text stringlengths 14 5.77M | meta dict | __index_level_0__ int64 0 9.97k ⌀ |
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declare module 'rw' {
var rw: any;
export default rw;
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 782 |
{"url":"http:\/\/nrich.maths.org\/public\/leg.php?code=-99&cl=3&cldcmpid=6310","text":"Search by Topic\n\nResources tagged with Working systematically similar to Addition Equation Sudoku:\n\nFilter by: Content type:\nStage:\nChallenge level:\n\nThere are 130 results\n\nBroad Topics > Using, Applying and Reasoning about Mathematics > Working systematically\n\nEquation Sudoku\n\nStage: 3, 4 and 5 Challenge Level:\n\nSolve the equations to identify the clue numbers in this Sudoku problem.\n\nStage: 3 and 4 Challenge Level:\n\nThis is a variation of sudoku which contains a set of special clue-numbers. Each set of 4 small digits stands for the numbers in the four cells of the grid adjacent to this set.\n\nStage: 3, 4 and 5 Challenge Level:\n\nYou need to find the values of the stars before you can apply normal Sudoku rules.\n\nStage: 3 and 4 Challenge Level:\n\nFour numbers on an intersection that need to be placed in the surrounding cells. That is all you need to know to solve this sudoku.\n\nIntersection Sudoku 1\n\nStage: 3 and 4 Challenge Level:\n\nA Sudoku with a twist.\n\nIntersection Sudoku 2\n\nStage: 3 and 4 Challenge Level:\n\nA Sudoku with a twist.\n\nOne Out One Under\n\nStage: 4 Challenge Level:\n\nImagine a stack of numbered cards with one on top. Discard the top, put the next card to the bottom and repeat continuously. Can you predict the last card?\n\nProduct Doubles Sudoku\n\nStage: 3 and 4 Challenge Level:\n\nEach clue number in this sudoku is the product of the two numbers in adjacent cells.\n\nTwin Corresponding Sudoku III\n\nStage: 3 and 4 Challenge Level:\n\nTwo sudokus in one. Challenge yourself to make the necessary connections.\n\nRectangle Outline Sudoku\n\nStage: 3 and 4 Challenge Level:\n\nEach of the main diagonals of this sudoku must contain the numbers 1 to 9 and each rectangle width the numbers 1 to 4.\n\nColour Islands Sudoku 2\n\nStage: 3, 4 and 5 Challenge Level:\n\nIn this Sudoku, there are three coloured \"islands\" in the 9x9 grid. Within each \"island\" EVERY group of nine cells that form a 3x3 square must contain the numbers 1 through 9.\n\nRatio Sudoku 2\n\nStage: 3 and 4 Challenge Level:\n\nA Sudoku with clues as ratios.\n\nPole Star Sudoku\n\nStage: 4 and 5 Challenge Level:\n\nA Sudoku based on clues that give the differences between adjacent cells.\n\nCorresponding Sudokus\n\nStage: 3, 4 and 5\n\nThis second Sudoku article discusses \"Corresponding Sudokus\" which are pairs of Sudokus with terms that can be matched using a substitution rule.\n\nBochap Sudoku\n\nStage: 3 and 4 Challenge Level:\n\nThis Sudoku combines all four arithmetic operations.\n\nWallpaper Sudoku\n\nStage: 3 and 4 Challenge Level:\n\nA Sudoku that uses transformations as supporting clues.\n\nRatio Sudoku 1\n\nStage: 3 and 4 Challenge Level:\n\nA Sudoku with clues as ratios.\n\nTwin Corresponding Sudokus II\n\nStage: 3 and 4 Challenge Level:\n\nTwo sudokus in one. Challenge yourself to make the necessary connections.\n\nIntersection Sums Sudoku\n\nStage: 2, 3 and 4 Challenge Level:\n\nA Sudoku with clues given as sums of entries.\n\nThe Naked Pair in Sudoku\n\nStage: 2, 3 and 4\n\nA particular technique for solving Sudoku puzzles, known as \"naked pair\", is explained in this easy-to-read article.\n\nConstellation Sudoku\n\nStage: 4 and 5 Challenge Level:\n\nSpecial clue numbers related to the difference between numbers in two adjacent cells and values of the stars in the \"constellation\" make this a doubly interesting problem.\n\nTwin Line-swapping Sudoku\n\nStage: 4 Challenge Level:\n\nA pair of Sudoku puzzles that together lead to a complete solution.\n\nIntegrated Product Sudoku\n\nStage: 3 and 4 Challenge Level:\n\nThis Sudoku puzzle can be solved with the help of small clue-numbers on the border lines between pairs of neighbouring squares of the grid.\n\nStage: 3 and 4 Challenge Level:\n\nFour small numbers give the clue to the contents of the four surrounding cells.\n\nTwin Corresponding Sudoku\n\nStage: 3, 4 and 5 Challenge Level:\n\nThis sudoku requires you to have \"double vision\" - two Sudoku's for the price of one\n\nRainstorm Sudoku\n\nStage: 4 Challenge Level:\n\nUse the clues about the shaded areas to help solve this sudoku\n\nSeasonal Twin Sudokus\n\nStage: 3 and 4 Challenge Level:\n\nThis pair of linked Sudokus matches letters with numbers and hides a seasonal greeting. Can you find it?\n\nIntegrated Sums Sudoku\n\nStage: 3 and 4 Challenge Level:\n\nThe puzzle can be solved with the help of small clue-numbers which are either placed on the border lines between selected pairs of neighbouring squares of the grid or placed after slash marks on. . . .\n\nTwin Chute-swapping Sudoku\n\nStage: 4 and 5 Challenge Level:\n\nA pair of Sudokus with lots in common. In fact they are the same problem but rearranged. Can you find how they relate to solve them both?\n\nGames Related to Nim\n\nStage: 1, 2, 3 and 4\n\nThis article for teachers describes several games, found on the site, all of which have a related structure that can be used to develop the skills of strategic planning.\n\nPole Star Sudoku 2\n\nStage: 3 and 4 Challenge Level:\n\nThis Sudoku, based on differences. Using the one clue number can you find the solution?\n\nRatio Sudoku 3\n\nStage: 3 and 4 Challenge Level:\n\nA Sudoku with clues as ratios or fractions.\n\nDiagonal Sums Sudoku\n\nStage: 2, 3 and 4 Challenge Level:\n\nSolve this Sudoku puzzle whose clues are in the form of sums of the numbers which should appear in diagonal opposite cells.\n\nAlphabetti Sudoku\n\nStage: 3 and 4 Challenge Level:\n\nThis Sudoku requires you to do some working backwards before working forwards.\n\nPlum Tree\n\nStage: 4 and 5 Challenge Level:\n\nLabel this plum tree graph to make it totally magic!\n\nMagic Caterpillars\n\nStage: 4 and 5 Challenge Level:\n\nLabel the joints and legs of these graph theory caterpillars so that the vertex sums are all equal.\n\nMagnetic Personality\n\nStage: 2, 3 and 4 Challenge Level:\n\n60 pieces and a challenge. What can you make and how many of the pieces can you use creating skeleton polyhedra?\n\nSandwiches\n\nStage: 2, 3, 4 and 5 Challenge Level:\n\nArrange the digits 1, 1, 2, 2, 3 and 3 so that between the two 1's there is one digit, between the two 2's there are two digits, and between the two 3's there are three digits.\n\nCinema Problem\n\nStage: 3 and 4 Challenge Level:\n\nA cinema has 100 seats. Show how it is possible to sell exactly 100 tickets and take exactly \u00a3100 if the prices are \u00a310 for adults, 50p for pensioners and 10p for children.\n\nAll-variables Sudoku\n\nStage: 3, 4 and 5 Challenge Level:\n\nThe challenge is to find the values of the variables if you are to solve this Sudoku.\n\nLCM Sudoku II\n\nStage: 3, 4 and 5 Challenge Level:\n\nYou are given the Lowest Common Multiples of sets of digits. Find the digits and then solve the Sudoku.\n\nOlympic Logic\n\nStage: 3 and 4 Challenge Level:\n\nCan you use your powers of logic and deduction to work out the missing information in these sporty situations?\n\nLOGO Challenge - Pentagram Pylons\n\nStage: 3, 4 and 5 Challenge Level:\n\nPentagram Pylons - can you elegantly recreate them? Or, the European flag in LOGO - what poses the greater problem?\n\nLCM Sudoku\n\nStage: 4 Challenge Level:\n\nHere is a Sudoku with a difference! Use information about lowest common multiples to help you solve it.\n\nMultiplication Equation Sudoku\n\nStage: 4 and 5 Challenge Level:\n\nThe puzzle can be solved by finding the values of the unknown digits (all indicated by asterisks) in the squares of the $9\\times9$ grid.\n\nInstant Insanity\n\nStage: 3, 4 and 5 Challenge Level:\n\nGiven the nets of 4 cubes with the faces coloured in 4 colours, build a tower so that on each vertical wall no colour is repeated, that is all 4 colours appear.\n\nIntroducing NRICH TWILGO\n\nStage: 1, 2, 3, 4 and 5 Challenge Level:\n\nWe're excited about this new program for drawing beautiful mathematical designs. Can you work out how we made our first few pictures and, even better, share your most elegant solutions with us?\n\nLOGO Challenge - Sequences and Pentagrams\n\nStage: 3, 4 and 5 Challenge Level:\n\nExplore this how this program produces the sequences it does. What are you controlling when you change the values of the variables?\n\nLatin Squares\n\nStage: 3, 4 and 5\n\nA Latin square of order n is an array of n symbols in which each symbol occurs exactly once in each row and exactly once in each column.\n\nProduct Sudoku\n\nStage: 3, 4 and 5 Challenge Level:\n\nThe clues for this Sudoku are the product of the numbers in adjacent squares.","date":"2016-05-25 09:26:56","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\": 0.5152940154075623, \"perplexity\": 2345.535723623802}, \"config\": {\"markdown_headings\": false, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2016-22\/segments\/1464049274324.89\/warc\/CC-MAIN-20160524002114-00029-ip-10-185-217-139.ec2.internal.warc.gz\"}"} | null | null |
Q: How to do vector based(polygon based) image proccessing in c# I'm looking for a a way that allows me to apply image processing techniques to a set of polygons that are defined by a set of points (x and y) and then apply the following opperations:
substract,erode,dialate,get all area that form a continues area connected to a particulair spot.
The images are monochrome (no colour data).
Anybody know what a way to do this? I have looked at gtk-sharp but that seems to miss some of these functions and most other libraries only take raster graphics as input.
Opencv does not seem to work so well with a set of points (or is there a way to fix this?).
What I'm looking for is something like this:
image image;
List<List<Point>> shapes;
List<Point> extract;
foreach(List<Point> shape in shapes){
image.add(shape)
}
image.remove(extract)
image.erode(5);
image.dialate(5)
image.getAllConnectedTo(0,0);
Anyone know a good library? Or a method to do this in general?
Note if anything is unclear feel free to ask so I can impprove it.
A: I used SharpDX library for 3D graphics. They also had 2D module and samples on GitHub. It seems easy to use and well documented. Check this sample.
Here is code snippet from sample:
void TessellationSink.AddTriangles(Triangle[] triangles)
{
// Add Tessellated triangles to the opened GeometrySink
foreach (var triangle in triangles)
{
GeometrySink.BeginFigure(triangle.Point1, FigureBegin.Filled);
GeometrySink.AddLine(triangle.Point2);
GeometrySink.AddLine(triangle.Point3);
GeometrySink.EndFigure(FigureEnd.Closed);
}
}
All drawing stuff executes in "forever" loop. So, you changes will be displayed on the screen after new render occurs.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 733 |
\section*{Abstract}
Video-based eye tracking is a valuable technique in various research fields. Numerous open-source eye tracking algorithms have been developed in recent years, primarily designed for general application with many different camera types. These algorithms do not, however, capitalize on the high frame rate of eye tracking cameras often employed in psychophysical studies. We present a pupil detection method that utilizes this high-speed property to obtain reliable predictions through recursive estimation about certain pupil characteristics in successive camera frames. These predictions are subsequently used to carry out novel image segmentation and classification routines to improve pupil detection performance. Based on results from hand-labelled eye images, our approach was found to have a greater detection rate, accuracy and speed compared to other recently published open-source pupil detection algorithms. The program's source code, together with a graphical user interface, can be downloaded at \url{https://github.com/tbrouns/eyestalker}.
\textbf{\textit{keywords:}} eye tracking, video-oculography, pupil detection, open source, algorithm, psychophysics, methodology
\section{Introduction}
The act of measuring the movement of the eye, known as eye tracking, enjoys a broad range of applications across various disciplines, from neuroscience and psychology to market research and industrial engineering \cite{duchowski2002}. Within neuroscience, eye tracking can serve as a method to diagnose neurological problems such as dyslexia, schizophrenia, Alzheimer's disease, attention deficit hyperactivity disorder and autism \cite{dolezal2015}, but can also be used as a research tool to study the visual, auditory \cite{volck2015} and vestibular systems \cite{allison1996}. Distinct eye tracking techniques have been developed over the years, each with their own set of advantages and disadvantages. In electro-oculography (EOG) small differences in the electric potential on the skin, caused by the retinal polarity, are measured with surface electrodes placed around the eye, which give an estimate of the eye position. The method is often employed in sleep studies, because it works when the eye is closed and one can measure for long stretches at a time without user discomfort. However, it suffers from poor accuracy and unreliability of vertical eye movement recordings \cite{heide1999}. Far greater resolution is achieved with the search coil technique, which is considered the gold standard for eye tracking \cite{geest2002}. It uses a small copper coil embedded in a contact lens that is placed on the eye. A voltage is induced in the coil in the presence of an external oscillating magnetic field. The voltage amplitude gives a measure of the eye position, because it is proportional to the orientation of the coil in the magnetic field. Like in EOG, this operational mechanism allows eye movement recording even when the eye is closed. An obvious drawback of the search coil method is the invasive nature of mounting and removing the lens. Furthermore, corneal irritation and erosion can occur in long recording sessions from wearing the coil \cite{heide1999}.
A compromise between the high accuracy of the search coil technique and the non-invasiveness of EOG can be offered by video-based eye tracking, also known as video-oculography (VOG). In VOG, the eye is tracked using a camera combined with computer vision algorithms. We can distinguish between two different approaches, which differ in how close the camera is removed from the eye and therefore require different image processing techniques. Much published work has focussed on developing eye tracking algorithms for camera systems that capture the whole head (e.g. webcam-based). These systems are particularly useful in human-computer interaction and are noted for their ease of use, but severely lack in accuracy, which makes them unsuitable for research purposes where eye tracking accuracy is crucial, such as saccade analysis. In order to compete with the search coil technique in terms of spatial resolution, the camera must be brought much closer to the eye. Generally speaking, eye movement is tracked in these close-up images of the eye by finding the pupil position through feature detection. Detection is aided by illumination of the eye with an infra-red (IR) light emitting diode (LED), which turns the pupil much brighter or darker, depending on whether the IR LED is close to or away from the camera's optical axis \cite{morimoto2000}, making the pupil a more clearly discernible feature in the image. The use of the IR light also means that the system is functional in the dark, which is often a requirement in experimental research.
Commercially available state-of-the-art eye trackers with the aforementioned VOG set-up are generally expensive, upwards of \$10.000. In addition to their price tag, another downside of commercial eye trackers is their use of proprietary software, which often prevents any customization to fit the specific needs of the user. Fortunately, a number of developers have been spurred on to produce low-cost hardware alternatives \cite{abbott2012}\cite{adiba2016}\cite{kim2014}\cite{li2006}\cite{mantiuk2012}\cite{putra2013}\cite{schneider2011}, publish their own sophisticated pupil detection algorithms \cite{li2005}\cite{lin2010}\cite{long2007}\cite{swirski2012}\cite{zhu1999} and make open-source VOG software publicly available \cite{Fuhl2015}\cite{Kassner2014}\cite{zimmermann2016}. Unfortunately, few implementations are suited for use in a wide variety of psychophysical experiments, which require both precision and speed. As technology advances, high-speed cameras with high-resolution sensors will become more affordable. This cutting-edge hardware should be accompanied by the appropriate software, capable of fully capitalizing on the improved camera features.
Here, we introduce an open-source pupil detection algorithm that aims to exploit the high frequency nature of many modern cameras ($\geq 200$ Hz). The method is based on the notion that the characteristics of the pupil can only change a relatively small amount between consecutive frames when recording at a high frame rate. This allows us to make accurate predictions about the pupil's characteristics for the next frame, where our search will be limited to features in the image that match these predictions. This method contrasts with many published pupil tracking algorithms where each camera frame is effectively treated as being independent from the one that preceded it. An advantage of algorithms that are built on this premise is their universal applicability. They work equally well with different cameras operating at varying speeds and on random collections of eye images. However, this property becomes irrelevant when users only utilize such algorithms in combination with high-speed cameras. The pupil detection method that we present in this text has been specifically tailored for individuals who use these types of powerful cameras under more ideal conditions (e.g. a controlled laboratory setting), giving them the opportunity to take greater advantage of their hardware and environment to improve eye tracking performance.
\section{Methods}
The pupil detection algorithm, which is illustrated in Figure \ref{fig:detection_method}, works by performing a number of distinct processing tasks in the following order:
\begin{enumerate} \itemsep0em
\setcounter{enumi}{-1}
\item Receive predicted pupil characteristics for current frame
\item Crop image to smaller search area depending on predicted size, shape and position of pupil
\item Update predicted pupil position following object recognition of the pupil through Haar-like feature detection, after removing corneal reflection interference
\item Detect all object boundaries, i.e. edges, in area around new position estimate by identifying brightness discontinuities with Canny edge detection
\item Select sub-set of detected edges that are at an acceptable distance from the position estimate
\item Thin Canny edges to minimum thickness using morphological operations
\item Segment edges at transition point between distinct features in the image
\begin{enumerate}
\item Choose non-branching path in edges that comes closest to predicted pupil circumference
\item Split edges at points of extremely high or low curvature
\item Reduce length of edges if they exceed pupil circumference estimate
\end{enumerate}
\item Classify edges into two classes: pupil contour edges and non-pupil contour edges
\item Fit one or more ellipses on edges categorized as pupil contour edges
\item Choose most optimal fit based on pupil characteristic predictions
\item Calculate new predictions for next frame
\end{enumerate}
\begin{figure}[ht]
\vspace*{-1.0cm}
\centering
\includegraphics[width=\textwidth]{detection_method_2.png}
\caption{Example of an image being sent through the pupil detection pipeline. The 0 to 10 labels refer to the processing steps as given in the main text. The Roman numerals indicate various explanatory images: \textit{I.} Raw image input in which the pupil position prediction is marked (teal cross). Position prediction is used to obtain the rectangular search area (blue outline). \textit{II.} Resulting image after cropping to search area. Detected position of corneal reflection (red rectangle) is accounted for during Haar-like feature detection of pupil (green rectangle). \textit{III.} Canny edge detection is performed in an area (blue rectangle in \textit{II}) whose dimensions and position are determined by combination of Haar-like feature detection and predicted pupil characteristics. Detected Canny edges are shown in red. \textit{IV.} Sub-set of Canny edges is selected (orange) based on their position. Remaining edges are discarded. \textit{V.} Selected edges are segmented if required. We then choose edge sections that belong to the pupil-iris contour (green), after classification. \textit{VI.} One or more ellipses are fitted on pupil-iris contour edges. Most optimal ellipse fit is accepted (teal), depending on predictions of pupil characteristics. Centre of chosen ellipse (cross) serves as pupil position. \textit{VII.} Ellipse fit characteristics are used to update pupil predictions for next frame.}
\label{fig:detection_method}
\end{figure}
\FloatBarrier
The algorithm relies on various parameters to carry out these steps, some of which depend on the image size and frame rate of the camera. The default parameter values given in the text are based on a reference eye image size of around 400 $\times$ 200 pixels and a frame rate of 250 Hz. Using these reference values, a few parameters should be scaled down or up for other image sizes or sampling rates. Furthermore, we will sometimes refer to a data set from which certain information has been extracted. This data set is a collection of more than 400,000 close-up images of the left eye of 12 different individuals, which were taken over the course of around 100 saccadic eye movements made by each person in various directions, while they were seated in a dark room. The pictures were made by a UI154xLE-M camera from IDS Imaging Development Systems (Obersulm, Germany), using infra-red illumination and with the reference camera settings specified earlier.
\FloatBarrier
\subsection{Feature value prediction}
The strength of the pupil detection algorithm is drawn from the use of predictions that are made about certain characteristics of the pupil in each camera frame. Consequently, we need a method that is able to make these predictions as accurately as possible, without being computationally demanding and the need to set many different parameters. For online estimation, a recursive estimation method is preferred over batch estimation, because it only requires the measurement from the current frame and the prediction from the previous one, which contrasts with batch estimation where significantly more data has to be processed to make the prediction. A popular type of recursive estimator is the Kalman filter, which has been extensively used in computer vision research \cite{wren1997} and has already been applied to eye tracking in the past \cite{abd2002} \cite{chi2014} \cite{zhang2006}, although limited to whole-face eye tracking. To my knowledge, this is the first application of recursive estimation in eye tracking using close-up images of the eye. Based on this previous work, the Kalman filter may seem like an obvious choice for our recursive estimator as well. However, because we are interested in estimating many different variables, the use of the Kalman filter is made difficult due to the exponentially increased complexity of setting the measurement and process noise covariances, which are integral to the functionality of the filter. Optimal parameter values would have to be found through tuning \cite{welch95}, a process that most likely has to be repeated for each new camera set-up. Since we want our algorithm to be easily implemented in other systems, we have opted to develop a more basic recursive estimation method, which may lack the accuracy of the Kalman filter, but requires fewer parameters to be set. Using our recursive estimator, we keep track of the pupil features that are listed in Table \ref{table:feature_variables}. After the pupil has been detected in a particular camera frame, each one of these features is updated with the following general formula:
\begin{equation} \label{eq:feature_update}
\hat{f}_{n+1} = \hat{f}_{n} + \alpha\Delta f_{n} + c_{n}p_{n}
\end{equation}
Here, $\hat{f}_{n}$ is the predicted feature value for the current frame $n$. This value is updated to obtain the prediction for the next frame, $\hat{f}_{n+1}$, by adding the difference $\Delta f_{n}$ between the measured and the predicted value (the prediction error), plus a momentum term, $p_{n}$. In every frame where we detect the pupil, we will possess a measured feature value $f_{n}$, which is used to calculate $\Delta f_{n}$ through:
\begin{equation}
\Delta f_{n} = f_{n} - \hat{f}_{n}
\end{equation}
\begin{table}
\centering
\begin{tabular}{r c l}
\textbf{Estimated features}
& \textbf{Symbol}
& \textbf{Description} \\
Position
& $\hat{s}$
& Position of the centre of the pupil in Cartesian coordinates \\
Circumference
& $\hat{C}$
& Circumference of pupil-iris boundary \\
Aspect ratio
& $\hat{AR}$
& Ratio between pupil major and minor axes \\
Width
& $\hat{W}$
& Width of pupil bounding box \\
Height
& $\hat{H}$
& Height of pupil bounding box \\
Angle
& $\hat{\theta}$
& Pupil rotation angle \\
Brightness/intensity
& $\hat{I}$
& Grey-scale value of the inner pupil-iris boundary \\
Radial gradient
& $\hat{G_{r}}$
& Radial image gradient of the pupil-iris boundary \\
Curvature
& $\hat{\kappa}$
& Signed curvature of pupil-iris boundary \\
\end{tabular}
\caption{Pupil characteristics that are estimated in successive frames with the recursive estimator.}
\label{table:feature_variables}
\end{table}
The value of $\Delta f_{n}$ is modified by the gain factor $\alpha$ in equation \ref{eq:feature_update}, which is a constant value between 0 and 1. The larger the gain, the more the estimation is based on the current measurement $f_{n}$. The value that we choose for $\alpha$ should depend on the feature in question. A feature such as pupil position should be updated relatively quickly, because it can shift rapidly on short timescales, requiring a larger $\alpha$. Significant changes in the pupil's size and shape, on the other hand, generally occur on longer timescales, allowing for a smaller gain, which helps reduce the influence of noise. If we were to just use $\Delta f_{n}$ to update $f_{n}$ then this approach would fail when the feature value monotonically increases, because our prediction will lag behind. For this reason, a momentum term $p_{n}$ has been added in equation \ref{eq:feature_update}, where $p_{n}$ is updated in every frame by:
\begin{equation} \label{eq:momentum_update}
p_{n+1} = p_{n}+ \alpha(\Delta f_{n} - p_{n})
\end{equation}
What $p_{n}$ tries to approximate is the prediction error $\Delta f_{n}$ of the feature. Ideally, $p_{n}$ causes $\Delta f_{n}$ to drop to zero, resulting in accurate predictions. The rate of change of $p_{n}$ is also reduced by the factor $\alpha$, with the same $\alpha$ value being used for both the prediction error and momentum. Furthermore, $p_{n}$ is modified by an additional factor $c_{n}$ in equation \ref{eq:feature_update}. This quantity gives a measure of the certainty of our prediction and plays an important role in other parts of the algorithm as well. This factor is added to the equation to avoid erratic behaviour when little information about the pupil is available. For example, when the pupil momentarily disappears from view because the eye is closed, we are unsure of its position and our prediction will most likely have a large error, $\Delta f_{n} \gg 0$. So when the pupil appears once again, $p_{n}$ will approximate this large error, resulting in an excessive rate of change, causing $\hat{f}_{n}$ to overshoot the actual value for the feature. To avoid this behaviour, we control the influence of momentum with the degree of certainty $c_{n}$ of our prediction, which is a value between 0 (low) and 1 (high). The value for $c_{n}$ is altered according to the rate of change of the feature. If the rate falls within an acceptable range, $c_{n}$ increases, otherwise it decreases. This is given by:
\begin{equation}
\Delta c_{n} = 1 - \frac{2} {1 + e^{k(\delta_{n} - \delta'_{\theta})}}
\end{equation}
Where $\Delta c_{n}$ indicates the change in certainty, which is calculated using a generalized logistic function that limits $\Delta c_{n}$ to values between $+1$ and $-1$. The variable $\delta_{n}$ tell us how much the feature value has shifted from one frame to the next. This quantity differs from $\Delta f_{n}$, because $\delta_{n}$ can be a relative change and is always positive, whereas $\Delta f_{n}$ is always an absolute change and can be negative or positive. The constant $\delta'_{\theta}$ is an upper limit of typical $\delta_{n}$ values and is determined empirically (see further down). If $\delta_{n}$ is below this threshold value of $\delta'_{\theta}$, the function will return a positive number, causing $c_{n}$ to increase. Alternatively, $c_{n}$ decreases when $\delta_{n} > \delta'_{\theta}$. The steepness $k$ of the curve is determined by:
\begin{equation}
k= \log\frac{\frac{1}{a} - 1}{b\delta'_{\theta}}
\end{equation}
Where we set $a = 0.99$ and $b = 0.50$, which translates to the logistic function reaching 99\% of its maximum value at $\delta_{n}=\frac{1}{2}\delta'_{\theta}$. These parameters are set in this way, because the certainty measure should not be biased towards ever smaller $\delta_{n}$ values, but should view all $\delta_{n}$ values that are physiologically feasible as equivalent. For example, how certain we are about the pupil's position during a fixation period should not automatically be greater than our level of certainty about its position during a saccadic eye movement. Both can realistically occur so should be treated as equal in our certainty calculation. We update the certainty $c_{n}$ by first computing an intermediate value:
\begin{equation} \label{eq:certainty_update}
c'_{n+1} = c'_{n} + \alpha \alpha_c \Delta c_{n}
\text{ with }
c'_{n} =
\begin{cases}
1 & \text{if } c'_{n} > 1\\
0 & \text{if } c'_{n} < 0\\
\end{cases}
\end{equation}
Where $\Delta c_{n}$ is modified with the product of $\alpha$ and a new gain factor $\alpha_{c}$. The certainty value that we use in equation \ref{eq:feature_update} is calculated through another logistic function:
\begin{equation}
c_{n} = \frac{1}{1 + e^{-\tau(c'_{n}-\frac{1}{2})}}
\end{equation}
The use of the logistic function has two purposes. First, the function bounds $c_{n}$ in the 0 to 1 range. Second, the function imposes some latency for changes in certainty from the limits, which is controlled by the constant $\tau$ factor (set to 10 by default). As a consequence, multiple precise measurements are required before the certainty starts to significantly increase from its minimum value, or multiple imprecise or non-detections before certainty drops down from the maximum. We essentially create three different states the algorithm can attain: a focussed state when certainty is high ($c \approx 1$), an exploratory state when it is low ($c \approx 0$), and a narrow transitional state in between. One reason for this design is that extreme values of $p_{n}$ are effectively ignored when the algorithm is still in the exploratory state. Another reason is that an all-or-nothing scheme is generally preferred for parts of the algorithm that rely on $c_{n}$ to modify specific parameters. This has the effect that some functions are turned on and off depending on the state the algorithm is in, which can boost performance.
In order to avoid having to specify a different set of parameters for each feature, we limit our features to two classes, namely position (S) and appearance (A). The position class only has the position prediction $\hat{s}$ as its member. All other features belong to the appearance class. This particular classification is chosen due to the different timescales these features change at, as mentioned before. The same $c_{n}$ and $\alpha$ values are shared between members of each class. We use $c_{S}$ and $\alpha_{S}$ for position, while $c_{A}$ and $\alpha_{A}$ are used for appearance (with $\alpha_{S} = 0.75$ and $\alpha_{A} = 0.40$). For the position class, $\delta_{n}$ is equal to the absolute change in position (i.e. displacement). For the appearance class, only changes in circumference, $C$, and aspect ratio, $AR$, are considered when calculating $\Delta c_{n}$. The average $\Delta c_{n}$ value between both is used to update $c_{A}$. We only consider these two features, because it can already be assumed that a given measurement is accurate if the measured size and shape of the pupil changes little between consecutive frames, making it redundant to check the change of other features (e.g. brightness) as well. The various $\delta_n$ values are given by:
\begin{align}
\delta_{s} &= \sqrt{(x_{n} - \hat{x}_{n})^2 + (y_{n} - \hat{y}_{n})^2} \\
\delta_{C} &= \frac{\abs{ C_{n} - \hat{C}_{n}}}{\max(C_{n},\hat{C}_{n})} \label{eq:change_C} \\
\delta_{AR} &= \ \abs{ AR_{n} - \hat{AR}_{n} } \label{eq:change_AR}
\end{align}
To obtain the threshold $\delta'_{\theta}$ values for $\delta_{s}$, $\delta_{AR}$ and $\delta_{C}$, we look at our data set to see what kind of values we typically obtain for these quantities. For each $\delta'_{\theta}$ type, we choose a threshold at which less than 1\% of measurements have a $\delta_{n}$ value that is larger than $\delta'_{\theta}$. We also determine another threshold, $\delta''_{\theta}$, which is similar to $\delta'_{\theta}$, but uses 0.1\% of measurements as the benchmark instead of 1\%. For our camera set-up, we set the lower thresholds to $\delta'_{\theta,s} = 3$, $\delta'_{\theta,C} = 0.03$, $\delta'_{\theta,AR} = 0.03$ and the upper thresholds to $\delta''_{\theta,s} = 6$, $\delta''_{\theta,C} = 0.12$, $\delta''_{\theta,AR} = 0.09$. When $\delta_{n} > \delta''_{\theta}$, we make the assumption that the measurement is very likely to have been significantly influenced by noise effects and is not a true physiological result. The $\delta''_{\theta}$ parameter is used at a later point. Both $\delta'_{\theta}$ and $\delta''_{\theta}$ need to be altered according to the frame-rate of the camera. The thresholds can be decreased when using a faster camera and should be increased with slower cameras.
Up until this point, we have not dealt with the issue of non-detections, i.e. frames in which the algorithm has not been able to find the pupil. In the case of a non-detection, measures of $f_{n}$ and $\delta_{n}$ are not available. This stops us from updating our feature predictions and certainty using equations \ref{eq:feature_update} and \ref{eq:certainty_update}. We deal with this problem in our certainty calculation by giving $\Delta c_{n}$ the minimum value of –1 when a non-detection occurs. For the feature predictions, this issue is resolved by introducing the average feature variable $\bar{f}_{n}$. This average feature variable is a quantity that our prediction $\hat{f}_{n}$ can fall back on when no immediate information about the pupil is available. We only calculate $\bar{f}_{n}$ for features belonging to the appearance class, since an average position is not a meaningful quantity in this context. For position, we instead rely on the detection algorithm to supply an estimation of the position prediction $\hat{s}_{n}$ during non-detections (see section \ref{sec:approximate_detection}). We update $\hat{f}_{n}$ during a non-detection according to:
\begin{equation}
\hat{f}_{n+1} = \hat{f}_{n} + \alpha_{A} \Delta \hat{f}_{n} + c_{A,n} p_{n}
\end{equation}
\begin{equation}
\Delta \hat{f}_{n} = \bar{f}_{n} - \hat{f}_{n}
\end{equation}
\noindent
The momentum term $p_{n}$ decays to zero by setting $\Delta \hat{f}_{n}$ to zero in equation \ref{eq:momentum_update}, leaving:
\begin{equation}
p_{n+1} = (1 - \alpha_{A})p_{n}
\end{equation}
\noindent
We update $\bar{f}_{n}$ in every frame in a similar fashion to $\hat{f}_{n}$, but without using the momentum term:
\begin{equation}
\bar{f}_{n+1} = \bar{f}_{n} + \alpha_{mean} \Delta \bar{f}_{n}
\end{equation}
\begin{equation}\label{eq:prediction_error_avg}
\Delta \bar{f}_{n} = \hat{f}_{n} - \bar{f}_{n}
\end{equation}
The gain factor of $\alpha_{mean}$ should be much smaller than $\alpha_{A}$ in order for $\bar{f}_{n}$ to keep track of an overall mean of $f_{n}$ (we use $\alpha_{mean} = 0.005$). During a non-detection, $\hat{f}_{n}$ in equation \ref{eq:prediction_error_avg} is replaced by a typical value that is initially assigned to $\hat{f}_{n}$ and $\bar{f}_{n}$ at the start of detection. The end result is that in every frame we have a prediction $\hat{f}$ for each type of feature, together with a measure $c$ of how certain we are about that prediction. These quantities play a role in almost every part of the pupil detection algorithm.
\FloatBarrier
\subsection{Search area}
The first application of our predictions and certainties is in reducing the processing area for pupil detection. By recognizing that the pupil can only translate and transform a limited amount between consecutive frames, we can narrow our search of the pupil to an area that is much smaller than the total size of the image. Obviously, the width $W_{AOI}$ and height $H_{AOI}$ of this area of interest (AOI) still need to be at least as large as the predicted width $\hat{W}$ and height $\hat{H}$ of the pupil, so we set:
\begin{equation}
W_{AOI} = \Delta L + \hat{W}
\end{equation}
\begin{equation}
H_{AOI} = \Delta L + \hat{H}
\end{equation}
Here, $\Delta L$ is some additional length that is added to our prediction for the pupil size to ensure that the pupil falls wholly in the AOI. This length is determined by how much the pupil could potentially move and resize between two successive frames:
\begin{equation}
\Delta L = \frac{\hat{C}\delta_{\theta,C}}{\pi} + 2 \delta_{\theta,s}
\end{equation}
\noindent
The variables $\delta_{\theta,s}$ and $\delta_{\theta,C}$ are derived from the threshold values $\delta''_{\theta}$ in the following way:
\begin{alignat}{2}
\delta_{\theta,s} &= (1 - c_{S}) (\delta_{max,s} - \delta''_{\theta,s}) &&+ \delta''_{\theta,s} \\
\delta_{\theta,C} &= (1 - c_{A}) (\delta_{max,C} - \delta''_{\theta,C}) &&+ \delta''_{\theta,C} \label{eq:delta_circumference}
\end{alignat}
The degree of certainty modifies the size of the AOI, with greater $c$ values resulting in smaller processing areas, up to a minimum value of $\delta_{\theta}$ and a maximum value of $\delta_{max}$. These maximum values are a type of theoretical upper limit and are given by:
\begin{align}
\delta_{max,s} &= \max(W_{img} - \hat{W}, H_{img} - \hat{H}) \\
\delta_{max,C} &= \max(\frac{C_{max} - \hat{C}}{C_{max}}, \frac{\hat{C} - C_{min}}{\hat{C}})
\end{align}
Where $W_{img}$ and $H_{img}$ respectively are the width and height of the input image. The maximum circumference change is calculated from the smallest possible pupil circumference ($C_{min}$) and the largest ($C_{max}$), which are empirically determined for our set-up.
\FloatBarrier
\subsection{Approximate detection}
\label{sec:approximate_detection}
For several parts of the algorithm, it is essential that a rough estimate of the pupil position is available. When $c_{S}$ is low, we cannot completely rely on $\hat{s}$ to provide this approximation, because it is most likely inaccurate. In that case, $\hat{s}$ is re-evaluated by convolving the image with a Haar-like feature detector \cite{viola2001}. This type of detector works by moving a Haar-like feature over the image and calculating the difference in the sum of all pixel values between the dark and light rectangular regions. This detection method is made computationally efficient by first calculating the integral image, which ensures $O(n)$ performance with respect to the number of input pixels.
An endless number of Haar-like features are available. In some other pupil detection algorithms \cite{swirski2012} \cite{Kassner2014}, the centre-surround feature is used to obtain an approximate position of the pupil (see Figure \ref{fig:haar_like_features}), where it is assumed that the maximum response is obtained if the dark region overlaps with the pupil. Here, on the other hand, we opt for a vertical line feature instead, for two reasons. First, the primary feature in the image that might fool a basic feature detector are the eye lashes, because they can sometimes be as dark as the pupil. However, they are also quasi horizontally homogeneous in their brightness. So by looking at the contrast between the central area and the two areas it is horizontally flanked by, we can better discriminate between pupil and eye lashes. Second, the contrast of a vertical line feature is not as negatively affected by eye lashes when the eye is partially closed compared to the centre-surround feature, since eye lashes are generally located directly above or below the pupil, which is mostly ignored by the vertical line feature when it is centred on the pupil.
\begin{figure}[ht]
\centering
\includegraphics[width=0.6\textwidth]{haar_like_features.png}
\caption{Two different Haar-like features. The vertical line feature is used in the pupil detection algorithm. Its dimensions are determined by pupil size predictions, $\hat{W}$ and $\hat{H}$. Three different values define the feature response, which are the summed pixel intensities of the left ($I_{l}$), centre ($I_{c}$) and right ($I_{r}$) areas.}
\label{fig:haar_like_features}
\end{figure}
As mentioned before, the Haar-like feature detection method works by just looking at the difference in total intensity between the dark and light regions, disregarding the absolute intensity of either area. This does mean, however, that a relatively bright feature can potentially mislead the feature detector if it happens to be flanked by an even brighter area. For this reason, we consider both the intensity of the dark region as well as the contrast between dark and light regions when calculating the Haar-like feature response, $F_{H}$:
\begin{equation}
F_{H} = -w_{1} \bar{i}_{c} + w_{2} (\frac{\bar{i}_{l} + \bar{i}_{r}}{2} - \bar{i}_{c})
\label{eq:haar_feature_response}
\end{equation}
We use the average pixel intensity $\bar{i}$ to calculate the response, instead of the summed intensity, because this makes the response invariant to the size of the fluctuating Haar-like feature area, which depends on the predicted width and height of the pupil. Both terms are modified by a weight constant $w$. We want to find the optimal combination of weights that leads to the greatest response for the pupil region and the smallest for non-pupil regions. From our data set, we obtain $\bar{i}$ values for the three regions at image locations that correspond with our measured pupil positions, but also at locations directly outside the pupil region (e.g. eye lashes). We then calculate $F_{H}$ for both groups and determine for which weight ratio the greatest separation between the two groups is found. This degree of separation is quantified by the test statistic of the two-sample Kolmogorov-Smirnov test, or \texttt{kstest2} in MATLAB\textsuperscript{\textregistered} (R2016a, The MathWorks, Natick, MA, United States). Greatest separation is obtained by finding the maximum of the test statistic with respect to the unconstrained weight values, using the MATLAB function \texttt{fminsearch}. This process is performed on a random sub-set of the data. A different sub-set is used for evaluation. The optimal weight ratio is found for:
\begin{equation*}
\frac{w_{1}}{w_{2}} = 3.3
\end{equation*}
One last confounding element that needs to be taken care of is the corneal reflection caused by the infra-red LED. This light can significantly disturb feature detection if it reflects off the cornea in front of the pupil, because it causes an increase in pupil brightness. We deal with this glint by first detecting it and then removing its influence if it overlaps with the centre rectangle of the vertical line feature. Glint detection is performed by convolving the AOI using the kernel given below in which every term is zero except in the centre and corners.
\begin{equation*}
kernel =
\begin{pmatrix}
-1 & 0 & \cdots & 0 & -1 \\
0 & \ddots & & & 0 \\
\vdots & & 1 & & \vdots \\
0 & & & \ddots & 0 \\
-1 & 0 & \cdots & 0 & -1 \\
\end{pmatrix}
\end{equation*}
A square kernel is used because the glint is often approximately circular. The width of this kernel, $W_{kernel}$, should be slightly larger than the diameter of the corneal reflection in the image. If the distance between IR-LED and eye varies little between individuals for the hardware set-up, which is generally the case, then this parameter can be set as a global constant. The point of maximum response after convolution is then assumed to coincide with the glint position. This method provides robust detection as long as the glint overlaps with the pupil in the image, which is sufficient for our purposes because its position is irrelevant to us when it does not obscure the pupil.
\begin{figure}[ht]
\centering
\includegraphics[width=0.4\textwidth]{haar_like_feature_glint.png}
\caption{Removal of glint influence from Haar-like feature. The dashed rectangle indicates the position of the corneal reflection. The quantity $I'_{glint}$ is the summed intensity of the section of the glint area that overlaps with the central area of the Haar-like feature. The intensity $I'_{glint}$ is subtracted from the summed intensity of the central area to obtain the corrected intensity, $I'_{c}$.}
\label{fig:haar_like_feature_glint}
\end{figure}
Once the glint has been detected, a square with the same size as the convolution kernel is centred on the glint position, $s_{glint}$. The summed intensity of the section of this square that overlaps with the dark area of the vertical line feature is subtracted from $I_{c}$ (see also Figure \ref{fig:haar_like_feature_glint}):
\begin{equation}
I'_{c} = I_{c} - I'_{glint}
\end{equation}
By doing the same for the surface areas, we can calculate a new average pixel intensity of the central region after glint removal:
\begin{equation}
\bar{i_{c}} = \frac{I'_{c}}{\hat{W} \times \hat{H} - A'_{glint}}
\end{equation}
\noindent
Where $A'_{glint}$ is the surface area of the region that corresponds with $I'_{glint}$.
The position $s_{H}$ of maximum Haar-like feature response can now be used to update $\hat{s}$. However, we must be cautious in using $s_{H}$, because it is more prone to errors than $\hat{s}$ when the certainty is high. Therefore, the greater the certainty, the less the new position prediction $\hat{s}_{new}$ will be based on $s_{H}$.
\begin{equation}
\hat{s}_{new} = s_{H} + c_{n}(\hat{s} - s_{H})
\end{equation}
Lastly, since these functions can potentially operate on the entire image (depending on the size of the AOI), it might be necessary to limit the number of iterations by down-sampling the image in order to achieve satisfactory computational speed. This should not appreciably affect accuracy of pupil detection, since we are only interested in an approximate position at this point. The original image resolution is immediately restored afterwards, before proceeding with the next processing steps. Further speed enhancement can be achieved during glint detection by only performing image convolution on pixels that have a brightness above a certain high threshold (e.g. $>200$ for 8-bit grayscale).
\FloatBarrier
\subsection{Canny edge detection}
\label{sec:canny_edge_detection}
After updating the approximate pupil position $\hat{s}$ in the previous step, Canny edge detection \cite{canny1986} is performed in an area with dimensions $W_{AOI} \times H_{AOI}$, centred around $\hat{s}$. The OpenCV \cite{opencv_library} implementation is chosen to carry out this task due to its computational efficiency. A Gaussian filter is applied beforehand. The Canny edge detector identifies and locates points of sharp changes in pixel intensity, which characterize boundaries of objects in the image \cite{maini2009}, and combines these points into thin line segments called edges. It transforms the AOI into a binary image, where all pixels that belong to an edge, also known as edge points, have been given a value of 1 and all other pixels a value of 0. Here, an individual edge is defined as any collection of 8-connected edge points.
\FloatBarrier
\subsection{Morphological operation}
In the next part of the algorithm, a number of processing steps are performed on the detected edges, which attempt to filter out any edge that does not belong to the pupil-iris boundary. In order to speed up and simplify some of these processes, all edges need to be thinned to the minimum amount of pixels required to define it. During Canny edge detection, non-maximum suppression will have already significantly sharpened the edges, but not yet sufficiently for our intentions. So two morphological operations are applied to the image, which are illustrated in Figure \ref{fig:morph}, that trim the edges to single pixel thickness. Any edge points erased by this operation are not entirely discarded, but are given a special tag instead. Before fitting an ellipse on a sub-set of the detected edges (see section \ref{sec:ellipse_fitting}), these removed edge points are restored in order to obtain a more accurate fit.
\begin{figure}[ht]
\centering
\includegraphics[width=0.5\textwidth]{morph.png}
\caption{Edges are thinned by removing pixels from the edge if their 8-connected neighbourhood matches at least one of these two patterns. Black and grey pixels represent the pixels that belong to an edge. The patterns can be freely rotated and mirrored. The left pattern trims diagonally oriented edges, whereas the right pattern trims horizontally or vertically (after \SI{90}{\degree} rotation) oriented edges.}
\label{fig:morph}
\end{figure}
\FloatBarrier
\subsection{Edge selection}
The first edge filter that is implemented is based on the fact that the pupil contour is likely to encircle or at least be close to the pupil position prediction $\hat{s}$. So we are going to select a sub-set of edges that are located at an acceptable distance from the pupil position estimate by sending out rays from $\hat{s}$ in eight directions, taking inspiration from the Starburst algorithm \cite{li2005}. The first edge that each ray encounters along its path is accepted, as well as any other edges that they come across after the first, provided they are within a radius:
\begin{equation*}
r = \frac{\Delta L}{2}
\end{equation*}
By allowing the ray to continue beyond $r$ when no edge was encountered, we make the method more robust against pupil size predictions that are too small. Furthermore, by accepting all encountered edges within a radius $r$, and not just the first edge a ray runs into, we avoid detection failure when $\hat{s}$ happens to be (partially) enclosed by non-pupil edges (e.g. glint contour). Furthermore, very small edges are ignored for edge selection. This threshold is determined by the edge window length parameter $N_{l}$, which serves multiple purposes in the algorithm (see section \ref{sec:curvature_segmentation} for its value).
\subsection{Edge classification}
\label{sec:edge_classification}
In the final step of our pupil detection algorithm, an ellipse needs to fitted on a combination of the previously selected edges. When multiple edges are available, it might be necessary to fit more than one ellipse and then choose the optimal one based on a few criteria described later (see section \ref{sec:ellipse_fitting}). We wish to avoid that scenario, however, because even though direct least squares fitting is relatively computationally inexpensive \cite{fitzgibbon1999} \cite{puatruaucean2012}, it can still be costly to fit many ellipses using this procedure in a single camera frame. Given $n$ edges, the maximum number of ellipses we would have to fit is equal to the total number of edge combinations:
\begin{equation} \label{eq:combinations}
\sum_{1 \leq k \leq n} \binom{n}{k} = 2^n - 1
\end{equation}
Which is the sum over all possible combination sizes $k$ chosen from $n$ (minus the null set). This value increases exponentially with respect to $n$, so it is important to keep the number of edges as small as possible. For this reason, a classification scheme is applied that divides edges into two categories: those that are part of the pupil-iris contour and those that are not. Edges that belong to the latter category are discarded. We base this classification on a number of different edge features, which are listed in Table \ref{table:edge_features}.
One edge feature is the length $L$ of an edge, which is approximated by the sum of the distances between 8-connected neighbours. If the relative arrangement of two neighbouring edge points is in one of the four cardinal directions then the distance between them is 1, if it is in one of the four intercardinal directions then it is $\sqrt{2}$. This length measure will be sufficiently accurate due to the morphological operations that have been applied, which have left the edges with minimum thickness.
The variance of radius $\sigma_{r}$ tells us about the orientation of an edge. It will be close to zero if the edge encircles $\hat{s}$ at a distance that is roughly constant, which we expect to be the case for the pupil boundary. Other edges are likely to have different orientations, leading to larger $\sigma_{r}$ values.
The gradient $G_{r}$ here is different from the one encountered during glint detection. This gradient is calculated at each edge point by looking at the direction vector between the edge point and the position estimate $\hat{s}$. The difference in intensity between two pixels that lie along this direction on opposite sides of the edge point is the measure for $G_{r}$ at that particular position. The pixel value closer to $\hat{s}$ is subtracted from the one further away, so that a larger $G_{r}$ is obtained for gradients that transition from dark to light the further we move outwards. By considering the radial gradient, we add additional weight to edges belonging to the pupil boundary, because it marks the transition from dark to light in the radial direction, unlike many other edges.
The intensity $I$ is not simply given by the mean grayscale value of all edge points, but is instead calculated while taking into account the direction the edge curves in. Only pixels that lie on the inside of the edge curve are considered for $I$. Of course, for the pupil these pixels will be relatively dark, but for other edges this is not necessarily the case. How the location of the inside of the curve is determined relates to how the curvature $\kappa$ is calculated, which is described at a later stage (see section \ref{sec:curvature_segmentation}).
\begin{table}[ht]
\resizebox{\textwidth}{!}{%
\begin{tabular}{ M{2cm} M{1cm} M{4cm} m{10cm} N}
\textbf{Edge features}
& \textbf{Symbol}
& \textbf{Feature value, $F_{i}$}
& \textbf{Description} \\
\hline
Length
& $L$
& $\displaystyle{\frac{\abs{ L - \hat{C} }}{\max(L,\hat{C})}}$
& Summed distance between neighbouring edge points
& \\ [25pt]
\hline
Radius
& $r$
& $\displaystyle{\frac{\abs{ r - \hat{C} / 2\pi }}{\max(r,\hat{C}/2\pi)}}$
& Mean distance from each edge point to $\hat{s}$
& \\ [25pt]
\hline
Variance of radius
& $\sigma_{r}$
& $\displaystyle{\frac{\sigma_{r}}{\hat{C}}}$
& Variance of the distance from each edge point to $\hat{s}$
& \\ [25pt]
\hline
Curvature
& $\kappa$
& $\displaystyle{\abs{ \kappa - \hat{\kappa} }}$
& Mean curvature of edge points
& \\ [25pt]
\hline
Radial gradient
& $G_{r}$
& $\displaystyle{\abs{ G_{r} - \hat{G}_{r} }}$
& Mean gradient of edge points calculated in radial direction from $\hat{s}$
& \\ [25pt]
\hline
Intensity
& $I$
& $\displaystyle{\abs{ I - \hat{I} }}$
& Mean brightness of pixels on inside of edge curve
& \\ [25pt]
\end{tabular}}
\caption{Edge features that the edge classification scheme is dependent on.}
\label{table:edge_features}
\end{table}
From the available edge feature values, higher-level feature values $F_{i}$ are constructed via the expressions given in Table \ref{table:edge_features}, which are made invariant to the image size. For each feature value $F_{i}$, a score is calculated through a function that is unique to each feature. We distinguish the two classes using a linear combination of all scores $S_{tot}$, where the classification is controlled by a constant threshold value. The total score value is computed by:
\begin{equation} \label{eq:score_function}
S_{tot} = \sum_{i}^{6} w_{i} g_{i}(F_{i})
\end{equation}
Where $g_{i}$ is a Gaussian function with a certain set of parameters unique to feature $i$, and $w_{i}$ is a weight that signifies the importance of that particular feature in classification. The choice for a Gaussian function is made because it can easily be scaled in accordance with the frame-rate. The standard deviation of the function is increased with smaller frame-rates, because our predictions are likely to be less accurate, causing feature values of pupil boundary edges to fall farther from zero.
We are interested in determining which range of feature values we typically get for pupil-iris contour edges. This frequency of occurrence is then used to obtain the Gaussian functions needed for equation \ref{eq:score_function}. The frequency data for each feature value are extracted from one half of our data set and plotted in Figure \ref{fig:edge_gaussians}. To clarify, one half of the data (randomly selected) is used to find the Gaussian functions and weights, while the other half is set aside for testing the classifier. This labelled data set was acquired by placing edges in one of the two classes depending on whether they were part of the accepted ellipse fit or not. This classification works, because pupil boundary edges are significantly more likely to be fitted by our ellipse fitting method compared to non-pupil edges. In Figure \ref{fig:edge_gaussians}, the fitted Gaussian functions are plotted in red. Each fit was constraint by setting the function's maximum to 1 and the position of the maximum to 0, which just leaves the standard deviation as a free parameter. The distributions of feature values $F_{\kappa}$ and $F_{\sigma_{r}}$ have been limited to edges with $F_{L} \leq 0.75$, because the curvature and variance of radius are only relevant if the edge is reasonably long. Furthermore, when calculating the scores for $\kappa$ and $\sigma_{r}$ their weights are modified with $F_{L}$ according to:
\begin{equation}
w' = (1 - \beta F_{L})w
\end{equation}
Where $\beta$ is another type of weight factor, whose value is determined alongside the other weights. We must also consider how certain we are about our predictions when calculating the scores. Every weight is therefore multiplied with the relevant certainty value. A summary of how each weight is calculated is given below.
\begin{align*}
w'_{L} &= c_{A}w_{L} \\
w'_{r} &= c_{S}w_{r} \\
w'_{\sigma_{r}} &= c_{S}(1 - \beta F_{L})w_{\sigma_{r}} \\
w'_{\kappa} &= c_{A}(1 - \beta F_{L})w_{\kappa} \\
w'_{G_{r}} &= c_{S}c_{A}w_{G_{r}} \\
w'_{I} &= c_{A}w_{I}
\end{align*}
\begin{figure}[ht]
\centering
\includegraphics[scale=1.10]{edge_gaussians.png}
\caption{For each of the six feature values given in Table \ref{table:edge_features}, a histogram (blue bars) is plotted with the fraction of edges that are pupil edges in each bin. So, each bin gives the estimated probability that an edge with that particular feature value belongs to the pupil edge. The smaller the feature value, the more likely it is that the corresponding edge belongs to the pupil-iris contour. Gaussian functions have been fitted on the data (red curves), which convert the feature value to a score.}
\label{fig:edge_gaussians}
\end{figure}
We can now train the classifier by assigning a weight to each feature value. To find the most optimal set of weights that best separates the two classes, we evaluate the degree of separation between them for a range of weight values. As in section \ref{sec:approximate_detection}, greatest separation between the two distributions is obtained by finding the maximum of the test statistic of the two-sample Kolmogorov-Smirnov test with respect to the weights. The weight configuration that corresponds with the greatest degree of separation is given in Table \ref{table:weight_factors}. The classifier is tested on the data that was not used to build the classifier. The resulting distributions are plotted in Figure \ref{fig:edge_classifier}, where the total score has been normalized to lie within 0 and 1. The vertical line marks the threshold position, $S_{\theta,edge}$, which corresponds with 99\% of pupil-iris contour edges being correctly classified ($S=0.38$). We make our threshold position dynamic with respect to the level of certainty, because we expect our score to be less accurate when the certainty becomes smaller. The actual threshold $S'_{\theta,edge}$ is then calculated by:
\begin{equation} \label{eq:score_certainty}
S'_{\theta,edge} = c_{S}c_{A}S_{\theta,edge}
\end{equation}
In Figure \ref{fig:eye_classification}, we see the effect of the edge classification method on a sample image in which the eye is partially closed. The end result is that the pupil-iris contour edges are successfully extracted from the collection of Canny edges in the image, with the corresponding ellipse fit providing an accurate measure of the pupil position.
\begin{table}[ht]
\centering
\begin{tabular}{M{1.5cm} M{1.5cm} N}
\textbf{Weight factor} & \textbf{Weight value} \\
\hline
$w_{L}$
& 0.7
&\\[7pt]
\hline
$w_{r}$
& 0.9
&\\[7pt]
\hline
$w_{\sigma_{r}}$
& 1.2
&\\[7pt]
\hline
$w_{\kappa}$
& 1.4
&\\[7pt]
\hline
$w_{G_{r}}$
& 0.7
&\\[7pt]
\hline
$w_{I}$
& 1.4
&\\[7pt]
\hline
$\beta$
& 0.9
&\\[7pt]
\end{tabular}
\caption{Optimal weight factors for maximum separation between the two edge distribution types.}
\label{table:weight_factors}
\end{table}
\begin{figure}[ht]
\centering
\includegraphics[width=0.8\textwidth]{edge_classifier.png}
\caption{Score histogram of edges classified as either belonging to the pupil-iris contour (blue) or some other feature (red). Shown is the greatest separation between the two classes that was obtained with the weights given in Table \ref{table:weight_factors}. The vertical line denotes the threshold score for which 99\% of pupil edges are correctly classified.}
\label{fig:edge_classifier}
\end{figure}
\begin{figure}[ht]
\centering
\includegraphics[width=\textwidth]{eye_classification.png}
\caption{Edge classification example. Red edges have not been chosen by the edge selection method. Orange edges were chosen, but were filtered out due to edge classification. Green edges have been categorized as pupil-iris contour edges by our classifier and an ellipse is subsequently fitted on them (white outline, right image). Centre of ellipse is marked by the white cross. Teal cross indicates predicted pupil position.}
\label{fig:eye_classification}
\end{figure}
\FloatBarrier
\subsection{Edge segmentation}
\label{sec:edge_segmentation}
Owing to our classifier, we are now able to distinguish between edges that make up the pupil outline and edges that do not. However, this leaves one important class of edges not dealt with, namely those that only partially belong to the pupil boundary. Before fitting an ellipse on these types of edges, we have to split them first at the transition point between pupil edge and some other feature (e.g. eyelid), otherwise the accuracy of the ellipse fit will be adversely affected. For this purpose, three novel edge segmentation techniques have been developed, which are applied \textit{before} edge classification is carried out.
\subsubsection{Path segmentation}
\label{sec:path_segmentation}
The first edge segmentation scheme is implemented based on the self-evident fact that the pupil is a non-intersecting closed shape. In other words, its perimeter is wholly described by a single non-branching path. So, if an edge consists of multiple branches, it definitely cannot exclusively be part of the pupil outline. Such edges are segmented by finding the non-branching path in the edge whose length comes closest to the circumference prediction $\hat{C}$ and separating it from the rest of the edge. To find this path, we will represent the edge as an undirected graph, which refers to the mathematical structure consisting of a collection of vertices (or nodes) joined by \textit{edges} with no orientation \cite[Chapter~6]{Newman2010}. To avoid confusion between the term edge here and how it was used before, we will use the term \textit{arc}, which is generally only used for directed graphs, to refer to connections between vertices in the network. The edge is converted to a graph through the following rules and definitions:
\begin{enumerate}[noitemsep]
\item A branch vertex is any edge point that has three or more 8-neighbours
\item A terminal vertex is any edge point that has exactly one 8-neighbour
\item An arc is a collection of edge points that connects two vertices
\item Vertices that are 8-neighbours are combined into a single vertex
\end{enumerate}
After creating the graph, we use a recursive implementation of depth-first search to find every possible simple path, which is any path that does not traverse the same arc or vertex more than once, and then select the most optimal one. We locate all paths starting from the terminal vertices first and keep track of the vertices we already started with so we do not end with them. This prevents acyclic path repetitions, because for our purposes a specific path between two vertices is the same in the reverse direction.
