denote by $\Eis : \mathcal{I}(\chi) \rightarrow C^\infty([G])$
the Eisenstein intertwiner, defined for $\Re(\chi)$ sufficiently
large by the convergent sum
$\Eis(f)(g) := \sum_{\gamma \in B \backslash G} f(\gamma g)$ and
in general by meromorphic continuation. As $f$ varies in a
holomorphic family, $\Eis(f)$ vanishes as $\chi$ tends to any
quadratic character (i.e., for $\chi^2$ trivial). For $\chi$
non-quadratic and $f \in \mathcal{I}(\chi)$, we set
$f^* := \Lambda(\chi^2,1) f$ and $\Eis^*(f) := \Eis(f^*)$.
Although $f^*$ itself is not defined when $\chi$ is quadratic,
we may interpret $\Eis^*(f)$ by analytic continuation.
For $f \in \mathcal{I}(\chi)$
varying holomorphically, $\Eis^*(f)$
extends holomorphically to all $\chi$
except for possible simple poles at $\chi^2 = |.|^{\pm 1}$.
\subsection{Generic representations and Fourier expansions}
\label{sec-5-3}
By a \emph{generic automorphic representation} $\sigma$ (always
of $G$), we mean either an (irreducible) cuspidal automorphic
subrepresentation of $L^2([G])$ or an Eisenstein representation
of the form $\Eis^*(\mathcal{I}(\chi))$ for some character
$\chi$ of $[A]$ with $\chi^2 \neq |.|^{\pm 1}$.
The local components $\sigma_\mathfrak{p}$
of any such representation are of Whittaker type.
We say
that $\sigma$ is \emph{standard} if it is either cuspidal or of
the form $\Eis^*(\mathcal{I}(\chi))$ with $\chi$ unitary.
Let $\sigma$ be a generic automorphic representation. Each
$\varphi \in \sigma$ admits a Fourier expansion
$\varphi(g) = \varphi_N(g) + \sum_{\alpha \in F^\times}
W_\varphi(a(\alpha) g)$, where $\varphi_N = 0$ unless $\sigma$
is Eisenstein and
$W_\varphi(g) := \int_{x \in \mathbb{A}/F} \varphi(n(x) \psi(-x)
g) \, d x$. If $\varphi$ is factorizable, then
we may write
$W_\varphi = \otimes_\mathfrak{p} W_{\varphi,\mathfrak{p}}
: g \mapsto \prod_{\mathfrak{p}}
W_{\varphi,\mathfrak{p}}(g_\mathfrak{p})$, where
$W_{\varphi,\mathfrak{p}}$
belongs to
$\mathcal{W}(\sigma_\mathfrak{p},\psi_\mathfrak{p})$
for all $\mathfrak{p}$
and is the normalized spherical
Whittaker function
for almost all finite $\mathfrak{p}$.
We denote by $\tilde{W}_\varphi$
the Whittaker function defined analogously,
but using the opposite character $\bar{\psi}$ in place of
$\psi$.
If $\sigma$ is irreducible,
then it is isomorphic
to its own contragredient;
the duality
may be given by the pairing
defined for
factorizable vectors by
\[
(\varphi, \tilde{\varphi})
\mapsto
\int_{A_\mathbb{A}}^{\reg}
\tilde{W}_{\tilde{\varphi}}
W_{\varphi}
:=
\frac{\Lambda^*(\ad \sigma,1)}{\xi(2)}
\prod_{\mathfrak{p}}
\frac{\zeta_{F_\mathfrak{p}}(2)}{L(\ad \sigma_\mathfrak{p},1)}
\int_{A_{\mathfrak{p}}}^{\reg}
\tilde{W}_{\tilde{\varphi}} W_{\varphi},
\]
so that the product is really a finite product.
We specialize now to the case
that
$\sigma$ is the Eisenstein representation
$\mathcal{I}(\chi)$
attached to some character
$\chi$ of $[A]$
with $\chi^2 \neq |.|^{\pm }$.
Let $f \in \mathcal{I}(\chi)$
be factorizable,
and write $f =\otimes_{\mathfrak{p}} f_\mathfrak{p}$ as above.
Let $S$ be a finite set of places of $F$
large enough that
for $\mathfrak{p} \notin S$,
$(F_\mathfrak{p},\psi_\mathfrak{p})$ is unramified
and $f_\mathfrak{p}$ is the normalized spherical element.
Then $W_{f_\mathfrak{p}^*}$ is normalized spherical
for $\mathfrak{p} \notin S$
(see \S\ref{sec:normalized-spherical-induced-rep}).
Thus for each
$g \in G_\mathbb{A}$,
we have
$W_{f_\mathfrak{p}^*}(g_\mathfrak{p}) = 1$
for almost all $\mathfrak{p}$.
Thus
\[
W_{\Eis^*(f)}(g) = \prod_\mathfrak{p}
W_{f_\mathfrak{p}^*}(g_\mathfrak{p}),
\]
with all but finitely many factors in the product equal to $1$.
(This identity follows first for $\chi$ non-quadratic,
then in general by continuity.)
The full Fourier expansion of the corresponding Eisenstein
series
reads
\begin{equation}\label{eqn:eis-FE}
\Eis^*(f)(g)
=
f^*(g)
+
M f^*(g)
+
\sum_{\alpha \in F^\times}
W_{\Eis^*(f)}(a(\alpha) g).
\end{equation}
(When $\chi$ is quadratic,
the sum of the terms $f^*(g) + M f^*(g)$
is interpreted via continuity from the non-quadratic case.)
By standard estimates for local Whittaker functions
(lemma \ref{lem:unif-polyn-type-hecke}),
it follows that for
$g = n(x) a(y) k$ ($x \in \mathbb{A}, y \in \mathbb{A}^\times, k
\in K)$
and fixed $B \geq 0$,
we have
\begin{equation}\label{eq:estimate-eisenstein-series-near-cusp}
\Eis^*(f)(g) =
|y|^{1/2} \chi(y) f^*(k) +
|y|^{1/2} \chi^{-1}(y) M f^*(k)
+ \operatorname{O}_{f,d}(|y|^{-d}),
\end{equation}
and similarly for archimedean derivatives. Since $w \in G$,
we have
$\Eis^*(f)(a(y)) = \Eis^*(f)(w a(y)) = \Eis^*(f)(a(1/y) w)$, so
the estimate \eqref{eq:estimate-eisenstein-series-near-cusp} is
useful both as $|y| \rightarrow \infty$ and as
$|y| \rightarrow 0$.
We note finally that
if $-1/2 < \Re(\chi) < 1/2$
and $\chi^2 \neq 1$,
then
for $f \in \mathcal{I}(\chi)$ and $\tilde{f} \in
\mathcal{I}(\chi^{-1})$,
we have by \eqref{eqn:local-parseval-f-vs-W-f} that
\begin{equation}\label{eqn:Eis-induced-vs-Whittaker-global-L-2}
\int_{B_\mathbb{A} \backslash G_\mathbb{A}}
\tilde{f}^*f^*
=
\int_{A_\mathbb{A}}^{\reg}
\tilde{W}_{\Eis^*(\tilde{f})}
W_{\Eis^*(f)}.
\end{equation}
\subsection{Hecke integrals}
\label{sec-5-4}
Let $\omega$ be a character of $[A]$.
We denote by
$\mathbb{C}(\omega)$ the corresponding one-dimensional
representation,
and adopt the same conventions as in the local case.
Let $\sigma$ be a generic automorphic representation of $G$,
$\omega$ a character of $[A]$, and $\varphi \in \sigma$.
The \emph{global Hecke integral} $\int_{[A]}^{\reg} \varphi
\omega$
is defined initially for $\Re(\omega)$ large enough
and extends meromorphically.
It unfolds to a product of local Hecke integrals:
if $\varphi \in \sigma$
is factorizable,
then
\begin{equation}\label{eqn:defn-global-hecke-integral-unfold}
\int_{[A]}^{\reg}
\varphi \omega
=
\int_{A_\mathbb{A}}^{\reg}
W_{\varphi} \omega
:=
\frac
{
\Lambda(\sigma \otimes \omega, 1/2)
}
{
\xi_F^*(1)
}
\prod_\mathfrak{p}
\frac{
\zeta_{F_\mathfrak{p}}(1)
}
{
L(\sigma_\mathfrak{p} \otimes \omega_\mathfrak{p}, 1/2)
}
\int_{A_\mathfrak{p}}^{\reg}
W_{\varphi,\mathfrak{p}}
\omega_\mathfrak{p},
\end{equation}
with almost all local factors
equal to $1$.
The ratio
\begin{equation}\label{eqn:global-hecke-integral-ratio}
\frac{\int_{[A]}^{\reg} \varphi \omega}{\Lambda(\sigma \otimes
\omega, 1/2)}
\end{equation}
extends to a holomorphic function of $\omega$.
If $\sigma$ is cuspidal,
then the regularization is not needed.
If $\sigma = \Eis^*(\mathcal{I}(\chi))$
is Eisenstein,
then we can define $\int_{[A]}^{\reg} \varphi \omega$
as the meromorphic continuation
from $\omega$ with large real part
of the convergent integral $\int_{[A]} (\varphi - \varphi_N)
\omega$.
If $\varphi = \Eis^*(f)$
where $f = f_\chi \in \mathcal{I}(\chi)$
varies holomorphically with respect to $\chi$,
then the ratio
\eqref{eqn:global-hecke-integral-ratio}
is jointly holomorphic in $\omega$ and $\chi$.
\subsection{Height function}\label{sec:height-function}
Recall the height function
$\htt : [G] \rightarrow \mathbb{R}_{>0}$ defined by
$\htt(g) := \sup |y|$, the supremum taken over all ways to write
$g = \gamma n a(y) k$ with
$\gamma \in G, n \in N_\mathbb{A}, y \in \mathbb{A}^\times, k
\in K$. The map $\htt$ is proper, its image is bounded below by
a positive quantity (see \cite[\S8]{MR0191899}), and we have
$\int_{[G]}\htt^\alpha < \infty$ precisely when $\alpha < 1$.
For instance, the estimate
\eqref{eq:estimate-eisenstein-series-near-cusp}
implies that if $\chi$ is a unitary character of $[A]$ and
$f \in \mathcal{I}(\chi)$, then $\Eis^*(f)$ is bounded by a
multiple of $\htt^{\alpha}$ for each $\alpha > 1/2$.
\subsection{Spectral decomposition}
\label{sec-5-5}
Let $\Psi_1, \Psi_2,\Psi \in C_c^\infty([G])$.
The Parseval relation for
$L^2([G])$ may be written
\begin{align*}
\int_{[G]} \Psi_1 \Psi_2
&=
\sum_{\omega^2 = 1}
\frac{
\int_{[G]}\Psi_1 \omega^{-1}(\det)
\int_{[G]}\omega(\det) \Psi_2
}
{
\vol([G])
}
\\
&\quad +
\sum_{\sigma:\text{cuspidal}}
\sum_{\varphi \in \mathcal{B}(\sigma)}
\frac{
\int_{[G]} \Psi_1 \tilde{\varphi}
\int_{[G]} \varphi \Psi_2
}
{
\int_{[G]} \tilde{\varphi} \varphi
}
\\
&\quad
+
\frac{1}{2}
\int_{\chi:\text{unitary}}
\sum_{f \in \mathcal{B}(\mathcal{I}(\chi))}
\frac{\int_{[G]} \Psi_1 \Eis(\tilde{f})
\int_{[G]} \Eis(f) \Psi_2
}
{
\int_{B_\mathbb{A} \backslash G_\mathbb{A}}
\tilde{f} f
},
\end{align*}
where
\begin{itemize}
\item $\omega$ runs over the quadratic characters
of $[A]$,
with $\omega(\det)(g) := \omega(\det(g))$,
\item
$\sigma$ runs over the cuspidal automorphic representations,
\item
$\chi$ runs over the unitary dual of $[A] \cong
\mathbb{A}^\times/F^\times$
with respect to the measure dual to the chosen measure on
$[A]$, and
\item
$\varphi$ (resp. $f$) runs over an orthonormal basis
consisting of $K$-isotypic vectors
for $\sigma$
(resp. $\mathcal{I}(\chi)$),
with inner product defined using $L^2([G])$
if $\sigma$ is cuspidal
and using $L^2(K)$ if $\sigma = \mathcal{I}(\chi)$.
We write $\tilde{\varphi}$ (resp. $\tilde{f}$)
for the corresponding element of the dual basis for
$\tilde{\sigma} = \sigma$ (resp. $\mathcal{I}(\chi^{-1})$),
with the duality normalized via the pairing given in the
respective denominators.
\end{itemize}
In other words,
the Fourier inversion formula
\begin{align}\label{eqn:fourier-inversion-L-2}
\Psi(g)
&=
\sum_{\omega^2 = 1}
\frac{
\int_{[G]}\Psi \omega^{-1}(\det)
}
{
\vol([G])
}
\omega(\det(g))
\\ \nonumber
&\quad +
\sum_{\sigma:\text{cuspidal}}
\sum_{\varphi \in \mathcal{B}(\sigma)}
\frac{
\int_{[G]} \Psi \tilde{\varphi}
}
{
\int_{[G]} \tilde{\varphi}\varphi
}
\varphi(g)
\\ \nonumber
&\quad
+
\frac{1}{2}
\int_{\chi:\text{unitary}}
\sum_{f \in \mathcal{B}(\mathcal{I}(\chi))}
\frac{\int_{[G]} \Psi \Eis(\tilde{f})
}
{
\int_{B_\mathbb{A} \backslash G_\mathbb{A}} \tilde{f} f
} \Eis(f)(g)
\end{align}
holds in an $L^2$-sense; the sums in
\eqref{eqn:fourier-inversion-L-2}
need not converge pointwise.
An obvious necessary condition for the inversion formula
\eqref{eqn:fourier-inversion-L-2} to hold pointwise in the
traditional sense is that for each unitary $\chi$ and
each $f \in \mathcal{I}(\chi)$, the
integral $\int_{[G]} \Psi \Eis(\tilde{f})$ converges absolutely.
In our applications, the slightly stronger condition that
$\Psi$ is smooth and every
archimedean derivative of $\Psi$ is bounded by a multiple of
$\htt^{\alpha}$ for some $\alpha < 1/2$ holds;
in particular, $\Psi \in L^2([G])$.
Arguing as in \cite[\S2.6.5]{michel-2009}
using the trace
class property on $L^2([G])$ of some inverse power of the
product of local Laplacians, one can show then that the RHS of
\eqref{eqn:fourier-inversion-L-2} converges uniformly on
compacta to a continuous function. On the other hand, the
$L^2$ theory implies that the difference between the two sides
of \eqref{eqn:fourier-inversion-L-2} is orthogonal to every
element of $C_c^\infty([G])$, hence vanishes. Thus
\eqref{eqn:fourier-inversion-L-2} gives a pointwise defined and
convergent expansion of such $\Psi$.
\subsection{Regularized adelic integrals}
\label{sec-9-2}
We recall, following
\cite{MR656029}, \cite[\S4.3]{michel-2009}
and \cite{MR3977317} with some minor modifications,
how to define
a regularized integral
of certain functions on $[G]$.
By a \emph{finite} function
$\phi : N_\mathbb{A} A \backslash G_\mathbb{A} \rightarrow
\mathbb{C}$ we mean a smooth function on $G_\mathbb{A}$ that
is finite in the sense of \S\ref{sec-9-1}
with respect to the left translation action of $[A]$;
equivalently, $\phi$
may be written in the form
\begin{equation}\label{eqn:phi-decompose-finite-N-A-K}
\phi(n a(y) k) = \sum_{i=1}^n \chi_i(y) \log^{m_i-1}|y|
\mathcal{K}_i(k)
\end{equation}
for some characters $\chi_i$ on $\mathbb{A}^\times/F^\times$,
positive integers $m_i$, and smooth functions $\mathcal{K}_i$ on
$K$. We may and shall assume that the $\mathcal{K}_i$ are not
identically zero and that each pair $(\chi_i,m_i)$ shows up at
most once, so that the decomposition
\eqref{eqn:phi-decompose-finite-N-A-K} of $\phi$ is unique up to
rearrangement. We say that a smooth function
$\Psi : [G] \rightarrow \mathbb{C}$ is of \emph{controlled
increase} if we can find a finite function $\phi$ on
$N_\mathbb{A} A \backslash G_\mathbb{A}$ so that for
$d \geq 0$ and all
$n \in N_\mathbb{A}, k \in K$ and $y \in \mathbb{A}^\times$ with
$|y| \geq 1$, the estimate
$\Psi(n a(y) k) = \phi(n a(y) k) + \operatorname{O}(|y|^{-d})$ holds, together
with the analogous estimates involving all archimedean
derivatives. The function $\phi$ is then uniquely determined;
we refer to it as the \emph{asymptotic part} of $\Psi$.
The characters $\chi_i$
that arise in the
decomposition \eqref{eqn:phi-decompose-finite-N-A-K}
are the exponents of $\phi$.
We will refer to these simply as the exponents
of $\Psi$.
For example, if $\chi$ is a character of $[A]$ with
$\chi^2 \neq |.|^{\pm}$ and $f$ is a nonzero element of
$\mathcal{I}(\chi)$, then the Eisenstein series $\Eis^*(f)$ is
of controlled increase with exponents
$|.|^{1/2} \chi$ and $|.|^{1/2} \chi^{-1}$.
Suppose $\Psi$ has controlled increase and that
the exponent $|.|$ does not occur. Following
\cite{MR656029} and \cite[\S4.3]{michel-2009},
we may then
define the regularized integral $\int_{[G]}^{\reg} \Psi$.
It may be characterized
as the unique $G$-invariant functional on the space of such $\Psi$
that extends integration on the subspace of integrable
functions. It may be defined using convolution or truncation
as in \S\ref{sec-9-1}, or Eisenstein series as in
\cite{MR656029} and \cite[\S4.3.5]{michel-2009}.
We briefly recall the last of these definitions. Split
the asymptotic part $\phi$ of $\Psi$ as the sum
$\phi_{>1/2} + \phi_{\leq 1/2}$ of two terms corresponding to
the exponents with real part indicated by the subscript. Using $\phi_{>1/2}$, we may construct
a linear combination of (derivatives of)
Eisenstein series $\mathcal{E}$ for which the difference
$\Psi - \mathcal{E}$ has controlled increase with exponents of
real part $\leq 1/2$.
That difference is then integrable,
and we take $\int_{[G]}^{\reg} \Psi := \int_{[G]}(\Psi - \mathcal{E})$.
\subsection{Regularized spectral decomposition}
\label{sec-9-3}
If $\Psi$ is a smooth function on $[G]$ of controlled increase
with exponents of real part strictly less than $1/2$, then
$\Psi$ and its archimedean derivatives are bounded by constant
multiples of $\htt^{\alpha}$ for some $\alpha <1/2$,
so the expansion
\eqref{eqn:fourier-inversion-L-2} is pointwise defined and
convergent, uniformly on compacta.
Suppose now that $\Psi$ has controlled increase and that
each exponent $\chi$ satisfies $\Re(\chi) \neq 1/2$,
hence in particular
$\chi^2 \neq |.|$. We may then split the asymptotic part $\phi$
as a sum $\phi_{>1/2} + \phi_{<1/2}$ and define
$\mathcal{E}$ using $\phi_{>1/2}$ as before. The difference
$\Psi - \mathcal{E}$ then has controlled increase with exponents
of real part $< 1/2$, hence admits a pointwise spectral
expansion, uniformly on compacta, with coefficients given by the
convergent integrals
$\int_{[G]} (\Psi - \mathcal{E}) \omega^{-1}(\det)$ and
$\int_{[G]} (\Psi - \mathcal{E}) \Eis(\tilde{f})$ and the
analogous integrals involving cusp forms. In fact, it follows
from representation-theoretic considerations
that
$\int_{[G]}^{\reg} \mathcal{E} \omega^{-1}(\det)
= \int_{[G]}^{\reg} \mathcal{E} \Eis(\tilde{f}) = 0$,
so that the spectral expansion of $\Phi$
may be written
\begin{align}\label{eqn:fourier-inversion-L-2-reg}
\Psi(g)
&=
\mathcal{E}(g)
+
\sum_{\omega^2 = 1}
\frac{
\int_{[G]}^{\reg}\Psi \omega^{-1}(\det)
}
{
\vol([G])
}
\omega(\det(g))
\\ \nonumber
&\quad
+
\sum_{\sigma:\text{cuspidal}}
\sum_{\varphi \in \mathcal{B}(\sigma)}
\frac{
\int_{[G]} \Psi \tilde{\varphi}
}
{
\int_{[G]} \varphi \tilde{\varphi}
}
\varphi(g)
\\ \nonumber
&\quad
+
\frac{1}{2}
\int_{\chi:\text{unitary}}
\sum_{f \in \mathcal{B}(\mathcal{I}(\chi))}
\frac{\int_{[G]}^{\reg} \Psi \Eis(\tilde{f})
}
{
\int_{B_\mathbb{A} \backslash G_\mathbb{A}} \tilde{f} f
} \Eis(f)(g).
\end{align}
Suppose now moreover
that $\Psi$ is orthogonal to
the one-dimensional subrepresentations of $L^2([G])$,
so that the sum over $\omega$ may be omitted.
The resulting expansion then involves only the standard \emph{generic}
automorphic representations.
It will be convenient
to rewrite that expansion in terms of Whittaker norms.
Using \eqref{eqn:Eis-induced-vs-Whittaker-global-L-2}
and the formula
$\int_{[G]}
\tilde{\varphi} \varphi = 2\int_{A_{\mathbb{A}}}^{\reg}
\tilde{W}_{\tilde{\varphi}} W_{\varphi}$
for cuspidal $\sigma$
(see \cite[\S3.2.2]{nelson-variance-II}
or \cite[Lem 2.2.3]{michel-2009})
gives the required identity,
which we restate in full for convenience:
\begin{theorem}\label{thm:reg-spect-pointwise}
Let $\Psi \in C^\infty([G])$
be of controlled increase,
with each exponent
$\chi$ satisfying
$\Re(\chi) \neq 1/2$
and $\chi^2 \neq |.|$,
and orthogonal to
the one-dimensional subrepresentations
$\mathbb{C} \omega(\det)$ of $L^2([G])$.
Let $\mathcal{E}$ denote the linear combination
of (derivatives) of Eisenstein series
attached to the
summand $\phi_{>1/2}$ of the asymptotic part $\phi$
of $\Psi$ obtained by collecting terms
involving characters of real part $> 1/2$.
We then have a pointwise defined and normally convergent
expansion
\begin{align*}
\Psi(g)
-
\mathcal{E}(g)
&=
\frac{1}{2}
\sum_{\sigma:\text{cuspidal}}
\sum_{\varphi \in \mathcal{B}(\sigma)}
\frac{
\int_{[G]} \Psi \tilde{\varphi}
}
{
\int_{A_{\mathbb{A}}}^{\reg} \tilde{W}_{\tilde{\varphi}} W_{\varphi}
}
\varphi(g)
\\
&\quad
+
\frac{1}{2}
\int_{\chi:\text{unitary}}
\sum_{f \in \mathcal{B}(\mathcal{I}(\chi))}
\frac{\int_{[G]}^{\reg} \Psi \Eis^*(\tilde{f})
}
{
\int_{A_{\mathbb{A}}}^{\reg}
\tilde{W}_{\Eis^*(\tilde{f})} W_{\Eis^*(f)}
}
\Eis^*(f)(g).
\end{align*}
\end{theorem}
We will often abbreviate the RHS of the above
decomposition to
\begin{equation}\label{eqn:spectral-decomp-reg-generic}
\int_{\sigma:\text{generic}}
\sum_{\varphi \in \mathcal{B}(\sigma)}
\frac{
\int_{[G]}^{\reg} \Psi \tilde{\varphi}
}
{
\int_{A_{\mathbb{A}}}^{\reg} \tilde{W}_{\tilde{\varphi}}W_{\varphi}
}
{
\varphi(g)
}
\end{equation}
or further to simply
\begin{equation}
\int_{\sigma:\text{generic}}
\Psi_{\sigma}.
\end{equation}
Note that the measures implicit in the integrals over $\sigma$
in \eqref{eqn:fourier-inversion-L-2-reg} and
\eqref{eqn:spectral-decomp-reg-generic} differ on the cuspidal
spectrum by the factor $1/2$.
It is useful to describe the rate of convergence a bit more
precisely. First
of all, for $\Psi$ as in Theorem
\ref{thm:reg-spect-pointwise}, the constant term
$\Psi_N(g) := \int_{x \in \mathbb{A}/F} \Psi(n(x) g) \, d x$
enjoys the normally convergent expansion
\begin{equation}
\Psi_N =
\mathcal{E}_N +
\int_{\sigma:\text{generic}}
\Psi_{\sigma,N},
\end{equation}
and thus likewise
\begin{equation}
\Psi = \Psi_N + \mathcal{E} - \mathcal{E}_N
+
\int_{\sigma:\text{generic}} (\Psi_\sigma - \Psi_{\sigma,N}).
\end{equation}
This last integral converges
uniformly near the cusp;
more precisely,
for $g = a(y) k$ with $y \in \mathbb{A}^\times, k \in K$,
we have
\begin{equation}\label{eqn:estimate-Psi-sigma-minus-constant-term}
\Psi_\sigma(g) - \Psi_{\sigma,N}(g)
\ll_{\Psi,d}
\frac{\min(|y|^{-1/2-\varepsilon}, |y|^{-d})}{C(\sigma)^d}
\end{equation}
for fixed $d \geq 0$. The bound
\eqref{eqn:estimate-Psi-sigma-minus-constant-term} may be
established first for $B$ sufficiently negative, then using
\cite[\S2.6.6]{michel-2009} and arguing as in
\S\ref{sec:integration-parts} or \cite[Lem 3.5.2]{michel-2009}
to save with respect to $C(\sigma)$. For Eisenstein $\sigma$,
we estimate each term of the Fourier expansion using
lemma \ref{lem:unif-polyn-type-hecke}
or \cite[(3.2.3)]{michel-2009} and
\cite[\S2, (S1d)]{michel-2009}. For cuspidal $\sigma$, we apply
the same argument for large $|y|$, while for small $|y|$ we
invoke the crude $L^\infty$-norm bound for $\Psi_{\sigma}$
following from \cite[\S2, (S2a), (S3b)]{michel-2009}.
\subsection{Rankin--Selberg integrals}
\label{sec-5-6}
For generic
automorphic representations
$\sigma_1, \sigma_2$ and characters $\chi$ of $[A]$ of large
enough real part, we define for
$\varphi_1 \in \sigma_1, \varphi_2 \in \sigma_2$ and
$f \in \mathcal{I}(\chi)$ the global Rankin--Selberg integral
$\int_{[G]}^{\reg} \varphi_1 \varphi_2 \Eis^*(f)$. If either
$\sigma_1$ or $\sigma_2$ is cuspidal, then the integral
converges absolutely. The Eisenstein case requires
regularization, for the details of which we refer to
\S\cite[\S4.4]{michel-2009}. The important feature for our
purposes is that in either case, the integral unfolds as the
Eulerian integral
$\int_{N_\mathbb{A} \backslash G_\mathbb{A}}^{\reg}
W_{\varphi_1} \tilde{W}_{\varphi_2} f^*$
(interpreted,
using the local unramified calculation of
\eqref{eqn:normalized-local-RS},
by analogy to
\eqref{eqn:defn-global-hecke-integral-unfold}).
It follows then from the corresponding
local results of \S\ref{sec-4-5} that as $f$ varies in
a holomorphic family, the ratio
\[
\frac{\int_{[G]}^{\reg} \varphi_1 \varphi_2 \Eis^*(f)
}{
\Lambda(\sigma_1 \otimes \sigma_2 \otimes \chi, 1/2)
}
\]
extends holomorphically
to all $\chi$.
\part{The basic identity}\label{part:basic-identity}
Here we formulate Theorem \ref{thm:basic-spectr-ident} precisely
(see \S\ref{sec:summary-main-results}) and give the proof.
\section{Local invariant functionals\label{sec:local-inv-func}}
\label{sec-10}
Let $F$ be a local field,
with notation as in \S\ref{sec-4}.
Let $s = (s_1,s_2,s_3) \in \mathbb{C}^3$.
We will eventually
take the limit as $s$ approaches the origin.
For each such $s$, we obtain a representation
\begin{equation}\label{eqn:Is1-Is2-Cs3}
\mathcal{I}(s_1) \otimes \mathcal{I}(s_2) \otimes
\mathbb{C}(s_3)
\end{equation}
of $G \times G \times A$.
(The notation $\mathcal{I}(s_j)$
is defined in \S\ref{sec:intro-representations},
$\mathbb{C}(s_3)$
in \S\ref{sec-4-3}.)
In the archimedean case, we take the completed tensor product,
as in \S\ref{sec:representation-pi-g};
in either case,
this representation identifies
with
the space $\mathcal{I}(s_1) \otimes \mathcal{I}(s_2)$ via the map
$f_1 \otimes f_2 \otimes 1 \mapsto f_1 \otimes f_2$,
which may in turn be defined as the
space of smooth functions $f : G \times G \rightarrow
\mathbb{C}$
satisfying $f(n(x_1) a(y_1) g_1, n(x_2) a(y_2) g_2)
= |y_1|^{1/2+s_1} |y_2|^{1/2+s_2}$.
We may speak, as in \S\ref{sec:families-vectors}, of holomorphic
families of vectors
$f[s] \in \mathcal{I}(s_1) \otimes \mathcal{I}(s_2) \otimes
\mathbb{C}(s_3)$ indexed by $s$ in an open subset on
$\mathbb{C}^3$.
We will define two
families of $A$-invariant functionals on the representation \eqref{eqn:Is1-Is2-Cs3},
corresponding to the ``strong Gelfand triple''
\cite{MR2373356}
formed by the two
sequences $G \times G \geq G \geq A$ and
$G \times G \geq A \times A \geq A$ of strong Gelfand pairs.
The special case $s = 0$ was sketched
in \S\ref{sec-6-3}.
\subsection{$G \times G \geq G \geq A$}
\label{sec-10-1}
\subsubsection{Definition and estimates}\label{sec:defn-ell-sigma-s-makes-sense}
Let $\sigma$ be a generic irreducible representation of $G$,
realized in its Whittaker model $\mathcal{W}(\sigma,\psi)$.
For almost all $s$, we
aim to define a continuous $A$-invariant map
\[
\ell_{\sigma,s} :
\mathcal{I}(s_1) \otimes \mathcal{I}(s_2)
\otimes \mathbb{C}(s_3) \rightarrow \mathbb{C}
\]
by the formula
\begin{equation}\label{eqn:defn-l-omega-s}
f_1 \otimes f_2
\mapsto
\sum_{W \in \mathcal{B}(\sigma)}
\frac{\int_{N \backslash G}^{\reg} \tilde{W} W_{f_1} f_2 \,
\int_A^{\reg} W |.|^{s_3}
}
{
\int_A^{\reg} \tilde{W} W
},
\end{equation}
with the integrals interpreted as in \S\ref{sec-4-3},
\S\ref{sec-4-5} and the sum as in \S\ref{sec-6-3}. More
precisely, we write $\sum_{W \in \mathcal{B}(\sigma)}$ as
shorthand for
$\sum_{f \in \mathcal{B}(\sigma), W := W_f \tilde{W} :=
\tilde{W}_{\tilde{f}}}$, where the orthonormal basis
$\mathcal{B}(\sigma)$ is as defined in
\S\ref{sec:norms-whit-type-reps}, and $\tilde{f}$ runs over the
corresponding dual basis for $\sigma$
(resp. $\mathcal{I}(\chi^{-1})$) for $\sigma$ square-integrable
(resp. $\sigma =\mathcal{I}(\chi)$), with the duality prescribed
by the denominator of \eqref{eqn:defn-l-omega-s} (or in the
induced case, equivalently by integration over $B \backslash G$
or $K$ -- see \eqref{eqn:local-parseval-f-vs-W-f}).
We pause to make sense of this definition.
By the local theory of Hecke and
Rankin--Selberg integrals recalled in
\S\ref{sec:local-hecke-basic} and \S\ref{sec:local-rs-basic},
together with the formula
\eqref{eqn:multiplicativity-L-factors-wrt-induction}
for $L$-factors of induced
representations,
we see
that the integrals
in the numerator of \eqref{eqn:defn-l-omega-s} are defined away
from the poles of the numerator of
\[
\mathcal{L}(\sigma,s)
:=
\frac{
L(\sigma, 1/2 + s_1 + s_2)L(\sigma, 1/2 - s_1 + s_2) L(\sigma,
1/2 + s_3)
}
{
L(\sigma \times \sigma, 1)
}.
\]
In the archimedean case, these integrals moreover define
continuous functionals.
It remains to make sense of the sum in
\eqref{eqn:defn-l-omega-s} over $W$.
We suppose henceforth that $\sigma \in G^\wedge_{\gen}$ is
$\vartheta$-tempered for some fixed $\vartheta > 0$, that $s$
lies in a fixed compact subset of $\mathbb{C}^3$
and
that $s$ is some fixed positive distance away from any pole of
$\mathcal{L}(\sigma,s)$.
\begin{lemma}
Let $f_1 \in \mathcal{I}(s_1), f_2 \in \mathcal{I}(s_2)$.
Then
for each fixed $d$,
\begin{equation}\label{eqn:sum-of-sobolev-norms-of-W-weighted-by-RS-int}
\sum_{W \in \mathcal{B}(\sigma)}
\left\lvert
\frac{\int_{N \backslash G}^{\reg} \tilde{W} W_{f_1} f_2
}
{
\int_A^{\reg} \tilde{W} W
}
\right\rvert
\mathcal{S}_d(W)
\ll
C(\sigma)^{-d} \mathcal{S}(f_1) \mathcal{S}(f_2) q^{\operatorname{O}(1)}.
\end{equation}
\end{lemma}
\begin{proof}
Let $W, \tilde{W}$ be as in the sum.
Let $f_3 \in \mathcal{B}(\sigma)$
be such that $W = W_{f_3}$,
and let $\tilde{f}_3$ denote the corresponding
dual basis element,
belonging to $\sigma$ in the square-integrable
case and to $\mathcal{I}(\chi^{-1})$ if $\sigma =
\mathcal{I}(\chi)$,
so that $\tilde{W} = \tilde{W}_{\tilde{f}_3}$.
We
recall from
\eqref{eqn:upper-bound-Mellin-transform-W}
that
\begin{equation}
\int_{N \backslash G}^{\reg}
\tilde{W} W_{f_1} f_2
\ll
\mathcal{S}_d(f_1)
\mathcal{S}_d(f_2) \mathcal{S}_d(\tilde{f}_3) q^{\operatorname{O}(1)}.
\end{equation}
We may strengthen this estimate
by integrating by parts with respect
to $\sigma$ (\S\ref{sec:integration-parts}),
giving for fixed $d,d'$ with $d'$ large in terms of $d$ that
\begin{equation}
\int_{N \backslash G}^{\reg}
\tilde{W} W_{f_1} f_2
\ll
C(\sigma )^{-d}
\mathcal{S}_{d'}(f_1)
\mathcal{S}_{d'}(f_2)
\mathcal{S}_{-2 d}(\tilde{f}_3)
q^{\operatorname{O}(1)}.
\end{equation}
By the construction of $\mathcal{B}(\sigma)$, we have
$\int_{A}^{\reg} \tilde{W} W = \int_{B \backslash G} \tilde{f}_3 f_3$,
$\mathcal{S}_{-2 d}(\tilde{f}_3) = N_{\sigma \nu}^{-2 d}$ and
$\mathcal{S}_d(f_3) = N_{\sigma \nu}^d$.
Thus
the LHS of
\eqref{eqn:sum-of-sobolev-norms-of-W-weighted-by-RS-int}
is majorized by
\[
C(\sigma)^{-d}
\mathcal{S}_{d'}(f_1)
\mathcal{S}_{d'}(f_2) q^{\operatorname{O}(1)} \sum_{\nu \in K^\wedge}
\dim(\sigma^\nu) N_{\sigma \nu}^{-d}.
\]
By the uniform trace class property
\eqref{eq:sum-N-sigma-nu-neg-d-0},
we may assume that $d$ is large enough that this last sum over $\nu$ is
$\operatorname{O}(1)$.
This completes the proof.
\end{proof}
We may thus define a linear map
\begin{equation}
\rho_{\sigma,s} : \mathcal{I}(s_1) \otimes \mathcal{I}(s_2) \rightarrow \mathcal{W}(\sigma,\psi)
\end{equation}
\begin{equation}
f_1 \otimes f_2
\mapsto
\sum_{W \in \mathcal{B}(\sigma)}
\frac{\int_{N \backslash G}^{\reg} \tilde{W} W_{f_1} f_2
}
{
\int_A^{\reg} \tilde{W} W
}
W
\end{equation}
satisfying $\mathcal{S}_d(\rho_{\sigma,s}(f)) \ll \mathcal{S}(f)$
for each fixed $d$.
This map may be characterized
by the property:
for any $\tilde{W} \in \mathcal{W}(\sigma,\bar{\psi})$,
\begin{equation}
\int_{A}^{\reg}
\tilde{W}
\rho_{\sigma,s}(f_1 \otimes f_2)
=
\int_{N \backslash G}^{\reg} \tilde{W} W_{f_1} f_2.
\end{equation}
From this property and the diagonal $G$-invariance of the
Rankin--Selberg integral, we see that $\rho_{\sigma,s}$ is
$G$-equivariant.
We now compose $\rho_{\sigma,s}$
with the Hecke integral
\begin{equation}
\mathcal{W}(\sigma,\psi) \otimes \mathbb{C}(s_3)
\rightarrow \mathbb{C}
\end{equation}
\begin{equation}
W \mapsto \int_A^{\reg} W |.|^{s_3}
\end{equation}
to obtain the required continuous $A$-invariant map
$\ell_{\sigma,s}$. We may use
\eqref{eqn:sum-of-sobolev-norms-of-W-weighted-by-RS-int} to
estimate $\rho_{\sigma,s}$; by combining with the estimate
\eqref{eqn:upper-bound-Mellin-transform-W} for local Hecke
integrals, we deduce that $\ell_{\sigma,s}$ satisfies for each
fixed $d$ the estimate
\begin{equation}\label{eq:crude-estimate-for-ell-sigma-s}
\ell_{\sigma,s}(f)
\ll
C(\sigma)^{-d}
\mathcal{S}(f) q^{\operatorname{O}(1)}.
\end{equation}
\subsubsection{}\label{sec:local-ell-sigma-s-where-defined-win}
It follows from the above discussion that if $\sigma$ is
$\vartheta$-tempered (\S\ref{sec:bounds-towards-raman}), then
$\ell_{\sigma,s}$ is defined whenever the real parts of
$s_1 + s_2, - s_1 + s_2$ and $s_3$ are at least
$-1/2 + \vartheta$. In particular, if $\sigma$ is
$1/6$-tempered, then $\ell_{\sigma,s}$ is defined for all $s$
satisfying
\begin{equation}\label{eqn:s1-s2-s3-one-sixth-initial-bounds}
|\Re(s_1)| < 1/6, \quad |\Re(s_2)| < 1/6, \quad \Re(s_3) > -1/6.
\end{equation}
Since $7/64 < 1/6$ (see \S\ref{sec:bounds-towards-raman}),
the poles of $\mathcal{L}(\sigma,s)$
will not play a significant role in our analysis.
\subsubsection{}
Recall from \eqref{eq:defn-f-asterisk} the notation
$f_i^* := \zeta_F(1 + 2 s_i) f_i$.
We define a normalized variant of $\ell_{\sigma,s}$
by the formula
\begin{equation}
\ell_{\sigma,s}^*(f_1 \otimes f_2)
:=
\ell_{\sigma,s}(f_1^* \otimes f_2^*).
\end{equation}
By the corresponding property
of local Hecke and Rankin--Selberg integrals,
the ratio $\ell_{\sigma,s}(f)/\mathcal{L}(\sigma,s)$ extends to
a holomorphic function of $s$, hence the ratio
$\ell_{\sigma,s}^*(f)/\mathcal{L}(\sigma,s)$ to a meromorphic
function of $s$.
\begin{lemma}\label{lem:unram-calc-ell-sigma-s}
If $(F,\psi)$ is unramified and $f = f_1 \otimes f_2$
with $f_1, f_2$ normalized spherical
(\S\ref{sec:normalized-spherical-induced-rep}),
then
the ratio $\ell_{\sigma,s}^*(f)/\mathcal{L}(\sigma,s)$
vanishes unless $\sigma$
is unramified, in which case it evaluates to $1$.
\end{lemma}
\begin{proof}
The vanishing is clear when $\sigma$ is ramified:
the image of $f$ in $\sigma$
is then $K$-invariant,
but $\sigma$ contains no $K$-invariant vectors.
Suppose that
$\sigma$ is unramified. We may assume that
$\mathcal{B}(\sigma)$ contains a $K$-invariant vector $W$,
with $\tilde{W}$ also $K$-invariant. The remaining basis
elements then do not contribute to the sum defining
$\ell_{\sigma,s}(f_1 \otimes f_2)$. Since the quantities
$W(1)$ and $\tilde{W}(1)$ are nonzero (see
\S\ref{sec:normalized-spherical-induced-rep}) and the ratio in question is
invariant under scaling $W$ and $\tilde{W}$, we may assume for
computational convenience that $W(1) = \tilde{W}(1) = 1$.
Then by the unramified calculations
recorded in
\S\ref{sec:normalized-spherical-induced-rep},
\S\ref{sec-4-3}
and
\S\ref{sec-4-5},
we have
$\int_A^{\reg} \tilde{W} W = L(\ad \sigma, 1)/\zeta_F(2)$
and
$\int_A^{\reg} W |.|^{s_3} = L(\sigma, 1/2 + s_3)/ \zeta_F(1)$
and
$W_{f_2}^*(1) = 1$ and
$\int_{N \backslash G}^{\reg} \tilde{W} W_{f_1^*} f_2^* =
L(\sigma \otimes \mathcal{I}(s_1), 1/2 + s_2) / \zeta_F(2)$.
The conclusion follows upon noting that
$L(\sigma \otimes \mathcal{I}(s_1), 1/2 + s_2) L(\sigma, 1/2 +
s_3)/ \zeta_F(1) L(\ad \sigma,1) = \mathcal{L}(\sigma,s)$.
\end{proof}
\subsubsection{Holomorphy of
$\ell_{\sigma,s}(f)$}\label{sec:holom-ell_s-sf}
In this section we verify that $\ell_{\sigma,s}(f)$ varies
holomorphically with respect to both $\sigma$ and $s$ away from
the poles of $\mathcal{L}(\sigma,s)$. We first record a
technical lemma related to the holomorphy of local Bessel
functions with respect to their index.
\begin{lemma}\label{lem:holom-bessel-index}
Fix $\nu \in K^\wedge$.
For $\chi \in A^\wedge$,
write $\sigma(\chi)$ for the generic irreducible
subquotient of $\mathcal{I}(\chi)$
and
$\mathcal{B}(\sigma(\chi)^\nu)$ for the set of all
$W \in \mathcal{B}(\sigma(\chi))$ having $K$-type $\nu$.
Then for $g_1, g_2 \in G$,
the sum
\begin{equation}\label{eqn:bessel-sigma-nu}
\sum_{W \in \mathcal{B}(\sigma(\chi)^\nu)}
\frac{\tilde{W}(g_1) W(g_2)}{\int_A ^{\reg} \tilde{W} W}
\end{equation}
varies holomorphically with respect to $\chi$.
\end{lemma}
\begin{proof}
Fix a component $\mathcal{X}$ of $A^\wedge$, i.e., a coset of
the subgroup $\{|.|^s : s \in \mathbb{C} \}$ of unramified
characters. Let $\chi_K$ denote the common restriction to
$A \cap K$ of elements $\chi$ of $\mathcal{X}$. Let
$\{\phi_i\}$ be an orthonormal basis for the $\nu$-isotypic
subspace of $\Ind_{A \cap K}^K(\chi_K) \subseteq L^2(K)$. For
each $\chi \in \mathcal{X}$, let $f_i[\chi]$
(resp. $\tilde{f}_i[\chi]$) denote the element of
$\mathcal{I}(\chi)$ (resp. $\mathcal{I}(\chi^{-1})$) whose
restriction to $K$ is $\phi_i$ (resp. the complex conjugate of
${\phi}_i$). For $g_1, g_2 \in G$, the sum
\begin{equation}\label{eqn:bessel-induced-chi}
\sum_{i}
\tilde{W}_{\tilde{f}_i[\chi]}(g_1) W_{f_i[\chi]}(g_2)
\end{equation}
is independent of the choice of basis, and varies
holomorphically with $\chi$. We claim that
\eqref{eqn:bessel-induced-chi} and \eqref{eqn:bessel-sigma-nu}
are equal. The claim implies the required conclusion.
The claim
clearly holds when $\mathcal{I}(\chi)$ is
irreducible, so suppose otherwise.
Then
for some choice of sign $\pm$,
$\sigma(\chi)$ is
isomorphic to (see \cite{MR3889963})
\begin{itemize}
\item the quotient $\mathcal{I}(\chi^{\mp})/ L$, where
$L$ denotes the finite-dimensional kernel of
$\tilde{f} \mapsto \tilde{W}_{\tilde{f}}$, and also to
\item the
subspace $L^\perp \subseteq \mathcal{I}(\chi^{\pm})$.
\end{itemize}
Suppose
for instance that $\chi^{\pm} = \chi$.
We may
assume the basis $\{\phi_i\}$
chosen so that $\tilde{f}_1[\chi], \dotsc, \tilde{f}_{\dim(L)}[\chi] \in L$
and
$f_{\dim(L)+1}[\chi],f_{\dim(L)+2}[\chi], \dotsc \in L^\perp$.
Then the
$i=1..\dim(L)$ summands
in \eqref{eqn:bessel-induced-chi} vanish,
while the remaining summands give \eqref{eqn:bessel-sigma-nu}.
The claim follows. A
similar argument applies if $\chi^{\pm} = \chi^{-1}$.
\end{proof}
\begin{lemma}
Let $U$ be an open subset of $\mathbb{C}^3$.
Let $f[s] \in \mathcal{I}(s_1) \otimes \mathcal{I}(s_2)$ be a
holomorphic family of vectors defined for $s \in U$. Let
$\mathcal{D} \subseteq G^\wedge_{\gen} \times U$ denote the
complement of the closure of the set of all pairs $(\sigma,s)$
for which $\mathcal{L}(\sigma,s)$ is infinite. Then
$\ell_{\sigma,s}(f[s])$ defines a holomorphic function on
$\mathcal{D}$.
\end{lemma}
Here the holomorphy means as in
\S\ref{sec-4-1} that
on the indicated domain $\mathcal{D}$,
\begin{itemize}
\item $\ell_{\sigma,s}(f[s])$ is holomorphic in $s$ for each
$\sigma$, and
\item $\ell_{\sigma(\chi),s}(f[s])$ is holomorphic in
$(\chi,s)$.
\end{itemize}
\begin{proof}
We may assume that $U = \mathbb{C}^3$ and that $f[s]$ is a
flat family. The Sobolev norms $\mathcal{S}_{d}(f[s])$ are
locally bounded in $s$, so the proof of the estimate
\eqref{eq:crude-estimate-for-ell-sigma-s} shows that the sum
defining $\ell_{\sigma,s}(f[s])$ converges locally uniformly.
Since each term varies holomorphically in $s$, we obtain the
required holomorphy of $\ell_{\sigma,s}(f[s])$ with respect to
$s$.
It remains to verify that $\ell_{\sigma(\chi),s}(f[s])$ is
holomorphic in $(\chi,s)$.
Let us fix
$\chi_0 \in A^\wedge$
and write $\chi = \chi_0 |.|^w$
with respect to the local coordinate $w \in \mathbb{C}$,
and fix $\nu \in K^\wedge$.
For $s \in \mathbb{C}^3$
and $w \in \mathbb{C}$,
we define $\Phi_0(s,w)$
like $\ell_{\sigma(\chi_0 |.|^w),s}(f_1 \otimes f_2)$,
but restricting
the sum to those $W$ having
$K$-type $\nu$.
In view of the locally uniform convergence
of the sum defining $\ell_{\sigma,s}$,
it is enough to verify
that $\Phi_0(s,w)$ is holomorphic
away from the poles
of $\mathcal{L}(\sigma(\chi_0 |.|^w), s)$.
To that end, we choose a sufficiently large real number $c$,
denote by $\Omega$ the set of all pairs $(s,w)$ for which
the real parts of $s_1,s_2,s_3,w$ have magnitude less than
$c$, choose a polynomial $P(s,w)$ so that
$P(s,w) \mathcal{L}(\mathcal{I}(\chi_0 |.|^w),s)$ is holomorphic on
$\Omega$, and set $\Phi(s,w) := P(s,w) \Phi_0(s,w)$. It is
enough then to verify that $\Phi(s,w)$ is holomorphic on
$\Omega$.
We know by \S\ref{sec:local-hecke-basic} and
\S\ref{sec:local-rs-basic} that on $\Omega$, the map
$s \mapsto \Phi(s,w)$ is holomorphic for each $w$. Moreover,
if $\Re(s_2)$ and $\Re(s_3)$ are large enough in terms of
$|\Re(s_1)|$ and $|\Re(w)|$, then the integral defining
$\Phi_0(s,w)$ converges absolutely, and so lemma
\ref{lem:holom-bessel-index} implies that $\Phi(s,w)$ is
holomorphic in both variables. Using the local functional
equations as in the proofs of lemmas
\ref{lem:unif-polyn-type-hecke} and \ref{lem:polyn-bound-rs},
we deduce that $\Phi(s,w)$ is holomorphic whenever
$|\Re(s_2)|$ and $|\Re(s_3)|$ are large enough in terms of
$|\Re(s_1)|$ and $|\Re(w)|$. In the archimedean case, we see
moreover by integrating by parts with respect to
$\mathcal{I}(s_2)$ and $\mathcal{I}(s_3)$
(\S\ref{sec:integration-parts})
that $\Phi(s,w)$
decays rapidly in vertical strips with respect to $s_2$ and
$s_3$. We may thus use Cauchy's theorem to express
$\Phi(s,w)$ for general $s_2,s_3$ as an alternating sum of
four contour integrals in which each of $\Re(s_2)$ and
$\Re(s_3)$ is sufficiently positive or negative. We thereby
deduce the holomorphy of $\Phi(s,w)$ for general arguments
$s_2, s_3$ from the case in which those arguments have large
real parts.
\end{proof}
\subsubsection{Holomorphy of pre-Kuznetsov
weights}\label{sec:holomorphy-of-pre-K-wts}
Let
$h : G_{\gen}^\wedge \rightarrow \mathbb{C}$ be a pre-Kuznetsov
weight (\S\ref{sec:pre-kuzn-weights}) with kernel
$\phi \in C_c^\infty(G)$.
We verify here that the formula defining
$h$ converges absolutely
and that $h$ is holomorphic
(in the sense of \ref{sec-4-1}).
Write $J_{\sigma,\nu}(g_1,g_2)$ for
the sum \eqref{eqn:bessel-sigma-nu}.
By regrouping the definition, we have
$h(\sigma) = \sum_{\nu \in K^\wedge} h^\nu(\sigma)$, where
$h^\nu(\sigma) := \int_{g \in G} J_{\sigma,\nu}(g,1) \phi(g)$.
By lemma \ref{eqn:bessel-sigma-nu},
we see that each
$h^\nu$ is holomorphic, so it suffices to verify for every compact
subset $\Sigma$ of $G^\wedge_{\gen}$ that
$\sup_{\sigma \in \Sigma} \sum_{\nu \in K^\wedge}
|h^\nu(\sigma)| < \infty$. We have
$h_\nu(\sigma) = \int_{g \in G} J_{\sigma,\nu}(g,1)
\phi_\nu(g)$, where $\phi_\nu$ denotes the $\nu$-isotypic
component of $\phi$ under right translation. In the
non-archimedean case, we have $\phi_\nu = 0$ for all but
finitely many $\nu$, so the required absolute convergence
follows from the continuity of $J_{\sigma,\nu}$. In the
archimedean case, the smoothness of $\phi$ implies that
$\|\phi_\nu \|_{L^\infty(G)} \ll_d (1 + c_\nu)^{-d}$ for each
fixed $d \geq 0$, where $c_\nu \geq 0$ denotes the Casimir
eigenvalue. By expanding $g = n(x) a(y) k$ in Iwasawa
coordinates, we reduce to showing that
$J_{\sigma,\nu}(a(y) k, 1) \ll (1 + c_\nu)^{\operatorname{O}(1)}$ for all
$k \in K$ and all $y$ belonging to a fixed compact subset of
$F^\times$. This last estimate follows (in stronger form) from
lemma \ref{lem:unif-polyn-type-hecke}.
An identical argument gives the locally uniform
convergence
of the sums \eqref{eqn:V-sig-proj-defn}.
\subsection{$G \times G \geq A \times A \geq A$}
\label{sec-10-2}
Let $\omega$ be a character of $A$.
For almost all $s$, we may define
an $A$-invariant map
\[
\ell_{\omega,s}
: \mathcal{I}(s_1) \otimes \mathcal{I}(s_2)
\otimes \mathbb{C}(s_3) \rightarrow \mathbb{C}
\]
\begin{equation}\label{eq:ell-omega-s-initial-defn}
f_1 \otimes f_2
\mapsto
\int_A^{\reg} W_{f_1} \omega
\int_A^{\reg} W_{f_2} \omega^{-1} |.|^{s_3}.
\end{equation}
The definition makes sense: for given $f_1, f_2$, the RHS of
\S\ref{eq:ell-omega-s-initial-defn} is defined away from the
poles of
the numerator of
\begin{align*}
\mathcal{L}(\omega,s)
&:=
\frac{
L(\mathcal{I}(s_1) \otimes \omega,1/2)
L(\mathcal{I}(s_2) \otimes \omega^{-1},1/2 + s_3)
}
{
\zeta_F(1)^2
}
\\
&=
\frac{ \prod_{\pm}
L(\omega,1/2 \pm s_1)
L(\omega^{-1},1/2 \pm s_2 + s_3)
}{
\zeta_F(1)^2
}
\end{align*}
(see \S\ref{sec:local-hecke-basic}). Moreover, in the
archimedean case, the integrals appearing on the RHS of
\eqref{eq:ell-omega-s-initial-defn} define continuous functionals
on $\mathcal{I}(s_1)$ and $\mathcal{I}(s_2)$, hence their
product extends to a continuous functional on the (completed)
tensor product.
In general, the Hecke integral estimate
\eqref{eqn:upper-bound-Mellin-transform-W}
and the reduction to pure tensors of
\S\ref{sec:reduct-pure-tens}
implies that if
$\sigma$ is $\operatorname{O}(1)$-tempered,
$s$ lies in a fixed compact subset of $\mathbb{C}^3$
and $s$ is some fixed positive distance
away from any pole of $\mathcal{L}(\omega,s)$,
then for each fixed $d$,
\begin{equation}\label{eq:crude-ell-omega-s-estimate}
\ell_{\omega,s}(f) \ll C(\omega)^{-d}
\mathcal{S}(f).
\end{equation}
As we did for $\ell_{\sigma,s}$, we define the normalized variant
\begin{equation}
\ell_{\omega,s}^*(f_1 \otimes f_2)
:=
\ell_{\omega,s}(f_1^* \otimes f_2^*).
\end{equation}
The ratio $\ell_{\omega,s}(f)/\mathcal{L}(\omega,s)$ extends to
a holomorphic function of $s$, the ratio
$\ell_{\omega,s}^*(f)/\mathcal{L}(\omega,s)$ to a meromorphic
one, and we have the expected unramified calculation, which
follows immediately from the corresponding calculation of
\S\ref{sec:local-hecke-basic}:
\begin{lemma}
If $(F,\psi)$ is unramified and $f = f_1 \otimes f_2$ with
$f_1,f_2$ normalized spherical, then the ratio
$\ell_{\omega,s}^*(f) / \mathcal{L}(\omega,s)$
vanishes
unless $\omega$ is unramified, in which case it evaluates to
$1$.
\end{lemma}
We note finally that if $\omega$ is
unitary, then $\ell_{\omega,s}$ is
defined whenever each of $\pm s_1$ and $\pm s_2 + s_3$ have real
parts at least $-1/2$,
as
follows from \eqref{eqn:s1-s2-s3-one-sixth-initial-bounds}.
\subsection{Variation with respect to the additive character}\label{sec:vari-with-resp}
It
is often convenient to assume that $\psi$ has a particular form
(e.g., unramified when $F$ is non-archimedean), so we record
here how the above definitions vary with $\psi$.
For $b \in F^\times$, set $\psi^b(x) := \psi(b x)$
and
let $\ell_{\sigma,s}^b$, $\ell_{\omega,s}^b$ be defined as
above, but using $\psi^b$.
We verify readily that
\begin{equation}
\ell_{\sigma,s}^b
= |b|^{3/2 - s_1 + s_2 - s_3}
\ell_{\sigma,s}, \quad
\ell_{\omega,s}^b
= |b|^{1 + s_1 + s_2 - s_3}
\ell_{\omega,s}.
\end{equation}
\section{Global invariant functionals\label{sec:global-inv-func}}
\label{sec-11}
Let $F$ be a number field,
with accompanying notation as in \S\ref{sec-5}.
Let $s \in \mathbb{C}^3$.
Then
$\mathcal{I}(s_1) \otimes \mathcal{I}(s_2) \otimes
\mathbb{C}(s_3)$ is an irreducible automorphic representation of
$G_\mathbb{A} \times G_\mathbb{A} \times A_\mathbb{A}$, given by
the restricted tensor product of the analogous local
representations. We will define two families of
$A_\mathbb{A}$-invariant functionals, the global analogues of
those of \S\ref{sec:local-inv-func}, and record their
factorizations into local functionals.
\subsection{$G \times G \geq G \geq A$}
\label{sec-11-1}
Let $\sigma$ be a generic automorphic representation.
For almost all $s$,
we define a functional
\[
\ell_{\sigma,s} : \mathcal{I}(s_1) \otimes \mathcal{I}(s_2)
\otimes
\mathbb{C}(s_3) \rightarrow \mathbb{C}
\]
\[
f_1 \otimes f_2
\mapsto \sum _{\varphi \in
\mathcal{B}(\sigma)}
\frac{
\int _{[G]}^{\reg} \Eis(f_1^*) \Eis(f_2^*) \tilde{\varphi}
\int _{[A]}^{\reg}
\varphi |.|^{s_3}
}{
\int_{A_{\mathbb{A}}}^{\reg}
\tilde{W}_{\tilde{\varphi}}
W_{\varphi}
},
\]
with the integrals interpreted as above.
Here
$\mathcal{B}(\sigma)$
is obtained by tensoring
the local bases
defined in \S\ref{sec:norms-whit-type-reps},
and as $\varphi$ traverses
$\mathcal{B}(\sigma)$
we let $\tilde{\varphi}$
traverse the corresponding dual basis,
with the duality normalized by the pairing given
in the denominator.
If $\sigma = \Eis^*(\mathcal{I}(\eta))$ with
$\eta$ quadratic, set $\mathcal{L}(\sigma, s) := 0$;
otherwise, set
\[
\mathcal{L}(\sigma,s)
:=
\frac{
\Lambda(\sigma, 1/2 + s_1 + s_2)
\Lambda (\sigma, 1/2 - s_1 + s_2) \Lambda (\sigma,
1/2 + s_3)
}
{
\Lambda^*(\sigma \times \sigma, 1)
}.
\]
\begin{remark}
If $\sigma$ is Eisenstein,
say
$\sigma = \Eis^*(\mathcal{I}(\chi))$,
then the quantity
\begin{equation}
\Lambda^*(\sigma \times \sigma, 1)
= \begin{cases}
\xi_F^*(1)^4 & \text{ if $\chi$ is quadratic,}\\
\xi_F^*(1)^2
\Lambda(\chi^2, 1) & \text{ otherwise}
\end{cases}
\end{equation}
does not vary continuously with respect to
$\chi \in [A]^\wedge$.
(Compare with \cite[\S2.2.2]{michel-2009}.)
The set of discontinuities consists of
the quadratic $\chi$, hence has measure zero. If $\chi = \eta
|.|^{i t}$ with
$\eta$
quadratic and
$t$ is a small nonzero real number, then
\begin{equation}
\Lambda^*(\sigma \times \sigma, 1)
\asymp
t^{-2}.
\end{equation}
These considerations motivate the
definition of $\mathcal{L}(\sigma,s)$,
which varies meromorphically with $\chi$
for $\sigma = \Eis^*(\mathcal{I}(\chi))$.
\end{remark}
We define $\mathcal{L}^{(S)}(\sigma,s)$
like $\mathcal{L}(\sigma,s)$,
but with the Euler products
over places not in $S$.
By the
unfolding of global Hecke and Rankin--Selberg integrals noted in
\S\ref{sec-5-4} and \S\ref{sec-5-6}, we see that for
factorizable factors,
\begin{equation}
\ell_{\sigma,s}(f_1 \otimes f_2)
=
\mathcal{L}(\sigma,s)
\prod_{\mathfrak{p}}
\mathcal{L}(\sigma_\mathfrak{p},s)^{-1}
\ell_{\sigma_\mathfrak{p},s}^*(f_{1 \mathfrak{p}} \otimes f_{2 \mathfrak{p}}),
\end{equation}
with the product really a finite product.
As $f$ varies
holomorphically, the ratio
$\ell_{\sigma,s}(f)/\mathcal{L}(\sigma,s)$ thus extends
holomorphically
to all $s$.
\subsection{$G \times G \geq A \times A \geq A$}
\label{sec-11-2}
Let $\omega$ be a character of $[A]$.
For almost all $s$, we may define
a functional
\[
\ell_{\omega,s} : \mathcal{I}(s_1) \otimes \mathcal{I}(s_2)
\otimes \mathbb{C}(s_3)
\rightarrow \mathbb{C}
\]
\[
f_1 \otimes f_2
\mapsto
\int_{[A]}^{\reg} \Eis^*(f_1) \omega
\int_{[A]}^{\reg} \Eis^*(f_2) \omega^{-1} |.|^{s_3},
\]
which unfolds
and then factors
on factorizable vectors
as the (finite) product
\[
\ell_{\omega,s}(f_1 \otimes f_2)
=
\mathcal{L}(\omega,s) \prod_{\mathfrak{p}}
\mathcal{L}(\omega_\mathfrak{p},s)^{-1}
\ell_{\omega_\mathfrak{p},s}^*(f_{1 \mathfrak{p}} \otimes f_{2 \mathfrak{p}}),
\]
where
\[
\mathcal{L}(\omega,s)
:=
\frac{ \prod_{\pm}
\Lambda (\omega,1/2 \pm s_1)
\Lambda (\omega^{-1},1/2 \pm s_2 + s_3)
}
{
(\xi_F(1)^*)^2
}.
\]
As $f$ varies holomorphically,
the ratio $\ell_{\omega,s}(f)/\mathcal{L}(\omega,s)$
thus extends
holomorphically to all
$\omega \in [A]^\wedge$ and $s$.
\section{Decompositions of global periods}
\label{sec-12}
We consider here
$s \in \mathbb{C}^3$ satisfying
the condition
\begin{equation}\label{eqn:assumptions-s-2}
\varepsilon_1 s_1 + \varepsilon_2 s_2 + \varepsilon_3 s_3
\notin \{0, \pm 1/2, \pm 1\}
\text{ for all }
0 \neq \varepsilon\in \{-1,0,1\}^3.
\end{equation}
We deduce from the estimate
\eqref{eq:estimate-eisenstein-series-near-cusp}
for the Eisenstein series
that the expression
\begin{equation}\label{eqn:prod-eis-and-y-s3}
\Eis^*(f_1)(a(y)) \Eis^*(f_2)(a(y)) |y|^{s_3}
\quad
(y \in \mathbb{A}^\times/F^\times)
\end{equation}
may be
approximated as $|y| \rightarrow \infty$
(resp. $|y| \rightarrow 0$)
by a linear combination
of the characters $|y|^{1 \pm s_1 \pm s_2 + s_3}$
(resp.
$|y|^{-1 \pm s_1 \pm s_2 + s_3}$).
Since
$\pm s_1 \pm s_2 \pm s_3 \neq 1$,
the trivial character never occurs,
so \eqref{eqn:prod-eis-and-y-s3} defines a strongly regularizable
function on $[A]$ (\S\ref{sec-9-1}).
We obtain
an $A_\mathbb{A}$-invariant functional
\[
\ell_s : \mathcal{I}(s_1) \otimes \mathcal{I}(s_2) \otimes
\mathbb{C}(s_3)
\rightarrow \mathbb{C}
\]
\[
f_1 \otimes f_2 \mapsto \int_{[A]}^{\reg} \Eis^*(f_1)
\Eis^*(f_2) |.|^{s_3},
\]
which we proceed to decompose in two ways.
\begin{remark}
The regularized integral
over $[A]$ considered above may be
defined concretely as the absolutely convergent integral
\begin{equation}
\int_{y \in [A]}
\left(
\int_{u \in \mathbb{A}^\times}
\Eis^*(f_1) \Eis^*(f_2) (a(y u))
\, d \nu(u)
\right)
|y|^{s_3}
\, \frac{d y}{|y|}
\end{equation}
for any finite measure $\nu$ on $\mathbb{A}^\times$ whose Mellin
transform vanishes at the characters
$|.|^{\pm 1 \pm s_1 \pm s_2}$ (with all $2^3$ possible sign
combinations).
\end{remark}
\subsection{$G \times G \geq G \geq A$}
\label{sec-12-1}
We phrase some estimates below in terms of ``generic complex
lines in $\mathbb{C}^3$ containing the origin.'' In each case,
one could take the line
$\{ (10^{-6} s_0, 10^{-12} s_0, 10^{- 18} s_0) \in \mathbb{C}^3
: s_0 \in \mathbb{C} \}$, for instance. Those estimates may be
formulated alternatively as bounds for the total polar multiplicity
at $0 \in \mathbb{C}^3$ of certain meromorphic functions.
\begin{theorem}\label{thm:GG-G-A}
Let $s \in \mathbb{C}^3$
satisfy $|\Re(s_i)| < 1/6$ for $i=1,2,3$
and
\eqref{eqn:assumptions-s-2}.
Then
\[
\ell_s
=
\int_{\sigma:\text{generic}}
\ell_{\sigma,s}
+
\sum_{i=1}^{7}
\ell^{\deg}_{i,s},
\]
where the integral
is taken over
standard generic automorphic
representations $\sigma$ as in \eqref{eqn:spectral-decomp-reg-generic}
and the ``degenerate maps'' $\ell^{\deg}_{i,s}$
are meromorphic families of functionals with the
following
properties:
\begin{itemize}
\item For $1 \leq i \leq 4$, $\ell^{\deg}_{i,s}$ factors
$\Delta G \times A$-equivariantly through
$\mathcal{I}(1/2 \pm s_1 \pm s_2) \otimes \mathbb{C}(s_3)$.
\item
$\ell^{\deg}_{5,s}$ is $N_\mathbb{A}$-invariant.
\item For $i=6,7$, $\ell^{\deg}_{i,s}$ factors
$\Delta G \times A$-equivariantly through
$\mathcal{I}(\pm (s_3 - 1/2)) \otimes \mathbb{C}(s_3)$.
\end{itemize}
Moreover,
for any holomorphic family $f[s] \in \mathcal{I}(s_1) \otimes
\mathcal{I}(s_2)
\otimes \mathbb{C}(s_3)$
defined for $s$ near zero,
we have
\begin{equation}\label{eqn:polar-bounds-for-degen-func}
\ell^{\deg}_{i,s}(f[s]) =
\begin{cases}
\operatorname{O}(|s|^{-4}) & \text{ for }i=1,2,3,4, \\
\operatorname{O}(|s|^{-5}) & \text{ for }i=6,7
\end{cases}
\end{equation}
for small $s \in \mathbb{C}^3$ in a generic complex line
containing the origin.
\end{theorem}
\begin{proof}
Let $f = f_1 \otimes f_2 \in \mathcal{I}(s_1)
\otimes \mathcal{I}(s_2)$.
Set $\Psi := \Eis^*(f_1) \Eis^*(f_2)$.
The regularizing Eisenstein series
as in \S\ref{sec-9-2}
is given by
\begin{equation}
\mathcal{E} := \Eis(f_1^* \cdot f_2^*) + \Eis(M f_1^* \cdot f_2^*) +
\Eis(f_1^* \cdot M f_2^*) + \Eis(M f_1^* \cdot M f_2^*).
\end{equation}
By Theorem \ref{thm:reg-spect-pointwise},
we have the normally convergent pointwise expansion
\[
\Psi - \mathcal{E} =
\int_{\sigma:\text{generic}}
\Psi_{\sigma},
\]
say.
We obtain
\[
\ell_s(f)
=
\sum_{i=1}^4 \ell^{\deg}_{i,s}(f) +
\int_{[A]}^{\reg}
\int_{\sigma:\text{generic}}
\Psi_{\sigma} |.|^{s_3},
\]
where the $\ell^{\deg}_{i,s}(f)$ are given by the regularized
integral over $[A]$ of the terms in the definition of
$\mathcal{E}$. The functionals $\ell^{\deg}_{i,s}$
have the required factorization property. The estimate
\eqref{eqn:polar-bounds-for-degen-func} for $i=1..4$ follows
by writing, e.g.,
\[
\Eis(f_1^* \cdot f_2^*) =
\frac{
\xi_F(1 + 2 s_1) \xi_F(1 + 2 s_2)
}
{
\xi_F(2 + 2 s_1 + 2 s_2)
}
\Eis^*(f_1 \cdot f_2),
\]
and recalling from
\S\ref{sec-5-4} that
\[
\int_{[A]}^{\reg} \Eis^*(f_1 \cdot f_2) |.|^{s_3}/\prod_{\pm }
\xi_F(1/2 \pm (1/2 + s_1 + s_2) + s_3)
\]
is holomorphic.
Continuing to assume
that $|\Re(s_1)|, |\Re(s_2)| < 1/6$,
we now change our assumptions
temporarily by supposing
that $\Re(s_3) > 1/2$.
Then for any standard generic automorphic representation
$\sigma$,
we have
\begin{equation}\label{eqn:ell-sigma-expand-with-large-s3}
\ell_{\sigma,s}(f) = \int_{[A]}
(\Psi_\sigma -
\Psi_{\sigma,N}) |.|^{s_3}.
\end{equation}
By \eqref{eqn:estimate-Psi-sigma-minus-constant-term},
the integrand in \eqref{eqn:ell-sigma-expand-with-large-s3}
is bounded for each fixed $d \geq 0$
by $C(\sigma)^{-d}$ times a fixed convergent integrand.
Taking
$d$
sufficiently large that
$\int_{\sigma} C(\sigma)^{-d} < \infty$
(see \cite[\S2.6.5]{michel-2009}),
we deduce
by interchanging summation with integration that
\begin{equation}\label{eqn:ell-with-five-degens}
\ell_s(f)
=
\sum_{i=1}^5 \ell^{\deg}_{i,s}(f)
+ \Phi(s)
\end{equation}
where, for $\Re(s_3) > 1/2$,
\begin{equation}\label{eqn:Phi-defn-integral-generic-stuff}
\Phi(s) := \int_{\sigma:\text{generic}} \ell_{\sigma,s}(f)
\end{equation}
and
\begin{equation}
\ell^{\deg}_{5,s}(f)
:=
\int_{[A]}^{\reg} \int_{\sigma:\text{generic}} \Psi_{\sigma,N} |.|^{s_3}.
\end{equation}
We may evaluate the functional $\ell^{\deg}_{5,s}$ explicitly
using truncation and that $\Psi_{\sigma,N} = 0$ unless
$\sigma$ is Eisenstein, but it suffices for our purposes to
note as claimed that this functional is
$N_\mathbb{A}$-invariant.
We now freeze the variables $s_1, s_2$
and view the above identity as one of meromorphic functions
of $s_3$ defined
initially for $\Re(s_3) > 1/2$.
We aim to meromorphically continue
$\Phi$ to the range
$-1/6 < \Re(s_3) < 1/2$
and to verify that a modified form of the identity \eqref{eqn:Phi-defn-integral-generic-stuff}
holds there.
We expand
\begin{equation}\label{eqn:expand-first-I}
\Phi(s)
=
\frac{1}{2}
\sum_{\sigma:\text{cuspidal}}
\ell_{\sigma,s}(f)
+
\frac{1}{2}
\int_{\chi:\text{unitary}}
\ell_{\mathcal{I}(\chi),s}(f).
\end{equation}
For cuspidal $\sigma$, the $L$-function
$\mathcal{L}(\sigma,s)$ is entire; by the convexity bound and
the estimate \eqref{eq:crude-estimate-for-ell-sigma-s}, we see
that $\ell_{\sigma,s}(f)$ decays faster than any power of
$C(\sigma)$, locally uniformly in $s$, and
extends to an entire function of $s$. Analogous assertions hold for
the individual Eisenstein contributions
$\ell_{\mathcal{I}(\chi), s}(f)$ except when $\chi$ is of the
form $|.|^t$, in which case we may encounter poles in the
indicated range. Indeed, we may write
$\ell_{\mathcal{I}(t),s}(f) = \Xi(t,s) h(t,s)$, where
\[
\Xi(t,s) :=
\prod_{\pm}
\frac{
\xi_F(\tfrac{1}{2} + s_1 + s_2 \pm
t) \xi_F(\tfrac{1}{2} - s_1 + s_2 \pm t) \xi_F(\tfrac{1}{2} +
s_3 \pm t)
}
{
\xi_F(1 \pm 2 t)
}
\]
and $h$ is an entire function
defined by a finite product of normalized
local integrals.
The poles of $\Xi(t,s)$
with $s_1, s_2$ as usual, $\Re(s_3) > -1/6$
and $\Re(t) = 0$
are at
$t = \pm (s_3 - 1/2)$.
We are led to the problem of determining the meromorphic
continuation of the integral
\[
I(s) :=
\int_{t \in i \mathbb{R}}
\ell_{\mathcal{I}(t),s}(f)
\, \frac{d t}{2 \pi i},
\]
defined initially for $\Re(s_3) > 1/2$. Let us first do this
under the assumption of GRH
(to avoid possible poles of $\xi_F(1 \pm 2 t)^{-1}$)
and working formally.
We fix $\varepsilon > 0$ sufficiently small, and consider $s_3$ of real part
in the interval $(1/2,1/2+\varepsilon)$. We shift the contour to
$\Re(t) = 2 \varepsilon$, passing a pole at $t = s_3 - 1/2$. The
resulting integral has no poles for $s_3$ of real part in
$(1/2-\varepsilon,1/2+\varepsilon)$. We obtain in this way the analytic
continuation of $I(s)$ to $1/2 - \varepsilon < \Re(s) < 1/2 + \varepsilon$.
We suppose next that $1/2 - \varepsilon < \Re(s) < 1/2$ and shift the
contour back to $\Re(t) = 0$, passing a pole at
$t = 1/2 - s_3$. We obtain in this way the analytic
continuation of $I(s)$ to $s_3$ of real part greater than
$-1/2$, and for $s_3$ of real part in the interval
$(-1/2,1/2)$ the formula
\begin{equation}\label{eq:formula-I-of-s-after-shift}
I(s)
=
\sum_{\pm}
\pm
\res_{t \rightarrow \pm(1/2 - s_3)}
\ell_{\mathcal{I}(t),s}(f)
+
\int_{t \in i \mathbb{R}}
\ell_{\mathcal{I}(t),s}(f)
\, \frac{d t}{2 \pi i}.
\end{equation}
The argument of the previous paragraph was not quite rigorous,
because we assumed GRH and did not justify the contour shifts. To
give a rigorous argument, we fix $T \geq 1$ sufficiently large
and restrict to $s_3$ with $|\Im(s_3)| < T/2$, say.
By the prime number theorem,
we may choose
$\varepsilon$ small enough that shifting $t$ from $i \mathbb{R}$ to
the piecewise-linear contour $\mathcal{C}$ with endpoints
$- i \infty, - i T, - i T + 2 \varepsilon, i T + 2 \varepsilon , i T, i\infty$
does not encounter any zeroes of $\xi_F(1 - 2 t)$.
We then argue as before but with $\{t : \Re(t) = 2 \varepsilon\}$
replaced
by $\mathcal{C}$.
Combining the formula \eqref{eq:formula-I-of-s-after-shift}
with our earlier remarks gives the required meromorphic continuation
of $\Phi$ and the identity, for $s_3$ of real part in $(-1/6,1/2)$,
\begin{equation}\label{eqn:expand-second-I}
\Phi(s)
=
\int_{\sigma:\text{generic}}
\ell_{\sigma,s}(f)
+
\sum_{i=6,7}
\ell^{\deg}_{i,s}(f),
\end{equation}
where the $\ell^{\deg}_{i,s}$
have the required factorization properties.
Since the estimate
\begin{equation}
\res_{t=\pm (s_3 - 1/2)} \Xi(t,s) \ll |s|^{-5}
\end{equation}
holds for small $s \in \mathbb{C}^3$ in a generic complex line
containing the origin,
the estimate \eqref{eqn:polar-bounds-for-degen-func}
follows for $i=6,7$.
We obtain the required identity by combining
\eqref{eqn:ell-with-five-degens}
and \eqref{eqn:expand-second-I}.
\end{proof}
\subsection{$G \times G \geq A \times A \geq A$}
\label{sec-12-2}
\begin{theorem}\label{thm:GG-AA-A}
Let $s \in \mathbb{C}^3$
satisfy $|\Re(s_i)| < 1/6$ for $i=1,2,3$
and
\eqref{eqn:assumptions-s-2}.
Then
\[
\ell_s
= \int_{\omega:\text{unitary}}
\ell_{\omega,s}
+ \sum_{i=8}^{15}
\ell^{\deg}_{i,s},
\]
where the integral is taken
over unitary characters $\omega$ of $[A]$
as in \eqref{eq:measure-on-dual-omega}
and the degenerate functionals
are given on $f = f_1 \otimes f_2$
by
\begin{equation}\label{deg:8}
f_1^*(1)
\int_{A_{\mathbb{A}}}^{\reg}
W_{\Eis^*(f_2)} |.|^{1/2+s_1+s_3},
\end{equation}
\begin{equation}\label{deg:9}
M f_1^*(1)
\int_{A_{\mathbb{A}}}^{\reg}
W_{\Eis^*(f_2)} |.|^{1/2-s_1+s_3},
\end{equation}
\begin{equation}\label{deg:10}
f_1^*(w) \int_{A_\mathbb{A}}^{\reg}
W_{\Eis^*(f_2)} |.|^{-1/2-s_1+s_3},
\end{equation}
\begin{equation}\label{deg:11}
M f_1^*(w) \int_{A_\mathbb{A}}^{\reg}
W_{\Eis^*(f_2)} |.|^{-1/2+s_1+s_3},
\end{equation}
together with the analogous quantities
obtained by swapping
$(f_1,s_1)$ with $(f_2,s_2)$.
\end{theorem}
\begin{proof}
Set $\varphi_i(y) := \Eis^*(f_i)(a(y))$.
By estimate \eqref{eq:estimate-eisenstein-series-near-cusp}
for the Eisenstein series,
we see that
$\varphi _i$
admits the finite expansions
\begin{equation}
\varphi_i(y)
\sim
\begin{cases}
\phi_i^\infty(y)
& \text{ as $|y| \rightarrow \infty$,}
\\
\phi_i^0(y)
& \text{ as $|y| \rightarrow 0$}
\end{cases}
\end{equation}
where
\begin{align*}
\phi_i^\infty(y) &:= |y|^{1/2+s_i} f_i^*(1)
+ |y|^{1/2-s_i} M f_i^*(1),
\\ \phi_i^0(y) &:=
|y|^{-1/2-s_i} f_i^*(w)
+ |y|^{-1/2+s_i} M f_i^*(w).
\end{align*}
Since regularizable finite functions
have vanishing regularized integral,
we have
\begin{align}\label{eqn:subtract-off-asymptotics-at-infi}
\int_{\mathbb{A}^\times/F^\times }^{\reg}
\varphi_1 \varphi_2 |.|^{s_3}
&=
\int_{\mathbb{A}^\times/F^\times}^{\reg}
(\varphi_1 - \phi_1^{\infty})
(\varphi_2 - \phi_2^\infty ) |.|^{s_3}
\\ \nonumber
&\quad
+
\int_{\mathbb{A}^\times/F^\times}^{\reg}
\phi_1^{\infty} \varphi_2 |.|^{s_3}
+
\int_{[A]}^{\reg}
\phi_2^{\infty} \varphi_1 |.|^{s_3}.
\end{align}
The second and third terms on the RHS of
\eqref{eqn:subtract-off-asymptotics-at-infi}
contribute the degenerate terms \eqref{deg:8},
\eqref{deg:9}
listed above plus their analogues with $(f_1,s_1)$ and $(f_2,s_2)$ swapped.
The first term
may be defined for (say) $\Re(s_3) \geq 20$
as an absolutely convergent integral,
then in general by meromorphic continuation.
Similarly,
the Mellin transform
(cf. \S\ref{sec-5-4})
\begin{equation}
\hat{\varphi}_i(\omega)
:=
\int_{y \in \mathbb{A}^\times/F^\times}^{\reg}
\varphi_i(\omega) \omega(y)
=
\int_{A_{\mathbb{A}}}^{\reg}
W_{\Eis^*(f_i)}
\omega
\end{equation}
may be defined for (say) $\Re(\omega) \geq 10$ via the
absolutely convergent integral of
$(\varphi_i - \phi_i^{\infty})\omega$.
By Mellin inversion,
we have
\[
\varphi_1(y)
-
\phi_1^{\infty}(y)
= \int_{\omega : \Re(\omega) = 10}
\hat{\varphi}_1(\omega)
\omega^{-1}(y).
\]
(The
crude estimate \eqref{eq:crude-ell-omega-s-estimate}
and the convexity bound for the $L$-values
give adequate decay at infinity
for $\hat{\varphi}_i$
to justify this expansion and subsequent contour shifts.)
Thus for $\Re(s_3) = 20$,
\begin{equation}
\int_{\mathbb{A}^\times/F^\times}^{\reg}
(\varphi_1 - \phi_1^{\infty})
(\varphi_2 - \phi_2^\infty ) |.|^{s_3}
=
\int_{\omega : \Re(\omega) = 10}
\hat{\varphi}_1(\omega)
\hat{\varphi}_2(\omega^{-1} |.|^{s_3}).
\end{equation}
We shift the $\omega$-contour to $\Re(\omega) = 0$,
passing poles
at $\omega = |.|^{1/2 \pm s_1}$
whose residues contribute
the degenerate terms
\eqref{deg:10},
\eqref{deg:11}.
We then take $\Re(s_3)$ nearly as small as we can
without passing a pole of the integrand
and shift to
$\Re(\omega) = \varepsilon$,
passing two more poles that contribute
the remaining degenerate terms.
We then take $\Re(s_3)$ close to $0$
and shift back to $\Re(\omega) = 0$,
giving the required identity.
\end{proof}
\subsection{Summary}
\label{sec:summary-main-results}
\begin{theorem}\label{thm:basic-identity-summary}
Suppose $s \in \mathbb{C}^3$
satisfies the hypotheses of
Theorem \ref{thm:GG-AA-A}.
Let $f = \otimes f_{\mathfrak{p}} \in \mathcal{I}(s_1) \otimes \mathcal{I}(s_2)$
be a factorizable vector.
Let $S$ be a finite set of places
of $F$
such that for each $\mathfrak{p} \notin S$,
we have that $(F_\mathfrak{p}, \psi_\mathfrak{p})$
is unramified and
$f_\mathfrak{p} = f_{1 \mathfrak{p}} \otimes f_{2
\mathfrak{p}}$
with $f_{1 \mathfrak{p}}, f_{2 \mathfrak{p}}$
normalized spherical.
Then
\begin{align}
\label{eqn:basic-moment-identity-2}
&\int_{\substack{
\sigma:\text{generic}, \\
\text{unram. outside $S$}
}}
\mathcal{L}^{(S)}(\sigma,s)
\prod_{\mathfrak{p} \in S}
\ell_{\sigma_\mathfrak{p},s}^*(f_\mathfrak{p})
+ \sum_{i=1}^{7} \ell^{\deg}_{i,s}(f_\mathfrak{p})
\\ \nonumber
&\quad
\int_{\substack{
\omega:\text{unitary}, \\
\text{unram. outside $S$}
}}
\mathcal{L}^{(S)}(\omega,s)
\prod_{\mathfrak{p} \in S}
\ell_{\omega_\mathfrak{p},s}^*(f)
+
\sum_{i=8}^{15} \ell^{\deg}_{i,s}(f),
\end{align}
where $\mathcal{L}^{(S)}(\sigma,s)$
and $\mathcal{L}^{(S)}(\omega,s)$
denote the corresponding partial Euler products.
\end{theorem}
\begin{proof}
We decompose $\ell_s(f)$ via Theorems \ref{thm:GG-G-A} and
\ref{thm:GG-AA-A} and unfold as in
\S\ref{sec:global-inv-func}.
\end{proof}
\begin{corollary}\label{cor:basic-identity-summary}
Let $f = \otimes f_{\mathfrak{p}} \in \mathcal{I}(0) \otimes \mathcal{I}(0)$
be a factorizable vector,
and let
$S$ be as in Theorem \ref{thm:basic-identity-summary}.
Then
the difference
\begin{equation}\label{eq:difference-two-moments}
\int_{\substack{
\sigma:\text{generic}, \\
\text{unram. outside $S$}
}}
\mathcal{L}^{(S)}(\sigma,0)
\prod_{\mathfrak{p} \in S}
\ell_{\sigma_\mathfrak{p},0}^*(f_\mathfrak{p})
-
\int_{\substack{
\omega:\text{unitary}, \\
\text{unram. outside $S$}
}}
\mathcal{L}^{(S)}(\omega,0)
\prod_{\mathfrak{p} \in S}
\ell_{\omega_\mathfrak{p},0}^*(f_\mathfrak{p})
\end{equation}
is equal to the limit
\begin{equation}\label{eq:difference-degen-terms}
\lim_{s \rightarrow 0} (\sum_{i=8}^{15} - \sum_{i=1}^7)
\ell^{\deg}_{i,s}(f[s])
\end{equation}
for any holomorphic family
$f[s] \in \mathcal{I}(s_1) \otimes \mathcal{I}(s_2)$, defined
for $s \in \mathbb{C}^3$ near the origin, with $f[0] = f$.
\end{corollary}
By combining the remarks of \S\ref{sec:vari-with-resp} with the
unramified calculations of \S\ref{sec-10-1} and
\S\ref{sec-10-2}, we see that the above results extend with
minor modification to the case that $\psi_\mathfrak{p}$ is
ramified for some finite $\mathfrak{p} \notin S$.
\begin{remark}
\label{rmk:}
One could likely evaluate the limit
\eqref{eq:difference-degen-terms} more explicitly as in
Motohashi's work, but it is more convenient in our experience
to work with directly with the individual degenerate functionals
$\ell_{i,s}^{\deg}$, each of which has clear
representation-theoretic significance.
\end{remark}
\subsection{Holomorphy}
\label{sec:holomorphy-global-functionals}
We record, for future reference,
some holomorphy properties implicit in the above arguments.
As in the local setting (\S\ref{sec:families-vectors}), we may
speak of holomorphic families
$f[s] \in \mathcal{I}(s_1) \otimes \mathcal{I}(s_2) \otimes
\mathbb{C}(s_3)$ indexed by $s$ in an open subset $U$ of
$\mathbb{C}^3$; this means that $f[s]$ varies pointwise
holomorphically, the seminorms $\mathcal{S}_d(f[s])$ defined in
\S\ref{sec:repr-adel-groups} are locally bounded in $s$, and,
locally in $s$, $f[s]$ is invariant by a compact open subgroup
of the finite adelic points of $\operatorname{PGL}_2 \times \operatorname{PGL}_2$. We say
that a family of (continuous) functionals $\rho_s$ indexed by
$s$ in $U$ varies holomorphically if $s \mapsto \rho_s(f[s])$ is
holomorphic for all holomorphic families $f[s]$ defined on an
open subset of $U$, or equivalently, if $\rho_s(f[s])$ varies
holomorphically for each flat family $f[s]$ defined on $U$.
\begin{lemma}
Each of the following functionals varies holomorphically
on $\{s \in \mathbb{C}^3 : |\Re(s_j)| < 1/6 \text{ for } j=1,2,3\}$:
\begin{equation}\label{eq:five-functionals-holomorphic}
\ell_s, \quad
\int_{\sigma:\text{generic}}
\ell_{\sigma,s},
\quad
\int_{\omega:\text{unitary}}
\ell_{\omega,s},
\quad
\sum_{i=1}^{7}
\ell_{i,s}^{\deg},
\quad
\sum_{i=8}^{15}
\ell_{i,s}^{\deg}.
\end{equation}
\end{lemma}
\begin{proof}
The holomorphy of $\ell_s$ on the indicated domain follows
from \eqref{eqn:assumptions-s-2}, lemma
\ref{lem:reg-int-holom-var}, and the holomorphic variation of
the asymptotic expansions near $0$ and $\infty$ of
\eqref{eqn:prod-eis-and-y-s3} as $f_1, f_2$ vary
holomorphically. The holomorphy of the next two functionals
in \eqref{eq:five-functionals-holomorphic} was implicit in the
proofs of Theorems \ref{thm:GG-G-A} and \ref{thm:GG-AA-A}.
The holomorphy of the last two then follows from the
identities proved in those theorems.
\end{proof}
\part{Analysis of local weights and degenerate terms}\label{part:analys-local-weights}
\section{Holomorphic families of
weights and vectors}
\subsection{Local}\label{sec:holom-famil-local}
Let $F$ be a local field,
with nontrivial unitary character $\psi$.
We fix a small neighborhood $\Omega$ of the origin in
$\mathbb{C}^2$. We let $s = (s_1,s_2) \in \Omega$ and set
$\pi := \mathcal{I}(s_1) \otimes \mathcal{I}(s_2)$.
For $f \in \mathcal{I}(s_1) \otimes \mathcal{I}(s_2)$
we define $W_f$ and $V_f$
as in \S\ref{sec-6-5}.
We aim to
generalize the results of \S\ref{sec-6} from $s=0$ to all
$s \in \Omega$.
No new ideas are required here,
but the formulas obtained are slightly more complicated.
We first generalize
the results of
\S\ref{sec-6-5}:
\begin{lemma}\label{lem:W_f-vs-V_f-general-s}~
\begin{enumerate}[(i)]
\item The set $\{W_f : f \in \pi \}$
contains $C_c^\infty(F^\times \times F^\times)$.
\item The set $\{V_f : f \in \pi \}$ contains
$C_c^\infty(N \backslash G, \psi)$.
\item
For $f \in \pi$,
we have
\begin{align}\label{eqn:W_f-via-V_f-general-s}
&W_f(t_1,t_2) \\ \nonumber
&=
|t_1|^{1/2-s_1} |t_2|^{1/2-s_2}
\int_{x \in F}
|1-x|^{2 s_1} |x|^{2 s_2}
V^\sharp[s](
x,
\frac{(t_1 + t_2) x - t_2 }{x(1-x)}
)
\,
\frac{d x}{|x(1-x)|},
\end{align}
where
\begin{equation}
V^\sharp[s](x,y)
:=
\int_{\xi \in F}
|\xi|^{-2 s_2}
V^\wedge[s](\xi,-x/\xi)
\psi(-\xi y)
\, d \xi
\end{equation}
with
\begin{equation}
V^\wedge[s](\xi,z)
:=
\int_{y \in F^\times}
|y|^{s_1-s_2}
V(a(y) n'(z))
\psi(\xi y)
\, \frac{d y}{|y|}.
\end{equation}
The integration in \eqref{eqn:W_f-via-V_f-general-s}
is understood as in lemma \ref{lem:W_f-via-V_f-0}.
\end{enumerate}
\end{lemma}
\begin{proof}
The first assertion is again
a consequence of standard properties
of the Kirillov model.
For the remaining assertions,
we fix a nonzero
test function $\phi_0 \in C_c^\infty(F^\times)$ supported
close enough to the identity that
$\int_{F} \phi_0 |.|^{2 s_1} \neq 0$ for all
$s \in \Omega$
and define $\Phi[s] \in \mathcal{S}(F^3)$ by
\begin{equation}\label{eq:defn-Phi-of-s-via-V-wedge-of-s}
\Phi[s](x,y,z)
:=
\frac{\phi_0(x-y z)}{ \int_{ F} \phi_0 |.|^{2 s_1} }
V^\wedge[s] (\frac{y}{x - y z}, z).
\end{equation}
and then $f[s] \in \mathcal{I}(s_1) \otimes \mathcal{I}(s_2)$
by
\begin{equation}\label{eq:defn-f-of-s-via-Phi-of-s}
f[s](g, n'(z)) := |\det g|^{1/2 + s_1} \int_{r \in F}
\Phi[s]( (0,r) g, z) |r|^{2 s_1} \, d r.
\end{equation}
The same calculations as in the proof of lemma
\ref{lem:many-V-f} confirm that $V_{f[s]} = V$, whence the
second assertion. The same calculations as in the proof of
lemma \ref{lem:W_f-via-V_f-0} lead to the required formula
for $W_f$ in terms of $V_f$.
\end{proof}
We next generalize and slightly refine
the results of
\S\ref{sec-6-6}:
\begin{theorem}\label{thm:refined-constr-adm-weight}
Let $\phi$, $h$
be as in Theorem \ref{thm:constr-admiss-weight}.
Fix a small neighborhood $\Omega$ of the origin in
$\mathbb{C}^3$.
Then we may find a holomorphic family
$f[s] \in \mathcal{I}(s_1) \otimes \mathcal{I}(s_2)$,
defined for $s = (s_1,s_2,s_3) \in \Omega$,
with the following properties:
\begin{enumerate}[(i)]
\item \label{item:ell-f-s-recovers-h} $\ell_{\sigma,s}(f[s]) = h(\sigma)$ for all
$s$.
\item \label{item:ell-omega-f-o-gives-tilde-h} $\ell_{\omega,0}(f[0]) = \tilde{h}(\omega)$
is as described in Theorem \ref{thm:constr-admiss-weight}.
\item \label{item:vanish-on-N-inv-funcs}
for each $s$
and every $N$-invariant functional $\ell : \pi \rightarrow
\mathbb{C}$,
we have $\ell(f[s]) = 0$.
\item \label{item:crude-bound-f-of-s-via-tilde-phi}
Suppose that $\phi$ has the form
$\phi(n(x) a(y) n'(z)) = \tilde{\phi}(x,y,z)$
for some $\tilde{\phi} \in \mathcal{S}(F^3)$.
Then for each fixed $d$ there is a fixed
$d'$
so that
for all $s \in \Omega$,
\begin{equation}\label{eq:estimate-S-f-s-via-tilde-phi}
\mathcal{S}_d(f[s]) \ll \mathcal{S}_{d'}(\tilde{\phi})
q^{\operatorname{O}(1)},
\end{equation}
where $q := 1$ if $F$ is archimedean and the Sobolev norms
$\mathcal{S}_d$ are as defined in
\S\ref{sec:constr-sobol-norms} and
\S\ref{sec:norms-schw-spac}.
\end{enumerate}
\end{theorem}
\begin{proof}
As in the proof of Theorem
\ref{thm:constr-admiss-weight},
we define
$V \in C_c^\infty(N \backslash G, \psi)$ by
\eqref{eqn:defn-V-via-phi} and then
$f_0[s] \in \mathcal{I}(s_1) \otimes \mathcal{I}(s_2)$ using
lemma \ref{lem:W_f-vs-V_f-general-s}, so that
$V = V_{f_0[s]}$.
We choose $\phi_1 \in C_c^\infty(A) \cong C_c^\infty(F^\times)$
supported close enough to the identity
that $\int_A \phi_1 |.|^{s_3} \neq 0$ for all $s \in \Omega$,
and set
\begin{equation}\label{eqn:f-s-via-phi1-iota-N-psi-f-0}
f[s] :=
\frac{1}{\int_A \phi_1 |.|^{s_3}}
\phi_1^{\iota} \ast N_\psi f_0[s]
\in \mathcal{I}(s_1) \otimes \mathcal{I}(s_2),
\end{equation}
with notation as in \eqref{eqn:defn-f-via-f0-basic-case}.
Assertion \eqref{item:ell-f-s-recovers-h}
holds
because $V = V_{f_0[s]}$.
The proof of assertion
\eqref{item:ell-omega-f-o-gives-tilde-h}
is the same calculation
as in the proof of
Theorem \ref{thm:constr-admiss-weight},
noting that
the present definitions specialize to the earlier ones
upon taking $s = 0$.
Assertion \eqref{item:vanish-on-N-inv-funcs}
follows from the definition
\eqref{eqn:f-s-via-phi1-iota-N-psi-f-0}
and the fact that $\phi_1$ is supported away from $0$.
For the proof of assertion
\eqref{item:crude-bound-f-of-s-via-tilde-phi}, it is
convenient to use the construction of $f[s]$ given by
\eqref{eq:defn-Phi-of-s-via-V-wedge-of-s} and
\eqref{eq:defn-f-of-s-via-Phi-of-s}. We may assume in the
non-archimedean case that $\phi_0$ is the normalized
characteristic function of $\mathfrak{o}^\times$. Since
$\Phi[s]$ is essentially a partial Fourier transform of
$\tilde{\phi}$,
we see that
\begin{equation}
\mathcal{S}_d(\Phi[s]) \ll \mathcal{S}_{d'}(\tilde{\phi}).
\end{equation}
with $d,d'$ as above. Similarly, it follows readily from
\eqref{eq:defn-f-of-s-via-Phi-of-s} that
\begin{equation}
\mathcal{S}_d(f[s]) \ll \mathcal{S}_{d'}(\Phi[s]) q^{\operatorname{O}(1)}.
\end{equation}
The required estimate \eqref{eq:estimate-S-f-s-via-tilde-phi}
follows.
\end{proof}
\subsection{Global}\label{sec:constr-glob-test}
Returning to the global setting, let $F$ be a number field,
let $S$ be a finite set of places of $F$ containing all
archimedean places,
and for each $\mathfrak{p} \in S$, let $h_\mathfrak{p}$
be a pre-Kuznetsov weight defined on the set of generic
irreducible representations $\sigma_\mathfrak{p}$
of $\operatorname{PGL}_2(F_\mathfrak{p})$.
Let $\Omega \subseteq \mathbb{C}^3$
be a small neighborhood of the origin. For $s \in \Omega$, let
$f[s] = \otimes f[s]_\mathfrak{p} \in \mathcal{I}(s_1) \otimes
\mathcal{I}(s_2) \otimes \mathbb{C}(s_3)$ denote the
factorizable vector such that
\begin{itemize}
\item
for $\mathfrak{p} \notin S$,
the local component
$f[s]_\mathfrak{p}$ is normalized spherical
(i.e., the unique
$\operatorname{PGL}_2(\mathfrak{o}_\mathfrak{p})^2$-invariant
vector with $f[s]_\mathfrak{p}(1) = 1$), while
\item for $\mathfrak{p} \in S$,
the local component $f[s]_\mathfrak{p}$ is as
constructed in Theorem \ref{thm:refined-constr-adm-weight}.
\end{itemize}
We refer subsequently to
$(f[s])_{s \in \Omega}$
as \emph{the holomorphic family
attached to the local weights
$(h_\mathfrak{p})_{\mathfrak{p} \in S}$}.
\section{Local estimates for long
families}\label{sec:crude-local-estim}
The results of this section may be used in place of lemma
\ref{lem:crude-lower-bound-individual} to ensure that our main
estimates depend polynomially upon auxiliary parameters. We
note that many of the estimates recorded this section may be strengthened
significantly via explicit calculation.
Let $F$ be a local field and $\psi$ a nontrivial unitary
character of $F$. If $F$ is non-archimedean, then we assume
that $\psi$ is unramified. Let $Q \geq 1$ be an element of the
value group of $F$. In the non-archimedean case, we assume that
$Q \geq q$.
We define $\phi \in C_c^\infty(G)$ by applying the recipe of
\S\ref{sec:constr-suit-weight} with $\chi$ the trivial
character. For example, for $F$ non-archimedean, we let
$J \leq G$ denote the set consisting of $g = n(x) a(y) n'(z)$
with $|x| \leq 1, |y| = 1, |z| \leq 1/Q$ and take for
$\phi \in C_c^\infty(G)$ the characteristic function of $J$. In
either case, we have $\phi(g) = \tilde{\phi}(x,y,z)$ where
$\tilde{\phi} \in \mathcal{S}(F^3)$ satisfies
\begin{equation}\label{eq:cal-S-tilde-phi-polyn}
\mathcal{S}_d(\tilde{\phi})
\ll Q^{\operatorname{O}(1)}
\end{equation}
for fixed $d$. Indeed, one can check that
$\mathcal{S}_d(\tilde{\phi}) \asymp
q^{\operatorname{O}(1)} Q^{d-1/2}$ for fixed $d \geq 0$.
Let $h$ denote the pre-Kuznetsov weight with kernel $\phi$
(\S\ref{sec:pre-kuzn-weights}).
Let $\tilde{h}$ denote its dual, as given by Theorem
\ref{thm:constr-admiss-weight}.
\begin{lemma}
Fix $\vartheta \in [0,1/2)$,
and let $\sigma \in G_{\gen}^\wedge$ be unitary.
\begin{enumerate}[(i)]
\item We have $h(\sigma) \geq 0$.
\item If $\sigma$ is $\vartheta$-tempered
and $C(\sigma) \leq Q$, then
$h(\sigma) \gg_{\vartheta} 1/Q$.
\end{enumerate}
\end{lemma}
\begin{proof}
The proof is similar to but simpler than that of Theorem
\ref{thm:lower-bounds-weights}. We discuss only the
non-archimedean case in detail, since the modifications
required for the archimedean case are exactly as in \S\ref{sec:lower-bounds-for-wts}.
We note first that $h(\sigma)$ is the sum of
$\vol(J) |W(1)|^2$ taken over $W$ in an orthonormal basis for
the space $\sigma^J$ of $J$-fixed vectors in $\sigma$; in
particular, $h(\sigma) \geq 0$. Suppose now that $\sigma$ is
$\vartheta$-tempered and $C(\sigma) \leq Q$. By newvector
theory \cite{MR0337789}, $\sigma$ then contains a $J$-invariant element $W$
with $W(1) = 1$. Moreover, as before, we have
$W(1) \gg_{\vartheta} \|W\|$. Since $\vol(J) \asymp 1/Q$, the
required lower
bound for $h(\sigma)$
follows.
\end{proof}
\begin{lemma}
Let $f[s] \in \mathcal{I}(s_1) \otimes \mathcal{I}(s_2)$
denote the holomorphic family,
defined for small $s \in \mathbb{C}^3$,
attached to $\phi$ and $h$
by Theorem \ref{thm:refined-constr-adm-weight}.
Then for each fixed $d$,
\begin{equation}
\mathcal{S}_d(f[s]) \ll Q^{\operatorname{O}(1)}.
\end{equation}
Moreover, for unitary $\omega \in A^\wedge$,
\begin{equation}
\tilde{h}(\omega) \ll Q^{\operatorname{O}(1)} C(\omega)^{-d}
\end{equation}
\end{lemma}
\begin{proof}
The first estimate is a consequence of
\eqref{eq:cal-S-tilde-phi-polyn} and
\eqref{eq:estimate-S-f-s-via-tilde-phi}. The second then
follows from \eqref{eq:crude-ell-omega-s-estimate}.
\end{proof}
\section{The first seven degenerate
terms}\label{sec:handl-degen-terms-1}
We give conditions on the local
weights $h_\mathfrak{p}$ under which the first seven degenerate
terms may be neglected. The informal content of these
conditions is that the weights ``vanish adequately near the
trivial representation.''
\begin{theorem}\label{thm:first-seven-degen}
Let $F$ be a number field.
Let $S$ be a finite set of places of $F$, containing all
archimedean places.
For each $\mathfrak{p} \in S$, let $h_\mathfrak{p}$
be a pre-Kuznetsov weight for $\operatorname{PGL}_2(F_\mathfrak{p})$.
Assume that
there exists either
\begin{enumerate}[(i)]
\item \label{item:first-seven-degen-finite-place}
a
finite place $\mathfrak{p} \in S$
such that $h_\mathfrak{p}$ vanishes
on the subset of unramified representations, or
\item \label{item:first-seven-degen-arch-place} an infinite place $\mathfrak{p} \in S$
for which $h_\mathfrak{p}$ is divisible
by the sixth power of the Casimir operator
$\mathcal{C}_\mathfrak{p}$
on $\operatorname{PGL}_2(F_\mathfrak{p})$,
i.e., there exists a pre-Kuznetsov weight
$h_\mathfrak{p}^0$ so that
$h_\mathfrak{p}(\sigma) = \lambda_{\sigma_\mathfrak{p}}^6
h_\mathfrak{p}^0(\sigma_\mathfrak{p})$
for all $\sigma_\mathfrak{p}$,
where
$\lambda_{\sigma_\mathfrak{p}}$
denotes the $\mathcal{C}_\mathfrak{p}$-eigenvalue.
\end{enumerate}
Let $\Omega \subseteq \mathbb{C}^3$ be a small neighborhood of
the origin, and let $(f[s])_{s \in \Omega}$ be the holomorphic
family attached to $(h_\mathfrak{p})_{\mathfrak{p} \in S}$.
Then for $i=1..7$, we have
\begin{equation}\label{eqn:first-seven-degen-tend-to-zero}
\lim_{s \rightarrow 0} \ell^{\deg}_{i,s}(f[s]) = 0,
\end{equation}
with the limit taken along $s$ in a generic complex
line in $\mathbb{C}^3$
containing the origin.
\end{theorem}
Note that, by assertion
\eqref{item:assertion-about-unramified-vanishing-of-h} of
Theorem \ref{thm:lower-bounds-weights}, the condition
\eqref{item:first-seven-degen-finite-place} of Theorem
\ref{thm:first-seven-degen} is satisfied (at some finite place)
for the weights relevant to our applications
(see \S\ref{sec-8}).
\begin{proof}
We observe first, by the stated properties
of $f[s]$
and the $N_{\mathbb{A}}$-invariance
of $\ell^{\deg}_{5,s}$,
that
$\ell^{\deg}_{5,s}(f[s]) = 0$.
It thus suffices to verify
\eqref{eqn:first-seven-degen-tend-to-zero}
for $i \in \{1,2,3,4,6,7\}$.
We will make use
of the factorization properties
stated in Theorem \ref{thm:GG-G-A},
namely,
that $\ell^{\deg}_{i,s}$
factors $\Delta G \times A$-equivariantly
through
$\mathcal{I}(t) \otimes \mathbb{C}(s_3)$
for some complex number $t = \pm 1/2 + \operatorname{O}(|s|)$.
For a complex number $t$
and a place $\mathfrak{p}$,
we denote by $\mathcal{I}_\mathfrak{p}(t)$
the corresponding induced
representation of $\operatorname{PGL}_2(F_\mathfrak{p})$,
so that $\mathcal{I}(t) = \otimes_{\mathfrak{p}}
\mathcal{I}_\mathfrak{p}(t)$.
Consider first the case that the condition
\eqref{item:first-seven-degen-finite-place} is satisfied for
some finite place $\mathfrak{p}$.
The space of
$\operatorname{PGL}_2(F_\mathfrak{p})$-invariant functionals
$\mathcal{I}_{\mathfrak{p}}(s_1) \otimes \mathcal{I}_{\mathfrak{p}}(s_2) \rightarrow
\mathcal{I}_{\mathfrak{p}}(t)$ is one-dimensional, and described explicitly
using the local Rankin--Selberg integral
$\mathcal{I}_{\mathfrak{p}}(s_1) \otimes \mathcal{I}_{\mathfrak{p}}(s_2) \otimes
\mathcal{I}_{\mathfrak{p}}(-t) \rightarrow \mathbb{C}$ or some
Laurent coefficient thereof
(see \S\ref{sec:local-rs-basic}).
The assumption on $h_\mathfrak{p}$ implies that any such
Rankin--Selberg integral vanishes at $f[s]_\mathfrak{p}$. It
follows that $f[s]_\mathfrak{p}$ lies in the kernel of any
invariant functional
$\mathcal{I}_{\mathfrak{p}}(s_1) \otimes \mathcal{I}_{\mathfrak{p}}(s_2) \rightarrow
\mathcal{I}_{\mathfrak{p}}(t)$, hence that $\ell^{\deg}_{i,s}(f) = 0$.
Consider next the case that the condition
\eqref{item:first-seven-degen-arch-place} is satisfied for
some infinite place $\mathfrak{p}$.
Let $f^0[s]$ be attached to $h^0$
in the same way that $f[s]$ was to $h$.
Then for the same reasons as in the previous case,
we have
$\ell^{\deg}_{i,s}(f[s])
= \lambda_t^6 \ell^{\deg}_{i,s}(f^0[s])$,
where $\lambda_t$ denotes the eigenvalue for
$\mathcal{C}_\mathfrak{p}$ on $\mathcal{I}(t)$. If
$t = \pm 1/2 + \operatorname{O}(|s|)$, then $\lambda_t = \operatorname{O}(|s|)$.
Since
$\ell^{\deg}_{i,s}(f^0[s]) = \operatorname{O}(|s|^{-5})$ (see
\eqref{eqn:polar-bounds-for-degen-func}), it follows that
$\ell^{\deg}_{i,s}(f[s]) = \operatorname{O}(|s|)$, whence the required
conclusion.
\end{proof}
\section{The remaining eight degenerate
terms}\label{sec:handl-degen-terms-2}
\begin{theorem}\label{thm:last-eight-degen-func}
Let $F$ be a number field, equipped with a nontrivial unitary
character $\psi$ of $\mathbb{A}/F$. Let $S = S_0 \cup S_1$,
$(h_\mathfrak{p})_{\mathfrak{p} \in S}$ and $Q$ be as in
\S\ref{sec:CI} or \S\ref{sec:PY1}. Let
$f[s] \in \mathcal{I}(s_1) \otimes \mathcal{I}(s_2) \otimes
\mathbb{C}(s_3)$ be the holomorphic family, defined for small
$s$, attached to $(h_\mathfrak{p})_{\mathfrak{p} \in S}$
in \S\ref{sec:constr-glob-test}.