A caveat should be added. The task of finding all possible simple paths in an undirected graph is NP-hard, since the more basic \textit{longest path problem} is already NP-hard \cite{Cormen2001}. However, we are dealing with relatively tiny networks here with usually only a few connections, which allows for this problem to be quickly solved.
\begin{figure}[ht]
\centering
\includegraphics[scale=1.40]{edge_graph.png}
\caption{\textit{(Left)} Pixel representation of a sample Canny edge, where each pixel has been given its own unique label. \textit{(Right)} Graph representation of the same edge. The pixels contained by the vertices (boxes) and the arcs (lines) are given in set notation. Black pixels or vertices indicate the desired path.}
\label{fig:edge_graph}
\end{figure}
In Figure \ref{fig:edge_graph}, a sample edge is depicted along with its graph representation. The structure of this graph is an exaggerated version of what one typically encounters. The total number of possible simple paths in this graph is equal to 139, which is still a manageable quantity, even when having to process multiple edges. However, this number can be brought down further, while also dealing with noise. We remove arcs from the edge if they contain fewer edge points than the edge window length $N_{l}$ when they are attached to a terminal vertex, are a self-loop or when they connect two vertices that are already connected by an arc, in which case the shortest of the two arcs is kept. As an example, when these filters are applied to the edge in Figure \ref{fig:edge_graph}, all grey edge pixels are removed.
Generally, several paths will be detected, one of which has to be accepted. Our decision is made based on a few criteria. Cyclic paths always take precedence over acyclic paths, since the ideal pupil outline is cyclic. However, if the pupil boundary is described by a cycle then it has to be wholly represented by that cycle, so the length $L$ of such a cyclic path has to match the expected circumference of the pupil boundary, where the acceptable range is given by:
\begin{equation*}
C_{min} \leq \hat{C}(1 - \delta_{\theta,C}) \leq L \leq \hat{C}(1 + \delta_{\theta,C}) \leq C_{max}
\end{equation*}
The lower limit of this criterion must not apply for acyclic paths, because multiple acyclic paths can make up the pupil boundary. We could enforce the upper limit, though, since the length of an acyclic path should not be greater than $C_{max}$ either, but this is rejected in favour of a different solution (see section \ref{sec:length_segmentation}). Finally, any arcs that were not filtered out, but also not included in the final accepted path are not discarded. Instead, new graphs are made using these remaining edges and the process is repeated.
\FloatBarrier
\subsubsection{Curvature segmentation}
\label{sec:curvature_segmentation}
Another property of the pupil-iris contour is its smooth curvature. Abrupt changes in curvature along its path are most likely caused by obstructions of the pupil periphery by the eyelid, eye lashes or corneal reflection from the IR-LED. Zhu et al. \cite{zhu1999} developed an algorithm that exploits the curvature characteristic of the pupil boundary in order to separate it from these types of artefacts. The location of an occlusion or \textit{breakpoint} in the pupil boundary is determined by checking if the curvature $\kappa$ at a certain point on the perimeter is above an upper threshold ($\kappa_{max}$) or below a lower threshold ($\kappa_{min}$). The pupil boundary is then segmented according to a number of heuristics that use the distances between detected breakpoints to detect different types of occlusions.
Inspired by this approach, we have developed our own segmentation algorithm based on edge curvature. The method that Zhu et al. used to calculate the curvature is repeated here, but with a few slight alterations. One change is made in the way direction coding is performed, because we are interested in calculating the curvature of Canny edges and not the boundary of a 'blob' of pixels that is generated after application of a brightness threshold. The method works by scanning the 8-connected environment of each edge point and assigning a vector label to the edge point depending on the relative position of its neighbour. For example, starting from the central edge point $P_{i}$ in Figure \ref{fig:edge_curvature}, we scan its 8-neighbours and locate $P_{i+1}$ in the south-east direction, which corresponds with a vector of ($\sqrt{.5}$,$\sqrt{.5}$) or a cardinal direction label of SE. We then do the same for $P_{i+1}$ and so forth, until every edge point has been given a label.
\begin{figure}[ht]
\centering
\includegraphics[scale=0.70]{edge_curvature.png}
\caption{Example of curvature calculation procedure. Each pixel in the edge is given a (cardinal) direction label, which corresponds with one of the vectors displayed on the right. Curvature is calculated at pixel $P_{i}$ by finding the angle between the vector tangents of the two windows on either side of it.}
\label{fig:edge_curvature}
\end{figure}
After obtaining all direction labels, the curvature $\kappa$ at each edge point is calculated. Theoretically, the (signed) curvature at a specific point on a curve is defined \cite{patrikalakis2009} as the rate of change of the tangential angle $\phi$ with respect to the arc length $s$. This exact definition of $\kappa$ is approximated by placing two windows on either side of the edge point we are calculating $\kappa$ for and finding the mean unit vector for each window, where the size of a window is equal to the edge window length $N_{l}$. The difference in tangential angles $\Delta \phi$ between the two mean vectors, divided by $N_{l}$, is then the signed curvature at that point:
\begin{equation*}
\kappa = \frac{d\phi}{ds} \approx \frac{\Delta \phi}{N_{l}}
\end{equation*}
When determining the curvature for $P_{i}$ in Figure \ref{fig:edge_curvature}, we calculate the first vector tangent $\boldsymbol{\vec{T}_1}$ for edge points running from $P_{i-5}$ to $P_{i-1}$ (i.e. $N_{l} = 5$ in this example) and the second $\boldsymbol{\vec{T}_2}$ for points from $P_{i+1}$ to $P_{i+5}$. The angle difference $\Delta \phi$ is then given by:
\begin{equation*}
\Delta \phi = \atantwo (T_{2,y},T_{2,x}) - \atantwo (T_{1,y},T_{1,x})
\text{ with }
\Delta \phi =
\begin{cases}
\Delta \phi - 2\pi & \text{if } \Delta \phi > \pi\\
\Delta \phi + 2\pi & \text{if } \Delta \phi < -\pi\\
\end{cases}
\end{equation*}
For edges belonging to the pupil boundary, we expect that each edge point has the same curvature sign, since the boundary is elliptical. However, whether the majority sign is negative or positive depends on the scanning direction, which we do not control here. In order to properly set a lower curvature threshold it is crucial that each edge has the same majority curvature sign. For this reason, we count the number of positive and negative curvatures in the edge. If there are more negative curvatures than positive ones, all signs are inverted.
\noindent
We also determine the vector difference $\Delta\boldsymbol{\vec{T}_1}$ between $\boldsymbol{\vec{T}_1}$ and $\boldsymbol{\vec{T}_2}$:
\begin{equation*}
\Delta\boldsymbol{\vec{T}_1} = \boldsymbol{\vec{T}_2} - \boldsymbol{\vec{T}_1}
\end{equation*}
\noindent
The vector $\Delta\boldsymbol{\vec{T}_1}$ points towards the centre of curvature. This direction is used when calculating the edge intensity in edge classification.
Now that we have a measure of the curvature for every edge point, we can use it to segment edges at the intersection between distinct features in the image. To do this, we do not use any of the heuristics developed by Zhu et al. Instead, our curvature segmentation algorithm works on the basis of only one simple rule: an edge is segmented at every breakpoint. This makes it far less specific and requires fewer parameters. The only parameters we need to specify are the values for $\kappa_{min}$ and $\kappa_{max}$. Like Zhu et al., we could set these to a constant value for a given pupil, with $\kappa_{min} = -\kappa_{max}$. However, this seems like a poor decision for two reasons. First of all, this heavily overestimates the expected lower curvature limit. Since the pupil contour is elliptical, the signed curvature should theoretically not become negative, as mentioned before. This means that $\kappa_{min}$ should be much closer to zero. Second of all, the expected values for $\kappa$ are very dependent on where the eye is looking. This is because the range of $\kappa$ is not only determined by the size of the pupil, but also on its shape. The circumference $C$ of the pupil is inversely proportional to the mean of $\kappa$. The aspect ratio $AR$, on the other hand, is inversely proportional to the size of the range of $\kappa$. When the eye is looking straight ahead, the pupil can be reasonably approximated by a circle (i.e. $AR = 1$), which means that $\kappa$ will be constant around the perimeter, so its range is minimal. However, when the eye is looking up, the pupil shape is more eccentric, so $AR$ will be smaller. This causes the curvature at the antipodal points of the semi-minor axis to decrease and the curvature at the semi-major axis antipodes to increase, broadening the range of $\kappa$.
\begin{figure}[ht]
\centering
\includegraphics[width=\textwidth]{curvature_3D_3.png}
\caption{\textit{(Left)} Neural network fit of maximum edge curvature as a function of pupil circumference and aspect ratio for an edge window length of 5. The function is represented by the mesh surface, which is fitted on the red data points, indicating local maxima. \textit{(Right)} Dependence of minimum and maximum edge curvature on circumference, aspect ratio and edge window length, $N_{l}$. Bottom three surfaces are for $\kappa_{min}$ and top three for $\kappa_{max}$.}
\label{fig:curvature_3D_3}
\end{figure}
To ensure that $\kappa_{min}$ and $\kappa_{max}$ depend on the size and shape of the pupil, these parameters are turned into dynamic thresholds that are automatically updated according to our predicted values $\hat{C}$ and $\hat{AR}$. The thresholds are based on the lower and upper limit of the $\kappa$ range we would expect to get for an ellipse with a circumference and aspect ratio equal to $\hat{C}$ and $\hat{AR}$. To find the relation between the range of $\kappa$ with respect to $C$ and $AR$, we measured the minimum and maximum $\kappa$ values of 3600 artificial pupils, which are solid black ellipses on a white background. These ellipses varied in circumference between 30 and 380 pixels, and in aspect ratio between 0.15 and 1.00. Furthermore, to investigate how the edge curvature depends on the curvature window size, the $\kappa$ range of each ellipse was evaluated multiple times in each frame for different $N_{l}$ values between 5 and 11. After obtaining all $\kappa_{min}$ and $\kappa_{max}$ data points, neural networks are trained to fit functions that take $C$ and $AR$ as their input and give $\kappa_{min}$ or $\kappa_{max}$ as their output. This task is performed by the MATLAB routine \texttt{fitnet} using Levenberg-Marquardt with 5 hidden nodes. The functions are only fitted on the largest $\kappa_{max}$ values or on the smallest $\kappa_{min}$ values that were measured in a specific circumference and aspect ratio bin. This is to ensure that we get the true curvature limit for the $C$ and $AR$ pair. A sample fit for $\kappa_{max}$ is displayed on the left side of Figure \ref{fig:curvature_3D_3}. The dependence of $\kappa_{min}$ and $\kappa_{max}$ on $N_{l}$ is shown on the right side of Figure \ref{fig:curvature_3D_3}. We can observe that the $\kappa$ range narrows with increase in $N_{l}$. This is because the greater the window size, the less refined our measurement is. We lose information about local curvature maxima or minima, causing them to be somewhat averaged out, which leads to a smaller $\kappa$ range. In this sense, we want to have the smallest possible value for $N_{l}$, but larger $N_{l}$ values make our $\kappa$ measurement more robust against noise. The default value for $N_{l}$ is set to 7, which was found to give good results for our set-up. However, it can be desirable to turn $N_{l}$ into a dynamic parameter that is reduced when $\hat{C}$ becomes smaller, because the smaller the pupil is, the more its contour edge will be affected by tiny interferences, which may go unnoticed when $N_{l}$ is set too high.
\FloatBarrier
The threshold values obtained here are curvature limits for ideal ellipses. In practice, these thresholds need to be offset (up and down) by a few degrees to account for inaccuracies in our predictions and for natural deviation of the pupil from the ellipse shape. The final thresholds are therefore determined by:
\begin{align}
\kappa_{max} &= h(\hat{C}(1 - \delta_{\theta,C}), \hat{AR} - \delta_{\theta,AR}) + offset \\
\kappa_{min} &= h(\hat{C}(1 - \delta_{\theta,C}), \hat{AR} - \delta_{\theta,AR}) - offset
\end{align}
\noindent
Where $h(C,AR)$ is the fitted function. The variable $\delta_{\theta,AR}$ is calculated in a similar way to $\delta_{\theta,C}$ (see equation \ref{eq:delta_circumference}):
\begin{equation} \label{eq:delta_aspect_ratio}
\delta_{\theta,AR} = (1 - c_{AR}) (1.0 - \delta''_{\theta,AR}) + \delta''_{\theta,AR} \\
\end{equation}
Figure \ref{fig:eye_curvature} shows the effect of curvature edge segmentation on a sample image in which the eye is partially closed. The pupil-iris outline is detected by Canny edge detection as one continuous edge that is also part of the upper eyelid. Through curvature segmentation we are able to split the edge up into two sections corresponding to the distinct features, after which the pupil edge can be successfully identified with edge classification.
\begin{figure}[ht]
\centering
\includegraphics[scale=0.80]{eye_curvature.png}
\caption{Curvature segmentation example. \textit{(Left)} Yellow edge is part of pupil boundary as well as eyelid. \textit{(Centre)} Edge is segmented at transition point between both features, resulting in new pupil edge section (green). \textit{(Right)} Ellipse (white outline) is fitted on segmented pupil edge. Centre of ellipse is marked by the white cross. Teal cross indicates predicted pupil position.}
\label{fig:eye_curvature}
\end{figure}
\FloatBarrier
\subsubsection{Length segmentation}
\label{sec:length_segmentation}
If the pupil circumference prediction in a given frame is accurate, then no edge that entirely resides on the pupil boundary should be longer than this prediction. When the edge length $L$ is greater than $\hat{C}$, we segment the edge so that one of the two parts has a length that is approximately equal to $\hat{C}$. The question, however, is where to make the separation. We make this decision by temporarily dividing the edge into three sections, which is graphically shown in Figure \ref{fig:length_segmentation}. The edge is cut in such a way that the length of the edge body (\textit{II}) plus the length of either of the two edge tails (\textit{I} or \textit{III}) is equal to $\hat{C}$, where $\hat{C} = 2\pi\hat{r}$ in the figure.
\begin{figure}[ht]
\centering
\includegraphics[scale=1.00]{length_segmentation.png}
\caption{Length segmentation procedure. See main text for further details.}
\label{fig:length_segmentation}
\end{figure}
We now investigate which of the two edge tails resembles the body the most with respect to its features. The closest match is re-attached to the central section, while the other one is left segmented. The level of similarity is quantified using the score functions we developed for edge classification, but instead of using the difference between an edge feature value and the corresponding predicted value to calculate the score, we will work with the difference between feature values of the tail compared to the body. The expressions for calculating $F_{i}$ and $w'_{i}$ are given in Table \ref{table:length_weights}. The length weight is set to zero, since $F_{L}$ is equal for both edge tails. We only use $F_{L}$ to modify the weights of $\sigma_{r}$ and $\kappa$. The certainty terms $c_{A}$ are set to 1 in the weight expressions, because we are not using any predicted values to calculate $F_{i}$, so their accuracy is irrelevant here. However, the accuracy of $r$, $\sigma_{r}$ and $G_{r}$ does depend on $c_{S}$, hence it remains. The scores for each tail are calculated using equation \ref{eq:score_function}, and the edge tail with the highest score is reconnected to the edge body. In Figure \ref{fig:length_segmentation}, we expect edge \textit{I} to have the highest score, since its $r$, $\kappa$ and $\sigma_{r}$ are very close to edge \textit{II}, so edge \textit{III} is left severed.
\begin{table}[ht]
\centering
\begin{tabular}{ M{2cm} M{4cm} M{4cm} N}
\textbf{Edge features} & \textbf{Feature value, $F_{i}$} & \textbf{Weight, $w'_{i}$} \\
\hline
Length
& $\displaystyle{\frac{\abs{ L_{1} - \hat{C} }}{\max(L_{1},\hat{C})}}$
& 0
& \\ [25pt]
\hline
Radius
& $\displaystyle{\frac{\abs{ r_{1} - r_{2} }}{\max(r_{1},r_{2})}}$
& $c_{S}w_{r}$
& \\ [25pt]
\hline
Variance of radius
& $\displaystyle{\frac{\abs{\sigma_{r,1} - \sigma_{r,2}}}{\hat{C}}}$
& $c_{S}(1 - \beta F_{L})w_{\sigma_{r}}$
& \\ [25pt]
\hline
Curvature
& $\displaystyle{\abs{ \kappa_{1} - \kappa_{2} }}$
& $(1 - \beta F_{L})w_{\kappa}$
& \\ [25pt]
\hline
Radial gradient
& $\displaystyle{\abs{ G_{r,1} - G_{r,2} }}$
& $c_{S}w_{G_{r}}$
& \\ [25pt]
\hline
Intensity
& $\displaystyle{\abs{ I_{1} - I_{2} }}$
& $w_{I}$
& \\ [25pt]
\end{tabular}
\caption{Edge features that are taken into account for length segmentation. Included are the expressions for the corresponding feature value and weight factor.}
\label{table:length_weights}
\end{table}
\FloatBarrier
\subsection{Ellipse fitting}
\label{sec:ellipse_fitting}
Having segmented and classified the edges obtained with Canny edge detection, we can now fit an ellipse on one or more of them, using one of several possible methods. Search- and voting-based schemes, such as Hough transform and Random Sample Consensus (RANSAC), are often implemented in pupil detection algorithms because they are robust to outliers, however they are also computationally expensive \cite{Fuhl2015}\cite{raguram2008}\cite{swirski2012}. Since we have performed numerous segmentation routines to remove any potential outliers, we instead choose the faster, but more sensitive, direct least squares fitting method \cite{fitzgibbon1999}. Owing to its computational efficiency, we are able to fit an ellipse multiple times in the same frame and then choose the most optimal one. However, recall equation \ref{eq:combinations}. We cannot have a large set of remaining edges, otherwise the number of possible edge combinations to fit an ellipse on will be overwhelming. So we choose up to a fixed number of available edges classified as lying on the pupil-iris contour, favouring edges with a higher score. The maximum number that we choose is set to 4 by default, which translates to 15 possible combinations.
Fitting that many ellipses in one frame is still too demanding, but we are going to reduce their numbers further by requiring that the expanse of each edge combination must reasonably correspond with our width and height predictions of the pupil. This also means that we require that a significant portion of the pupil-iris contour is visible in the image and has been detected before fitting an ellipse, because the fit will be gravely inaccurate otherwise. We calculate the range of the edge combination by finding the minimum and maximum $x$ and $y$-positions in the collection of edge points and set the following criterion:
\begin{alignat*}{2}
0.3(\hat{W} - \Delta l) &\leq x_{max} - x_{min} &&\leq \hat{W} + \Delta l \\
0.3(\hat{H} - \Delta l) &\leq y_{max} - y_{min} &&\leq \hat{H} + \Delta l
\end{alignat*}
\noindent
Where:
\begin{equation}
\Delta l = \frac{\hat{C}\delta_{\theta,C}}{\pi}
\end{equation}
These thresholds ensure that cases where the vast majority of the pupil is obscured by the eyelid are ignored. A limit is also imposed on how many ellipses we are allowed to fit in one frame, which is fixed at 6 by default. If there are more possible fits available, then we choose the edge combinations with a combined length that is closest to $\hat{C}$.
We fit an ellipse on each edge combination that remains. The properties of every ellipse are subsequently calculated from the general equation of the ellipse using rotation transformation \cite{zhu1999}. This includes the position, semi-major and minor axes, rotation angle and bounding box dimensions. The circumference is computed by Ramanujan's second approximation \cite{ramanujan1914}. Immediately after obtaining these characteristics, each ellipse passes through a series of filters that will judge the size, shape and quality of the fit. The circumference of the ellipse has to fall within the $C_{min}$ and $C_{max}$ limits and its aspect ratio should be larger than $AR_{min}$. These bounds have been empirically determined from our data set ($C_{min} = 60$, $C_{max} = 290$ and $AR_{min} = 0.4$). Furthermore, we observe that the limit of $C$ is dependent on $AR$, which is expected because the larger an ellipse is on the surface of a sphere, the greater its polar angle needs to be to have an equally low aspect ratio when projected on the 2D plane as a smaller ellipse. A linear function is created that acts as a circumference threshold, which is given by:
\begin{equation}
C'_{max} = k(AR - 1) + C_{max}
\end{equation}
Where $k$ is assigned a value of 154. The circumference of the fit has to be less than the circumference limit calculated by this function. Besides looking at the absolute values of the size and shape of the pupil, we also inspect how much these features deviate from the predictions, which is quantified by $\delta_{C}$ and $\delta_{AR}$ calculated through equations \ref{eq:change_C} and \ref{eq:change_AR}. We establish the criterion that $\delta_{C}$ and $\delta_{AR}$ are not allowed to respectively exceed $\delta_{\theta,C}$ and $\delta_{\theta,AR}$ (see equations \ref{eq:delta_circumference} and \ref{eq:delta_aspect_ratio}), otherwise the fit is rejected. Another filter examines the number of points that the ellipse was fitted on with respect to the fit's circumference. We cannot accept a large ellipse fit on a tiny set of edge points even when these points are adequately spread out. Such a fit is most likely erroneous and influenced by noisy edges. We therefore enforce the following threshold for the edge length:
\begin{equation*}
L \geq 0.3\hat{C}(1-\delta_{\theta,\hat{C}})
\end{equation*}
The last filter works on the basis of the error between the fit and fitted points. For ellipse fits on the pupil-iris edge, we expect a small fit error for all fitted edge points, since we are able to adequately approximate the pupil shape by an ellipse \cite{wyatt1995}. So a large fit error for a particular edge point would indicate that the ellipse is not at all or not entirely fitted on the pupil boundary. The accuracy of the pupil fit can already significantly diminish if the edge combination contains just a few outliers, because of the high sensitivity of direct least squares compared to other, slower, ellipse fitting methods \cite{Halir98}. For this reason, only ellipse fits where all edge points in the set have a small fit error should be accepted. On the other hand, we do not want to reject a fit because of just one outlier. As a compromise, we instead consider the fit errors of the $0.05C$ largest outliers of the set. If their average fit error is above a given threshold, the ellipse fit is rejected.
We also investigate how the absolute fit errors change with respect to the circumference of the ellipse fit by analyzing the fit errors of around 44,000 ellipse fits. Only fits that the pupil detection algorithm has classified as acceptable pupil fits are considered here. We observe that $\epsilon$ is directly proportional to $C$ with larger errors being measured at larger circumferences. Since we do not want to adapt our fit error threshold according to $C$, we remove this linear relationship by using the relative fit error $\epsilon_{rel}$ instead, which is given by:
\begin{equation*}
\epsilon_{rel} = \frac{\epsilon_{abs} - \alpha}{C}
\end{equation*}
Where $\alpha$ (y-intercept) is a constant found by linear regression ($\alpha = -0.56$). The dependence of $\epsilon$ on $C$ disappears when using the relative error. This allows us to set a constant error threshold, which is assigned a value of 0.6.
In many cases, only one ellipse fit remains at this stage which we then consider to represent the pupil. However, it also frequently occurs that there are still a few ellipse fits left to choose from. The final choice is made through a similar strategy employed during edge classification. Each ellipse is assigned a score based on a number of its features, after which the ellipse with the highest score is selected as the pupil representation. The relevant features are listed in Table \ref{table:fit_features}. Once again, the feature values are made invariant to the image size. The labelled data set was acquired by marking accepted fits as pupil fits and any other ones as non-pupil fits.
\begin{table}[ht]
\centering
\resizebox{\textwidth}{!}{%
\begin{tabular}{ M{3cm} M{2cm} M{3.5cm} M{2.5cm} m{5cm} N }
\textbf{Fit features}
& \textbf{Symbol}
& \textbf{Feature value, $F_{i}$}
& \textbf{Weight factor}
& \textbf{Description}
&\\[20pt]
\hline
Circumference
& $C$
& $\displaystyle{\frac{\abs{ C - \hat{C} }}{\max(C,\hat{C})}}$
& $c_{A}$
& Circumference of ellipse fit in pixels
&\\[25pt]
\hline
Aspect ratio
& $AR$
& $\displaystyle{\abs{ AR - \hat{AR} }}$
& $c_{A}$
& Aspect ratio of ellipse fit
&\\[25pt]
\hline
Edge length
& $L$
& $\displaystyle{\frac{\abs{ L - \hat{L} }}{\max(L,\hat{L})}}$
& $c_{A}$
& Combined length of fitted edges in pixels
&\\[25pt]
\hline
Fit error
& $\epsilon$
& $\epsilon$
& -
& Relative error between the ellipse fit and the edge points
&\\[25pt]
\hline
Rotation angle
& $\theta$
& $\displaystyle{\abs{ \theta - \hat{\theta} }}$
& $c_{A}(1 - \rho(AR))$
& Rotation angle of ellipse fit in radians
&\\[25pt]
\end{tabular}}
\caption{Ellipse fit features that the fit classification scheme is based on.}
\label{table:fit_features}
\end{table}
Following the same steps as with edge classification, we split our data set into two halves and determine the Gaussian functions and weights from one half, and test the classifier with the other half. The frequency data is plotted in Figure \ref{fig:fit_gaussians}, together with the corresponding Gaussians. The weights are modified by the degree of certainty as indicated in Table \ref{table:fit_features}. An additional factor is added to the rotation angle's weight, because the significance of $\theta$ should decrease the more circular the pupil becomes.
Once more the two-sample Kolmogorov-Smirnov test is used to obtain the greatest degree of separation between the two classes. The optimal weights are given in Table \ref{table:fit_weights}. The distributions of the two classes are plotted in Figure \ref{fig:fit_classifier}. We choose the ellipse fit with the highest score as the pupil representation, but also any fit that has a score within a certain distance $\Delta S_{\theta,fit}$ from the highest score. This parameter $\Delta S_{\theta,fit}$ is given a value of 0.10. The average characteristics of all accepted fits then corresponds with our final pupil measurement for the current frame.
\begin{figure}[ht]
\centering
\includegraphics[scale=1.20]{fit_gaussians.png}
\caption{For each of the six feature values given in Table \ref{table:fit_features}, a histogram (blue bars) is plotted with the fraction of ellipse fits that are pupil fits in each bin, similarly to Figure \ref{fig:edge_gaussians}. Gaussian functions have been fitted on the data (red curves), which convert the feature value to a score.}
\label{fig:fit_gaussians}
\end{figure}
\begin{table}[ht]
\centering
\begin{tabular}{M{1.5cm} M{1.5cm} N}
\textbf{Weight factor} & \textbf{Weight value} \\
\hline
$w_{C}$
& 0.4
&\\[7pt]
\hline
$w_{AR}$
& 0.6
&\\[7pt]
\hline
$w_{L}$
& 1.6
&\\[7pt]
\hline
$w_{\epsilon}$
& 0.9
&\\[7pt]
\hline
$w_{\theta}$
& 1.5
&\\[7pt]
\hline
$\rho$
& 0.7
&\\[7pt]
\end{tabular}
\caption{Optimal weight factors for maximum separation between the two fit distribution types.}
\label{table:fit_weights}
\end{table}
\begin{figure}[ht]
\centering
\includegraphics[width=0.8\textwidth]{fit_classifier.png}
\caption{Score histogram of ellipse fits classified as either entirely belonging to the pupil (blue) or not (red). Shown is the greatest separation between the two classes that was obtained with the weights given in Table \ref{table:fit_weights}. The vertical line denotes the score for which 99\% of pupil fits are correctly classified.}
\label{fig:fit_classifier}
\end{figure}
\section{Evaluation}
The performance of the presented pupil detection algorithm, which is given the name EyeStalker, is assessed by determining its effectiveness of locating the pupil centre in images where the position of the pupil has been manually determined. This performance is then compared with two other open-source eye tracking algorithms, PupilLabs \cite{Kassner2014} and ExCuSe \cite{Fuhl2015} (latest versions as of February 2017). A number of different hand-labelled data sets are available \cite{swirski2012} \cite{Kassner2014}, but these are either random collections of eye images or have been recorded with low frame-rate cameras (around 25 Hz). It is a prerequisite for EyeStalker that the algorithm is applied on high frequency image data that is ordered in sequence. For this reason, a new hand-labelled dataset is created that consists of 5000 images of the pupil during 49 saccadic eye movements from 12 different individuals (between 3 to 5 saccades per person), which is a sub-set of the data set referred to in Methods. Especially challenging trials were selected, including images where the pupil is notably obstructed by the eyelid or glint, as well as images of highly eccentric or tiny pupils.
A tool to manually detect the pupil was developed in MATLAB. The program works by fitting an ellipse by hand on the pupil boundary. The centre of the ellipse then corresponds with the centre of the pupil. The ellipse is translated, rotated and reshaped using keyboard controls until a good fit is achieved. The image is resized to fit the entire monitor screen, allowing for more precise measurements. The precision of this method is determined by manually detecting the pupil in 12 images 12 times and calculating the variation in the measurement. Each of these images shows the pupil of a different person and was hand-picked for their higher detection difficulty. The 144 detections are done in a random order and the shape, angle and position of the ellipse is reset each time a new image is displayed on screen. From this analysis, a standard deviation of \SI{0.6}{\degree} is found for manual detection. This labelling tool together with the raw eye images and hand-labelled pupil coordinates are publicly available for download.\footnote{\url{https://drive.google.com/open?id=0Bw57olSwQ4EbUWV5ajNKeG93NEk}}
The hand-labelled data set only features frames that are part of a saccadic eye movement plus a short fixation period before and after. On average, this comes down to around 100 frames per trial. However, when applying the three algorithms on the data, we include a longer preceding fixation period, adding between 20 to 200 additional frames to the start of the trial, which are not included in our analysis. These extra prior detections are required for EyeStalker to already acquire an adequately accurate measure of $\bar{f}_{n}$ before the actual measurement begins. In normal circumstances, more accurate information will always be available, because a test subject will have been wearing the eye tracking device for a relatively long time before the start of a recording.
As a constraint, all trials are processed in a single run, which means that parameters are kept to their default values and not altered between individuals. In the EyeStalker and PupilLabs algorithms, the parameters for Canny edge detection are set equal to one another to achieve a fairer comparison (ExCuSe uses an automatized method to set these parameters). The performance of each algorithm is evaluated by calculating the detection rate and error for different error thresholds. The results are shown in Figure \ref{fig:p1_results_performance}. It is clear that EyeStalker not only achieves a greater detection rate for all error thresholds compared to the other two approaches, but also a smaller detection error. The much lower detection rate of ExCuSe can possibly be attributed to the fact that the algorithm is specifically tailored for eye tracking in real-life scenarios and thus functions relatively poorly in controlled environments.
The performance of PupilLabs comes closer to that of EyeStalker, but requires considerably more processing time, which can be inferred from Figure \ref{fig:p1_results_speed}. The figure shows detection durations for every frame, which were processed using a C++ implementation of each algorithm combined in a single application on a triple-core 3.3 GHz CPU running Linux. EyeStalker is significantly faster than both of the other two algorithms. PupilLabs is surprisingly slow, but the timing agrees with their reported processing pipeline latency of 45 ms \cite{Kassner2014}, although it is also possible that our implementation or hardware was suboptimal, since their commercially available hardware reportedly has a latency of 5.7 ms.\footnote{\url{https://pupil-labs.com/}} The average computational time of EyeStalker is 2.1 ms, but this duration can potentially double during periods of low certainty. The vast majority of processing time is spent on approximate detection (section \ref{sec:approximate_detection}) and Canny edge detection (section \ref{sec:canny_edge_detection}).
\begin{figure}[ht]
\centering
\includegraphics[scale=1.00]{p1_results_performance.png}
\caption{Performance of our pupil detection algorithm (EyeStalker) on 5000 test images compared to two other open-source algorithms, PupilLabs and ExCuSe. The detection rate and error are evaluated as a function of the allowable error threshold. EyeStalker not only achieves a significantly higher detection rate, but is also able to achieve this with an overall greater accuracy.}
\label{fig:p1_results_performance}
\end{figure}
\begin{figure}[ht]
\centering
\includegraphics[scale=1.00]{p1_results_speed.png}
\caption{Histograms of the processing time for each of the 5000 test images are plotted for the three algorithms. Our pupil detection algorithm (EyeStalker) achieves an average computation time of 2.1 ms per frame.}
\label{fig:p1_results_speed}
\end{figure}
\section{Discussion}
A novel eye tracking algorithm has been presented that is designed for use with high-speed cameras, relying on estimations of pupil characteristics to carry out sophisticated feature-based pupil detection techniques that are both fast and robust. Based on ground truth hand-labelled data, the method was found to surpass other recently published open-source algorithms in terms of detection rate, accuracy and speed. To further verify the efficiency of the algorithm, it should be tested with multiple camera set-ups using different combinations of operational speed and image quality. It is expected that performance will increase with higher frame rate cameras since the pupil feature predictions will become more accurate.
Additional enhancements can be made to the algorithm itself. The implemented recursive estimation method is sufficiently accurate, but using a more established design (e.g. Kalman filter) will most likely lead to better predictions, though requiring more effort to set-up. This could significantly improve pupil detection, since the predictions are used in many different parts of the algorithm. One processing step that does not use any predictions, however, is Canny edge detection, but the estimated pupil location can be utilized to tune the method to specifically detect the pupil-iris edge. In its current state, Canny edge detection does not make any distinctions between edges based on their orientation or gradient direction. It can be made more discerning by calculating the radial gradient outwards from the predicted pupil position, which puts more weight on the pupil contour and less on edges that are not oriented in the tangential direction (e.g. eye lashes). Furthermore, by using the signed gradient, we only consider edge points that are darker closer to the origin point and lighter on the other side, which holds true for the pupil perimeter. An additional improvement can be made in the approximate detection of the pupil position. Even though Haar-like feature detection is made significantly more computationally efficient with the calculation of the integral image compared to more naive methods, it is still quite demanding relative to other processing steps. Since we only require an approximate pupil location, an alternative approach is to train a convolution neural network on the eye image data, which would subsequently be able to rapidly supply a rough position estimate in each frame.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 9,476 |
Conway is a town in Northampton County, North Carolina, United States. The population was 836 at the 2010 census. It is part of the Roanoke Rapids, North Carolina Micropolitan Statistical Area.
Geography
Conway is located at (36.437687, -77.226782).
According to the United States Census Bureau, the town has a total area of , all land.
Demographics
2020 census
As of the 2020 United States census, there were 752 people, 306 households, and 221 families residing in the town.
2010 census
As of the 2010 United States Census, there were 836 people living in the town. The racial makeup of the town was 48.6% White, 47.7% Black and 1.3% from two or more races. 2.4% were Hispanic or Latino of any race.
2000 census
As of the census of 2000, there were 734 people, 328 households, and 205 families living in the town. The population density was 406.0 people per square mile (156.6/km2). There were 356 housing units at an average density of 196.9 per square mile (75.9/km2). The racial makeup of the town was 65.53% White, 33.24% African American, 0.54% Native American, and 0.68% from two or more races. Hispanic or Latino of any race were 0.27% of the population.
There were 328 households, out of which 27.7% had children under the age of 18 living with them, 41.8% were married couples living together, 18.0% had a female householder with no husband present, and 37.5% were non-families. 35.1% of all households were made up of individuals, and 14.9% had someone living alone who was 65 years of age or older. The average household size was 2.24 and the average family size was 2.88.
In the town, the population was spread out, with 24.5% under the age of 18, 7.2% from 18 to 24, 23.7% from 25 to 44, 25.6% from 45 to 64, and 18.9% who were 65 years of age or older. The median age was 42 years. For every 100 females, there were 75.6 males. For every 100 females age 18 and over, there were 66.9 males.
The median income for a household in the town was $23,250, and the median income for a family was $27,386. Males had a median income of $26,932 versus $21,538 for females. The per capita income for the town was $14,969. About 24.9% of families and 24.1% of the population were below the poverty line, including 36.3% of those under age 18 and 21.1% of those age 65 or over.
Notable people
Stu Flythe, former Major League Baseball player
Fred Vinson, NBA player
References
Towns in Northampton County, North Carolina
Towns in North Carolina
Roanoke Rapids, North Carolina micropolitan area | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 8,260 |
\section{Introduction}
The study of zeros of polynomials and its derivatives have been of interest for long. A well known result relating these refer to Gauss and Lucas' theorem which states that the critical points of a polynomial lie in the convex hull formed by the zeros of the polynomial. There have been several extensions and refinements of this result. For instance, Pereira \cite{pereira} and Malamud \cite{malamud} independently extended this result to relating the zeros and critical points with a doubly stochastic matrix. For a deeper discussion on this topic, we refer the reader to classical texts \cite{marden} and \cite{rahman}.
In the theory of random polynomials, one natural way of constructing model is imposing randomness to the coefficients of the polynomials. Then it is of interest to understand the limiting behavior of zero sets when the degree of polynomials goes to infinity. This question was answered by Kabluchko and Zaparozhets in the case that coefficients are taken to be scaled i.i.d. random variables, see \cite{kab_zap}. They showed that the limiting empirical measure of zeros is radially symmetric and depends only on the variance profile of the coefficients. Following the result of Kabluchko and Zaparozhets, one can observe that the empirical measure of zeros of higher derivatives of these polynomials converge to the same measure as that of the zeros of these polynomials.
It is to be noted that when the zeros are all taken to be on the real line, this phenomenon holds for any suitable deterministic sequence of polynomials. This follows from the elementary property that the zeros and critical points of the polynomial interlace. However this phenomenon cease to hold at this generality when the zeros are allowed to be complex numbers. A simple example can be constructed by choosing $P_n(z)=z^n-1$, where the zeros are uniformly distributed on the unit circle, whereas the critical points are accumulated at the origin. For more examples, where this phenomenon does not hold, see the discussion in \cite{Re16b}.
It was first conjectured by Pemantle and Rivin in \cite{PR13} that the limiting empirical measures of zero set and critical point coincide for general random polynomials. Moreover in \cite{PR13}, they studied a model having zeros chosen by i.i.d. random variables and established this phenomenon when the measure $\mu$ from which the zeros are chosen has finite 1-energy. In \cite{sneha}, Subramanian showed a similar result in the case of general probability measure $\mu$ supported on unit circle. Kabluchko extended this phenomenon for general probability measure $\mu$, see \cite{Ka15}.
In addition, the case that random zeros interact with each other has also been studied. One of the important examples is the characteristic polynomial of some random matrix model. For instance, in \cite{orourke}, O'Rourke showed this phenomenon in the case of characteristic polynomials of circular ensembles. Dennis and Hannay studied the association of critical points of the characteristic polynomial of Ginibre random matrix to its zeros, see \cite{hannay_dennis}. Hanin in \cite{hanin1}, studied the correlation functions of critical points and zeros of spherical polynomial ensembles. It is pertinent to note that whenever the support of the limiting measure of the zeros does not divide the complex plane into disconnected components this phenomenon holds without invoking any randomness, see \cite[Example 4]{eremenko}. As a consequence, this phenomenon holds for the characteristic polynomial of Ginibre random matrix.
It is natural to study this phenomenon for the higher derivatives too. We remark that in the case that the limiting empirical measure of zero set is purely atomic, this phenomenon trivially holds. For non-atomic case, Hu and Chang in \cite{chang_hu} studied this problem when the zeros of polynomials are in a bounded strip.
Our first purpose of this paper is to extend some known results of this phenomena from critical points to zeros of higher derivatives. See Theorem~\ref{iid zero k-th} for the generalization of result in \cite{Ka15} and Theorem~\ref{Ber zero k-th},~\ref{Ber zero array} for those of theorems in \cite{Re16b}.
Our second aim is to extend the class of random polynomials in which this phenomenon occurs by virtue of introducing randomness to a given deterministic polynomials. More precisely, we will impose randomness in such a way as to exclude one zero at random (Theorem~\ref{remove zero}) or to include finite random zeros (Theorem~\ref{t:add}). We remark that Theorem~\ref{remove zero} affirmatively settles the conjecture posed in \cite{Re16b}, when measure is non-atomic. In the last section, we will utilize methods used above to show that this phenomenon also occurs in the case when zeros of the polynomials are given by the 2D Coulomb gas density with general external potential, see Theorem~\ref{Coulomb critical}.
\subsection{Symbols and notation}
\hfill
\medskip
Throughout this paper, we will use the following symbols and notation. Let $\mathbb{D}_r$ be the disk of radius $r$, centered at origin. For any polynomial
$P$, we denote by $\mathcal Z(P)$ the multi-set of zeros of $P$ and $\mathcal M(P)$ the uniform probability measure supported on $\mathcal Z(P)$. We write $\delta_a$ the Dirac measure supported at $a$. We recall preliminary definitions introduced in \cite{Re16b}.
\begin{definition}[$\mu$-distributed sequence/triangular array]
Let $\{a_n\}_{n\ge 1}$ $($resp, $\{a_{n,i}\}_{n\ge1;1\le i\le n})$ be a sequence $($resp., triangular array$\,)$ of complex numbers. If the measure $\frac{1}{n}\sum_{i=1}^n \delta_{a_i}$ $($resp., $\frac{1}{n}\sum_{i=1}^n \delta_{a_{n,i}})$ converges weakly to a probability measure $\mu$, we call such a sequence $($resp, triangular array$\,)$ to be \textit{$\mu$-distributed}.
\end{definition}
\begin{definition}[$\log$-Ces\'{a}ro bounded sequence/triangular array]
We say a sequence $($resp, triangular array$\,)$ of complex numbers $\{a_n\}_{n\ge 1}$ $($resp, $\{a_{n,i}\}_{n\ge1;1\le i\le n})$ is \textit{log-Ces\'{a}ro-bounded} if the Ces\'{a}ro means of the positive part of their logarithms are bounded, i.e., the sequence \linebreak $\left \{\frac{1}{n}\sum_{i=1}^n \log_+|a_i| \right \}_{n \ge 1}$, $($resp, $ \left \{\frac{1}{n}\sum_{i=1}^n \log_+|a_{n,i}|\right \}_{n \ge 1})$ is bounded.
\end{definition}
\subsection{Results}
\hfill
\medskip
Our first result deals with the random polynomial whose zeros are chosen to be i.i.d. random variables. Note that the empirical measure of these zeros converge to the probability measure from which the random variables are drawn. We show that the same phenomenon appear for the zeros of any derivative of these random polynomials. This result generalizes the result of Kabluchko \cite{Ka15} which was established for zeros of first derivative (critical points).
\begin{thm} \label{iid zero k-th}
Let $\{ z_i \}_{i\ge1}$ be i.i.d. random variables distributed according to $\mu$, where $\mu$ is an arbitrary probability measure on $\mathbb{C}$. For each $n\in \mathbb N$, let
\begin{equation}
\displaystyle P_n(z):=(z-z_1)\cdots(z-z_n).
\end{equation}
Then for any $k \in \mathbb N$, $\mathcal{M}(P^{(k)}_n) \rightarrow \mu$ in probability.
\end{thm}
We show that the same result holds when the zeros of polynomials are independently sampled, from two deterministic sequences (triangular arrays), see Theorem \ref{Ber zero k-th} (Theorem \ref{Ber zero array}) below. We remark that Theorem \ref{Ber zero k-th} can be utilized to verify this phenomenon in the case that the deterministic zeros are perturbed independently at random. For example, choose $a_i=z_i+\sigma_iX_i$ and $b_i=z_i-\sigma_iX_i$, where $\{z_i\}_{i\geq1}$ is a $\mu$-distributed deterministic sequence, $X_i$'s are i.i.d. symmetric random variables and $\{\sigma_i\}_{i\geq1}$ is a sequence of positive numbers converging to $0$. This is stated as Corollary \ref{perturb zer}.
These results in the case of zeros of the first derivative were shown in \cite{Re16b}.
\begin{thm} \label{Ber zero k-th}
Let $\{ a_i \}_{i \ge 1}$ and $\{ b_i \}_{i \ge 1}$ be two $\mu$-distributed, log-Ces\'{a}ro bounded sequences of complex numbers. Suppose that $a_i \neq b_i$ for infinitely many $i$. Let $\{\xi_i\}_{i \ge 1}$ be a sequence of independent random variables such that $\xi_i=a_i$ or $\xi_i=b_i$ with equal probability. For each $n \in \mathbb N$, let
$$
\displaystyle P_n(z):=(z-\xi_1) \cdots(z-\xi_n).
$$
Then $\mathcal{M}(P_n) \rightarrow \mu$ almost surely and $\mathcal{M}(P^{(k)}_n) \rightarrow \mu$ in probability for any $k \in \mathbb N$.
\end{thm}
\begin{cor} \label{perturb zer}
Let $\{ z_i \}_{\i \ge 1}$ be a $\mu$-distributed log-Ces\'{a}ro bounded sequences of complex numbers. For a non-zero i.i.d. sequence of symmetric random variables $\{ X_i \}$ satisfying $\mathbf{E}[|X_1|] < \infty$, let
$$
\displaystyle P_n(z):= \prod_{i=1}^{n} (z-z_i+\sigma_i X_i),
$$
where $\{ \sigma_i \}_{i\ge1}$ is a decreasing sequence of real number satisfying $\lim_i \sigma_i=0$. Then $\mathcal{M}(P_n) \rightarrow \mu$ almost surely and $\mathcal{M}(P^{(k)}_n) \rightarrow \mu$ in probability for any $k \in \mathbb N$.
\end{cor}
\begin{thm}\label{Ber zero array}
Let $\{ a_{i,j} \}_{i \ge 1 ; 1 \le j \le i}$ and $\{ b_{i,j} \}_{i \ge 1 ; 1 \le j \le i}$ be two $\mu$-distributed and log-Ces\'{a}ro bounded triangular arrays of complex numbers satisfying $\sum_{i=1}^n \log_{+} \frac{1}{|a_{n,i}-b_{n,i}|}=o(n^2).$ For each $i \ge 1$, let $\{\xi_{i,j}\}_{j \le i}$ be a sequence of independent random variables such that $\xi_{i,j}=a_{i,j}$ or $\xi_{i,j}=b_{i,j}$ with equal probability. For each $n \in \mathbb N$, let
$$
\displaystyle P_n(z):=(z-\xi_{n,1}) \cdots(z-\xi_{n,n}).
$$
Then $\mathcal{M}(P_n) \rightarrow \mu$ almost surely and $\mathcal{M}(P^{(k)}_n) \rightarrow \mu$ in probability for any $k \in \mathbb N$.
\end{thm}
Our next result deals with a question appeared in \cite[Conjecture 2.14]{Re16b} which states that the same phenomenon will happen when a zero is removed uniformly at random from a deterministic sequence of polynomials. We resolve this conjecture positively when the empirical measure of zeros of the polynomials converge to a non-atomic probability measure.
\begin{thm} \label{remove zero}
Suppose $\{z_i\}_{i \ge 0}$ is a $\mu$-distributed, log-Ces\'{a}ro bounded sequence of complex number, where $\mu$ is a non-atomic probability measure on $\mathbb{C}$. For each $n \in \mathbb N$, let
$$\displaystyle P_n(z)=\frac{(z-z_0)(z-z_1)\dots (z-z_n)}{z-z_{s_n}}, $$
where $s_n$ is a random number distributed uniformly on the set $\{0,1,\cdots,n\}$. Then $\mathcal{M}(P_n) \rightarrow \mu$ almost surely and $\mathcal{M}(P'_n) \rightarrow \mu$ in probability.
\end{thm}
We now consider sequence of polynomials whose zeros are deterministic except for finite ones. Further we assume that the zero set $\{z_{n,i}\}_{n \ge 1, i \le n}$ is $\mu$-distributed triangular array of complex numbers. For such polynomials, we show that the empirical measure of zeros of higher derivatives (up-to the number of random zeros) converge to the same limiting measure as that of the zeros of these polynomials. We remark that it can be interpreted as a random perturbation of polynomials where the perturbed polynomial is obtained by multiplying with a random factor, which strengthens Theorem 2.1 in \cite{orourke_williams}.
Before stating our results we introduce the assumptions on the random zeros of our polynomials. Consider a random vector $(X_{n,1}, \dots, X_{n,k})$, where $X_{n,j}$'s are complex-valued random variables distributed according to the joint probability density function $\nu_n(w_1, \dots, w_k)$. Here, we assume that for any $k$, there exist positive constants $C_1,C_2>0$ and $a \in [0,1)$, which does not depend on $n,i$ such that $\nu_n$ satisfies the following conditions.
\begin{equation}\label{c:add1}
\int_{\mathbb{C}^k} \sum_{i=1}^k\log_{+}|w_i| \nu_n(w_1,w_2,\dots w_k)dw_1 dw_2 \dots dw_k \le C_1 <\infty;
\end{equation}
\begin{equation}\label{c:add2}
\frac{\sup_{w_i \in \mathbb{C}} \nu_n(w_1, \dots, w_k)}{ \int_{\mathbb{C}} \nu_n(w_1, \dots, w_k) dw_i } \le C_2 \exp\big( n^a \big) \quad \mbox{for all} \quad w_1,\dots, w_{i-1},w_{i+1}, \dots, w_k \in \mathbb{C};
\end{equation}
\begin{equation}\label{c:add3}
\lim_{ r \to \infty} \limsup_{n \to \infty}\mathbf{P} \left( \max_{1 \le i \le k}|X_{n,i}| \ge r \right) = 0.
\end{equation}
Notice that \eqref{c:add1} ensures that $X_{n,1}, \dots, X_{n,k}$ have finite $\log_+$-moments. Note also that \eqref{c:add2} has the following probabilistic interpretation: for any given complex numbers $$w_1, \dots, w_{i-1}, w_{i+1}, \dots w_k, $$ the conditional density of $w_i$ is sub-exponentially bounded.
\begin{thm} \label{t: genadd}
For fixed $k \in \mathbb{N}$ and each $n\in \mathbb N$, suppose that sequence of complex-valued random vector $(X_{n,1}, \dots, X_{n,k})$ with joint probability density $\nu_n(w_1, \dots, w_k)$ satisfies \eqref{c:add1}, \eqref{c:add2} and \eqref{c:add3}. Let
\begin{equation}
\displaystyle P_n(z):=(z-z_{n,1})\cdots(z-z_{n,n})(z-X_{n,1}) \cdots (z-X_{n,k}).
\end{equation}
Then $\mathcal{M}(P_n) \rightarrow \mu$ and $\mathcal{M}(P^{(\ell)}_n) \rightarrow \mu$ in probability for any $1\le \ell \le k$.
\end{thm}
We remark that one of simple examples of probability distributions satisfying above conditions \eqref{c:add1}, \eqref{c:add2} and \eqref{c:add3} is the mutually independent random variables with bounded densities. We state this specific case as the following corollary.
\begin{cor} \label{t:add}
Suppose that $\{z_i\}_{i \ge1}$ is a log-Ces\'{a}ro bounded $\mu$-distributed sequence of complex numbers, where $\mu$ is any probability measure on $\mathbb{C}$. Let $k \in \mathbb{N}$ and $X_1, \dots X_k$ be independent complex-valued random variables according to bounded density $\nu_1, \dots, \nu_k$ on $\mathbb{C}$, respectively. For each $n\in \mathbb N$, let
$$ \displaystyle P_n(z):=(z-z_1)\cdots(z-z_n)(z-X_1)\cdots(z-X_k). $$
Then $\mathcal{M}(P_n) \rightarrow \mu$ and $\mathcal{M}(P^{(\ell)}_n) \rightarrow \mu$ in probability for any $1\le \ell \le k$.
\end{cor}
As a consequence of Corollary~\ref{t:add}, Theorem~\ref{iid zero k-th} can be obtained in a special case, when the measure $\mu$ has bounded density and satisfy $\int_{\mathbb{C}}\log_{+}|z|d\mu(z) < \infty$. This is obtained by conditioning on all the zeros except for the first $k$ of them.
We further extend this phenomenon in the case where the zeros of the random polynomials follow 2D Coulomb gas density. For the benefit of the reader we recall some definitions and existing results concerning 2D Coulomb gases. For a fixed positive value $\beta$ and given external field $Q: \mathbb{C} \rightarrow \mathbb{R}$, let $\mathbf{P}_n^{\beta}$ be the point process distributed as
\begin{equation*}
d\mathbf{P}_n^{\beta}(\zeta_1, \cdots, \zeta_n)=
\frac{1}{ Z_n^{\beta}} \prod_{j,k:j<k}|\zeta_j-\zeta_k|^{2\beta}e^{-\beta n \sum_i Q(\zeta_i)} d\mbox{vol}_{2n},
\end{equation*}
where $Z_n^{\beta}$ stands for the partition function and $d\mbox{vol}_{2n}$ is the Lebesgue measure in $\mathbb{R}^{2n}$.
As the number of particles goes to infinity, the system $\{\zeta_i \}_{1 \le i \le n}$ tends to be concentrated in a certain compact set $S$ called the droplet. One of the well-known examples is the complex Ginibre ensemble, in which $\beta=1$ and $Q(z)=|z|^2$. In this case the droplet is given as $S= \{ z : |z| \le1 \}$.
In general, Hedenmalm and Makarov showed that under the mild assumptions on $Q$, the empirical measure of the system $\{\zeta_i \}_{1 \le i \le n}$ converges weakly to the equilibrium measure given by weighted (logarithmic) potential. See \cite{HM13} for more details. Also when $Q$ satisfies some regularity conditions in a neighborhood of $S$, the limiting equilibrium measure $\sigma_Q$ is absolutely continuous with respect to Lebesgue measure $dm$, and takes the following explicit form:
\begin{equation}\label{e:Q}
d\sigma_Q(z)=\frac{1}{4\pi}\,\chi_S \cdot \Delta Q(z) dm(z).
\end{equation}
Before we state our theorem below, we introduce the assumptions on the external potential $Q$. One of the main ingredients in proving Theorem~\ref{Coulomb critical} is a certain type of concentration inequality for 2D Coulomb gas due to Chafa\"{i}, Hardy and Ma\"{i}da, see \cite{CHM17}. Therefore, we also consider the same assumptions on $Q$ as follows.
\noindent$\bullet$ \textbf{Assumptions (A0).}
\begin{enumerate}
\item $Q$ is finite on a set of positive capacity and $C^2$-differentiable;
\item $\displaystyle \liminf\limits_{|z|\rightarrow \infty} \frac{Q(z)}{|z|^2}>0$;
\item $\displaystyle \limsup\limits_{|z|\rightarrow \infty} \frac{1}{Q(z)}\sup\limits_{|w-z|<1} \Delta Q(w) <4$.
\end{enumerate}
For basic notions in logarithmic potential theory, we refer the reader to \cite{ST97}. For instance, the potentials $Q(z)=|z|^{2\alpha}$ ($\alpha \ge 1$) satisfies \textbf{(A0)}. Note that (1) implies that $Q$ is admissible and the Boltzmann-Shannon entropy
$S(\sigma_{Q})= -\int \frac{d\sigma_{Q}}{dx} \log \frac{d\sigma_{Q}}{dx} dx$
of equilibrium measure $\sigma_Q$ is finite.
We remark that as the authors pointed out, the assumptions \textbf{(A0)} can be weakened as follows, see \cite[Remark 1.10]{CHM17}.
\noindent$\bullet$ \textbf{Assumptions (A1).}
\begin{enumerate}[(i)]
\item $Q$ is finite on a set of positive Lebesgue measure and $\sigma_{Q}$ is of the form \eqref{e:Q};
\item $\displaystyle \liminf\limits_{|z|\rightarrow \infty} \frac{Q(z)}{|z|^2}>0$;
\item $Q$ can be decomposed as $Q=\tilde{Q}+h$, where $\tilde{Q}$ is twice differentiable function satisfying condition \textbf{(A0)}-(3), and $h$ is a super-harmonic function.
\end{enumerate}
Note that by (i) and (iii), $\sigma_{Q}$ has bounded density inside the support, which implies $S(\sigma_{Q})$ is finite. We remark that if $Q$ has a Lipschitz continuous derivative, then $\sigma_{Q}$ is of the form \eqref{e:Q}, see e.g., \cite{CHM17}. In the case of radially symmetric potentials given as $Q(z)=g(|z|)$ for some $g: \mathbb{R}_+ \rightarrow \mathbb{R}$, the following condition implies (i): $Q$ is finite on a set of positive Lebesgue measure and $r g'(r)$ is increasing on $\mathbb{R}_+$ (or $g$ is convex on $\mathbb{R}_+$), see e.g., \cite[IV.6]{ST97}
We emphasize that under our assumption, the ``shape'' of droplets is not restricted to a simply connected domain. For example our theorem cover the case that the external potential is given by ``Mittag-Leffler'' potential
$Q(z)=|z|^{2\alpha}-2\nu\log|z|$, $(\alpha \ge 1,\nu > 0)$. In this case, the droplet is given by annulus where its modulus depends on $\alpha, \nu$. We remark that these cases are not covered by Eremenko's result, see \cite[Example 4]{eremenko}.
\begin{thm} \label{Coulomb critical}
For any $\beta>0$ and any external potential $Q$ satisfying \rm{\textbf{(A1)}}, let $\{\zeta_i \}_{1 \le i \le n}$ be the corresponding 2D Coulomb gas ensemble, and define
$$
P_n(z):=(z-\zeta_1)\cdots(z-\zeta_n).
$$
Then for any $k \in \mathbb{N}$,
\begin{equation*}
\mathcal M \big( P^{(k)}_n \big) \rightarrow \frac{1}{4\pi}\,\chi_S \cdot \Delta Q(z) dm(z) \quad \text{in probability.}
\end{equation*}
\end{thm}
\section{Outline of method}
To prove our results, we follow the potential theoretic approach introduced by Kabluchko in \cite{Ka15}. For given polynomial $P_n(z)$ with $\mathcal Z(P)=\{ w_1, \cdots, w_n \}$ and $1 \le k \le n$, let us define
\begin{equation}\label{L_n^k}
\displaystyle L^k_n(z):=\frac{1}{k!} \frac{P^{(k)}_n(z)}{P_n(z)} = \sum_{1\le i_1 < i_2<\dots<i_k \le n} \frac{1}{z-w_{i_1}}\dots\frac{1}{z-w_{i_k}}.
\end{equation}
Note that $L_n^k$ is the product of logarithmic derivatives of $P_n,P_n^{(1)},\dots,P_n^{(k-1)}$, with a scaling of $1/k!$. For fixed $k \in \mathbb{N}$ and any $f \in C_c^{\infty}(\mathbb{C})$ whose support is contained in $\mathbb{D}_r$, we define $
f_n(z):=\frac{1}{n}(\log|L_n^k(z)|)\Delta f(z)
$. Then by applying Green's theorem twice, we have
$$
\displaystyle \frac{1}{2\pi}\int_{\mathbb{D}_r} f(z) \Delta \frac{1}{n} \log |L_n^k(z)| =\int_{\mathbb{D}_r} f_n(z)dm(z).
$$
Here $\Delta \frac{1}{n} \log |L_n^k(z)|$ is interpreted in the sense of distributions. Notice that,
$$
\frac{1}{2\pi}\int_{\mathbb{D}_r} f(z) \Delta \frac{1}{n} \log |L_n^k(z)|=\frac{1}{n} \sum_{j=1}^{n} f(w_j)-\frac{1}{n} \sum_{j=1}^{n-k} f(\xi^k_j),
$$
where $\{ \xi_j^k : 1 \le j \le n-k \}= \mathcal{Z}(P_n^{(k)})$.
Therefore to verify the concentration of empirical measures $\mathcal{M}(P_n)$ and $\mathcal{M}(P_n^{(k)})$, it is enough to show that $\int_{\mathbb{D}_r} f_n(z)dm(z)$ converges to $0$. To ensure the tightness, we recall a lemma of Tao and Vu.
\begin{lem} {\rm \cite[Lemma 3.1]{TVK10}}. \label{TVK-lem}
Let $(X,\mathcal{A},\nu)$ be a finite measure space and $f_n: X \rightarrow \mathbb{R}, n \ge 1$ be random functions which are defined over a probability space $(\Omega, \mathcal{B},\mathbf{P})$ and jointly measurable with respect to $\mathcal{A} \otimes \mathcal{B}$. Assume that :
\begin{enumerate}
\item For $\nu-$a.e. $x\in X$ we have $f_n(x) \rightarrow 0$ in probability, as $n \rightarrow \infty$;
\item For some $\delta>0$, the sequence $\int_X |f_n(x)|^{1+\delta}d\nu(x)$ is tight.
\end{enumerate}
Then, $\int_X f_n(x)d\nu(x) \rightarrow 0$ in probability, as $n \rightarrow 0$.
\end{lem}
By Lemma~\ref{TVK-lem}, it is enough to show that the following two statements hold.
\begin{equation}\label{A12}
\frac{1}{n}\log\left|L^k_n(z)\right| \rightarrow 0 \quad \mbox{in probability for Lebesgue a.e. }z \in \mathbb{C};
\end{equation}
\begin{equation}\label{A3}
\text{the sequence} \quad \left \{\int_{\mathbb{D}^r}\frac{1}{n^2}\log^2 \left|L^k_n(z)\right|dm(z) \right \}_{n\ge 1} \quad \text{is tight.}
\end{equation}
\section{Controlling large values of $L_n^k$ and tightness}
As we explained above in Section 2, all we need to show is the upper and lower estimate of $L_n^k$ and \eqref{A3} for given sequence of random polynomials $P_n$. In this section, we prove the following two lemmas which provide the upper estimate and tightness which can be applied for every cases in our theorems. The following lemma is counterpart of \cite[Lemma 4.2]{Re16b}.
\begin{lem} \label{A1 Re16b}
Let $\{ a_{i,j} \}_{i \ge1; 1 \le j \le i}$ be any triangular array of numbers. Define $$\tilde{L}_n^{k}(z)=\sum\limits_{1 \le j_1 < \cdots < j_k \le n } \frac{1}{|z-a_{n,j_1}|} \frac{1}{|z-a_{n,j_2}|} \cdots \frac{1}{|z-a_{n,j_k}|}.$$
Then for Lebesgue a.e. $z \in \mathbb{C}$,
$$
\displaystyle \limsup_{n \rightarrow \infty} \frac{1}{n} \log \tilde{L}_n^{k}(z) \le 0.
$$
\end{lem}
\begin{proof} First, notice that $\log\tilde{L}_n^{k}(z) \leq k\log\tilde{L}_n^{1}(z)$.
Now it is enough to bound $\log\tilde{L}_n^{1}(z)$ which follows from the proof of Lemma 4.2 in \cite{Re16b}.
\end{proof}
\begin{lem}
Let $\{ a_{i,j} \}_{i\ge1; 1\le j \le n} $ be a log-Ces\'{a}ro bounded triangular array of numbers. Define $P_n(z)= \prod_{j=1}^n (z-a_{n,j})$. Then, for any $r>0$ and $k \in \mathbb{N}$, the sequence
$$
\left\{ \frac{1}{n^2} \int_{\mathbb{D}_r} \log^2 \left| \frac{P_n^{(k)}(z)}{P_n(z)} \right| dm(z)\right\}_{n\ge k}
$$
is bounded.
\end{lem}
\begin{proof}
Notice that
$$\int_{\mathbb{D}_r} \log^2 \left| \frac{P_n^{(k)}(z)}{P_n(z)} \right| dm(z)=\int_{\mathbb{D}_r} \log_+^2 \left| \frac{P_n^{(k)}(z)}{P_n(z)} \right| dm(z)+\int_{\mathbb{D}_r} \log_-^2 \left| \frac{P_n^{(k)}(z)}{P_n(z)} \right| dm(z).$$
We now analyse above positive and negative parts of the logarithm separately. For the positive part, observe that by Cauchy-Schwarz inequality, we obtain
\begin{align*}
\int_{\mathbb{D}_r} \log_+^2 \left| \frac{P_n^{(k)}(z)}{P_n(z)} \right| dm(z)
&= \int_{\mathbb{D}_r} \log_+^2 \left| \frac{P_n^{(k)}(z)}{P_n^{(k-1)}(z)}\dots\frac{P^{(1)}_n(z)}{P_n(z)} \right| dm(z),\\
&\leq \int_{\mathbb{D}_r} \left( \log_+ \left| \frac{P_n^{(k)}(z)}{P_n^{(k-1)}(z)}\right|+ \cdots + \log_+ \left| \frac{P^{(1)}_n(z)}{P_n(z)} \right| \right)^2 dm(z),
\\
&\leq k \int_{\mathbb{D}_r} \log_+^2 \left| \frac{P_n^{(k)}(z)}{P_n^{(k-1)}(z)}\right|+ \cdots + \log_+^2 \left| \frac{P_n^{(1)}(z)}{P_n(z)} \right| dm(z).
\end{align*}
Note that for any $0 \le j \le k-1$, each term in the above expression is bounded as
\begin{align*}
&\int_{\mathbb{D}_r} \log_+^2 \left| \frac{P_n^{(j+1)}(z)}{P^{(j)}_n(z)} \right| dm(z)
= \int_{\mathbb{D}_r} \log_+^2 \left| \sum_{\omega: P^{(j)}_n(\omega)=0} \frac{1}{z-\omega} \right| dm(z),
\\
\leq& \int_{\mathbb{D}_r} \left(\log(n-j)+\sum_{\omega: P^{(j)}_n(\omega)=0} \log_+ \left| \frac{1}{z-\omega} \right| \right)^2 dm(z),
\\
\leq& (n-j+1)\int_{\mathbb{D}_r} \left( \log^2(n-j)+\sum_{\omega: P^{(j)}_n(\omega)=0} \log_+^2 \left| \frac{1}{z-\omega} \right| \right) dm(z),
\\
=& (n-j+1)\left( \pi r^2 \log^2(n-j)+\int_{\mathbb{D}_r} \sum_{\omega: P^{(j)}_n(\omega)=0} \log_-^2 \left| z-\omega \right| dm(z) \right),
\end{align*}
where in the first inequality we have used $\log_+( \sum_{k=1}^n a_k) \le \log n + \sum_{k=1}^n \log_+ a_k$ for any $n \in \mathbb{N}$ and $a_1,\dots,a_n \in \mathbb{C}$, see \cite[Remark 3.2]{Re16b} for instance. By the translation invariance of Lebesgue measure,
\begin{align}\begin{split}\label{123}
&\int_{\mathbb{D}_r}\log_-^2|z-\xi|dm(z)=\int_{\mathbb{D}_r+\xi}\log_-^2|z|dm(z) \\ &\le \sup_{K \subset \mathbb{C}} \int_K\log_-^2|z|dm(z)= \int_{\mathbb{D}_1}\log_-^2|z|dm(z)< \infty.
\end{split}\end{align}
Therefore, we conclude that $ \left\{ \frac{1}{n^2} \int_{\mathbb{D}_r} \log_+^2 \left| \frac{P_n^{(k)}(z)}{P_n(z)} \right| dm(z)\right\}_{n\ge k}$ is bounded.
Now we show that the negative part is also bounded. Note that using $\log_-|ab| \le \log_-|a| + \log_-|b|$ for any $a,b \in \mathbb{C}$ we have
$$
\frac{1}{n^2}\int_{\mathbb{D}_r}\log^2_-\left| \frac{P_n^{(k)}(z)}{P_n(z)} \right| dm(z) \le \frac{1}{n^2} \int_{\mathbb{D}_r}\log^2_-\left| P_n^{(k)}(z) \right| + \log^2_-\left| \frac{1}{P_n(z)} \right| dm(z).
$$
As in the same way above, we get the following inequality for the first term:
$$
\frac{1}{n^2} \int_{\mathbb{D}_r}\log^2_-\left| P_n^{(k)}(z) \right| dm(z) \le \frac{n-k}{n^2}\int_{\mathbb{D}_r}\sum_{w\in \mathcal Z(P_n^{(k)})} \log_-^2|z-w|dm(z).
$$
which is uniformly bounded in $n$ by \eqref{123}. Also, we have
\begin{align*}
\frac{1}{n^2}\int_{\mathbb{D}_r} \log^2_-\left| \frac{1}{P_n(z)} \right| dm(z)=& \frac{1}{n^2}\int_{\mathbb{D}_r} \log^2_+\left| P_n(z) \right| dm(z),
\\
\leq& \frac{1}{n^2} \int_{\mathbb{D}_r} \left( \sum_{j=1}^n \log_+\left| z-a_{n,j} \right| \right)^2 dm(z),
\\
\leq& \int_{\mathbb{D}_r} \left( \log 2 +\log_+|z|+\frac{1}{n}\sum_{j=1}^n \log_+\left|a_{n,j} \right| \right)^2 dm(z).
\end{align*}
Now the lemma follows from the fact that $\{ a_{n,j} \}$ is log-Ces\'{a}ro bounded.
\end{proof}
\section{Controlling small values of $L_n^k$.}
\subsection{Proof of Theorem~\ref{iid zero k-th}, ~\ref{Ber zero k-th} and ~\ref{Ber zero array}.}
\hfill
\medskip
In this subsection, we present the proof of Theorem~\ref{iid zero k-th}, ~\ref{Ber zero k-th} and ~\ref{Ber zero array}.
Controlling the small values of $L_n^1$ were given in \cite[Lemma 2.6.]{Ka15} and \cite[Lemma 3.3]{Re16b}. We will use these lemmas and the induction argument to control the small values of $L_n(z)$ in the corresponding theorems.
\subsubsection{Proof of Theorem~\ref{iid zero k-th}}
\hfill
\medskip
Recall that under the conditions of Theorem~\ref{iid zero k-th}, zeros of random polynomials $P_n(z)$ are given by i.i.d. random variables $\{ z_i \}_{i\ge1}$ distributed according to $\mu$. First we introduce the following lemma due to Kabluchko.
\begin{lem}{\rm \cite[Lemma 2.6]{Ka15}}
For Lebesgue a.e. $z \in \mathbb{C}$,
\begin{equation}\label{e:L_n^1:1}
\lim_{n\rightarrow \infty }\mathbf{P}\left(|L^1_n(z)| < e^{-n \varepsilon}\right) =0
\end{equation}
for any $\varepsilon>0$, where
$L^1_n(z):=\frac{P'_n(z)}{P_n(z)}$.
Here $P_n(z)$'s are random polynomials in Theorem \ref{iid zero k-th}.
\end{lem}
Recall that $L_n^k$ is given as \eqref{L_n^k} for each $k \le n$. Now, to complete the proof of Theorem~\ref{iid zero k-th}, all we need to show is the following lemma.
\begin{lem}
For Lebesgue a.e. $z \in \mathbb{C}$,
\begin{equation}\label{e:L_n^k:1}
\lim_{n\rightarrow \infty }\mathbf{P}\left(|L^k_n(z)| < e^{-n \varepsilon}\right) =0
\end{equation}
for every $k \in \mathbb{N}$ and $\varepsilon>0$, where $L^k_n(z):=\frac{1}{k!} \frac{P^{(k)}_n(z)}{P_n(z)}$.
Here $P_n(z)$'s are random polynomials in Theorem \ref{iid zero k-th}.
\end{lem}
\begin{proof}
When $\mu$ is purely atomic measure, the proof of \eqref{e:L_n^k:1} is straightforward so without loss of generality we can assume that $\mu$ is not purely atomic. Note that in the case $k=1$, \eqref{e:L_n^k:1} follows from \cite[Lemma 2.6.]{Ka15}. Suppose that \eqref{e:L_n^k:1} holds for some $k \ge 1$. First, we decompose $\mu$ by $\mu_1+\mu_2$ such that $\mu_1$ is purely atomic, and $\mu_2$ is non-atomic. Then $0 \le x:=\mu_1(\mathbb{C})<1$ since $\mu$ is not purely atomic. Let $E:=supp(\mu_1)$.
For $j,m<n$, define a random variable $L^{k,j}_n(z)$ by
$$ L^{m,j}_n(z):=\sum_{1\le i_1 < \dots < i_m \le n, i_\ell \neq j \forall \ell} \frac{1}{z-z_{i_1}} \dots \frac{1}{z-z_{i_m}}.$$
Note that $L^{m,j}_n(z)$ is independent of $z_j$.
Fix $z \in \mathbb{C}$ which satisfies \eqref{e:L_n^k:1} for every $k \le k_0$ and $\varepsilon>0$. First, for any $\ell \in \mathbb{N}$, by definition of $L_n^{m,j}(z)$ we have
\begin{equation}\label{1}
L^{k_0+1}_{n+\ell}(z)=\frac{1}{z-z_j} L^{k_0,j}_{n+\ell}(z) + L^{k_0+1,j}_{n+\ell}(z) \quad \mbox{for all} \quad 1\le j \le \ell.
\end{equation}
For $1\le j \le \ell$, let $\Omega$ be the sample space and denote
\begin{align*}
A_j&:= \{\omega \in \Omega : |L^{k_0,j}_{n+\ell}(z)| < e^{-n\varepsilon} \}; \\
B_j&:= \{\omega \in \Omega : z_j \notin E \}; \\
C&:= \{\omega \in \Omega : |L^{k_0+1}_{n+\ell}(z)| < e^{-2n\varepsilon} \},
\end{align*}
for fixed $\varepsilon>0$. Note that
$$ \displaystyle \Omega = \left(\cup_{j=1}^\ell A_j \right) \cup \left(\cup_{j=1}^\ell(A^c_j \cap B_j)\right) \cup \left(\cap_{j=1}^\ell B_j^c\right), $$
which implies
\begin{equation*} C \subset \big(\cup_{j=1}^\ell A_j \big) \cup \big(\cup_{j=1}^\ell(A^c_j \cap B_j \cap C)\big) \cup \big(\cap_{j=1}^\ell B_j^c\big).
\end{equation*}
Therefore, we immediately obtain
\begin{equation}\label{2}
\displaystyle \mathbf{P}(C)\le \displaystyle \sum_{j=1}^\ell \mathbf{P}(A_j)+\sum_{j=1}^\ell \mathbf{P}(A_j^c \cap B_j \cap C) + \mathbf{P}(\cap_{j=1}^\ell B_j^c).
\end{equation}
Note that, since $L^{k_0,j}_{n+\ell}(z)$ and $L^{k_0}_{n+\ell-1}(z)$ are identically distributed, we have
\begin{equation}\label{3}
\mathbf{P}(A_j) = \mathbf{P}\left(|L^{k_0}_{n+\ell-1}(z)|<e^{-n\varepsilon}\right) \quad \mbox{for all} \quad 1 \le j \le \ell.
\end{equation}
which converge to $0$ as $n$ goes to $\infty$, for all $1 \le j \le \ell$ by the induction hypothesis. For the second term, using \eqref{1}, we get
\begin{align} \begin{split}
&\mathbf{P}(A_j^c \cap B_j \cap C)
\\
=& \mathbf{P} \left(z_j \notin E, \left|L^{k_0,j}_{n+\ell}(z)\right| \ge e^{-n\varepsilon}, \left|\frac{1}{z-z_j} L^{k_0,j}_{n+\ell}(z) +L^{k_0+1,j}_{n+\ell}(z)\right|<e^{-2n\varepsilon} \right),
\\
=& \mathbf{P} \left (z_j \notin E, \left|L^{k_0,j}_{n+\ell}(z) \right| \ge e^{-n\varepsilon}, \left|\frac{1}{z-z_j} +\frac{L^{k_0+1,j}_{n+\ell}(z)}{ L^{k_0,j}_{n+\ell}(z)}\right|<\frac{e^{-2n\varepsilon}}{ L^{k_0,j}_{n+\ell}(z)} \right) ,
\\
\le& \mathbf{P} \left(z_j \notin E, \left|\frac{1}{z-z_j} +\frac{L^{k_0+1,j}_{n+\ell}(z)}{ L^{k_0,j}_{n+\ell}(z)}\right| < e^{-n\varepsilon} \right). \label{4}
\end{split}
\end{align}
Let $\nu_z(dw)$ be a distribution of $\frac{1}{z-z_1} 1_{\{z_1 \notin E\}}$. Then since $\nu_z(dw)$ is non-atomic measure, we have
\begin{equation}\label{5}
\lim_{r \downarrow 0} \sup_{z_0 \in \mathbb{C}} \nu_z(B(z_0,r)) =0.
\end{equation}
Using \eqref{5} to \eqref{4}, we obtain
\begin{align*}
\mathbf{P}(A_j^c \cap B_j \cap C) &\le \mathbf{P} \left(z_j \notin E, \left|\frac{1}{z-z_j} +\frac{L^{k_0+1,j}_{n+\ell}(z)}{ L^{k_0,j}_{n+\ell}(z)}\right| < e^{-n\varepsilon}\right),
\\
&= \mathbf{E}\left[\nu_z\left(B \big(\frac{L^{k_0+1,j}_{n+\ell}(z)}{ L^{k_0,j}_{n+\ell}(z)},e^{-n\varepsilon} \big) \right) \right],
\\
&\le \sup_{z_0 \in \mathbb{C}} \nu_z(B(z_0,e^{-n\varepsilon})).
\end{align*}
Also, since $z_i$ are i.i.d. we have
\begin{equation}\label{6}
\mathbf{P}(\cap_{j=1}^\ell B_j^c) = \prod_{j=1}^\ell \mathbf{P}(z_j \in E) = x^\ell.
\end{equation}
Combining all \eqref{3}, \eqref{5}, \eqref{6} and \eqref{2}, we obtain
$$
\limsup_{n\rightarrow \infty}\mathbf{P}(|L^{k_0+1}_n(z)|<e^{-2n\varepsilon})= \limsup_{n \rightarrow \infty} \mathbf{P}(C) \le x^\ell.
$$
for arbitrarily $\ell \in \mathbb{N}$. Now, \eqref{1} for $k=k_0+1$ follows from the fact that $x<1$. This finishes the proof.
\end{proof}
\subsubsection{Proof of Theorem~\ref{Ber zero k-th} and Theorem~\ref{Ber zero array}}
\hfill
\medskip
In this section we will prove Theorem~\ref{Ber zero k-th} and Theorem~\ref{Ber zero array} at once. Let $\{a_{n,i}\}$ and $\{b_{n,i}\}$ be two $\mu$-distributed log-Ces\'aro bounded triangular arrays of complex numbers. Here we assumed in Theorem~\ref{Ber zero array} that $\sum_{i=1}^n \log_+ \frac{1}{|a_{n,i}-b_{n,i}|} = o(n^2)$. Note that this condition is used only for the case $k=1$. Therefore it suffices to prove Theorem~\ref{Ber zero array} and the proof of Theorem~\ref{Ber zero k-th} follows from similar argument. Recall that zeros of random polynomials $P_n(z)$ are given by the random sequence $\xi_{n,i}$ where $\xi_{n,i}=a_{n,i}$ or $b_{n,i}$ with equal probability. \\
Now following lemma completes the proof of Theorem~\ref{Ber zero array}.
\begin{lem}\label{l:two2}
For a.e. $z \in \mathbb{C}$,
\begin{equation}\label{e:two2}
\lim_{n\rightarrow \infty }\mathbf{P} \left( \left|L^k_n(z)\right| < e^{-n \varepsilon} \right) =0
\end{equation}
for all $k \in \mathbb{N}$ and $\varepsilon>0$, where $L^k_n(z):=\frac{1}{k!} \frac{P^{(k)}_n(z)}{P_n(z)}$.
Here $P_n(z)$'s are random polynomials in Theorem~\ref{Ber zero array}.
\end{lem}
\begin{proof}
Note that the case $k=1$ is given by \cite[Lemma 4.3 and Lemma 4.4]{Re16b}. In particular, for the proof of \cite[Lemma 4.4]{Re16b} the author has loosened the conditions $\sum_{i=1}^n \log_+ \frac{1}{|a_{n,i}-b_{n,i}|} = o(n^2)$ to the following condition :
\begin{equation}\label{e:cond}
\lim_{\varepsilon \downarrow 0} \liminf_{n \to \infty }\frac{ |\{ i: \log_+ \frac{1}{|a_{n,i}-b_{n,i}|} \le \varepsilon n \}|}{n} \ge \frac{3}{4}.
\end{equation}
Also, \eqref{e:two2} holds for any $z \in \mathbb{C}$ that does not agree with any $a_{n,i}$ or $b_{n,i}$ in the triangular array.
We will prove Lemma \ref{l:two2} by using induction argument on $k$.
Let $$\Omega= \left \{ \xi_n=\{\xi_{n,i}\}_{i\leq n} \, | \, \xi_{n,i} = a_{n,i} \mbox{ or } b_{n,i}, \, n \in \mathbb{N} \right \}
$$ be the sample space of all possible finite sequences $\{\xi_{n,i}\}_{i\leq n}$ equipped with uniform probability measure $\mathbf{P}$. By \eqref{e:cond}, we have
$$ |\{ i: a_{n,i} \neq b_{n,i}, i \le n \}| \to \infty \quad \mbox{as} \quad n \to \infty. $$
Thus, for any $\ell \in \mathbb{N}$, we may assume that $a_{n,i} \neq b_{n,i}$ for $1 \le i \le \ell$ for large $n$, since finite permutations on sequences does not affect \eqref{e:two2}.
Fix $z \in \mathbb{C}$, $\ell,k \in \mathbb{N}$, and let $\mathcal{N}_n=\mathcal{N}_n(z)$ be a set of finite sequence $\xi_n=\{\xi_{n,i}\}_{i \le n}$ such that
$$\mathcal{N}_n:= \left \{ \xi_n \in \Omega \, |\, \ \, \left|L^{k}_n(z) \right| < e^{-n\varepsilon} \right \}.
$$
Note that for any $n \in \mathbb{N}$ and $z \in \mathbb{C}$, the value of $L_n^{k}(z)$ depends only on $\xi_n$. Set
\begin{equation}\label{e:7}
\mathcal{N}_n^\ell := \left\{ \xi_n \in \mathcal{N}_{n} \, |\, \exists \eta_n \in \mathcal{N}_n \mbox{ such that }\eta_n \neq \xi_n, \eta_{n,i} = \xi_{n,i} \mbox{ for all } l < i \le \ell \right\}.
\end{equation}
Note that if there exist two sequences $\xi_n, \tilde{\xi}_n$ in $\mathcal{N}_n \backslash \mathcal{N}^\ell_n$ with $\xi_{n,i}=\tilde{\xi}_{n,i}$ for $1 \le i \le \ell$, then $\xi_n=\tilde{\xi}_n$ by the construction. Therefore for each given sequence of tails $\{\xi_{n,i} \}_{\ell <i \le n}$, there is at most one sample contained in $\mathcal{N}_n \backslash \mathcal{N}^\ell_n$, which implies
\begin{equation}\label{e:11}
\mathbf{P}(\mathcal{N}_n \backslash \mathcal{N}^\ell_n) \le 2^{-\ell}.
\end{equation}
For any $n \in \mathbb{N}$ and $\xi_n \in \mathcal{N}^\ell_n$, using the definition we can find a sequence $\eta_n \in \mathcal{N}^\ell_n$ such that $\eta_n \neq \xi_n$ and $\eta_{n,i} = \xi_{n,i}$ for all $l<i \le \ell$. Note that since $\xi_n, \, \eta_n \in \mathcal{N}_n$,
\begin{equation}\label{e:3}
\left|L^k_n(z) (\xi_n ) \right|<e^{-n\varepsilon}, \quad \quad \left|L^k_n(z)(\eta_n ) \right|<e^{-n\varepsilon}.
\end{equation}
Let $F=\{ i_j \}_{1 \le j \le m}$ be the set of index such that $\xi_{n,i_j} \neq \eta_{n,i_j}$. Without loss of generality, we may assume that $1 \le i_1<i_2<...<i_m \le \ell$.
For $1\le p \le m$, let us denote
$$\alpha_p:= \sum_{1 \le j_1<\dots<j_p \le m} \frac{1}{z-\xi_{n,i_{j_1}}}\dots\frac{1}{z-\xi_{n,i_{j_p}}}, \quad \beta_p:=\sum_{1 \le j_1<\dots <j_p \le m} \frac{1}{z-\eta_{n,i_{j_1}}}\dots\frac{1}{z-\eta_{n,i_{j_p}}},
$$
and $\alpha_0=\beta_0=1$.
Note that, by construction,
$$
\displaystyle \sum_{p=0}^m \alpha_p w^{m-p}=\prod_{j=1}^m \left(w+ \frac{1}{z-\xi_{n,i_j}}\right), \quad \sum_{p=0}^m \beta_p w^{m-p}=\prod_{j=1}^m \left(w+ \frac{1}{z-\eta_{n,i_j}}\right),
$$
which implies that two polynomials $\sum_{p=0}^m \alpha_p w^{m-p}$ and $\sum_{p=0}^m \beta_p w^{m-p}$ cannot be the same as polynomials in `$w$'. Therefore, $\alpha_p \neq \beta_p$ for at least one of $1 \le p \le m$.
For each $k_1 \in \mathbb{N}$ and $E \subset \{1,2,\dots,n\}$, let us define $L^{k_1,E}_n(z)$ by
$$L^{k_1,E}_n(z)=L^{k_1,E}_n(z)(\xi_n) :=\sum_{1\le i_1 < \dots < i_{k_1} \le n, i_j \notin E\, \forall j} \frac{1}{z-\xi_{n,i_1}} \dots \frac{1}{z-\xi_{n,i_k}}.
$$
Note that $L^{k_1,E}_n(z)$ is independent of $\{\xi_{n,i}\}_{i \in E}$ by the independence of $\xi_{n,i}$. Also by definition of $\alpha_p$, $\beta_p$ and $F$, we have the following decomposition of $L^{k}_n(z)$:
\begin{equation}
\label{e:4}
L^{k}_n(z)(\xi_n )=\sum_{p=0}^m \alpha_p L^{k-p,F}_{n}(z)(\xi_n ), \quad L^{k}_n(z)(\eta_n )=\sum_{p=0}^m \beta_p L^{k-p,F}_{n}(z)(\eta_n ).
\end{equation}
Notice that since $\xi_{n,i} = \eta_{n,i}$ for all $i \notin F$, $L^{k-p,F}_{n}(z)(\xi_n)= L^{k-p,F}_{n}(z)(\eta_n)$ for all $0 \le p \le m$. Therefore by \eqref{e:3} and \eqref{e:4}, we get
\begin{align}
\begin{split}\label{e:5}
2e^{-n\varepsilon} &\quad \ge\quad \left |L_n^{k}(z)(\xi_n)-L_n^{k}(z)(\eta_n) \right|, \\
&\quad = \quad \left |\sum_{p=0}^m \alpha_p L^{k-p,F}_{n}(z)(\xi_n) - \sum_{p=0}^m \beta_p L^{k-p,F}_{n}(z)(\eta_n)\right|, \\
&\quad = \quad \left |\sum_{p=1}^m (\alpha_p- \beta_p) L^{k-p,F}_{n}(z)(\xi_n) \right|,
\end{split}\end{align}
for sufficiently large $n$.
Let $m_0:= \max\{1 \le p \le m: \alpha_p \neq \beta_p \}$, $m_1:= \min\{1 \le p \le m: \alpha_p \neq \beta_p \}$ and $\kappa_1,\dots,\kappa_q$ be zeros of polynomial
$$ f(w):=\sum_{p=m_1}^{m_0} (\alpha_p - \beta_p)w^{m_0 - p}. $$
Note that $q \le m-1$ since $q=\deg f = m_0 - m_1 \le m-1$. Also the fact that $f(0)= \alpha_{m_0} - \beta_{m_0} \neq 0$ implies $\kappa_i \neq 0$ for every $1 \le i \le q$. Let $\gamma_i= z+ \frac{1}{\kappa_i}$ for each $1 \le i \le q$. Then we obtain
\begin{equation}
f(w)= \prod_{i=1}^q \left(w+\frac{1}{z-\gamma_i}\right), \quad \alpha_p-\beta_p= L_q^p(\{\gamma_i\}_{i \le q}).
\end{equation}
Now let us define a random sequence $\{\nu_{n,i}\}_{i \le n-m+q}$ by
\begin{equation}\label{d:nu}
\nu_{n,i} =\begin{cases} \gamma_i &\mbox{for } \quad 1 \le i \le q \\
\xi'_{i-q} & \mbox{for } \quad q < i \le n-m+q,
\end{cases}
\end{equation}
where the random sequence $\{\xi'_i\}_{1 \le i \le n-m}$ is constructed as follows:
\begin{itemize}
\item For $1 \le i \le \ell-m$, by shortening the sequence $\{\xi_i\}_{i \in \{1,\dots,\ell\}} $ to $\{\xi_i\}_{i \in \{1,\dots,\ell\} \backslash F}$;
\item For $i > \ell-m$, pick $a_{n,i+m}$ or $b_{n,i+m}$ with equal probability.
\end{itemize}
Combining all above, we obtain
\begin{align}
\begin{split}\label{e:6}
\left|\sum_{p=1}^m (\alpha_p-\beta_p) L^{k_0-p,F}_{n}(z) \left(\xi_n \right) \right|
&\quad =\quad \left|\sum_{p=m_1}^{m_0} (\alpha_p-\beta_p) L^{k_0-p,F}_{n}(z) \left(\{\xi_{n,i}\}_{i \le n} \right) \right| ,
\\
&\quad =\quad \left|\sum_{p=0}^q L_q^p (z)\left(\{\gamma_i\}_{i \le q}\right) L_n^{k-m_1-p,F}(z)\left(\{\xi_{n,i}\}_{i \le n} \right)\right| ,
\\
&\quad =\quad \left|\sum_{p=0}^q L_q^p (z)\left(\{\gamma_i\}_{i \le q}\right) L_{n-m}^{k-m_1-p}(z)(\{\xi'_i\}_{i \le n-m}) \right|,
\\
&\quad =\quad \left|L^{k-m_1} _{n-m+q}(z)\left(\{\nu_{n,i}\}\right) \right|.
\end{split}
\end{align}
Note that since $\ell$ is fixed number, there are only finite cases of $\{\nu_{n,i}\}_{i \le n-m+q}$. More precisely, since the deterministic part of $\{\nu_{n,i}\}_{i \le n-m+q}$ is determined by the values of $\xi_{n,1},\dots,\xi_{n,\ell}$ and $\eta_{n,1},\dots,\eta_{n,\ell}$, the number of cases is less than $2^{2\ell}$ for each $n$. Now we inductively define $\mathcal{Z}_{k,n,\ell}$ and $C_{k,n,\ell}$ as follows.
\begin{enumerate}[(i)]
\item First we set
$$C_{1,n,\ell}:= \{ \{(a_{n,i},b_{n,i}) \}_{i \le n} \}\quad \mbox{and} \quad \mathcal{Z}_{1,n,\ell}:= \{a_{n,i},b_{n,i} : \{(a_{n,i},b_{n,i}) \}_{i \le n} \in C_{1,n,\ell} \}.$$
\item We denote by $C_{2,n,\ell}$ the collection of $ \{\nu_{n,i}\}$, where $\nu$ is a sequence of 2-vector defined by \eqref{d:nu} with the sequence of 2-vector $\xi \in C_{1,n,\ell} $ and set
$$\mathcal{Z}_{2,n,\ell}:= \mathcal{Z}_{1,n,\ell} \cup \{ \nu^1_{n,i}, \nu^2_{n,i} : \{\nu_{n,i}\} = \{ (\nu^1_{n,i}, \nu^2_{n,i})\} \in C_{1,n,\ell} \}. $$
\item Finally, for any $k \ge 1$, we define $C_{k+1,n,\ell}$ as the collection of $ \{\nu_{n,i}\}$, where $\nu$ is a sequence of 2-vector defined by \eqref{d:nu} with the sequence of 2-vector $\xi \in C_{k,n,\ell} $ and set
$$\mathcal{Z}_{k+1,n,\ell}:=\mathcal{Z}_{k,n,\ell} \cup \{ \nu^1_{n,i}, \nu^2_{n,i} : \{\nu_{n,i}\} = \{ (\nu^1_{n,i}, \nu^2_{n,i})\} \in C_{k+1,n,\ell} \}. $$
\end{enumerate}
Note that for each $k,n,\ell \in \mathbb{N}$, $|C_{k,n,\ell}| \le 2^{2k\ell}$ since there is at most $2^{2 \ell}$ choice of $\{\nu_{n,i} \}$ for each sequence of 2-vector. Thus, $Z_{k,n,\ell}$ is also finite set. Let
$$ \mathcal{Z}_{k} := \bigcup_{n=1}^\infty \bigcup_{\ell=1}^\infty \mathcal{Z}_{k,n,\ell},$$
which will be exceptional set of \eqref{e:two2} for $k$. Note that $\mathcal{Z}_{k} \subset \mathbb{C}$ is countable set since each $\mathcal{Z}_{k,n,\ell}$ is finite. Also, $\mathcal{Z}_{k}$ is increasing in $k$ since $\mathcal{Z}_{k,n,\ell}$ is increasing in $k$ for each $n$, $\ell$. Now we are ready to state our induction hypothesis. We claim that for any $\{a_{n,i}\}$ and $\{b_{n,i}\}$ satisfying \eqref{e:cond}, \eqref{e:two2} holds for any $k \in \mathbb{N}$ and $z \in \mathcal{Z}_k^c$. Let us emphasize that we already proved the case $k=1$ at the beginning of this proof. Assume that our claim is valid for $k=1,2,\dots,k_0 - 1$. Fix $w \in \mathcal{Z}^c_{k_0}$ and $\ell \in \mathbb{N}$, and define $\mathcal{N}_n$ and $\mathcal{N}_n^\ell$ as above. Note that it suffices to show that
$$ \lim_{n \to \infty} \mathbf{P}(\mathcal{N}_n ) = \lim_{n \to \infty} \mathbf{P}(\mathcal{N}_n(w) )=0 $$
By \eqref{e:11}, \eqref{e:5} and \eqref{e:6} we obtain
\begin{align*}
&\limsup_{n \rightarrow \infty} \mathbf{P}(\mathcal{N}_n) \le \limsup_{n \rightarrow \infty} \mathbf{P}(\mathcal{N}_n \backslash \mathcal{N}_n^\ell ) + \limsup_{n \rightarrow \infty} \mathbf{P}( \mathcal{N}_n^\ell ) \le 2^{-\ell} + \limsup_{n \rightarrow \infty} \mathbf{P}( \mathcal{N}_n^\ell ) ,\\
&\le 2^{-\ell} + \limsup_{n \to \infty} \mathbf{P}\left(|L_n^{k_0}(w)(\xi_n)|<e^{-n \varepsilon} \right),\\
&\le 2^{-\ell} + \limsup_{n \to \infty}\mathbf{P}\left(\left|L^{k_0-m_1}_{n-m+q}(w)\left(\nu_n \right)\right| < 2e^{-n\varepsilon} \mbox{ for some } \nu_n=\{\nu_{n,i}\} \in C_{2,n,\ell} \right), \\
&\le 2^{-\ell} + \sum_{\nu_n \in C_{2,n,\ell}} \limsup_{n \to \infty}\mathbf{P}\left(\left|L^{k_0-m_1}_{n-m+q}(w)\left(\nu_n \right)\right| < 2e^{-n\varepsilon} \right).
\end{align*}
By the definition of $C_{k_0,n,\ell}$ and induction hypothesis, we first have $|C_{2,n,l}| \le 2^{2 \ell}$.
For the sequence of 2-vector $\xi= \{\xi_n \}_{n \ge 1} = \{ (\xi^1_{n,i}, \xi^2_{n,i} ) \}_{i \le n, n \ge 1}$, let us denote by $\mathcal{Z}_{k,n,\ell}(\xi_n)$ the set defined same as $\mathcal{Z}_{k,n,\ell}$, except that $C_{1,n,\ell}= \{ \xi_n \}$. Then we observe that for any $m \ge 1$ and $\nu_n \in C_{2,n,\ell}$,
$$\mathcal{Z}_{k_0-m,n,\ell}(\nu_n) \subset \mathcal{Z}_{k_0-m+1,n,\ell} \subset \mathcal{Z}_{k_0,n,\ell}.$$
Thus, we conclude that since $w \notin \mathcal{Z}_{k_0}$ and $m_1 \ge 1$,
$$ \lim_{n \to \infty}\mathbf{P}\left(\left|L^{k_0-m_1}_{n-m+q}(w)\left(\nu_n \right)\right| < 2e^{-n\varepsilon} \right)=0 \quad \mbox{for any} \quad \nu_n \in C_{2,n,\ell}.$$
Therefore,
$$ \limsup_{n \to \infty} \mathbf{P}(\mathcal{N}_n^\ell)=0.$$
Now the claim follows from the fact that $\ell$ is arbitrary, which completes the proof.
\end{proof}
\subsection{Proof of Theorem~\ref{remove zero} and ~\ref{t: genadd}} \label{1.5,6}
\hfill
\medskip
In this subsection, we prove Theorem~\ref{remove zero} and ~\ref{t: genadd}. Recall that in these cases, considering random polynomials are constructed as giving randomness to deterministic ones.
\subsubsection{Proof of Theorem~\ref{remove zero}}
\hfill
\medskip
Let $\{a_{n,i}\}_{n\ge 1, 0\le i \le n}$ be a triangular array defined by
$a_{n,i}:=1-\delta_{s_n}(i)$, where $s_n$ is a random number distributed uniformly on the set $\{0,1,\cdots,n\}$. Assume that $\mu$ is non-atomic probability measure. Let $\{z_n\}_{n \ge 0}$ be a $\mu$-distributed and log-Cesaro bounded sequence.
\begin{lem}
For Lebesgue a.e. $z \in \mathbb{C}$,
\begin{equation}
\lim_{n\rightarrow \infty }\mathbf{P}(|L^1_n(z)| < e^{-n \varepsilon}) =0
\end{equation}
for every $\varepsilon>0$, where $L^1_n$ is given as
$$ L^1_n(z):= \frac{a_{n,0}}{z-z_0} + \frac{a_{n,1}}{z-z_1} + \dots + \frac{a_{n,n}}{z-z_n}.
$$
\end{lem}
\begin{proof} Since $\mu$ is non-atomic, it suffices to show that if
$\limsup_{n\rightarrow \infty} \mathbf{P}(|L_n(z)|<e^{-n\varepsilon})>0$ for some $\varepsilon>0, z\in \mathbb{C},$ there exist $x\in \mathbb{C}$ with $\mu(x)>0$. Suppose that
$$\displaystyle \limsup_{n\rightarrow \infty} \mathbf{P}(|L_n(z)|<e^{-n\varepsilon})=3\delta>0
$$
for some fixed $\varepsilon>0, z\in \mathbb{C}$.
Then, there exists a subsequence $\{n_k\}_{k\ge1}$ satisfying
\begin{equation}\label{A2:1}
\displaystyle \mathbf{P} \left(|L_{n_k}(z)|<e^{-n_k\varepsilon}\right)>2\delta
\end{equation}
Since the value $L_n(z)$ is just determined by $s_n \in \{0,1,\dots,n\}$, \eqref{A2:1} implies that
\begin{equation}
\label{A2:2}
\left| 0\le i\le n_k : \big|\sum_{j=0}^{n_k} \frac{1}{z-z_j} - \frac{1}{z-z_i} \big|<e^{-n_k\varepsilon} \right| > 2\delta (n_k+1).
\end{equation}
Let
$$N_k:=\left|0\le i\le n_k : \left|\sum \limits_{j=0}^{n_k} \frac{1}{z-z_j} - \frac{1}{z-z_i} \right|<e^{-n_k\varepsilon} \right|,
$$
and $w_{k,1},w_{k_2},\dots,w_{k,N_k}$ are those values of $z_i$'s, i.e.,
$$
\displaystyle \left|\sum \limits_{j=0}^{n_k} \frac{1}{z-z_j} - \frac{1}{z-w_{k,j}} \right|<e^{-n_k\varepsilon}, \quad 1 \le j \le N_k.
$$
Note that $w_{k,i}$ may have same values. Without loss of generality, we may assume $|w_{k,1}| \le |w_{k,2}| \le \dots \le |w_{k,N_k}|.$ Then by \eqref{A2:2}, for all $1\le i_1,i_2 \le N_k$,
\begin{equation}\label{A2:3}
\displaystyle \left|\frac{1}{z-w_{k,i_1}}-\frac{1}{z-w_{k,i_2}}\right| < 2e^{-n_k \varepsilon}.
\end{equation}
We will use the following inequality:
\begin{equation}\label{log ineq.}
\displaystyle \frac{1}{n_k}\sum_{i=1}^{n_k} \log_+ |z_i| \ge \frac{1}{n_k}\sum_{i=1}^{N_k} \log_+ |w_{k,i}| \ge \frac{N_k}{n_k} \log_+ |w_{n,1}| \ge 2\delta \log_+|w_{n,1}|.
\end{equation}
Notice that since $\{z_k\}$ is log Ces\'{a}ro bounded, $\log_+|w_{n,1}|$ is also bounded.
Let $B_k:={\bar B(w_{k,1},r_k)}$ be closed ball centered at $w_{k,1}$ with radius
$$\displaystyle r_k:=\frac{2\exp(-{n_k} \varepsilon){R_k}^2}{1-2R_k \exp(-{n_k}\varepsilon)}, \quad R_k:=|z-w_{k,1}|.
$$
By \eqref{log ineq.}, $R_k$ is bounded, which implies $1>r_k>0$ for large $k$. Therefore, we may assume that $1>r_k>0$ for all $k\ge 1$ without loss of generality.
Note that if $w \notin B_k$,
\begin{equation}\label{A2:4}
\begin{array}{lcl}
\displaystyle \left|\frac{1}{z-w_{k,1}}-\frac{1}{z-w}\right|&=& \displaystyle \left|\frac{w-w_{k,1}}{(z-w_{k,1})(z-w_k)}\right| ,
\\
\\
&\ge& \displaystyle \left|\frac{r_k}{(z-w_{k,1})(z-w)}\right| \ge \frac{r_k}{R_k(R_k+r_k)} \ge 2e^{-n_k \varepsilon}.
\end{array}
\end{equation}
\\
Then by \eqref{A2:3} and \eqref{A2:4}, $w_{k,i} \in B_k$ for all $1 \le i \le N_k$.
Let $\displaystyle \mu_k:= \frac{1}{n_k+1} \sum_{i=0}^{n_k} \delta_{z_i}.$ Then, $\mu_k(B_k)=\frac{N_k}{n_k+1} \ge 2\delta$ and $\mu_k \rightarrow \mu$ weakly. Note that if $\mu_k \rightarrow \mu$ weakly, $\displaystyle \limsup_{k\rightarrow \infty}\mu_k(C) \le \mu(C)$ for all closed set C and $\displaystyle \liminf_{k\rightarrow \infty}\mu_k(U) \ge \mu(U)$ for all open set U.
We can find $\tilde{R}>0$ satisfying $\mu(B(0,\tilde{R}-2)) > 1-\delta/3.$ Since $\displaystyle \liminf_{k\rightarrow \infty}\mu_k(B(0,\tilde{R}-2)) \ge \mu(B(0,\tilde{R}-2))$, $\mu_k(B(0,\tilde{R}-2)) > 1-\delta/2.$ for large k. So $B_k \cap B(0,\tilde{R}-2) \neq \emptyset$. Since $r_k<1$, we can say that $B_k \subset \bar{B}(0,\tilde{R}):=K$ for large k.
Suppose $\mu$ has no point measure. Then, for all $x\in K$, there exist $\varepsilon=\varepsilon(x)>0$ such that $\mu(\bar{B}(x,\varepsilon(x)))<\delta.$ $\{B(x,\frac{\varepsilon(x)}{2})\}_{x\in K}$ makes open cover of compact set K and there exist finite open cover $\{B(x_i,\frac{\varepsilon(x_i)}{2})\}_{i\le N}.$
Let $\varepsilon=\min\{ \varepsilon(x_i)/2 : i \le N\}$. $r_k<\varepsilon$ for large k since $r_k \downarrow 0.$ For each k with $r_k<\varepsilon$ , there exists $i\le N$ satisfying $B_k \cap B(x_i, \varepsilon(x_i)/2) \neq \emptyset$. So, $B_k \subset \bar{B}(x_i,\varepsilon(x_i))$ for some $i$. Therefore, there exists $i\le N$ such that
\begin{equation}\label{A2:5}
\displaystyle B_k \subset \bar{B}(x_i,\varepsilon(x_i))=:C\mbox{ for infinitely many }k.
\end{equation}
Therefore, $2\delta \le \limsup_k \mu_k(C) \le \mu(C) < \delta$ is contradiction, where we used \eqref{A2:5} for the first inequality, and the fact $C$ is closed and $\mu_k \rightarrow \mu$ for the second inequality. Therefore, $\mu$ has point mass and the conclusion of the lemma follows.
\end{proof}
\subsubsection{Proof of Theorem~\ref{t: genadd}}
\hfill
\medskip
Now we prove Theorem~\ref{t: genadd}. Recall that sequence of complex-valued random vector $(X_{n,1}, \dots, X_{n,k})$ with joint probability density $\nu_n(w_1, \dots, w_k)$ satisfies \eqref{c:add1}, \eqref{c:add2} and \eqref{c:add3}. The following lemma shows that \eqref{A12} holds for $L_n^1(z)$.
\begin{lem}\label{l:add1}
For Lebesgue a.e. $z \in \mathbb{C}$,
\begin{equation}\label{e:L_n^k:2}
\lim_{n\rightarrow \infty }\mathbf{P}(|L^1_n(z)| < e^{-n \varepsilon}) =0
\end{equation}
for every $\varepsilon>0$, where
$L^1_n(z):=\frac{P'_n(z)}{P_n(z)}$. Here $P_n(z)$'s are random polynomials given as
\begin{equation}\label{pn}
\displaystyle P_n(z):=(z-z_{n,1})\cdots(z-z_{n,n})(z-X_{n,1}) \cdots (z-X_{n,k}).
\end{equation}.
\end{lem}
\begin{proof}
First we claim that it suffices to show the case $k=1$. Assume that $\eqref{e:L_n^k:2}$ holds for $k=1$. Now for any fixed $X_{n,1}, \dots, X_{n,k-1}$ with general $k \ge 2$, we can define $w_{n+k-1,i}= z_{n,i}$ for $1 \le i \le n$, $w_{n+k-1,n+j} = X_{n,j}$ for $1 \le j \le k-1$ and $Y_{n,1}=X_{n+k,1}$ so that $\{w_{n,i}\}_{n \in \mathbb{N}, 1 \le i \le n}$ is $\mu$-distributed and $Y_{n,1}$ satisfies \eqref{c:add2} and \eqref{c:add3}. Thus we have $\eqref{e:L_n^k:2}$ for any fixed $X_{n,1}, \dots, X_{n,k-1}$, which implies $\eqref{e:L_n^k:2}$. So we can assume $k=1$ without loss of generality.
Fix $\delta >0$ and $z \in \mathbb{C}$ satisfying $z \neq z_{n,i}$ for any $n \in \mathbb{N}$ and $1 \le i \le n$. By \eqref{c:add3}, we have constants $r_\delta>0$ and $N_\delta \in \mathbb{N}$ such that
$$ \mathbf{P}(|X_{n,1}| \ge r_\delta - |z|) \le \frac{\delta}{2} \quad \mbox{for} \quad n \ge N_\delta.
$$
Let $\theta^{(n)}_z(w)dw$ be the conditional density of $\frac{1}{z-X_{n,1}} \textbf{1}_{\{ |X_{n,1}| \le r_\delta - |z| \}}$ given the others. Note that by \eqref{c:add2} we have
$$ \theta^{(n)}_z(w) \le \frac{|z-X_{n,1}|^2 \sup_{w_i \in \mathbb{C}} \nu_n(w)}{\int_{\mathbb{C}} \nu_n(w) dw } \le C_2 e^{cn^a}|z-X_{n,1}|^2 \le C_2 r_{\delta}^2 e^{cn^a}.$$
By definition of $L_n^1$, we may write $$L_n^1(z)=\frac{1}{z-X_{n,1}} + \sum_{i=1}^n \frac{1}{z-z_i} .$$
Let us denote
$$ Y_n := \frac{1}{z-X_{n,1}} \quad \mbox{and} \quad x_n:= \sum_{i=1}^n \frac{1}{z-z_i},
$$
i.e., $L_n^1(z) = Y_n + x_n$. Then for large $n$ satisfying $n \ge N_\delta$ and $\pi C_2 r_\delta^2 e^{-2n\varepsilon+cn^a} \le \delta/2$, we have
\begin{align*}
\mathbf{P}(L_n^1(z) \le e^{-n\varepsilon}) &\le \mathbf{P}(|X_{n,1}| \ge r_\delta - |z|) + \mathbf{P}( |Y_n+x_n| \le e^{-n\varepsilon} ; |X_{n,1}| \le r_\delta - |z|) ,\\ &\le \frac{\delta}{2} + \sup_{x \in \mathbb{C}} \mathbf{P}( |Y_n| \le B(-x,e^{-n\varepsilon});|X_{n,1}| \le r_\delta - |z|), \\
&\le \frac{\delta}{2} + C_3 e^{-2n\varepsilon+cn^a} \le \delta.
\end{align*}
for some constant $C_3$. Since $\delta>0$ is arbitrary, we obtain \eqref{e:L_n^k:2}.
\end{proof}
Now we are ready to use induction on $k$ to obatin \eqref{A12}. The following lemma completes the proof of Theorem~\ref{t: genadd}.
\begin{lem}\label{l:add2}
Let $k \in \mathbb{N}$ be the number of $X_{n,j}$'s. For Lebesgue a.e. $z \in \mathbb{C}$, we have
\begin{equation}\label{e:L_n^k:3}
\lim_{n\rightarrow \infty }\mathbf{P}(|L^\ell_n(z)| < e^{-n \varepsilon}) =0
\end{equation}
for every $\varepsilon>0$ and $\ell \le k$, where $L^\ell_n(z):=\frac{1}{\ell!} \frac{P^{(\ell)}_n(z)}{P_n(z)}$.
Here $P_n(z)$'s are random polynomials defined by \eqref{pn}.
\end{lem}
\begin{proof} Repeating the argument of Lemma \ref{l:add1}, we can obtain that it suffices to prove Lemma \ref{l:add2} for the case $\ell=k$.
Define
$$
Q_n(z):= (z-z_{n,1})\cdots(z-z_{n,n})(z-X_{n,1})\cdots(z-X_{n,k-1})= \frac{P_n(z)}{z-X_{n,k}}.
$$ Then we observe that
$$
L_n^k(z) = \frac{1}{z-X_{n,k}} M^{k-1}_n(z) + M^k_n(z),
$$
where $ M^l_n(z):= \frac{1}{l!}\frac{Q_n^{(l)}(z)}{Q_n(z)}$ for $l \in \mathbb{N}$. Thus, for any $\varepsilon,\delta>0$ and any $z \in \mathbb{C}$ satisfying $z \neq z_{n,i}$ for all $n,i$, we have
\begin{align*}
\mathbf{P}\left(|L^k_n(z)| < e^{-2n \varepsilon}\right) &=\mathbf{P} \left( \left| \frac{M^{k-1}_n(z)}{z-X_{n,k}}+M^k_n(z) \right| < e^{-2n \varepsilon}\right) ,\\ &\le \mathbf{P} \left( \left| \frac{1}{z-X_{n,k}}+\frac{M^k_n(z)}{M^{k-1}_n(z)} \right| < e^{-n \varepsilon} ; |X_{n,k}| \le r_\delta - |z| \right) ,
\\
&+\mathbf{P}(|M_n^{k-1}(z)| < e^{-n\varepsilon}) + \mathbf{P}(|X_{n,k}| \ge r_\delta - |z|),
\end{align*}
where $r_\delta>0$ is a constant in the proof of Lemma \ref{l:add1}. Note that by induction hypothesis and definition of $r_\delta$ we have
$$ \lim_{n \to \infty}\mathbf{P}\left(|M_n^{k-1}(z)| < e^{-n\varepsilon}\right) = 0 \quad \mbox{and} \quad \limsup_{n \to \infty} \mathbf{P}\left(|X_{n,k}| \ge r_\delta - |z|\right) \le \frac{\delta}{2}.$$
Set
$$ Y_n:= \frac{1}{z-X_{n,k}} \textbf{1}_{\{|X_{n,k}| \le r_\delta -z\} }. $$
Then by \eqref{c:add2}, we obtain
$$ \theta_z(w) \le C_3 r_{\delta}^2 n^a $$
for density $\theta_z(w)$ of $Y_n$, and
$$ \mathbf{P} \left( \left| \frac{1}{z-X_{n,k}}+\frac{M^k_n(z)}{M^{k-1}_n(z)} \right| < e^{-n \varepsilon} ; |X_{n,k}| \le r_\delta - |z| \right) \le \sup_{x \in \mathbb{C}} \mathbf{P}(|Y_n+x| < e^{-n\varepsilon}), $$
which goes to $0$ as $n \rightarrow \infty$. Therefore, we have
$$ \limsup_{n \to \infty}\mathbf{P}\left(|L^k_n(z)| < e^{-2n \varepsilon}\right) \le \delta. $$
Now lemma follows from the fact that $\delta$ is an arbitrary constant.
\end{proof}
\section{Application to 2D Coulomb gas ensembles}
In this section, we prove Theorem~\ref{Coulomb critical}. Recall that the joint probability density of 2D Coulomb gas ensemble is given as
\begin{equation*}
d\mathbf{P}_n^{\beta}(\zeta_1, \cdots, \zeta_n)=
\frac{1}{ Z_n^{\beta}} \prod_{j,k:j<k}|\zeta_j-\zeta_k|^{2\beta}e^{-\beta n \sum_j Q(\zeta_j)} d\mbox{vol}_{2n},
\end{equation*}
where $\beta>0$ is inverse temperature and $Q: \mathbb{C} \rightarrow \mathbb{R}$ is external potential satisfying the assumptions \textbf{(A1)} in Section 1.
First we recall some definitions and properties of equilibrium measure which we will use in this section.
For any probability measure $\mu$, logarithmic potential $U^{\mu} : \mathbb{C} \rightarrow (-\infty, \infty]$ is defined by
$ U^{\mu}(\zeta):=\int_{\mathbb{C}}\log \frac{1}{|\zeta-\eta|^2}d\mu(\eta) $
and logarithmic energy $I[\mu]$ is given as
$ I[\mu]:=\int_{\mathbb{C}^2}\log \frac{1}{|\zeta-\eta|^2}d\mu(\zeta)d\mu(\eta)=
\int_{\mathbb{C}} U^{\mu}(\zeta) d\mu(\zeta).$
A subset $\mathcal{N}$ of $\mathbb{C}$ is said to be polar if $I[\mu]=\infty$ for all compactly supported probability measures with $\mbox{supp} \,(\mu) \in \mathcal{N}$.
We say that some property holds {\it{quasi-everywhere (q.e)}} on $E \subset \mathbb{C}$ if it holds everywhere on $E$ except some Borel polar set. Note that every Borel probability measure with finite logarithmic energy assigns zero Lebesgue measure
to Borel polar sets.
For given admissible potential $Q$, the {\it{weighted logarithmic energy}} $I_Q[\mu]$ for each probability measure $\mu$ is defined as
$$
I_Q[\mu]:=\iint_{\mathbb{C}^2}\log \frac{1}{|\zeta-\eta|^2}d\mu(\zeta)d\mu(\eta)+2\int_{\mathbb{C}} Q d\mu.
$$
The following theorem is a collection of properties of equilibrium measure.
\begin{thm} \label{eq msr} {\rm\cite[Chap I Theorem 1.3]{ST97}}
Suppose that the potential $Q$ is admissible and let $$I_{Q} := \inf \left\{ I_Q[\mu] \right\}, $$ where infimum is over all probability measures on $\mathbb{C}$.
Then the following properties hold.
\begin{enumerate}
\item There exists the unique probability measure $ \sigma_{Q} $ such that $$I_Q \left[\sigma_{Q} \right]=I_{Q}. $$
\item
$ I_Q\left[\sigma_{Q} \right]=I_Q$ is finite.
\item
$I[\sigma_{Q}]=I_Q-2 \int_{\mathbb{C}}Qd\sigma_{Q}$ is finite.
\item
$S_{Q}:= \mbox{supp} \,(\sigma_{Q}) $ is compact.
\item
Let
$$
F_{Q}:=I_Q-\int Q d\sigma_{Q}.
$$
Then
$$
U^{ \sigma_{Q}}(\zeta) +Q(\zeta) \ge F_{Q} \quad \text{holds for q.e. } \zeta \in \mathbb{C}.
$$
and
$$
U^{\sigma_{Q}}(\zeta)+Q(\zeta) = F_{Q} \quad \text{holds for q.e. } \zeta \in S_{Q}.
$$
\end{enumerate}
\end{thm}
The measure $ \sigma_{Q} $ is called the \textit{equilibrium measure} associated with $Q$. The constant $F_Q$ is called the \textit{modified Robin constant} for $Q$.
Using the notion of equilibrium measure, we immediately obtain the following lemma.
\begin{lem} For Lebesgue a.e. $\zeta \in \mathbb{C}$, there exists a positive constant $C>0$ satisfying
\begin{equation}\label{ub condi den}
\frac{ e^{-\beta n \left( Q(\zeta)-2\int_{\mathbb{C}}\log|\zeta-z| d\sigma_Q(z) \right) } }{ \int_{\mathbb{C}} e^{-\beta n \left( Q(\zeta)-2\int_{\mathbb{C}}\log|\zeta-z| d\sigma_Q(z) \right) } d\zeta } < C.
\end{equation}
\end{lem}
\begin{proof}
Note that by Theorem~\ref{eq msr}(3), the Borel polar set has Lebesgue measure zero. Therefore by Theorem~\ref{eq msr}(5), we have
$$
Q(\zeta)-2\int_{\mathbb{C}}\log|\zeta-z| d\sigma_Q(z)= U^{\sigma_{Q}}(\zeta)+Q(\zeta) \ge F_{Q},
$$
for a.e. $\zeta \in \mathbb{C}$ and
$$
Q(\zeta)-2\int_{\mathbb{C}}\log|\zeta-z| d\sigma_Q(z)= U^{\sigma_{Q}}(\zeta)+Q(\zeta) = F_{Q},
$$
for a.e. $\zeta \in S_Q$, which implies
\begin{align*}
\frac{ e^{-\beta n \left( Q(\zeta)-2\int_{\mathbb{C}}\log|\zeta-z| d\sigma_Q(z) \right) } }{ \int_{\mathbb{C}} e^{-\beta n \left( Q(\zeta)-2\int_{\mathbb{C}}\log|\zeta-z| d\sigma_Q(z) \right) } d\zeta } &\le \frac{ e^{-\beta n F_Q } }{ \int_{S_Q} e^{-\beta n F_Q } d\zeta } \le \frac{1}{m(S_Q)},
\end{align*}
where $m$ denotes the Lebesgue measure in $\mathbb{C}$. Therefore, Theorem~\ref{eq msr}(4) concludes the lemma.
\end{proof}
We denote by $W_p$ the \textit{Wasserstein distance} of order $p$.
In particular, if $p=1$, by Kantorovich-Rubinstein dual representation, we have
$$
W_1(\mu, \nu)= \sup_{||f||_{\rm Lip} \le 1} \int f(x) (\mu-\nu) (dx), \quad ||f||_{\rm Lip} := \sup_{x \neq y} \frac{|f(x)-f(y)|}{|x-y|}.
$$
Let $\mu_n$ be the empirical measure of Coulomb gas ensemble, i.e.,
$\mu_n = \frac{1}{n} \sum_{j=1}^{n} \delta_{\zeta_{j}}.$ The following concentration inequality is due to Chafa\"i, Hardy and Ma\"ida.
\begin{prop} \label{Concent ineq.}{\rm \cite[Theorem 1.5.]{CHM17}}
There exists a constant $a'>0$ such that for any $n \ge 2$, and $r >0$,
\begin{equation}\label{e:wd}
\mathbf{P}_n^{\beta} \left( W_1(\mu_n, \sigma_Q)\ge r \right) \le e^{-a' n^2 r^2 }.