Then
\begin{equation}\label{eq:last-eight-degen-func}
\lim_{s \rightarrow 0}
\sum_{i=8}^{15}
\ell^{\deg}_{i,s}(f[s])
\ll
Q^\varepsilon.
\end{equation}
In the setting of
\S\ref{sec:CI},
the implied constant
depends polynomially
upon $\sigma_0$.
\end{theorem}
The proof is given in \S\ref{sec:proof-theor-degen-2}
after several preliminaries.
\begin{remark}
It should be possible to refine this estimate to an asymptotic
formula for the LHS of \eqref{eq:last-eight-degen-func};
compare with \cite{MR3394377}.
\end{remark}
\subsection{Factorization}\label{sec:main-term-degen-func-factorization}
Take $s \in \mathbb{C}^3$ small
and satisfying \eqref{eqn:assumptions-s-2}.
We introduce the temporary notation
\[
Z(u_1,u_2,u_3)
:=
\frac{\zeta_F^{(S)}( u_1) \zeta_F^{(S)}( u_2)
\zeta_F^{(S)}( u_3)}{\zeta_F^{(S),*}(1)}.
\]
Take $f = f_1 \otimes f_2$
with $f_i = \otimes f_{i \mathfrak{p}} \in \mathcal{I}(s_i)$
unramified outside $S$.
Then by \S\ref{sec-5-4}, the $\ell^{\deg}_{i,s}(f)$ ($i=8..15$)
factor as
\begin{equation}\label{eq:factor-degeg-8}
Z(1+ 2 s_1, 1+ s_1 + s_2 + s_3,
1+ s_1 - s_2 - s_3)
\prod_{\mathfrak{p} \in S}
f_{1 \mathfrak{p}}^*(1)
\int_{A_\mathfrak{p}}
W_{f_{2 \mathfrak{p}}^*} |.|^{1/2+s_1+s_3}
\end{equation}
\begin{equation}
Z(1 - 2 s_1,1 - s_1 + s_2 + s_3,1 - s_1 - s_2 + s_3)
\prod_{\mathfrak{p} \in S}
M f_{1 \mathfrak{p}}^*(1)
\int_{A_\mathfrak{p}}
W_{f_{2 \mathfrak{p}}^*} |.|^{1/2-s_1+s_3}
\end{equation}
\begin{equation}
Z(1 + 2 s_1, -s_1 + s_2 + s_3,-s_1 - s_2 + s_3)
\prod_{\mathfrak{p} \in S}
f_{1 \mathfrak{p}}^*(w)
\int_{A_\mathfrak{p}}^{\reg}
W_{f_{2 \mathfrak{p}}^*} |.|^{-1/2-s_1+s_3}
\end{equation}
\begin{equation}\label{eq:factor-degeg-11}
Z(1 - 2 s_1, s_1 + s_2 + s_3, s_1 - s_2 + s_3)
\prod_{\mathfrak{p} \in S}
M f_{1 \mathfrak{p}}^*(w)
\int_{A_\mathfrak{p}}^{\reg}
W_{f_{2 \mathfrak{p}}^*} |.|^{-1/2-s_1+s_3},
\end{equation}
together with four similar terms
obtained by swapping $(f_1,s_1)$ with $(f_2,s_2)$.
\subsection{Evaluation of local functionals}
Let $(F,\psi)$ be a local field.
Take $s \in \mathbb{C}^3$ small
and satisfying \eqref{eqn:assumptions-s-2}.
We consider the eight $A$-invariant functionals on
$\mathcal{I}(s_1) \otimes \mathcal{I}(s_2) \otimes
\mathbb{C}(s_3)$ defined
by sending $f = f_1 \otimes f_2$ to the local
integrals implicit above, namely,
\begin{equation}\label{eqn:four-functionals-degen-local}
f_1(1) \int_A W_{f_2} |.|^{1/2 + s_1 + s_3},
\quad M f_1(1) \int_A W_{f_2} |.|^{1/2 - s_1 + s_3},
\end{equation}
\begin{equation}\label{eqn:four-functionals-degen-local-2}
f_1(w) \int_A^{\reg} W_{f_2} |.|^{-1/2 - s_1 + s_3},
\quad
M f_1(w) \int_A^{\reg} W_{f_2} |.|^{-1/2 + s_1 + s_3},
\end{equation}
together with the analogous quantities obtained by swapping the
indices $1$ and $2$. For notational simplicity, we have
replaced $f_i^*$ with $f_i$ (see \eqref{eq:defn-f-asterisk});
this has no effect on the estimation to be carried out because
$\zeta_F(1+2 s_i) \asymp 1$ for small $s_i$. We aim to evaluate
the quantities \eqref{eqn:four-functionals-degen-local},
\eqref{eqn:four-functionals-degen-local-2} in a manner more
convenient for estimation.
For $s \in \mathbb{C}^3$ near the origin,
$f = \mathcal{I}(s_1) \otimes \mathcal{I}(s_2)$
and $\nu \in \mathbb{C}^2$,
we define the double Hecke integral
\begin{equation}
D_f(\nu) := \int_{A \times A}^{\reg} W_f |.|^{\nu_1} \otimes |.|^{\nu_2}
\end{equation}
and its normalized variant (cf. \S\ref{sec-4-3})
\begin{equation}\label{eqn:normalized-double-hecke}
D_f^*(\nu) :=
\frac{D_f(\nu)}{L(\mathcal{I}(s_1), 1/2 + \nu_1)
L(\mathcal{I}(s_2), 1/2 + \nu_2)},
\end{equation}
\begin{lemma}\label{lem:eval-local-funcs-via-D-f}
For $f = f_1 \otimes f_2$,
the quantities listed in \eqref{eqn:four-functionals-degen-local}
and \eqref{eqn:four-functionals-degen-local-2}
are respectively equal to
\begin{equation}
\beta_1(s)
D_f^*(-1/2 - s_1, 1/2 + s_1 + s_3),
\end{equation}
\begin{equation}
\beta_2(s)
D_f^*(-1/2 + s_1, 1/2 - s_1 + s_3),
\end{equation}
\begin{equation}
\beta_3(s)
D_f^*(1/2+s_1, -1/2 - s_1 + s_3),
\end{equation}
\begin{equation}
\beta_4(s) D_f^*(1/2-s_1, -1/2 + s_1 + s_3),
\end{equation}
with
\begin{align*}
\beta_1(s)
&:=
\frac{
L(\mathcal{I}(s_1), 1 + s_1)
L(\mathcal{I}(s_2), 1 + s_1 + s_3)
}{
\varepsilon(\mathcal{I}(s_1),\psi,1+s_1),
},
\\
\beta_2(s) &:=
\gamma(\psi,1 - 2 s_1)
\frac{
L(\mathcal{I}(s_1), 1 - s_1)
L(\mathcal{I}(s_2), 1 - s_1 + s_3)
}{
\varepsilon(\mathcal{I}(s_1),\psi,1-s_1),
}
\\
\beta_3(s) &:= L(\mathcal{I}(s_1), 1 + s_1)
L(\mathcal{I}(s_2), - s_1 + s_3),
\\
\beta_4(s) &:=
\gamma(\psi,1 - 2 s_1)
L(\mathcal{I}(s_1), 1 - s_1)
L(\mathcal{I}(s_2), s_1 + s_3),
\end{align*}
\end{lemma}
\begin{proof}
By the formulas
\eqref{eq:W-f-viz-f} and \eqref{eqn:W-Mf-vs-W-f}
and Fourier inversion
(see \eqref{eq:fourier-inversion-for-whittaker-intertwiner}),
we have
\begin{equation}
f_1(w) = \int_{A}
W_{f_1} |.|^{1/2 + s_1},
\end{equation}
\begin{equation}
M f_1(w) =
\gamma(\psi,1 - 2 s_1)
\int_{A}
W_{f_1} |.|^{1/2 - s_1}.
\end{equation}
Thus
\begin{equation}
f_1(1) = \int_{A}
W_{w f_1} |.|^{1/2 + s_1},
\end{equation}
\begin{equation}
M f_1(1) =
\gamma(\psi,1 - 2 s_1)
\int_{A}
W_{w f_1} |.|^{1/2 - s_1}.
\end{equation}
To express these last two Hecke
integrals in terms of $W_{f_1}$, we
invoke the local functional equation
(see \eqref{eq:31})
\begin{equation}
\int_A W_{w f_1} |.|^{1/2 \pm s_1}
=
\lim_{u \rightarrow 0}
\frac{1}{\gamma(\psi,\mathcal{I}(s_1), 1 + u \pm s_1)}
\int_A^{\reg} W_{f_1} |.|^{- 1/2 - u \mp s_1}.
\end{equation}
Here we need to take a limit because the
latter Hecke integral may have a
pole at $u = 0$, compensated for by the pole of the
$\gamma$-factor
\[
\gamma(\psi,\mathcal{I}(s_1), 1 + u \pm s_1)
=
\gamma (\psi,1 + u)
\gamma (\psi,1 + u \pm 2 s_1).
\]
The required identities follow readily.
\end{proof}
\subsection{Reduction to local estimates}
\begin{definition}\label{defn:D_f-of-nu}
Let $F$ be a local field.
For a holomorphic family
$f[s] \in \mathcal{I}(s_1) \otimes \mathcal{I}(s_2) \otimes
\mathbb{C}(s_3)$ defined for $s \in \mathbb{C}^3$ near the
origin and $\alpha > 0$ sufficiently small, define
\[
\mathcal{N}_\alpha(f) :=
\sup |D_{f[s]}^*(\nu)|,
\]
with the supremum
taken over all $s$ and $\nu$
such that for some choice of sign $\pm$, each of the quantities
\[
s_1, s_2, s_3, \nu_1 \pm 1/2, \nu_2 \mp 1/2
\]
is bounded in magnitude by $\alpha$.
\end{definition}
We note, by \S\ref{sec-4-3},
that $\mathcal{N}_\alpha(f)$ is finite.
\begin{proposition}\label{lem:degen-reduce-to-local}
Let $F$ be a number field. Fix a nontrivial unitary character
$\psi$ of $\mathbb{A}/F$.
Let $S$, $(h_\mathfrak{p})_{\mathfrak{p} \in S}$
and $f[s]$
be as in \S\ref{sec:constr-glob-test}.
Then for any small $\alpha > 0$,
\begin{equation}\label{eqn:reduction-bounbd-last-eight-degen-funcs}
\lim_{s \rightarrow 0}
\sum_{i=8}^{15}
\ell^{\deg}_{i,s}(f[s])
\ll_{\alpha}
\exp(\operatorname{O}_{\alpha}(\# S))
\prod_{\mathfrak{p} \in S}
\mathcal{N}_\alpha(f_\mathfrak{p}).
\end{equation}
\end{proposition}
\begin{proof}
Write $\Phi(s)$
for the LHS of
\eqref{eqn:reduction-bounbd-last-eight-degen-funcs}.
Recall from \S\ref{sec:holomorphy-global-functionals}
that $\Phi$ is holomorphic near
$s = 0$.
By Cauchy's theorem, it will suffice to estimate
$\Phi(s)$ for $s$ belonging to a fixed small circle
$\mathcal{C}$ in a generic complex line containing the origin.
For concreteness, take
\begin{equation}
\mathcal{C} = \{
(10^{-6} s_0, 10^{-12} s_0, 10^{- 18} s_0) \in \mathbb{C}^3 :
s_0 \in \mathbb{C}, |s_0| = \alpha
\}.
\end{equation}
The products of zeta functions $Z(\dotsb)$ appearing in
\eqref{eq:factor-degeg-8}--\eqref{eq:factor-degeg-11}, as well
as the quantities $\beta_i(s)$ of lemma
\ref{lem:eval-local-funcs-via-D-f}, are $\ll_\alpha 1$ for
$s \in \mathcal{C}$. By another application of Cauchy's
theorem, our task reduces to bounding the normalized Hecke
integrals $D_{f[s]}^*(\nu)$ in a small neighborhood of the
relevant points.
We conclude by the definition of $\mathcal{N}_\alpha(f_\mathfrak{p})$.
\end{proof}
\subsection{Local estimates}
We next obtain local estimates
for the weights relevant
to our applications.
Let $\alpha > 0$ be small.
\begin{lemma}\label{lem:estimate-N-alpha-case-of-interest}
Let $F$ be a non-archimedean local field equipped with an
unramified additive character $\psi$. Let $\chi$ be a
ramified character of $F^\times$, with conductor
$Q := C(\chi)$. Let $h$ be the pre-Kuznetsov weight attached
to $\Sigma_F(\chi)$ in \S\ref{sec:constr-suit-weight}. Let
$f[s] \in \mathcal{I}(s_1) \otimes \mathcal{I}(s_2) \otimes
\mathbb{C}(s_3)$ be the holomorphic family, defined for $s$
near zero, that is attached to $h$ by Theorem
\ref{thm:refined-constr-adm-weight}. Then
\begin{equation}
\mathcal{N}_\alpha(f) \ll Q^{\operatorname{O}(\alpha)}.
\end{equation}
\end{lemma}
\begin{proof}
Expanding the construction \eqref{eqn:f-s-via-phi1-iota-N-psi-f-0}
of $f[s]$
and
calculating as in the proof of Theorem
\ref{thm:constr-admiss-weight}
(specifically,
\eqref{eqn:h-tilde-of-omega-via-h-sharp}),
we see that
\begin{equation}
D_{f[s]}(\nu)
=
\frac{\int_A \phi_1 |.|^{\nu_1 + \nu_2}}{
\int_A \phi_1 |.|^{s_3}
}
D_0,
\end{equation}
where
\begin{equation}
D_0 :=
\int_{t \in F^\times}
W_{f_0[s]}(1-t, t) |1-t|^{\nu_1} |t|^{\nu_2}
\, \frac{d t}{|t(1-t)|}.
\end{equation}
By \eqref{eqn:W_f-via-V_f-general-s},
we have
\begin{align}
&W_{f_0[s]}(1-t, t)
\\ \nonumber
&=
|1-t|^{1/2-s_1} |t|^{1/2-s_2}
\int_{x \in F}
|1-x|^{2 s_1} |x|^{2 s_2}
V^\sharp[s](
x,
\frac{ x - t }{x(1-x)}
)
\,
\frac{d x}{|x(1-x)|}.
\end{align}
The calculation thus far has been general.
Recall now that $F$ is non-archimedean,
$\chi$ is ramified and $\psi$ is unramified.
Arguing
as in the calculation of
\eqref{eqn:V-sharp-non-arch-evald-0-0},
we have $V^\wedge[s](\xi,z) = V^\wedge[0](\xi,z)$
(because $V(a(y) n'(z))$ is supported on $|y| = 1$)
and $V^\sharp[x](x,y)
= Q^{-2 s_2}
V^\sharp[0](x,y)$ (because
$V^\wedge[s](\xi,-x/\xi)$
is supported on $|\xi| = Q$),
hence by Fourier inversion,
\begin{equation}\label{eqn:V-sharp-of-s-explicit-non-arch}
V^\sharp[s](x,y)
= Q^{- 2 s_2}
1_{|x| \leq 1}
1_{|y| = 1}
\chi(y).
\end{equation}
Substituting \eqref{eqn:V-sharp-of-s-explicit-non-arch}
above yields an explicit
formula for $D_0$
as a double integral over $x$ and $t$.
The substitution $(x,t) \mapsto (1-x,1-t)$
swaps the roles of $(\nu_1,s_1)$
and $(\nu_2,s_2)$,
so as in the proof of lemma \ref{prop:non-arch-estimates-key}, we may decompose
\[
D_0 = D_1 + D_2 -
D_3,
\]
where
\begin{itemize}
\item $D_2$ denotes the contribution from when
$|1-x| = |1-t| = 1$,
\item $D_3$ that from when $|1-x| = |1-t| = |x| = |t| = 1$, and
\item $D_1$ denotes the contribution from $|x| = |t| = 1$, or
equivalently, that from $|1 - x| = |1-t| = 1$ but with
$(s_1,\nu_1)$ and $(s_2,\nu_2)$ swapped,
and with everything multiplied by $\chi(-1)$.
\end{itemize}
We change coordinates to
$(u,v)$ as before,
with $x = 1/v, t = 1/u v$,
and dyadically
decompose
according to the values $U,V \in \{1,q, q^2, \dotsc \}$
for $|u|,|v|$.
We obtain in this way
for $i=1,2$ that
\begin{equation}\label{eqn:D-i-formula-yay}
D_i
=
\chi(\pm 1)
Q^{-2 s_2}
\sum_{U,V \in \{1, q, q^2, \dotsc \}}
U^{- 1/2 + s_i - \nu_i}
V^{-1/2 - s_i - \nu_i}
D_{U,V}
\end{equation}
and that
\begin{equation}
D_3
= Q^{-2 s_2} D_{1,1},
\end{equation}
where
\begin{equation}\label{eqn:D-1-defn}
D_{U,V}
=
\int_{\substack{
u,v \in F: \\
|u - 1| = |u| = U, \\
|v - 1| = |v| = V, \\
|u v - 1| = U V
}
}
\chi \left( \frac{1 - 1/u}{1 - 1/v} \right)
\, \frac{d u \, d v}{ |u v| }.
\end{equation}
The sum \eqref{eqn:D-i-formula-yay} converges absolutely
for small $\nu_1, \nu_2$,
and is understood in general by meromorphic continuation.
The integrals $D_{U,V}$ are evaluated below in lemma
\ref{lem:exhaustive-char-computation}.
Substituting that evaluation
and summing some geometric series
gives an evaluation of the $D_i$.
The essential feature of this evaluation
is that
$D_{U,V}$ vanishes
unless
\begin{itemize}
\item $(U,V) = (Q/q,Q/q)$,
\item $U = Q/q$ and $V \geq Q$
or $V = Q/q$ and $U \geq Q$, or
\item $U, V \geq Q$.
\end{itemize}
Moreover, the
value of $D_{U,V}$
does not vary
within each of these three cases,
and has size $\ll U V/Q^2$.
If we write
$D_{Q/q,Q/q} = c_0/q^2$
and, for $U, V \geq q$,
$D_{Q/q,V} = D_{U, Q/q}
= c_1/q$
and $D_{U,V} = c_2$,
then
$c_0,c_1,c_2 \ll 1$
and
we have
for $i=1,2$ that
\begin{equation}
D_i
=
\chi(\pm 1)
Q^{-1 - 2 \nu_i - 2 s_2}
\left(
\begin{gathered}
\frac{c_0}{q^2}
q^{1 + 2 \nu_i}
+ \frac{c_1}{q}
\left(
\frac{q^{1/2 + s_i + \nu_i}}{
1 - q^{-1/2 + s_i - \nu_i}
}
+ \frac{q^{1/2 - s_i + \nu_i}}{
1 - q^{-1/2 - s_i - \nu_i}
}
\right)
\\
+
c_2
\frac{ 1 }{
(1 - q^{-1/2+s_i-\nu_i})
(1 - q^{-1/2-s_i-\nu_i})
}
\end{gathered}
\right)
\end{equation}
and that
\begin{equation}
D_3
=
Q^{-2 s_2}
\begin{cases}
c_2 & \text{ if } Q = q, \\
0 & \text{ if } Q> q.
\end{cases}
\end{equation}
Dividing by
$L(\mathcal{I}(s_1), 1/2 + \nu_1) L(\mathcal{I}(s_2), 1/2 +
\nu_2)$ has the effect of clearing all denominators.
The required estimate follows readily;
note that $Q^{-1 - 2 \nu_i}, Q^{- 2 s_2} \ll Q^{\operatorname{O}(\alpha)}$ for the
indicated ranges.
\end{proof}
\begin{lemma}\label{lem:local-estimates-final-degen-terms-long-families-crude}
Let $F, \psi, Q, \phi$ and $h$ be as in
\S\ref{sec:crude-local-estim}.
Let
$f[s] \in \mathcal{I}(s_1) \otimes \mathcal{I}(s_2) \otimes
\mathbb{C}(s_3)$ be the holomorphic family, defined for $s$
near zero, that is attached to $h$ by Theorem
\ref{thm:refined-constr-adm-weight}. Then
\begin{equation}
\mathcal{N}_\alpha(f) \ll Q^{\operatorname{O}(1)}.
\end{equation}
\end{lemma}
\begin{proof}
The required estimate follows
from \eqref{eqn:upper-bound-Mellin-transform-W},
\S\ref{sec:reduct-pure-tens} and Cauchy's theorem.
\end{proof}
\subsection{Proof of Theorem \ref{thm:last-eight-degen-func}}\label{sec:proof-theor-degen-2}
By Proposition \ref{lem:degen-reduce-to-local} and the
consequence $\exp(\# S) \ll Q^{\varepsilon}$ of the divisor bound, we
reduce to estimating $\mathcal{N}_\alpha(f_\mathfrak{p})$ for
$\mathfrak{p} \in S$.
For
$\mathfrak{p} \in S_1$, we apply lemma
\ref{lem:estimate-N-alpha-case-of-interest} with $\alpha$
sufficiently small.
For $\mathfrak{p} \in S_0$:
\begin{itemize}
\item in the setting of \S\ref{sec:CI},
we apply
lemma
\ref{lem:local-estimates-final-degen-terms-long-families-crude},
taking there for ``$Q$''
the analytic conductor of the local component
of $\sigma_0$ at $\mathfrak{p}$;
\item
in the setting of \S\ref{sec:PY1},
it suffices to note
that $\mathcal{N}_\alpha(f_\mathfrak{p})$ is finite.
\end{itemize}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 3,337 |
{"url":"https:\/\/www.math.tifrbng.res.in\/events\/c1-a-regularity-for-degenerate-fully-nonlinear-elliptic-equations-with-neumann-boundary-conditions","text":"Personal tools\n\nTheme for TIFR Centre For Applicable Mathematics, Bangalore\n\nAbstract:\u00a0In this talk, we present a proof of\u00a0 $$C^{1, \\alpha}$$ regularity up to the boundary for a class of degenerate fully nonlinear elliptic equations with Neumann boundary conditions. The proof is achieved via compactness arguments combined with new boundary H\\\"{o}lder estimates for equations which are uniformly elliptic when the gradient is either small or large.","date":"2019-12-06 01:37:03","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\": 0.20854374766349792, \"perplexity\": 345.5369806164913}, \"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-2019-51\/segments\/1575540482954.0\/warc\/CC-MAIN-20191206000309-20191206024309-00294.warc.gz\"}"} | null | null |
Camoensia es un género de plantas fanerógamas de la familia Fabaceae. Comprende 4 especies descritas y de estas, solo 2 aceptadas.
Taxonomía
El género fue descrito por Welw. ex Benth. & Hook.f. y publicado en Genera Plantarum 1: 557. 1865. La especie tipo es: Dicrocaulon pearsonii N.E. Br.
Especies aceptadas
A continuación se brinda un listado de las especies del género Camoensia aceptadas hasta julio de 2014, ordenadas alfabéticamente. Para cada una se indica el nombre binomial seguido del autor, abreviado según las convenciones y usos.
Camoensia brevicalyx Benth.
Camoensia scandens (Welw.) J.B.Gillett
Referencias
Enlaces externos
Sophoreae | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 5,716 |
\section{Introduction}
\noindent Let $ \lambda_D(z) \vert dz \vert $ denote the Poincar\'e metric on a hyperbolic domain $ D \subset \C $. To quickly recall the construction of this metric, note that there exists a holomorphic covering from the unit disc $ \mathbb{D} $ to $D$,
\[
\pi : \mathbb{D} \rightarrow D
\]
whose deck transformations form a Fuchsian group $ G $ that acts on $ \mathbb{D} $. The Poincar\'e metric on $ \mathbb{D} $,
\[
\lambda_{\mathbb{D}}(z)\vert dz \vert = \frac{\vert dz \vert}{1 - \vert z \vert^2}
\]
is invariant under ${\rm Aut}(\mathbb{D}) $ (hence in particular $ G $) and therefore, for $z \in D$, the prescription
\begin{equation}\label{Eq:PullBackPoincare}
\lambda_D(\pi(z)) \vert \pi^{\prime}(z) \vert = \lambda_{\mathbb{D}}(z)
\end{equation}
defines the Poincar\'e metric $ \lambda_D $ on $ D $ in an unambiguous manner. For an arbitrary $ z \in D $, we may choose the covering projection so that
$ \pi(0) = z $ and hence (\ref{Eq:PullBackPoincare}) implies that
\[
\lambda_D(z) = \vert \pi^{\prime}(0) \vert^{-1}.
\]
Solynin \cite{SolyninMultiplicativity}, \cite{Solynin2} proved the following remarkable relation between the Poincar\'e metrics on a pair of hyperbolic domains and those on their union and intersection:
\medskip
\noindent {\it Let $ D_1, D_2 \subset \C $ be domains such that $ D_1 \cap D_2 \neq \emptyset $. Suppose that $ D_1 \cup D_2 $ is hyperbolic. Then
\[
\lambda_{D_1 \cap D_2}(z) \lambda_{D_1 \cup D_2}(z) \leq \lambda_{D_1}(z) \lambda_{D_2}(z)
\]
for all $ z \in D_1 \cap D_2 $. If equality holds at one point $ z_0 \in D_1 \cap D_2$, then $ D_1 \subset D_2 $ or $ D_2 \subset D_1 $ and in this case, equality holds for all points $ z \in D_1 \cap D_2 $.}
\medskip
A direct proof of this was given by Kraus-Roth \cite{OliverRothMultiplicativity} that relied on a computation reminiscent of the classical Ahlfors lemma and the fact that
\begin{equation}
\Delta \log \lambda_D(z) = 4 \lambda_D^2(z),
\end{equation}
which precisely means $ \lambda_D(z) \vert dz \vert $ has constant curvature $ -4 $ on $ D $. Solynin's result follows from a comparison result for solutions to
non-linear elliptic PDE's of the form
\begin{equation}\label{Eq:EllipticPDE}
\Delta u - \mu(u) - f = 0,
\end{equation}
where $ \mu : \mathbb{R} \rightarrow \mathbb{R} $ and $ f : D \rightarrow \mathbb{R} $ are suitable continuous non-negative functions that satisfy an additional convexity condition. Clearly, this reduces to the curvature equation (\ref{Eq:EllipticPDE}) by writing $ u = \log \lambda_D $ and letting $ \mu(x) = 4 e^{2x} $ and $ f \equiv 0 $.
\medskip
The purpose of this note is to prove an analogue of Solynin's theorem for the Carath\'{e}odory metric on planar domains. Let $ D \subset \C $ be a domain that admits at least one non-constant bounded holomorphic function. Recall that for
$ z \in D $, the Carath\'eodory metric $ c_D(z) \vert dz \vert $ is defined by
\[
c_D(z) \vert dz \vert = \sup\{\vert f^{\prime}(z) \vert : f : D \rightarrow \mathbb{D} \,\, \text{holomorphic and} \, \, f(z) = 0\}.
\]
This is a distance-decreasing (and hence conformal) metric in the sense that if $h : U \rightarrow V$ is a holomorphic map between a pair of planar domains $U, V$, then
\[
c_V(h(z)) \lvert h'(z) \rvert \le c_U(z), \;\; z \in U.
\]
If $U \subset V$, applying this to the inclusion $i : U \rightarrow V$ shows that the Carath\'{e}odory metric is monotonic as a function of the domain, i.e., $c_V(z) \le c_U(z)$ for $z \in U$. Furthermore, for each $\zeta \in D$, there is a unique holomorphic map $f_{\zeta} : D \rightarrow \mathbb{D}$ that realizes the supremum in the definition of $c_D(z)$ -- this is the Ahlfors map.
\section{Statement and proof of the main result}
\begin{thm}
Let $ D_1, D_2 \subset \C$ be smoothly bounded domains with $ D_1 \cap D_2 \neq \emptyset $. Then there exists a constant $ C = C(D_1, D_2) > 0 $ such that
\[
c_{D_1 \cap D_2}(z) \cdot c_{D_1 \cup D_2}(z) \leq C\, c_{D_1}(z) \cdot c_{D_2}(z)
\]
for all $z \in D_1 \cap D_2$.
\end{thm}
Note that $D_1 \cap D_2$ may possibly have several components. The notation $c_{D_1 \cap D_2}(z)$ refers to the Carath\'{e}odory metric on that component of $D_1 \cap D_2$ which contains a given $z \in D_1 \cap D_2$. Furthermore, the constant $C > 0$ is independent of which component of $D_1 \cap D_2$ is being considered and only depends on $D_1$ and $D_2$. The proof, which is inspired by \cite{OliverRothMultiplicativity}, uses the following known supplementary properties of the Carath\'{e}odory metric:
\medskip
First, $c_D(z)$ is continuous and $\log c_D(z)$ is subharmonic on $D$ -- see \cite{AhlforsAndBeurling} for instance. The possibility that $\log c_D(z) \equiv -\infty$ can be ruled out for bounded domain since for every $\zeta \in D$, the affine map $f(z) = (z - \zeta)/a$ vanishes at $\zeta$ and maps $D$ into the unit disc $\mathbb D$ for every positive $a$ that is bigger than the diameter of $D$. The fact that $f'(z) \equiv 1/a > 0$ shows that $c_D(\zeta) > 0$ and hence
$\log c_D(\zeta) > -\infty$.
\medskip
Second, Suita \cite{SuitaI} showed that $c_D(z)$ is real analytic in fact and hence we may speak of its curvature
\[
\kappa_D(z) = -c_D^{-2}(z) \Delta \log c_D(z)
\]
in the usual sense. The subharmonicity of $\log c_D(z)$ already implies that $\kappa_D \le 0$ everywhere on $D$, but by using the method of supporting metrics, Suita \cite{SuitaI} was also able to prove a much stronger inequality namely,
$\kappa_D \le -4$ on $D$. Following this line of inquiry further, Suita \cite{SuitaII} (see \cite{BurbeaPaper} as well) showed that if the boundary of $D$ consists of finitely many Jordan curves, the assumption that $\kappa_D(\zeta) = -4$ for some point $\zeta \in D$, implies that $D$ is conformally equivalent to $\mathbb D$.
\medskip
Finally, it is known that this metric admits a localization near $C^{\infty}$-smooth boundary points -- see for example \cite{InvariantMetricJarnicki} which contains a proof for the case of strongly pseudoconvex points that works verbatim in the planar case too. That is, if $p \in \partial D$ is a $C^{\infty}$-smooth boundary point of a bounded domain $D \subset C$, then for a small enough neighbourhood $U$ of $p$ in $\mathbb C$,
\[
\lim_{z \rightarrow p} \frac{c_{U \cap D}(z)}{c_D(z)} = 1.
\]
Using this, it was shown in \cite{SV} that the curvature $\kappa_D(z)$ of the
Carath\'{e}odory metric approaches $-4$ near each $C^{\infty}$-smooth boundary point of a bounded domain $D \subset \mathbb C$. The point here being that as one moves nearer to such a point, the metric begins to look more and more like the Carath\'{e}odory metric on $\mathbb D$ -- the use of the scaling principle makes all this precise. It follows on any smoothly bounded planar domain $D$,
$ \kappa_D \approx -4 $ for points close to the boundary and for those that are at a fixed positive distance away from it, there is a large negative lower bound for $\kappa_D$ as a result of its continuity. Hence for every such $ D \subset \C$, there is a constant $ C = C(D) > 0$ such that
\[
-C \leq \kappa_D(z) \leq -4
\]
for all $ z \in D $. Another consequence of the localization principle is that $c_D(z) \rightarrow +\infty$ as $z$ approaches a smooth boundary point. Indeed, near such a point, $c_D(z) \approx c_{U \cap D}(z)$ and the latter is the same as the Carath\'{e}odory metric on the disc $\mathbb D$ (since $U \cap D$ can be taken to be simply connected) which blows up near every point on $\partial \mathbb D$.
\medskip
To prove the theorem, let $ \kappa_1 $ and $ \kappa_2 $ be the curvatures of the Carath\'eodory metric $ c_{D_1}(z) \lvert dz \rvert $ and $ c_{D_2}(z) \lvert dz \rvert $ on $ D_1 $ and $ D_2 $ respectively.
\medskip
Consider the metric
\[
c(z) \lvert dz \rvert = \frac{c_{D_1}(z)\cdot c_{D_2}(z)}{c_{D_1 \cup D_2}(z)} \lvert dz \rvert
\]
on the possibly disconnected open set $ D_1 \cap D_2 $. What follows applies to each component of $D_1 \cap D_2$ without any regard to its analytic or topological properties and hence we will continue to write $c(z) \lvert dz \rvert$ to denote this metric on any given component. Its curvature is
\begin{equation}
\begin{split}
\kappa(z) &= -c^{-2}(z) \Delta \log c(z)\\
&= -c^{-2}(z)\left( \Delta \log c_{D_1}(z) + \Delta \log c_{D_2}(z) - \Delta \log c_{D_1 \cup D_2}(z) \right)\\
&= I_1 + I_2 + I_{12},
\end{split}
\end{equation}
where
\[
I_1 = -c^{-2}(z) \Delta \log c_{D_1}(z),
\]
\[
I_2 = -c^{-2}(z) \Delta \log c_{D_2}(z)
\]
and
\[
I_{12} = c^{-2}(z)\Delta \log c_{D_1 \cup D_2}(z).
\]
Note that $ I_{12} \geq 0$ since $ \log c_{D_1 \cup D_2}(z) $ is subharmonic and hence
\[
\kappa(z) \geq I_1 + I_2.
\]
To analyze each of these terms, note that $ c_{D_1} \geq c_{D_1 \cup D_2} $ and $ c_{D_2} \geq c_{D_1 \cup D_2} $. Combining this with the fact that the curvature of the Carath\'eodory metric is negative everywhere, it follows that
\[
I_1 \geq \kappa_1(z)
\]
and
\[
I_2 \geq \kappa_2(z).
\]
Hence there is a constant $ C = C(D_1, D_2) > 0 $ such that
\[
\kappa(z) \geq \kappa_1(z) + \kappa_2(z) > -C
\]
for all $ z \in D_1 \cap D_2 $.
\medskip
Let $U$ be a component of $D_1 \cap D_2$. Since $D_1, D_2$ have smooth boundaries, $U$ has finite connectivity, say $m \ge 1$ and is non-degenerate in the sense that the interior of its closure coincides with itself. In particular, its boundary cannot contain isolated points. Let $U_{\epsilon}$ be an $\epsilon$-thickening of $U$. For all sufficiently small $\epsilon > 0$, $U_{\epsilon}$ also has connectivity $m$ and is non-degenerate. Furthermore, $U_{\epsilon} \rightarrow U$ in the sense of Carath\'{e}odory as $\epsilon \rightarrow 0$.
\medskip
For a fixed $\epsilon > 0$, let $c_{\epsilon}(z) \vert dz \vert$ be the
Carath\'{e}odory metric on $U_{\epsilon}$. Consider the function
\[
u(z) = \log\left(\frac{c_{\epsilon}(z)}{\sqrt{C\big/ 4 }\,c(z)}\right)
\]
on $U$. Since $ U$ is compactly contained in $ U_{\epsilon} $,
$ c_{\epsilon} $ is bounded on $U$ and if $ \xi \in \partial U $, then as
$z \rightarrow \xi$ within $U$,
\[
\lim_{z \to \xi} \frac{c_{D_1}(z)\cdot c_{D_2}(z)}{c_{D_1 \cup D_2}(z)} \geq \lim_{z \to \xi}c_{D_2}(z) = + \infty,
\]
where the inequality follows from the monotonocity of the Carath\'{e}odory metric, i.e., $ c_{D_1}(z) \geq c_{D_1 \cup D_2}(z) $. The fact that $\xi$ is a smooth boundary point of $D_2$ implies that $c_{D_2}$ blows up near it and this means that $ u(z) \to - \infty $ at $\partial U$. Therefore, $u$ attains a maximum at some point $z_0 \in U$. As a result,
\begin{equation}
\begin{split}
0 \geq \Delta u(z_0) &= \Delta \log c_{\epsilon}(z_0) - \Delta \log c(z_0)\\
&\geq - \kappa_{{\epsilon}}(z_0)c_{\epsilon}^2(z_0) - C c^2(z_0)\\
&\geq 4 c^{2}_{\epsilon}(z_0) - C c^{2}(z_0),
\end{split}
\end{equation}
where $ \kappa_{\epsilon} $ is the curvature of $c_{\epsilon}(z) \vert dz \vert$.
It follows that
\[
u(z_0) = \log \left(\frac{c_{\epsilon}(z_0)}{\sqrt{C \big/4} \, c(z_0)}\right) \leq 0.
\]
For an arbitrary $ z \in U$, $ u(z) \leq u(z_0) \leq 0 $ and this gives
\[
\log\left(\frac{c_{\epsilon}(z)}{\sqrt{C \big/4} \, c(z)}\right) \leq 0
\]
or
\[
c_{\epsilon}(z) \leq \sqrt{C \big/4}\, c(z)
\]
which is same as
\[
c_{\epsilon}(z)\cdot c_{D_1 \cup D_2}(z) \leq \sqrt{C \big/4}\,c_{D_1}(z)\cdot c_{D_2}(z)
\]
and this holds for all $z \in U$. It remains to show that $c_{\epsilon} \rightarrow c_U$ pointwise on $U$ for then we can pass to the limit as $\epsilon \rightarrow 0$, keeping in mind that $C$ is independent of $\epsilon$, to get
\[
c_U(z)\cdot c_{D_1 \cup D_2}(z) \leq \sqrt{C \big/4}\,c_{D_1}(z)\cdot c_{D_2}(z)
\]
as claimed. Fix $p \in U$. To show that $c_{\epsilon}(p) \rightarrow c_U(p)$, it suffices to prove that
$\vert f'_{\epsilon}(p) \vert \rightarrow \vert f'(p) \vert$ as $\epsilon \rightarrow 0$, where $f_{\epsilon} : U_{\epsilon} \rightarrow \mathbb D$ and
$f : U \rightarrow \mathbb D$ are the respective Ahlfors maps for the domains $U_{\epsilon}$ and $U$ at $p$. What this is really asking for is the continuous dependence of the Ahflors maps on the domains. But this is addressed in \cite{Younsi} -- indeed, Theorem 3.2 therein can be applied as the domains $U_{\epsilon}, U$ have the same connectivity by construction and the
$U_{\epsilon}$'s decrease to $U$ as $\epsilon \rightarrow 0$. The nuance, in this theorem, about the base point being the point at infinity can be arranged by sending $p \mapsto \infty$ by the map $T(z) = 1/(z-p)$ and working with the domains $T(U_{\epsilon})$ and $T(U)$. This completes the proof.
\medskip
\noindent {\it Question:} Let $\Omega \subset \mathbb C$ be a domain on which the Carath\'{e}odory metric $c_{\Omega}(z) \vert dz \vert$ is not degenerate. Does its curvature $\kappa_{\Omega}$ admit a global lower bound?
\begin{bibdiv}
\begin{biblist}
\bib{AhlforsAndBeurling}{article}{
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date={1950},
ISSN={0001-5962},
journal={Acta Math.},
volume={83},
pages={101\ndash 129},
url={https://doi.org/10.1007/BF02392634},
review={\MR{0036841}},
}
\bib{BurbeaPaper}{article}{
author={Burbea, Jacob},
title={The curvatures of the analytic capacity},
date={1977},
ISSN={0025-5645},
journal={J. Math. Soc. Japan},
volume={29},
number={4},
pages={755\ndash 761},
url={https://doi.org/10.2969/jmsj/02940755},
review={\MR{0460624}},
}
\bib{InvariantMetricJarnicki}{book}{
author={Jarnicki, Marek},
author={Pflug, Peter},
title={Invariant distances and metrics in complex analysis},
edition={extended},
series={De Gruyter Expositions in Mathematics},
publisher={Walter de Gruyter GmbH \& Co. KG, Berlin},
date={2013},
volume={9},
ISBN={978-3-11-025043-5; 978-3-11-025386-3},
url={https://doi.org/10.1515/9783110253863},
review={\MR{3114789}},
}
\bib{OliverRothMultiplicativity}{article}{
author={Kraus, Daniela},
author={Roth, Oliver},
title={Strong submultiplicativity of the {P}oincar\'{e} metric},
date={2016},
ISSN={0971-3611},
journal={J. Anal.},
volume={24},
number={1},
pages={39\ndash 50},
url={https://doi.org/10.1007/s41478-016-0006-5},
review={\MR{3755807}},
}
\bib{SV}{article}{
author={Sarkar, Amar~Deep},
author={Verma, Kaushal},
title={Boundary behaviour of some conformal invariants on planar
domains, preprint available at https://arxiv.org/pdf/1904.06867.pdf},
}
\bib{SolyninMultiplicativity}{article}{
author={Solynin, A.~Yu.},
title={Ordering of sets, hyperbolic metric, and harmonic measure},
date={1997},
ISSN={0373-2703},
journal={Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov.
(POMI)},
volume={237},
number={Anal. Teor. Chisel i Teor. Funkts. 14},
pages={129\ndash 147, 230},
url={https://doi.org/10.1007/BF02172470},
review={\MR{1691288}},
}
\bib{Solynin2}{article}{
author={Solynin, A.~Yu.},
title={Elliptic operators and {C}hoquet capacities},
date={2009},
ISSN={0373-2703},
journal={Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov.
(POMI)},
volume={371},
number={Analiticheskaya Teoriya Chisel i Teoriya Funktsi\u{\i}. 24},
pages={149\ndash 156, 178\ndash 179},
url={https://doi.org/10.1007/s10958-010-9861-9},
review={\MR{2749230}},
}
\bib{SuitaI}{article}{
author={Suita, Nobuyuki},
title={On a metric induced by analytic capacity},
date={1973},
journal={K\=odai Math. Sem. Rep.},
volume={25},
pages={215\ndash 218},
}
\bib{SuitaII}{article}{
author={Suita, Nobuyuki},
title={On a metric induced by analytic capacity. {II}},
date={1976},
ISSN={0023-2599},
journal={K\=odai Math. Sem. Rep.},
volume={27},
number={1-2},
pages={159\ndash 162},
url={http://projecteuclid.org/euclid.kmj/1138847170},
review={\MR{0404603}},
}
\bib{Younsi}{article}{
author={Younsi, Malik},
title={On the analytic and {C}auchy capacities},
date={2018},
ISSN={0021-7670},
journal={J. Anal. Math.},
volume={135},
number={1},
pages={185\ndash 202},
url={https://doi.org/10.1007/s11854-018-0028-9},
review={\MR{3827348}},
}
\end{biblist}
\end{bibdiv}
\end{document}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 3,522 |
{"url":"http:\/\/www.physicsforums.com\/showthread.php?p=4239999","text":"# Find the value of x1^6 +x2^6 of this quadratic equation without solving it\n\nby chloe1995\n P: 2 1. The problem statement, all variables and given\/known data Solve for $x_1^6+x_2^6$ for the following quadratic equation where $x_1$ and $x_2$ are the two real roots and $x_1 > x_2$, without solving the equation. $25x^2-5\\sqrt{76}x+15=0$ 2. Relevant equations 3. The attempt at a solution I tried factoring it and I got $(-5x+\\sqrt{19})^2-4=0$ What can I do afterwards that does not constitute as solving the equation? Thanks.\n HW Helper Thanks P: 4,274 Notice that 5 can be factored from the quadratic without changing the roots. Also, you haven't truly factored the quadratic, you have merely re-written it.\nPF Patron\nHW Helper\nEmeritus\nP: 7,077\n Quote by chloe1995 1. The problem statement, all variables and given\/known data Solve for $x_1^6+x_2^6$ for the following quadratic equation where $x_1$ and $x_2$ are the two real roots and $x_1 > x_2$, without solving the equation. $25x^2-5\\sqrt{76}x+15=0$ 2. Relevant equations 3. The attempt at a solution I tried factoring it and I got $(-5x+\\sqrt{19})^2-4=0$ What can I do afterwards that does not constitute as solving the equation? Thanks.\nHello chloe1995. Welcome to PF !\n\nSuppose that x1 and x2 are the solutions to the quadratic equation, $\\displaystyle \\ \\ ax^2+bx+c=0\\ .$\n\nThen $\\displaystyle \\ \\ x_1 + x_2 = -\\frac{b}{a}\\ \\$ and $\\displaystyle \\ \\ x_1\\cdot x_2=\\frac{c}{a}\\ .\\$\n\nP: 2\n\n## Find the value of x1^6 +x2^6 of this quadratic equation without solving it\n\n Quote by SteamKing Notice that 5 can be factored from the quadratic without changing the roots. Also, you haven't truly factored the quadratic, you have merely re-written it.\nOops! I meant completing the square.\n\n Quote by SammyS Hello chloe1995. Welcome to PF ! Suppose that x1 and x2 are the solutions to the quadratic equation, $\\displaystyle \\ \\ ax^2+bx+c=0\\ .$ Then $\\displaystyle \\ \\ x_1 + x_2 = -\\frac{b}{a}\\ \\$ and $\\displaystyle \\ \\ x_1\\cdot x_2=\\frac{c}{a}\\ .\\$\nThank you.\n PF Patron HW Helper Sci Advisor Emeritus P: 7,077 So, Have you managed to solve the problem?\n\n Related Discussions Precalculus Mathematics Homework 4 Calculus & Beyond Homework 1 Precalculus Mathematics Homework 2 Precalculus Mathematics Homework 5 Precalculus Mathematics Homework 2","date":"2013-12-08 09:04:17","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\": 0.530903160572052, \"perplexity\": 668.4381505416058}, \"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-2013-48\/segments\/1386163057372\/warc\/CC-MAIN-20131204131737-00065-ip-10-33-133-15.ec2.internal.warc.gz\"}"} | null | null |
{"url":"https:\/\/math.stackexchange.com\/questions\/691956\/finding-the-roc-and-multiplicity-of-a-function","text":"# Finding the ROC and multiplicity of a function\n\nConsider Newton's method for finding the root of f(x) = x-sin(x). Run it on Matlab to find what is the rate of convergence. What is the value $\\lambda$ such that $|x_{n+1} - \\alpha| = \\lambda|x_n - \\alpha|$ where $\\alpha$ is the root. Find the multiplicity m of this root $\\alpha$.\n\nUsing Newton's method for finding roots (octave) I have found that this f(x) converges at 2.0236e-08, and it does so in close to 50 iterations. My code is as follows $x = x - ((x-\\sin(x))\/(1-\\cos(x)))$\n\nHow do I go about solving for $\\lambda$ and the multiplicity of m?\n\nThanks for your time!\n\n\u2022 If you have the numbers still, then you can examine $$\\lambda=\\frac{|x_{n+1}-\\alpha|}{|x_n-\\alpha|}$$ and see what they look like. (The equality you gave suggests they should all be the same, although I would expect a limiting value.) If you don't still have the numbers, then I guess you get to regenerate the numbers. \u2013\u00a0tabstop Feb 27 '14 at 0:57\n\nWe have $f(x) = x- \\sin x~$ and it is clear that $x = 0$ is a root.\n\n\u2022 $f(x) = x- \\sin x~$ and $~f(0) = 0$\n\u2022 $f'(x) = 1 - \\cos x$ and $~f'(0) = 0$\n\u2022 $f''(x) = \\sin x~$ and $~f''(0) = 0$\n\u2022 $f'''(x) = \\cos x~$ and $~f'''(0) = 1$\n\nHence, we have a triple root, so $m = 3$.\n\nWe can now write (do you know where this result comes from):\n\n$$\\lambda = \\dfrac{m-1}{m} = \\dfrac{2}{3}$$\n\nNow, if we use $x_0 = 1$ and want to figure out how many steps it will take to get eight digits of accuracy, we have:\n\n$$\\left(\\dfrac{2}{3}\\right)^n < \\dfrac{1}{2} 10^{-8} \\implies n \\ge 48$$\n\nCompare this to your numerical results using $e_{n+1} \\approx \\dfrac{2}{3} e_n$. In other words, use your numerical results and add a column which shows $~|x_{n+1} - \\alpha| = \\lambda|x_n - \\alpha|~$ where $\\alpha$ is the root\n\nPlease fill in the details.","date":"2019-06-20 15:56:52","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\": 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\": 0.8733858466148376, \"perplexity\": 199.99426714518304}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.3, \"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-2019-26\/segments\/1560627999261.43\/warc\/CC-MAIN-20190620145650-20190620171650-00531.warc.gz\"}"} | null | null |
{"url":"https:\/\/www.askiitians.com\/forums\/Vectors\/dear-sir-pls-help-me-solve-this-question-if-a-b-c_100600.htm","text":"# Dear sir pls help me solve this question.If A,B,C are any 3 vectors then show that -\u00a0Avector cross (bvector +cvector)=Avector cross bvector+Avector cross C vector.\n\nApoorva Arora IIT Roorkee\n8 years ago\nThis is a rule of cross product which all vectors follow\nTo prove the rule$A+(B\\times C)=A\\times B+A\\times C$\nyou need to put\n$A=a_{1}\\hat{i}+a_{2}\\hat{j}+a_{3}\\hat{k}$\n$B=b_{1}\\hat{i}+b_{2}\\hat{j}+b_{3}\\hat{k}$\n$C=c_{1}\\hat{i}+c_{2}\\hat{j}+c_{3}\\hat{k}$\nand just do the cross product with both the methods i.e. LHS and RHS and hence prove that both result in the same answer\n\nThanks and Regards\nApoorva Arora\nIIT Roorkee","date":"2022-06-26 20:07:53","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\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 4, \"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\": 0.8831307291984558, \"perplexity\": 4477.264758368188}, \"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-2022-27\/segments\/1656103271864.14\/warc\/CC-MAIN-20220626192142-20220626222142-00745.warc.gz\"}"} | null | null |
\section{Introduction}
\label{sec:introduction}
\IEEEPARstart{M}{any} signals such as audio signals (music, speech, bird songs) \cite{gribonval2003harmonic},
medical data (electrocardiogram, thoracic and abdominal movement signals),
can be modeled as a superimposition of amplitude- and frequency-modulated (AM-FM) modes \cite{Lin2016, Herry2017}, and are therefore called \emph{multicomponent signals} (MCSs).
Time-frequency analysis is often used to deal with such signals \cite{Flandrin1998,Boashash2003,stankovic2014time}, since their modes are associated with curves in the \emph{time-frequency} (TF) plane, called \emph{ridges}.
To extract these ridges is essential when one is interested in separating a MCS into its constituent modes \cite{Auger2013}. For that purpose,
several TF-based techniques were developed \cite{Carmona1997}\cite{Carmona1999}
using the idea that the ridges correspond to local maxima along the frequency axis
of the modulus of some \emph{time-frequency representation} (TFR).
Indeed, it was shown in \cite{stankovic2001measure,stankovic2001performance}
that these maxima approximate the \emph{instantaneous frequency} (IF) of the modes,
the quality of approximation depending on the noise level and on the length of the analysis window. This concept has then been used on various types of TFRs such as for instance the \emph{short-time Fourier transform} (STFT) \cite{meignen2018retrieval,colominas2020fully},
or \emph{synchrosqueezed transforms} either based on the continuous wavelet transform \cite{Daubechies2011} or STFT \cite{Thakur2013,oberlin2013,behera2018theoretical}.
TF ridges are also used in demodulation approaches
still developed for the purpose of \emph{mode retrieval} (MR) \cite{Li2012,meignen2017demodulation}.
An alternative TF-based MR technique, not explicitly using ridge detection, consists of finding locally the linear chirp that fits the best the STFT of the signal \cite{colominas2019time}.
The main limitation of the above techniques is that at high noise level,
the local maxima along the frequency axis that define the ridges in the noiseless case may no longer exist.
In this regard, the analysis of these maxima proposed in \cite{stankovic2001measure} assumes a low noise level. In case of high noise level, a study was carried out on the TFR associated with the \emph{Wigner-Ville distribution} (WVD) in \cite{djurovic2004algorithm}, basically remarking that peak searching based techniques are not able to cope with multicomponent or monocomponent signals contaminated by strong noise. The key ideas of the algorithm proposed in \cite{djurovic2004algorithm} were first that, if the WVD maximum at the considered time instant is not at the IF point, there is a high probability that the IF is at a point having one of the largest WVD values. The second argument was based on the assumption that the IF variations between two consecutive points is not extremely large.
Following \cite{djurovic2004algorithm}, it seems to be possible to follow the ridges by considering
local maxima and then by chaining them using some proximity criterion in the TF plane.
However, as we will see, when one considers STFT as TFR, the noise can generate zeros in the vicinity
of true IF location splitting the ridge into two chains of local maxima.
It is therefore illusory to try and build the ridges by simply chaining local maxima in the TF plane.
The second argument used in \cite{djurovic2004algorithm}
is however very interesting in that even, at a high noise level, the local maxima associated with the signal
components are close in the TF plane.
Though there exist many different techniques to extract the ridges in the TF plane, mainly using
optimization procedures as in \cite{Carmona1999,zhu2019two}, they rely on an initial so-called \emph{skeleton of the transform} which is usually not available in heavy noise situations.
Indeed, to reconstruct the modes, these approaches use modified versions of
\emph{least-square minimizations} in which the data terms are ridge points, supposed to be reliable enough otherwise the algorithms fail. However, to assign a TF point to a specific
ridge is probably one of the most complicated aspect of RD and should be handled with care.
To perform this kind of initial estimation of ridge points, one needs to carefully analyze the behavior of the coefficients of the TFR
in the vicinity of local maxima.
In the present paper, to accurately assign a TF point to a ridge, we will make the
assumption that the modes are not crossing, our goal being to improve RD in the TF plane in very
noisy situations in a fully adaptive way. It is however possible to deal with
crossing modes by imposing regularity constraints on the extracted modes in the TF plane \cite{Thakur2013,Carmona1999,Carmona1997}, or by analyzing the signal in the time domain
using a parametric approach \cite{chen2017intrinsic}, but this is not the scope of the present paper.
Indeed, we aim to propose a fully adaptive RD that is very competitive in noisy situations.
For that purpose, we will first analyze carefully
the energy in the TF plane in the vicinity of local maxima.
In particular, we will see that some of these local maxima can be
gathered into so-called \emph{relevant ridge portions} (RRPs) that will be the basis to our new approach. As already mentioned in the context of WVD \cite{djurovic2004algorithm},
the effect of noise on TF ridges is to split them into ridge portions that need to be identified and then gathered together when they correspond to the same mode. This is in essence the goal we pursue in the present paper.
The paper is organized as follows: in Section \ref{sec:notations}, we recall basic notations regarding STFT and MCSs as well the most commonly used TF-based RD technique. Since the latter depends on several arbitrary parameters, we recall in Section \ref{sec:adpative_ridge}, how it can be made more adaptive by using a local chirp rate
estimate, as recently proposed in \cite{colominas2020fully}.
Then, taking into account that, in noisy situations, a mode cannot be associated with a single ridge
computed from chaining local maxima in the TF plane,
we alternatively consider that it corresponds to a set of so-called
\emph{relevant ridge portions} (RRPs), which are defined
in Section \ref{sec:RRPs}.
This helps us define a new RD technique in Section \ref{sec:newrd}.
Section \ref{sec:res} is then devoted to the comparison of the proposed new technique to state-of-the-art TF-based methods for RD and MR, highlighting the improvement brought by the former especially in the presence of heavy noise. An application to the analysis of gravitational-wave concludes the paper.
\section{Definitions and Notations}
\label{sec:notations}
\subsection{Short-Time Fourier Transform}
\label{sec:STFT}
Let $\tilde{f}$ be a discrete signal of length $L$ altered
by an additive noise $\varepsilon$, and such that $\tilde{f}[n] = \tilde{f}(\frac{n}{L})$:
\begin{equation}
\label{def:ftilde}
\tilde{f} := f + \varepsilon,
\end{equation}
and $g$ a discrete real window supported on $[-\frac{M}{L},\frac{M}{L}]$. In that context, we define the \emph{short time Fourier transform} (STFT) as follows:
\begin{equation}
\label{def:STFT}
V_{\tilde{f}}^{g} [m,k] := \sum_{n = 0}^{N-1} \tilde{f}[n + m - M]g[n - M] e^{-2i\pi \frac{k}{N}(n - M)},
\end{equation}
with $2M+1\leq N$, where $N$ is the number of frequency bins, the index $k$
corresponding to the frequency $k \frac{L}{N}$. The STFT of the signal $ \tilde{f}$
is invertible, provided $g[0] \neq 0$, since one has:
\begin{equation}
\label{def:inv_STFT}
\tilde{f}[n] = \frac{1}{g[0]N} \sum_{k=0}^{N-1} V_{\tilde{f}}^{g}[n,k].
\end{equation}
\subsection{Multicomponent Signal Definition}
\label{sec:def_MCS}
In this paper, we will intensively study MCSs defined as a superimposition of AM-FM components or modes:
\begin{equation}
\label{def:MCS}
f[n] = \sum \limits_{p=1}^P f_p[n]~~~~\textrm{with}~~
f_p[n] = A_p[n] e^{i 2\pi \phi_p[n]},
\end{equation}
for some finite $P \in \mathbb{N}$, $A_p[n]$ and $\phi'_p[n]$
being respectively the instantaneous amplitude (IA) and frequency (IF) of
$f_p$ satisfying: $A_p[n]>0,\phi'_p[n]>0$ and
$\phi'_{p+1}[n]>\phi'_p[n]$ for each time index $n$.
We also assume that $A_p$ is differentiable with $|A_p'[n]|$ small compared with $\phi_p'[n]$, that the modes are separated with resolution
$\Delta$ and their modulations are bounded by $B_f$, i.e. for each time index $n$,
\begin{equation}
\begin{aligned}
\label{def:sep}
\forall \ 1 \leq p \leq P-1, \ \phi_{p+1}'[n] - \phi_{p}'[n] > 2 \Delta\\
\forall \ 1 \leq p \leq P, \ |\phi_p''[n]| \leq B_f.
\end{aligned}
\end{equation}
\subsection{Classical Ridge Detection}
\label{sec:ridge}
A commonly used RD approach was originally proposed in \cite{Carmona1997},
and then used in \cite{Daubechies2011}. The goal is to compute an integer estimate $\varphi_p[n]$ of $\phi_p'[n] N/L$ by extracting, on the spectrogram, a ridge corresponding to mode $p$. This is actually done
by computing
\begin{align}
\label{eq:Optimization_discrete}
\max_{\bm{\varphi}} \sum_{\tiny \begin{array}{c}1 \leq p \leq P\\ 0\leq n \leq L-1 \end{array}}
| V_{f}^g \left [ n,\varphi_p[n] \right ] |^2- \alpha ( \frac{\Delta^1 \varphi_p[n]L^2}{N})^2 - &\beta ( \frac{\Delta^2 \varphi_p[n]L^3}{N} )^2,
\end{align}
with $\bm{\varphi} = (\varphi_p)_{p=1,\cdots,P}$ where $\varphi_p : \{0,\cdots,L-1\} \mapsto \{0,\cdots,N-1\}$, $\alpha$ and $\beta$ both positive, and in which
$\frac{\Delta^1 \varphi_p[n] L^2}{N} = \frac{(\varphi_p[n+1] - \varphi_p[n])L^2}{N}$
and
$\frac{\Delta^2 \varphi_p[n]L^3}{N} = \frac{(\varphi_p[n+1] - 2\varphi_p[n] + \varphi_p[n-1])L^3}{N}$, are approximations of $\phi''_p[n]$ and $\phi'''_p[n]$.
Taking into account that regularization terms associated with $\alpha$ and $\beta$ are not relevant when a ridge is associated with a local maximum of the STFT magnitude and that to consider such penalization parameters in noisy situations leads to
inaccurate IF estimation \cite{meignen2017demodulation}, one can alternatively
put a bound on the frequency modulation allowed while extracting the ridges, and then replace \eqref{eq:Optimization_discrete} by a \emph{peeling algorithm} where
a first mode is extracted as follows \cite{colominas2020fully}:
\begin{equation}
\label{eq:Optimization_bound}
\max_{\varphi_1} \sum_{n = 0}^{L-1} | V_{f}^g \left [n,\varphi_1[n] \right ] |^2, \quad \text{s.t. } |\Delta^1 \varphi_1[n]| \frac{L^2}{N} \leq B_f.
\end{equation}
Then, after $\varphi_1$ is computed, one defines $V_{f,0}^g := V_{f}^g$, and RD continues replacing $V_{f,0}^g$ by:
\begin{align}
\label{def_x1}
V_{f,1}^g [n,k] :=
\left \{\begin{array}{l}
0 \text{, if } \varphi_1[n] - \lceil \frac{\Delta N}{L} \rceil \leq k \leq \varphi_1[n]+ \lceil \frac{\Delta N}{L} \rceil \\
V_{f,0}^g [n,k] \text{, otherwise,}
\end{array}
\right .
\end{align}
in which $\lceil X \rceil$ is the smallest integer larger than $X$.
This then enables the computation of $\varphi_2$ replacing $V_{f}^g$
by $V_{f,1}^g$ in \eqref{eq:Optimization_bound}, and then the definition of
$V_{ f,2}^g$ from $V_{f,1}^g$ the same way as $V_{f,1}^g$
from $V_{f,0}^g$. Such a procedure is iterated until $P$ ridges $(\varphi_p)_{p=1,\cdots,P}$ are extracted.
In practice, to compute a candidate for $\varphi_1$, one first fixes
an initial time index $n_0$, computes
\begin{equation}
\label{eq:max_magnitude_freq}
k_0 := \mathop{\textrm{argmax}}_{0 \leq k \leq N-1} |V_f^g [n_0,k]|,
\end{equation}
and puts $\varphi_1[n_0] := k_0$. Then, to define $\varphi_1$ on
$\{n_0+1,\cdots,L-1\}$, one uses the following recurring principle
starting from $n = n_0$:
\begin{equation}
\label{eq:mb_interval0}
\varphi_1[n+1] := \mathop{\text{argmax}}_{k} \left \{ |V_f^g[n+1,k]|,
\varphi_1[n] - \lceil \frac{NB_f}{L^2} \rceil \leq k \leq \varphi_1[n] + \lceil \frac{NB_f}{L^2} \rceil
\right \}.
\end{equation}
The same principle is applied on $\{0,\cdots,n_0-1\}$, again starting from $n = n_0$ but replacing $n+1$ by $n-1$ in \eqref{eq:mb_interval0}.
Finally, other initialization time indices than $n_0$ are considered
to define other candidates for $\varphi_1$, the ridge finally kept
being the one among all candidates, maximizing the energy in the TF plane,
i.e. $\sum\limits_n |V_{f}^g [n,\varphi_1[n]]|^2$.
This RD will be called \emph{Simple Ridge Detection} (S-RD) in the sequel.
There are however two strong limitations to S-RD.
The first one is that, in a noisy context, each mode cannot be associated at each time instant with a local maximum of $|V_{\tilde f}^g|$ along the frequency axis.
As a result, S-RD enforces the extraction of $P$ ridges even when
these are not associated with modes.
This is illustrated in Fig. \ref{Fig1}, in which we first display, in Fig. \ref{Fig1} (a), the magnitude of the STFT of a linear chirp along with the three largest local maximum along the frequency axis at each time instant: we see that these maxima are not chained in the TF plane.
To have a clearer view of the effect of noise on the ridge we display in Fig. \ref{Fig1} (b), a zoom on a region containing the expected ridge location and corresponding to the situation where a global maximum along the frequency axis is split into two local maxima. In such a case, it transpires that the noise generates a zero of the STFT close to the location of the IFs of the modes, surrounded by two local maxima. This is the reason why any technique defining the ridge by following the local maxima along the frequency axis will result in poor IF estimation.
Such a paradigm is however the basis to RD technique like S-RD, and one of the goal of the present paper is to circumvent this limitation. Note that this figure also illustrates the dependency of RD techniques
such as S-RD on the initialization time $n_0$.
The second important drawback of S-RD is that the upper bound $B_f$
for the modulation is fixed a priori:
S-RD does not take into account the value of the modulation and even not
its sign. To deal with this issue is also one of the aim of the present paper.
In this regard, we first recall in the following section how to introduce some kind of adaptivity in RD by removing the dependence of S-RD on upper frequency bound $B_f$, as proposed in \cite{colominas2020fully}.
\begin{figure*}[!htb]
\centering
\begin{minipage}{0.5\linewidth}
\begin{tabular}{c}
\includegraphics[width=\textwidth,height = 4.5 cm] {F1_zeros_LC.pdf}\\
\hspace{0.6 cm} (a)
\end{tabular}
\end{minipage}%
\begin{minipage}{0.5\linewidth}
\begin{tabular}{c}
\includegraphics[width=\textwidth,height = 5.2 cm] {F1_bifurcation}\\
\hspace{0.6 cm} (b)
\end{tabular}
\end{minipage}
\caption{(a): Magnitude of the STFT of a noisy linear chirp along with its maxima along the frequency axis;
(b): Zoom on the effect of noise on local maxima; Cyan coefficients are among the two largest coefficients along the frequency axis, red ones correspond to the third largest along the frequency axis.}
\label{Fig1}
\end{figure*}
\section{Fully Adaptive Ridge Detection}
\label{sec:adpative_ridge}
To circumvent the lack of adaptivity in S-RD, a novel approach called \emph{modulation based ridge detection} (MB-RD) was proposed in \cite{colominas2020fully}.
In a nutshell, this approach considers the complex approximation
of the modulation of the mode used in the definition of the
\emph{second order synchrosqueezing transform} and defined as \cite{behera2018theoretical}:
\begin{equation}
\label{def:secondorder}
\tilde q_{f}[n,k]
=
\frac{1}{2i \pi} \frac{V_{f}^{g''}[n,k] V_{f}^g[n,k]- (V_{f}^{g'}[n,k])^2}
{V_{f}^{tg}[n,k] V_{f}^{g'}[n,k] - V_{f} ^{tg'}[n, k] V_{f}^g[n, k]},
\end{equation}
in which $V_{f}^{g'}, V_{f}^{tg}, V_{f}^{g''}, V_{f}^{tg'}$
are respectively STFTs of $f$ computed with windows $n \mapsto g'[n], (tg)[n], g''[n]$ and $(tg')[n]$.
In that context, $\hat q_{f}[n,k] = \Re \left \{ \tilde q_{f}[n,k] \right \}$ consists of an estimate of the modulation of the closest mode to $[n,k]$ in the TF plane. To extract the first ridge MB-RD uses the same recurring principle
as S-RD introduced in Section \ref{sec:ridge}, but, instead of defining
$\varphi_1$ with $B_f$ as upper bound for the modulation, MB-RD uses $\hat q_f$
and a constant $C > 0$, meaning \eqref{eq:mb_interval0} is replaced
by:
\begin{equation}
\label{eq:mb_interval}
\begin{aligned}
\varphi_1[n+1] := \\
\mathop{\text{argmax}}_{k} \left \{ |V_f^g[n+1,k]|,
\varphi_1[n] + \lceil \hat q_f[n,\varphi_1[n]]\frac{N}{L^2} \rfloor - C \leq k \leq
\varphi_1[n] + \lceil \hat q_f[n,\varphi_1[n]]\frac{N}{L^2} \rfloor + C \right \},
\end{aligned}
\end{equation}
in which $\lceil X \rfloor$ is the closest integer to $X$,
the user-defined constant $C$ compensating for potential
estimation errors.
The motivation for \eqref{eq:mb_interval} lies in the fact that $\hat q_f$
is a first order estimate of $\phi_1''$ corresponding to the local orientation
of the ridge.
Though MB-RD improves S-RD in that it is more adaptive,
both techniques are based on the assumption that a
mode generates a significant local maximum along the frequency axis
at each time instant, which is not the case in heavy noise situations. Another limitation is that since the ridges are
extracted one after the other using a \emph{peeling algorithm} \cite{colominas2020fully}, if RD fails for one mode then the detection process will also most probably fail for the next ones.
To deal with this issue, we propose, in the following section, to introduce the concept of \emph{relevant ridge portions} (RRPs) that will be subsequently used to define our new RD technique. It will consist in computing all the ridges simultaneously, thus getting rid of the traditional peeling algorithm.
\section{Definition of Relevant Ridge Portions}
\label{sec:RRPs}
Let us consider, for the sake of simplicity that $g$ is the Gaussian window
$g[n] = e^{-\pi \frac{n^2}{\sigma^2 L^2}}$, for which it can be shown that if
$f$ is a linear chirp with constant amplitude $A$, i.e. $f[n] = A e^{2i \pi \phi[n]}$
with $\phi''$ a constant function, one has \cite{behera2018theoretical}:
\begin{equation}
\label{eq:STFT_linear_chirp}
\begin{aligned}
|V_{f}^g[n,k]| \approx A L \sigma (1+\sigma^4 \phi''[n]^2)^{-\frac{1}{4}}
e^{-\pi \frac{\sigma^2(k\frac{L}{N}-\phi'[n])^2}{1+\sigma^4\phi''[n]^2}},
\end{aligned}
\end{equation}
whose standard deviation is:
\begin{equation}
\label{def:std-LC}
\begin{aligned}
\frac{1}{\sqrt{2 \pi} \sigma} \sqrt{1+ \sigma^4 \phi''[n]^2}
\approx std_{LC}[n,\varphi[n]] := \frac{1}{\sqrt{2 \pi} \sigma} \sqrt{1+ \sigma^4 \hat q_{f}[n,\varphi[n]]^2}.
\end{aligned}
\end{equation}
We then define an interval corresponding to this standard deviation
around $\varphi[n]$, the global maximum of $|V_{f}^g|$ along the frequency axis at time index $n$, namely:
\begin{equation}
\label{def:interval_STD}
I_{LC}[n,\varphi[n]] = \left [ \lfloor \varphi[n] - std_{LC}[n,\varphi[n]]\frac{N}{L}
\rfloor , \lceil \varphi[n] + std_{LC}[n,\varphi[n]]\frac{N}{L} \rceil \right ].
\end{equation}
For a MCS defined as in \eqref{def:MCS},
at each time instant $n$, $|V_f^g|$ admits $P$ local
maxima along the frequency axis, and for each of them, one can
define an interval $I_{LC}$ by considering that each mode can be locally approximated
by a linear chirp. Now, if some noise is added to $f$ to obtain $\tilde f$,
some other local maxima not associated with a mode arise in the TF plane.
As the magnitude of $|V_{\tilde f}^g|$ at a local maximum
is usually larger when it corresponds
to a mode rather than to noise, we strengthen these maxima by first computing at each local maximum $[n,k_0]$, the interval $I_{LC}[n,k_0]$, and then by defining the auxiliary variable:
\begin{equation}
\label{def:S_LC}
S_{LC} [n,k_0] = \sum_{k \in I_{LC}[n,k_0]} |V_{\tilde f}^g[n,k]|^2.
\end{equation}
Using the variable $S_{LC}$, we then construct $P$ ridge
portions starting at time index $n_0$, by considering
$(\varphi_p[n_0])_{p=1,\cdots,P}$ such that $(S_{LC} [n_0,\varphi_p[n_0]])_{p=1,\cdots,P}$ are the $P$ largest
local maxima at time index $n_0$.
From each of these points, we define ridge portions
using a variant of \eqref{eq:mb_interval}, by introducing first
$\psi_p[n_0+1] := \varphi_p[n_0] + \lceil \hat q_{\tilde f}[n_0,\varphi_p[n_0]] \frac{N}{L^2} \rfloor$, and then
\begin{equation}
\label{eq:mb_interval_bis}
\varphi_p[n_0+1] := \mathop{\text{argmin}}_{k}
\left \{
|k-\psi_p[n_0+1]|, \textrm{ s.t. } S_{LC}[n_0+1,k] \textrm{ is a local maximum}
\right \}.
\end{equation}
The differences with \eqref{eq:mb_interval} is that we use
$S_{LC}$ instead of $V_{\tilde f}^g$ and that, for mode $p$,
we look for the closest maximum to $\psi_p[n_0+1]$ along the frequency axis of
$S_{LC}$, thus avoiding the use of an extra parameter $C$.
Then $[n_0+1,\varphi_p[n_0+1]]$ belongs to the ridge starting at $[n_0,\varphi_p[n_0]]$ only if $S_{LC}[n_0+1,\varphi_p[n_0+1]]$ is among the $2P+1$ largest maxima of $S_{LC}$ at time index $n_0+1$, otherwise the ridge construction is stopped. To consider the $2P+1$ largest maxima is motivated by the
fact that when a local maximum corresponding to a ridge is destroyed by the
noise it often gives rise to two maxima as illustrated in Fig. \ref{Fig1} (b), therefore the $2P$. largest maxima not to miss any information. Then, we
add one to take into account the fact that locally
some maxima related to noise may be larger than some others related to signal.
The procedure is then iterated forward and backward (from $n_0$) until the construction of each of the $P$ ridge portions is stopped.
Thus, with such a formulation, each initialization point $n_0$ leads to
$P$ ridge portions, with a priori different lengths. It is important to
mention that, contrary to MB-RD and S-RD,
we do not construct the ridges one by one using a so-called
\emph{peeling algorithm} as in \cite{colominas2020fully,colominas2019time,Daubechies2011,behera2018theoretical}, but alternatively construct
$P$ ridge portions at a time, and do not impose that the ridge portions starting at time $n_0$ last for the whole time span.
Now, investigating the stability of these ridge portions with
respect to the initialization point $n_0$, we consider other initialization points around $n_0$, and compute the associated $P$ ridge portions using the just described procedure. If a ridge portion computed at time index $n_0$ is present in the set
of the $P$ ridge portions for $s$ successive initialization time indices
including $n_0$, the latter consists of a \emph{relevant ridge portion} (RRP)
at scale $s$.
\section{RD Based on RRPs}
\label{sec:newrd}
In this section, we are going to build a new RD technique based
on RRPs. For that purpose, we first introduce a procedure to gather
the RRPs associated with the same mode, and then explain
how to build RD from these gathered RRPs.
\subsection{Gathering RRPs Based on TF Location}
\label{sec:gather_RRPs}
Our motivation to gather RRPs is that though heavy noise splits the ridge associated with a mode in many different RRPs, these remain close to each
other in the TF plane, following \cite{djurovic2004algorithm} in that matter.
Our goal is thus to associate to a mode a set of
RRPs by defining the notion of \emph{intersection} for $RRPs$.
For that purpose, let us denote $({\cal R}_j)_{j \in J}$ the set of RRPs.
For each ${\cal R}_i$, we denote $[n_0,k_0]$ and $[n_1,k_1]$ the beginning and the end of ${\cal R}_i$, then define $I_{PH}[k] =
\left [\lfloor k - \frac{3N}{L \sqrt{2\pi} \sigma}\rfloor,
\lceil k + \frac{3N}{L \sqrt{2\pi} \sigma} \rceil \right ]$ ($PH$ standing for \emph{pure harmonic}). Using these notations,
we define the neighborhood of ${\cal R}_i$, as follows:
\begin{equation}
\label{def:NRi}
\begin{aligned}
{\cal NR}_i =
\left \{ [n,l], n \in [n_0-\Delta t,n_0], l \in I_{PH}[k_0] \right \}
\bigcup
\left \{ [n,l], [n,k] \in {\cal R}_i \textrm{ and }
l \in I_{PH}[k] \right \}\\
\bigcup
\left \{ [n,l], n \in [n_1,n_1 + \Delta t], l \in I_{PH}[k_1] \right \}.
\end{aligned}
\end{equation}
At each location on a RRP, the frequency neighborhood consists of the
interval corresponding to 3 times the standard deviation computed assuming
no modulation (because to take into account the modulation in that context
may lead to instabilities),
and at each end of an RRP we extend this neighborhood using the time parameter
$\Delta t$, which corresponds to the maximal time distance, measured in number
of time bins, between successive RRPs associated with the same mode.
In that context, we say that ${\cal R}_i$ and ${\cal R}_j$ are
intersecting if ${\cal NR}_i \bigcap {\cal NR}_j \neq \emptyset$.
All the intersecting RRPs are then gathered together
to obtain a set $({\cal M}_q)_{q \in Q}$, that is each ${\cal M}_q$ is the union of some RRPs, and we then define
the energy of ${\cal M}_q$ as:
\begin{equation}
\label{def:val_ridge}
R ({\cal M}_q) =
\sum\limits_{[m,l] \in {\mathcal M}_q} S_{LC}[m,l].
\end{equation}
All the points in ${\cal M}_q$ are then given the energy $R ({\cal M}_q)$.
Finally, we only keep in ${\cal M}_q$ only one TF point per time index:
if there are several TF points in ${\cal M}_q$ associated with one
time index we only keep in ${\cal M}_q$ the one associated with the
most energetic RRP (the energy of ${\cal R}_i$ being defined as $R ({\cal R}_i)$).
\subsection{Definition of New RD Technique}
\label{sec:newrd1}
For the sake of simplicity let us now assume that the set
${\cal M} = ({\cal M}_q)_{q \in Q}$ is ranked according to decreasing energies.
Starting from $q=0$, we consider the first index $q_0$ such that $P$ elements in $({\cal M}_q)_{q=0,\cdots,q_0}$ contains some points associated with
the same time index.
Let us denote $({\cal M}_p^0)_{p=1,\cdots,P}$, the corresponding elements of $\cal M$, ranked according to increasing frequencies.
To obtain a first approximation of the $p^{\textrm{th}}$ ridge,
we compute the following polynomial approximation:
\begin{equation}
\label{def:least_square}
D_{p}^0 = \mathop{\textrm{argmin}}_{D}
\sum_{[n,k] \in {\cal M}_{p}^0}
|k - D(\frac{n}{L})|^2 R ({\cal M}_p^0)^2 , \ p = 1,\cdots,P,
\end{equation}
where $D$ is some polynomial of degree $d$. We then associate with each
polynomial $D_p^0$ a TF region defined as:
\begin{equation}
\label{def:domain}
\begin{aligned}
{\cal TF}_{D_p^0} = \left \{ [n,k], \ 0 \leq n \leq L-1, \ k \in I_{PH}(D_p^0(\frac{n}{L})),
\right \}
\end{aligned}
\end{equation}
in which $I_{PH}(D_p^0(\frac{n}{L}))$ corresponds to the definition given in the previous section except we use round brackets instead of square brackets since $D_p^0(\frac{n}{L})$ is not located on the frequency grid.
If the TF regions $({\cal TF}_{D_p^0})_{p=1,\cdots,P}$
intersect some elements in ${\cal M}$ other than $({\cal M}_p^0)_{p=1,\cdots,P}$ then the corresponding RRPs are added, with the corresponding energy, in minimization \eqref{def:least_square}, and
updated approximation polynomials are computed. Such a procedure is iterated until no new elements in ${\cal M}$ are
intersected by the updated $({\cal TF}_{D_p^0})_{p=1,\cdots,P}$ (
for the sake of simplicity, we still denote by $(D_p^0)_{p=1,\cdots,P}$
the set of polynomials obtained after these iterations).
Finally, we define the energy corresponding to these polynomial approximations as follows:
\begin{equation}
\label{def:energies}
E^0 :=
\sum_{p=1}^P \sum_{n=0}^{L-1} S_{LC}(n,D^0_(\frac{n}{L})),
\end{equation}
in which the round brackets are used to mean $D^0_p(\frac{n}{L})$ is not necessarily on the frequency grid.
The set $(D_p^0)_{p=1,\cdots,P}$ consists of a first approximation for the $P$ ridges. Then, we consider the next index $q_1$ larger than $q_0$ such that there exists
a subset $({\cal M}_p^1)_{p=1,\cdots,P}$ of
$({\cal M}_q)_{q=0,\cdots,q_1}$ having one time index in common and different from
$({\cal M}_p^0)_{p=1,\cdots,P}$. Solving the updated optimization problem for each $p$:
\begin{equation}
\label{def:least_square0}
D_{p}^1 = \mathop{\textrm{argmin}}_{D}
\left ( R ({\cal M}_{p}^0) \right )^2 \sum_{[n,k] \in {\cal M}_{p}^0}
|k - D(\frac{n}{L})|^2
+ \left ( R ({\cal M}_{p}^1 ) \right )^2 \sum_{[n,k] \in {\cal M}_{p}^1}
|k - D(\frac{n}{L})|^2,
\end{equation}
and then iterating as explained before, we get an updated set of approximation polynomials which we call $(D_p^1)_{p=1,\cdots,P}$,
to which we associate the energy $E^1$ following \eqref{def:energies}.
If $E^1 > E^0$, then the approximation polynomials become
the set $(D_p^1)_{p=1,\cdots,P}$, otherwise one keeps the original set $(D_p^0)_{p=1,\cdots,P}$.
Such a procedure is iterated until all the elements in $\cal M$ have been considered, and we end up with a set of polynomials called $(D_p^{fin})_{p=1,\cdots,P}$. We finally rerun this procedure
starting with $(D_p^{fin})_{p=1,\cdots,P}$ as initial
polynomials values and replacing $I_{PH}(D^0(\frac{n}{L}))$
by $\tilde I_{LC}(n,D_p^{fin}(\frac{n}{L}))$, defined by:
\begin{equation}
\label{def:interval_STD0}
\tilde I_{LC}(n,D_p^{fin}(\frac{n}{L})) = \left [ \lfloor D_p^{fin}(\frac{n}{L})) - 3 std_{LC}(n,D_p^{fin}(\frac{n}{L})) \frac{N}{L}
\rfloor , \lceil D_p^{fin}(\frac{n}{L})) + 3 std_{LC}(n,D_p^{fin}(\frac{n}{L}))\frac{N}{L} \rceil \right ],
\end{equation}
with $std_{LC}(n,D_p^{fin}(\frac{n}{L})) := \frac{1}{\sqrt{2 \pi} \sigma} \sqrt{1+ \sigma^4 (D_p^{fin})'(\frac{n}{L})^2}$.
This post-processing step enables to take into account some ridge portions that were wrongly left apart.
Note that if the final set of polynomials obtained at the end the procedure is such that the polynomials are intersecting, we consider instead the set
of non intersecting polynomials obtained with the just described procedure and associated with the largest energy.
For the sake of simplicity, we still denote by $(D_p^{fin})_{p=1,\cdots,P}$
this final set of approximation polynomials.
In the sequel, we denote $[F_p^-[n],F_p^+[n]]:= \tilde I_{LC}(n,D_p^{fin}(\frac{n}{L}))$, and we omit $p$ in the case of a monocomponent signal. Furthermore, the set of RRPs used in the construction of $(D_p^{fin})_{p=1,\cdots,P}$ are denoted by $({\cal M}_p^{fin})_{p=1,\cdots,P}$.
From now on, the proposed RD method is called RRP-RD.
Note that for particular applications, in particular when the
phases of the modes contain fast oscillations, polynomial approximation may be replaced by spline approximation, as illustrated in Section \ref{sec:res_gravit} on a gravitational wave signal.
\begin{figure*}[!htb]
\centering
\begin{minipage}{0.32\linewidth}
\begin{tabular}{c}
\includegraphics[width=\textwidth,height = 4 cm] {F2_TFR_LC}\\
\hspace{0.6 cm} (a)
\end{tabular}
\end{minipage}
\begin{minipage}{0.32\linewidth}
\begin{tabular}{c}
\includegraphics[width=\textwidth,height = 4 cm] {F2_TFR_cos}\\
\hspace{0.6 cm} (b)
\end{tabular}
\end{minipage}
\begin{minipage}{0.32\linewidth}
\begin{tabular}{c}
\includegraphics[width=\textwidth,height = 4 cm] {F2_TFR_exp}\\
\hspace{0.6 cm} (c)
\end{tabular}
\end{minipage}
\centering
\begin{minipage}{0.32\linewidth}
\begin{tabular}{c}
\includegraphics[width=\textwidth,height = 4 cm] {F2_RD_LC}\\
\hspace{0.6 cm} (d)
\end{tabular}
\end{minipage}%
\begin{minipage}{0.32\linewidth}
\begin{tabular}{c}
\includegraphics[width=\textwidth,height = 4 cm] {F2_RD_cos}\\
\hspace{0.6 cm} (e)
\end{tabular}
\end{minipage}
\begin{minipage}{0.32\linewidth}
\begin{tabular}{c}
\includegraphics[width=\textwidth,height = 4 cm] {F2_RD_exp}\\
\hspace{0.6 cm} (f)
\end{tabular}
\end{minipage}
\caption{(a): STFT of a noisy linear chirp (SNR = -10 dB);
(b): STFT of a noisy mode with cosine phase (SNR = -10 dB);
(c): STFT of a noisy mode with exponential phase (SNR = -10 dB);
(d): ridge $D^{fin}$ detected for the signal displayed in (a), with the set
of RRPs ${\cal M}^{fin}$ used in the computation of the ridge $D^{fin}$,
along with $F^-[n]$ and $F^+[n]$ for each $n$; (e): same as (d) but for the signal
whose STFT is displayed in (b);
(f): same as (d) but for the signal whose STFT is displayed in (c)}
\label{Fig2}
\end{figure*}
We display in Fig. \ref{Fig2} an illustration of
the proposed RD on a noisy linear chirp, a noisy mode with
cosine phase, and a noisy mode with exponential phase (displayed on the first
row of that figure). In each case, the noise is a complex Gaussian noise and the noise level corresponds to an input SNR of $-10$ dB
(the input SNR is defined as
$SNR(f,\tilde f) = 20 \log_{10} \left (\frac{\|f \|_2}{\|\tilde f -f \|_2} \right )$).
On the second row of Fig. \ref{Fig2}, we display $D^{fin}$
along with the RRPs used for its construction, namely ${\cal M}^{fin}$,
and also the interval $[F^-[n],F^+[n]]$ for each $n$.
These first illustrations show that the proposed procedure seems to be efficient
in very noisy situations ;
this will be further quantified in Section
\ref{sec:res}.
We also illustrate the procedure on the two mode signals of
Fig. \ref{Fig3}, which consist either of two linear chirps, two modes with cosine phase, and
a last signal made of a linear chirp plus an exponential chirp. The input SNR is still fixed
to $-10$ dB. We again notice that RRP-RD seems to be well adapted to deal with MCS in the presence of heavy noise, regardless of the modulation of the modes.
\begin{figure*}[!htb]
\centering
\begin{minipage}{0.32\linewidth}
\begin{tabular}{c}
\includegraphics[width=\textwidth,height = 4 cm] {F3_TFR_LC}\\
\hspace{0.6 cm} (a)
\end{tabular}
\end{minipage}
\begin{minipage}{0.32\linewidth}
\begin{tabular}{c}
\includegraphics[width=\textwidth,height = 4 cm] {F3_TFR_cos_lim}\\
\hspace{0.6 cm} (b)
\end{tabular}
\end{minipage}
\begin{minipage}{0.32\linewidth}
\begin{tabular}{c}
\includegraphics[width=\textwidth,height = 4 cm] {F3_TFR_exp_LC}\\
\hspace{0.6 cm} (c)
\end{tabular}
\end{minipage}
\centering
\begin{minipage}{0.32\linewidth}
\begin{tabular}{c}
\includegraphics[width=\textwidth,height = 4 cm] {F3_RD_LC}\\
\hspace{0.6 cm} (d)
\end{tabular}
\end{minipage}%
\begin{minipage}{0.32\linewidth}
\begin{tabular}{c}
\includegraphics[width=\textwidth,height = 4 cm] {F3_RD_cos_lim}\\
\hspace{0.6 cm} (e)
\end{tabular}
\end{minipage}
\begin{minipage}{0.32\linewidth}
\begin{tabular}{c}
\includegraphics[width=\textwidth,height = 4 cm] {F3_RD_exp_LC}\\
\hspace{0.6 cm} (f)
\end{tabular}
\end{minipage}
\caption{(a): STFT of two noisy linear chirps (SNR = -10 dB);
(b): STFT of two noisy modes with cosine phases, with different modulation (SNR = -10 dB);
(c): STFT of a signal made of a linear chirp and a mode with exponential phase (SNR = -10 dB);
(d): ridges $(D_p^{fin})_{p=1,2}$ detected for the signal displayed in (a),
along with $({\cal M}_p^{fin})_{p=1,2}$. We also display the interval $[F_p^-[n],F_p^+[n]]$, for each $n$ and $p$;
(e): same as (d) but for the signal whose STFT is displayed in (b);
(f): same as (d) but for the signal whose STFT is displayed in (c).}
\label{Fig3}
\end{figure*}
\subsection{Mode Reconstruction}
\label{sec:mode_rec}
Having defined RRP-RD, we explain how we proceed with mode reconstruction. A simple strategy consists of summing the coefficients in
$[F_p^-[n],F_p^+[n]]$ for each $n$, namely
\begin{equation}
\label{def:inv_STFT_loc}
f_p [n] \approx \frac{1}{g[0]N} \sum_{k \in [F_p^-[n],F_p^-[n]]} V_{\tilde{f}}^{g}[n,k].
\end{equation}
To take into account the fact that the intervals $[F_p^-[n],F_p^+[n]]$ and $[F_{p+1}^-[n],F_{p+1}^+[n]]$ may intersect for some time
index, in such instances these intervals are
replaced by $[F_p^-[n],\frac{F_{p}^+[n]+F_{p+1}^-[n]}{2}]$
and $[\frac{F_{p}^+[n]+F_{p+1}^-[n]}{2},F_{p+1}^+[n]]$ respectively.
This reconstruction procedure is denoted by RRP-MR in the sequel.
An alternative technique for mode reconstruction was
recently proposed in \cite{laurent2020novel}, and is
based on a linear chirp approximation for the mode.
In our context, the technique proposed
in \cite{laurent2020novel} would consider $k_0 := \lfloor D^{fin}_p[n] \frac{N}{L} \rceil$, and then the following approximation for the STFT
of $f_p$ (see \cite{laurent2020novel} for details):
\begin{equation}
\label{eq:approx_Vfp}
V_{f_p}^g [n,k]
\approx V_{\tilde f}^g [n,k_0] e^{\frac{\pi\sigma^2(1 + i (D^{fin}_p)'(\frac{n}{L}) \sigma^2)}{1 + (D^{fin}_p)'(\frac{n}{L})^2 \sigma^4} \left [ \frac{L(k_0-k)}{N} ( \frac{L(k_0+k)}{N} - 2 D^{fin}_p (\frac{n}{L})) \right ]}.
\end{equation}
If one denotes $\tilde V_{f_p}^g$ the estimation of $V_{f_p}^g$ given by \eqref{eq:approx_Vfp}, the retrieval of $f_p$ can be carried out through:
\begin{equation}
\label{eq:retrieve-Mode}
f_p[n] \approx \frac{1}{g(0)N} \sum_{k=0}^{N-1} \tilde V_{f_p}^g [n,k].
\end{equation}
This technique applied to RRP-RD will be denoted by RRP-MR-LCR (LCR standing for \emph{linear chirp reconstruction}).
To compare RRP-MR to reconstruction based on S-RD and MB-RD,
it is natural to consider for each time $n$ the interval:
\begin{equation}
\label{def:interval_STD1}
\begin{aligned}
\bar I_{LC}[n,\varphi_p[n]] &=
\left [ \lfloor \varphi_p [n] - 3std_{LC}[n,\varphi_p[n] \frac{N}{L} \rfloor , \lceil \varphi_p[n] + 3std_{LC}[n,\varphi_p [n]] \frac{N}{L} \rceil \right ]\ \\
& := [\overline{F}_p^-[n],\overline{F}_p^+[n]],
\end{aligned}
\end{equation}
which is the same as \eqref{def:interval_STD0}, except
that $D_p^{fin}$ is replaced by $\varphi_p$, and that the modulation is estimated by $\hat q_{\tilde f}[n,\varphi_p[n]]$
instead of $(D_p^{fin})'$.
Then the reconstruction of mode $p$ is carried out by means of the formula:
\begin{equation}
\label{def:inv_STFT_loc1}
f_p [n] \approx \frac{1}{g[0]N} \sum_{k \in [\overline{F}_p^-[n],\overline{F}_p^+[n]]}
V_{\tilde{f}}^{g}[n,k].
\end{equation}
For the sake of a fair comparison with RRP-MR, if $[\overline{F}_p^-[n],\overline{F}_p^+[n]]$ intersects $[\overline{F}_{p+1}^-[n],\overline{F}_{p+1}^+[n]]$ these are replaced by $[\overline{F}_p^-[n],\frac{\overline{F}_p^+[n]+
\overline{F}_{p+1}^-[n]}{2}]$ and $[\frac{\overline{F}_p^+[n]+\overline{F}_{p+1}^-[n]}{2},\overline{F}_{p+1}^+[n]]$ respectively.
This type of reconstruction technique used
with S-RD or MB-RD are called S-MR and MB-MR respectively.
\section{Numerical Applications}
\label{sec:res}
In this section, we first investigate the quality of RRP-RD compared with S-RD and MB-RD on multicomponent
simulated signals, in Section \ref{sec:res_ridge}.
Then, we assess the quality of the just described
mode reconstruction techniques in Section \ref{sec:res_rec}, on the same
simulated signals. We finally investigate the behavior
of RRP-RD on a gravitational-wave signal in Section \ref{sec:res_gravit} (all the Matlab programs enabling the figures reproduction is available at
https://github.com/Nils-Laurent/RRP-RD).
Note that to compute the STFT in all cases, we use a Gaussian window such that its standard deviation corresponds to
the minimal R\'enyi entropy \cite{baraniuk2001measuring},
which is proved to be a good trade-off minimizing interference between the modes in the TF plane \cite{meignen2020use}.
We are aware of recent works on adaptive window
determination as developed in \cite{li2020adaptive,li2019adaptive}, but though to choose the window adaptively may ease ridge determination, adaptation require the knowledge of a rough version of the ridges, which is very critical in noisy situations.
\subsection{Comparison on RRP-RD, S-RD and MB-RD on Simulated Signals}
\label{sec:res_ridge}
Our goal in this section is to show that RRP-RD is much more relevant in noisy situations than S-RD and MB-RD. For that purpose we compute RD results for the signals
of Fig. \ref{Fig3}, when the input SNR varies
between -10 dB and 4 dB. We choose to consider only low input
SNRs because at higher SNRs the modes are associated with a
local maximum at each time instant, and the ridge detection is less challenging.
When the signal is made of two linear chirps as in Fig. \ref{Fig3} (a),
the detection results depicted in Fig. \ref{Fig4} (a) tell
us that RRP-RD performs much better than S-RD and MB-RD. Note that the approximation polynomials used in RRP-RD are both with degree 5 (to reduce their orders does not significantly change the results).
For that example $L=4096$ and $\Delta t$ is set to $20$, meaning the maximal time between two ridge portions associated with the same mode is $\frac{20}{4096}$ seconds and $s$ defining the RRPs is set to $8$. Comparing the results associated with S-RD and MB-RD,
we notice that the former behaves better than the
latter at low noise level and similarly at a higher noise level.
Such a behavior is related to the fact
that the modulation operator $\hat q_{\tilde f}$ is not
accurate at locations where the ridge is split, and
MB-RD fails to follow the different ridge
portions corresponding to a mode (in these simulations,
$C$ is set to $2$). On the contrary, since S-RD
uses the fixed modulation parameter $B_f$ (here set to $10$), it is able to follow ridge portions that are not necessarily connected which explains why it works better that MB-RD at high noise level.
On that example, for positive input SNRs, since S-RD and MB-RD lead to the same results, local maxima along the frequency axis of the spectrogram associated with each mode exist at each time instant: the gain in output SNR brought by RRP-RD arises from polynomial approximation.
For negative input SNRs, such maxima no longer exist for each
time instant and the gain brought by RRP-RD is related to
the relevance of the grouping of RRPs.
\begin{figure*}[!htb]
\centering
\begin{minipage}{0.32\linewidth}
\begin{tabular}{c}
\includegraphics[width=\textwidth,height = 4 cm] {F4_RD_LC}\\
\hspace{0.6 cm} (a)
\end{tabular}
\end{minipage}
\begin{minipage}{0.32\linewidth}
\begin{tabular}{c}
\includegraphics[width=\textwidth,height = 4 cm] {F4_RD_cos}\\
\hspace{0.6 cm} (b)
\end{tabular}
\end{minipage}
\begin{minipage}{0.32\linewidth}
\begin{tabular}{c}
\includegraphics[width=\textwidth,height = 4 cm] {F4_RD_exp}\\
\hspace{0.6 cm} (c)
\end{tabular}
\end{minipage}
\caption{(a): Comparison between S-RD, MB-RD and RRP-RD, for
the signal of Fig. \ref{Fig3} (a): for each mode $p = 1, 2$,
computation of output SNR between IF $\phi_p'$ and estimated IF with respect to input SNR (the results are averaged over 30 realizations of the noise); (b): same as (a) but for the signal of Fig. \ref{Fig3} (b); (c): same as (a) but for the signal of Fig. \ref{Fig3} (c)}
\label{Fig4}
\end{figure*}
Now, analyzing RD results for the signal of Fig. \ref{Fig3} (b),
we remark that the conclusions for mode $f_2$ are similar
to those for linear chirps. Indeed, at low input SNRs
RRP-RD performs much better than the other two techniques since it better copes with the absence of a local maximum along the frequency axis close to the IF locations of the modes.
On the contrary, when such maxima exist, namely for input SNRs such that
S-RD and MB-RD coincide, the gain in terms of output SNR
with RRP-RD is less important than in the case of linear
chirps since to approximate a cosine phase with a polynomial
of degree 5 leads to larger errors.
As for mode $f_1$, which is much more modulated that
$f_2$, we see that at a high noise level, RRP-RD is still much better than the other two techniques, but when the noise level decreases, S-RD behaves better than RRP-RD due to inaccuracy in IF approximation using a polynomial of degree 5.
In such a case, since the modulation
operator $\hat q_{\tilde f}$ is more sensitive to
noise when the mode is more modulated, MB-RD behaves worse
than the other two techniques.
Finally, regarding the signal of Fig. \ref{Fig3} (c),
we only comment on the results related to the mode with exponential phase, for which we again notice that RRP-RD behaves much better than the other two tested methods at low input SNR, meaning the grouping of RRPs is still performing well in that case.
When the noise level decreases, namely when $f_2$ generates a
local maximum along the frequency axis at each time
instant, RRP-RD and S-RD behaves similarly (an exponential phase can be accurately approximated by a polynomial with degree 5).
On the contrary, the modulation operator $\hat q_{\tilde f}$ is not accurate enough to follow the local maxima along the frequency axis of the spectrogram of $f_2$.
\begin{figure*}[!htb]
\centering
\begin{minipage}{0.32\linewidth}
\begin{tabular}{c}
\includegraphics[width=\textwidth,height = 4 cm] {F5_MR_LC}\\
\hspace{0.6 cm} (a)
\end{tabular}
\end{minipage}
\begin{minipage}{0.32\linewidth}
\begin{tabular}{c}
\includegraphics[width=\textwidth,height = 4 cm] {F5_MR_cos}\\
\hspace{0.6 cm} (b)
\end{tabular}
\end{minipage}
\begin{minipage}{0.32\linewidth}
\begin{tabular}{c}
\includegraphics[width=\textwidth,height = 4 cm] {F5_MR_exp}\\
\hspace{0.6 cm} (c)
\end{tabular}
\end{minipage}
\caption{(a): For each mode $p = 1, 2$, output SNR between mode
$f_p$ of signal of Fig. \ref{Fig3} (a) and reconstructed mode for
each methods, namely S-MR, MB-MR, RRP-MR, and RRP-MR-LCR (the results
are averaged over 30 noise realizations);
(b): same but with signal of Fig. \ref{Fig3} (b);
(c): same but with signal of Fig. \ref{Fig3} (c);}
\label{Fig5}
\end{figure*}
\subsection{Comparison of the Mode Reconstruction Techniques}
\label{sec:res_rec}
In this section, we investigate the quality of mode reconstrutcion techniques S-MR, MB-MR, RRP-MR, and RRP-MR-LCR still for the
signals of Fig. \ref{Fig3}.
Looking at the results of Fig. \ref{Fig5} (a) related to the signal
of Fig. \ref{Fig3} (a), it transpires that while the RRP-RD is much
better than S-RD and MB-RD at high noise levels this is not reflected
in mode reconstruction. This is due to the fact that too
much noise is included in intervals $[F_p^-[n],F_p^+[n]]$ and $[\overline{F}_p^-[n],\overline{F}_p^+[n]]$ for each $n$, and
consequently the coefficients used for reconstrution are not relevant.
Alternatively, when one considers RRP-MR-LCR, the results are significantly improved, meaning that at low SNR one had
rather use the information on the ridge to reconstruct rather than summing the coefficients in the vicinity of the ridge.
Such a conclusion remains true for the mode $f_2$ of the signal
of Fig. \ref{Fig3} (b), and also for the mode $f_1$ of that signal, but only at low input SNRs. Indeed, at high input SNRs, RRP-MR-LCR is hampered
by inaccuracy in phase approximation with a polynomial of degree 5,
which entails larger errors in IF and CR estimations in
the linear chirp approximation involved in this technique.
Finally, for the signal of Fig. \ref{Fig3} (c), the reconstruction
results are always much better with RRP-MR-LCR than with the
other tested techniques (the ridge approximation provided by RRP-RD for the exponential chirp being of good quality).
To conclude on that part, we should say that to reconstruct the modes in very noisy scenarios, one had rather use the information on the ridge to construct
a linear chirp approximation than sum the coefficients in the TF plane.
\subsection{Application to Gravitational-Wave Signals}
\label{sec:res_gravit}
In this section, we investigate the applicability of RRP-RD and RRP-MR-LCR to a transient gravitational-wave signal,
generated by the coalescence of two stellar-mass black holes.
This event, called \textbf{GW150914}, was detected by the LIGO detector Hanford, Washington and closely matches the waveform Albert Einstein predicted almost 100 years ago in his general relativity theory for the inspiral, the merger of a pair of black holes and the ringdown
of the resulting single black hole \cite{Abbott2016}.
The observed signal has a length of 3441 samples in $T = 0.21$ seconds.
\begin{figure*}[!htb]
\begin{minipage}{0.24\linewidth}
\begin{tabular}{c}
\includegraphics[width=\textwidth,height = 4 cm] {F_GW_STFT}\\
\hspace{0.6 cm} (a)
\end{tabular}
\end{minipage}
\begin{minipage}{0.24\linewidth}
\begin{tabular}{c}
\includegraphics[width=\textwidth,height = 4 cm] {F_GW_RD}\\
\hspace{0.6 cm} (b)
\end{tabular}
\end{minipage}
\begin{minipage}{0.24\linewidth}
\begin{tabular}{c}
\includegraphics[width=\textwidth,height = 4 cm] {F_GW_LCR_STFT}\\
\hspace{0.6 cm} (c)
\end{tabular}
\end{minipage}
\begin{minipage}{0.24\linewidth}
\begin{tabular}{c}
\includegraphics[width=\textwidth,height = 4 cm] {F_GW_NR}\\
\hspace{0.6 cm} (d)
\end{tabular}
\end{minipage}
\caption{(a): STFT of the Hanford signal; (b): Spline associated with RRP-RD (tol = 3) along with the ridge portion ${\cal M}^{fin}$; (c): Denoised STFT used by RRP-MR-LCR technique; (d):
Signal reconstructed using RRP-MR-LCR along with the one predicted by numerical relativity.}
\label{Fig6}
\end{figure*}
We first display in Fig. \ref{Fig6} (a), the STFT of such a signal.
Since the latter is first very slightly modulated during the inspiral
phase, then behaves like an exponential chirp during the merging and
as a fast decreasing phase during the ringdown, to try to approximate
the instantaneous frequency with a polynomial in RRP-RD is not appropriate. Therefore, keeping the same formalism we use spline approximation instead.
This means that, after the merging step, we consider the cubic spline
$s^0$
\begin{equation}
\label{def:spline}
s^0 = \mathop{\textrm{argmin}}_{s \in S_{{\cal M}^0}}
\int_0^T (s''(t))^2 dt,
\end{equation}
in which $S_{{\cal M}^0}$ is the spline space defined at the knots
${\cal M}^0$ (here we omit the subscript $p$ because we look for a single mode), under the constraints:
\begin{equation}
\label{def:spline_constraints}
\sum_{[n,k] \in {\cal M}^0} |k - s(\frac{n}{L})|^2 R({\cal M}^0)^2 \leq D,
\end{equation}
where $D$ is a tolerance parameter that can easily related to
a number of frequency bins:
$tol = \frac{1}{R({\cal M}^0)} \sqrt{\frac {D}{\#{\cal M}^0}}$, where $\#{\cal M}^0$ denotes the cardinal of ${\cal M}^0$, should be of the order of a few frequency bins.
Then, the same type of approach as in the polynomial approximation is carried out, integrating new ridge portions in the minimization process,
to obtain in the end the spline $s^{fin}$ corresponding to
the set of ridge portions ${\cal M}^{fin}$. The spline obtained using RRP-RD
along with the set ${\cal M}^{fin}$ corresponding to the STFT of Fig.
\ref{Fig6} (a) is displayed in Fig. \ref{Fig6} (b). Then, at each point on the spline, one can
define a denoised STFT based on the linear chirp approximation
following \eqref{eq:approx_Vfp} in which $D_p^{fin}$ is replaced
by $s^{fin}$. Such a STFT is displayed in Fig. \ref{Fig6} (c).
Finally, the signal reconstructed with RRP-MR-LCR is
displayed in Fig. \ref{Fig6} (d) along with the signal
given by the numerical relativity \cite{abbott2016gw151226}, showing a great similarity between the two signals.
In this respect, to investigate the quality of mode reconstruction, we compute the SNR between the signals obtained with the different reconstruction techniques and that given by the numerical relativity. The results are displayed in Table \ref{Table1}, showing that RRP-MR-LCR behaves slightly better than
RRP-MR, and that each method is only slightly sensitive to the parameter
tol.
\begin{table}[ht]
\centering
\begin{tabular}{|l|c|c|c|}
\hline
$tol$ & 1 & 2 & 3\\\hline
\textbf{RRP-MR} & 7.6097 & 7.6403 & 7.6669\\\hline
\textbf{RRP-MR-LCR} & 8.4620 & 8.6645 & 8.7498\\\hline
\end{tabular}
\caption{SNR between the signals obtained with RRP-MR or RRP-MCR-LR and the signal given by the numerical relativity}
\label{Table1}
\end{table}
The results obtained here are interesting in that they reflect that RRP-RD
enables the detection of the ringdown. Similar results were
obtained using higher order syncrosqueezed STFT \cite{pham2017high} or
second order synchrosqueezed continuous wavelet transform
\cite{meignen2019synchrosqueezing}, but we here show that if one
uses a robust ridge detector as RRP-RD, it is not necessary to reassign
the transform to detect the ringdown.
\section{Conclusion}
In this paper, we have introduced a novel technique to detect the ridges made by the modes of a multicomponent signal in the time-frequency plane.
Our focus was to design the technique in such a way that it enables the computation of the ridges in very noisy situations. For that purpose we have brought about the notion of relevant ridge portions, which we subsequently used in our ridge detector. The proposed technique show significant improvements over state-of-the-art methods based
on time-frequency representations on simulated signals, and is also
interesting to analyze gravitational-wave signals. Some remaining limitations of the present work that it cannot deal with crossing modes and requires
that the number of modes is fixed for the whole signal duration. In a near a future, we will investigate how to adapt our algorithm to such situations.
\bibliographystyle{IEEEtran}
| {
"redpajama_set_name": "RedPajamaArXiv"
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{"url":"https:\/\/proofwiki.org\/wiki\/Definition:Independent_Subgroups\/Definition_2","text":"# Definition:Independent Subgroups\/Definition 2\n\n## Definition\n\nLet $G$ be a group whose identity is $e$.\n\nLet $\\sequence {H_n}$ be a sequence of subgroups of $G$.\n\nThe subgroups $H_1, H_2, \\ldots, H_n$ are independent if and only if:\n\n$\\displaystyle \\forall k \\in \\set {2, 3, \\ldots, n}: \\paren {\\prod_{j \\mathop = 1}^{k - 1} H_j} \\cap H_k = \\set e$\n\nThat is, the product of any elements from different $H_k$ instances forms the identity if and only if all of those elements are the identity.","date":"2021-04-11 19:14:25","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\": 0.9790533781051636, \"perplexity\": 218.88942876107348}, \"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-2021-17\/segments\/1618038064898.14\/warc\/CC-MAIN-20210411174053-20210411204053-00387.warc.gz\"}"} | null | null |
{"url":"https:\/\/hindimaintutorial.in\/double-sort-solution-codeforces\/","text":"# Double Sort solution codeforces\n\n## Double Sort solution codeforces\n\nYou are given two arrays\u00a0aa\u00a0and\u00a0bb, both consisting of\u00a0nn\u00a0integers.\n\nIn one move, you can choose two indices\u00a0ii\u00a0and\u00a0jj\u00a0(1i,jn1\u2264i,j\u2264niji\u2260j) and swap\u00a0aiai\u00a0with\u00a0ajaj\u00a0and\u00a0bibi\u00a0with\u00a0bjbj. You have to perform the swap in both arrays.\n\nYou are allowed to perform at most\u00a0104104\u00a0moves (possibly, zero). Can you make both arrays sorted in a non-decreasing order at the end? If you can, print any sequence of moves that makes both arrays sorted.\n\nInput\n\nThe first line contains a single integer\u00a0tt\u00a0(1t1001\u2264t\u2264100)\u00a0\u2014 the number of testcases.\n\nThe first line of each testcase contains a single integer\u00a0nn\u00a0(2n1002\u2264n\u2264100)\u00a0\u2014 the number of elements in both arrays.\n\nThe second line contains\u00a0nn\u00a0integers\u00a0a1,a2,,ana1,a2,\u2026,an\u00a0(1ain1\u2264ai\u2264n)\u00a0\u2014 the first array.\n\nThe third line contains\u00a0nn\u00a0integers\u00a0b1,b2,,bnb1,b2,\u2026,bn\u00a0(1bin1\u2264bi\u2264n)\u00a0\u2014 the second array.\n\nOutput\n\nFor each testcase, print the answer. If it\u2019s impossible to make both arrays sorted in a non-decreasing order in at most\u00a0104104\u00a0moves, print\u00a0-1. Otherwise, first, print the number of moves\u00a0kk\u00a0(0k104)(0\u2264k\u2264104). Then print\u00a0ii\u00a0and\u00a0jj\u00a0for each move\u00a0(1i,jn(1\u2264i,j\u2264nij)i\u2260j).\n\nIf there are multiple answers, then print any of them. You don\u2019t have to minimize the number of moves.\n\nExample\ninput\n\nCopy\n3\n2\n1 2\n1 2\n2\n2 1\n1 2\n4\n2 3 1 2\n2 3 2 3\n\noutput\n\nCopy\n0\n-1\n3\n3 1\n3 2\n4 3","date":"2022-06-27 21:11:59","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\": 0.40648406744003296, \"perplexity\": 457.1096874164328}, \"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-2022-27\/segments\/1656103341778.23\/warc\/CC-MAIN-20220627195131-20220627225131-00174.warc.gz\"}"} | null | null |
This is a list of Fellows of the Royal Society elected in 1993.