\end{equation}
\end{prop}
Using this concentration inequality, we prove following lemma.
\begin{lem} \label{condi den}
There exists $\varepsilon>0$ such that for Lebesgue a.e. $\zeta \in \mathbb{C}$,
\begin{equation}\label{e:1}
\lim_{n \to \infty} \mathbf{P}_n^{\beta} \left( \frac{e^{-\beta n Q(\zeta)} \prod_{j=1}^{n-1} \left| \zeta-\zeta_j \right|^{2\beta} }{ \int_{\mathbb{C}} e^{-\beta n Q(\zeta)} \prod_{j=1}^{n-1} \left| \zeta-\zeta_j \right|^{2\beta}d\zeta } \le \exp \left( n^{1-\frac{\varepsilon}{2}} \right) \right)= 1.
\end{equation}
\end{lem}
\begin{proof}
For some small $\varepsilon>0$, set
\begin{align*}
f_{n,\zeta}(z)&:= \log (n^{1/2-2\varepsilon}|z-\zeta|)_+ - (1/2 -2\varepsilon) \log n; \\
&= \begin{cases} \log |z-\zeta|, &\text{if}\quad |z-\zeta| \ge n^{-1/2+2\varepsilon} \\ (-1/2+2\varepsilon)\log n, &\text{if}\quad |z-\zeta| < n^{-1/2+2\varepsilon}. \end{cases}
\end{align*}
Using the concentration inequality \eqref{e:wd} with the choice $r = \frac{1}{2}n^{-1/2+\varepsilon}$, we obtain
$$\mathbf{P}_n^{\beta}\left(W_1(\mu_n, \sigma_Q) \ge \frac{1}{2}n^{-1/2+\varepsilon} \right) \le e^{-2a n^{1+2\varepsilon} }, $$
for some positive constant $a>0$. Set $\mu'_n:= \frac{1}{n-1} \sum_{j=1}^{n-1} \delta_{\zeta_j}$. Then we have
$$\mathbf{P}_n^{\beta}\left(W_1(\mu'_n, \sigma_Q) \ge n^{-1/2+\varepsilon} \right) \le e^{-a n^{1+2\varepsilon} }, $$
for large $n$. Indeed, we have $W_1(\mu_n,\mu'_n) \le \frac{1}{n(n-1)} \sum_{j=1}^{n-1} |\zeta_j - \zeta_{n}| \to 0$ as $n \to \infty$ in probability, which follows from the tightness property of $\zeta_j$, see \cite[Theorem 1.12]{CHM17}. Thus, using $\| f_{n,\zeta} \|_{\rm Lip} = n^{1/2-2\varepsilon}$ for every $\zeta \in \mathbb{C}$, we obtain
$$ \mathbf{P}_n^{\beta} \left( \sup_{\zeta \in \mathbb{C}} \left| \int f_{n,\zeta}(z) \mu'_n(dz) - \int f_{n,\zeta}(z) \sigma_Q(dz) \right| \ge n^{-\varepsilon} \right) \le e^{-a n^{1+2\varepsilon} }. $$
Also recall that conditions \textbf{(A1)}-(i),(iii) deduce the boundedness of the density of $\sigma_Q$. Thus, using the definition of $f_{n,\zeta}$ we have
\begin{align*}
\int \left( f_{n,\zeta}(z)- \log|z-\zeta| \right) \sigma_Q(dz) \le \int_{B(\zeta,n^{-1/2+2\varepsilon})} -\log|z-\zeta| \sigma_Q(dz) \le c \, n^{-1+5\varepsilon}.
\end{align*}
Combining these estimates with the fact that $f_{n,\zeta}(z) \ge \log|z-\zeta|$, we conclude that for large $n$,
\begin{align*}
&\mathbf{P}_n^{\beta} \left( \sup_{\zeta \in \mathbb{C}} \int \log|z-\zeta| \mu'_n(dz) - \int \log|z-\zeta| \sigma_Q(dz) \ge 2n^{-\varepsilon} \right) \label{e:5.3.1}
\\
\le& \mathbf{P}_n^{\beta} \left( \sup_{\zeta \in \mathbb{C}} \int \log|z-\zeta| \mu'_n(dz) - \int f_{n,\zeta}(z) \sigma_Q(dz) \ge 2n^{-\varepsilon} - c \, n^{-1+5\varepsilon} \right) , \nonumber
\\
\le& \mathbf{P}_n^{\beta}\left(\sup_{\zeta \in \mathbb{C}} \int f_{n,\zeta}(z) \mu'_n(dz) - \int f_{n,\zeta}(z) \sigma_Q(dz) \ge n^{-\varepsilon} \right) \le e^{-an^{1+2\varepsilon}}. \nonumber
\end{align*}
Then by Theorem~\ref{eq msr}(5) and the fact that
\begin{align*}
e^{-\beta nQ(\zeta)} \prod_{j=1}^{n} |\zeta - \zeta_j|^{2\beta} = \exp \left[ -\beta n \left\{Q(\zeta) + \int \log \frac{1}{|z-\zeta|^2} \mu_n(dz) \right\} \right],
\end{align*}
we have
\begin{equation}\label{e:5.3.2}
\mathbf{P}_n^{\beta} \left( \sup_{\zeta \in \mathbb{C}} e^{-\beta nQ(\zeta)} \prod_{j=1}^{n} |\zeta - \zeta_j|^{2\beta} \ge \exp(-n \beta F_Q + \beta n^{1-\varepsilon}) \right) \le e^{-an^{1+2\varepsilon}}.
\end{equation}
On the other hand, since $\int_{\mathbb{C}} f_{n,\zeta}(z) - \log|z-\zeta| \sigma_Q(d\zeta) \le c \,n^{-1+5\varepsilon} $, we have
\begin{align}
\begin{split}\label{e:5.3}
&\int_{S_Q} \int_{\mathbb{C}} f_{n,\zeta}(z) - \log|z-\zeta| \mu'_n(dz) \sigma_Q(d\zeta) \\
&= \int_{\mathbb{C}} \int_{S_Q} f_{n,\zeta}(z) - \log|z-\zeta| \sigma_Q(d\zeta) \mu'_n(dz), \\
&\le \int_\mathbb{C} cn^{-1+5\varepsilon} \mu'_n(dz) = c \, n^{-1+5\varepsilon}.
\end{split}
\end{align}
Set
$$
A:=\left\{ \zeta\in S_Q : \int_{\mathbb{C}} \left| \log|z-\zeta| - f_{n,\zeta}(z)\right| \mu'_n(dz) \ge n^{-1+6\varepsilon} \right\}.
$$
Then by Chebyshev inequality and \eqref{e:5.3}, we have
\begin{align*}
\lim_{n \to \infty} \sigma_Q(A) \le \lim_{n \to \infty} \frac{c \, n^{-1+5\varepsilon}}{n^{-1+6\varepsilon}}=0.
\end{align*}
Note that we may assume that $m(A) \le m(S_Q)/2$ since $\sigma_Q$ is absolutely continuous with respect to Lebesgue measure.
Therefore, we obtain that for large $n$,
\begin{align*}
&\mathbf{P}_n^{\beta} \left( \int_{\mathbb{C}} e^{-\beta n Q(\zeta)} \prod_{j=1}^{n-1} \left| \zeta-\zeta_j \right|^{2\beta}d\zeta \le \int_{S_Q\setminus A} e^{-\beta n F_Q - \beta n ^{1-\varepsilon} - c\beta n^{5\varepsilon} } d\zeta \right) \\
\le& \mathbf{P}_n^{\beta} \left( \int_{S_Q \setminus A} e^{-\beta n Q(\zeta)} \prod_{j=1}^{n-1} \left| \zeta-\zeta_j \right|^{2\beta}d\zeta \le \int_{S_Q \setminus A} e^{-\beta n F_Q - \beta n ^{1-\varepsilon}- c\beta n^{5\varepsilon} } d\zeta \right),
\\
\le& \mathbf{P}_n^{\beta} \left( \inf_{\zeta \in \mathbb{C}} e^{-\beta n Q(\zeta)} \prod_{j=1}^{n-1} \left| \zeta-\zeta_j \right|^{2\beta} \le e^{-\beta n F_Q - \beta n^{1-\varepsilon} -c\beta n^{5\varepsilon} } \right),
\\ \le& \mathbf{P}_n^{\beta}\left(\sup_{\zeta \in \mathbb{C}} \left| \int f_{n,\zeta}(z) \mu'_n(dz) - \int \log|z-\zeta|\sigma_Q(dz)\right| \ge n^{-\varepsilon} + cn^{-1+5\varepsilon} \right), \\
\le& \mathbf{P}_n^{\beta}\left(\sup_{\zeta \in \mathbb{C}} \left| \int f_{n,\zeta}(z) \mu'_n(dz) - \int f_{n,\zeta}(z) \sigma_Q(dz) \right| \ge n^{-\varepsilon} \right)
\le e^{-an^{1+2\varepsilon}}.
\end{align*}
This and \eqref{e:5.3.2} proves the lemma.
\end{proof}
\begin{proof}[Proof of Theorem~\ref{Coulomb critical}]
Fix $k \in \mathbb{N}$. For $n \in \mathbb{N}$, let us denote by $\zeta_{n}, \dots, \zeta_{n}$ the $n$-th Coulomb gas ensembles. Set $z_{n,i}= \zeta_{i}$ for $1 \le i \le n-k$ and $Y_{n,i}=\zeta_{n-k+i}$ for $1 \le i \le k$. Recall that all we need to show is
\begin{equation} \label{Coulomb eq 123}
\lim_{n \to \infty} \mathbf{P}(|L_n^k(z)| < e^{-n\varepsilon})= 0, \quad L^k_n(z):=\frac{1}{k!} \frac{P^{(k)}_n(z)}{P_n(z)},
\end{equation}
where $P_n(z):=(z-\zeta_1)\cdots(z-\zeta_n)$.
For each $0 \le i \le k-1$, let $\mathcal{N}_{n,i}$ be the subset of sample space, which satisfies
$$ \sup_{\zeta \in \mathbb{C}} \left[ \frac{e^{-\beta n Q(\zeta)} \prod_{j=1, j \neq n-i}^{n} \left| \zeta-\zeta_j \right|^{2\beta} }{ \int_{\mathbb{C}} e^{-\beta n Q(z)} \prod_{j=1}^{n-1} \left| z-\zeta_j \right|^{2\beta}dz } \right] > \exp \left( n^{1-\frac{\epsilon}{2}} \right)
$$
and set $\mathcal{N}_n:= \cup_{i=0}^{k-1} \mathcal{N}_{n,i}$. Then by Lemma~\ref{condi den},
$$\mathbf{P}_n^{\beta}(\mathcal{N}_n) \le \sum_{i=0}^{k-1} \mathbf{P}_n^{\beta}(\mathcal{N}_{n,i}) \le k e^{-an^{1+2\varepsilon}},
$$
which implies $\lim_{n \to \infty} \mathbf{P}_n^{\beta}(\mathcal N_n)=0$.
In the case of $\mathcal{N}^c_n$, we verify \eqref{Coulomb eq 123} as a consequence of Lemma~\ref{l:add2}. Therefore all we need to check is \eqref{c:add1}, \eqref{c:add2}, and \eqref{c:add3} for Coulomb gas ensemble. First note that \eqref{c:add2} is obtained from the construction of $\mathcal{N}_n$. To establish \eqref{c:add1}, notice that
$$ \log_+ |\zeta| e^{-\beta n Q(\zeta)} \le e^{-\beta n Q_0(\zeta)}, $$
where $Q_0(\zeta) = Q(\zeta) - |\zeta|$. Then by the assumption \textbf{(A1)}-(ii), the partition function of the Coulomb gas with potential $Q_0$ is also finite, which implies \eqref{c:add2}.
Finally, \eqref{c:add3} follows from the well-known tightness of Coulomb gas ensemble, see e.g., \cite[Theorem 1.12]{CHM17}.
Combining these, we obtain that
$$\lim_{n \to \infty} \mathbf{P}(|L_n^k(z)| < e^{-n\varepsilon} ; \mathcal{N}_n^c) = 0. $$
Therefore we conclude
$$ \lim_{n \to \infty} \mathbf{P}(|L_n^k(z)| < e^{-n\varepsilon}) \le \lim_{n \to \infty} [\mathbf{P}(|L_n^k(z)| < e^{-n\varepsilon} ; \mathcal{N}_n^c) + \mathbf{P}(\mathcal N_n)] = 0,$$
which completes the proof.
\end{proof}
\section{Questions}
\begin{enumerate}
\item It is expected that the Theorem \ref{remove zero} hold for any probability measure $\mu$. Does the proof of Theorem \ref{remove zero} extend to the case when the limiting measure $\mu$ has atoms as well?
\item In the spirit of \cite{hanin1,hanin2}, one may ask if it is possible to show that most of the zeros have a critical point with in a distance of $O(\frac{1}{n})$. Establishing this would imply a natural pairing between zeros and critical points. Once the pairing is established, it will be of interest to study how the sum of pairwise distances (matching distance) behaves with $n$. This was studied in the case when all the zeros are real in \cite[Chapter-4]{Re16a}.
\item Extending the previous question, it is pertinent to ask about the density of the distances between zeros and critical points in the scale of $\frac{1}{n}$. A particular case being the study of critical points of characteristic polynomial of Haar distributed unitary matrix. In this case one can notice that the law of zeros and hence for the critical points is rotationally invariant. The characteristic polynomial of CUE random matrix is believed to model Riemann zeta function and this problem is of interest in the study of critical points of Riemann zeta function. For more on this problem see \cite{CUE_Riemann_zeta} and references there in. \\
\end{enumerate}
\noindent\textbf{Acknowledgments:} The authors would like to thank the organizers of the Second ZiF Summer School on Randomness in Physics and Mathematics, held at Bielefeld in August 2016, where this work was initiated.
\bibliographystyle{abbrv}
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Q: To improve my productivity: how to lock the /etc/hosts file on Linux with a passcode? I would like to lock the /etc/hosts file somehow in a way that only someone else can unlock it, possibly using a lock code.
I would then give the passcode to someone else.
I'm running Ubuntu 10.10.
A: Create an account for the other person and add it to the admin group (gives sudo access). Take yourself out of the admin group, so you can't use sudo.
A: As far as I know, Pass codes on files nearly always (well every time I have seen) uses third party software in order to encrypt the file.
I do not know of any software that will accomplish what you want as the file needs to be read normally by the OS. If you were to some how encrypt it, then someone else could always just delete it and recreate it.
What you want to do is to look in to standard file level protection and only give certain groups access to the file - such as Root/Administrators. You can read up about Linux permissions using chmod here.
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} | 8,033 |
{"url":"https:\/\/www.nag.com\/numeric\/py\/nagdoc_latest\/naginterfaces.library.blas.dsysrc.html","text":"# naginterfaces.library.blas.dsysrc\u00b6\n\nnaginterfaces.library.blas.dsysrc(uplo, pivot, direct, k1, k2, c, s, a)[source]\n\ndsysrc performs an orthogonal similarity transformation (as a sequence of plane rotations) of a real symmetric matrix.\n\nFor full information please refer to the NAG Library document for f06qm\n\nhttps:\/\/www.nag.com\/numeric\/nl\/nagdoc_28.7\/flhtml\/f06\/f06qmf.html\n\nParameters\nuplostr, length 1\n\nSpecifies whether the upper or lower triangular part of is stored.\n\nThe upper triangular part of is stored.\n\nThe lower triangular part of is stored.\n\npivotstr, length 1\n\nSpecifies the plane rotated by .\n\n(variable pivot)\n\nrotates the plane.\n\n(top pivot)\n\nrotates the plane.\n\n(bottom pivot)\n\nrotates the plane.\n\ndirectstr, length 1\n\nSpecifies the sequence direction.\n\n(forward sequence)\n\n.\n\n(backward sequence)\n\n.\n\nk1int\n\nThe value .\n\nk2int\n\nThe value .\n\ncfloat, array-like, shape\n\nmust hold , the cosine of the rotation , for .\n\nsfloat, array-like, shape\n\nmust hold , the sine of the rotation , for .\n\nafloat, array-like, shape\n\nThe symmetric matrix .\n\nReturns\nafloat, ndarray, shape\n\nThe transformed matrix .\n\nRaises\nNagValueError\n(errno )\n\nOn entry, error in parameter .\n\nConstraint: or .\n\n(errno )\n\nOn entry, error in parameter .\n\nConstraint: , or .\n\n(errno )\n\nOn entry, error in parameter .\n\nConstraint: or .\n\n(errno )\n\nOn entry, error in parameter .\n\nConstraint: .\n\nNotes\n\nNo equivalent traditional C interface for this routine exists in the NAG Library.\n\ndsysrc performs the transformation\n\nwhere is an real symmetric matrix, and is a real orthogonal matrix defined as a sequence of plane rotations, , applied in planes to .\n\nThe plane rotation part of is assumed to have the form","date":"2023-01-30 15:07:58","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 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\": 0.8886542916297913, \"perplexity\": 11289.150262141942}, \"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-2023-06\/segments\/1674764499819.32\/warc\/CC-MAIN-20230130133622-20230130163622-00002.warc.gz\"}"} | null | null |
Rollin' and Tumblin' (ursprünglich Roll and Tumble Blues) ist ein Bluessong, der von Hambone Willie Newbern geschrieben und von diesem erstmals 1929 aufgenommen wurde. Er wurde durch zahlreiche Coverversionen, etwa von Muddy Waters, innerhalb, aber auch außerhalb des Bluesgenres zu einem Klassiker und Standard. Während der British Invasion, bzw. der britischen Bluesrock-Explosion der 1960er-Jahre wurde er von vielen rockorientierten Bands wie etwa Cream oder Canned Heat live und im Studio gespielt.
Ursprünge
Die Originalfassung des Stückes wurde 1929 von Hambone Willie Newbern als Roll and Tumble Blues aufgenommen. Die Aufnahme fand am 14. März 1929 bei Okeh Records statt. Der Song scheint seinen Ursprung bei Minglewood Blues von Gus Cannon zu finden.
Newberns Originalfassung ist ein Bluessong in A-Dur mit einem Standardreimschema. Der Text handelt von einer zerbrochenen Liebschaft. Der Song ist einer von nur sechs überlieferten Songs von Newbern, die allesamt in dieser einen Session im März 1929 aufgenommen wurden. Roll and Tumble Blues wurde so gut angenommen, dass er schon damals von vielen Delta-Blues-Musikern in ihr Repertoire übernommen wurde.
Resonanz in der Bluesmusik
Der Song wurde nicht nur von vielen Bluesmusikern gecovert, sondern auch bearbeitet, sodass viele Musiker dieser Zeit ihre eigene Version des Liedes präsentierten. Eine der berühmtesten Versionen von Rollin' and Tumblin' ist If I Had Possession over Judgement Day von Robert Johnson, welches offenkundig von Rollin' and Tumblin' inspiriert und beeinflusst wurde. Sleepy John Estes adaptierte Newberns Song mit Brownsville Blues und The Girl I Love, She Got Long Curly Hair gleich zweimal. Auch Sunnyland Slim (Goin' Back to Memphis), Charley Patton (Banty Blues), John Lee Hooker (Rollin' Blues), Willie Dixon (Down in the Bottom, aufgenommen von Howlin' Wolf) und Johnny Shines (Red Sun) wurden von dem Stück zu eigenen Songs angeregt.
Der eigentliche Song Rollin' and Tumblin' wurde in den 1950er-Jahren vor allem durch gleich zwei Versionen von Muddy Waters populär, der das Stück innerhalb von nur einem Monat für zwei Labels (Aristocrat und Parkway) mit unterschiedlichen Begleitmusikern aufnahm. Die Parkway-Fassung enthält u. a. Little Walter und Leroy Foster. Letzterem wurde 2020 mit dem Song ein Eintrag in der Blues Hall of Fame gewidmet.
Rollin' and Tumblin' als Standardstück im Bluesrock
In den frühen 1960er-Jahren mischten zuerst britische Bands wie Cream und Canned Heat Blues intensiv mit dem neuen Rock 'n' Roll. Eines der größten Idole dieser Bands war Muddy Waters, die Ikone des Chicago Blues. Den Bands waren Waters' Fassungen von Rollin' and Tumblin' bekannt, und aufgrund seines schnellen Rhythmus eignete sich das Lied gut für eine Mischung mit Rockmusik. Wiederum inspiriert von den Fassungen von Cream und Canned Heat nahmen rockorientierte Bands wie etwa Grateful Dead das Stück ebenfalls auf.
2004 nahm Eric Clapton, der bereits bei der Cream-Version mitgewirkt hatte und den Song auf dem Album Unplugged wieder spielte, drei Versionen der Johnson-Abwandlung If I Had Possesion Over Judgement Day für das Album Me and Mr. Johnson und die DVD Sessions for Robert J auf, die sich weltweit über zwei Millionen Mal verkauften. Während der Cream-Reunion-Konzerte 2005 wurde die Originalversion aufgeführt.
2006 wurde das Stück Gegenstand einer Kontroverse, als Bob Dylan es für sein Album Modern Times aufnahm und die Autorenschaft des Liedes für sich beanspruchte. Weil er unter anderem auch bei anderen Songs des Albums Textstellen aus Songtexten oder eben auch Gedichten anderer Autoren in seine Liedtexte einfließen ließ, ohne die ursprüngliche Quelle zu nennen, wurde Dylan des Plagiats beschuldigt. Tatsächlich ähnelt Dylans Arrangement von Rollin' and Tumblin' sehr den Fassungen von Muddy Waters, allerdings führt Dylan einen komplett neuen Text ein. In der Tat kann man Dylans Fassung zu den oben genannten Songs zählen, die von Rollin' and Tumblin' beeinflusst wurden, aber nicht zwingend ein direktes Cover darstellen. Der Unterschied zu etwa Red Sun von Johnny Shines liegt zum größten Teil darin, dass Dylan den Originaltitel beibehielt.
Weblinks
Vollständige Liste der Fassungen des Songs
Internetartikel über den Song
Einzelnachweise
Blues-Titel
Lied 1929
Lied in der Blues Hall of Fame
Muddy Waters
Eric-Clapton-Lied
Bob-Dylan-Lied | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 4,147 |
\section{Introduction}
We study here theoretical aspects of generation and amplification of microwave
(millimeter waves) radiation by traveling wave tubes. Generally speaking,
generation and amplification of electromagnetic radiation can be produced by
enormous variety of devices of different designs depending on the frequency of
the radiation and its power. For light such devices are lasers; we remind that
the term laser is an acronym for Light Amplification by Stimulated Emission of
Radiation. For microwaves, which are of our special interest here, there is a
large class of amplifying devices including maser, a predecessor of the laser,
magnetrons, klystrons, traveling wave tubes, crossed-field amplifiers and gyrotrons.
In the case of lasers, as suggested by its very name, the general principle of
the amplification is based on the stimulated emission resulting from certain
atomic transitions. "Lasers come in a great variety of forms, using many
different laser materials, many different atomic systems, and many different
kinds of pumping or excitation techniques. The beams of radiation that lasers
emit or amplify have remarkable properties of directionality, spectral purity,
and intensity.", \cite[p. 2]{Siegman}. An important and defining property of
laser radiation is its coherency, that is its monochromaticity. For
amplification the coherency means that in a narrow frequency band the output
signal, after being amplified, reproduces pretty accurately the shape of the
input signal but with a substantial increase in amplitude. Coherent
amplification combined with a feedback allows to produce highly directional
and highly monochromatic beams. Observe that atomic transitions of the laser
medium constitute a fundamental basis of amplification, that is the
amplification mechanism is fixed by the nature, so to speak. There is extended
literature on the theory of lasers, see for instance \cite[4.1]{Fox},
\cite{Loudon}, \cite{Siegman}. Its basic phenomenological elements include:
(i) Einstein's treatment of the spontaneous and stimulated emission,
\cite[4.1]{Fox}, and (ii) operation principle based on interaction between the
laser (gain) medium and electromagnetic modes of a cavity containing this
medium. More detailed and fundamental theory that can justify the laser
phenomenology involves quantum optics (electronics), \cite{Loudon}, \cite{Fox}.
In the case of microwaves the radiation is produced by microwave vacuum
electronic devices, known formerly as \emph{microwave tubes}. These devices
use free electrons in a vacuum to convert energy from a DC power source to an
RF (radio frequency) signal. In other words, as a result of interaction
between the electron beam and properly designed structure the kinetic energy
of the electrons is converted into electromagnetic energy stored in the field,
\cite{Gilm1}, \cite[2.2]{Nusinovich}, \cite[4]{SchaB}, \cite{Tsimring}.
\emph{The key operational principle of any microwave device is a positive
feedback interaction between coherent radiation by electrons radiating in
phase on one hand and on the other hand electron bunching caused by radiation
on the stream of electrons. The electron bunching associated with acceleration
and deceleration of groups of electrons along the beam constitutes the
physical mechanism of radiation generation and its amplification.}
An important class of microwave devices uses as its operation principle the
\emph{Cherenkov radiation} generated by charged particles propagating in or
near a medium supporting slow waves with phase velocity comparable with the
particle velocity. Traveling wave tubes, the main subject of our studies here,
belongs to this class.
Traveling wave tubes (TWT) are used widely in many areas including satellite
communication and radar systems. Typical TWT consists of an elongated vacuum
tube containing an electron beam which passes down the middle of an RF circuit
(a \emph{slow-wave structure}). The operation principle of a TWT is as
follows. At one end of the TWT structure, the RF circuit is fed with a
low-powered radio signal to be amplified. As the RF signal travels along the
tube at near the same speed as the electron beam, the electromagnetic field
acts upon the beam and causes electron bunching with consequent formation of
the so-called \emph{space-charge wave}. The electromagnetic field associated
with the space-charge wave induces more current back into the RF circuit, thus
enhancing the bunching, and so on. The EM field thus builds up and is
amplified as it passes down the structure until a saturation regime is reached
and a large RF signal is collected at the output. The role of the slow-wave
structure is to slow down the electromagnetic wave to match up with the
velocity of the electrons in the beam, usually a small fraction of the speed
of light. Such a synchronism is required for effective in phase interaction
between the structure and the beam with optimal extraction of the kinetic
energy of the electrons. A typical slow-wave structure is the helix, which
reduces the speed of propagation according to its pitch. Further details on
the design and operation of TWT can be found in \cite{Gilm1}, \cite{PierTWT},
\cite{Tsimring}, \cite[4]{Nusinovich}.
An effective mathematical model for a TWT interacting with an electron beam
was introduced by J. R. Pierce, \cite{PierTWT}, \cite[I]{Pier51}. This model
is the simplest one that accounts for wave amplification along the structure,
energy extraction from the electron beam and its conversion into microwave
radiation in the TWT, see also \cite[4]{SchaB}, \cite{Gilm1}, \cite{Gilm},
\cite{Tsimring} and \cite[4]{Nusinovich}. In Section \ref{SectionPierce}, we
provide for precise description of the model as presented in \cite[I]{Pier51}.
The mentioned presentation is a time domain model, in contrast to other
presentations dealing with the frequency domain counterpart. Though simple,
the Pierce model allows for adequate estimates of the gain and it was used
effectively in designing working TWTs in the fifties. This model captures
remarkably well significant features of wave amplification and the beam-wave
energy transfer, and is still in use for basic design estimates.
The model presented by Pierce is one-dimensional and\ consists of (i) an ideal
linear representation of the electron beam and (ii) a lossless transmission
line (TL) representing the waveguide structure. The transmission line is
assumed to be homogeneous, that is, with uniformly distributed capacitance and
inductance. To overcome the Pierce theory limitations far more sophisticated
nonlinear theories have been developed to model very involved physics of the
electron beam and slow-wave structures, \cite{SchaB}, \cite{Gilm},
\cite{Tsimring}. Needless to say that those theories are far more complex and
often require a massive computer work.
In this paper we advance the Pierce theory to a theory that, while keeping its
simplicity and constructiveness, allows for more complex slow-wave structures.
We start by developing a Lagrangian field framework for the original Pierce
model. Such framework allows for extension of the model in two directions:
\textit{a)} we can replace the transmission line by a multi-transmission line
(MTL) and \textit{b)} we can dispense with the homogeneity assumption, thus
considering general nonhomogeneous systems consisting of a multi-transmission
line (MTL) coupled to an electron beam. We refer to such a system as a MTLB
system. Extension to multiple transmission lines is motivated by the fact that
general MTLs can approximate with desired accuracy real waveguided structures
which can be homogeneous (uniform) as well as inhomogeneous (nonuniform),
\cite{Nitsch}, \cite{Paul}, \cite{SchwiE}.
One of the advantages of the Lagrangian formulation is that conservation laws
and explicit expressions for the conserved quantities and their fluxes can be
obtained at once from the Noether theorem. We would like to point out that
though conservation laws do follow from the Euler-Lagrange evolution equations
there is no systematic way to extract them from those equations. In addition
to that, since all the information about dynamics is encoded in the scalar
Lagrange function we can trace the amplification mechanisms and the properties
of the energy transfer from the electron beam to the microwave radiation to
certain terms in the Lagrangian density.
For homogeneous MTLB systems, we study the amplification phenomenon by
considering the exponentially growing eigenmodes and associated complex-valued
wave numbers for the field equations, just as in the original Pierce theory,
\cite{PierTWT}, \cite{PierW}, \cite{Pier51}. We provide also a rigorous proof
of the fact that, on the growing mode, the energy always flows in the expected
direction, \textit{i.e.} from the beam to the MTL. In this case, the
eigenmodes analysis can be carried out analytically, providing for explicit
expressions for their energy density and energy flux distributions as well as
sufficient conditions for the existence of amplification regimes (growing
modes). The analysis includes derivation of a special canonical form of the
dispersion relation having a remarkable feature: one of its two terms depends
only on the MTL, whereas another one depends only the beam parameters. Such a
special factorization and separation of variables simplifies the analysis significantly.
As to inhomogeneous MTLB systems, they are by far more involved compared to
homogeneous ones. In particular, for periodic MTLB systems the dispersion
relations are not polynomial and that requires to turn to the most general
form of the Floquet theory, \cite[II, III]{YakSta1}. For this general case we
provide the first step towards a systematic study, namely we transform the
Euler-Lagrange field equations\ into the canonical Hamiltonian form using
basics of the de Donder-Weyl theory, \cite[4.2]{Rund}. This particular
Hamiltonian form consists of a system of equations which is of first order in
the spatial variable, thus providing the basis for the most effective use of
the Floquet theory, \cite[II, III]{YakSta1} in the study of periodic
structures. Detailed development of the Floquet theory for periodic MTLB
system requires to overcome a number of technical difficulties and it is left
for future studies.
One of the features of the proposed here phenomenological approach is that it
captures the electron bunching as a physical mechanism of amplification in
some form. Consequently, our analysis is a valuable source of a solid
information on the electron bunching.
The structure of the paper is as follows: In Section \ref{SectMainResults} we
briefly summarize our main results. Section \ref{SectionPierce} is devoted to
the description of Pierce's model for beam-TL interaction as presented in
\cite[I]{Pier51}. Section \ref{SectLagrangian} deals with the Lagrangian
approach to the model, including generalizations to both non-homogeneous and
multiple transmission lines. In the following Section
\ref{SectAmplificationGeneral copy(1)}, we explore the amplification mechanism
in the MTLB system as linked to instabilities in the dynamics of the beam. The
appropriate mathematical setting, in particular the Hamiltonian structure of
the model aimed at the study of eigenmodes in the periodic case is the subject
of Section \ref{SectHamiltonian}. In Section \ref{AmplMTL-beam}, we focus on
the detailed study of growing modes for the homogeneous MTLB system. Section
\ref{EnergyConsEx} deals with the questions of \ general energy conservation
and energy transfer between the beam and the MTL on the growing mode. In
Section \ref{PierceRev copy(1)} we make apparent how our general approach
allows to easily recover some of the original Pierce's results.
Finally, in Section \ref{MathSubj} we collect some technically involved
subjects which have been deferred there to avoid distracting the reader from
the main flow of ideas.
\section{Main results\label{SectMainResults}}
One of the goals of this work is to identify the mathematical mechanism of
amplification in MLTB systems. This goal has been accomplished by the
construction of a Lagrangian field theory of MLTB systems that underlines
their physical properties. Leaving detailed developments of this theory to the
following sections we simply identify here the key term of the system
Lagrangian responsible for amplification. This term quite expectedly is
associated with the electron beam and is described by the following expressio
\begin{equation}
\mathcal{L}_{\mathrm{b}}=\frac{\xi}{2}\left( \partial_{t}q+u_{0}\partial
_{z}q\right) ^{2},\qquad\xi=\frac{4\pi}{\omega_{\mathrm{p}}^{2}\sigma}>0,
\label{potenergyterm
\end{equation}
where $t$ and $z$ are, respectively, time and longitudinal variable,
$q=q(t,z)$ is the charge ("smoothed-out jelly of charge", \cite[I]{Pier51})
flowing through the beam. $\sigma$ and $u_{0}$ stand, respectively, for the
cross section and the electron velocity and $\omega_{\mathrm{p}}$ is the
plasma frequency. According to the general theory, we can identify the kinetic
and potential energies of the beam by expanding the expression
(\ref{potenergyterm}), that is
\[
\mathcal{L}_{\mathrm{b}}=\frac{\xi}{2}\left( \partial_{t}q\right) ^{2}+\xi
u_{0}\partial_{t}q\partial_{z}q+\frac{\xi}{2}u_{0}^{2}\left( \partial
_{z}q\right) ^{2},
\]
where \emph{the potential energy of the beam }$-\frac{\xi}{2}\left(
u_{0}\partial_{z}q\right) ^{2}$\emph{ is a negative quantity. This is a
marked feature distinguishing MLTB from common oscillatory systems, in which
the potential energy is always positive}. \emph{The negative sign of this
potential energy term is ultimately responsible for system instability and
consequent amplification}.
Indeed, a typical oscillatory system has a positive potential energy
manifested in forces that move the system toward its equilibrium state. The
simplest examples are given by a linear mass-spring system or its electric
analog - a simple electric $LC$ oscillatory circuit. The corresponding
Lagrangians ar
\[
\mathcal{L}_{\mathrm{1}}(x,x^{\prime})=\frac{1}{2}mx^{\prime2}-\frac{1
{2}kx^{2};\qquad\mathcal{L}_{\mathrm{2}}(q,q^{\prime})=\frac{1}{2}Lq^{\prime
2}-\frac{1}{2C}q^{2},
\]
where $m$ is the mass of the point, $k$ is the elastic Hooke constant of the
spring and $q$ is the charge in the capacitor. Such forces result in a stable
motion with oscillatory energy transfer between its kinetic and potential
forms. A qualitatively different picture occurs when the potential energy is
negative, as in $\mathcal{L}_{\mathrm{b}}.$ In this case resulting forces move
the system away from the equilibrium at an exponentially growing rate. Such
situation corresponds to having a negative Hooke constant $k$ in
$\mathcal{L}_{\mathrm{1}}$ or a negative capacitance in $\mathcal{L
_{\mathrm{2}}$ above. Interestingly, Pierce has observed an effective negative
capacitance in his studies of a transmission line interacting with the
electron beam, \cite{Pier51}.
Another marked feature of the term $\mathcal{L}_{\mathrm{b}}$ in
(\ref{potenergyterm}) is its degeneration as quadatric form manifested as a
perfect square trinomial expression or, alternatively, as a precise gyrotropic
term. According to the general theory of unstable regimes, \cite{YakSta1},
this kind of degeneration is a necessary condition for instability arising
under proper perturbations. From the point of view of the second order partial
differential equation describing the beam dynamics this degeneracy is
manifested as parabolicity compared to hyperbolicity occurring for common wave motion.
The power and efficiency of the Lagrangian approach is further demonstrated by
an exhaustive analysis of amplification regimes for a general homogeneous MTLB
system, including precise conditions under which amplification takes place. In
particular, if $\ 0\leq v_{1}\leq v_{2}\leq...\leq v_{n}$ \ denote the
characteristic velocities of the MTL as an independent system, we show that
there is always an amplifying regime if $u_{0}\leq v_{1}$. If $u_{0}>v_{1}$,
we show that amplification occurs only for sufficiently small $\xi$ in
(\ref{potenergyterm}). We also provide a transparent form of the dispersion
relation for a general homogeneous MTLB system, including possible
degenerations, as well as an asymptotic analysis of the amplification factor
as $\ $the beam parameter $\xi$ defined in (\ref{potenergyterm}) becomes
arbitrarily small or large. The limits $\xi$ $\rightarrow0$ and $\xi
\rightarrow\infty$ correspond to high, respectively small electron density of
the beam. In \cite{Pier51}, Pierce deals with large values of $\xi,$ which
allows him to simplify the dispersion relation to an exactly solvable third
degree equation for the forward eigenmodes. We review Pierce's result in the
light of our approach.
Yet another benefit of our Lagrangian approach is an exhaustive analysis of
the energetic issues, including the overall energy conservation and energy
transfer between the MTL and the beam. This analysis yields explicit
expressions for the power $P_{\mathrm{B}\rightarrow\mathrm{MTL}}$ flowing from
the beam to the MTL for an exponentially growing solution of the for
\begin{equation}
Q(z,t)=\widehat{Q}\mathrm{e}^{-\mathrm{i(}\omega t-k_{0}z)},\qquad
q(z,t)=\widehat{q}\mathrm{e}^{-\mathrm{i(}\omega t-k_{0}z)},\RIfM@\expandafter\text@\else\expandafter\mbox\fi{\qquad
}\operatorname{Im}k_{0}<0,
\end{equation}
where $Q$ is the coordinate describing the MTL and $q$ is the one describing
the beam. Namely, the following formula hold
\begin{equation}
\left\langle P_{\mathrm{B}\rightarrow\mathrm{MTL}}\right\rangle (z)=-\left[
\omega\xi\left\vert k_{0}\right\vert ^{2}\left\vert \widehat{q}\right\vert
^{2}(\operatorname*{Re}v_{0}-u_{0})\operatorname{Im}v_{0}\right]
\mathrm{e}^{-2\left( \operatorname{Im}k_{0}\right) z},\RIfM@\expandafter\text@\else\expandafter\mbox\fi{\qquad
v_{0}=\frac{\omega}{k_{0}}. \label{IntroAmplFormula
\end{equation}
We show that in the above formula the constant in front of the exponential is
indeed \emph{positive,} meaning that the energy flows from the beam to the
MTL. Formula (\ref{IntroAmplFormula}) indicates also that the power
transferred to the MTL increases exponentially in the direction of the
electron flow. The opposite is true of the evanescent wave when the power
flows to the beam and decreases exponentially in the $+z$ direction.
\subsection{Negative potential energy and general gain media}
Looking at the above analysis we can identify two main features of the
Lagrangian providing for the amplification in the MTLB system. The first one
is the fact that the beam potential energy is negative and unbounded from
below. This feature of the electron beam Lagrangian clearly indicates that the
model is an ideal one with the negative potential energy term representing
effectively an inexhaustible source of energy. This energy can be converted
into another form of energy such as energy of electromagnetic radiation. Such
ideal model can be suitable for describing the amplification and gain up to
the point of saturation. The saturation can conceivably be modeled
phenomenologically by introducing an additional positive potential energy term
into the beam Lagrangian represented by a higher order polynomial with a small
coefficient. That would make the theory nonlinear, of course.
The second feature of the Lagrangian providing for amplification is a
particular degeneracy of the expression (\ref{potenergyterm}) for the
Lagrangian and its role in the system stability. More precisely, such term
makes the system unstable under proper perturbations, as discussed in detail
in Section \ref{ChgWave}. It is a well known fact from the Floquet theory of
periodic Hamiltonian systems that such degeneration is indeed necessary in
order to have unstable perturbations, \cite{YakSta1}
The association of the amplification and gain with a negative potential energy
term in a system Lagrangian can be a general way to model gain media.
Interestingly, the phenomenon of negative energy waves in inhomogeneous
plasmas is well known and understood at phenomenological level, see for
instance, \cite[7.7]{Bellan}, \cite[1.3]{Hasegawa}, \cite[3.1]{Melrose}. The
explanation provided in the cited references is essentially that in the
approximate phenomenological model the wave-energy density corresponds to the
change in the total system energy density in a more detailed theory. Such
negative energy waves typically occur when the system is near equilibrium with
a steady-state flow velocity and there exists a mode that reduces the average
kinetic energy of the particles to a value below the initial equilibrium
value. Importantly, concepts of negative energy waves and gain media are
intimately related to the instability.
It is instructive to compare and contrast the developed here approach for
modeling the gain medium by a negative potential energy term with the
conventional approach that represent the gain medium as a system with negative
absorption. As an important and relevant example of later let us consider
colisionless plasma in a weak external electric field $E=E_{0}e^{-i(\omega
t-kx)}$ described in \cite[3]{LiP}. The interactions in such a plasma are
non-local and consequently the plasma permittivity depends on the both on
$\omega$ and $k$ (the so-called spatial dispersion), and it has non-zero
imaginary part resulting in dissipation. The imaginary part of permittivity
and the energy dissipation are given respectively by formula
\begin{equation}
\epsilon^{\prime\prime}=-\frac{4\pi^{2}e^{2}m}{k^{2}}\left[ \frac{\partial
f}{\partial p}\right] _{v=\omega/k};\qquad Q=\frac{\omega}{8\pi
\epsilon^{\prime\prime}\left\vert E\right\vert ^{2}=-\left\vert E\right\vert
^{2}\frac{\pi me^{2}\omega}{2k^{2}}\left[ \frac{\partial f}{\partial
p}\right] _{v=\omega/k}, \label{LandauDis
\end{equation}
where $m,e$ are the electron mass (respectively charge) and $f$ is the
momentum distribution function of the stationary plasma. If the plasma is
isotropic (that is, the distribution function of momenta only depends on
$\left\vert p\right\vert $ or, in the one-dimensional case, is an even
function), it can be shown that $Q>0$, (\cite[3.30]{LiP}), and consequently
the plasma absorbs energy from the field, a phenomenon called \textsl{Landau
damping}. However, in the presence of anisotropy, the sign of $\left[
\frac{\partial f}{\partial p}\right] _{v=\omega/k}$ and hence that of $Q$
might be reversed yielding a net flow of energy from the electrons to the
field and providing an example of gain medium. It is intuitively clear from
(\ref{LandauDis}) that the net energy flux depends on the relative number of
electrons with the velocity larger/smaller than the phase velocity of the wave.
Main differences\ between our approach for modeling gain in the MTLB system
and the conventional approach for modeling gain in the plasma example
described above are as follows. The conventional approach is based
fundamentally on the concept of open system and the gain medium is not modeled
explicitly but rather by its effect on the system. In our approach the beam
interacting with the electric field form a conservative system and the gain
medium is modeled explicitly as the beam term in the system Lagrangian with a
negative potential component. Yet another difference is that, in the MTLB
system, the gain occurs for the space charge wave velocities larger or smaller
than the wave phase velocity.
In fact, a causal dissipative system can always be extended uniquely to a
properly constructed conservative system, \cite{FigSch1}, \cite{FigShi1},
\cite{FigSch2}. It is an interesting question then whether one can carry out
similar construction for the gain medium. Answering this question is not in
the scope of this paper but we intend to look at this subject in our future work.
\section{Pierce's model\label{SectionPierce}}
In \cite[I]{Pier51}, J.R. Pierce presented a linear, one-dimensional model for
the description of the \ interaction of an electron beam with a surrounding
waveguide. The model is based on the following assumptions.
\textbf{Assumption I}. \textit{The modulation of both the electron velocity
and the current on the beam (so called a.c. components) are small compared to
the average or unperturbed velocity and current}.
This assumption justifies the linearization of the equations around the
unperturbed regime. Let the total velocity of the electrons be $u_{0}+v,$
where $u_{0\RIfM@\expandafter\text@\else\expandafter\mbox\fi{ }}$is the average velocity and $v$ is a small perturbation.
Analogously, let $\rho_{0}+\rho$ be the total electron density (per unit
volume) where $\rho_{0}$ is the unperturbed density and $\rho$ is the
perturbation. Let $\sigma$ be the cross section of the beam. Then, the total
current flowing is $I_{T}=I_{0}+I_{\mathrm{b}}$, where $I_{0}=\sigma\rho
_{0}u_{0}$ is the d.c. current and the perturbation is given b
\begin{equation}
I_{\mathrm{b}}=\sigma\left( \rho u_{0}+v\rho_{0}+\rho v\right) .
\label{beam1
\end{equation}
Linearization around the d.c. regime gets rid of the term $\rho v,$ which is
quadratic in the perturbations. Thus we tak
\begin{equation}
I_{\mathrm{b}}=\sigma\left( \rho u_{0}+v\rho_{0}\right) \label{beam1bis
\end{equation}
in what follows. The linearized conservation of charge equation read
\begin{equation}
\frac{\partial\rho}{\partial t}+\frac{\partial i}{\partial z}=\frac
{\partial\rho}{\partial t}+\frac{1}{\sigma}\frac{\partial I_{\mathrm{b}
}{\partial z}=0, \label{beambis
\end{equation}
where $t$ represents time, $z$ is the longitudinal variable and $i$ is the
current density, $i=I_{\mathrm{b}}/\sigma$.
\textbf{Assumption II}. \textit{The beam is thought of as a continuous medium
(electron jelly) with no internal stress and a unique volumetric force acting
along it, namely the one resulting from the axial component of the electric
field associated to the signal on the waveguide.}
It is further assumed that the charge/mass ratio in the electron jelly is
precisely $e/m,$ $e=-\left\vert e\right\vert $ \ being the electron charge and
$m$ being the electron mass. Therefore, if $E=E_{z}$ \ is the axial component
of the field, the motion equation for the medium reads
\begin{equation}
\frac{\partial v}{\partial t}+(u_{0}+v)\frac{\partial v}{\partial z}=\frac
{e}{m}E, \label{beam3
\end{equation}
where, on the left-hand side, we have used the usual Eulerian expression for
the acceleration in terms of the velocity field $v(z,t).$ Upon linearization,
the term $v\frac{\partial v}{\partial z}$ is dropped, thus yieldin
\begin{equation}
\frac{\partial v}{\partial t}+u_{0}\frac{\partial v}{\partial z}=\frac{e}{m}E.
\label{beam3bis
\end{equation}
Notice that in Pierce's original paper, \cite[I]{Pier51}, the charge of the
electron is denoted by $-e,$ whereas here it is just $e$.
Actually, the full blown Pierce model, as presented in the book \cite{PierTWT
, also includes the effect of electron-electron repulsion in the beam (so
called space charge effects); see also \cite{Gilm},\cite{Tsimring}. Here we do
not include such effect for the sake of simplicity, but we advance that this
can be done and we plan to report on this issue in the future.
Taking the derivatives of (\ref{beam1bis}) with respect to $t$ and $z$ \ we
obtain the following expressions for $\partial v/\partial t$ and $\partial
v/\partial z:
\begin{equation}
\frac{\partial v}{\partial t}=\frac{1}{\sigma\rho_{0}}\frac{\partial I_{b
}{\partial t}-\frac{u_{0}}{\rho_{0}}\frac{\partial\rho}{\partial t
;\qquad\frac{\partial v}{\partial z}=\frac{1}{\sigma\rho_{0}}\frac{\partial
I_{b}}{\partial z}-\frac{u_{0}}{\rho_{0}}\frac{\partial\rho}{\partial z}.
\label{beamfdef
\end{equation}
We use (\ref{beambis}) \ to express $\partial\rho/\partial t$ in terms of
$\partial I_{b}/\partial z$ in the first of the above relations and
differentiate the resulting relation with respect to $t$ thus yieldin
\begin{equation}
\frac{\partial^{2}v}{\partial t^{2}}=\frac{1}{\sigma\rho_{0}}\frac
{\partial^{2}I_{b}}{\partial t^{2}}+\frac{u_{0}}{\sigma\rho_{0}}\frac
{\partial^{2}I_{b}}{\partial z\partial t}. \label{equpierce1
\end{equation}
Next, we differentiate the second relation in (\ref{beamfdef}) with respect to
$t,$ expressing again $\partial\rho/\partial t$ in terms of $\partial
I_{b}/\partial z$ . We obtai
\begin{equation}
\frac{\partial^{2}v}{\partial z\partial t}=\frac{1}{\sigma\rho_{0}
\frac{\partial^{2}I_{b}}{\partial z\partial t}+\frac{u_{0}}{\sigma\rho_{0
}\frac{\partial^{2}I_{b}}{\partial z^{2}}. \label{equpierce2
\end{equation}
On the other hand, differentiating (\ref{beam3bis}) with respect to $t$ \ we
ge
\begin{equation}
\frac{\partial^{2}v}{\partial t^{2}}+u_{0}\frac{\partial^{2}v}{\partial
z\partial t}=\frac{e}{m}\frac{\partial E}{\partial t}. \label{equpierce3
\end{equation}
Finally, we replace the second derivatives in (\ref{equpierce3}) through their
expressions in (\ref{equpierce1}) and (\ref{equpierce2}), yielding a second
order equation for the beam curren
\begin{equation}
\partial_{t}^{2}I_{\mathrm{b}}+2u_{0}\partial_{t}\partial_{z}I_{\mathrm{b
}+u_{0}^{2}\partial_{z}^{2}I_{\mathrm{b}}=\sigma\frac{e}{m}\rho_{0
\partial_{t}E \label{beam4
\end{equation}
(here and in what follows, we use $\partial_{t}^{2}$ for $\partial/\partial
t^{2},$ $\partial_{z}^{2}$ for $\partial/\partial z^{2},$ etc. for the sake of
brevity). Next, Pierce considers the reciprocal action of the electron beam on
the transmission line (TL).
\textbf{Assumption III}. \textit{The action of the beam onto the waveguide
amounts to a shunt current instantaneously induced on the line. This current
is equal in absolute value and opposite to the current on the beam.}
According to this assumption, the usual transmission line (telegraph)
equations are modified so as to include an additional source term,
\cite[I]{Pier51},
\begin{equation}
\partial_{z}I=-C\partial_{t}V-\partial_{z}I_{\mathrm{b}},\qquad\partial
_{z}V=-L\partial_{t}I.\label{traneq1
\end{equation}
Here, as usual, $I=I\left( t,z\right) $ and $V=V\left( t,z\right) $ denote
respectively the current through the inductive element and the voltage on the
shunt capacitive element of the TL, $C>0$ and $L>0$ are respectively the shunt
capacitance and inductance per unit of length. Note also that in the equations
(\ref{traneq1}) $\partial_{z}I$ and $\partial_{z}V$ are respectively the
current through the shunt capacitive element and the voltage drop on the
inductive element of the TL per unit length. The addition of the source term
$-\partial_{z}I_{\mathrm{b}}$ can be justified under the assumption of
quasi-stationarity of the process: the charge wave on the beam "mirrors" onto
the line. One of the lumped elements in the discretization of such excited TL
is represented in Fig. \ref{Circuit}. Induced current can be thought of as a
distributed shunt current source
\begin{figure}[ptb
\centering
\ifcase\msipdfoutput
\includegraphics[
height=2.2935in,
width=4.2843in
{CircuitoNuevo.eps
\else
\includegraphics[
height=2.2935in,
width=4.2843in
{D:/alatex/Preparation/Reyes/lagsys/arxiv2/graphics/CircuitoNuevo__1.pdf
\fi
\caption{Discrete element of the TL-beam system in Pierce's model. The arrows
represent shunt current induced on the capacitor.
\label{Circuit
\end{figure}
The axial component of the electric field associated to the waveguide is
related to the TL voltage:
\begin{equation}
E\left( t,z\right) =-\partial_{z}V\left( t,z\right) . \label{traneq3
\end{equation}
Plugging the above expression into (\ref{beam4}), we arrive at the equatio
\begin{equation}
\partial_{t}^{2}I_{\mathrm{b}}+2u_{0}\partial_{t}\partial_{z}I_{\mathrm{b
}+u_{0}^{2}\partial_{z}^{2}I_{\mathrm{b}}=-\sigma\frac{e}{m}\rho_{0
\partial_{t}\partial_{z}V. \label{traneq4
\end{equation}
Thus, according to \cite[I]{Pier51}, the equations (\ref{traneq1}) and
(\ref{traneq4}) constitute a model of the interactive TL-beam (TLB) system.
Some comments are in order. In more recent literature, improved versions of
the linear Pierce model have been considered, see e.g. \cite[4]{Nusinovich}.
These versions account for finer features such as bunching saturation, or
retain the nonlinearity present in the original versions of equations
(\ref{beam1}) and (\ref{beam3}), etc. Although such enriched models are
undoubtedly more realistic and numerical computations based on them might
provide a better agreement with experiment, they hardly allow for analytical
treatment. In particular, they do not possess a Lagrangian structure. Pierce's
model, though simple, already captures the mechanism of amplification and, as
mentioned in the Introduction, can be generalized to the case of MTLB systems,
and allows for a thorough mathematical analysis in all cases. Taking into
account the fact that real wave guides can be approximated, in principle, by
an MTL with any degree of accuracy, \cite{Nitsch}, \cite{Paul}, \cite{SchwiE},
such generalization opens new perspectives in design optimization, which is
the ultimate goal of our study.
\section{Lagrangian formulation of Pierce's model\label{SectLagrangian}}
In this section we construct a Lagrangian field theory underlying the Pierce
model. The Lagrangian theory provides a deeper insight into mathematical
mechanism of amplification and energy transfer from the electron beam to the radiation.
\subsection{The Lagrangian\label{Lagrangian}}
The linear system of equations (\ref{traneq1})-(\ref{traneq4}) arises as
Euler-Lagrange equations associated to certain quadratic Lagrangian. To see
this, let us first introduce the charge variables $Q$ and $q$ related
respectively to the currents $I$ and $I_{\mathrm{b}}$ b
\begin{equation}
I=\partial_{t}Q,\qquad I_{\mathrm{b}}=\partial_{t}q. \label{tranbe1
\end{equation}
Thus the variables $Q,q$ represent the amount of charge traversing the
cross-section of the line (respectively the beam) at the point $z$ within the
time interval $(t_{0},t),$ where $t_{0}$ is some fixed reference time. Then
the TLB system (\ref{traneq1}) and (\ref{traneq4}) takes the for
\begin{equation}
\partial_{z}Q=-CV-\partial_{z}q,\qquad\partial_{z}V=-L\partial_{t}^{2}Q,
\label{tranbe2
\end{equation
\begin{equation}
\left( \partial_{t}+u_{0}\partial_{z}\right) ^{2}q=-\frac{\sigma
\omega_{\mathrm{p}}^{2}}{4\pi}\partial_{z}V, \label{tranbe3
\end{equation}
where $\omega_{\mathrm{p}}$ is the \emph{plasma frequency} defined (in
Gaussian units) b
\begin{equation}
\omega_{\mathrm{p}}^{2}=\frac{4\pi e\rho_{0}}{m}, \label{tranbe4
\end{equation}
\cite[2.2]{DavNP}.
Since it is not any harder to deal with inhomogeneous (in particular,
periodic) TLs, we suppose from now on that $C$ and $L$ can be position
dependent, that i
\begin{equation}
C=C\left( z\right) ,\qquad L=L\left( z\right) . \label{tranbe5
\end{equation}
Notice that the first equation in (\ref{tranbe2}) readily implies the
following representation for $V$
\begin{equation}
V=-C^{-1}\partial_{z}(Q+q). \label{tranbe6
\end{equation}
Inserting the above expression for $V$ into the second equation in
(\ref{tranbe2}) and into the equation (\ref{tranbe3}) yield the following TLB
evolution equations for the charges:
\begin{equation}
L\partial_{t}^{2}Q-\partial_{z}\left[ C^{-1}\partial_{z}\right] \left(
Q+q\right) =0, \label{tranbe7
\end{equation
\begin{equation}
\xi\left( \partial_{t}+u_{0}\partial_{z}\right) ^{2}q-\partial_{z}\left[
C^{-1}\partial_{z}\right] \left( Q+q\right) =0,\quad\xi=\frac{4\pi
{\omega_{\mathrm{p}}^{2}\sigma}=\frac{m}{\sigma e\rho_{0}}>0. \label{tranbe8
\end{equation}
We observe now that the above evolution equations are the Euler-Lagrange
equations for the following Lagrangia
\begin{equation}
\mathcal{L(}z,\partial_{t}Q,\partial_{z}Q,\partial_{t}q,\partial
_{z}q\mathcal{)}=\frac{L}{2}\left( \partial_{t}Q\right) ^{2}-\frac{1
{2}C^{-1}\left( \partial_{z}Q+\partial_{z}q\right) ^{2}+\frac{\xi}{2}\left(
\partial_{t}q+u_{0}\partial_{z}q\right) ^{2}. \label{tranbe9
\end{equation}
Indeed, for a general\ Lagrangian density $\mathcal{L}=\mathcal{L}\left(
t,z;Q,\partial_{t}Q,\partial_{z}Q;q,\partial_{t}q,\partial_{z}q\right) ,$ the
Euler-Lagrange equations take the for
\begin{equation}
\partial_{t}\frac{\partial\mathcal{L}}{\partial(\partial_{t}Q)}+\partial
_{z}\frac{\partial\mathcal{L}}{\partial\left( \partial_{z}Q\right)
-\frac{\partial\mathcal{L}}{\partial Q}=0,\RIfM@\expandafter\text@\else\expandafter\mbox\fi{ \ \ \ \ }\partial_{t
\frac{\partial\mathcal{L}}{\partial(\partial_{t}q)}+\partial_{z}\frac
{\partial\mathcal{L}}{\partial\left( \partial_{z}q\right) }-\frac
{\partial\mathcal{L}}{\partial q}=0\RIfM@\expandafter\text@\else\expandafter\mbox\fi{\ .} \label{tranbe10
\end{equation}
A straightforward computation confirms that the application of equation
(\ref{tranbe10}) to the Lagrangian defined by (\ref{tranbe9}) indeed yields
the TLB evolution equations (\ref{tranbe7}) and (\ref{tranbe8}). Pierce's
original equations are obtained as a particular case, when $C,$ $L$ are
constant along the line.
As to the units of $\mathcal{L}$, they are energy/length, as expected for a
Lagrangian density. In this respect, we remind that we are using Gaussian
units and charge$^{2}=$ force$\times$length$^{2}$, in agreement with the
Gaussian version of Coulomb's law, $F=q_{1}q_{2}/r^{2}.$
Let us make a final observation: \ It is assumed that the current induced by
the beam onto the TL is due to the fact that the charge on the beam perfectly
"mirrors" onto the waveguide. This assumption can be justified as an
approximation in the "quasistatic" regime, in the spirit of Ramo's Theorem,
\cite{Ra}, \cite{Tsimring}. According to some authors, e.g. R. Kompfner,
\cite{Kom} or J.H. Booske, \cite[4]{Nusinovich}, in dealing with real devices
a coefficient $\varkappa\in(0,1)$ must be included in front of $\partial
_{z}I_{b}$ in (\ref{traneq1}) (accordingly in (\ref{tranbe2})) to account for
the real induced current, the case $\varkappa=1$ being regarded as ideal. The
Lagrangian approach can easily handle the general case. However, in order to
keep the exposition as simple as possible, we only consider the ideal situation.
\subsection{Generalization to multiple transmission lines}
It is known that fairly general wave guides can be well approximated by
multiple transmission lines, MTL, \cite{Paul}. The corresponding
generalization of Pierce's model is straightforward thanks to our Lagrangian
formulation. Indeed, suppose that we have $n+1$ conductors, one of them being
grounded, say the $(n+1)$-th. \ We denote by $V(z,t)=\left\{ V_{i
(z,t)\right\} _{i=1\ldots n}$ the $n$-dimensional vector-column of voltages
\ on \ the first $n$ conductors with respect to the ground and \ by
$I(z,t)=\left\{ I_{i}(z,t)\right\} _{i=1\ldots n}$ the vector-column of
currents flowing on them and se
\[
Q(z,t)=\left\{ Q_{i}(z,t)\right\} _{i=1\ldots n},\RIfM@\expandafter\text@\else\expandafter\mbox\fi{ \qquad}Q_{i}(z,t)
{\displaystyle\int\limits^{t}}
I_{i}(z,s)ds.
\]
Let $L=L(z),$ $C=C(z)$ be the $n\times n$ matrices of self- and mutual
inductance and capacity. As it is well known, they are positive symmetric
(Hermitian). A natural generalization of (\ref{tranbe9}) is provided b
\begin{equation}
\mathcal{L}=\frac{1}{2}\left\{ \left( \partial_{t}Q,L\partial_{t}Q\right)
-\left( \partial_{z}Q+\partial_{z}qB,C^{-1}\left[ \partial_{z}Q+\partial
_{z}qB\right] \right) \right\} +\frac{\xi}{2}\left( \partial_{t
q+u_{0}\partial_{z}q\right) ^{2}, \label{mtraneq1
\end{equation}
where $($ $,$ $)$ stands for the scalar product in $\Re^{n}$ and $B$ is the
$n$-dimensional vector-column with all components being the unity, that is
\begin{equation}
B=\left( 1,1,\ldots1\right) ^{\mathrm{T}}. \label{mtraneq1a
\end{equation}
The corresponding Euler-Lagrange second order system i
\begin{gather}
L\partial_{t}^{2}Q-\partial_{z}\left[ C^{-1}(\partial_{z}Q+\partial
_{z}qB)\right] =0;\label{MTLEuler-Larange}\\
\xi\left[ \partial_{t}^{2}q+2u_{0}\partial_{t}\partial_{z}q+u_{0}^{2
\partial_{z}^{2}q\right] -\left( B^{\mathrm{T}},\partial_{z}\left[
C^{-1}(\partial_{z}Q+\partial_{z}qB)\right] \right) =0.\nonumber
\end{gather}
The generalized telegraph equations, equivalent to the first equation above,
adopt the for
\begin{equation}
\partial_{z}I=-C\partial_{t}V-\partial_{z}I_{b}B;\qquad\partial_{z
V=-L\partial_{t}I. \label{TelegMulti
\end{equation}
Our choice of the vector $B$ assumes, besides perfect induction, a symmetry in
the interaction between the beam and the different lines. A more realistic
approach might include coefficients $\varkappa_{i}\in(0,1)$ in vector $B$ to
account for non-symmetric interaction. As we already mentioned in Subsection
\ref{SectLagrangian}, such effects can be easily handled by our approach.
Observe that if we remove the beam\ from the system by setting $q=0,$ our
model \ is in full agreement with well established models for the interaction
of several lines, derived from Maxwell's equations under reasonable
assumptions. See, for example, \cite[2]{Nitsch}, \cite[1.4.1]{Paul} for models
of interacting TLs.
To summarize: from now on, by MTLB system we mean the field Lagrangian system
governed by the Lagrangian $\mathcal{L}$ in (\ref{mtraneq1}) and the
corresponding Euler-Lagrange field equations (\ref{MTLEuler-Larange}).
\section{The beam as a source of amplification. The role of instability
\label{SectAmplificationGeneral copy(1)}}
Evidently, the beam is the sole source of energy in the MTLB system and the
ultimate responsible for the presence of exponentially growing modes. In this
section we identify and analyze the mathematical mechanism underlying amplification.
To trace the amplification to the beam we view the Lagrangian (\ref{mtraneq1})
as a perturbation of the Lagrangian $\mathcal{L}_{\mathrm{b}}$ for the
isolated beam defined b
\begin{equation}
\mathcal{L}_{\mathrm{b}}=\frac{1}{2}\left( \partial_{t}q+u_{0}\partial
_{z}q\right) ^{2}=\frac{1}{2}\left[ \left( \partial_{t}q\right)
^{2}+2u_{0}\partial_{t}q\partial_{z}q+u_{0}^{2}\left( \partial_{z}q\right)
^{2}\right] . \label{Lagrbeam
\end{equation}
We introduce the equivalent Lagrangian $\widetilde{\mathcal{L}}=\frac{1}{\xi
}\mathcal{L}$, where $\mathcal{L}$ is as in (\ref{mtraneq1}),\textit{ i.e.
\begin{gather}
\widetilde{\mathcal{L}}=\mathcal{L}_{b}+\varepsilon\mathcal{L}^{\prime
=\frac{1}{2}\left( \partial_{t}q+u_{0}\partial_{z}q\right) ^{2
+\label{FullLagrangian}\\
+\frac{\varepsilon}{2}\left\{ \left( \partial_{t}Q,L\partial_{t}Q\right)
-\left( \partial_{z}Q+\partial_{z}qB,C^{-1}\left[ \partial_{z}Q+\partial
_{z}qB\right] \right) \right\} ,\nonumber
\end{gather}
and $\varepsilon=1/\xi$.\ Small values of $\xi$ defined by (\ref{tranbe8}) and
consequently large values $\varepsilon$ correspond to strong coupling and
regimes where the beam effectively feeds its energy into transmission lines in
the form of EM field. The EM field energy gain originates in the beam as an
infinite reservoir of the potential energy $-\frac{1}{2}(u_{0}\partial
_{z}q)^{2}$. Importantly, the potential energy is negative unlike in
oscillatory systems. For small coupling as we will show no energy transfer
might occur from the beam to the EM field. This perturbation analysis suggests
to consider first the beam as an isolated system.
\subsection{Charge wave dynamics\label{ChgWave}}
In this subsection, we investigate beam charge dynamics as an isolated system,
described by (\ref{Lagrbeam}). We already mentioned the role of the term
$u_{0}^{2}\left( \partial_{z}q\right) ^{2}$ as a source of energy. This term
is responsible for the system instability manifesting itself by exponentially
growing solutions of the associated E-L equations. The gyrotropic term
$u_{0}\partial_{t}q\partial_{z}q$ in the Lagrangian provides for stabilizing
effect. As we will see, for the Lagrangian (\ref{Lagrbeam}) the balance
between instability and stability is struck exactly in the margin. Namely, a
small perturbation of this Lagrangian can make the system either stable or unstable.
The beam Lagrangian $\mathcal{L}_{\mathrm{b}}$ is quadratic in $(\partial
_{t}q,\partial_{z}q)$, see \ Section \ref{AppQuadLag}, and has the following
structur
\begin{equation}
\mathcal{L}_{\mathrm{b}}=\frac{1}{2}\alpha(\partial_{t}q)^{2}+\theta
\partial_{t}q\partial_{z}q-\frac{1}{2}\eta(\partial_{z}q)^{2}=(\partial
_{t}q,\partial_{z}q)^{\mathrm{T}}M(\partial_{t}q,\partial_{z}q),
\label{beamlag1
\end{equation}
wher
\begin{equation}
M=\left[
\begin{array}
[c]{ll
\alpha & \theta\\
\theta & -\eta
\end{array}
\right] =\left[
\begin{array}
[c]{ll
1 & u_{0}\\
u_{0} & u_{0}^{2
\end{array}
\right] ,\RIfM@\expandafter\text@\else\expandafter\mbox\fi{ \ \ (thus }\alpha=1,\ \theta=u_{0},\ \eta=-u_{0}^{2}).
\label{beamlag2
\end{equation}
The corresponding Euler-Lagrange equation (\ref{qulaq4}) is
\begin{equation}
\left( \partial_{t}+u_{0}\partial_{z}\right) ^{2}q=0. \label{BeamEq
\end{equation}
Applying the general formulas (\ref{encoq4a})-(\ref{encoq4b}) for the energy
$H$ and its flux $S$ we obtai
\begin{equation}
H_{\mathrm{b}}\left[ q\right] =\frac{1}{2}\left( \partial_{t}q\right)
^{2}-\frac{u_{0}^{2}}{2}\left( \partial_{z}q\right) ^{2}, \label{BeamEnergy
\end{equation
\begin{equation}
S_{\mathrm{b}}\left[ q\right] =\partial_{t}q\left( u_{0}\partial_{t
q+u_{0}^{2}\partial_{z}q\right) =u_{0}\partial_{t}q\left( \partial
_{t}q+u_{0}\partial_{z}q\right) =u_{0}\left( \partial_{t}q\right)
^{2}+u_{0}^{2}\partial_{t}q\partial_{z}q. \label{BeamFlux
\end{equation}
Since $\mathcal{L}_{\mathrm{b}}$ does not depend on time explicitly,
conservation of energy takes place, (\ref{gblag5})
\begin{equation}
\frac{\partial H_{\mathrm{b}}}{\partial t}+\frac{\partial S_{\mathrm{b}
}{\partial z}=0. \label{consenergy
\end{equation}
\subsubsection{Eigenmodes and stability issues.}
Since the beam parameters are constant in space we can make use of the
dispersion relation to study the eigenmodes. Thus, if we try solutions of the
form $q(z,t)=\mathrm{e}^{-\mathrm{i}(\omega t-kz)}$ in (\ref{BeamEq}), we ge
\begin{equation}
\omega^{2}-2u_{0}\omega k+u_{0}^{2}k^{2}=\left( \omega-u_{0}k\right) ^{2}=0,
\label{disprelbeam
\end{equation}
hence $k_{\omega}=\omega/u_{0}$ is a double real root. The corresponding
eigenmodes are $q_{1}(z,t)=\mathrm{e}^{\mathrm{i}\left( k_{\omega}z-\omega
t\right) }$\ and $q_{2}(z,t)=z\mathrm{e}^{\mathrm{i}\left( k_{\omega
}z-\omega t\right) }$ or their real valued counterparts
\[
v_{1}(z,t)=\cos\left( kz-\omega t\right) ,\qquad v_{2}(z,t)=z\cos\left(
kz-\omega t\right) .
\]
The associated energy flux is, according to (\ref{BeamFlux})
\begin{equation}
S_{\mathrm{b}}\left[ v_{1}\right] =0,\qquad S_{\mathrm{b}}\left[
v_{2}\right] =-u_{0}^{2}z\omega\sin\left( kz-\omega t\right) \cos\left(
kz-\omega t\right) . \label{fluxformulas
\end{equation}
To make useful inference related to conservation laws it is common to use the
following\ time-averaging operation. Namely, for a (locally integrable)
function $f$ defined on $[0,\infty)$ we introduc
\begin{equation}
\left\langle f\right\rangle =\lim_{T\rightarrow\infty}\frac{1}{T
{\displaystyle\int_{0}^{T}}
f(t)\,\mathrm{d}t. \label{encoq5
\end{equation}
This time-averaging operation has the following properties. If $f$ is a smooth
and bounded function on $[0,\infty)$, the
\begin{equation}
\left\langle \frac{df}{dt}\right\rangle =\lim_{T\rightarrow\infty}\frac{1}{T
{\displaystyle\int_{0}^{T}}
\frac{df}{dt}\,\mathrm{d}t=\lim_{T\rightarrow\infty}\frac{1}{T}\left[
f(T)-f(0)\right] =0. \label{prop1ave
\end{equation}
Differentiation with respect to parameters commutes with the time-averaging
operation. Namely, if $f$ also depends (smoothly) on some parameter $z$ the
following identity hold
\begin{equation}
\left\langle \partial_{z}f\right\rangle =\partial_{z}\left\langle
f\right\rangle . \label{prop2ave
\end{equation}
Taking time average on both sides \ of the conservation law (\ref{consenergy})
and using the above properties of averaging, we conclude that
\[
\left\langle S_{\mathrm{b}}\left[ v_{2}\right] \right\rangle
(z)=\mathrm{const}.
\]
On the other hand, it \ follows from (\ref{fluxformulas}) that $\left\langle
S_{\mathrm{b}}\left[ v_{2}\right] \right\rangle (0)=0$. Hence $\left\langle
S_{\mathrm{b}}\left[ v_{2}\right] \right\rangle (z)=0.$
From the stability point of view, this situation is a very degenerate one. To
illustrate this point, let us introduce a special form perturbation in the
beam dispersion relation (\ref{disprelbeam})
\[
\omega^{2}-2\alpha u_{0}\omega k+u_{0}^{2}k^{2}=0\RIfM@\expandafter\text@\else\expandafter\mbox\fi{\quad or\quad}\left(
\omega-\alpha u_{0}k\right) ^{2}=\left( \alpha^{2}-1\right) u_{0}^{2
k^{2},
\]
where $\alpha$ is a real number. Our situation corresponds to $\alpha=1.$ Let
us consider the behavior close to $\alpha=1$. The quadratic equation above has
the following solution
\[
k_{\omega}(\alpha)=\frac{\omega}{u_{0}}(\alpha\pm\sqrt{\alpha^{2}-1}).
\]
Notice that if $\alpha^{2}<1$ the above solutions become complex conjugate,
whereas if $\alpha^{2}>1$ they are real distinct. $\alpha=1$ \ corresponds to
a double real solution, already showing the degeneracy.
An important subject of our interest is the analysis of MTL structures in
which the parameters vary periodically in $z$. The Floquet theory and, in
particular, the Floquet multipliers are the mathematical objects that deal
with such situations \textit{par excellence}. As explained above, we may
consider the coupled system as a perturbation of the beam. Consequently, it is
instructive to take a look at the isolated beam in the light of Floquet theory
with arbitrary period (eventually dictated by the period of the structure).
The Floquet multipliers with period unity are $\rho_{\omega}(\alpha
)=\mathrm{e}^{\mathrm{i}k_{\omega}(\alpha)}$. It is clear that in the case
$\alpha^{2}<1$ they are symmetrically located with respect to the unit circle
in the complex plane. The solution corresponding to the multiplier outside the
circle is a growing wave, whereas the one corresponding to the multiplier
inside the circle is an evanescent one. In the opposite case $\alpha^{2}>1$,
both roots are located on the unit circle, and the corresponding modes are
purely oscillatory. We refer to these two qualitatively different
perturbations as respectively unstable and stable. Aiming at amplification by
coupling the beam to a MTL, the unstable situation is the one to be favoured.
The special perturbation of the beam equation considered above was for
illustration purposes to see the degenerate stability properties of the system
under perturbation of its parameters. For the MTLB system, however, it is the
MTL the one that plays the role of perturbation.\ In Section
\ref{AmplMTL-beam}, we prove that the desired instability and resulting
amplification for spatially homogeneous MTL is achieved by sufficiently strong
coupling (small values of $\xi$). An extension of this result to the case of
periodic MTLs is left for a forthcoming publication.
Additional insight into the mathematical mechanism of amplification associated
with instability can be gained by looking at the nature of the partial
differential equations involved. Indeed, the equatio
\begin{equation}
\left( \partial_{t}^{2}+2\alpha u_{0}\partial_{t}\partial_{z}+u_{0
^{2}\partial_{z}^{2}\right) q=0
\end{equation}
is\textbf{\ }hyperbolic if $\alpha^{2}>1$. In this case, there are two
propagation velocities $v^{\pm}(\alpha)$ of the same sign, namel
\[
v^{\pm}(\alpha)=\frac{\omega}{k_{\omega}^{\pm}(\alpha)}=-u_{0}(\alpha\mp
\sqrt{\alpha^{2}-1}),
\]
and the general solution has the for
\[
q(z,t)=q_{1}(z-v^{+}t)+q_{2}(z-v^{-}t).
\]
Therefore, any solution which is bounded in time (as it is the case for
harmonic in time solutions) is automatically bounded in space. In other words,
no harmonic in time regime can be exponentially growing in space.
\ In the critical case, $\alpha=1,$ the equation is of the parabolic type.
Changing variables $(z,t)\rightarrow(\xi,\eta)$ with $\xi=x-u_{0}t$,
$\eta=ax+bt$ with $b+au_{0}\neq0,$ it can be easily checked that the general
solution in this case i
\[
q(z,t)=zF(z-u_{0}t)+G(z-u_{0}t)=t\widetilde{F}(z-u_{0}t)+\widetilde{G
(z-u_{0}t),
\]
where $F,G,\widetilde{F},\widetilde{G}$ \ are arbitrary functions. In
particular any travelling wave with velocity $u_{0}$ is a solution. Again here
we see that bounded in time dependence can be accompanied by at most linear
growth in space.
If $\alpha^{2}<1$ we are dealing with the elliptic case where there is no
propagation. This is the only case allowing for exponential amplification.
Indeed, a linear change of variables $(z,t)\rightarrow(\xi=az+bt,\eta=cz+dt)$
transforms the equation into the Laplace equatio
\[
u_{\xi\xi}+u_{\eta\eta}=0,
\]
which admits real solutions of the form $u(\xi,\eta)=\mathrm{e}^{k\xi
\cos(\omega\eta)$, $u(\xi,\eta)=e^{k\xi}\sin(\omega\eta),$ etc.
\section{Hamiltonian structure of the MTLB system\label{SectHamiltonian}}
In order to study the MTLB system, in particular the associated modes, their
stability and the amplification phenomenon, we make use of the Hamiltonian
structure associated to the Lagrangian (\ref{FullLagrangian}). More precisely,
we use a version of Hamiltonian formalism that treats the space and time
variables on the same footing, known as de Donder-Weyl formalism. For reader's
\ convenience, we have gathered the basic information about this topic in
Section \ref{AppdeDonder}. As usual, \ the passage from Lagrangian to
Hamiltonian point of view allows to cast the second-order Euler-Lagrange
system of equations (\ref{MTLEuler-Larange}) in the form of a first order
system, either with respect to $t$ or with respect to $z.$
To comply with notations of Section \ref{AppdeDonder}, from now on we put
$\mathsf{q}_{1}=Q$, $\mathsf{q}_{2}=q$, $\mathsf{q}=(\mathsf{q}_{1
,\mathsf{q}_{2})^{^{\mathrm{T}}}$. The Lagrangian$\ \widetilde{\mathcal{L}}$
in (\ref{FullLagrangian}) is quadratic in its variables $\left( \partial
_{t}Q,\partial_{t}q,\partial_{z}Q,\partial_{z}q\right) $, that is in $\left(
\partial_{t}\mathsf{q},\partial_{z}\mathsf{q}\right) ^{^{\mathrm{T}}}$ in the
new notation. Indeed
\begin{equation}
\widetilde{\mathcal{L}}=\frac{1}{2}\partial_{t}\mathsf{q}^{\mathrm{T}
\alpha\partial_{t}\mathsf{q}+\partial_{t}\mathsf{q}^{\mathrm{T}}\theta
\partial_{z}\mathsf{q}-\frac{1}{2}\partial_{z}\mathsf{q}^{\mathrm{T}
\eta\partial_{z}\mathsf{q}, \label{Lagqq1
\end{equation}
wher
\begin{equation}
\alpha=\left[
\begin{array}
[c]{cc
\varepsilon L & 0\\
0 & 1
\end{array}
\right] ,\RIfM@\expandafter\text@\else\expandafter\mbox\fi{\quad}\theta=\left[
\begin{array}
[c]{cc
0 & 0\\
0 & u_{0
\end{array}
\right] ,\RIfM@\expandafter\text@\else\expandafter\mbox\fi{\quad}\eta=\left[
\begin{array}
[c]{cc
\varepsilon C^{-1} & \varepsilon C^{-1}B\\
\varepsilon B^{\mathrm{T}}C^{-1} & \varepsilon B^{\mathrm{T}}C^{-1}B-u_{0}^{2
\end{array}
\right] \RIfM@\expandafter\text@\else\expandafter\mbox\fi{,} \label{Lagqq2
\end{equation}
or, using a block matrix
\begin{equation}
\widetilde{\mathcal{L}}=\frac{1}{2}\mathsf{u}^{\mathrm{T}}M_{\mathrm{L
}\mathsf{u},\RIfM@\expandafter\text@\else\expandafter\mbox\fi{\quad with}\qquad M_{\mathrm{L}}=\left[
\begin{array}
[c]{ll
\alpha & \theta\\
\theta & -\eta
\end{array}
\right] ,\qquad\mathsf{u}=\left[
\begin{array}
[c]{l
\partial_{t}\mathsf{q}\\
\partial_{z}\mathsf{q
\end{array}
\right] . \label{Lagqq3
\end{equation}
Let us introduce the vector of canonical momenta $\mathsf{p}=(\mathsf{p
_{t},\mathsf{p}_{z})^{^{\mathrm{T}}},$ related to the vector $\mathsf{u}$
above by means o
\begin{equation}
\mathsf{p}=M_{\mathrm{L}}\mathsf{u,} \label{momentum
\end{equation}
where $M_{\mathrm{L}}$ is as in \ref{Lagqq3}. In the following result, we
express the dynamics of our system in terms of the variables $\mathsf{p}_{z}$
and $\partial_{t}\mathsf{q}$.
\begin{theorem}
The second order Euler-Lagrange system (\ref{MTLEuler-Larange}) is equivalent
to the $2n-$first order system
\begin{equation}
\tilde{J}\partial_{z}V=\mathrm{i}\partial_{t}\tilde{M}V,\qquad V=\left[
\begin{array}
[c]{l
\mathsf{p}_{z}\\
\partial_{t}\mathsf{q
\end{array}
\right] , \label{HamEq1
\end{equation}
wher
\begin{equation}
\tilde{J}=\left[
\begin{array}
[c]{cc
0 & \mathrm{i}\mathbf{1}\\
\mathrm{i}\mathbf{1} & 0
\end{array}
\right] ,\qquad\tilde{M}=\tilde{M}\left( z\right) =\left[
\begin{array}
[c]{ll
-\eta\left( z\right) ^{-1} & \eta\left( z\right) ^{-1}\theta\\
\theta\eta\left( z\right) ^{-1} & -\alpha\left( z\right) -\theta
\eta\left( z\right) ^{-1}\theta
\end{array}
\right] . \label{aeta4
\end{equation}
\end{theorem}
\begin{proof}
The derivation of de Donder-Weyl version of Hamilton equations in the variable
$z$ for general quadratic Lagrangians is described in Section \ref{AppQuadLag
. In particular, for $\widetilde{\mathcal{L}}$ \ defined by (\ref{Lagqq1}) the
Hamiltonian $H_{\mathrm{DW}}\left( \mathsf{p}\right) $ does not depend
explicitly on $\mathsf{q}$ an
\[
H_{\mathrm{DW}}\left( \mathsf{p}\right) =\widetilde{\mathcal{L}}\left(
\mathsf{u}\right) ,
\]
where $\mathsf{p}$ is linked to $\mathsf{u}$ as in (\ref{momentum}).
Equivalently
\begin{equation}
H_{\mathrm{DW}}(\mathsf{p})=\frac{1}{2}\mathsf{p}^{\mathrm{T}}M_{\mathrm{L
}^{-1}\mathsf{p}. \label{Lagqq4
\end{equation}
According to (\ref{mhzne2}), (\ref{mhzne3}), in the variables $(\mathsf{p
_{z},\partial_{t}\mathsf{q})^{^{\mathrm{T}}}$, the corresponding first-order
system is precisely (\ref{HamEq1}), \ with $\ \tilde{J}$ \ and $\ \tilde{M}$
\ as in (\ref{aeta4}).
\end{proof}
We recall that $\tilde{J}$ \ and $\ \tilde{M}$ are, respectively,
antihermitian and hermitian,\textit{ i.e.
\[
\tilde{J}^{\ast}=-\tilde{J},\qquad\tilde{M}^{\ast}=\tilde{M}.
\]
Consider now\ a time harmonic solution of the form $\mathsf{q(
z,t)=\widehat{\mathsf{q}}\left( z\right) \mathrm{e}^{-\mathrm{i}\omega t}.$
In this case,
\begin{equation}
V(z,t)=\left[
\begin{array}
[c]{l
\mathsf{p}_{z}\\
\partial_{t}\mathsf{q
\end{array}
\right] =\hat{V}\left( z\right) \mathrm{e}^{-\mathrm{i}\omega t},\qquad
\hat{V}(z)=\left[
\begin{array}
[c]{l
\widehat{\mathsf{p}}_{z}\left( z\right) \\
-\mathrm{i}\omega\widehat{\mathsf{q}}\left( z\right)
\end{array}
\right] \label{aeta4a
\end{equation}
and the Hamiltonian equation (\ref{HamEq1}) is reduced t
\begin{equation}
\tilde{J}\partial_{z}\hat{V}=\omega\tilde{M}\hat{V}. \label{aeta3
\end{equation}
Notice that the equation (\ref{aeta3}) for $\hat{V}$ is Hamiltonian according
to the definition in Section \ref{AppCanHam}, and the conservation law
(\ref{Jzhz10}) applies, yielding
\begin{gather}
\hat{V}^{\ast}\tilde{J}\hat{V}=\mathrm{i}\left[ \widehat{\mathsf{p}
_{z}^{\ast}\left( -\mathrm{i}\omega\widehat{\mathsf{q}}\left( z\right)
\right) +\left( -\mathrm{i}\omega\widehat{\mathsf{q}}\left( z\right)
\right) ^{\ast}\widehat{\mathsf{p}}_{z}\right] =2\mathrm{i
\operatorname*{Re}\left\{ \left( -\mathrm{i\omega}\widehat{\mathsf{q
}\left( z\right) \right) ^{\ast}\widehat{\mathsf{p}}_{z}\right\}
\label{aeta5}\\
=-2\mathrm{i}\omega\operatorname{Im}\left\{ \left( \widehat{\mathsf{q
}\left( z\right) \right) ^{\ast}\left[ \theta\left( -\mathrm{i
\omega\right) \widehat{\mathsf{q}}\left( z\right) -\eta\left( z\right)
\partial_{z}\widehat{\mathsf{q}}\left( z\right) \right] \right\}
=\operatorname*{constant}.\nonumber
\end{gather}
Later on, in Section \ref{SubsEnergyExchange}, we will see how the above
conservation law relates to energy flux constancy.
\section{Amplification for the homogeneous case\label{AmplMTL-beam}}
This subsection is devoted to the analysis of the amplification regime
associated with a single exponentially growing mode in the case of an
homogeneous MTLB system, that is with parameters not varying with $z$. For
real $\omega$ we seek solutions of (\ref{MTLEuler-Larange}) in the for
\begin{equation}
Q(z,t)=\widehat{Q}\mathrm{e}^{-\mathrm{i}(\omega t-kz)},\qquad
q(z,t)=\widehat{q}\mathrm{e}^{-\mathrm{i}(\omega t-kz)},
\label{PlaneWavesStructure
\end{equation}
where $\widehat{q}$ and $k$ are complex constants and $\widehat{Q}$ is a
complex vector. We show that, under certain conditions, there is a solution
with genuinely complex, that is, non real wave number $k$.
Let us recall that the eigenvelocities of the MTL are the roots of the
equatio
\[
\left\vert C^{-1}-v^{2}L\right\vert =0,
\]
\cite{Paul}, \cite{Nitsch}. \ Since both $L$ and $C$ are positive definite,
the symmetric $n\times n$ matrix $L^{-1/2}C^{-1}L^{-1/2}$ has positive
eigenvalues $0<\lambda_{1}\leq\lambda_{2}\leq...\leq\lambda_{n}$, where
multiple eigenvalues are repeated according to their multiplicity. Then the
MTL has characteristic velocities are precisel
\begin{equation}
\pm v_{i},\RIfM@\expandafter\text@\else\expandafter\mbox\fi{ where }v_{i}^{2}=\lambda_{i}. \label{chav1
\end{equation}
\begin{theorem}
\label{TeoremAmplification}Let $u_{0},\xi>0.$ If either
\end{theorem}
\begin{enumerate}
\item[(i)] $0<u_{0}\leq v_{1}$ \textit{or}
\item[(ii)] $v_{1}<u_{0}$ \textit{and} $\xi>0$ \textit{is sufficiently small,}
\noindent\textit{then for each real }$\omega$\textit{ there are exactly two
genuinely complex conjugate values }$k_{0}$\textit{ and }$k_{0}^{\ast
$\textit{ such that (\ref{PlaneWavesStructure}) is a non-trivial solution of
equations (\ref{MTLEuler-Larange}).}
\end{enumerate}
Hence, assuming $\operatorname*{Im}k_{0}<0$ we have the associated solution
\begin{equation}
Q(z,t)=A(z)\mathrm{e}^{-\mathrm{i}\omega t}\mathrm{e}^{-(\operatorname*{Im
k_{0})z},\qquad q(z,t)=B(z)\mathrm{e}^{-\mathrm{i}\omega t}\mathrm{e
^{-(\operatorname*{Im}k_{0})z},\qquad A(z),B(z)\neq0, \label{chav2
\end{equation}
that grows exponentially in the $+z$ direction, whereas the solution
associated with $k_{0}^{\ast}$ decays exponentially.
We sketch the proof, deferring the mathematical details to section
\ref{AppAmplification}. Substituting the expressions
(\ref{PlaneWavesStructure}) into the system (\ref{MTLEuler-Larange}), we
obtain the following linear algebraic system of $n+1$ equations for
$\widehat{Q},\widehat{q}$
\begin{equation}
\left[
\begin{array}
[c]{cc
-v^{2}L+C^{-1} & D\\
D^{T} & d-\xi(v-u_{0})^{2
\end{array}
\right] \left[
\begin{array}
[c]{c
\widehat{Q}\\
\widehat{q
\end{array}
\right] =\left[
\begin{array}
[c]{c
0\\
0
\end{array}
\right] ,\quad\RIfM@\expandafter\text@\else\expandafter\mbox\fi{where }v=\frac{\omega}{k} \label{Systemforv
\end{equation}
an
\begin{equation}
D=(D_{i}),\qquad D_{i}
{\displaystyle\sum\limits_{j}}
(C^{-1})_{ij},\qquad d
{\displaystyle\sum\limits_{i}}
D_{i}. \label{ddcj1
\end{equation}
For the sake of brevity, we denot
\begin{equation}
A(v)=-v^{2}L+C^{-1},\qquad\widetilde{A}(v)=\left[
\begin{array}
[c]{cc
A(v) & D\\
D^{T} & d-\xi(v-u_{0})^{2
\end{array}
\right] . \label{ddcj2
\end{equation}
The system (\ref{Systemforv}) has nontrivial solutions if and only if
$\ \left\vert \widetilde{A}(v)\right\vert =0$. The corresponding polynomial
equation of degree $2n+2$ is the dispersion relation of our system written in
terms of the velocity $v$. In Section \ref{AppAmplification} we prove in full
detail that the equation $\left\vert \widetilde{A}(v)\right\vert =0$ has
exactly one pair of complex conjugate solutions if either (i) or (ii) holds.
Here we outline the main ideas of the proof.
First of all, if $\left\vert A(v)\right\vert \neq0$, the following
\emph{canonical factorization }takes plac
\begin{equation}
\left\vert \widetilde{A}(v)\right\vert =\left\vert A(v)\right\vert \left[
d-\xi(v-u_{0})^{2}-D^{T}(A(v))^{-1}D\right] . \label{CanFact
\end{equation}
The values of $v$ such that $\left\vert A(v)\right\vert =0$ are precisely the
eigenvelocities $\pm v_{i}$ of the waveguide. Therefore, the roots of
$\left\vert \widetilde{A}(v)\right\vert =0$ different from $\pm v_{i}$,
$i=1,2,...n$ \ are the roots of the equatio
\begin{equation}
-\xi(v-u_{0})^{2}=R(v),\RIfM@\expandafter\text@\else\expandafter\mbox\fi{ where }R(v)=D^{T}(A(v))^{-1}D-d,
\label{DispRelGen
\end{equation}
in which the two components of the system enter separately. The rational
function $R(v)$ in (\ref{DispRelGen}) contains the relevant information about
the MTL whereas the left hand side depends only on the beam parameters. In
what follows we refer to function $R(v)$ as \emph{MTL characteristic
function}. It can be explicitly written in terms of the characteristic
velocities
\begin{equation}
R(v)
{\displaystyle\sum\limits_{i=1}^{n}}
\frac{\widetilde{D}_{i}^{2}}{v_{i}^{2}-v^{2}}-d, \label{R
\end{equation}
where $\widetilde{D}_{i}$ are constants related to $D_{i}$, see Section
\ref{AppAmplification}. The graph of the MTL characteristic function $R$ is
symmetric with respect to the vertical axis and is made up of branches, \ a
central one with the minimum at $(0,0)$, a number of increasing \ branches for
$v>0$ and decreasing for $v<0$. One can readily see that $\lim_{v\rightarrow
\infty}R(v)=-d$. In addition to that, the graph of $R$ has vertical asymptotes
at $v=\pm v_{i}$ if at least one of the associated $\widetilde{D}_{j}$ does
not vanish. The number of the asymptotes varies between $2$ and $2n$. The
left-hand side in (\ref{DispRelGen}) is a parabola with vertex at $(u_{0},0)$.
Figure \ref{FigAmpl} shows the graph of $R$ and that of the parabola
$\ y=-\xi(v-u_{0})^{2}$ with the following inductance and capacity matrice
\[
L=\left[
\begin{array}
[c]{ccc
4 & 1 & 1/2\\
1 & 5 & 2\\
1/2 & 2 & 2
\end{array}
\right] ;\qquad C=\left[
\begin{array}
[c]{ccc
2 & 1 & 2\\
1 & 4 & 0\\
2 & 0 & 1
\end{array}
\right] .
\]
The approximate \ values of the characteristic velocities are: $v_{1}=0.18357$
and $v_{2}=0.42383$. In Figure \ref{FigAmpl} (a), $u_{0}=0.18$ and $\xi=2$; in
Figure \ref{FigAmpl} (b), $u_{0}=0.8$ and $\xi=18$.
It is important to observe that the parabola always intersects all the
branches of $R$ except for the central one. For small $\xi$ each branch is
intersected only once, and consequently the number of real roots of the
equation (\ref{DispRelGen}) is exactly the number of asymptotes, as in Figure
\ref{FigAmpl} (a) above. For large $\xi$ however the number of real roots can
exceed the number of asymptotes as in Figure \ref{FigAmpl} (b), where a large
value of $\xi$ produces three points of intersection with the far right branch
of the graph of $R$. Moreover, if $\ u_{0}\leq v_{1}$ (geometrically, the
vertex of the parabola lies between the vertical axis and the first
asymptote), then clearly the number of real roots equals the number of
asymptotes irrespective of the value of $\xi>0$. These facts can be proved
rigorously based on monotonicity properties, but their geometric
interpretation is so transparent that a quick look at Figure \ref{FigAmpl} is
quite convincing
\begin{figure}[ptb
\centering
\ifcase\msipdfoutput
\includegraphics[
height=2.4794in,
width=6.4264in
{Fig12Amp.eps
\else
\includegraphics[
height=2.4794in,
width=6.4264in
{D:/alatex/Preparation/Reyes/lagsys/arxiv2/graphics/Fig12Amp__2.pdf
\fi
\caption{(a) $u_{0}<v_{1}$: the parabola $y=-\xi(v-u_{0})^{2}$ (dashed line)
intersects each branch of $y=R(v)$ (where $R$ is as in (\ref{R})) just once.
Four real roots. \ (b) $u_{0}>v_{1}:$ for large $\xi,$ the parabola intersects
one of the branches of $y=R(v)$ three times. Six real roots.
\label{FigAmpl
\end{figure}
\ In the generic case, the roots of the dispersion relation $\left\vert
\widetilde{A}(v)\right\vert =0$ are exactly those of equation
(\ref{DispRelGen}), but in general some of the $v_{i}$ can also be roots.
Whenever some $v_{i}$ is a real root (maybe multiple) of $\left\vert
\widetilde{A}(v)\right\vert =0$, the number of asymptotes in the graph of $R$
is reduced by the corresponding amount. The same is true of the number of real
roots of (\ref{DispRelGen}) under either (i) or (ii). This fact follows from
factorization (\ref{CanFact}). The main point is that in all cases the total
number of real roots of $\left\vert \widetilde{A}(v)\right\vert =0$ \ is $2n$
\ if either condition (i) or (ii) above holds. We thus conclude that under (i)
or (ii) there is necessarily a \emph{unique} pair of complex conjugate roots.
The detailed proof of these facts is provided in Section
\ref{AppAmplification}.
It is not difficult to estimate how small $\xi$ should be in condition (ii)
above. In the case $n=1$ and $u_{0}>v_{1}$ there is a precise criterion on
$\xi$, namely amplification takes place i
\begin{equation}
\xi<\xi_{0}:=\frac{L\gamma^{2}}{1-\gamma^{2/3}};\qquad\gamma=\frac{v_{1
}{u_{0}}=\frac{1}{u_{0}\sqrt{LC}}. \label{xismall
\end{equation}
A simple sufficient condition can be also given for $n>1$. For example, one
can just impose that the left branch of the parabola at $v=0$ be flatter than
the flattest point of the graph of $R$ on $(v_{1},u_{0}).$This leads t
\begin{equation}
\xi<\widetilde{\xi}_{0}:=\frac{\min_{v\in(v_{1},u_{0})}R^{\prime}(v)}{2u_{0}}.
\label{xismallbis
\end{equation}
Observe that both $\xi_{0}$ and $\widetilde{\xi}_{0}$ vanish as $u_{0
\rightarrow\infty$, as expected. The value of $\widetilde{\xi}_{0}$ is not
sharp but we did not make an effort to find one.
Thus, under the above assumptions the system exhibits spatially exponentially
growing, as well as exponentially decaying \ time harmonic regimes. Using the
terminology of dynamical systems, if we restrict to time harmonic evolutions
$z\rightarrow X,$ where $X=\left\{ e^{-i\omega t}\widetilde{Q},\quad
\widetilde{Q}\i
\mathbb{C}
^{n+1}\right\} ,$ there is a subspace of data \ inducing an individual
exponential dichotomy for $X,$ both for forward and backward in $z$
evolutions, \cite[XIII]{Ha}. The subspace is determined by the solutions
$\widetilde{Q}=(\widehat{Q},\widehat{q})$ of the system (\ref{Systemforv})
with $v$ being the corresponding complex solution of $\left\vert
\widetilde{A}(v)\right\vert =0.$
\subsection{Asymptotic behavior of the amplification factor as $\xi
\rightarrow0$ and as $\xi\rightarrow\infty$.\label{BehAmplificationFactor}}
Let $k_{0}$ denote the complex root with $\operatorname*{Im}k_{0}<0$ whose
existence we proved in the previous section under appropriate conditions. It
is interesting to study the asymptotics of the "amplification factor"
$-\operatorname*{Im}k_{0}$ as the beam parameter $\xi\rightarrow0,$ as well as
its behavior when $\xi\rightarrow\infty.$ A careful analysis shows (see
Section \ref{AppAmplification}) that, if we denote by $v_{0}=\omega/k_{0}$ the
corresponding velocity with $\operatorname*{Im}v_{0}>0$, then
\[
\operatorname*{Im}v_{0}=\sqrt{K^{\prime}\xi+o(\xi)}=\sqrt{K^{\prime}}\sqrt
{\xi}+o(\sqrt{\xi})\RIfM@\expandafter\text@\else\expandafter\mbox\fi{ \ as }\xi\rightarrow0,
\]
where $K^{\prime}$ depends only on $L,C,u_{0}.$ As a consequence
\begin{equation}
-\operatorname*{Im}k_{0}=\frac{\operatorname*{Im}v_{0}}{\RIfM@\expandafter\text@\else\expandafter\mbox\fi{\ \ }\left\vert
v_{0}\right\vert ^{2}}\sim\frac{K^{\prime\prime}}{\sqrt{\xi}\RIfM@\expandafter\text@\else\expandafter\mbox\fi{\
}\ \RIfM@\expandafter\text@\else\expandafter\mbox\fi{\ as \ }\xi\rightarrow0;\ \ K^{\prime\prime}>0. \label{AmpFactorAt0
\end{equation}
The conclusion is that, in this model, the amplification factor can be
indefinitely improved by reducing $\xi.$ According to (\ref{tranbe8}), this
amounts to increasing $\sigma\rho_{0},$ the linear electron density of the beam.
On the other hand, the limit $\xi\rightarrow\infty$ makes sense only if
$0<u_{0}\leq v_{1}$. In the case of one line and $u_{0}=v_{1}$, it can be
proved tha
\begin{equation}
-\operatorname*{Im}k_{0}=\frac{\operatorname*{Im}v_{0}}{\RIfM@\expandafter\text@\else\expandafter\mbox\fi{\ \ }\left\vert
v_{0}\right\vert ^{2}}\sim\frac{K^{\prime\prime\prime}}{\sqrt[3]{\xi
}\ \RIfM@\expandafter\text@\else\expandafter\mbox\fi{\ as \ }\xi\rightarrow\infty;\ \ K^{\prime\prime\prime}>0,
\label{AmpFactorAtInfty
\end{equation}
see Section \ref{AppAmplification}.
The regime considered by Pierce corresponds to the latter situation, in which
there are two real solutions (for $v$) close to $\pm u_{0,}$ and two complex
conjugate with real part close to $u_{0}$; see Section \ref{PierceRev copy(1)
. The situation is similar for $u_{0}<v_{1}$, but in this case
$-\operatorname*{Im}k_{0}$ has a finite positive limit as $\xi\rightarrow
\infty$.
\section{Energy conservation and transfer\label{EnergyConsEx}}
The conservation laws for our system can be obtained via Noether theorem,
\cite[38.2-3]{GelFom}, \cite[13.7]{Gold}.
\begin{theorem}
Conservation of energy \ for the system (\ref{MTLEuler-Larange}) holds in the
for
\begin{equation}
\partial_{t}H+\partial_{z}S=0, \label{encoq2
\end{equation}
where the total energy $H$ and the total energy flux $S$ are given by
\begin{equation}
H=\frac{1}{2}\partial_{t}\mathsf{q}^{\mathrm{T}}\alpha\partial_{t
\mathsf{q}+\frac{1}{2}\partial_{z}\mathsf{q}^{\mathrm{T}}\eta\partial
_{z}\mathsf{q}; \label{encoq4a
\end{equation
\begin{equation}
S=\partial_{t}\mathsf{q}^{\mathrm{T}}\theta\partial_{t}\mathsf{q}-\partial
_{t}\mathsf{q}^{\mathrm{T}}\eta\partial_{z}\mathsf{q}=\partial_{t
\mathsf{q}^{\mathrm{T}}\left( \theta\partial_{t}\mathsf{q}-\eta\partial
_{z}\mathsf{q}\right) =\partial_{t}\mathsf{q}^{\mathrm{T}}\mathsf{p}_{z}.
\label{encoq4b
\end{equation}
\end{theorem}
\begin{proof}
The Lagrangian density $\mathcal{L}$ does not depend explicitly on $t$ (this
is a consequence of the closedness of the system), therefore by the fields
version of Noether theorem, \cite[38.2-3]{GelFom}, \cite[13.7]{Gold},
conservation of energy (\ref{encoq2}) holds, with energy and the energy flux
densities given b
\begin{equation}
H
{\displaystyle\sum_{j}}
\frac{\partial\mathcal{L}}{\partial(\partial_{t}\mathsf{q}_{j})}\partial
_{t}\mathsf{q}_{j}-\mathcal{L},\quad S
{\displaystyle\sum_{j}}
\frac{\partial\mathcal{L}}{\partial(\partial_{z}\mathsf{q}_{j})}\partial
_{t}\mathsf{q}_{j}. \label{encoq1
\end{equation}
A straightforward computation yields the expressions of $H$ and $S$ given in
(\ref{encoq4a}), (\ref{encoq4b}). In (\ref{encoq4b}), $\mathsf{p}_{z}$ is the
canonical momentum defined in (\ref{qulaq5}), Section \ref{AppQuadLag}.
\end{proof}
Consider now a real time harmonic eigenmod
\begin{equation}
\mathsf{q}\left( t,z\right) =\operatorname{Re}\left\{ \mathsf{\hat{q
}\left( z\right) \mathrm{e}^{-\mathrm{i}\omega t}\right\} ,\RIfM@\expandafter\text@\else\expandafter\mbox\fi{ with a
complex valued }\mathsf{\hat{q}}\left( z\right) , \label{encoq4c
\end{equation}
which solves the Euler-Lagrange equation (\ref{qulaq4}). Notice that
$\left\langle \mathsf{q}\right\rangle (z)=0$, where $\left\langle
\cdot\right\rangle $ is the time average operation defined in (\ref{encoq5}).
However, i
\begin{equation}
a\left( t\right) =\operatorname{Re}\left\{ \hat{a}\mathrm{e}^{-\mathrm{i
\omega t}\right\} ,\RIfM@\expandafter\text@\else\expandafter\mbox\fi{ with a complex valued }\hat{a} \label{encoq5a
\end{equation}
and $b\left( t\right) $ is defined by a similar formula then we hav
\begin{equation}
\left\langle ab\right\rangle =\frac{1}{2}\operatorname{Re}\left\{ \hat
{a}^{\ast}\hat{b}\right\} . \label{encoq5b
\end{equation}
Applying the averaging operation $\left\langle \cdot\right\rangle $ to the
conservation law (\ref{encoq2}) for a time harmonic eigenmode $q$ as in
(\ref{encoq4c}) and using (\ref{prop1ave}) and (\ref{prop2ave}), we obtai
\begin{equation}
\partial_{z}\left\langle S\right\rangle \left( z\right) =0\RIfM@\expandafter\text@\else\expandafter\mbox\fi{ implying
}\left\langle S\right\rangle \left( z\right) =\operatorname*{constant}.
\label{encoq6
\end{equation}
On the other hand, $S$ defined by (\ref{encoq4b}) can be written as the
product of two real time harmonic functions
\begin{equation}
S(t,z)=\operatorname{Re}\left\{ \widehat{A}(z)\mathrm{e}^{-\mathrm{i}\omega
t}\right\} \operatorname{Re}\left\{ \widehat{B}(z)\mathrm{e}^{-\mathrm{i
\omega t}\right\} , \label{encoq6b
\end{equation}
wher
\begin{equation}
\widehat{A}(z)=-i\omega\mathsf{\hat{q}}\left( z\right) ;\qquad
\widehat{B}(z)=-i\omega\theta\mathsf{\hat{q}}(z)-\eta\partial_{z
\mathsf{\hat{q}}(z). \label{encoq6c
\end{equation}
Using (\ref{encoq5b}) we obtain the energy flux conservation law in the form
\begin{equation}
\left\langle S\right\rangle \left( z\right) =\frac{1}{2}\operatorname{Re
\left\{ \left\langle \left( -i\omega\mathsf{\hat{q}}\right) ^{\ast}\left(
-i\omega\theta\mathsf{\hat{q}}-\eta\partial_{z}\mathsf{\hat{q}}\right)
\right\rangle \right\} =\frac{1}{2}\operatorname{Re}\left\{ \left\langle
\left( -i\omega\mathsf{\hat{q}}\right) ^{\ast}\mathsf{\hat{p}
_{z}\right\rangle \right\} =\operatorname*{constant}. \label{encoq6a
\end{equation}
Constancy of $\left\langle S\right\rangle \left( z\right) $ is related to
the constancy of the symplectic square of the solution of the Hamiltonian
system satisfied b
\begin{equation}
\widehat{V}(z)=\left[
\begin{array}
[c]{l
\widehat{\mathsf{p}}_{z}\\
-i\omega\widehat{\mathsf{q}
\end{array}
\right] , \label{encoq7
\end{equation}
see formula (\ref{aeta5}). Indeed
\begin{equation}
V^{\ast}\tilde{J}V=2i\operatorname*{Re}\left\{ \left( -i\omega
\mathsf{\hat{q}}\left( z\right) \right) ^{\ast}\left[ -\mathrm{i
\omega\theta\mathsf{\hat{q}}\left( z\right) -\eta\partial_{z}\mathsf{\hat
{q}}\left( z\right) \right] \right\} =\operatorname*{constant
=4\mathrm{i}\left\langle S\right\rangle \left( z\right) . \label{encoq8
\end{equation}
\subsection{Energy exchange between subsystems\label{SubsEnergyExchange}}
This section deals with the balance of energy between the two subsystems
making up our system: the beam and the MTL. As already pointed out by Pierce
in \cite[p. 635]{Pier51} an amplification regime assumes that the energy
extracted from the beam is stored in the EM field. In other words, the net
flux of energy must have a definite sign.\ Pierce tacitly considers this
condition as an additional one to be imposed on top of other conditions
ensuring the existence of an exponentially growing solution. We show below
that in fact this condition is automatically satisfied for exponentially
growing solutions.
When computing the energy flux between the beam and the MTL we take advantage
of our Lagrangian setting. This setting allows for a systematic derivation of
expressions for energies and fluxes satisfying \textit{a priori} the
fundamental conservation laws. We proceed using the results from Section
\ref{AppEnergyExchange} for a more general coupled system.
First, we should split the Lagrangian into two parts $\mathcal{L=L
_{1}\mathcal{+L}_{2}$ corresponding to the MTL and the beam. Namely
\begin{align}
\mathcal{L}_{1}(Q_{t},Q_{;z}) & =\frac{1}{2}\left( \partial_{t
Q,L\partial_{t}Q\right) ^{2}-\frac{1}{2}\left( \partial_{_{;z}
Q,C^{-1}\partial_{_{;z}}Q\right) ^{2};\label{LLQq1}\\
\mathcal{L}_{2}(q_{t},q_{z}) & =\frac{\xi}{2}(\partial_{t}q+u_{0
\partial_{z}q)^{2},\RIfM@\expandafter\text@\else\expandafter\mbox\fi{ where }\partial_{;z}Q=\partial_{z}Q+B\partial
_{z}q.\nonumber
\end{align}
The above Lagrangian has the structure of (\ref{gblag1}), with
$\ B=(1,1...1)^{\mathrm{T}}$. Our first result concerning energy flows is
contained in the following
\begin{theorem}
The instantaneous power by unit length supplied by the beam to the MTL is
given by
\begin{equation}
P_{\mathrm{B}\rightarrow\mathrm{MTL}}=\partial_{t}\left[ \frac{1}{2}\left(
CV,V\right) +\frac{1}{2}\left( LI,I\right) \right] +\partial_{z}(I,V)
\label{exprpower1
\end{equation}
where, as usual, $(,)$ stands for the scalar product.
\end{theorem}
\begin{proof}
According to (\ref{gblag11}), the power $P_{\mathrm{B}\rightarrow\mathrm{MTL
}$ flowing from the beam to the (unit length of) the MTL is given b
\begin{gather}
P_{\mathrm{B}\rightarrow\mathrm{MTL}}=-\frac{\partial\mathcal{L}_{1}
{\partial(\partial_{;z}Q)}\partial_{tz}^{2}q=\partial_{;z}Q^{T}C^{-1
B\partial_{tz}^{2}q=\partial_{z}I_{b
{\displaystyle\sum\limits_{i}}
D_{i}\partial_{;z}Q_{i},\label{LLQq2}\\
\RIfM@\expandafter\text@\else\expandafter\mbox\fi{where }D_{i}
{\displaystyle\sum\limits_{j}}
(C^{-1})_{ij}.\nonumber
\end{gather}
Using (\ref{TelegMulti}) we recast the expression for $P_{\mathrm{B
\rightarrow\mathrm{MTL}}$ in terms of currents and voltages. Indeed, the
voltage $V$ \ is given b
\begin{equation}
V=-C^{-1}(\partial_{z}Q+\partial_{z}q). \label{LLQq3
\end{equation}
Then we notice that
\
{\displaystyle\sum\limits_{i}}
D_{i}\partial_{;z}Q_{i}
{\displaystyle\sum\limits_{i}}
{\displaystyle\sum\limits_{j}}
(C^{-1})_{ij}(\partial_{z}Q_{i}+\partial_{z}qB_{i})=
{\displaystyle\sum\limits_{j}}
V_{j
\]
and hence, according to (\ref{TelegMulti}),
\begin{gather}
P_{\mathrm{B}\rightarrow\mathrm{MTL}}=
{\displaystyle\sum\limits_{j}}
\partial_{z}I_{b}V_{j}=-\left( \partial_{z}I_{b}B,V\right) =\left(
C\partial_{t}V,V\right) +\left( \partial_{z}I,V\right) =\label{LLQq4}\\
=\partial_{t}\left[ \frac{1}{2}\left( CV,V\right) \right] +\partial
_{z}\left( I,V\right) -\left( I,\partial_{z}V\right) =\partial_{t}\left[
\frac{1}{2}\left( CV,V\right) \right] +\left( L\partial_{t}I,I\right)
+\partial_{z}\left( I,V\right) =\nonumber\\
=\partial_{t}\left[ \frac{1}{2}\left( CV,V\right) +\frac{1}{2}\left(
LI,I\right) \right] +\partial_{z}(I,V).\nonumber
\end{gather}
\end{proof}
The first two terms in (\ref{exprpower1}) correspond to $\partial_{t}H$ where
\begin{equation}
H=\frac{1}{2}(CV,V)+\frac{1}{2}(LI,I) \label{LLQq5
\end{equation}
is the density of the total energy stored in the shunt capacitors and the
inductances per unit length. The last term in $P_{\mathrm{B}\rightarrow
\mathrm{MTL}}$ represents the divergence of the energy flux, $S=(I,V).$ In the
particular case of one line, we recover the usual expressions for the
corresponding quantities
\begin{equation}
P_{\mathrm{B}\rightarrow\mathrm{MTL}}=\partial_{t}\left[ \frac{1}{2
CV^{2}\right] +\partial_{t}\left[ \frac{1}{2}LI^{2}\right] +\partial
_{z}(IV). \label{PowerOneLine
\end{equation}
Our next result deals with the direction on the (time averaged) power flow.
\begin{theorem}
Let $k_{0},v_{0}$ denote the complex values of the wave number and the
velocity for the unique \emph{\ }exponentially growing solution according to
Theorem \ref{TeoremAmplification}. Then, the following formula holds for the
time average of the power
\begin{equation}
\left\langle P_{\mathrm{B}\rightarrow\mathrm{MTL}}\right\rangle (z)=-\left[
\omega\xi\left\vert k_{0}\right\vert ^{2}\left\vert \widehat{q}\right\vert
^{2}(\operatorname*{Re}v_{0}-u_{0})\operatorname{Im}v_{0}\right]
\mathrm{e}^{-2\left( \operatorname{Im}k_{0}\right) z}. \label{exprpower2
\end{equation}
Moreover, $\left\langle P_{\mathrm{B}\rightarrow\mathrm{MTL}}\right\rangle
(z)>0$ for all $z.$Thus, the power on the growing solution flows from the beam
to the MTL.
\end{theorem}
\begin{proof}
First, observe that for real time harmonic solutions $Q$ and $q$ of the for
\begin{equation}
Q=\operatorname*{Re}\left( \widehat{Q}\mathrm{e}^{i(kz-\omega t)}\right)
,\qquad q=\operatorname*{Re}\left( \widehat{q}\mathrm{e}^{i(kz-\omega
t)}\right) , \label{gblag16
\end{equation}
where $\widehat{Q}_{,}\widehat{q}$ are complex constants, the expression for
$P_{\mathrm{B}\rightarrow\mathrm{MTL}}$ can be written in the for
\begin{equation}
P_{\mathrm{B}\rightarrow\mathrm{MTL}}=\partial_{;z}Q^{T}C^{-1}B\partial
_{tz}^{2}q=\operatorname*{Re}(\widehat{a}(z)e^{-i\omega t})\operatorname*{Re
(\widehat{b}(z)e^{-i\omega t}), \label{gblag17
\end{equation}
wher
\begin{equation}
\widehat{a}(z)=ike^{ikz}(\widehat{Q}+B\widehat{q})^{T}C^{-1};\qquad
\widehat{b}(z)=\omega k\widehat{q}e^{ikz}B. \label{gbalg18
\end{equation}
Applying formula (\ref{encoq5b}) for time average, we get
\begin{equation}
\left\langle P_{\mathrm{B}\rightarrow\mathrm{MTL}}\right\rangle (z)=\frac
{\omega}{2}\mathrm{e}^{-2\left( \operatorname{Im}k\right) z
\operatorname{Im}\left\{ \left\vert k\right\vert ^{2}\left( \widehat{Q
+B\widehat{q}\right) ^{\ast T}C^{-1}B\widehat{q}\right\} . \label{Pow1
\end{equation}
Suppose now that $k_{0}$ is the complex root providing amplification, that is,
in the notation of Subsection \ref{BehAmplificationFactor}, $k_{0
=\omega/v_{0}$ with $\operatorname{Im}k_{0}<0.$ Then, $v_{0}$ is a root of the
system (\ref{Systemforv}) and therefore, in the notation of Section
\ref{AmplMTL-beam} and returning to the variable $k$
\[
k_{0}^{2}(\widehat{Q}^{\mathrm{T}}D+\widehat{q}d)=\xi(\omega-k_{0}u_{0
)^{2}\widehat{q}.
\]
Taking complex conjugate in the above equation and observing that $C^{-1}B=D$
and $B^{\mathrm{T}}C^{-1}B=d,$we can rewrite (\ref{Pow1}) in the for
\begin{align}
\left\langle P_{\mathrm{B}\rightarrow\mathrm{MTL}}\right\rangle (z) &
=\frac{\omega\xi}{2}\mathrm{e}^{-2\left( \operatorname{Im}k_{0}\right)
z}\operatorname{Im}\left\{ \frac{\left\vert k_{0}\right\vert ^{2}
{k_{0}^{\ast2}}\left( \omega-u_{0}k_{0}^{\ast}\right) ^{2}\left\vert
\widehat{q}\right\vert ^{2}\right\} \label{Pow2}\\
& =\frac{\omega\xi\left\vert k_{0}\right\vert ^{2}\left\vert \widehat{q
\right\vert ^{2}u_{0}^{2}}{2}\mathrm{e}^{-2\left( \operatorname{Im
k_{0}\right) z}\operatorname{Im}\left\{ \left( \frac{k_{b}-k_{0}^{\ast
}{k_{0}^{\ast}}\right) ^{2}\right\} ,\qquad k_{b}=\frac{\omega}{u_{0
}.\nonumber
\end{align}
In terms of velocities, we hav
\begin{equation}
\operatorname{Im}\left( \frac{k_{b}-k_{0}^{\ast}}{k_{0}^{\ast}}\right)
^{2}=\operatorname{Im}\left( \frac{v_{0}^{\ast}}{u_{0}}-1\right) ^{2
=-\frac{2}{u_{0}^{2}}(\operatorname*{Re}v_{0}-u_{0})\operatorname{Im}v_{0}.
\label{Pow3
\end{equation}
Since we are assuming $\operatorname{Im}v_{0}>0,$ we see from formula
(\ref{Pow2}) that $\left\langle P_{\mathrm{B}\rightarrow\mathrm{MTL
}\right\rangle (z)\geq0$ for all $z$ exactly if $\operatorname*{Re}v_{0}\leq
u_{0}$. But this is always the case, as it follows from (\ref{Vieta1}) and
(\ref{Asymmetry}). Formula (\ref{exprpower2}) follows at once from
(\ref{Pow2}) and (\ref{Pow3}).
\end{proof}
Observe also that, since $\operatorname{Im}k_{0}<0$, formula (\ref{exprpower2
) implies that $\left\langle P_{\mathrm{B}\rightarrow\mathrm{MTL
}\right\rangle $ increases in the $+z$ direction. For the evanescent wave,
corresponding to the value $k_{0}^{\ast},$ we have exactly the opposite
situation: the energy flows from the MTL to the beam and the power flux
decreases in the $+z$ direction.
\section{The Pierce model revisited\label{PierceRev copy(1)}}
Let us examine Pierce's original results in the light of our general theory.
They correspond to $n=1$, hence $d=D=C^{-1}$ and the dispersion relation
$\left\vert \widetilde{A}(v)\right\vert =0$ become
\begin{equation}
\left( -v^{2}L+C^{-1}\right) \left[ C^{-1}-\xi(v-u_{0})^{2}\right]
-C^{-2}=0, \label{PierRev1
\end{equation}
which, in terms of $\ k=\omega/v,$ read
\begin{equation}
-L\omega^{2}k^{2}+\xi(\omega-ku_{0})^{2}(LC\omega^{2}-k^{2})=0.