^ "Fellowship of the Royal Society 1660-2015". London: Royal Society. Archived from the original on 2015-07-15.
^ Huppert, H. E.; Peake, N. (2001). "David George Crighton. 15 November 1942 - 12 April 2000: Elected F.R.S. 1993". Biographical Memoirs of Fellows of the Royal Society. 47: 105. doi:10.1098/rsbm.2001.0007.
^ Stoneham, M.; Buckley-Golder, I. (2010). "Geoffrey Dearnaley. 22 June 1930 -- 5 May 2009". Biographical Memoirs of Fellows of the Royal Society. 56: 41. doi:10.1098/rsbm.2010.0001.
^ EVANS, Sir Martin (John). ukwhoswho.com. Who's Who. 2015 (online Oxford University Press ed.). A & C Black, an imprint of Bloomsbury Publishing plc.
^ Dyson, J.E.; Lynden Bell, D. (1999). "Franz Daniel Kahn. 13 May 1926 -- 8 February 1998: Elected F.R.S. 1993". Biographical Memoirs of Fellows of the Royal Society. 45: 255. doi:10.1098/rsbm.1999.0017.
^ "Emeritus Professor Trevor Lamb FRS". London: Royal Society. Archived from the original on 2015-11-17.
^ Fortey, R.A. (1999). "Colin Patterson. 13 October 1933--9 March 1998: Elected F.R.S. 1993". Biographical Memoirs of Fellows of the Royal Society. 45: 365. doi:10.1098/rsbm.1999.0025.
^ Anon (2014). "Colin Trevor Pillinger". Physics Today. doi:10.1063/PT.5.6059. ISSN 1945-0699.
^ "Dr Bruce Alberts CBE ForMemRS". London: Royal Society. Archived from the original on 2015-11-17.
^ "Professor Jean-Marie Lehn ForMemRS". London: Royal Society. Archived from the original on 2015-11-17.
^ Crow, J. F. (1997). "Motoo Kimura. 13 November 1924--13 November 1994: Elected For.Mem.R.S. 1993". Biographical Memoirs of Fellows of the Royal Society. 43: 255. doi:10.1098/rsbm.1997.0014.
^ Terzian, Y. (2010). "Edwin Ernest Salpeter. 3 December 1924 -- 26 November 2008". Biographical Memoirs of Fellows of the Royal Society. 56: 391. doi:10.1098/rsbm.2010.0005. | {
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What Seahawks coach Pete Carroll is saying about the Eagles | Early Birds
by EJ Smith, Updated: January 2, 2020
Seahawks coach Pete Carroll congratulates strong safety Quandre Diggs (37) and cornerback Shaquill Griffin (26) after Diggs recovered a fumble against the Eagles at Lincoln Financial Field in November.
Tim Tai / File Photograph
Good morning, Eagles fans. Hopefully your 2020 is off to a great start and your resolutions aren't already abandoned. The Eagles are one day closer to Sunday's wild-card playoff game against the Seattle Seahawks. The team had a walk-through yesterday, and will hit the practice field again this afternoon.
The Eagles will no doubt be watching the two teams' regular-season matchup from earlier this season, but a lot has changed. Just ask Seahawks coach Pete Carroll. More on that later.
If you like what you're reading, tell your friends it's free to sign up here. I want to know what you think, what we should add, and what you want to read, so send me feedback by email or on Twitter @EJSmith94.
— EJ Smith (earlybirds@inquirer.com)
Brian Blanco / AP
Seahawks coach Pete Carroll talks to the media after a game against the Carolina Panthers on Dec. 15.
An outsider's perspective
When addressing his players this week, Seahawks coach Pete Carroll said he makes sure to remind them of what the Eagles achieved two seasons ago.
"They're a championship team," Carroll said during a conference call with reporters on Wednesday. "They've shown that makeup and I'm sure it comes right from the top."
With the Eagles and Seahawks set to face off for the second time this season, both sides are rewatching the previous game trying to gain an edge. The only catch: A lot has changed since the Eagles' 17-9 loss to Seattle on Nov. 24. Greg Ward was getting his first action of the season, Miles Sanders was still developing into a main running back, and Andre Dillard was playing right tackle for the first (and possibly only) time in his career.
What has caught Carroll's attention is how the Eagles have played during the team's four-game winning streak to end the regular season, which featured four victories over four division opponents in order to run the table and win the NFC East.
"They've really rallied well," Carroll said. "They've played like a championship team down the stretch, and to put together the four games to win it and to ice it with a big win over Dallas, it just shows that their leadership from the coaches on down is really strong. I mean, I really admirably watched them hang tough and put together a great finish to the year."
The Eagles may be playing better, but their injury situation hasn't improved since the team's last meeting at the Linc. Brandon Brooks missed the first game, but is now headed for shoulder surgery after being placed on injured reserve. Lane Johnson also missed the game, and will likely be questionable with a high ankle sprain this time around.
Sanders is dealing with an ankle sprain of his own, and missed practice on Wednesday.
"Some of the names are different, but the style of play and the way Coach called the offense and the defense and how they do their stuff doesn't look different," Carroll said. "I don't know what Miles Sanders' deal is, but Boston Scott is so much in the same vein, the same style, type of guys who are really exciting football players, they know how to use their guys, so we're just going to expect them to do as they do."
HEATHER KHALIFA / Staff Photographer
Eagles quarterback Carson Wentz throws for a touchdown to tight end Josh Perkins against the Giants.
What you need to know about the Eagles
Carson Wentz is in unfamiliar ground as the Eagles head into the playoffs, but he doesn't have the luxury of acting like it, writes Les Bowen.
Even though the Eagles' secondary has struggled at times since the team last played the Seahawks, the group has confidence in knowing it fared well against Russell Wilson earlier this year.
Josh McCown is on a playoff team for just the second time in his 16-year career, writes Bowen.
The Eagles brought back Shelton Gibson, a former fifth-round pick of the team, on Wednesday. Domowitch details where the wide receiver has been since being released by the Eagles in August.
From the mailbag
The Eagles selected a defensive end in the fourth round in successive years. Who do you expect to make a greater contribution down the road, Josh Sweat or Shareef Miller? And will either reach the level of Derek Barnett? Or even Brandon Graham? — From Scott B., via email.
Good question, Scott. Thanks for e-mailing in. I think Josh Sweat has shown a good amount of potential in his limited role this season, so I'd go with him. Not to say Shareef Miller won't be a good pro, but we haven't seen enough of him to this point to really know, just like we didn't see much from Sweat last year. This season, Sweat has four sacks. It usually takes mid-to-late round picks a few years to show you what they can be.
Another reason I'm going with Sweat, beyond production, is measurables. Brandon Graham is a guy who has overcome not being the biggest, longest guy at his position and is still productive, but Josh Sweat has those physical tools. He's 6-foot-5, 250 pounds with a 7-foot wingspan. It's hard to teach that! From what I can tell, Sweat has put in a lot of work to get stronger, and has been healthy this season after concerns about his knee caused his draft stock to fall. I could see him becoming a bigger part of the team's defensive end rotation in the next season or two.
Posted: January 2, 2020 - 6:13 AM
EJ Smith | @EJSmith94 | ESmith@inquirer.com
Eagles' most important offseason dates leading into free agency | Early Birds
EJ Smith
Why the Eagles and Roseman passed on Minkah Fitzpatrick and Jalen Ramsey | Early Birds
Eagles' offseason wish list on defense | Early Birds
Eagles' offseason wish list on offense | Early Birds
What Eagles' Brandon Brooks learned about himself this season | Early Birds
Eagles speak out about how the 2019 squad will be remembered | Early Birds
Erin McCarthy | {
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} | 3,205 |
Table of Contents
Title Page
Copyright Page
Dedication
CHAPTER ONE
CHAPTER TWO
CHAPTER THREE
CHAPTER FOUR
CHAPTER FIVE
CHAPTER SIX
CHAPTER SEVEN
CHAPTER EIGHT
CHAPTER NINE
CHAPTER TEN
CHAPTER ELEVEN
CHAPTER TWELVE
CHAPTER THIRTEEN
CHAPTER FOURTEEN
CHAPTER FIFTEEN
CHAPTER SIXTEEN
CHAPTER SEVENTEEN
CHAPTER EIGHTEEN
CHAPTER NINETEEN
CHAPTER TWENTY
Teaser chapter
_**Nora Roberts**_
HOT ICE
SACRED SINS
BRAZEN VIRTUE
SWEET REVENGE
PUBLIC SECRETS
GENUINE LIES
CARNAL INNOCENCE
DIVINE EVIL
HONEST ILLUSIONS
PRIVATE SCANDALS
HIDDEN RICHES
TRUE BETRAYALS
MONTANA SKY
SANCTUARY
HOMEPORT
THE REEF
RIVER'S END
CAROLINA MOON
THE VILLA
MIDNIGHT BAYOU
THREE FATES
BIRTHRIGHT
NORTHERN LIGHTS
BLUE SMOKE
ANGELS FALL
HIGH NOON
TRIBUTE
BLACK HILLS
_**Series**_
_Born in Trilogy_
BORN IN FIRE
BORN IN ICE
BORN IN SHAME
_Dream Trilogy_
DARING TO DREAM
HOLDING THE DREAM
FINDING THE DREAM
_Chesapeake Bay Saga_
SEA SWEPT
RISING TIDES
INNER HARBOR
CHESAPEAKE BLUE
_Gallaghers of Ardmore Trilogy_
JEWELS OF THE SUN
TEARS OF THE MOON
HEART OF THE SEA
_Three Sisters Island Trilogy_
DANCE UPON THE AIR
HEAVEN AND EARTH
FACE THE FIRE
_Key Trilogy_
KEY OF LIGHT
KEY OF KNOWLEDGE
KEY OF VALOR
_In the Garden Trilogy_
BLUE DAHLIA
BLACK ROSE
RED LILY
_Circle Trilogy_
MORRIGAN'S CROSS
DANCE OF THE GODS
VALLEY OF SILENCE
_Sign of Seven Trilogy_
BLOOD BROTHERS
THE HOLLOW
THE PAGAN STONE
_Bride Quartet_
VISION IN WHITE
BED OF ROSES
_**Nora Roberts & J. D. Robb**_
REMEMBER WHEN
_**J. D. Robb**_
NAKED IN DEATH
GLORY IN DEATH
IMMORTAL IN DEATH
RAPTURE IN DEATH
CEREMONY IN DEATH
VENGEANCE IN DEATH
HOLIDAY IN DEATH
CONSPIRACY IN DEATH
LOYALTY IN DEATH
WITNESS IN DEATH
JUDGMENT IN DEATH
BETRAYAL IN DEATH
SEDUCTION IN DEATH
REUNION IN DEATH
PURITY IN DEATH
PORTRAIT IN DEATH
IMITATION IN DEATH
DIVIDED IN DEATH
VISIONS IN DEATH
SURVIVOR IN DEATH
ORIGIN IN DEATH
MEMORY IN DEATH
BORN IN DEATH
INNOCENT IN DEATH
CREATION IN DEATH
STRANGERS IN DEATH
SALVATION IN DEATH
PROMISES IN DEATH
_**Anthologies**_
FROM THE HEART
A LITTLE MAGIC
A LITTLE FATE
MOON SHADOWS
_(with Jill Gregory, Ruth Ryan Langan, and Marianne Willman)_
_The Once Upon Series_
_(with Jill Gregory, Ruth Ryan Langan, and Marianne Willman)_
ONCE UPON A CASTLE
ONCE UPON A STAR
ONCE UPON A DREAM
ONCE UPON A ROSE
ONCE UPON A KISS
ONCE UPON A MIDNIGHT
SILENT NIGHT
_(with Susan Plunkett, Dee Holmes, and Claire Cross)_
OUT OF THIS WORLD
_(with Laurell K. Hamilton, Susan Krinard, and Maggie Shayne)_
BUMP IN THE NIGHT
_(with Mary Blayney, Ruth Ryan Langan, and Mary Kay McComas)_
DEAD OF NIGHT
_(with Mary Blayney, Ruth Ryan Langan, and Mary Kay McComas)_
THREE IN DEATH
SUITE 606
_(with Mary Blayney, Ruth Ryan Langan, and Mary Kay McComas)_
_**Also available . . .**_
THE OFFICIAL NORA ROBERTS COMPANION
_(edited by Denise Little and Laura Hayden)_
**THE BERKLEY PUBLISHING GROUP**
**Published by the Penguin Group**
**Penguin Group (USA) Inc.**
**375 Hudson Street, New York, New York 10014, USA**
Penguin Group (Canada), 90 Eglinton Avenue East, Suite 700, Toronto, Ontario M4P 2Y3, Canada
(a division of Pearson Penguin Canada Inc.)
Penguin Books Ltd., 80 Strand, London WC2R 0RL, England
Penguin Group Ireland, 25 St. Stephen's Green, Dublin 2, Ireland (a division of Penguin Books Ltd.)
Penguin Group (Australia), 250 Camberwell Road, Camberwell, Victoria 3124, Australia
(a division of Pearson Australia Group Pty. Ltd.)
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(a division of Pearson New Zealand Ltd.)
Penguin Books (South Africa) (Pty.) Ltd., 24 Sturdee Avenue, Rosebank, Johannesburg 2196,
South Africa
Penguin Books Ltd., Registered Offices: 80 Strand, London WC2R 0RL, England
This book is an original publication of The Berkley Publishing Group.
This is a work of fiction. Names, characters, places, and incidents either are the product of the author's imagination or are used fictitiously, and any resemblance to actual persons, living or dead, business establishments, events, or locales is entirely coincidental. The publisher does not have any control over and does not assume responsibility for author or third-party websites or their content.
BED OF ROSES
Copyright © 2009 by Nora Roberts.
__
All rights reserved.
No part of this book may be reproduced, scanned, or distributed in any printed or electronic form
without permission. Please do not participate in or encourage piracy of copyrighted materials in
violation of the author's rights. Purchase only authorized editions.
BERKLEY® is a registered trademark of Penguin Group (USA) Inc.
The "B" design is a trademark of Penguin Group (USA) Inc.
PRINTING HISTORY
Berkley trade paperback edition / November 2009
Library of Congress Cataloging-in-Publication Data
Roberts, Nora.
Bed of roses / Nora Roberts.—Berkley trade paperback ed.
p. cm.
eISBN : 978-1-101-14894-5
1. Florists—Fiction. 2. Architects—Fiction. I. Title.
PS3568.O243B43 2009
813'.54—dc22
2009019178
<http://us.penguingroup.com>
_For girlfriends_
And 'tis my faith that every flower
Enjoys the air it breathes.
—WORDSWORTH
Love is like a friendship caught on fire.
—BRUCE LEE
PROLOGUE
_R_ OMANCE, IN EMMALINE'S OPINION, MADE BEING A WOMAN special. Romance made every woman beautiful, and every man a prince. A woman with romance in her life lived as grandly as a queen, because her heart was treasured.
Flowers, candlelight, long walks in the moonlight in a secluded garden . . . just the idea brought on a sigh. _Dancing_ in the moonlight in a secluded garden, now _that_ reached the height of romantic on her scale.
She could imagine it, the scent of summer roses, the music drifting out of the open windows of a ballroom, the way the light turned the edges of everything silver, like in the movies. The way her heart would beat (the way it beat now as she imagined it).
She longed to dance in the moonlight in a secluded garden.
She was eleven.
Because she could see so clearly how it should be— _would_ be—she described the scene, every detail, to her best friends.
When they had sleepovers, they talked and talked for hours about everything, and listened to music or watched movies. They could stay up as long as they wanted, even _all_ night. Though none of them had managed to. Yet.
When they had a sleepover at Parker's, they were allowed to sit or play on the terrace outside her bedroom until midnight if the weather was okay for it. In the spring, her favorite time there, she loved to stand on the bedroom terrace, smell the gardens of the Brown Estate and the green from the grass if the gardener had cut it that day.
Mrs. Grady, the housekeeper, would bring the cookies and milk. Or sometimes cupcakes. And Mrs. Brown would come in now and then to see what they were up to.
But mostly, it was just the four of them.
"When I'm a successful businesswoman living in New York, I won't have time for romance." Laurel, her own sunny blond hair streaked with green from a lime Kool Aid treatment, worked her fashion sense on Mackensie's bright red.
"But you _have_ to have romance," Emma insisted.
"Uh-uh." With her tongue caught in her teeth, Laurel tirelessly twined another section of Mac's hair into a long, thin braid. "I'm going to be like my aunt Jennifer. She tells my mother how she doesn't have time for marriage, and she doesn't need a man to be complete and stuff. She lives on the Upper East Side and goes to parties with Madonna. My dad says she's a ballbuster. So I'm going to be a ballbuster and go to parties with Madonna."
"As if." Mac snorted. The quick tug on the braid only made Mac giggle. "Dancing's fun, and I guess romance is okay as long as it doesn't make you stupid. Romance is all my mother thinks about. Except money. I guess it's both. It's like, how can she get romance and money at the same time."
"That's not really romance." But Emma rubbed her hand on Mac's leg as she said it. "I think romance is when you just do things for each other because you're in love. I wish we were old enough to be in love." Emma sighed, hugely. "I think it must feel really good."
"We should kiss a boy and see what it's like."
Everyone stopped to stare at Parker. She lay belly-down on her bed, watching her friends play Hair Salon. "We should pick a boy and get him to kiss us. We're almost twelve. We need to try it and see if we like it."
Laurel narrowed her eyes. "Like an experiment?"
"But who would we kiss?" Emma wondered.
"We'll make a list." Parker rolled across the bed to grab her newest notebook from her nightstand. This one featured a pair of pink toe shoes on the cover. "We'll write down all the boys we know, then which ones we think might be okay to kiss. And why or why not."
"That doesn't really sound romantic."
Parker gave Emma a small smile. "We have to start somewhere, and lists always help. Now, I don't think we can use relatives. I mean like Del," she said, speaking of her brother, "or either of Emma's brothers. Besides, Emma's brothers are way too old."
She opened the book to a fresh page. "So—"
"Sometimes they stick their tongue in your mouth."
Mac's statement brought on squeals, gags, more giggles.
Parker slid off the bed to sit on the floor beside Emma. "Okay, after we make the master list, we can divide it. Yes and No. Then we pick from the Yes list. If we get the boy we pick to kiss us, we have to tell what it was like. And if he puts his tongue in your mouth, we have to know what that's like."
"What if we pick one and he doesn't want to kiss us?"
"Em?" Securing the last braid, Laurel shook her head. "A boy's going to want to kiss you for sure. You're really pretty, and you talk to them like they're regular. Some of the girls get all stupid around boys, but you don't. Plus you're starting to get breasts."
"Boys like breasts," Mac said wisely. "Anyway, if he won't kiss you, you just kiss him. I don't think it's that big a deal anyway."
Emma thought it was, or should be.
But they wrote down the list, and just the act of it made them all laugh. Laurel and Mac acted out how one boy or another might approach the moment, and _that_ had them rolling on the floor until Mr. Fish, the cat, stalked out of the bedroom to curl up in Parker's sitting room.
Parker tucked the notebook away when Mrs. Grady came in with cookies and milk. Then the idea of playing Girl Band had them all pawing through Parker's closet and dressers to find the right pieces for stage gear.
They fell asleep on the floor, across the bed. Curled up, sprawled out.
Emma woke before sunrise. The room was dark but for the glow of Parker's night-light, and the stream from the moon through her windows.
Someone had covered her with a light blanket and tucked a pillow under her head. Someone always did when they had sleepovers.
The moonlight drew her, and, half dreaming still, she walked to the terrace doors and out. Cool air, scented by roses, brushed her cheeks.
She looked out over the silver-edged gardens where spring lived in soft colors, sweet shapes. She could almost hear the music, almost see herself dancing among the roses and azaleas, the peonies that still held their petals and perfume in tight balls.
She could almost see the shape of her partner, the one who spun her in the dance. The waltz, she thought with a sigh. It should be a waltz, like in a storybook.
That was romance, she thought, and closed her eyes to breathe in the night air.
One day, she promised herself, she'd know what it was like.
CHAPTER ONE
_S_ INCE DETAILS CROWDED HER MIND, MANY OF THEM BLURRY, Emma checked her appointment book over her first cup of coffee. The back-to-back consults gave her nearly as much of a boost as the strong, sweet coffee. Basking in it, she leaned back in the chair in her cozy office to read over the side notes she'd added to each client.
In her experience, the personality of the couple—or often, more accurately, the bride—helped her determine the tone of the consult, the direction they'd pursue. To Emma's way of thinking, flowers were the heart of a wedding. Whether they were elegant or fun, elaborate or simple, the flowers were the romance.
It was her job to give the client all the heart and romance they desired.
She sighed, stretched, then smiled at the vase of petite roses on her desk. Spring, she thought, was the best. The wedding season kicked into high gear—which meant busy days and long nights designing, arranging, creating not only for _this_ spring's weddings, but also next.
She loved the continuity as much as the work itself.
That's what Vows had given her and her three best friends. Continuity, rewarding work, and that sense of personal accomplishment. And she got to play with flowers, live with flowers, practically swim in flowers every day.
Thoughtfully, she examined her hands, and the little nicks and tiny cuts. Some days she thought of them as battle scars, and others as medals of honor. This morning she just wished she'd remembered to fit in a manicure.
She glanced at the time, calculated. Boosted again, she sprang up. Detouring into her bedroom, she grabbed a scarlet hoodie to zip over her pjs. There was time to walk to the main house before she dressed and prepared for the day. At the main house Mrs. Grady would have breakfast, so Emma wouldn't have to forage or cook for herself.
Her life, she thought as she jogged downstairs, brimmed with lovely perks.
She passed through the living room she used as a reception and consult area, and took a quick scan around as she headed for the door. She'd freshen up the flowers on display before the first meeting, but oh, hadn't those stargazer lilies opened beautifully?
She stepped out of what had been a guest house on the Brown Estate but was now her home and the base for Centerpiece—her part of Vows.
She took a deep breath of spring air. And shivered.
Damn it, why couldn't it be warmer? It was April, for God's sake. It was daffodil time. Look how cheerful the pansies she'd potted up looked. She refused to let a chilly morning—and okay, it was starting to drizzle on top of it—spoil her mood.
She hunched inside the hoodie, stuck the hand not holding her coffee mug in her pocket, and began to walk to the main house.
Things were coming back to life all around her, she reminded herself. If you looked closely enough you could see the promise of green on the trees, the hint of what would be delicate blooms of dogwood and cherry blossoms. Those daffodils wanted to pop, and the crocuses already had. Maybe there'd be another spring snow, but the worst was over.
Soon it would be time to dig in the dirt, to bring some of her beauties out of the greenhouse and put them on display. She added the bouquets, the swags and garlands, but nothing beat Mother Nature for providing the most poignant landscape for a wedding.
And nothing, in her opinion, beat the Brown Estate for showing it off.
The gardens, showpieces even now, would soon explode with color, bloom, scent, inviting people to stroll along the curving paths, or sit on a bench, relax in sun or shade. Parker put her in charge—as much as Parker could put anyone else in charge—of overseeing them, so every year she got to play, planting something new, or supervising the landscape team.
The terraces and patios created lovely outdoor living spaces, perfect for weddings and events. Poolside receptions, terrace receptions, ceremonies under the rose arbor or the pergola, or perhaps down by the pond under a willow.
We've got it all, she thought.
The house itself? Could anything be more graceful, more beautiful? The wonderful soft blue, those warm touches of yellow and cream. All the varied rooflines, the arching windows, the lacy balconies added up to elegant charm. And really, the entrance portico was made for crowding with lush greenery or elaborate colors and textures.
As a child she'd thought of it as a fairyland, complete with castle.
Now it was home.
She veered toward the pool house, where her partner Mac lived and kept her photography studio. Even as she aimed for it, the door opened. Emma beamed a smile, shot out a wave to the lanky man with shaggy hair and a tweed jacket who came out.
"Morning, Carter!"
"Hi, Emma."
Carter's family and hers had been friends almost as long as she could remember. Now, Carter Maguire, former Yale prof and current professor of English lit at their high school alma mater, was engaged to one of her best friends in the world.
Life wasn't just good, Emma thought. It was a freaking bed of roses.
Riding on that, she all but danced to Carter, tugged him down by his lapel as she angled up on her toes and kissed him noisily.
"Wow," he said, and blushed a little.
"Hey." Mackensie, her eyes sleepy, her cap of red hair bright in the gloom, leaned on the doorjamb. "Are you trying to make time with my guy?"
"If only. I'd steal him away but you've dazzled and vamped him."
"Damn right."
"Well." Carter offered them both a flustered smile. "This is a really nice start to my day. The staff meeting I'm headed to won't be half as enjoyable."
"Call in sick." Mac all but purred it. "I'll give you something enjoyable."
"Hah. Well. Anyway. Bye."
Emma grinned at his back as he hurried off to his car. "God, he is so _cute_."
"He really is."
"And look at you, Happy Girl."
"Happy Engaged Girl. Want to see my ring again?"
"Oooh," Emma said obligingly when Mac wiggled her fingers. "Ahhh."
"Are you going for breakfast?"
"That's the plan."
"Wait." Mac leaned in, grabbed a jacket, then pulled the door closed behind her. "I didn't have anything but coffee yet, so . . ." As they fell into step together, Mac frowned. "That's my mug."
"Do you want it back now?"
"I know why I'm cheerful this crappy morning, and it's the same reason I haven't had time for breakfast. It's called Let's Share the Shower."
"Happy Girl is also Bragging Bitch."
"And proud of it. Why are you so cheerful? Got a man in your house?"
"Sadly no. But I have five consults booked today. Which is a great start to the week, and comes on the tail of the lovely end to last week with yesterday's tea party wedding. It was really sweet, wasn't it?"
"Our sexagenarian couple exchanging vows and celebrating surrounded by his kids, her kids, grandchildren. Not just sweet, but also reassuring. Second time around for both of them, and there they are, ready to do it again, willing to share and blend. I got some really great shots. Anyway, I think those crazy kids are going to make it."
"Speaking of crazy kids, we really have to talk about your flowers. December may be far away—she says shivering—but it comes fast, as you well know."
"I haven't even decided on the look for the engagement shots yet. Or looked at dresses, or thought about colors."
"I look good in jewel tones," Emma said and fluttered her lashes.
"You look good in burlap. Talk about bragging bitches." Mac opened the door to the mudroom, and since Mrs. Grady was back from her winter vacation, remembered to wipe her feet. "As soon as I find the dress, we'll brainstorm the rest."
"You're the first one of us to get married. To have your wedding here."
"Yeah. It's going to be interesting to see how we manage to run the wedding and be _in_ the wedding."
"You know you can count on Parker to figure out the logistics. If anyone can make it run smoothly, it's Parker."
They walked into the kitchen, and chaos.
While the equitable Maureen Grady worked at the stove, movements efficient, face placid, Parker and Laurel faced off across the room.
"It has to be done," Parker insisted.
"Bullshit, bullshit, bullshit."
"Laurel, this is business. In business you serve the client."
"Let me tell you what I'd like to serve the client."
"Just stop." Parker, her rich brown hair sleeked back in a tail, was already dressed in a meet-the-client suit of midnight blue. Eyes of nearly the same color flashed hot with impatience. "Look, I've already put together a list of her choices, the number of guests, her colors, her floral selections. You don't even have to speak to her. I'll liaise."
"Now let me tell you what you can do with your list."
"The bride—"
"The bride is an asshole. The bride is an idiot, a whiny baby bitch who made it very clear nearly one year ago that she neither needed nor wanted my particular services. The bride can bite me because she's not biting any of my cake now that she's realized her own stupidity."
In the cotton pajama pants and tank she'd slept in, her hair still in sleep tufts, Laurel dropped onto a chair in the breakfast nook.
"You need to calm down." Parker bent down to pick up a file. Probably tossed on the floor by Laurel, Emma mused. "Everything you need is in here." Parker laid the file on the table. "I've already assured the bride we'll accommodate her, so—"
"So you design and bake a four-layer wedding cake between now and Saturday, and a groom's cake, and a selection of desserts. To serve two hundred people. You do that with no previous preparation, and when you've got three other events over the weekend, and an evening event in three days."
Her face set in mutinous lines, Laurel picked up the file and deliberately dropped it on the floor.
"Now you're acting like a child."
"Fine. I'm a child."
"Girls, your little friends have come to play." Mrs. Grady sang it out, her tone overly sweet, her eyes laughing.
"Ah, I hear my mom calling me," Emma said and started to ease out of the room.
"No, you don't!" Laurel jumped up. "Just listen to this! The Folk-Harrigan wedding. Saturday, evening event. You'll remember, I'm sure, how the bride sniffed at the very idea of Icings at Vows providing the cake or any of the desserts. How she _sneered_ at me and my suggestions and insisted her cousin, a pastry chef in New York, who studied in Paris and designed cakes for _important_ affairs, would be handling all the desserts.
"Do you remember what she said to me?"
"Ah." Emma shifted because Laurel's finger pointed at her heart. "Not in the exact words."
"Well, I do. She said she was sure—and said it with that sneer—she was sure I could handle _most_ affairs well enough, but she wanted the _best_ for her wedding. She said that to my face."
"Which was rude, no question," Parker began.
"I'm not finished," Laurel said between her teeth. "Now, at the eleventh hour, it seems her brilliant cousin has run off with one of her—the cousin's—clients. Scandal, scandal, as said client met brilliant cousin when he commissioned her to design a cake for _his_ engagement party. Now they're MIA and the bride wants me to step in and save her day."
"Which is what we do here. Laurel—"
"I'm not asking you." She flicked her fingers at Parker, zeroed in on Mac and Emma. "I'm asking them."
"What? Did you say something?" Mac offered a toothy smile. "Sorry, I must've gotten water in my ears from the shower. Can't hear a thing."
"Coward. Em?"
"Ah . . ."
"Breakfast!" Mrs. Grady circled a finger in the air. "Everybody sit down. Egg white omelettes on toasted brown bread. Sit, sit. Eat."
"I'm not eating until—"
"Let's just sit." Interrupting Laurel's next tirade, Emma tried a soothing tone. "Give me a minute to think. Let's just all sit down and . . . Oh, Mrs. G, that looks fabulous." She grabbed two plates, thinking of them as shields as she crossed to the breakfast nook and scooted in. "Let's remember we're a team," she began.
"You're not the one being insulted and overworked."
"Actually, I am. Or have been. Whitney Folk puts the _zilla_ in Bridezilla. I could relay my personal nightmares with her, but that's a story for another day."
"I've got some of my own," Mac put in.
"So your hearing's back," Laurel muttered.
"She's rude, demanding, spoiled, difficult, and unpleasant," Emma continued. "Usually when we plan the event, even with the problems that can come up and the general weirdness of some couples, I like to think we're helping them showcase a day that begins their happy ever after. With this one? I'd be surprised if they make it two years. She was rude to you, and I don't think it was a sneer, I think it was a smirk. I don't like her."
Obviously pleased with the support, Laurel sent her own smirk toward Parker, then began to eat.
"That being said, we're a team. And clients, even smirky bitch clients, have to be served. Those are good reasons to do this," Emma said while Laurel scowled at her. "But there's a better one. You'll show her rude, smirky, flat, bony ass what a really brilliant pastry chef can do, and under pressure."
"Parker already tried that one on me."
"Oh." Emma sampled a skinny sliver of her omelette. "Well, it's true."
"I could bake her man-stealing cousin into the ground."
"No question. Personally, I think she should grovel, at least a little."
"I like groveling." Laurel considered it. "And begging."
"I might be able to arrange for some of each." Parker lifted her coffee. "I also informed her that in order to accommodate her on such short notice we would require an additional fee. I added twenty-five percent. She grabbed it like a lifeline, and actually wept in gratitude."
A new light beamed in Laurel's bluebell eyes. "She cried?"
Parker inclined her head, and cocked an eyebrow at Laurel. "So?"
"While the crying part warms me inside, she'll still have to take what I give her, and like it."
"Absolutely."
"You just let me know what you decide on when you decide on it," Emma told her. "I'll work in the flowers and decor for the table." She sent a sympathetic smile at Parker. "What time did she call you with all this?"
"Three twenty A.M."
Laurel reached over, gave Parker's hand a pat. "Sorry."
"That's my part of the deal. We'll get through it. We always do."
_T_ HEY ALWAYS DID, EMMA THOUGHT AS SHE REFRESHED HER LIVING room arrangements. She trusted they always would. She glanced at the photograph she kept in a simple white frame, one of three young girls playing Wedding Day in a summer garden. She'd been bride that day, and had held the bouquet of weeds and wildflowers, worn the lace veil. And had been just as charmed and delighted as her friends when the blue butterfly landed on the dandelion in her bouquet.
Mac had been there, too, of course. Behind the camera, capturing the moment. Emma considered it a not-so-small miracle that they'd turned what had been a favored childhood game of make-believe into a thriving business.
No dandelions these days, she thought as she fluffed pillows. But how many times had she seen that same delighted, dazzled look on a bride's face when she'd offered her a bouquet she'd made for her? Just for her.
She hoped the meeting about to begin would end in a wedding next spring with just that dazzled look on the bride's face.
She arranged her files, her albums, her books, then moved to the mirror to check her hair, her makeup, the line of the jacket and pants she'd changed into.
Presentation, she thought, was a priority of Vows.
She turned from the mirror to answer her phone with a cheerful, "Centerpiece of Vows. Yes, hello, Roseanne. Of course I remember you. October wedding, right? No, it's not too early to make those decisions."
As she spoke, Emma took a notebook out of her desk, flipped it open. "We can set up a consultation next week if that works for you. Can you bring a photo of your dress? Great. And if you've selected the attendants' dresses, or their colors . . . ? Mmm-hmm. I'll help you with all of that. How about next Monday at two?"
She logged in the appointment, then glanced over her shoulder as she heard a car pull up.
A client on the phone, another coming to the door.
God, she _loved_ spring!
_E_ MMA SHOWED HER LAST CLIENT OF THE DAY THROUGH THE DISPLAY area where she kept silk arrangements and bouquets as well as various samples on tables and shelves.
"I made this up when you e-mailed me the photo of your dress, and gave me the basic idea of your colors and your favorite flowers. I know you'd talked about preferring a large cascade bouquet, but . . ."
Emma took the bouquet of lilies and roses, tied with white pearl-studded ribbon off the shelf. "I just wanted you to see this before you made a firm decision."
"It's beautiful, plus my favorite flowers. But it doesn't seem, I don't know, big enough."
"With the lines of your dress, the column of the skirt, and the beautiful beadwork on the bodice, the more contemporary bouquet could be stunning. I want you to have exactly what you want, Miranda. This sample is closer to what you have in mind."
Emma took a cascade from the shelf.
"Oh, it's like a garden!"
"Yes, it is. Let me show you a couple of photos." She opened the folder on the counter, took out two.
"It's my dress! With the bouquets."
"My partner Mac is a whiz with Photoshop. These give you a good idea how each style looks with your dress. There's no wrong choice. It's your day, and every detail should be exactly what you want."
"But you're right, aren't you?" Miranda studied both pictures. "The big one sort of, well, overwhelms the dress. But the other, it's like it was made for it. It's elegant, but it's still romantic. It is romantic, isn't it?"
"I think so. The lilies, with that blush of pink against the white roses, and the touches of pale green. The trail of the white ribbon, the glow of the pearls. I thought, if you liked it, we might do just the lilies for your attendants, maybe with a pink ribbon."
"I think . . ." Miranda carried the sample bouquet over to the old-fashioned cheval glass that stood in the corner. Her smile bloomed like the flowers as she studied herself. "I think it looks like some really creative fairies made it. And I love it."
Emma noted it down in her book. "I'm glad you do. We'll work around that, sort of spiraling out from the bouquets. I'll put clear vases on the head table, so the bouquets will not only stay fresh, but serve as part of the decor during the reception. Now, for your tossing bouquet, I was thinking just the white roses, smaller scale like this." Emma took down another sample. "Tied with pink and white ribbons."
"That would be perfect. This is turning out to be so much easier than I thought."
Pleased, Emma made another note. "The flowers are important, but they should also be fun. No wrong choices, remember. From everything you've told me, I see the feel of the wedding as modern romance."
"Yes, that's exactly what I'm after."
"Your niece, the flower girl, is five, right?"
"She just turned five last month. She's really excited about scattering rose petals down the aisle."
"I bet." Emma crossed the idea of a pomander off her mental list. "We could use this style basket, covered with white satin, trimmed in baby roses, trailing the pink and white ribbons again. Pink and white rose petals. We could do a halo for her, pink and white baby roses again. Depending on her dress, and what you like, we can keep it simple, or we can trail ribbons down the back."
"The ribbons, absolutely. She's really girly. She'll be thrilled." Miranda took the sample halo Emma offered. "Oh, Emma. It's like a little crown! Princessy."
"Exactly." When Miranda lifted it onto her own head, Emma laughed. "A girly five year-old will be in heaven. And you'll be her favorite aunt for life."
"She'll look so sweet. Yes, yes, to everything. Basket, halo, ribbons, roses, colors."
"Great. You're making it easy for me. Now you've got your mothers and your grandmothers. We could do corsages, wrist or pin-on, using the roses or the lilies or both. But—"
Smiling, Miranda set the halo down again. "Every time you say 'but' it turns out fantastic. So, but?"
"I thought we could update the classic tussy-mussy."
"I have no idea what that is."
"It's a small bouquet, like this, carried in a little holder to keep the flowers fresh. We'd put display stands on the tables by their places, which would also dress up their tables, just a little more than the others. We'd use the lilies and roses, in miniature, but maybe reverse the colors. Pink roses, white lilies, those touches of pale green. Or if that didn't go with their dresses, all white. Small, not quite delicate. I'd use something like this very simple silver holder, nothing ornate. Then we could have them engraved with the wedding date, or your names, their names."
"It's like their own bouquets. Like a miniature of mine. Oh, my mother will . . ."
When Miranda's eyes filled, Emma reached over and picked up the box of tissue she kept handy.
"Thanks. I want them. I have to think about the monogramming. I'd like to talk that over with Brian."
"Plenty of time."
"But I want them. The reverse, I think, because it makes them more theirs. I'm going to sit down here a minute."
Emma went with her to the little seating area, put the tissue box where Miranda could reach. "It's going to be beautiful."
"I know. I can see it. I can already see it, and we haven't even started on the arrangements and centerpieces and, oh, everything else. But I can see it. I have to tell you something."
"Sure."
"My sister—my maid of honor? She really pushed for us to book Felfoot. It's been _the_ place in Greenwich, you know, and it is beautiful."
"It's gorgeous, and they always do a fabulous job."
"But Brian and I just fell for this place. The look of it, the feel of it, the way the four of you work together. It felt right for us. Every time I come here, or meet with one of you, I know we were right. We're going to have the most amazing wedding. Sorry," she said, dabbing at her eyes again.
"Don't be." Emma took a tissue for herself. "I'm flattered, and nothing makes me happier than to have a bride sit here and cry happy tears. How about a glass of champagne to smooth things out before we start on the boutonnieres?"
"Seriously? Emmaline, if I wasn't madly in love with Brian, I'd ask _you_ to marry me."
With a laugh, Emma rose. "I'll be right back."
_L_ ATER, EMMA SAW OFF HER EXCITED BRIDE AND, COMFORTABLY tired, settled down with a short pot of coffee in her office. Miranda was right, she thought as she keyed in all the details. She was going to have the most amazing wedding. An abundance of flowers, a contemporary look with romantic touches. Candles and the sheen and shimmer of ribbons and gauze. Pinks and whites with pops of bold blues and greens for contrast and interest. Sleek silver and clear glass for accents. Long lines, and the whimsy of fairy lights.
As she drafted out the itemized contract, she congratulated herself on a very productive day. And since she'd spend most of the next working on the arrangements for their midweek evening event, she considered making it an early night.
She'd resist going over and seeing what Mrs. G had for dinner, make herself a salad, maybe some pasta. Curl up with a movie or her stack of magazines, call her mother. She could get everything done, have a relaxing evening, and be in bed by eleven.
As she proofed the contract, her phone let out the quick two rings that signaled her personal line. She glanced at the readout, smiled.
"Hi, Sam."
"Hello, Beautiful. What are you doing home when you should be out with me?"
"I'm working."
"It's after six. Pack it in, honey. Adam and Vicki are having a party. We can go grab some dinner first. I'll pick you up in an hour."
"Whoa, wait. I told Vicki tonight just wasn't good for me. I was booked solid today, and still have about another hour before—"
"You've got to eat, right? And if you've been working all day you deserve to play. Come play with me."
"That's sweet, but—"
"Don't make me go to the party by myself. We'll swing by, have a drink, a couple laughs, leave whenever you want. Don't break my heart, Emma."
She cast her eyes up to the ceiling and saw her early night go up in smoke. "I can't make dinner, but I could meet you there around eight."
"I can pick you up at eight."
Then angle to come in when you bring me home, she thought. And that's not happening. "I'll meet you. That way if I need to go and you're having fun, you can stay."
"If that's the best I can get, I'll take it. I'll see you there."
CHAPTER TWO
_S_ HE LIKED PARTIES, EMMA REMINDED HERSELF. SHE LIKED people and conversation. She enjoyed picking the right outfit, doing her makeup, fussing with her hair.
She was a girl.
She liked Adam and Vicki—and had, in fact, introduced them four years ago when it had become clear she and Adam made better friends than lovers.
Vows had done their wedding.
She liked Sam, she thought with a sigh as she pulled up in front of the contemporary two-story, then flipped down her visor mirror to check her makeup.
She enjoyed going out with Sam—to dinner, to a party, to a concert. The problem was the spark-o-meter. When she'd met him, he'd hit a solid seven, with upward potential. In addition, she'd found him smart and funny, appreciated his smooth good looks. But the first-date kiss had dropped to a measly two on the spark-o-meter.
Not his fault, she admitted as she got out of the car. _It_ just wasn't there. She'd given it a shot. A few more kisses—kissing was one of her favorite things. But they'd never risen over the two—and that was being generous.
It wasn't easy to tell a man you had no intention of sleeping with him. Feelings and egos were at stake. But she'd done it. The problem, as she saw it, was he didn't really believe her.
Maybe she'd find someone to introduce him to at the party.
She stepped inside, into the music, the voices, the lights—and felt an immediate lift of mood. She really did like parties.
After one quick scan, she saw a dozen people she knew.
She kissed cheeks, exchanged hugs, and kept moving in a search for her host and hostess. When she spotted a distant cousin by marriage she shot out a wave. Addison, she mused, and signaled that she'd be back around to say hi. Single, fun loving, stunning. Yes, she could see Addison and Sam hitting it off.
She'd make sure she introduced them.
She found Vicki in the kitchen area of the generous great room, talking to friends while she refreshed a tray of party food.
"Emma! I didn't think you were going to make it."
"It's going to be practically a hit-and-run. You look great."
"So do you. Oh, thank you!" She took the bouquet of candy-striped tulips Emma offered. "They're beautiful."
"I'm in a 'Damn it, it's spring' frame of mind. These said I'm right. Can I give you a hand?"
"Absolutely not. Let me get you a glass of wine."
"Half a glass. I'm driving, and I really can't stay very long."
"Half a glass of cab." Vicki laid the flowers on the counter to free her hands. "Did you come alone?"
"Actually, I'm sort of meeting Sam."
"Oh," Vicki said, drawing out the syllable.
"Not really, no."
"Oh."
"Listen. Here, let me do that," she said when Vicki got out a vase for the flowers. Lowering her voice, Emma continued as she dealt with the flowers. "What do you think about Addison and Sam?"
"Are they an item? I didn't realize—"
"No. I was just speculating. I think they'd like each other."
"Sure. I suppose. You look so good together. You and Sam."
Emma made a noncommittal sound. "Where's Adam? I didn't see him in the mob."
"Probably out on the deck having a beer with Jack."
"Jack's here?" Emma kept her hands busy and her tone casual. "I'll have to say hi."
"They were talking baseball, last I heard. You know how they are."
She knew exactly. She'd known Jack Cooke for over a decade, since he and Parker's brother, Delaney, had roomed together at Yale. And Jack had spent a lot of time at the Brown Estate. He'd ultimately moved to Greenwich and opened his small, exclusive architecture firm.
He'd been a rock, she remembered, when Parker and Del's parents had been killed in a private plane crash. And when they'd decided to start the business, he'd been a lifesaver by designing the remodels of the pool house and guest house to accommodate the needs of the company.
He was practically family.
Yes, she'd make sure to say hi before she left.
She turned with the glass of wine in her hand just as Sam made his way into the room. He was _so_ good-looking, she thought. Tall and built, with that perpetual twinkle in his eyes. Maybe just a _tiny_ bit studied, with his hair always perfectly styled, his clothes always exactly right, but—
"There she is. Hi, Vic." He passed Vicki a very nice bottle of cabernet—exactly the right thing—kissed her cheek, then gave Emma a warm, warm smile. "Just who I've been looking for."
He caught Emma up in an enthusiastic kiss that barely bumped the pleasant level on her scale.
She managed to ease back an inch and get her free hand on his chest in case he got it into his head to kiss her again. She smiled up at him, added a friendly laugh. "Hi, Sam."
Jack, dark blond hair tousled from the evening breeze, leather jacket open over faded jeans, walked in from the deck. His eyebrows rose at Emma; his lips curved. "Hey, Em. Don't let me interrupt."
"Jack." She nudged Sam back another inch. "You know Sam, don't you?"
"Sure. How's it going?"
"Good." Sam shifted, draped his arm over Emma's shoulders. "You?"
"Can't complain." He took a chip, shoveled it into salsa. "How are things back on the farm?" he asked Emma.
"We're busy. Spring's all about weddings."
"Spring's all about baseball. I saw your mother the other day. She remains the most beautiful woman ever created."
Emma's casual smile warmed like sunlight. "True."
"She still refuses to leave your father for me, but hope springs. Anyway, see you later. Sam."
As Jack walked off, Sam shifted. Knowing the dance well, Emma shifted in turn—so she avoided being trapped between him and the counter. "I'd forgotten how many mutual friends Vicki, Adam, and I have. I know almost everyone here. I need to touch some bases. Oh, and there's someone I really want you to meet."
Cheerfully, she took Sam's hand. "You don't know my cousin, Addison, do you?"
"I don't think so."
"I haven't seen her in months. Let's track her down so I can introduce you."
She pulled him into the heart of the party.
_J_ ACK SCOOPED UP A HANDFUL OF NUTS AND CHATTED WITH A group of friends. And watched Emma lead the Young Executive at Play through the crowd. She looked . . . freaking amazing, he thought.
Not just the sexy, sloe-eyed, curvy, golden-skinned, masses of curling hair, soft, full-lipped amazing. That was killer enough. But you had to add in the heat and light she just seemed to emanate. She made one hell of a package.
And, he reminded himself, she was his best friend's honorary sister.
In any case, it was rare to see her when she wasn't with her regular gang of girls, some of her family, surrounded by people. Or, like now, with some guy.
When a woman looked like Emmaline Grant, there was always some guy.
Still, it never hurt to look. He was a man who appreciated lines and curves—in buildings and in women. In his estimation, Emma was pretty much architecturally perfect. So he popped nuts, pretended to listen to the conversation, and watched her slide and sway through the room.
Looked casual, he observed, the way she'd stop, exchange greetings, pause, laugh or smile. But he'd made a kind of study of her over the years. She moved with purpose.
Curiosity piqued, Jack eased away from the group, merged with another to keep her in his eyeline.
The some guy—Sam—did a lot of back stroking, shoulder draping. She did plenty of smiling at him, laughing up at him from under that thicket of lashes she owned. But oh yeah, her body language—he'd made a study of her body—wasn't signaling reception.
He heard her call out _Addison!_ and follow up with that sizzle-in-the-blood laugh of hers before she grabbed a very fine-looking blonde in a hug.
They chattered, beaming at each other the way women did, holding each other at arm's length to take the survey before—no doubt—they told each other how great they looked.
_You look fabulous. Have you lost weight? I love your hair._ From his observations, that particular female ritual had some variations, but the theme remained the same.
Then Emma angled herself in a way that put the some guy and the blonde face-to-face.
He got it then, by the way she sidled back an inch or two, then waved a hand in the air before giving the some guy a pat on his arm. She wanted to ditch the some guy, and thought the blonde would distract him.
When she melted away in the direction of the kitchen, Jack lifted his beer in toast.
Well played, Emmaline, he thought. Well played.
_H_ E CUT OUT EARLY. HE HAD AN EIGHT O'CLOCK BREAKFAST meeting and a day packed with site visits and inspections. Somewhere in there, or the day after, he needed to carve out some time at the drawing board to work up some ideas for the addition Mac wanted on her studio now that she and Carter were engaged and living together.
He could see how to do it, without insulting the lines and form of the building. But he wanted to get it down on paper, play with it awhile before he showed Mac anything.
He hadn't quite gotten used to the idea of Mac getting married—and to Carter. You had to like Carter, Jack thought. He'd barely blipped on Jack's radar when he and Del and Carter had been at Yale together. But you had to like the guy.
Plus, he put a real light in Mac's eyes. That counted big.
With the radio blasting, he turned over in his head various ideas for adding on the space so Carter had a home office to do . . . whatever English professors did in home offices.
As he drove, the rain that had come and gone throughout the day came back in the form of a thin snow. April in New England, he thought.
His headlights washed over the car sitting on the shoulder of the road, and the woman standing in front of the lifted hood, her hands fisted on her hips.
He pulled over, got out, then, sliding his hands into his pockets, sauntered over to Emma. "Long time no see."
"Damn it. It just died. Stopped." She waved her arms in frustration so he took a cautious step back to avoid getting clocked with the flashlight she gripped in one hand. "And it's snowing. Do you _see_ this?"
"So it is. Did you check your gas gauge?"
"I didn't run out of gas. I'm not a moron. It's the battery, or the carburetor. Or one of those hose things. Or belt things."
"Well, that narrows it down."
She huffed out a breath. "Damn it, Jack, I'm a florist, not a mechanic."
That got a laugh out of him. "Good one. Did you call for road service?"
"I'm going to, but I thought I should at least look in there in case it was something simple and obvious. Why don't they make what's in there simple and obvious for people who drive cars?"
"Why do flowers have strange Latin names nobody can pronounce? These are questions. Let me take a look." He held out a hand for the flashlight. "Jesus, Emma, you're freezing."
"I'd have worn something warmer if I'd known I'd end up standing on the side of the road in the middle of the stupid night in a snowstorm."
"It's barely snowing." He stripped off his jacket, passed it to her.
"Thanks."
She bundled into it while he bent under the hood. "When's the last time you had this serviced?"
"I don't know. Some time."
He glanced back at her, a dry look out of smoky gray eyes. "Some time looks to have been the other side of never. Your battery cables are corroded."
"What does that mean?" She stepped up, stuck her head under the hood along with him. "Can you fix it?"
"I can . . ."
He turned his head toward her, and she turned hers toward him. All he could see were those brown velvet eyes, and for a moment, he simply lost the power of speech.
"What?" she said, and her breath whispered warm over his lips.
"What?" What the hell was he doing? He leaned back, out of the danger zone. "What . . . What I can do is give you a jump that should get you home."
"Oh. Okay. Good. That's good."
"Then you've got to get this thing in for service."
"Absolutely. First thing. Promise."
Her voice jumped a bit and reminded him it was cold. "Go ahead and get in the car, and I'll hook it up. Don't start it, don't touch anything in there, until I tell you."
He pulled his car around so it was nose-to-nose with hers. As he got his jumper cables, she got out of the car again. "I want to see what you do," she explained. "In case I ever have to do it."
"Okay. Jumper cables, batteries. You have your positive and your negative. You don't want to get them mixed up because if you hook them up wrong you'll—"
He clamped one onto the battery, then made a strangling noise and began to shake. Instead of squealing, she laughed and smacked his arm. "Idiot. I have brothers. I know your games."
"Your brothers should've shown you how to jump-start a car."
"I think they sort of did, but I ignored them. I have a set of those in the trunk, along with other emergency stuff. But I never had to use any of it. Under yours is shinier than mine," she added as she frowned at his engine.
"I suspect the pit of hell is shinier than yours."
She puffed out a breath. "Now that I've seen it, I can't argue."
"Get in, turn it over."
"Turn what over? Kidding," she said.
"Ha. If and when it starts, don't turn it off."
"Got it." In the car, she held up crossed fingers, turned the key. The engine coughed, hacked—made him wince—then rumbled to life.
She stuck her head out the window and beamed at him. "It worked!"
He had an errant thought that with that much power, her smile could have sparked a hundred dead batteries. "We'll let it juice up a few minutes, then I'll follow you home."
"You don't have to do that. It's out of your way."
"I'll follow you home so I know you didn't conk out on the way."
"Thanks, Jack. God knows how long I'd've been out here if you hadn't come along. I was cursing myself for going to that damn party when all I wanted to do tonight was zone out with a movie and go to bed early."
"So why'd you go?"
"Because I'm weak." She shrugged. "Sam really didn't want to go alone, and, well, I like a party, so I figured it wouldn't hurt to meet him there and hang out for an hour."
"Uh-huh. How'd it work out with him and the blonde?"
"Sorry?"
"The blonde you palmed him off on."
"I didn't palm him off." Her gaze slid away, then rolled back to his. "Okay, I did, but only because I thought they'd like each other. Which they did. I'd've considered that good deed worth coming out tonight. Except I ended up broken down on the side of the road. It seems unfair. And mildly embarrassing since you noticed."
"On the contrary, I was impressed. That and the salsa were my favorite parts of the evening. I'm going to take the cables off. Let's see if she holds a charge. If we're good, wait until I'm in my car before you pull out."
"Okay. Jack? I owe you."
"Yeah, you do." He gave her a grin before he walked off.
When her car continued to run, he shut her hood then his own. Once he'd tossed the jumper cables back in his trunk, he got behind the wheel and flashed his lights to signal her to go.
He followed her through the lace of the light snow, and tried not to think of that moment under the hood when her breath had brushed warm over his lips.
She gave a friendly toot of her horn when she reached the private road for the Brown Estate. He eased over, stopped. He watched her taillights shimmer in the dark, then disappear around the bend that led to the guest house.
Then he sat a little while longer, in the dark, before turning the car around and heading home.
_I_ N HER REARVIEW MIRROR, EMMA SAW JACK STOP AT THE MOUTH of the drive. She hesitated, wondering if she should've asked if he wanted to come down, have some coffee before he doubled back and drove home.
She probably should have—least she could do—but it was too late now. And all for the best, no question.
It wasn't wise to entertain a family friend who banged a booming ten on your spark o-meter, alone, late at night. Especially when you still have some belly vibes going from a ridiculous moment under the hood of a car when you'd nearly humiliated yourself by moving on him.
That would never do.
She wished she could go by and talk over the whole stupid mess with Parker or Laurel or Mac—better yet, all three of them. But that, too, wouldn't do. Some things couldn't be shared even with the best friends in the world. Especially since it was clear Jack and Mac had gotten snuggly once upon a time.
She suspected that Jack got snuggly with a lot of women.
Not that she held it against him, she thought as she parked. She liked the company of men. She liked sex. Sometimes one led to the other.
Besides, how could you find the love of your life if you didn't look for him?
She turned off the car, bit her lip, then turned the key again. It made very unhappy noises, seemed undecided, then fired.
That had to be a good sign, she decided, then switched it off again. But she'd take it into the shop as soon as she could. She'd have to ask Parker about mechanics, as Parker knew everything.
Inside the house, she got herself a bottle of water to take upstairs. Thanks to Sam and the stupid battery she wouldn't make it to bed by the righteous hour of eleven, but she could get there by midnight. Which meant she had no excuse to miss the early workout she'd planned for the morning.
No excuse, she lectured herself.
She set the water on her bedside table by a little vase of freesia and started to undress. Then realized she was still wearing Jack's jacket.
"Oh, damn it."
It smelled so good, she thought. Leather and Jack. And that wasn't a scent that was going to give her quiet dreams. She carried it across the room, laid it over the back of a chair. Now she had to get it back to him, but she'd worry about that later.
One of the girls might be going into town for something and could drop it off. It wasn't cowardly to pass the task off. It was efficient.
Cowardice had nothing to do with it. She saw Jack all the time. _All_ the time. She just didn't see the point in making a special trip if someone else was already going. Surely he had another jacket. It wasn't like he needed that particular one immediately. If it was so important, why hadn't he taken it back?
It was his own fault.
And hadn't she said she'd worry about it later?
She changed into a nightshirt then went into the bathroom to begin her nightly ritual. Makeup off, skin toned and moisturized, teeth and hair brushed. The routine and her pretty bathroom usually relaxed her. She loved the happy colors, her sweet slipper tub, the shelf of pale green bottles that held whatever flowers she had handy.
Miniature daffodils now, to celebrate spring. But their cheerful faces seemed to smirk at her. Irritated, she flipped off the light.
She continued the ritual by removing the small mountain of throw pillows from the bed, setting aside the embroidered shams, fluffing up her sleep pillows. She slid under the duvet, snuggled in to enjoy the feel of smooth, soft sheets against her skin, the dreamy scent of freesia perfuming the air, and . . .
_Shit!_ She could still smell his jacket.
Sighing, she flopped over on her back.
So what? So what if she had lusty thoughts about her best friend's brother's best friend? It wasn't a crime. Lusty thoughts were absolutely reasonable and normal. In fact, lusty thoughts were good things. Healthy things. She _liked_ having lusty thoughts.
Why wouldn't a normal woman have lusty thoughts about a sexy, gorgeous man with a great body and eyes that were like smoke wrapped up in fog?
She'd be crazy not to have them.
Acting on them, now _that_ would be crazy. But she was perfectly entitled to have them.
She wondered what he'd have done if she'd moved in that last inch under the hood of the car and planted one on him?
Being a man, he'd have moved in right back, she imagined. And they'd have spent a very interesting few minutes smolder ing on the side of the road in the lightly falling snow. Bodies heating, hearts pounding with the snow showering over them and . . .
No, no, she was romanticizing it. Why did she always do that, always move from healthy lust to romance? That was her problem, and certainly rooted in the wonderfully romantic love story of her parents. How could she not want what they had?
Put it aside, she ordered herself. She wasn't going to find happy ever after with Jack. Stick with lust.
So they'd have gotten all hot and tangled on the side of the road. But. After that impulsive and no doubt spark-loaded kiss, they'd have been awkward and embarrassed with each other.
Then they'd have had to apologize to each other, or try to make some kind of a joke out of it. Everything would be weird and strained.
The simple fact was it was too late to act on the lust. They were friends, the next thing to family. You didn't hit on friends and family. She was better off, tons better off, keeping her thoughts to herself and continuing to look for the real thing. For love.
The sort that lasted lifetimes.
CHAPTER THREE
_F_ ILLED WITH RESENTMENT AND SELF-PITY, EMMA TRUDGED UP to the home gym at the main house. Its design reflected Parker's efficient style and unassailable taste, both of which Emma bitterly detested at that moment.
CNN muttered away on the flat screen while Parker, her phone's earbud in place, racked up her miles on the elliptical. Emma scowled at the Bowflex as she stripped off her sweatshirt. She turned her back on it and the recumbent bike, on the rack of free weights, the shelf of DVDs with their perky or earnest instructors who might take her through a session of yoga or pilates, torture her with the exercise ball, or intimidate her with tai chi.
She unrolled one of the mats, sat down with the intention of doing some warm-up stretches. And just lay down.
"Morning." Parker glanced at her as she continued to pump along. "Late night?"
"How long have you been on that thing?"
"You want it? I'm nearly done. I'm just hitting my cooldown."
"I hate this room. A torture chamber with shiny floors and pretty paint is still a torture chamber."
"You'll feel better after you do a mile or two."
"Why?" From her prone position, Emma threw up her hands. "Who says? Who decided that people all of a sudden have to do miles every damn day, or that twisting themselves into unnatural shapes is good for them? I think it's the people who sell this hideous equipment, and the ones who design all the cute little outfits like the one you're wearing."
Emma narrowed her eyes at Parker's slate-colored cropped pants and perky pink and gray top. "How many of those cute little outfits do you own?"
"Thousands," Parker said dryly.
"See? And if they hadn't convinced you to do miles and twist yourself into unnatural shapes—and look good doing it—you wouldn't have spent all that money on those cute little outfits. You could've donated it to a worthy cause instead."
"But these yoga pants make my ass look great."
"They really do. But nobody's seeing your ass but me, so what's the point?"
"Personal satisfaction." Parker slowed, stopped. Hopping off, she plucked out one of the alcohol wipes to wipe down the machine. "What's wrong, Em?"
"I told you. I hate this room and all it stands for."
"So you've said before. But I know that tone. You're irritable, and you almost never are."
"I'm as irritable as anybody."
"No." Parker got her towel, mopped her face, then drank from her water bottle. "You're nearly always cheerful, optimistic, and good-natured, even when you bitch."
"I am? God, that must be annoying."
"Hardly ever." Moving to the Bowflex, Parker began to do some upper body exercise she made look smooth and easy. Emma knew it was neither. When she felt another pop of resentment, she sat up.
"I am irritable. I'm filled with irritable this morning. Last night—"
She broke off when Laurel came in, her hair bundled up, her trim body in a sports bra and bike shorts. "I'm switching off CNN," she announced, "because I just don't care." She snagged the remote, switched from TV to hard, pounding rock.
"Turn it down at least," Parker ordered. "Emma's about to tell us why she's full of irritable this morning."
"Em's never full of irritable." Laurel got a mat, unrolled it onto the floor. "It's annoying."
"See?" Since she was already on the floor, Emma decided she might as well stretch. "My best friends, and all these years you've let me go around annoying people."
"It probably only annoys us." Laurel started a set of crunches. "We're around you more than anyone else."
"That's true. In that case, screw you. God, _God_ , do the two of you really do this every day?"
"Parker's every day, as she's obsessive. I'm a three-day-a-week girl. Four if I'm feeling frisky. This is usually an off day, but I came up with a design for the crying bride and it motored me up."
"Have you got something you can show me?" Parker demanded.
"See, obsessive." Laurel switched to roll-ups. "Later. Now I want to hear about the irritable."
"How can you do that?" Being full of irritable, Emma snarled. "It's like somebody's pulling you up with an invisible rope."
"Abs of steel, baby."
"I hate you."
"Who could blame you? I deduce irritable equals man," Laurel continued. "So I require all details."
"Actually—"
"Jeez, what is this? Ladies Day at the Brown Gym?" Mac strolled in, stripping off a hooded sweatshirt.
"I think it's Snowcones in Hell Day." Laurel paused. "What are you doing here?"
"I come here sometimes."
"You look at a picture of here sometimes and consider that a workout."
"I've turned over a new leaf. For my health."
"Bullshit," Laurel said, grinning.
"Okay, bullshit. I'm pretty sure I'm going with strapless for the wedding gown. I want amazing arms and shoulders." Turning to the mirror, Mac flexed. "I have good arms and shoulders, but that's not enough." She let out a sigh as she wiggled out of sweat-pants. "And I'm becoming an obsessed, fussy bride. I hate me."
"But you'll be an obsessed, fussy bride who looks fabulous in her wedding dress. Here," Parker said, "see what I'm doing."
Mac frowned. "I see it, but I don't think I'll like it."
"You just keep it steady and smooth. I'm going to cut back the resistance a bit."
"Are you intimating I'm a weenie?"
"I'm avoiding all the moaning and crying you'd do tomorrow if you started at my level. I do this three times a week."
"You do have really good arms and shoulders."
"Plus I have it on good authority my ass looks great in these pants. Okay, smooth and steady. Fifteen reps, set of three." Parker gave Mac a pat. "Now, hopefully that's the last interruption. Emma, you have the floor."
"She's already on the floor," Mac pointed out.
"Shh. Emma's irritable this morning because . . ."
"I went over to Adam and Vicki's last night—the MacMillians?—which I hadn't planned on because yesterday was a full book and today's another. I'd had a really good day, especially the last consult, and spent time writing up the contracts and notes, decided I'd make a little dinner, have a movie, an early night."
"Who called and talked you into going out with him?" Mac asked as she frowned her way through the first set.
"Sam."
"Sam's the hot computer nerd who defies that oxymoron despite—or maybe because of—the Buddy Holly glasses."
"No." Emma shook her head at Laurel. "That's Ben. Sam's the ad exec with the great smile."
"The one you decided not to date anymore," Parker added.
"Yes. And it wasn't actually a date. I said no to dinner, no to him picking me up. But . . . okay I caved on the party, and agreed to meet him there. I told him I wasn't going to sleep with him—full disclosure—two weeks ago. But I don't think he believes me. But Addison was there—third cousin, I think, my father's side. She's great, and just exactly his type. So I got to introduce them, and that was good."
"We should offer a matchmaking package," Laurel suggested, and started on leg lifts. "Even if we launched it just with the guys Emma wants to brush off, we could double our business."
"Brush off has negative connotations. I redirect. Anyway Jack was there."
"Our Jack?" Parker asked.
"Yeah, which turned out to be lucky for me. I ducked out early, and halfway home, my car conks. Just cough, choke, die. And it's snowing, and it's dark, I'm _freezing_ , and that stretch of road is deserted, of course."
As the leg lifts didn't look horrible, Emma shifted to mirror Laurel's movements.
"You really need to get OnStar installed," Parker told her. "I'll get you the information."
"Don't you think that's kind of creepy?" Mac huffed a little, pumping through the third set. "Having them know exactly where you are. And I think, I really think, they can hear you, even when you don't push the button. They're listening. Yes, they are."
"Because they love hearing people sing off-key with the radio. It must brighten their day. Who did you call?" Parker asked Emma.
"As it turned out, I didn't have to call anyone. Jack came along before I could. So, he takes a look, and it's the battery. He jumps it. Oh, and he lent me his jacket, which I forgot to give back. So instead of having a nice quiet evening, I'm dodging Sam's lips, trying to redirect him, standing in the freezing cold on the side of the road when all I wanted was a big salad and a romantic movie. Now I have to get my car in the shop, and make a trip to Jack's to return his jacket. And I'm completely swamped today. Just can't do it. So, irritable because . . ."
She hedged, just a little, as she rolled over to do the other leg. "I didn't sleep well worrying about getting everything done today and kicking myself for getting talked into going out in the first place."
She huffed out a breath. "And now that I said all that, it doesn't seem worth getting upset about."
"Breakdowns are always a bitch," Laurel said. "Breakdowns at night, in the snow? Serious pisser. You get a pass on the irritable."
"Jack had to point out that it was my own fault, and it's worse because, yes, it was, since I haven't had the car serviced. Ever. And that was annoying. But he did save the day, plus the jacket. Plus, he followed me home to make sure I got here. Anyway, that's all done. Now I have to hassle with having somebody check out the car and do whatever it is they do. I've got guys in the family who could probably take care of most of it, but I don't want yet another lecture on how I neglect my car, blah blah. So, Parker, where should I take it?"
"I know, I know!" Mac puffed, then stopped her reps. "You should take it in to that guy who towed my mother's car for me last winter. The one Del likes? Anybody who can basically tell Linda to stick it when she's on a rant gets my vote."
"Agreed," Parker said. "And he does get the Delaney Brown stamp of approval. Del's a maniac about who touches his cars. Kavanaugh's. I'll get you the number and the address."
"Malcolm Kavanaugh's the owner," Mac added. "Very hot."
"Really? Well, maybe a faulty battery's not such a bad thing. I'll try to get it in next week. Meanwhile, is anyone going into town, anywhere near Jack's office? I really have to stick here today."
"Give it back to him Saturday," Parker suggested. "He's on the list for the evening event."
"Oh. Fine." Emma looked with avid dislike at the elliptical. "Since I'm here, I might as well work up a sweat."
"How about me?" Mac demanded. "Am I cut yet?"
"The improvement's astounding. Biceps curls," Parker ordered. "I'll show you."
_B_ Y NINE, EMMA WAS SHOWERED, DRESSED, AND WHERE SHE wanted to be. At her work counter, surrounded by flowers.
To celebrate their parents' fiftieth anniversary, the clients wanted Emma to re-create the couple's wedding and backyard garden reception. Then kick it up a notch.
She had copies of snapshots from the wedding album pinned to a board, had added some concept sketches and diagrams, a list of flowers, receptacles, accessories. On another board she'd pinned Laurel's sketch of the elegantly simple three-tiered wedding cake ringed with bright yellow daffodils and pale pink tulips. Beside it was a photograph of the cake topper the family had commissioned, replicating the couple on their wedding day, down to the lace hemming the bell of the bride's tea-length skirt.
Fifty years together, she thought as she studied the photos. All those days and nights, birthdays and Christmases. The births, the deaths, the arguments, the laughter.
It was, to her, more romantic than windswept moors and fairy castles.
She'd give them their garden. A world of gardens.
She started with daffodils, potting them in long, moss-lined troughs, mixing in tulips and hyacinths, narcissus. Here and there she added trails of periwinkle. A half dozen times she filled a rolling cart, wheeled it back to her cooler.
She mixed gallons of flower food and water, filling tall glass cylinders. She stripped stems, cut them under running water and began arranging larkspur, stock, snapdragons, airy clouds of baby's breath, lacy asparagus fern. Soft colors and bold, she'd mass them at various heights to create the illusion of a spring garden.
Time ticked away.
She paused long enough to roll her shoulders, circle her neck, flex her fingers.
Using the foam holder she'd soaked, she circled it with lemon leaf to create a base she glossed with leaf shine.
She gathered roses for her holding bucket, stripped stems, barely bothered to curse when she nicked herself, cutting the stems to length to make the first of fifty reproductions of the bouquet the bride had carried a half century before.
She worked from the center out, painstakingly locking each stem in the form with adhesive. Stripping, cutting, adding—and appreciating the bride's choice of multicolored roses.
Pretty, Emma thought, happy. And when she tucked the holder in the squat glass vase, she thought: lovely.
"Only forty-nine to go."
She decided she'd start on that forty-nine after she took a break.
After carting bags of floral debris out to her composters, she scrubbed the green off her fingers and from under her nails at her work sink.
To reward herself for the morning's work, she took a Diet Coke and a plate of pasta salad out on her side patio. Her gardens couldn't compete—yet—with the one she was creating. But her happy couple had been married in southern Virginia. Give me a few weeks, she mused, pleased to see the green spears of spring bulbs, the freshening foliage of perennials.
Last night's snow was just a memory under blue skies and almost balmy temperatures.
She spotted Parker with a group of people—one of the day's potential clients doing the tour—crossing one of the terraces at the main house. Parker gestured toward the pergola, the rose arbor. The clients would have to imagine the abundance of white roses, the lushness of wisteria, but Emma knew the urns she'd planted with pansies and trailing vinca showed off very well. At the pond dotted with lily pads, the willows were just beginning to green.
She wondered if the prospective bride and groom would one day have a busy florist creating fifty bouquets to commemorate their marriage. Would they have children, grandchildren, great-grandchildren who loved them enough to want to give them that celebration?
With a small groan for muscles aching from the morning's exercise and the morning's work, she propped her feet on the chair across from her, lifted her face to the sun, and shut her eyes.
She smelled earth, the tang of mulch, heard a bird chittering its pleasure in the day.
"You've got to stop slaving away like this."
She jerked up—had she fallen asleep?—and blinked at Jack. Mind blank, she watched him pluck a curl of pasta from her plate, pop it into his mouth. "Good. Got any more?"
"What? Oh God!" Panicked, she looked at her watch, then breathed a sigh of relief. "I must've dozed off, but only for a couple minutes. I have forty nine bouquets left to make."
His brows drew together over smoky eyes. "You're having a wedding with forty-nine brides?"
"Hmm. No." She shook her head to clear the cobwebs. "Fiftieth anniversary, and a re-creation of the bridal bouquet for every year. What are you doing here?"
"I need my jacket."
"Oh, right. Sorry I forgot to give it back to you last night."
"No problem. I had an appointment down the road." He took another twirl of pasta. "Do you have any more of this? I missed lunch."
"Yeah, sure. I owe you lunch at least. Sit down. I'll get you a plate."
"I'll take it, and I wouldn't mind a hit of caffeine. Hot or cold."
"No problem." Studying him, she pushed at hair that escaped pins. "You look a little whipped."
"Busy morning. And I've got another site to visit in about forty-five minutes. You were between the two, so . . ."
"That's handy. Be right back."
He was whipped, he thought, and stretched out his legs. Not so much from the work, or the in-your-face with an inspector that morning. Which he would've handled better if he hadn't been sleep-deprived. Tossing and turning and trying to block out sex dreams of a Spanish-eyed lady would whip anyone.
So, of course, he had to be stupid and masochistic, and drop by with the excuse of the jacket.
Who knew how sexy she looked when she slept in the sunlight?
He did, now. It wasn't going to give him easier dreams.
The thing to do was get over it. He should make a date with a blonde or a redhead. Several dates with several blondes and/or redheads until he managed to put Emma back on the No Trespassing list.
Where she belonged.
She came out, his jacket over her arm, a tray in her hands.
She had, he thought, the kind of beauty that just slammed a man's throat shut. And when she smiled, the way she did now, it blew through him like a bolt of lightning.
He tried to build a No Trespassing sign in his head.
"I had some of my aunt Terry's olive bread," she told him. "It's great. I went with cold caffeine."
"That does the job. Thanks."
"No problem. And it's nice to have company on a break." She sat again. "What are you working on?"
"I'm juggling a few things." He bit into the bread. "You're right. It's great."
"Aunt Terry's secret recipe. You said you had a job near here?"
"A couple. The one I'm heading to's a never-ending. The client started out two years ago wanting a kitchen remodel, which moved into a complete reno of the master bath, which now includes a Japanese soaking tub, a sunken whirlpool, and a steam shower big enough for six friends."
She wiggled her brows over those gorgeous eyes, then took a bite of pasta. "Fancy."
"I kept waiting for her to ask if we could extend the addition a little more for the lap pool. But she turned her focus outside. She decided she wants a summer kitchen by the pool. She saw one in a magazine. She can't live without it."
"How does anyone?"
He smiled and ate. "She's twenty-six. Her husband's fifty-eight, rolling in it and happy to indulge her every whim. She has a lot of whim."
"I'm sure he loves her, and if he can afford it, why not make her happy?"
Jack merely shrugged. "Fine by me. It keeps me in beer and nachos."
"You're cynical." She pointed at him with her fork before she stabbed more pasta. "You see her as the bimbo trophy wife and him as the middle-aged dumbass."
"I bet his first wife does, but I see them as clients."
"I don't think age should factor into love or marriage. It's about the two people in it, and how they feel about each other. Maybe she makes him feel young and vital, and opened something new inside him. If it was just sex, why marry her?"
"I'll just say a woman who looks like she does has great powers of persuasion."
"That may be, but we've done a lot of weddings here where there's been a significant age difference."
He wagged his fork, then stabbed more pasta in a mirror of her move. "A wedding isn't a marriage."
She sat back, drummed her fingers. "Okay, you're right. But a wedding's a prelude, it's the symbolic and ritualistic beginning of the marriage, so—"
"They got married in Vegas."
He continued to eat, face bland as he watched her try not to laugh.
"Many people get married in Vegas. That doesn't mean they won't have many happy and fulfilling years together."
"By a transvestite Elvis impersonator."
"Okay, now you're making things up. But even if you're not, that kind of . . . choice shows a sense of humor and fun, which, I happen to believe, are important elements for a successful marriage."
"Good save. Great pasta." He glanced over to where Parker sat with potential clients on the main terrace. "Business seems to be clicking along."
"Five events this week on-site, and a bridal shower we coordinated off-site."
"Yeah, I'll be here for the one Saturday evening."
"Friend of Bride or Groom?"
"Groom. The bride's a monster."
"God, she really is." Emma leaned back and laughed. "She brought me a picture of her best friend's bouquet. Not because she wanted me to duplicate it, which she certainly did not. Hers is a completely different style, but she'd counted the roses, and told me she wanted at least one more in hers—and warned me she'd be counting them."
"She will, too. And I can pretty much guarantee no matter how good a job you do, she'll find fault."
"Yeah, we've figured that out. It's part of the job around here. You get monsters and angels and everything in between. But I don't have to think about her today. Today's a happy day."
He knew she meant it. She looked relaxed, and had a glow about her. Then again, she usually did. "Because you have fifty bouquets to make?"
"That, and knowing the bride of fifty years is going to love them. Fifty years. Can you imagine?"
"I can't imagine fifty years of anything."
"That's not true. You must imagine what you build lasting fifty years. Hopefully much longer."
"Point," he agreed. "But that's building."
"So's marriage. It's building lives. It takes work, care, maintenance. And our anniversary couple proves it can be done. And now I have to get back to them. Break's over for me."
"Me, too. I'll get this for you." He loaded up the tray, lifted it as they rose. "You're working alone today? Where are your elves?"
"They'll be here tomorrow. And there will be chaos as we start on the flowers for the weekend events. Today it's just me, about three thousand roses, and blissful quiet." She opened the door for him.
"Three _thousand_? Are you serious? Your fingers will fall off."
"I have very strong fingers. And if I need it, one of the pals will come by for a couple hours and help strip stems."
He set the tray on her kitchen counter, thinking, as he always did, that her place smelled like a meadow. "Good luck with that. Thanks for lunch."
"You're welcome." She walked him to the door where he stopped.
"What about your car?"
"Oh. Parker gave me the name of a mechanic, a place. Kavanaugh's. I'm going to call."
"He's good. Call soon. I'll see you Saturday."
He imagined her going back to her roses as he walked to his car. Of sitting, for hours, drenched in their scent, cleaning stems of thorns then . . . doing whatever it was she did, he decided, to make what women who took the plunge carried.
And he thought of how she'd looked when he'd come upon her, sitting in the sunlight, face tipped up, eyes closed, those luscious lips of hers just slightly curved as if she dreamed of something very pleasant. All that hair bundled up and slim dangles of silver at her ears.
He'd thought, briefly but actively, about just leaning down and taking that mouth with his. He could've played it light, made some crack about Sleeping Beauty. She had a sense of humor, so maybe she'd have gotten a kick out of it.
She also had a temper, he mused. She didn't cut it loose often, but she had one.
It didn't matter either way, he reminded himself, as he'd missed that opportunity. The bevy of blondes and redheads was a better idea than scratching this increasingly annoying itch where Emma was concerned.
Friends were friends, lovers were lovers. You could make a friend out of a lover, but you were on boggy ground when you made a lover out of a friend.
He was nearly to the job site when he realized he'd left his jacket on her patio.
"Shit. _Shit_."
Now he was like one of those idiots who deliberately left something at a woman's place so he had an excuse to go back and try to score. And that wasn't it.
Was it?
Shit. Maybe it was.
CHAPTER FOUR
_A_ T TWO FIFTEEN ON SATURDAY, EMMA HAD HER TROOPS LINED up to transform the event rooms from the cheerful Caribbean themed daytime wedding into what she privately thought of as the Paris Explodes event.
"Everything goes." Emma rolled to the toes of her move fast sneakers. "The bride wants all the remaining baskets, vases, centerpieces. We'll help them load up whatever hasn't already been given to guests. Beach and Tiffany, strip the garlands and swags, inside and out. Start with the portico, then move inside. Tink, you and I will start the changeover in the Grand Hall. When the portico's ready to be dressed, let me know. The bride's and groom's suites have already been changed over. New bride's due at three thirty for hair, makeup, dressing, and photos in her suite. We need the entrance, foyer, staircase complete by three twenty, and the Grand Hall complete by four. Terraces, pergola, and patios by four forty-five, Ballroom complete by five forty five. If you need extra hands get me or Parker. Let's do this."
With Tink beside her, Emma shot off like a bullet. Tink, she knew, was reliable when she wanted to be—which was about seventy-five percent of the time. But Emma only had to show her or explain something to her once. She was a talented florist, again when she wanted to be. And was, to Emma's mind, almost spookily strong.
Tiny and toned, her wildly chopped boot-black hair liberally streaked with cotton-candy pink for spring, Tink attacked the mantel dressing like a whirlwind.
They stripped, boxed, dragged, hefted, and hauled candles of mango orange and surf white, garlands of bougainvillaea, pots of ferns and palm trees.
Tink snapped the gum she was never without and wrinkled her nose so the silver hoop in it glinted. "If you're going to want palm trees and shit, why don't you just go to the beach?"
"If they did, we wouldn't get paid to create the beach."
"Good point."
When she got the signal, Emma deserted the hall for the portico. She twined and draped and swagged miles of white tulle, acres of white roses to create a regal entryway for the bride and her guests. Colorful pots of hibiscus and orchids made way for enormous white urns filled with a forest of lilacs.
"Bride and Groom One and all guests checked out," Parker told her. She stood in her simple gray suit, her BlackBerry in one hand, her beeper hooked to her pocket, and her earbud dangling. "My God, Emma, this looks amazing."
"Yeah, it's coming along. She balked on the lilacs—too simple a flower, according to Monster Bride, but I found a picture that convinced her." She stepped back, nodded. "Okay, yeah. Excellent."
"She's due in twenty."
"We'll make it."
Emma hustled inside to where Tink and Tiffany worked on the staircase. More tulle, more white roses, these twined with fairy lights, with long swags of roses dripping down every ten inches. Perfect.
"Okay, Beach, entry and gift table arrangements. We can haul over the first of the Grand Hall pieces, too."
"I can get you Carter." Parker tapped her beeper. "I drafted him to help in the Ballroom, but I can spare him."
"Handy to have Mac hooked with a strong, willing back. I'll take him."
With the gangly Carter and her fireplug Beach, Emma transported pots, vases, baskets, greenery, garland, swags, and candles.
"MB's pulling in." Parker's voice sounded through Emma's headset and made her snort. Monster Bride.
She put the finishing touches on the mantel, lush with white and silver candles, white roses, and lavender lisianthus, before making the dash to wade into the outdoor arrangements.
She set more lilacs in more urns, muscled enormous silver baskets filled with calla lilies in eggplant and snowy white, hung cones of flowers dripping with silver ribbon on the white-draped aisle chairs, and guzzled water like a dying woman.
"Man, is this the best you can do?"
Rubbing the aching small of her back, Emma turned to Jack.
He stood, hands in the pockets of a gorgeous gray suit jacket, eyes shaded against the beaming sunlight by Oakleys.
"Well, she wanted simple."
He laughed, shook his head. "It looks amazing, and somehow elaborately French."
"Yes." She pointed a finger at him. "Exactly my plan. Wait!" Panic leaped in her chest like a terrier after a bone. "What are you doing here? What time is it? We can't be that far behind. Parker would—" She broke off as she checked her watch. "Oh, thank God. You're really early."
"Yeah. Parker mentioned to Del since I was coming, maybe I could make it early and pitch in. So I'm here to pitch."
"Come with me. Tink! I need to get the bouquets. Finish up—ten minutes—then start on the Ballroom."
"On it."
"You can help me load. I'm heading over to get them now," she said into her headset. "Oh, slip a Xanax in her champagne, Parker. I can't move any faster. Ten minutes. Have Mac stall her."
Moving at a jog now, she reached the van she used for transport, then jumped behind the wheel.
"Do you do that often?" Jack asked her. "Drug the bride?"
"We never do it, but we want to with some of them. And really, we'd be doing everyone a favor. This one wants her bouquet and she wants it now because if she doesn't _love_ it, there's going to be hell to pay. Laurel breezed by earlier and told me Mac told her the MB made her hairdresser cry and had a fight with her MOH. Parker smoothed it out, of course."
"MB?"
"Think about it," Emma suggested, and jumped out of the van to dash into her workshop.
He did as he followed her inside. "Mean Bitch. Monster Bitch. No, Monster Bride."
"Ding, ding, ding." She hauled open the door of her cooler. "Everything on the right goes. One rose cascade bouquet, twelve, count them twelve, attendant bouquets." She tapped one of the boxes. "Do you know what this is?"
"A bouquet. A purplish sort of thing. Pretty cool looking, actually. I've never seen anything like it."
"It's kale."
"Get out."
"Ornamental kale, variegated purple and green. Bride's colors are purple and silver. We've used a lot of silver accents and tones from pale orchid to deep eggplant, with lots of white and green in the arrangements."
"Son of a bitch. Cabbage bouquets. You didn't tell her what it is."
"Only after I made her fall in love with it. Okay, bouquets, corsages, boutonnieres, both the pomanders—she has two flower girls, two halos of white roses and lavender, and holding vases. Check, check, double check. Let's load them up."
"Do you ever get sick of flowers?" he asked her as they carried boxed bouquets.
"Absolutely not. Do you smell that lavender? Those roses?"
"Impossible not to, under the circumstances. So, a guy's taking you out. First date or some special deal, and he brings you flowers. You're not like: Oh, flowers. Great."
"I'd think he was very thoughtful. God, every muscle in my body is begging for a glass of wine and a hot bath." She stretched her back when Jack closed the cargo doors. "Okay, let's go knock the MB's socks off. Oh wait. Your jacket. The one you lent me. It's inside."
"I'll get it later. So, did she get one more rose than her friend?"
Emma blanked for a moment, then remembered telling him about the bouquets. "Ten more. She'll bow to me before I'm done with her. Yes, Parker, yes, I'm on my way." Even as she spoke, her beeper sounded. "Now what? Can you read that? I can't get to it while I'm driving. It's hooked to my skirt, right under the jacket on your side."
He lifted the hem of the jacket, and his fingers brushed her skin just above her waist as he tilted the beeper. She thought, uh-oh, and kept her eyes straight ahead.
"It says DTMB! Mac."
"DTMB?" His knuckles continued to rest there, just above her waist. Very distracting. "Ah . . . Death to Monster Bride."
"Any answer? Suggestions on the method maybe?"
She managed a smile. "Not at this time. Thanks."
"Nice jacket," he said and smoothed it back into place.
She stopped in front of the house. "If you help me haul all this up, I won't tell Parker or give you grief when you sneak off to the Grand Hall for a beer before the wedding."
"That's a deal."
With her, he carried boxes into the foyer. He stopped a moment, took a survey. "You do good work. If she doesn't bow to you, she's a bigger idiot than I already think she is."
"Shh!" She stifled a laugh, rolled her eyes. "You don't know who's wandering around from the immediate family or wedding party at this stage."
"She knows I can't stand her. I told her."
"Oh, Jack." She did laugh now as she hurried up the steps. "Don't do or say anything to set her off. Consider the Wrath of Parker before you speak."
Emma balanced the box she carried and opened the door to the Bride's Suite.
"There you are. Finally! Emmaline, really, how am I supposed to take my formal portraits without my bouquet? And now my nerves are just _shot_! You know I wanted to see it early enough so you could make changes if I wanted them. Do you know what time it is? Do you?"
"I'm sorry, I didn't hear a word you said. I'm just dazzled. Whitney, you look absolutely spectacular."
That much, at least, was true. With miles of skirt, a universe of pearls and beads sparkling on the train, the bodice, and her expertly low-lighted blond hair swept up and crowned with a tiara, Monster Bride was magnificent.
"Thank you, but I've been a wreck worrying about the bouquet. If it's not perfect—"
"I think it's exactly what you hoped for." Carefully, Emma lifted the massive cascade of white roses from the box. She did a mental C-jump when the bride's eyes popped wide, but kept her tone professional. "I tweaked the temperature so the roses would just be partially open. And just hints of green and the silver beads to set off the blooms. I know you talked about trails of silver ribbons, but I really think that would take away from the flowers, and the shape. But I can add it in no time if you still want it."
"The silver would add a sparkle, but . . . Maybe you're right." Whitney reached out to take the bouquet.
Nearby the mother of the bride pressed her palms together as if in prayer and lifted them to her lips.
Always a good sign.
Whitney turned, studied herself in the full-length mirror. And smiled. Emma stepped beside her to whisper in her ear. And the smile widened.
"You can count them later," Emma suggested. "Now I'll turn you over to Mac."
"Let's try between the windows over here, Whitney. The light's wonderful." Mac gave Emma a thumbs-up behind the bride's back.
"Now, ladies," Emma said, "it's your turn."
She distributed bouquets, corsages, set out the holding vases, then put the MOG in charge of the pomanders and flower girls.
She stepped out again, glanced at Jack. "Whew."
"The 'maybe you're right'? From her, that's a bow."
"Understood. I can take it from here. Go get that beer. Carter's around here somewhere. Corrupt him."
"I try, but he's a hard nut to crack."
"Boutonnieres," she said, already on the move again. "Then I need to check on the Ballroom." She looked at her watch. "We're right on schedule, so thanks. I'd be running behind if you hadn't helped me load and haul."
"I can take up the boutonnieres. It'd give me a chance to see Justin, make bad jokes about balls and chains."
"Good idea. Do that." With the few minutes of time that bought her, she opted to swing through the Grand Hall, out onto the terrace.
Satisfied after a few tweaks, she climbed up to the Ballroom where her team was well underway. Emma pushed up her sleeves and dived in.
While she worked, Parker gave periodic updates, and started the countdown in her ear.
_Guests still trickling in. Most are seated or on the terrace._
_Formal prewedding shots complete. Mac's on the move._
_Grandparents escorted in two minutes. I'm bringing the boys down. Laurel, get ready for the pass-off._
"Roger that," Laurel said dryly. "Em, cake's assembled and ready for the table decor anytime."
_Boys passed off to Laurel,_ Parker announced a moment later as Emma finished with a stand of hydrangeas. _MOG escorted by BOG in one. MOB on deck. Escort is BOB. Queuing up attendants. Music change on my mark._
Emma walked back to the entrance doors, shut her eyes for ten seconds, then opened them to take in the entire space. She drew a breath in, let a breath out.
Paris Explodes, she thought, but it did so in lush style. Whites, silvers, purples, touches of green to set them off spilled, spread, speared, and shimmered under a perfect April sky. She watched the groom and his party take their places in front of a pergola simply smothered in flowers.
"Guys, we rule. We _kill_. You're done. Hit the kitchen for food and drink."
Alone, she took one last circuit of the room as Parker signaled the attendants to _go!_ one by one. Then Emma sighed, rubbed her back, the back of her neck, her hands. And went to change into her heels as Parker gave the MB her cue.
_J_ ACK DIDN'T KNOW HOW THEY PULLED IT OFF, EVERY TIME, ALL the time. He'd been drafted to lend a hand now and again at an event. Hauling and lifting, bartending, even bussing tables in a pinch. As payment invariably included great food, drinks, and music, he never minded.
But he still didn't know how they managed to pull it all together.
Parker consistently managed to be everywhere at once, and so subtly he suspected no one really noticed she might be prepping the best man on his toast one minute and passing out a pack of tissues to the mother of the bride the next while coordinating the service of the meal in the Grand Hall like a general coordinating troops during battle.
Mac popped up all over the place, too, and was just as cagey about it as she shot candids of the wedding party or the guests, or maneuvered the bride and groom into a quick posed photo.
Laurel streamed in and out, signaled, he supposed, through the headset they all wore, or by some sort of hand signal. Maybe mental telepathy. He wouldn't discount that one.
And Emma, of course, on the spot when a guest spilled wine on the tablecloth, or when the bored ring bearer started to poke at one of the flower girls.
He doubted anyone noticed or understood there were four women literally holding everything together, juggling all the balls and passing them to each other with the grace and skill of NFL quarterbacks.
Just as he imagined no one knew the logistics and sheer timing involved in leading the guests from the Hall to the Ballroom. He lingered while Emma and her team along with Laurel swarmed on the head table to gather up the bouquets and holding vases.
"Need any help?" he asked her.
"Hmm? No, thanks, we've got it. Tink, six on either side, baskets on the end. Everything else stays in place for two hours here before undressing and loading. Beach, Tiff, snuff the candles, leave the overheads on half."
"I can get that," Tink said when Emma took the bride's bouquet.
"One bruised rose and she'll go on attack. Better she rips my throat out than yours. Let's go, first dance is starting."
While the flowers headed up the back stairs, Jack wandered to the main. He slipped into the Ballroom in the middle of the first official dance. The bride and groom chose what he considered the overused and overorchestrated "I Will Always Love You," while people stood in the flower-drenched Ballroom or sat at one of the tables strategically arranged around the dance floor.
The terrace doors stood open, inviting guests to stroll outside. He thought he'd do just that once he got a glass of wine.
When he saw Emma ducking out again, he adjusted his plan. Carrying two glasses of wine, he went down the back stairs.
She sat on the second level, and popped up like a spring when she heard his footsteps. "Oh, it's only you." She sank back down on the steps.
"Only me is bearing wine."
She sighed, circled her head on her neck. "We at Vows frown on drinking on the job. But . . . I'll lecture myself tomorrow. Hand it over."
He sat down beside her, gave her the glass. "How's it going?"
"I should ask you. You're a guest."
"From the guest point of view, it's a smash. Everything looks great, tastes great, smells great. People are having fun and have no idea the whole business is clicking along on a timetable that would make a Swiss train conductor weep in admiration."
"Exactly what we're after." She sipped the wine, shut her eyes. "Oh God, that's good."
"How's the MB behaving?"
"She's actually not too bad. It's hard to be bitchy when everyone's telling you how beautiful you look, how happy they are for you. She actually did count the roses in her bouquet, so that made her happy. Parker's smoothed over a couple of potential crises, and Mac actually got a nod of approval over the B and G shots. If Laurel's cake and dessert table pass muster, I'd say we hit all the hot spots."
"Did she do those little crème brûlées?"
"Oh, yeah."
"You're gold. Lot of buzz on the flowers."
"Really?"
"I actually heard gasps a few times—the good kind."
She rolled her shoulders. "Then it's all worth it."
"Here."
He boosted himself up a stair, straddled her from behind, and dug his fingers into her shoulders.
"You don't have to . . . Never mind." She leaned back into his hands. "Carry on."
"You've got some concrete in here, Em."
"I've got about a sixty-hour week in there."
"And three thousand roses."
"Oh, adding the other events, we could double that. Easily."
He worked his thumbs up the back of her neck, made her groan. And as his stomach knotted in response, realized he wasn't doing himself any favors. "So . . . how'd the fiftieth go?"
"It was lovely, really lovely. Four generations. Mac got some wonderful pictures. When the anniversary couple had their first dance, there wasn't a dry eye in the house. It goes down as one of my all-time favorite events."
She sighed again. "You have to stop that. Between the wine and your magic hands I'm going to end up taking a nap right here on the steps."
"Aren't you done?"
"Not even close. I have to get the tossing bouquet, help out with the cake service. Then there's the bubbles, which we hope to do outside. In an hour, we'll start breaking down the Grand Hall, boxing centerpieces and arrangements."
Her voice went a little thick, a little sleepy when he kneaded her neck. "Um . . . Loading up those, and the gifts. Loading up the outdoor arrangements. We have an afternoon event tomorrow, so we'll break down the Ballroom, too."
He tortured himself, running his hands down her biceps, back up to her shoulders. "Then you should relax while you can."
"And you should be upstairs enjoying the party."
"I like it here."
"So do I, which makes you a bad influence with your wine and staircase massages. I have to get back up, relieve Laurel on patrol." She reached back, patted his hand before she rose. "Cake cutting in thirty."
He got to his feet as she started up. "What kind of cake?"
She stopped, turned, and ended up on level with him. Her eyes, those deep velvet eyes, looked sleepy to match her voice. "She's calling it her Parisian Spring. It's this gorgeous pale lavender blue covered with white roses, sprigs of lilac, with this soft milk chocolate ribboning and—"
"I was more about what's inside."
"Oh, it's her genoise with Italian meringue buttercream. You don't want to miss it."
"It may beat out the crème brûlée." She smelled like flowers. He couldn't say which ones. She was a mysterious and lush bouquet. Her eyes were dark and soft and deep, and her mouth . . . Wouldn't it taste every bit as rich as Laurel's cake?
The hell with it.
"Okay, this is probably out of line, so apologies in advance."
He took her shoulders again, eased her to him. Those dark, soft, deep eyes widened in surprise an instant before his lips took hers.
She didn't jerk away, or laugh it off as a joke. Instead she made the same sort of sound she had when he'd rubbed her neck—just a little breathier.
Her hands clamped on his hips, and those luscious lips of hers parted.
Like her scent, her flavor was mysterious and essentially female. Dark and warm and sensual. When her hands moved up his back, he took more. Just a little more.
Then he changed angles, took more still, and pleasure hummed in her throat.
He thought of just snatching her up, carrying her off to whatever dark room he could find to finish what a moment of impulse had begun.
The beeper at her waist sounded, and both of them jolted. She made a strangled sound, then managed, "Oh. Well." In a jerky move she unclipped the beeper, stared at it. "Parker. Um. I have to go. I have to . . . go," she said, then turned and bolted up the stairs.
Alone, he lowered to the stairs again and finished off his neglected wine in two long gulps. He decided he'd skip the rest of the reception, and take a long walk outside instead.
_E_ MMA COULD ONLY BE GRATEFUL WORK KEPT HER TOO BUSY TO actually think. She helped clean up an incident involving the ring bearer and chocolate éclairs, delivered the tossing bouquet, rearranged the decor on the cake table to ease the serving, then began the stripping down of the Grand Hall.
She readied centerpieces and other arrangements for transport and supervised the loading of them for the proper recipients.
When the bubbles were blown and the last dance finished, she began the same process on the patios and terraces.
She didn't see a trace of Jack.
"Everything okay?" Laurel asked her.
"What? Yes. Sure. Everything went great. I'm just tired."
"Right there with you. At least tomorrow's event will be a breeze after today. Have you seen Jack?"
"What?" She jumped like a thief at the shrill of an alarm. "Why?"
"I lost track of him. I planned to bribe him with pastries to help with the breakdown. I guess he skipped."
"I guess. I wasn't paying attention."
_Liar, liar_. Why was she lying to her friend? It couldn't be a good sign.
"Parker and Mac are seeing off the stragglers," Laurel commented. "They'll do the security check. Do you want me to help you cart these to your place?"
"No, I've got it." Emma loaded the last of the leftovers she'd put back in the cooler. She'd donate the bulk to the local hospital, take the rest apart and make smaller arrangements to put around her place, and her friends'.
She closed the cargo doors. "See you in the morning."
She drove the van home, reversed the process and carried flowers and garlands into her cooler.
No matter how firmly she ordered her mind to stay calm and blank, it just kept opening up to one single thought.
Jack kissed her.
What did it mean?
Why should it _mean_ anything?
A kiss was just that. It had just been a product of the moment. Nothing more.
She readied for bed, trying to convince herself it was nothing more.
But when a kiss blew right off the spark-o-meter, blasted through the scale, it was hard to describe it as "nothing more."
Something else was what it was, she admitted. And she didn't know what to do about it. That was frustrating because she _always_ knew what to do when it came to men and kisses and sparks. She just knew.
She climbed into bed telling herself since she'd never be able to sleep, she'd just lie there in the dark until she came up with a solution.
And she dropped away in seconds, pushed off the edge by sheer exhaustion.
CHAPTER FIVE
_E_ MMA GOT THROUGH THE SUNDAY EVENT AND HER MONDAY consults and adjusted the arrangements for some upcoming events due to changes of bridal minds.
She canceled two dates with two perfectly nice men she now had no desire to spend evenings with. She filled those evenings by doing inventory and ordering ribbons, pins, containers, forms.
And wondering if she should call Jack and make some light, breezy comment about the kiss—or pretend it never happened.
She alternated between the top options and a third, which involved going over to his house and jumping him. So she ended up doing nothing but tying herself into knots over it.
Annoyed with herself, she arrived early for a scheduled afternoon staff meeting. She cut through Laurel's kitchen, where her friend was arranging a plate of cookies beside a small fruit and cheese platter.
"I'm out of Diet Coke," Emma announced and opened the fridge to take one. "I'm out of almost everything because I keep forgetting my car battery is dead as disco."
"Did you call the garage?"
"That, at least, I remembered to do about ten minutes ago. When I confessed—under expert interrogation by the guy—that I've owned the car for four years, have never taken it in for a tune-up, couldn't remember exactly the last time, if ever, I've had the oil changed or some computer chip check job thing and other car business I don't remember now, he said he'd have it picked up, taken in."
Pouting a little, she popped the top and drank straight from the can. "I sort of felt as if I'd been holding my car hostage and he's releasing it. He made me feel like even more of an idiot than Jack did. I want a cookie."
"Help yourself."
Emma picked one up.
"Now I'm going to be without a car until he decides to give it back. If he does, and I'm not entirely sure he intends to."
"You've been without a car for over a week because your battery's dead."
"True, but I had the illusion of a car because it was sitting there. I guess I need to take the van and go to the grocery store, and the zillion other places I've put off going. I'm actually afraid to, as it occurred to me I've had the van for a year more than the car. It may rebel next."
Laurel tossed some pretty pastel mints on the cookie tray. "I know it's a crazy idea, but maybe once you get your car back, you can have the garage service the van."
Emma nibbled at the cookie. "The car guy tossed that idea in the hat. I need consolation. How about dinner and movie night?"
"Don't you have a date?"
"I canceled. I'm not in the mood."
Laurel blew hair out of her eyes, the better to stare in shock. " _You're_ not in the mood for a date?"
"I have to get an early start tomorrow. Six hand-tied bouquets, and the bride's makes seven. That's a good six, seven hours of work. I have Tink coming in for half a day, so it cuts it back, but there's all the rest to put together for the Friday night event. And I spent most of the morning processing the flowers."
"That's never stopped you before. Are you sure you're feeling all right? You've been just a shade off."
"No, I'm fine. I'm good. I'm just not . . . in the mood for men."
"That couldn't include me." Delaney Brown walked in, lifted Emma off her feet to give her a resounding kiss. "Mmm. Sugar cookie."
Emma laughed. "Get your own."
He plucked one from the tray, grinned at Laurel. "Consider it part of my fee."
Going from experience, Laurel got out a Ziploc bag and began to fill it with cookies. "Are you in on the meeting?"
"No. I just had some legal business to go over with Parks."
Since it was there and so was he, Del went to the coffeepot.
He and Parker shared the dark brown hair, the dark blue eyes. What Laurel would have called their refined features were just a little more roughly carved on him. In the smoke gray pin-striped suit, Italian shoes, and Hermès tie, he looked every bit the successful Connecticut lawyer. The scion of the Connecticut Browns.
With the food prep complete, Laurel untied her baker's apron and hung it on a peg.
Del leaned on the counter. "I hear you kicked some ass with the Folk wedding last weekend."
"Do you know them?" Emma asked.
"Her parents are clients. I haven't had the pleasure—though from what Jack says that may be overstating—of meeting the new Mrs. Harrigan."
"You will when they file for divorce," Laurel said.
"Always the optimist."
"She's a nightmare. She sent Parker a critique list this morning. E-mailed from Paris. From her honeymoon."
"You're kidding!" Stunned, Emma gaped at Laurel. "It was perfect. Everything was perfect."
"The champagne could've been colder, the wait service faster, the sky bluer, and the grass greener."
"Well, she's just a bitch. After I gave her ten more roses. Not one, but _ten_." Emma shook her head. "It doesn't matter. Everyone who was there, and who was an actual human, knows it was perfect. She can't spoil it."
"That's my girl." Del toasted her with his coffee.
"Anyway, speaking of Jack, have you seen him? I mean, will you be seeing him?"
"Tomorrow, actually. We're heading into the city to catch the Yankees."
"Maybe you could take him his jacket. He left his jacket. Or I forgot to give it back. Anyway, I have his jacket, and he probably wants it. I can go get it. It's in my office. I can just go get it."
"I'll go by and get it on my way out."
"Good. That'd be great. Since you're seeing him anyway."
"No problem. I'd better get going." He picked up the bag, shook it lightly at Laurel. "Thanks for the cookies."
"A baker's dozen, including the one you ate, will be deducted from your fee."
He shot Laurel a grin, and sauntered out.
Laurel waited a few beats then pointed at Emma. "Jack."
"What?"
"Jack."
"No," Emma said slowly, laying her hand between her breasts. "Emma. Em-ma."
"Don't be funny, I can see right through you. You said 'any way' three times in under a minute."
"No, I didn't." Maybe she had. "And so what?"
"So, what's going on with you and Jack?"
"Nothing. Absolutely nothing. Don't be ridiculous." She felt the lie burning her tongue. "You can't say anything to anyone."
"If I can't say anything, it's not nothing."
"It is nothing. It's probably nothing. I'm overreacting. Damn it." Emma popped the half a cookie she had left in her mouth all at once.
"You're eating like a normal person. Something is wrong in the Emma-verse. Spill."
"Swear first. You won't say anything to Parker or Mac."
"You drive a hard bargain." Laurel swiped a fingertip diagonally across her breasts, then pointed it to the ceiling. "Sworn."
"He kissed me. Or we kissed each other. But he started it, and I don't know what would've happened next because Parker beeped me. I had to go, then he left. So, that's it."
"Wait, I lost the sense of hearing right after you said Jack kissed you."
"Cut it out. This is serious." She bit her lip. "Or it's not. Is it?"
"This isn't like you, Em. You are the goddess of handling men and romantic or sexual situations."
"I _know_. It's just this is Jack. It's not supposed to be . . ." She waved her arms in the air. "Something to handle. I'm making too much of it. It was just a moment, just the circumstances. Just a thing. Now it's done, so it's not a thing."
"Emma, you tend to romanticize men, potential relationships, but you never get flustered over them. You're flustered."
"Because it's Jack! What if you were standing around, minding your own business, baking, and Jack came in and kissed you stupid. Or Del did. You'd be flustered."
"The only reason either of them come in here is to mooch baked goods. As Del just demonstrated. When did this happen? The night you broke down?"
"No. It almost did. There was a second there . . . I think because there was a second there, it just led into it happening. During the reception Saturday."
"Right, right, you said Parker beeped you. Well, how was it? How did it rank on the patented Emmaline Grant spark-o-meter?"
Emma let out a breath, pointed her thumb up, then swiped a hand through an imaginary line. "Slapped the top of the red zone before it broke the meter."
With her lips pursed, Laurel nodded. "I always suspected that about Jack. He has that red zone vibe about him. What are you going to do about it?"
"I don't know. I haven't decided. It's thrown me off. I need to get my balance back, then figure out what to do. Or not do."
"Then you have to tell me, and also let me know when the gag order is lifted."
"All right, but meanwhile, not a word." Emma picked up the cheese tray. "Let's go be businesswomen."
Vows housed its conference room in what had been the library. The books remained, framing the room and giving way in spaces for photos and mementos. The room maintained its warmth, its elegance, even as it served for business.
Parker sat at the big inlaid table, laptop and BlackBerry at the ready. As the morning client meetings and tours were complete for the day, she'd hung her suit jacket on the back of the chair. Mac sat across from her, long legs stretched out, wearing the jeans and sweater that served her for her workday.
When Emma set the tray on the table, Mac levered herself up to snag a cluster of grapes. "You guys are late."
"Del stopped by the kitchen. Before we start business, who's up for dinner and a movie night?"
"Me, me!" Mac shot up a hand. "Carter has a teacher thing, and that saves me from working until he gets back. I put in a full one today."
"As it happens, my calendar is clear." Laurel laid the cookie plate beside the platter.
Parker merely picked up the house phone, pressed a button. "Hey, Mrs. G, can you handle the four of us for dinner? That'd be great. Thanks." She hung up. "We'll have chicken and like it."
"Works for me." Mac bit into a grape.
"All right then, the first order of business is Whitney Folk Harrigan, aka Monster Bride. As Laurel knows, I received an e-mail from her wherein she lists several bullet points addressing what she feels we could improve."
"Bitch." Mac leaned up this time to spread some goat cheese on a rosemary cracker. "We kicked severe ass on that event."
"We should've kicked her severe ass," Laurel commented.
"Whitney feels, in no particular order of importance, that . . ." Parker opened a file to read from the e-mail she'd printed out. "The champagne was inadequately chilled, the service during dinner was slow, the gardens lacked enough color and bloom, the photographer spent more time than she deems necessary on the wedding party when the bride deserved more attention, and the offerings on the dessert table weren't as varied or as well presented as she'd hoped. She adds that she felt rushed and/or neglected by the wedding planner during some parts of the event. She hopes we'll take these criticisms in the spirit with which they're offered."
"To which I respond . . ." Mac shot up a middle finger.
"Succinct." Parker nodded. "However, I responded with our thanks for her comments, and our hopes that she and Justin enjoy Paris."
"Panderer," Laurel muttered.
"You bet. I could've responded with: Dear Whitney, you're full of shit. Which was my first thought. I restrained it. I have, however, upgraded her to Monster Bitch Bride."
"She must be a genuinely unhappy person. Seriously," Emma said when her friends just looked at her. "Anyone who could take a wedding day like we provided for her and pick it apart is just innately unhappy. I'd feel sorry for her if I wasn't so mad. I will feel sorry for her when I stop being mad."
"Well, mad, sorry, or fuck you, the upside is we've had four new tours booked through that event. And I expect more."
"Parks said fuck." Mac grinned and ate another grape. "She's very mad."
"I'll get over it, especially if we book four more events as a result of the stupendous job we did on Saturday. For now, I'm putting Whitney in my newly designed Closet of Doom, where everything makes her look fat, all the patterns are polka dots, and the color choices are puce or dead-flesh beige."
"That's really mean," Laurel commented. "I like it."
"Moving on," Parker continued. "Del and I met about some of the legal and financial issues of the business. The partnership agreement is coming up for renewal, which includes the percentage funneled back into Vows from the individual arms for outside events. If anyone wants to discuss changes to the agreement, including the percentages, the floor's open."
"It's working, isn't it?" Emma glanced around at her partners. "I don't think any of us really imagined we'd build what we've built when we started Vows. Not just financially, which is certainly more than I'd have made by now if I'd been able to open my own shop. But, Monster Bitch Bride aside, the reputation we've earned, together and individually. The percentage is fair, and the fact is, the cut Del takes for his part of the estate is way below what he could've asked. We're all doing what we love with people we love. And we're making a good living at it."
"I think what Em's saying is: Sign me up." Mac popped another grape. "I say ditto."
"I'm right there," Laurel added. "Is there any reason to change anything?" she asked Parker.
"Not from my perspective, but as Del advised—in his legal function—each of you should read over the agreement again, and voice any reservations, make any suggestions before we renew."
"I suggest we have Del draw up the papers, sign them, then open a bottle of Dom."
Mac pointed at Emma in agreement. "Seconded."
"And the 'ayes' have it," Laurel announced.
"I'll let him know. I've also had a discussion with our accountant."
"Better you than me," Laurel said.
"Much better." Parker smiled and sipped some water. "We've had a strong first quarter, and are on track to increase our net profit by about twelve percent over last year. I'm advised we should consider rolling a portion of the net back into the business. So, if any or all of you have a need, whim, or selfish desire for additional equipment, or ideas on what Vows could use as a whole, we can work out what we should spend our money on, and how much we should spend."
Emma shot her hand up before anyone could speak. "I've been thinking about this, especially after I looked at my books for the last quarter. We have our biggest event, to date, next spring with the Seaman wedding. The flowers alone are going to outstrip the capacity of my cooler, so we'll need to rent another for several days. I may be able to find a used one for a cost that could make it more practical, in the long term, than renting."
"That's good." Parker made a note. "Get some prices."
"This may be the time," Emma continued, "considering that event, and the increase we're seeing in business, to buy some of the other equipment we usually rent. The additional outdoor seating, for instance. Then, when we do an outside event, _we_ rent it to the client and pocket the fee. And—"
"You really have been thinking," Mac commented.
"I really have. Since Mac's already planning to add on to her place, increasing the upstairs living area to accommodate true love, why not add on to the work space, the studio space at the same time? She needs more storage space, a real dressing room instead of the little powder room. And while I'm rolling, the mudroom off Laurel's kitchen is really redundant, as we have one off the main kitchen. If that was converted, she could have an auxiliary kitchen in there, another oven, another cooler, more storage."
"We'll just let Emma do the talking," Laurel put in.
"And Parker needs a computerized security system so she can monitor all the public areas of the house."
Parker waited a beat. "I think you've spent that net profit increase several times."
"Spending money's the fun part of earning it. You be Parker, and that'll keep us from going wild. But I really think we ought to do at least some of those things, and put the others on the list for as soon as possible down the road."
"Being Parker then, I'll say the cooler makes sense. See what you can find. Since we'd need to talk to Jack on how to work the cooler into your space, we can ask him to give us an idea how to add on to Mac's studio, and refit the mudroom."
She made more notes as she spoke. "I'd thought of the furniture buy already, and I've started researching the cost there. I'll get projections so we know where we stand on all of this, then we can decide which makes the most sense first."
Nodding, she flipped over to the next order of business.
"Now, upcoming events that will help pay for our hopes and dreams. The commitment ceremony. They got their vows and the script for the ceremony to me today. Friday evening ceremony with, after a coin toss, Allison, now known as Bride One, arriving at three thirty, and Marlene, now Bride Two, at four. Bride One takes Bridal Suite, Bride Two Groom's Suite. As they share a MOH, she's going to float between the suites. Bride One's brother is BM, so we'll use the second floor family parlor for him, and the FOBs, as needed. BM will stand on B-One's side during the ceremony, MOH on B-Two's."
"Wait." Mac held up a finger as she keyed the details into her laptop. "Okay."
"These ladies know exactly what they want and stick to a plan, so they've been extremely easy to deal with on my end. MOB-One and siblings of B-Two aren't particularly happy with the formalization of this relationship, but are cooperating. Mac, you may have to work to get the shots the clients hope for that include them."
"No problem."
"Good. Emma, flowers?"
"They wanted unconventional, but feminine. Neither wanted to carry a bouquet, so we've gone with a headpiece for Allison and flower combs for Marlene. A halo for the MOH who'll carry four white roses. They'll exchange single white roses during the ceremony, right after the lighting of the unity candle. And each will give her mother a rose. White rose boutonnieres for the men. It should be very pretty."
Emma scrolled over to arrangements as she sipped her Diet Coke. "They wanted an airy, meadowy look for arrangements and centerpieces. I'm using a lot of baby's breath and painted daisies, Shastas and gerberas, branches of blooming cherry, wild strawberries, and so on. Minimal tulle, and garlands I'm doing like daisy chains. Bud vases for the roses during the reception.
"A lot of fairy lights and candles, Grand Hall and Ballroom, with a continuation of the natural look for arrangements. It'll be simple and very sweet, I think. If one of you can help me transport, I can do the setup solo."
"I can do that," Laurel told her. "The cake's the vanilla sponge with raspberry mousse filling, topped with Italian meringue. They wanted simple flowers there, too, echoing Emma's. I don't need to add those to the cake until around five, so I'm clear for setup. Otherwise, they want assorted cookies and pastel mints."
"We have the standard Friday night itinerary," Parker added, "excluding bouquet and garter toss. Rehearsal Thursday afternoon, so if there are any glitches, we'll deal with them then. Saturday," she began.
_W_ HENEVER EMMA THOUGHT OF HER PARENTS, HOW THEY MET, fell in love, it ran through her mind like a fairy tale.
Once upon a time there was a young woman from Guadala jara who traveled across the continent to the great city of New York to work in the business of her uncle, to tend the homes and children of people who needed or wanted their homes and children tended. But Lucia longed for other things, a pretty home instead of a noisy apartment, trees and flowers instead of pavement. She worked hard, and dreamed of one day having her own place, a little shop perhaps, where she would sell pretty things.
One day her uncle told her of a man he knew who lived miles away in a place called Connecticut. The man had lost his wife, and so his young son had no mother. The man had left the city for a quieter life—and, perhaps, Lucia thought, because the memories were too painful in the home he'd shared with his wife. Because he wrote books, he needed a quiet place, and because he often traveled, he needed someone he could trust with his little boy. The woman who had done these things for the three years since the sad death of his wife wished to move back to New York.
So Lucia took a great leap, and moved out of the city and into the grand house of Phillip Grant and his son, Aaron.
The man was handsome as a prince, and she saw he loved his son. But there was a sorrow in his eyes that touched her heart. The child had had so many changes in his short four years, she understood his shyness with her. She cooked their meals and tended the house, and looked after Aaron while the man wrote his book.
She fell in love with the boy, and he with her. He was not always good, but Lucia would have been sad if he had been. In the evenings, she and Phillip would often talk about Aaron, or books, or ordinary things. She would miss the talks—she would miss him—when he went away for business.
There were times when she looked out the window to watch Phillip play with Aaron, and her heart yearned.
She didn't know he often did the same. For he'd fallen in love with her, as she had with him. He was afraid to tell her, lest she leave them. And she feared to tell him in case he sent her away.
But one day, in the spring, under the arching blooms of a cherry tree while the little boy they both loved played on the swing, Phillip took Lucia's hand in his. And kissed her.
When the leaves of the trees turned vivid with autumn, they were married. And lived happily ever after.
Was it any wonder, Emma thought as she pulled her van into the crowded double drive of her parents' home on Sunday evening, that she was a born romantic? How could anyone grow up with that story, with those people, and not want some of the same for herself?
Her parents had loved each other for thirty-five years, had raised four children in the sprawling old Victorian. They'd built a good life there, a solid and enduring one.
She had no intention of settling for less for herself.
She got the arrangement she'd made out of the van, and hurried across the walk for the family dinner. She was late, she thought, but she'd warned them she would be. Cradling the vase in the crook of her arm, she pushed open the door and walked into a house saturated with the color her mother couldn't live without.
And as she hurried back toward the dining room, she moved into the noise as colorful as the paints and fabrics.
The big table held her parents, her two brothers, her sister, her sisters-in-law, her brother-in-law, her nieces and nephews—and enough food to feed the small army they made.
"Mama." She went to Lucia first, kissed her cheek before setting the flowers on the buffet and rounding the table to kiss Phillip. "Papa."
" _Now_ it's family dinner." Lucia's voice still held the heat and music of Mexico. "Sit before all the little piggies eat all the food."
Emma's oldest nephew made oinking noises and grinned as she took her seat beside him. She took the platter Aaron passed her. "I'm starving." She nodded, gestured a go-ahead as her brother Matthew lifted a bottle of wine. "Everybody talk so I can catch up."
"Big news first." Across the table her sister, Celia, took her husband's hand. Before she could speak, Lucia let out a happy cry.
"You're pregnant!"
Celia laughed. "So much for surprises. Rob and I are expecting number three—and the absolute final addition—in November."
Congratulations erupted, and the youngest member of the family banged her spoon enthusiastically on her high chair as Lucia leaped up to embrace her daughter and her son-in-law. "Oh, there's no happier news than a baby. Phillip, we're having another baby."
"Careful. The last time you told me that, Emmaline came along nine months later."
With a laugh, Lucia went over to wrap her arms around his neck from behind, press her cheek to his. "Now the children do all the hard work, and we just get to play."
"Em hasn't done her part yet," Matthew pointed out and wiggled his eyebrows at her.
"She's waiting for a man as handsome as her father, and not so annoying as her brother." Lucia sent Matthew an arch look. "They don't grow on trees."
Emma smirked at her brother and cut her first sliver of roast pork. "And I'm still touring the orchards," she said sweetly.
She lingered after the others to take a walk around the gardens with her father. She'd learned about flowers and plants, had come to love them under his guidance.
"How's the book going?" she asked him.
"Crap."
She laughed. "So you always say."
"Because it's always true at this stage." He wrapped an arm around her waist as they walked. "But family dinners and digging in the dirt help me put the crap aside awhile. Then it's never quite as bad as I thought when I get back to it. And how are you, pretty girl?"
"Good. Really good. We stay busy. We had a meeting earlier in the week because profits are up, and all I could think was how lucky we are—I am—doing work we love, being able to do it with the best friends I've ever had. You and Mama always said to find what we loved, and we'd work well and happily. I did."
She turned as her mother crossed the lawn carrying a jacket. "It's chilly, Phillip. Do you want to catch cold so I have to listen to you complain?"
"You uncovered my plan." He let his wife bundle him into the jacket.
"I saw Pam yesterday," she spoke of Carter's mother. "She's so excited about the wedding. It's lovely for me, too, having two of my favorite people fall in love. Pam was a good friend to me, always, and a champion when some were scandalized your father would marry the help."
"They didn't see how clever I was to get all the labor for free."
"The practical Yankee." Lucia snuggled up against his side. "Such a slave driver."
Look at them, Emma thought. How perfectly they fit. "Jack told me the other day you were the most beautiful woman ever created, and he's waiting to run off with you."
"Remind me to beat him up the next time I see him," Phillip said.
"He's the most charming flirt. Maybe I'll make you fight for me." Lucia tipped her face up to Phillip's.
"How about a foot rub instead?"
"We have a deal. Emmaline, when you find a man who gives you a good foot rub, look closely. Many flaws are outweighed by that single skill."
"I'll keep it in mind. Meanwhile, I should go." She opened her arms to embrace them both. "Love you."
Emma glanced back as she walked away, and watched her father take her mother's hand under the arching branches of the cherry tree with its blooms still tightly closed.
And kiss her.
No, she thought, it was no wonder she was a born romantic. No wonder she wanted that, some part of that, for her own.
She got in the van and thought about the kiss on the back stairs.
Maybe it was only flirtation or curiosity. Maybe it was just chemistry. But she'd be damned if she'd pretend it didn't happen. Or let him pretend.
It was time to deal with it.
CHAPTER SIX
_I_ N HIS OFFICE ON THE SECOND FLOOR OF THE OLD TOWNHOUSE he'd remodeled, Jack refined a concept on his computer. He considered the addition to Mac's studio after-hours work, and since neither she nor Carter were in any particular hurry, he could fiddle, reimagine, and revise the overall structure and every fussy detail.
Now that Parker wanted a second concept to include additions on both the first and second floors, he needed to revisualize not only the details and design, but the entire flow. It was smarter, in his opinion, to do it all at once, even if it did mean scrapping his original concept.
He toyed with lines and flow, the play of light as part of the increased space that would remain studio. With refitting the current powder room and storage and increasing the square footage of both, he could widen the bath, add a shower—something he thought they'd appreciate down the road—give Mac the client dressing area she wanted, and double her current storage space.
Carter's study on the second floor . . .
He sat back, guzzled some water, and tried to think like an English professor. What would his wants and needs be for work space? Efficiency, and a traditional bent—it being Carter. Built-ins along the wall for books. Make that two walls.
Breakfronts, he decided, shifting in his own U-shaped work space to try a quick hand sketch. Cabinets beneath for holding office supplies, student files.
Nothing slick, nothing sleek. Not Carter.
Dark wood, he thought, an Old English look. But generous windows to match the rest of the building. Angle the roof to break up the lines. A couple of skylights. Frame out this wall to form an alcove. Add interest, create a sitting area.
A place a guy could escape to when his wife was pissed at him, or when he just wanted an afternoon nap.
Put an atrium door here, and add a terrace—small scale. Maybe a guy wanted a brandy and cigar. It could happen.
He paused a moment, tuned back in to the game he had on the flat-screen to his left. While his thoughts brewed in the back of his mind, he watched the Phillies strike out the Red Sox in order.
That sucked.
He turned back to the drawing. And thought: Emma.
Cursing, he tunneled a hand through his hair. He'd been doing a damn good job of not letting her in. He was good at compartmentalizing. Work, ball game, the occasional toggle over to check other scores. Emma was in another compartment, and that one was supposed to stay shut.
He didn't want to think about her. It did no good to think about her. He'd made a mistake, obviously, but it wasn't earth-shattering. He'd kissed the girl, that's all.
A hell of a kiss, he thought now. Still, just one of those things, just one of those moments. A few more days to let the reverberations die down, and things could get back to normal.
She wasn't the type of woman to hold it against him.
Besides, she'd been right there with him. He scowled, guzzled more water. Yeah, damn right she had. So what was she all bent out of shape about?
They were grown ups; they'd kissed each other. End of story.
If she was waiting for him to apologize, she could keep waiting. She'd just have to deal with it—and him. He and Del were tight, and he was friends, good friends, with the other members of the Quartet. Added to it, with the remodeling Parker was talking about, he'd be spending more time on the estate for the next several months.
He dragged his hand through his hair again. Okay, that being the case, they'd both have to deal with it.
"Hell."
He scrubbed his hands over his face, then ordered himself to push his brain back into work. Frowning, he studied the bare bones of his design. Then narrowed his eyes.
"Wait a minute, wait a minute."
If he canted the whole thing, angled it, cantilevering the study, he'd create a back patio area, partially covered. It would give them the outdoor living space they lacked, privacy, a potential little garden area or shrubbery. Emma would have ideas on that.
It would add interest to the shape and lines of the building, and increase usable space without significantly adding on to the cost of the build.
"You're a genius, Cooke."
As he began to plot it out, someone knocked on the back door.
Mind still on the drawing, he rose to walk through the main living area of his quarters over his firm. And assuming it was Del or one of his other friends—and hoping they brought their own beer—he opened the door that led into his kitchen.
She stood in the glimmer of porch light and smelled like moonlit meadows.
"Emma."
"I want to talk to you." She breezed right by him, tossed her hair back, pivoted. "Are you alone?"
"Ah . . . yeah."
"Good. What the hell is wrong with you?"
"Give me a context."
"Don't try to be funny. I'm not in the mood for funny. You go flirty on me, jumping my car, rubbing my shoulders, eating my pasta, lending me your jacket, and then—"
"I guess I could've just waved as I passed you on the side of the road. Or let you shiver until you turned blue. And I was hungry."
"It's all of a piece." She snapped it out then strode through the kitchen into his wide hallway with her hands waving in the air. "And you conveniently left out the shoulder rubbing and the 'and then.' "
He saw no choice but to tag after her. "You looked stressed and knotted up. You were okay with it at the time."
Spinning around, she narrowed those brown velvet eyes. "And then?"
"Okay, there was an 'and then.' You were there, I was there, so 'and then.' It's not like I jumped you or you tried to fight me off. We just . . ." _Kissed_ suddenly sounded too important. "Locked lips for a minute."
"Locked lips. Are you twelve? You kissed me."
"We kissed each other."
"You started it."
He smiled. "Are you twelve?"
She made a low hissing sound that had the back of his neck prickling. "You made the move, Jack. _You_ brought me wine, _you_ got all cozy on the stairs, rubbing my shoulders. _You_ kissed me."
"Guilty, all counts. You kissed me right back. Then you went tearing off like I'd drawn blood."
"Parker beeped me. I was _working_. You poofed. And you've stayed poofed since."
"Poofed? I left. You ran off like the hounds of hell were on your heels, and Whitney irritates the shit out of me. So I left. And, strangely, I have a job—just like you—and I've spent the last week working. Not poofing. Jesus, I can't believe I said poofing." He had to drag in a breath. "Look, let's sit down."
"I don't want to sit down. I'm too mad to sit down. You don't just do that then walk away."
Since she pointed an accusatory finger at him, he pointed right back. "You walked away."
"That's not what I mean, and you know it. Beeper, Parker, work." She threw her hands in the air again. "I didn't _go_ anywhere. I just left because the MBB decided she had to inspect the tossing bouquet before she'd deign to toss it, and insisted it had to be right then and there. She irritates the shit out of everyone, but I didn't just leave."
She gave him a little shove, palm to chest. "You did. It was rude."
"God. Are you going to scold me now? Wait, you already are. I kissed you. I confess. You have that mouth, and I wanted it—was pretty clear about that." His eyes sparked, storm clouds full of thunder and electric light. "You didn't scream for help so I took it. Hang me."
"It's not about the kiss. It is, but it isn't. It's about the why and the after that and the what."
He stared at her. "What?"
"Yes! I'm entitled to some sort of reasonable answer."
"Where, you forgot where, so I'll insert that one. Where is the reasonable question? Find it, and I'll do what I can with a reasonable answer. Thereto."
She smoldered. He hadn't known a woman could actually smolder. God, it was sexy.
"If you can't discuss this like an adult, then—"
"Screw it."
If he was going to be damned for it once, he might as well be damned for it twice. He grabbed her, jerking her forward and up to her toes. The sound she made might have been the beginning of what, or why, but before she could finish the word he plundered her mouth. He used his teeth, one quick, impatient bite, that had her lips parting in surprise or response. He wasn't in the mood to care which, not when his tongue found hers, not when the taste of her sizzled along his senses like a wire in the blood.
His hands tangled in the wild glory of her hair, tugging so her head dipped back.
Stop. She meant to say it. She meant to do it. But it was like being drenched in summer. In the heat and the wet. Every sensible thought melted away as her body leaped from temper to shock to fevered response.
When he lifted his head, said her name, she only shook her head and dragged him back.
For one wild moment his hands were everywhere, inciting, igniting, until she could barely get her breath.
"Let me—" He fumbled with the buttons of her shirt.
"Okay." She'd let him do pretty much anything.
When his hand covered her racing heart, she pulled him to the floor.
Smooth flesh, hard muscle, and a mouth mad with hunger. She arched under him, rolled over him. Yanked his T-shirt up and away to scrape her teeth over his chest. With a groan, he dragged her back up to ravish her mouth, her throat, with a frenzied desperation that matched the rush of hers.
Half mad, he flipped her onto her back, ready to rip her clothes away. Her elbow smacked the floor with a sound like a gunshot. Stars burst in front of her eyes.
"Oh! God!"
"What? Emma. Shit. Fuck. I'm sorry. Let me see."
"No. Wait." Dazed, tingling, and not a little stupefied, she managed to sit up. "Funny bone. Ha-ha. Oh, God," she said again.
"I'm sorry. I'm sorry. Here." He started to rub her forearm to help with the needles and pins he imagined were stabbing her, and struggling to steady his breathing, wheezed.
"You're laughing."
"No. No. I'm too overcome with lust and passion to draw a clear breath."
"You're laughing." She jabbed him in the chest with the index finger of her good arm.
"No. I'm fighting manfully not to." Which was, he mused, likely the first time he'd done so while sporting a massive hard-on. "Is it better? Any better?" he asked, and made the mistake of looking over, and into her eyes.
The laugh sparkled in them, like gold over brown. He lost the fight, simply collapsed and gave in to the belly laugh. "Really sorry."
"Why? When you showed such exquisite finesse."
"Yeah, that's what they all say. You're the one who headed for the floor when I've got a perfectly good couch ten feet away, and a damn fine bed up those stairs. But no, you can't control yourself long enough to let me get us to a soft surface."
"Only a wimp requires a soft surface for sex."
He shifted his gaze over with a slow, hot smile. "I ain't no wimp, sister." He sat up. "Let's try take two."
"Wait." She slapped a hand on his chest. "Mmm, nice pecs, by the way. But wait." Lifting her still tingling arm she pushed back her hair. "Jack, what are we doing?"
"If I have to explain it, I'm doing it wrong."
"No, really. I mean . . ." She glanced down at her open shirt, and the lacy white bra perkily peeking out. "Look at us. Look at me."
"Believe me, I was. Am. Want to keep doing that. You have this seriously crazy body. I just want to—"
"Yes, I get that. Back at you, but, Jack, we can't just . . . We got off the track here."
"Down the track, heading for home, from my viewpoint. Give me five minutes to mesh viewpoints. One. Give me one."
"It would probably take under thirty seconds. But no," she added when he grinned. "Really. We can't just do this, like this. Or at all. Maybe." Everything inside her hitched and sparked and _wanted_.
"I'm not sure. We need to think, muse, mull, maybe ponder and brood. Jack, we're friends."
"I'm feeling pretty damn friendly."
Her eyes went soft as she reached out to lay her hand on his cheek. "We're friends."
"We are."
"More, we have friends who are friends. So many connections. So as much as I'd like to say 'what the hell, let's try out that couch, then the bed and maybe take round three on the floor—' "
"Emmaline." His eyes were deep, dark smoke. "You're killing me."
"Sex isn't a kiss on the back stairs. Even a really great kiss on the back stairs. So we have to think and so on before we decide. I refuse to not be friends with you, Jack, just because right now I really want you naked. You're important."
He heaved a sigh. "I wish you hadn't said that. You're important. You always have been."
"Then let's take a little time and think this through." She eased back and began to button her shirt.
"You don't know how sorry I am to see you do that."
"Yes, I do. About as sorry as I am to do it. Don't get up," she said, and got to her feet, picked up the purse she'd dropped when he'd grabbed her. "If it's any consolation, I'm going to have a miserable night thinking about what would've happened if we hadn't stopped to think."
"It isn't, because I'm going to have the same."
"Well." She glanced back as she headed for the door. "You started it."
_I_ N THE MORNING, AFTER THE PREDICTED MISERABLE NIGHT, Emma wanted the comfort of pals and Mrs. Grady's pancakes. She bargained with herself. She could have the pals, no question, but she could only have the pancakes if she first faced the dreaded home gym.
She dragged on her gear and began the resented, caffeine-deprived trudge to the main house. On the way, she veered toward Mac's studio. She could see no good reason why her friend shouldn't suffer along with her.
Without thinking she walked right in, angled toward the kitchen. There was Mac, in cotton boxers and a tank, leaning against the counter with a wide grin and a cup of coffee. And Carter opposite her, mirroring the pose and the grin, in his tweed jacket.
She should've knocked, Emma thought instantly. She had to remember to start knocking now that Carter lived here, too.
Mac glanced her way, lifted her cup in casual greeting. "Hey."
"Sorry."
"Are you out of coffee again?"
"No, I—"
"There's plenty," Carter told her. "I made a full pot."
Emma gave him a sorrowful look. "I don't know why you have to marry her instead of me."
The tips of his ears went a little pink, but he shrugged. "Well, maybe if things don't work out . . ."
"He thinks he's cute," Mac said dryly. "And damn it, he's right." She stepped over, gave his tie a tug.
The kiss was light and sweet, to Emma's eye. The kind of morning kiss between lovers who knew there would be time, lots of time, for deeper, hotter kisses.
She envied the light and sweet outrageously.
"Go to school, Professor. Enlighten young minds."
"That's the plan." He picked up his briefcase, brushed his hand over Mac's bright hair. "See you tonight. Bye, Emma."
"Bye."
He opened the door, glanced back, and rapped his elbow on the jamb. "Damn it," he muttered, and closed the door behind him.
"He does that about every third time he . . . What's with you?" Mac demanded. "You went all blushy."
"Nothing." But she caught herself rubbing her own elbow and remembering. "Nothing. I just stopped by on my way over to the torture chamber. I plan on begging Mrs. G for pancakes after I've suffered."
"Give me two minutes to change."
While Mac dashed upstairs, Emma paced. There had to be a simple, subtle, sensible way to explain to Mac what had happened with Jack. What was happening with Jack. To ask her for dispensation from the no-sleeping-with-friends'-exes rule.
Mac and Jack were friends, so that had to be a point. And more, bigger, huge, was the fact that Mac was madly and totally in love with Carter. She was getting married, for God's sake. What kind of friend would hold another friend to the no-exes rule when she was getting married to Mr. Adorable?
It was just selfish and narrow-minded and mean.
"Let's go before I change my mind." With a hoodie flopping open over a sports bra and bike pants, Mac jogged into the kitchen. "I can feel my bis and tris beefing up. Killer arms, you are mine!"
"Why do you have to be that way?" Emma demanded.
"Way? What?"
"We've been friends since we were _babies_. I don't know why you'd be so hard-assed about this when you don't want him."
"Who? Carter? Yes, I do. You didn't have any coffee this morning, did you?"
"If I have coffee, my brain wakes up enough to find reasons not to work out. And that's not the point."
"Okay. Why are you mad at me?"
"I'm not mad at you. You're the one who's mad at me."
"Then say you're sorry and all's forgiven." Mac opened the door and sailed out.
"Why should I be sorry? I stopped." Emma slammed the door behind them.
"Stopped what?"
"Stopped . . ." Groaning, she pressed her fingers to her eyes. "It's caffeine deprivation. My mind's blurring. I'm starting in the middle. Or maybe the end."
"I demand to know why I'm mad at you so I can put some effort into it. You bitch."
Emma sucked in a breath, held it. "I kissed Jack. Or he kissed me. He started it. And then he poofed, so I went over there to give him a piece of my mind, and he did it again. Then I did it again. Then we were rolling around on the floor and clothes were coming off until I rapped my elbow. Really hard. And it brought me back to my senses. So I stopped and you've got no reason to be mad."
Mac, who'd been gaping at Emma since the first sentence, just kept gaping. "What? What?" She banged her palm on one ear, shook her head as if to shake out water. _"What?"_
"I'm not saying it all again. The point is I stopped, and I said I'm sorry."
"To Jack?"
"No—well, yes—but to you. I'm telling you I'm sorry."
"Why?"
"For God's sake, Mac, the _rule_."
"Okay." Mac stopped, fisted her hands on her hips and stared off into space. "No, I'm still confused. So let's try this." She made exaggerated wiping gestures with both hands. "There's the board, and it's all cleaned off. Let's start fresh. You and Jack—wow—one minute to absorb . . . Done. You and Jack shared a big sloppy."
"It wasn't sloppy. He's an excellent kisser, as you very well know."
"I do?"
"And I'm not sorry for that one. Not really, because it was completely out of the blue. All right, not completely, since I got the vibes when we were under the hood."
"Hood? What . . . Oh, the car. God, only someone who's known you forever could interpret half of what you're saying."
"But I wasn't expecting him to bring me a glass of wine while I was taking a quick break, just sitting on the back stairs, minding my own business."
"Wine, back stairs," Mac mumbled. "MBB. The wedding."
"Then he gave me a shoulder rub, so I _should've_ known, but I was going. I was going back to the reception and then we were standing there and he kissed me. Then Parker beeped me, and I had to go, and I realized what I'd done. It's not really a betrayal, not really. You have Carter."
"What do _I_ have to do with this?"
"But I didn't sleep with him, and that's the fine point of it."
A bird winged by, singing like a mad thing. Without sparing it a glance, Emma slapped her hands on her hips and scowled. "The kissing came as a surprise, both times. And the rolling around was just in the heat of the moment. I stopped, so I didn't—technically—break the rule, but I'm apologizing anyway."
"I'll happily accept your apology if you'll just tell me what the hell I have to do with this!"
"The ex rule."
"The . . . Oh, the _EX_ rule. Still confused as to my . . . Wait. You think Jack and I were . . . You think I had sex with Jack? Jack Cooke?"
"Of course, Jack Cooke."
"I never had sex with Jack."
Emma poked her. "Yes, you did."
Mac poked her back. "No, I didn't, and I ought to know who I did or didn't have sex with, and Jack and I never did the deed. We never even got close to doing the deed. I have not rolled around on the floor removing clothes with Jack."
"But . . ." Baffled, almost weak with it, Emma dropped her arms to her side. "But when he first started coming home with Del, for vacations and holidays during college, the two of you . . ."
"Flirted. Period. Start and stop. We never hit the sheets, or the floor, or the wall, or any other surface together in any way approaching nakedness. Clear?"
"I always thought . . ."
Mac quirked her eyebrows. "You could've asked."
"No, because, _damn_ it, I wanted to flirt with him, and you already were, so I couldn't, and I thought what I thought. And then when it was obvious you were just friends again, the rule went into effect. I thought."
"You've had a thing for Jack, all this time?"
"On and off. I channeled it into other areas, or restrained it, due to the rule. But recently it's gotten more problematic, the channeling and the restraining. God." Emma slapped her hands over her face. "I'm an idiot."
"You slut." Face stern, Mac folded her arms over her chest. "You almost had sex with a man I never had sex with. What kind of friend are you?"
Emma hung her head as her lips twitched. "I said I was sorry."
"I may forgive you, but only after you tell me all—coherently and in minute detail." Grabbing Emma's arm, Mac jogged the rest of the way to the house. "Which means after coffee, which means after workout."
"We could skip the workout and go straight to coffee."
"No, I'm pumped to pump." Mac led the way through the side door of the main house and toward the stairs. When they reached the third floor, Laurel and Parker came out of the gym. "Em kissed Jack and they almost had sex."
"What?" Two voices spoke in unison.
"I can't talk about it now. I haven't had coffee. I can't talk about it until I do, and unless there are pancakes." Snarling with dislike, Emma stalked to the elliptical.
"Pancakes. I'll tell Mrs. G." Laurel dashed away.
"Jack? Jack Cooke?" Parker said.
Mac flexed her arms and headed to the Bowflex. "That's what I said."
_W_ HEN THEY SAT IN THE BREAKFAST NOOK, AND EMMA CLUTCHED her first cup of coffee, Mac raised a hand. "Let me tell the first part, because it'll be faster and you'll still have your normal complement of brain cells at the end. So, Emma had the hots for Jack, but thought Jack and I had a thing, including sex, in the way back, so sticking to the No-Ex Rule, she suffered in silence."
"I didn't suffer."
"I'm telling this part. Then during the MBB's reception, Jack did the 'oh, you're so stressed, let me rub your shoulders,' then laid a big wet one on her. Then Parker beeped her."
"That's what was wrong with you. Thanks, Mrs. G." Parker smiled at Mrs. Grady and took one of the pancakes from the platter the housekeeper set on the table.
"So last night, after waiting over a week, she went by his place to give him the what-for. One thing led to another, and they ended up rolling around on the floor naked."
"Half. It wasn't even half naked. It was maybe a quarter naked," Emma calculated. "At the most."
"This morning she apologized to me for nearly having sex with my imaginary ex."
"As well she should," Mrs. Grady put in. "No friend poaches another friend's man, even if she's kicked him to the curb."
"It just sort of happened," Emma began and hunched under Mrs. Grady's cool stare. "I said I was sorry, and I stopped before we actually . . ."
"That's because you're a good girl with an honest heart. Eat some of that fruit now. It's fresh. Sex is better when you're eating healthy."
"Yes, ma'am." Emma stabbed a little chunk of pineapple.
"I don't get why you ever thought Mac had been sleeping with Jack in the first place." Laurel dumped syrup on her pancakes. "If she had, she'd have bragged about it and talked about it until we all wanted her dead."
"No, I wouldn't."
"In the way back you would have."
Mac considered. "Yes, that's true. In the way back I would have. I've evolved."
"How hot are the hots?" Parker wanted to know.
"Extremely. He hit high prior to the back stairs. After, he set a record."
Nodding, Parker ate. "He's an exceptional kisser."
"He really is. He . . . How do you know?" When Parker just smiled, Emma's jaw dropped. " _You?_ You and Jack? When? How?"
"I think it's disgusting," Mac muttered. "Yet another best pal moving on my imaginary ex."
"Two kisses, my first year at Yale, after we ran into each other at a party and he walked me back to the dorm. It was nice. Very nice. But as exceptional a kisser as he is, it was too much like kissing my brother. And as exceptional a kisser as I am, I believe he felt it was too much like kissing his sister. And that's how we left it. I gather that wasn't an issue for you and Jack."
"We're nowhere in the vicinity of brother- or sisterhood. Why didn't you ever tell us you kissed Jack?"
"I didn't realize we were supposed to report on every man we've ever kissed. But I could make a list."
Emma laughed. "I bet you could. Laurel? Any Jack incidents to report?"
"I'm feeling very annoyed and deprived that I have none. Even imaginary. It seems like he could've hit on me at least once in all this time. The bastard. How about you, Mrs. G?"
"A very nice one under the mistletoe a few Christmases back. But being the love them and leave them type, I let him off easy so as not to break his heart."
"I'd say Em plans to take him down, and take him down hard." Mac arched her eyebrows. "And that he doesn't have a prayer against the awesome power of Emmaline."
"I don't know. I need to think. It's complicated. He's a friend. Our friend. And he's Del's best friend. Del's your brother," she said to Parker, "and the next thing to a brother to the rest of us. And we're all friends, _and_ business partners. Del's our lawyer, and Jack helps out when we need him. Plus he's designing the remodeling. We have all these connections, and they're all tangled up."
"And nothing tangles up the tangles like sex," Mac put in.
"Exactly. What if we end up having this thing, then the thing goes south. Then we're awkward with each other, and that makes the rest of us awkward with the rest of us. We have a kind of balance, don't we? Sex isn't worth upsetting the balance."
"You wouldn't be doing it right then," Mrs. Grady commented, and shook her head. "Youth thinks too damn much. I'm going to start the wash."
Emma sulked over her pancakes. "She thinks I'm being an idiot, but I just don't want anyone to get hurt."
"Then set the ground rules going in. What you each expect from each other, and how you'll handle any complications."
"What kind of ground rules?"
Parker shrugged. "That's for you to decide, Em."
CHAPTER SEVEN
_A_ T HER WORKTABLE WITH A SOOTHING NEW AGE MIX IN THE background, Emma processed a delivery. For the midweek, off-site bridal shower, she'd opted for fun and female. The gerbera daisies were just the ticket.
Visualizing the finished arrangement, she cut the lower inch of the stems under water. Fresh and pretty, she thought as she transferred the daisies to her solution of water, flower food, and preservatives.
She carried the first batch to the cooler for rehydrating. As she started on the next batch, she heard Parker call out to her.
"Back here!"
Parker came in, took a look at the flowers, foliage, buckets, tools. "McNickey bridal shower?"
"Yes. Just look at the color of these gerberas. From soft to vibrant. They're going to be perfect."
"What are you doing with them?"
"For the centerpiece, a trio of topiaries in pots I'm covering with lemon leaf. I'll work in some waxflower and acacia, add some sheer ribbon. The client wants a couple others, a more elaborate arrangement for her entry table, another with candles to put in her fireplace, and something delicate, fragrant, and pretty for the powder room. I need to get them all processed before my eleven o'clock consult. It's moving along."
"Festive and female." Parker scanned the work space. "I know you've got a pretty full slate. Can you squeeze another off-site event in?"
"When?"
"Next Thursday. I know," she said as Emma slid over a cool stare. "The potential client called the main number, and since I knew you were elbow deep in a delivery I didn't transfer it. She was at the Folk-Harrigan wedding. Tells me she just couldn't get over the flowers—which is another score for us over MBB."
"You're using that to seduce me."
"Yes, I am. She'd planned to just go buy some cut flowers and do some vases, but now that she's seen your work, she's obsessed. She can't get over how beautiful they were."
"Stop it."
"How gorgeous and creative and perfect."
"Damn you, Parker."
"She can't sleep or eat or function in any normal fashion now that she's seen what can be done with flowers."
"I hate you. What kind of event, and how much is she after?"
Parker's smile managed to be both smug and sympathetic. Emma considered it a major skill.
"A baby shower, and it sounds similar to what you're doing here. Except for the fireplace arrangement. Very girly—the baby's a girl—so she's looking for a lot of pink. But told me she'd trust your judgment."
"It's cutting it close. I have to see what my wholesaler can do. And I'd have to take a look at next week."
"I already did. Your Monday's solid, but you have a block Tuesday afternoon. You start designing Wednesday for Friday's event, Thursday for Saturday's. You've got Tink coming in to help those two days, so is it realistic the two of you could add this in? It's her daughter-in-law," Parker added. "And her first grandchild."
Emma sighed. "You knew that would do it."
"Yes, I did." She patted Emma's shoulder, unrepentant. "If you need it, you can call in Tiffany or Beach."
"Tink and I can handle it." Emma carried the next batch to the cooler, then came back to finish. "I'll call the client as soon as I'm done here, make sure we understand what she's after. Then I'll make sure I can get it."
"I put her name and number on your desk."
"Of course you did. It's going to cost you."
"What's your price?"
"The garage called. My car's finished, but I can't get in to pick it up today. And tomorrow's nearly as full."
"I'll take care of it."
"Knew you would." Looking at what crowded her plate, Emma rubbed the back of her neck. "The hour you save me can go toward the expectant grandma."
"I'll get back to her, take her off her tenterhooks, and let her know you'll be in touch. And speaking of touch, have you talked to Jack?"
"No. I'm in the mulling and musing stage. If I talked to him I'd start thinking how much I'd like to jump him or be jumped by him. Which, of course, since I brought it up, I'm thinking about right now."
"Should I give you a moment of privacy?"
"Very funny. I told him we needed to stop and think, so I'm stopping and thinking." Her brow creased and she made her voice prim. "Sex isn't everything."
"Since you have more of it—and offers for more of it—than I do, I'll bow to your superior knowledge."
"That's because I'm not intimidating." She flicked Parker a glance. "I didn't mean that as an insult."
"I don't mind being intimidating. It saves time. Which," she added with a look at her watch, "I have to consider now. I'm meeting a bride in town. Mac's got a delivery to make. I'll run and catch her before she goes, have her drop me off at the garage. I should be back by four. Don't forget we have an evening consult tonight. Six thirty."
"I've got it on my appointment book."
"I'll see you then. Thanks, Emma. Really," Parker added as she hurried out again.
Alone, Emma cleaned off her work area before reaching for the Neosporin she used like other women used hand cream. With her latest nicks and scratches tended, she set up for her consult.
Satisfied with the selection of arrangements, photo albums, and magazines, she called the number Parker had left her—and made a grandmother-to-be very happy. As they spoke she took notes, made calculations on the number of baby roses, mini calla lilies. Pink for the roses, white for the callas. More calculations as she designed the larger arrangement in her head. Eggplant callas, Bianca roses, pink spray roses.
Sweet, female, but with elegant touches—if she read the client correctly. She added to her notes, jotted down the time and place for delivery, and promised the client an e-mail contract and itemization by midafternoon.
Gauging the time, she put in a hurried call to her wholesaler, then popped up to peel off her work clothes and suit up.
While she freshened her makeup, she wondered if Jack was musing and mulling.
On impulse, she dashed to her computer to send him an e-mail.
_I'm still thinking. Are you?_
She hit Send before she could change her mind.
_I_ N HIS OFFICE, JACK CHECKED THE CHANGES HIS ASSOCIATE had done. The new construction project continued to be tweaked as the clients waffled. They wanted stately, he thought, and they'd gotten it. They'd also wanted six fireplaces. Until they'd decided they needed nine. And an elevator.
The latest change involved enclosing the projected swimming pool for year-round use and attaching it to the house via a breezeway.
Nice job, Chip, he thought even as he made a couple of small changes. He studied the result, then the drawings submitted by the structural engineer.
Good, he decided. Very, very good. The dignity of Georgian Colonial wasn't compromised. And the client could do laps in January.
Everybody's happy.
He started to send an e-mail clearing the drawings for submission to the client, and noticed the mail from Emma.
He clicked it open, read the single line.
Was she kidding?
Every thought that didn't revolve around her—particularly a naked her—was a struggle. Everything he'd done that morning had taken twice as long as it should have _because_ he was thinking.
No point in telling her that, he decided. So just how did he answer? He angled his head, and smiled as he hit Reply.
_I'm thinking you should come over tonight wearing nothing but a trench coat and elbow pads._
After he clicked Send, he sat back and imagined—very well—what Emma might look like in a trench coat. And maybe really high heels, he thought. Red ones. And once he'd loosened the belt of the coat, he'd—
"Got the go to come on back."
With his mind still opening a trench coat—short, black—Jack stared at Del.
"So hey, where the hell are you?"
"Ah . . . just work. Drawings." Shit. Casually, he hoped, he brought up his screen saver. "No work for you?"
"I'm on my way to the courthouse, and you have better coffee."
Del strolled over to the setup on the counter, and helped himself. "Ready to lose?"
"Lose what?"
"It's Poker Night, pal, and I'm feeling lucky."
"Poker Night."
Eyebrows lifting, Del studied him. "What the hell are you working on? You look like you've just shifted dimensions."
"It just shows my uncanny ability to focus on the job at hand. Which I'll be doing with poker tonight. You'll have to do a lot more than feel lucky to win."
"Side bet. A hundred."
"Done."
Del toasted him, drank. "How's it going on the additions for the Quartet?"
"I've got something I like for Mac and Carter. I just want to refine it a little more."
"Good. Are you working on Emma?"
"What? Am I what?"
"Emma. The second cooler?"
"Not yet. It . . . shouldn't be complicated." Then why was it? Jack wondered. Why did he feel like he was lying to his closest friend?
"Simple works. I've got to go be a lawyer." Del set the mug down, started to the door. "See you tonight. Oh, and try not to cry when you pay me the hundred. It's embarrassing."
Jack shot up a middle finger, so Del walked away laughing.
Jack waited ten full seconds, ear cocked for any sound of return before bringing up his e-mail again.
No reply, yet, from Emma.
How could he have forgotten it was Poker Night? That sort of thing was engraved on his brain. Pizza, beer, cigars, cards. Men only. A tradition, maybe even a ritual, that he and Del had established when they'd still been in college.
Poker Night was sacred.
What if she said she'd be there? That she'd be knocking on his door tonight?
He thought of Emma in a black trench coat and red high heels.
He thought of good friends, cold beer, and a hot deck of cards.
Of course, he thought, there was only one answer. If she got back to him and said she'd come by, he'd simply explain.
He'd tell Del he'd come down with a violent case of stomach flu.
No man living or dead would blame him.
_M_ AC GLANCED OVER AT PARKER AS SHE DROVE TOWARD GREENWICH. "Okay, it's just you and me. What do you really think of Emma and Jack?"
"They're both adults, single, healthy."
"Uh-huh. What do you really think of Emma and Jack?"
Parker let out a sigh that ended on a reluctant laugh. "That I never saw it coming, and I thought I was good at that kind of thing. And if it feels this weird to me, it must feel a lot weirder for them."
"Weird bad?"
"No. No. Just odd. There's the four of us, and the two of them—Jack and Del. Together it's the six of us. Well, seven with Carter, but this is all rooted in pre-Carter. We've been in and out of each other's lives and business for years. Forever for the four of us and Del, and for what, a dozen years with Jack? When you think of a man as a brother, it's an adjustment to realize not everyone in that same network feels the same. It's almost as strange as it would be if one of us really disliked him."
"That's what's hanging Em up."
"I got that."
"They get all sexy, and that's good, but then the heat backs off. Maybe it backs off for one of them before the other. That's awkward." Mac checked her mirrors before changing lanes. "Does the one who's still warmed up get their feelings hurt, or feel sort of betrayed?"
"Feelings are feelings. I don't understand why people blame other people for what they feel."
"Maybe not, but they do. And Emma is the softest of soft touches. She's a whiz at handling men—I bow in awe—but she really _feels_ for them if she doesn't . . . feel for them. You know what I mean."
"Yeah." Because they approached the garage, Parked slipped back into the shoes she'd slipped off when she'd gotten in the car. "She'll end up going out with a guy a second, third, fourth time even when she figured out from the first date she wasn't interested. She doesn't want to hurt his feelings."
"Still, she dates more than the three of us put together. Pre-Carter," Mac added. "And she nearly always manages to shake a man off without denting his ego. I tell you, she's skilled."
"The trouble is, she's closer to Jack. She loves him."
"You think—"
"We _all_ love him," Parker finished.
"Oh, that way. True."
"It has to be hard to break off a relationship with someone you really care about. And being Emma, she's trying to work that part of it out before they up the relationship. Hurting him isn't an option for her."
Mac considered as she waited at a light. "Sometimes I wish I was as genuinely nice as Emma. But not very often. It's too much work."
"You have your moments. Me? I'm intimidating."
Mac snorted. "Oh yeah, you scare the shit out of me, Parks." She eased through the light. "But you are pretty scary when you put the Parker Brown of the Connecticut Browns cloak on. And if you give it that little swirl, many fall dead."
"Not dead. Temporarily stunned perhaps."
"You knocked Linda cold," Mac commented, speaking of her mother.
"You handled that yourself. You stood up to her."
Mac shook her head. "I'd stood up to her before. Maybe not like this last time, not as tall and straight. But if I started it, you finished her off for me. You add in Carter, and the fact that as, God, _kind_ as he is, he's not susceptible to her bullshit—then the fact she's getting pampered by her rich fiancé in New York? My life's gotten a lot smoother."
"Has she contacted you since?"
"Funny you should ask. This morning, in fact, and as if we'd never had that really ugly last scene. She and Ari have decided to elope. Sort of. Those crazy kids are jetting off to Lake Como next month, and they'll be married at the villa of one of Ari's dear friends once Linda's planned all the details—which is her version of eloping, I guess."
"Oh God, if you say George Clooney, I'm going to go."
"If only. I don't think we're invited anyway. She mostly called to make sure I understood she's doing a _lot_ better than Vows for her wedding."
"What did you say?"
_"Buona fortuna."_
"You did?"
"I did. It felt good. And I actually meant it. I do wish her luck. If she's happy with this Ari, she'll leave me the hell alone. So . . ." She turned, turned again, and pulled into the lot of Kavanaugh's. "It's all good. Do you want me to wait, just in case?"
"No, you go on. I'll see you back at the house for tonight's consult."
Parker got out, adjusted her grip on her portfolio bag as she checked the time. Right on schedule.
She scanned the long building that housed what appeared to be offices attached to a large garage. She heard the _whoosh_ of some sort of compressor as she approached, and saw through the open garage doors the legs, hips, and most of the torso of the mechanic who worked on a car on a lift.
She caught glimpses of shelves, which she assumed held parts and other paraphernalia, racks of tools. Tanks, hoses.
She smelled oil and sweat, not offensive to her mind. Work odors, productive scents. She approved of them, especially since she saw Emma's car sitting in the lot, very clean and very shiny.
Curious, she detoured to it. The chrome glinted in the sunlight, and through the window she noted the signs of meticulous detailing.
If, she mused, the car ran as good as it looked, she'd bring hers here instead of to the dealer for its next regular service.
She crossed the lot toward the office to settle the bill and get the key.
Inside, a woman with hair more orange than red sat on a stool at the short leg of an L-shaped counter, pecking with two fingers at the keyboard of a computer.
Her brow furrowed, her mouth twisted in a way that told Parker the computer was not her friend.
She stopped, sized Parker up over the top of a pair of bright green cheaters. "Help you?"
"Yes, thanks. I'm here to pick up Emmaline Grant's car."
"You Parker Brown?"
"Yes."
"She called, said you'd be coming to get it."
When the woman made no move, just continued to stare over the tops of her glasses, Parker smiled politely. "Would you like to see some identification?"
"No. She said what you looked like when I asked, and you look like what she said."
"Well then, if I could see the bill?"
"I'm working on it." The woman shifted on the stool, pecked at the keys again. "You can sit right down there. It won't take me long. Take less time if I could just write it out on an invoice pad, but Mal has to have it this way."
"All right."
"Vending machines through that door there if you want something to drink."
Parker thought of her client, and the distance to the bridal boutique, the traffic. "You said it wouldn't take long."
"It won't. I'm just saying . . . What does this demon from hell want from me?" The woman raked long red nails through her orange frizzy hair. "Why won't it just spit the damn thing out?"
"May I . . ." Parker leaned over the counter, scanned the screen. "I think I see the problem. Just point and click here, with the mouse." She tapped the screen. "Good. Now see where it says Print? Click that. There you go. Now click on Okay."
Parker leaned back as the printer clicked into life. "There you go."
"Click this, click that. I can never remember which click comes first." But she looked over the counter and smiled for the first time. Her eyes were as bold and engaging a green as the frames of her cheaters. "Appreciate it."
"No problem."
Parker took the bill, sighed a little as she ran down the work. New battery, tune-up, timing, oil change, fan belts, tire rotation, brake pads. "I don't see the charge for the detailing."
"No charge. First-time customer. Complimentary."
"Very nice." Parker paid the bill, then tucked her copy in a pocket of her bag. She took the key. "Thank you."
"Welcome. Come back when you need to."
"I believe I will."
Outside, she walked toward Emma's car, clicking the key lock as she went.
"Hey, hey, hold it."
She stopped, turned. She recognized the legs, hips, torso she'd seen under the belly of the car in the garage. This view added chest and shoulders. The light spring breeze fluttered through dark hair—that needed a trim—disordered either from work or carelessness. She supposed it suited the strong, sharp lines of his face, and the dark stubble that indicated he hadn't picked up a razor in a day or two.
She took it all in quickly, just as she took in the hard set of his mouth and the hot green of eyes that transmitted temper.
She'd have looked down her nose if she hadn't been forced to look up when he stopped in front of her. She angled her head up, met his eyes with hers, and said in her coolest tone, "Yes?"
"You think all it takes is a key and a driver's license?"
"I beg your pardon?"
"Your battery cables were covered with corrosion, your oil was sludge. Your tires were low and your brake pads damn near shot. I bet you slather yourself with some fancy cream every day of your life."
"Excuse me?"
"But you can't bother to get your car serviced. Lady, this car was a disgrace. You probably spent more on those shoes than you have on maintaining it."
Her shoes? Her shoes were none of his damn business. But she kept her tone bland—insultingly bland. "I appreciate that you have passion for your work, but I doubt your boss would approve of the way you speak to customers."
"I am the boss, and I'm fine with it."
"I see. Well, Mr. Kavanaugh, you have an interesting business manner. Now, if you'll excuse me."
"There's no excuse for the way you've neglected this vehicle. I've got it up and running for you, Ms. Grant, but—"
"Brown," she interrupted. "That's Ms. Brown."
He narrowed his eyes as he studied her face. "Del's sister. Should've seen it. Who's Emmaline Grant?"
"My business partner."
"Fine. Pass on what I said to her. It's a good car. It deserves better."
"Be sure I will."
She reached for the door, but he beat her to it, opened it for her. She got in, placed her bag on the seat beside her, fastened her seat belt. Then froze the air between them with a "Thank you."
He grinned, fast as a lightning strike. "You mean go to hell. Drive safe," he added and shut the door.
She turned the key, found herself mildly disappointed when the engine purred like a kitten. As she drove away, she glanced in the rearview mirror, saw him standing, hip-shot, watching her.
Rude, she thought—absurdly rude, really. But he apparently knew how to do his job.
When she parked near the bridal boutique where she intended to meet her client, Parker pulled out her BlackBerry to e-mail Emma.
_Em. Car is done. Looks and runs better than it has since you bought it. You owe me more than the bill. Will discuss tonight. P_
_A_ T HOME, EMMA USED THE TIME BETWEEN APPOINTMENTS TO write itemized contracts. She loved the choices made by her last client, a December bride. Color, color, and more color, she thought. All that hot and bold would be a pleasure to work with in winter.
She sent the contract to the client for approval, copied Parker for Vows' files. She smiled when she spotted an e-mail from Jack. Then snorted out a laugh as she read it.
"Trench coat and elbow pads. Good one. Let's see . . ."
_You'll need to choose between my red lace elbow pads and the black velvet set. Or I can just surprise you. I'll try them on later with my collection of trench coats. I have a particular favorite. It's black and has a shine so it always looks . . . wet._
_Unfortunately tonight won't work for me. But that gives us both more time to think._
"That ought to give you a moment or two," Emma murmured, and hit Send.
CHAPTER EIGHT
_A_ T SIX, EMMA WALKED INTO THE KITCHEN FROM THE MUDROOM as Parker walked in from the hall.
"Good timing. Hi, Mrs. G."
"Grilled chicken Caesars," Mrs. Grady announced. "Use the breakfast nook. I'm not setting up the dining room when you girls are going to be coming in and out and picking."
"Yes, ma'am. I worked through lunch. I'm starved."
"Have a glass of wine with it." Mrs. Grady jerked her head toward Parker. "This one's in a mood."
"I'm not in any particular mood." But Parker took one of the glasses of wine Mrs. Grady poured. "Your bill."
Emma glanced at the bottom line, winced. "Ouch. I guess I deserve it."
"Maybe so. But _I_ didn't deserve the angry lecture from the proprietor who assumed I was you."
"Uh-oh. What hospital is he in? I should send flowers."
"He survived, unscathed. Partially because I was on a schedule and didn't have time to hurt him. Your car was also detailed, expertly, inside and out—gratis to first-time customers. Which counted in his favor. Marginally."
Pausing, Parker took another sip of wine. "Mrs. G, you know everyone."
"Whether I want to or not. Sit. Eat." When they had, Mrs. Grady plopped down on one of the counter stools with her own glass of wine. "You want to know about young Malcolm Kavanaugh. Bit of a wild one. Army brat. His father died overseas when he was a boy. Ten or twelve, I think, which may account for the bit of wild. His ma had a hard time keeping him in line. She used to waitress at Artie's, the place on the avenue. He'd be her brother, Artie would, and why she moved here when she lost her husband."
Mrs. Grady took a sip of wine, and settled back a bit to tell the rest. "As you may know, Artie Frank is a complete asshole, and his wife is a prissy snob of a woman. What I heard was Artie decided to take the boy in hand, and the boy did his level best to snap that hand off at the wrist. And good for him," she added with some relish. "He went off, the boy did, to race cars or motorcycles or something like. Did some stunt work in the movies, I believe. Did well enough for himself, from what I'm told. And made sure his ma got a piece of the pie he was making."
"Well. That speaks well of him, I suppose," Parker allowed.
"Got busted up on a stunt, and got some kind of settlement out of it. He used it to buy the garage out on Route One, about three years ago. Bought his ma a little house as well. He's built up a nice business, from what I'm told, and still has a bit of the wild in him."
"I'll assume he's built up his business through his skill with engines and not through his skill with customer relations."
"Put your back up," Emma commented.
"I'll get over it, as long as he does the job well." Parker glanced over as Laurel came in. "Cutting it close."
"Coffee and cookies are set up. Some of us don't have time to sit around eating and gossiping before a consult." Laurel frowned as she combed her fingers through her hair. "Plus you're having wine."
"Parker was in a mood because—"
"I heard all about that already." Laurel poured herself a scant half glass. "I want new juice. What's the current situation with Jack?"
"I think we're having virtual sex. We're still in the early stages of foreplay, so I'm not sure where it's going."
"I've never had cyber sex. I've never liked anyone enough to have cyber sex." Laurel cocked her head as she considered. "And that sounds odd. I like a guy well enough to have actual sex, but not virtual?"
"Because it's a game." Emma got up to give Laurel the remaining half of her salad. "You might like a man enough to go to bed with him, but you might not want to play with him."
"That makes weird sense." With a nod, Laurel stabbed at the salad. "You always make weird sense when it comes to men."
"And obviously she likes Jack enough to play with him," Parker added.
"Jack's got a sense of fun, which is one of the things I've always liked about him. And found attractive." Emma's lips curved in a slow, easy smile. "We'll see how much we like playing games."
_I_ N THE PARLOR, OVER COFFEE AND LAUREL'S MACAROONS, PARKER led the consult with the engaged couple and their mothers. "As I explained to Mandy and Seth, Vows will tailor our services to suit your needs. As much or as little as you want. Our goal, together and individually, is to give you the perfect wedding. _Your_ perfect wedding. Now, when we spoke last, you hadn't chosen a date, but had decided you wanted evening and outdoors."
Emma listened with half an ear as dates were discussed.
She wondered if Jack had gotten her e-mail yet.
The bride wanted romance. Didn't they all? Emma thought, but perked up when she said she'd be wearing her grandmother's wedding gown.
"I have a photo," Mandy announced, "but Seth isn't allowed to see. So . . ."
"Seth, would you like a beer?"
He looked over at Laurel, grinned. "I would."
"Why don't you come with me? I'll set you up. When you've finished the beer we should be ready for you again."
"Thanks." Mandy reached into a large folder when Laurel led Seth out. "I know it's probably silly—"
"Not at all." Parker held out a hand for the photo, and her polite expression turned radiant. "Oh. Oh, it's gorgeous. It's just stunning. Late thirties, early forties?"
"You're good," the mother of the bride said. "My parents were married in 1941. She was just eighteen."
"Ever since I was a little girl I've talked about wearing Nana's wedding gown when I got married. It needs to be fitted, and a little repair, but Nana's taken good care of it."
"Do you have a seamstress in mind?"
"We've spoken to Esther Brightman."
As she studied the photo, Parker nodded approval. "She's a genius, and exactly who I'd recommend for this. Mandy, you're going to look absolutely amazing. And we could, if you want, build the entire wedding around this dress. Vintage glamour with class, romance with style. Tails rather than the more expected tux for the groom and groomsmen."
"Oh, wow. Wow. Would he go for that?" she asked her future mother-in-law.
"He'll go for anything you want, honey. Personally, I love the idea. We'd want to find vintage dresses, or the vintage style for the bridal party."
Emma studied the photo when it came to her. Fluid, she thought, Deco-inspired lines, with a sheen that said silk. She lifted her gaze to study Mandy, and decided the new bride would wear the gown as beautifully as her grandmother had. "I can replicate the bouquet," she said half to herself.
"What?" Mandy cut herself off in midsentence and swung her attention to Emma.
"The bouquet—if you wanted—I can replicate it. Look how clever she was, how smart to offset the long, fluid lines of the gown with the oversized crescent of calla lilies. Do you have the veil and headpiece?"
"Yes."
"From what I can see, she had it trimmed with lily-of-the-valley. I can do that, if it appeals to you. I just wanted to mention that before Seth comes back. Something you can think about."
"I love it! Mom?"
"My mother will be a puddle. So will I. I love it, too."
"We'll talk about it in more detail when we do our individual consult. Meanwhile, when you select the dresses for the bridesmaids, if you can get pictures then I can get copies made or you can scan them and send them in an e-mail so I can see what kind of flowers she chose for them."
Emma handed the photo back to Mandy. "You'd better put that away."
"Mac, why don't you give Mandy an overview of the photography?"
"First, I want to duplicate the pose in your grandmother's formal portrait. It's classic and gorgeous. But tonight, we should talk about what you'd like for your engagement portraits."
They moved from stage to stage, step to step, with a rhythm they'd developed over the years. As they discussed photography, cakes, food, Emma jotted down key words that would help her create a picture of the bride, the groom, and what they envisioned.
And if her thoughts veered in Jack's direction a few times, she reminded herself she excelled at multitasking.
By the time she and her partners walked the clients to the door, she was ready to duck out and see if Jack had answered her e-mail.
"Good job," she said. "I'm going to go home and start a file for the event. So—"
"There's something else," Parker interrupted. "When I was at the boutique today, I found Mac's dress."
"You what?" Mac blinked at her. " _My_ dress?"
"I know you, and what you're looking for. And since it was right there, saying I'm Mac's, I used our connections and brought it home for approval. Maybe I'm wrong, but I thought at least you'd want to try it on."
"You brought home a wedding dress for me to try on?" Eyes narrowed, Mac pointed at Parker. "Aren't you the one who's always telling brides they might try on a hundred dresses before they find the one?"
"Yes. You're not most brides. You know immediately what works and what doesn't. If it doesn't, no harm done. Why don't we go take a look? It's up in the Bride's Suite."
"Oh, we _have_ to see." Thrilled with the idea, Emma grabbed Mac's hand and tugged. "Wait, we need champagne. Which Parker would have thought of already."
"Mrs. G will have it up there by now."
"Champagne and a potential wedding dress?" Mac mused. "What are we waiting for? No hurt feelings if I don't like it," she added as they started up the stairs.
"Absolutely not. If you don't it would only tell me how vastly superior my taste is to yours." With the faintest of smirks, Parker opened the door to the Bride's Suite where Mrs. Grady poured flutes of champagne.
"Heard you coming." And she winked at Parker as Mac simply stared at the gown hanging from the hook.
"It's beautiful," Mac murmured. "It's . . ."
"Strapless, which I think will suit you," Parker continued. "And the slight A-line will flatter your build. I know you were leaning toward something completely unadorned, but I think you're wrong. The tissue organza over the silk adds romance, softens the lines. You're angular. And the back?"
Parker lifted it off the hook, turned it around.
"I love it!" Emma pushed forward. "The ruffle train, out of the organza! It's fabulous, just a little flirty. Plus the way it should drape over your butt—"
"Will actually give you one," Laurel finished. "Try it on, or I will."
"Give me a second, this is a moment. Okay, there's the moment." And Mac unhooked her pants. As she stripped down, Emma circled a finger.
"Turn your back to the mirror. You don't want to see yourself putting it on. You want the _pow_ effect once you're in it."
"Dropping your clothes where you stand." Mrs. Grady shook her head as she scooped them up. "Just as you always have. Well, help her into it," she ordered, and stood back, smiled.
"Oh. I'm going to cry." Emma sniffled while Parker fastened the gown in place.
"They didn't have your size, so it's a little big."
"That's what I'm here for." Mrs. Grady picked up her pin cushion. "We'll nip and tuck a bit here and there so it shows better on you. It's a shame you've always been such an ugly thing."
"Insult me, but don't stick me."
"That'll do for now." Mrs. Grady stepped around to fuss a little with the bodice, then reached up to smooth Mac's bright red hair. "We have to work with what we've got."
"Count to three, Mac, then turn and look." Emma pressed both hands to her lips. "Just look at you."
"Okay." Mac took in a breath, let it out, then turned toward the cheval glass where she'd watched so many brides study their reflections. The only thing she could say was "Oh!"
"And that says it all." Laurel blinked at tears. "It's . . . it. You're it in it."
"It's . . . I'm . . . Holy shit, I'm a bride." Mac's fingers fluttered up to her heart as she angled herself. "Oh, check out the back. It's fun, and female, and I _do_ have an ass." In the glass, her gaze shifted to Parker's. "Parks."
"Am I good or am I good?"
"You're the best. This is my wedding dress. Aw, Mrs. G."
Mrs. Grady dabbed her eyes. "I'm just shedding a tear of joy that I won't have four spinsters on my hands."
"Flowers in your hair. A wide floral headband instead of a veil," Emma suggested.
"Really?" Pursing her lips, Mac studied herself, imagined. "That could work. That could work well."
"I'll show you some ideas. And you know, I think with the lines of the dress, I'd like to see a long sweep of a bouquet, probably hand tied. Maybe arm-carried." Emma angled one arm, swept her hand down to demonstrate. "Or a cascade, but with a waterfall effect. Rich, warm autumn colors, and . . . I'm getting ahead of myself."
"No. God, we're planning my wedding. I think I need that drink."
Retrieving Mac's flute, Laurel stepped to her. "It sure looks better on you than any of our old Wedding Day costumes."
"Plus, it doesn't itch."
"I'm going to make you one hell of a cake."
"Oh man, I'm watering up again."
"Turn around, all of you," Mrs. Grady ordered as she took a camera out of her pocket. "Our redhead's not the only one who can take a picture. Glasses up. There's my girls," she murmured, and captured the moment.
_W_ HILE THE LADIES DRANK CHAMPAGNE AND DISCUSSED WEDDING flowers, Jack popped a beer and prepared to fleece friends at Texas Hold 'Em.
And tried not to think about Emma and her latest e-mail.
"Since it's Carter's first official Poker Night, let's try not to humiliate him." Del clapped a friendly hand on Carter's shoulder. "Taking his money's one thing, embarrassing him's another."
"I'll be gentle," Jack promised.
"I could just watch."
"Now where's the fun and profit in that. For us?" Del asked.
"Ha," Carter managed.
They mingled around Del's lower level. A boy's dream space, in Jack's opinion, with its antique bar that had once served pints in Galway, its slate pool table, its flat-screen TV—an auxiliary to the even bigger one in the media room on the other side of the house. It boasted a vintage jukebox, video games, and two classic pinball machines. Leather chairs, sofas that could take a beating. And a Vegas-style poker table just waiting for action.
No wonder he and Del were friends.
"If you were a girl," Jack said to Del, "I'd marry you."
"No. You'd just have sex with me then never call me."
"You're probably right."
Since it was there, Jack snagged a slice of pizza. Skinning friends was hungry work. As he ate he considered the group. Two lawyers, the professor, the architect, the surgeon, the landscape designer—and as he watched the last player come through the door—the mechanic.
Interesting group, he thought. It fluctuated from time to time with a new addition, like Carter, or when one of them couldn't make it. The tradition of Poker Night had begun when he and Del had met in college. The faces might change off and on, but the foundation remained.
Eat, drink, tell lies, talk sports. And try to win money from your friends.
"We're all here. Want a beer, Mal?" Del asked.
"I'm breathing. How's it going?" Mal said to Jack.
"Well enough. The new blood's Carter Maguire. Carter, Malcolm Kavanaugh."
Mal nodded. "Hey."
"Nice to meet you. Kavanaugh? The mechanic?"
"Guilty."
"You towed my future mother-in-law's car."
"Yeah? Did she want me to?"
"No. Linda Barrington."
Mal narrowed his eyes. "Okay. Yeah. The BMW convertible. The 128i."
"Um. I guess."
"Nice ride. Interesting woman." Mal smirked as he lifted his beer again. "Good luck with that."
"The daughter doesn't take after the mother," Del put in.
"Lucky for you," Mal said to Carter. "I met her—the daughter. Mackensie, right? She's hot. She does the bride thing with the Cobalt I just serviced."
"Emma," Del added.
"Right. She ought to be arrested for vehicular abuse. I met your sister when she picked it up," he told Del, and grinned. "She's hot, too. Even when she gives you the deep freeze."
"So . . . Emma didn't pick up her car?"
Mal glanced at Jack. "No, the other one did. _Ms. Brown_." He took a hit of his beer. "The one who says 'excuse me' and means 'fuck you.' "
"That would be Parker," Del confirmed.
"Does the car abuser look as good as the other two?"
"They all look good," Jack murmured.
"Sorry I missed her."
"Before I have to punch Mal for thinking lascivious thoughts about my sisters—biological and honorary," Del said, "let's play cards."
"Be right there." As the others wandered to the table, Jack pulled out his phone to check his e-mails.
_I_ T WAS NEARLY MIDNIGHT WHEN EMMA GOT HOME. ONCE they'd started talking plans and ideas for Mac's wedding, time whizzed.
She all but bounced into the house, energized by the evening, and just a little giddy on champagne.
Mac's wedding.
She could already see how utterly perfect the bride would be in her gorgeous gown, a waterfall of flowers in her arms. And she, Parker, and Laurel, triple maids of honor. Russet for her, autumn gold for Parker, pumpkin for Laurel. And oh, the flowers she'd do with that rich palette of fall.
It would be a challenge, Emma thought as she started upstairs. Parker had been right to point that out so they could begin to plan how it could and would be done. Running a wedding was one thing. Running it and being part of it was another.
They'd need extra help, more subs, but they'd not only do it, they'd knock it out of the park.
Cruising on the mood, she began her nightly ritual. When her bed was turned down, she nodded, smoothed the sheets. There, she'd shown a very mature restraint. An evening with friends—business and pleasure—and no neglecting of her nighttime routine.
It proved she was a sensible adult.
Crossing the fingers of both hands, she dashed from her bedroom to her office to bring up her e-mail.
"There, I knew it."
She clicked open Jack's latest message.
_Now you're playing dirty. Thanks._
_I like surprises. I especially like unwrapping them, so I look forward to helping you out of your coat. I like to take my time with surprises, build anticipation. So I'm going to unwrap you very slowly. Inch by inch._
"Oh," she said, "my."
_And when I have, I'm going to want to take a good, long look. Before I touch. Inch by inch._
_When, Emma?_
"How about right now?"
She closed her eyes and imagined Jack slipping her out of the slick black coat she didn't even own. In a room shimmering with candlelight. Music playing, low and hot—so you felt the bass beat in the blood.
His eyes, dangerous as hellsmoke, gliding over her until heat drenched her skin. Then his hands, strong, sure, slow, following that path of heat, easing the velvet on her elbows down until . . .
"That's just silly." She straightened in her chair.
Silly, maybe, she thought, but she'd managed to stir herself up. Or he had.
Time to respond in kind.
_I like to play, and I don't mind getting dirty._
_Surprises are fun, and being the surprise can be even better. When I am, sometimes I like being unwrapped slowly. Fingertips patiently untying the bow, then hands carefully, very carefully, folding back that wrapping to get to what's waiting inside._
_And other times I want those fingers, those hands, to just_ rip _through the barriers. Fast and greedy, and maybe a little rough._
_Soon, Jack._
Not _if_ any longer, she thought.
Just when.
_W_ ITH HER THREE TOPIARIES FINISHED AND TINK DEEP INTO processing another delivery, Emma took a quick look at her notes and sketches.
"Six hand-tied bouquets including the bride's tossing bouquet for Friday's event. Six pedestal arrangements, eighteen centerpieces, white rose ball, garlands, and swags for the pergola." She muttered her way down the list. "I'll need you at least three hours tomorrow. Four would be better."
"I've got a date tonight, and I'm looking to get lucky." Fingers busy, Tink snapped her gum. "I could be here around noon."
"If you can stick till four, that ought to do it. Another four on Thursday. Five if you want it. I've got Tiffany coming in Thursday, and Beach can give me all day Friday. I can use whatever time you can give me Friday morning. We can start dressing for Friday's event at three. Saturday's another twofer. We need to start by eight for the first. That's A.M., Tink."
Tink rolled her eyes, and kept stripping thorns.
"We break down the first at three thirty, and need the second fully dressed by five thirty. Sunday, we have a big one, a single starting at four. So we'll need to start at ten or ten thirty."
"I'll try to squeeze what there is of my life in there," Tink said dolefully.
"You'll manage. I'll take what you've processed back to the cooler and get the stock we need for the arrangements." As she picked up the first container and turned, Jack walked in.
"Oh . . . Hi."
"Hi back. How's it going, Tink?"
"Emma drives the slaves."
"Yes, she is abused constantly," Emma said. "You can there-there her while I haul these back to the cooler."
God, she thought, he looked so _good_ in his fieldwork clothes, the boots, the faded jeans, the shirt rolled up to the elbows.
She wished she could take just one quick bite.
"Why don't I give you a hand?" He hefted another tub and started back to the cooler.
"We're a little crazy this week," Emma told him. "A midweek off site, and four events over the weekend. Sunday's wedding is a monster—in a good way." She set her tub down, gestured where Jack should place his. "Now I need to—"
He spun her around, boosted her up to her toes in one fast move. Her arms locked around his neck in a combination of instinct and answer even as his mouth laid claim to hers.
The wild, rich perfume of flowers saturated the air just as need and pleasure saturated her body. Greed and urgency swam through her blood.
Not just one bite, she thought, and not quick. She wanted gulp after gulp.
"Does that door lock from the inside?"
She tunneled her fingers through his hair to bring his mouth back to hers. "What door?"
"Emma, you're killing me. Let me just—"
"Oh, that door. No. Wait. Damn it. Just one more." She caught his face in her hands this time, let herself simply sink into the kiss, the perfume, the greed. Then eased back.
"We can't. Tink. And . . ." Regretfully, she blew out a breath as she glanced around. "There really isn't room in here."
"When is she leaving? I'll come back."
"I don't know, exactly, but . . . Wait."
Now he took her face, met her eyes. "Why?"
"I . . . I can't think of a good reason, but that may be because I lost many thousands of brain cells during that kiss. I can't remember if I have any evening appointments. My mind's wiped clean."
"I'm coming back at seven. I'll bring food. Unless you call me and say otherwise. Seven, here."
"Okay. All right. I'll check my book when I regain the power of cogent thought. But—"
"Seven," he repeated and kissed her again. "If we need to talk, we'll talk."
"It may have to be in short, declarative sentences and words of one or two syllables."
"We can do that." His grin shot fresh heat straight to her belly. "Do you need anything out of here?"
"Yes, but I can't remember what. Give me a second." She pushed her hands through her hair, closed her eyes. "All right, yeah. Those, those. Then you've really got to go away. I can't work if I'm thinking about you, this. Sex. Any of it."
"Tell me about it. Seven," he repeated, and helped her carry out the flowers.
"I'll, uh, get back to you on that," she told him when he set the flowers in her work area. "When I'm not so . . . busy."
"Great." The warm gray eyes lingered on her just a moment longer. "See you, Tink."
"You bet." Tink clipped another few stems while Jack left, then slid them into their holding tub. "So, when did you and Jack start doing it?"
"Doing what? Oh. Tink." Shaking her head, Emma turned to her shelves to select the proper container for the fireplace arrangement she had planned. "We're not."
"If you tell me he didn't plant a big yummy one on you back there, I'm going to call you a liar."
"I don't understand why you . . ." Stupid, Emma told herself, then reached for her flower foam. "How do you know?"
"Because your eyes were still glazed when you came back, and he looked like a guy who'd only gotten a few nibbles when he's ready for a great big bite."
"Bite. Ha-ha."
"Why aren't you doing it? He's prime."
"I'm—we're . . . You know, sex doesn't fluster me. I mean talking about sex, because if actually having sex doesn't fluster you at least a little, you're missing something. But this flusters me."
As she continued to work, Tink nodded sagely. "Moving from friends to friends with benefits has the advantage of knowing who the hell you're getting naked with."
"There's that. But it could be awkward, right? After."
"Only if one of you's an asshole about it." She gave her gum another cheerful snap. "So, my advice—don't be an asshole."
"On some odd level that's actually wise." Emma set the foam to soak. "I need to check something in my appointment book."
"Okay. I'd schedule that nookie in for tonight," Tink called after her. "You'll be the happy flower lady tomorrow."
And there's another point, Emma thought.
She saw by her book she'd left the evening open. She'd marked the date with a large X after five o'clock, her way of warning herself not to get talked into going out. Too much work lined up for a date.
But this wasn't actually a date, she decided. He'd come by, bring food, and then . . . they'd see. She didn't have to change or think about what she should wear or . . .
Who was she kidding? Of course she'd worry about what to wear. There was no way whatever was going to happen with Jack was going to happen while she was wearing her work clothes and her nails were green from stems and foliage.
Plus, she'd need fresh flowers and candles in the bedroom. And she'd be more relaxed if she could take a nice bubble bath. Choosing an outfit was a vital element in an evening like this, not just what went on top, but what was under it.
She closed the book.
When she thought it all through, a not-actual date required more work than an actual one.
She hurried back to her flowers. She had to finish her workday, give the client her best. Then she needed plenty of time before seven to make everything perfect, without making it obvious she'd gone to any trouble at all.
CHAPTER NINE
_S_ HE SETTLED ON A DRESS IN A BREEZY PRINT. CASUAL, EMMA determined, simple and almost sweet with the little cropped sweater she paired with it.
And what she wore under it was lethal.
Pleased with the results, she did a final turn in the mirror before giving the bedroom a close inspection. Candles for soft, romantic light, lilies and roses for romantic scents. The CD player set on low with a quiet, romantic mix ready to play.
Pillows plumped, shades drawn.
It was, she decided, a female den of seduction. She was damn proud of it.
Now all she needed was the man.
She walked downstairs to make sure everything was ready on that front. Wine, glasses, candles, flowers. Music again, still low but more upbeat than the mix waiting upstairs. She turned it on, adjusted the volume, then circled around lighting the candles.
They'd have some wine, she thought, and talk. Then a meal and more conversation. They'd never had problems with conversation. Even though they knew where the evening was headed—maybe because they knew—they'd be able to talk, relax, just enjoy each other's company before they—
She spun around when the door opened, giddy nerves dancing. And Laurel walked in.
"Hey, Em, can I get you to put together a couple of . . ." Laurel stopped, lifted her eyebrows as she looked around the room. "You've got a date. You have a sex date."
"What? What's wrong with you? Where do you come up with—"
"How long have I known you? This side of forever? You put out new candles. You have foreplay music on."
"I put out new candles all the time, and I happen to like this mix."
"Let me see your underwear."
Emma choked out a laugh. "No. You want me to make up a couple what?"
"That can wait. I have twenty bucks that says you have on the sexing underwear." Laurel strode over, started to tug at the bodice of Emma's dress—and got her hand slapped away.
"Cut it out."
"You took a bath in the tonight's-the-night bubbles." Laurel sniffed. "I can smell it."
"So what? I often have dates. Sometimes I have sex dates. I'm a grown woman. I can't help it if you haven't had sex in six months."
"Five months, two weeks, three days. But who's counting?" Laurel stopped again, sucked in an exaggerated breath as she pointed at Emma. "You have a sex date with Jack."
"Stop it. Will you stop it? You're freaking me out."
"When is he getting here? What's the plan?"
"Soon, and I'm still working on the plan. But it doesn't include you being here. At all. Go away now."
Ignoring the order, Laurel folded her arms. "Is it the white 'I'm a good girl but I can be bad' underwear or the black 'I'm only wearing this so you can rip it off me, big boy' underwear? I need to know."
Emma cast her eyes to the heavens. "It's the red with the black roses."
"We may need to call the paramedics. If you're functional tomorrow, can you make me up three mini arrangements? Just mixed spring types? I have a consult and little springy flowers would set the mood for what I think the client wants."
"Sure. Go home."
"I'm going, I'm going."
"You're stopping at Mac's to tell her before you go home and tell Parker."
Laurel paused at the door, flicked back the hair that fell over her cheek. "Duh. And I'm going to ask Mrs. G if she'll make frittatas for breakfast so we can fuel up while you give us all the details."
"I have a full day tomorrow."
"Me, too. Seven A.M., food and sex recap. Good luck tonight."
Resigned, Emma let out a sigh and decided she wouldn't wait for Jack to have a glass of wine. The trouble with friends, she thought as she went to the kitchen, was they knew you too well. Sex date, foreplay music, sexing underwear. No secrets among . . .
She stopped with the bottle in hand. Jack was a friend. Jack knew her very well. Wouldn't he . . . ? What if he . . . ?
"Oh, shit!"
She poured a very large glass of wine. Before she could take the first sip, she heard the knock on her door.
"Too late," she murmured. "Too late to change a thing. Time to see what happens, and deal with it."
She set the wine down, went to the door.
He'd changed, too, she noted. Khakis instead of jeans, a crisp shirt instead of a chambray. He carried a large take-out bag from her favorite Chinese restaurant, and a bottle of her preferred cabernet.
Sweet, Emma thought. And certainly another advantage of being friends.
"When you said you'd bring food you meant it." She took the bag from him. "Thanks."
"You like a little—and that's usually very little—of everything. So I got a variety." He cupped the back of her neck, leaned in to kiss her. "Hi again."
"Hi back again. I just poured myself a glass of wine. Why don't I make it two?"
"I'd say yes. How'd the work go?" he asked when he followed her to the kitchen. "You were pretty much buried in it when I was here earlier."
"We got it done. The next few days are wall-to-wall, but we'll get that done, too." She poured a second glass, offered it. "How about your summer kitchen?"
"It's going to rock. I don't know how much use the clients will get out of it, but it's going to look great. I'll need to talk to you about the work here. Your second cooler. I dropped some preliminary sketches at Parker's when I was by before, for the changes there, and Mac's plans are finished. After spending a little time in your cooler today, it's easy to see why you need another one. I like your dress."
"Thanks." Watching him, she sipped her wine. "I guess we've got other things to talk about, too."
"Where do you want to start?"
"I keep thinking it's a lot, but I realized it really comes down to two things, and they both grow out of one root. We're friends. We are friends, aren't we, Jack?"
"We're friends, Emma."
"So the first thing is I think friends should tell each other the truth. Be honest. If we realize, after tonight, that it's just not what we expected—or if either of us feel like, well, that was nice, but I'm finished—we should be able to say so. No hard feelings."
Reasonable, straightforward, and without sticky edges or invisible strings. Perfect. "I can go with that."
"The second is staying friends." Worry wove through the words as she watched him. "That's the most important thing. Whatever happens, however it works out, we need to promise each other we'll be friends. Not just for you and me, but for everyone we're connected to. We can say it's just sex, Jack, but sex isn't a just. Or it shouldn't be. We like each other. We care about each other. I don't want anything to change that."
He brushed a hand down her hair. "Blood oath or pinky swear?" he asked and made her laugh. "I can promise you that, Emma. Because you're right. Friends." He eased over to kiss her cheeks, one, then the other, before rubbing his lips lightly over hers.
"Friends." She repeated the gesture so they stood, lips a breath apart, eyes locked. "Jack? How did we ever keep from doing this all these years?"
"Hell if I know." He touched his lips to hers again, then took her hand. "We were at the beach," he said as he led her to the stairs.
"What?"
"We'd gone to the beach for a week. All of us. A friend of Del's lent us his place—his parents' place, I guess—in the Hamptons. It was the summer before you started this place."
"Yes. I remember. We had the best time."
"One morning early, I couldn't sleep, so I walked down to the beach. And I saw you. For a minute—just a second or two, really—I didn't realize it was you. You were wearing this long scarf thing tied around your waist, lots of wild colors, and it blew around your legs. You had on a red bathing suit under it."
"You . . ." She literally had to catch her breath. "You remember what I was wearing?"
"Yes, I do. And I remember your hair was longer than it is now, halfway down your back. All those mad curls flying. Bare feet. All that golden skin, wild colors, mad curls. My heart just stopped. I thought: That's the most beautiful woman I've ever seen. And I wanted that woman, in a way I'd never wanted one before."
He stopped, turned a little as she simply stared at him. "Then I saw it was you. You walked off, down the beach, the surf foaming up over your bare feet, your ankles, your calves. And I wanted you. I thought I'd lost my mind."
She wouldn't be able to catch her breath much longer, she realized. Wouldn't be able to think. Wouldn't want to be able to think.
"If you'd walked down to me, looked at me the way you're looking at me now, you'd have had me."
"Worth waiting for." He kissed her long, slow, deep, then walked with her into the bedroom. "Nice," he said, noting the flowers, the candles.
"Even friends should fuss a little, I think." Because it would calm her, and set the mood, she picked up the lighter, wandered the room setting candles to flame.
"Nicer." He smiled when she switched on the music.
She turned to him, with the room between them. "I'm going to be honest with you, Jack—as promised. I have a weakness for romance, the trappings, the gestures. I also have a weakness for passion, the quick and the crazed. I'll take you either way. And tonight, you can take me, any way you want."
With those words, with Emma standing in candlelight, he was utterly seduced.
He crossed to her, and she to him so they met in the center of the room. He combed his fingers through her hair, drawing it back from her face, lowering his lips to hers slowly. Tonight, he would do all in his power to exploit all her weaknesses.
She gave, her body soft in surrender to echo the kiss. Warmth layered on warmth, longing wrapped in anticipation. When he swept her up to carry her to the bed, those dark eyes went slumberous.
"I want to touch you everywhere I've dreamed of touching you." Slowly, he slid his hand under her dress, along her thigh. "Everywhere."
He kissed her again, hints of greed now, of possession, while his fingers feathered over her skin, over the lace that barely covered her. She bowed up at his touch, offering more.
His lips trailed down her throat in whispers as he slid the sweater down her arms. Then in a fast, rough move, he flipped her over to graze his teeth over her shoulder. When he straddled her to ease down the zipper at the back of her dress, she looked over her shoulder. Her smile was full of secrets.
"Need any help?"
"I think I've got it."
"I think you do. Since I'm not in a position to do it myself, take off your shirt."
He unbuttoned it, peeled it off while she watched him. "I've always liked watching you shirtless around here in the summer. I like this even better." She rolled over again. "Undress me, Jack, and touch me. Everywhere."
She moved under him, lazy, teasing motions as he drew the dress over her head, and felt the sizzle of pleasure as his gaze traveled over her.
"You're spectacular." He traced the edges of red lace, the tiny black petals. "This may take a while."
"No rush."
When he lowered his lips again, she let herself steep in the sensation of being explored.
Inch by inch, he'd said, and he was a man of his word. He touched, he tasted, he lingered until her quivers became trembles and the perfumed air thickened.
Generous curves, skin gold in candlelight, her hair spread out in lush coils of black silk. He'd thought her beautiful, always, but tonight she was a banquet willing to allow him a feast.
Every time he came back to those soft, lush lips, she gave a little more. He guided her up, slowly, slowly, felt her rise and rise, then crest and break.
Sensation drenched her, sweet and hot and lovely.
"My turn." She pushed herself up to link her arms around his neck, to fix her mouth to his.
She shifted, nudging him over and back. Now she explored, strong shoulders, hard chest, firm belly. And teased his zipper down to free him.
"I'd better—"
"I'll take care of it."
She took a condom out of her nightstand and took her time pleasuring him in the act of protection. Her hands, her lips set every muscle quivering until he gripped her hair, dragged her up. "Now."
"Now."
She slid down, bowed up. And took him into her.
The shiver ran through her, bright, silver-edged—a shimmering in the blood—as she began to move. Slowly, to draw out every drop of pleasure, with her eyes on his.
He gripped her hips, fighting to let her set the torturous pace. As her hands ran down her body in glorious abandon, he ached from the sight of her. Her skin glowed, like gold dust set to flame with her black velvet eyes shimmering in the flickering light. His pulse beat in wild drums while she took her fill. And fisting around him, she shuddered over the edge.
He levered up, rolled her to her back. On her gasp he pushed her knees up. "My turn."
He let control snap.
The sleepy, shimmering pleasure flashed to frenzy. She cried out from the shock of it as he drove her in fast, powerful thrusts. Lost, thrilled, she met the unreasoned demand beat for beat. The orgasm ripped through her, filled her, then hollowed her out.
She lay helpless, quivering even as he took more and reached his own.
He collapsed on her, undone. He felt her quaking beneath him, felt the hammer strikes of her heart, and still her hand came up to stroke his back in a gesture of affection that was so utterly Emma.
Jack closed his eyes a moment. He'd lost his wind, had probably lost his mind. He lay, breathing her in, absorbing the way her body, completely relaxed now, felt under his.
"Well, since we promised to be honest," he began, "I have to tell you that didn't do much for me."
Under him she laughed and pinched his ass. "Yeah, it's a shame. I guess we just don't have any chemistry."
He grinned, lifted his head. "No chemistry. That's why we blew up the lab."
"Lab, hell. We leveled the building." She sighed, long and deep as she stroked her hands down. "God, you've got a nice ass. If I may say so."
"You may, and, baby, you, too."
She smiled up at him. "Look at us."
He kissed her, softly, then again with light affection. "Are you hungry? I'm starving. How do you feel about cold Chinese?"
"I feel perfectly fine about it."
_T_ HEY ATE AT HER KITCHEN COUNTER, DIGGING NOODLES, SWEET and sour pork, and Kung Pao chicken right out of the cartons.
"Why do you eat like that?" he asked.
"Like what?"
"In microscopic bites."
"Well." She worked her way through a single noodle as he topped off her wineglass. "It started as a way to needle my brothers, and became a habit. Whenever we'd get a treat, ice cream or candy, whatever, they'd just scarf theirs down. It drove them crazy that I'd have some of mine left. So I started eating even slower so I'd have _more_ left and make them crazier. Anyway, I eat less and enjoy it more this way."
"I bet." Jack purposely shoveled a huge forkful of noodles into his mouth. "You know, your family's part of your appeal."
"Is it?"
"Your family's probably part of the reason you're appealing, but I meant they're all . . . great," he decided for lack of better. "They're great."
"I'm lucky. Of the four—well, six of us counting you and Del—I'm the only one with the whole shot. The Browns were amazing. You didn't know them very well, but I grew up here almost as much as at home. And they were amazing. It was devastating for all of us when they died."
"Del was wrecked. I liked them a lot. They were fun, interesting people. Involved people. Losing your parents so suddenly, both of them, out of the blue, it has to be the worst. Divorce is hard on a kid, but . . ."
"It is hard. It was tough on Mac when we were little, then it happened again. And again. For Laurel I think it came out of nowhere. She was a teenager, and suddenly her parents are splitting up and then they're not, then they're whatever they are. She hardly ever sees them. It couldn't have been easy for you, either."
"It was rough, but it could've been a lot rougher." He shrugged and ate. It wasn't something he liked to dwell on. Why dwell on something painful that couldn't be changed? "Both my parents made a real effort not to play tug-of-war with me, and they managed to keep it civilized. Eventually, they figured out how to be friendly."
"They're both nice people, and they both love you. It makes a difference."
"We do okay." And he'd learned "okay" sometimes had to be good enough. "Plus I think we do better with the distance. My mother has her second family, my father his." His tone was a shrug, despite the fact he'd never reconciled himself to the ease with which they'd gone their separate ways, made their separate lives. "It got smoother all around when I went off to college. Smoother yet when I decided to move here."
He studied her as he drank some wine. "Your family, on the other hand, is like one of those rubber band balls you make, all twisted together into a solid core." He considered for a moment. "Are you going to tell them about this?"
She blinked. "Ah. I don't know. If they ask me, but I don't know why any of them would."
"Could be sticky."
"They like you. And they know I've had sex. They might be surprised. I mean, I'm surprised. But I don't see anyone having a problem with it."
"Good. That's good."
"The girls are fine with it."
"The girls?" Those smoky eyes widened. "You told the others we were going to sleep together?"
"We're girls, Jack," she said dryly.
"Right."
"Plus I thought, before, that you and Mac had been together."
"Whoa."
"Well, I thought you had, so I had to say something to her because of the Rule, and by the time we got that straightened out, everybody knew I was thinking about you and sex in the same sentence."
"I never slept with Mac."
"I know that now. I didn't, however, know you kissed Parker."
"That was a long time ago. And it wasn't really . . . Okay, it was, but it didn't work." He dug out more pork.
" _And_ you kissed Mrs. G. You man-slut."
"Now that might've worked. I don't think we gave it enough time."
She grinned at him, poked at some chicken. "What does Del think?"
"About me kissing Mrs. Grady?"
"No. You and me. This."
"I don't know. I'm not a girl."
She paused with the glass halfway to her lips. "You haven't talked to him about it? He's your best friend."
"My best friend is going to want to kick my ass for thinking about touching you, much less doing what we just did upstairs."
"He, too, knows I've had sex."
"I'm not sure that's true. He puts that in another dimension. The other-dimension Emma has sex." Jack shook his head. "You, not so much."
"If we're going to be together in bed, I'm not going to treat it like some illicit affair. He'll find out. You'd better say something to him before he does. Because if you don't, and he does, he _will_ kick your ass."
"I'll figure it out. There's just one more thing, since we're on all this. Since we're together like this, I'd like to know that we're not together with anyone else like this. Is that a problem?"
She sipped her wine wondering why he'd have to ask. "Blood oath or pinky swear?" When he laughed, she took another sip. "If I'm sleeping with a man, I don't see anyone else. It's not only rude and against my principles, but it's too much trouble."
"Good. So it's you and me."
"It's you and me," she repeated.
"I have to be on-site at seven."
Here it comes, she thought. Early day tomorrow, honey. It was great. I'll call you.
"Any objection if I stay, since I'd need to get up at about five?"
Her lips curved. "No objection."
_J_ ACK DISCOVERED WHEN THEY FINALLY SLIPPED TOWARD SLEEP that Emma was a snuggler. The sort of woman who burrowed in and wrapped around.
He was generally a man who liked his space. Space kept a man from getting tangled up—literally and metaphorically.
But he found, under the circumstances, he didn't really mind.
She fell asleep like a stone dropped in a pond. Up and moving one minute, submerged the next. He was a drifter, with the movie reel of the day's events and the previews of the next running through his mind as his body settled down.
So he drifted, with Emma's head nestled in the curve of his shoulder, her arm flung around his waist, and her leg twined between his.
He woke, in nearly the same position, about six hours later to the beep of his cell phone's alarm. And as he woke to the scent of her hair, she was his first conscious thought.
His attempt to ease away without waking her resulted in causing her to snuggle closer. Even as his body cheerfully responded, he tried to nudge her away.
She said, _"Hmmmm?"_
"Sorry. I've got to get going."
"Time's it?"
"Just after five."
She sighed again, then lifted her mouth to brush his lips with hers. "I've got about an hour. Too bad you don't."
He'd managed to shift her so they were front to-front, and her hand was making slow, lazy circles over his ass.
"There are two things I'm finding really convenient at the moment."
"What?"
"Being the boss, so I don't get fired for being late. Even more, my own habit of keeping spare work clothes in the trunk. If I leave right from here, I've got most of an hour."
"Convenient. Want coffee?"
"That, too," he said, and rolled on top of her.
CHAPTER TEN
_W_ HILE TIFFANY PROCESSED ANOTHER DELIVERY, EMMA COMPLETED the third hand-tied bouquet. She loved the combination of frilly tulips with the ranunculus and hydrangea. And though wiring the tiny crystals among the blooms abused her fingers, she knew she'd been right to suggest it. As she had with the strips of lace, the studs of pearls securing the stems.
With the steps, the details, the precision required, even with her experience each bouquet took nearly an hour to create. Wasn't she lucky, she thought, that she enjoyed every minute of it?
There wasn't a better job in the world, as far as she was concerned. And just now, as she began the painstaking assembly of the next bouquet, with Tiffany working quietly at the other end of the counter, with music and perfume winding in the air, she considered herself the luckiest woman on the planet.
She turned the flowers in her hand, adding tulips at varying heights, adjusting, interspersing the ranunculus to create the shape she wanted. She added the beads, pleased with the touch of glitter, and time clicked away.
"Do you want me to start on the centerpieces?"
"Hmm?" Emma glanced up. "Oh. Sorry, off in another world. What did you say?"
"It's really beautiful. All the textures." As she admired the work, Tiffany gulped down water. "You've got one more to go after that. I'd start it, but I'm not as good at the hand tied. I can get the centerpieces started though. I've got the list and the design."
"Go ahead." Emma used a cable tie to secure the stems, clipped the excess plastic with her wire cutters. "Tink should be here . . . Well, she's already late, so she should be here." She exchanged cutters for clippers and began trimming the stems. "If you take the centerpieces, I'll get her started on the standing arrangements."
Emma wrapped the stems in lace, anchored the lace with pearl corsage pins. Once the bouquet was in its holding vase and in the cooler, she washed her hands—again—rubbed in Neosporin—again—then set to work on the final hand-tied.
When Tink wandered in, guzzling from a bottle of Mountain Dew, Emma merely lifted her eyebrows.
"You're late," Tink said, "blah, blah, blah. I'll stay late if you need me." And yawned. "Didn't get to bed—well, to sleep—until after three. This guy? Jake? He's Iron Man, in all good ways. Then this morning . . ." She trailed off, blowing a streak of pink out of her eyes as she angled her head. "Somebody else got lucky last night. Jack, right? Hey, Jake and Jack. Cool."
"I managed to get lucky and finish four hand-tieds. If you want to make enough to keep yourself in Mountain Dew, you'd better get started."
"No problem. Is he as good as he looks?"
"I'm not complaining, am I?"
"Who's Jack?" Tiffany wanted to know.
"You know. Jack of the excellent ass and smoky eyes." Tink stepped over to wash her hands.
" _That_ Jack?" Gaping, Tiffany stopped with a hydrangea in her hand. "Wow. Where have I been?"
"It's still breaking news, so you're pretty up to date. You going back for more?" Tink asked Emma.
"Work," Emma muttered. "We're working here."
"She's going back for more," Tink concluded. "Nice bouquet," she added. "The tulips look like they come from the Planet Zorth, but in a romantic way. What am I on first?"
"The standing arrangements for the terraces. You need—"
"Hydrangeas, the tulips, ranunculus," Tink began, and rattling off the rest of the flowers and foliage, reminded Emma why she kept her on.
At five, she let Tiffany go and, leaving Tink working magic with flowers, took a break to rest her hands and clear her head. She stepped outside to stroll toward Mac's studio.
Her friend came out, a camera bag slung on her shoulder, a can of Diet Coke in her hand.
"Five thirty rehearsal," Emma called out.
"Just heading that way." Mac detoured toward Emma.
"You can tell the bride the flowers for tomorrow are amazing, if I do say so myself." When they met halfway, Emma stopped, stretched her back. "Long day, and a longer one coming."
"I heard a rumor Mrs. G's making lasagna. Big rafts of lasagna. Carter and I plan to pig out."
"I'm there. In fact, the thought of lasagna inspires me. Tink's finishing up her part. I'll give you and Parker a hand with the rehearsal, indulge, then put in an hour or two later tonight."
"There's a plan."
Emma looked down at her work clothes. "How bad am I?"
Mac took a survey while she chugged her drink. "You look like a woman who's put in a long day. The bride will be thrilled with you."
"I say you're right. I don't want to clean up, then have to change again." She hooked her arm through Mac's free one as they started toward the house. "You know what I was thinking today? I'm the luckiest woman in the world."
"Jack was that good?"
Snorting out a laugh, Emma bumped Mac's hip with hers. "Yes, but besides that. I'm tired, my hands hurt, but I spent all day doing what I love. I got a call this afternoon after my flowers got to the off-site, the baby shower? The client just bubbled at me over the phone, just had to call me as soon as she saw the flowers to tell me how fabulous they were. Who else gets what we get, Mac?" She sighed and lifted her face to the sun. "We have such happy jobs."
"While I agree, in general, here's what I love about you. You can forget or ignore all the Monster Brides, all the Insane Mothers, Drunken Groomsmen, Bitchy Bridesmaids, and remember all the good stuff."
"It's mostly good stuff."
"It is. Despite the nightmare of an engagement shoot I did today. The happy couple had a vicious fight before I'd taken the first frame. My ears, they still ring."
"I hate when that happens."
"You? Screams, tears, storming out, storming back. Accusations, threats, ultimatums. More tears, apologies, wrecked makeup, shame, and horrible embarrassment. Screwed up my day good and proper. Plus, due to red, puffy eyes, we had to reschedule."
"Still, drama adds interest to the day. Then there's that." Emma gestured to where tomorrow's groom swept tomorrow's bride up for a spin on the walk to the house.
"Shit. They're early. Don't stop, don't stop," Mac muttered as she shoved the drink at Emma and yanked her camera out of the bag.
"They're anxious to get going," Emma murmured. "And they're happy."
"Plus fairly adorable," Mac added as she managed to zoom in for a couple of candids. "And speaking of adorable, look who just pulled up."
"Oh." Spotting Jack's car, Emma instinctively brushed at her hair.
"He's seen you look a lot worse."
"Thanks very much. We both had a pretty full day, so I didn't expect . . ."
He looked so good, khakis today and a crisp pin-striped shirt, which meant client meetings and office work rather than construction sites. The easy gait, the burnished hair shining in the sun, the quick, killer smile all added up to . . . yum.
"My ass looks fat in these pants," she hissed to Mac. "I don't care because they're for work, but—"
"Your ass doesn't look fat in those. I'd tell you if it did. The red sweats with the cropped legs? Your ass looks fat in those."
"Remind me to burn them." Emma passed the drink back to Mac, then tuned up her smile as Jack crossed to them.
"Ladies."
"Man," Mac responded. "I've got to get to work. Later."
She loped off.
"Rehearsal," Emma explained.
"Are you in on that?"
"Just as backup. Are you done for the day?"
"Yeah. I had to make a stop at a client's not far from here, so I . . . Am I in the way?"
"No. No." Flustered, she pushed at her hair again. "I was just taking a break, walking over to the rehearsal in case they needed me for anything."
He slid his hands in his pockets. "We're being weird with each other."
"God. Yes. We are. Let's stop. Here." She rose to her toes, kissed him firmly. "I'm glad you came by. I've been at it since about eight, and wanted a break. Mrs. G's making lasagna. Do you want in on that?"
"Oh yeah."
"Then why don't you go charm her, have a beer, and I'll see you inside when we're finished."
"I'll do that." He caught her chin in his hand, leaned down to kiss her again. "You smell like your work. It's nice. I'll see you inside."
As they separated, her smile bloomed.
_E_ MMA WALKED INTO THE HOUSE TO THE GOOD, RICH SCENTS OF dinner and Mrs. Grady's big, bawdy laugh. The combination boosted her already happy mood. She heard Jack relaying what seemed to be the tail end of a work story.
"Then, when she clued in, she says, 'Oh, well. Can't you just _move_ the door?' "
"She did not."
"Would I lie to you?"
"Every day and twice on Sunday. Are you moving the door?"
"We're moving the door, which will cost her about twice as much as the armoire she fell in love with. But client is king."
He took a sip from his beer, and his gaze shifted toward Emma when she walked in. "How'd it go?"
"Easy and fun, which is always a good sign for the real thing. They're trusting luck and the weather forecaster on tomorrow's predicted rain holding off until late evening, and going without the tents. So, fingers crossed on that."
As she would in her own home, Emma got out a glass for wine. "They're off to the rehearsal dinner. But I think we've got the better deal here." She sniffed the air. "It smells great, Mrs. G."
"Table's set," Mrs. Grady said as she tossed a salad. "You'll eat in the dining room like the civilized."
"Parker and Mac will be right along. I haven't seen Laurel."
"She's fiddling in her own kitchen, and knows what time I'm serving."
"I'll give her a heads-up."
"All right then. Jack, make yourself useful since you're mooching, and put this salad on the table."
"Yes, ma'am. Hey, Carter."
"Hello, Jack. They're right behind me, Mrs. G."
She gave Carter a steely stare. "Did you teach anything useful today?"
"I like to think so."
"Did you wash your hands?" she demanded.
"Yes, ma'am."
"Then take that wine in and go sit down. And no picking until everyone's seated."
She served family style in the big dining room with its lofty ceiling and generous windows. Because it was Grady's Rule, cell phones were turned off, and Parker left her BlackBerry in the kitchen.
"Sunday Bride's aunt stopped by," Parker began. "She brought the chuppah, she just finished making it last night. It's a work of art. I'm keeping it upstairs. Emma, you may want to take a look at it, in case you feel you should tweak any of the arrangements. Carter, you're teaching the aunt's sister-in-law's older boy. David Cohen."
"David? He's a bright kid, who's currently using most of his creativity to cut up in class. Just last week he gave a report on _Of Mice and Men_ in the style of a stand-up comic."
"How'd he do?" Mac asked him.
"I'm not sure how Steinbeck would've felt about it, but I gave him an A."
"It's such a sad book. Why do we have to read so many sad books in school?" Emma wondered.
"We're reading _The Princess Bride_ in my freshman class now."
"Why didn't I have teachers like you? I like happy books, and happy endings. And look at you, with your own Buttercup."
Mac rolled her eyes. "Yeah, that's me. I'm a real Buttercup. Tomorrow's event has a nice fairy-tale feel, though. All those fairy lights and candles, all white flowers."
"Tink complained she was going snow-blind. But they're beautiful. A couple more hours tonight, and they're done. All the hand-tying and wiring makes this one very labor-intensive. Plus." She held up a hand sporting new nicks and scratches. "Ouch."
"You wouldn't consider being a florist a dangerous career." Jack took her hand, studied it. "But you've got the battle scars." And kissed her knuckles.
There was a long beat of silence, speculative stares.
"Stop," he ordered with a half laugh.
"You've got to expect it." Still watching them, Laurel stabbed into her salad. "We're making adjustments here. I think you should lay one on her, right here, so we can use the visual to help us adjust."
"Wait! Wait!" Mac waved a hand. "Let me get my camera."
"Pass the lasagna," Jack said.
Leaning back, Parker sipped her wine. "For all we know, the two of them are just having a joke at our expense. Pretending to be involved, then laughing at us behind our backs when we buy in to it."
"Oooh," Mac murmured. "You're good."
"I am," Parker agreed. "But really, it's not like either of them are the shy type. Certainly not too shy for one little PDA, and among friends, too." She shrugged as a smile tugged at her lips. "So I'm leaning toward practical joke."
"Kiss the girl," Mrs. Grady told him, "or this bunch won't give you any peace."
"Or lasagna," Laurel decided. "Kiss!" She clapped her hands together. "Kiss!"
Mac picked up the chant. Even when she elbowed Carter he just laughed and shook his head.
Giving up, Jack turned to a laughing Emma, pulled her over and gave her a kiss that brought cheers and applause from the table.
"Looks like somebody's having a party and forgot to invite me."
The noise died away as everyone turned to the doorway, and Del. He stared at Jack, lifting a hand to stop Parker when she started to get to her feet.
"What the hell's going on?"
"We're having dinner." Laurel spoke coolly. "If you want some you'll need to get a plate."
"No, thanks," he said, just as coolly. "Parker, I've got some paperwork to go over with you. We'll take care of it another time since you're in the middle of something that's apparently none of my business."
"Del—"
"You and I." He interrupted his sister, never taking his eyes off Jack. "We'll deal later, too."
When he strode out, Parker released a long sigh. "You didn't tell him."
"I was still figuring out how to . . . No," Jack said. "No, I didn't. I need to go straighten this out," he told Emma.
"I'll go with you. I can—"
"No, better not. It may take a while, so . . . I'll call you tomorrow." He pushed back from the table. "Sorry."
When he left, Emma managed nearly ten seconds. "I have to at least try." She popped up, rushed after Jack.
"He looked pretty steamed," Mac said.
"Of course, he's steamed. His perfect balance has been shifted." Laurel shrugged when Parker snapped a look at her. "That's part of it. And that part's only worse because Jack didn't tell him. He's got a right to be steamed."
"I could go after them," Carter suggested. "Try to mediate."
"Mediators often get punched in the face by both parties."
He smiled humorlessly at Mac. "Wouldn't be the first time."
"No, let them hash it out." Parker sighed again. "That's what friends do."
_B_ ECAUSE EMMA'S CONCERN HELD HIM UP A GOOD TEN MINUTES, Jack didn't catch up with Del on the estate. But he knew where he'd go. Home, where he could curse, snarl, and brood in private.
He knocked, and had no doubt Del would open the door. For one thing, he had a key, and they both knew he'd use it if necessary. But more, Delaney Brown wasn't one for avoiding confrontation.
When Del yanked open the door, Jack looked him in the eye. "You swing at me, I'll swing back. We'll both get bloody, and won't resolve anything."
"Fuck you, Jack."
"Okay, fuck me. Fuck you, Del, for being an ass about—"
He took the punch to the face—because he hadn't seen it coming—and returned it. They stood there, in the doorway, mouths bleeding.
Jack swiped at his. "Do you want to beat the hell out of each other inside or out?"
"I want to know what the hell you were doing with your hands on Emma."
"Do you want to hear about that inside or out?"
Del merely turned, and stalked back to his great room for a beer. "How long have you been moving in on her?"
"I didn't move in on her. If anything we moved in on each other. For Christ's sake, Del, she's a grown woman, she makes her own choices. It's not like I twirled my moustache and stole her virginity."
"Watch it," Del warned, then the temper in his eyes went lethal. "You slept with her?"
"Let's back it up." Not a good start, Cooke, he thought. Not the best of springboards. "Let's just back it up."
"Yes or no, goddamn it."
"Yes, goddamn it. I slept with her, she slept with me. We slept with each other."
Something murderous flashed in Del's eyes. "I ought to beat you senseless."
"You can try. We'll both end up in the ER. And when I get out, I'm still going to sleep with her." Something equally deadly flared in Jack's. "It's none of your fucking business."
"The hell it isn't."
Because he felt he had more strikes on the wrong side of the column than Del, Jack nodded. "Okay, given the circumstances, it's your business. But it's not your right to tell either of us who to be with."
"How long?"
"It just happened. It just started turning on me, on us, I guess, the last couple of weeks."
"A couple of weeks." Del bit off the words "And you didn't say anything to me about it."
"No, I didn't, mostly to try to avoid getting punched in the face." Jack yanked open the fridge, got out a beer. "I knew you wouldn't like it, and I hadn't figured out how to explain."
"You didn't have any trouble explaining it to everyone else, apparently."
"No, I didn't, but then everyone else wasn't going to smash their bare fists into my face because I'm sleeping with a beautiful, interesting, _willing_ woman."
"She's not any woman. She's Emma."
"I know that." Frustration piqued to beat down the anger. "I know who she is, and I know how you feel about her. About all of them. Which is why I kept my hands off her until . . . recently," he finished, and held the cold bottle to his throbbing jaw. "I've always had a thing for her, but I set it aside. 'Just don't go there, Jack.' Because you wouldn't like it, Del. You're my closest friend."
"You've had a thing for a lot of women."
"That's right," Jack said evenly.
"Emma isn't the type you sleep with until you catch wind of something new. She's the kind you make promises to, make plans with."
"For God's sake, Del, I'm just getting used to . . ." He didn't make plans or promises—ever. Plans changed, didn't they? Promises got broken. Keeping it loose was keeping it honest.
"We were together one night. We're still figuring things out. And cut me a small break here. However many women I've been with I've never lied to them or treated them with anything but respect."
"April Westford."
"Jesus, Del, we were in grad school, and she was stalking me. She was a lunatic. She tried to break into our house. She keyed my car. She keyed _your_ car."
Del paused, took a swig of beer. "All right, you've got a point with that one. Emma's different. She's different."
"That small break, Del? I know she's different. Do you think I don't care about her? That it's just the sex?" Unable to stand still, Jack paced from the bar to the counter and back again. It unnerved him, the depth of the caring. It was twisted up enough already without his best friend going off about promises, about Emma being different.
"I've always cared about Emma. About all of them. You know that. You damn well know that."
"Have you had sex with the rest of them, too?"
Jack took a long sip, and thought the hell with it. "I kissed your sister. Parker, since right now you're thinking of all of them as your sister. Back in college, after we ran into each other at a party."
"You hit on Parker?" It wasn't temper now but sheer shock that radiated. "Do I even know who you are?"
"I didn't hit on her. We bumped lips. It seemed like the thing to do at the time. Then, since it felt like kissing _my_ sister, and she had pretty much the same reaction, we had a good laugh about it, and that was that."
"Did you try out Mac next? Laurel?"