\label{PierRev2
\end{equation}
After elementary algebraic transformations the above equation turns int
\begin{equation}
u_{0}^{2}k^{4}-2u_{0}\omega k^{3}+\left[ 1+\frac{L}{\xi}-LCu_{0}^{2}\right]
\omega^{2}k^{2}+2LCu_{0}\omega^{3}k-LC\omega^{4}=0, \label{PierRev3
\end{equation}
which is precisely the fourth order equation in \cite[ (1.16)]{Pier51}
The TL has only two characteristic velocities, namely $\pm v_{1}=\pm
1/\sqrt{LC}$ which are not solutions of (\ref{PierRev1}). The graph of the
characteristic function $R$ has only two vertical asymptotes at $v=\pm v_{1}$.
The special regime considered in \cite{Pier51} corresponds to taking large
$\xi$, and $u_{0}=v_{1}$. As we know, in this case amplification occurs for
any $\xi>0$. For small values of the paramete
\begin{equation}
k_{p}=\frac{\omega_{p}}{u_{0}}=\frac{1}{u_{0}}\sqrt{\frac{4\pi}{\sigma\xi}},
\label{kpuu1
\end{equation}
Pierce asserts that $k\simeq k_{\mathrm{b}}=\omega/u_{0}$ for the forward
unattenuated wave. In terms of velocities this means that for large values of
$\xi$ the positive real solution $v_{1}^{+}$ is very close to $u_{0}$. The
graph in Figure \ref{PierceAmp} refers to this situation and it clearly shows
that indeed $v_{1}^{+}\simeq u_{0}$ and $-v_{1}^{-}$ $\simeq-u_{0}$ for large
$\ \xi$ (the parabola becomes very narrow and the right and left branches of
the graph of $R$ are intersected close to the asymptotes)
\begin{figure}[ptb
\centering
\ifcase\msipdfoutput
\includegraphics[
height=2.4102in,
width=3.2543in
{PierceAmp.eps
\else
\includegraphics[
height=2.4102in,
width=3.2543in
{D:/alatex/Preparation/Reyes/lagsys/arxiv2/graphics/PierceAmp__3.pdf
\fi
\caption{Pierce's dispersion relation for $u_{0}=v_{1}$: $\ $For large $\xi,$
the parabola $y=-\xi(v-u_{0})^{2}$ is very narrow and intersects the graph of
$y=R(v)$ close to the asymptotes: $v_{1}^{+},v_{1}^{-}\approx u_{0}.$
\label{PierceAmp
\end{figure}
Consequently, the identit
\begin{equation}
2\operatorname*{Re}v_{0}+v_{1}^{+}+v_{1}^{-}=2u_{0} \label{kpuu2
\end{equation}
implies that $\operatorname*{Re}v_{0}$, $\operatorname*{Re}v_{0}^{\ast}\simeq
u_{0}$. Therefore, three solutions have real part close to $u_{0}$ and the
remaining real solution is close to $-u_{0}.$The latter corresponds to the
backward wave. In terms of the wavenumber, three solutions have real part
close to $k_{\mathrm{b}}$. By looking for solutions (in $k)$ to the fourth
order equation (\ref{PierRev3}) in the for
\[
k=k_{\mathrm{b}}+\mathrm{i}\delta,
\]
with small (compared to $k_{\mathrm{b}}$) complex $\delta$, Pierce gets rid of
the backward wave. The dispersion relation (\ref{PierRev2}) in terms of
$\delta$ read
\begin{equation}
\left( \mathrm{i}\delta\right) ^{3}\left( 2+\mathrm{i}\delta k_{\mathrm{b
}^{-1}\right) =-L\xi^{-1}k_{\mathrm{b}}^{2}\left( 1+\mathrm{i}\delta
k_{\mathrm{b}}^{-1}\right) ^{2}. \label{kp7uu3
\end{equation}
Neglecting $\mathrm{i}\delta/k_{\mathrm{b}}$ we arrive at Pierce's third
degree equation for $\delta$
\begin{equation}
\delta^{3}=-\frac{Lk_{\mathrm{b}}^{2}\xi^{-1}}{2}\mathrm{i}, \label{kpuu4
\end{equation}
which has three complex roots
\begin{equation}
\delta_{1}=c\mathrm{i},\qquad\delta_{2}=c\left( -\sqrt{3}-\mathrm{i}\right)
/2,\qquad\delta_{3}=c\left( \sqrt{3}-\mathrm{i}\right) /2,\RIfM@\expandafter\text@\else\expandafter\mbox\fi{ where
}c=\sqrt[3]{Lk_{\mathrm{b}}^{2}\xi^{-1}/2}, \label{kpuu5
\end{equation}
corresponding respectively to the unattenuated wave faster than the natural
phase velocity of the circuit ($v_{1}^{+}>v_{1}=u_{0})$, the increasing and
the decreasing waves.
It is clear from the analysis in Section \ref{AppAmplification} that in case
of several identical, non-interacting TLs, only two asymptotes are present in
the graph of $R$. This fact suggests that we can replace such a system by a
single effective line, with modified parameters, interacting with the beam.
Indeed, let $C=\widehat{C}\mathrm{Id}_{n}$, $L=\widehat{L}\mathrm{Id}_{n}$
with $n\geq2$. Then, there are exactly two characteristic velocities $\pm
v_{1}$ where $v_{1}=1/\sqrt{\widehat{L}\widehat{C}}$. According to Section
\ref{AppAmplification} the latter are necessarily characteristic velocities of
the entire system, of multiplicity $n-1$ each. Using the notation from that
section, we hav
\[
D=\widehat{C}^{-1}(1,1,...1)^{T},\RIfM@\expandafter\text@\else\expandafter\mbox\fi{ \ \ \ }d=n\widehat{C}^{-1},\RIfM@\expandafter\text@\else\expandafter\mbox\fi{
\ \ \ \ }\widetilde{D}=\widehat{L}^{-1/2}\widehat{C}^{-1}(1,1,...1)^{T
.\RIfM@\expandafter\text@\else\expandafter\mbox\fi{\
\]
The MTL characteristic function $R(v)$ has the explicit expressio
\begin{equation}
R(v)=\frac{n\widehat{L}^{-1}\widehat{C}^{-2}}{v_{1}^{2}-v^{2}}-n\widehat{C
^{-1}. \label{kpuu6
\end{equation}
If we choose $\widetilde{C}=\widehat{C}/n$ and $\widetilde{L}=n\widehat{L},$
the above function coincides with the characteristic function for one line
with parameters $\widetilde{C}$ and $\widetilde{L},
\[
R(v)=\frac{\widetilde{L}^{-1}\widetilde{C}^{-2}}{v_{1}^{2}-v^{2
}-\widetilde{C}^{-1}.
\]
Since amplification depends only on the complex root of the dispersion
relation, which is a root of the canonical dispersion relation, amplification
factors also coincide.
Actually, a more general assertion holds:
\begin{theorem}
Let $C$ and $L$ be the capacity, respectively inductance matrices of an
$n$-lines MTL. If
\begin{equation}
LC=v_{1}^{-2}Id, \label{linesequivalence
\end{equation}
then the canonical dispersion relation of the system consisting of \ the MTL
and a given beam coincides with the canonical dispersion relation of the
system consisting of a single transmission line with parameters $\widetilde{L
,\widetilde{C}$ defined by
\[
\widetilde{C}^{-1}=\sum_{i,j=1}^{n}(C^{-1})_{ij},\qquad\widetilde{L
=v_{1}^{-2}\widetilde{C}^{-1
\]
and the same beam. Consequently, the amplification factors of both systems coincide.
\end{theorem}
It should be noted that the multiple line system and the reduced (one line)
system above are not equivalent in all respects. Actually, the multiline
system admits oscillatory modes with eigenvelocity $\pm v_{1},$ whereas the
equivalent one-line system does not. However, the exponentially growing and
evanescent modes coincide, as well as the two purely oscillatory modes with
eigenvelocities different from $\pm v_{1}.$ The proof of the above theorem is
a straightforward generalization of the case of identical lines and we omit it.
A different reduction can be achieved by suitably modifying the beam. Suppose
we have $n$ identical, uncoupled lines as before. Dividing \ the dispersion
relation (\ref{kpuu6}) by $n,$ we conclude that the interaction of the system
with a beam with parameters $\left( \xi,u_{0}\right) $ is equivalent to the
interaction of one line with parameters $\widehat{L}$, $\widehat{C}$ with a
beam with parameters $\left( \xi/n,u_{0}\right) $. The asymptotic formula
(\ref{AmpFactorAt0}) then implies that the amplification factor grows like
$\sqrt{n}$ as $n\rightarrow\infty$.
\section{Mathematical subjects\label{MathSubj}}
\subsection{ de Donder-Weyl version of the Hamiltonian
formalism\label{AppdeDonder}}
In this section we introduce basic settings of the de Donder-Weyl (DW) version
of the Hamilton equations which treats the time and space variable in equal
manner just as the Lagrangian approach which constitutes its basis. The DW
theory is a generalization of the standard Hamiltonian formalism and the
Hamilton-Jacobi theory, \cite[4.2]{Rund} that has the advantage of requiring a
finite-dimensional phase space. We do not use any significant results of the
DW theory but rather take advantage of its set up that allows to treat the
time $t$ and the space variable $z$ on equal footing. We remind that the
standard Hamilton-Jacobi theory gives preferential treatment to time $t$.
Let us consider a system $\mathsf{q}=\left\{ \mathsf{q}_{j}\left(
t,z\right) ,\ j=1,\ldots n\right\} $ of real valued fields depending on time
$t$ and one-dimensional space variable $z$. Suppose it has a Lagrangian
density of the for
\begin{equation}
\mathcal{L}=\mathcal{L}\left( t,z,\mathsf{q},\mathsf{q}_{,t},\mathsf{q
_{,z}\right) ,\RIfM@\expandafter\text@\else\expandafter\mbox\fi{ where }\mathsf{q}_{,t}=\partial_{t}\mathsf{q
,\ \mathsf{q}_{,z}=\partial_{z}\mathsf{q}. \label{donwe1
\end{equation}
The corresponding Euler-Lagrange equations are, \cite[4.16]{GelFom
\begin{equation}
\frac{\partial\mathcal{L}}{\partial\mathsf{q}}-\partial_{t}\frac
{\partial\mathcal{L}}{\partial\mathsf{q}_{,t}}-\partial_{z}\frac
{\partial\mathcal{L}}{\partial\mathsf{q}_{,z}}=0. \label{donwe2
\end{equation}
Evidently, (\ref{donwe2}) is a system of second order partial differential
equations for $\mathsf{q}$ as a function of $t,z$. It can be recast as a first
order partial differential system with respect to time $t$ or with respect to
the space variable $z$ using a generalization of the standard Hamiltonian
formalism known as de Donder-Weyl (DW) theory. Thus, following the DW theory
we introduce two canonical momenta densities $\mathsf{p}_{t}$ and
$\mathsf{p}_{z}$ and the DW Hamiltonian density $\mathcal{H}$ by the formula
\begin{gather}
\mathsf{p}_{t}=\frac{\partial\mathcal{L}}{\partial\mathsf{q}_{,t}}\left(
t,z,\mathsf{q},\mathsf{q}_{,t},\mathsf{q}_{,z}\right) ,\label{donwe3a}\\
\mathsf{p}_{z}=\frac{\partial\mathcal{L}}{\partial\mathsf{q}_{,z}}\left(
t,z,\mathsf{q},\mathsf{q}_{,t},\mathsf{q}_{,z}\right) ,\label{donwe3b}\\
\mathcal{H}_{\mathrm{DW}}=\mathcal{H}_{\mathrm{DW}}\left( t,z,\mathsf{q
,\mathsf{p}_{t},\mathsf{p}_{z}\right) =\mathsf{p}_{t}^{\mathrm{T}
\mathsf{q}_{,t}+\mathsf{p}_{z}^{\mathrm{T}}\mathsf{q}_{,z}-\mathcal{L}\left(
t,z,\mathsf{q},\mathsf{q}_{,t},\mathsf{q}_{,z}\right) , \label{donwe3c
\end{gather}
where $\mathsf{q}_{,t}$ and $\mathsf{q}_{,z}$ are supposed to be found from
respective equations (\ref{donwe3a})-(\ref{donwe3b}) and to be substituted in
the right-hand side for the second equation in (\ref{donwe3c}). Then the
corresponding DW version of the Hamilton equations ar
\begin{gather}
\partial_{t}\mathsf{q}=\frac{\partial\mathcal{H}_{\mathrm{DW}}}{\partial
\mathsf{p}_{t}}\left( t,z,\mathsf{q},\mathsf{p}_{t},\mathsf{p}_{z}\right)
,\label{donwe4a}\\
\partial_{z}\mathsf{q}=\frac{\partial\mathcal{H}_{\mathrm{DW}}}{\partial
\mathsf{p}_{z}}\left( t,z,\mathsf{q},\mathsf{p}_{t},\mathsf{p}_{z}\right)
,\label{donwe4b}\\
\partial_{t}\mathsf{p}_{t}+\partial_{z}\mathsf{p}_{z}=-\frac{\partial
\mathcal{H}_{\mathrm{DW}}}{\partial\mathsf{q}}\left( t,z,\mathsf{q
,\mathsf{p}_{t},\mathsf{p}_{z}\right) , \label{donwe4c
\end{gather}
and this system of $3n$ first order equations is equivalent to the
Euler-Lagrange system (\ref{donwe2}).
One can solve the system (\ref{donwe3a}) -(\ref{donwe3b}) for $\mathsf{q
_{,t}$ and $\mathsf{q}_{,z}$ in terms of the momenta, obtaining
representations
\begin{equation}
\mathsf{q}_{,t}=G_{t}\left( t,z,\mathsf{q},\mathsf{p}_{t},\mathsf{p
_{z}\right) ,\qquad\mathsf{q}_{,z}=G_{z}\left( t,z,\mathsf{q},\mathsf{p
_{t},\mathsf{p}_{z}\right) , \label{donwe5a
\end{equation}
for some functions $G_{t}$ and $G_{z}$. Solving for $\mathsf{p}_{t}$ in the
first and for $\mathsf{p}_{z}$ in the second, we get
\begin{equation}
\mathsf{p}_{t}=K_{t}\left( t,z,\mathsf{q},\mathsf{q}_{,t},\mathsf{p
_{z}\right) ,\qquad\mathsf{p}_{z}=K_{z}\left( t,z,\mathsf{q},\mathsf{p
_{t},\mathsf{q}_{,z}\right) , \label{donwe5b
\end{equation}
for some functions $K_{t}$ and $K_{z}$.
To obtain the first order partial differential equations with respect to $t$
we consider the pair $\mathsf{p}_{t},\mathsf{q}$ and using equations
(\ref{donwe4a}) and (\ref{donwe4c}) we ge
\begin{gather}
\partial_{t}\mathsf{q}=\frac{\partial H_{\mathrm{DW}}}{\partial\mathsf{p}_{t
}\left( t,z,\mathsf{q},\mathsf{p}_{t},\mathsf{p}_{z}\right) =F_{\mathsf{q
}\left( t,z,\mathsf{q},\mathsf{q}_{,z},\mathsf{p}_{t}\right)
,\label{donwe6a}\\
\partial_{t}\mathsf{p}_{t}=-\partial_{z}\mathsf{p}_{z}-\frac{\partial
H_{\mathrm{DW}}}{\partial\mathsf{q}}\left( t,z,\mathsf{q},\mathsf{p
_{t},\mathsf{p}_{z}\right) =F_{\mathsf{p}}\left( t,z,\mathsf{q
,\mathsf{q}_{,z},\mathsf{q}_{,zz},\mathsf{p}_{t},\mathsf{p}_{t,z}\right)
,\nonumber
\end{gather}
where the expressions $F_{\mathsf{q}}$ and $F_{\mathsf{p}}$ are obtained by
replacing $\mathsf{p}_{z}$ in (\ref{donwe6a}) by its representation
(\ref{donwe5b}). Observe that the system of partial differential equations
(\ref{donwe6a}) for $\mathsf{p}_{t}$ and $\mathsf{q}$ is of the first order
with respect to time $t$.
To obtain the first order partial differential equations with respect to $z$
we consider the pair $\mathsf{p}_{z},\mathsf{q}$ and\ proceed just as in the
previous case with using the equations (\ref{donwe4b}) and (\ref{donwe4c}) to
ge
\begin{gather}
\partial_{z}\mathsf{q}=\frac{\partial H_{\mathrm{DW}}}{\partial\mathsf{p}_{z
}\left( t,z,\mathsf{q},\mathsf{p}_{t},\mathsf{p}_{z}\right) =\widetilde{F
_{\mathsf{q}}\left( t,z,\mathsf{q},\mathsf{q}_{,t},\mathsf{p}_{z}\right)
,\label{donwe6b}\\
\partial_{z}\mathsf{p}_{z}=-\partial_{t}\mathsf{p}_{t}-\frac{\partial
H_{\mathrm{DW}}}{\partial\mathsf{q}}\left( t,z,\mathsf{q},\mathsf{p
_{t},\mathsf{p}_{z}\right) =\widetilde{F}_{\mathsf{p}}\left( t,z,\mathsf{q
,\mathsf{q}_{,t},\mathsf{q}_{,tt},\mathsf{p}_{z},\mathsf{p}_{z,t}\right)
,\nonumber
\end{gather}
where the expressions $\widetilde{F}_{\mathsf{q}}$ and $\widetilde{F
_{\mathsf{p}}$ are determined by replacing $\mathsf{p}_{t}$ in the relevant
expressions in (\ref{donwe6b}) by its representation (\ref{donwe5b}). Observe
that the system of partial differential equations (\ref{donwe6b}) for
$\mathsf{p}_{z}$ and $\mathsf{q}$ is of the first order with respect to the
space variable $z$.
Summing up, we have proved the following
\begin{theorem}
The second order Euler-Lagrange system (\ref{donwe2}) is equivalent to either
the first order system (\ref{donwe6a}) for $\mathsf{q}$ and $\mathsf{p}_{t}$
or the first order system (\ref{donwe6b}) for $\mathsf{q}$ and $\mathsf{p
_{z}$.
\end{theorem}
\subsection{ Quadratic Lagrangian densities\label{AppQuadLag}}
In this section we present some results concerning a special family of
Lagrangians, namely those quadratic in the derivatives (and independent both
of coordinates and the fields). \ This kind of Lagrangians often appear in
practice, in particular in the TL-beam interaction system. Thus, let us
consider a quadratic Lagrangian density of the for
\begin{equation}
\mathcal{L(}\mathsf{q}_{,t},\mathsf{q}_{,z})=\frac{1}{2}\partial_{t
\mathsf{q}^{\mathrm{T}}\alpha\partial_{t}\mathsf{q}+\partial_{t
\mathsf{q}^{\mathrm{T}}\theta\partial_{z}\mathsf{q}-\frac{1}{2}\partial
_{z}\mathsf{q}^{\mathrm{T}}\eta\partial_{z}\mathsf{q}, \label{qulaq1
\end{equation}
where $\mathsf{q}=\left\{ \mathsf{q}_{j}\left( t,z\right) ,\ j=1,\ldots
n\right\} $ are real valued fields depending on time $t$ and one-dimensional
space variable $z$, $\mathsf{q}_{,t}=\partial_{t}\mathsf{q,}$ $\mathsf{q
_{,z}=\partial_{z}\mathsf{q}$ and $\alpha\left( t,z\right) $, $\eta\left(
t,z\right) $, $\theta\left( t,z\right) $ are symmetric $n\times n$ matrices
with real entries, that is
\begin{equation}
\alpha^{\mathrm{T}}=\alpha,\qquad\eta^{\mathrm{T}}=\eta,\qquad\theta
^{\mathrm{T}}=\theta. \label{qulaq2
\end{equation}
The Lagrangian density (\ref{qulaq1}) can be recast into the following form,
involving a block matrix
\begin{equation}
\mathcal{L}=\frac{1}{2}\mathsf{u}^{\mathrm{T}}M_{\mathrm{L}}\mathsf{u;}\qquad
M_{\mathrm{L}}=\left[
\begin{array}
[c]{ll
\alpha & \theta\\
\theta & -\eta
\end{array}
\right] ,\qquad\mathsf{u}=\left[
\begin{array}
[c]{l
\partial_{t}\mathsf{q}\\
\partial_{z}\mathsf{q
\end{array}
\right] . \label{qulaq3
\end{equation}
The Euler-Lagrange equation (\ref{donwe2}) for this Lagrangian is
\begin{equation}
\left[ \partial_{t}\alpha\partial_{t}+\partial_{t}\theta\partial_{z
+\partial_{z}\theta\partial_{t}-\partial_{z}\eta\partial_{z}\right]
\mathsf{q}=0. \label{qulaq4
\end{equation}
Now we would like to use the DW Hamiltonian approach from the previous section
to recast the second order differential $n\times n$ system (\ref{qulaq4}) into
first order ones with respect to $t$ and with respect to $z$ as well. With
that in mind we introduce the canonical momenta as in (\ref{donwe3a
)-(\ref{donwe3b}
\begin{equation}
\mathsf{p}_{t}=\frac{\partial\mathcal{L}}{\partial\mathsf{q}_{,t}
=\alpha\partial_{t}\mathsf{q}+\theta\partial_{z}\mathsf{q},\qquad
\mathsf{p}_{z}=\frac{\partial\mathcal{L}}{\partial\mathsf{q}_{,z}
=\theta\partial_{t}\mathsf{q}-\eta\partial_{z}\mathsf{q}, \label{qulaq5
\end{equation}
which can be recast a
\begin{equation}
\mathsf{p}=\left[
\begin{array}
[c]{l
\mathsf{p}_{t}\\
\mathsf{p}_{z
\end{array}
\right] =\left[
\begin{array}
[c]{ll
\alpha & \theta\\
\theta & -\eta
\end{array}
\right] \left[
\begin{array}
[c]{l
\partial_{t}\mathsf{q}\\
\partial_{z}\mathsf{q
\end{array}
\right] =M_{\mathrm{L}}\mathsf{u}, \label{qulaq5a
\end{equation}
o
\begin{equation}
\left[
\begin{array}
[c]{l
\partial_{t}\mathsf{q}\\
\partial_{z}\mathsf{q
\end{array}
\right] =\mathsf{u}=M_{\mathrm{L}}^{-1}\mathsf{p}=M_{\mathrm{L}}^{-1}\left[
\begin{array}
[c]{l
\mathsf{p}_{t}\\
\mathsf{p}_{z
\end{array}
\right] . \label{qulaq5b
\end{equation}
Notice that the difference in signs in expressions for momenta $\mathsf{p
_{t}$ and $\mathsf{p}_{z}$ in (\ref{qulaq5}) is due to difference in signs for
matrices $\alpha$ and $\eta$ as they enter the expressions for the kinetic and
potential energies in the Lagrangian density defined by (\ref{qulaq1}).
Solving equations (\ref{qulaq5a}) for $\partial_{t}\mathsf{q}$ and
$\partial_{z}\mathsf{q}$ we obtai
\begin{equation}
\partial_{t}\mathsf{q}=\alpha^{-1}\left( \mathsf{p}_{t}-\theta\partial
_{z}\mathsf{q}\right) ,\qquad\partial_{z}\mathsf{q}=\eta^{-1}\left(
\theta\partial_{t}\mathsf{q}-\mathsf{p}_{z}\right) . \label{qulaq5c
\end{equation}
Using (\ref{qulaq1}) and (\ref{qulaq5}) we get the following identity
\begin{gather}
\mathsf{p}_{t}^{\mathrm{T}}\partial_{t}\mathsf{q}+\mathsf{p}_{z}^{\mathrm{T
}\partial_{z}\mathsf{q}=\partial_{t}\mathsf{q}^{\mathrm{T}}\mathsf{p
_{t}+\partial_{z}\mathsf{q}^{\mathrm{T}}\mathsf{p}_{z}=\label{qulaq5ca}\\
=\partial_{t}\mathsf{q}^{\mathrm{T}}\left( \alpha\partial_{t}\mathsf{q
+\theta\partial_{z}\mathsf{q}\right) +\partial_{z}\mathsf{q}^{\mathrm{T
}\left( \theta\partial_{t}\mathsf{q}-\eta\partial_{z}\mathsf{q}\right)
=2L.\nonumber
\end{gather}
Then in view of (\ref{qulaq5ca}) the general DW Hamiltonian $\mathcal{H
_{\mathrm{DW}}$ defined by (\ref{donwe3c}) takes here the for
\begin{equation}
\mathcal{H}_{\mathrm{DW}}=\mathsf{p}_{t}^{\mathrm{T}}\partial_{t
\mathsf{q}+\mathsf{p}_{z}^{\mathrm{T}}\partial_{z}\mathsf{q}-\mathcal{L
=\mathcal{L}=\frac{1}{2}\partial_{t}\mathsf{q}^{\mathrm{T}}\alpha\partial
_{t}\mathsf{q}+\partial_{t}\mathsf{q}^{\mathrm{T}}\theta\partial_{z
\mathsf{q}-\frac{1}{2}\partial_{z}\mathsf{q}^{\mathrm{T}}\eta\partial
_{z}\mathsf{q}. \label{qulaq5cb
\end{equation}
Another way to obtain a representation for the DW Hamiltonian is to use
(\ref{qulaq5b}) yieldin
\begin{equation}
\mathcal{H}_{\mathrm{DW}}=\mathsf{p}^{\mathrm{T}}\mathsf{u}-\frac{1
{2}\mathsf{u}^{\mathrm{T}}M_{\mathrm{L}}\mathsf{u}=\mathsf{u}^{\mathrm{T
}M_{\mathrm{L}}\mathsf{u}-\frac{1}{2}\mathsf{u}^{\mathrm{T}}M_{\mathrm{L
}\mathsf{u}=\frac{1}{2}\mathsf{u}^{\mathrm{T}}M_{\mathrm{L}}\mathsf{u
=\mathcal{L}=\frac{1}{2}\mathsf{p}^{\mathrm{T}}M_{\mathrm{L}}^{-1}\mathsf{p}.
\label{qulaq5d
\end{equation}
Observe that the DW Hamiltonian $\mathcal{H}$ equals the Lagrangian
$\mathcal{L}$ at the corresponding point, that i
\begin{equation}
\mathcal{H}_{\mathrm{DW}}=\mathcal{H}_{\mathrm{DW}}\left( \mathsf{p}\right)
=\mathcal{L}\left( \mathsf{u}\right) =\mathcal{L},\RIfM@\expandafter\text@\else\expandafter\mbox\fi{ where
\mathsf{p}=M_{\mathrm{L}}\mathsf{u} \label{qulaq5da
\end{equation}
(actually, this is a general property of the Legendre transform of homogeneous
quadratic polynomials). The equation (\ref{donwe4c}) takes here the for
\begin{equation}
\partial_{t}\mathsf{p}_{t}+\partial_{z}\mathsf{p}_{z}=0. \label{qulaq5e
\end{equation}
To obtain the first order equations with respect to $t$ we pick the pair
$\mathsf{p}_{t},\mathsf{q}$. We use equations (\ref{qulaq5e}) and
(\ref{qulaq5c}) for respectively $\partial_{t}\mathsf{p}_{t}$ and
$\partial_{t}\mathsf{q}$. We eliminate $\mathsf{p}_{z}$ in (\ref{qulaq5e}) by
using its representation (\ref{qulaq5}) getting the syste
\begin{gather}
\partial_{t}\mathsf{p}_{t}=-\partial_{z}\mathsf{p}_{z}=-\partial_{z
\theta\partial_{t}\mathsf{q}+\partial_{z}\eta\partial_{z}\mathsf{q
=-\partial_{z}\theta\alpha^{-1}\left( \mathsf{p}_{t}-\theta\partial
_{z}\mathsf{q}\right) +\partial_{z}\eta\partial_{z}\mathsf{q},
\label{qulaq6a}\\
\partial_{t}\mathsf{q}=\alpha^{-1}\left( \mathsf{p}_{t}-\theta\partial
_{z}\mathsf{q}\right) . \label{qulaq6b
\end{gather}
Observe that we used equation (\ref{qulaq6b}) to get the right-hand side of
equation (\ref{qulaq6a}). The above system can be written in matrix form
\begin{equation}
\partial_{t}\left[
\begin{array}
[c]{l
\mathsf{p}_{t}\\
\mathsf{q
\end{array}
\right] =\left[
\begin{array}
[c]{ll
-\partial_{z}\theta\alpha^{-1} & \partial_{z}\eta\partial_{z}+\partial
_{z}\theta\alpha^{-1}\theta\partial_{z}\\
\alpha^{-1} & -\alpha^{-1}\theta\partial_{z
\end{array}
\right] \left[
\begin{array}
[c]{l
\mathsf{p}_{t}\\
\mathsf{q
\end{array}
\right] . \label{qulaq6c
\end{equation}
One can recast the above system into a canonical Hamiltonian form by using the
following symplectic matri
\begin{equation}
J=\left[
\begin{array}
[c]{cc
0 & -\mathbf{1}\\
\mathbf{1} & 0
\end{array}
\right] ,\quad J^{2}=-\mathbf{1},\quad J=-J^{\mathrm{T}}. \label{qulaq6d
\end{equation}
Namely
\begin{equation}
\partial_{t}V=JM_{\mathrm{Ht}}V,\quad V=\left[
\begin{array}
[c]{l
\mathsf{p}_{t}\\
\mathsf{q
\end{array}
\right] \label{qulaq6e
\end{equation}
wher
\begin{gather}
M_{\mathrm{Ht}}=\left[
\begin{array}
[c]{ll
\alpha^{-1} & -\alpha^{-1}\theta\partial_{z}\\
\partial_{z}\theta\alpha^{-1} & -\partial_{z}\theta\alpha^{-1}\theta
\partial_{z}-\partial_{z}\eta\partial_{z
\end{array}
\right] =\label{qulaq6f}\\
=\left[
\begin{array}
[c]{cc
\mathbf{1} & 0\\
\partial_{z}\theta & \mathbf{1
\end{array}
\right] \left[
\begin{array}
[c]{cc
\alpha^{-1} & 0\\
0 & -\partial_{z}\eta\partial_{z
\end{array}
\right] \left[
\begin{array}
[c]{cc
\mathbf{1} & -\theta\partial_{z}\\
0 & \mathbf{1
\end{array}
\right] .\nonumber
\end{gather}
To obtain the first order equations with respect to $z$ we pick the pair
$\mathsf{p}_{z},\mathsf{q}$. We use equations (\ref{qulaq5e}) and
(\ref{qulaq5c}) for respectively $\partial_{z}\mathsf{p}_{z}$ and
$\partial_{z}\mathsf{q}$. We eliminate $\mathsf{p}_{t}$ in (\ref{qulaq5e}) by
using its representation (\ref{qulaq5}) getting the syste
\begin{gather}
\partial_{z}\mathsf{p}_{z}=-\partial_{t}\mathsf{p}_{t}=-\partial_{t}\left(
\alpha\partial_{t}\mathsf{q}+\theta\partial_{z}\mathsf{q}\right)
=-\partial_{t}\alpha\partial_{t}\mathsf{q}-\partial_{t}\theta\eta^{-1}\left(
\theta\partial_{t}\mathsf{q}-\mathsf{p}_{z}\right) ,\label{qulaq7}\\
\partial_{z}\mathsf{q}=\eta^{-1}\left( \theta\partial_{t}\mathsf{q
-\mathsf{p}_{z}\right) . \label{qulaq7a
\end{gather}
Observe that we used equation (\ref{qulaq7a}) to get the right-hand side of
equation (\ref{qulaq7}). The above system can be written as
\begin{equation}
\partial_{z}\left[
\begin{array}
[c]{l
\mathsf{p}_{z}\\
\mathsf{q
\end{array}
\right] =\left[
\begin{array}
[c]{ll
\partial_{t}\theta\eta^{-1} & -\partial_{t}\alpha\partial_{t}-\partial
_{t}\theta\eta^{-1}\theta\partial_{t}\\
-\eta^{-1} & \eta^{-1}\theta\partial_{t
\end{array}
\right] \left[
\begin{array}
[c]{l
\mathsf{p}_{z}\\
\mathsf{q
\end{array}
\right] . \label{qulaq7b
\end{equation}
The system (\ref{qulaq7b}) can be transformed into the following canonical
Hamiltonian for
\begin{equation}
\partial_{z}V=JM_{\mathrm{Hz}}V,\quad V=\left[
\begin{array}
[c]{l
\mathsf{p}_{z}\\
\mathsf{q
\end{array}
\right] , \label{qulaq7c
\end{equation}
wher
\begin{gather}
M_{\mathrm{Hz}}=\left[
\begin{array}
[c]{ll
-\eta^{-1} & \eta^{-1}\theta\partial_{t}\\
-\partial_{t}\theta\eta^{-1} & \partial_{t}\alpha\partial_{t}+\partial
_{t}\theta\eta^{-1}\theta\partial_{t
\end{array}
\right] =\label{qulaq7d}\\
=\left[
\begin{array}
[c]{cc
\mathbf{1} & 0\\
\partial_{t}\theta & \mathbf{1
\end{array}
\right] \left[
\begin{array}
[c]{ll
-\eta^{-1} & 0\\
0 & \partial_{t}\alpha\partial_{t
\end{array}
\right] \left[
\begin{array}
[c]{cc
\mathbf{1} & -\theta\partial_{t}\\
0 & \mathbf{1
\end{array}
\right] .\nonumber
\end{gather}
Comparing expressions (\ref{qulaq6f}) and (\ref{qulaq7d}) we observe a
noticeable difference in signs that is explained by the difference in signs in
the expressions for the kinetic and potential energies in the Lagrangian
density defined by (\ref{qulaq1}).
We can transform the system (\ref{qulaq7c})-(\ref{qulaq7d}) further yet into
another form intimately related to the energy conservation law. For that we
begin with the identit
\begin{gather}
M_{\mathrm{Hz}}=\left[
\begin{array}
[c]{ll
-\eta^{-1} & \eta^{-1}\theta\partial_{t}\\
-\partial_{t}\theta\eta^{-1} & \partial_{t}\alpha\partial_{t}+\partial
_{t}\theta\eta^{-1}\theta\partial_{t
\end{array}
\right] =\label{mhzne1}\\
=\left[
\begin{array}
[c]{cc
\mathbf{1} & 0\\
0 & -\partial_{t
\end{array}
\right] \left[
\begin{array}
[c]{ll
-\eta^{-1} & \eta^{-1}\theta\\
\theta\eta^{-1} & -\alpha-\theta\eta^{-1}\theta
\end{array}
\right] \left[
\begin{array}
[c]{cc
\mathbf{1} & 0\\
0 & \partial_{t
\end{array}
\right] .\nonumber
\end{gather}
Based on (\ref{mhzne1}), the system (\ref{qulaq7c})-(\ref{qulaq7d}) can be
recast into the following "Hamiltonian" form
\begin{equation}
\tilde{J}\partial_{z}V=\mathrm{i}\partial_{t}\tilde{M}V,\quad V=\left[
\begin{array}
[c]{l
\mathsf{p}_{z}\\
\partial_{t}\mathsf{q
\end{array}
\right] , \label{mhzne2
\end{equation}
wher
\begin{equation}
\tilde{J}=\left[
\begin{array}
[c]{cc
0 & \mathrm{i}\mathbf{1}\\
\mathrm{i}\mathbf{1} & 0
\end{array}
\right] ,\qquad\tilde{M}=\left[
\begin{array}
[c]{ll
-\eta^{-1} & \eta^{-1}\theta\\
\theta\eta^{-1} & -\alpha-\theta\eta^{-1}\theta
\end{array}
\right] . \label{mhzne3
\end{equation}
When deriving the Hamiltonian equation (\ref{mhzne2})-(\ref{mhzne3}) we used
the following identity relating $\tilde{J}$ and $J$ defined in (\ref{qulaq6d}
\begin{equation}
\left[
\begin{array}
[c]{cc
\mathbf{1} & 0\\
0 & \partial_{t
\end{array}
\right] J\left[
\begin{array}
[c]{cc
\mathbf{1} & 0\\
0 & -\partial_{t
\end{array}
\right] =-\mathrm{i}\partial_{t}\left[
\begin{array}
[c]{cc
0 & \mathrm{i}\mathbf{1}\\
\mathrm{i}\mathbf{1} & 0
\end{array}
\right] =-\mathrm{i}\partial_{t}\tilde{J} \label{mhzne4a
\end{equation}
Note that the matrices $\tilde{J}$ and $\tilde{M}$ are respectively
antihermitian and hermitian, that i
\begin{equation}
\tilde{J}^{\ast}=-\tilde{J},\qquad\tilde{M}^{\ast}=\tilde{M}. \label{mhzne4
\end{equation}
Notice also that the definitions of $V$ and $\tilde{J}$ in (\ref{mhzne2
)-(\ref{mhzne3}) imply the identit
\begin{equation}
V^{\ast}\tilde{J}V=\mathrm{i}\left[ \mathsf{p}_{z}^{\ast}\partial
_{t}\mathsf{q}+\left( \partial_{t}\mathsf{q}\right) ^{\ast}\mathsf{p
_{z}\right] =2\mathrm{i\operatorname{Re}}\left\{ \left( \partial
_{t}\mathsf{q}\right) ^{\ast}\mathsf{p}_{z}\right\} , \label{mhzne5
\end{equation}
which via the theory of Hamiltonian equations can be associated with the
energy conservation law as we show below.
\subsection{Canonical and Hamilton equations\label{AppCanHam}}
In this section we provide a concise review of canonical and Hamilton
equations following \cite[II.3.1-4]{YakSta1}. By \emph{canonical} we call an
equation of the for
\begin{equation}
\tilde{J}\frac{dz}{dt}=\tilde{H}\left( t\right) z, \label{Jzhz1
\end{equation}
where $\tilde{H}\left( t\right) $ is a $2n\times2n$ symmetric matrix valued
function with real entries and $\tilde{J}$ is a constant $2n\times2n$
nondegenerate skew-symmetric matrix with real entries, that i
\begin{equation}
\tilde{H}^{\mathrm{T}}\left( t\right) =\tilde{H}\left( t\right)
,\quad\tilde{J}^{\mathrm{T}}=-\tilde{J},\quad\left\vert \tilde{J}\right\vert
\neq0. \label{Jzhz2
\end{equation}
The matrix $\tilde{H}\left( t\right) $ in (\ref{Jzhz1}) is a called
"Hamiltonian" of the equation. A standard form of $2n\times2n$ nondegenerate
skew-symmetric matrix $J$ \ i
\begin{equation}
J_{2n}=\left[
\begin{array}
[c]{cc
0 & -\mathbf{1}_{n}\\
\mathbf{1}_{n} & 0
\end{array}
\right] . \label{Jzhz3
\end{equation}
The canonical equation (\ref{Jzhz1}) can be always reduced to the special for
\begin{equation}
J_{2n}\frac{dx}{dt}=H\left( t\right) x, \label{Jzhz4
\end{equation}
by means of a linear change of variables,\textit{\ i.e.}
\begin{equation}
x=Sz,\qquad\tilde{J}=S^{\mathrm{T}}J_{2n}S,\qquad\tilde{H}\left( t\right)
=S^{\mathrm{T}}H\left( t\right) S \label{Jzhz5
\end{equation}
for some real nondegenerate $2n\times2n$ matrix $S$.
We call an equation \emph{Hamiltonian} if it is of the form (\ref{Jzhz1}) and
(i) $\tilde{H}\left( t\right) $ is a Hermitian matrix with complex valued
entries; (ii) $\tilde{J}$ is a constant nondegenerate antihermitian matrix,
that i
\begin{equation}
\tilde{H}^{\ast}\left( t\right) =\tilde{H}\left( t\right) ,\qquad\tilde
{J}^{\ast}=-\tilde{J},\qquad\left\vert \tilde{J}\right\vert \neq0.
\label{Jzhz6
\end{equation}
Canonical equations are of course Hamiltonian. A Hamiltonian equation
(\ref{Jzhz1}) can be always reduced by a transformation $x=Sz$ \ with a
nondegenerate $S$ \ to the following special for
\begin{equation}
-iG_{0}\frac{dx}{dt}=H_{0}\left( t\right) x, \label{Jzhz7
\end{equation}
where $H_{0}\left( t\right) $ is a Hermitian matrix an
\begin{equation}
G_{0}=\left[
\begin{array}
[c]{cc
\mathbf{1}_{p} & 0\\
0 & -\mathbf{1}_{q
\end{array}
\right] ,\RIfM@\expandafter\text@\else\expandafter\mbox\fi{ where }p+q=2n. \label{Jzhz8
\end{equation}
Any matrix solution $Z\left( t\right) $ to the Hamiltonian equation
(\ref{Jzhz1}) satisfies the identity, \cite[II.3.4]{YakSta1}
\begin{equation}
Z\left( t\right) ^{\ast}\tilde{J}Z\left( t\right) =\tilde{J},
\label{Jzhz9
\end{equation}
and for any two vector solutions $z_{1}\left( t\right) $ and $z_{2}\left(
t\right) $ there hold
\begin{equation}
\left( z_{1}\left( t\right) ,\tilde{J}z_{2}\left( t\right) \right)
=\left[ z_{1}\left( t\right) \right] ^{\ast}\tilde{J}z_{2}\left(
t\right) =\operatorname*{constant}. \label{Jzhz10
\end{equation}
(so called Poincar\'{e} invariant).
\subsection{Energy exchange between subsystems\label{AppEnergyExchange}}
In this section we derive a general formula for the energy flux between two
systems constituting a closed conservative system described by the Lagrangian
$\mathcal{L=L(}\mathsf{q}_{t},\mathsf{q}_{z})$ With the MTLB Lagrangian in
mind let us put $\mathsf{q}=(Q,q)$ and assume that $\mathcal{L} $ can be split
a
\begin{equation}
\mathcal{L}=\mathcal{L}_{1}\left( \partial_{t}Q,\partial_{;z}Q\right)
+\mathcal{L}_{2}\left( \partial_{t}q,\partial_{z}q\right) , \label{gblag1
\end{equation}
where
\[
\partial_{;z}Q=\partial_{z}Q+B\partial_{z}q
\]
and $B$ is a fixed matrix. \ The Lagrangian $\mathcal{L}$ of the general form
(\ref{gblag1}) describes two coupled interacting systems. The special form of
coupling via the modified derivative $\partial_{;z}Q$ in (\ref{gblag1})
resembles the minimal coupling in the charge gauge theory. The variable $q$
plays the role of the gauge field potential and $B$ \ plays the role of
coupling constant.
The corresponding Euler-Lagrange equations are (\ref{gblag3}), (\ref{gblag4}
\begin{equation}
\partial_{t}\frac{\partial\mathcal{L}_{1}}{\partial\partial_{t}Q}+\partial
_{z}\frac{\partial\mathcal{L}_{1}}{\partial\partial_{;z}Q}=0, \label{gblag3
\end{equation
\begin{equation}
\partial_{t}\frac{\partial\mathcal{L}_{2}}{\partial\partial_{t}q}+\partial
_{z}\left[ \frac{\partial\mathcal{L}_{2}}{\partial\partial_{z}q
+\frac{\partial\mathcal{L}_{1}}{\partial\partial_{;z}Q}B\right] =0,
\label{gblag4
\end{equation}
where the derivative $\frac{\partial\mathcal{L}}{\partial Q}$ of the scalar
function $\mathcal{L}$ with respect to a column-vector $Q$ is understood as a
row-vector of the same dimension.
Recall now that the energy conservation law for the entire system has the
form, \cite[38.2-3]{GelFom}, \cite[13.7]{Gold
\begin{equation}
\partial_{t}H+\partial_{z}S=0, \label{gblag5
\end{equation}
where $H$ and $S$ are the energy and energy flux densities defined b
\begin{equation}
H=H_{1}+H_{2},\qquad S=S_{1}+S_{2}, \label{gblag6
\end{equation}
with the following expressions for the individual energies and energy fluxe
\begin{equation}
H_{1}=\frac{\partial\mathcal{L}_{1}}{\partial\partial_{t}Q}\partial
_{t}Q-\mathcal{L}_{1}\left( \partial_{t}Q,\partial_{;z}Q\right) ,\qquad
S_{1}=\frac{\partial\mathcal{L}_{1}}{\partial\partial_{;z}Q}\partial_{t}Q,
\label{gblag7
\end{equation
\begin{equation}
H_{2}=\frac{\partial\mathcal{L}_{2}}{\partial\partial_{t}q}\partial
_{t}q-\mathcal{L}_{2}\left( \partial_{t}q,\partial_{z}q\right) ,\qquad
S_{2}=\left[ \frac{\partial\mathcal{L}_{2}}{\partial\partial_{z}q
+\frac{\partial\mathcal{L}_{1}}{\partial\partial_{;z}Q}B\right] \partial
_{t}q. \label{gbalg8
\end{equation}
The above expressions imply the following identities for the first syste
\begin{gather}
\partial_{t}H_{1}=\frac{\partial\mathcal{L}_{1}}{\partial\partial_{t
Q}\partial_{t}^{2}Q+\partial_{t}\left( \frac{\partial\mathcal{L}_{1
}{\partial\partial_{t}Q}\right) \partial_{t}Q-\frac{\partial\mathcal{L}_{1
}{\partial\partial_{t}Q}\partial_{t}^{2}Q-\frac{\partial\mathcal{L}_{1
}{\partial\partial_{;z}Q}\left( \partial_{tz}^{2}Q+B\partial_{tz
^{2}q\right) =\label{gblag9}\\
=\partial_{t}\left( \frac{\partial\mathcal{L}_{1}}{\partial\partial_{t
Q}\right) \partial_{t}Q-\frac{\partial\mathcal{L}_{1}}{\partial\partial
_{;z}Q}\left( \partial_{tz}^{2}Q+B\partial_{tz}^{2}q\right) ,\nonumber
\end{gather
\begin{equation}
\partial_{z}S_{1}=\partial_{z}\left( \frac{\partial\mathcal{L}_{1}
{\partial\partial_{;z}Q}\right) \partial_{t}Q+\frac{\partial\mathcal{L}_{1
}{\partial\partial_{;z}Q}\partial_{tz}^{2}Q. \label{gblag10
\end{equation}
The equations (\ref{gblag9}), (\ref{gblag10}), combined with the
Euler-Lagrange equations (\ref{gblag3}), yield the following energy
conservation law for the first syste
\begin{equation}
\partial_{t}H_{1}+\partial_{z}S_{1}=-\frac{\partial\mathcal{L}_{1}
{\partial\partial_{;z}Q}B\partial_{tz}^{2}q, \label{gblag11
\end{equation}
where the right-hand side of (\ref{gblag11}) can be interpreted as the power
flow density from the second system into the first one.
Carrying out similar computations for the second system we obtai
\begin{gather}
\partial_{t}H_{2}=\frac{\partial\mathcal{L}_{2}}{\partial\partial_{t
q}\partial_{t}^{2}q+\partial_{t}\left( \frac{\partial\mathcal{L}_{2
}{\partial\partial_{t}q}\right) \partial_{t}q-\frac{\partial\mathcal{L}_{2
}{\partial\partial_{t}q}\partial_{t}^{2}q-\frac{\partial\mathcal{L}_{2
}{\partial\partial_{z}q}\partial_{tz}^{2}q=\label{gblag12}\\
=\partial_{t}\left( \frac{\partial\mathcal{L}_{2}}{\partial\partial_{t
q}\right) \partial_{t}q-\frac{\partial\mathcal{L}_{2}}{\partial\partial_{z
q}\partial_{tz}^{2}q,\nonumber
\end{gather
\begin{equation}
\partial_{z}S_{2}=\partial_{z}\left[ \frac{\partial\mathcal{L}_{2}
{\partial\partial_{z}q}+\frac{\partial\mathcal{L}_{1}}{\partial\partial_{;z
Q}B\right] \partial_{t}q+\left[ \frac{\partial\mathcal{L}_{2}
{\partial\partial_{z}q}+\frac{\partial\mathcal{L}_{1}}{\partial\partial_{;z
Q}B\right] \partial_{tz}^{2}q. \label{gblag13
\end{equation}
Combining equations (\ref{gblag12}) and (\ref{gblag13}) with the
Euler-Lagrange equations (\ref{gblag4}) for the second system we obtain the
following conservation la
\begin{equation}
\partial_{t}H_{2}+\partial_{z}S_{2}=\frac{\partial\mathcal{L}_{1}
{\partial\partial_{;z}Q}B\partial_{tz}^{2}q, \label{gblag14
\end{equation}
where the right-hand side of (\ref{gblag14}) can be interpreted as the power
density flow transferred from the first system into the second one.
Notice that relations (\ref{gblag11}) and (\ref{gblag14}) have right-hand
sides of the same magnitude and opposite signs. This can be viewed as a
manifestation of the conservation of energy for the entire system. Indeed we
recover (\ref{gblag5}) by adding (\ref{gblag11}) and (\ref{gblag14}).
\subsection{Amplification for homogeneous MTLB systems:
proofs.\label{AppAmplification}}
This section contains rigorous formulations and proofs of the assertions made
in Section \ref{AmplMTL-beam}.
\begin{theorem}
Let $\ $the hypotheses in Theorem \ref{TeoremAmplification} hold. Then, there
is a unique pair of complex conjugate solutions $v_{0},$ $v_{0}^{\ast}$ of the
equation $\left\vert \widetilde{A}(v)\right\vert =0$, where $\widetilde{A}(v)$
is defined in (\ref{ddcj1}), (\ref{ddcj2}).
\end{theorem}
\begin{proof}
By our assumption, the equation $\left\vert A(v)\right\vert =\left\vert
-v^{2}L+C^{-1}\right\vert =0$ has exactly $2n$ real roots, $\pm v_{1},\pm
v_{2},...\pm v_{n},$ with $v_{i}>0$ $(\lambda_{i}=v_{i}^{2})$. We assume in
what follows that they are ordered: $0<v_{1}\leq v_{2}\leq...\leq v_{n}$ and
each root is repeated a number of times equal to its multiplicity. \ If
$\left\vert A(v)\right\vert \neq0,$ the following decomposition hold
\begin{equation}
\left\vert \widetilde{A}(v)\right\vert =\left\vert A(v)\right\vert \left[
d-\xi(v-u_{0})^{2}-D^{T}(A(v))^{-1}D\right] . \label{CanonicalFact
\end{equation}
This follows from the following more general fact: if $M$ is a square block
matrix of the for
\[
M=\left[
\begin{array}
[c]{cc
A_{1} & A_{2}\\
A_{3} & A_{4
\end{array}
\right] ,
\]
where $A_{1},A_{4}$ are square matrices with $\left\vert A_{1}\right\vert
\neq0$, the
\[
\left\vert M\right\vert =\left\vert A_{1}\right\vert \left\vert A_{4
-A_{3}A_{1}^{-1}A_{2}\right\vert ,
\]
see e.g. \cite[Lemma 2.8.6, page 108]{Bern}. Observe that in our case
$A_{2}=D$ is a column matrix and $A_{3}=D^{\mathrm{T}}$ is a row matrix. Then,
if $\left\vert A(v)\right\vert \neq0$, $v$ is a root of $\left\vert
\widetilde{A}(v)\right\vert =0$ if and only if it is a root of the equation
\[
-\xi(v-u_{0})^{2}=D^{T}(A(v))^{-1}D-d=:R(v).
\]
The function $R(v)$ above turns out to have very nice properties. A well known
fact from linear algebra concerning simultaneous diagonalization of two
quadratic forms, one of which is positive, assures that there exists a
non-degenerate matrix $P$ such tha
\[
P^{T}A(v)P=\mathrm{diag}\RIfM@\expandafter\text@\else\expandafter\mbox\fi{ }(v):=\left[
\begin{array}
[c]{cccc
v_{1}^{2}-v^{2} & 0 & \cdots & 0\\
0 & v_{2}^{2}-v^{2} & \cdots & 0\\
\vdots & \vdots & \ddots & 0\\
0 & 0 & 0 & v_{n}^{2}-v^{2
\end{array}
\right] .
\]
Consequently
\[
D^{T}(A(v))^{-1}D=\widetilde{D}^{T}\left[
\begin{array}
[c]{cccc
\frac{1}{v_{1}^{2}-v^{2}} & 0 & \cdots & 0\\
0 & \frac{1}{v_{2}^{2}-v^{2}} & \cdots & 0\\
\vdots & \vdots & \ddots & 0\\
0 & 0 & 0 & \frac{1}{v_{n}^{2}-v^{2}
\end{array}
\right] \widetilde{D}
{\displaystyle\sum\limits_{1}^{n}}
\frac{\widetilde{D}_{i}^{2}}{v_{i}^{2}-v^{2}},
\]
where $\widetilde{D}=P^{T}D$. $\ $Therefore
\[
R(v)
{\displaystyle\sum\limits_{1}^{n}}
\frac{\widetilde{D}_{i}^{2}}{v_{i}^{2}-v^{2}}-d
\]
is a rational function defined on the set $\left\{ v:\RIfM@\expandafter\text@\else\expandafter\mbox\fi{ }\left\vert
A(v)\right\vert \neq0\right\} $. It is immediately seen that $R$ is an even
function, exhibiting vertical asymptotes at $v=\pm v_{i}$ if at least one of
the $\widetilde{D}_{k}$ associated to $v_{i}$ does not vanish ($v_{i}$ may be
a multiple root). For $v>0,$ each branch between two consecutive asymptotes is
increasing and they are decreasing for $v<0.$ Moreover, $\lim_{v\rightarrow
\infty}R(v)=-d$ . Also
\begin{gather*}
R(0)+d=D^{T}(A(0))^{-1}D=D^{T}CD
{\displaystyle\sum\limits_{i=1}^{n}}
{\displaystyle\sum\limits_{j=1}^{n}}
C_{ij}D_{i}D_{j}=\\
{\displaystyle\sum\limits_{i=1}^{n}}
{\displaystyle\sum\limits_{j=1}^{n}}
C_{ij}\left[
{\displaystyle\sum\limits_{k=1}^{n}}
(C^{-1})_{ik}\right] \left[
{\displaystyle\sum\limits_{r=1}^{n}}
(C^{-1})_{jr}\right] =\\
{\displaystyle\sum\limits_{k=1}^{n}}
{\displaystyle\sum\limits_{r=1}^{n}}
\left[
{\displaystyle\sum\limits_{i=1}^{n}}
(C^{-1})_{ik
{\displaystyle\sum\limits_{j=1}^{n}}
C_{ij}(C^{-1})_{jr}\right]
{\displaystyle\sum\limits_{k=1}^{n}}
{\displaystyle\sum\limits_{r=1}^{n}}
\left[
{\displaystyle\sum\limits_{i=1}^{n}}
(C^{-1})_{ik}\delta_{ir}\right] \\
{\displaystyle\sum\limits_{k=1}^{n}}
{\displaystyle\sum\limits_{r=1}^{n}}
(C^{-1})_{rk}
{\displaystyle\sum\limits_{k=1}^{n}}
D_{k}=d;
\end{gather*}
hence $R(0)=0.$ Since $C^{-1}$ is non-degenerate, $D\neq0$. Moreover, since
the matrix $P$ is non-degenerate, we have $\widetilde{D}\neq0.$ Therefore, the
graph has at least two vertical asymptotes and always exhibits a central
symmetric branch with the minimum at the point $(0,0).$ The number of real
roots of the equatio
\[
-\xi(v-u_{0})^{2}=R(v)
\]
is the number of intersection points of the parabola $y=f(v):=-\xi
(v-u_{0})^{2}$ and the graph of $R.$ For $\xi$ small, it is exactly the number
of monotonic branches (all branches, except for the central one), which
coincides with the number of asymptotes. This number is always between $2$ and
$2n,$ depending on the number of vanishing $\widetilde{D}_{i}$ and on the
possible multiple roots; a precise description is given below. Moreover, it is
easily seen that whenever $u_{0}\in(0,v_{1}],$ the number of intersection
points is equal to the number of asymptotes irrespective of the value of
$\xi>0$; see Figure \ref{FigAmpl} (a), whereas $\xi$ small is needed
otherwise; indeed, in Figure \ref{FigAmpl} (b) a large value of $\xi$ produces
three points of intersection with the far right branch of the graph of $R$,
making the total number of intersection points exceed by two the number of
asymptotes. If either (i) or (ii) holds, the intersections are transversal,
hence the roots are simple. The previous assertions follow easily and
rigorously from the monotonicity properties of $R$ and $f$ \ but their clear
geometric meaning makes a lengthy proof unnecessary.
So far, we have considered the real roots of the equation$\left\vert
\widetilde{A}(v)\right\vert =0$ in the set $\left\{ v:\det A(v)\neq0\right\}
$. Next, we consider the possible roots of the equation in the complementary
set $\left\{ \pm v_{1},\pm v_{2},...\pm v_{n}\right\} $. Multiplying the
matrix $\widetilde{A}(v)$ by $\widehat{P}^{T}$ from the left and by
$\widehat{P}$ from the right, wher
\[
\widehat{P}=\left[
\begin{array}
[c]{cc
P & 0\\
0 & 1
\end{array}
\right] ,
\]
there follows that the equation $\ \left\vert \widetilde{A}(v)\right\vert =0$
is equivalent to the equatio
\[
\Delta(v):=\left\vert
\begin{array}
[c]{cc
\begin{array}
[c]{cccc
v_{1}^{2}-v^{2} & 0 & .. & 0\\
0 & v_{2}^{2}-v^{2} & .. & \vdots\\
\vdots & \vdots & .. & 0\\
0 & .. & 0 & v_{n}^{2}-v^{2
\end{array}
& \widetilde{D}\\
\widetilde{D}^{T} & d-\xi(v-u_{0})^{2
\end{array}
\right\vert =0,
\]
where, as before, $\widetilde{D}=P^{T}D.$ Let us analyze under what condition
$\pm v_{i}$ are roots of the equation $\Delta(v)=0$. Expanding the determinant
with respect to the last column, and then the $n$-th order minor corresponding
to $\widetilde{D}_{i}$ with respect to its $i$-th column, we get the
expressio
\begin{gather}
\Delta(v)=-\widetilde{D}_{1}^{2}\left\vert
\begin{array}
[c]{cccc
v_{2}^{2}-v^{2} & 0 & .. & 0\\
0 & v_{3}^{2}-v^{2} & .. & \vdots\\
\vdots & \vdots & .. & 0\\
0 & .. & 0 & v_{n}^{2}-v^{2
\end{array}
\right\vert -\widetilde{D}_{2}^{2}\left\vert
\begin{array}
[c]{cccc
v_{1}^{2}-v^{2} & 0 & .. & 0\\
0 & v_{3}^{2}-v^{2} & .. & \vdots\\
\vdots & \vdots & .. & 0\\
0 & .. & 0 & v_{n}^{2}-v^{2
\end{array}
\right\vert -...\label{BigDeterminant}\\
-\widetilde{D}_{n}^{2}\left\vert
\begin{array}
[c]{cccc
v_{1}^{2}-v^{2} & 0 & .. & 0\\
0 & v_{2}^{2}-v^{2} & .. & \vdots\\
\vdots & \vdots & .. & 0\\
0 & \cdots & 0 & v_{n-1}^{2}-v^{2
\end{array}
\right\vert +\left[ d-\xi(v-u_{0})^{2}\right] \left\vert
\begin{array}
[c]{cccc
v_{1}^{2}-v^{2} & 0 & .. & 0\\
0 & v_{2}^{2}-v^{2} & .. & \vdots\\
\vdots & \vdots & .. & 0\\
0 & .. & 0 & v_{n}^{2}-v^{2
\end{array}
\right\vert ,\nonumber
\end{gather}
that is
\begin{equation}
\Delta(v)
{\displaystyle\prod\limits_{i=1}^{n}}
(v_{i}^{2}-v^{2})\left[ d-\xi(v-u_{0})^{2}\right] -\sum_{i=1}^{n
\widetilde{D_{i}}^{2
{\displaystyle\prod\limits_{j\neq i}}
(v_{j}^{2}-v^{2}). \label{BigDetermBIS
\end{equation}
We note in passing that the factorization (\ref{CanonicalFact}) is easily
obtained from the above expression by extracting the factor
\[
\left\vert A(v)\right\vert
{\displaystyle\prod\limits_{i=1}^{n}}
(v_{i}^{2}-v^{2})
\]
under the assumption $v\neq v_{i}.$
Assume first that $\pm v_{i}$ are simple roots of $\left\vert A(v)\right\vert
=0,$ that is, that the binomial $v_{i}^{2}-v^{2}$ appears only once in the
matrix $\mathrm{diag}(v)$. Then, there follows that $\Delta(v_{i})=0$ if and
only if $\widetilde{D}_{i}=0$. Whenever this condition holds, the partial
fraction $\widetilde{D}_{i}^{2}/(v_{i}^{2}-v^{2})$ in the expression of $R$
disappears and the number of asymptotes is reduced by two. The number of real
roots is thus increased by two ( $\pm v_{i})$ and reduced by two, leaving the
total number of roots unaffected.
Let us next consider the case of a multiple root. Assume that $v_{i
=v_{i+1}=...=v_{i+k-1}$, hence the binomial $v_{i}^{2}-v^{2}$ appears $k$
times in $\mathrm{diag}(v)$, $k>1$. Then $\pm v_{i}$ are necessarily roots of
$\Delta(v)=0,$ as it can be readily seen from (\ref{BigDetermBIS}). As for
their multiplicity, there are two cases:
\begin{enumerate}
\item[a)] multiplicity $=k,$ if all of $\widetilde{D}_{i},\widetilde{D
_{i+1},...\widetilde{D}_{i+k-1}$ are zero, since in this case all non-zero
terms in (\ref{BigDetermBIS}) contain $k$ times the factor $v_{i}^{2}-v^{2}$;
\item[b)] multiplicity $=k-1,$ if not all of $\widetilde{D}_{i},\widetilde{D
_{i+1},...\widetilde{D}_{i+k-1}$ are zero, since in this case the terms in
(\ref{BigDetermBIS}) corresponding to the non-zero $\widetilde{D}_{r}$ \ with
$r\in\left\{ i,i+1,...i+k-1\right\} $ contain the factor $v_{i}^{2}-v^{2}$
only $k-1$ times, while the rest contain it $k$ times.
\end{enumerate}
In case (a), all the fractions with denominator $v_{i}^{2}-v^{2}$ are missing
in the rational function $R,$ with consequent reduction of the number of
asymptotes (with respect to the total possible number $2n$) by $2k$, which is
precisely the number of additional roots, counting their multiplicity. Thus
the total number of real roots is unaffected.
In case (b), there is one fraction with denominator $v_{i}^{2}-v^{2}.$Thus the
total number of asymptotes is reduced by $2k-2,$ which is the number of
additional roots, counting their multiplicity.
Summing up, the total number of real roots of $\Delta(v)=0$ (counting their
multiplicity) is exactly $2n$ under our assumptions. Since the total number of
roots of $\Delta(v)=0$ is $2n+2,$ there is necessarily one and only one pair
of complex conjugate roots, thus proving the assertion.
\end{proof}
The following theorem deals with the behavior of amplification as
$\xi\rightarrow0$ and as $\xi\rightarrow\infty$.
\begin{theorem}
Let $v_{0},\overline{v_{0}}$ with $\operatorname*{Im}v_{0}>0$ denote the
unique pair of complex conjugate roots of the equation $\left\vert
\widetilde{A}(v)\right\vert =0$ under the assumptions of Theorem
\ref{TeoremAmplification}. Let $k_{0}=\omega/v_{0}.$ Then
\begin{equation}
-\operatorname*{Im}k_{0}\sim\frac{C}{\sqrt{\xi}}\RIfM@\expandafter\text@\else\expandafter\mbox\fi{ \ as }\xi
\rightarrow0,\RIfM@\expandafter\text@\else\expandafter\mbox\fi{ }C>0. \label{AsBehxitozero
\end{equation}
Under the additional assumption $u_{0}=v_{1}$ we also hav
\begin{equation}
-\operatorname*{Im}k_{0}\sim\frac{C^{\prime}}{\sqrt[3]{\xi}}\RIfM@\expandafter\text@\else\expandafter\mbox\fi{ \ as
\xi\rightarrow\infty,\RIfM@\expandafter\text@\else\expandafter\mbox\fi{ }C^{\prime}>0. \label{AsBehxitoinfty
\end{equation}
\end{theorem}
\begin{proof}
The idea of the proof is to use very detailed information about real roots in
combination with well known Vieta's formulas relating the roots to the
coefficients of the corresponding polynomial. Let us first prove
(\ref{AsBehxitozero}). Denote the $2n$ real roots of the equation $\left\vert
\widetilde{A}(v)\right\vert =0$ by $v_{1}^{+}$, $v_{2}^{+},...$ $v_{n}^{+}$ ;
$-v_{1}^{-},-v_{2}^{-},...-v_{n}^{-}$, where $v_{i}^{+},v_{i}^{-}>0$ and
$0<v_{1}^{+}\leq v_{2}^{+}\leq...\leq$ $v_{n}^{+}$, $0<v_{1}^{-}\leq v_{2
^{-}\leq...\leq v_{n}^{-}$.$\ $The roots are repeated according to their
multiplicity and some of them may coincide with some $v_{i}$; see the proof of
Theorem \ref{TeoremAmplification}. If $n>1,$ the roots $v_{i}^{+}$and
$-v_{i}^{-}$ with $i=1,2,...(n-1)$ \ lie in the interval $\left[ -v_{n
,v_{n}\right] $ for any value of \ $\xi>0$ \ (recall that by $v_{i}$ we
denote the characteristic velocities of the MTL), whereas $v_{n}^{+}$ and
$-v_{n}^{-},$ which correspond to the points of intersection of the parabola
$y=-\xi(v-u_{0})^{2}$ with the farthest right and farthest left branches of
$y=R(v),$ lie outside of this very interval.
The extreme roots $v_{n}^{+}$ and $-v_{n}^{-}$ approach $+\infty$
(respectively $-\infty$) as $\xi\rightarrow0.$ This can be proved as follows:
the parabola $y=-\xi(v-u_{0})^{2}$ is decreasing for $v>u_{0},$ its
intersection with the horizontal asymptote of $R,$ $y=-d,$ is $v^{\ast
=u_{0}+\sqrt{d/\xi}$ and $R(v)<-d$ for $v>v_{n}.$ Therefore, $v_{n
^{+}>v^{\ast}\rightarrow+\infty$ as $\xi\rightarrow0.$ A similar argument can
be applied to $-v_{n}^{-}.$ In order to establish the asymptotic behavior of
$\operatorname{Im}v_{0}$ we will make use of Vieta's formulas, relating the
roots of a polynomial to its coefficients.
We start by observing that $\left\vert \widetilde{A}(v)\right\vert $ is a
polynomial in $v$ of degree $2n+2:
\[
\left\vert \widetilde{A}(v)\right\vert =a_{2n+2}v^{2n+2}+a_{2n+1
v^{2n+1}+...a_{1}v+a_{0.
\]
The coefficients $a_{2n+2},a_{2n+1}$ and $a_{0}$ can be easily computed in
terms of the parameters. Indeed, $a_{0}=\left\vert \widetilde{A}(0)\right\vert
,$ which can be computed by adding the first $n$ rows of $\widetilde{A}(0)$
and subtracting the result from the last. Recalling that $D_{i}
{\displaystyle\sum\limits_{j}}
(C^{-1})_{ij}$ and that $d
{\displaystyle\sum_{i}}
D_{i},$ we obtai
\[
a_{0}=\left\vert \widetilde{A}(0)\right\vert =\left\vert
\begin{array}
[c]{cc
C^{-1} & D\\
0 & -\xi u_{0}^{2
\end{array}
\right\vert =-\xi u_{0}^{2}\left\vert C^{-1}\right\vert .
\]
The only addends in $\left\vert \widetilde{A}(v)\right\vert $ yielding powers
$v^{2n+2}$ or $v^{2n+1}$ are those coming from the product $\left\vert
-v^{2}L+C^{-1}\right\vert \left[ d-\xi(v-u_{0})^{2}\right] $. $\ $Clearly,
the relevant terms ar
\[
(-1)^{n}\left\vert L\right\vert v^{2n}\left[ d-\xi(v-u_{0})^{2}\right]
+...=(-1)^{n+1}\xi\left\vert L\right\vert v^{2n+2}+2(-1)^{n}\xi u_{0
\left\vert L\right\vert v^{2n+1}+...
\]
where the dots stand for lower order in $v$ terms. Consequently
\[
a_{2n+2}=(-1)^{n+1}\xi\left\vert L\right\vert ;\RIfM@\expandafter\text@\else\expandafter\mbox\fi{ \ \ \ \ \
a_{2n+1}=2(-1)^{n}\xi u_{0}\left\vert L\right\vert .
\]
Vieta's formulas then impl
\begin{align}
2\operatorname*{Re}v_{0}
{\displaystyle\sum\limits_{i=1}^{n}}
v_{i}^{+}
{\displaystyle\sum\limits_{i=1}^{n}}
v_{i}^{-} & =-\frac{a_{2n+1}}{a_{2n+2}}=2u_{0}\label{Vieta1}\\
(-1)^{n}\left\vert v_{0}\right\vert ^{2
{\displaystyle\prod\limits_{i=1}^{n}}
v_{i}^{+}v_{i}^{-} & =\frac{a_{0}}{a_{2n+2}}=(-1)^{n}\frac{u_{0
^{2}\left\vert C^{-1}\right\vert }{\left\vert L\right\vert }=(-1)^{n
\frac{u_{0}^{2}}{\left\vert LC\right\vert } \label{Vieta2
\end{align}
Next, we study the behavior as $\xi\rightarrow0$ of both the sum and the
product of the real roots. In the asymptotic formulas below, $K_{1
,K_{2},K_{1}^{\prime},K_{2}^{\prime}$ etc. denote positive constants depending
on $L,C,u_{0}$ but not on $\xi$.
Let $n>1$ and suppose that the graph of $R$ has more than two asymptotes.
First of all, we note that, as $\xi\rightarrow0,$ the parabola becomes flat
and the roots $v_{i}^{+},-v_{i}^{-}$ with $i=1,2,...n-1$ become symmetric due
to the symmetry of the graph of $R$. More precisely, if we denote by
$\widehat{v}_{k}^{+},-\widehat{v}_{k}^{-}$ with $k\in\left\{
1,2,...n-1\right\} $ the abscissas of the points on the $k$-th right
(respectively, $k$-th left) branch of the graph of $R$ for which
$R(\widehat{v}_{k}^{+})=R($ $-\widehat{v}_{k}^{-})=0,$ then clearly $v_{k
^{+}\left( \xi\right) \rightarrow\widehat{v}_{k}^{+},v_{k}^{-}\left(
\xi\right) \rightarrow\widehat{v}_{k}^{-}$ and $\widehat{v}_{k
^{+}=\widehat{v_{k}}^{-}.$ Moreover, since the branches of $R$ are strictly
increasing for $v>0$ and strictly decreasing for $v<0$, $v_{k}^{+}\left(
\xi\right) -\widehat{v}_{k}^{+}\sim-A_{k}\xi,$ $-v_{k}^{-}\left( \xi\right)
+\widehat{v}_{k}^{-}\sim B_{k}\xi$ \ as $\xi\rightarrow0$, with $A_{k
,B_{k}>0.$We also note the following fact, which is used in the proof of
Section \ref{SubsEnergyExchange} and \ is a simple consequence of the lack of
symmetry of the parabola $y=-\xi(v-u_{0})^{2}$ with respect to the vertical
axis: if $v_{k}^{+},-v_{k}^{-}$ is a pair of real roots not belonging to the
set $\left\{ \pm v_{1},\pm v_{2},...\pm v_{n}\right\} $ (and there is at
least one such pair, see the proof of Theorem \ref{TeoremAmplification}), the
\begin{equation}
v_{k}^{+}\left( \xi\right) -v_{k}^{-}\left( \xi\right) >0.
\label{Asymmetry
\end{equation}
Thus in particular $B_{k}>A_{k}$ in the above asymptotic relations.This
inequality can be easily seen on the graph and given a simple analytical proof.
The roots belonging to the set $\left\{ \pm v_{1},\pm v_{2},...\pm
v_{n}\right\} $ are symmetric and do not contribute to their sum. Therefore
\begin{equation
{\displaystyle\sum\limits_{i=1}^{n-1}}
v_{i}^{+}(\xi)
{\displaystyle\sum\limits_{i=1}^{n-1}}
v_{i}^{-}(\xi)=K_{1}\xi+o(\xi)\RIfM@\expandafter\text@\else\expandafter\mbox\fi{ as }\xi\rightarrow0.
\label{SmallRootsSum
\end{equation}
As for the product of roots, we hav
\begin{equation
{\displaystyle\prod\limits_{i=1}^{n-1}}
v_{i}^{+}(\xi)v_{i}^{-}(\xi)=(-1)^{n}K_{2}+K_{3}\xi+o(\xi)\RIfM@\expandafter\text@\else\expandafter\mbox\fi{\ as
\xi\rightarrow0. \label{SmallRootsProduct
\end{equation}
If there are only two asymptotes, then $v_{1}=v_{2}=...=v_{n-1}$ and the
left-hand side in (\ref{SmallRootsSum}) is zero. Also, the left-hand side in
(\ref{SmallRootsProduct}) is the constant $(-1)^{n}K_{2}$ \ Thus this case can
be formally included in (\ref{SmallRootsSum}) and (\ref{SmallRootsProduct}) by
allowing $K_{1}$ and $K_{3}$ to vanish.
Let us now consider the extreme roots. As we noted, $v_{n}^{+},v_{n
^{-}\rightarrow+\infty.$ More precisely, since we hav
\begin{equation}
-\xi(v_{n}^{+}-u_{0})^{2}=R(v_{n}^{+})\rightarrow-d\RIfM@\expandafter\text@\else\expandafter\mbox\fi{ \ as }\xi
\rightarrow0, \label{asymptequality
\end{equation}
then necessarily $\lim_{\xi\rightarrow0}\xi(v_{n}^{+}-u_{0})^{2}=d>0$ and thu
\begin{equation}
v_{n}^{+}(\xi)=\sqrt{\frac{d}{\xi}}+u_{0}+E(\xi),\RIfM@\expandafter\text@\else\expandafter\mbox\fi{ where }E(\xi)=o\left(
\sqrt{\frac{1}{\xi}}\right) \RIfM@\expandafter\text@\else\expandafter\mbox\fi{ \ as }\xi\rightarrow0. \label{beh1
\end{equation}
We need further refinement in the asymptotics of $E(\xi)$ as $\xi
\rightarrow0.$ To this end, we recall tha
\begin{equation}
R(v)+d\sim-\frac{A}{v^{2}}\RIfM@\expandafter\text@\else\expandafter\mbox\fi{ \ as }v\rightarrow\infty,\RIfM@\expandafter\text@\else\expandafter\mbox\fi{\quad}A>0.
\label{beh3
\end{equation}
Replacing $v_{n}^{+}(\xi)$ in (\ref{asymptequality}) by the expression
(\ref{beh1}) and using (\ref{beh3}), we arrive a
\[
2E(\xi)\sqrt{\frac{d}{\xi}}+E(\xi)^{2}\rightarrow A\RIfM@\expandafter\text@\else\expandafter\mbox\fi{ \ as }\xi
\rightarrow0,
\]
which implie
\[
E(\xi)=K_{3}\sqrt{\xi}+o(\sqrt{\xi})\RIfM@\expandafter\text@\else\expandafter\mbox\fi{ \ as }\xi\rightarrow0.
\]
Summing up, we have the following asymptotic representation for $v_{n}^{+}:
\begin{equation}
v_{n}^{+}(\xi)=\sqrt{\frac{d}{\xi}}+u_{0}+K_{3}\sqrt{\xi}+o(\sqrt{\xi})\RIfM@\expandafter\text@\else\expandafter\mbox\fi{
\ as }\xi\rightarrow0. \label{behdef
\end{equation}
An analogous representation takes place for $v_{n}^{-}:
\begin{equation}
-v_{n}^{-}(\xi)=-\sqrt{\frac{d}{\xi}}+u_{0}-K_{3}\sqrt{\xi}+o(\sqrt{\xi
})\RIfM@\expandafter\text@\else\expandafter\mbox\fi{ \ as }\xi\rightarrow0. \label{behdefbis
\end{equation}
Plugging (\ref{behdef}), (\ref{behdefbis}), (\ref{SmallRootsSum}) and
(\ref{SmallRootsProduct}) into (\ref{Vieta1}) and (\ref{Vieta2}) yields
\begin{equation}
\operatorname*{Re}v_{0}=o\left( \sqrt{\xi}\right) \RIfM@\expandafter\text@\else\expandafter\mbox\fi{ };\RIfM@\expandafter\text@\else\expandafter\mbox\fi{
\ \ }\left\vert v_{0}\right\vert ^{2}=K_{4}\xi+o(\xi)\RIfM@\expandafter\text@\else\expandafter\mbox\fi{ \ as
\xi\rightarrow0.
\end{equation}
As a consequence
\[
\operatorname*{Im}v_{0}=\sqrt{K_{4}\xi+o(\xi)}=\sqrt{K_{4}}\sqrt{\xi
+o(\sqrt{\xi})\RIfM@\expandafter\text@\else\expandafter\mbox\fi{ as }\xi\rightarrow0
\]
and, finally
\[
-\operatorname*{Im}k_{0}=\frac{\operatorname*{Im}v_{0}}{\RIfM@\expandafter\text@\else\expandafter\mbox\fi{\ \ }\left\vert
v_{0}\right\vert ^{2}}\sim\frac{K_{5}}{\sqrt{\xi}\RIfM@\expandafter\text@\else\expandafter\mbox\fi{\ }}\RIfM@\expandafter\text@\else\expandafter\mbox\fi{\ as \
\xi\rightarrow0,
\]
thus proving (\ref{AsBehxitozero}) for $n>1$. If $n=1$, (\ref{behdef}) and
(\ref{behdefbis}) hold \ and plugging into (\ref{Vieta1}) and (\ref{Vieta2})
yields the same result.
We turn now to the proof of (\ref{AsBehxitoinfty}), restricting ourselves to
the case of just one line$;$ the case of several lines can be handled in a
similar fashion. First of all, it is clear that $v_{1}^{+}\downarrow u_{0}$
and $-v_{1}^{-}\uparrow-u_{0}$ as $\xi\rightarrow\infty$. This can be
rigorously proved in a way, similar to the above proof of the fact that
$v_{n}^{+}\uparrow\infty,$ $v_{n}^{-}\downarrow-\infty$ as $\xi\rightarrow0$.
Pu
\begin{equation}
v_{1}^{+}(\xi)=u_{0}+G(\xi);\RIfM@\expandafter\text@\else\expandafter\mbox\fi{\quad}G(\xi)>0,\RIfM@\expandafter\text@\else\expandafter\mbox\fi{\quad}G(\xi
)\rightarrow0\RIfM@\expandafter\text@\else\expandafter\mbox\fi{ as }\xi\rightarrow\infty. \label{PierceRegime1
\end{equation}
Near $u_{0}=v_{1}$ we hav
\begin{equation}
R(v)\sim\frac{A}{u_{0}-v}\RIfM@\expandafter\text@\else\expandafter\mbox\fi{ as }v\rightarrow u_{0}^{+}\RIfM@\expandafter\text@\else\expandafter\mbox\fi{ \ with }A>0.
\label{PierceRegime2
\end{equation}
After use of (\ref{PierceRegime1}) and (\ref{PierceRegime2}), the equatio
\[
-\xi(v_{1}^{+}-u_{0})^{2}=R(v_{1}^{+})
\]
yields the following asymptotic relation
\[
\xi G(\xi)^{3}\sim A\RIfM@\expandafter\text@\else\expandafter\mbox\fi{\ as }\xi\rightarrow\infty,
\]
that is
\[
G(\xi)\sim\frac{K_{1}^{\prime}}{\sqrt[3]{\xi}}\RIfM@\expandafter\text@\else\expandafter\mbox\fi{ as }\xi\rightarrow
\infty.
\]
An analogous formula takes place for the negative root
\[
v_{1}^{-}(\xi)=u_{0}+H(\xi)\ \RIfM@\expandafter\text@\else\expandafter\mbox\fi{\ with }H(\xi)\sim\frac{K_{2}^{\prime
}{\sqrt[3]{\xi}}\RIfM@\expandafter\text@\else\expandafter\mbox\fi{\ as }\xi\rightarrow\infty.
\]
Therefore
\[
v_{1}^{+}(\xi)-v_{1}^{-}(\xi)\sim\frac{K_{3}^{\prime}}{\sqrt[3]{\xi
}\RIfM@\expandafter\text@\else\expandafter\mbox\fi{\ as }\xi\rightarrow\infty.
\]
Applying again (\ref{Vieta1}) and (\ref{Vieta2}) we obtai
\begin{equation}
\operatorname{Re}v_{0}=u_{0}+\frac{K_{4}^{\prime}}{\sqrt[3]{\xi}}+o\left(
\frac{1}{\sqrt[3]{\xi}}\right) ,\RIfM@\expandafter\text@\else\expandafter\mbox\fi{\qquad}\left\vert v_{0}\right\vert
\rightarrow1/\sqrt{LC}+\frac{K_{5}^{\prime}}{\sqrt[3]{\xi}}+o\left( \frac
{1}{\sqrt[3]{\xi}}\right) \RIfM@\expandafter\text@\else\expandafter\mbox\fi{\ as }\xi\rightarrow\infty.
\label{PierceRegime3
\end{equation}
Recall that $v_{1}=1/\sqrt{LC}=u_{0}.$ The last two relations imply
$\operatorname*{Im}v_{0}\sim K_{6}^{\prime}/\sqrt[3]{\xi}$. Finally
\[
-\operatorname*{Im}k_{0}=\frac{\operatorname*{Im}v_{0}}{\RIfM@\expandafter\text@\else\expandafter\mbox\fi{\ \ }\left\vert
v_{0}\right\vert ^{2}}\sim\frac{K_{7}^{\prime}}{\sqrt[3]{\xi}}\RIfM@\expandafter\text@\else\expandafter\mbox\fi{\qquad as
}\xi\rightarrow\infty.
\]
as was to be proved.
\end{proof}
One can also verify that if $u_{0}<1/\sqrt{LC}$ then both $\operatorname*{Im
v_{0}$ and $\operatorname*{Im}k_{0}$ have a finite, nonzero limit as
$\xi\rightarrow\infty.$
\textbf{Acknowledgment:} This research was supported by AFOSR MURI Grant
FA9550-12-1-0489 administered through the University of New Mexico. The
authors are grateful to F. Capolino and A. Tamma for helpful discussions.
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| {
"redpajama_set_name": "RedPajamaArXiv"
} | 958 |
Category: FED
Impact of Interest Rates on P2P Lenders
Every time there are talks of a rise in interest rates, financial institutions go into a frenzy. The 2008 recession had forced central banks around the world to drop their rates to historical lows. But with the global economy stabilizing, the US Fed and other central banks have started tightening rates. We will be covering […]
Every time there are talks of a rise in interest rates, financial institutions go into a frenzy. The 2008 recession had forced central banks around the world to drop their rates to historical lows. But with the global economy stabilizing, the US Fed and other central banks have started tightening rates. We will be covering effects of the rising interest rates on three of the biggest P2P markets in the world: the UK, US, and China.
Last year, Bank of England (BoE) reduced the interest rate by 0.25 percent along with a sling of monetary policies to encourage growth. But in October of this year, BoE increased the interest rate, though only marginally by 0.25%. Both of these times, banks and other financial institutions immediately reviewed the mortgage and saving rates, and the effect was passed on to customers. But the change is too small to have any damaging effect on P2P lenders.
Moreover, P2P lenders do not fall directly under the scope of BoE, hence, are not immediately affected by the BoE base rate. Lender return is directly proportional to the demand and supply in the market. For the short term, P2P lenders will not be making any headline-grabbing changes to their rate structure. If the rate does end up crossing 2%, the P2P market will definitely ring up some changes in their rate structure.
Zopa, the pioneering UK-based P2P platform, was able to thrive even when rates were above 5%. But the real reason P2P gained a substantial foothold in the UK was because of the low yield environment. With saving rates below inflation, money lost its value over a period of time when held in savings accounts. This prompted people to go for P2P platforms as alternative saving options even though it had a way higher percentage of risk attached to it. If rates do reach a certain level, the lender might go for the safer option of saving accounts returns rather than the high-risk high-profit approach offered by P2P platforms. But if P2P lending is considered an investment, the yield will always be attractive for a certain class of investors.
During the last decade, the US economy endured the worst financial recession in 80 years. And to counter that, the Fed dropped the rates to 0-0.25% for an extended period of time. But with US unemployment rates at multi-decade lows, the rate has been slowly yet steadily hiked to a range between 1%-1.25%.
Impact of increase in the interest rates
Lending Club, the largest P2P platform in the United States, has mentioned in its prospectus the adverse effect hike in interest rate will have on its business
"… fluctuations in the interest rate environment may discourage investors and borrowers from participating in our marketplace, which may adversely affect our business."
Many retail investors use the P2P platform as an investment option. Since they lend directly to the borrowers, the non-involvement of the intermediaries allows them to enjoy handsome returns as compared to returns they get for depositing into saving accounts.
As the chart above clearly depicts, a number of delinquencies are directly impacted by the increase in the interest rate. The higher the interest rates, the higher are delinquencies. Thus, the low rate environment had led to lower delinquencies. But now, there is a growing fear that a hike in interest rates will be detrimental to peer-to-peer lending platforms.
Ben McLannahan of the Financial Times believes that an increase in Fed rates can actually devastate the nascent peer-to-peer lending industry.
Increasing default rates for peer-to-peer lending notes might trigger an exodus by institutional investors, thus triggering liquidity concerns and
Traditional investment options would see returns revert to historical means, rendering moot the allure of higher returns offered by the p2p industry. (Source)
China is one of the biggest markets in terms of size and population. Even new regulations and the crackdown by authorities cannot halt China's P2P market juggernaut. In March this year, it crossed $130 billion (Rmb 900bn), and it is widely expected that lending will cross Rmb 1 trillion mark this year.
With outstanding loans weighing heavy on the balance sheets of lenders, the regulatory body PBOC is worried this will lead to yet another economic recession, and that is the reason it is clamping down on the P2P industry. For the most part of its journey, the P2P industry in China has been unregulated. The sudden onslaught of regulators led to a major decline in the number of players. Case in point: There were 2,281 P2P platforms in the country, and now the number has come down to 331.
The basic interest rate is hovering around 4.35 percent. Chinese regulators believe the global economic situation is still shaky, therefore, raising the open market rate is much more appropriate than raising the benchmark rates. In February this year, PBOC raised the interest rate on open market operation to reverse repurchase agreements (repos) by 10 basis points, and furthermore, it increased the lending rates on standing lending facility (SLF). The overnight rate for the SLF loan was raised to 3.1 percent from 2.75 percent. The SLF rate acts as a de facto ceiling for interbank lending.
A change in short and medium lending rates were executed to address the rapid build-up in debt and to safeguard the economy against the consequent financial risk. Increase in interest rates along with regulatory reforms being put in place by PBOC will help the industry to grow in a safe and structured environment.
Rising interest rates pose a threat to the P2P industry as the initial adoption was triggered by the low yield offered by traditional banks. P2P lenders were able to circumvent this and offer a win-win for both borrowers and lenders. With interest rates inching up, many feel the P2P lenders will be facing higher delinquency and lower investor interest. But this does not take into account the evolution of the industry; the players have been extremely nimble and understand the predicament facing them. Many have tightened lending norms, and almost all players now have long-term institutional money on board. With credit algorithms leveraging artificial intelligence and machine learning, profitability for the nascent industry is not a distant dream. In such a scenario, interest rates offered by P2P lenders will adapt to the demand-supply ratio in the market.
Written by Heena Dhir.
Author Allen TaylorPosted on December 20, 2017 Categories Analysis, Artificial Intelligence, Bank of England, china p2p, defaults, Featured, FED, interbank lending, Interest Rates, investors, lendingclub, loan delinquencies, P2P lenders, p2p lending, p2p uk, PBOC, Regulation, Zopa
Friday December 1 2017, Daily News Digest
News Comments Today's main news: Affirm partners with Shopify Plus.The Fed is thinking about starting a cryptocurrency.Payday lending group sues Consumer Financial Protection Bureau (CFPB).Funding Circle gets first IFISA sign-up within 15 minutes of opening.Assetz Capital to launch IFISA with manual lending.KappAhl to offer mobile payments in store with Klarna. Today's main analysis: How should […]
Today's main news: Affirm partners with Shopify Plus.The Fed is thinking about starting a cryptocurrency.Payday lending group sues Consumer Financial Protection Bureau (CFPB).Funding Circle gets first IFISA sign-up within 15 minutes of opening.Assetz Capital to launch IFISA with manual lending.KappAhl to offer mobile payments in store with Klarna.
Today's main analysis: How should equity investors position for a secular rates uptrend?
Today's thought-provoking articles: Amazon, Google, and the disruption of small business lending.Australia treasurer visits new Prospa office.Canadian activists demand fair banking.Etherecash gets 40K contributors during pre-ICO.
Affirm partners with Shopify Plus. AT: "This is a smart partnership."
Affirm app gives loans for designer jeans, and more. AT: "Mostly a rehash of previously reported information about Affirm."
How should equity investors position for a secular rates uptrend? AT: "Today's must-read report."
The Fed is thinking about its own cryptocurrency. AT: "If governments everywhere created their own cryptocurrencies, there would be a global realignment of economies. There'd be no reason to peg state currencies to foreign money unless a country only had fiat currency that tended to fluctuate wildly. It could even lead to the dollar and other fluctuating currencies pegging once again to gold. What would that then do to lending? It would like benefit alternative lenders a great deal."
Amazon, Google, and the disruption of small business lending. AT: "I still can't see Google getting in on the lending business. Perhaps, if Amazon did, and Apple and Facebook followed, then Google would feel compelled to do so to remain competitive. Other than that, I can't see it."
Community Financial Services Association of America sues the CFPB. AT: "This time, the legal action is going the other way. I can see a shift in priorities for the CFPB coming soon."
Judge refuses to block Mulvaney appointment.
Payday lender Curo Group seeks to raise $100M in public raise.
Marlette Funding president talks about personal loan market.
Startups experiment with no-interest loans.
YieldStreet CEo is excited about the future.
Innovative approaches to expanding home availability, affordability. AT: "A good read. Lend Academy."
PwC says asset managers are laggards in digital technology.
Lincoln Financial Network launches new wealth management platform.
Banks resist pressure to raise rates.
Regional banks consider advisory services.
KeyBank partners with Snapsheet.
Concord president, COO to lead marketplace lending conference panel.
LendingTree logo to be on two NBA jerseys.
How to do dodge family loan drama.
Funding Circle IFISA gets first sign up within 15 minutes.
Assetz Capital to offer IFISA with manual lending.
Digital banking is a tough road to profits.
P2P Global Investments speeds up timetable for returns.
How LendInvest plans to fill the portfolio landlord shortage.
3 reasons to consider alternative lending.
Did Chinese fintech grow up in 2017?
Chinese shadow banking is a big threat to economic growth.
Ant Financial, QCash win fintech awards.
KappAhl hooks up with Klarna for in-store mobile payments.
Deposit Solutions raises $20M from existing investors.
Why Nordea is establishing a fintech fund.
Nordea invests in Betalo.
PSD2 delays fintech disruption.
Etherecash receives support from 40K contributors in pre-ICO. AT: "Congratulations."
Aussie treasurer visits Prospa office.
KrazyBee expands.
Disruption to affect bank operating income.
Mobile wallets gaining ground.
Activists demand fair banking for low-income people.
Mobetize CEO to speak at British Columbia FinTech Day.
Affirm Joins Forces With Shopify Plus to Help High Growth Retailers Rapidly Scale Online Store Sales (BusinessWire), Rated: AAA
Affirm App Gives Loans For Designer Jeans, Holiday Flights And More (International Business Times), Rated: A
Expert Commentary: how should equity investors position for a secular uptrend in rates? (INTL FCStone Email), Rated: AAA
Dudley Says Fed Has Started Thinking About Official Digital Currency (WSJ), Rated: AAA
THE FINTECH EFFECT: AMAZON, GOOGLE, AND THE DISRUPTION OF SMALL BUSINESS LENDING (The Boss Magazine), Rated: AAA
Payday lending group plans to sue the Consumer Financial Protection Bureau (USA Today), Rated: AAA
Federal judge refuses to block Trump's designation of Mulvaney as interim head of CFPB (Legal NewsLine), Rated: A
Payday lender going public as new sheriff takes over at CFPB (Seeking Alpha), Rated: B
Marlette Funding President Offers Insight on Personal Loan Market (LendEDU), Rated: A
For Workers In A Pinch, Start-Ups Experiment With No-Interest Loans (Forbes), Rated: A
YieldStreet CEO Milind Mehere: Excited about Growth and YieldStreet's Future (Crowdfund Insider), Rated: A
Innovative Approaches to Expanding Home Availability and Affordability (Lend Academy), Rated: A
PWC CHARGE: ASSET MANAGERS ARE DIGITAL TECH 'LAGGARDS' (AllAboutAlpha), Rated: A
Lincoln Financial Network Launches Integrated Technology Platform to Drive Greater Client Engagement and Collaboration in Financial Planning (BusinessWire), Rated: A
Banks resist pressure to raise rates, but for how long? (American Banker), Rated: A
U.S. regional banks delve deeper into advisory services to boost growth (NASDAQ), Rated: A
KeyBank Forms Strategic Partnership With Snapsheet To Provide Powerful Insurance Claims Payment Solutions (PR Newswire), Rated: A
Concord President & COO Shaun O'Neill to Lead Panel at Marketplace Lending Conference in New York City (PRWeb), Rated: B
LendingTree Logo To Appear On Greensboro Swarm Jerseys As Part of Company's Partnership With Hornets (NBA.com), Rated: B
Family loans: How to dodge the drama (Work IT, SOVA), Rated: B
Funding Circle IFISA motors ahead with instant sign-ups (P2P Finance News), Rated: AAA
Assetz Capital Announcement: Soon to Launch IFISA Set to Include Manual Lending (Crowdfund Insider), Rated: AAA
Digital banking: a tough way to make money (Financial Times), Rated: A
P2PGI unveils new strategy that speeds up timetable for target returns (P2P Finance News), Rated: A
LendInvest: How our BTL launch will fill the portfolio landlord lending gap (Mortgage Solutions), Rated: A
Three reasons why investors must consider alternative lending (Money Observer), Rated: A
Was 2017 the year that Chinese fintech grew up? (Ecns.cn), Rated: A
China leverage and shadow banking biggest threats to growth (The Asset), Rated: A
Ant Financial and QCash scoop FT fintech awards (Financial Times), Rated: B
KappAhl first to offer mobile payments in store with Klarna (NB Herard), Rated: AAA
Fintech company Deposit Solutions raises $ 20 million from existing investors (Tech.eu), Rated: A
Why Scandinavia's Biggest Bank Is Setting Up Its Own Fintech Startup Fund (Forbes), Rated: A
Nordea invests in fintech company Betalo (Nordea), Rated: B
PSD2 BRINGS DELAYED DISRUPTION TO THE FINTECH ECOSYSTEM (TechSavvy), Rated: A
Blockchain P2P Lending, Sending, and Spending: Etherecash Garners Support from Over 40,000 Contributors During Pre-ICO (Digital Journal), Rated: AAA
Treasurer of Australia Scott Morrison Visits Online Lender Prospa's New Office in Darlinghurst (Crowdfund Insider), Rated: AAA
KrazyBee looks to expand in Tamil Nadu (The Times of India), Rated: A
Asian banks' operating income could be hit by fintech disruption (Channel News Asia), Rated: A
Mobile wallets taking hold in Asia (The Asset), Rated: A
Activists across Canada demand fair banking for low-income people (TheStar), Rated: AAA
Mobetize' CEO to speak at first BC Tech Association FinTech Day (Globe Newswire), Rated: B
Affirm, Inc., the company started by Max Levchin to provide fair and honest consumer financing, today announced it has joined the Shopify Plus Technology Partner Program to help more retailers quickly scale their online store sales by giving their customers a quick and easy alternative to credit cards.
Affirm's chief of staff and head of international expansion, Ryan Metcalf, told International Business Times the startup works with 1,200 retailers nationwide and issued $1 billion worth of loans in 2017.
"We are able to approve 126 percent more people than industry averages. A large portion of these people have no access to credit or if they do they are being mispriced in the market because their FICO score is outdated," Metcalf said. "Around one in 10 Americans have 'unscorable' credit reports. That's around 30 million people. So we're also able to offer credit to those people as well."
According to the Fair Isaac Corporation's data, 20 percent of American credit card owners are ranked as "subprime" because their FICO score is 600 or lower.
In Four Interest Rate Myths, I made a theoretical case for higher rates by debunking the New Gospel of the New Normal. The Slow Agony of and Old Bull highlighted seven signs that the bond bull market was already over. This report discusses the most important question facing market participants for the next five years – how should equity investors position for a secular uptrend in rates?
The report reviews the performance of U.S. sectors, currencies, and international indices during prior hiking cycles and their recent correlations with yields. Six conclusions emerge:
Almost tautologically, bond proxies have consistently underperformed during prior hiking cycles.
Currently, only two sectors are positively related to interest rates: financials and energy. Since their valuations remain below average, these sectors are a cheap option against the risk of rising rates.
Investors should monitor the correlation between yields and tech stocks: higher rates would kill the bull market if the correlation between tech stocks and bond yields turned negative.
U.S. stocks and the dollar index have tended to fall in prior hiking cycles.
Korean equities and, to a lesser extent, Japanese stocks have outperformed during prior hiking cycles.
The performance of emerging markets and commodity-driven markets is mixed: they have outperformed massively during in the 2003-2006 cycles, but have suffered during the hiking cycles of the 90s.
Read the full report here.
Federal Reserve Bank of New York President William Dudley said Wednesday the U.S. central bank is beginning to explore whether it could adopt its own digital currency, in an appearance at Rutgers University where he also expressed optimism about the economy.
Bitcoin is "really more of a speculative activity," Mr. Dudley said. But he said aspects of the technology are interesting and worthy of attention. "It's premature to be talking about the Federal Reserve offering digital currencies, but it is something we are starting to think about," he said.
Some academics have called for the Fed to offer its own digital currency. They believe it would afford the central bank better control over the economy by tweaking interest rates at the consumer level, bypassing fickle financial markets that often work at cross-purposes with Fed policy aims.
If banking bigwigs and fintech entrepreneurs have seemed a bit queasy since October's Lendit Europe conference, they might be blaming Karen Mills for daring to illuminate the elephants in the room: Amazon and Google, and their ability to disrupt the small business lending industry.
Mills, a former White House administrator for small business and current Harvard Business Review fellow, succinctly pointed out the obvious. With the tremendous amount of financial and personal data these behemoths collect, a broadening of scope into small business lending may be inevitable.
While Google hasn't made any notable overtures into the lending business yet, Amazon launched its lending business to support its merchants in 2012. As reported by Bloomberg, the retailer issued $1 billion in loans in the 12 months between May 2016 and June 2017. To date, they have extended $3 billion to over 20,000 small businesses here, as well as in the U.K. and Japan.
The Magnificence of Micro Loans
Merchant services provider Square has given its merchants loans of over $1.5 billion since its in inception in 2014, and PayPal's Working Capital program has loaned over 115,000 global businesses a total of $3 billion.
Amazon and Square merchants repay the loans automatically based on the amount of sales they make. PayPal's maximum small business loan amount is 30 percent of a merchant's annual PayPal sales, not to exceed $97,000 for the first loan.
Small Business Domination
Small businesses account for roughly 99.9 percent of all businesses in the U.S., and are responsible for 61.8 percent of the new jobs established from Q1 1993 to Q3 2016. About 80 percent of the nation's 29.6 million small businesses are nonemployers.
The Community Financial Services Association of America plans to challenge one of the federal watchdog's signature achievements could signal how the consumer bureau's previous enforcement policies will shift under new Trump administration leadership.
The anticipated battle would target a new rule that was indeed published in the Federal Register on Nov. 17, capping a contentious 18-month public comment and lobbying battle between the payday loan industry and consumer advocates.
Federal budget director Mick Mulvaney, installed by Trump as the bureau's acting director, has been critical of the payday lending rule and has received campaign backing from the industry. He received $31,700 in 2015-2016 federal campaign cycle contributions from payday lenders, ranking ninth among all congressional recipients, according to data analyzed by the Center for Responsive Politics.
A federal judge on Tuesday rejected arguments by Leandra English, who was named the deputy director of the Consumer Financial Protection Bureau by outgoing director Richard Cordray, in a lawsuit she brought over the agency's interim leadership.
Judge Timothy J. Kelly for the U.S. District Court for the District of Columbia, according to a minute order and entry on the case docket, denied English's emergency motion for temporary restraining order after a motion hearing held Tuesday.
English filed her lawsuit Sunday night in attempt to block President Donald Trump's naming of Office of Management and Budget Director Mick Mulvaney as the bureau's acting director.
In a minute order filed Wednesday, Kelly said the parties will meet, confer and submit by Dec. 1 a joint proposed schedule for briefing the merits and/or for briefing a preliminary injunction, or separate schedules.
Curo Group is looking to raise about $100M with the sale of 6.7M shares at hoped-for range of $14-$16 each. Prospectus here
Q: So we know that Fintech personal loan lenders are starting to attract more consumers and take up more of the market. How do you expect traditional banks to react to this over the next couple of years if the trend continues?
A: You're starting to see banks wake up to this new way of lending. They've been impacted by a regulation-focused environment in recent years, driving them towards a compliance mindset. However, banks are starting to think of ways to grow their consumer lending businesses, and technology is a big part of this.
Q: What sort of future do you see for blockchain technology in the Fintech personal loanmarket? What sort of challenge would its implementation pose to Fintech lenders?
A: One use that I could see for Fintech lending is creating a more secured identity verification process for the customer. From the recent Equifax news, you have a single source of data where all relevant info is in one location, and a breach creates both chaos as well as problems with trust. Distributed ledger tech creates an interesting opportunity to limit this concern, but it's going to take a long time before it can be implemented fully.
Dave is part of a new crop of financial technology companies that are trying to help consumers avoid nasty overdraft fees, as well as payday loans, pawn shops and other expensive forms of debt, via zero-interest loans. They're going after workers who may struggle to make ends meet, but who could benefit from a minor influx of cash at the right time.
Dave analyzes a consumer's bank account history to issue warnings about potential overdrafts up to seven days in advance. Then, for users who still find they're in a pinch, it may approve a loan of up to $75. Dave doesn't charge interest, but the app costs $1 a month and users are asked to leave a tip on advances. The Mark Cuban-backed service has amassed 100,000 users since it launched in April.
In 2016, financial institutions hauled in $33.3 billion on overdraft fees alone, according to Moebs Services, an economic research firm.
Dave, in addition to companies like Even and Earnin (formerly Activehours), are attempting to do away with the high interest rates and fees that they say put a financial institution's incentives in contrast with those of the borrower. Their answer: Small, zero-interest advances on a person's next paycheck with no hidden or punitive fees.
According to one study of low and moderate income families, household income spiked — or fell — by more than 25% in six months out of every year.
Launched in NYC in 2015, YieldStreet aims to allow people to invest in alternative investments that are backed by real collateral. With a world-class advisory board which recently added three new members Ron Suber (Prosper Group), Mitch Jacobs (On Deck)Alexandra Wilkis Wilson (Gilt Group) and a growing leadership team, including Volfi Mizrahi who just joined as Managing Director of Originations and Ivor Wolk as General Counsel, the platform's growth is undeniable.
Erin: On what other elements of your YieldStreet street vision are you currently working?
Milind: Continuing to expand our product and audience offering – AutoInvest will let users choose their investment preferences such as asset class, yield and duration, then the algorithm our platform uses will match them as offerings become available. In 2018 we hope to open the platform to non-accredited investors, and we are working to provide liquidity on our platform, as well as creating products for the Financial Advisor/RIA market and IRA market.
Erin: How do you expect YieldStreet to grow? How do you source deals?
We work with a network of originators and asset managers, as well as many funds (from $50M to $10B) in the private credit space.
Erin: What lessons from Yodle — from its beginnings to its $342M sale to web.com in 2016 — have you applied to YieldStreet?
Milind: We have been incredibly efficient at YieldStreet because of that. We have just raised $3.7M in seed capital to reach $200M in originations, where some of our peers have raised anywhere from 6x-25x to achieve the same results. Yodle taught me to be extremely disciplined about where to invest and when.
Erin: What are YieldStreet's future plans for growth by 2018? by 2020? by 2025? How do you predict the sector will change and be disrupted?
Milind: According to a recent report by PricewaterhouseCoopers (PwC), the asset management industry is set for "transformational change" and booming growth in the next decade. Alternative asset classes, such as real estate and private debt are expected to grow to about $21.1 trillion by 2025.
It seems like almost every day I see a story about increasing real estate prices in the major metropolitan areas of the US. Prices in cities like San Francisco, New York, Seattle, Washington DC have made homeownership unobtainable for many people.
SoFi comes to mind with their jumbo mortgage which allows borrowers to put just 10% down and offers loans up to $3 million.
Landed is taking a different approach. I spoke with Alex Lofton who is Head of Growth and Co-founder at the company. They first came on my radar this summer when TechCrunch profiled them. They are similar to companies like Unison (who recently was on the Lend Academy podcast) and Point with a slight twist. Currently, the company focuses on teachers to help purchase a home, providing up to 50% of the down payment. Like other similar products, Landed participates in either the upside or downside when the home is sold.
In filling out the particulars of this claim the authors of the new report make four more specific points: one, asset management is a buyers' market and will become more so, in large part because "institutional investors have the tools to differentiate alpha and beta," and they want to pay for the former not the latter. They also say that asset managers have been filling gaps in the financial system that emerged in the wake of the global financial crisis – they'll need to capitalize on and expand these once-niche markets. Thirdly, while they make the common point that traditional active managers feel a squeeze between passive management on the one hand and alternatives on the other, they go further in that direction than other analysts have, saying that the way to react to this squeeze is not to try to beat back the competing forces but to join them, to turn a management firm into a "multi-asset solutions firm."
But perhaps the most surprising of the four points is the contention that asset management has been a refuge of digital technology "laggards," and that this will change in the near future, as "technology giants … enter the sector, flexing their data analytics and distribution muscle. The race is on."
Lincoln Financial Network (LFN), the retail wealth management affiliate of Lincoln Financial Group (NYSE:LNC), today announced that it has successfully launched a meaningful enhancement to its fully integrated wealth management platform for financial advisors and their clients – Automated Account Opening (AAO). AAO encompasses a full suite of new capabilities, integrated tools, and client-servicing solutions that will increase client satisfaction and collaboration with advisors.
Online banks have been aggressively raising the rates they pay on consumer deposits, and that is putting pressure on mainstream banks to consider following suit or risk losing valuable deposits to their more nimble competitors.
A recent survey of 100 banks conducted by MoneyRates.com found that online banks such as Ally Bank, Goldman Sachs' GS Bank and Sallie Mae Bank are paying significantly higher rates on savings and money market accounts than their brick-and-mortar counterparts.
Smaller banks, like their bigger Wall Street rivals, have aggressively cut costs since the 2008 financial crisis and trusted ultra-low interest rates to increase loan volumes.
U.S. Bancorp, BB&T Corp, SunTrust Banks Inc, Fifth Third Bancorp, KeyCorp and Citizens Financial Group Inc together earned $6.97 billion in non-interest income in the third quarter, up 10.6 percent from a year earlier and 15.2 percent from the second quarter.
That compares with growth in net interest income of 7.7 percent and 2 percent, respectively.
RICH PICKINGS
The number of millionaires in the United States is at the highest since Chicago-based research company Spectrem Group started measuring it in 2004, but thresholds of – for example $250,000 to invest – mean many are too small to get personal attention from the big Wall Street firms.
Born between the early 1960s and 2000, Americans from Generations X and Y who have an average annual income of about $200,000, account for 18 percent of millionaires compared with 8 percent in 2012.
Yet only 58 percent have financial advisers compared to 72 percent five years ago, according to a study by Fidelity Investment.
KeyCorp (NYSE: KEY) announced today its strategic investment and partnership with Snapsheet, an innovator of self-service claims solutions for insurance carriers. This investment follows the joint launch and announcement of Snapsheet Transactions, a payment platform on the back end of Snapsheet's existing claims solution.
Snapsheet Transactions provides carriers with a payment hub that features a variety of payment options, without adding complexity or risk to insurance carriers' back-end processes. Key and Snapsheet will continue to partner with each other to support the rollout and execution of enhancements and innovations related to Snapsheet Transactions.
Shaun O'Neill, President and COO of Concord Servicing Corporation, a leading force in the portfolio servicing and financial technology industry, has been invited to serve as moderator of a finance-related panel during the upcoming Information Management Network's 3rd Annual Investors' Conference on Marketplace Lending. O'Neill's panel will focus on the highly topical "Trends and Best Practices for Loan Servicing" during the conference, to be held December 1st at the Marriott New York Downtown, in New York City.
The Charlotte Hornets, Greensboro Swarm and LendingTree announced today that the LendingTree logo will appear on the jerseys of the Swarm as part of the Founding Level Partnership announced earlier this month between the Hornets and LendingTree.
There are advantages of a family loan for a borrower: no credit check, low or no interest and flexible payback terms.
Family loans may also come with tax considerations, whether the lender charges interest or not. Charge zero interest, and you may face a gift tax; a borrower who receives a gift may have to report it as taxable income. Tack on an interest charge and you must follow IRS-specified guidelines for the rate you charge and report it as income.
BORROWERS: EXHAUST OTHER OPTIONS FIRST
When weighing the pros and cons of a family loan, also consider alternative options, including a personal loan borrowed from a bank, credit union or online lender that can be used for any purpose.
Personal loans from credit unions and online lenders typically have more flexible qualification requirements than a bank loan.
LENDERS: ASSESS THE REASON FOR THE REQUEST
If you are lending the money, try to set your emotions aside and look at the reason for the loan. Has your family member been rejected by banks and other lenders? If so, why? Will your loan help promote good financial decisions?
Funding Circle has begun rolling out its Innovative Finance ISA (IFISA) to investors and had a customer sign up within 15 minutes.
The peer-to-peer business lending giant started emailing users on Thursday morning, in order of when they opened accounts and started investing.
The IFISA account is a flexi-ISA, meaning you can withdraw any available funds without affecting your annual £20,000 ISA subscription limit, providing you transfer them back in by the end of the tax year.
The online lender reported that the IFISA will be launched next month, with users able to use their £20,000 annual tax-free allowance on the Assetz Capital platform. Users will be able to transfer in past years' ISA savings from their cash and shares ISAs. Assetz Capital also noted that new and existing investors will be able to open an IFISA wrapper on the platform and then invest into any automated Assetz investment account. The IFISA is also set to include the popular Manual Loan Investment Account (MLIA) in the New Year.
It's been a busy period for the UK's fledgling digital banks. Since January, eight UK digital banks have collectively raised $600m and two challenger banks were acquired for $2B+. Digital banks have built out the tech, landed banking licenses, and started winning customers – but they have arrived at a 'now what' moment. How can they capture a large enough customer base to validate their significant collective investment?
Monzo reported that its prepaid card scheme loses around £50 per active customer per year, and other digital banks face similar costs. While on the one hand the cost to acquire these current account customers is not very high, given the 'buzz' around the sector and banks' word of mouth-driven growth – these current accounts, with their low average balances, are also inherently unprofitable. So it's a steep climb for digital banks to recoup their operational costs, much less make a lot of money per customer.
P2P GLOBAL Investments (P2PGI) has brought forward its timetable for reaching its target returns of six to eight per cent after unveiling its new portfolio strategy on Thursday morning.
The investment trust said it now expects to provide a dividend of at least 15p per quarter by the end of the second quarter of 2018, which analysts say reflects an annualised yield of 7.8 per cent.
The prospects of the dinner party landlord, who picked up a property or two during the boom years, have been dented by moves like the additional rate of stamp duty on second homes and the changes to mortgage interest tax relief.
In contrast, it's the professionals who are best placed to adjust their budgets and ride out such changes. These are the investors who spend their working hours – rather than just their spare time – focused on running their property businesses.
Countrywide's letting index in August flagged up the fact that the number of homes on the market to tenants has jumped by 171,000 over the last two years, despite the number of landlords falling by 154,000 over the same period.
With cash held at the bank slowly being eroded by inflation, many investors have been attracted to the enhanced return prospects offered by alternative – or 'peer-to-peer' – lending.
Alternative lending is very interesting from this perspective, as it is one of the few income options available to retail investors that may be shielded from market volatility. This has grown in importance recently as many markets are currently trading at historically high valuations. Markets follow a supply and demand dynamic and the traditional asset classes are definitely vulnerable to sudden downside pressures in stressed market environments.
While investments in the Chinese fintech sector tripled to almost 10 billion US dollars in 2016 compared to the year before, 2017 has seen a significant drop in corporate fintech investments across Asia. KPMG reports that corporates have only put 840 million US dollars into the sector in 2017, compared to 6.8 billion US dollars in 2016.
Decline of P2P, robo-advisors
One other area that has struggled in 2017 has been robo-advisors. In 2016, China Merchants Securities predicted that by 2020, some 5.22 trillion yuan (758 billion US dollars) worth of assets would be managed by robot financiers.
FINANCIAL system leverage and shadow banking pose the biggest threat to China's economic growth, according to a live poll of attendees at the Fitch on China Forum.
The forum was organized by The Asset in association with Fitch Ratings and held on November 30 at the Four Seasons Hotel in Hong Kong.
Source: The Asset
A big Chinese group and a US not-for-profit have triumphed in the second annual Financial Times fintech awards, with Ant Financial taking the "impact" prize and QCash winning for "innovation".
KappAhl is the first major fashion chain to offer its customers digital payment solutions in stores via their smartphones. Customers will have the option to make their purchases with Klarna In-Store, paying either on the spot or upon invoice.
This new payment solution will become one of the cornerstones in KappAhl's digital transformation, with customers in stores benefitting from the same payment options that they have in Shop Online.
The service has been rolled out gradually and, as of 1 December, will be available in all 173 KappAhl and Newbie stores in Sweden. From 1 December, the service will be available in all 96 Norwegian stores, and, from 4 December, in all 58 stores in Finland.
Deposit Solutions, a German fintech company, has raised $20 million in a round led by e.Ventures and Greycroft, both existing shareholders.
The new funds will be used to grow the Hamburg-based company's Open Banking platform for savings deposits for both B2B and B2C services, and to expand internationally. Its APIs allow banks to connect to the platform to build and offer deposit services. It has partnered with more than 50 banks.
The bank has announced that it's setting up Nordea Ventures, to make strategic investments in fintech start-ups.
A case in point is Tink the Swedish-based fintech company, where Nordea provided capital and advice and integrated some of Tink's own technology into its own digital products while preserving Tink's name and brand.
Tink's app helps consumers to aggregate financial transactions in one place, to compare and switch mortgages to a partner bank or open a savings account, for instance. Another Tink app for banks and payment services like Klarna provides account aggregation and payment capabilities.
Nordea is investing in the fintech company Betalo. This takes our partnership with the Swedish company to the next level after a cooperation agreement was signed in March 2017.
A new EU-directive is about to force banks to open up their data vaults and allow third parties to access their user data. Nordea has chosen to embrace the change with open eyes, and a fintech startup predicts tough competition embarking on the opportunities it brings along.
The release of bank data is bound to cause a stir in an otherwise traditional and established sector. One of the incumbents that have already made an imprint is the fintech startup Spiir.
American tech giants might end up owning the financial space. Rune Mai looks to China to catch a glimpse of what the financial future might hold. The retail giant Alibaba owns half the payment market here with an all-encompassing app that offers everything from dating, financing to shopping.
Nordea is more inspired than afraid of Amazon. The bank has more than 10 million customers in the Nordic region, and they have decided to face the coming change with open eyes. They are actively pursuing a first mover strategy and has allocated more than 100 people to ready themselves for the coming digital disruption.
A little over three weeks are left in the Etherecash token sale and it's been a fantastic run so far; the success they have seen comes after a big appearance at the World Blockchain Summit, Dubai, which was closely followed by a heated Pre-ICO.
The platform is the remedy to the overly-complex and lengthy process of getting a traditional bank account, and will provide access to finances through a cryptocurrency-backed P2P (Peer-to-Peer) fiat currency loan marketplace. P2P loans are backed by the borrower's own crypto-wealth allowing them to borrow up to 80 percent of their wallet's value.
On top of this, once the crypto debit card is available, users will be able to store multiple types of cryptocurrency on it, allowing them to shop anywhere and everywhere as they please, even abroad.
Based on the Ethereum standard token ERC20, purchasable with Bitcoin or Ethereum, the exciting ICO Launch began 15th November, 2017 – ending December 19th, 2017.
Prospa, an online lender serving SMEs in Australia, had a visit from the Honorable Scott Morrison yesterday. The Treasurer of Australia help to open up Prospa's new high tech Darlinghurst office, which apparently is quite large extending over two floors housing a team of 150.
Prospa expects to add another 50 hires over the next 12 months as it accommodates platform growth.
Online lender KrazyBee says it is rapidly expanding its business in Tamil Nadu and its focus in the state will be on solving unique needs of the student community.
KrazyBee, which earlier operated in five cities (Bengaluru, Hyderabad, Pune, Vellore and Mysore), said that is expanding aggressively in over 11 cities, including Chennai. With more than four lakh registered student borrowers on its platform, KrazyBee says it currently processes over 3,000 loan applications and disburse around 1,700 loans per day.
Asian banks that do not take any action against the rise of financial technology (fintech) could see their operating income take a hit, said the Monetary Authority of Singapore (MAS) on Thursday (Nov 30) in its latest Financial Stability Review.
For lenders in Singapore that do nothing to stave off the disruption, that could mean a 5 per cent loss in operating income over the next five years, the central bank warned.
WHILE the development of digital payments started with the launch of the first universal credit card in the 1950s, the space has rapidly evolved, and now the mantle is being passed to e-wallets, otherwise known as mobile wallets.
In 2014, credit and debit cards accounted for more than half of e-commerce payments in terms of transaction value. However, that share is predicted to drop to 49% in 2019 as mobile wallet options start to gain ground, according to a report by the United Nations Conference on Trade and Development.
At the Toronto rally held outside Finance Minister Bill Morneau's constituency office, a 46-year-old man was holding the loan he got in August from a payday loan company and was trying to get pedestrians to look at it.
He took out a $5,500 loan to pay his rent in August, to be paid back at 60 per cent interest by 2020.
Don is a member of the grassroots activist group called Association of Community Organizations for Reform Now (ACORN), and one of thousands of people who, on Tuesday, rallied across Canada demanding fair banking.
Mobetize Corp. (OTCQB:MPAY), a leading fintech service provider for payments, remittances and mobile banking solutions, today announced CEO Ajay Hans will be the keynote speaker at BC Tech's Fintech Day event on December 5.