His eyes went hard and hot; his fingers itched to make another fist. "Oh yeah, I went through them all. That's what I do. I go through women like they're bags of chips then litter the streets with what's left of them. What the fuck do you take me for?"
"Right now, I don't know. You should've told me you were thinking about Emma that way."
"Oh yeah, I can see that. 'Hey, Del, I'm thinking about having sex with Emma. What do you think?' "
It wasn't temper that leaped back on Del's face, nor was it shock now. It was ice, and to Jack's mind, that was worse.
"Let's try it this way. How would you feel if you'd walked in tonight? Try that on, Jack."
"I'd be pissed. I'd feel betrayed. You want me to say I fucked it up? I fucked it up. But every way I look at it ends up like this. You think I don't know how it is for you? The position you took on when your parents died? And what they all mean to you? Every one of them. I was there with you through it, Del."
"This doesn't have anything to do with—"
"Everything does, Del." Jack paused a moment, spoke more calmly now. "I know it doesn't matter that Emma has a family. She's yours."
Some of the ice thawed. "Remember that. And remember this. If you hurt her, I'll hurt you."
"That's fair. Are we okay on this?"
"Not yet."
"Let me know when we are." Jack set down the half-finished beer.
_W_ ITH NO CHOICE, EMMA BUCKLED DOWN TO FINISH THE WORK for Friday's event. She and her full crew began early Friday morning designing and creating the flowers for the other weekend events.
Late in the afternoon, she began shifting flowers from the cooler, putting others in, loading the van so her team could start dressing the house and terraces.
Once the reception was under way, she'd come back and finish what was left on her own.
Just prior to the bride's arrival, she and Beach filled the portico urns with enormous white hydrangeas. "Gorgeous. Perfect. Go on in and help Tiffany with the foyer. I'll go work with Tink around back."
She made the dash, calculating the time, checking other pots and arrangements along the route. On the terrace, she climbed the ladder to hook the white rose ball in the center of the pergola.
"I didn't think I was going to like it." Tink hauled the standing arrangements into place. "White's so, you know, _white_. But it's really interesting, and sort of magical. Hiya, Jack. Gee, who punched you?"
"Del and I punched each other. Just something we do every so often."
"For God's sake."
If he'd expected Emma to get fluttery about his bruised jaw, he was disappointed. Annoyance in every movement, she climbed down the ladder, set her hands on her hips. "Why is it men think beating on each other fixes anything?"
"Why do women think eating chocolate does? It's the nature of the beast."
"Tink, let's finish the swags. Chocolate at least makes you feel good," Emma said as she continued to work. "A fist in the face doesn't. And did it fix things?"
"Not completely. But it's a start."
"Is he all right?" She pressed her lips together as she glanced back at Jack. "I know Parker tried to call him, but he's been in court all day."
"He hit me first." Jack took the ladder from her, moved it where she pointed, then tapped his swollen lip. "Ouch."
With a roll of her eyes, Emma gave him a very light kiss. "I don't have time to feel sorry for you right now, but I promise to make time later if you want to stay."
"I was just going to drop by, let you know things are . . . not quite, then get out of the way. I know you're slammed through the weekend."
"I am, and you can probably find something a lot better to do than hang around here."
He'd feel guilty, just a little miserable, still somewhat pissed, she thought. It called, to her mind, for friends and family.
"But . . . you could hang around here. Or with Carter, or at my place. If you want. I'm going to duck out during the reception and finish up some things for tomorrow."
"Why don't we play it by ear?"
"That's fine." She stepped back, studied the pergola, then hooked an arm through Jack's. "What do you think?"
"That I didn't know there were so many white flowers in the world. It's elegant and fanciful at the same time."
"Exactly." She turned toward him, brushed her fingers through his hair and her lips at the corner of his abused mouth. "I need to go check the Grand Hall and the Ballroom."
"Maybe I'll see if Carter can come out and play."
"I'll see you later, if . . ."
"If," he agreed, then risked the pain for a more serious kiss. "Okay. I'll see you later."
She laughed, and made the dash inside.
CHAPTER ELEVEN
_A_ T THE END OF THE NIGHT, WITH HER COOLER FILLED WITH bouquets, centerpieces, and arrangements for the rest of the weekend—and the full knowledge she'd have to be up by six to complete more—Emma made it as far as the sofa before she dropped.
"You're actually going to do all of this again tomorrow," Jack said. "Twice."
"Mmm-hmm."
"And one more time on Sunday."
"Uh-huh. I need to get in a solid two hours on Sunday's, in the morning before dressing the first event. But the team can finish up the rest of Sunday's while I'm dealing with Saturday's. Both Saturday's."
"I've helped out a few times, but never actually . . . It's every weekend?"
"It slows down some in the winter." She snuggled in a little, toed off her shoes. "April through June are the prime months, with another big jump in September and October. But basically? Yes, every weekend."
"I took a look at your cooler when you were working. You definitely need that second one."
"I really do. When we started, none of us imagined we'd get this big. No, that's wrong. Parker did." It made her smile to think of it. "Parker always did. I just figured I'd be able to make a living wage doing what I liked." Relaxing inch by inch, she curled aching toes. "I never thought we'd get to a point where we're all juggling events and duties, clients, subs. It's amazing."
"You could use more help."
"Probably. It's the same for you, really, isn't it?" When he lifted her feet onto his lap, rubbed those cramped toes and tired arches, her eyes drifted shut. "I remember when you started your firm. It was basically you. Now you have staff, associates. If you're not working on drawings, you're on-site or meeting with clients. When it's your company, it's a whole lot different from punching time."
She opened her eyes again, met his gaze. "And every time you hire somebody—even when it's the best thing, the right thing, to do for yourself and your business—it feels like giving just a little bit of it away."
"I had myself talked in and talked out of hiring Chip a dozen times, just for that reason. The same with Janis, then Michelle. Now I've taken on a summer intern."
"That's great. God, doesn't that make us the older generation? That's hard to deal with."
"He's twenty-one. Just. I felt ancient when I interviewed him. What time do you have to start tomorrow?"
"Let me think . . . Six, I guess. Six thirty maybe."
"I should let you get some sleep." In an absent gesture, he ran a hand up and down her calf. "You're pretty tied up for the weekend. If you're up for it, we could go out Monday."
"Out? Like out there?" She waved a hand in the air. "Where there are places where people bring you food, and possibly entertainment?"
He smiled. "Dinner and a movie sound good?"
"Dinner and a movie? It sounds like whole buckets of good."
"Then I'll grab a bucket and pick you up Monday, about six thirty?"
"It works for me. Really works. I have a question." She stretched luxuriously as she sat up. "You stuck around here until after midnight, and now you're going to go home so I can get some sleep?"
"You put in a long one." He gave her calf a quick squeeze. "You must be tired."
"Not that tired," she said, and, grabbing a fistful of his shirt, pulled him down with her.
_M_ ONDAY EVENING, LAUREL WALKED HER CONSULT CLIENTS TO the door. September's bride and groom took away a container holding a variety of cake samples. But she knew they'd decided on the Italian cream cake. Just as she knew the bride was leaning toward her Royal Fantasy design, and the groom her Mosaic Splendor.
The bride would win, she had no doubt, but it was nice to have a man take a genuine interest in the details.
Plus she'd talk the bride into having a groom's cake in a mosaic design that complemented the wedding cake.
Everybody wins, she thought.
"Just let me know when you make up your minds, and don't worry about changing those minds. There's plenty of time." She kept the easy smile on her face, the breezy manner intact even when she saw Del coming up the walk.
He projected successful lawyer, she thought, in his perfectly cut suit, his perfect briefcase, his handsome shoes.
"Parker's in her office," she told him. "I think she's clear."
"Okay." He came in, shut the door. "Hey," he said when she started up the stairs. "Are you not speaking to me?"
She flicked a glance back at him. "I just did."
"Barely. I'm the one who should be pissed off here. You don't have anything to be snotty about."
"I'm being snotty?" She paused, waited for him to join her on the stairs.
"I don't expect my friends and family to lie to me, or lie by omission. And when they do—"
She poked a finger, hard, into his shoulder, then held it up. "Number one, I didn't know _you_ didn't know. Neither did Parker or Mac or Carter. Or Emma, for that matter. So that's between you and Jack. Second," she continued, poking him again when he started to speak. "I agree with you."
"If you'd take a minute to . . . You agree with me?"
"Yes, I do. And in your place I'd have been hurt and pissed off. Jack should have told you he and Emma were involved."
"Well, okay. Thanks—or sorry. Whichever you prefer."
"However."
"Shit."
"However," she repeated. "You might want to ask yourself why your best friend didn't tell you. And you might want to look back at the way you handled the other night, how you came across as a tight-ass having a sulk."
"Wait a damn minute."
"That's the way I see it, just as I see—even if I don't agree—why Jack didn't tell you. You'd have gone all Delaney Brown on him."
"Just what does that mean?"
"If you don't know, telling you won't make any difference."
He grabbed her hand to stop her as she moved on. "That's such a cop-out."
"Fine. Delaney Brown disapproves. Delaney Brown knows best. Delaney Brown will manipulate and maneuver until he positions you where he wants—for your own good."
"That's cold, Laurel."
She sighed, softened. "No, it's not. Not really. Because you really do have the best interest of your friends and family at heart. You're just always so damn sure, Del, that you know what that is."
"Are you going to stand there and tell me you think what's going on with Emma and Jack is the best thing, for either of them?"
"I don't know." She lifted her hands, palms up. "I don't pretend to know. All I know is that, for the moment, they're enjoying each other."
"It doesn't even weird you out? It doesn't make you feel as if you've stepped into an alternate reality?"
She had to laugh. "Not exactly. It's a little—"
"It's like—what if I suddenly put moves on you? I just decide, hey, I'd like to have sex with Laurel."
The soft hardened; the laughter died. "You're such an idiot."
"What? _What?_ " he demanded as she stormed away up the stairs. "It is an alternate reality," he muttered, and climbed the rest of the way to his sister's office.
She sat at her desk, where he'd expected to find her, talking on her headset as she worked at the computer. "That's just exactly right. I knew I could count on you. They'll need two hundred and fifty. You can deliver them to me, here, and I'll take it from there. Thank you, so much. You, too. Bye."
She pulled off the headset. "I just ordered two hundred and fifty rubber duckies."
"Because?"
"The client wants them swimming in the pool on her wedding day." She sat back, sipped from her bottle of water, and gave him a long, sympathetic look. "How are you doing?"
"I've been better, I've been worse. Laurel just agreed Jack was an asshole for not telling me, but apparently that's my fault because I'm Delaney Brown. Do I manipulate people?"
She studied him carefully. "Is that a trick question?"
"Damn it." He dumped his briefcase on the desk then walked to her coffee setup.
"Okay, serious question. Yes, of course you do. So do I. We're problem solvers, and good at finding solutions and answers. When we do, we do what we can to move people toward those solutions and answers."
He turned back to her, scanning her face. "Do I manipulate you, Parks?"
"Del, if you hadn't manipulated me, to some extent, regarding the estate, how you intended to set it up after Mom and Dad died, I wouldn't have just ordered two hundred and fifty rubber duckies. I wouldn't have the business. None of us would."
"That's not the kind of thing I mean."
"Would you, have you ever pushed me into doing something I didn't want to do—genuinely didn't want—and did you push because it was what you wanted? No. I'm sorry you found out about Jack and Emma the way you did. But I think the situation's a little strange for all of us. None of us saw it coming. I don't think Jack and Emma saw it coming."
"I can't get used to it." He sat, sipped his coffee. "By the time I do, it'll probably be over anyway."
"Aren't you the romantic?"
He shrugged. "Jack's never been serious about a woman. He's not a dog—exactly—but he's not the long haul guy either. He wouldn't hurt her on purpose. He's not made like that. But . . ."
"Maybe you should have a little more faith in your two friends." She sat back, swiveling side-to-side in the chair. "I think things happen between people for a reason. Otherwise I couldn't do what I do every day. Sometimes it works out, sometimes it doesn't, but there's always a reason."
"Under that you're telling me to stop being such a hard-ass, and just be a friend."
"Yes." She smiled at him. "That's my answer, my solution, and where I'm trying to manipulate you. How am I doing?"
"Pretty well. I guess I should stop by and see Emma."
"It would be nice."
"Let's go over these papers first." He opened his briefcase.
Twenty minutes later, he gave a quick knuckle rap on Em-ma's door, then pushed it open. "Em?"
He heard the music, or what he thought of as her work music—harps and flutes—so walked back to her work area. She sat at the counter, arranging little pink rosebuds in a white basket.
"Em."
She jumped, swung around. "Scared me. I didn't hear you."
"Because I'm interrupting."
"Just getting a head start on some arrangements for a baby shower this week. Del." She got up. "How mad at me are you?"
"Zero. Less than zero." He found himself ashamed she would think otherwise. "I'm at about seven out of ten with Jack, but that's an improvement."
"I should point out that when Jack's sleeping with me I'm also sleeping with him."
"Maybe we could just find a code word for that. Like you and Jack are writing a novel together, or doing lab work."
"Are you mad because we're doing lab work, or because we didn't tell you?"
"He didn't tell me. Anyway, it's mixed. I'm trying to come to terms with the lab work, and I'm pissed he didn't tell me you and he were . . ."
"Lining up the test tubes? Labeling the petri dishes?"
Frowning, he slid his hands into his pockets. "I don't like the lab work code after all. I just want you to be okay, and happy."
"I am okay. I am happy. Even though I know the two of you punched each other over it. Actually, maybe that makes me happier. It's always flattering to have guys punch each other in the face over me."
"It was an impulse of the moment."
She stepped to him, reached up so she could frame his face, and brushed her lips over his. "Try not to do it again. It involves two of my favorite faces. Let's go sit out on the back patio, drink some lemonade, and be friends."
"Okay."
_W_ HILE THEY DID, JACK TOOK A SEAT IN MAC'S STUDIO AND unrolled the plans for the proposed addition.
"It's the same design I e-mailed you, but with more detail, and the couple of changes you wanted."
"Look, Carter! You have your own room."
He danced his fingers over Mac's bright cap of hair. "I was kind of hoping we'd still share one."
Mac laughed, leaned closer to the plans. "Just look at my dressing room. Well, client dressing room. And God, I _love_ the patio space we'll get. Want a beer, Jack?"
"No, thanks. Got anything soft?"
"Sure. Diet."
"Crap. Water."
When she went into the kitchen, Jack pointed out details to Carter. "These built-ins will give you plenty of shelves for books, or whatever you want. For files, for supplies."
"What's this? A fireplace?"
"One of Mac's changes. She said every professor worth his PhD should have a fireplace in his study. It's a small gas log unit. It'll also provide an additional heat source for the room."
Carter glanced over as Mac came back with a bottle of water and two beers. "You got me a fireplace."
"I did. It must be love." She kissed him lightly, then bent to pick up their three-legged cat, Triad.
It must be, Jack thought when she sat and the cat curled in her lap.
While they discussed details, choices of materials, he wondered what it was like to feel that connection with and that certainty about another person.
No doubt in their minds, he mused, that this was _the_ one. The one to make a home with, build a future with, maybe have kids with. Share a cat with.
How did they know? Or at least believe enough to risk it?
It was, for him, one of life's great mysteries.
"When can we start?" Mac demanded.
"I'll submit for permit tomorrow. Do you have a contractor in mind?"
"Um . . . the company we used on the initial remodel was good. Are they still available?"
"I ran it by him. I can contact him tomorrow, ask him to submit a bid."
"You're the man, Jack." Mac gave him a friendly punch in the arm. "Do you want to stay for dinner? We're making pasta. I can call and see if Emma's interested."
"Thanks, but we're going out."
"Aw."
"Stop." But he shook his head and laughed.
"I can't help it if I find it adorable that my pals are getting all cozy."
"We're going to grab some dinner and catch a flick."
"Aw."
He laughed again. "I'm getting out of here. See you on Poker Night, Carter. Prepare to lose."
"I could just hand you the money now, save time."
"Tempting, but I prefer the satisfaction of skinning you at the table. I'll get you that bid," he added as he headed for the door. "You keep that copy of the plans."
He heard Mac's "uh-oh" an instant before he spotted Del.
They stopped, about five feet apart.
"Wait!" Mac called out. "If you're going to punch each other again, I want my camera."
"I'll shut her up," Carter promised.
"Hey! Wait! I was serious," she managed before Carter dragged her back inside.
Jack jammed his hands into his pockets. "This is just fucking stupid."
"Maybe. Probably."
"Look, we punched each other, we each said our piece. We had a beer. According to the rules, that should about cover it."
"We didn't take in a sporting event."
Jack felt some of the tension in his shoulders ease. That was more like Del. "Can we do that tomorrow? I've got a date."
"What happened to bros before hos?"
A smile spread amiably over Jack's face. "Did you just call Emma a ho?"
Del's mouth opened and closed before he dragged a hand through his hair. "You see the complications here? I just called Emma a ho because I wasn't thinking of Emma as Emma, and I was being a smart-ass."
"Yeah, well, I know that. Otherwise I'd've had to punch you in the face again. The Yankees have a home game tomorrow night."
"You drive."
"Uh-uh. We get Carlos. I spring for the car service. You spring for the tip and the beer. We split the dogs."
"All right." Del considered a moment. "Would you punch me in the face over her?"
"I already did."
"That wasn't about her."
Point taken, Jack thought. "I don't know."
"That's a good answer," Del decided. "I'll see you tomor row."
_S_ INCE DINNER—BISTRO FARE—AND A MOVIE—ACTION FLICK—worked so well, they made a second official Monday night date. Full schedules prevented any appreciable time together between, but they managed what they termed a friendly booty call and a few teasing e-mails.
Emma wasn't sure if their current relationship led off with sex or friendship, but it felt as if both of them were trying to find a happy balance between the two.
She was nearly finished dressing for the evening when Parker came in and called up the stairs.
"Be right down. I've got the flowers you wanted in the back, in a holding vase. Though I still don't see why you have to go watch people make wedding favors."
"The MOB wants me to stop by, give it all the once-over. So I stop by, give it all the once-over. It shouldn't take that much time."
"I'd have saved you some of that time and dropped them by, but I got hung up with my last consult of the day." Emma dashed downstairs, stopped, did a runway turn. "How do I look?"
"Gorgeous. One expects no less."
Emma laughed. "The hair up works, right? Just a little messy and ready to tumble."
"It works. So does the dress. That deep red really suits you. And let me add, the workouts are paying off."
"Yeah, I hate that part because it means I have to keep it up. Wrap or sweater?" she asked, holding a choice in either hand.
"Where are you going?"
"Art opening. Local artist, modern."
"The wrap's more arty, and aren't you clever?"
"Am I?"
"Most people will be in black, so that red dress is going to pop. You could give lessons."
"If you're going to dress up, might as well get noticed, right? How about the shoes?"
Parker considered the peep-toe spikes with their sexy ankle straps. "Killers. Nobody with a Y chromosome is going to look at the paintings."
"I've only got one Y chromosome in mind."
"You look happy, Emma."
"It's hard not to, because I am. I'm involved with an interesting man who makes me laugh _and_ makes me tingle, one who actually listens to what I have to say, and who knows me well enough I can be myself without any of the filters. And the same goes for him. I know he's fun, funny, smart, not afraid to work, values his friends, is obsessed with sports. And . . . well, all the things you just know when you've been around someone for a dozen years the way we have."
She led the way to her work area. "Some people might think that takes the discovery or the excitement out of things, but it doesn't. There's always something new, and there's the stability of real understanding. I can be comfortable and excited around him at the same time.
"I went with the pink tulips and the mini iris. It's cheerful, female, springy."
"Yes, it's perfect." Parker waited while Emma took them out of the vase, adjusted the sheer white ribbon.
"I could add some lisianthus if you want it fuller."
"No, it's great. Just right. Emma," Parker began as her friend coned the arrangement in clear, glossy paper, "do either of you know you're in love with him?"
"What? No. I never said . . . Of course, I love Jack. We _all_ love Jack."
"We all didn't put on a red dress and sexy shoes to spend the evening with him."
"Oh, well that's just . . . I'm going out."
"It's not just that. Em, you're going out with Jack. You're sleeping with Jack. Which is what I figured was what, more or less. But I listened to you just now, I watched your face just now. And, honey, I know you. You're in love."
"Why do you have to say that?" Distress covered Emma's face. "It's just the sort of thing that's going to mess with my head, and make everything all sticky and awkward."
Brow lifted, Parker angled her head. "Since when have you thought of being in love as sticky and awkward?"
"Since Jack. I'm okay with the way things are now. I'm better than okay. I'm in an exciting relationship with an exciting man and I don't . . . I don't expect it to be anything else. Because that's not Jack. He isn't the kind who thinks about what we'll be doing five years from now. Or five weeks from now. It's . . . just now."
"You know, it's odd that you and Del, who are closer to him than anyone, both have such little confidence in him."
"It's not that. It's just that in this particular area, Jack's not looking for . . . permanent."
"What about you?"
"I'm going to enjoy the moment." She said it with a decisive nod. "I'm not going to be in love with him, because we both know what'll happen if I am. I'll start romanticizing it, and him, and us, and wishing he'd . . ."
She trailed off, pressed a hand to her belly. "Parker, I know what it's like to have someone feel that way about me, when I don't feel that way. It's just as awful for the one who's not in love as it is for the one who is."
She shook her head. "No, I'm not going there. We've only been seeing each other like this for a little while. I'm not going there."
"All right." To soothe, Parker stroked a hand over Emma's shoulder. "If you're happy, I'm happy."
"I am."
"I'd better run. Thanks for putting these together."
"Never a problem."
"I'll see you tomorrow. Follow-up consult on the Seaman wedding."
"I've got it in my book. I know they want to walk around the gardens, see them now to project what they'll want in those areas next April. I'm going to dress a couple of the urns with Nikko blue hydrangeas I've been coaxing along in the greenhouse. They're lush, and should give a good show. I've got a couple other tricks up my sleeve, too," she added as she walked to the door with Parker.
"You always do. Have a good time tonight."
"I will."
Emma closed the door, then just leaned back against it.
She could fool herself, she admitted. She could certainly fool Jack. But she could never fool Parker.
Of course she was in love with Jack. She'd probably been in love with Jack for years, and simply convinced herself it was lust. The lust had been bad enough, but love? Deadly.
She knew exactly what she wanted from love—from the down into the bones, rooted in the heart, blooming through the body love. She wanted forever.
She wanted the day after day, night after night, year after year, the home, the family, the fights, the support, the sex, the everything.
She'd always known what she wanted in a partner, in a lover, in the father of her children.
But why did it have to be Jack?
Why, when she finally felt all the things she'd waited all of her life to feel, did it have to be for a man she knew so well? Well enough to understand he was someone who wanted his own space, his own direction, who considered marriage a gamble with long odds?
She knew all those things about him, and still she'd fallen.
If he knew, he'd be . . . appalled? she wondered. No, that was probably too strong. Concerned, sorry—which was worse. He'd be kind, and he'd pull the plug gently.
And that was mortifying.
There was no reason he had to know. It was only a problem if she let it be a problem.
So, no problem, she decided.
She was as skilled at handling men as she was at handling flowers. They'd go on just as they were, and if it got to a point where it caused her pain instead of pleasure, she'd be the one to pull the plug.
Then she'd get over it.
She pushed away from the door to wander into the kitchen for a glass of water. Her throat felt dry and a little raw.
She'd get over him, she assured herself. What was the point in worrying about that now when they were still together?
Or . . . she could make him fall in love with her. If she knew how to keep a man from falling for her—or nudge him into falling out if he thought he was falling in—why couldn't she make one fall in?
"Wait, I'm confusing myself."
She took a breath, took a sip.
"If I make him fall in love with me, is it real? God, this is too much to think about. I'm going out to an opening. That's it, that's all."
The knock on the door brought relief. Now she could stop thinking, stop worrying all this to pieces.
They'd go out. They'd enjoy each other. Whatever happened next, happened.
CHAPTER TWELVE
_S_ ATISFACTION, EMMA DECIDED, WENT A LONG WAY TO STAMPING out worry. The look in Jack's eyes when she opened the door was exactly what she'd aimed for.
"I need a moment of silence," he told her, "to offer up thanks."
She gave him a slow, sultry smile. "Then let me say you're welcome. Do you want to come in?"
Closing the distance, he trailed his fingers over her shoulder, down her arm. Those smoky eyes stayed fixed on hers. "I'm just having this thought about how I come in and we forget about the opening."
"Oh no." She nudged him back, and stepped out. Handing him the wrap, she turned her back, glanced around as he draped it over her shoulders. "You promised me strange paintings, lousy wine, and soggy canapes."
"We could go back inside." He leaned down to nuzzle her neck. "I'll sketch some erotic drawings, we'll drink good wine, and call out for pizza."
"Choices, choices," she said as they walked to his car. "Art opening now, erotic sketches later."
"If we must." But he stopped at the car to draw her into a luxurious kiss. "I like the way you look, which is amazing."
"That was the plan." She stroked her hand over the slate gray sweater he wore under a leather jacket. "I like the way you look, Jack."
"Since we look so good I guess we'd better go be seen." When he got behind the wheel he sent her an easy smile. "How was the weekend?"
"Jammed, as advertised. And successful, since Parker talked the clients into renting the tents for Saturday. When it rained, everybody stayed dry. Even better, we scrabbled around for more candles and some of my emergency supply of flowers so we had all this soft light and fragrance while the rain pattered on the tent. It was really lovely."
"I wondered how that worked out. I was out on new construction Saturday afternoon, and we didn't. Stay dry, that is."
"I like spring rains. The way they sound, the way they smell. Not all brides feel the same, but we managed to make this one really happy. And how was Poker Night?"
He scowled at the road as his headlights cut through the dark. "I don't want to talk about it."
She laughed. "I heard Carter cleaned your clock."
"The guy hustled us with all that 'I'm not much of a card player' routine, and that open, honest face. He's a shark."
"Yes, oh yes, Carter's a real shark."
"You haven't played cards with him. Believe it."
"Sore loser."
"Damn right."
Amused, she leaned back in the seat. "So, tell me a little about this artist."
"Ah . . . yeah, I should do that." During a beat of silence, he tapped his fingers on the steering wheel. "A friend of a client. I think I mentioned that."
"You did." She'd meant the art itself, but she caught enough in his tone to zero in. "And a friend of yours?"
"Sort of. We went out a couple of times. A few times. Maybe several."
"Ah. I see." Though her interest spiked, she kept her tone casual. "She's an ex."
"Not exactly. We weren't . . . It was more we hooked up for a few weeks. More than a year ago. Closer to two, actually. It was just a thing, then it wasn't."
His uneasiness struck her as both interesting and flattering. "If you're looking at this as boggy ground, Jack, you don't need to. I've had my suspicions you've slept with other women."
"It's true. I have. And Kellye—she spells it with an 'e' on the end—is one of them. She's . . . interesting."
"And artistic."
His lips twitched, intriguing her. "You be the judge."
"So, why did the thing stop being a thing, or is that too awkward a question?"
"It got a little too intense for me. She's an intense sort, and high-maintenance."
"Required too much attention?" Emma asked, with just a hint of cool.
" _Required_ is a good word for it. Anyway, it stopped being a thing."
"But you stayed friendly."
"Not so much. But I ran into her a couple months ago, and we were okay. Then she got in touch about her opening, and I figured it wouldn't hurt to go. Especially since you're here to protect me."
"Do you often need protection from women?"
"All the time," he said and amused her again.
"Don't you worry." She patted his hand on the gear shift. "I'm here for you."
After he'd parked, they walked through the cool spring evening in a breeze that fluttered the ends of her wrap. The little shops she enjoyed browsing were already closed, but the bistros did brisk business. A number of diners braved the chill for a chance to eat outside with candles flickering on tables.
She smelled roses and red sauce.
"You know what I haven't done for you?" Emma began.
"I have a list, but I figured I'd work up to some of the more interesting items."
She poked him with her elbow. "Cook. I'm a good cook when I have time. I'll have to seduce you with my fajitas."
"Anytime, anywhere." He stopped in front of the gallery. "Here we are. Are you sure you wouldn't rather cook?"
"Art," she said, and breezed inside.
No, not really, she thought immediately. The first thing she saw other than a number of people standing around looking intense was a large white canvas with a single, wide, blurry line of black running down the center.
"Is it a tire tread? A single tire tread on a white road, or a division of . . . something?"
"It's a black line on a white canvas. And we're going to need drinks," Jack decided.
"Mmm-hmm."
While he left her to find some, Emma wandered. She studied another canvas holding a twisted black chain with two broken links titled _Freedom_. Another boasted what seemed to be a number of black dots, which on closer inspection proved to be a scattering of lowercase letters.
"Fascinating, isn't it?" A man in dark-framed glasses and a black turtleneck stepped up beside her. "The emotion, the chaos."
"Uh-huh."
"The minimalist approach to intensity and confusion. It's brilliant. I could study this one for hours, and see something different each time."
"It depends on how you arrange the letters."
He beamed at her. "Exactly! I'm Jasper."
"Emma."
"Have you seen _Birth_?"
"Not firsthand."
"I believe it's her best work. It's just over there. I'd love to hear what you think."
He touched a hand to her elbow—testing, she knew—as he gestured. "Can I get you some wine?"
"Actually . . . I have some," she said when Jack joined them and offered her a glass. "Jack, this is Jasper. We were admiring _Babel_ ," she added when she found the title.
"A confusion of language," Jack supposed and dropped a light, possessive hand on Emma's shoulder.
"Yes, of course. If you'll excuse me."
"Busted his bubble," Jack added when Jasper slunk off. Testing the very bad wine, he studied the canvas. "It's like one of those magnet kits people buy for their refrigerator."
"Thank God. Thank God. I thought you actually saw something."
"Or somebody dropped the Scrabble tiles."
"Stop." She had to suck in a breath to stop a laugh. "Jasper finds it brilliant in its minimalist chaos."
"Well, that's Jasper for you. Why don't we just—"
"Jack!"
Emma turned to see a six-foot redhead, arms outflung, burst through the crowd. She wore snug black that showed miles of legs, a pencil-thin body offset by high, firm breasts that almost poured out of the scooped-neck of her top. She jangled from the clanging of a dozen silver bangles on her arm.
And nearly mowed Emma down as she threw her arms around Jack to fix her murderous red mouth to his.
The best Emma could do was grab Jack's wineglass before it upended.
"I knew you'd come." Her voice was low, and nearly a sob. "You don't know what it means to me. You can't know."
"Ah," he said.
"Most of these people, they don't _know_ me. They haven't been _in_ me."
Jesus. Christ. "Okay. Let's just . . ." He tried untangling himself, but her arms tightened around his neck like a garrote. "I wanted to stop by and say congratulations. Let me introduce you to . . . Kellye, you're cutting off my oxygen."
"I've _missed_ you. And tonight means so much, now so much more." Dramatic tears glimmered in her eyes; her lips quivered with emotion. "I know I can get through tonight, the stress, the _demands_ , now that you're here. Oh, Jack, Jack, stay close to me. Stay close."
Any closer, he thought, and he _would_ be in her. "Kellye, this is Emmaline." Desperate now, Jack gripped Kellye's wrists to unlock them from his neck. "Emma . . ."
"It's wonderful to meet you." Cheerful, enthusiastic, Emma offered a hand. "You must—"
Kellye stumbled back as if stabbed, then whirled on Jack. "How dare you! How could you? You'd bring _her_ here? Throw _her_ in my face? Bastard!" She ran, shoving her way through the fascinated crowd.
"Okay, this was fun. Let's go." Jack grabbed Emma's hand and pulled her to the door. "Mistake. Big mistake," he said when he managed a good gulp of fresh air. "I think she punctured my tonsils with her tongue. You didn't protect me."
"I failed you. I'm so ashamed."
He narrowed his eyes as he pulled her along the sidewalk. "And you think that was funny."
"I'm a bitch, too. Coldhearted. More shame." She had to stop, just stop and howl with laughter. "God, Jack! What were you thinking?"
"When a woman has the power to puncture a man's tonsils with her tongue, he stops thinking. She also has this trick where she . . . And I almost said that out loud." He dragged a hand through his hair as he studied her glowing face. "We've been friends too long. It's dangerous."
"In the spirit of friendship, I'm going to buy you a drink. You deserve it." She took his hand. "I didn't believe you when you said she got too intense and so on. I figured you were just being the usual no-commitment guy. But _intense_ is way too quiet a word for her. Plus, her art is ridiculous. She really ought to hook up with Jasper. He'd adore her."
"Let's drive across town for that drink," he suggested. "I don't want to chance running into her again." He opened the car door for her. "You weren't the least bit embarrassed by that."
"No. I have a high embarrassment threshold. If she'd been remotely sincere, I'd have felt sorry for her. But she's as fake as her art. And probably just as odd."
He considered as he walked around to get in the driver's side. "Why do you say that? About her being fake?"
"It was all about the drama, and her in the center of it. She may feel something for you, but she feels a lot more for herself. And she saw me, before she jumped you. She knew you'd brought me with you, so she put on a show."
"Deliberately embarrassed herself? Why would anyone do that?"
"She wasn't embarrassed, she was revved." She angled her head, looking into his baffled eyes. "Men really don't see things like that, do they? It's so interesting. Jack, she was the star of her own romantic tragedy, and she fed on every moment. I bet she sells more of that nonsense she calls art tonight because of it."
When he drove in silence for the next few moments, she winced. "And all that really hammered your ego."
"Scratched it, superficially. I'm weighing that against knowing I didn't somehow give her the wrong signal and actually deserve that entertaining little show." He shrugged. "I'll take the scratch."
"You're better off. So . . . any other ex we-had-a-thing you want me to meet?"
"Absolutely not." He glanced at her, and the streetlights sheened over the golds and bronzes in his hair. "But I do want to say that, for the most part, the women I've dated have been sane."
"That speaks well of you."
_T_ HEY CHOSE A LITTLE BISTRO AND SHARED A PLATE OF ALFREDO. She relaxed him, he thought, which was odd, as he'd always considered himself fairly relaxed to begin with. But spending time with her, just talking about anything that came to mind, made any problem or concern he might be dealing with in some corner of his brain vanish.
Odder still was being excited and relaxed around a woman at the same time. He couldn't remember having that combination of sensations around anyone but Emma.
"How come," he wondered, "in all the years I've known you, you've never cooked for me?"
She wound a solitary noodle on her fork. "How come in all the years I've known you, you never took me to bed?"
"Aha. So you only cook for men when you get sex."
"It's a good policy." She smiled, her eyes laughing as she nibbled away at the noodle. "I go to a lot of trouble when I cook. It ought to be worth it."
"How about tomorrow? I can make it worth it."
"I bet you can, but tomorrow won't work. No time to market. I'm very fussy about my ingredients. Wednesday's a little tight, but—"
"I have a business thing Wednesday night."
"Okay, next week's better anyway. Unlike Parker, I don't carry my schedule in my head backed up by the BlackBerry attached to my hand, but I think . . . Oh. Cinco de Mayo. It's nearly the fifth of May. Big family deal—you remember, you've come before."
"Biggest blast-out party of the year."
"A Grant family tradition. Talk about cooking. Let me check my book and all of that, and we'll figure it out."
She sat back with her wine. "It's almost May. That's the best month."
"For weddings?"
"Well, it's a big one for that, but I'm thinking in general. Azaleas, peonies, lilacs, wisteria. Everything starts budding and blooming. And I can start planting some annuals. Mrs. G will put in her little kitchen garden. Everything starts over or comes back. What's your favorite?"
"July. A weekend at the beach—sun, sand, surf. Baseball's cruising. Long days, grills smoking."
"Mmm, all good, too. All very good. The smell of the grass right after you mow it."
"I don't have grass to mow."
"City boy," she said, pointing at him.
"My lot in life."
As they both toyed with the pasta, she leaned in. The conversations humming around them barely registered. "Did you ever consider living in New York?"
"Considered. But I like it here. For living, and for the work. And I'm close enough to go in and catch the Yankees, the Knicks, the Giants, the Rangers."
"I've heard rumors about ballet, opera, theater there, too."
"Really?" He sent her an exaggerated look of puzzlement. "That's weird."
"You, Jack, are such a guy."
"Guilty."
"You know, I don't think I've ever asked you, why architecture?"
"My mother claims I started building duplexes when I was two. I guess it stuck. I like figuring out how to use space, or change an existing structure. How can you use it better? Are you going to live in it, work in it, play in it? What's around the space, what's the purpose? What are the best and most interesting or practical materials? Who's the client and what are they really after? Not all that different, in some way, than what you do."
"Only yours last longer."
"I have to admit I'd have a hard time seeing my work fade and die off. It doesn't bother you?"
She pinched off a knuckle-sized piece of bread. "There's something about the transience, you could say. The fact that it's only temporary that makes it more immediate, more personal. A flower blooms and you think, oh, pretty. Or you design and create a bouquet, and think, oh, stunning. I'm not sure the impact and emotion would be the same if you didn't know it was only temporary. A building needs to last; its gardens need to cycle."
"What about landscape design. Ever consider it?"
"Probably more briefly than you did New York. I like working in the garden, out in the air, the sun, seeing what I put in come back the next year, or bloom all through the spring and summer. But every time I get a delivery from my wholesaler it's like being handed a whole new box of toys."
Her face went dreamy. "And every time I hand a bride her bouquet, see her reaction, or watch wedding guests look at the arrangements, I get to think: I did that. And even if I've made the same arrangement before, it's never exactly the same. So it's new, every time."
"And new never gets boring. Before I met you, I figured florists mostly stuck flowers in vases."
"Before I met you, I figured architects mostly sat at drawing boards. Look what we learn."
"A few weeks ago, I never imagined we'd be sitting here like this." He put his hand over hers, fingers lightly skimming while his eyes looked into hers. "And that I'd know before the night was over I'd be finding out what's under that really amazing dress."
"A few weeks ago . . ." Under the table, she slid her foot slowly up his leg. "I never imagined I'd be putting on this dress for the express purpose of you getting me out of it. Which is why . . ."
She leaned closer so the candlelight danced gold in her eyes, so her lips nearly brushed his. "There's nothing under it."
He continued to stare at her, into the warmth and the wicked. Then shot up his free hand. "Check!"
_H_ E HAD TO CONCENTRATE ON HIS DRIVING, PARTICULARLY SINCE he attempted to break the land speed records. She drove him crazy, the way she cocked her seat back, crossed those gorgeous bare legs so that the dress slithered enticingly up her thighs.
She leaned forward—oh yes, deliberately, he knew—so that in the second he dared take his eyes off the road he had a delectable view of her breasts rising against that sexy red.
She fiddled with the radio, cocked her head long enough to send him a feline, female smile, then leaned back again. Re-crossed her legs. The dress snuck up another half inch.
He worried he might drool.
Whatever she'd put on the radio came to him only in bass. Pumping, throbbing bass. The rest was white noise, static in the brain.
"You're risking lives here," he told her, and only made her laugh.
"I could make it more dangerous. I could tell you what I want you to do to me. How I want you to take me. I'm in the mood to be taken. To be used." She trailed a finger up and down the center of her body. "A few weeks ago, or longer than that, did you ever imagine taking me, Jack? Using me?"
"Yes. The first time was after that morning I saw you on the beach. Only, when I imagined it, it was night, and I walked down and pulled you into the water, into surf. I could taste your skin and the salt. I had your breasts in my hands, in my mouth, while the water beat over us. I took you on the wet sand while the waves crashed, until all you could say was my name."
"That's a long time ago." Her voice went thick. "A long time to imagine. I know one thing. We really need to go back to the beach."
The laugh should've eased some of the ache, but only increased it. Another first, Jack concluded: A woman who could make him laugh and burn at the same time.
He whipped the car off the road and onto the long drive of the Brown Estate.
There were lights glowing on the third floor, both wings of the main house, and the glimmer of them in Mac's studio. And there, thank God, the shine of Emma's porch light, and the lamp she'd left on low inside.
He hit the release for his seat belt even as he hit the brakes. Before she could do the same, he managed to shift toward her, grab hold of her and let his mouth ravish hers.
He molded her breasts, gave himself the pleasure of riding his hands up those legs, under that seductive red.
She closed her teeth over his tongue, a quick, erotic trap, and struggled with his fly.
He managed to yank down one shoulder of her dress before he rammed his knee into the gear shift.
"Ouch," she said on a breathless laugh. "We'll have to add knee pads to the elbow pads."
"Damn car's too small. We'd better get inside before we hurt ourselves."
Her hands gripped his jacket, yanked to bring him back for one more wild kiss. "Hurry."
They shoved out of opposite sides of the car, then bolted for each other. Another breathless laugh, a desperate moan, sounded in the silence. They stumbled, grappled, and groped as their mouths clashed.
She yanked and tugged his jacket away as they circled up the walk like a pair of mad dancers. When they reached the door she simply shoved him back against it. Her mouth warred with his, breaking only so she could drag his sweater up, nails scraping flesh before she tossed it aside.
The heels and the angle brought her mouth level with his jaw. She bit it as she whipped the belt out of his pants, and tossed that aside as well.
Jack fumbled behind him for the doorknob, and they both lurched inside. Now he pushed her back to the door, yanked her arms over her head and handcuffed her wrists with his hand. Keeping her trapped, he shoved her skirt up and found her. Just her, already hot for him, already wet. And her gasp ended on a cry when he drove her hard and fast to climax.
"How much can you take?" he demanded.
Breath ragged, body still erupting, she met his eyes. "All you've got."
He drove her up again, beyond moans and cries, storming her system with his hands, with his mouth. Heat sheathed her, slicked her skin as he dragged the dress down to free her breasts, to feed on them. Everything she wanted, more than she could imagine, rough and urgent, he used and exploited her body.
Owned her, she thought. Did he know? Could he know?
Want was enough, to want like this, be wanted like this. She would make it enough. And wanting him, craving him, she braced against the door and wrapped a leg around his waist.
"Give me more."
She consumed him, in that moment before he plunged inside her, the look, the feel, the taste of her consumed him. Then with a new kind of madness, he took her against the door, battering them both while her hair tumbled out of its pins, while she said his name over and over.
Release was both brutal and glorious.
He wasn't entirely sure he was still standing, or that his heart would ever beat normally again. It continued to jackhammer in his chest, making the basic act of breathing a challenge.
"Are we still alive?" he managed.
"I . . . I don't think I could feel like this if I wasn't. But I do think my life passed before my eyes at one point."
"Was I there?"
"In every scene."
He gave himself another minute, then eased back. He was indeed still standing, he noted. And so was she—flushed and glowing, and naked but for a pair of sky-high sexy heels.
"God, Emma, you're . . . There are no words." He had to touch again, but this time almost reverently. "We're not going to make it upstairs yet."
"Okay." When he gripped her hips, lifted, she boosted up to wrap both legs around his waist. "Can you make it as far as the couch?"
"I'm going to give it a try." He carried her there where they could fall in a tangled heap.
_T_ WO HOURS LATER, WHEN THEY FINALLY MADE IT UPSTAIRS, they slept.
She dreamed, and in the dream they danced in the garden, in the moonlight. The air was soft with spring and scented by roses. Moon and stars silvered the flowers that bloomed everywhere. Her fingers twined with his as they glided and turned. Then he brought hers to his lips to kiss.
When she looked up, when she smiled, she saw the words in his eyes even before he spoke them.
"I love you, Emma."
In the dream her heart bloomed like the flowers.
CHAPTER THIRTEEN
_I_ N PREPARATION FOR THE SEAMAN MEETING, EMMA FILLED THE entrance urns with her big pots of hydrangeas. The intense blue created such a strong statement, she thought, dramatic, romantic, and eye-catching. Since the bride's colors were blue and peach she hoped the hydrangeas would fit the bill for the initial impact.
Humming, she went back to her van to unload the pots of white tulips—the bride's favorite—that would line the steps. A sweeter image than the hot blue, softer, more delicate. A nice mix, to her mind, of texture, shape, and style.
A taste, she thought, of things to come.
"Em!"
Bent over between the urns, her arms full of tulips, Emma turned her head. And Mac snapped her camera. "Looking good."
"The flowers are. I hope to look better before the consult. Our biggest client to date requires careful grooming." She placed the pots. "All around."
In a suit as boldly green as her eyes, Mac stood, legs spread, feet planted. "Not much time left to beautify."
"Nearly done. This is the last." With her system bursting with flowers and scents, Emma took a deep breath. "God! What a gorgeous day."
"You're pretty chirpy."
"I had a really good date last night." After stepping back to examine the portico, she hooked an arm with Mac's. "It had everything. Comedy, drama, conversation, sex. I feel . . . energized."
"And look starry-eyed."
"Maybe." Briefly, she dipped her head to Mac's shoulder. "I know it's too soon, and we're not even talking about—or anywhere near—the serious L. But . . . Mac, you know how I always had this fantasy about the moonlit night, the stars—"
"Dancing in the garden." Instinctively, Mac slid an arm around Emma's waist. "Sure, since we were kids."
"I dreamed it last night, and it was Jack. I was dancing with Jack. It's the first time I ever had the dream, or imagined it where I knew who I danced with. Don't you think that means something?"
"You're in love with him."
"That's what Parker said last night before I went out, and of course, I'm all no, no, I'm not. And, of course, as usual, she's right. Am I crazy?"
"Who said love was sane? You've sort of been there before."
"Sort of been," Emma agreed. "Wanted to be, hoped to be. But now that I am, it's more than I imagined. And I imagined a lot." Emma sidestepped, pivoted, pirouetted. "It makes me happy."
"Are you going to tell him?"
"God, no. He'd freak. You know Jack."
"Yes," Mac said carefully, "I know Jack."
"It makes me happy," Emma repeated as she laid a hand on her heart. "I can stay there for now. He has feelings for me. You know when a man has feelings for you."
"True enough."
"So I'm going to be happy and believe he'll fall in love with me."
"Emma, solid truth? I don't know how he could resist you. You're good together, that's easy to see. If you're happy, I'm happy."
But Emma knew Mac's tones, her expressions, her heart. "You're worried I'll get hurt. I can hear it in your voice. Because, well, we know Jack. Mac, you didn't want to fall in love with Carter."
"You've got me there." Mac's lips curved as she danced her fingers at the ends of Emma's hair. "I didn't, but I did, so I should stop being so cynical."
"Good. Now I've got to stop standing around and go transform into a professional. Tell Parker I'm done, will you, and I'll be back in twenty."
"Will do." And with concern showing now, Mac watched her friend rush off.
_A_ N HOUR LATER, DRESSED IN A TRIM SUIT AND LOW HEELS, Emma took the lead in escorting the future bride, her eagle-eyed mother, and the mother's fascinated sister around the gardens.
"You can see what we'll have blooming next spring, and I realize the gardens aren't as flush as you need or want."
"They just can't wait until May or June," Kathryn Seaman muttered.
"Mom, let's not go there again."
"It is, however, prime time for tulips—which I know you favor," Emma said to Jessica. "We'll plant more this fall, white tulips, and peach tulips—you'll have a flood of them, and blue hyacinths. We'll also fill in with white containers of peach roses, delphinium, snapdragons, stock, the hydrangeas. All in your colors, popped out by the white. I plan to back this area here with a screen covered with roses."
She turned her smile on Kathryn. "I promise you, it'll be like a fantasy garden, and as full and lush and romantic as anything you could wish for your daughter's wedding."
"Well, I've seen your work so I'm going to take your word." Kathryn nodded to Mac. "The engagement photos were everything you said they'd be."
"It helps to have two gorgeous people wildly in love."
"We had so much fun, too." Jessica beamed at Mac. "Plus, I felt like a storybook princess."
"You looked like one," her mother said. "All right, let's talk about the terraces."
"If you remember from the sketches at the proposal," Emma began, and led the way.
"I've seen your work as well." Adele, the bride's aunt, scanned the terraces. "I've been to three weddings here, and all were beautifully done."
"Thank you." Parker added a polite smile to the acknowledgment.
"Actually, what you've done here, built here, has inspired me to look into plans for doing something similar. We live part of the year in Jamaica. A destination wedding spot. And a perfect place for a good, upscale, all-inclusive wedding company."
"You're serious about that?" Kathryn asked her.
"I've been looking into it, and getting more serious. My husband's going to retire," Adele told Parker. "And we plan to spend even more time in our winter home there. It would be an excellent investment, I think, and something fun."
She gave Emma a twinkling smile and a wink. "Now, if I could lure you away with the promise of unlimited tropical flowers and balmy island breezes, I'd have my first real building block."
"Tempting," Emma said in the same light tone, "but Centerpiece of Vows keeps me busy. If you move forward with your plans, I'm sure any of us will be happy to answer any questions you might have. Now, for this area . . ."
_A_ FTER THE MEETING ALL FOUR WOMEN COLLAPSED IN THE parlor.
"God." Laurel stretched out her legs. "That woman sure knows how to put you through your paces. I feel like we had the event instead of just talking it through. Again."
"Unless there are any objections, I'd like to black out the Friday and Sunday around the event. The size and scope of this wedding will more than make up for that lost revenue, plus the publicity and the word of mouth will bring in more." Parker toed off her shoes. "That would give us the full week to focus exclusively on this."
"Thank God." Emma heaved a long, relieved sigh. "The amount of flowers and landscaping, the type of bouquets and arrangements, centerpieces, swags, garlands, ornamental trees? I'd have to hire more designers to get it done. But with that full week on the single event, I think I can stick with the usual team. I can add someone else if need be for the actual dressing, but I'd really prefer to do as much of this as I can personally, and with the people I know."
"I'm right there with Emma," Laurel said. "The cakes, the dessert bar, the personalized chocolates, they're all on the elaborate and labor-intensive side. If I had the full week on nothing else, I'd actually get a couple hours' sleep."
"Make it three for three." Mac raised a hand. "They want full photo documentation of the rehearsal, and the rehearsal dinner, so if we had another event on Friday, I'd have to assign a photographer to that as I'd have to cover the Seamans. As it is I'm putting two more on the event itself, plus two videogra phers. Keeping Sunday black means we don't have to kill ourselves and our subs breaking down, and redressing."
"Which doesn't even begin to address what they expect of you," Emma said to Parker.
"So we're agreed. And," Parker added, "I'll let the MOB know we're clearing our decks for wedding week so we can give her daughter's wedding all our time, attention, and skill. She'll like that."
"She likes us," Emma pointed out. "The concept of a company founded and run by four women appeals to her."
"And her sister. Who else did the sneaky Adele try to lure to Jamaica?" Laurel asked.
All four women raised hands.
"And she didn't even realize it was rude," Parker added. " _Our_ business. It's not like we're employees. We own it."
"Rude, yes, but I don't think she meant any harm." Emma shrugged. "I elect to be flattered. She considers my flowers fabulous, Laurel's cakes and pastries superb, Parker's coordinating unmatched. Added to that, Mac blew it out of the park with the engagement photos."
"I did," Mac agreed. "I really did."
"Let's all take a moment to congratulate ourselves on our brilliance and talent." Parker offered a toast with her bottle of water. "Then get back to work."
"If we're taking a moment, I'd like to thank Emma for last night's entertainment."
Emma sent Laurel a blank look. "Sorry?"
"I happened to be taking a little air on my terrace last night before settling in for the night, and noticed a car barreling down the drive. For a minute I thought, uh-oh, something happened. But no, not quite yet."
"Oh my God." Emma slapped her hands over her eyes. "Oh my God."
"When no one immediately jumped out gushing blood, or jumped out at all, I actually considered running down, prepared to do triage. But momentarily both car doors flew open. Emma out of one, Jack out of the other."
"You _watched_?"
Laurel snorted. "Duh."
"More," Mac demanded. "We must have more."
"And more you will have. They fell on each other like animals."
"Oh, we did . . . too," Emma recalled.
"Then it's the classic back against the door."
"Oh, it's been so long since I had the back against the door," Parker said with a delicate shiver for emphasis. "Too long."
"From my view, Jack's got the move down cold. Or hot, I should say. But our girl holds her own. Or was it his?"
"Jesus, Laurel!"
"She wrestles his jacket off, tosses it. Rips his sweater off, heaves it away."
"Oh boy, oh boy, oh boy!" Mac said.
"But the gold medal move was the belt. She whips that belt off—" Laurel flicked an arm through the air to demonstrate. "Then lets it fly."
"I think I need another bottle of water."
"Unfortunately, Parker, they took it inside."
"Killjoys," Mac muttered.
"The rest was left to my very . . . fluid imagination. So I want to thank our own Emmaline for the view from my balcony seat. Sister, stand up and take a bow."
To enthusiastic applause, Emma did just that. "Now I'll leave you and Peeping Thomasina to your salacious thoughts. I'm going to work."
"Back against the door," Parker murmured. "I'm small enough to be jealous."
"If I were small enough, I'd be jealous of her having her back against anything. But it's okay, because I've declared myself in a sex moratorium."
"A sex moratorium?" Mac repeated, turning to Laurel.
"That's right. I'm in a sex moratorium so I can be in a dating moratorium, because for the last couple of months dating's just been irritating." Laurel lifted her shoulders, let them fall. "Why do something that irritates me?"
"For the sex?" Mac suggested.
Eyes slitted, Laurel shot a finger at her friend. "You're only saying that because you're getting laid regularly."
"Yes." Mac considered, nodded. "Yes, I am getting laid regularly."
"It's rude to brag to those of us who are not," Parker pointed out.
"But I'm getting laid with love." Mac drew out the final word so Laurel laughed.
"Now you're just getting sickening."
"I'm not the only one, at least on one side. Emma said you were right, Parks. She's in love with Jack."
"Of course she's in love with Jack," Laurel interrupted. "She wouldn't have slept with him otherwise."
"Um, I hate to disillusion you, Bright Eyes, but Emma's had sex with men she wasn't in love with. And," Mac added, "has gently refused to have sex with more men than the three of us combined have scored."
"My point exactly. What happens when the four of us go to a club, for instance? Four very hot chicks? We get some hits, naturally. But Emma? They swarm like wasps."
"I don't see—"
"I do." Parker nodded. "She doesn't have to sleep with someone just because she's attracted. She can and does pick and choose. And she's picky and choosy rather than promiscuous. If it were just lust, she could and would answer that call elsewhere, because to answer it with Jack is complicated, and risky."
"Which is the reason she waited so long to act on it," Mac pointed out. "I don't see . . . Yes, I do," she corrected. "Damn it, I hate when I don't have a chance to be right before you're right."
"Now that she's realized what I could've told her weeks ago, I wonder what she'll do."
"She had her dancing in the garden dream," Mac told them, "and it was with Jack."
"Okay that's serious. Not just in love," Laurel said, "but _in love_."
"She's okay with it. She's going to enjoy the moment."
No one spoke.
"I think," Parker said carefully, "love is never wrong. Whether it's for the moment, or it's forever."
"We all know Emma's always wanted forever," Mac pointed out.
"But you can't have forever unless you take the moment."
"And if it doesn't work?" Laurel looked at her two friends. "That's what we're here for."
_I_ N HER OFFICE, EMMA CAUGHT UP ON PAPERWORK WHILE SHE let a facial mask deep clean and hydrate. Just how many women were lucky enough to be able to deal with skin care _and_ generate invoices at the same time? In their bare feet, with Norah Jones crooning out of the speakers?
And how many of those who might be lucky enough had also had crazed jungle sex—twice—with an amazing man the night before?
Not many, she'd wager. Not many at all.
While the mask worked its magic, she placed an order with her suppliers for flower foam, plastic ties, wire, clear and colored stones, then did a cruise through to see what might be on sale, or on special, and added liquid foam and foam sheets and three dozen light bases.
That would hold her for a while, she thought, placed the order, then brought up her wholesale candle supplier to see what they had to offer.
"Knock, knock! Emmaline! Are you home?"
"Mom? Up here." She saved her shopping cart, before pushing away from the desk. She met her mother coming up the stairs. "Hi!"
"Hi, my baby. Your face is very pink."
"I . . . Oh, I forgot." Laughing, Emma tapped her fingers on her cheek. "It needs to come off. I started on candles and got caught up." She detoured to the bathroom to wash off the mask. "Playing hooky?"
"I worked this morning, and am now free as a bird so came by to see my daughter before I go home." Lucia picked up the jar of mask. "Is this good?"
"You tell me. It's the first time I've tried it." Emma finished splashing cool water on her face, then patted it dry.
Lucia pursed her lips. "You're too beautiful for me to know if it's because of the lucky genes I passed to you or from the jar."
Emma grinned. Studying her face in the mirror over the sink, she poked lightly at her cheeks, her chin. "Feels good though. That's a plus."
"You have a glow," Lucia added while Emma sprayed on toner, followed up with moisturizer. "But from what I hear that's not from the jar either."
"Lucky genes?"
"Lucky something. Your cousin Dana stopped in the bookstore this morning. It seems her good friend Livvy . . . You know Livvy a little."
"Yes, a little."
"Livvy was out with a new boyfriend, having dinner, and who did she spot in a quiet corner across the room sharing wine, pasta, and intimate conversation with a certain handsome architect?"
Emma fluttered her lashes. "How many guesses?"
Lucia raised and lowered her eyebrows.
"Let's go downstairs and get something to drink. Coffee, or something cold?"
"Something cold."
"Jack and I went to an art opening," Emma began as they started down. "A really terrible art opening, which is actually a good story."
"You can come back to that. Tell me about the wine and pasta."
"We had wine and pasta after we left the opening." In the kitchen, Emma got down glasses, filled them with ice.
"You're being evasive."
"Yes." With a laugh, Emma sliced a lemon. "Which is silly, since you've obviously figured out Jack and I are dating."
"Are you evading because you think I won't approve?"
"No. Maybe." Emma opened the sparkling water her mother liked, poured it over ice, added slices of lemon.
"Are you happy? I already see the answer on your face, but you can answer yes or no."
"Yes."
"Then why would I disapprove of anything that made you happy?"
"It's sort of odd, isn't it though? After all this time?"
"Some things take time, some don't." Lucia turned into the living room, sat on the sofa. "I love this little room. All the colors, the scents. I know it's a place that makes you happy."
Emma came over, sat beside her mother. "It does."
"You're happy in your work, your life, your home. And that helps a mother—even of a grown woman—sleep well at night. Now, if you're happy with a man I happen to like quite a bit, I'm happy, too. You need to bring him to dinner."
"Oh, Mom. We're just . . . dating."
"He's been to dinner before."
"Yes. Yes. Del's friend Jack has been to dinner, to some cookouts, to some parties at the house. But you're not asking me to bring Del's friend to dinner."
"Suddenly he can't eat my cooking or have a beer with your father? You understand, _nina_ , I know what 'dating' means in this case?"
"Yes."
"He should come for Cinco de Mayo. All your friends should come. We'll put the pork on the grill, and not Jack."
"Okay. I'm in love with him, Mama."
"Yes, baby." Lucia drew Emma's head to her shoulder. "I know your face."
"He's not in love with me."
"Then he's not as smart as I think he is."
"He cares. You know that. He cares, and there's a really big attraction. On both sides. But he's not in love with me. Yet."
"That's my girl," Lucia said.
"Do you think it's . . . underhanded to deliberately set out to make a man fall in love with you?"
"Do you intend to lie, to pretend to be what you're not, to cheat, make promises you won't keep?"
"No. Of course not."
"Then how could it be underhanded? If I hadn't made your father fall in love with me, we wouldn't be sitting here in your pretty little room."
"You . . . Really?"
"Oh, I was so in love. Hopelessly, or so I thought. He was so handsome, so kind, so sweet and funny with his little boy. So lonely. He treated me well, with respect, with honor—and as we grew to know each other, with friendship. And I wanted him to _sweep_ me away, to see me as a woman, to take me into his bed, even if it was just for a night."
Inside her chest, Emma's romantic heart simply soared. "Oh, Mama."
"What? You think you invented this? The needs, the wants? I was young and he was above me in station. The wealth, the position, these were barriers—at least I thought so. But I could dream.
"And maybe a little more than dream," Lucia added with a secret smile. "I tried to look my best, to cook meals he especially liked, to listen when he needed a friend. That's what I knew how to do. And I would make sure, when he was going out, that his tie wasn't quite straight—even when it was—so I'd have to fix it. I still do," she murmured. "I still want to. I knew there was something—I could feel it, I could see it in his eyes—something more than the bond over the little boy we both loved, something more than friendship and respect. All I could do was show him, in little ways, that I was his."
"Mama, that's so . . . You never told me all this before."
"I never needed to. Your papa, he was careful with me, so careful not to touch my hand too long, hold my gaze too long. Until that day I stood under the cherry blossoms, and I saw him walking to me. I saw him coming to me, and what was in his eyes. My heart."
Lucia pressed her hand to it. "Ah! It fell, right at his feet. How could he not know? And knowing, his heart fell beside mine."
"It's what I want."
"Of course."
Emma had to blink tears away. "I don't think fixing Jack's tie is going to do it."
"The little things, Emma. The gestures, the moments. And the big. I let him see my heart. I gave it to him, even when I believed he couldn't or wouldn't take it. I gave it anyway—a gift. Even if he broke it. I was very brave. Love is very brave."
"I'm not as brave as you."
"I think you're wrong." Lucia wrapped her arm around Emma's shoulders in a hug. "Very wrong. But now it's new, isn't it? New and bright and happy. Enjoy it."
"I am."
"And bring him to the party."
"All right."
"Now, I'm going home to let you get back to work. Do you have a date?"
"Not tonight. We had a long consult today—the Seaman wedding."
Lucia's eyes danced. "Ah, the _big_ one."
"The big one. And I have paperwork, ordering, planning to get to tonight, and a full day tomorrow. He has a business thing tomorrow night, but he's going to try to come by after and . . ."
"I know what _and_ is," Lucia said with a laugh. "Get a good night's sleep tonight then." She patted Emma's knee, rose.
"I'm so glad you came by." Standing, Emma wrapped her mother in a hard hug. "Kiss Papa for me."
"For you and for me. I think he'll take me out to dinner tonight, and we'll share wine and pasta and intimate conversation. To show we haven't lost our touch."
"As if ever."
Emma leaned on the doorjamb, waved her mother off. Then instead of going back into work, left the door open to the spring air and took a walk around the gardens.
Tight buds, fresh blossoms, tender shoots. The beginning of a new cycle, she thought. She wandered back to her greenhouses, gave herself the pleasure of puttering. Seeds she'd planted over the winter were now young plants, and doing nicely. She'd begin to harden them off in the next few days, she decided.
She circled back around, stopped to fill the bird feeders she shared with Mac. The air had already started to cool by the time she went back in. When the sun set, she thought, it would be chilly.
On impulse, she got out a pot. Then minced, chopped, poured, tossed in cubes of herbs she'd frozen the summer before. With a kettle of soup simmering, she went back up to finish her orders.
An hour later, she came down to stir, then glanced toward the window as she heard a car. Surprised, pleased, she hurried to the door to greet Jack.
"Well, hi."
"I had a meeting, and managed to wrap it up early. I left my jacket here again, so I thought I'd swing by on my way . . . You're cooking?"
"I took a walk, and it started cooling off, which put me in the mood for kitchen sink soup. There's plenty, if you're interested."
"Actually, I was . . . There's a ball game on tonight, so—"
"I have a television." She stepped in, straightened his tie, with a secret smile. "I allow it to broadcast ball games."
"Really?"
She gave his tie a little tug. "You can taste the soup. If it doesn't appeal, I'll get your jacket and you can watch the ball game at home."
She strolled off, went back to stirring. When he followed, she glanced over her shoulder. "Lean over, open up."
He did just that so she held the tasting spoon to his lips.
"It's good." His eyebrows lifted in surprise. "It's damn good. How come I never knew you could make soup?"
"You never stopped by to get your jacket after you wrapped up a meeting early. Do you want to stay for dinner?"
"Yeah. Thanks."
"It needs about an hour more. Why don't you open a bottle of cab?"
"Okay." Now he leaned down, kissed her. Paused, kissed her again, softly, slowly. "I'm glad I swung by."
"Me, too."
CHAPTER FOURTEEN
_T_ HE MEXICAN AND AMERICAN FLAGS FLEW THEIR PROUD colors as Emma's Mexican mother and Yankee father combined cultures to celebrate Cinco de Mayo.
Every year the expansive grounds offered games, from lawn bowling and badminton to moon bounces and waterslides. Friends, relatives, and neighbors played and competed while others crowded at picnic tables, diving into platters of pork and chicken, warm tortillas, bowls of red beans or chilis, guacamole or salsa hot enough to scorch the throat.
There were gallons of lemonade, Negra Modelo, Corona, tequila, and frosty margaritas to put out the fire.
Whenever he'd managed to drop by on the fifth of May, Jack had always been amazed at the number of people the Grants managed to feed. And the choices of fajitas and burgers, black beans and rice or potato salad. Flan or apple pie.
He supposed the food was just a symbol of how completely Phillip and Lucia blended.
He sipped his beer and watched some of the guests dance to the trio of guitars and marimbas.
Beside him, Del took a pull on his own beer. "Hell of a party."
"They pull out all the stops."
"So, is it weird being here this year with the hosts' baby girl?"
Jack started to deny it as a matter of principle. But hell, it was Del. "Little bit. But so far, nobody's called for the rope."
"It's still early."
"Brown, you're a comfort to me. Is it my imagination or are there about twice as many kids as there were last year? Year before," he remembered. "I couldn't make it last year."
"Might be. I don't think they're all related. I heard Celia's pregnant again though."
"Yeah, Emma mentioned it. You're here stag?"
"Yeah." Del smiled slowly. "You never know, do you? Check out the blonde in the blue dress. Those are some nice pins she's got."
"Yeah. I always thought Laurel had great legs."
Del choked on his beer. "That's not . . . Oh," he managed when she turned, laughed, and he got a better look. "Not used to seeing her in a dress, I guess." Very deliberately he turned in the opposite direction. "Anyway, there are a bevy of sultry brunettes, cool blondes, and a sprinkle of hot redheads. Many of whom are unattached. But I guess the days of scoping the field are over for you."
"I'm dating, not blind or dead." The idea put an itch between Jack's shoulder blades.
"Where is Em?"
"She went to help somebody with something food related.
We're not joined at the hip."
Del lifted an eyebrow. "Okay."
"I have friends, she has friends, and some of them happen to be mutual. We don't have to walk in step at a party."
"Right." Del took another contemplative sip of his beer.
"So . . . would the guy she's currently kissing on the mouth be her friend, your friend, or a mutual?"
Jack swung around, caught the end of the kiss between Emma and some Nordic god type. She laughed, and her hands gestured expressively before she grabbed one of Thor's and pulled him over to a group of people.
"Looks like he's not one of yours," Del commented.
"Why don't you . . ." He cut off the suggestion he had in mind as Lucia stopped in front of them. "You two should be eating instead of just standing here looking handsome."
"I'm considering all options," Del told her. "There are big decisions to be made, all the way down to apple pie or flan."
"There's also strawberry shortcake and empanadas."
"You see? Not to be taken lightly."
"You should sample each, then decide. Look here!" She beamed smiles and threw out her hands as Mac and Carter walked to them. "Mackensie, you made it."
"Sorry we're so late. The shoot ran a little longer than I hoped." She kissed Lucia's cheek.
"You're here, that's what counts. And you!" Lucia threw her arms around Carter for a hug.
Carter lifted her an inch off the grass in a gesture of long-term affection.
"It's been years since you came for Cinco de Mayo."
Carter grinned. "It's bigger."
"Because there are more of us. Your mama and papa are here, with Diane's children. Sherry and Nick are here, too," she said, speaking of his younger sister. "Diane and Sam should be here soon. Mac, your future mother in-law tells me the wedding plans are going well."
"They're clicking along."
"Let me see your ring again. Ah!" She twinkled a smile at Carter after examining the diamond on Mac's hand. "Very nicely done. Come, Celia hasn't seen it yet. Carter," she called as she pulled Mac away, "get food, get drink."
Instead, Carter stood where he was. "I haven't been back for one of these in . . . it must be ten years. I'd forgotten. It's like a carnival."
"The best in the county," Del commented. "The Grants either know or are related to everybody. Including, it seems, our mechanic and poker buddy. Hey, Mal."
"Hey." In dark shades, worn jeans, and a black T-shirt, he strolled over. He carried two beers by the neck. "Want one, Maverick?" he said to Carter.
"Sure. I didn't realize you knew the Grants."
"They've been bringing their cars in for service or repair for the last six, eight months. Before you know it, you're telling Lucia your life story, eating her corn bread and wishing she'd dump her husband and run off with you to Maui."
"Ain't that the truth," Jack said.
"She said I should come by after work, backyard deal for Cinco de Mayo. I figure a cookout, maybe on the fancy side, considering, some Mexican beer, tortillas." He shook his head. "Is anybody not here?"
"I think they covered everybody."
"Sorry that took so long." Emma hurried up, a margarita in her hand. "There were circumstances."
"Yeah, I saw one of them."
After giving Jack a puzzled smile, she turned to Malcolm. "Hi, I'm Emmaline."
"You're the Cobalt."
"I . . ." Her eyes widened, then filled with contrition. "Yes. You must be Malcolm."
"Mal." He gave her a long, head-to-toe scoping out. "You know, it's a good thing you look like your mother, who I hope to marry. Otherwise I'd replay the ass-kicking I gave your partner when I thought she was you."
"And I'd deserve it. Even though I learned my lesson, and I'm being much more conscientious. You did a great job. You have serious skills. I wonder if you'd have time to service my van if I bring it in next week."
"You don't just look like her, do you?"
Emma smiled as she sipped the margarita. "You need a plate," she told him, "and a great deal of food."
"Why don't you show me where—" Mal cut himself off when he caught the warning in Jack's eyes, and the casual and proprietary stroke of his hand down Emma's hair. "Right. Maybe I'll just go graze awhile."
"I'll do the same," Carter decided.
Del's lips quirked. "Looks like I'm empty." He jiggled the beer bottle. "Em, who's the long brunette? Pink top, skinny jeans?"
"Ah . . . Paige. Paige Haviller."
"Single?"
"Yes."
"See you later."
"He should've asked me if she had any brains," Emma said as Del strolled away. "He'll be bored in thirty minutes or less."
"Depends on what they're doing for thirty minutes."
She laughed up at him. "I suppose it does." She slipped a hand into his to squeeze. "It's a good day, isn't it?"
"I can never figure out how they pull this off."
"They work for weeks, and hire a platoon to help set up the games and activities. And Parker helps coordinate. Speaking of which, I—"
"Who was the guy?"
"The guy? There are a lot of them. Give me some hints."
"The one you were kissing a little while ago."
"Bigger hint."
That crawled into his spleen. "The one who looked like the prince of Denmark."
"The prince of . . . Oh, you must mean Marshall. One of the circumstances why I was so long getting back."
"So I saw."
She cocked her head, and the faintest frown line formed between her eyebrows. "He was late getting here. With his wife and their new baby boy. After he came out to get me, I went in to fuss over the baby for a while. Problem?"
"No." Idiot. "Del was yanking my chain, and I walked into it. And the mixed metaphor. Let's rewind. Speaking of which?"
"We dated a little, a few years ago, Marshall and I. I introduced him to his wife. We did their wedding about eighteen months ago."
"Got it. Apologies."
She smiled a little. "He didn't grab my ass like a certain crazy artist grabbed yours."
"His loss."
"Why don't we mingle, be sociable?"
"Good idea."
"Oh," she said as they started to walk, "speaking of which, I had a thought. Since I have several errands in town tomorrow, if I stayed at your place tonight, I'd be in town. Parker rode in with me, as we both needed to get here early to help, but she can ride back with Laurel. It would save me from going back and forth."
"Stay at my place?"
She lifted her brows, and the eyes under them went cool. "I could bunk on the couch if you don't want company."
"No. I just assumed you'd need to get home after this. You usually start pretty early in the morning."
"Tomorrow I'm starting in town, not quite as early. But if it's a problem—"
"No." He stopped, turning her so they faced each other. "It's fine. It's good. But don't you need some things—for tomorrow?"
"I put some things in my car when I had the thought."
"Then we're set." He leaned down to kiss her.
"Looks like you need another beer."
Then jerked back at her father's voice.
Phillip smiled. Casually, from the looks of it, Jack thought. Unless you were the one who'd just made arrangements to sleep with his daughter.
"Negra Modelo, right?" Phillip offered one.
"Yeah, thanks. Great party, as always."
"My favorite of the year." Phillip laid an arm around Emma's shoulders. Casually, affectionately. Territorially. "We started the tradition the spring Lucia was pregnant with Matthew. Friends, family, children. Now our children are grown and making families of their own."
"You're feeling sentimental," Emma said, and tipping her face up, brushed her lips over his jaw.
"I still see you running on the lawn with your friends, trying so hard to win prizes at ring toss, or to break one of the pinatas. Like your mother, you bring the color and the life."
"Papa."
Phillip shifted his gaze, directly into Jack's. "It's a lucky man who's offered that color and life. And a wise one who values it."
"Papa," she repeated, but in a warning tone now.
"A man only gets so many treasures," he said, and tapped her on the nose with his finger. "I'm going to check the grill. I don't trust your brothers or your uncles for long. Jack," he added with a nod before he walked away.
"Sorry. He can't help it."
"It's okay. Did I sweat through my shirt?"
Laughing, she hooked an arm around Jack's waist. "No. Why don't we go show those kids how to break a pinata?"
_L_ ATER, THEY FLOPPED DOWN ON THE GRASS TO WATCH SOME OF the teenagers in an impromptu game of soccer. Parker joined them, slipping off her sandals, smoothing down the skirt of her sundress.
"Night soccer," Jack commented. "Not your usual."
"Do you play?" Emma asked him.
"Not my game. Give me a bat, a football, a hoop. But I like to watch."
"You like to watch anything where a ball's involved." Mac dropped down beside them, tugged Carter down with her. "Ate much too much. It just kept being there."
"Oh, that's just pitiful," Emma muttered when the ball was intercepted. "Does he think it has eyes, radar?"
"You like soccer?"
She glanced at Jack. "Girls' Varsity at the Academy. All-State."
"Seriously?"
"Cocaptains," she added, wagging her thumb between herself and Parker.
"They were vicious." Laurel knelt on the grass beside Parker. "Mac and I would go to the games, and pity the opposition. Go on." She elbowed Parker. "Go on out and kick some ass."
"Hmm. Want to?" Emma asked Parker.
"Em, it's been a decade."
Emma boosted up to her knees so she could slap her hands on her hips. "Are you saying we're too old to take those losers and weak feet? Are you saying you have lost—your—edge?"
"Oh, hell. One goal."
"Let's score."
Like Parker, she slipped out of her sandals.
Fascinated, Jack watched the two women in their pretty spring dresses approach the field.
There was discussion, some hoots, a few catcalls.
"What's up?" Mal sauntered over to study the two groups.
"Emma and Parker are going to kick some soccer ass," Laurel told him.
"No kidding? This ought to be interesting."
They took position on the grass in the floodlights, with Emma and Parker's team set to receive. The women glanced at each other, then Emma held up three fingers, then two. Parker laughed, shrugged.
The ball sailed through the air. Emma two-fisted it to Parker, who took it on the bounce, and dodged her way through three opponents with a blur of footwork that had the earlier catcalls turning to cheers.
She pivoted, feinted, then bulleted the ball cross-field to where Emma sprang to receive. She scored with a blurring banana kick that left the goalie openmouthed.
In unison, she and Parker shot up both arms and screamed.
"They always did that," Mac told the group. "No modesty at all. Go Robins!"
"Girls' soccer team," Carter explained. "State bird."
When Parker started to leave the field, Emma grabbed her arm. Jack heard her say, "One more."
Parker shook her head; Emma persisted. Parker gripped her skirt, held it out, and whatever Emma said in response made her former cocaptain laugh.
They took defense against an opposing team who had considerably more respect now. They fought, blocking, rejecting, pushing their opponents back.
Jack's grin spread when Emma shoulder tackled an opponent. And looked gorgeous doing it, he realized—and just a little fierce. A fresh wave of lust curled in his belly as she charged the player in possession. Her slide tackle—Jesus, just look at her!—had the teenager off balance with his instep pass.
On alert, Parker leaped at the next hard, high kick, skirts flying as she sprang and executed a dead-on header.
"Well, well," Mal murmured.
"Interception!" Laurel cried out when Emma trapped the ball. "Woo!"
Emma avoided her opponents' attempts to regain the ball with a quick cut back. She bicycle kicked the ball back to Parker, who shot it between the goalie's legs.
Hands up, a scream, and Parker slung an arm around Emma's shoulders.
"Done?"
"Oh, so very done." Emma sucked in a breath. "No longer seventeen, but still. Felt righteous."
"Let's leave winners." They held up joined hands, bowed to applause, then deserted the field.
"Baby," Jack said as he grabbed Emma's hand to pull her back down to the grass, "you're a killer."
"Oh yeah." And she reached out for the bottle of water Mac offered. Before she could drink, her mouth was busy with Jack's.
The kiss earned more applause.
"I'm a slave," he murmured against her lips, "to a woman who can pull off an accurate bicycle kick."
"Really?" She scraped her teeth lightly over his bottom lip. "You ought to see my instep drive."
"Anytime. Anywhere."
At the edge of the field, Mal cut across Parker's path, offered one of the two beers he held. "Want?"
"No. Thanks."
Moving around him, she pulled a bottle of water out of one of the ice tubs.
"What gym do you use, Legs?"
She opened the bottle. "My own."
"Figures. You've got some moves. Play anything else?"
She took a slow sip of water. "Piano."
As she strolled away, he watched her over a lazy pull of his beer.
_L_ ATER, LAUREL SAT ON THE GRANTS' FRONT PORCH STEPS, elbows braced behind her, eyes half closed. The quiet rolled over her, as did the smell of the grass, the front garden. The spring stars showered down.
She heard the footsteps, kept her eyes closed. And hoped whatever guest was leaving would keep moving, and let her keep her solitude.
"Are you all right?"
No such luck, she thought, and opened her eyes to look at Del. "Yeah. I'm just sitting here."
"So I see."
He sat beside her.
"I said my bye-byes. Parker's still inside—or outside—doing the Parker check to make sure nothing else has to be done. I had too much tequila to care if something else has to be done."
He gave her a closer study. "I'll drive you home."
"I gave my keys to Parker. She's driving both of us home. No rescue required, sir."
"Okay. So I heard the Robins made a comeback earlier. Sorry I missed it."
"They ruled, as ever. I guess you were otherwise occupied." She looked behind her, side to side, movements exaggerated. "Alone, Delaney? With all these pickings today? Can't believe the Robins scored and you're not gonna."
"I didn't come to score."
She made a _pffft_ sound and gave him a shove.
His lips quirked into a reluctant smile. "Honey, you're toasted."
"Yes, I am. I'm gonna be so pissed off at me tomorrow, but right now? Feels good. Can't remember the last time I had too much tequila, or too much anything. Coulda scored."
"Sorry?"
"And I don't mean soccer." Cracking herself up, she shoved him again. "Very cute guy named . . . something made the play. But I'm in a sexual morit . . . morat . . . Wait. Sexual mor-a-tori-um," she said, enunciating each syllable.
Still smiling, he tucked her sunny swing of hair behind her ear. "Are you?"
"Yes, I am. I am toasted and I am in the thing I just said and don't want to have to say again." She shook back the hair he'd just smoothed, gave him a tipsy smile. "Not planning on making a play, are you?"
His smile dropped away. "No."
She _pffft_ 'd again, leaned back, then flicked her hand several times in dismissal. "Move along."
"I'll just sit here until Parker comes out."
"Mr. Brown, Delaney Brown, do you ever get tired of saving people?"
"I'm not saving you. I'm just sitting here."
Yeah, she thought, just sitting. On a beautiful spring night, under a shower of stars, with the scent of the first roses sweetening the air.
_E_ MMA PARKED HER CAR BEHIND JACK'S, RETRIEVED HER OVERSIZED purse. She got out, popped the trunk, then smiled as he reached in to retrieve her overnight case.
"No comments about what the hell's in this thing?"
"Actually, I thought it would be a lot heavier."
"I restrained myself. I never asked what time you have to get started tomorrow."
"About eight. Not too early."
She linked her hand with his, added a playful swing of arms. "I'll repay your hospitality and fix breakfast. If you have anything to fix."
"I probably do." They walked up the steps to the back door of the apartment above his office.
"It makes it easy, doesn't it, to live where you work? Though I sometimes think we end up working more than we would if we had more defined lines. I love this building. It's got character."
"I fell for it," he told her as he unlocked the door.
"It suits you. The character and tradition on the outside, the clean lines and balanced flow of space inside," she added as she stepped into his kitchen.
"Speaking of clean lines and flow, I'm still trying to find words over the soccer exhibition."
"That impulse is probably going to have my quads crying tomorrow."
"I think your quads can take it. Have I told you I have a weakness for women in sports?"
She walked with him through the apartment to the bedroom. "You didn't have to. I know you have a weakness for women and a weakness for sports."
"Put them together, and I'm gone."
"And a slave to the female bicycle kick." She lifted to her toes, pecked his lips with hers. "You should've seen me in my soccer uniform."
"Do you still have it?"
She laughed, and setting her overnight on the bed, unzipped it. "As a matter of fact."
"In there?"
"Afraid not. But I do have this . . ." She pulled out something very sheer, very short, very black. "If you're interested."
"I think this is going down as a perfect day."
_I_ N THE MORNING, SHE FIXED FRENCH TOAST, AND DID SOMETHING crispy and mildly sweet to an apple she'd cut into slices.
"This is great. Flower artist, soccer champ, kitchen wizard."
"I am many things." She sat across from him in the alcove he used for dining. She thought the space needed flowers, something bold and bright in a copper vase. "And you're now out of eggs, and very low on milk. I'm actually doing some marketing today if you want me to pick some up, or anything else."
She saw the hitch, the hesitation before he spoke.
"No, that's okay. I need to make a run later in the week. How're the quads?"
"Fine." She ordered herself not to make an issue out of his reluctance to have her pick up a damn carton of eggs for him. "I guess the bastard elliptical is doing its job. How do you keep in shape?"
"I use the gym three or four times a week, play basketball, that sort of thing."
She sent him a slitted-eye, accusatory stare. "I bet you like it. The gym."
"Yeah, I do."
"So does Parker. I think you're both sick."
"Keeping in shape is sick?"
"No, _liking_ what goes into keeping in shape is sick. I get doing it, but it should be considered a chore, a duty, a necessary evil. Like brussels sprouts."
Amusement warmed his eyes. "Brussels sprouts are evil?"
"Of course they are. Everyone knows this, even if they won't admit it. They're little green balls of evil. Just like squats are a form of torture designed by people who don't need to do squats in the first place. Bastards."
"I find your philosophy on fitness and nutrition fascinating."
"Honesty can be fascinating." She savored the last sip of her coffee. "At least when summer hits I can use the pool. That's sensible, and it's fun. Well, I should go up and shower since I slaved away over a hot stove while you had yours. I'll make it quick so I don't hold you up." She glanced back at the clock on that hot stove. "Really quick."
"Ah . . . listen, you don't have to rush. You can just lock up the back when you leave."
Pleased, she smiled. "Then I'll have another cup of coffee first."
It allowed her to linger a little, over the coffee, then over the shower. Wrapped in a towel, she slicked cream over her skin, then opened the moisturizer for her face.
As she started on her makeup, she saw Jack step in, saw in the mirror the way his gaze skimmed over the scatter of her tubes and pots on the bathroom counter. He barely missed a beat, but there was no mistaking the unease in his eyes—and no denying the hurt in her heart.
"I gotta go." The brush of his hand down her damp hair was sweet, as was the kiss. "See you later?"
"Sure."
Alone, she finished her makeup, her hair. She dressed, and she packed.
When she was done she went back into the bathroom, viciously scrubbed the sink, the counter until she was sure she'd left no trace of her or her things in his space.
"No need to panic, Jack," she mumbled. "All clear. All yours."
On the way out, she stopped and left a note on his kitchen board.
_Jack—forgot I'm booked tonight. We'll catch up later. Emma_
She needed a break.
She tested the back door to make sure it locked behind her, carried her case down to her car. Once she got behind the wheel, she flipped open her phone and called Parker.
"Hey, Emma, I'm on the other line with—"
"I'll be quick. Can we have a girl night tonight?"
"What's wrong?"
"Nothing. Really. I just need girl night."
"In or out?"
"In. I don't want to go out."
"I'll take care of it."
"Thanks. I'll be home in a couple of hours."
Emma closed the phone.
Friends, she thought. Girlfriends. They never let you down.
CHAPTER FIFTEEN
_"I_ OVERREACTED."
After a full day of work, during which she'd replayed dozens of Jack details, Emma settled down.
"We'll be the judge of that." Laurel took her place in the third-floor parlor, then bit into a slice of Mrs. Grady's exceptional homemade pizza.
"He didn't do anything wrong. He didn't even say anything wrong. I'm annoyed with myself."
"Okay, but you tend to be annoyed with yourself instead of anybody else. Even when the anybody else deserves it." Mac poured a glass of wine, offered the bottle to Laurel.
"Nope. Detoxifying massive quantities of tequila. It could take days."
"I don't do that." Emma scowled over her pizza. "That makes me sound like a weenie."
"You're not a weenie. You're just tolerant, and you have a sympathetic nature." Since Emma held up her glass, Mac filled it. "So when you get annoyed with somebody, you mean it."
"I'm not a pushover," Emma replied.
"Just because you're not as mean as we are, doesn't mean you're a pushover," Laurel pointed out.
"I can be mean."
"You can," Mac agreed and gave Emma a bolstering pat on the shoulder. "You have the tools, you have the skills. Mostly you don't have the heart for it."
"I—"
"Being innately nice isn't a character flaw," Parker interrupted. "I like to think we're all innately nice."
"Except for me." Laurel held up her Diet Coke.
"Yes, except for you. Why don't you just tell us what upset you, Emma?"
"It's going to sound stupid, even petty." She brooded into her wine, then down at the candy pink polish on her toes while her friends waited. "It's just that he's so protective of his space, his place. He doesn't actually say anything, but there's this invisible boundary around his area. Except he did say it before. You remember, Mac."
"Give me a hint."
"When you decided to reorganize your bedroom last winter. The closet thing. You got crazed because Carter left some of his things at your place. And Jack came over, and he agreed with you. He said all those things about what happens when you let somebody you're involved with stake territory."
"He was joking, mostly. You got mad," Mac remembered. "Walked out."
"He said that women start leaving their things all over the bathroom counter, and then they want a drawer. And before you know it, they take over. As if wanting to leave a toothbrush means you're ready to register at Tiffany."
"He freaked because you wanted to leave a toothbrush at his place?" Laurel demanded.
"No. Yes. Not exactly, because I never said anything about a damn toothbrush. Look, it's like this. Even if we're out somewhere and his place is closer, we come back here. Last night, I asked if I could stay at his place because I needed to be in town in the morning anyway, and he . . . he hesitated."
"Maybe his place wasn't in girl-friendly condition," Mac suggested. "He had to think if he'd left any dirty socks or _Big Jugs_ magazines lying around, or if he'd changed the sheets in the last decade."
"It wasn't that. His place is always neat, which may be part of the thing. He likes everything where it is. Like Parker."
"Hey."
"Well, you do," Emma said, but with a smile that held both love and apology. "It's just the nature. The thing is, you'd be okay if a guy slept over, maybe left a toothbrush. You'd just put the toothbrush in some proper space."
"Which guy? Can I have a name, an address, a photograph?"
Emma relaxed enough to laugh. "In theory. Anyway, over breakfast I mentioned I was hitting the market, and since he was out of eggs and milk, I could pick some up for him. And there it was again. That same sort of uh-oh before the no, thanks. But the killer was when he came upstairs. I was putting on my makeup and, beat me with a stick, had my stuff out on the counter. And he got this look. Annoyed and . . . wary. I told you it was going to sound stupid."
"It doesn't," Parker corrected. "It made you feel unwelcome and intrusive."
"Yes." Emma shut her eyes. "Exactly. I don't think he meant to, or that he's even fully aware, but—"
"It doesn't matter. In fact, the unconscious slight's worse."
"Yes!" Emma repeated, and shot Parker a grateful look. "Thank you."
"What did you do about it?" Laurel demanded.
"Do?"
"Yes, do, Em. Such as tell him to get over himself, it's a toothbrush or a tube of mascara."
"He went to work and I spent a half hour making sure I hadn't left so much as a flake of that mascara in his precious space."
"Oh yeah, that'll teach him," Laurel added. "I'd've stripped off my bra, left it hanging over his shower, left him a sarcastic love note in lipstick on the mirror. Oh, oh, and I'd have gone out and bought the economy-sized box of tampons and left them on the counter. _That_ would get the point across."
"Wouldn't that be making his point?"
"No, because he has no point. You're sleeping together. Whoever's bed is in play, the other party requires some of the basics on hand. Do you get wigged out when he leaves his toothbrush or his razor at your place?"
"He doesn't. Ever."
"Oh, come on. Don't tell me he never forgets to—"
"Never."
"Well, Jesus." Laurel slumped back. "Obsessive much?"
Mac raised her hand, offered a sheepish smile. "I'm just going to say I was kind of that way. Not as—okay, obsessive. I would forget things or leave things at Carter's, and he'd do the same. But that's what started me off that day you're talking about, Em. His jacket, his shaving kit, his whatever, mixed up with my stuff. It wasn't the stuff, it was what it meant. He's here. He's really here, and it's not just sex. It's not just casual. It's real." Mac shrugged, spread her hands. "I panicked. I had this amazing man in love with me, and I was scared. Jack's probably feeling some of that."
"I haven't said anything about love."
"Maybe you should." Parker shifted to tuck up her feet. "It's easier to know how the cards should be played when they're on the table. If he doesn't know what you're feeling, Emma, how can he take those feelings into consideration?"
"I don't want him to take my feelings into consideration. I want him to feel what he feels, be what he is. If he didn't and wasn't, I wouldn't be in love with him in the first place." She sighed and took a sip of wine. "Why did I ever think being in love would be wonderful?"
"It is once you work out the kinks," Mac told her.
"Part of the problem is I already know him so well I pick up on all the little . . ." She huffed out a breath, sipped more wine. "I think I have to stop being so sensitive, and stop romanticizing everything."
"You have to feel what you feel, be what you are."
When Parker tossed her words back at her, Emma blinked. "I guess I do, don't I? And I guess I should probably have an actual conversation with Jack about this."
"I like my economy-sized box of tampons better. It requires no words." Laurel shrugged. "But if you've got to be all mature about it."
"I don't really want to, but I got tired of sulking about halfway through the day. I might as well see how a reasonable conversation works out. Next week, I think. Maybe we both need a little space."
"We should have a man-free, work free night once a month."
"We pretty much do," Mac reminded Laurel.
"But that's because it just happens, which is good. But now that half of us are hooked up with men, we should formalize it. An estrogen revival."
"No men, no work." Emma nodded. "That sounds—"
Parker's phone beeped. She glanced at the display. "Willow Moran, first Saturday in June. Shouldn't take long. Hi, Willow!" she said cheerfully as she rose and stepped out of the room. "No, no problem at all. That's what I'm here for."
"Well, almost no work. And more pizza for me." Laurel took a second slice.
Despite a few interruptions, Emma thought the evening had been just what she'd needed. A little space, a little time with friends. She let herself into her house feeling pleasantly tired. As she started upstairs she went through her schedule for the next few days. She'd barely have time to catch her breath, she realized. And that, too, was just what she needed.
After crossing the room, she picked up the phone she'd deliberately left behind and saw she had a voice mail from Jack. Her spirits took a quick jump. So quick, she told herself to set the phone down again. It couldn't be anything urgent or he'd have called the main house.
It could wait until morning.
And who was she kidding?
She sat on the side of the bed to listen.
_Hi. Sorry I missed you. Listen, Del and I are going to work on our further corruption of Carter and drag him to a game on Sunday. I thought I might come by sometime on Saturday. Maybe I can give you a hand. I could return this morning's favor and fix you breakfast before we kidnap Carter. Give me a call when you get a chance. I'm going to work on some drawings for your place, so . . . Thinking about you._
_What are you wearing?_
It made her laugh. He always could make her laugh, she thought. It was a nice message. Considerate, affectionate, funny.
What else did she want?
Everything, she admitted. She wanted it all.
_S_ HE LET IT WAIT. EMMA TOLD HERSELF SHE WAS JUST TOO BUSY for that mature conversation. May meant a full slate of weddings, bridal showers, and Mother's Day. When she wasn't neck deep in flowers, she was planning the next design.
With her schedule, it simply made more sense for Jack to come to her, when it worked for both of them. She told herself to be grateful she was involved with a man who didn't complain about her working weekends, the long hours—and who could be counted on to lend a hand if he was around.
On a stormy afternoon in May she worked alone. Blessedly. Her ears might have been ringing from the echoes of Tink's and Tiffany's chatter, but now the rolls of thunder, the whoosh of rain and wind soothed.
She finished the maid of honor's bouquet, then stood for a moment to stretch. Turning, she jumped like a rabbit when she saw Jack.
Her squeal of shock bubbled into laughter as she slammed her hand to her heart. "God! You scared me."
"Sorry. Sorry. I knocked, yelled out, but it's hard to hear over the wrath of God."
"You're soaked."
"It's probably because of the rain." He ran a hand through his hair, scattering drops. "Killed my last on-site meeting, so I took a chance and swung by. Nice," he added, nodding at the bouquet.
"It is, isn't it? I was just about to put it in the cooler and start on the bride's. Why don't you get some coffee, dry off."
"Exactly what I hoped to hear." He stepped up to kiss her, brush a hand down her back. "I brought the drawings over for you to look at. When you get a chance. Weather permitting, they'll start on Mac's place Monday morning. Early. Be prepared."
"That's exciting. Do they know?"
"I stopped in the studio first. You want coffee?"
"No, thanks."
She made the trip to the cooler and back, then settled down with the flowers, her tools, and the picture she wanted to create in her head.
She glanced up when he came back in. "I've never really watched you work, on this part. Will it bug you?"
"No. Sit down. Talk to me."
"I saw your sister today."
"Oh?"
"We ran into each other in town. Don't you need a picture or a sketch?"
"I often use both, but this one's . . ." She tapped a finger on her temple. "White spray roses, this pale viburnum for accents. Slight cascade, which will be both sweet and romantic when I coax these majolicas into full bloom."
He watched as she clipped and wired, and thunder boomed. "I thought you said it was a bouquet."
"It is."
"Why the vase?"
"I've soaked the foam, attached the holder. See this part?" She angled the vase. "I keep that anchored in the vase so I can work the flowers in, get the right shape, the right cascade."
"What do you do when you have the others working in here with you?"
"Hmm?"
"What, you're all lined up here? Assembly-line method?"
"Yes, but no. We're all sort of lined up here, but we'd all be working on whatever arrangement I assigned. It's not like I do so much, then pass the bouquet to Tink."
She worked on in the quiet punctuated by thunder and rain.
"You need an L-shaped in here." He scanned the space again, the tools, the holding tubs. "Maybe a U's better. With over- and under-counter bins and drawers. You were primarily solo when I initially designed this space. You've outgrown it. Plus you need space under for a rolling bin, for your compost, another for nonbio waste. Do you ever have clients back here when you're working, or one of the others is working?"
She sucked the thumb a stray thorn pricked. "Sometimes, sure."
"Okay."
He got up, leaving Emma frowning after him.
He came back, soaked again, with a notebook she assumed he'd gotten out of his car. "Just keep working," he told her. "I just want to draw up some adjustments for what I've already done. We're going to want to move that wall."
"Move?" Her attention arrowed to him. "The wall?"
"Bump it out, open up your work and display areas. Better flow, and more efficient work space. Too much for a solo operation, but . . . Sorry." He glanced up from his drawing. "Thinking out loud. Annoying."
"No, it's fine." And a little strange, she thought, for them to be working together on a stormy afternoon.
They worked in silence for a time, though she discovered he was a mutterer with a pencil in his hand. She didn't mind it, and found it surprising that there were still things to learn about him.
When she'd finished, she lifted the bouquet out, turned it to study it from every angle. And caught him watching her. "It'll look fuller and softer when the roses open."
"You work fast."
"This sort isn't especially labor intensive." She rose, turned to the full-length mirror. "The dress has a lot of detail, very intricate, so this simpler, softer bouquet will suit it. No ribbons, nothing trailing, just the subtle cascade. Held here, waist high, both hands. It's going to . . ."
Her eyes met his in the mirror, and she caught the faint frown in his. "Don't worry, Jack. I'm not practicing."
"Huh?"
"I need to put these in the cooler."
When she carried them back, placed them, he spoke from the doorway. "I was thinking that the white looked good on you—with you? Whatever it would be. But everything does. And that you never wear flowers. It's probably too clichéd for you. So maybe I made a mistake."
She stood, surrounded by scent and blossom. "A mistake?"
"Yeah. I'll be back in a minute."
She shook her head when he walked off again. She stepped out, closed the cooler. She'd need to clean off her workstation, then she should go over her notes for the next day.
"I always try out the bouquets," she said when she heard him come back, "to make sure they're comfortable to hold, that the shape and the use of color and texture work."
"Sure. I get it. I pick up a hammer at least once on every job, just to get a feel for the building. I get it, Emma."
"Okay then, I just wanted . . ." She trailed off when she turned and saw the long, slim box in his hand. "Oh."
"I had a meeting in town, and I saw this. It sort of yelled out of the display window, 'Hey, Jack, Emma needs me.' And I thought, yeah, she does. So . . ."
"You brought me a present," she said when he handed it to her.
"You said you liked getting flowers."
She opened the box. "Oh, Jack."
The bracelet burst with color, bold jewel-toned stones, each a small, perfect rose.
"But you don't wear flowers."
Surprise and delight clear on her face, she looked up. "I will now. It's beautiful. Just beautiful." She took it out, laid it across her wrist. "I'm dazzled."
"I know the feeling. Here, the jeweler showed me how it works. The clasp slides in here, so you don't see it."
"Thank you. It's . . . Oh, look at my hands."
He took them, stained and scratched from her work, and brought them to his lips. "I do. A lot."
"I snap at you, and you give me flowers." She slid into his arms. "I'll have to snap at you more often." On a sigh, she closed her eyes. "The rain's stopped," she murmured, then leaned back. "I need to clean up a little, then go help with tonight's rehearsal. But after, we could have a drink, maybe something to eat out on the patio. If you want to stay."
"I want to stay." A sudden intensity darkened his eyes as they roamed her face. "Emma. I don't think I've told you enough that I care about you."
"I know you do." She rose up to kiss him softly. "I know."
_L_ ATER, WHEN SHE'D LEFT FOR THE MAIN HOUSE, HE ROOTED through her supplies and found what he needed to toss a quick meal together. It wasn't as if he couldn't cook when he needed to, he thought. Or that he expected her to cook for him when they stayed in.
As they did more often, he realized.
He could even put a pretty damn good meal together, the benefit of once dating a sous chef.
A little garlic and olive oil, some herbs and chopped tomatoes and they'd have some pasta. No big deal.
He'd made her breakfast before, hadn't he?
Once.
Why did he suddenly feel he was taking advantage of her, taking her for granted, the way he'd often thought others did?
He knew why. He knew exactly why, he admitted as he minced and chopped.
The look on her face when their eyes had met in the mirror, just that split second of hurt before irritation had smothered it.
_I'm not practicing_.
He _had_ been thinking of the flowers, the bracelet. But she hadn't been completely wrong in her instincts. On some level he had been . . . uneasy. Or . . . hell if he knew. But the sight of her holding the bouquet had given him a—jolt, he admitted. Just for a second.
And he'd hurt her, bruised her feelings. The last thing in the world he wanted to do was hurt her.
She'd forgiven him, or let it go, or pushed it aside. Not because of the bracelet, he thought. She wasn't the type to angle for gifts, or to sulk over a slight.
She was . . . Emma.
Maybe he had taken her for granted here and there. That would stop now that he recognized it. He'd be more careful, that was all. Just because they'd been seeing each other for . . .
The shock had him nicking his thumb. Seven weeks. No, nearly eight, which was the same thing as two months. And that was practically an entire season.
A quarter of a year.
It had been a very long time since he'd been able to measure the time in months he'd been exclusively with one woman.
In a couple of weeks they'd have been together throughout spring, and starting into summer.
And he was okay with it, he realized. More than okay with it.
There was no one else he wanted to be with.
It felt good. Whatever the hell it meant, it felt good to know she'd come back soon and they'd share a meal out on her patio.
He poured himself a glass of wine as he began to sauté garlic. "Here's to the rest of the spring," he said, lifting his glass, "and right through summer."
_"R_ ED ALERT!" ATOP THE LADDER, HER HANDS FULL OF DELICATE garlands, Emma craned her neck to read the display on the beeper hooked to her pants. "Crap. Crap. Red alert. Beach, you'll need to finish the garland. Tiff, swags. Tink, ride herd."
As she scrambled down, Jack stepped forward to spot her. "Careful. It's not a national emergency."
"It is when Parker issues a red. Come with me. Sometimes an extra pair of hands, especially male, can come in handy. If it's just a girl thing, maybe you could come back, help cover chairs. Damn it. I was on schedule."
"You'll make it."
She moved like lightning, across the terrace, up the steps—that still needed to be dressed—and through the door to the corridor outside the Bride's Suite.
Straight into hysteria.
The small mob of people crammed the hall, all in various states of dress. Voices pitched toward the register only dogs could hear. Tears flowed like wine.
In the midst, Parker stood like a cool island in stormy seas. But Emma recognized the fraying of desperation around the edges.
"Everyone, everyone! Everything is going to be fine. But you have to calm down, and listen. Please, Mrs. Carstairs, please sit down here. Sit down now, take a breath."
"But my baby, my baby."
Carter nudged his way forward—a brave soul—and took the weeping woman by the arm. "Here now, have a seat."
"Something has to be done. Something has to be done."
Emma recognized the mother of the bride. She wasn't crying—yet—but her face approached the color of ripe beets. Even as Emma moved in to take her, or whoever needed it most, off Parker's hands, a shrill whistle cut the air into shocked silence.
"Okay, everybody, just stop!" Laurel ordered. She wore a white bib apron smeared with what looked to be raspberry sauce.
Parker plowed into the opening. "Mr. Carstairs, why don't you sit down with your wife a moment? Groom, if you and your party would go back to your suite, Carter will give you a hand. Mrs. Princeton, Laurel's going to take you and your husband downstairs. You'll have some tea. Give me fifteen minutes. Jack, could you go with Laurel? We'll bring Mr. and Mrs. Carstairs some tea up here."
"Any chance of scotch?" Mr. Princeton asked.
"Absolutely. Just tell Jack what you'd like. Emma, I could use you in the Bride's Suite. Fifteen minutes, everyone. Just stay calm."
"What's the story?" Emma demanded.
"Quick update. Two of the bridesmaids are severely hung-over, and one was puking heroically in the bathroom moments ago. MOG had a meltdown when she went in to see her son in the Groom's Suite, which annoyed MOB—they don't get along particularly well. Words were exchanged, tempers flared, and continued to flare as the women battled their way to the Bride's Suite. The drama apparently sent the MOH, who's eight months pregnant, into labor."
"Oh my God. She's in labor? Now?"
"It's Braxton Hicks." Parker's face was a study of sheer determination and unassailable will. "It's going to be Braxton Hicks. Her husband called the doctor, and the MOH convinced him to let us time the contractions for now. Mac and the bride and the rest of the party, not currently puking or moaning, are with her. She and the bride are the only ones keeping their heads. Besides Mac. So."
Parker sucked in a breath, opened the door of the Bride's Suite.
The MOH lay propped on the little sofa, pale, but apparently calm with the bride—a hairdresser's cape over her corset and garters—kneeling beside her. Across the room, Mac offered a cool compress to a bridesmaid.
"How are you doing?" Parker asked as she moved briskly toward the pregnant woman. "Do you want your husband?"
"No. Let him stay with Pete. I'm okay, really. Haven't had anything in the last ten minutes."
"Nearly twelve now," the bride told her and held up the stopwatch.
"Maggie, I'm so sorry."
"Stop saying that." The bride gave her friend a shoulder rub. "Everything's going to be fine."
"You should finish getting your hair and your makeup. You should—"
"It can wait. Everything can just wait."
"Actually, it's a good idea," Parker said in a tone that managed to be brisk, businesslike, and cheerful all at once. "If you're not comfortable here, Jeannie, we can move you to my room. It's quieter."
"No, I'm fine here, really. And I'd like to watch. I think he's gone back to sleep." She patted the mound of her belly. "Honestly. Jan's in worse shape than I am."
"I'm an idiot." The attendant with the pale green complexion closed her eyes. "Maggie, just shoot me."
"I'm going to have some tea and toast sent up. It should help. Meanwhile, Emma and Mac are here to help out. I'll be back in two minutes. Any more contractions," Parker said quietly to Emma, "beep me."
"Believe it. Come on, Maggie, let's make you gorgeous." She drew Maggie to her feet, passed her to the hairdresser. With the stopwatch in hand, Emma settled down by the expectant mother. "So, Jeannie, it's a boy?"
"Yes, our first. I've got another four weeks. I had a checkup Thursday. Everything's fine. We're fine. How's my mother?"
It took Emma a moment to remember Jeannie was the groom's sister. "She's fine. Excited and emotional, of course, but—"
"She's a wreck." Jeannie laughed. "One look at Pete in his tux and she dissolved. We heard the wails in here."
"Which, of course, set my mother off," Maggie said from the salon chair. "Then they're at each other like pit bulls. Jan's tossing it in the bathroom and Shannon's curled in a ball."
"Better now." Shannon, a little brunette currently sipping what looked like ginger ale, waved from her own chair.
"Chrissy's good, so she took the kids outside for just a bit. She should be back by now."
Judging things were under control in this area, Emma glanced at Maggie. "Looks like we've cleared the fifteen-minute mark on baby. If Shannon's up to it, she can take over the timer, and I can go find Chrissy and the kids. Bridesmaid, flower girl, ring bearer?"
"Please. Thanks so much. This is all just crazy."
"We've had crazier." She gave the stopwatch to Shannon, took one more look at Jeannie. The color was back in her cheeks. If anything, she looked serene. "Mac, you've got the fort?"
"No problem. Hey, let's take some pictures!"
"You're a cruel woman," Jan muttered.
Emma dashed out. She spotted the MOG on the terrace, sobbing into a tissue while her husband patted her shoulder and said, "Come on, Edie. For God's sake."
She detoured and headed for the main stairs. Parker was already charging back up. "Status?"
"I think we're down to yellow status. No more contractions, one hangover well on the mend, the other—hard to tell. The bride's in hair, and I'm off to round up the last attendant and the kids."
"In the kitchen having cookies and milk. If you could take the FG and RB, send the BA up. Mrs. G's putting tea and toast together. I want to check on the groom, and let the expectant daddy know everything's okay."
"On my way. The MOG's on the terrace, wildly weeping."
Parker set her jaw. "I'll deal with her."
"Good luck." Emma hurried down, swung toward the kitchen just as Jack came in from the direction of the Grand Hall.
"Please tell me there's not a woman delivering a baby upstairs."
"That crisis, it seems, has passed."
"Well, thank you, Jesus."
"POB?"
"Huh?"
"Parents of Bride?"
"Carter's got them. It seems he teaches a nephew. And the mom's repairing her makeup or something."
"Good. I've got to get the last BA, send her up and take over with the FG and RB."
His brow furrowed, then he gave up on the code. "Whatever you say."
Pausing, Emma considered him. "You're pretty good with kids, as I recall."
"I'm okay. They're just short."
"If you can take the RB—the boy, he's five—and entertain him for about fifteen minutes, it would help. You can deliver him to the Groom's Suite as soon as we get the all clear. I'll take the girl up, help get her dressed." She glanced at her beeper with some trepidation when it signaled. Then blew out a breath. "Yellow and holding. Good."
"Don't these kids have parents?" he asked as he followed her toward the kitchen.
"Yes, and both are in the wedding party. They're brother and sister, twins. The BA with them is Mom. The dad's a groomsman, so you can take the RB up in ten or fifteen. Just give everything a few more minutes to smooth out. Once I get the FG settled, I need to get back out and finish dressing the outside areas. So—"
She broke off, fixed a big, happy smile on her face before she pushed into the kitchen.
In an hour, the bride and attendants were beautified, the groom and his men polished. While Mac organized the separate parties for formal photos, and Parker kept the respective mothers at a distance, Emma finished the outside decor.
"Want a job?" she asked Jack as he helped cover the last row of chairs.
"So absolutely not. I don't know how you do this every weekend."
She attached cones holding the palest of pink peonies to selected chairs. "It's never boring. Tink, I've got to run home and change. Guests are arriving."
"We're good here."
"Parker estimates we'll only be about ten minutes late, which is a miracle. There's food for all of you in the kitchen when we're done. I'm back in fifteen. Jack, go have a drink."
"I plan to."
She was back in twelve, having traded her work clothes for a quiet black suit. She pinned boutonnieres while Parker's voice sounded in her headset. "We're a go in the Bride's Suite. Cuing music. Ushers to start escort."
She listened to the countdown as she brushed lapels, joked with the groom. She spotted Parker arranging the parents, and Mac getting into position for shots.
She took a moment, just one, to admire the view outside. The crisp white covers on the chairs served as a perfect backdrop for the flowers. All the greens and pinks, from the palest to the deepest, blooming against the shimmer of tulle and lace.
Then the moment was over as the groom took his place, and the mothers—one teary, the other maybe just a little tipsy on scotch—were escorted to their seats.
She turned to gather the bouquets and pass them out as Parker lined up the ladies.
"You all look so beautiful. Still holding, Jeannie?"
"He's awake, but behaving."
"Maggie, you're just stunning."
"Oh, don't." The bride waved a hand in front of her face. "I didn't think I'd get all choked up, but I'm right on the edge. I'm about to give my new mother-in-law a run for her money."
"One breath in, one breath out," Parker ordered. "Slow and easy."
"Okay. Okay. Parker, if I ever need to wage war, you're my general. Emma, the flowers are . . . Breathe in, breathe out. Daddy."
"Don't you start." He gave her hand a squeeze. "Do you want me to walk you down while I'm blubbering like a baby?"
"Here now." Parker reached under the veil, gently dabbed at Maggie's eyes. "Head up, and smile. Okay, number one, you're on."
"See you on the other side, Mags." Jan, still a bit pale but beaming, started her walk.
"And two . . . Go."
With her job done for the moment, Emma stepped back while Parker ran the show.
"Have to admit," Jack said from beside her, "I didn't think you were going to pull this one off. Not this smooth. I'm not only impressed, I'm very nearly awestruck."
"We've had a lot worse than this."
"Uh-oh," he said when her eyes filled.
"I know. Sometimes they just hit me. I think it was the way the bride handled herself—crisis by crisis—then started to crumble at her big moment. But she's holding on. Just look at that smile. And look at him look at her." She sighed. "Sometimes they just hit me," she repeated.
"I think you've earned this." Jack held out a glass of wine.
"Oh boy, have I. Thanks."
She hooked her arm through his, tipped her head toward his shoulder. And watched the wedding.
CHAPTER SIXTEEN
_P_ OST-EVENT, THEY TOOK A MOMENT TO UNWIND IN THE FAMILY parlor. Appreciating every moment, Emma sipped her second glass of wine of the evening.
"No visible hitches." She rolled her shoulders, curled and uncurled her bare toes. "And that's what counts. I expect the wedding party will be telling stories of hangovers, spatting mothers, and baby alert for weeks. But that's the sort of thing that makes every wedding unique."
"I wouldn't have believed anyone could cry, almost without pause, for nearly six hours." Laurel popped a couple of aspirin, chased them down with fizzy water. "You'd think it was her son's funeral instead of his wedding."
"I'm going to have to Photoshop the hell out of the MOG's photos. And even then . . ." Mac shrugged. "I think it's a brave bride who takes on a mother-in-law who literally howled during the I do's."
Tossing back her head, Mac gave a terrifyingly accurate rendition of Mrs. Carstair's wail.
"My head," Laurel muttered. "My head."
From his perch on the arm of the sofa, Carter laughed at Mac even as he gave Laurel's shoulder a comforting pat. "I don't know about the rest of you, but that woman scared me."
"I think part of it was the upcoming grandchild. It's all just too much for her."
"Then somebody should've slipped her a Valium," Laurel said to Emma. "And I'm not really kidding. I kept waiting for her to throw herself on the wedding cake—like it was a pyre."
"Oh man, what a shot that would've been." Mac sighed. "Regrets."
"Carter, Jack." Parker lifted her bottle of water. "You were a huge help. If I'd known the MOG was a wailer, I'd have taken steps beforehand, but she was fine at rehearsal. Even bubbly."
"I bet someone slipped her drugs," Laurel said.
"What sort of steps?" Jack wondered.
"Oh, there are all sorts of tricks of the trade." Parker's smile hinted at secrets. "I may not have been able to keep her from blubbering all during the ceremony, but I'd have kept her from upsetting the bride and groom during dressing. If Pete and Maggie hadn't kept their heads, we'd have had a disaster on our hands. Keeping the overly emotional types busy, giving them little assignments usually works."
"I know that's what kept me from crying," Jack told her.
"We'll have to muddle through without the reserve troops tomorrow." Mac gave Carter a friendly kick from her chair. "They're deserting us for the Yankees."
"And speaking of tomorrow, I'm going up to fall flat so I can get up for it." Laurel rose. " 'Night, kids."
"There's our cue. Let's pack it in, Professor. God, my feet are killing me."
Carter turned his back, gestured to it. With a laugh, Mac boosted herself on. "Now this is love," she said, planting a noisy kiss on the top of his head. "Him for the offer, and me for trusting Professor Grace not to trip and drop me. See you tomorrow. Giddyup!"
"God, they're cute." Emma smiled after them. "Even Scary Linda can't dull their shine."
"She called Mac this morning," Parker told her.
"Hell."
"Told Mac she'd changed her mind, and expected Mac and Carter to be at her wedding, in Italy, next week. The usual drama and guilt trip when Mac told her it wasn't possible for her to fly to Italy on such short notice."
"Mac didn't say anything about it to me."
"She didn't want to get into it with the event. Linda, of course, called just as Mac was getting her gear packed for the morning wedding. But the point is, you're right, she can't dull the shine. Before Carter, a call like that would've sent Mac into the blue. It wasn't pleasant, but she got through it, set it aside."
"The Power of Carter defeats the Power of Linda. I owe him a big kiss."
"I'll see him tomorrow if you want to give it to me," Jack suggested.
She leaned over, gave him a prim peck.
"Kinda stingy."
"He belongs to a friend. Okay, getting up, going home."
"Eight o'clock briefing," Parker reminded her.
"Yeah, yeah." She smothered a yawn. "How do you feel about piggybacks?" she asked Jack.
"I like this way better." In a deliberately dramatic move, he swept her up.
"Wow. Me, too. 'Night, Parker."
"Good night." And just a little wistfully, Parker watched Jack Rhett Butler Emma out of the parlor.
"Great exit." Delighted, Emma pressed her lips to Jack's cheek. "You don't have to carry me all the way back."
"You think I'm going to let Carter show me up? You know nothing about true competition. It's good to see Mac look so happy," he added. "I've been around a few times when Linda did a number on her. Tough to watch."
"I know." Idly, Emma fluttered her fingers through Jack's sun-streaked hair. "She's the only person I actually and actively dislike. I used to try to find excuses for her, then I realized there just aren't any."
"She hit on me once."
Emma's head jerked up. "What? Mac's _mother_ hit on you?"
"Long time ago. Actually there was another time not all that long ago. So that makes two hits. First time I was still in college, spending a couple of weeks here during the summer break. We were all going to a party, and I said I'd swing by and pick up Mac. She didn't have a car back then. So her mother came to the door, and gave me the kind of once-over you don't generally get from mothers, then sort of backed me into a corner until Mac got down. It was . . . interesting, and yeah, scary. Scary Linda. Good name."
"What were you, twenty? She should be ashamed. Arrested. Something. Now I dislike her more. I didn't think it was possible."
"I survived. But if she tries it again, I'm counting on you to protect me. And a lot better than you did with Scary Kellye."
"One of these days I'm going to tell her what I think of her. Linda, not Kellye. And if she actually shows up at Mac's wedding and tries to pull something, I might get violent."
"Can I watch?"
Emma laid her head back down on his shoulder. "I'm calling my mother tomorrow, just to tell her she's wonderful." She kissed his cheek again. "And so are you. This is the first time I've ever been carried through the moonlight."
"Actually, it's overcast."
She smiled. "Not from where I'm sitting."
_J_ ACK STUDIED HIS HOLE CARDS. POKER NIGHT HAD BEEN GOOD to him, so far, but the pair of deuces didn't look promising. He checked, waited while the bet walked around the table. When it got to Doctor Rod, he tossed in twenty-five. Beside him, Mal folded. Del tossed in his chips. Landscape Frank did the same. Lawyer Henry folded.
Jack debated briefly, and coughed up the twenty-five.
Del burned the top card, then turned over the flop. Ace of clubs, ten of diamonds, four of diamonds.
Possible flush, possible straight. And he had a crap pair of deuces.
He checked.
Rod went another twenty-five.
Carter folded, Del and Frank met the bet.
Stupid, Jack thought, but he just had a feeling. Sometimes feelings were worth twenty-five.
He added his chips to the pot.
Del buried a card, turned the next up. Two of diamonds.
Now that was interesting. Still, knowing how Rod played, he checked.
Rod bet another twenty-five, with Del raising it twenty-five more.
Frank folded. Jack thought about trip deuces. But he still had a feeling.
He tossed in the fifty.
"Glad it didn't scare you off. I'm looking to score here. Need to sweeten the pot." Rod grinned. "I just got engaged."
Del glanced over. "Seriously? We're dropping like flies."
"Congratulations," Carter said.
"Thanks. Raise it back fifty more. I figured, what the hell am I waiting for? So I took the jump. Shell's all about taking a look at your sister's place. Maybe you can get me the Poker Buddy discount."
"Not a chance." Del counted out chips. "But I'll see your fifty. Seeing as it's probably the end of poker and cigars for you."
"Hell, Shell's not that way. Bet's to you, Jack."
Pocket aces, probably. Rod never bluffed, or he sucked at it so wide you saw through it like a plate glass window. Pocket aces or a couple of pretty diamonds. Still . . .
"I'll stick. Consider it an engagement present."
"Appreciate it. We're looking at next June. Shell wants the big splash. I figured, hey, we'll just fly down to some island over the winter, get some sun, get some surf, get married. But she wants the big deal."
"And so it begins," Mal said in funereal tones.
"You're having the big deal, right, Carter?"
"Mac's in the business. They do a great job. Make it really special. Personalized."
"Don't sweat it," Mal said to Rod. "You won't have any say in it anyway. Just learn to repeat 'sure, baby' whenever she asks if you like something, want something, will do something."
"A lot you know. You've never been there."
"Nearly was. I didn't say 'sure, baby' enough." Mal examined the tip of his cigar. "Fortunately."
"I'm going to like being married." Rod nudged his glasses back up his nose. "Settled in, settled down. I guess you're heading in that direction, Jack."
"What?"
"You've been tight with the hot florist for a while now. Off the market."
Del clamped his cigar in his teeth. "Are we playing poker, or should we start talking about where Rod's going to register? Three players in for the river."
Del turned over the last card, but Jack was too busy staring at Rod to notice.
"My bet. And I'm all in."
"That's interesting, Rod." Expression bland, Del puffed on his cigar. "I'll cover it. How about it, Jack? You sticking or folding?"
"What?"
"Bet's to you, brother."
"Right." Off the market? What did that _mean_? He took a slow sip of beer, ordered himself to focus. And saw the river card was the deuce of hearts.
"I'll call."
"I got myself three bullets."
"And a GSW," Del told him, flipping his cards over. "Because I've got two sparkling diamonds, just like the one you put on your sweetheart's finger. King high flush."
"Son of a bitch. I figured you for the tens."
"Figured wrong. Jack?"
"What?"
"Jesus, Jack, show your cards or toss them in."
"Sorry." He shook himself back. "Real sorry about the GSW and the sparkles. But I've got these two little deuces, that add up to four of a kind. I believe that's my pot."
"You pulled a fourth deuce in the fucking river?" Rod shook his head. "You're one lucky bastard."
"Yeah. One lucky bastard."
_A_ FTER THE GAME, WHEN JACK HAD THE WINNER'S SHARE OF everyone's fifty-dollar entry fee in his pocket, he lingered with Del on the back deck.
"Since you're having another beer, you're figuring on flopping here?"
"Thinking about it," Jack said.
"You make the coffee in the morning."
"I've got an early meeting, so the coffee's going on about six."
"Fine. I've got a divorce deposition. Man, I hate it when a friend pressures me into handling a divorce. I hate fucking di vorce cases."
"What friend?"
"You don't know her. We dated off and on some back in high school. She ended up marrying this guy, moving to New Haven about five years ago. Two kids."
With a shake of his head he took a short pull of his beer. "Now they've decided they can't stand the sight of each other, and she's moved back here, staying with her parents until she figures out what the hell she wants to do. He's pissed because she wants to live back here and it complicates visitation." He tipped the bottle to the left. "She's pissed because she put her career on hold to take the Mommy Track." Then tipped it to the right. "He didn't appreciate her enough, she didn't understand the pressure he was under. The usual."
"I thought you weren't going to handle any more divorces."
"A woman whose breasts you've once fondled comes into your office asking for help, it's tough to say no."
"That's true. It doesn't happen often in my line of work, but it's true."
Del shot him a smirk over another sip of beer. "Maybe I've just fondled more breasts than you have."
"We could have a contest."
"If you can remember all the breasts you've had in your hands, you haven't had enough of them."
Jack laughed, tipped back in his chair. "We should go to Vegas."
"For the breasts?"
"For . . . Vegas. A couple of days at the casinos, followed by a titty bar. So, yes, breasts would be involved. Just hang out for a couple days."
"You hate Vegas."
" _Hate_ 's a strong word. No, better, we could go to St. Martin or St. Barts. Something. Play the tables, scope the beach. Go deep-sea fishing."
Del's eyebrows rose. "You want to fish? To my knowledge you've never so much as held a fishing rod."
"There's always a first time."
"Itchy feet?"
"Just thinking about getting away for a few days. Summer's coming. I got locked in last winter with work, and had to cut the week at Vail down to three days. So we can make up for it."
"I could probably stretch a long weekend."
"Good. We'll do that." Satisfied, Jack took another pull on his beer. "Weird about Rod."
"What?"
"Getting engaged. It came out of the blue."
"He's been with Shelly a couple of years. Not so blue."
"He's never made any marriage noises," Jack insisted. "I didn't figure him for it. I mean, a guy like Carter, yeah. He's the type. Come home from work every night, put on the slippers."
"Slippers?"
"You know what I mean. Come home, make a little dinner, pet the three-legged cat, watch some tube, maybe bang Mac if the mood's right."
"You know I try not to think about Mac and banging in the same sentence."
"Get up the next day, do it again," Jack continued in a tone that edged toward a rant. "Add a couple of kids along the way, maybe a one-eyed dog to go with the three-legged cat. Bang less because now you've got kids running around. Deep-sea fishing and titty bars are a thing of the past because now you've got nightmare trips to the mall and daycare and a freaking minivan and college funds. And Christ!" He threw up both hands. "Christ, now you're forty and coaching Little League and you've probably got a gut because who the hell has time to go to the gym when you've got to stop by the market and pick up bread and milk. Then you blink and you're fucking fifty and falling asleep in the Barcalounger watching reruns of _Law and Order_."
Del said nothing for a minute, just continued to study Jack's face. "That's an interesting roundup of the next twenty years of Carter's life. I hope they named one of the kids after me."
"That's the way it goes, isn't it?" What was this panic, this spurt of it rising up in his chest? He didn't want to think about it. "The good part is Mac won't be coming to you to file for divorce because it'll probably work for them. And she's not the type to freak out because he's heading out to Poker Night or hit him with the 'you never take me anywhere' routine."
"And Emma is?"
"What? No. I'm not talking about Emma."
"No?"
"No." Jack took a deliberate breath, found himself mildly shocked by his own babble. "Things with Emma are fine. They're good. I'm just talking in general."
"And in general, marriage is Barcaloungers and minivans, and the end of life as we know it?"
"Could be a La-Z-Boy and a station wagon. I think they're going to make a comeback. The point is, Mac and Carter will do okay with that. So . . . good for them. Not everybody can make it work."
"Depends on the dynamic, for one thing."
"Dynamics change. That's why you're doing a deposition tomorrow." Calmer now, he shrugged. "People change, and the elements, circumstances, situation all evolve."
"Yeah, they do. And the ones who want it enough keep working at it through the evolutions."
Puzzled, and unaccountably annoyed, he scowled at Del. "Suddenly you're a fan of marriage?"
"I've never been an opponent. I come from a long line of married couples. I figure it takes a lot of guts or blind faith to go into it, and a lot of work and considerable flexibility to stay in it. Considering Mac and Carter, and their backgrounds, I'd say she's the guts, he's the blind faith. It's a good combination."
Del paused, considered his beer. "Are you in love with Emma?"
Panic spurted again. He washed it back with beer. "I said this wasn't about her. Us. Any of that."
"And that's bullshit, Jack. We're sitting here having a last beer after a night where you came out on top and I hit near the bottom. Instead of ragging me, you're talking about marriage, and deep-sea fishing. Neither of which have ever been of particular interest to you."
"We're dropping like flies. You said it yourself."
"Sure I did. And we are. Tony's coming up on three, maybe it's four years now. Frank took the plunge last year, Rod's engaged. Add in Carter. I'm not involved with anyone in particular right now, and neither's Mal as far as I know. That leaves you, and Emma. Given that, it'd be surprising if Rod's little announcement didn't get your gears turning."
"Maybe I'm starting to wonder about her expectations, that's all. She's in the marriage business."
"No, she's in the wedding business."
"Okay, good point. She's from a big family. A big, tight, apparently happy family. And while weddings and marriages are different things, one leads to the other. One of her best friends since childhood is getting married. You know how those four are, Del. They're like a fist. The fingers may wiggle individually, but they come out of the same hand. Just like you said you and Mal are in the field, from what I can tell so are Laurel and Parker. But Mac? That shifts things. Now one of my poker buddies is going to be talking wedding plans with them. _That_ shifts things."
He gestured with his beer. "If _I'm_ thinking about it, it's a sure bet she is."
"You could do something radical and have an actual conversation with her about it."
"If you have a conversation about it, it takes you a step closer."
"Or it takes you a step back. Which way do you want to head, Jack?"
"See, you're asking me." To emphasize the point, Jack shot a finger at Del. "She sure as hell will. What am I supposed to say?"
"Again, radical. How about the truth?"
"I don't know the truth." Okay, he thought, that's the source of the panic. "Why do you think I'm freaked out?"
"I guess you have to figure it out. You never answered the lead question. Are you in love with her?"
"How the hell does anybody know that? More, how do they know they're going to stay that way?"
"Guts, blind faith. You've got it or you don't. But from where I'm sitting, brother, the only person putting pressure on you is you." Crossing his ankles, Del polished off his beer. "Something to think about."
"I don't want to hurt her. I don't want to let her down."
Listen to yourself, Del thought. You're already sunk and don't know it. "I don't want to see that happen either," he said casually. "Because I'd hate having to kick your ass."
"What you'd hate is for me to kick yours if you tried."
There followed the more comfortable interlude of insults over the last beer.
_B_ ECAUSE HE WANTED TO KEEP A CLOSE EYE ON MAC'S ADDITION, Jack tried to swing by the job site every day. It gave him a spectator seat to The Life of Mac and Carter.
Every morning he'd catch sight of them in the kitchen—one of them feeding the cat, the other pouring coffee. At some point, Carter would clear out with his laptop case, and Mac would get to work in the studio.
If his swing-by came in the afternoon, he might see Carter walking back from the main house—but never, he noted, when Mac was with a client. The guy must have radar, Jack concluded.
Occasionally one or both of them came out to check the progress, ask questions, offer him coffee or a cold drink, depending on the time of day he dropped by.
The rhythm fascinated him enough that he stopped Carter one morning.
"School's out, right?"
"The summer of fun has begun."
"So I notice you head over to the big house most days."
"It's a little crowded in the studio right now. And noisy." Carter glanced back toward the buzz of saws, the thwack of nail guns. "I teach teenagers, so I have a high tolerance for confusion, and still I don't know how she works with the noise. It doesn't seem to bother her."
"What the hell are you doing all day? Plotting pop quizzes for next fall?"
"The beauty of the pop quiz is that it can be repeated endlessly through the years. I have files."
"Yeah, I bet. So?"
"Actually, I'm using one of the guest rooms as a temporary study. It's quiet, and Mrs. Grady feeds me."
"You're studying?"
Carter shifted his feet, a tell Jack recognized as mild to middling embarrassment. "I'm sort of working on a book."
"No shit?"
"It may be shit. Parts of it probably are. But I thought I'd take the summer to find out."
"That's great. How do you know when she's cleared out—the clients? Does she call over, tell you it's safe to come home?"
"She's trying to schedule clients in the morning, whenever she's doing a shoot here, and shifting most consults over to the main house while the construction's going on. I just check her book for the day, so I don't come back during a shoot, break the mood or her concentration. It's a pretty simple system."
"It seems to be working for you."
"Speaking of work, I didn't expect all this to move so fast." Carter gestured toward the studio. "Every day there's something new."
"Weather holds and the inspections pass, it'll keep moving. It's a good crew. They should—Sorry," he said when his phone rang.
"Go ahead. I'd better get started."
He pulled out his phone as Carter walked off. "Cooke. Yeah, I'm on the Brown site." As he spoke, Jack moved away from the noise. "No, we can't just . . . If that's what they want we'll need to draw up the changes and get a revised permit."
He listened, continued to walk.
His job visits also gave him a clear idea of Emma's basic routine. Clients came and went like clockwork in the beginning of the week. Midweek, she'd take deliveries. Boxes and boxes of flowers. She'd be working with them now, he thought. Early start, on her own. Tink or one of the others would probably come in later, do whatever they did.
In the middle of the day, if she could manage it, she'd take a break and sit out on her patio. If he was on-site, he'd squeeze in the time to sit out with her awhile.
How could a man resist Emma sitting in the sunlight?
And there she was now, he realized. Not on the patio, but kneeling on the ground, her hair bundled under a hat while she turned dirt with a garden spade.
"Tell them two to three weeks," he said, and she turned, tipped up the brim of her hat and smiled at him. "I'm heading out from here in a few minutes. I'll talk it over with the job boss. I'll be in the office in a couple hours. No problem."
He flipped the phone closed, scanned the flats of plants. "Don't you have enough flowers?"
"Never. I wanted to plug in some more annuals here in front. It makes a nice show from the event areas."
He crouched, kissed her. "You make a nice show. I figured you'd be working inside."
"I couldn't resist, and this won't take long. I'll put in an extra hour at the end of the day if I need to."
"Busy after the end of the day?"
She cocked her head, slanted him a killer look from under the brim of her hat. "That depends on the offer."
"How about we go into New York for dinner? Someplace where the waiters are snobs, the food's overpriced, and you look so beautiful I don't notice either."
"I'm definitely not busy at the end of the day."
"Good. I'll pick you up about seven."
"I'll be ready. And since you're here." She wrapped her arms around his neck, and took his mouth in a deep, dreamy kiss. "That should hold you," she murmured.
"Pack a bag."
"What?"
"Pack what you need for overnight and we'll get a hotel suite in New York. Make a night of it."
"Really?" She did a quick dance in place. "Give me ten seconds and I'll pack right now."
"Then we're on."
"I have to be back early, but—"
"So do I." This time he kissed her, catching her face with his hands, drawing it out. "That should hold _you_. Seven," he said, and rose.
Pleased with his idea and her reaction, he drew his phone out as he walked to his truck, and got his assistant busy making reservations.
CHAPTER SEVENTEEN
_"I_ TOLD HIM I COULD PACK IN TEN SECONDS. I'M SUCH A LIAR." With the workday scrubbed off and every inch of her creamed and scented, Emma folded a shirt into her overnight case. "Obviously the coming home clothes aren't a real issue, but . . ."
She turned, held up a silky white gown for Parker's opinion. "What do you think?"
"It's gorgeous." Stepping forward, Parker brushed a finger over the delicate lace that framed the bodice. "When did you get this?"
"Last winter. I couldn't resist it, and I told myself I'd wear it just for me, whenever. Of course, I didn't. Haven't. It has this little matching robe. I love lush hotel robes, but this is romantic. I feel like I want to have something romantic to put on after dinner."
"Then it's perfect."
"I don't even know where we're going, where we're staying. I love that. Love the feeling of being whisked away." She did a quick spin then laid the peignoir in her bag. "I want champagne and candlelight, and some ridiculously indulgent dessert. And I want him to look at me in the candlelight and tell me he loves me. I can't help it."
"Why should you?"
"Because it should be enough to be whisked away, to be with a man who'd plan a night like this. He makes me happy. That should be enough."
As Emma continued to pack, Parker stepped forward to rub her shoulders. "It's not as if you're setting limits for yourself, Emma. If you feel you have to."
"I'm not doing that. I don't think I'm doing that. I know I've had some ups and downs about this, so I'm trying to adjust my expectations. And do what I said I'd do when we started." Reaching back, she laid her hand on Parker's, squeezed. "Just enjoy and take things as they come. I've been in love with him for so long, but that's my deal. In reality we've only been together a couple of months. There's no rush."
"Emma, as long as I've known you—which is forever—you've never been afraid to say how you feel. Why are you afraid to tell Jack?"
Emma closed her case. "If he's not ready, and telling him made him feel obliged to step back, to just be friends again? I don't think I could stand it, Parker." She turned, faced her friend. "I guess I'm not ready to risk what we have. Not yet. So I'm going to enjoy our night away, and not put any added weight on it.
"God, I've got to get dressed. Okay, I'll be back by eight, eight thirty at the latest. But if for some reason we get stuck in traffic—"
"I'll call Tink, force her to get out of bed. I know how. She'll take the morning delivery and start processing."
"Good." Confident in Parker's abilities, Emma wiggled into the dress. "But I'll be back." She turned so Parker could do up the zipper.
"I love this color. Citrine. It's annoying to know it would make me sallow. It just makes you glow." She met Emma's eyes in the mirror, then wrapped her arm around her friend's waist and hugged. "Have a great time."
"Can't miss."
Twenty minutes later when she opened the door, Jack took one look and grinned. "This is an excellent idea. I should've had this idea long before. You look absolutely stunning."
"Snobby waiter and overpriced-food worthy?"
"More than." He took her hand, kissed her wrist where the bracelet he'd given her sparkled.
Even the drive into New York struck her as perfect, whether they whizzed along or crept through a snarl of traffic. The light softening toward balmy evening, she thought, and the whole night ahead.
"I always think I'm going to get into the city more often," she told him. "To play or to shop, to check out the florists and markets. But I don't nearly as much as I'd like. So every trip in is exciting."
"You haven't even asked where we're going."
"It doesn't matter. I love the surprise, the spontaneity. So much of what I do—you, too, actually—has to run on a schedule. So this? This is like a magic minivacation. If you promise to buy me champagne, I'll have it all."
"All you want."
When he pulled up in front of the Waldorf, she lifted her eyebrows. "And the excellent ideas keep coming."
"I thought you'd like the traditional."
"You thought right."
She waited on the sidewalk while the doorman took their bags, then she reached for Jack's hand. "Thank you, in advance, for a lovely evening."
"You're welcome, in advance. I'm just going to check in, have them take the bags up. The restaurant's about three blocks from here."
"Can we walk? It's beautiful out."
"Sure. Give me five minutes."
She wandered the lobby, entertaining herself with the shop windows, the lavish flower displays, the people swarming in, swarming out, until he joined her. He skimmed a hand down her back.
"Ready?"
"Absolutely." She put her hand in his again to walk out on Park Avenue. "I had a cousin who got married at the Waldorf—before Vows, of course. Huge, ultrafancy formal affair as many of the Grants' affairs are prone to be. I was fourteen, and very impressed. I still remember the flowers. Acres of flowers. Yellow roses the feature. Her bridesmaids were in yellow, too, and looked like sticks of butter, but oh, the flowers. They'd done this elaborate arbor of yellow roses and wisteria right there in the ballroom. It must have taken an army of florists. But it's what I remember best, so it must've been worth it."
She smiled at him. "What struck you most about a building that left that kind of impression on you?"
"There've been a few." He turned east at the corner, strolling while New York rushed around them. "But honestly? One of my strongest impressions was the first time I saw the Brown Estate."
"Really?"
"Plenty of mansions where I grew up in Newport, and some incredible architecture. But there was something—is something—about the estate that stands out. Its balance and lines, its understated grandeur, the confidence that combines dignity with touches of fanciful."
"That's it exactly," she agreed. "Fanciful dignity."
"When you walk in the main house, there's an immediate impression that people live there. Really live, and more, the people who live there love the house, and the land. All of it. It remains one of my favorite places in Greenwich."
"It's certainly one of mine."
He turned again, to open the door of the restaurant. The minute she stepped inside, Emma felt the pace, the rush drop away. Even the air seemed to hush.
"Nice job, Mr. Cooke," she said quietly.
The maitre d' inclined his elegant head. _"Bonjour, mademoiselle, monsieur."_
"Cooke," Jack said in a James Brown deadpan that had Emma biting the inside of her cheek to smother a laugh. "Jackson Cooke."
"Mr. Cooke, _bien sûr_ , right this way."
He led them through elaborate flower displays and flickering candles, around the gleam of silver and glint of crystal on snowy white linen. They were seated with all expected pomp and offered a cocktail.
"The lady prefers champagne."
"Very good. I'll inform your sommelier. Enjoy your evening."
"I already am." Emma leaned toward Jack. "Very much."
"Heads turned when you walked through."
She sent him that smile—that sexy, sultry smile. "We're a very attractive couple."
"And now, every man in this place envies me."
"I'm enjoying the evening even more. Don't let me interrupt."
He glanced over at the approach of the sommelier. "Let me get back to you."
When he'd ordered a bottle that met with the wine steward's lofty approval, Jack laid his hand over Emma's. "Now, where was I?"
"Making me feel incredibly special."
"An easy job considering what I've got to work with."
"Now you're turning my head. Do go on."
He laughed, kissed her hand. "I love being with you. You're a lift to the day, Emma."
What did it say about her, she wondered, that "love being with you" made her heart jump? "Why don't you tell me about the rest of your day?"
"Well, I solved the mystery of Carter."
"There was a mystery?"
"Where does he go, what does he do?" Jack began, and told her the studio routine he'd observed. "I'm only around for short periods," he continued, "but those short periods range from morning to late afternoon, so my canny observations have taken in a variety of slices of the pie of their day."
"And what were your conclusions?"
"No conclusions, but many theories. Was he slinking off to have a torrid affair with Mrs. Grady, or indulging in a desperate and downward cycle of online gambling on his laptop?"
"He could do both."
"He could; he's an efficient sort." Jack paused to approve the label on the bottle presented to him. "The lady will taste."
As the uncorking ritual began, Jack leaned closer to Emma. "And there, our beloved Mackensie, unaware, trusting, slaving away. Could the seemingly innocent and affable Carter Maguire have these shameful secrets? I had to know."
"You put on a disguise and followed him to the house?"
"Considered and rejected." He waited while the sommelier poured a taste of the champagne into Emma's flute. She sipped, paused, then sent the man a smile that melted the dignified ice. "It's wonderful. Thank you."
"A pleasure, _mademoiselle_." He poured the rest expertly. "I hope you'll enjoy every sip. _Monsieur_." He replaced the bottle in its bucket, bowed away.
"All right, how did you solve the mystery of Carter?"
"Give me a minute, I lost my train with the spillover dazzle. Oh yeah, my method was ingenious. I asked him."
"Diabolical."
"He's writing a book. Which, you already knew," Jack concluded.
"I see them every day, or nearly. Mac told me, but your method was a lot more fun. He's been writing it on and off for years, when he can squeeze in the time. Mac gave him a nudge to work on it this summer instead of teaching summer classes. I think he's good."
"You've read it?"
"Not what he's working on, but he's had some short stories and essays published."
"He has? He's never mentioned it. Another mystery of Carter."
"I don't think you ever learn everything about anyone, no matter how long you know them, or how well. There's always another pocket somewhere."
"I guess we're proof of that."
Her eyes smiled and warmed as she took another sip of champagne. "I guess we are."
_"T_ HE WAITERS AREN'T SNOOTY ENOUGH. YOU'VE CHARMED THEM so they want to please you."
Emma took a scant spoonful of the chocolate souffle she'd asked to share. "I believe they achieved the perfect level of snoot." She slipped the souffle between her lips. Her quiet moan spoke volumes. "This is every bit as good as Laurel's, and hers is the best I've ever tasted."
" _Tasted_ is the operative word. Why don't you actually eat it?"
"I'm savoring." She scooped up another smidgen. "We did have five courses." She sighed over her coffee. "I feel like I've had a little trip to Paris."
He traced his finger over the back of her hand. She never wore rings, he thought. Because of her work, and because she didn't want to draw attention to her hands.
Odd he felt they were one of the most compelling aspects of her.
"Have you been?"
"To Paris?" She savored another stingy bite of souffle. "Once when I was too young to remember, but there's a picture of Mama pushing me in my stroller down the Champs-Élysées. I went again when I was thirteen, with Parker and her parents, Laurel and Mac and Del. At the last minute Linda said Mac couldn't go, over some slight or infraction. It was awful. But Parker's mom went over and fixed it. She'd never say how. We had the best time. A few days in Paris then two amazing weeks in Provence."
She allowed herself another spoonful. "Have you?"
"A couple times. Del and I did the backpack through Europe thing the summer of our junior year in college. That was an experience."
"Oh, I remember. All the postcards and pictures, the funny e-mails from cyber cafes. We were going to do it, the four of us. But when the Browns died . . . It was too much, and so many things to deal with. And Parker channeled everything into putting together a business model for Vows. We just never got around to it."
She sat back. "I really can't eat another bite."
He signaled for the check. "Show me one of your pockets."
"My pockets?"
"One of those things I don't know about you."
"Oh." Laughing, she sipped her coffee. "Hmm, let's see. I know. You may not be aware that I was the Fairfield County Spelling Bee Champion."
"Get out. Really?"
"Yes, I was. In fact, I went all the way to the state competition, where I was this close . . ." She held up her thumb and finger, a fraction apart. " _This_ close to winning, when I was eliminated."
"What was the word?"
"Autocephalous."
His eyes slitted. "Is that a real word?"
"From the Greek, meaning being independent of external authority, particularly patriarchal." She spelled it out. "Except under pressure, I spelled it with an _e_ for the second _a_ , and that was that. I remain, however, a killer at Scrabble."
"I'm better at math," he told her.
She leaned forward. "Now, let me see one of yours."
"It's pretty good." He tucked his credit card in the leather folder discreetly placed at his elbow. "Nearly up there with spelling bee champion."
"I'll be the judge."
"I was Curly in my high school's production of _Oklahoma!_ "
"Seriously?" She pointed at him. "I've heard you sing. You're good. But I didn't know you had any interest in acting."
"None. I was interested in Zoe Malloy, who was up for the part of Laurey. Crazy about her. So I put it all out there for 'Surrey with the Fringe on Top,' and got the part."
"Did you get Zoe?"
"I did. For a few shining weeks. Then, unlike Curly and Laurey, we parted. And that was the end of my acting career."
"I bet you made a great cowboy."
He sent her a quick, teasing grin. "Well, Zoe certainly thought so."
With the bill addressed, he rose, held out a hand for hers.
"Let's walk the long way around." She laced her fingers with his. "I bet it's a beautiful night."
It was. Warm and sparkling so even the traffic jamming the streets glittered and gleamed. They strolled, winding their way around the blocks and back to the grand front entrance of the hotel.
People swept in and out, in business suits, in jeans, in evening clothes. "Always busy," she said. "Like a movie where no one ever says 'cut.' "
"Do you want a drink before we go up?"
"Mmm, no." She tipped her head toward his shoulder as they walked to the elevators. "I've got everything I want."
In the elevator she turned into his arms, tipped her face up to his. Her pulse rate climbed as the car did, up and up, level by level.
When he opened the door, she stepped into candlelight. On the white-draped table a silver bucket held a bottle of champagne. A single red rose speared from a slim vase while around the room tea lights flickered in clear glass. Music drifted, whisper soft.
"Oh, Jack."
"How did this get here?"
Laughing, she took his face in her hands. "You've just bumped this up from great date to dream date. This is amazing. How did you manage it?"
"I arranged for the maitre d' to alert the hotel when they brought the check. Planning isn't just your business."
"Well, I like your plan." She kissed him, lingered for another. "A lot."
"I had a feeling. Should I open the bottle?"
"Absolutely." She wandered to the window. "Look at the view. Everything's still so bright and busy, and here we are."
The bottle opened with a sophisticated _pop!_ When he'd poured the glasses and joined her, she tapped hers to his. "To excellent planning."
"Tell me something else." He touched her hair, just a skim of the fingers. "Something new."
"Another pocket?"
"I've discovered the spelling bee champ, the ace soccer player. These are interesting facets."
"I think we've covered all my hidden skills." She reached out, trailed a fingertip down his tie. "I wonder if you can handle the dark side."
"Try me."
"Sometimes when I'm alone at night, after a long day . . . especially if I'm feeling unsettled. Or on edge—" She broke off, lifted her glass for a sip. "I'm not sure if I should confess this one."
"You're among friends."
"True. Still, not many men really understand some of a woman's needs. And some just can't deal with the fact that there are certain needs they can't meet."
He took a long drink. "Okay, I don't know whether to be scared or fascinated."
"I once asked a man I was seeing to join me one evening for this particular activity. He wasn't ready for it. I've never asked another."
"Does it involve tools? I'm good with tools."
She shook her head and strolled over to top off her glass, then held up the bottle in invitation.
"What I do is . . ." She poured bubbling wine into his glass. "First, I'll take a big glass of wine up to my bedroom, then I'll light candles. I'll put on something soft and comfortable, something that makes me feel relaxed. Feel . . . female. Then I get into bed with all the pillows arranged just so, because I'm about to take a journey just for myself. And when I'm ready . . . When I'm just sinking in . . . I watch my DVD of _Truly, Madly, Deeply_."
"You watch porn?"
"It's not porn." Laughing, she gave his arm a quick slap. "It's an amazing love story. Juliet Stevenson is devastated when the man she loves, Alan Rickman, dies. She's overwhelmed with grief. Oh, it's painful to watch." Eyes radiating emotion, she laid a hand just under her throat. "I cry buckets. Then he comes back as a ghost. He loves her so much. It rips your heart out, and it makes you laugh."
"Rips out your heart and makes you laugh?"
"Yes. Men never get that. I'm not going to tell you the whole thing, just that it's wrenching and charming and sad and affirming. It's unspeakably romantic."
"And that's what you do, secretly, in your bed at night, when you're alone."
"It is. Hundreds of times. I've had to replace the DVD twice."
Obviously baffled, he studied her as he drank champagne. "A dead guy's romantic?"
"Hello? Alan Rickman. And yes, in this case, it's wonderfully romantic. After I watch it—and finish crying—I sleep like a baby."
"What about _Die Hard_? He's in _Die Hard_. Now that's a movie you can watch a hundred times. Maybe we should do a double feature some time. If you can handle that."
"Yippee-ki-yay."
He grinned at her. "Pick a night next week, and you're on. But there has to be popcorn. You can't watch _Die Hard_ without popcorn."
"Fair enough. Then we'll see what you're made of." She brushed her lips to his. "I'm going to change. It won't take me long. Maybe you should bring the champagne into the bedroom."
"Maybe I should."
In the bedroom he took off his jacket and tie, and thought about her. Thought about the surprises and facets and layers of her.
It was odd, really, to think you knew someone inside and out, and discover there was more to learn. And the more you learned the more you wanted to know.
On impulse, he took the rose from the vase and laid it on a pillow.
When she stepped out into the candlelight, he lost his breath. Black hair tumbling over white silk, smooth skin gold against white lace. And those eyes, he thought, deep and dark, looking into his.
"You said something about dream date," he managed.
"I wanted to do my part."
The silk flowed over her curves as she walked to him, and as she lifted her arms to wind them around his neck in a way that was so essentially Emma, her scent shimmered in the air like the candlelight.
"Did I thank you for dinner?"
"You did."
"Well . . ." She scraped her teeth over his bottom lip—lightly, lightly—before the kiss. "Thanks again. And the champagne? Did I thank you for that?"
"As I recall."
"Just in case." On a sigh her mouth met his. "And the candlelight, the rose, the long walk, the view." Her body moved against his, leading him into a slow, circling dance.
"You're welcome."
He drew her in, closer still, so her body pressed to his. Time spun out as they circled, as mouth clung to mouth, as heart beat to heart.
She drew in his scent, his flavor. So familiar and still so new. Her fingers trailed up into hair bronzed and gilded by the sun, then curled, tugged to bring him just a little closer.
They slid down together onto smooth white sheets, and into the perfume of a single red rose. More sighs now, more dreamy movements. A caress, a tender touch, shimmered over her skin. She stroked his face, opened—body and heart—as she found, with him, passion wrapped in the shimmer of romance.
Here was all she wanted, had ever wished for. The sweetness and the heat. And as she gave, more and more, she filled until she was dizzy with love.
His flesh to her flesh, so warm, brought her a quiet joy even as pulses spiked. His lips pressed to her heart as it beat for him.
Did he know it? Couldn't he feel it?
And when he took her up, slowly up, his name—just his name—bloomed in that heart.
She clouded his mind like a silver mist, sparkled in his blood like champagne. Every languorous move, every whisper, every touch seduced, entranced.
When she broke for him, rising up like a wave, she breathed his name. And she smiled.
Something inside him stumbled.
"You're so beautiful," he murmured. "Impossibly beautiful."
"I feel beautiful when you look at me."
He skimmed his fingers over her breast, watched her eyes glow with fresh pleasure. He lowered his mouth, a gentle taste with teeth and tongue, and felt her body quiver with fresh need.
"I want you." Her breath caught as she arched under him. "You're what I want, Jack."
She surrounded him, taking him in, moving with him in slow, savoring beats. Surrounded, he lost himself in her.
_S_ ATED, HE RESTED HIS CHEEK ON HER BREAST, LET HIS MIND drift.
"No chance of playing hooky tomorrow and staying right here?"
"Mmm." Her fingers threaded through his hair. "Not this time. But what a nice thought."
"The way things stand we'll have to get up at dawn."
"I find I often do better on no sleep than with a few stingy hours."
He lifted his head, smiled at her. "That's funny. I was thinking the same thing."
"It would be a shame to waste the rest of that champagne, and those lovely chocolate-covered strawberries."
"Criminal. Stay right here. Don't move. I'll get them."
She stretched, sighed. "I'm not going anywhere."
CHAPTER EIGHTEEN
_F_ IVE MINUTES AFTER EMMA GOT HOME, MAC CAME THROUGH the door.
"I waited until he left," Mac called out as she climbed the stairs. "That's herculean restraint." She scowled when she walked into Emma's bedroom. "You're unpacking. Putting everything away. I hate this level of efficiency. Why can't at least one of you be a slob like me?"
"You're not a slob. You're just a bit relaxed with your personal space."
"Hey, I like that. Relaxed with my personal space. Okay, enough about me. Tell me all. I left my own lover to his lonely bowl of corn flakes."
With last night's dress in her hands, Emma spun a happy circle. "It was fabulous. Every minute."
"Deets, deets, deets."
"An elegant French restaurant, champagne, a suite at the Waldorf."
"God, that's all so you. Snazzy date wise. Casual date-wise, maybe moonlight picnic at the beach, red wine, candles tucked into little shells."
Emma closed her empty suitcase. "Why aren't I dating you?"
"We'd make a lovely couple, it's true." Draping an arm over Emma's shoulder, Mac turned to the mirror and studied—Emma in her trim jeans and soft shirt, and herself in the cotton pants and T-shirt she'd slept in. "Stunning, really. Well, we can keep it in reserve if things don't work out otherwise."
"Always good to have a backup. Oh, God, Mac, it was the most perfect night." She turned, squeezed Mac into a hug before doing another spin. "We didn't sleep. At all. It's amazing, really, that we have so much to talk about, to find out about each other still. We talked all through dinner, then took a long walk. And he'd had them bring up champagne and light candles, put on music."
"Wow."
"We drank more champagne and we talked, and we made love. It was so romantic." On a humming sound, she closed her eyes, hugged herself. "Then we talked and drank more champagne, and we made love again. We had breakfast by candlelight and—"
"Made love again."
"We did. We drove home through horrible traffic with the top down, and the traffic didn't matter. Nothing did. Nothing could." She gave herself another hug. "Mac? I'm a happy person most of the time."
"Yeah, it can be annoying."
"I know, but too bad. Anyway, I'm a happy person, but I never knew I could be this happy. I didn't know I could feel like this. Like I just want to jump and dance and spin and sing. Like Julie Andrews on a mountaintop."
"Okay, but don't do that because that's seriously annoying."
"I know, so I'm only doing it on the inside. All the times I imagined what it would be like to be crazy in love, I never knew."
She dropped down on the bed, grinned up at the ceiling. "Do you feel this way all the time? With Carter?"
Mac flopped down beside her. "I never thought I'd be in love. Not really. I never imagined it the way you always did, or looked for it. In some ways it snuck up on me, and in others it fell on me like a ton of bricks. It's still a shock to the system to realize I have this inside me—not the spinning and singing part, because even inside it would annoy me. But I've got the jumping and dancing going on. And somebody has it inside for _me_. Talk about shocking."
Emma reached out to take Mac's hand. "I don't know if Jack has it back for me, not the way I do for him. But I know he cares. I know he feels. And I have so much, Mac. I have to believe all this love I have will . . . take root, I guess. I thought I loved him before, but now I think that was a kind of infatuation mixed up inside lust. Because this is different."
"Can you tell him?"
"I would've said no, even a couple of days ago. Don't want to ruin anything, don't want to tip the scales. Actually did say no when Parker and I talked about it. But now I think I can. I think I should. I just have to figure out how and when."
"It scared me, when Carter told me he loved me. Don't be upset if it scares him a little, at least at first."
"I don't think you tell someone you love them because you expect something. I think you tell them because you have something to give."
"You unpack as soon as you get home from a trip. You have a happy nature. And you're wise about love. I'm surprised the three of us haven't ganged up and beaten the hell out of you regularly."
"You can't. You love me."
Mac turned so they faced each other. "We do. I'm pulling for you, Em. We all are."
"Then how can I go wrong?"
_T_ HE KNOCK INTERRUPTED EMMA HALFWAY THROUGH PROCESSING the morning delivery. Grumbling only a little, she left the flowers holding. She winced when she saw Kathryn Seaman and her sister through the glass. Wet and messy weren't ways to impress important clients.
Trapped, she fixed an easy smile on her face and opened the door. "Mrs. Seaman, Mrs. Lattimer, how nice to see you."
"I apologize for dropping in on you this way, but Jessica and her girls decided on the bridesmaids' dresses. I wanted to bring you the swatch of the material."
"That's perfect. Please come in. Can I get you something to drink? Maybe some sun tea? It's a warm day."
"I'd love some," Adele said immediately. "If it's no bother."
"Not at all. Why don't you sit down, be comfortable? I'll just be a minute."
Tea, Emma thought as she hurried into the kitchen. Lemon slices, the good glasses. Crap, crap. A little plate of cookies. Thank God for Laurel's emergency tin. She scrambled everything onto a tray, shoved at her hair.
She pulled her emergency lip gloss out of a kitchen drawer, glided some on, then pinched her cheeks.
As that was the best she could manage under the circumstances, she took two deep breaths to make sure she looked un-rushed. She strolled back in to find both women wandering her greeting area.
"Kate told me what a pretty setup you had here. She was right."
"Thank you."
"And your private rooms are upstairs?"
"Yes. It's not only convenient, but very comfortable."
"I noticed your partner—Mackensie—is expanding her studio."
"Yes." Emma poured the tea, then continued to stand as neither woman seemed inclined to sit. "Mac's getting married this December, and they'll need more room in their private space, so they're expanding the studio space as well."
"Isn't that exciting?" Sipping tea, Adele continued to wander, fluffing at flowers, studying photos. "Planning a wedding for one of your own."
"It really is. We've all been friends since we were children."
"I noticed the photo here. Is that you, and two of your partners?"
"Yes, Laurel and Parker. We loved playing Wedding Day," Emma told her as she smiled at the photo. "I was the bride that day, and Mac, in a glimpse of the future, official photographer. She'll tell you it was that moment—the blue butterfly moment—when she knew she wanted to be a photographer."
"It's charming." Kathryn turned to Emma. "We've interrupted your work, and are taking up entirely too much of your time."
"It's always lovely to have an unexpected break."
"I hope you mean that," Adele put in, "because I'm dying to see where you work. Are you arranging today? Making bouquets?"
"Ah . . . actually I'm processing a morning delivery, which is why I'm a little messy."
"I'm shameless, and I'm going to ask if I can see where you work."
"Oh. Of course." She shot a look at Kathryn. "Don't panic."
"I've seen where you work."
"Yes, but not while I was working," Emma pointed out as she led the way. "Processing is . . . Well, as you can see." She gestured to her work counter.
"Just look at the flowers!" Flushed with excitement, Adele moved forward. "Oh, and smell the peonies."
"The bride's favorite," Emma told her. "We'll be using this wonderful rich red for her bouquet, contrasted with the bold pinks down to the palest blush. It'll be hand-tied with wine-colored ribbon and candy pink studs. The attendants will carry smaller versions, in the pinks."
"And you keep them in these buckets?"
"In a solution that hydrates and feeds. It's an important step to keep them fresh, and to help them last after the event. I'll keep them in the cooler until we're ready to start designing."
"How do you—"
"Adele." Kathryn clucked her tongue. "You're interrogating again."
"All right, all right. I'm full of questions, I know. But I'm very serious about launching a wedding planning company in Jamaica." Nodding, Adele scanned the area again. "It seems you have a perfect arrangement here, so there's little hope in luring you away."
"But I'm happy to answer questions. Still, for an overview of a business model, Parker's your girl."
"We're going to get out of your way." Kathryn reached in her bag. "The swatch."
"Oh, what a beautiful color. Like a spring leaf through a drop of dew. Perfect for a fairy-tale wedding." She turned to her display and chose a white silk tulip. "See how the white just gleams against this watery green?"
"Yes. Yes, I do. As soon as the final designs are approved, we'll send you the sketches. Thank you, Emma, for the time."
"We're all here to make certain Jessica has the perfect day."
"You see." Adele poked her sister's arm. "That's exactly the sort of attitude I want to offer. In fact, I think The Perfect Day would be a wonderful name for the business."
"I like it," Emma told her.
"If you change your mind, you've got my card," she reminded Emma. "I'll promise you ten percent over what you make annually now."
_"I_ 'M TRYING NOT TO BE ANNOYED SHE'D TRY TO STEAL YOU. Again." Parker slipped off her shoes after the second of two full consults.
"How much did she offer you to move to Jamaica?" Emma asked.
"Carte blanche, which I told her was a rudimentary mistake. No one's worth a blank check, especially when you're designing a business model."
"She's rolling in it," Laurel pointed out. "And yes, I know that doesn't matter on a practical, business level. But she's used to rolling in it."
"She has a good concept. An exclusive and inclusive wedding company in a popular destination wedding site. And she's smart to try to hook people with solid experience. But she's got to create a budget, and stick."
"Then why aren't we doing it?" Mac wanted to know. "I don't mean let's all pack up and move to Jamaica or Aruba, or wherever, but a branch of Vows in some exotic locale? We'd kill."
"I'll kill _you_." Laurel formed a gun with her thumb and finger, and went bang. "Haven't we got enough work?"
"I've thought about it."
Laurel gaped at Parker. "Let me reload."
"Just a loose outline, for the future."
"When they perfect human cloning."
"A franchise rather than a branch," Parker explained. "With very specific requirements. But I haven't worked out all the details or kinks. If and when I do, we'll all talk it through. And we'll all have to agree. But for now, yes, we do have enough work. Except for the third week in August. We're blank."
"I saw that. I meant to ask you about it," Emma continued as she stretched out some kinks in the small of her back. "I figured I'd forgotten to plug something in."
"No, we don't have an event that week because I blacked it out. I can change that if nobody's interested in taking a week at the beach."
There was a moment of stunned silence, then three women leaped up to do a happy dance. Laurel snatched Parker's hand and pulled her up to join them.
"I take it you're interested."
"Can we pack now? Can we? Can we?" Mac demanded.
"Sunscreen, a bikini, and a blender for margaritas. What else do you need?" Laurel swung Parker around. "Vacation!"
"Where?" Emma asked. "What beach?"
"Who cares?" Laurel flopped down on the couch again. "It's the beach. It's a week without fondant or sugar paste. I wipe a tear from my cheek."
"The Hamptons. Del bought a house."
"Del bought a house in the Hamptons?" Mac lifted her fists in the air. "Go, Del."
"Actually, Brown LLC bought it. That's what some of the paperwork he's been bringing over was about. A property came up. It's a good investment. I didn't say anything, in case it fell through. But it's a done deal now. So, we'll all pack ourselves off to the beach for a week the end of August."
"All?" Laurel echoed.
"The four of us, Carter, Del, Jack, of course. It's six bedrooms, eight baths. Plenty of room for everyone."
"Does Jack know?" Emma wondered.
"He knows Del was looking at the property, but not about August. We both felt there wasn't a point in talking about taking the week if we didn't go through with it. Now we have."
"I have to go tell Carter. Yay!" Mac gave Parker a smacking kiss before she rushed out.
"This is so great. I'm going to go put it on my calendar, with lots of little hearts and shiny suns. Moonlight walks on the beach." Emma hugged Parker. "It's nearly as perfect as dancing in a moonlit garden. I'm going to call Jack."
When they were alone, Parker looked at Laurel. "Is anything wrong?"
"What? No. God, what could be wrong. Beach, a week. I think I'm in shock. We need new beach clothes."
"Damn right."
Laurel pushed up. "Let's go shopping."
_W_ HEN INSPIRATION STRUCK, EMMA RAN WITH IT. IT TOOK SOME juggling and a client flexible enough to bump up a consult by an hour, but she managed to clear her Monday afternoon.
She planned to surprise Jack with a twist on their usual Monday night date.
On the way out she stopped by the main house and tracked down Parker in the office.
Parker paced, her headset in place, and rolled her eyes when Emma came in.
"I'm sure Kevin's mother didn't mean to be critical or insulting. You're absolutely right, it is your wedding, your day, your choice. You're entitled to . . . Yes, he is very sweet, Dawn, and extremely well behaved. I know . . . I know."
Parker closed her eyes, mimed strangling herself for Emma's benefit.
"Ah, why don't you let me take care of this for you? It would take the stress off you and Kevin. And sometimes an outside party is better able to explain and . . . I'm sure she doesn't. Yes, of course. I'd be angry, too. But—But . . . Dawn!" Her tone sharpened just a fraction, enough, Emma knew, to shut down whatever rant the bride might be on. "You have to remember, above anything else, any detail, any complication or disagreement, the day and everything about it is for you and Kevin. And you have to remember, I'm here to see that you and Kevin have the day you want."
This time Parker shot her gaze to the ceiling. "Why don't you and Kevin go out and have a nice dinner tonight, just the two of you? I can make a reservation for you wherever . . . I love that restaurant." Parker scribbled down a name on a pad. "Say seven? I'll take care of that for you right now. And I'll speak to his mother this evening. By tomorrow, everything will be fine. Don't worry about a thing. I'll talk to you soon. Yes, Dawn, that's what I'm here for. Good. Great. Mmm-hmm. Bye."
She held up a finger. "One more minute." Once she'd contacted the bride's choice of restaurant, wrangled a reservation, she pulled off the headset.
Parker took a breath, let out a short but enthusiastic scream, then nodded. "Better. Much better."
"Dawn's having a problem with her soon-to-be mother-in-law?"
"Yes. Oddly, the MOG doesn't understand or approve of the bride's choice of ring bearer."
"It's really not her—"
"Which is Beans, the bride's Boston bull terrier."
"Oh, I'd forgotten about that one." Emma's brow creased. "Wait. Did I know about that one?"
"Probably not, as she only told me a couple of days ago. The MOG thinks it's silly, undignified, and embarrassing. And said so in very clear terms. The bride's decided her future mother-in-law is a dog hater."
"Is he wearing a tux?"
Parker's lips twitched "At this point, just a bow tie. She wants the dog, she gets the dog. So I'll ask the MOG to have a drink with me—as such matters are best done in person and with alcohol—and smooth this over."
"Good luck with that. I'm heading into town. I'm going to surprise Jack, cook him dinner, so I won't be back until morning. But I'm also going to see if you and Laurel left any sexy summer clothes anywhere in Greenwich."
"There may be a halter top left. Possibly one pair of sandals."
"I'll find them. I'm going to the market, and by the nursery. Is there anything you need? I can drop it back by in the morning."
"Are you going by the bookstore?"
"I'm going to town; what would my mother say if I didn't drop in?"
"Right. She's got a book I ordered."
"I'll get it for you. If you think of anything else, just call my cell."
"Have fun." As Emma left, Parker looked at her BlackBerry. Sighed. And picked it up to call Kevin's mother.
_D_ ELIGHTED TO HAVE A FEW HOURS OUT AND ABOUT, EMMA stopped at the nursery first. She gave herself permission to just wander and enjoy before settling down to the business of selection.
She loved the smells—the earth, the plants, the green—so much she had to order herself not to just buy some of everything. But she promised herself she'd take another swing through in the morning and pick up a few more plants for the estate.
For now, she debated on pots while envisioning Jack's back porch entrance. She found two slim urns in a rusted bronze color she decided would be perfect flanking his kitchen door.
"Nina?" She signaled to the manager. "I'm going to take these two."
"They're great, aren't they?"
"They are. Can you have them loaded in my car? It's right out front. And the potting soil? I'm just going to pick out the plants."
"Take your time."
She found exactly what she wanted, sticking with deep reds and purples with a few sparks of gold to set them off.
"Gorgeous," Nina commented when Emma pushed her cart through toward the cashier. "Strong colors, great textures. And that heliotrope smells wonderful. Is this for a wedding?"
"No, actually they're a gift for a friend."
"Lucky friend. Everything's loaded."
"Thanks."
In town, she wandered the shops, treated herself to new sandals, a breezy skirt, and thinking of the long-ago summer, a boldly printed scarf to use as a beach wrap.
She swung into the bookstore, waved to the clerk ringing up a sale at the counter.
"Hi, Emma! Your mom's in the back."
"Thanks."
She found her mother opening a recent delivery of books. The minute she saw Emma, Lucia set the shipment aside. "Now this is the best kind of surprise."
"I've been out spending money." Emma leaned over the box to kiss Lucia's cheek.
"My favorite activity. Almost. Did you buy something that made you so happy, or . . ." She tapped a finger on Emma's bracelet. "Are you just happy?"
"Both. I'm going to cook dinner for Jack, so I still have to go to the market. But I found the cutest sandals, which—of course—I had to wear out."
Emma did a pivot, a turn, showing them off.
"They _are_ cute."
"And . . ." Emma flicked her index fingers at her new gold dangles to make them sway.
"Ah, pretty."
"Plus a wonderful summer skirt just covered with red poppies. A couple of tops, a scarf, and . . . so on."
"That's my girl. I saw Jack this morning. I thought he said you were going to the movies tonight."
"Change of plans. I'm going to make him your flank steak. Mrs. G had one in the freezer so I begged it from her and it's been marinating all night. It's out in the car in a cooler. I thought I'd do those roasted fingerling potatoes with rosemary, maybe asparagus, a nice chunk of bread with dipping oil. What do you think?"
"Very manly."
"Good, that was the idea. I couldn't bring myself to hit Laurel up for a dessert. She's swamped. I thought maybe just ice cream and berries."
"A manly and thoughtful meal. Is this an occasion?"
"Partly to thank him for the incredible night in New York, and the rest . . . I'm going to tell him, Mama. I'm going to tell him how I feel about him, that I love him. It seems almost wrong to have all this"—she pressed a hand to her heart—"and not tell him."
"Love is brave," Lucia reminded her. "I know when he says your name, he looks happy. I'm glad you told me. Now I can think good thoughts for you, for both of you, tonight."
"I'll take them. Oh, and you have a book for Parker. I told her I'd pick it up for her."
"I'll get it for you." Lucia wrapped an arm around Emma's waist as she walked her out of the storeroom. "You'll call me tomorrow? I want to know how your dinner went."
"I'll call you, first thing."
"Emma?"
Emma looked over, smiled at the pretty brunette she desperately tried to place. "Hi."
"It is you! Oh, hi, Emma!"
Emma found herself gripped in an enthusiastic hug and rocked side to side. Baffled, she gave the girl a friendly squeeze in return as she shot questioning looks at her mother.
"Rachel, you're home from college." Lucia beamed as she gave her daughter hints. "It seems like last week Emma was heading out to babysit for you."
"I know. I can hardly—"
"Rachel? Rachel Monning?" Emma pulled her back, stared into bright blue eyes. "Oh my God. _Look_ at you. I didn't recognize you. You're grown-up and gorgeous. When did you stop being twelve?"
"A while ago. It's just been so long, between this and that and college. Oh, Emma, you look awesome. You always did. I can't believe I ran into you this way. I was actually going to call you."
"You're in college now? Home for the summer?"
"Yes. One more year. I'm working at Estervil, in public relations. It's my day off and I stopped in because I needed a book. A wedding planning book. I'm engaged!"
She held out her hand to show off the sparkle of her diamond.
"Engaged?" Emma pushed through the moment of speechless shock. "But you were playing with your Barbies ten minutes ago."
"I think it's closer to ten years." Rachel's face lit up with her laugh. "You have to meet Drew. He's amazing. Of course you'll meet him. We're going to get married next summer, after I graduate, and I really want to have you do the flowers, and, well, everything. My mother says Vows is _the_ place. Can you believe it? I'm getting married, and you'll make my bouquet. You used to make those Kleenex bouquets for me, and now, it'll be real."
She felt the jab straight in the belly, hated herself for it, but felt it. "I'm so happy for you. When did this happen?"
"Two weeks, three days and . . ." Rachel checked her watch. "Sixteen hours ago. Oh, I wish I had more time, but I have to get the book and run or I'll be late." She hugged Emma again. "I'll call and we'll talk flowers and cakes and, oh God, everything. Bye! Bye, Mrs. Grant. I'll see you soon."
"Rachel Monning's getting married."
"Yes." Lucia patted Emma's shoulder. "She is."
"I used to babysit for her. I used to French braid her hair and let her stay up past her bedtime. Now I'm going to do her wedding flowers. Good God, Mama."
"There there," Lucia said and didn't bother to mask a chuckle. "Aren't you about to spend the evening with a wonderful man?"
"Yes. Right. I get it. Everyone takes different directions. But . . . Good God."
She managed to put babysitting and weddings aside to finish her shopping. She'd barely stepped out of the market before being hailed again.
_"Buenos tardes, bonita!"_
"Rico." Instead of a hug she had both cheeks affectionately kissed. "How are you?"
"Better for seeing you."
"Why aren't you flying somewhere fabulous?"
"Just back from a run to Italy. The owner took his family to Tuscany for a little R and R."
"Ah, the hard life of the private pilot. And how's Brenna?"
"We broke up a couple of months ago."
"Oh, I'm sorry. I hadn't heard."
"The way it goes." He shrugged. "Let me carry those for you." He took her grocery bags, peeked in as he walked her to her car. "Looks like good eating, and a lot better than the Hungry Man dinner I have on tap."
"Oh, poor thing." She laughed at him then unlocked the passenger side door. "Just in here. I'm already pretty loaded in the back."
"So I see," he said as he glanced at the plants and bags in the backseat. "It looks like you've got a busy evening planned, but if you want to change your mind, I'll take you to dinner." He trailed a flirtatious finger down her arm. "Or better yet, give you that flying lesson we used to talk about."
"Thanks, Rico, but I'm seeing someone."
"It ought to be me. Feel free to change your mind about that, too—anytime—and give me a call."
"If I do you'll be the first." She brushed her lips over his cheek before rounding the hood to her door. "Do you remember Jill Burke?"
"Ah . . . little blonde, big laugh."
"Yes. She's single again, too."
"Is that so?"
"You should call her. I bet she'd love a flying lesson."
His grin flashed adding a sparkle to his eyes and reminding her why she'd enjoyed spending time with him. She got in, and sent him a wave as she drove away.
Considering the planters, plants, groceries, Emma parked in the back of Jack's building and as close to the steps as she could manage. She angled her head as she studied the little kitchen deck, then nodded. The planters would do very well there, very well indeed.
Eager to get started, she walked around to the front of the building to use the main entrance. The beveled glass in the door and the tall front windows brought in pretty light, adding a sense of style and comfort to the reception area. He'd been right to keep it cozy rather than sleek, she thought. It projected calm and quiet dignity, while she knew in the individual offices and planning rooms chaos often reigned.
"Hi, Michelle."
"Emma." The woman working on a computer at a ruthlessly organized desk stopped to shift her chair. "How are you?"
"I'm great. How are you feeling?"
"Twenty-nine weeks and counting." Michelle patted her baby belly. "We're perfect. I _love_ your sandals."
"Me, too. I just bought them."
"They're great. Monday night date, right?"
"Exactly."
"You're a little early, aren't you?"
"New plan. Is Jack busy? I haven't actually told him the new plan."
"He's not back yet. Running late, glitch on a site. Not very happy with the subs or the new county inspector, or, well, anything just at the moment."
"Oh." Emma winced. "Well, my new plan is either very good or very bad under those circumstances."
"Can you share?"
"Sure. I thought I'd cook dinner, surprise him with that and some planters for his little deck. Dinner and a movie at home, instead of going out."
"If you want my opinion, it's inspired. I think he'd be thrilled to have a home-cooked meal after the day he's put in. You can call and check, but he may be in round three with the building inspector."
"Why don't we just let that play out? The problem is, Michelle, I don't have a key."
There was a beat, just a quick bump of surprise. "Oh, well, that's no problem." Michelle opened a drawer of her desk to fish out a spare set.
"Are you sure it's okay?" And how mortifying is it, Emma thought, to have to ask?
"I can't think of why it wouldn't be. You and Jack have been friends for years, and now you're . . ."
"Yes, we are," Emma said, deliberately bright. "Second problem? The two planters I bought weigh about fifty pounds each."
"Chip's in the back. I'll send him out."
"Thanks, Michelle," Emma said as she took the keys. "You're a lifesaver."
She closed her hand around the keys as she started around to the back again. No point, she told herself, in feeling embarrassed. No point in feeling slighted that the man she'd been sleeping with for nearly three months—and had known for more than a decade—hadn't bothered to give her a key.
It wasn't symbolic, for God's sake. He wasn't locking her out. He was just . . .
It didn't matter. She would forge ahead with her plans for the evening. Give him flowers, cook him dinner, and tell him she loved him.
And, damn it, she was going to ask for a key.
CHAPTER NINETEEN
_S_ HE SPENT A HAPPY HOUR PUTTING AWAY GROCERIES, ARRANGING the sunflowers she'd brought from her stock for his kitchen counter, then prepping the planters.
She'd been right, she thought, about how perfect they'd be flanking the door. Deep, bold spots of color, she decided as she tucked red salvia behind purple heliotrope. The combination of plants she'd chosen would give him color and bloom all season, and be even showier when the lobelia spilled and the sweet alys sum foamed over the lip.
A nice welcome home, she thought, every time he walked up the stairs. And, she thought with a little smile, a living reminder of the woman who'd laid out that welcome.
Sitting back on her heels, she studied the result. "Gorgeous, if I do say so myself."
After stacking the empty pots and cell packs, she shifted to duplicate the arrangement in the second urn.
She wondered if he had a watering can, then decided probably not. She should've thought of that, but they'd make do until he got one. Happy to have her hands in dirt, she hummed along with the radio she'd switched on. His front entrance planters needed more zip, she mused as she worked. She'd try to pick up a few more things in the next week or so.
When she'd finished, she swept up the spilled dirt, then carried the plastic trays and pots, her gardening tools down to her car. Brushing off her hands she looked up to admire the work.
Flowers, she'd always thought, were an essential element of home. Now he had them. And, she'd always believed, flowers planted with love bloomed more beautifully. If true, these would be spectacular right up to the first hard frost.
When she checked the time, she dashed back up the stairs. She needed to wash up and start on dinner, especially since she'd decided to add an appetizer to the menu.
_D_ IRTY, SWEATY, AND STILL PISSED OFF DUE TO THE DISAPPEARING plumber and a rookie building inspector with an attitude, Jack turned toward the rear of his offices.
He wanted a shower, a beer, maybe a handful of aspirin. If the general contractor wasn't going to fire the asshole plumber—who also happened to be his brother-in-law—then _he_ could explain the delay to the client. And _he_ could take on the building inspector who decided to throw his weight around because a doorway was a damn seven-eighths of an inch off.
Okay, maybe the aspirin, the shower, then the drink.
Maybe that would smooth out a day that had begun with a call at six A.M. from a client with a tape measure who'd gone ballistic because the framing for his service bar came in at five feet eight inches instead of six feet.
Not that he blamed the client. He'd felt ballistic himself. Six feet on the plans meant six feet on the job, not whatever the sub decided would do.
And, Jack thought as he tried to roll the worst of the tension out of his shoulders, the day had just gone downhill from there. If he was going to put in a twelve-hour day, at least he wanted to finish up feeling he'd accomplished something instead of just riding around the goddamn county putting out fires.
He made the last turn, telling himself to be grateful he was home, where, since the office was now closed, nobody—please God—was going to ask him to fix anything, negotiate anything, or argue about anything.
When he spotted Emma's car he struggled to think past the headache. Had he mixed things up? Had they planned to meet in town, go from there?
No, no, dinner, maybe a movie—which he'd intended to switch to carry-out, possibly a DVD, and that _after_ he'd had a chance to cool off and settle down. Except he'd forgotten to call her about that as he'd been hip-deep in crises and complaints.
But if she was in town somewhere, he could just . . .
His mind switched gears as he noticed his back door open to the screen, and the pots of flowers beside it. He sat where he was a moment, then tossed his sunglasses on the dash. When he stepped out of the truck, he heard the music pouring through the screen door.
Where the hell did the plants come from? he wondered as fresh irritation banged against an already full-blown headache. And why the hell was his door open?
He wanted air-conditioning, a cool shower, and five damn minutes to shake off the worst of the day. Now he had flowers he'd have to remember to water, music blasting, and somebody who'd require attention and conversation in his house.
He trudged up the steps, scowled at the plants, pushed through the screen door.
And there she was, singing along with the radio—which was blasting through his aching head, cooking something on his stove when he'd set his system on take-out pizza, and his spare keys sat on the counter beside a vase of enormous sunflowers that made his eyes throb.
She shook the frying pan with one hand, reached for a glass of wine with the other—then saw him.
"Oh!" She laughed when her hand jerked on the handle of the pan. "I didn't hear you."
"Not surprising, as you're entertaining the neighborhood with . . . Jesus, is that ABBA?"
"What? Oh, the music. It is loud." She gave the pan another shake before adjusting the heat under it. With an easy side step, she picked up the remote, lowered the volume on the stereo. "Cooking music. I thought I'd surprise you with a ready-made meal. These scallops just need another minute. The sauce is already done, so you can have a little something right away. How about a glass of wine?"
"No. Thanks." He reached over her head into the cabinet for a bottle of aspirin.
"Hard day." In sympathy, she rubbed a hand down his arm as he fought open the bottle. "Michelle told me. Why don't you sit down for a minute, get your bearings?"
"I'm filthy. I need a shower."
"Well, you're right about that." She rose on her toes to brush a light kiss on his lips. "I'll get you some ice water."
"I can get it." He moved past her to the refrigerator. "Michelle gave you the key?"
"She said you were stuck out on a job, and having a bad day. I had the food out in the car, so . . ." She shook the pan again, turned off the flame. "I've got a flank steak marinating. Red meat ought to help your headache. You can just clean up and relax. Or I can hold dinner awhile if you want to stretch out until you feel better."
"What is all this, Emma?" Even at the lower volume, the music scraped against his nerves. He grabbed the remote, turned it off. "Did you haul those pots up here?"
"Chip did the heavy work. I had the best time picking out the urns, the plants." She sprinkled the scallops with a mixture of cilantro, garlic, and lime, poured on the sauce she'd prepared. "They really pop against the house, don't they? I wanted to do something to thank you for New York, and when inspiration hit, I juggled a few things and hit the road."
She set the empty bowl in the sink, turned. Her smile faded. "And I miscalculated, didn't I?"
"It's been a lousy day, that's all."
"Which I've added to, clearly."
"Yes. No." He pressed his fingers to the drill trying to bore through his temple. "It's been a bad day. I just need to smooth out some. You should've called if you wanted to . . . do this."
Without thinking, out of sheer habit, he picked up the spare keys and shoved them in his pocket.
He might as well have slapped her.
"Don't worry, Jack, I didn't hang anything of mine in a closet, put anything in a drawer. My toothbrush is still in my bag."
"What the hell are you talking about?"
"My trespassing only went as far as the kitchen, and it won't happen again. I didn't run out and make a copy of your precious keys, and I hope you won't give Michelle any grief for giving them to me."
"Give me a small break, Emma."
"Give _you_ a break? Do you have any idea how humiliating it was to have to tell her I didn't have a key? To know we've been sleeping together since April and I can't be trusted."
"It has nothing to do with trust. I just never—"
"Bullshit, Jack. Just bullshit. Every time I stay here—which is very rare because it's _your_ space, I have to make sure I don't leave so much as a stray hairpin behind because, dear God, what's next? An actual hairbrush? A shirt? Before you know it I'll actually feel welcome here."
"You are welcome here. Don't be ridiculous. I don't want to fight with you."
"Too bad, because I want to fight with you. You're irritated because I'm here, because I invaded your space, made myself at home. And that tells me I'm wasting my time, I'm wasting my feelings, because I deserve better than that."
"Look, Emma, all this just caught me at a bad time."
"It's not the time, Jack, not just the time. It's always. You don't let me in here because that's too close to a commitment for you."
"Jesus, Emma, I am committed. There's no one else. There hasn't been anyone else since I touched you."
"It's not about someone else. It's about you and me. It's about wanting me, but only on your terms, on your—your blueprint," she said waving her hands in the air. "As long as we stick to that, no problem. But that's not going to work for me anymore. It's not going to work when I can't pick up a quart of milk for you or leave a damn lipstick on your bathroom counter. Or give you some damn plants without pissing you off."
"Milk? What milk? Jesus Christ, I don't know what you're talking about."
"It's not going to work when cooking you a fucking meal is like a criminal act." She snatched up the plate of scallops, tossed it into the sink with a crash of stoneware.
"Okay, that's enough."
"No, it isn't enough." She whirled, shoved him back with both hands as tears of anger and heartbreak clouded her eyes, thickened her voice. "And I'm not going to settle for what isn't enough. I'm in love with you, and I want you to love me. I want a life with you. Marriage and babies and a _future_. So this? This isn't enough, not nearly. It turns out you were right, Jack. Absolutely right. Give them an inch, they'll take a mile."
"What? How? Wait."
"But don't worry, no need to run for the hills. I'm responsible for my own feelings, my own needs, my own choices. And I'm done here. I'm done with this."
"Hold it." He wondered his head didn't explode. Maybe it already had. "Wait a damn minute so I can think."
"Time's up, thinking's over. Don't touch me now," she warned when he started toward her. "Don't even think about putting a hand on me. You had your chance. I'd have given you everything I had. If you'd needed more, I'd have found it, and given you that. It's the way I love. It's the only way I know how. But I can't give where it's not wanted and valued. Where I'm not."
"Be pissed off." He snapped it out. "Break dishes. But don't stand there and tell me I don't want you, don't value you."
"Not the way I want or need. And trying not to want, Jack? Trying not to love you the only way I know how to love? It's breaking my heart." She grabbed her bag. "Stay away from me."
He slapped a hand on the screen door to stop her. "I want you to sit down. You're not the only one with things to say."
"I don't care what you want. I'm done caring. I said stay away from me."
She looked up at him then. It wasn't temper or heat in her eyes. Those he would've ignored until they'd burned this out. But he had no power against her pain.
"Emma. Please."
She only shook her head, and, pushing past him, ran to her car.
_S_ HE DIDN'T KNOW HOW SHE MANAGED TO DAM THE TEARS. SHE only knew she couldn't see through them and she had to get home. She needed home. Her hands wanted to shake so she gripped the wheel tighter. Every breath hurt. How was that possible? How could the simple act of drawing breath burn? She heard herself moan, and pressed her lips together to hold back the next. It sounded like a wounded animal.
She wouldn't let herself feel that. Not now. Not yet.
Ignoring the cheerful ringtones of her phone, she kept her eyes focused on the road.
The dam collapsed; the tears broke through when she turned into the drive. She swiped at them, a fast, impatient hand until she'd navigated along the curve, parked.
Now the trembling came, so that she shook as she stumbled from the car, up the walk. She made it inside, safe, home, before the first sob took her.
"Emma?" Parker's voice carried down the stairs. "What are you doing back so early? I thought you were—"
Through the flood of tears, Emma saw Parker rush down the stairs. "Parker."
Then there were arms around her, strong and tight. "Oh, Emma. Oh, baby. Come on now, come with me."
"What's all this commotion? What's . . . Is she hurt?" Like Parker, Mrs. Grady hurried forward.
"Not that way. I'm going to take her upstairs. Can you call Mac?"
"I'll see to it. There now, lamb." Mrs. Grady stroked a hand down Emma's hair. "You're home now. We'll take care of everything. Go on with Parker."
"I can't stop. I can't make it stop."
"You don't have to stop." With an arm around Emma's waist, Parker led her upstairs. "Cry all you want, as long as you need. We'll go up to the parlor. To our place."
As they started up to the third floor, Laurel bolted down. Saying nothing, she simply wrapped an arm around Emma from the other side.
"How could I be so stupid?"
"You weren't," Parker murmured. "You aren't."
"I'll get her some water," Laurel said, and Parker nodded as she led Emma to the couch.
"It hurts, so much. So much. How can anyone stand it?"
"I don't know."
When they sat, Emma curled up, laid her head in Parker's lap.
"I had to get home. I just had to get home."
"You're home now." Laurel sat on the floor, pushed tissues into Emma's hand.
Burying her face in them, Emma sobbed out the pain and grief throbbing in her chest, twisting in her belly. Raw sobs scorched her throat until there were none left. Still, tears spilled down her cheeks.
"It feels like some horrible illness." She squeezed her eyes shut for a moment. "Like I may never be well again."
"Drink a little water. It'll help." Parker eased her up. "And these aspirin."
"It's like a terrible flu." Emma sipped water, took a breath, then swallowed the aspirin Parker handed her. "The kind where even when it's over, you're weak and sick and helpless."
"There's tea and soup." Like Laurel, Mac sat on the floor. "Mrs. G brought it up."
"Not yet. Thanks. Not yet."
"This wasn't just a fight," Laurel said.
"No. Not just a fight." Exhausted, she rested her head on Parker's shoulder. "Is it worse, do you think, since it's my own fault?"
"Don't you dare blame yourself." Laurel squeezed Emma's leg. "Don't you dare."
"I'm not letting him off the hook, believe me. But I got myself into it. And tonight, especially tonight, I worked myself up to wanting—expecting," she corrected, "things that weren't going to happen. I know him, and still I jumped off the cliff."
"Can you tell us what happened?" Mac asked her.
"Yeah."
"Take a little tea first." Laurel held out the cup.
After one sip, Emma blew out a breath. "There's whiskey in here."
"Mrs. G said to drink it. It'll help."
"Tastes like medicine. And I guess it is." Emma took another sip. "I crossed his lines, I guess you could say. I don't find those lines acceptable. So we're done. We have to be done because I can't feel this way."
"What are the lines?" Parker asked.
"He doesn't make room." Emma shook her head. "I wanted to do something for him. Part of it was certainly for me, but I wanted to do something special. So I went by the nursery," she began.
When she finished the tea, the ache throbbed behind a thin cushion. "I had this moment, when I had to tell Michelle I didn't have a key. Part of me stepped back, said: Stop."
"What the hell for?" Laurel demanded.
"And that's what the rest of me said. We were together, a couple. And under that, good friends. What could be wrong with going into his place to surprise him with dinner? But I knew. That other part of me knew. Maybe it was a test. I don't know. I don't care. And maybe it was worse—the buildup, the crash—because I'd run into Rachel Monning at the bookstore. Do you remember her, Parker? I babysat her."
"Yes, vaguely."
"She's getting married."
"You _babysat_ for her?" Laurel held up her hands. "They're letting twelve-year-olds get married?"
"She's in college. Graduating next year, followed by her wedding. Which she wants here, by the way. And when I got over the genuine shock, all I could think was, I want that. I want what this girl I _babysat_ has. Damn it, I want what I see on her face. All that joy, that confidence, that eagerness to start a life with the man I love. Why shouldn't I want that? Why aren't I entitled to that? Wanting marriage is as legitimate as not wanting it."
"Preaching to the choir," Mac reminded her.
"Well, I do want it. I want the promise and the work and the children and all of it. All of it. I know I want the fairy tale, too. Dancing in the moonlit garden, but that's just . . . Well, it's like a bouquet or a beautiful cake. It's a symbol. I want what it symbolizes. He doesn't." She leaned back, closed her eyes a moment. "Neither of us is wrong. We just don't want the same thing."
"Did he say that? That he doesn't want what you want?"
"He was angry to find me in his house," she said to Parker. "Not even angry. Worse. Annoyed. I'd been presumptuous."
"Oh, for God's sake," Mac muttered.
"Well, I had presumed. I presumed he'd be pleased to see me, to have me willing to fuss over him a bit after he'd had a long, hard day. I had my copy of _Truly, Madly, Deeply_ with me. We joked about doing a double feature so he could see why I loved it, and we'd pair it up with _Die Hard_."
"Alan Rickman." Laurel nodded.
"Exactly. I had sunflowers, and the planters—God they're really beautiful—and I'd nearly finished making the appetizer when he came in. I just bubbled along for a while. Let me get you some wine, why don't you relax? God! What a moron. Then it got through, loud and clear. He . . . picked up the spare keys, and put them in his pocket."
"That's cold," Laurel said with quiet fury. "That's fucking cold."
"His keys," Emma stated. "His right. So I told him what I thought, what I felt, and that I was finished trying _not_ to want and _not_ to feel. I told him I was in love with him. And all he could really say to that is to give him a minute to think."
"There's your moron."
Emma nearly managed a smile at Mac's disgusted tone. "I got the 'caught him off guard, wasn't expecting.' Even the 'caught me at a bad time.' "
"Oh my God."
"That was before I told him I was in love with him, but it doesn't matter. So I ended it, and I walked out. It hurts. I think it's going to hurt a really long time."
"He called," Mac told her.
"I don't want to talk to him."
"Figured that. He wanted to make sure you were here, that you got home. I'm not taking his side, believe me, but he sounded pretty shaken up."
"I don't care. I don't want to care. If I forgive him now, if I go back—settle for what he can give me—I'll lose myself. I have to get over him first." She curled up again. "I just need to get over him. I don't want to see him or talk to him until I do. Or at least until I feel stronger."
"Then you won't. I'm going to reschedule your consults for tomorrow."
"Oh, Parker—"
"You need a day off."
"To wallow?"
"Yes. Now you need a long, hot bath, and we're going to heat up that soup. Then after your second cry—there will be another."
"Yeah." Emma sighed. "There will."
"After that, we're going to tuck you into bed. You're going to sleep until you wake up."
"I'm still going to be in love with him when I wake up."
"Yes," Parker agreed.
"And it's still going to hurt."
"Yes."
"But I'll be a little bit stronger."
"You will."
"I'll fix the bath. I have a formula." Mac rose, then leaned over and kissed Emma's cheek. "We're all here."
"I'll take care of the soup, and I'll ask Mrs. Grady to make a batch of her fabulous french fries. I know it's a cliché." Laurel gave Emma's leg another squeeze. "But it's a cliché for a reason."
"Thanks." She closed her eyes, reached for Parker's hand when they were alone. "I knew you'd be here."
"Always."
"Oh, God. Parker. Oh, God, here comes the second one now."
"It's okay," Parker soothed, and rubbed Emma's back as she wept. "It's okay."
_W_ HILE EMMA WEPT, JACK KNOCKED ON DEL'S DOOR. HE HAD TO do something or he'd drive over to Emma's. If she hadn't made it clear he wasn't wanted—and she had—Mac had made it double.
Del pulled open the door. "What's up? Jesus, Jack, you look like shit."
"It goes with how I feel."
Del's brow creased. "Oh man, if you're coming over here to cry in your beer over a fight with Emma—"
"It wasn't a fight. Not . . . just a fight."
Del took a harder look, stepped back. "Let's have a beer."
Jack shut the door behind him, then noticed Del's jacket and tie. "You're going out?"
"I was heading that way in a while. Get the beer. I have to make a call."
"I should say it's no big deal, it can wait. But I'm not going to."
"Get the beer. I'll be out in a minute."
Jack got two beers and went out on the back deck. But instead of sitting he walked to the rail and stared out at the dark. He tried to remember if he'd ever felt this bad before. He decided other than waking up in the hospital with a concussion, a broken arm, and a couple of cracked ribs after a car wreck, the answer was no.
And even then, the seriously bad had been only physical.
No, he thought, he remembered feeling like this before, nearly exactly like this. Sick and baffled and confused. When his parents had sat him down, so civilized, to tell him they were getting a divorce.
You're not to blame, they'd told him. We still love you, and always will. But . . .
In that moment his world had turned upside down. So why was this worse somehow? Why was it worse to realize that Emma could and would walk away from him? Could and would, he thought, because he'd made her feel _less_ when he should have done everything in his power to make her feel _more_.
He heard the door open. "Thanks," he said as Del came out. "Really."
"I should say it's no big deal, but I'm not going to."
Jack managed a weak laugh. "God, Del, I fucked up. I fucked it up and I'm not even sure exactly how. But what I know is I hurt her. I really hurt her, so you're welcome to kick my ass as promised. But you'll have to wait until I'm finished doing it."
"I can wait."
"She said she's in love with me."
Del took a pull on his beer. "You're not an idiot, Jack. Are you going to stand there and tell me you didn't know?"
"Not completely, or altogether. It's all just happened, and . . . No, I'm not an idiot, and I knew we were heading toward something. That. But then there's this leap, and I'm flat-footed. Can't keep up, can't figure out how to deal with it, or what to say, and she's so hurt, so hurt and pissed off she won't give me a chance. She hardly ever gets mad. You know how she is. She hardly ever blows, and when she does, you don't have a prayer."
"Why did she blow?"
He went back for the beer, but still didn't sit. "I had a pisser of a day, Del. I'm talking the kind of day that makes hell look like Disney World. I'm filthy and pissed off and have a motherfucker of a stress headache. I pull up, and she's there. In the house."
"I didn't know you gave her a key. Major step for you, Cooke."
"I didn't. I hadn't. She got it from Michelle."
"Uh-oh. Infiltrated the front lines, did she?"
Jack stopped, stared. "Is that how I am? Come on."
"It's exactly how you are, with women."
"And that makes me, what, a monster, a psycho?"
Del hitched a hip onto the deck rail. "No, a little phobic, maybe. So?"
"So, I'm filthy and my mood matches it, and she's there. She's made these pots for the deck. What are you laughing at?"
"Just imagining your shock and dismay."
"Well, Jesus, she's cooking, and there's flowers, and the music's blasting, and my head's screaming. God, if I could rewind it, I would. I would. I'd never hurt her."
"I know."
"She's hurt and pissed because . . . I'm being a prick. No question, but instead of having a fight, maybe yelling at each other for a while, clearing the air, it turns." Because the headache wanted to bully its way back, Jack rubbed the cold bottle over his temple. "It turns and dives south. It's how I don't trust her, and she's not welcome in my house. How she's not going to settle. She's in love with me, and she wants . . ."
"What does she want?"
"What do you think? Marriage, kids, the whole ball. I'm trying to keep up, trying to keep my head from just blasting off my shoulders and _think_ , but she won't give me time. She won't let me deal with what she just said. She's done with me, with us. I broke her heart. She cried. She's crying."
Her face flashed back into his mind until he was sick with regret. "I just want her to sit down, to wait a minute, and sit down. Just until I can get my breath, until I can think. She won't. She told me to stay away from her. I'd rather she'd shot me than look at me the way she did when she told me to stay away from her."
"Is that it?" Del asked after a moment.
"That's not enough?"
"I asked you once before, and you didn't answer. I'll ask you again. Yes or no this time. Are you in love with her?"
"Okay." He took a long drink of beer. "Yes. I guess it took an ass-kicking to shake it out of me, but yes. I'm in love with her. But—"
"Do you want to fix it?"
"I just said I was in love with her. Why wouldn't I want to fix it?"
"You want to know how?"
"Goddamn it, Del." He drank again. "Yes, since you're so fucking smart. How do I fix it?"
"Crawl."
Jack blew out a breath. "I can do that."
CHAPTER TWENTY
_H_ E STARTED CRAWLING IN THE MORNING. HE HAD THE SPEECH he'd edited, revised, and expanded most of the night in his head. The trick, as far as he could tell, would be getting her to listen to him.
She'd listen, he told himself as he turned into the Brown Estate. She was Emma. No one was more kind, more open-hearted, than Emma—and wasn't that only one of the dozens of reasons he loved her?
He'd been an idiot, but she'd forgive him. She had to forgive him because . . . she was Emma.
Still his stomach clutched when he saw her car parked at the main house. She hadn't gone home.
He wouldn't just be facing her, he thought with genuine, back-sweating fear, but all of them. The four of them, with Mrs. Grady for backup.
They'd roast his balls.
He deserved it, no question. But, dear God, did it have to be the four of them? Fucking A.
"Strap it on, Cooke," he muttered, and got out of the truck.
As he walked to the door, he wondered if the condemned walking the last mile experienced this same feeling of doom and dull terror.
"Get a grip, get a freaking grip. They can't kill you."
Maim possibly, verbally assault most definitely. But they couldn't kill him.
He started to open the door out of habit, then realized as a persona non grata he wasn't entitled. He rang the bell.
He thought he could get around Mrs. G. She liked him—really liked him. He could throw himself on her mercy, then . . .
Parker answered. No one, he thought, absolutely no one got around Parker Brown.
"Uh," he said.
"Hello, Jack."
"I want—need—to see Emma. To apologize for . . . everything. If I could talk to her for a few minutes and—"
"No."
Such a small word, he thought, so coolly delivered. "Parker, I just want to—"
"No, Jack. She's sleeping."
"I can come back, or wait, or—"
"No."
"Is that all you're going to say to me? Just no?"
"No," she said again without any hint of irony or humor. "It's not all we're going to say."
Mac and Laurel stepped up behind her. As battle plans went, he had to admit it was superior. No choice but surrender.
"Whatever you're going to say, I deserve. You want me to say I was wrong? I was wrong. That I was an idiot? I was. That—"
"I was thinking more along the lines of selfish prick," Laurel commented.
"That, too. Maybe there were reasons, maybe there were circumstances, but they don't matter. Certainly not to you."
"They really don't." Mac eased forward a step. "Not when you hurt the best person we know."
"I can't fix it, I can't make up for it if you don't let me talk to her."
"She doesn't want to talk to you. She doesn't want to see you," Parker said. "Not now. I can't say I'm sorry you're hurting, too. I can see you are, but I can't say I'm sorry for it. Not now. Now, this is about Emma, not about you. She needs time, and she needs you to leave her alone. So that's what you're going to do."
"For how long?"
"As long as it takes."
"Parker, if you'd just listen—"
"No."
As he stared at her, Carter started down the hall from the kitchen. Carter shot him one brief, sympathetic look, then turned around and walked back again.
So much for male solidarity.
"You can't just close the door."
"I can, and I will. But I'll give you something first, because I love you, Jack."
"Oh God, Parker." Why not just roast his balls? he thought. It couldn't be more painful.
"I love you. You're not just _like_ a brother to me, you _are_ a brother to me. To us. So, I'll give you something. I'll forgive you eventually."
"I'm not on board with that," Laurel told him. "I have reservations."
"I'll forgive you," Parker continued, "and we'll be friends again. But more importantly, Emma will forgive you. She'll find a way. Until she does, until she's ready, you're going to leave her alone. You're not going to call her, or contact her, or try to see her. We're not going to tell her you came here this morning, unless she asks. We won't lie to her."
"You can't come here, Jack." The slightest hint of sympathy eked into Mac's voice. "If there's any problem or question with the work on the studio, we'll handle it by phone. But you can't come here until Emma's okay with it."
"How are you supposed to know when that is?" he demanded. "Is she just going to say, 'Hey, I'm okay if Jack comes around'?"
"We'll know," Laurel said simply.
"If you care about her, you'll give her all the time she needs. I need your word."
He dragged a hand through his hair as Parker waited. "All right. You, all of you, know her better than anyone. You say this is what she needs, okay, it's what she needs. You've got my word I'll leave her alone until . . . until."
"And, Jack," Parker added, "you'll take that time for yourself, too. Time to think about what you really want, really need. I want your word on one more thing."
"Want me to sign in blood?"
"A promise will do. When she's ready, I'll call you. I'll do that for you—and for her—but only if you promise to come here and talk to me before you go to her."
"All right. I promise. Can you just get in touch once in a while, let me know how she is? What she's—"
"No. Good-bye, Jack." Parker closed the door, quietly, in his face.
On the other side of the door, Mac heaved out a breath. "It's not being disloyal to say I have to feel a little bit sorry for him. I know what it's like to be a complete jerk about this kind of thing. Having someone love you and being an ass."
Laurel nodded. "Yeah, you do. Take a minute to feel a little bit sorry for him." She waited, glanced at her watch. "Done?"
"Yeah, pretty much."
"I guess I'll take a minute, too, because the guy looked rough." Laurel glanced toward the steps. "But she's had it rougher. We should go check on her."
"I will. I think we need to stick to routine as much as we can," Parker added. "She'll only feel worse if things get too backed up, if it affects the business. So for now, we work—and if we do get backed up or hit snags, let's try to keep her out of it until she's steadier."
"If we need an extra hand with anything, we can ask Carter. My guy is the best."
"Do you ever get tired of bragging about that?" Laurel asked Mac.
Mac considered. "Really don't." She slung an arm around Laurel's shoulders. "I guess that's why I feel a little bit for Jack, and a whole lot for Emma. Love can really screw you up before you figure out how to live with it. And once you do? You wonder how the hell you ever lived without it. I think I need to go give Carter a real kick-in-the-ass kiss. I'll check back in this afternoon," Mac added as she started toward the kitchen. "Call if she needs me sooner."
" 'Love can really screw you up before you figure out how to live with it.' " Laurel pursed her lips. "You know, we could put that on the web page."
"It has a ring."
"She's right about Carter. He's the best. But that man is not coming in my kitchen when I'm working. I don't want to have to hurt him, Parker. Let me know if Em needs another shoulder, or you need a soldier on the front line of the bride wars."
With a nod, Parker started up the steps.
_U_ PSTAIRS, EMMA ORDERED HERSELF TO GET OUT OF BED, TO stop lying there feeling sorry for herself. Instead, she hugged a pillow close and stared at the ceiling.
Her friends had drawn the curtains over the windows so the room would stay dark and quiet. They'd tucked her in like an invalid, with extra pillows, a vase of freesia on the nightstand. They'd sat with her until she'd slept.
She should be ashamed, she told herself. Ashamed of being so needy, so weak. But she could only be grateful they'd been there, they'd understood what she'd needed.
But now it was another day. She needed to move on, needed to deal with reality. Broken hearts healed. Maybe the cracks were always there, like thin scars, but they healed. People lived and worked, laughed and ate, walked and talked with those cracks.
For many, even the scars healed and they loved again.
But how many of those people had the one who'd broken their heart so entrenched in their life that they had to see him over and over again? For how many was that person like a thread that was so woven into the tapestry of their every day that to pull it out meant everything else unraveled?
She didn't have the option of shutting Jack outside the structure of her life. Of not seeing him again, or only seeing him at specified times.
That was why office romances were so fraught with pitfalls, she decided. When they went bad, you had to face the pain every day. Nine to five, five days a week. Or you quit, you transferred, moved to another city. You escaped so you could heal and go on.
Not an option for her because . . .
Jamaica. Adele's offer.
Not just another office, another city, but another country. A completely fresh start. She could continue to do the work she loved, but be a new person. No complicated relationships, no interwoven ties. No Jack to face whenever he dropped by the house, or whenever they happened to be in the market at the same time. Invited to the same party.
No looks of sympathy from the scores of people who'd know she had those cracks on her heart.
She could do good work. All those tropical flowers. A perpetual spring and summer. A little house on the beach, maybe, where she could listen to the waves every night.
Alone.
She shifted when she heard the door ease open.
"I'm awake."
"Coffee." Parker crossed to the bed, offered the cup and saucer. "I brought it just in case."
"Thanks. Thanks, Parker."
"How about some breakfast?" Moving briskly now, Parker walked over to open the drapes, let in the light.
"Just not hungry."
"Okay." Parker sat on the side of the bed, brushed the hair back from Emma's cheek. "Did you sleep?"
"I did, actually. I guess it was an escape route, and I took it. I feel sort of musty and dull now. And stupid. I'm not suffering from some fatal disease. I don't have broken bones or internal bleeding. No one died, for God's sake. And I can't even talk myself into getting out of bed."
"It's been less than a day."
"You're going to tell me to give myself time. It'll get better."
"It will. Some people say divorce can be like death. I think that's true. And I think something like this, when the love is so big, so deep, it's the same." Parker's eyes, warm and blue, radiated sympathy. "There has to be grief."
"Why can't I just be mad? Why can't I just be pissed off? The son of a bitch, the bastard, whatever. Can't I skip off the grief part and just hate him? We can all go out, get drunk, and trash him?"
"Not you, Emma. If I thought you could do it, if I thought it would help, we'd blow off the day, get drunk, and start the trashing right now."
"You would." Finding a smile, finally, Emma sat back against the pillows and studied her friend's face. "You know what I was lying here in my ocean of self-pity thinking right before you came in?"
"What?"
"That I should take Adele's offer. I could go to Jamaica, relocate, help her launch her business. I'd be good at it. I know how to set it up, handle the reins. Or at least find the right people to handle the various reins. It would be a fresh start for me, and I could make it work. I could make it shine."
"You could." Rising, Parker walked to the window again, adjusted the curtains. "It's a big decision to make, especially when you're in emotional upheaval."
"I've been asking myself how, for God's sake, how can I deal with seeing Jack all the time? Here, in town, at events. He's invited to one of our events every month or so. We all know so many of the same people, our lives are so interlinked. Even when I get to the point where I can think about him, about us, without . . ."
She had to pause, dig for control. "Without wanting to cry, how can I handle all of that? I knew it could be this way, I knew it going in, but . . ."
"But." Parker nodded, turned back.
"So I was lying here imagining taking the offer, starting fresh, building something new. The beach, the weather, a new challenge to focus on. I considered it for about five minutes. No, probably closer to three. This is home, this is family, this is you, this is us. This is me. So I'll have to figure out how to deal with it."
"I can be really pissed off at him for bringing you to the point you'd have considered that for even three minutes."
"But if I'd decided it was best for me, you'd have let me go."
"I'd have tried to talk you out of it. I'd have done spreadsheets, bullet points, graphs, charts, and many, many lists. With a DVD."
Tears spilled over again. "I love you so much, Parker."
Parker sat again, wrapped her arms around Emma and held tight.
"I'm going to get up, take a shower, get dressed. I'm going to start figuring out how to deal with it."
"Okay."
_S_ HE GOT THROUGH THE DAY, AND THE NEXT. SHE BUILT ARRANGEMENTS, created bouquets, met with clients. She cried, and when her mother came by to be with her, she cried some more. But she dried the tears, and got through the day.
She dealt with crises, managed to handle her team's spoken and unspoken sympathy when they dressed an event. She watched brides carrying her flowers walk to the men they loved.
She lived and worked, laughed and ate, walked and talked.
Even though there was a void inside her nothing seemed to fill, she forgave him.
She came into the midweek briefing a few minutes late. "Sorry. I wanted to wait for the delivery for Friday night's event. I've got Tiffany processing, but I wanted to check the callas. We'll be using a lot of Green Goddess and I wanted to check the tone with the orchids before she started."
She went to the sideboard, chose a Diet Pepsi. "What did I miss?"
"Nothing yet. Actually, you can start," Parker told her. "Since Friday's our biggest event this week, and the flowers just arrived. Any problems?"
"With the flowers, no. Everything came in, and looks good. The bride wanted ultracontemporary, with a touch of funk. Green calla lilies, the cymbidiums—which are very cool in a yellow-green shade—with some white Eucharist lilies to pop the colors, in a hand-tied bouquet. Her ten, yes ten, attendants will carry three hand-tied Green Goddess callas. Small bouquet of Eucharist lilies, and a hair clip of orchids for the flower girl. Rather than corsages or tussy-mussies, the MOB and MOG will each carry a single orchid. Vases for all will be on the tables at dinner and reception."
Emma scrolled down on her laptop. "We have the Green Goddess again for the entrance urns, with horsetail bamboo, the orchids, trails of hanging amaranthus and . . ."
She tipped the top of the computer down. "I need to step out of business mode for a few minutes. First just to say I love you, and I don't know what I'd have done without all of you the past week or so. You must've gotten sick of me moping and whining at first—"
"I did." Laurel rose her hand, waved it, and made Emma laugh. "Actually, your moping is substandard and your whining needs considerable work. I hope you'll do better in the future."
"I can only strive. Meanwhile, I'm done. I'm okay. I have to assume, since Jack hasn't dropped by, hasn't tried to call me, or e-mail or send up a smoke signal, you warned him off."
"Yes," Parker confirmed, "we did."
"Thanks for that, too. I needed the time and distance to work the whole thing out and, well, level off. Since I haven't seen a sign of Del either, I'm going to assume you asked him to steer clear for a while."
"It seemed better all around," Mac said.
"You're probably right. But the fact is we're all friends. We're family. We've got to get back to being those things. So if you've worked out an all-clear signal, you can send it. Jack and I can clear the air, if it needs to be cleared, and we can all get back to normal."
"If you're sure you're ready."
She nodded at Parker. "Yes, I'm sure. So, moving to the foyer," she began.
_J_ ACK SLID INTO A BOOTH AT COFFEE TALK. "THANKS FOR MEETING me, Carter."
"I feel like a spy. Like a double agent." Carter considered his green tea. "I kind of like it."
"So, how's she doing? What's she doing? What's going on? Anything, Carter, just anything. It's been ten days. I can't talk to her, see her, text her, e-mail her. How long am I supposed to . . ." He trailed off, frowned. "Is that me?"
"Yeah, that's you."
"Jesus Christ, I can't stand to be around myself." He glanced up at the waitress. "Morphine. A double."
"Ha-ha," she said.
"Try the tea," Carter suggested.
"I'm not quite that bad. Yet. Coffee, regular. How is she, Carter?"
"She's okay. There's a lot of work right now. June is . . . It's insane, actually. She's putting in a lot of hours. They all are. And she spends a lot of time at home. One of them usually goes over, at least for a while, in the evenings. Her mother came over, and I know that was pretty emotional. Mac told me. That's the double-agent part. Emma doesn't talk about any of this with me. I'm not the enemy, exactly, but . . ."
"I get it. I haven't gone by the bookstore either because I don't think Lucia wants to see me. I feel like I should be wearing a sign."
Caught between annoyance and misery, Jack slumped back in his seat. "Del can't go over there either. Parker decree. God, it's not like I cheated on her or smacked her around or . . . And yes, I'm trying to justify. How can I tell her I'm sorry if I can't talk to her?"
"You can practice what you're going to say when you can say it."
"I've been doing a lot of that. Is it like this for you, Carter?"
"Actually, I'm allowed to talk to Mac."
"I meant—"
"I know. Yes, it's like that. She's the light. Before, you can fumble around in the dark, or manage in the dim. You don't even know it's dim because that's the way it's always been. But then, she's the light. Everything changes."
"If the light shuts off, or worse, if you're stupid enough to shut it off yourself, it's a hell of a lot darker than it was before."
Carter shifted forward. "I think, to get the light back, you have to give her a reason. What you say is one part, but what you do, that's the big one. I think."
Jack nodded, then pulled out his phone when it signaled. "It's Parker. Okay. Okay. Yeah?" he said when he answered. "Is she—What? Sorry. Okay. Thanks. Parker—Okay. I'll be there."
He closed the phone. "They opened the door. I have to go, Carter. There are things I need to—"
"Go ahead. I'll get this."
"Thanks. God, I feel a little sick. You could wish me a whole shitload of luck."
"A whole shitload of luck, Jack."
"I think I'll need it." He shoved out, strode quickly to the door.
Jack arrived at the main house at exactly the time Parker specified. He didn't want to piss her off. Twilight fell softly, sweet with the perfume of flowers. His palms were sweaty.
For the second time in more years than he could count, he rang the bell.
She answered. The gray suit, and the smooth roll of hair at the nape of her neck told him she hadn't changed from work mode. One look at her—so neat, so fresh, so lovely, made him realize how much he'd missed her.
"Hello, Parker."
"Come in, Jack."
"I wondered if I'd ever hear you say that again."
"She's ready to talk to you, so I'm ready to let you talk to her."
"Are you and I never going to be friends again?"
She looked at him, then cupped his face, kissed him lightly. "You look terrible. That goes in your favor."
"Before I talk to Emma, I want to tell you, it would've killed me to lose you. You, Laurel, Mac. It would've killed me."
This time she put her arms around him, let him hold on. "Family forgives." She gave him a squeeze before stepping back.
"What choice do we have? I'm going to give you two options, Jack, and you'll pick when you go to Emma. The first. If you don't love her—"
"Parker, I—"
"No, you don't tell me. If you don't love her, if you can't give her what she needs and wants—not just for her, but for yourself—make it a clean break. She's already forgiven you, and she'll accept it. Don't promise her what you can't give or don't want. She'd never get over that, and you'll never be happy. Second option. If you love her, if you can give her what she needs and wants—not just for her, but for yourself—I can tell you what to do, what will make the difference."
"Then tell me."
_S_ HE WORKED LATE AND ALONE, AS SHE DID MOST NIGHTS NOW. That would have to stop soon, Emma thought. She missed people, conversations, movement. She was nearly ready to step outside the safety zone again. Clear the air, she decided, say what she had to say, then get back to being Emma.
She missed Emma, too, she realized.
She took the finished work to the cooler, then came back to clean her station.
The knock stopped her. She knew before she walked out it would be Jack. No one was more efficient than Parker.
He held an armload of bold red dahlias—and her heart twisted.
"Hello, Jack."
"Emma." He let out a breath. "Emma," he said again. "I realize it's shallow. Bringing flowers to clear the way, but—"
"They're beautiful. Thank you. Come on in."
"There's so much I want to say."
"I need to put these in water." She turned, went into the kitchen for a vase, a jug of the food she kept mixed, her snips. "I understand there are things you want to say, but there are things I need to say first."
"All right."
She began to clip the stems under water. "First, I want to apologize."
"Don't." Temper licked around the edges of his tone. "Don't do that."
"I'm going to apologize for the way I acted, for what I said. First, because when I got over myself I realized you were exhausted, upset, not feeling well, and I had—very deliberately—crossed a line."
"I don't want a damn apology."
"You're getting one, so deal with it. I was angry because you didn't give me what I wanted." She arranged the flowers, stem by stem. "I should've respected your boundaries; I didn't. You were unkind, so that's on you, but I pushed. That's on me. But the biggest issue here is we promised each other we'd stay friends, and I didn't keep that promise. I broke my word, and I'm sorry."
She looked at him now. "I'm so sorry for that, Jack."
"Fine. Are you done?"
"Not quite. I'm still your friend. I just needed some time to get back to that. It's important to me that we're still friends."
"Emma." He started to lay his hand on hers on the counter, but she slid it away, fussed with the flowers.
"These really are beautiful. Where'd you get them?"
"Your wholesaler. I called and begged, and told them they were for you."
She smiled, but kept her hand out of reach. "There. How can we not be friends when you'd think to do something like that? I don't want any hard feelings between us. We still care about each other. We'll just put the rest behind us."
"That's what you want?"
"Yes, it's what I want."
"Okay then. I guess we get to talk about what I want now. Let's take a walk. I want some air to start with."
"Sure." Proud of herself, she put away her snips, her jug.
The minute they stepped outside, she put her hands in her pockets. She could do this, she thought. She was doing it, and doing it well. But she couldn't if he touched her. She wasn't ready for that, not yet.
"That night," he began, "I was exhausted and pissed off, and all the rest. But you weren't wrong in the things you said. I didn't realize it, about myself. Not really. That I put those shields up or restrictions on. I've thought about that since, about why. The best I can figure is how when my parents split, and I'd stay with my father, there'd be stuff—from other women. In the bathroom, or around. It bothered me. They were split, but . . ."
"They were your parents. Of course it bothered you."
"I never got over the divorce."
"Oh, Jack."
"Another cliché, but there it is. I was a kid, and oblivious, then suddenly . . . They loved each other once, were happy. Then they didn't and they weren't."
"It's never that easy, that cut and dried."
"That's logic and reason. It's not what I felt. It's come home to me recently that they were able to behave civilly, able to make good, happy lives separately without waging war or making me a casualty. And I took that and turned it on its head. Don't make promises, don't build a future because feelings can change and they can end."
"They can. You're not wrong, but—"
"But," he interrupted. "Let me say it. Let me say it to you. But if you can't trust yourself and your own feelings, and you can't take a chance on that, what's the damn point? It's a leap, and I figure if you take that leap, if you say this is it, you have to mean it. You'd better be sure because it's not just you. It's not just for now. You have to believe to make the leap."
"You're right. I understand better now why things . . . Well, why."
"Maybe we both do. I'm sorry I made you feel unwelcome. Sorry you now feel you crossed a line by trying to do something for me. Something I should've appreciated. Do appreciate," he corrected. "I've been watering the planters."
"That's good."
"You were . . . God, I've missed you so much. I can't think of all the things I've worked out to say, practiced saying. I can't think because I'm looking at you, Emma. You were right. I didn't value you enough. Give me another chance. Please, give me another chance."
"Jack, we can't go back and—"
"Not back, forward." He took her arm then, shifted so they were face-to-face. "Forward. Emma, have some pity. Give me another chance. I don't want anyone but you. I need your . . . light," he said remembering Carter's word. "I need your heart and your laugh. Your body, your brain. Don't shut me out, Emma."
"Starting from here, when we both want—both need—different things . . . It wouldn't be right for either of us. I can't do it."
When her eyes filled, he drew her in.
"Let me do it. Let me take the leap. Emma, because with you, I believe. With you, it's not just now. It's tomorrow and whatever comes with it. I love you. I love you."
When the first tear spilled, he moved with her. "I love you. I'm so in love with you that I didn't see it. I couldn't see it because it's everything. You're everything. Stay with me, Emma, be with me."
"I am with you. I want . . . What are you doing?"
"I'm dancing with you." He brought the hand he held to his lips. "In the garden, in the moonlight."
Her heart shuddered, swelled. And all the cracks filled. "Jack."
"And I'm telling you I love you. I'm asking you to make a life with me." He kissed her while they circled, swayed. "I'm asking you to give me what I need, what I want even though it took me too much time to figure it out. I'm asking you to marry me."
"Marry you?"
"Marry me." The leap was so easy, the landing smooth and right. "Live with me. Wake up with me, plant flowers for me that you'll probably have to remind me to water. We'll make plans, and change them as we go. We'll make a future. I'll give you everything I've got, and if you need more, I'll find it and give it to you."
She heard her own words come back to her in the perfumed air, under the moonlight while the man she loved turned her in a waltz.
"I think you just did. You just gave me a dream."
"Say yes."
"You're sure?"
"How well do you know me?"
Smiling, she blinked away tears. "Pretty well."
"Would I ask you to marry me if I wasn't sure?"
"No. No, you wouldn't. How well do you know me, Jack?"
"Pretty well."
She brought her lips to his, lingered through the joy. "Then you know my answer."
_O_ N THE THIRD FLOOR TERRACE, THE THREE WOMEN STOOD watching, their arms around each other's waists. Behind them, Mrs. Grady sighed.
When Mac sniffled, Parker reached in her pocket for a pack of tissues. She handed one to Mac, to Laurel, to Mrs. Grady, then took one for herself.
"It's beautiful," Mac managed. "They're beautiful. Look at the light, the silver cast to the light, and the shadows of the flowers, the gleam of them, and the silhouette Emma and Jack make."
"You're thinking in pictures." Laurel wiped her eyes. "That's serious romance there."
"Not just pictures. Moments. That's Emma's moment. Her blue butterfly. We probably shouldn't be watching. If they see us, it'll spoil it."
"They can't see anything but each other." Parker took Mac's hand, then Laurel's, and smiled when she felt Mrs. Grady's rest on her shoulder.
The moment was just as it should be.
So they watched as Emma danced in the soft June night, in the moonlight, in the garden, with the man she loved.
• • •
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SAVOR _the_ MOMENT
COMING IN MAY 2010
FROM BERKLEY BOOKS
# PROLOGUE
_A_ S THE CLOCK TICKED DOWN ON HER SENIOR YEAR IN HIGH school, Laurel McBane learned one indisputable fact.
Prom was hell.
For weeks all anyone wanted to talk about was who might ask who, who did ask who—and who asked some other who, thereby inciting misery and hysteria.
Girls, to her mind, suffered an agony of suspense and an embarrassing passivity during prom season. The halls, classrooms, and quad throbbed with emotion running the gamut from giddy euphoric—because some guy asked them to some overhyped dance—to bitter tears—because some guy didn't.
The entire cycle revolved around "some guy," a condition she believed both stupid and demoralizing.
And after that, the hysteria continued, even escalated, with the hunt for a dress, for shoes; the intense debate about updos versus down-dos. Limos, after parties, hotel suites—the yes, no, maybe of sex.
She would have skipped the whole thing if her friends, especially Parker Right-of-Passage Brown, hadn't ganged up on her.
Now her savings account—all those hard-earned dollars and cents from countless hours waiting tables—reeled in shock at the withdrawals for a dress she'd probably never wear again, for the shoes, the bag, and all the rest.
She could lay all that on her friends' heads, too. She'd gotten caught up shopping with Parker, Emmaline, and Mackensie, and spent more than she should have.
The idea, gently broached by Emma, of asking her parents to spring for the dress wasn't an option, not to Laurel's mind. A point of pride, maybe, but money in the McBane household had become a very sore subject since her father's dicey investments fiasco and the little matter of the IRS audit.
No way she'd ask either of them. She earned her own money, and had for several years now.
She told herself it didn't matter. She didn't have close to enough saved for the tuition for the Culinary Institute, or for the living expenses in New York, despite the hours she'd put in after school and on weekends at the restaurant. The cost of looking great for one night didn't change that one way or the other and—and what the hell, she did look great.
She fixed on her earrings while across the room—Parker's bedroom—Parker and Emma experimented with ways to prom-up the hair Mac had impulsively hacked off to resemble what Laurel thought of as Julius Caesar takes the Rubicon. They tried various pins, sparkle dust, and jeweled clips in what was left of Mac's flame-red hair while the three of them talked nonstop, and Aerosmith rocked out of the CD player.
She liked listening to them like this, when she was a little bit apart. Maybe especially now, when she _felt_ a little bit apart. They'd been friends all their lives, and now, rite of passage or not, things were changing. In the fall Parker and Emma would head off to college. Mac would be working and squeezing in a few courses on photography.
And with the dream of the Culinary Institute poofed due to finances and her parents' most recent marital implosion, she'd settle for community college part-time. Business courses, she supposed. She'd have to be practical. Realistic.
And she wasn't going to think about it now. She might as well enjoy the moment, and this ritual Parker, in her Parker way, had arranged.
Parker and Emma might be going to prom at the Academy while she and Mac went to theirs at the public high school, but they had this time together, getting dressed and made-up. Downstairs Parker's and Emma's parents hung out, and there'd be dozens of pictures, and "oh, look at our girls!" hugs, and probably some shiny eyes.
Mac's mother was too self-involved to care about her daughter's senior prom, which, Linda being Linda, could only be a good thing. And her own parents? Well, they were too steeped in their own lives, their own problems, for it to matter where she was or what she did tonight.
She was used to it. Had even come to prefer it.
"Just the fairy dust sparkles," Mac decided, tipping her head from side to side to judge. "It's kind of Tinkerbelly. In a cool way."
"I think you're right." Parker, her straight-as-rain brown hair a glossy waterfall down her back, nodded. "It's waif with an edge. What do you think, Em?"
"I think we need to play up the eyes more, go dramatic." Emma's eyes, a deep, dreamy brown, narrowed in thought. "I can do this."
"Have at it." Mac shrugged. "But don't take forever, okay? I still have to set up for our group shot."
"We're on schedule." Parker checked her watch. "We've still got thirty minutes before . . ." She turned, caught sight of Laurel. "Hey. You look awesome!"
"Oh, you really do!" Emma clapped her hands together. "I _knew_ that was the dress. The shimmery pink makes your eyes even bluer."
"I guess."
"Need one more thing." Parker hurried to her dresser, opened a drawer on her jewelry box. "This hair clip."
Laurel, a slim girl in shimmery pink, her sun-shot hair done—at Emma's insistence—in long, loose sausage curls, shrugged. "Whatever."
Parker held it against Laurel's hair at different angles. "Cheer up," she ordered. "You're going to have fun."
God, get over yourself, Laurel! "I know. Sorry. It'd be more fun if the four of us were going to the same dance, especially since we all look seriously awesome."
"Yeah, it would." Parker decided to draw some of the curls from the sides to clip them in the back. "But we'll meet up after and party. When we're done we'll come back here and tell each other everything. Here, take a look."
She turned Laurel to the mirror, and the girls studied themselves and each other.
"I do look great," Laurel said and made Parker laugh.
After the most perfunctory of knocks, the door opened. Mrs. Grady, the Browns' longtime housekeeper, put her hands on her hips to take a survey.
"You'll do," she said, "which you should after all this fuss. Finish up with it and get yourselves downstairs for pictures. You." She pointed a finger at Laurel. "I need a word with you, young lady."
"What did I do?" Laurel demanded, looking from friend to friend as Mrs. Grady strode away. "I didn't do anything." But since Mrs. G's word was law, Laurel rushed after her.
In the family sitting room, Mrs. G turned, arms folded. Lecture mode, Laurel thought as her heart tripped. And she cast her mind back looking for an infraction that might have earned her one from the woman who'd been more of a mother to her than her own through her teenage years.
"So," Mrs. Grady began as Laurel hurried in, "I guess you think you're all grown up now."
"I—"
"Well, you're not. But you're getting there. The four of you've been running around here since you were in diapers. Some of that's going to change, with all of you going your own ways. At least for a time. Birds tell me your way's to New York and that fancy baking school."
Her heart took another trip, then suffered the pinprick of a deflated dream. "No, I'm, ah, keeping my job at the restaurant and I'm going to try to take some courses at the—"
"No, you're not." Again, Mrs. G pointed a finger. "Now, a girl your age in New York City best be smart and best be careful. And from what I'm told, if you want to make it at that school you have to work hard. It's more than making pretty frostings and cookies."
"It's one of the best, but—"
"Then you'll be one of the best." Mrs. G reached into her pocket. She pulled out a check to Laurel. "That'll cover the first semester, the tuition, a decent place to live, and enough food to keep body and soul together. You make good use of it, girl, or you'll answer to me. If you do what I expect you're capable of, we'll talk about the next term when the time comes."
Stunned, Laurel stared at the check in her hand. "You can't—I can't—"
"I can and you will. That's that."
"But—"
"Didn't I just say that's that? If you let me down, there'll be hell to pay, I promise you. Parker and Emma are going off to college, and Mackensie's dead set on working full time with her photography. You've got a different path, so you'll take it. It's what you want, isn't it?"
"More than anything." Tears stung her eyes, burned her throat. "Mrs. G, I don't know what to say. I'll pay you back. I'll—"
"Damn right, you will. You'll pay me back by making something of yourself. It's up to you now."
Laurel threw her arms around Mrs. Grady, clung. "You won't be sorry. I'll make you proud."
"I believe you will. There now. Go finish getting ready."
Laurel held on another moment. "I'll never forget this," she whispered. "Never. Thank you. Thank you, thank you!"
She rushed for the door, anxious to share the news with her friends, then turned, young, radiant. "I can't wait to start."
# CHAPTER ONE
_A_ LONE, WITH NORAH JONES WHISPERING THROUGH THE IPOD, Laurel transformed a panel of fondant into a swatch of elegant, edible lace. She didn't hear the music, used it more to fill the air than as entertainment, while she painstakingly pieced the completed panel onto the second tier of four.
She stepped back to eye the results, to circle, to search for flaws. Vows' clients expected perfect, and that's exactly what she intended to deliver. Satisfied, she nodded, and picked up a bottle of water to sip while she stretched her back.
"Two down, two to go."
She glanced toward the board, where she'd pinned various samples of antique lace and the final sketched design for the cake Friday evening's bride had approved.
She had three more designs to complete—two for Saturday, one for Sunday—but that was nothing new. June at Vows, the wedding and event business she ran with her friends, was prime time.
In a handful of years, they'd turned an idea into a thriving enterprise. Sometimes just a little too thriving, she mused, which was why she was making fondant lace at nearly one in the morning.
It was a very good thing, she decided. She loved the work.
They all had their passions. Emma had the flowers, Mac the photography, Parker the details. And she had the cakes. And the pastries, she thought, and the chocolates. But the cakes stood as the crowning touch.
She got back to it, began to roll out the next panel. Following habit, she'd clipped her sunny blond hair up and back out of her way. Cornstarch dusted the baker's apron she wore over cotton pants and T-shirt, and the slide-on kitchen shoes kept her feet as comfortable as possible after hours of standing. Her hands, strong from years of kneading, rolling, lifting, were capable and quick. As she began the next pattern, her sharp-featured, angular face set in serious lines.
Perfection wasn't simply a goal when it came to her art. For Icing at Vows it was a necessity. The wedding cake was more than baking and piping, sugar paste and filling. Just as the wedding photos Mac took were more than pictures, and the arrangements and bouquets Emma created more than flowers. The details and schedules and wishes Parker put together were, in the end, bigger than the sum of their parts.
Together, the elements became a once-in-a-lifetime event, and the celebration of the journey two people chose to make together.
Romantic, certainly, and Laurel believed in romance. In theory, anyway. More, she believed in symbols and celebrations. And in a really fabulous cake.
Her expression softened into pleasure as she completed the third tier, and her deep blue eyes warmed as she glanced over to see Parker hovering in the doorway.
"Why aren't you in bed?"
"Details." Parker circled a finger over her own head. "Couldn't settle. How long have you been at this tonight?"
"Awhile. I need to finish it so it can set overnight. Plus I have the two Saturday cakes to assemble and decorate tomor row."
"Want company?"
They knew each other well enough that if Laurel said no, it was understood and there'd be no offense. And often, when deep in work, _no_ was the answer.
"Sure."
"I love the design." Parker, as Laurel had, circled the cake. "The delicacy of the white on white, the interest of the different heights of each tier—and the intricacy of each. They really do look like different panels of lace. Old-fashioned, vintage. That's our bride's theme. You've nailed it with this."
"We're going to do pale blue ribbon around the pedestal," Laurel said as she started on the next panel. "And Emma's going to scatter white rose petals at the base. It's going to be a winner."
"The bride's been good to work with."
Comfortable in her pajamas, her long brown hair loose rather than in its work mode of sleek tail or smooth chignon, Parker put on the kettle for tea. One of the perks of running the business out of her home, and of having Laurel living there—with Emma and Parker right on the estate as well—was these late-night visits.
"She knows her mind," Laurel commented, choosing a tool to scallop the edges of the panel. "But she's open to suggestion, and so far hasn't been insane. If she makes it through the next twenty-four that way, she'll definitely earn Vows' coveted Good Bride status."
"They looked happy and relaxed tonight at rehearsal, and that's a good sign."
"Mmm-hmm." Laurel continued the pattern with precisely placed eyelets and dots. "So, again, why aren't you in bed?"
Parker sighed as she heated a little teapot. "I think I was having a moment. I was unwinding with a glass of wine out on my terrace. I could see Mac's place, and Emma's. The lights were on in both houses, and I could smell the gardens. It was so quiet, so pretty. The lights went off—Emma's first, and a little while after, Mac's. I thought about how we're planning Mac's wedding, and that Emma just got engaged. And about all the times we played Wedding Day, the four of us, when we were kids. Now it's real. I sat there in the quiet, and the dark, and found myself wishing my parents could be here to see it. To see what we've done here, and who we are now. I got stuck"—she paused to measure out tea—"between being sad they're gone and being happy because I know they'd be proud of me. Of us."
"I think about them a lot. We all do." Laurel continued to work. "Because they were such an essential part of our lives, and because there are so many memories of them here. So I know what you mean by being stuck."
"They'd get a kick out of Mac and Carter, out of Emma and Jack, wouldn't they?"
"Yeah, they would. And what we've done here, Parker? It rocks. They'd get a kick out of that, too."
"I'm lucky you were up working." Parker poured hot water into the pot. "You've settled me down."
"Here to serve. I'll tell you who else is lucky, and that's Friday's Bride. Because, this cake?" She blew stray hair out of her eyes as she nodded smugly. "It kicks major ass. And when I do the crown, angels will weep with joy."
Parker set the pot aside to steep. "Really, Laurel, you need to take more pride in your work."
Laurel grinned. "Screw the tea. I'm nearly done here. Pour me a glass of wine."
_I_ N THE MORNING, AFTER A SOLID six HOURS OF SLEEP, LAUREL got in a quick session at the gym before dressing for the workday. She'd be chained to her kitchen for the bulk of it, but before that routine began, there was the summit meeting that prefaced every event.
Laurel dashed downstairs from her third-floor wing to the main level of the sprawling house, and back to the family kitchen, where Mrs. Grady put a fruit platter together.
"Morning, Mrs. G."
Mrs. Grady arched her eyebrows. "You look feisty."
"Feel feisty. Feel righteous." Laurel fisted both hands, flexed her muscles. "Want coffee. Much."
"Parker's taken the coffee up already. You can take this fruit, and the pastries. Eat some of that fruit. A day shouldn't start with a danish."
"Yes, ma'am. Anyone else here yet?"
"Not yet, but I saw Jack's truck leave a bit ago, and I expect Carter will be along giving me the puppy eyes in hopes of a decent breakfast."
"I'll get out of the way." Laurel grabbed the platters, balancing them with the expertise of the waitress she'd been once upon a time.
She carried them up to the library, which now served as Vows' conference room. Parker sat at the big table, with the coffee service on the breakfront. Her BlackBerry, as always, remained within easy reach. The sleek ponytail left her face unframed, and the crisp white shirt transmitted business mode as she sipped coffee and studied data on her laptop with midnight blue eyes Laurel knew missed nothing.
"Provisions," Laurel announced. She set the trays down, then tucked her chin-length swing of hair behind her ears before she obeyed Mrs. Grady and fixed herself a little bowl of berries. "Missed you in the gym this morning. What time did you get up?"
"Six, which was a good thing, since Saturday afternoon's bride called just after seven. Her father tripped over the cat and may have broken his nose."
"Uh-oh."
"She's worried about him, but nearly equally worried about how he's going to look for the wedding, and in the photographs. I'm going to call the makeup artist to see what she thinks can be done."
"Sorry about the FOB's bad luck, but if that's the biggest problem this weekend, we're in good shape."
Parker shot out a finger. "Don't jinx it."
Mac strolled in, long and lean in jeans and a black T-shirt. "Hello, pals of mine."
Laurel squinted at her friend's easy smile and slumberous green eyes. "You had morning sex."
"I had stupendous morning sex, thank you." Mac poured herself coffee, grabbed a muffin. "And you?"
"Bitch."
With a laugh, Mac dropped down in her chair, stretched out her legs. "I'll take my morning exercise over your treadmill and Bowflex."
"Mean, nasty bitch," Laurel said and popped a raspberry.
"I love summer, when the love of my life doesn't have to get up and out early to enlighten young minds." Mac opened her own laptop. "Now I'm primed, in all possible ways, for business."
"Saturday afternoon's FOB may have broken his nose," Parker told her.
"Bummer." Mac's brow creased. "I can do a lot with Photoshop if they want me to, but it's kind of a cheat. What is, is. And it makes an amusing memory, in my opinion."
"We'll see what the bride's opinion is once he gets back from the doctor." Parker glanced over as Emma rushed in.
"I'm not late. There's twenty seconds left." Black curls bouncing, she scooted to the coffee station. "I fell back to sleep. After."
"Oh, I hate you, too," Laurel muttered. "We need a new rule. No bragging about sex at business meetings when half of us aren't getting any."
"Seconded," Parker said immediately.
"Aww." Laughing, Emma scooped some fruit into a bowl.
"Saturday afternoon's FOB may have a broken nose."
"Aww," Emma repeated, with genuine concern at Mac's announcement.
"We'll deal with it when we have more details, but however it turns out, it really falls to Mac and me. I'll keep you updated," Parker said to Mac. "Tonight's event. All out-of-town attendants, relatives, and guests have arrived. The bride, the MOB, and the attendants are due here at three for hair and makeup. The MOG has her own salon date and is due by four, with the FOG. FOB will arrive with his daughter. We'll keep him happy and occupied until it's time for the formal shots that include him. Mac?"
"The bride's dress is a beaut. Vintage romance. I'll be playing that up."
As Mac outlined her plans and timetable, Laurel rose for a second cup of coffee. She made notes here and there, continued to do so when Emma took over. As the bulk of Laurel's job was complete, she'd fill in when and where she was needed.
It was a routine they'd perfected since Vows had gone from concept to reality.
"Laurel," Parker said.
"The cake's finished and it's a wowzer. It's heavy, so I'll need help from the subs transferring it to Reception, but the design doesn't require any on-site assembly. I'll need you to do the ribbon and white rose petals, Emma, once it's transferred, but that's it until it's time to serve. They opted against a groom's cake, and went for a selection of mini-pastries and heart-shaped chocolates. They're done, too, and we'll serve them on white china lined with lace doilies to mirror the design of the cake. The cake table linen is pale blue, eyelet lace. Cake knife and server, provided by the B & G. They were her grandmother's, so we'll keep our eye on them.
"I'm going to be working on Saturday's cakes most of today, but should be freed up by four if anyone needs me. During the last set, the subs will put leftover cake in the takeaway boxes and tie them with blue ribbon we've had engraved with the B & G's names and the date. Same goes if there are any leftover chocolates or pastries. Mac, I'd like a picture of the cake for my files. I haven't done this design before."
"Check."
"And Emma, I need the flowers for Saturday night's cake. Can you bring them to me when you come to dress today's event?"
"No problem."
"On the personal front?" Mac lifted a hand for attention. "No one's mentioned that my mother's latest wedding is tomorrow, in Italy. Which is, thankfully, many, many miles away from our happy home here in Greenwich, Connecticut. I got a call from her just after five this morning, as Linda doesn't get the concept of time zones and, well, let's face it, doesn't give a shit anyway."
"Why didn't you just let it ring?" Laurel demanded even as Emma reached over to rub Mac's leg in sympathy.
"Because she'd just keep calling back, and I'm trying to deal with her. On my terms, for a change." Mac raked her fingers through the bold red of her gamine cap of hair. "There were, as expected, tears and recriminations, as she's decided she really wants me there. Since I have no intention of hopping on a plane, particularly when I have an event tonight, two tomorrow, and another on Sunday, to see her get married for the fourth time, she's not speaking to me."
"If only it would last."
"Laurel," Parker murmured.
"I mean it. You got to give her a piece of your mind," she reminded Parker. "I didn't. I can only let it fester."
"Which I appreciate," Mac said. "Sincerely. But as you can see, I'm not in a funk, I'm not swimming in guilt or even marginally pissed off. I think there's an advantage to finding a guy who's sensible, loving, and just really solid. An advantage over and above really terrific morning sex. Each one of you has been on my side when I've had to deal with Linda, you've tried to help me through her demands and basic insanity. I guess Carter just helped tip the scales, and now I can deal with it. I wanted to tell you."
"I'd have morning sex with him myself, just for that."
"Hands off, McBane. But I appreciate the sentiment. So." She rose. "I want to get some work done before I need to focus on today's event. I'll swing by and get some shots of the cake."
"Hang on, I'll go with you." Emma pushed up. "I'll be back with the team shortly and I'll drop the flowers off for you, Laurel."
When they'd gone, Laurel sat another moment. "She really meant it."
"Yes, she really did."
"And she's right." Laurel took a last moment to sit back and relax with her coffee. "Carter's the one who turned the key in the lock. I wonder what it's like to have a man who can do that, who can help that way without pushing. Who can love you that way. I guess when it comes down to it, I envy her that even more than the sex."
Shrugging, Laurel rose. "I'd better get to work."
_L_ AUREL DIDN'T HAVE TIME TO THINK ABOUT MEN OVER THE next couple of days. She didn't have the time, or the energy, to think about love and romance. She might have been neck-deep in weddings, but that was business—and the business of weddings demanded focus and precision.
Her Antique Lace cake, which had taken her nearly three days to create, had its moment in the spotlight before being disassembled and devoured. Saturday afternoon featured her whimsical Pastel Petals with its hundreds of embossed, gum-paste rose petals, and Saturday evening her Rose Garden, where tiers of bold red roses layered with tiers of vanilla-bean cake and silky buttercream frosting.
For Sunday afternoon's smaller, more casual event, the bride had chosen Summer Berries. Laurel had done the baking, the filling, the assembly, and the basketweave frosting. Now, even as the bride and groom exchanged vows on the terrace outside, Laurel completed the project by arranging the fresh fruit and mint leaves on the tiers.
Behind her, the subs completed table decorations for the wedding brunch. She wore a baker's apron over a suit nearly the same color as the raspberries she selected.
Stepping back, she studied the lines and balance, then chose a bunch of champagne grapes to drape over a tier.
"Looks tasty."
Her eyebrows drew together as she grouped stemmed cherries. Interruptions while she worked were common, but that didn't mean she had to like them. Added to it, she hadn't expected Parker's brother to drop by during an event.
Then again, she reminded herself, he came and went as he pleased.
But when she spotted his hand reaching for one of her containers, she slapped it away smartly.
"Hands off."
"Like you're going to miss a couple blackberries."
"I don't know where your hands have been." She set a trio of mint leaves, and didn't bother, yet, to spare him a glance. "What do you want? We're working."
"Me, too. More or less. Lawyer capacity. I had some paperwork to drop off."
He handled all their legal dealings, both individually and as a business. She knew, very well, that he put in long hours on their behalf, and often on his own time. But if she didn't jab at him, she'd break long-standing tradition.
"And timed it so you could mooch from catering."
"There ought to be some perks. Brunch deal?"
She gave in and turned. His choice of jeans and a T-shirt didn't make him less of an Ivy League lawyer, not to her mind. Delaney Brown of the Connecticut Browns, she thought. Tall, appealingly rangy, his dense brown hair just a smidge longer than lawyerly fashion might dictate.
Did he do that on purpose? She imagined so, as he was a man who always had a plan. He shared those deep, midnight blue eyes with Parker, but though Laurel had known him all her life, she could rarely read what was behind them.
He was, in her opinion, too handsome for his own good, too smooth for anyone else's. He was also unflinchingly loyal, quietly generous, and annoyingly overprotective.
He smiled at her now, quick and easy, with a disarming flash of humor she imagined served as a lethal weapon in court. Or the bedroom.
"Cold poached salmon, mini chicken florentine, grilled summer vegetables, potato pancakes, a variety of quiches, caviar with full accompaniment, assorted pastries and breads along with a fruit and cheese display, followed by the poppyseed cake with orange marmalade filling and Grand Mariner buttercream frosting, topped with fresh fruit."
"Sign me up."
"I expect you can sweet-talk the caterers," she said. She rolled her shoulders, circled her head on her neck as she chose the next berries.
"Something hurt?"
"The basketweave's a killer on the neck and shoulders."
His hands lifted, then retreated to his pockets. "Are Jack and Carter around?"
"Somewhere. I haven't seen them today."
"Maybe I'll go hunt them down."
"Mmm-hmm."
But he wandered across the room to the windows and looked down at the flower-decked terrace, the white slippered chairs, the pretty bride turned toward the smiling groom.
"They're doing the ring thing," Del called out.
"So Parker just told me." Laurel tapped her headset. "I'm set. Emma, the cake's ready for you."
She balanced the top layer with an offset stem loaded with blackberries. "Five-minute warning," she announced, and began loading her bin with the remaining fruit. "Let's get the champagne poured, the Bloody Marys and mimosas mixed. Light the candles, please." She started to lift the bin, but Del beat her to it.
"I'll carry it."
She shrugged, and moved over to hit the switch for the background music that would play until the orchestra took over.
They started down the back stairs, passing uniformed wait-staff on their way up with hors d'oeuvres for the brief cocktail mixer designed to keep guests happy while Mac took the for mals of the bride and groom, the wedding party, and family.
Laurel swung into her kitchen, where the caterers ran full steam. Used to the chaos, she slid through, got a small bowl and scooped out fruit. She passed it to Del.
"Thanks."
"Just stay out of the way . . . Yes, they're ready," she said to Parker through the headset. "Yeah, in thirty. In place." She glanced over at the caterers. "On schedule. Oh, Del's here. Uh-huh."
He watched her, leaning on the counter and eating berries as she stripped off her apron. "Okay, heading out now."
Del pushed off the counter to follow her as she headed through the mudroom that would soon be transformed into her extra cooler and storage area. She pulled the clip out of her hair, tossed it aside, and shook her hair into place as she stepped outside.
"Where are we going?"
"I'm going to help escort guests inside. You're going away, somewhere."
"I like it here."
It was her turn to smile. "Parker said to get rid of you until it's time to clean up. Go find your little friends, Del, and if you're good boys you'll be fed later."
"Fine, but if I get roped into cleanup, I want some of that cake."
They separated, him strolling toward the remodeled pool house that served as Mac's studio and home, her striding toward the terrace where the bride and groom exchanged their first married kiss.
Laurel glanced back once, just once. She'd known him all her life—that was fate, she supposed. But it was her own fault, and her own problem, that she'd been in love with him nearly as long.
She allowed herself one sigh before fixing a bright, professional smile on her face to lend a hand herding the celebrants into Reception.
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GMH2 (Geo Maya Hair 2) is a more advanced version of GMH script, specialized in hair modeling & stylizing, developed by Phung Dinh Dzung at Thunder Cloud Studio. User can easily model complex hair styles using traditional polygon modeling method and convert to Maya Hair using GMH2. 11 Aug GMH2 (Geo Maya Hair 2) is an advanced version of GMH script, specialized in hair modeling and stylizing, developed by Phung Dinh Dzung at. GMH2 - Geo Maya Hair User Group has members.
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Home » COMICS » National Book Lovers Day: Eleven Graphic Memoirs You Must Read
National Book Lovers Day: Eleven Graphic Memoirs You Must Read
BY Roman Colombo August 9, 2019February 11, 2021
It's National Book Lovers Day, so, to celebrate, we're taking a look at ten original graphic memoirs that either already have a lasting impact on graphic novel literature or will have one. The popularity of these memoirs has increased a lot over the past few years, but they have a long history. As a graphic novel literature teacher, I can't stress the importance of these books enough. The reason why the people have started taking the comic book medium more seriously is because of the graphic memoir—and many of the works on this list. From these, texts like Watchmen, The Dark Knight Returns, Sandman, and, especially now, Infinity Gauntlet were looked at in a new light. Before they were superhero comics. Now they're intricate moral and societal portrayals with complex characters. These eleven graphic memoirs are not only influential but great books too. So, in no particular order…
Graphic Memoirs from Two Godfathers of the Comic Industry
The importance of these two men can't be overstated. One of them revolutionized the comic industry, co-creating hundreds of characters we still love today, and injecting much-needed pathos into the superhero tale. The other is so synonymous with greatness, the industry's top award is named after him. If you want to learn more about the minds behind the makers, these two graphic memoirs are a good place to start.
Amazing, Fantastic, Incredible: A Marvelous Memoir
Stan Lee with Peter David and Colleen Doran
(Gallery 13 Press)
From a poor childhood to a World War II propaganda writer, to editor-in-chief of the biggest comic house in the industry, this is the story of Stan Lee. Out of all the graphic memoirs on this list, this is the only one that looks and feels like a superhero story—and it should. Stan "the Man" is as mythical and legendary as the superheroes he helped create. His story needs to feel like one of his old comics. Colleen Doran provides that feeling. With a little help from one of the industry's best writers, Peter David, Amazing, Fantastic, Incredible: A Marvelous Memoir feels more like an origin story than it does an autobiography.
Life, In Pictures
(W.W. Norton and Company)
We're talking about graphic memoirs, but that is an offshoot of graphic novels—and Will Eisner was the first person in the industry to create one (A Contract with God). His work on The Spirit, a 1940's comic series, dabbled in themes and experimented with form in a way later creators would be praised for, especially Alan Moore and Frank Miller (In fact, Frank Miller directed The Spirit film.) And since this is Will Eisner we're talking about, his graphic memoir is uniquely different from all the other memoirs on this list. Instead of a linear narrative, he offers short episodes throughout his life. The result is actually quite powerful. Have you ever had one of those moments, sitting with a grandparent, and they launch into a story about their past that is amazing to hear? Well, that's what Life in Pictures is like.
The Classic Graphic Memoirs
These three works are assigned in classrooms across the country, from High School all the way through the Ph.D. (I know because I've assigned them to my classes.) But it's not just in graphic novel lit courses we see them. Fun Home is assigned regularly in women's studies classes. Maus in Holocaust history and literature. Persepolis in Middle Eastern history and literature. I've met middle school teachers who assigned all three texts, despite some very heavy material for an eleven or twelve-year-old. Back when comics were seen as childish and low-brow, these graphic memoirs the "exceptions" to the rule. Sure, they're comics, but they "rise above" the form. But they aren't great "despite" being comics. They're great because they are comics.
Fun Home: A Family Tragicomic
(Mariner Books)
Bechdel wrote two graphic memoirs: Fun Home and Are You My Mother: A Comic Drama. They are both wonderful, but Fun Home was a true sensation when it came out. Don't let the title fool you. Fun Home is bleak. Bechdel explores her relationship with her cold and distant father: an English teacher and funeral home (or, as Bechdel's family calls it, fun home) director. Bechdel came out as a lesbian in college, and then her father came out as gay—sadly, he died shortly after, leaving Alison with more questions than answers. Fun Home is how she processes these questions, and it makes for one of the most powerful graphic memoirs of all time.
The Complete Maus: A Survivor's Tale
(Pantheon)
Maus is actually two graphic memoirs, split between book one, My Father Bleeds History, and book two, And Here My Trouble Begins. As far as graphic memoirs go, the complexity of Maus is both brilliant and captivating. First, it's two books—no, not just the two installments, it's simultaneously a graphic memoir and a graphic biography. Spiegelman accounts his father's experience before, during, and after the Holocaust, but he also explores his relationship with his father.
The book is most known for depicting the Jewish characters as mice (hence the German word for mice as the title), and the German characters as cats. This decision was only a small part of what makes Maus so unique. Even among other Holocaust stories, Maus is very different. Spiegelman doesn't try to make his father look like the suffering saint. Instead, he's honest about how flawed his father was, even showing how other Holocaust survivors disliked him. He doesn't allow us to use the horrendous experience to explain away his father's actions later in life. It's raw, it's rough, and it's real.
The Complete Persepolis
Like Maus, Persepolis is two graphic memoirs: The Story of a Childhood and The Story of a Return. Satrapi's art style is overly cartoonish in stark black and white. Meanwhile, her memoir focuses on growing up in Iran during the civil war in the 1980s. All Marjane wants to do is listen to American music and be a kid, but the country is changing. Women can no longer be unaccompanied by a man. They can no longer drive. American music, movies, and everything is outlawed. Clothing becomes more restricted. Her mother and father are liberal Iranians, along with her family—and people start disappearing.
Her parents are forced to send her away to Europe, where she'll be safe. After her teenage years in France, she returns home—but realizes you can't ever really go home. She's changed too much. She doesn't recognize Iran anymore. We see stories coming out of the Middle East often, but Persepolis gives a perspective from a child, with the style of a kids' cartoon. It's a surreal experience.
Soon-to-be Classic Graphic Memoirs
While the last three graphic memoirs have established themselves as classics, these next three are right on the line of being a classic.
David Small
David Small is mostly known for children's books, but Stitches is definitely not one of them. In this memoir, Small tells the story about a devastating moment in his life, waking up at fourteen with a vocal cord removed and rendered mute. While going through the dark days of adolescence, he also has to undergo more surgeries to get his voice back. This is a harrowing story of recovery—physically and psychologically. As hard as it is, it's one of the graphic memoirs on this list that young adults should certainly read. While others explore teenage years, like Persepolis, but Stitches gets more into the psyche and personal troubles of those formative years unlike any other.
Craig Thompson
(Drawn + Quarterly)
Blankets is a love story…set in the Midwest during a harsh winter and a militantly fundamental Christian church. But Craig manages to find his first love, Raina, during Church Camp. Blankets is a great graphic novel for anyone struggling with not fitting in with their community. Both Craig and Raina dream of escape. They struggle with religious questions. It's also a story that's very familiar. The excitement of first love and the pain of first heartbreak. But the way Thompson tells his story makes it one of those graphic memoirs that feels like it's a part of you when you're finished.
March Trilogy
Congressman John Lewis with Andrew Ayden and Nate Powell
(Top Shelf Productions)
When Congressman John Lewis decided to finally write about his experiences during the Civil Rights era, everyone was looking forward to that book. But what no one expected was that Lewis chose to tell his story in three graphic memoirs. A Congressman wrote a comic book. And not just any Congressman—a living legend and hero. And it's a really powerful series too. March starts as an intimate story of John Lewis's childhood and ends as an inspirational rallying cry to stand up and fight for America. Every American citizen should—nay, has a duty—to read March.
The Future Classic Graphic Memoirs
These next three are pretty recent, and though not as well-knows, probably will be soon, especially since once is by an icon in both the Sci-Fi community and the LGBTQ+ community.
The Bride Was A Boy
Chii
(Seven Seas)
The Bride Was A Boy is an upbeat memoir about a transgender woman finding the love of her life while in transition…it wasn't her plan. Chii was assigned male at birth. In her memoir, she describes her struggles with gender identity and sexuality growing up—and her husband who is adorably in love with her. Reading this love story will make you long to find a relationship this genuine and hopeful. Out of all the graphic memoirs on this list, The Boy Was A Bride will leave you smiling the most.
My Lesbian Experience with Loneliness
Kabi Nagata
There are many graphic memoirs exploring sexuality, but Nagata also details her fight with depression. It's this aspect that we need more of. Nagata tackles her mental disorders head on. All the anxiety, loneliness, depression, etc. We see how depression makes it hard for her to pursue her art—one of the most relatable things that many creators feel—and the crippling doubt. However, as serious as these topics are, the memoir is very uplifting. If you struggle with depression, regardless of gender or sexuality, My Lesbian Experience With Loneliness will help you understand it more.
They Called Us Enemy
George Takei with Justin Eisinger, Steven Scott, and Harmony Becker
Takei might best be known as Sulu in the original Star Trek series and movies, but he's also a Japanese-American citizen who speaks openly about his experience as a child in a Japanese internment camp, which the U.S. government called "relocation centers." We often use Holocaust stories, like Maus, to discuss the horrors of—in technical terms—concentration camps, but because those stories take place in Germany, they feel isolated and far away. They Call Us Enemy shows an America that is ugly and heartless. It shows the inhumanity of putting people behind barbed wire, in cages. It should also stop anyone from excusing the practice for not being "as bad" as the concentration camps of Germany. Not as bad is still bad. And yes, this book came out right as America is once again forcing an ethnic group into camps, or, as officially called, "detention centers" instead of "relocation centers." Takei's memoir shows, you can change the terms, but a camp is a camp is a camp. Sadly, in the future, we'll have graphic memoirs similar to They Called Us Enemy about children locked away. They could even have the same title.
Go Read Some!
There are hundreds of graphic memoirs. Choosing eleven was painful (the list started as ten). If you want to check out more, Book Riot has a great list of graphic memoirs. There are also a lot coming out. If you're looking for a good mix of diverse voices, check out The Conscious Kid's list too. If you haven't read any graphic memoirs, go for one. Because of the sequential art, graphic memoirs feel much more personal, and we connect more strongly to the subjects too.
Side Story
One year, I assigned to my class Maus, V for Vendetta, and X-Men: Endangered Species + X-Men: Messiah Complex. They were the first three texts we read in the class and my students asked me why I overloaded them with depictions of concentration camps. It was entirely by accident, but it did explain why they were all so depressed by the time we finished Messiah Complex. Oops.
(Featured Image from Blankets by Craig Thompson)
COMICSIndependent
Roman Colombo finished his MFA in 2010 and now teaches writing and graphic novel literature at various Philadelphia colleges. His first novel, Trading Saints for Sinners, was published in 2014. He's currently working on his next novel and hoping to find an agent soon.
George TakeiGraphic MemoirsNational Book Lovers Daystan leeWill Eisner
Great News Marvel® Fans: CGC Comics Announces Special Marvel Labels
Marvel Entertainment, LLC the owners of Marvel Comics®, recently licensed Certified Guaranty Company® (CGC®) with rights to create certification labels that will feature some of the most iconic characters of Marvel. Marvel legend characters that […]
Serial Box: Partnering with Marvel to Tell New Serialized Stories of your Comic Book Superheroes
Serial Box is a publishing firm that has managed to gain the public's attention by telling stories in several new ways. The firm is now focusing on telling stories that focus on superheroes with its […]
Marvel Fans, Celebrate! CGC comics has announced Special Marvel Labels
Certified Guaranty Company, i.e, CGC is a grading company for comics. They announced that Marvel Entertainment LLC has granted it rights to create a lineup of certification labels featuring the most iconic Marvel characters – […] | {
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//
// HHPlayerNavView.h
//
// Copyright (c) 2013 Wanqiang Ji
//
// Permission is hereby granted, free of charge, to any person obtaining a
// copy of this software and associated documentation files (the "Software"),
// to deal in the Software without restriction, including without limitation
// the rights to use, copy, modify, merge, publish, distribute, sublicense,
// and/or sell copies of the Software, and to permit persons to whom the
// Software is furnished to do so, subject to the following conditions:
//
// The above copyright notice and this permission notice shall be included in
// all copies or substantial portions of the Software.
//
// THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
// IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
// FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE
// AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER
// LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING
// FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS
// IN THE SOFTWARE.
//
#import <UIKit/UIKit.h>
/**
* The navigation view of the Player.
*/
@interface HHPlayerNavView : UIView
{
@private
UIImageView *_bgImgView;
UILabel *_titleLabel;
}
/**
* Returns a button which location at the left.
*/
@property (nonatomic, readonly) UIButton *backButton;
/**
* The title will display in center.
*/
@property (nonatomic, copy) NSString *title;
/**
* Set the background image.
*
* @param img background image
*/
- (void)setBackgroundImage:(UIImage *)img;
/**
* Set the back button's image
*
* @param nImg normal image
* @param hImg highlighted image
*/
- (void)setBackButtonNormalImage:(UIImage *)nImg highlightedImage:(UIImage *)hImg;
@end
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{"url":"http:\/\/scholar.cnki.net\/WebPress\/brief.aspx?dbcode=SJDM","text":"\u4f5c\u8005\uff1aMathieu Lewin\u00a0,\u00a0Elliott H. Lieb\u00a0,\u00a0Robert Seiringer \u6765\u6e90\uff1a[J].CEDRAM \u6458\u8981\uff1aIn this paper we define and study the classical Uniform Electron Gas (UEG), a system of infinitely many electrons whose density is constant everywhere in space. The UEG is defined differently from Jellium, which has a positive constant background but no constraint on the density....\n \u4f5c\u8005\uff1aSt\u00e9phane Druel\u00a0,\u00a0Henri Guenancia \u6765\u6e90\uff1a[J].CEDRAM \u6458\u8981\uff1aIn this paper we show that any smoothable complex projective variety, smooth in codimension two, with klt singularities and numerically trivial canonical class admits a finite cover, \u00e9tale in codimension one, that decomposes as a product of an abelian variety, and singular a...\n \u4f5c\u8005\uff1aDelphine Pol \u6765\u6e90\uff1a[J].Annales de l'institut Fourier(IF 0.527),\u00a02018,\u00a0Vol.68\u00a0(2),\u00a0pp.725-766CEDRAM \u6458\u8981\uff1aWe consider the germ of a reduced curve, possibly reducible. F. Delgado de la Mata proved that such a curve is Gorenstein if and only if its semigroup of values is symmetrical. We extend here this symmetry property to any fractional ideal of a Gorenstein curve. We then focus...\n \u4f5c\u8005\uff1aStergios Antonakoudis\u00a0,\u00a0Javier Aramayona\u00a0,\u00a0Juan Souto \u6765\u6e90\uff1a[J].Annales de l'institut Fourier(IF 0.527),\u00a02018,\u00a0Vol.68\u00a0(1),\u00a0pp.217-228CEDRAM \u6458\u8981\uff1aWe prove that every non-constant holomorphic map $\\mathcal{M}_{g,p}\\rightarrow \\mathcal{M}_{g^{\\prime },p^{\\prime }}$ between moduli spaces of Riemann surfaces is a forgetful map, provided that $g\\ge 6$ and $g^{\\prime }\\le 2g-2$.\n \u4f5c\u8005\uff1aAdriana A. Albarrac\u00edn-Mantilla\u00a0,\u00a0Edwin Le\u00f3n-Cardenal \u6765\u6e90\uff1a[J].Journal de th\u00e9orie des nombres de Bordeaux,\u00a02018,\u00a0Vol.30\u00a0(1),\u00a0pp.331-354CEDRAM \u6458\u8981\uff1aWe study the twisted local zeta function associated to a polynomial in two variables with coefficients in a non-Archimedean local field of arbitrary characteristic. 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What movies are you playing next weekend?
We usually do not finalize our movie lineup until Monday afternoon before the following weekend. This is due mainly to 2 reasons: first, we want to keep the best films available on the screen which depends upon how they performed in attendance on the previous weekend, and second, we have to negotiate the best film rental terms with the movie studios in order to keep our costs lower and be able to keep our admission prices low. The longer the negotiation continues, the more leverage the drive-in has against the movie studio. Sometimes we will not have our confirmed movie bookings until Tuesday afternoon prior to the following weekend. The movie lineup is sent out on Wednesday via email to those who have subscribed to our drive-in newsletter, after we have received the verifications from the film studios for all of our movies.
Why do you pair certain movies together?
We always try to pair two movies together that complement each other. It is not as easy as you may think. A first run feature can be very expensive, and the second feature must cost less so that can afford to show two movies for one low price. Two first run movies on the same screen would be financially impossible to support. At the same time, if the second movie is to old, no one will want to stay to see it even as a second feature. What helps is when the same movie studio has two films which are compatible with each other in release close to the same time of year, and they will allow us to pair them together for a reduced rental fee.
Are people allowed to bring their pets to the drive-in?
We currently do allow pets at the drive-in, but we will ask people with unruly pets to leave if they are disturbing or intimidating other customers. If a dog is likely to bark around strangers, or is menacing when people walk near to their vehicle, it is best not to bring that pet to the drive-in. Pets should remain leashed at all times while at the drive-in, and when duty calls, the pet owner should clean up after their pet. We do provide a pet walking area between screen 2 and screen 3.
How early should we arrive at the drive-in?
The gates are open at 7:30 pm. First show starts at dusk and it is best to arrive at least a half hour before dusk.
What happens when it rains at the drive-in?
We show the movies, rain or clear. Once you enter the theatre, you are assuming the risk that it may rain at some point during the evening. We do not give rain checks or refunds due to rain. The only exception would be dense fog, which obscures the ability to see the movie on the screen, as a dense fog will absorb to much light to be able to view the movie. Passes will only be given to those patrons in possession of their ticket stubs in the event of fog cancellation.
How do we listen to the movie at the drive-in?
This drive-in uses state of the art FM stereo broadcasts to deliver the sound for the movies on each screen. You simply tune your FM stereo in your car to the designated frequency, and listen to the movie from the comfort of your own vehicle. Under normal operating conditions, this will not drain or harm your car battery. You may also bring a battery operated portable radio with you, and listen from outside of your vehicle sitting in lawn chairs.
Can we switch from one screen to another to watch a different second feature than the one our original screen?
Not unless you want to exit the theatre and purchase new tickets at the box office. There are several reasons for this. Each pair of movies is owned by different film studios. When you purchase your original ticket, the money from that sale goes directly to the film studio which owns the movies being shown on the screen that you have a ticket for. If you switch to another screen, the other film studio will not receive any compensation for the viewing of their movie. This is not only a violation of our licensing agreement with the film studio to show their films, but it is also a violation of the copyright laws which protect every movie being shown in our drive-in. Purchasing tickets at the Evergreen Drive In permits our guests to view only the movies shown on the screen where their tickets specify. Any guests who switch screens or attempt to view a movie without possession of the proper tickets will forfeit their right to remain on the property.
We want to maintain a safe and family friendly environment for all of our guests. | {
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// Copyright (c) 2010-2013 SharpDX - Alexandre Mutel
//
// Permission is hereby granted, free of charge, to any person obtaining a copy
// of this software and associated documentation files (the "Software"), to deal
// in the Software without restriction, including without limitation the rights
// to use, copy, modify, merge, publish, distribute, sublicense, and/or sell
// copies of the Software, and to permit persons to whom the Software is
// furnished to do so, subject to the following conditions:
//
// The above copyright notice and this permission notice shall be included in
// all copies or substantial portions of the Software.
//
// THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
// IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
// FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE
// AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER
// LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM,
// OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN
// THE SOFTWARE.
using System;
namespace SharpDX.Direct3D11
{
public partial class ComputeShaderStage
{
/// <summary>
/// Gets an array of views for an unordered resource.
/// </summary>
/// <remarks>
/// Any returned interfaces will have their reference count incremented by one. Applications should call IUnknown::Release on the returned interfaces when they are no longer needed to avoid memory leaks.
/// </remarks>
/// <param name="startSlot">Index of the first element in the zero-based array to return (ranges from 0 to D3D11_PS_CS_UAV_REGISTER_COUNT - 1). </param>
/// <param name="count">Number of views to get (ranges from 0 to D3D11_PS_CS_UAV_REGISTER_COUNT - StartSlot). </param>
/// <unmanaged>void CSGetUnorderedAccessViews([In] int StartSlot,[In] int NumUAVs,[Out, Buffer] ID3D11UnorderedAccessView** ppUnorderedAccessViews)</unmanaged>
public UnorderedAccessView[] GetUnorderedAccessViews(int startSlot, int count)
{
var temp = new UnorderedAccessView[count];
GetUnorderedAccessViews(startSlot, count, temp);
return temp;
}
/// <summary>
/// Sets an array of views for an unordered resource.
/// </summary>
/// <remarks>
/// </remarks>
/// <param name="startSlot">Index of the first element in the zero-based array to begin setting. </param>
/// <param name="unorderedAccessView">A reference to an <see cref="SharpDX.Direct3D11.UnorderedAccessView"/> references to be set by the method. </param>
/// <unmanaged>void CSSetUnorderedAccessViews([In] int StartSlot,[In] int NumUAVs,[In, Buffer] const ID3D11UnorderedAccessView** ppUnorderedAccessViews,[In, Buffer] const int* pUAVInitialCounts)</unmanaged>
public void SetUnorderedAccessView(int startSlot, SharpDX.Direct3D11.UnorderedAccessView unorderedAccessView)
{
SetUnorderedAccessView(startSlot, unorderedAccessView, -1);
}
/// <summary>
/// Sets an array of views for an unordered resource.
/// </summary>
/// <remarks>
/// </remarks>
/// <param name="startSlot">Index of the first element in the zero-based array to begin setting. </param>
/// <param name="unorderedAccessView">A reference to an <see cref="SharpDX.Direct3D11.UnorderedAccessView"/> references to be set by the method. </param>
/// <param name="uavInitialCount">An Append/Consume buffer offsets. A value of -1 indicates the current offset should be kept. Any other values set the hidden counter for that Appendable/Consumable UAV. uAVInitialCount is only relevant for UAVs which have the <see cref="SharpDX.Direct3D11.UnorderedAccessViewBufferFlags"/> flag, otherwise the argument is ignored. </param>
/// <unmanaged>void CSSetUnorderedAccessViews([In] int StartSlot,[In] int NumUAVs,[In, Buffer] const ID3D11UnorderedAccessView** ppUnorderedAccessViews,[In, Buffer] const int* pUAVInitialCounts)</unmanaged>
public void SetUnorderedAccessView(int startSlot, SharpDX.Direct3D11.UnorderedAccessView unorderedAccessView, int uavInitialCount)
{
SetUnorderedAccessViews(startSlot, new[] { unorderedAccessView }, new[] { uavInitialCount });
}
/// <summary>
/// Sets an array of views for an unordered resource.
/// </summary>
/// <remarks>
/// </remarks>
/// <param name="startSlot">Index of the first element in the zero-based array to begin setting. </param>
/// <param name="unorderedAccessViews">A reference to an array of <see cref="SharpDX.Direct3D11.UnorderedAccessView"/> references to be set by the method. </param>
/// <unmanaged>void CSSetUnorderedAccessViews([In] int StartSlot,[In] int NumUAVs,[In, Buffer] const ID3D11UnorderedAccessView** ppUnorderedAccessViews,[In, Buffer] const int* pUAVInitialCounts)</unmanaged>
public void SetUnorderedAccessViews(int startSlot, params SharpDX.Direct3D11.UnorderedAccessView[] unorderedAccessViews)
{
var uavInitialCounts = new int[unorderedAccessViews.Length];
for (int i = 0; i < unorderedAccessViews.Length; i++)
uavInitialCounts[i] = -1;
SetUnorderedAccessViews(startSlot, unorderedAccessViews, uavInitialCounts);
}
/// <summary>
/// Sets an array of views for an unordered resource.
/// </summary>
/// <remarks>
/// </remarks>
/// <param name="startSlot">Index of the first element in the zero-based array to begin setting. </param>
/// <param name="unorderedAccessViews">A reference to an array of <see cref="SharpDX.Direct3D11.UnorderedAccessView"/> references to be set by the method. </param>
/// <param name="uavInitialCounts">An array of Append/Consume buffer offsets. A value of -1 indicates the current offset should be kept. Any other values set the hidden counter for that Appendable/Consumable UAV. pUAVInitialCounts is only relevant for UAVs which have the <see cref="SharpDX.Direct3D11.UnorderedAccessViewBufferFlags"/> flag, otherwise the argument is ignored. </param>
/// <unmanaged>void CSSetUnorderedAccessViews([In] int StartSlot,[In] int NumUAVs,[In, Buffer] const ID3D11UnorderedAccessView** ppUnorderedAccessViews,[In, Buffer] const int* pUAVInitialCounts)</unmanaged>
public unsafe void SetUnorderedAccessViews(int startSlot, SharpDX.Direct3D11.UnorderedAccessView[] unorderedAccessViews, int[] uavInitialCounts)
{
var unorderedAccessViewsOut_ = (IntPtr*)0;
if (unorderedAccessViews != null)
{
IntPtr* unorderedAccessViewsOut__ = stackalloc IntPtr[unorderedAccessViews.Length];
unorderedAccessViewsOut_ = unorderedAccessViewsOut__;
for (int i = 0; i < unorderedAccessViews.Length; i++)
unorderedAccessViewsOut_[i] = (unorderedAccessViews[i] == null) ? IntPtr.Zero : unorderedAccessViews[i].NativePointer;
}
fixed (void* puav = uavInitialCounts)
SetUnorderedAccessViews(startSlot, unorderedAccessViews != null ? unorderedAccessViews.Length : 0, (IntPtr)unorderedAccessViewsOut_, (IntPtr)puav);
}
}
} | {
"redpajama_set_name": "RedPajamaGithub"
} | 9,910 |
{"url":"http:\/\/www.ii.uni.wroc.pl\/~nivelle\/publications\/index.html","text":"Publications\n\n2016\n\nSubsumption Algorithms for Three-Valued Geometric Resolution\n\nIn the implementation of geometric resolution, the most costly operation is subsumption (or matching). In the matching problem, one has to decide for a three-valued, geometric formula, whether this formula is false in a given interpretation. The formula contains only atoms with variables, equality, and existential quantifiers. The interpretation contains only atoms with constants, which are assumed to be distinct.\n\nBecause matching is not restricted by term structure, matching for geometric resolution is a hard problem. We translate the matching problem into a generalized constraint satisfaction problem, and give an algorithm that solves it efficiently. The algorithm uses learning teachings, similar to clause learning in propositional logic. After that, we adapt the algorithm in such a way that it finds solutions that use a minimal subset of the interpretation.\n\nThe techniques presented in this paper may also have applications in constraint solving.\n\nThe paper is accepted at IJCAR 2016. A preprint is available here. I also gave talk at our local seminar about this topic.\n\n2014?\n\nTheorem Proving for Classical Logic with Partial Functions by Reduction to Kleene Logic\n\nPartial functions are abundant in mathematics and program specifications. Despite this, their importance has been mostly ignored in automated theorem proving. In this paper, we develop a theorem proving strategy for Partial Classical Logic (PCL). Proof search takes place in Kleene Logic. We show that PCL theories can be translated into equivalent sets of formulas in Kleene logic. For proof search in Kleene logic, we use a three-valued adaptation of geometric resolution. We prove that the procedure is sound and complete.\n\nThe paper has been accepted for publication in the Journal of Logic and Computation. This is a preprint.\n\n2011\n\nClassical Logic with Partial Functions\n\nIn this paper, I introduce a semantics for classical logic with partial functions, in which ill-typed formulas are guaranteed to have no truth value, so that they cannot be used in any form of reasoning. The semantics makes it possible to mix reasoning about types and preconditions with reasoning about other properties. In this way, it is possible to treat partial functions that have preconditions of unlimited complexity. We show that, in spite of its increased complexity, the semantics is still a natural generalization of first-order logic with simple types. If one does not use the increased expressivity, the type system is not stronger than classical logic with simple types. I define two sequent calculi for the new semantics, and prove that they are sound and complete. The first calculus follows the semantics closely, and hence its completeness proof is fairly straightforward. The second calculus is further away from the semantics, but more suitable for practical use because it has better proof theoretic properties. Its completeness can be shown by proving that proofs from the first calculus can be translated.\n\nThis is a preprint of the paper. The final version was published in the Journal of Automated Reasoning, and can be obtained from Springer .\n\n2010\n\nClassical Logic with Partial Functions\n\nWe introduce a semantics for classical logic with partial functions. We believe that this semantics is natural. When a formula contains a subterm in which a function is applied outside of its domain, our semantics ensures that the formula has no truth-value, so that it cannot be used for reasoning. The semantics relies on order of formulas. In this way, it is able to ensure that functions and predicates are properly declared before they are used. We define a sequent calculus for the semantics, and prove that this calculus is sound and complete for the semantics. The paper appeared in the proceedings of IJCAR 2010.\n\nThis is a preprint.\n\n2008\n\nA small Framework for Proof Checking\n\nWe describe a small framework in which first-order theorem provers can be used for the verification of mathematical theories. The verification language is designed in such a way that the use of higher-order constructs is minimized. In this way, we expect to be able to take advantage of the first order theorem prover as much as possible.\n\nps or pdf .\n\n2006\n\nGeometric Resolution: A Proof procedure based on Finite Model Search\n\nWe present a proof procedure that is complete for first-order logic, but which can also be used when searching for finite models. The procedure uses a normal form which is based on geometric formulas. For this reason we call the procedure geometric resolution. We expect that the procedure can be used as an efficient proof search procedure for first-order logic. In addition, the procedure can be implemented in such a way that it is complete for finding finite models. Download ps or pdf .\n\nUsing Resolution as a Decision Procedure (Habilitation Thesis)\n\nThis habilitation thesis is based on five papers that deal with resolution decision procedures. Resolution is a well-known technique for first-order theorem proving. It is complete, but it does not terminate in general, when there exists no proof. However, in many cases, resolution can be modified in such a way that it becomes a decision procedure for certain subclasses of first-order logic.\n\nWe have studied several aspects related to the use of resolution as decision procedure. The first two papers introduce resolution decision procedures for the guarded fragment with and without equality. The third paper builds on the decision procedure for the guarded fragment. Using the decision procedure, modal logics can be decided by translation into the guarded fragment with the relational translation. We have extended the standard relational translation of modal logics, so that more modal logics can be translated into the guarded fragment. In particular, modal logic S4 can now be translated. Before, only translations into extensions of the guarded fragment existed. The fourth paper shows that a standard refinement of resolution, which uses a liftable order, can be used to obtain a decison procedure for the E-plus class. This was posed as an open problem in (Resolution Methods for the Decision Problem, C. Fermueller, A. Leitsch, T. Tammet, N. Zamov, Springer Verlag 1993)\n\nThe fifth paper introduces a resolution-based decision procedure for the two-variable fragment with equality. The procedure consists of three stages: Saturation under resolution, elimination of equality, and one more time saturation under resolution.\n\nIn addition to the main habilitation thesis, one has to submit 5 additional publications (sonstige Arbeiten). They are collected in a supplement, which you may download either as ps or pdf .\n\n2003\n\nDeciding Regular Grammar Logics with Converse through First-Order Logic\n\n(Joint work with Stephane Demri, LSV Paris, France)\n\nWe provide a simple translation of the satisfiability problem for regular grammar logics with converse into $\\GF^2,$ which is the intersection of the guarded fragment and the $2$-variable fragment of first-order logic. This translation is theoretically interesting because it translates modal logics with certain frame conditions into first-order logic, without explicitly expressing the frame conditions. A consequence of the translation is that the general satisfiability problem for regular grammar logics with converse is in EXPTIME. This extends a previous result of the first author for grammar logics without converse. Using the same method, we show how some other modal logics can be naturally translated into $\\GF^2,$ including nominal tense logics and intuitionistic logic. In our view, the results in this paper show that the natural first-order fragment corresponding to regular grammar logics is simply $\\GF^2$ without extra machinery such as fixed point-operators. Download as ps or pdf .\n\nImplementing the Clausal Normal Form Transformation with Proof Generation\n\nThe paper describes how I intend to implement the clausal normal form transformation with proof generation described in the paper below. The paper introduces a convenient datastructure for sequent proofs, which allows easy proof checking and proof normalization. Download ps or pdf .\n\nTranslation of Resolution Proofs into Short First-Order Proofs without Choice Axioms\n\nWe present a way of transforming a resolution proof containing Skolemization into a natural deduction proof of the same formula but not using Skolemization. The size of the proof increases only moderately (polynomially). This makes it possible to translate the output of a resolution theorem prover into a purely first-order proof that is moderate in size. Download draft as ps or pdf .\n\nSubsumption of concepts in FL0 for (cyclic) terminologies with respect to descriptive semantics is PSPACE-complete.\n\nThe paper solves an open problem concerning the complexities of the simple description logic FL0. FL0 is the description logic which allows only conjunction and universal value restriction. We show that subsumption in this logic is PSPACE-complete under the descriptive semantics. The descriptive semantics is the semantics that one obtains if one interprets definitions as simple first-order equivalences. (The other two possibilities are to interpret definitions as least fixed points, or greatest fixed points) Download ps or pdf .\n\nDeciding the Guarded Fragments by Resolution\n\n(joint work with Maarten de Rijke, University of Amsterdam)\n\nWe give resolution based decision procedures for both the strictly guarded fragment and the loosely guarded fragment of first-order logic. We prove that the decision procedures are in 2EXPTIME, which is theoretically optimal. Download ps or pdf . The article in the JSC contains a few crucial printer errors which are corrected here.\n\n2002\n\nAutomated Proof Construction in Type Theory using Resolution\n\n(joint work with Marc Bezem, Dimitri Hendriks)\n\nWe provide techniques to integrate resolution logic with equality in type theory. The results may be rendered as follows:\n1. A clausification procedure in type theory, equipped with a correctness proof, all encoded using higher-order primitive recursion.\n2. A novel representation of clauses in minimal logic such that the $\\lambda$-representation of resolution steps is linear in the size of the premisses.\n3. Availability of the power of resolution theorem provers in interactive proof construction systems based on type theory\nps or pdf .\n\nExtraction of Proofs from the Clausal Normal Form Transformation\n\nWe give techniques for extracting proofs from the Clausal Normal Form transformation. We discuss and solve three technical problems:\n1. How to handle the introduction of definitions and Skolem functions.\n2. How to generate short (linear size) proofs.\n3. How to handle optimized Skolemization. We reduce it to standard Skolemization.\n\n2001\n\nSplitting through New Proposition Symbols\n\nThis paper presents ways of simulating splitting through new propositional symbols. The contribution of the paper is that the method presented here does not loose backward redundancy. We give a general method of obtaining a calculus with splitting by adjoining splitting symbols to an arbitrary resolution refinement. We then prove a relative completeness theorem: Every saturated set of clauses of the resolution calculus with adjoined proposition symbols contains a saturated set of the original resolution refinement without adjoined symbols. Download as ps or pdf .\n\n2000\n\nDatastructures for Resolution\n\n(This is unpublished)\n\nThe paper presents the benchmark tests out of which Bliksem is grown. The first part compares various ways of implementing terms relative to unification and matching. The second part compares different ways of implementing substitutions. The third part compares three ways of implementing discrimination trees. The paper ends with a description of a subsumption algorithm. Download ps or pdf .\n\n1999\n\nTranslation of S4 and K5 into GF and 2VAR\n\n(This is unpublished)\n\nThis short note gives a method of translating S4 and K5 into the intersection of GF and 2VAR through the introduction of new symbols. Download ps or pdf .\n\nA Superposition Decision Procedure for the Guarded Fragment with Equality\n\nWe give a decision procedure for the guarded fragment with equality. The procedure is based on resolution with superposition. The relevance of the guarded fragment lies in the fact that many modal logics can be translated into it. In this way the guarded fragment acts as a framework explaining some of the nice properties of these modal logics. By constructing an implementable decision procedure for the guarded fragments we define an effective procedure for deciding these modal logics. It is surprising to see that one does not need any sophisticated simplification and redundancy elimination method to make superposition terminate on the class of clauses that is obtained from the clausification of guarded formulas. Yet the decision procedure obtained is optimal with regard to time complexity.\n\n1997\n\nA Classification of Non-Liftable Orders for Resolution\n\nThis paper tries to solve the problem of completeness of resolution restricted by non-liftable orders that cannot be modelled by the resolution game. Various types of non-liftability are distinguished and combined with various corresponding types of subsumption. All but one of the meaningful combinations are solved. The remaining case is still open.\n\n1995\n\nThesis (October 1995, Delft University of Technology, the Netherlands)\n\nThe resolution game\n\nMain contribution of the thesis is the resolution game . The resolution game is a game that can be played between two players, based on a set of ordered clauses. The opponent tries to derive the empty clause, using ordered resolution and factoring. The defender tries to disturb the opponent by changing the order at arbitrary moments. The main theorem states that under certain restrictions, the opponent has a winning strategy (i.e. can always derive the empty clause) if and only if the clause set is unsatisfiable.\n\nNon-liftable Orders\n\nThe resolution game can be used for proving completeness of resolution with non-liftable orders. Non-liftable orders are orders that are not preserved by substition, i.e. they do not always satisfy A < B implies Theta(A) < Theta(B). The standard method of proving completeness (making use of lifting) does not work for such orders. With the resolution game, completeness can be proven for some such orders.\n\nResolution Based Decision Procedures\n\nThe main application of non-liftable orders is in resolution based decision procedures . These are resolution restrictions that are guaranteed to terminate on certain subsets of first-order logic. In my thesis, two decision procedures are developed, one for the E+ -class, and one for the two-variable fragment (without equality).\n\nMaslov's Method\n\nAnother (smaller) contribution of the thesis is that it clarifies the relation between Maslov's inverse method and resolution. Most resolution calculi work top-down. They decompose a formula into clauses, resolving upon conflicts between clauses.\n\nUsing Maslov's method, it is very easy to obtain a resolution calculus for every logic that has a cut-free, complete sequent calculus. However, the resolution calculi that are obtained in this way are bottom-up, i.e. working from the axioms towards the goal.\n\nIn my thesis, I show that many sequent calculi have a reversal , which is obtained by exchanging axioms and goal, and reversing the direction of the rules. 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\section{Introduction}
The regularization theory for plurisubharmonic functions on a compact complex manifold has been well-developed so far. Let $M$ be a compact complex manifold and $\gamma$ a $(1,1)$ form on $M$. Let $\varphi$ be a $\gamma$-plurisubharmonic function on $M$ (i.e. $\gamma+\sqrt{-1}\partial\bar{\partial}\varphi\ge 0$ on $M$).
Roughly speaking, we hope to find a family of smooth functions $\varphi_{j}$ decreasing to $\varphi$ where each $\varphi_j$ is still $\gamma$-plurisubharmonic (or has very small loss of Hessian).
Before we survey any previous result, we would like to remark that classical version of such questions on $\mathbb{C}^n$ is quite easy. Indeed, if $\varphi$ is a plurisubharmonic function on $\mathbb{C}^n$, we could simply define $\varphi_{\eps}(z)=\int_{\mathbb{C}^n}\varphi(z+\eps w)\chi(|w|)dV(w)$, where $\chi$ is a standard smoothing kernel. Then one can easily verify that $\varphi_{\eps}$ would decrease to $\varphi$ and $\varphi_{\eps}$ is also plurisubharmonic.
Such an approach even generalizes to the case when the underlying manifold $M$ is homogeneous. Indeed, it is possible to define ``translation with length $h$" on homogeneous manifolds and one can define $\varphi_{\eps}$ by taking the average over all translations of $\varphi$ with length $h$. One may refer to \cite{EGZ3} for more details of this argument.
It is clear that we can no longer use the above definition for $\varphi_{\eps}$ on manifolds. To find such approximations, either we can try to define $\varphi_{\eps}$ locally using the above formula and try to glue them together, or we can try to find a global expression to replace the above formula. Blocki and Kolodziej \cite{BK} took the first approach for bounded plurisubharmonic functions. They used convolution to define $\varphi_{\eps}$ locally, and try to glue them together. The major progress was later made by Demailly \cite{Demailly}, \cite{DP}. What he had in mind was mainly about applications to algebraic geometry. Another application of his regularization theory was to prove the H\"older continuity of solutions to Monge-Ampere equations when the right hand side is in $L^p$ with $p>1$.
Let $(M,\omega_0)$ be a compact K\"ahler manifold. Consider the complex Monge-Ampere equation:
\begin{equation*}
(\omega_0+\sqrt{-1}\partial\bar{\partial}\varphi)^n=e^F\omega_0^n,\,\,\sup_M\varphi=0.
\end{equation*}
We assume that $e^F\in L^p(\omega_0^n)$ for some $p>1$. In the pioneering work by Kolodziej \cite{K1}, he proved that $\varphi$ has uniform $L^{\infty}$ bound (a bound for $\int e^F|F|^p\omega_0^n$ for $p>n$ is already sufficient).
Later on, Kolodziej \cite{K2} also proved that the solution to complex Monge-Ampere is H\"older continuous when the right hand side is in $L^p$ for some $p>1$. Then S. Dinew \cite{Dinew1} proved that the H\"older exponent actually depends only on $p$ and $n$, under a positivity assumption on bisectional curvature.
In order to show that the solution is H\"older continuous, the proof in \cite{DPGDHKZ} have two main ingredients (very roughly speaking):
\begin{enumerate}
\item (Stability Estimate)\\ Let $v$ be any bounded $\omega_0$-psh function, then one has
\begin{equation*}
\sup_M(v-\varphi)\le C||(v-\varphi)^+||_{L^1}^{\mu},
\end{equation*}
for some $\mu>0$, and $x^+=\max(x,0)$.
\item (Upper Regularization) \\Consider $\varphi_{\eps}(z)=\int_{\zeta\in T_zM}\varphi(\exp_z(\eps \zeta))\chi(|\zeta|^2)dV_z(\zeta)$, and show that $\omega_0+\sqrt{-1}\partial\bar{\partial}\varphi_{\eps}\ge -\eps^{\alpha}\omega_0$, $||\varphi_{\eps}-\varphi||_{L^1}\le C\eps^{\beta}$.
\end{enumerate}
Once we have (1) and (2), we can take $v=\frac{\varphi_{\eps}}{1+\eps^{\alpha}}$ in the Stability Estimate, and obtain that $\varphi_{\eps}-\varphi\le C\eps^{\gamma}$. On the other hand, $\Delta \varphi\ge -n$ since $\varphi$ is $\omega_0$-psh, we will be able to get $\sup_{z'\in B_{\frac{\eps}{2}}(z)}\varphi(z')-\varphi(z)\le C\eps^{\gamma}$, and this will imply $\varphi$ is in $C^{\gamma}$.
The goal of this present work is to make a first attempt to generalize such results to more general Hessian equations. We will consider equations of the following form:
\begin{equation}\label{1.1New}
f(\lambda[h_{\varphi}])=e^{F},\,\,\lambda[h_{\varphi}]\in\Gamma.
\end{equation}
In the above, $\varphi$ is a real valued function on $M$ and $(h_{\varphi})_j^i=\sum_kg^{i\bar{k}}(g_{j\bar{k}}+\varphi_{j\bar{k}})$ and $\lambda[h_{\varphi}]$ means the eigenvalues of $\{(h_{\varphi})_j^i\}_{1\le i,\,j\le n}$.
In the above, $f(\lambda_1,\lambda_2,\cdots,\lambda_n)$ is a function defined on a cone $\Gamma\subset \mathbb{R}^n$. Following \cite{GPT}, we make the following assumptions on the cone $\Gamma$ as well as the function $f$:
\begin{enumerate}
\item $\Gamma$ and $f$ are symmetric in $\lambda$, which means they are invariant under swapping $\lambda_i$ and $\lambda_j$.
\item $\Gamma_n\subset\Gamma\subset\Gamma_+$, where $\Gamma_n=\{\lambda\in \mathbb{R}^n:\lambda_i>0,\,1\le i\le n\}$ and $\Gamma_+=\{\lambda\in\mathbb{R}^n:\sum_i\lambda_i>0\}$. $\Gamma$ is convex.
\item $\frac{\partial f}{\partial \lambda_i}>0$, $1\le i\le n$ and $f$ is concave on $\Gamma$.
\item There exists $c_0>0$ such that if we put $F(h)=f(\lambda(h))$ where $h$ is a positive definite $n\times n$ Hermitian matrix, then $ \mbox{det} \frac{\partial F}{\partial h_{i\bar{j}}}\ge c_0>0$.
\item There exists $C_0>0$ such that for $\lambda\in\Gamma_n$, $\sum_{i=1}^n\lambda_i\frac{\partial f}{\partial\lambda_i}\le C_0f$.
\end{enumerate}
The examples which one would like to study first is the $\sigma_m$ equation (with $1\le m\le n$), where $f(\lambda)=\sigma_m^{\frac{1}{m}}(\lambda)$ where \sloppy $\sigma_m(\lambda)=\sum_{1\le i_1<i_2<\cdots <i_m\le n}\lambda_{i_1}\cdots \lambda_{i_m}$, defined on the cone $\Gamma_m=\{\lambda\in\mathbb{R}^n:\sigma_i(\lambda)>0,\,1\le i\le m\}$.
Note that examples like $f(\lambda)=\big(\frac{\sigma_k}{\sigma_m}\big)^{\frac{1}{k-m}}$, $1\le m\le k-1$ would not satisfy the assumption (4) above, but $f(\lambda)=\big(\frac{\sigma_k}{\sigma_m}\big)^{\frac{1}{k-m}}+c\sigma_l^{\frac{1}{l}}$ with $c>0$ does satisfy the assumption (4).
Due to the main result of \cite{GPT}, $\varphi$ is bounded in $L^{\infty}$. In order to study the H\"older continuity, we need to establish the Stability Result and Upper Regularization result for (\ref{1.1New}), similar to the approach for complex Monge-Ampere equations.
The stability result for a general Hessian equations can be obtained following a PDE approach developed in \cite{WWZ0}, \cite{WWZ} and \cite{GPT}. It is done in Section \ref{Stability}. It can be stated as follows:
\begin{prop}\label{p1.2}
(Lemma \ref{l3.7}) Let $\delta>0$ and let $v$ be a bounded function such that $\lambda\big(\frac{\delta}{2}I+(h_v)_j^i\big)\in \Gamma$. Let $\varphi$ solve \ref{1.1New} with $e^{nF}\in L^{p_0}(\omega_0^n)$ for some $p_0>1$. Let $s_0>0$ be suitably large (compared with $\delta$) such that
\begin{enumerate}
\item $s_0\ge 2\delta||v||_{L^{\infty}}$;
\item $\int_M((1-\delta)v-\varphi-s_0)^+e^{nF}\omega_0^n\le \delta^{n+1}$.
\end{enumerate}
Then for any $\mu<\frac{1}{nq_0}$ (where $q_0=\frac{p_0}{p_0-1}$),
\begin{equation*}
\sup_M(v-\varphi)\le s_0+C_1s_0^{-\mu}||(v-\varphi)^+||_{L^1}^{\mu}.
\end{equation*}
\end{prop}
The next step is to find a suitable upper approximation $\varphi_{\eps}$. We need to choose $\varphi_{\eps}$ so that:
\begin{enumerate}
\item ($\varphi_{\eps}$ is almost $\Gamma$-admissible)\\There exists $\gamma_1>0$ such that $\frac{C\eps^{\gamma_1}}{2}I+(h_{\varphi_{\eps}})_j^i\in \Gamma$ in the viscosity sense. (Actually one can show that $(\varphi_{\eps})_{i\bar{j}}$ is bounded below for each $\eps>0$)
\item ($\varphi_{\eps}$ approximates $\varphi$ in $L^1$) \\There exists $\gamma_2>0$ such that $||\varphi_{\eps}-\varphi||_{L^1}\le C\eps^{\gamma_2}$.
\item ($\varphi_{\eps}$ approximates $\varphi$ from above) \\$\varphi_{\eps}(z)\ge \sup_{z'\in B_{\eps}(z)}\varphi(z')-C\eps$.
\end{enumerate}
In this work, we try a sup convolution to construct the approximation, namely we define
\begin{equation}\label{1.3N}
\varphi_{\eps}(z)=\sup_{\xi\in T_zM}\big(\varphi(\exp_z(\xi))+\eps-\frac{1}{\eps}|\xi|_z^2)\big).
\end{equation}
In the above, $\exp_z(\xi)$ is the exponential map defined using $\omega_0$, and $|\xi|_z$ is the length of the tangent vector (at $z$).
Unfortunately, at this moment, we need to require $M$ to have nonnegative holomorphic bisectional curvature. With this assumption, one has:
\begin{prop}\label{p1.4}
(Theorem \ref{hessian L1 estimate}) Assume that $M$ has nonnegative holomorphic bisectional curvature. Let $\varphi$ be $\Gamma$-admissible and bounded in the viscosity sense. Define $\varphi_{\eps}$ according to (\ref{1.3N}), then (1), (2), (3) above holds.
\end{prop}
The requirement of being compact K\"ahler and having nonnegative holomorphic bisectional curvature is rather restrictive. Mok \cite{Mok} has a uniformization theorem which says that its universal cover is isometrically biholomorphic to $(\mathbb{C}^k,g_0)\times (\mathbb{P}^{N_1},\theta_1)\times\cdots(\mathbb{P}^{N_l},\theta_l)\times (M_1,g_1)\times\cdots(M_p,g_p)$.
In the above, $g_0$ is the standard Euclidean metric. $M_1$, $\cdots$, $M_p$ are Hermitian symmetric spaces with $g_1,\cdots g_p$ being canonical metrics on them, and $\theta_1$, $\cdots$, $\theta_l$ are metrics on $\mathbb{P}^{N_1}$, $\cdots$, $\mathbb{P}^{N_l}$ carry nonnegative holomorphic bisectional curvature.
In our work, the only reason we need this additional curvature assumption is to estimate the lower bound of complex Hessian for $\varphi_{\eps}$ (i.e. to show $\varphi_{\eps}$ is almost $\Gamma$-admissible in the viscosity sense).
We hope to relax or remove such curvature assumptions in the future.
\iffalse
Let $M$ be a compact K\"ahler manifold with a K\"ahler form which can be expressed in a coordinate as:
\begin{equation*}
\omega_0 =\frac{i}{2}\sum_{k,j}g_{k\bar{j}}dz^k \wedge d\bar{z}^j.
\end{equation*}
Let $\varphi$ be a upper-semi-continuous function defined on $M$. Define $(h_{\varphi})^i_j=g^{i\bar{l}}(g+dd^c \varphi)_{j\bar{l}}$. We want to consider the H\"older continuity of the solution to the following equation:
\begin{equation}\label{main equation}
f(\lambda[h_{\varphi}])=e^h.
\end{equation}
$f$ satisfies the following conditions:\\
(1) $f$ is defined on a symmetric convex cone $\Gamma$.\\
(2) $f$ is concave, and is bounded from below by $-C_0$ on $\Gamma$.\\
(3) $\frac{\partial}{\partial i}f>0$ for each $0\le i\le n$.\\
(4) $det \frac{\partial F}{\partial h_{ij}} \ge c>0$, where $F(h_{\phi})=f(\lambda[h_{\varphi}])$\\
(5) $\sum_i \lambda_i \frac{\partial f}{\partial \lambda_i}\le C_0 f$.\\
$\varphi$ is call an admissible solution if $\lambda(dd^c\varphi+\omega)\in \Gamma$. One example of $f$ that satisfies the above conditions is the $\sum_k$ equations.\\
\fi
Combining Proposition \ref{p1.2} and \ref{p1.4}, we can show that solutions to (\ref{1.1New}) with $e^F\in L^{p_0}$ for some $p_0>1$ is H\"older continuous. Indeed, we take $v=\frac{1}{1+\frac{C\eps^{\gamma_1}}{2}}\varphi_{\eps}$, then $v$ would satisfy the assumptions in Proposition \ref{p1.2}. Then we can take $\delta=C\eps^{\gamma_1}$ and we can take $s_0=\eps^a$ for some $a>0$ and small. This is possible due to some additional estimates regarding (1) and (2) in Proposition \ref{p1.2}. So the estimate would give us that:
\begin{equation*}
\sup_M(v-\varphi)\le C\eps^b.
\end{equation*}
Since $\varphi_{\eps}$ is uniformly bounded, this easily translates to:
\begin{equation*}
\sup_{B_{\eps}(z)}\varphi(z')\le \sup_M(\varphi_{\eps}-\varphi)+C\eps\le C'\eps^{b'}.
\end{equation*}
This would imply that $\varphi$ is H\"older continuous.
More precisely, we have:
\begin{thm}
(Theorem \ref{t4.1})
Let $(M,\omega_0)$ be a compact K\"ahler manifold with nonnegative holomorphic bisectional curvature. Let $\varphi$ be a solution to (\ref{1.1New}) with $f$ satisfying the structural assumptions and the right hand side $e^F\in L^{p}(\omega_0^n)$ for some $p>n$. Then for any $\gamma<\frac{p-n}{2(p-n+pn)}$ we have that $||\varphi||_{C^{\gamma}}\le C'$, with $C'$ depends on the background metric, $p_0$, $n$, as well as $||e^F||_{L^{p}}$ as well as the choice of $\gamma$.
\end{thm}
This theorem corresponds to Theorem \ref{t4.1}, with $p_0=\frac{p}{n}$.
If we specialize to the $\sigma_m$ equation by taking $f(\lambda)=\sigma_m^{\frac{1}{m}}(\lambda)$, then we have:
\begin{cor}
Let $(M,\omega_0)$ be a compact K\"ahler manifold with nonnegative holomorphic bisectional curvature. Let $\varphi$ be an $m$-subharmonic solution to the $\sigma_m$ equation:
\begin{equation*}
(\omega_0+\sqrt{-1}\partial\bar{\partial}\varphi)^m\wedge \omega_0^{n-m}=e^F\omega_0^n,\,\,\,\lambda\big(g^{i\bar{k}}(g_{\varphi})_{j\bar{k}}\big)_j^i\in\Gamma_m.
\end{equation*}
Assume that $e^F\in L^p(\omega_0^n)$ for some $p>\frac{n}{m}$, then for any $\gamma<\frac{mp-n}{2(mp-n+mpn)}$, one has $||\varphi||_{C^{\gamma}}\le C'$, where $C'$ depends additionally on the background metric and $||e^F||_{L^p(\omega_0^n)}$ and also the choice of $\gamma$.
\end{cor}
As far as we know, the previous results regarding H\"older continuity to solutions of $\sigma_m$-equation are for domains of $\mathbb{C}^n$ and $\omega_0$ is the standard Euclidean metric in $\mathbb{C}^n$, see \cite{BAZ} and \cite{NC}.
The organization of the paper is as follows.
In Section 2, we develop the theory for upper approximation for bounded functions for which the comlex Hessian lies in a general cone $\Gamma$, assuming the holomorphic bisectional curvature of the underlying manifold is nonnegative.
In Section 3, we use the approximations found in section 2 to show the H\"older continuity of solutions to complex Hessian equations.
\iffalse
There are some previous works related to this problem. With $f\in L^p$ for $p>1$, Kolodziej showed that the admissible solution to the complex Monge-Ampere equation has bounded $L^{\infty}$ norm and has a control on the modulus of continuity in \cite{K1}. Then he showed in \cite{K2} that under the same assumption one can get the estimate of the H\"older norm of the solution. Later it was shown by Dinew that under the additional assumption that the orthogonal bisectional curvature is non-negative, the H\"older constant depends only on the dimension of the manifold and $p$. Then it was shown in the \cite{DPGDHKZ} that the H\"older constant can be estimated if we have bounded geometry.
For the complex $\sum_k$ equation, it is proved in \cite{DK} that an $L^{\infty}$ estimate still holds. Besides, \cite{BAZ} proved a H\"older continuous subsolution theory. For the general complex hessian equation, it was proved in \cite{GP} that an $L^{\infty}$ estimate still holds.\\
In general, in order to construct the H\"older estimate, we need two steps: \\
(1) Defining proper approximation functions.\\
(2) Proving stability results\\.
For the first step, we hope that the approximation functions have good hessian estimates. A usual way to do this is using a convolution of $\varphi$ (see \cite{Demailly} and \cite{DP}) or take maximum of $\varphi$ in small balls (see \cite{K2}). These papers all deal with pluri-subharmonic functions. The difficulty of applying this method to the $m-$subharmonic functions and even the admissible solutions of general complex hessian equations is that the plurisubharmonic functions are invariant under the change of coordinate but functions in other cases are not. Instead, we try to use an approximation method similar to the method used in \cite{JLS} for viscosity solutions:
\begin{defn}\label{uepsilon}
For each $\eps\le \eps_0$, we define
\begin{equation}
\varphi_{\eps}(x)=\sup_{|\xi|\le C_0}(\varphi(exp_x(\xi))+\eps-\frac{1}{\eps}|\xi|^2),
\end{equation}
where $\xi\in T_x M$ and both $\eps_0$ and $C_0$ are some small constants.
\end{defn}
For the second step...
\fi
\section{Approximation using sup-convolution}
In this section, we verify that the $\varphi_{\eps}$ defined by (\ref{1.3N}) will be a good upper approximation of $\varphi$, satisfying (1), (2) and (3) above (\ref{1.3N}). More precisely, we have the following theorem:
\begin{thm}\label{hessian L1 estimate}
Suppose that $(M,\omega_0)$ is a K\"ahler manifold such that the holomorphic bisectional curvature is non-negative. Let $\varphi\in C(M)$ and $\lambda[h_{\varphi}]\in \Gamma$ in the viscosity sense, as defined by Definition \ref{d2.3}.
Define $\varphi_{\eps}$ as in (\ref{1.3N}). Then we have:\\
$(i)$ $\varphi_{\eps}$ is semi-convex and $|\varphi_{\eps}|\le C$ for a constant $C$ depending only on $||\varphi||_{L^{\infty}}$. \\
$(ii)$ There exists a constant $C$ depending on $||\varphi||_{\infty}$ and $(M,\omega_0)$ such that:
\begin{equation*}
\lambda(g^{i\bar{k}} (\varphi_{\eps})_{i\bar{k}} +(1+C\eps^{\frac{1}{2}})I)\in \Gamma.
\end{equation*}
$(iii)$ There exists a constant $C$ depending on $||\varphi||_{\infty}$ and $(M,\omega_0)$ such that:
\begin{equation*}
||\varphi_{\eps}-\varphi||_{L^1}\le C \eps^{\frac{1}{4}}.
\end{equation*}
Moreover, if we have that $\varphi$ is H\"older continuous with a H\"older exponent $\gamma_0$, then we can improve the above results as follows:\\
$(ii')$ There exists a constant $C$ depending on $||\varphi||_{C^{0,\gamma_0}}$ and $(M,\omega_0)$ such that:
\begin{equation*}
\lambda(dd^c \varphi_{\eps} +(1+C\eps^{\frac{1+\gamma_0}{2-\gamma_0}})\omega_0)\in \Gamma.
\end{equation*}
$(iii')$ There exists a constant $C$ depending on $||\varphi||_{C^{0,\gamma_0}}$ and $(M,\omega_0)$ such that:
\begin{equation*}
||\varphi_{\eps}-\varphi||_{L^1}\le C \eps^{\frac{1}{2(2-\gamma_0)}}.
\end{equation*}
\end{thm}
Demaily used a special holomorphic coordinate in his paper \cite{DP} to simplify the calculation. We give the following definition in light of his idea.
\begin{defn}\label{d2.2}
We call a holomorphic coordinate $(z)$ normalized coordinate if in this coordinate the K\"ahler form has the following expansion:
\begin{equation*}
(\omega_0)_{i\bar{k}}=\delta_{i k} -c_{i k \alpha \beta} z_{\alpha} \bar{z}_{\beta}+O(|z|^3)
\end{equation*}
\end{defn}
In \cite{DP}, Demailly define the following approximation for an $\omega_0$-psh function $\varphi$:
\begin{equation*}
\Psi(z,w)=\int_{\zeta\in T_zM}\psi(exph_z(w\zeta))\chi(|\zeta|^2)d\lambda(\zeta).
\end{equation*}
In the above, $\chi$ is a smoothing kernel with compact support. exph is a modification of the usual exponential map which erases all the non-holomorphic terms with respect to $\zeta$ in the Taylor expansion of $\exp_z(\zeta)$ at $\zeta=0$. The estimates in \cite{DP} depends crucially on that $\sqrt{-1}\partial\bar{\partial}\psi\ge 0$, and does not seem to generalize to when it may have negative eigenvalues.
\iffalse With the above definition, he obtained the following estimates (equation (4.3) in \cite{DP}): for any $(\theta, \eta)\in T_xM\times \mathbb{C}$, and with $\tau=\theta+\eta\zeta+O(|w|)$,
\begin{equation*}
\begin{split}
&\frac{\sqrt{-1}}{\pi}\partial\bar{\partial}\Psi_{(x,w)}[\theta,\eta]^2\ge\frac{1}{\pi}|w|^2\int_{\mathbb{C}^n}-\chi_1(|\zeta|^2)\sum_{j,k,l,m}\frac{\partial^2\psi}{\partial\bar{z}_l\partial z_m}(exph_x(w\zeta))\\
&\times \big(c_{jklm}+\frac{1}{|w|^2}\delta_{jm}\delta_{kl}\big)\tau_j\bar{\tau}_kd\lambda(\zeta)-K'(|\theta||\eta|+|\eta|^2).
\end{split}
\end{equation*}
\fi
Now we work with (\ref{1.3N}) and our first step is to get a lower bound of the complex Hessian. In this work, we will mostly work with the notion of viscosity solutions. Following \cite{JLS}, we can define:
\begin{defn}\label{d2.3}
Let $\varphi\in C(M)$, we say that $\lambda[h_{\varphi}]\in\Gamma$ in the viscosity sense, if for all $x_0\in M$ and all $P\in C^2(M)$ touching $\varphi$ from above at $x_0$ ($P(x_0)=\varphi(x_0)$, $P(x)\ge \varphi(x)$ in a neighborhood of $x_0$), we have
\begin{equation*}
\lambda[h_P](x_0)=\lambda\big((g^{i\bar{k}}(g_{j\bar{k}}+\partial_j\partial_{\bar{k}}P))_j^i\big)(x_0)\in \Gamma.
\end{equation*}
\end{defn}
\begin{rem}
We note that if $\varphi\in C^2(M)$ with $\lambda[h_{\varphi}]\in\Gamma$ pointwise, then $\lambda[h_{\varphi}]\in\Gamma$ also in viscosity sense. Indeed, if $P$ touches $\varphi$ from above at $x_0$, then we would have $P_{i\bar{j}}(x_0)\ge \varphi_{i\bar{j}}(x_0)$. Hence if we take eigenvalues, $\lambda_i(\lambda[h_P])\ge \lambda_i(\lambda[h_{\varphi}])$, $1\le i\le n$. But $\Gamma$ is a convex cone containing $\Gamma_n:=\{\lambda\in\mathbb{R}^n:\lambda_i>0\}$, we get $\lambda[h_P]\in\Gamma$.
\end{rem}
First we show that $\lambda[h_{\varphi}]\in\Gamma$ in the viscosity sense translate to a similar condition for $\varphi_{\eps}$. We start with the following simple observation.
\begin{lem}\label{l2.5}
\begin{enumerate}
\item
Let $\varphi\in C(M)$ and we define $\varphi_{\eps}$ according to (\ref{1.3N}). Let $x_0\in M$ and $P$ be a $C^2$ function defined in a neighborhood of $x_0$ touching $\varphi_{\eps}$ from above. Assume that $\varphi_{\eps}(x_0)=\varphi(\exp_{x_0}(\xi_0))+\eps-\frac{1}{\eps}|\xi_0|^2_{x_0}$ for some $\xi_0\in T_{x_0}(M)$. Let $\xi(z)\in T_zM$ be a smooth vector field defined in a neighborhood of $x_0$ with $\xi(x_0)=\xi_0$. Define $\phi(z)=\exp_z(\xi(z))$, then $z\mapsto P-\eps+\frac{1}{\eps}|\xi(z)|^2_z$ touches $\varphi\circ \phi$ from above at $z_0$.
\item If the map $\phi$ is holomorphic with $D_z\phi(x_0)$ being invertible, then $w\mapsto (P-\eps+\frac{1}{\eps}|\xi|^2)\circ \phi^{-1}(w)$ touches $\varphi$ from above at $\exp_{x_0}(\xi_0)$.
\end{enumerate}
\end{lem}
\begin{proof}
The proof is rather straightforward out of the definition. For part (1), let $P$ touch $\varphi_{\eps}$ from above will lead to the following string of inequalities:
\begin{equation*}
P(z)\ge \varphi_{\eps}(z)\ge \varphi(\exp_z(\xi(z)))+\eps-\frac{1}{\eps}|\xi(z)|^2_z,
\end{equation*}
with equality achieved at $x_0$.
This just means $P(z)+\frac{1}{\eps}|\xi(z)|^2_z-\eps$ touches $\varphi\circ \phi$ from above.
For part (2), the additional assumption on $\phi$ implies that $\phi$ is actually biholomorphic between a neighborhood of $x_0$ and a neighborhood of $w_0:=\exp_{x_0}(\xi_0)$.
\end{proof}
As a direct consequence, we have
\begin{cor}\label{c2.6}
Assume that $\lambda[h_{\varphi}]\in\Gamma$ in the viscosity sense. Let $\xi(z)$ and $\phi(z)$ satisfy the assumptions in point (2) of Lemma \ref{l2.5}. Put $w_0=\exp_{z_0}(\xi_0)$ and $Q(w)=(P-\eps+\frac{1}{\eps}|\xi(z)|^2_z)\circ \phi^{-1}(w)$. Then $\lambda(g^{i\bar{k}}(g_{j\bar{k}}+\partial_{w_j\bar{w}_k}Q)(w_0)\in \Gamma$.
\end{cor}
In order to get the estimate of Hessian for $\varphi_{\eps}$ (in the viscosity sense), we need to translate Corollary \ref{c2.6} to a statement at $z_0$. For this we have:
\begin{lem}\label{l2.7}
$\lambda(g^{i\bar{k}}(g_{j\bar{k}}+\partial_{w_j\bar{w}_k}Q))(w_0)$ is the same as the set of roots of
\begin{equation*}
f(\lambda)= \mbox{det} \big((\lambda-1) \sum_{i,j}g_{i\bar{j}}(w_0)\frac{\partial \phi_i}{\partial z_a}\frac{\partial \phi_{\bar{j}}}{\partial\bar{z}_b}(x_0)-(P+\eps-\frac{1}{\eps}|\xi(z)|^2_z)_{z_a\bar{z}_b}(x_0)\big)_{1\le a,b\le n}.
\end{equation*}
\end{lem}
\begin{proof}
First we observe that
\begin{equation}\label{2.8NN}
Q_{i\bar{j}}(w_0)=\frac{\partial(\phi^{-1})_a}{\partial w_i}\frac{\partial(\phi^{-1})_{\bar{b}}}{\partial\bar{w}_j}(P-\eps+\frac{1}{\eps}|\xi(z)|^2_z)_{z_a\bar{z}_b}(x_0).
\end{equation}
On the other hand, $\lambda(g^{i\bar{k}}(g_{j\bar{k}}+\partial_{w_j\bar{w}_k}Q))(w_0)$ is the set of roots for:
\begin{equation*}
\begin{split}
& \mbox{det}\big(\lambda \delta_{ij}-g^{i\bar{k}}(g_{j\bar{k}}+\partial_{j\bar{k}}Q)(w_0)\big)= \mbox{det}\big((\lambda-1) \delta_{ij}-g^{i\bar{k}}\partial_{j\bar{k}}Q(w_0)\big)\\
&= \mbox{det}(g^{-1}) \mbox{det}\big((\lambda-1)g_{i\bar{j}}(w_0)-Q_{i\bar{j}}(w_0)\big)\\
&= \mbox{det}(g^{-1})| \mbox{det}(D_w\phi^{-1})(w_0)|^2 \mbox{det}\big((\lambda-1)\sum_{i,j}g_{i\bar{j}}(w_0)\frac{\partial\phi_i}{\partial z_a}\frac{\partial\phi_{\bar{j}}}{\partial\bar{z}_b}(z_0)\\
&-(P+\eps-\frac{1}{\eps}|\xi(z)|^2_z)_{z_a\bar{z}_b}\big).
\end{split}
\end{equation*}
In the last equality, we used (\ref{2.8NN}).
\end{proof}
Since we want to ``translate" from $w_0=\exp_{z_0}(\xi_0)$, we wish to take the map $\phi$ to satisfy:
\begin{equation}\label{2.9N}
\sum_{i,j}g_{i\bar{j}}(w_0)\frac{\partial\phi_i}{\partial z_a}\frac{\partial\phi_{\bar{j}}}{\partial\bar{z}_b}(z_0)=g_{a\bar{b}}(x_0),\,\,\phi \text{ holomorphic},\,\, \mbox{det} D_z\phi(x_0)\neq 0.
\end{equation}
Next let us explain how to achieve (\ref{2.9N}). First let us observe that $w_0$ and $x_0$ are actually very close if $\eps$ is small enough. Indeed,
\begin{lem}\label{l2.10}
Let $\varphi\in C(M)$ and $\varphi_{\eps}$ be as defined by (\ref{1.3N}). Let $\xi_0\in M$ and $\xi_0\in T_{x_0}M$. We assume that $\xi_0$ achieves the sup in the definition of $\varphi_{\eps}$, then
\begin{equation*}
|\xi_0|_{x_0}\le (2||\varphi||_{L^{\infty}}\eps)^{\frac{1}{2}}.
\end{equation*}
In particular, $d(w_0,x_0)\le C\eps^{\frac{1}{2}}$.
If we assume additionally that $\varphi\in C^{\alpha}(M)$, then
\begin{equation*}
|\xi_0|_{x_0}\le (\eps[\varphi]_{\alpha})^{\frac{1}{2-\alpha}},
\end{equation*}
where $[\varphi]_{\alpha}=\sup_{x,y\in M}\frac{|\varphi(x)-\varphi(y)|}{d(x,y)^{\alpha}}$.
\end{lem}
\begin{proof}
Since $\xi_0$ achieves the sup, we have:
\begin{equation*}
\varphi(\exp_{x_0}(\xi_0))+\eps-\frac{1}{\eps}|\xi_0|^2_{x_0}\ge \varphi(x_0)+\eps-0.
\end{equation*}
In the right hand side above, we are taking $\xi=0$. If $\varphi$ does not have any estimate on the modulus of continuity, we can only estimate $\varphi(\exp_{x_0}(\xi_0))-\varphi(x_0)$ by $2||\varphi||_{L^{\infty}}$. If $\varphi\in C^{\alpha}$, then we have
\begin{equation*}
\varphi(\exp_{x_0}(\xi_0))-\varphi(x_0)\le [\varphi]_{\alpha}d(\exp_{x_0}(\xi_0),x_0)^{\alpha}\le [\varphi]_{\alpha}|\xi_0|_{x_0}^{\alpha}.
\end{equation*}
\end{proof}
As a direct consequence of this, we note that $\varphi_{\eps}$ is semi-convex:
\begin{lem}\label{l2.11New}
Define $\varphi_{\eps}$ according to (\ref{1.3N}). Then $\varphi_{\eps}$ is semi-convex for $\eps$ small enough, in the sense that for any $z_0\in M$, there is a coordinate neighborhood of $z_0$ such that $\varphi_{\eps}(z)+C_{\eps}|z-z_0|^2$ is locally convex near $z_0$.
In particular, $\varphi_{\eps}$ is second order differentiable a.e.
\end{lem}
\begin{proof}
Since for any $z\in M$, $D_{\xi}\exp_z(\xi)|_{\xi=0}$ is invertible, we see that, by implicity function theorem, there exists a neighborhood $U_0$ of $z_0$, such that for any $z\in U_0,\,w\in U_0$, there is a unique $\xi\in T_zM$ such that $\exp_z(\xi)=w$. We can denote $\xi=\log_z(w)$ and we may assume that this map is smooth.
Because of Lemma \ref{l2.10}, we have
\begin{equation*}
\varphi_{\eps}(z)=\sup_{|\xi|_z\le (2||\varphi||_{L^{\infty}})^{\frac{1}{2}}}(\varphi(\exp_z\xi)+\eps-\frac{1}{\eps}|\xi|^2)=\sup_{w\in U_0}\big(\varphi(w)+\eps-\frac{1}{\eps}|\log_z(w)|^2_z\big).
\end{equation*}
We observe that there exists $C_{\eps}>0$ such that
\begin{equation*}
D_z^2\big(\varphi(w)+\eps-\frac{1}{\eps}|\log_z(w)|^2_z\big)\ge -C_{\eps}I,\,\,\,\text{for any $w\in U_0$.}
\end{equation*}
This essentially follows from that $(z,w)\mapsto -\frac{1}{\eps}|\log_zw|^2_z$ is jointly smooth.
Therefore, taking the sup with respect to $w$ will make it semi-convex in $z$ with the same lower bound of the Hessian.
\end{proof}
Next we are going to take the normal coordinate centered at $x_0$ as in Definition \ref{d2.2}. Without loss of generality, we can assume that $x_0$ is represented by $z=0$ under this coordinate. Let $N$ be an $n\times n$ matrix such that:
\begin{equation}\label{2.11N}
\sum_{i,j}N_{ia}\bar{N}_{jb}g_{i\bar{j}}(w_0)=\delta_{ab}=g_{a\bar{b}}(x_0).
\end{equation}
Then under this coordinate, we define
\begin{equation}\label{2.11}
\phi(z)=\exp_{x_0}(\xi_0)+N\cdot z.
\end{equation}
We need to make sure that one can define $\xi(z)$ which satisfies $\exp_z(\xi(z))=\phi(z)$:
\begin{lem}\label{l2.13}
For all $\eps>0$ small enough, there exists a smooth vector field $z\mapsto \xi(z)\in T_zM$ defined in a neighborhood of $x_0$ with $\xi(x_0)=\xi_0$ such that $\exp_z(\xi(z))=\phi(z)$.
\end{lem}
\begin{proof}
This follows from the implicit function theorem.
Indeed, let $U_0$ and $V_0$ be neighborhoods of $0$ and $\xi_0$ respectively (in $\mathbb{C}^n$), then we may consider the nonlinear map (where the exponential map is represented using the above normal coordinates):
\begin{equation*}
F:U_0\times V_0\rightarrow \mathbb{C}^n,\,\,\,(z,\xi)\mapsto \exp_z(\xi)-\exp_{x_0}(\xi_0)-N\cdot z.
\end{equation*}
It is clear that $F(0,\xi_0)=0$. Also we find that $D_{\xi}F(0,\xi_0)=D_{\xi}(\exp_{x_0}(\xi))|_{\xi=\xi_0}$. Note that $D_{\xi}(\exp_{x_0}(\xi))|_{\xi=0}=id$, we see that $D_{\xi}F(0,\xi_0)$ will be invertible if $\eps$ is small enough, thanks to Lemma \ref{l2.10}. Then we can apply the implicity function theorem to define $\xi(z)$ with $F(z,\xi(z))=0$.
\end{proof}
Combining Corollary \ref{c2.6} and Lemma \ref{l2.7}, we can conclude that:
\begin{prop}\label{p2.14}
Let $\varphi\in C(M)$ and $\lambda[h_{\varphi}]\in \Gamma$ in the viscosity sense. Define $\varphi_{\eps}$ according to (\ref{1.3N}). Let $\phi$ be given by (\ref{2.11}) and $\xi(z)$ be given by Lemma \ref{l2.13}. Let $P\in C^2(M)$ and touch $\varphi_{\eps}$ from above at $x_0\in M$, then
\begin{equation*}
\lambda\big(g^{i\bar{k}}(g_{j\bar{k}}+\partial_{z_j\bar{z}_k}P+(\frac{1}{\eps}|\xi(z)|^2_z)_{z_j\bar{z}_k})(x_0)\big)\in\Gamma.
\end{equation*}
\end{prop}
At this moment, it is clear that we have to estimate the lower bound of the complex Hessian for $\frac{1}{\eps}|\xi(z)|^2_z$ at $x_0$.
We will do it now.
First,
\begin{equation}\label{2.15N}
\begin{split}
&(|\xi(z)|^2)_{i\bar{j}}(x_0)=(g_{a\bar{b}}(z)\xi_a(z)\bar{\xi}_{\bar{b}}(z))_{i\bar{j}}=g_{a\bar{b},i\bar{j}}(x_0)\xi_a\bar{\xi}_{\bar{b}}(x_0)+g_{a\bar{b}}(x_0)(\xi_a\bar{\xi}_{\bar{b}})_{i\bar{j}}(x_0)\\
&+g_{a\bar{b},i}(x_0)(\xi_a\bar{\xi}_{\bar{b}})_{\bar{j}}(x_0)+g_{a\bar{b},\bar{j}}(x_0)(\xi_a\bar{\xi}_{\bar{b}})_i(x_0)=-c_{abij}\xi_a\bar{\xi}_{\bar{b}}(x_0)\\
&+\xi_{a,i\bar{j}}(x_0)\bar{\xi}_{\bar{a}}+\xi_a(x_0)\bar{\xi}_{\bar{a},i\bar{j}}(x_0)+\xi_{a,i}(x_0)\bar{\xi}_{\bar{a},\bar{j}}(x_0)+\xi_{a,\bar{j}}(x_0)\xi_{a,i}(x_0).
\end{split}
\end{equation}
In the above calculation, we used that $g_{a\bar{b}}(x_0)=\delta_{ab}$, $g_{a\bar{b},i}(x_0)=0$. Next we need to estimate $D\xi$ and $D^2\xi$.
For this we have:
\begin{lem}
Let $\xi(z)$ be defined by $\exp_z(\xi(z))=\phi(z)$ with $\phi(z)=\exp_{x_0}(\xi_0)+N\cdot z$ in a neighborhood of $x_0$, where we have taken normal coordinates centered at $x_0$.
Then there is a constant $C>0$ depending only on the background metric, such that
\begin{equation*}
|D\xi|(x_0)\le C|\xi_0|_{x_0}^2,\,\,\,|D^2\xi|(x_0)\le C|\xi_0|_{x_0}^2.
\end{equation*}
\end{lem}
\begin{proof}
The result follows from differentiating the equality $\exp_z(\xi(z))=\phi(z)$. Differentiate once, we get:
\begin{equation}\label{2.17}
D_z\phi=D_z\exp_z(\xi)|_{\xi=\xi(z)}+D_{\xi}\exp_z(\xi)|_{\xi=\xi(z)}\cdot D_z\xi(z).
\end{equation}
Differentiate in $z$ again:
\begin{equation}\label{2.18}
\begin{split}
&D_z^2\phi=D_z^2\exp_z(\xi)|_{\xi=\xi(z)}+2D^2_{z\xi}\exp_z(\xi)|_{\xi=\xi(z)}\cdot D_z\xi(z)\\
&+(D_z\xi)^T\cdot D^2_{\xi\xi}\exp_z(\xi)|_{\xi=\xi(z)}\cdot D_z\xi+D_{\xi}\exp_z(\xi)\cdot D_z^2\xi(z).
\end{split}
\end{equation}
Now we evaluate (\ref{2.17}) at $z=0$, so that we get:
\begin{equation*}
N=D_z\exp_z(\xi)|_{z=0,\xi=\xi_0}+D_{\xi}\exp_z(\xi)|_{z=0,\xi=\xi_0}D_z\xi(z)|_{z=0}.
\end{equation*}
From Lemma \ref{l2.19} below, we have that:
\begin{equation*}
|D_z\exp_z(\xi)|_{z=0,\xi=\xi_0}-I|\le C|\xi_0|^2_{x_0},\,\,\,|D_{\xi}\exp_z(\xi)|_{z=0,\xi=\xi_0}-I|\le C|\xi_0|^2.
\end{equation*}
On the other hand, we use (\ref{2.11N}), and note that under normal coordinates, at $x_0$ we have:
\begin{equation*}
|g_{i\bar{j}}(w_0)-\delta_{ij}|\le Cd(w_0,x_0)^2\le C|\xi_0|^2_{x_0},
\end{equation*}
This would imply that we could choose $N$ so that $|N-I|\le C|\xi|_{x_0}^2$.
Hence
\begin{equation*}
D_z\xi|_{z=x_0}=(D_\xi\exp_z\xi)^{-1}|_{z=0,\xi=\xi_0}\big(N-D_z\exp_z(\xi)|_{z=0,\xi=\xi_0}\big).
\end{equation*}
Hence we can obtain that $|D_z\xi(x_0)|\le C|\xi_0|_{x_0}^2$.
To estimate $D_z^2\xi$, we use (\ref{2.18}). We note that, $D_z^2\phi(z)=0$ and Lemma \ref{l2.19} gives $|D_z^2\exp_z(\xi)|_{z=0,\xi=\xi_0}|\le C|\xi_0|^2$. Also we just need that $D_{z\xi}\exp_z(\xi)|_{z=0,\xi=\xi_0}$ and $D_{\xi\xi}^2\exp_z(\xi)|_{z=0,\xi=\xi_0}$ is bounded.
\end{proof}
In the above, we used the following lemma about the Taylor expansion of the exponential map:
\begin{lem}\label{l2.19}
The following Taylor expansion holds for expential map on a K\"ahler manifold, near $z=0$, $\xi=0$ under normal coordinates centered at $z=0$:
\begin{equation*}
\exp_z(\xi)_m=z_m+\xi_m+\sum_{j,k,l}c_{jklm}(\frac{1}{2}\bar{z}_k+\frac{1}{6}\bar{\xi}_k)\xi_j\xi_l+O(|\xi|^2(|z|^2+|\xi|^2)).
\end{equation*}
\end{lem}
\begin{proof}
This is the equation (2.7) in \cite{DP} and a detailed proof can be found in Section 2 of \cite{DP}.
\end{proof}
Using (\ref{2.15N}), we see that:
\begin{lem}\label{l2.20}
There exists a constant $C>0$ depending only on the background metric such that:
\begin{equation*}
|(|\xi(z)|^2_z)_{z_i\bar{z_j}}(x_0)+\sum_{a,b}c_{abij}\xi_{0,a}\bar{\xi}_{0,\bar{b}}|\le C|\xi_0|_{x_0}^3.
\end{equation*}
\end{lem}
Combining Lemma \ref{l2.10}, Proposition \ref{p2.14} and Lemma \ref{l2.20}, we conclude the following corollary:
\begin{cor}\label{c2.22N}
Let $\varphi\in C(M)$ and $\lambda[h_{\varphi}]\in\Gamma$ in the viscosity sense. Define $\varphi_{\eps}$ according to (\ref{1.3N}). Let $P\in C^2(M)$ and touch $\varphi_{\eps}$ at $x_0\in M$. Assume that $M$ has nonnegative holomorphic bisection curvature at $x_0\in M$, in the sense that $c_{ijkl}\eta_i\bar{\eta}_j\xi_k\bar{\xi}_l\ge 0$. Then we have:
\begin{equation*}
\lambda\big(g^{i\bar{k}}(g_{j\bar{k}}+\partial_{z_j\bar{z}_k}P+C\eps^{\frac{1}{2}}g_{j\bar{k}})\big)(x_0)\in\Gamma.
\end{equation*}
Here the constant $C$ depends on the background metric and $||\varphi||_{L^{\infty}}$.
If in addition, we assume that $\varphi\in C^{\alpha}$, we can get:
\begin{equation*}
\lambda\big(g^{i\bar{k}}(g_{j\bar{k}}+\partial_{z_j\bar{z}_k}P+C\eps^{\frac{1+\alpha}{2-\alpha}}g_{j\bar{k}})\big)(x_0)\in\Gamma.
\end{equation*}
Here $C$ depends on the background metric and $C^{\alpha}$ norm of $\varphi$.
\end{cor}
\begin{proof}
First let us make the curvature assumption but do not assume $\varphi\in C^{\alpha}$. Then Lemma \ref{l2.10} tells us that $|\xi_0|_{x_0}\le C\eps^{\frac{1}{2}}$. Hence Proposition \ref{p2.14} would give us that:
\begin{equation*}
\lambda\big(g^{i\bar{k}}(g_{j\bar{k}}+C\eps^{\frac{1}{2}}g_{j\bar{k}}+\partial_{z_j\bar{z}_k}P-\sum_{a,b}c_{abjk}\xi_{0,a}\bar{\xi}_{0,\bar{b}})\big)(x_0)\in \Gamma.
\end{equation*}
The curvature assumption tells us that $\big(\sum_{a,b}c_{abij}\xi_{0,a}\bar{\xi}_{0,\bar{b}}\big)_{1\le i,j\le n}$ is semi-positive definite. It follows that
\begin{equation*}
\lambda\big(g^{i\bar{k}}(g_{j\bar{k}}+C\eps^{\frac{1}{2}}g_{j\bar{k}}+\partial_{z_j\bar{z}_k}P)\big)(x_0)\in\Gamma.
\end{equation*}
If we assume that $\varphi\in C^{\alpha}(M)$, then we can estimate:
\begin{equation*}
\frac{1}{\eps}(|\xi(z)|^2_z)_{z_i\bar{z_j}}\le \frac{C|\xi_0|^3_{x_0}}{\eps}\delta_{ij}\le C\eps^{-1}(\eps[\varphi]_{\alpha})^{\frac{3}{2-\alpha}}\le C'\eps^{\frac{1+\alpha}{2-\alpha}}.
\end{equation*}
Here we used Lemma \ref{l2.10} and Lemma \ref{l2.20}.
\end{proof}
\begin{rem}
This is the only place in the whole proof that we use the curvature assumption.
\end{rem}
From now on, let us estimate the $L^1$ difference between $\varphi_{\eps}$ and $\varphi$.
For this we need a version of mean value inequality on manifolds which is in light of \cite{C}.
\begin{prop}\label{mean value inequality with a error term}
Let $u$ be a upper semicontinuous function defined on an open subset $U$ of a Riemannian manifold ($M,g$). Suppose that $\Delta_g u \ge 0$, $u$ is bounded and the Ricci curvature is bounded on $M$. Then there exists a constant $C$ which is a uniform constant depending on the bound of $||u||_{L^{\infty}}$ and $(M,g)$ such that:
\begin{equation*}
u(x)\le \frac{1}{\alpha(2n)r^{2n}}\int_{B_r(x)}udvol_g+Cr,
\end{equation*}
for all $B_r(x)\subset U$. Here $\alpha(2n)$ is the volume of the unit ball in $\mathbb{R}^{2n}$.
\end{prop}
\iffalse
\begin{lem}\label{family of normalized coordinates}
For any $x\in M$, there exists a neighbourhood $U_x$ and a family of normalized coordinates, where $\phi_y$ are the corresponding coordinate maps
for $y\in U_x$, such that $\phi_y$ depends smoothly on $y$.
\end{lem}
\begin{proof}
For any $x\in M$, we can first construct a normalized coordinate $(z)$ around $x$ following \cite{DP}. Then there exists a neighbourhood $U_x$ of $x$ such that there exists a smooth matrix-valued function $Q_w$ defined for each $w\in U_x$. $Q_w$ satisfies that:
\begin{equation*}
Q_w g(w) Q_w^* =Id
\end{equation*}
Fix $w$, we can define a new coordinate $\widetilde{z}$ by $z=Q_w \widetilde{z}+w$.
In the coordinate $\widetilde{z}$, the K\"ahler form has the following expansion:
\begin{equation*}
\begin{split}
&(\omega_0)_{l\bar{m}}=\delta_{l\bar{m}}+\sum_j(a_{jlm}\widetilde{z}_j +\bar{a}_{jml}\bar{\widetilde{z}}_j) +\sum_{j,k}(b_{jklm}\widetilde{z}_j \widetilde{z}_k \\
&+\bar{b}_{jkml}\bar{\widetilde{z}}_j \bar{\widetilde{z}}_k)-\sum_{j,k}c_{jklm}\widetilde{z}_j \bar{\widetilde{z}}_k +O(|\widetilde{z}|^3).
\end{split}
\end{equation*}
In order to get a normalized coordinate we need to make the following transformation:
\begin{equation*}
z'_m= \widetilde{z}_m-\frac{1}{4}\sum (a_{jlm}+a_{ljm})\widetilde{z}_j \widetilde{z}_l -\frac{1}{3}\sum b_{jklm}\widetilde{z}_j \widetilde{z}_k \widetilde{z}_l.
\end{equation*}
It is easy to verify that $(z')$ is a normalized coordinate.
In order to prove that $(z')$ is a smooth family of normalized coordinates, it suffices to prove that $a_{jlm}$ and $a_{ljm}$ and $b_{jklm}$ are smooth with respect to $w$. We prove for $a_{jlm}$ as an example, the rest are similar.
According to the definition, we have that:
\begin{equation*}
\begin{split}
a_{jlm}&=\frac{\partial}{\partial \widetilde{z}_j}(\omega_w)_{l \bar{m}}|_{\widetilde{z}=0}=\frac{\partial}{\partial \widetilde{z}_j}((Q_w \omega Q_w^*)_{lm})|_{\widetilde{z}=0}\\
&=(Q_w \frac{\partial \omega}{\partial \widetilde{z}_j}Q_w^*)_{lm}|_{z=w} =(Q_w \frac{\partial \omega(w)}{\partial z_a} Q_w^*)_{lm}(
Q_w)^l_j
\end{split}
\end{equation*}
Here $\omega_w$ means the expression of $\omega_0$ in the coordinate of $\widetilde{z}$. From the above formula we have that $a_{jlm}$ depends smoothly on $w$ because $Q_w$ is smooth. This concludes the proof of the lemma.
\end{proof}
\begin{lem}\label{find normalized coordinates}
There exists uniform constants $R$ and $C$ depending only on $(M, \omega_0)$. For any $x\in M$ and $y\in M$ such that $d(x,y)<R$, we can find a normalized coordinate around $x$ such that in this coordinate there exists a matrix $N$ satisfying that:
\begin{equation*}
g(y)=N^{-1} N^{-1*}.
\end{equation*}
and $|N-Id|\le C|y|^2$.
Here we identify $y$ with its corresponding point in the coordinate. Then $|y|$ is the norm of $y$ in this coordinate.
\end{lem}
\begin{proof}
It suffices to prove the lemma locally and then the lemma follows from a standard covering argument.
For any $x\in M$, let $(\phi_w)$ be a family of normalized coordinate maps given by the Lemma \ref{family of normalized coordinates} We define
\begin{equation}\label{h}
h_{ij}(Q,x,z)=((Q\circ \phi_x)_* g -Id)_{ij}(z)
\end{equation}
for $Q\in U(n)$, $x\in U_x$ and $z\in B_{\delta}(0)\subset C^n$. From the above formula, we have that $h_{ij}$ is smooth and satisfies that:
\begin{equation*}
\begin{split}
&h_{ij}(Q,x,0)=0 \\
&h_{ij,\xi}(Q,x,0)=0 \\
&|h_{ij,kl}|\le C_0,
\end{split}
\end{equation*}
Here $\xi$ is any unit vector in $C^n$ and $C_0$ is a constant and $k,l\in \{1,2,...,n\}$. By integration we can get that
\begin{equation}\label{estimate of h}
|h_{ij}(Q,x,\xi)|\le C|\xi|^2.
\end{equation}
Then we fix $Q\in U(n)$ such that $(Q\circ \phi_x)_* g(y)$ is diagonalized. So $N^{-1}$ can be define as $((Q\circ \phi_x)_* g(y))^{\frac{1}{2}}$ by taking the square root of the corresponding diagonal component. From the quations(\ref{h}) and (\ref{estimate of h}) we have that $|N^{-1}-Id|\le C|y|^2$. Since $N^{-1}$ is diagonalized, we also have that $|N-Id|\le C |y|^2$. Then the normalized coordinate required by the lemma can be taken as $Q\circ \phi_x$.
\end{proof}
\fi
\begin{proof}
Fix $r_0>0$ such that $B_{r_0}(x)\subset U$. Since $u$ is upper semicontinuous, there exists a sequence of continuous functions $\{u_k\}$ such that $u\le u_k$ and $\lim_{k\rightarrow \infty}u_k=u$. For example, we could take $u_k(x)=\sup_{y\in M}\big(u(y)+\frac{1}{k}-kd_g^2(x,y)\big)$.
Let us assume that $u<-1$ for the moment so that we can assume that each $u_k<0$.
Let $0<r\le r_0$ and $h_k$ be the harmonic extension of $u_k$ on the $B_{r}(x)$. By the maximum principle, we have that $0>h_k\ge u$. This is because $h_k=u_k\ge u$ on $\partial B_{r}(x_0)$ and $\Delta_g(h_k-u)\le 0$ in $B_{r}(x_0)$.
By the Cheng-Yau harnack inequality (\cite{SY}, Chapter 1, Section 3, Theorem 3.1 ), we have that:
\begin{equation*}
|\nabla \log(-h_k)|_{B_{\frac{r}{2}}(x)}\le C_r,
\end{equation*}
where $C_r$ depends on the Ricci curvature and $r$.
Then we have that for any $s<\frac{r}{2}$,
\begin{equation*}
\inf_{B_s(x)}h_k \ge e^{C_rs}\sup_{B_s(x)}h_k\ge e^{C_rs}\varphi(x).
\end{equation*}
Combining this with the almost monotonicity Lemma \ref{monotonicity lemma} we have that:
\begin{equation*}
\begin{split}
r^{1-2n}\int_{\partial B_r(x)}h_k &\ge s^{1-2n}\int_{\partial B_s(x)}h_k -Cr\\
&\ge e^{C_rs}s^{1-2n}vol(\partial B_s(x))\varphi(x)-Cr.
\end{split}
\end{equation*}
Hence, we have that:
\begin{equation*}
\begin{split}
&\frac{1}{vol(\partial B_r(x))}\int_{\partial B_r(x)}u =\lim_{k\rightarrow \infty}\frac{1}{vol(\partial B_r(x))}\int_{\partial B_r(x)}h_k \\
&\ge \lim_{s\rightarrow 0+}\frac{s^{1-2n}vol(\partial B_s(x))e^{C_rs}u(x)-Cr}{r^{1-2n}vol(\partial B_r(x))}\\
&\ge \frac{2n\alpha(2n)r^{2n-1}}{vol(\partial B_r(x))}u(x)-Cr.
\end{split}
\end{equation*}
Here we use the Lemma \ref{estimate of the volume of balls} below.
So we have that, for any $0<r\le r_0$:
\begin{equation*}
2n \alpha(2n) r^{2n-1}u(x)\le \int_{\partial B_r(x)}u +Cr^{2n}.
\end{equation*}
Now we integrate in $r$, we get:
\begin{equation}\label{2.25NNN}
\alpha(2n)r_0^{2n}u(x)\le \int_{B_{r_0}(x)}u-Cr_0^{2n+1}.
\end{equation}
So we have finished the proof if we assume that $\varphi\le -1$. In general, we consider $\tilde{u}:=u-(||u||_{L^{\infty}}+1)$ so that $\tilde{u}\le -1$. Applying (\ref{2.25NNN}), we see that
\begin{equation*}
\alpha(2n)r_0^{2n}\tilde{u}(x)\le \int_{B_{r_0}(x)}\tilde{u}dvol_g+Cr_0^{2n+1}.
\end{equation*}
Translating back to $u$, we get:
\begin{equation*}
\begin{split}
&\alpha(2n)r_0^{2n}u(x)\le \int_{B_{r_0}(x)}udvol_g+(||u||_{L^{\infty}}+1)(\alpha(2n)r_0^{2n}-vol(B_{r_0}(x)))\\
&+Cr_0^{2n+1}\le C'r_0^{2n+1}.
\end{split}
\end{equation*}
\end{proof}
\begin{lem}\label{estimate of the volume of balls}
Let $(M,g)$ be a compact Riemannian manifold with real dimension $n$. Then there exist $r_0>0$ and a constant $C$ depending only on the manifold such that for any $y\in M$ and $r<r_0$, we have the estimate:
\begin{equation*}
\begin{split}
&|vol(\partial B_r(y))-n\alpha(n)r^{n-1}|\le Cr^n,\\
& |vol(B_r(y))-\alpha(n)r^{n}|\le C r^{n+1}.
\end{split}
\end{equation*}
\end{lem}
\begin{proof}
We want to compare the volume form induced by the K\"ahler form with the Euclidean volume form in the normal coordinates.
Using the covering argument, it suffices to prove that for any $x\in M$, the conclusion of the lemma holds uniformly near $x$ with some $r_0$ and $C$.
We just need to prove the first inequality, as the second inequality would follow from the first by integrating in $r$. First we find, by using the Area formula:
\begin{equation*}
vol(\partial B_r(x))=\int_{S^{n-1}}|J_{\xi\mapsto \exp_x(r\xi)}|d\mathcal{H}^{n-1}(\xi).
\end{equation*}
In the above, $\partial B_r$ is the image of the unit sphere (in $T_xM$) under the map $S^{n-1}\ni \xi\mapsto \exp_x\xi$. $\mathcal{H}^{n-1}$ is the Hausdorff measure induced by the metric $g(x)$ on $T_xM$. Also $J_{\xi\mapsto \exp_x(r\xi)}$ is the Jacobian of the map, which is calculated by taking an othornormal frame of $S^{n-1}$ denoted as $v_1\wedge v_2\cdots\wedge v_{n-1}$, then $J$ is the sup of $|(d_{\xi}(\exp(r\xi))v_1)\wedge \cdots \wedge (d_{\xi}(\exp(r\xi))v_{n-1}|$ over all orthonormal frames of $T_{\xi}S^{n-1}$. Note that $d_{\xi}(\exp_z(\xi))|_{\xi=0}=id$, which has Jacobian equaling $1$, hence
\begin{equation*}
||d_{\zeta}\exp_z(\zeta)|_{\zeta=r\xi}-id||\le Cr,\,\,\xi\in S^{n-1}.
\end{equation*} hence we see that:
\begin{equation*}
J_{\xi\mapsto \exp_x(r\xi)}=r^{n-1}(1+O(r)).
\end{equation*}
Therefore,
\begin{equation*}
vol(\partial B_r(x))=\int_{S^{n-1}}r^{n-1}(1+O(1))d\mathcal{H}^{n-1}(\xi)=r^{n-1}vol(S^{n-1},g_E)+O(r^n).
\end{equation*}
Here $g_E$ just means the Euclidean metric. But $vol(S^{n-1},g_E)=n\alpha(n)$.
\iffalse
First we define for any $w$ in a neighborhood of $x$ a map $\phi_w$ as an isometry between $T_w M$ and $C^n$ which sends the origin to the origin. We can assume that $\phi_w$ depends smoothly on $w$. Let $log_w$ be a local inverse map of $exp_w$ near the origin. Then we define a function:
\begin{equation*}
f(z,w)=\frac{\omega^n (exp_x(z))}{(\phi_w \circ log_w)^* \omega_0^n (\phi_w \circ log_w \circ exp_x(z))},
\end{equation*}
where $\omega_0$ is the standard K\"ahler form on $C^n$. From the definition, we can see that $f$ is smooth. Then we define
\begin{equation*}
\xi=\phi_w\circ log_w \circ exp_x (z)
\end{equation*}
Then we can change the coordinate to express f as $g(\xi,w)=f(z,w)$. Since $g(0,w)=1$ and $g$ is smooth, we have that
\begin{equation}\label{estimate between volume forms}
|g(\xi,w)-1|\le M|\xi|.
\end{equation}
Then we can integrate $\omega_0^n$ and $((exp_w)^* \omega)^n$ in the ball of radius $r$ in normal coordinate at $w$. Note that this ball corresponds to $B_{r}(w)$. Using the Inequality \ref{estimate between volume forms} we can prove the lemma.
\fi
\end{proof}
Another lemma that is used in the proof of Propositon \ref{mean value inequality with a error term} is:
\begin{lem}\label{monotonicity lemma}
Let $M$ be a closed manifold of dimension $2n$ with bounded Ricci curvature. Let $u$ be a bounded negative harmonic function defined on an open subset of $M$. Then we have that for $0<s<r$:
\begin{equation*}
r^{1-2n}\int_{\partial B_r(x)}\varphi -s^{1-2n}\int_{\partial B_s(x)}\varphi \ge -Cr,
\end{equation*}
where $C$ is a uniform constant depending on $||\varphi||_{\infty}$ and $(M,\omega_0)$.
\end{lem}
\begin{proof}
Using the Laplacian comparision theorem, we have that $H\le H'$, where $H$ is the mean curvature of a geodesic sphere on $M$ and $H'$ is the mean curvature of a geodesic sphere on a space-form whose metric can be written in a normal coordinate as:
\begin{equation*}
g_R=dr^2 + R^2 sinh^2(\frac{r}{R})d S_{2n-1}^2,
\end{equation*}
Here $R$ only depends on the lower bound of the Ricci curvature on $M$.
By calculation, we have that
\begin{equation*}
H'(r)=\frac{(2n-1)cosh(\frac{r}{R})}{R sinh(\frac{r}{R})}=\frac{(2n-1)}{r}+O(1)
\end{equation*}
So we have that:
\begin{equation*}
\begin{split}
\frac{d}{dt}(r^{1-2n}\int_{\partial B_r}\varphi)&=(1-2n)r^{-2n}\int_{\partial B_r}\varphi +r^{1-2n}\int_{\partial B_r}\varphi H \\
&\le (1-2n)r^{-2n}\int_{\partial B_r}\varphi +r^{1-2n}\int_{\partial B_r}\varphi H' \\
&\le (1-2n)r^{-2n}\int_{\partial B_r}\varphi +r^{1-2n}\int_{\partial B_r}\varphi(\frac{2n-1}{r}+O(1))\\
&=O(1)
\end{split}
\end{equation*}
The conclusion of the lemma follows by integrating the above formula from $s$ to $r$.
\end{proof}
\iffalse
Next we need to prove a version of the mean value inequality.
\begin{lem}\label{mean value inequality with a error term}
Let $\varphi$ be a upper semicontinuous function defined on an open subset of $M$. Suppose that $\Delta \varphi \ge 0$, $\varphi<-\eps_0$ for some $\eps_0>0$ and the Ricci curvature is bounded on $M$. Then there exists a constant $C$ which is a uniform constant depending on the bound of $||\varphi||_{\infty}$ and $(M,\omega_0)$ such that:
\begin{equation*}
\varphi(x)\le \frac{1}{2n \alpha(2n)r^{2n}}\int_{B_r(x)}\varphi dArea +C r
\end{equation*}
\end{lem}
\begin{proof}
Fix $r>0$. Since $\varphi$ is upper semicontinuous function and $\varphi<-\eps<0$, there exists a sequence of continuous functions $\{\varphi_k\}$ such that $\varphi\le \varphi_k<0$ and $\lim_{k\rightarrow \infty}\varphi_k=\varphi$. Let $h_k$ be the harmonic extension of $\varphi_k$ on the $B_r(x)$. By the maximum principle, we have that $0>h_k>\varphi_k$. By the Cheng-Yau harnack inequality, we have that:
\begin{equation*}
|\nabla log(-h_k)|_{B_{\frac{r}{2}}(x)}\le C,
\end{equation*}
where $C$ depends on the Ricci curvature and $r$.
Then we have that for any $s<\frac{r}{2}$,
\begin{equation*}
\inf_{B_s(x)}h_k \ge e^{Cs}\sup_{B_s(x)}h_k\ge e^{Cs}\varphi(x).
\end{equation*}
Combining this with the almost monotonicity Lemma \ref{monotonicity lemma} we have that:
\begin{equation*}
\begin{split}
r^{1-2n}\int_{\partial B_r(x)}h_k &\ge s^{1-2n}\int_{\partial B_s(x)}h_k -Cr\\
&\ge e^{Cs}s^{1-2n}Vol(\partial B_s(x))\varphi(x)-Cr.
\end{split}
\end{equation*}
Hence, we have that:
\begin{equation*}
\begin{split}
&\frac{1}{Vol(\partial B_r(x))}\int_{\partial B_r(x)}\varphi =\lim_{k\rightarrow \infty}\frac{1}{Vol(\partial B_r(x))}\int_{\partial B_r(x)}h_k \\
&\ge \lim_{s\rightarrow 0+}\frac{s^{1-2n}Vol(\partial B_s(x))e^{Cs}\varphi(x)-Cr}{r^{1-2n}Vol(\partial B_r(x))}\\
&\ge \frac{2n\alpha(2n)r^{2n-1}}{Vol(\partial B_r(x))}\varphi(x)-Cr.
\end{split}
\end{equation*}
Here we use the Lemma \ref{estimate of the volume of balls}.
So we have that:
\begin{equation*}
2n \alpha(2n) r^{2n-1}\varphi(x)\le \int_{\partial B_r(x)}\varphi -Cr^{2n}
\end{equation*}
The conclusion follows by integrative the above formula.
\end{proof}
\fi
With the help of Proposition \ref{mean value inequality with a error term}, we are ready to estimate $\varphi_{\eps}-\varphi$ in $L^1$.
First, we observe that by taking $\xi=0$:
\begin{equation*}
\varphi_{\eps}(z)=\sup_{\xi\in T_zM}\big(\varphi(\exp_z(\xi))+\eps-\frac{1}{\eps}|\xi|^2_z)\ge \varphi(z)+\eps.
\end{equation*}
Hence it will suffice to estimate $\int_M(\varphi_{\eps}-\varphi)\omega_0^n$.
We will do it in the following lemma:
\begin{lem}\label{l2.29}
Let $\varphi\in C(M)$ and satisfy $\lambda[h_{\varphi}]\in\Gamma$ in the viscosity sense. Let $\varphi_{\eps}$ be defined as (\ref{1.3N}), then one has:
\begin{equation*}
||\varphi_{\eps}-\varphi||_{L^1}\le C\eps^{\frac{1}{4}}.
\end{equation*}
Here $C$ depends only on the background metric and also $||\varphi||_{L^{\infty}}$.
If in addition, we assume that $\varphi\in C^{\gamma}$, then one has
\begin{equation*}
||\varphi_{\eps}-\varphi||_{L^1}\le C\eps^{\frac{1}{2(2-\gamma)}},
\end{equation*}
where $C$ depends only on the background metric and also $||\varphi||_{C^{\gamma}}.$
\end{lem}
\begin{proof}
We can find a cover $\{U_{\alpha}\}$ of $M$ such that $U_{\alpha}=B_{\sigma_0}(x_{\alpha})$ where $\sigma_0$ is selected such that $B_{2\sigma_0}(x_{\alpha})$ is contained in a normal coordinate neighborhood centered at $x_{\alpha}$. Moreover, there exists a K\"ahler potential $\phi_{\alpha}$ defined in $B_{2\sigma_0}(x_{\alpha})$. Namely $\omega_0=\sqrt{-1}\partial\bar{\partial}\rho_{\alpha}$ on $U_{\alpha}$.
We can also assume that $\rho_{\alpha}\le 0$ and both $|\rho_{\alpha}|_{C^0}$ and $|\rho_{\alpha}|_{C^{\alpha}}$ are uniformly bounded.
Since $\lambda[h_{\varphi}]\in\Gamma$ in the viscosity sense, we see that $\lambda\big(g^{i\bar{k}}((\rho_{\alpha})_{j\bar{k}}+\varphi_{j\bar{k}})\big)\in\Gamma$ in the viscosity sense. Since $\Gamma\subset \Gamma_+=\{\lambda\in\mathbb{R}^n:\sum_i\lambda_i>0\}$, we see that
\begin{equation*}
\Delta(\varphi+\rho_{\alpha})>0\,\,\,\text{ on $U_{\alpha}$.}
\end{equation*}
Now we wish to apply the Proposition \ref{mean value inequality with a error term} to $\varphi+\rho_{\alpha}$. For any $x\in M$, we can find $\xi_x\in T_xM$ with $|\xi_x|_x\le (2||\varphi||_{L^{\infty}}\eps)^{\frac{1}{2}}$ by Lemma \ref{l2.10}. Let $0<\beta<\frac{1}{2}$ be a constant we will choose later, we have
\begin{equation}\label{uepsilon mean value inequality}
\begin{split}
&\varphi_{\eps}(x) +\rho_{\alpha}(\exp_x(\xi_x))\le (\varphi+\rho_{\alpha})(\exp_x(\xi_x))+\eps \\
&\le \frac{1}{\alpha(2n)\eps^{2n\beta}}\int_{B_{\eps^{\beta}}(\exp_x(\xi_x))}(\varphi(y)+\rho_{\alpha}(y)) +C\eps^{\beta} \\
&\le \frac{1}{\alpha(2n)\eps^{2n\beta}}\int_{B_{\eps^{\beta}-C_0\eps^{\frac{1}{2}}}(x)}\varphi(y) +\frac{1}{\alpha(2n)\eps^{2n\beta}}\int_{B_{\eps^{\beta}}(\exp_x(\xi_x))}\rho_{\alpha}(y)\\
&-\rho_{\alpha}(\exp_x(\xi_x))+\rho_{\alpha}(\exp_x(\xi_x)) +C \eps^{\beta} \\
&\le \frac{1}{\alpha(2n)\eps^{2n\beta}}\int_{B_{\eps^{\beta}-C_0\eps^{\frac{1}{2}}}(x)} \varphi(y)+\rho_{\alpha}(\exp_x(\xi_x))+C'\eps^{\beta}.
\end{split}
\end{equation}
In the above, the constants $C$ and $C'$ depends only on $||\varphi||_{L^{\infty}}$ and the background metric.
In the first line above, we used the definition of $\varphi_{\eps}$ given by (\ref{1.3N}).
In the second line, we used that $B_{\eps^{\beta}-C_0\eps^{\frac{1}{2}}}(x)\subset B_{\eps^{\beta}}(\exp_x(\xi_x))$, which again follows from Lemma \ref{l2.10}. Also $\varphi$ is negative.
In the last inequality, we used that for any $y\in B_{\eps^{\beta}}(\exp_x(\xi_x))$, $|\rho_{\alpha}(y)-\rho_{\alpha}(\exp_x(\xi_x))|\le \sup|\nabla\rho_{\alpha}|\eps^{\beta}$, hence
\begin{equation*}
\begin{split}
&\frac{1}{\alpha(2n)\eps^{2n\beta}}\int_{B_{\eps^{\beta}}(\exp_x(\xi_x))}\rho_{\alpha}(y)-\rho_{\alpha}(\exp_x(\xi_x))\\
&=\frac{1}{\alpha(2n)\eps^{2n\beta}}\int_{B_{\eps^{\beta}}(\exp_x(\xi_x))}(\rho_{\alpha}(y)-\rho_{\alpha}(\exp_x(\xi_x)))+\big(\frac{vol(B_{\eps^{\beta}}(\exp_x(\xi_x)))}{\alpha(2n)\eps^{2n\beta}}-1\big)\rho_{\alpha}(\exp_x(\xi_x))\\
&\le C(|\rho_{\alpha}|_{C^1})\eps^{\beta}.
\end{split}
\end{equation*}
Here we also used Lemma \ref{estimate of the volume of balls}.
The result of (\ref{uepsilon mean value inequality}) is that:
\begin{equation}\label{2.31}
\varphi_{\eps}(x)\le \frac{1}{\alpha(2n)\eps^{2n\beta}}\int_{B_{\eps^{\beta}-C_0\eps^{\frac{1}{2}}}(x)}\varphi(y)\omega_0^n(y)+C\eps^{\beta}.
\end{equation}
Here $C$ depends only on the background metric and $||\varphi||_{L^{\infty}}$.
Now we can integrate on $M$ to get:
\begin{equation*}
\begin{split}
&\int_M\varphi_{\eps}(x)\omega_0^n(x)\le \frac{1}{\alpha(2n)\eps^{2n\beta}}\int_M\omega_0^n(x)\int_{B_{\eps^{\beta}-C_0\eps^{\frac{1}{2}}}(x)}\varphi(y)\omega_0^n(y)+Cvol(M)\eps^{\beta}\\
&=\int_M\varphi(y)\frac{vol(B_{\eps^{\beta}-C_0\eps^{\frac{1}{2}}}(y))}{\alpha(2n)\eps^{2n\beta}}\omega_0^n(y)+C_1\eps^{\beta}\le \inf_{y\in M}\frac{vol(B_{\eps^{\beta}-C_0\eps^{\frac{1}{2}}}(y))}{\alpha(2n)\eps^{2n\beta}}\int_M\varphi(y)\omega_0^n(y)+C_1\eps^{\beta}\\
&\le \frac{(\eps^{\beta}-C_0\eps^{\frac{1}{2}})^{2n}}{\eps^{2n\beta}}\int_M\varphi(y)\omega_0^n(y)+C_2\eps^{\beta}.
\end{split}
\end{equation*}
Therefore
\begin{equation*}
\int_M(\varphi_{\eps}-\varphi)\omega_0^n(x)\le \big(1-(1-C_0\eps^{\frac{1}{2}-\beta})^{2n}\big)\int_M\varphi(y)\omega_0^n(y)+C_2\eps^{\beta}\le C_3\eps^{\frac{1}{2}-\beta}+C_2\eps^{\beta}.
\end{equation*}
Hence we may take $\beta=\frac{1}{4}$ to obtain that $\int_M(\varphi_{\eps}-\varphi)\omega_0^n\le C\eps^{\frac{1}{4}}$.
Next if $\varphi\in C^{\gamma}$, the calculation is very similar except that we have a better estimate for $\xi_x$: $|\xi_x|_x\le C\eps^{\frac{1}{2-\gamma}}$.
Then in \ref{uepsilon mean value inequality}, the range for $\beta$ should be $0<\beta<\frac{1}{2-\gamma}$. Also instead of (\ref{2.31}), one has instead:
\begin{equation*}
\varphi_{\eps}(x)\le \frac{1}{\alpha(2n)\eps^{2n\beta}}\int_{B_{\eps^{\beta}-C_1\eps^{\frac{1}{2-\gamma}}}(x)}\varphi(y)\omega_0^n(y)+C\eps^{\beta}.
\end{equation*}
Then we will make the choice that $\beta=\frac{1}{2(2-\gamma)}$ to obtain that:
\begin{equation*}
\int_M(\varphi_{\eps}-\varphi)\omega_0^n\le C\eps^{\frac{1}{2(2-\gamma)}}.
\end{equation*}
\iffalse
In the first line we use the Lemma \ref{mean value inequality with a error term}. From now on we take a constant $\beta=\frac{1}{4}$.
Then we have that:
\begin{equation*}
\varphi_{\eps}(x)\le \frac{1}{\alpha(2n)r^{2n}}\int_{B_{\eps^{\beta}-C \eps^{\frac{1}{2}}}(x)}\varphi(y) +C(|\phi|_{C^{1}},|\phi|_{C^)})\eps^{\frac{1}{4}}
\end{equation*}
For any $x\in M$, we can find some $\alpha$ such that $x\in U_{\alpha}$. Then we can use the above calculation with $\phi$ replaced by $\phi_{\alpha}$. Note that both $|\phi_{\alpha}|_{C^0}$ and $|\phi_{\alpha}|_{C^{1}}$ are uniformly bounded. We have that:
\begin{equation*}
\varphi_{\eps}(x)\le \frac{1}{\alpha(2n)r^{2n}}\int_{B_{\eps^{\beta}-C\eps^{\frac{1}{2}}(x)}}u(y) +C\eps^{\frac{1}{4}}.
\end{equation*}
Then we can integrate the above formula on $M$ to get that:
\begin{equation*}
\begin{split}
\int_M \varphi_{\eps}&\le \frac{1}{\alpha(2n)r^{2n}}\int_m \omega_0^n(x)\int_{B_{\eps^{\beta}-C\eps^{\frac{1}{2}}}(x)}\varphi(y)\omega_0^n(y) +C \eps^{\frac{1}{4}}Vol(M) \\
&=\frac{1}{\alpha(2n)r^{2n}}\int_M \varphi(y)\omega_0^n (y) \int_{B_{\eps^{\beta}-C\eps^{\frac{1}{2}}}(y)}\omega_0^n(x) +C\eps^{\frac{1}{4}} Vol(M)\\
&\le \inf_y \frac{Vol(B_{\eps^{\beta_0}-C\eps^{\frac{1}{2}}})}{\alpha(2n)r^{2n}}\int_M \varphi(y)\omega_0^n(y) +C\eps^{\frac{1}{4}}Vol(M).
\end{split}
\end{equation*}
Then following the Lemma \ref{estimate of the volume of balls} and the assumption that $\varphi$ is bounded, we have that:
\begin{equation*}
\int_M \varphi_{\eps}\le \int_M \varphi +C\eps^{\frac{1}{4}}.
\end{equation*}
\fi
\end{proof}
Now we are in a position to prove Theorem \ref{uepsilon mean value inequality}.
\begin{proof}
(of Theorem \ref{uepsilon mean value inequality})
The proof follows from combining the lemmas obtained in this section. Indeed, point (i) follows from Lemma \ref{l2.11New}. Point (ii) and (ii)' follow from Corollary \ref{c2.22N}. Lemma \ref{l2.29} would give us point (iii) and (iii)'.
\end{proof}
\iffalse
\begin{thm}
Suppose that $(M,\omega_0)$ is a K\"ahler manifold such that the holomorphic bisectional curvature is non-negative. Define $\varphi_{\eps}$ as in the Definition \ref{uepsilon} with $\varphi$ an admissible solution to the Equation \ref{main equation} which is bounded and upper semicontinuous. Then we have:\\
$(i)$ $\varphi_{\eps}$ is continuous and $|\varphi_{\eps}|\le C$ for a constant $C$ depending on $||\varphi||_{\infty}$. \\
$(ii)$ There exists a constant $C$ depending on $||\varphi||_{\infty}$ and $(M,\omega_0)$ such that:
\begin{equation*}
\lambda(g^{i\bar{k}} (\varphi_{\eps})_{i\bar{k}} +(1+C\eps^{\frac{1}{2}})I)\in \Gamma.
\end{equation*}
$(iii)$ There exists a constant $C$ depending on $||\varphi||_{\infty}$ and $(M,\omega_0)$ such that:
\begin{equation*}
||\varphi_{\eps}-\varphi||_{L^1}\le C \eps^{\frac{1}{4}}.
\end{equation*}
Moreover, if we have that $\varphi$ is H\"older continuous with a H\"older exponent $\gamma_0$, then we can improve the above results as follows:\\
$(ii')$ There exists a constant $C$ depending on $||\varphi||_{C^{0,\gamma_0}}$ and $(M,\omega_0)$ such that:
\begin{equation*}
\lambda(dd^c \varphi_{\eps} +(1+C\eps^{\frac{1+\gamma_0}{2-\gamma_0}})\omega_0)\in \Gamma.
\end{equation*}
$(iii')$ There exists a constant $C$ depending on $||\varphi||_{C^{0,\gamma_0}}$ and $(M,\omega_0)$ such that:
\begin{equation*}
||\varphi_{\eps}-\varphi||_{L^1}\le C \eps^{\frac{1}{2(2-\gamma_0)}}.
\end{equation*}
\end{thm}
\begin{proof}
In the definition of $\varphi_{\eps}(x)$, we can assume that the supreme is taken in the interior of the $B_{\eps_0}$, which we denote as $\xi_x$, if we make $\eps_0$ to be small enough. \\
$(i)$ $\varphi_{\eps}$ is continuous.
\begin{equation*}
\begin{split}
\varphi_{\eps}(x)&=\varphi(exp_x(\xi_x))+\eps - \frac{1}{\eps}|\xi_x|^2 \\
&=\varphi(exp_y(\xi_{xy}))+\eps -\frac{1}{\eps}|\xi_{xy}|^2 +\frac{1}{\eps}|\xi_{xy}|^2 -\frac{1}{\eps}|\xi_x|^2 \\
&\le \varphi_{\eps}(y)+\frac{1}{\eps}|\xi_{xy}|^2 -\frac{1}{\eps}|\xi_x|^2,
\end{split}
\end{equation*}
where $\xi_{xy}$ is defined by $exp_y (\xi_{xy})=exp_x(\xi_x)$. Note that $\xi_{xy}$ converges to $\xi_x$ as $y$ goes to $x$. we have that
\begin{equation*}
\lim_{y \rightarrow x} \varphi_{\eps}(y)\ge u_{\eps}(x),
\end{equation*}
so $\varphi_{\eps}$ is lower semi-continuous. We use a similar way to prove that $\varphi_{\eps}$ is also upper semi-continuous.
\begin{equation}\label{upper semi-continuous}
\begin{split}
\varphi_{\eps}(x) &\ge u(exp_x(\xi_{yx}))+\eps -\frac{1}{\eps} \\
&=\varphi(exp_y(\xi_y))+\eps -\frac{1}{\eps}|\xi_y|^2 +\frac{1}{\eps}|\xi_y|^2-\frac{1}{\eps}|\xi_{yx}|^2 \\
&= \varphi_{\eps}(y)+\frac{1}{\eps}|\xi_y|^2 -\frac{1}{\eps}|\xi_{yx}|^2,
\end{split}
\end{equation}
where $\xi_{yx}$ is selected such that $exp_x(\xi_{yx})=exp_y(\xi_y)$. For any sequence $\{y_k\}$ converging to $x$, for any $\eps_1>0$, $|\xi_{y_k x}|=o(1)$ which is small with respect to $\eps_1$. For those $k$ such that $|\xi_{y_k}|\ge \eps_1$, if $k$ is big enough, $d(y_k,x)<< \eps_1$, so $||\xi_{y_k x}|-|\xi_{y_k}||$ is very small. As a result, by letting $y\rightarrow x$ in the equation \ref{upper semi-continuous}, we have that $u_{\eps}$ is upper-semicontinuous as well.\\
$(ii)$ For any $\eps \le \eps_0$,
\begin{equation*}
\lambda(dd^c \varphi_{\eps} +(1+C\eps^{\frac{1}{2}})\omega_0)\in \Gamma.
\end{equation*}
For any function $P\in C^2$ touching $\varphi_{\eps}$ from above at $x$, (note that P may be only defined near x) we have that:
\begin{equation}\label{touch above}
\begin{split}
P(x)&=\varphi_{\eps}(x)=\varphi(exp_x(\xi_x))+\eps -\frac{1}{\eps}|\xi_x|^2\\
P(y)&\ge \varphi_\eps(y)\ge \varphi(exp_y(\xi_x))+\eps-\frac{1}{\eps}|\xi_x|^2.
\end{split}
\end{equation}
Let $(z^i)$ be the normalized coordinate around $x$ given in the Lemma \ref{find normalized coordinates} where $y=exp_x(\xi_x)$. Let $N$ be the matrix given in the same lemma. Then define
\begin{equation}\label{phi}
\phi (z) =Nz+exp_x(\xi_x).
\end{equation}
We can calculate the third order expansion of $exp$:
\begin{equation*}
exp_z(\xi)_m=z_m+\xi_m +\sum_{j,k,l} c_{jklm}(\frac{1}{2}\bar{z_k} +\frac{1}{6}\bar{\xi})\xi_j\xi_l +O(|\xi|^2(|z|+|\xi|)^2),
\end{equation*}
where $c_{jklm}$ is the curvature.
Note that $\nabla_{\xi}exp_0 =Id +O(|\xi_x|^2)$. Since $|\xi_x|$ goes to zero as $\eps$ goes to zero, we can assume that $\nabla_{\xi}exp_0$ is invertible. So by the implicit function theorem, there is a local section of TM, $\xi$ which is the solution to the equations:
\begin{equation*}
exp_z(\xi(z))=\phi(z)
\end{equation*}
and $\xi(0)=\xi_x$.
Then we define a function $Q$ such that $P(z)=Q(\phi(z))$. Denote $h(z)=Q(z)-\eps +\frac{1}{\eps}|\xi|_{\phi^{-1}(z)}^2$. Then the Equations \ref{touch above} implies that $h$ touches $\varphi$ from above at $exp_x(\xi_x)$.
Since $\varphi$ is an admissible solution to the Equation \ref{main equation}, we have that:
\begin{equation*}
\lambda(g(\phi(x))^{-1} Hessh +I)\in \Gamma.
\end{equation*}
First, according to the definition of $P$ and the holomorphicity of $\phi$,
\begin{equation*}
HessQ=\partial \phi^{-1} HessP \partial \phi^{-1*}.
\end{equation*}
Second, we estimate $\frac{1}{\eps}dd^c|\xi|^2(\phi^{-1}(z))$
according to the calculation of Demailly, we have that:
\begin{equation*}
|\xi|^2=\sum |\xi^m|^2- c_{jklm}z_j \bar{z_k}\xi^l \bar{\xi^m} +O(|z|^3)|\xi|^2.
\end{equation*}
So we have that:
\begin{equation*}
(|\xi|^2)_{\alpha \bar{\beta}}(0)=\xi^m_{\alpha \bar{\beta}}\xi^{\bar{m}} + \xi^m_{\bar{\beta}}\xi^{\bar{m}}_{\alpha} + \xi^m \xi^{\bar{m}}_{\alpha \bar{\beta}}+ \xi^m_{\alpha} \xi^{\bar{m}}_{\bar{\beta}}- c_{\alpha \beta l m }\xi^l \xi^{\bar{m}}.
\end{equation*}
From the Equation \ref{phi}, we have that
\begin{equation}\label{first derivative}
\nabla_z exp +\nabla_{\xi}exp \nabla \xi =N(exp_x(\xi_x))=Id +O(|\xi_x|^2).
\end{equation}
We have that
\begin{equation*}
\nabla \xi (0) = (\nabla_{\xi}exp)^{-1}(Id +O(|\xi_x|^2) -\nabla_z exp)= O(|\xi_x|^2).
\end{equation*}
Take another derivative with respect to $z$ at the Equation \ref{first derivative}, we have that
\begin{equation*}
\nabla_z\nabla_z exp+ \nabla_{\xi}\nabla_z exp \nabla \xi + \nabla_z\nabla_{\xi}exp \nabla \xi + \nabla_{\xi}\nabla_{\xi}exp \nabla \xi \nabla \xi + \nabla_{\xi}exp \nabla^2 \xi =0.
\end{equation*}
So we have that
\begin{equation*}
\nabla^2 \xi(0) =O(|\xi_x|^2).
\end{equation*}
Then we have that
\begin{equation*}
(|\xi|^2)_{\alpha \bar{\beta}}(0)=O(|\xi_x|^3)-c_{\alpha \beta l m}\xi_x^l \xi_x^{\bar{m}}.
\end{equation*}
Since the holomorphic bisectional curvature is non-negative, we have that
\begin{equation*}
\frac{1}{\eps}dd^c(|\xi|^2)(0)\le \frac{1}{\eps}O(|\xi_x|^3).
\end{equation*}
Note that
\begin{equation}\label{estimate of xix}
\varphi_{\eps}(x)=\varphi(\phi(x))+\eps -\frac{1}{\eps}|\xi_x|^2\ge \varphi(x)+\eps.
\end{equation}
We have that $|\xi_x|^2\le C \eps$, where $C$ depends on $||\varphi||_{\infty}$. Then we have that:
\begin{equation*}
\frac{|\xi_x|^3}{\eps}\le C \eps^{\frac{1}{2}},
\end{equation*}
where $C$ depends on $||\varphi||_{\infty}$ and $(M,\omega_0)$. So we have that:
\begin{equation*}
\frac{1}{\eps}dd^c(|\xi|^2)(0)\le C \eps^{\frac{1}{2}}\omega.
\end{equation*}
In conclusion, we have that
\begin{equation*}
\begin{split}
\lambda(g(\phi(x))^{-1}Hess h +I) & \le \lambda (N^* N \partial \phi^{-1}HessP \partial \phi^{-1*} + C\eps^{\frac{1}{2}}I + I) \\
&= \lambda (HessP +(1+C\eps^{\frac{1}{2}})I)
\end{split}
\end{equation*}
So we have that $\lambda (HessP +(1+C\eps^{\frac{1}{2}})Id)\in \Gamma$. \\
$(iii)$ $||\varphi-\varphi_{\eps}||_1\le C \eps^{\frac{1}{4}}$. \\
We can find a cover $\{U_{\alpha}\}$ of $M$ such that $U_{\alpha}=B_{\sum_0}(x_{\alpha})$ where $\sum_0$ is selected such that $B_{2\sum_0}(x_{\alpha})$ is a normal ball and for any $\alpha$, there exists a K\"ahler potential $\phi_{\alpha}$ defined in $B_{2\sum_0}(x_{\alpha})$. We can also assume that $\phi_{\alpha}\le 0$ and both $|\phi_{\alpha}|_{C^0}$ and $|\phi_{\alpha}|_{C^{\alpha}}$ are uniformly bounded. Since $\Delta(\varphi+\phi_{\alpha})\ge 0$, we have that:
\begin{equation}\label{uepsilon mean value inequality}
\begin{split}
&\varphi_{\eps}(x) +\phi_{\alpha}(exp_x(\xi_x)) \le \frac{1}{\alpha(2n)\eps^{2n\beta}}\int_{B_{\eps^{\beta}-C\eps^{\frac{1}{2}}}}(\varphi(y)+\phi_{\alpha}(y)) +C\eps^{\beta} \\
&\le \frac{1}{\alpha(2n)\eps^{2n\beta}}\int_{B_{\eps^{\beta}-C\eps^{\frac{1}{2}}}(x)}\varphi(y) +\frac{1}{\alpha(2n)\eps^{2n\beta}}\int_{B_{\eps^{\beta}-C\eps^{\frac{1}{2}}}(x)}\phi_{\alpha}(y)\\
&-\phi_{\alpha}(exp_x(\xi_x))+\phi_{\alpha}(exp_x(\xi_x)) +C \eps^{\beta} \\
&\le \frac{1}{\alpha(2n)\eps^{2n\beta}}\int_{B_{\eps^{\beta}-C\eps^{\frac{1}{2}}}(x)} \varphi(y)\\
&+\frac{Vol(B_{\eps^{\beta}-C\eps^{\frac{1}{2}}})}{\alpha(2n)\eps^{2n\beta}}|\phi_{\alpha}|_{C^{1}(B_{\eps^{\beta}-C\eps^{\frac{1}{2}}}(x))}2(\eps^{\beta}-C\eps^{\frac{1}{2}}) \\
&+\phi_{\alpha}(exp_x(\xi_x))+C\eps^{\beta} +C \eps^{\frac{1}{2}-\beta}\\
&\le \frac{1}{\alpha(2n)\eps^{2n\beta}}\int_{B_{\eps^{\beta}-C\eps^{\frac{1}{2}}}(x)}\varphi(y) +\phi_{\alpha}(exp_x(\xi_x))+C\eps^{\beta} +C\eps^{\frac{1}{2}-\beta}.\\
\end{split}
\end{equation}
In the first line we use the Lemma \ref{mean value inequality with a error term}. From now on we take a constant $\beta=\frac{1}{4}$.
Then we have that:
\begin{equation*}
\varphi_{\eps}(x)\le \frac{1}{\alpha(2n)r^{2n}}\int_{B_{\eps^{\beta}-C \eps^{\frac{1}{2}}}(x)}\varphi(y) +C(|\phi|_{C^{1}},|\phi|_{C^)})\eps^{\frac{1}{4}}
\end{equation*}
For any $x\in M$, we can find some $\alpha$ such that $x\in U_{\alpha}$. Then we can use the above calculation with $\phi$ replaced by $\phi_{\alpha}$. Note that both $|\phi_{\alpha}|_{C^0}$ and $|\phi_{\alpha}|_{C^{1}}$ are uniformly bounded. We have that:
\begin{equation*}
\varphi_{\eps}(x)\le \frac{1}{\alpha(2n)r^{2n}}\int_{B_{\eps^{\beta}-C\eps^{\frac{1}{2}}(x)}}u(y) +C\eps^{\frac{1}{4}}.
\end{equation*}
Then we can integrate the above formula on $M$ to get that:
\begin{equation*}
\begin{split}
\int_M \varphi_{\eps}&\le \frac{1}{\alpha(2n)r^{2n}}\int_m \omega_0^n(x)\int_{B_{\eps^{\beta}-C\eps^{\frac{1}{2}}}(x)}\varphi(y)\omega_0^n(y) +C \eps^{\frac{1}{4}}Vol(M) \\
&=\frac{1}{\alpha(2n)r^{2n}}\int_M \varphi(y)\omega_0^n (y) \int_{B_{\eps^{\beta}-C\eps^{\frac{1}{2}}}(y)}\omega_0^n(x) +C\eps^{\frac{1}{4}} Vol(M)\\
&\le \inf_y \frac{Vol(B_{\eps^{\beta_0}-C\eps^{\frac{1}{2}}})}{\alpha(2n)r^{2n}}\int_M \varphi(y)\omega_0^n(y) +C\eps^{\frac{1}{4}}Vol(M).
\end{split}
\end{equation*}
Then following the Lemma \ref{estimate of the volume of balls} and the assumption that $\varphi$ is bounded, we have that:
\begin{equation*}
\int_M \varphi_{\eps}\le \int_M \varphi +C\eps^{\frac{1}{4}}.
\end{equation*}
Note that $\varphi_{\eps}\ge \varphi$, we have proved that:
\begin{equation*}
||\varphi_{\eps}-\varphi||_1 \le C\eps^{\frac{1}{4}}.
\end{equation*}
Next we assume that $\varphi$ is H\"older continuous. We can use similar methods as above to prove $(ii')$ and $(iii')$. We just state the main differences here. In the Inequality \ref{estimate of xix}, we can use the H\"older continuity of $u$ to get a better estimate of $|\xi_x|$:
\begin{equation*}
|\xi_x|\le C\eps^{\frac{1}{2-\gamma_0}}
\end{equation*}
In the Inequality \ref{uepsilon mean value inequality}, we integrate on the ball $B_{\eps^{\beta}-C\eps^{\frac{1}{2-\gamma_0}}}$ instead. Then we choose $\beta=\frac{1}{2(2-\gamma_0)}$.
\end{proof}
\fi
\section{Stability estimates of Hessian equations}
\label{Stability}
Next we improve the estimate of $\varphi_{\eps}-\varphi$ from $L^1$ norm to the $L^{\infty}$ norm.
\iffalse
\begin{defn}
Let $v\in C(M)$. We say that $\frac{\delta}{2}I+(h_v)_j^i\in\Gamma$ in the sense of viscosity, if for any point $p\in M$, any local $C^2$ function $\eta$ touching $v$ from above at point $p$, we have:
\begin{equation*}
\frac{\delta}{2}I+(h_{\eta})_j^i:=\frac{\delta}{2}I+g^{i\bar{k}}(g_{j\bar{k}}+\eta_{j\bar{k}})\in\Gamma.
\end{equation*}
\end{defn}
\fi
First we can prove the following proposition.
\begin{prop}\label{p3.2}
Let $v\in C(M)$, $s>0$, $\delta>0$and $\lambda\big(g^{i\bar{k}}((1+\frac{\delta}{2})g_{j\bar{k}}+v_{j\bar{k}})\big)_j^i\in\Gamma$ in the viscosity sense as in Definition \ref{d2.3}. Define $A_{\delta,s}=\int_M ((1-\delta)v-\varphi-s)^+ e^{nF} \omega_0^n$.\\
Then there is a constant $\beta_n>0$, and $C>0$, depending only on the background metric and structural constant of f, such that:
\begin{equation*}
\int_M exp(\frac{\beta_n(((1-\delta)v-\phi-s)^+)^{\frac{n+1}{n}}}{A_{\delta,s}^{\frac{1}{n}}})\omega_0^n \le C exp(C\delta^{-(n+1)}A_{\delta,s}).
\end{equation*}
\end{prop}
\begin{proof}
We consider the function
\begin{equation*}
\Phi=\eps_1((1-\delta)v-\varphi-s)-(-\psi+\Lambda)^{\alpha}.
\end{equation*}
Here $\psi$ is defined as the solution to:
\begin{equation*}
\begin{split}
&(\omega_0+\sqrt{-1}\partial\bar{\partial}\psi)^n=\frac{((1-\delta)v-\varphi-s)^+e^{nF}}{A_{\delta,s}}\omega_0^n,\\
&\sup_M\psi=0.
\end{split}
\end{equation*}
Since $v$ is only continuous, there is no guarantee that $\psi$ is $C^2$. To do it rigorously, we need to take $v_j$ to be a sequence of smooth functions which converges to $v$ uniformly, and $\eta_j(x)$ be a sequence of smooth and strictly positive functions which converges to $x^+$. Then we consider instead $\psi_j$, which solves
\begin{equation*}
(\omega_0+\sqrt{-1}\partial\bar{\partial}\psi_j)^n=\frac{\eta_j((1-\delta)v_j-\varphi-s)}{A_{j,\delta,s}}\omega_0^n.
\end{equation*}
Here $A_{j,\delta,s}=\int_M\eta_j((1-\delta)v_j-\varphi-s)^+e^{nF}\omega^n$. Then we can consider $\Phi_j=\eps_0((1-\delta)v-\varphi-s)-(-\psi_j+\Lambda)^{\alpha}$.
For now, let us assume that $\psi$ is $C^2$ so as to justify the following argument. One can consider $\psi_j$ above in order to be completely rigorous (but argument is the same).
Since $\Phi$ is continuous, it achieves maximum on $M$, say, at $p_0$, and we denote:
\begin{equation*}
M_0:=\max_M\eps_0\big((1-\delta)v-\varphi-s)-(-\psi+\Lambda)^{\alpha_1}.
\end{equation*}
Our goal is to show that $M_0\le 0$. If not, namely $M_0>0$, we can do the following argument.
At point $p$, the function $Q:=\frac{1}{1-\delta}\big(\varphi+s+\frac{M_0+(-\psi+\Lambda)^{\frac{n}{n+1}}}{\eps_1}\big)$ touches $v$ from above at $p$. Hence we get:
\begin{equation*}\label{0.18NNN}
\lambda((\frac{\delta}{2}+1)I+g^{i\bar{k}}Q_{j\bar{k}})\in\Gamma.
\end{equation*}
As before, we consider the linearized operator defined as:
\begin{equation*}
Lv=\frac{\partial F}{\partial h_{ij}}((h_{\varphi})_j^i)g^{i\bar{k}}v_{j\bar{k}}.
\end{equation*}
We consider normal coordinate at $p$ so that $g_{i\bar{j}}=\delta_{ij}$ and $(h_{\varphi})_j^i$ is diagnal at $p$. Then we have: $\frac{\partial F}{\partial h_{ij}}(p)=\frac{\partial f}{\partial \lambda_i}(\lambda(h_{\varphi}))\delta_{ij}$. Then we can compute:
\begin{equation*}
L(\frac{Q}{1+\frac{\delta}{2}})=-\sum_i\frac{\partial f}{\partial \lambda_i}+\sum_i\frac{\partial f}{\partial \lambda_i}(\lambda(h_{\varphi}))(h_{\frac{Q}{1+\frac{\delta}{2}}})_{i\bar{i}}\ge -\sum_i\frac{\partial f}{\partial \lambda_i}+\sum_i\frac{\partial f}{\partial \lambda_i}(\lambda(h_{\varphi}))\mu_i.
\end{equation*}
In the above, $\mu_i$ are the eigenvalues of $h_{\frac{Q}{1+\frac{\delta}{2}}}$, in the increasing order. The last inequality used Horn-Shur Lemma \ref{l3.2} (see \cite{Horn}). Now we know that $\mu=(\mu_1,\cdots,\mu_n)\in\Gamma$, thanks to (\ref{0.18NNN}).
Hence
\begin{equation*}
\sum_i\frac{\partial f}{\partial \lambda_i}(\lambda(h_{\varphi}))\mu_i=\frac{d}{dt}|_{t=0}(f(\lambda +t\mu)\ge 0.
\end{equation*}
It is due to that $t\mapsto f(\lambda+t\mu)$ is a concave function bounded from below.
Hence we get
\begin{equation*}
L(\frac{Q}{1+0.5\delta})\ge -\sum_i\frac{\partial f}{\partial \lambda_i}.
\end{equation*}
On the other hand, we can compute:
\begin{equation*}
\begin{split}
&Q_{j\bar{k}}=\frac{1}{1-\delta}(\varphi_{j\bar{k}}+\eps_0^{-1}\frac{n}{n+1}(-\psi+\Lambda)^{-\frac{1}{n+1}}(-\psi_{j\bar{k}})-\frac{n}{(n+1)^2}(-\psi+\Lambda)^{-\frac{n+2}{n+1}}\psi_j\psi_{\bar{k}})\\
&=\frac{1}{1-\delta}\big((g_{\varphi})_{j\bar{k}}-\eps_0^{-1}\frac{n}{n+1}(-\psi+\Lambda)^{-\frac{1}{n+1}}(g_{\psi})_{j\bar{k}}-(1-\eps_0^{-1}\frac{n}{n+1}(-\psi+\Lambda)^{-\frac{1}{n+1}})g_{j\bar{k}}\\
&-\frac{n}{(n+1)^2}(-\psi+\Lambda)^{-\frac{n+2}{n+1}}\psi_j\psi_{\bar{k}})
\end{split}
\end{equation*}
So that (after dropping the term $\psi_j\psi_{\bar{k}}$ and replace $(-\psi+\Lambda)^{-\frac{1}{n+1}}$ by $\Lambda^{-\frac{1}{n+1}}$)
\begin{equation*}
\begin{split}
&-(1+0.5\delta)\sum_i\frac{\partial f}{\partial \lambda_i}\le LQ\le \frac{1}{1-\delta}\big(\sum_i\frac{\partial f}{\partial \lambda_i}\lambda_i-\eps_0^{-1}\frac{n}{n+1}(-\psi+\Lambda)^{-\frac{1}{n+1}}\frac{\partial F}{\partial h_{ij}}(g_{\psi})_{j\bar{i}}\\
&-\frac{1}{1-\delta}(1-\eps_0^{-1}\frac{n}{n+1}\Lambda^{-\frac{1}{n+1}})\sum_i\frac{\partial f}{\partial\lambda_i}.
\end{split}
\end{equation*}
Next, you need to choose $\Lambda$ large enough so that
\begin{equation*}
\eps_0^{-1}\frac{n}{n+1}\Lambda^{-\frac{1}{n+1}}=0.1\delta.
\end{equation*}
Then from above, we get:
\begin{equation*}
\begin{split}
&0\le \sum_i\frac{\partial f}{\partial \lambda_i}\lambda_i-\eps_0^{-1}\frac{n}{n+1}(-\psi+\Lambda)^{-\frac{1}{n+1}}\frac{\partial F}{\partial h_{ij}}(g_{\psi})_{j\bar{i}}\\
&\le C_0f-\eps_0^{-1}\frac{n}{n+1}(-\psi+\Lambda)^{-\frac{1}{n+1}}\big(\frac{((1-\delta)v-u-s)^+e^{nF}}{A_{\delta,s}}\big)^{\frac{1}{n}}.
\end{split}
\end{equation*}
Since $M_0>0$, it would imply that:
\begin{equation*}
(((1-\delta)v-\varphi-s)^+)^{\frac{1}{n}}(-\psi+\Lambda)^{-\frac{1}{n+1}}>\eps_0^{-\frac{1}{n}}.
\end{equation*}
Therefore, if $\eps_0$ is small enough so that
\begin{equation*}
C_0-\eps_0^{-\frac{n+1}{n}}A_{\delta,s}^{-\frac{1}{n}}<0,
\end{equation*}
it would not happen.
\end{proof}
In the above, we used the following Horn-Shur Lemma:
\begin{lem}\label{l3.2}
(\cite{Horn})
Let $A$ be a Hermitian matrix, then the vector $(a_{1\bar{1}},a_{2\bar{2}},\cdots,a_{n\bar{n}})$ formed by the diagnal entries is contained in the convex envelope of $\{(\mu_{\sigma(1)},\mu_{\sigma(2)},\cdots,\mu_{\sigma(n)})\}_{\sigma\in S_n}$. Here $\mu_i$ are the eigenvalues of $A$ and $\sigma$ denotes permutations.
\end{lem}
Next we need to estimate the upper level set $\{(1-\delta)v-\varphi-s>0\}$, to the effect that De Giorge's lemma applies. For this we have the following lemma:
\begin{lem}\label{l3.3}
Let $0<\delta<1$ and $v$ be as assumed in Proposition \ref{p3.2} and $\varphi$ solves $f(\lambda(h_{\varphi}))=e^F$ with $e^{nF}\in L^{p_0}(\omega_0^n)$ for some $p_0>1$. Define $\Omega_{\delta,s}=\{(1-\delta)v-\varphi-s>0\}$. Assume that $\delta$ and $s$ are chosen so that:
\begin{equation*}
\int_M((1-\delta)v-\varphi-s)^+e^{nF}\omega_0^n\le \delta^{n+1}.
\end{equation*}
Then for any $r>0$ and any $\mu<\frac{1}{nq_0}$, we have:
\begin{equation*}
rvol(\Omega_{\delta,s+r})\le B_0vol(\Omega_{\delta,s})^{1+\mu}.
\end{equation*}
Here $q_0$ is the dual exponent to $p_0$, namely $\frac{1}{q_0}=1-\frac{1}{p_0}$. Here $B_0$ depends only on the background metric, $p_0$ and choice of $\mu$.
\end{lem}
\begin{proof}
we see Proposition \ref{p3.2} implies that for any $p\ge 1$, we have:
\begin{equation*}
\int_{\Omega_{\delta,s}}((1-\delta)v-\varphi-s)^{\frac{n+1}{n}p}\omega_0^n\le C_1e^{C_0\delta^{-(n+1)}A_{\delta,s}}A_{\delta,s}^{\frac{p}{n}}.
\end{equation*}
In the above, $C_1=C_0p!$.
If we put $p'=\frac{n+1}{n}p$, we get that
\begin{equation}\label{0.0.5}
||((1-\delta)v-\varphi-s)^+||_{L^{p}}\le C_2e^{C_0\delta^{-(n+1)}A_{\delta,s}}A_{\delta,s}^{\frac{1}{n+1}}.
\end{equation}
Here $C_2$ depends on $p$ but $C_0$ doesn't.
Denote $h=((1-\delta)v-\varphi-s)^+$ for the moment.
Now we use the assumption that $e^{nF}\in L^{p_0}(\omega_0^n)$ for some $p_0>1$ to obtain that:
\begin{equation*}
\begin{split}
&A_{\delta,s}=\int_{\Omega_{\delta,s}}he^{nF}\omega_0^n\le \big(\int_{\Omega_s}h^{q_0}\omega_0^n\big)^{\frac{1}{q_0}}||e^{nF}||_{L^{p_0}(\omega_0^n)}\le ||h||_{L^{\beta q_0}(\omega_0^n)}vol(\Omega_{\delta,s})^{\frac{1}{q_0}(1-\frac{1}{\beta})}||e^{nF}||_{L^{p_0}(\omega_0^n)}\\
&\le C_2e^{C_0\delta^{-(n+1)}A_{\delta,s}}A_{\delta,s}^{\frac{1}{n+1}}vol(\Omega_{\delta,s})^{\frac{1}{q_0}(1-\frac{1}{\beta})}||e^{nF}||_{L^{p_0}(\omega_0^n)}.
\end{split}
\end{equation*}
That is, we have
\begin{equation*}
A_{\delta,s}^{\frac{1}{n+1}}\le C_2^{\frac{1}{n}}e^{\frac{C_0}{n}\delta^{-(n+1)}A_{\delta,s}}vol(\Omega_{\delta,s})^{\frac{1}{nq_0}(1-\frac{1}{\beta})}||e^{nF}||_{L^{p_0}(\omega_0^n)}^{\frac{1}{n}}.
\end{equation*}
On the other hand,
\begin{equation*}
\begin{split}
&||h||_{L^1(\omega_0^n)}\le ||h||_{L^{\beta}(\omega_0^n)}vol(\Omega_{\delta,s})^{1-\frac{1}{\beta}}\le C_2e^{C_0\delta^{-(n+1)}A_{\delta,s}}A_{\delta,s}^{\frac{1}{n+1}}vol(\Omega_{\delta,s})^{1-\frac{1}{\beta}}\\
&\le C_2^{1+\frac{1}{n}}e^{(1+\frac{1}{n})C_0\delta^{-(n+1)}A_{\delta,s}}vol(\Omega_{\delta,s})^{(1+\frac{1}{nq_0})(1-\frac{1}{\beta})}||e^{nF}||_{L^{p_0}(\omega_0^n)}.
\end{split}
\end{equation*}
Note that
\begin{equation*}
||h||_{L^1(\omega_0^n)}\ge rvol(\Omega_{\delta,s+r}).
\end{equation*}
So that we have
\begin{equation*}
rvol(\Omega_{\delta,s+r})\le C_3vol(\Omega_{\delta,s})^{(1+\frac{1}{nq_0})(1-\frac{1}{\beta})}.
\end{equation*}
Here for any $\mu<\frac{1}{nq_0}$, one could take $\beta$ so that $(1+\frac{1}{nq_0})(1-\frac{1}{\beta})=1+\mu$.
\end{proof}
On the other hand, we have the following lemma of De Giorge:
\begin{lem}\label{l3.4}
Let $\phi:[0,\infty)\rightarrow [0,\infty)$ be an decreasing and continuous function, such that there exists $\mu>0$, $B_0>0$, $s_0\ge 0$, such that for any $r>0$, and any $s\ge s_0$, one has:
\begin{equation*}
r\phi(s+r)\le B_0\phi(s)^{1+\mu}.
\end{equation*}
Then $\phi(s)\equiv 0$ for $s\ge s_0+\frac{2B_0\phi(s_0)^{\mu}}{1-2^{-\mu}}.$
\end{lem}
\begin{proof}
We can choose a sequence $\{s_k\}_{k\ge 1}$ by induction:
\begin{equation*}
s_{k+1}-s_k=2B_0\phi(s_k)^{\mu}.
\end{equation*}
Then we choose $s=s_k$, $r=s_{k+1}-s_k$, we see that:
\begin{equation*}
\phi(s_{k+1})\le \frac{B_0\phi(s_k)^{1+\mu}}{s_{k+1}-s_k}\le \frac{1}{2}\phi(s_k).
\end{equation*}
That is, $\phi(s_k)\le 2^{-k}\phi(s_0)$. Hence
\begin{equation*}
0\le s_{k+1}-s_k\le 2B_0\phi(s_0)^{\mu}2^{-k\mu}.
\end{equation*}
This implies that $s_k$ is increasing and bounded from above, hence must converge to some $s_{\infty}$. Moreover,
\begin{equation*}
s_{\infty}-s_0=\sum_{k=0}^{\infty}(s_{k+1}-s_k)\le \sum_{k=0}^{\infty}2B_0\phi(s_0)^{\mu}2^{-k\mu}=\frac{2B_0\phi(s_0)^{\mu}}{1-2^{-\mu}}.
\end{equation*}
\end{proof}
Combining Lemma \ref{l3.3} and Lemma \ref{l3.4}, we immediately have the following corollary:
\begin{cor}\label{c3.5}
Let $v$ and $\varphi$ be as in previous lemma. Let $\delta>0$, $s_0>0$ be chosen so that:
\begin{equation*}
A_{\delta,s_0}=\int_M((1-\delta)v-\varphi-s_0)^+e^{nF}\omega_0^n\le \delta^{n+1},
\end{equation*}
then for any $\mu<\frac{1}{nq_0}$,
\begin{equation*}
\sup_M((1-\delta)v-\varphi)\le s_0+C_0vol(\{1-\delta)v-\varphi>s_0\})^{\mu}.
\end{equation*}
Here $C_0$ depends on $\mu$, $||e^{nF}||_{L^{p_0}}$ but not on $\delta$.
\end{cor}
\begin{proof}
Put $\phi(s)=vol(\Omega_{\delta,s})$. Then $\phi(s)$ would satisfy the assumptions of Lemma \ref{l3.4}, according to Lemma \ref{l3.3}. Then we see that $\phi(s)\equiv 0$ for $s\ge s_0+\frac{2B_0\phi(s_0)^{\mu}}{1-2^{-\mu}}$. This means $(1-\delta)v-\varphi\le s_0+\frac{2B_0\phi(s_0)^{\mu}}{1-2^{-\mu}}$.
\end{proof}
Next we need to get rid of $\delta$ in the above estimate. For this let us assume that $v\le 0$ and is bounded. Then in the left hand side, we have
\begin{equation*}
\sup_M(v-\varphi)\le \sup_M((1-\delta)v-\varphi).
\end{equation*}
On the other hand, we note that:
\begin{equation*}
vol(\Omega_{\delta,s_0})\le \frac{1}{s_0}\int_{\Omega_{\delta,s_0}}((1-\delta)v-\varphi)^+\omega_0^n\le \frac{1}{s_0}\big(||(v-\varphi)^+||_{L^1}+\delta||v||_{L^{\infty}}vol(\Omega_{\delta,s_0}).
\end{equation*}
Therefore, if $\delta||v||_{L^{\infty}}\le \frac{1}{2}$, we immediately conclude that:
\begin{equation}\label{3.6N}
vol(\Omega_{\delta,s_0})\le \frac{2}{s_0}||(v-\varphi)^+||_{L^1}.
\end{equation}
In view of (\ref{3.6N}) and Corollary \ref{c3.5}, we conclude that:
\begin{lem}\label{l3.7}
Let $v$ and $\varphi$ be as in Proposition \ref{p3.2}. We assume additionally that $v\le 0$ and is bounded. Assume also that $s_0$ and $\delta$ are chosen so that:
\begin{enumerate}
\item $s_0\ge 2\delta||v||_{L^{\infty}}$,
\item $A_{\delta,s_0}\le \delta^{n+1}$,
\end{enumerate}
then for any $\mu<\frac{1}{q_0n}$, we have:
\begin{equation}\label{3.8N}
\sup_M(v-\varphi)\le s_0+C_1s_0^{-\mu}||(v-\varphi)^+||_{L^1}^{\mu}.
\end{equation}
\end{lem}
Define $s_*(\delta)$ to be the infimum (with fixed $\delta$) of $s_0$ which satisfies the assumptions (1) and (2) put forth in Lemma \ref{l3.7}.
Note that the left hand side of (\ref{3.8N}) does not depend on $s_0$, we get:
\begin{equation*}
\sup_M(v-\varphi)\le \inf_{s_0\ge s_*(\delta)}(s_0+C_1s_0^{-\mu}||(v-\varphi)^+||_{L^1}^{\mu}).
\end{equation*}
At this point, it is clear that we must have an estimate (from above) of $s_*(\delta)$, and we have:
\begin{lem}\label{l3.9}
Let $\delta>0$ and $v$ be as in Proposition \ref{p3.2}. We also assume that $v$ is bounded. Define $s_*(\delta)$ to be the infimum of the set of $s_0$ such that $A_{\delta,s_0}\le \delta^{n+1}$ and $s_0\ge 2\delta||v||_{L^{\infty}}$, then for any $\beta>1$,
\begin{equation*}
s_*(\delta)\le \max(2\delta||v||_{L^{\infty}},C_2\delta^{-\frac{q_0n}{1-\beta^{-1}}}||(v-\varphi)^+||_{L^1}\big).
\end{equation*}
Here $C_2$ depends on $||e^{nF}||_{L^{p_0}}$, $\beta$, but is independent of $\delta$.
\end{lem}
\begin{proof}
Note that $A_{\delta,s_0}$ depends continuously on $s_0$, we see that, for $s_0=s_*(\delta)$, either you have $A_{\delta,s_0}=\delta^{n+1}$ or $s_0=2\delta||v||_{L^{\infty}}$.
If $s_*(\delta)=2\delta||v||_{L^{\infty}}$, then we are already done.
If $A_{\delta,s_*(\delta)}=\delta^{n+1}$, then from Proposition \ref{p3.2}, we see that for any $q>1$,
\begin{equation*}
A_{\delta,s_*}^{-\frac{q}{n}}\int_M(((1-\delta)v-\varphi-s)^+)^{\frac{n+1}{n}q}\omega_0^n\le C\int_M\exp\big(\frac{\beta_n((1-\delta)v-\varphi-s_*)^{\frac{n+1}{n}}}{A_{\delta,s_*}^{\frac{1}{n}}}\big)\omega_0^n\le C'.
\end{equation*}
This gives:
\begin{equation}\label{3.10}
||((1-\delta)v-\varphi-s_*)^+||_{L^q}\le C_4 A_{\delta,s_*}^{\frac{1}{n+1}}.
\end{equation}
Here $C_4$ depends on $q$ but is independent of $\delta$.
Therefore
\begin{equation*}
\begin{split}
&A_{\delta,s_*}=\int_{\Omega_{\delta,s_*}}((1-\delta)v-\varphi-s)^+e^{nF}\omega_0^n\le ||e^{nF}||_{L^{p_0}}||((1-\delta)v-\varphi-s_*)^+||_{L^{q_0}}\\
&\le ||e^{nF}||_{L^{p_0}}||((1-\delta)v-\varphi-s_*)^+||_{L^{\beta q_0}}vol(\Omega_{\delta,s_*})^{\frac{1}{q_0}(1-\beta^{-1})}\\
&\le ||e^{nF}||_{L^{p_0}}C(\beta,p_0)A_{\delta,s_*}^{\frac{1}{n+1}}(\frac{2}{s_*}||(v-\varphi)^+||_{L^1})^{\frac{1}{q_0}(1-\beta^{-1})}.
\end{split}
\end{equation*}
In the above, we used (\ref{3.6N}) and (\ref{3.10}). Keeping in mind that $A_{\delta,s_*}=\delta^{n+1}$, we get that
\begin{equation*}
s_*\le C\delta^{-\frac{nq_0}{1-\beta^{-1}}}||(v-\varphi)^+||_{L^1}.
\end{equation*}
\end{proof}
As a corollary of the above estimate, we see that:
\begin{cor}\label{stability}
Let $\varphi$ be a bounded negative solution to $f(\lambda(h_{\varphi}))=e^{F}$ with $e^{nF}\in L^{p_0}(\omega_0^n)$ for some $p_0>1$.
Let $\{\varphi_{\eps}\}_{0<\eps<1}$ be a family of upper approximations of $\varphi$ such that $\varphi_{\eps}<0$, $\varphi_{\eps}$ uniformly bounded, and for some $\gamma_2>0$, $0<\gamma_1<\frac{\gamma_2}{q_0n}$,
\begin{enumerate}
\item There exists $C_{3}>0$ independent of $\eps$ such that $\frac{C_{1.5}\eps^{\gamma_1}}{2}I+g^{i\bar{k}}(g_{j\bar{k}}+(\varphi_{\eps})_{j\bar{k}})\in \Gamma$ in the viscosity sense;
\item There exists $C_4>0$ independent of $\eps$ such that $||\varphi_{\eps}-\varphi||_{L^1}\le C\eps^{\gamma_2}.$
\end{enumerate}
Then for each $\gamma<\min(\gamma_1,\gamma_2-q_0n\gamma_1,\frac{\gamma_2}{1+q_0n})$, there exists $C_5>0$ such that
\begin{equation*}
\sup_M(\varphi_{\eps}-\varphi)\le C_5\eps^{\gamma}.
\end{equation*}
\end{cor}
\begin{proof}
Apply Lemma \ref{l3.7} with $\delta=C_3\eps^{\gamma_1}$, we find that
\begin{equation}\label{3.12NN}
\sup_M(\varphi_{\eps}-\varphi)\le s_0+C_1s_0^{-\mu}(C_4\eps^{\gamma_2})^{\mu},
\end{equation}
for any $s\ge s_*(\delta)$. (Of course here we have $\delta=C_{3}\eps^{\gamma_1}$.)
We would like to obtain an optimal estimate for $\sup_M(\varphi_{\eps}-\varphi)$ by minimizing the right hand side of (\ref{3.12NN}) in terms of $s_0$, with $s_0$ subject to the constraint that $s_0\ge s_*(C_3\eps^{\gamma_1})$.
Denote $M_1=\sup_{\eps>0}||\varphi_{\eps}||_{L^{\infty}}$.
Lemma \ref{l3.9} gives us that:
\begin{equation}
\begin{split}
&s_*(C_3\eps^{\gamma_1})\le \max(2(C_3\eps^{\gamma_1})M_1,
C_2(C_3\eps^{\gamma_1})^{-\frac{q_0n}{1-\beta^{-1}}}C_4\eps^{\gamma_2})\\
&\le C_{4.5}\eps^{\min(\gamma_1,-\frac{q_0n\gamma_1}{1-\beta^{-1}}+\gamma_2)},\,\,\,C_{4.5}=\max(2C_3M_1,\,C_2C_3^{-\frac{q_0n}{1-\beta^{-1}}}C_4).
\end{split}
\end{equation}
On the other hand, the right hand side of (\ref{3.12NN}) is monotone decreasing for $s_0\in (0,(C_1C_4^{\mu})^{\frac{1}{1+\mu}}\eps^{\frac{\gamma_2\mu}{1+\mu}})$ and is monotone increasing for \sloppy $s_0\in ((C_1C_4^{\mu})^{\frac{1}{1+\mu}}\eps^{\frac{\gamma_2\mu}{1+\mu}},+\infty)$.
Denote $\gamma_3=\min(\gamma_1,-\frac{q_0n\gamma_1}{1-\beta^{-1}}+\gamma_2)$ and $C_{4.7}=(C_1C_4^{\mu})^{\frac{1}{1+\mu}}$. There are two cases to consider:
\begin{enumerate}
\item If $\gamma_3>\frac{\gamma_2\mu}{1+\mu}$, then the choice of $s_0=C_{4.7}\eps^{\frac{\gamma_2\mu}{1+\mu}}$ will be $\ge s_*(C_3\eps^{\gamma_1})$, and we are allowed to use (\ref{3.12NN}) to conclude that
\begin{equation*}
\sup_M(\varphi_{\eps}-\varphi)\le C_{4.9}\eps^{\frac{\gamma_2\mu}{1+\mu}}.
\end{equation*}
\item If $\gamma_3\le \frac{\gamma_2\mu}{1+\mu}$, then we may not be able to choose $s_0=C_{4.7}\eps^{\frac{\gamma_2\mu}{1+\mu}}$. In this case, we just choose $s_0=C_{4.5}\eps^{\gamma_3}$ in the estimate (\ref{3.12NN}) to conclude that:
\begin{equation*}
\sup_M(\varphi_{\eps}-\varphi)\le C_{4.91}\eps^{\gamma_3}.
\end{equation*}
\end{enumerate}
Combining both cases, we get
\begin{equation*}
\sup_M(\varphi_{\eps}-\varphi)\le C_{4.92}\eps^{\min(\gamma_3,\frac{\gamma_2\mu}{1+\mu})}=C_{4.92}\eps^{\min(\gamma_1,\gamma_2-\frac{q_0n\gamma_1}{1-\beta^{-1}},\frac{\gamma_2\mu}{1+\mu})}.
\end{equation*}
Note that one can choose arbitrary $\beta>1$ and arbitrary $\mu<\frac{1}{q_0n}$, so the H\"older exponent can be made as close to $\min(\gamma_1,\gamma_2-q_0n\gamma_1,\frac{\gamma_2}{1+q_0n})$ as one desires.
\end{proof}
\iffalse
\section{Stability estimates of Hessian equations}
Next we improve the estimate of $u_{\eps}-u$ from $L^1$ norm to the $L^{\infty}$ norm.
First let me be clear what is meant by ``admissible in the viscosity sense".
\begin{defn}
Let $v\in C(M)$. We say that $\frac{\delta}{2}I+(h_v)_j^i\in\Gamma$ in the sense of viscosity, if for any point $p\in M$, any local $C^2$ function $\eta$ touching $v$ from above at point $p$, we have:
\begin{equation*}
\frac{\delta}{2}I+(h_{\eta})_j^i:=\frac{\delta}{2}I+g^{i\bar{k}}(g_{j\bar{k}}+\eta_{j\bar{k}})\in\Gamma.
\end{equation*}
\end{defn}
Then we can prove the following proposition.
\begin{prop}
Let $v\in C(M)$, $s>0$, $\delta>0$and $\frac{\delta}{2}I+(h_v)_j^i\in\Gamma$ in the viscosity sense as defined above. Define $A_{\delta,s}=\int_M ((1-\delta)v-\phi-s)^+ e^{nF} \omega^n$.\\
Then there is a constant $\beta_n>0$, and $C>0$, depending only on the background metric and structural constant of f, such that:
\begin{equation*}
\int_M exp(\beta_n)(\frac{(((1-\delta)v-\phi-s)^+)^{\frac{n+1}{n}}}{A_{\delta,s}^{\frac{1}{n}}})\omega^n \le C exp(C\delta^{-(n+1)}A_{\delta,s}).
\end{equation*}
\end{prop}
\begin{proof}
We consider the function
\begin{equation*}
\Phi=\eps_1((1-\delta)v-u-s)-(-\psi+\Lambda)^{\alpha}.
\end{equation*}
Here $\psi$ is defined as the solution to:
\begin{equation*}
\begin{split}
&(\omega+\sqrt{-1}\partial\bar{\partial}\psi)^n=\frac{((1-\delta)v-u-s)^+e^{nF}}{A_{\delta,s}}\omega^n,\\
&\sup_M\psi=0.
\end{split}
\end{equation*}
Since $v$ is only continuous, there is no guarantee that $\psi$ is $C^2$. To do it rigorously, we need to take $v_j$ to be a sequence of smooth functions which converges to $v$ uniformly, and $\eta_j(x)$ be a sequence of smooth and strictly positive functions which converges to $x^+$. Then we consider instead $\psi_j$, which solves
\begin{equation*}
(\omega+\sqrt{-1}\partial\bar{\partial}\psi_j)^n=\frac{\eta_j((1-\delta)v_j-u-s)}{A_{j,\delta,s}}\omega_0^n.
\end{equation*}
Here $A_{j,\delta,s}=\int_M\eta_j((1-\delta)v_j-u-s)^+e^{nF}\omega^n$. Then we can consider $\Phi_j=\eps_0((1-\delta)v-u-s)-(-\psi_j+\Lambda)^{\alpha}$.
For now, let us assume that $\psi$ is $C^2$ so as to justify the following argument. One can consider $\psi_j$ above in order to be completely rigorous (but argument is the same).
Since $\Phi$ is continuous, it achieves maximum on $M$, say, at $p_0$, and we denote:
\begin{equation*}
M_0:=\max_M\eps_0\big((1-\delta)v-u-s)-(-\psi+\Lambda)^{\alpha_1}.
\end{equation*}
Our goal is to show that $M_0\le 0$. If not, namely $M_0>0$, we can do the following argument.
At point $p$, the function $Q:=\frac{1}{1-\delta}\big(u+s+\frac{M_0+(-\psi+\Lambda)^{\frac{n}{n+1}}}{\eps_1}\big)$ touches $v$ from above at $p$. Hence we get:
\begin{equation*}\label{0.18NNN}
\lambda((\frac{\delta}{2}+1)I+g^{i\bar{k}}Q_{j\bar{k}})\in\Gamma.
\end{equation*}
As before, we consider the linearized operator defined as:
\begin{equation*}
Lv=\frac{\partial F}{\partial h_{ij}}((h_{u})_j^i)g^{i\bar{k}}v_{j\bar{k}}.
\end{equation*}
We consider normal coordinate at $p$ so that $g_{i\bar{j}}=\delta_{ij}$ and $(h_{u})_j^i$ is diagnal at $p$. Then we have: $\frac{\partial F}{\partial h_{ij}}(p)=\frac{\partial f}{\partial \lambda_i}(\lambda(h_{u}))\delta_{ij}$. Then we can compute:
\begin{equation*}
L(\frac{Q}{1+\frac{\delta}{2}})=-\sum_i\frac{\partial f}{\partial \lambda_i}+\sum_i\frac{\partial f}{\partial \lambda_i}(\lambda(h_{u}))(h_{\frac{Q}{1+\frac{\delta}{2}}})_{i\bar{i}}\ge -\sum_i\frac{\partial f}{\partial \lambda_i}+\sum_i\frac{\partial f}{\partial \lambda_i}(\lambda(h_{u}))\mu_i.
\end{equation*}
In the above, $\mu_i$ are the eigenvalues of $h_{\frac{Q}{1+\frac{\delta}{2}}}$, in the increasing order. The last inequality used Horn-Shur lemma. Now we know that $\mu=(\mu_1,\cdots,\mu_n)\in\Gamma$, thanks to (\ref{0.18NNN}).
Hence
\begin{equation*}
\sum_i\frac{\partial f}{\partial \lambda_i}(\lambda(h_{\varphi}))\mu_i=\frac{d}{dt}|_{t=0}(f(\lambda +t\mu)\ge 0.
\end{equation*}
It is due to that $t\mapsto f(\lambda+t\mu)$ is a concave function bounded from below.
Hence we get
\begin{equation*}
L(\frac{Q}{1+0.5\delta})\ge -\sum_i\frac{\partial f}{\partial \lambda_i}.
\end{equation*}
On the other hand, we can compute:
\begin{equation*}
\begin{split}
&Q_{j\bar{k}}=\frac{1}{1-\delta}(u_{j\bar{k}}+\eps_0^{-1}\frac{n}{n+1}(-\psi+\Lambda)^{-\frac{1}{n+1}}(-\psi_{j\bar{k}})-\frac{n}{(n+1)^2}(-\psi+\Lambda)^{-\frac{n+2}{n+1}}\psi_j\psi_{\bar{k}})\\
&=\frac{1}{1-\delta}\big((g_{u})_{j\bar{k}}-\eps_0^{-1}\frac{n}{n+1}(-\psi+\Lambda)^{-\frac{1}{n+1}}(g_{\psi})_{j\bar{k}}-(1-\eps_0^{-1}\frac{n}{n+1}(-\psi+\Lambda)^{-\frac{1}{n+1}})g_{j\bar{k}}\\
&-\frac{n}{(n+1)^2}(-\psi+\Lambda)^{-\frac{n+2}{n+1}}\psi_j\psi_{\bar{k}})
\end{split}
\end{equation*}
So that (after dropping the term $\psi_j\psi_{\bar{k}}$ and replace $(-\psi+\Lambda)^{-\frac{1}{n+1}}$ by $\Lambda^{-\frac{1}{n+1}}$)
\begin{equation*}
\begin{split}
&-(1+0.5\delta)\sum_i\frac{\partial f}{\partial \lambda_i}\le LQ\le \frac{1}{1-\delta}\big(\sum_i\frac{\partial f}{\partial \lambda_i}\lambda_i-\eps_0^{-1}\frac{n}{n+1}(-\psi+\Lambda)^{-\frac{1}{n+1}}\frac{\partial F}{\partial h_{ij}}(g_{\psi})_{j\bar{i}}\\
&-\frac{1}{1-\delta}(1-\eps_0^{-1}\frac{n}{n+1}\Lambda^{-\frac{1}{n+1}})\sum_i\frac{\partial f}{\partial\lambda_i}.
\end{split}
\end{equation*}
Next, you need to choose $\Lambda$ large enough so that
\begin{equation*}
\eps_0^{-1}\frac{n}{n+1}\Lambda^{-\frac{1}{n+1}}=0.1\delta.
\end{equation*}
Then from above, we get:
\begin{equation*}
\begin{split}
&0\le \sum_i\frac{\partial f}{\partial \lambda_i}\lambda_i-\eps_0^{-1}\frac{n}{n+1}(-\psi+\Lambda)^{-\frac{1}{n+1}}\frac{\partial F}{\partial h_{ij}}(g_{\psi})_{j\bar{i}}\\
&\le C_0f-\eps_0^{-1}\frac{n}{n+1}(-\psi+\Lambda)^{-\frac{1}{n+1}}\big(\frac{((1-\delta)v-u-s)^+e^{nF}}{A_{\delta,s}}\big)^{\frac{1}{n}}.
\end{split}
\end{equation*}
Since $M_0>0$, it would imply that:
\begin{equation*}
(((1-\delta)v-u-s)^+)^{\frac{1}{n}}(-\psi+\Lambda)^{-\frac{1}{n+1}}>\eps_0^{-\frac{1}{n}}.
\end{equation*}
Therefore, if $\eps_0$ is small enough so that
\begin{equation*}
C_0-\eps_0^{-\frac{n+1}{n}}A_{\delta,s}^{-\frac{1}{n}}<0,
\end{equation*}
it would not happen.
\end{proof}
From this, we still have Corollary \ref{cor0.21} and Lemma \ref{l0.22}, since we only need the inequality from Proposition \ref{p0.20}.
Let $0<\gamma<\frac{1}{2}$ to be determined.
Now we should choose $v=\varphi_{\eps}$, and $\delta=2C_3\eps^{\gamma}$. Then according to Lemma \ref{l0.22}, we get
\begin{equation*}
s_*(\delta)\le \max(C_4\eps^{\gamma},C_5
\eps^{-\gamma\frac{q_0n}{1-\frac{1}{\beta}}+\alpha\frac{1}{q_0}(1-\frac{1}{\beta})})<\eps^b.
\end{equation*}
Here $b>0$ is sufficiently small. Then we choose $s_0=\eps^b$ in Corollary \ref{cor0.21}, we get (by choosing $b<\alpha$)
\begin{equation*}
\sup(\varphi_{\eps}-\varphi)\le \eps^b+C\eps^{-\mu b+\mu\alpha}\le \eps^c.
\end{equation*}
As desired...\\
\fi
We can now prove the H\"older continuity of solutions:
\begin{thm}\label{t4.1}
Let $(M,\omega_0)$ be a compact K\"ahler manifold with nonnegative holomorphic bisectional curvature. Let $\varphi$ be a solution to the Hessian equation:
\begin{equation*}
f(\lambda[h_{\varphi}])=e^F,\,\,\,\lambda[h_{\varphi}]\in \Gamma,\,\,\,\sup_M\varphi=-1,
\end{equation*}
where $f$ and $\Gamma$ satisfies the structural assumptions put forth after (\ref{1.1New}). Assume that $e^{nF}\in L^{p_0}(\omega_0^n)$ for some $p_0>1$. Then for any $\mu<\frac{1}{2(1+q_0n)}$, $||\varphi||_{C^{\mu}}\le C$. Here $q_0=\frac{p_0}{p_0-1}$ and $C$ depends on the structural constants of $f$, $||e^F||_{L^{p_0n}}$ and the background metric.
\iffalse
Let $c$ be a positive lower bounded of $det\frac{\partial F}{\partial h_{ij}}$ given in the conditions satisfied by $f$. Let $q_0$ be given in the Corollary \ref{stability}. Suppose that the holomorphic bisectional curvature is non-negative. Then for any $\gamma_0 \in (0,\frac{2}{5+4q_0 n +\sqrt{16 q_0^2 n^2 +32 q_0 n +17}})$, we have that $||\varphi||_{C^{0,\gamma_0}}\le C$, where the constant $C$ depends on $q_0, c, M, \omega_0, ||e^F||_{L^{p_0}}$.
\fi
\end{thm}
\begin{proof}
Let $\frac{1}{2}<\beta\le 1$, then we have:
\begin{equation*}
\varphi_{\eps}(z)\ge\sup_{|\xi|_z<\eps^{\beta}}\big(\varphi(\exp_z(\xi))+\eps-\frac{1}{\eps}|\xi|_z^2\big)\ge \sup_{d(w,z)<\eps^{\beta}}\varphi(w)+\eps-\eps^{2\beta-1}.
\end{equation*}
On the other hand, using \ref{stability}, we see that
\begin{equation*}
\sup_{d(w,z)<\eps^{\beta}}\varphi(w)-\varphi(z)\le \eps^{2\beta-1}+C\eps^{\gamma}.
\end{equation*}
Hence we may take $\beta=\frac{1+\gamma}{2}$, we see that:
\begin{equation*}
\sup_{d(w,z)<\eps^{\frac{1+\gamma}{2}}}\varphi(w)-\varphi(z)\le C\eps^{\gamma}=C(\eps^{\frac{1+\gamma}{2}})^{\frac{2\gamma}{1+\gamma}}.
\end{equation*}
This will imply that $\varphi$ is H\"older continuous with exponent $\frac{2\gamma}{1+\gamma}$.
According to Corollary \ref{stability}, we can take any $\gamma<\min(\gamma_1,\gamma_2-q_0n\gamma_1,\frac{\gamma_2}{1+q_0n})$.
On the other hand, from Theorem \ref{hessian L1 estimate}, we see that we could take $\gamma_1=\frac{1}{2}$, $\gamma_2=\frac{1}{4}$ (with no initial assumption on the continuity of $\varphi$). So the range of $\gamma$ given by Corollary \ref{stability} says:
\begin{equation*}
\gamma<\sup_{0<\gamma_1\le \frac{1}{2}}\min(\gamma_1,\frac{1}{4}-q_0n\gamma_1,\frac{1}{4(1+q_0n)}).
\end{equation*}
We see that the range of $\gamma$ is $\gamma<\frac{1}{4(1+q_0n)}$.
Therefore the range of H\"older exponent is $<\frac{2}{4(1+q_0n)+1}$.
Next we observe that this H\"older exponent can be improved by an iteration process, using the second part of Theorem \ref{hessian L1 estimate}. To be more specific, we already obtained some H\"older exponent $\mu_0$, then the second part of Theorem \ref{hessian L1 estimate} would give an improved estimate for both $\gamma_1$ and $\gamma_2$ in Corollary \ref{stability}, hence will give us an improved $\gamma$, hence an improved H\"older exponent. Now we do the detailed calculation.
We already found that $\varphi$ has H\"older continuity for all $\mu<\frac{2}{4(1+q_0n)+1}$. We denote $\mu_0=\frac{2}{4(1+q_0n)+1}$. In general, let us denote $0<\mu_k<1$ to be the upper bound of the H\"older exponent. Then (ii)' of Theorem \ref{hessian L1 estimate} gives us that $\gamma_1$ can be improved to be $\frac{1+\mu_k}{2-\mu_k}$ and part (iii)' of the same theorem implies that $\gamma_2$ can be improved to be $\frac{1}{2(2-\mu_k)}$. So the upper bound of $\gamma$ given by Corollary \ref{stability} is given by:
\begin{equation*}
\gamma<\sup_{0<\gamma_1\le \frac{1+\mu_k}{2-\mu_k}}\min(\gamma_1,\frac{1}{2(2-\mu_k)}-q_0n\gamma_1,\frac{1}{2(2-\mu_k)(1+q_0n)}).
\end{equation*}
Hence the upper bound of $\gamma$ is just $\frac{1}{2(2-\mu_k)(1+q_0n)}$. Therefore, the new upper bound of H\"older exponent can be calculated by:
\begin{equation*}
\mu_{k+1}=\frac{2\times \frac{1}{2(2-\mu_k)(1+q_0n)}}{1+\frac{1}{2(2-\mu_k)(1+q_0n)}}=\frac{2}{2(2-\mu_k)(1+q_0n)+1},\,\,\,\mu_0=\frac{2}{4(1+q_0n)+1}.
\end{equation*}
One can verify that $\mu_k\le \mu_{k+1}$ and $\mu_k$ converges to $\frac{1}{2(1+q_0n)}$.
\end{proof}
\iffalse
\begin{proof}
We prove the theorem by iteration. First we prove that $\varphi$ is H\"older continuous for some H\"older exponent.
\begin{equation*}
\begin{split}
\varphi_{\eps}(x)&=\sup(\varphi(exp_x(\xi))+\eps -\frac{1}{\eps}|\xi|^2) \\
&\ge \varphi(y)+\eps -\frac{1}{\eps}|log_x(y)|^2
\end{split}
\end{equation*}
Integrate $y$ over $B_{\eps}(x)$ and divide by the volume of $B_{\eps}(x)$. We have that:
\begin{equation*}\label{mean value inequality inverse direction}
\begin{split}
\varphi_{\eps}(x)&\ge \frac{1}{Vol(B_{\eps}(x))}\int_{B_{\eps}(x)}(\varphi(y)+\eps-\frac{1}{\eps}|log_x(y)|^2)dy\\
&\ge \frac{1}{Vol(B_{\eps}(x))}\int_{B_{\eps}(x)}\varphi
(y) -C\eps.
\end{split}
\end{equation*}
Let $\gamma_1$, $\gamma_2$ and $\gamma$ be given in the Corollary \ref{stability}. Using the result of \cite{GP} we have that $|\varphi|\le C$ for some constant $C$. Then we can apply the Theorem \ref{hessian L1 estimate} to get that we can assume $\gamma_1\in (0,\frac{1}{2}]$ and $\gamma_2\in (0,\frac{1}{4}]$ whose exact values will be determined later to make $\gamma$ as big as possible. Using the Corollary \ref{stability}, we have that
\begin{equation*}
\begin{split}
&\gamma< \sup_{0 < \gamma_1\le \frac{1}{2}, 0< \gamma_2 \le \frac{1}{4}} \min(\gamma_1, \gamma_2-q_0n\gamma_1, \frac{\gamma_2}{1+q_0n}) \\
&=\sup_{0 < \gamma_1\le \frac{1}{2}} \min(\gamma_1, \frac{1}{4}-q_0n\gamma_1, \frac{1}{4(1+q_0n)})\\
&=\frac{1}{4(1+q_0n)}
\end{split}
\end{equation*}
We can have that for any $z\in M$ such that $d(x,z)=\eps$:
\begin{equation}\label{gamma2gamma0}
\begin{split}
\varphi(x)&\ge \frac{1}{Vol(B_{\eps^{\beta_0}}(x))}\int_{B_{\eps^{\beta_0}}(x)}\varphi(y)-C\eps^{\gamma \beta_0} \\
&\ge \frac{1}{Vol(B_{\eps^{\beta_0}}(x))}\int_{B_{\eps^{\beta_0}+
\eps}(z)}\varphi(y)-C\eps^{\gamma \beta_0} \\
&\ge \frac{\alpha(2n)(\eps^{\beta_0}+\eps)^{2n}}{Vol(B_{\eps^{\beta_0}}(x))}\frac{\int_{B_{\eps^{\beta_0}+\eps}(z)}\varphi(y)}{\alpha(2n)(\eps^{\beta_0}+\eps)^{2n}} -C\eps^{\gamma \beta_0}\\
&\ge \frac{\alpha(2n)(\eps^{\beta_0}+\eps)^{2n}}{Vol(B_{\eps^{\beta_0}}(x))}\varphi(z)-C\eps^{\gamma \beta_0} \\
&\ge \varphi(z)-C\eps^{\beta_0}-C\eps^{1-\beta_0}-C\eps^{\beta_0 \gamma}\\
&=\varphi(z) -C\eps^{\frac{\gamma}{1+\gamma}}.
\end{split}
\end{equation}
In the last line we take $\beta_0=\frac{1}{1+\gamma}$. Thus we prove that $\varphi$ is H\"older continuous with any exponent $\gamma_0=\frac{\gamma}{1+\gamma}<\frac{1}{4(1+q_0n)+1}$.\\
Next we want to use the iteration to improve the H\"older exponent. In the $k-$th iteration step, we denote $(\gamma_1)_k$ (resp. $(\gamma_2)_k$, $(\gamma)_k$ and $(\gamma_0)_k$) as a number we get such that $\gamma_1$ (resp. $\gamma_2$, $\gamma$, $\gamma_0$) can be taken as any positive number smaller than $(\gamma_1)_k$ (resp. $(\gamma_2)_k$, $(\gamma)_k$ and $(\gamma_0)_k$). From the Inequality \ref{gamma2gamma0}, we have that:
\begin{equation}\label{4.1}
\begin{split}
&(\gamma_0)_k=\frac{(\gamma)_k}{1+(\gamma)_k} \\
&(\gamma_0)_{k+1}=\frac{(\gamma)_{k+1}}{1+(\gamma)_{k+1}}.
\end{split}
\end{equation}
From the Theorem \ref{hessian L1 estimate}, we have that:
\begin{equation}\label{gamma1 gamma2}
\begin{split}
&(\gamma_1)_{k+1}=\frac{1+(\gamma_0)_k}{2-(\gamma_0)_k} \\
&(\gamma_2)_{k+1}=\frac{1}{2(2-(\gamma_0)_k)}.
\end{split}
\end{equation}
Then we can use the Corollary \ref{stability} to get that:
\begin{equation}\label{gamma}
\begin{split}
&(\gamma)_{k+1}=\sup_{0<s_1<(\gamma_1)_{k+1}, 0<s_2<(\gamma_2)_{k+1}} \min(s_1,s_2-q_0 n s_1, \frac{s_2}{1+q_0n}) \\
&=\sup_{0<s_1<(\gamma_1)_{k+1}} \min (s_1, (\gamma_2)_{k+1}-q_0 n s_1, \frac{(\gamma_2)_{k+1}}{1+q_0 n})
\end{split}
\end{equation}
Using the Equations \ref{gamma1 gamma2} we have that:
\begin{equation*}
\frac{(\gamma_2)_{k+1}}{1+q_0 n} <(\gamma_1)_{k+1}.
\end{equation*}
Thus the sup in the Equation \ref{gamma} is achieved when $s_1=\frac{(\gamma_2)_{k+1}}{1+q_0 n}$ and
\begin{equation}\label{4.2}
(\gamma)_{k+1}=\frac{(\gamma_2)_{k+1}}{1+q_0 n}.
\end{equation}
Combining the Equations \ref{4.1}, the Equation \ref{gamma1 gamma2} and the Equation \ref{4.2}, we have that:
\begin{equation}\label{main iteration}
\begin{split}
& (\gamma)_{k+1}=\frac{1}{2(2-\frac{(\gamma)_k}{1+(\gamma)_k})(1+q_0 n)} \\
& (\gamma_0)_{k+1}=1-\frac{1}{\frac{1}{2(2-(\gamma_0)_k)(1+q_0 n)}+1}
\end{split}
\end{equation}
We can calculate that $(\gamma)_1=\frac{1}{4(1+q_0 n)}$ and $(\gamma)_2=\frac{1}{2(2-(\gamma_0)_1)(1+q_0 n)}$. So we have that:
\begin{equation}\label{increasing 1 step}
(\gamma)_1 < (\gamma)_2.
\end{equation}
From the Equation \ref{main iteration}, we have that $(\gamma)_{k+1}=f((\gamma)_k)$ and $f$ is increasing on $(0,1)$. Combining this and the Inequality \ref{increasing 1 step}, we have that
\begin{equation}\label{increasing gemma step}
(\gamma)_k < (\gamma)_{k+1}
\end{equation}
for each $k$. Since
\begin{equation*}
(\gamma_0)_k=g((\gamma)_k)=\frac{(\gamma)_k}{1+(\gamma)_k}
\end{equation*}
and $g$ is an increasing function on $(0,1)$, combining the Inequality \ref{increasing gemma step} we have that:
\begin{equation*}
(\gamma_0)_k <(\gamma_0)_{k+1}.
\end{equation*}
Note that $(\gamma_0)_k\in (0,1)$ so they are bounded. Then we can take limit and define:
\begin{equation*}
\gamma_0 =\lim_{k\rightarrow \infty} (\gamma_0)_k
\end{equation*}
Let $k \rightarrow \infty $ in the second equation in the Equations \ref{main equation} to get that:
\begin{equation*}
\gamma_0 =1-\frac{1}{\frac{1}{2(2-\gamma_0)(1+q_0 n)}+1}
\end{equation*}
The only solution of this equation that lies in $[0,1]$ is: $\frac{2}{5+4q_0 n +\sqrt{16 q_0^2 n^2 +32 q_0 n +17}}$. So we have concluded the Theorem.
\end{proof}
\fi
\section{Bibliography}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 2,769 |
Q: How to center my text vertically in a circular div with css I am wondering what I am neglecting in my css, which is preventing the text from being properly centered vertically? Below is my current code, and here is a fiddle.
Here is my html
<div class="donateButton"><a href="#">Donate</a></div>
This is my CSS
.donateButton {
width:100px;
height:100px;
background-color:#FE6D4C;
-moz-border-radius: 50px;
-webkit-border-radius: 50px;
border-radius: 50px;
margin:0 auto 30px;
text-align:center;
}
.donateButton:hover {
background-color:#09C;
}
.donateButton a {
display:block;
width:100%;
color:#FFFFFF;
font-size:1em;
text-decoration:none;
padding-top:50%;
line-height:1em;
margin-top:0.5em;
}
Thanks in advance for any insight.
I also want to make this responsive eventually, but i guess that will be another question :)
A: If you're sure that the text will be just one line you can do this:
.donateButton a {
display:block;
width:100%;
color:#FFFFFF;
font-size:1em;
text-decoration:none;
line-height:100px;
}
A: The simplest way of doing this is to remove your padding-top from the text and set the line height to 100px, the same height as your div.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 4,333 |
By Julia Goldberg
COVID-19 by the numbers
Since the start of the new year, New Mexico has added 3,571 new COVID-19 cases, with 1,286 on Friday, 1,252 on Saturday and 1,033 on Sunday, bringing the total number of cases so far to 146,394. Of those, the health department has designated 68,876 as recovered. Santa Fe County had 133 additional cases during those three days (59, 49 and 25, respectively).
There have been 74 additional deaths over the last three days (25, 32 and 17), with three from Santa Fe County among residents at the Vista Hermosa facility: a woman in her 90s who was hospitalized and had underlying conditions and a man in his 80s with underlying conditions, both on Jan. 2; and yesterday, a woman in her 80s. There have now been 2,551 total fatalities statewide. As of yesterday, 716 people were hospitalized with COVID-19.
You can read all of SFR's COVID-19 coverage here. If you've had experiences with COVID-19, we would like to hear from you.
New study details Santa Fe's extreme housing shortage
"Housing affordability is a key issue facing employers, prospective businesses, city leaders, state officials and local Santa Feans," Susan Orth, president of the Santa Fe Association of Realtors, writes in the introduction to a new study that provides a comprehensive look at the issue for the city and county. "Growing smart will be key as we face the extreme housing and rental shortages in our community with limited resources to address the issue," she notes. Partners on the study included affordable housing, homelessness and business representatives. A few key takeaways include: In both the county and city, the majority of homes are owner occupied (70% vs. 62%, respectively). More than a third of Santa Fe residents commute out and 37.8% of its workers commute in for work. The Santa Fe Metropolitan Statistical Area runs short on rental units by 7,343 and, in 2019, Santa Fe rental vacancy rates were 2.3%, with average rent prices of $1,038. The study also notes what it describes as an "overlooked" market condition for Santa Fe's rental situation: the conversion of rentals to condominiums. The report uses 2019 as a baseline year, given that 2020 data was severely impacted by the COVID-19 pandemic. "Santa Fe, like the rest of the nation, will enter 2021 with economic uncertainty due to the [COVID-19] pandemic's looming presence," the report notes. "The housing market will continue to see-saw as waves of coronavirus infections pop up throughout the country."
Blazing new ground
Santa Fe County's first female fire chief, Jackie Lindsey, will start her new job mid-January. She didn't set out to become the first female fire chief but, rather, pursued firefighting as a passion, she tells the Albuquerque Journal. "It is important to society, more than ever before, for our young women, and girls, to see that there are no boundaries," she says. "And you do everything you can to get that—you're going to run into obstacles, but every person that's tried to achieve something is running into obstacles. So, I think don't let it be an obstacle." Lindsey has been a firefighter since graduating from the University of New Mexico, working for the city of Albuquerque before going to study at the Naval Postgraduate School. She also served as cabinet secretary for New Mexico's Homeland Security and Emergency Management department, and is currently on the FEMA National Advisory Council.
Fall-out for Acoma Pueblo
The New York Times delves into Indian Health Service's decision last November to slash funding at the Acoma-Cañoncito-Laguna hospital in Acoma Pueblo. As Acoma Pueblo Gov. Brian Vallo recently told SFR, that decision led his pueblo to work with the state rather than IHS on vaccine distribution. The service reduction at the hospital—in the middle of the pandemic—had dire consequences for tribal members reliant on it, resulting in people having to travel an hour away to Albuquerque for medical care. The situation highlighted systemic problems between tribal and federal governments, such as IHS' "decades-long weaknesses," the Times writes, which has "contributed to disproportionately high infections and death rates among Native Americans." Greg Smith, a lobbyist for the Pueblo of Acoma, says the decision to cut services resulted in 30 doctors and nurses either quitting or retiring, and reflects IHS' dislike for small and less cost-effective hospitals. "From a public health standpoint, how does it make any sense to close a 24-hour facility that is serving a vulnerable population during a global pandemic and effectively replace it with two daytime clinics?" Smith asks.
In Georgia O'Keeffe—A Life Well Lived, photographer Malcolm Varon captured O'Keeffe as she neared her 90th birthday, showcasing her homes and companions at Ghost Ranch and Abiquiú, as well as the landscapes that inspired her. Varon's book includes both his own reflections on his time with O'Keeffe, as well as those from the Georgia O'Keeffe Museum Director Cody Hartley and O'Keeffe scholar Barbara Buhler Lynes. KSFR's MK Mendoza speaks with Varon about this new book in a recent edition of "Wake Up Call."
2021 Indigenous Playlist
Cowboys & Indians magazine features two New Mexico musicians in its "Real Deal" column recommending Indigenous American musicians to check out in 2021. Albuquerque's Shelley Morningsong (Cheyenne), the magazine says, "should be on your playlist and live-performance bucket list." Morningsong won the Native American Music Awards Artist of the Year in 2019 and Best Blues Recording for her album Simple Truth. Guitarist Levi Platero (Navajo) also makes the list. Platero began performing as a member of his family's band blues rock band "The Plateros" in 2004 before venturing out on as a solo artist; his single "Take Me Back" won the 2016 New Mexico Music Awards for Best Blues.
Santa Fe made Mansion Global's "listing of the day" for Jan. 1 with a $3 million former gallery that features 5,300 square feet of showroom space attached to a "cozy" one-bedroom casita. Sotheby's listing agent Darlene Streit tells the magazine the property owner has another gallery nearby and used this space for special events and exhibits. Streit says the property "could be turned into a fabulous residence for someone, or it could be live/work or an actual gallery again. It has that flexibility." Indeed: The property has been staged to look like a home, and the listing counts three bedrooms and five bathrooms. The 512-square-foot casita has one bedroom and one full bathroom. Location: East side, just a stone's throw from Canyon Road.
High and dry
Today will be mostly sunny and windy, with a high near 48 degrees and north wind 5 to 15 mph becoming south in the afternoon. From the looks of it, that's the week in a nutshell with our next (slim) chance for precipitation currently on tap for next weekend…a lifetime away.
Thanks for reading! The Word totally relates to this New Yorker "Hibernation Insomnia" essay, even though it is satire told from a bear's point of view.
Read More by Julia Goldberg
Julia Goldberg is a senior correspondent at SFR, covering politics, technology and other topics. She previously served as SFR editor from December 2000 through April 2011, taught journalism and creative writing full-time at Santa Fe University of Art and Design and is the author of Inside Story: Everyone's Guide to Reporting and Writing.
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