Author Allen TaylorPosted on December 1, 2017 Categories affirm, alternative lending, Amazon, Ant Financial, asset management, Assetz Capital, Banking, Banks, Betalo, CFPB, Concord, cryptocurrency, Daily News Digest, Dave, Deposit Solutions, digital banks, Earnin, Etherecash, Even, Featured, FED, fintech, Funding Circle, Google, ICO, IFISA, insurtech, KeyBank, Klarna, KrazyBee, Landed, Lend Academy, lend to family, LendingTree, LendInvest, Lincoln Financial Network, marketplace lending, Marlette Funding, Mobetize, mobile wallets, News, no-interest loans, Nordea, P2P Global Investments, payday lending, private equity investing, Prospa, PSD2, PwC, QCash, Shadow Banking, small business lending, Snapsheet, SoFi, startups, wealth management, YieldStreet
Friday November 24 2017, Daily News Digest
News Comments Today's main news: Revolut sings 1 millionth customer. KBRA assigns preliminary ratings to Lending Club's Consumer Loan Underlying Bond Credit Trust 2017-P2. Funding Circle to launch Isa. Orca is launching investment platform. Chinese regulators investigating potential Qudian data leak. China cracks down on shadow banking. China tells provincial goverments to halt microlender approvals. Swiss consortium adopts single digital identity for […]
Today's main news: Revolut sings 1 millionth customer. KBRA assigns preliminary ratings to Lending Club's Consumer Loan Underlying Bond Credit Trust 2017-P2. Funding Circle to launch Isa. Orca is launching investment platform. Chinese regulators investigating potential Qudian data leak. China cracks down on shadow banking. China tells provincial goverments to halt microlender approvals. Swiss consortium adopts single digital identity for online purchases. YES Bank diversifies funding.
Today's main analysis: Peter Renton's quarterly MPL results.
Today's thought-provoking articles: How many borrowers can Marcus count on? China's assault on microlenders impacts U.S. Qudian's fast track from darling to dog. Investors divide over P2P lending. ICO risks.
Naysayers question whether Marcus can rely on borrowers. AT: "It's possible that Goldman Sachs has underestimated the risks, but I doubt it."
Peter Renton's MPL results for Q3. AT: "I wonder why Lending Club's investments are struggling while Prosper's are doing well. The interesting part of this read are the new investments for this quarter."
KBRA assigns preliminary ratings to Lending Club's CLUB Credit Trust 2017-P2.
China's war on online loans comes to the U.S. AT: "Naturally, the stock prices are struggling for microlenders listed on the New York Stock Exchange, which is almost all of the big ones."
Joseph Otting has a lot on his plate as the new Comptroller.
One-third of small business owners work half of major holidays.
Consumers accuse Victory Park Capital of financing illegal tribal lending.
Payday lenders attempt legislative run around.
How the Fed can help families living paycheck to paycheck.
More on the Cleveland Fed study.
SoFi bought six-second ad for Thanksgiving Day football game.
Revolut signs millionth customer. AT: "Congratulations, a huge milestone indeed."
Funding Circle to launch Isa.
Funding Circle borrowers favor joining European Free Trade Agreement.
Orca to roll out diversified P2P portfolios.
Fountain nabs seed investment.
ThinCats rebrands.
How the FCA's guidance paper misses the mark.
Three-quarters of advisers not threatened by robo-advice.
An investigation into a possible Qudian data leak. AT: "This may shed some light on why New York law firm Faruqi & Faruqi recently opened an investigation into Qudian."
China cracks down on shadow banking.
China asks provincial governments to halt microlender approvals. AT: "This is a slight nuance from targeting microlenders themselves. If the provincial governments are using microlenders as fronts for their own pocketbooks, the supreme authority may not take too kindly to that."
Assault on microlenders threatens U.S. IPO listings. AT: "Chinese lenders need U.S. capital more than Wall Street needs Chinese companies. But if China wants to compete on the world stage for capital fundraising, they'll need to consider the protection of consumer interests."
Qudian's fast track from darling to dog.
The secret to Alibaba's success: Debt.
WeLab plans $500M IPO.
The real target of China's crackdown: Local governments.
Mobile payment users exceed 520M.
How fintechs create an alternative capital market.
Swiss consortium adopts single digital identity.
Interview with Lendoit CEO.
The 5-year-old Bondora portfolio.
Finbee moves into Czech market.
Investors divide over P2P lending.
ICO regulation and risks.
FintruX Network makes unsecured loans secure.
The Banking and Financial Services Commission of Inquiry.
The age of algorithmic advice.
YES Bank diversifies funding.
BankBazaar CEO recognized.
Millennials happy to take advice from robots.
LALA World issues ICO for migrants and the unbanked.
Goldman Sachs Faces Doubts About Loss Rates at New Online Lender (Newsmax Finance), Rated: AAA
My Quarterly Marketplace Lending Results – Q3 2017 (Lend Academy), Rated: AAA
KBRA Assigns Preliminary Ratings to Consumer Loan Underlying Bond (CLUB) Credit Trust 2017-P2 (BusinessWire), Rated: AAA
China War on Online Loans Makes Waves in New York: QuickTake Q&A (Bloomberg), Rated: A
Joseph Otting Has a Lot on His Plate as the New Comptroller (Crowdfund Insider), Rated: A
One-Third of Small Business Owners Work Half of the Major Holidays (Small Business Trends), Rated: A
Consumers Say Hedge Fund Financed Illegal Tribal Lending (Law360), Rated: A
Payday Lenders Try Legislative Run Around State Laws, CFPB Regulation (Chicago Crusader), Rated: A
How the Fed Can Help Families Living Paycheck to Paycheck (Real Clear Markets), Rated: A
Peer pressure (BreakingViews), Rated: A
SoFi Among Companies To Buy Six-Second Ads During Fox' Thanksgiving Game (Sports Business Daily), Rated: B
Fintech group Revolut signs up its millionth customer (Irish Times), Rated: AAA
Funding Circle to become latest P2P platform to launch Isa (Moneywise), Rated: AAA
Funding Circle borrowers back joining European Free Trade Agreement post-Brexit (P2P Finance News), Rated: A
Orca to unveil diversified P2P portfolios for investors (P2P Finance News), Rated: AAA
Digital wealth manager start-up Fountain secures seed investment (AltFi), Rated: A
Peer to Peer Lender ThinCats to Rebrand as Next Phase of SME Funding in UK (Crowdfund Insider), Rated: A
Robo-guidance or electric dreams? (FT Adviser), Rated: A
Three-quarters of advisers unthreatened by robo-advice (Financial Reporter), Rated: A
China Regulators, Police Probe Qudian Client Data Leak (Bloomberg), Rated: AAA
China Commences Crackdown on Shadow Banking (The Epoch Times), Rated: AAA
China Urges Local Governments To Halt New Internet Microlender Approvals (PYMNTS), Rated: AAA
China's Micro-Lender Assault Threatens Path to U.S. Listings (Bloomberg), Rated: AAA
China Online Lender Qudian's Fast Track From NYSE Darling To Dog (Forbes), Rated: AAA
Debt: The secret sauce of Alibaba's Singles Day success (Technode), Rated: A
Credit Suisse-Backed Online Lender to Plan $ 500 Million IPO (Bloomberg), Rated: A
An Overdone Payday Mayday (Bloomberg), Rated: A
Mobile payment users in China exceed 520m (GB Times), Rated: A
How fintech companies create an alternative capital market in China (The Asset), Rated: A
Swiss Consortium Adopts Single Digital Identity For Online Purchases (PYMNTS), Rated: AAA
Exclusive Interview with Lendoit CEO Ori Erez (Chipin), Rated: A
My P2P Lending Investment Portfolio at Bondora is now 5 Years Old (P2P-Banking), Rated: A
Finbee Expands into Czech Market (P2P-Banking), Rated: A
Investors divide in peer-to-peer lending (Silicon Republic), Rated: AAA
Initial coin offerings: regulation and the risks (Lexology), Rated: AAA
FintruX Network: Making Unsecured Loans Highly Secure (BTCManager), Rated: A
Financial services industry to get its Groundhog Day commission of inquiry (Financial Review), Rated: A
The age of algorithmic advice (Financial Standard), Rated: A
YES Bank diversifies funding sources (The Asset), Rated: AAA
BankBazaar CEO Honored at India FinTech Awards 2017 (Finovate), Rated: B
Millennials happy to take financial advice from robots (IOL), Rated: AAA
Upcoming ICO for Global Migrants and Their Unbanked Families (Digital Journal), Rated: AAA
As Goldman Sachs Group Inc. lends more money to Main Street, one question won't go away: How many borrowers will pay them back?
A recent example it gave suggests the firm expects loan losses to be lower than what some rivals are seeing, and half of what many credit-card lenders experienced the last time the economy went south.
The bank is counting on its consumer push to deliver $1 billion in revenue growth over the next three years. While the firm looks to attract borrowers with better credit than many rivals, others think it may be underestimating the risks of a business where it's the upstart.
If you have been reading these posts in the past year or so you will have noticed a steady decline in my returns, primarily caused by underperformance in my LendingClub accounts.
Earlier this year I adjusted my strategy and started investing across the entire risk spectrum but it is a bit like steering a battleship. Given my many thousands of notes it takes a while for any changes to show up in my portfolio returns.
My trailing 12 month returns for the year ended September 30, 2017 across all my accounts was 6.64%.
Source: Lend Academy
My main LendingClub account has performed poorly over the past 12 months. My TTM return is at a paltry 1.64%, my lowest return ever. All of my LendingClub accounts are below 5% and all have shown reduced returns over the past year.
Prosper continues to perform quite well. My three accounts are all returning between 7% and 8% which I consider quite respectable. My average interest rate of the loans I have invested in is just under 20% but returns have been quite consistent recently in the 7-8% range.
PeerStreet is a real estate platform focused on fix and flip properties. These are short term loans, typically between 6 and 24 months, and they are backed by the property. I use their automated investment tool to invest in only those loans that are paying 8% or more, up to a 75% LTV and a duration up to 24 months.
My first new entrant this quarter is AlphaFlow. They are a real estate platform that build diversified portfolios of fix and flip properties for you. What I like about AlphaFlow is that they deploy your money quickly, my entire investment was fully deployed in a matter of days. And they diversify across 75-100 properties, my own portfolio currently has 83 investments in 22 states with an average LTV of 68%.
Finally, as I do every quarter I want to end by highlighting the net interest number which for the last 12 months stands at $46,631.
Get the lowdown on the full range of Peter Renton investments here.
Kroll Bond Rating Agency (KBRA) assigns preliminary ratings to three classes of notes issued by Consumer Loan Underlying Bond (CLUB) Credit Trust 2017-P2 ("CLUB 2017-P2"). This is a $330.0 million consumer loan ABS transaction that is expected to close December 6, 2017.
Preliminary Ratings Assigned: Consumer Loan Underlying Bond (CLUB) Credit Trust 2017-P2
Class Preliminary Rating Expected Initial Class Principal
A A- (sf) $239,400,000
B BBB (sf) $34,600,000
*C BB (sf) $56,000,000
This transaction is LendingClub Corporation's ("LendingClub" or the "Company") third rated sponsored securitization and the second sponsored securitization consisting of "prime" unsecured consumer loans facilitated by LendingClub's proprietary technology platform supporting an online marketplace that connects borrowers and investors by offering a variety of loan products originated by issuing banks through the platform, www.lendingclub.com (the "LendingClub Platform" or the "Platform").
The transaction has initial credit enhancement levels of 35.45%, 26.05% and 10.83% for the Class A, Class B and Class C notes, respectively.
Chinese President Xi Jinping's campaign to reduce risk in the financial system is being felt in New York. The assault on the sector threatens to stymie any new listings of such lenders on New York's stock exchange — as well as spelling trouble for investors in the handful of companies that have already listed.
Joseph Otting, a former banker and CEO of OneWest Bank, was approved by the Senate in a party line vote last week to take over the helm at the Office of the Comptroller of the Currency (OCC).
If Otting decides to stand up to the banking hyperbole it won't be an easy task.
All of this begs the question: who will gain if Fintech is allowed to compete with banks?
One-third of small business owners work at least three of the six major holidays in the US.
Kabbage's new survey reveals several work/life balance issues related to the sacrifices small business owners are willing to make. The research involved surveying 400 small business owners, with 67 percent stating they expect to increase revenues by the end of the year. More than half of the small business owners interviewed said they anticipate an increase in revenue of 10 percent or higher.
The survey found that 60 percent of small business owners only take one full vacation a year, while 23 percent take less than two holidays off annually. Furthermore, when on holiday, 75 percent of small business owners continue working.
Vermont residents on Tuesday hit a hedge fund with a proposed class action in federal court alleging it helped concoct a sham tribal payday lending scheme meant to skirt laws preventing companies from charging consumers exorbitant interest rates while hiding behind tribal sovereign immunity.
Plaintiffs Jessica Gingras and Angela Given accused the firm, Victory Park Capital Advisors LLC, of striking a deal with payday lender Plain Green and the Chippewa-Cree Tribe of the Rocky Boy's Reservation to use the tribe's name in exchange for a small…
The same deception that hides the real cost of predatory, consumer loans is reflected in the title of pending legislation in both the House of Representatives and in the Senate. The Protecting Consumers' Access to Credit Act of 2017 (H.R. 3299 and S. 1624) would allow payday lenders, high-cost online lenders, and other predatory lenders to partner with banks to make loans that surpass existing state interest rate limits.
The next Chairman of the Federal Reserve System (Fed) confronts a deep and growing problem: rising inequality. A new Fed Chair could combat this problem in an unexpected way by implementing real-time payments. The few days between checks clearing are a major driver of why it is so expensive to be poor. They are also unnecessary given technology and easily removable with some regulatory will. Real-time payments could save billions of dollars for American families living paycheck to paycheck.
The check casher costs $20, but two overdrafts cost $70. Check cashing is a $2 billion a year business and represents yet another cost born by those who have less.
The technology for real-time payments has been around for a long time. The United Kingdom adopted real-time payments in 2008. Japan, Poland, Mexico and South Africa all have the technology in place today. Financial technology (FinTech) firms like PayPal are offering real-time payments for customers who exist on both ends of their system. But unless your employer will migrate to using a FinTech for payroll, you need the banking system to modernize.
The Federal Reserve's eggheads are usually a pretty reliable bunch. So when researchers at the central bank's Cleveland branch recently published a study asserting that peer-to-peer loans were defaulting at rates reminiscent of subprime mortgages a decade ago, it seemed to confirm the worst fears about the budding online-lending market. But industry critics and academics questioned the researchers' data, forcing the Fed to pull the paper.
It's not easy to come by good data for this nascent field of finance, which makes the botched study all the more regrettable.
Duracell and personal finance company SoFi have "snapped up" some of the six-second spots Fox has set aside for its Thanksgiving broadcast of Vikings-Lions, while Disney will "air a mini trailer for 'Star Wars: Episode VIII The Last Jedi,'" according to Anthony Crupi
Revolut, an app-based banking alternative which has over 50,000 customers in Ireland, has now signed up 1 million customers globally and claims it has saved users over £120 million (€134 million) in fees.
London-based Revolut said it is now signing up between 3,000 and 3,500 new users every day, an increase of 50 per cent growth from three months ago.
Users have now made over 42 million transactions since the company officially launched in July 2015 with a total transaction volume of $6.1 billion.
In an email to its customers seen by Moneywise and confirmed directly with Funding Circle, the provider says it will allow existing customers to invest in an Isa from Thursday 30 November.
It has yet to announce a launch date for new customers and says this is because it is anticipating strong demand for the product. For the same reason, customers will not be able to transfer existing Isas to Funding Circle when the product is launched.
Customers must deposit at least £1,000 to open an Isa.
MORE THAN half of small business owners want the UK to join the European Free Trade Agreement (EFTA) once Brexit is complete, Funding Circle research has found.
A survey of 1,254 borrowers on the peer-to-peer lending platform found 57 per cent would support EFTA, also known as the 'Norway option,' as it provides a regional free trade area comprising of Iceland, Liechtenstein, Norway, and Switzerland.
PEER-TO-PEER analysis firm Orca is set to launch an investment platform.
The proposition will automatically build portfolios of P2P investments across more than 50 per cent of the market.
The portfolios would include major lenders across the consumer, business and property lending space such as Zopa, Funding Circle and Assetz Capital.
Fountain, a digital wealth management platform aiming to "empower" investors to achieve their financial goals has secured seed round investment.
The cash, an undisclosed sum, came from a number of City figures led by Patrick Day, chairman of Day Cooper Day, a specialist pensions provider.
ThinCats unveiled a new brand last week at an event attended by more than 100 business leaders. The gathering took place at the National Space Centre in Leicester but the new branding will not be officially launched until mid-December.
Effectively, FG17/8 is the new bible for everyone interested in developing a new automated (digital /robo /telephone-based) advice solution. Or it is a checklist for those who have already trodden down this well-worn path.
Do note though – as if you did not already know – the paper "contains general guidance and is not binding", is not "exhaustive", must not be read in isolation of the handbook, and does not address any potential changes that might arise from the implementation of the Insurance Distribution Directive. (Heaven forbid anyone would actually take any accountability for what is between the covers).
Two years. Two years. To pull together in one document the working practices that professional firms already follow with their eyes closed?
New research shows that 78% of financial advisers are confident robo-advice offers no threat to their business, despite nearly half expecting more demand for robo-advice over the next 12 months.
The research from Aegon found that the degree of concern felt by advisers correlates to the typical size of their client portfolios, with advisers whose client portfolios are at the lower end of the scale more alert to the threat from the lower cost option of robo-advice.
For advisers with client portfolios of more than £200k, 88% feel it offers no threat to their business, and even for portfolios of up to £100k, the figure remains high at 73%.
While the majority of advisers believe robo-advice is no threat to their business, a third (31%) do point to robo-advice and similar digital services as one of the top challenges to the wider industry over the coming two years, a little behind Brexit (40%).
Chinese regulators and police are investigating a potential leak of data from online lender Qudian Inc., according to people with knowledge of the matter.
Officials are probing allegations that data from more than a million students who are clients of Beijing-based Qudian was leaked and possibly sold online, said the people, who asked not to be named discussing private information.
The probe's initial findings show that at least part of the leaked data match information clients had provided to Qudian, the people said. Investigators are checking whether the data came from Qudian, if the company was aware of the breach, and whether it took necessary measures to ensure the safety of personal information it collects.
Chinese regulators introduced major rules on Nov. 17—the scale of which has been compared to the U.S. Dodd-Frank Act—to unify regulations for the asset managementindustry and curtail shadow banking activities.
The rules are broad-based, covering China's $15 trillion of asset management products issued by all financial institutions.
For example, the rules will prohibit asset managers from promising guaranteed rates of return to investors, and require issuers to set aside 10 percent of their fees from managing client assets in escrow, to serve as a buffer against losses.
For publicly offered funds, total assets cannot exceed 140 percent of the funds' net asset value. The same ratio is set at 200 percent for privately offered funds.
China is regulating micro loans on the internet, with a high-level Chinese government agency issuing a notice urging provincial governments to halt approval of new web-based online lenders.
The firms are lending to consumers in China that have been turned down by Chinese banks. However, interest rates on these tiny loans can be very high — something borrowers don't realize.
According to the International Financial News, China plans to purge the country's 157 online micro-lenders, leaving only large state-owned companies and the biggest internet firms intact with licenses. Few of the existing lenders will survive, said the newspaper, which is managed by the official People's Daily.
A comprehensive cleansing of the industry, which offers almost immediate unsecured loans over the Internet, often at high interest rates, would escalate earlier moves to crack down on the sector and its estimated $152 billion of loans. News that China has halted further approvals for online micro-lenders has already pummeled the New York shares of firms like Qudian Inc. and PPDAI Group Inc.
"It would seem to be an enormous, enormous risk to try an IPO with that hanging over your head," said Christopher Balding, an associate professor at Peking University HSBC School of Business. "It would most likely put a halt to any IPO plans of these companies now."
The listing of online lender Qudian at the New York Stock Exchange on Oct. 18 heralded the birth of a new China billionaire, 34-year-old chairman and CEO Luo Min. The stock rose by as much as 43% that day, giving Luo a fortune worth $2.2 billion amid optimism about industry prospects.
Five weeks later, more than half of Qudian's value has been wiped out and he's on the verge of dropping from the ranks of the world's billionaires altogether. Qudian fell 16% last night and at yesterday's closing price, Luo's fortune (which he shares in a trust with family) was worth $1.02 billion. Investors in other China fintech stocks got socked yesterday, too. Jingpu Technology plunged 12.9% to $5.75, way below it IPO price of $8 from last week. China Rapid Finance fell 6% yesterday and PPDai fell a whopping 24%.
One of most notable online lending players aptly named Huabei (花呗, Just Spend) comes from the company that invented Singles Day—Alibaba.
To help them give away money to uncle Jack Ma, as hand-choppers have joked, this year Huabei has raised its credit limit to almost 80 percent during the promotion activities before Singles Day, allowing users to spend an extra RMB 2200 on average.
Huabei is the credit card of millennials, it targets the young and the unbanked. According to a report published recently, 86% of Huabei users belong to the generations born after the 80s and 90s (in Chinese). The fact that the 60% of them never owned a credit card is a good illustrator why online lending has experienced such a meteoric rise in China.
According to Huabei data, 38% of users choose to repay their debt in 12 monthly installments (in Chinese).
WeLab Ltd. has picked banks to advise on a Hong Kong initial public offering that could raise about $500 million, according to people with knowledge of the matter. The China-focused lender, whose backers also include billionaire Li Ka-shing, is aiming to list as soon as next year, the people said, asking not to be identified because the information is private.
Stop panicking about China's online lenders. The real target of the crackdown is rogue local governments.
Financial News said government entities can't issue new licenses for internet micro-lending beyond the 157 institutions that already have them. The consequences were immediate: Zhejiang Busen Garments Co., for one, said in a filing Thursday it's terminating plans to set up an online lender.
As of September, there were 8,610 micro-lenders with 970 billion yuan ($147 billion) of loans outstanding. Many of those weren't licensed by national regulators such as the People's Bank of China or the China Banking Regulatory Commission, which have strict rules.
Rather, authorization was handed out by local governments, most of which have no fintech expertise, to companies claiming to be affiliated with state-owned enterprises.
Ant Financial, Alibaba's financial affiliate, has announced that China now has more than 520m mobile payment users, reports state-owned news agency Xinhua.
A report released by the People's Bank of China detailing the country's payment system in the second quarter of 2017, notes that Chinese banks dealt with 8.6bn payments from mobile services during that period – up 33.84 percent from last year.
The combined value of mobile payments increased by 33.8 percent to 39.2tn yuan (around US$6tn).
IN China, an alternative capital market is taking shape with the rise of fintech companies, where fintechs are the intermediaries linking borrower and lenders. Moreover, fintechs are edging into the credit rating space, leveraging on their big data capabilities.
One core competence of fintech companies is their IT stability in the areas of payments and cloud computation. The strength of their IT infrastructure makes the technology players resilient under extreme conditions. During the recent Singles' Day sale on November 11 – China's online shopping bonanza equivalent to that of the US' Black Friday – Alibaba's Alipay processed a peak of 256,000 transactions per second and Alibaba Cloud processed as many as 42 million instructions per second.
A consortium of nine large companies — including UBS, Credit Suisse, Swisscom, Swiss Post, SIX, Raiffeisen, Swiss Railways, Zuercher Kantonalbank and Mobiliar — will enable Swiss consumers to use a single digital identity when making eCommerce purchases.
According to a report in Reuters, the idea behind the project is to get to a point where consumers can use one login to make purchases at shops, buy train tickets and engage in banking activities online. The group aims to create a joint venture in 2018.
Lendoit is a Decentralized P2P lending platform, which connects borrowers and lenders from all over the world in a trusted, fast and easy way using the advantages of Smart Contracts and the Blockchain technology.
What do you think is the biggest problem Lendoit will solve and why is it important?
The lending industry is not efficient because it's controlled by centralized financial organizations that set the interest rates according to their own interest. It's not fair that honest borrower from Brazil is paying 60% interest rate while borrower from Japan pays around 1%.
Lendoit uses three types of scoring:
Local rating provided by a local supplier from the borrower's state. Lendoit is working to create cooperation with some entities in various countries to provide this service.
International scoring providers that are using innovative methods such as scanning social networks and scanning the borrower's e-mail.
Lendoit is working to create cooperation with these International entities.We have already signed / in the process of signing with several companies in the scoring area, such as FriendlyScore, BLOOM, LENNO, and others, as noted in Lendoit's WhitePaper.
In the Lendoit eco-system platform, there is a special Smart Contract: a Reputation contract that retroactively checks each borrower who takes a loan, and set reputation score according to his or her historical activities within the platform
5 years have passed since I first started to invest into p2p lending at Bondora in October 2012. I still have 604 loans in my Bondora portfolio with an outstanding principal of 7,467 Euro at an average interest rate of 23.78%. Of these 2,746 Euro are in current loans, 778 Euro in overdue loans and 3,941 Euro in 60+ days overdue loans.
Bondora shows a net return of 19.0% for my portfolio. In my own calculations, using XIRR in Excel, assuming that 30% of my 60+days overdue and 15% of my overdue loans will not be recovered, my ROI calculations result in 17.2% return. Even if I assume total loss on all outstanding loans that are 60+days overdue my ROI calculation results in 15.6%.
Source: P2P-Banking
FinBee, a Lithuania based p2p lending platform, has started to expand internationally by launching in the Czech Republic. By 2020, FinBee plans to begin operations in another two European countries.
FinBee will provide personal lending services for residents of the Czech Republic as well as for investors from across the entire European Union.
Banks – local banks, in particular – have traditionally been the main and sometimes the only source of external capital for SMEs. However, increasing regulatory requirements have lowered the probability for SMEs to obtain access to bank financing.
P2P lending is part of the wider universe of crowdfunding. This is a bigger market than many people expect. For example, a 2016 paper for the European Commission reported that crowdfunding expanded by 167pc in 2014 and reached $16.2bn. North America remains the largest market ($9.5bn), followed by Asia ($3.4bn) and Europe ($3.3bn). While there are no accurate figures on the Irish market, Orca Money reports that the UK P2P market had £9.6bn cumulative lending since 2010, £1bn of which was in Q1 2017. In 2016, Orca Money reported that the UK P2P market comprised 177,000 retail investors with consumer (46pc), business (35pc) and property (19pc) borrowers.
P2P platforms have been very cautious about the loans they offer to investors, with most of them being classified as low-risk. This has resulted in low default rates and acceptable positive returns for investors. The potential for positive returns has attracted institutional and professional investors (eg investment banks, venture capitalist etc) into the game and created a disproportionate capital supply and demand. Such a trend is particularly visible in the US and UK, the two largest P2P markets, but it has recently emerged in smaller markets like Australia and New Zealand and is likely to occur, to a greater or lesser extent, in all regulated markets, including Ireland.
The lack of a clear regulation has arguably prevented the growth of the Irish P2P lending market by discouraging both investors and small businesses to participate. A clear regulatory framework is necessary to ensure transparency and to increase investors' confidence in P2P lending markets.
On 12 September 2017, FCA published a consumer warning on initial coin offerings (ICOs), stating that they are 'very high-risk, speculative investments', and that 'there is a good chance of losing your whole stake' as a purchaser.
Earlier in September, the People's Bank of China had denounced ICOs as 'illegal fundraising' and issued a ban that caused the value of cryptocurrencies such as Bitcoin to plummet. The following day, Canadian regulators accepted a firm offering ICOs into its regulatory sandbox as part of its broad goal of supporting innovative fintech projects. The European Securities and Markets Authority has been the latest to denounce ICOs, echoing the FCA's warning to consumers that ICOs are 'very risky and highly speculative investments.'
By applying the conditions from SEC vs Howey, the US Supreme Court test for determining whether transactions qualify as investment contracts (and by extension, securities), the investigation found that the tokens emergent from the DAO's ICO are securities and thus could fall within the US regulatory perimeter.
The SEC made the classification by fulfilling the following criteria from the Howey test:
Investment of money
Reasonable expectation of profits
Derived from the managerial efforts of others
Investor voting rights were limited
The FintruX Network has been established to transform unsecured loans to highly secured loan without any hurdles to borrowers and investors. The platform has unique blockchain approach of global P2P lending highways which proposed to raise $30 million by selling digital tokens.
The FintruX Network aims to enhance credit enhancements by introducing cascading levels which involves:
Additional collateral
A local third-party guarantor
Cross-collateralization
Fintrux ultimate protection reserve
But it was probably not as long as the minimum two years contemplated by O'Sullivan and the Greens for the proposed Banking and Financial Services Commission of Inquiry.
It should not be a problem if the three judges have no background or experience in fintech, cryptocurrencies, blockchain, peer-to-peer lending, equity crowd funding and payment systems riding off messaging services such as those offered by WeChat, Facebook, Apple and Google.
After all, this is not about the future. This inquiry is about spending more than $200 million looking in the rear view mirror.
Futurist and chief executive of global consultancy firm Tomorrow, Mike Walsh, told the 2017 Financial Planning Association Professionals Congress that sweeping technological change driven by complex algorithms is nothing to fear as it's simply "not unique."
Walsh said financial planners' fear-based thinking that technology will replace jobs must shift to ask how will jobs need to change.
INDIA's fifth-largest private sector bank, YES Bank, is raising a total of US$400 million in two transactions in the offshore syndicated loan markets as it further diversifies its funding sources.
The first transaction is a five-year loan amounting to US$250 million raised from a group of Taiwanese banks, led by CTBC Bank, Bank of Taiwan, Mega International Commercial Bank and Land Bank of Taiwan. The deal was upsized from the initial target of US$200 million as YES Bank exercised the green shoe option following an oversubscription of US$355 million from 13 other banks.
Adhil Shetty, CEO of BankBazaar, was recognized by the India FinTech Awards 2017 earlier this month. Shetty was named Fintech Leader of the Year at the event, which featured more than 200 attendees, more than 40 speakers, and 20 shortlisted startups from six countries.
Millennials are not only developing a healthy appetite for financial advice, they are also more likely to trust digital advice from automated investment services than older generations.
Results from the study showed that in Europe 32% of online adults between the ages of 18 and 37 say they "rely on financial advice from professionals", compared with 29% of older generations.
At least two-thirds of US Millennials were willing to share personal data in order to obtain better service from their financial institution.
Only 38% of US Millennials are confident that a bank or credit union will offer them valuable financial advice, compared with 46% of their older counterparts.
The migrant and their unbanked families in emerging and frontier markets have been suppressed for the longest time without any access to basic services, financial or otherwise. Approximately 2.4 billion people in poverty worldwide are often excluded from free movement or basic rights which often leads them to corruption and crime, including slavery, human trafficking and in extreme cases, death. Migrants far too often are denied basic financial tools.
LALA World ("LALA") is a wholesome ecosystem for the unbanked, starting with the migrants and their families back home. The base of this ecosystem is the LALA Wallet platform. By creating a whole new peer-to-peer infrastructure, LALA aims to revolutionize the way individuals, small businesses and micro-entrepreneurs transact, make domestic and cross-border payments, borrow money and associated products like insurances, cards, wealth and other general banking products.
LALA World Products from their Ecosystem
LALA Transfer – A Peer-to-Peer local and global remittance backed by crypto as well as fiat.
LALA Bill Pay – Local and International bill payments for you and your family.
LALA Lends – Domestic and International peer-to-peer lending via crypto and fiat, individual and small businesses.
LALA Card – Crypto and Fiat card synced to your Wallet and usable at millions of PoS globally.
LALA Kit – Contains a mobile phone with pre-loaded LALA Wallet, LALA Insurance, LALA Card, partners' products, etc.
ICO Pre-sale – Nov. 25-Dec. 15, 2017 (discounts available).
ICO – Jan. 5-Feb. 5, 2018
Author Allen TaylorPosted on November 24, 2017 Categories Alibaba, alternative capital, Bankbazaar, Bondora, Daily News Digest, Digital Identity Verification, Events, FCA, Featured, FED, Finbee, FintruX Network, Fountain, Funding Circle, Huabei, ICO, IPO, ISA, KBRA, LALA World, lawsuits, Lend Academy, Lending Club, Lendoit, Marcus, marketplace lending, microlenders, Millennials, Mobile payments, OCC, Online Lending, Orca, p2p china, p2p lending, payday lenders, Qudian, Regulation, Revolut, robo-advice, Shadow Banking, SMBs, SoFi, ThinCats, unbanked, unsecured lending, Victory Park Capital, WeLab, Yes Bank
Tuesday November 21 2017, Daily News Digest
News Comments Today's main news: Experian buys, integrates Clarity Services. Think Finance files for bankruptcy. PayPal offers robo-investing. Assetz Capital achieves 1.5M GBP funding through Seedrs. Elevate launches industry research repository. Nav Athwal steps down as CEO of RealtyShares. Ping An Insurance prepares for Lufax IPO. TransferWise doubles revenue. Today's main analysis: LendingTree releases monthly mortgage offer report. Today's thought-provoking articles: The […]
Today's main news: Experian buys, integrates Clarity Services. Think Finance files for bankruptcy. PayPal offers robo-investing. Assetz Capital achieves 1.5M GBP funding through Seedrs. Elevate launches industry research repository. Nav Athwal steps down as CEO of RealtyShares. Ping An Insurance prepares for Lufax IPO. TransferWise doubles revenue.
Today's main analysis: LendingTree releases monthly mortgage offer report.
Today's thought-provoking articles: The Cleveland Fed retracts P2P lending report. Why the Cleveland Fed pulled their report on P2P lending. Why Heartland Bank is giving up on traditional banking to pursue digital banking. Inflation and counterfeit credit.
Experian buys, integrates with Clarity Services. AT: "Experian and Clarity Services have had some relationships since as early as 2012. Clarity now has a tagline on its website which reads 'A part of experian'."
Think Finance files for Chapter 11. AT: "It's difficult to imagine that this doesn't have something to do with the recent Consumer Financial Protection Bureau's targeting of Think Finance for deceiving consumers into paying invalid debts."
PayPal to offer robo-investing. AT: "This is big news for PayPal customers. The payments service has increasingly become more like a traditional finance company with acquisitions and interest in technology that will expand its core business. A welcome development indeed."
Elevate launches industry research respository. AT: "This promises to be a great resource."
The Cleveland Fed retracts P2P lending report. AT: "Lend Academy weighs in on this move by stating that the report does make some important points."
Why the Cleveland Fed pulled its online lending report. AT: "A very interesting read with some great insight. A must-read."
LendingTree releases monthly mortgage offer report. AT: "Another must-read report from LendingTree."
Inflation and counterfeit credit. AT: "Very interesting."
RealtyShares CEO exits, to remain on board.
Leading RealtyShares into the next chapter. AT: "Blog post written by Nav Athwal."
Kabbage releases research on building a successful small business.
RealtyMogul tops $300M in real estate funding. AT: "Congratulations."
RealtyMogul wins gold in 2017 Stevie Awards for Women in Business.
The postal banking solution. AT: "The biggest problem with this is that the postal service is shrinking due to an increase in digital communications."
Fighting financial advice fees with robos.
InterNex Capital launches Velocity.
Wells Fargo adds overdraft protection to Rewind.
Intrinio adds NASDAQ data.
Global Debt Registry unveils collateral pledge blockchain proof of concept.
Midwest on the tech investor radar.
Consumers get help with credit card debt.
Atomist announces new automation platform for developers.
Mphasis highlights top cognitive intelligence trends.
Assetz Capital raises 1.5M GBP through Seedrs. AT: "Congrats."
Bots advise NatWest customers.
Monzo plans crowdfunding campaign to give customers deeper sense of ownership.
Speed-e-loans.com used 1.2M GBP pension liberation scheme to pay debts.
Ping An Insurance prepares for Lufax IPO. AT: "Lufax could do as well as Qudian on its lunge out of the gate."
Ping An Insurance sells technology, lightens assets.
TransferWise doubles revenues for last fiscal year. AT: "Congratulations. We're expecting great things from TransferWise."
Eidoo launches ICO engine for startups.
BNP Paribas buys 10% stake in Caple.
Why Heartland Bank is giving up traditional banking for digital.
Where things stand with the new P2P lending norms.
How fintech is shaping the future of financial services.
How two cities thrive in fintech.
Integration of Clarity Services Inc by Experian (Experian Email), Rated: A
Think Finance Files for Chapter 11 (U.S. Bankruptcy Court), Rated: AAA
PayPal in robo-investing venture (The Australian), Rated: AAA
Elevate's Center for the New Middle Class Launches Industry Research Repository (BusinessWire), Rated: AAA
The Cleveland Fed Retracts Their Report on "P2P Lending" (Lend Academy), Rated: AAA
Why The Cleveland Fed Pulled Their Online Lending Study (PYMNTS), Rated: AAA
LendingTree Releases Monthly Mortgage Offer Report for October (Business Insider), Rated: AAA
Inflation and Counterfeit Credit (GoldSeek), Rated: AAA
Nav Athwal Exits CEO Role at RealtyShares (Crowdfund Insider), Rated: AAA
Leading RealtyShares Into Its Next Chapter (RealtyShares), Rated: A
Kabbage Releases Research: On Building Successful Small Businesses (Crowdfund Insider), Rated: A
RealtyMogul Tops $ 300 Million in Real Estate Funding (Crowdfund Insider), Rated: A
RealtyMogul Wins Gold in 2017 Stevie Awards for Women in Business (BusinessWire), Rated: A
The Postal Banking Solution (Jacobin Magazine), Rated: A
Fighting the fees: robo-advisers win followers, if only for lower costs (Pittsburgh Post-Gazette), Rated: A
InterNex Capital Launches "Velocity" (PR Newswire), Rated: A
Wells Fargo adds overdraft protection with Rewind (Bankrate), Rated: B
Fintech seeks to 'democratize data' with Nasdaq's help (American Banker), Rated: A
Global Debt Registry Unveils Collateral Pledge Blockchain Proof of Concept (Global Debt Registry), Rated: A
Midwest is on the rise in attracting tech investors' interest (Crain's Cleveland Business), Rated: A
Getting Help with Credit Card Debt (Business Insider), Rated: A
Atomist announces new automation platform targeted at developers (SD Times), Rated: B
Mphasis' top trends in Cognitive Intelligence in 2018 (CIO), Rated: B
Assetz Capital Achieves £1.5 Million Through Latest Seedrs Funding Round (Crowdfund Insider), Rated: AAA
Banking bots advise NatWest customers on investments (ITPro), Rated: AAA
Monzo plans crowdfunding push to deepen ties with customers (Financial Times), Rated: A
Payday loan firm used £1.2m pension liberation scheme to pay debts (Citywire), Rated: A
Ping An Insurance Preparing For IPO Of Online Financing Platform Lufax (Asian Review), Rated: AAA
Ping An Insurance sells tech, lightens assets to aid returns (Asian Review), Rated: A
TransferWise reports doubled revenue for last fiscal year (AltFi), Rated: AAA
Swiss Platform Launches ICO Engine to Host Start-ups (TheStreet), Rated: A
BNP Paribas buys 10% stake in SME credit specialist Caple (AltFi), Rated: A
CEO Jeff Greenslade details why Heartland Bank is moving aggressively into digital banking (Interest), Rated: AAA
Lower interest rates for borrowers in offing? Read where things stand (Financial Express), Rated: A
How Fintech is Shaping Up the Future of Financial Services (Dekh News), Rated: A
A tale of how two cities thrive in FinTech world (Khaleej Times), Rated: A
Dear Clarity Services Inc Supplier:
As a supplier to Clarity Services Inc, we are writing to formally notify you that as of October 6, 2017, Clarity Services Inc has been purchased by Experian Holdings, Inc.
Effective January 1, 2018, purchases and invoice payments will be processed by Experian's centralized Procurement and Accounting departments.
Source: Experian
Read the court brief here.
The payments company was connecting its website and smartphone apps with those of Acorns Grow, a five-year-old automated savings and investment service, the two companies said on Monday.
PayPal users would be able to use their accounts to make contributions to Acorns and would be able to monitor and manage their Acorns investments from the PayPal app, said Joanna Lambert, the company's vice-president of consumer financial services.
PayPal is rolling out the Acorns offerings in phases, with the first batch of users getting access on Monday and all US users by early 2018.
The Center for the New Middle Class, a research-focused body developed by Elevate to engage and educate the public about the growing needs of individuals who do not have access to traditional credit options, today announced it has launched an industry research repository for researchers, reporters, policy makers and the general public. Known as the Resource Database, it is a curated collection of the best research on non-prime Americans and their challenges, attitudes, and needs.
In addition to containing external research and editorial content from sources such as Pew, the National Bureau of Economic Research and "The Atlantic," the database will house research and commentary from the Center for the New Middle Class regarding economic conditions that affect America's New Middle Class.
By visiting the database here users can search for entries, filter the results, and see the full bibliographic reference of information provided.
The shame of all this is that all the sensational headlines have already been written and confirmed in many people's minds the supposed shady nature of our industry. It would have been far better for everyone if the authors of this report had done their homework and produced a thoroughly researched report in the first place.
As Todd Baker pointed out, "we really should know which online lenders are adding to consumer financial health and which ones are detracting from it."
"[These borrowers] are not underbanked, they're sort of overbanked," observed Yuliya Demyanyk, a Cleveland Fed economist and co-author of the report. "Defaults on [marketplace] loans have been increasing at an alarming rate, resembling pre-2007 crisis increases in sub-prime mortgage defaults, where loans of each vintage perform worse than those of prior origination years."
The authors of the Nov. 9 report "have received several questions about the composition of the underlying data set they used in their analysis," the Cleveland Fed said on its website, and are "revising their paper to further clarify the data sample they used" and will post the new version as soon as it's ready.
So, what happened?
One theory is that they may have stretched the definition of online lending so far as to make an accurate and credible apples-to-apples comparison implausible.
Karen Webster spoke with Lending Club's head of government relations, Richard Neiman, to get a better sense of the source of discrepancy, since even Googling the definition of marketplace loan, Webster commented, might have saved the Cleveland Fed economists a lot of grief.
"The industry is so big now," Neiman said, "that it is not easy for policymakers to fully understand the divergences between different platforms, the different products, the different modeling and the differences in levels of transparency that are now defined as online lending."
LendingTree, the nation's leading online loan marketplace, today released its first monthly Mortgage Offers Report which analyzes data from actual loan terms offered to borrowers on LendingTree.com by lenders on LendingTree's network.
October's best loan offers for borrowers with the best profiles had an average APR of 3.75% for purchase and 3.70% for refinance, on conforming 30-year loans.
For the average borrower, purchase APRs for conforming 30-yr fixed loans offered on LendingTree's platform were down 3 bps month over month, to 4.31%, the lowest since November 2016. In contrast, the loan note rate of 4.18% was up 7 bps to the highest since July.
Consumers with the highest credit scores (760+) saw an average APR offer of 4.18% vs 4.44% for consumers with scores of 680-719. The APR spread of 22 bps between these score ranges was 1 bps lower than in September. The spread represents nearly $12,600 in additional costs for borrowers with lower credit scores over 30-years for the average purchase loan amount of $228,730. Additional costs are due to higher interest rates, larger fees or a combination of the two.
Refinance APRs for conforming 30-yr fixed loans were up 10 bps to 4.26%. The credit score bracket spread widened to 16bps from 15 bps, nearly $7,500 in extra costs over the life of the loan for lower credit score borrowers given an average refinance loan of $235,844.
The average proposed down payment for purchase mortgages have been rising for 7 months and reached $59,680 in October.
Average monthly payments were little changed at just over $1,100 for both purchase and refinance. The credit score bucket spread was $241 for purchase and just $77 for refinance.
Source: Business Insider
Let's take a look at an often-repeated idea that is popular in the gold and alternative investing communities. The government possesses a printing press. Therefore, it will never default. It will just inflate its way out of the debt. It will devalue the dollar.
The government does not set the value of the dollar. And it has no mechanism to set it. So, logically, it has no mechanism to reset it. It cannot devalue it. In the same way, you cannot lower yourself down by your bootstraps since you are not lifting yourself up by them in the first place.
We must emphatically state that the government does not print. It borrows. Congress does not have a printing press, to create greenbacks. It has a Treasury that can sell bonds to cover whatever payments the government is obligated to make that it has not got tax revenues for. Over the past year, for example, the government increased its debt by over 630 billion dollars.
Like any bank, the Fed borrows to fund its purchases of interest-paying assets. It earns a spread between what it pays (currently about 1.25%) and what its asset portfolio pays (over 2%). The commercial banks currently deposit over $2.1 trillion in excess reserves, and the Fed's total liabilities are over $4.4 trillion including Federal Reserve Notes (on which the Fed pays zero). Unlike any commercial bank, there is a law that obligates us to treat the Fed's liabilities as if they were money.
We are working on the problem that all borrowing and lending uses the dollar. We offer gold financing, simplified and a yield on gold, paid in gold.
Source: GoldSeek
Right now speculative mania is occurring in crypto currencies so that may (but not necessarily, beware correlation!) shunt such capital flows away from gold. As to default risk, there are signs of rising stress in high yield credit markets, but it's early yet.
Nav Athwal, one of the more prominent founders in the real estate crowdfunding space, has announced his decision to step down from the CEO role at RealtyShares, a platform he founded four years ago.
Ed Forst, RealtyShares Board Member and former CEO of Cushman and Wakefield, has been selected as the interim CEO while the company searches for a permanent replacement.
Crowdfund Insider spoke to Athwal regarding his decision to change his leadership role at RealtyShares and he explained he would continue to be engaged with the company;
"RealtyShares is in the strongest position it's ever been in. The company is moving from the build phase to the scale phase of its lifecycle. To best position RealtyShares for the future, I made the decision to transition out of my role as CEO to a new role on the Board of Directors. I asked Ed Forst to take on the role of interim CEO, while we look for a permanent CEO who will fit the culture and profile we're seeking. I am still very much a part of RealtyShares and will be actively involved in strategic decision-making. I am looking forward to supporting the company in this new capacity and getting back into principal real estate investing and agribusiness. As I begin to work on additional projects, I will be sure to let you know."
I started RealtyShares four years ago with the idea of creating a company that would make real estate more accessible, efficient, and transparent. RealtyShares has come a long way since those early days in my living room. It is now a 100-person operation and the leading platform for online real estate investing and capital formation.
My primary focus has always been to best position RealtyShares for future success. RealtyShares is now at an inflection point. I will remain on the Board of Directors supporting the company as it continues on its journey to build a global marketplace for real estate investing.
Kabbage Inc., a global financial services, tech and data platform serving small businesses, released new data reporting on the similarities that connect all small business owners (SBOs), including personal sacrifices, professional challenges and growth expectations. Featuring responses from 400 SBOs, the data shows more than 67 percent expect to increase revenues by the end of 2017, with more than half anticipating an increase of 10 percent or higher.
In partnership with Bredin, a leading small-business market research firm, Kabbage polled small business owners across industries, including retail, education, manufacturing, food and beverage, healthcare, automotive, energy and finance.
RealtyMogul has topped $300 million in total deal flow.
RealtyMogul has garnered over 140,000 investors, received over $300 million invested into deals presented on its platform and returned $65 million to its investors since its inception in 2012.
RealtyMogul has been awarded the Gold Stevie Award in the Consumer Services category during the 14th annual Stevie Awards for Women in Business.
The Stevie Awards for Women in Business are the world's top honors for female entrepreneurs, executives, employees and the organizations they run. All individuals and organizations worldwide are eligible to submit nominations – public and private, for-profit and non-profit, large and small. The 2017 awards received entries from 25 nations and territories.
Fewer than two thousand people live in Bluff, but any one of them can walk into the post office and cash a check or apply for a loan.
Meanwhile, the United States is riddled with what are called banking deserts — inhabited areas, many of them urban, where residents have no access to a bank.
One in four US households is unbanked or underbanked, meaning they're fully or partially boxed out of traditional financial services. Those 68 million people represent a growing market for payday loan sharks, and spend an average of 10 percent of their yearly income on the high interest and fees that go with alternative financial services — roughly the same proportion they spend on food.
But there's a collective solution to the banking desert: we could set up a public postal banking system like New Zealand's.
But one day last year Mr. Hansen was complaining to his mother, an avid investor, about the high fees he was paying on his investment account. She suggested he look into an online investment company called Betterment that markets itself as a low-fee alternative to traditional financial advisers.
Some of the most popular robo-advisers — such as Betterment, Wealthfront and Charles Schwab's Intelligent Portfolio — use exchange-traded funds to keep costs low. Betterment charges an annual fee of 0.25 percent of the account value. Wealthfront charges no fee for accounts $10,000 or less. Schwab's robo-advising platform limits its fees to the operating expenses included in the ETF, which range between 0.07 percent and 0.21 percent of the fund balance.
Citizens Bank, a Providence, R.I.-based bank that claims the third-largest deposit market share in the Pittsburgh region, introduced a new digital investment and advisory platform on its online banking home page in September.
A July report by S&P Global Market Intelligence predicts that digital advice assets will grow from $98 billion at the end of 2016 to $460 billion at the end of 2021.
Digital asset-based lender, InterNex Capital ("InterNex") is pleased to announce their Velocity platform for small and medium-sized businesses. Velocity provides borrowers on demand liquidity through the InterNex Line of Credit and delivers real-time access for working capital management. Velocity empowers accelerated growth and powerful analytics, traditionally only available to large enterprises.
On Nov, 14th, Wells Fargo announced Overdraft Rewind, a new feature to help customers avoid fees for overdrawing their checking accounts right before payday, when overdrafts most commonly occur.
Going forward, the bank will not charge overdraft or insufficient funds fees if a direct deposit large enough to cover those charges is received by 9 a.m. local time the day after the account goes negative.
You don't need to opt in; if you have a Wells Fargo account, you're covered by default.
The fintech startup Intrinio has partnered with Nasdaq to include the exchange's real-time data feeds in its financial data marketplace.
Global Debt Registry (GDR), the asset certainty company, today announced it has developed a collateral pledge registry, the first of its kind in the structured credit space, using Hyperledger Fabric, one of the Hyperledger blockchain framework implementations hosted by The Linux Foundation.
Among those quoted in the story is Mark Kvamme, a top venture capitalist in Silicon Valley who now heads Columbus-based Drive Capital. The firm has raised $550 million and invested in 26 companies, betting, The Timessays, that "the middle of America amounts to an undervalued asset, rich in markets, new business ideas and budding entrepreneurs."
Even so, three-quarters of all venture capital invested in America goes to California, New York and Massachusetts, the National Venture Capital Association estimates, and Ohio gets less than 1%.
For the many Americans who face unmanageable credit card debt, it's time to get their financial lives in order, says Andrew Housser, co-founder and CEO of Freedom Debt Relief– and if they need outside help, time to know how to find the right firm.
People looking for a trustworthy debt relief organization to help win the battle against debt can ask Housser's seven questions:
Does the company provide actual consultations and free advice to consumers?
Does the company provide educational material, including budgeting and financial advice, free of charge?
What is the background of the firm's management team?
How long has the company been in business?
Request and review the company's dropout and completion rates.
What are the fees, and how will the firm assess them?
How will the company help with creditor calls?
Atomist is formally launching with the Developer Automation Platform, an open source client and API. The new Development Automation Platform is designed to bring automation into the development and delivery process so that developers can focus on more important tasks.
Companies such as NVIDIA, Pivotal, Kyyti Group, Marlette Funding and Barclays Africa use Atomist for automation.
Smart environments with Pervasive Human and Machine Networks
Predictive Analytics driven Customer 360
Artificial Intelligence driven Multi-structured analytics – Cognitive Intelligence can enable insurance companies in analysing contact centre as well as chat data interactions in real time to predict propensity for fraud based on voice, video and text analysis and correlating the same with other similar fraudulent customer behaviors. The long term objective in such scenarios is to build machine learning based intelligent systems which learn on an ongoing basis based on historical pattern based analysis of billions of user and machine data points and predicts events.
Immersive Multi-modal User Experiences
Less than a month after launching its latest equity crowdfunding campaign on Seedrs, peer-to-peer lending platform Assetz Capital has successfully secured £1.5 million from more than 700 investors. The online lender took to the funding portal to raise £1 million for expansion.
NatWest has launched a chatbot that allows customers to seek financial advice from the comfort of their sofa.
The bot will determine the best way for customers to invest their money by asking questions such as what they want to achieve from investments, their current financial situation, what they can afford, their debts and other personal information, plus their attitude to risk.
It will then suggest ISA products they could consider for investment, plus how much they should consider investing and the most effective way to use their ISA allowance.
Banking app Monzo is planning to launch one of the UK's largest crowdfunding efforts next year to give customers in the fast-growing bank what it calls "a greater share of ownership".
Tom Blomfield, chief executive, said the start-up, which raised £71m in a private equity fundraising earlier this month, intends to make a further cash call of between £10m and £30m in 2018.
However, the main driver of the crowdfunding push would be "to enable the customer base to own part of the bank", Mr Blomfield said, adding that crowdfunding created a "genuine sense of ownership".
Three directors of an insolvent payday loan firm which received cash from pension liberation schemes have been disqualified.
Speed-e-Loans.com (SEL), used £1.2 million from private investors via the schemes to meet its existing debts.
Directors Philip Miller, Robert Alan Davies and Daniel Jonathan Miller have been banned from acting as directors for nine, six and five years respectively for breaching fiduciary duties and the duties of care, skill and diligence.
Ping An Insurance Group is working towards an initial public offering for its peer-to-peer online financing platform Lufax Holding, the Chinese insurer's chief operating officer said on Monday.
She didn't provide a timeline or size for the planned public offering. The company was valued at $18.5 billion after a round of fundraising last year.
Ping An Insurance Group will go light on assets and target technology exports as its next mainstay, China's leading financial conglomerate said Monday.
The London-based firm's annual revenue has grown by 140 per cent since last year, coming in at £67m this year compared to £28m in 2016. Its audited results show an adjusted operating profit of £2m, and an overall profit for the fiscal year of £7.4m.
As a result, TransferWise's operating loss has decreased from £17m in 2016 to £56,000 in 2017.
Switzerland headquartered Eidoo has today officially launched a initial coin offering (ICO)engine that allows companies and startups to host ICOs through the Eidoo mobile app.
According to data provider Coinschedule, $3.3 billion has been raised in more than 200 ICOsover the past 12 months alone and the popularity of this innovative ICO form of crowdfunding shows no signs of relenting.
Through the alliance and a 10 per cent stake in Caple, BNP Paribas Asset Management (BNPP AM) is establishing a new platform to offer alternative credit to European SMEs.
Greenslade also reiterated Heartland's forecast for June 2018 year net profit after tax of between $65 million and $68 million, an increase of up to 12% from $60.8 million in the June 2017 year.
"While some understandably lament the decline of branch banking, the bricks and mortar approach is not something we can compete in and is showing signs of obsolescence," said Greenslade.
He pointed to Heartland's digital services offering livestock finance, personal loans, SME working capital finance and deposits, noting "significant opportunity" to build on these.
Heartland has just six branches in Takapuna on Auckland's North Shore, Hamilton, Tauranga, Wellington, Riccarton in Christchurch, and Ashburton.
The Reserve Bank of India's (RBI) guidelines for peer-to-peer (P2P) lending will attract more people to these platforms and help bring down the interest rates for borrowers, Bhavin Patel, founder and CEO of LenDen Club, told Shritama Bose. Players in the segment have sought the regulator's clarifications on the permissibility of institutional lenders on P2P platforms, he added.
Will the RBI guidelines have an impact on the lending rates?
In the current scenario, the P2P lending market is a minuscule percentage of the huge lending market in India. But, due to the sentimental impact of the regulation, many more lenders may take to P2P lending, resulting in higher liquidity on such platforms. This will eventually lead to reduction of the interest rates offered to borrowers in this segment.
Traditional banking institutions have changed very little in the last hundred years. Most offer online banking and mobile apps these days, but behind the scenes, very little has changed.
In total, $49.7 billion was invested in fintech between 2012-16, which indicates just how important fintech is.
Fintech innovators are more cost-effective than traditional lenders. Their technology and business models are low cost. A traditional lender may have operating costs of around 7% compared to an online lender whose operating costs are as low as 2%.
A staggering 38% of customers no longer visit banks. The rise of online banks has proven that traditional bank branches are not essential. Online lenders such as Atom have become firmly embedded in the banking ecosystem.
Two cities thrive in this new world. Singapore and Abu Dhabi. Long before renewable energy was pervasive, Abu Dhabi had established Masdar. It has brought in the magic of the Louvre from Paris. Singapore has consistently been ahead of the curve of change. The world's largest vertical botanical garden paves the way for urban farming. The Marina Reservoir is a masterpiece of engineering and vision that turned an inlet of the sea into a strategically critical freshwater resource for the "Little Red Dot".
Recently, the Abu Dhabi Global Market (ADGM) held its inaugural FinTech Abu Dhabi Summit.
Two major announcements were made at the event. The first was the launch of the ADGM FinTech Innovation Centre by the first half of 2018.
The second was about a collaboration. ADGM and Plug and Play, the world's largest startup accelerator based in Silicon Valley, signed a new partnership to launch a startup acceleration programme in Abu Dhabi, focused on FinTech. The programme, first of its kind in the Mena region, will be housed within the ADGM FinTech Innovation Centre. The partnership was signed by Ahmed Al Sayegh, chairman of ADGM, and Saeed Amidi, CEO, Plug and Play. Some of Plug and Play's success stories include Google, Paypal, Dropbox and Lending Club.
Author Allen TaylorPosted on November 21, 2017 Categories Artificial Intelligence, Assetz Capital, Atomist, Banking, Blockchain, BNP Paribas, Caple, Clarity Services, Credit, credit card debt, Daily News Digest, Eidoo, elevate credit, experian, Featured, FED, financial advice, fintech, fraud, Global Debt Registry, Heartland Bank, ICO, Inflation, InterNex Capital, Intrinio, IPO, kabbage, LendingTree, Lufax, Monzo, mortgage, Mphasis, NASDAQ, NatWest, News, p2p India, p2p lending, paypal, Ping An, RealtyMogul, RealtyShares, Rewind, robo-advice, robo-investing, Seedrs, SMEs, speed-e-loans, startups, Think Finance, TransferWise, Velocity, Wells Fargo
Monday November 20 2017, Daily News Digest
News Comments Today's main news: PayPal to sell $6B in consumer loans. Cleveland Fed retracts study on P2P lending. China Citic, Baidu launch direct bank. Flender to expand into eastern Europe, Spain. Douugh partners with Choice Financial. Today's main analysis: Orchard Platform says how hurricanes affect unsecured consumer loans. Is LendingClub shifting to higher quality borrowers permanently? Today's thought-provoking articles: […]
Today's main news: PayPal to sell $6B in consumer loans. Cleveland Fed retracts study on P2P lending. China Citic, Baidu launch direct bank. Flender to expand into eastern Europe, Spain. Douugh partners with Choice Financial.
Today's main analysis: Orchard Platform says how hurricanes affect unsecured consumer loans. Is LendingClub shifting to higher quality borrowers permanently?
Today's thought-provoking articles: Why the Cleveland Fed should retract online lending study. Deep dive into Marcus. Are banks, credit unions prepared for the new mobile era? Unsecured consumer lending is booming in China. How fintechs simplify digital lending.
PayPal to sell $6B in consumer loans to Synchrony Financial. AT: "This transaction actually pivots PayPal's relationship with its credit issuer from Comenity to Synchony Bank."
Orchard Platform analyzes how hurricanes affect consumer lending. AT: "This really should come as no surprise, but an increase of 3x in the non-performing rate is huge, and in such a short time could have a devastating effect on the industry."
Why the Cleveland Fed should retract its online lending study. AT: "The Fed did in fact take down its study, so we can expect an updated study in the near future."
Cleveland Fed pulls online lending study from website.
Marcus deep dive. AT: "Marcus has already become a force to reckon with for the industry, and it will likely be even more so now that Goldman Sachs has been emboldened by early success."
Is LendingClub permanently shifting toward higher quality borrowers? AT: "It appears to be so, and I'll say it's a natural progression. That's where the long-term profits are."
Are banks, credit unions prepared for the new mobile era? AT: "Interesting read. And it's even more interesting that young smartphone users are concerned about how much time they spend on their phones while those 55+ are not. Will banks and credit unions begin to target older customers for mobile products?"
How PeerStreet is changing real estate investing.
What to know about Mulvaney if he is appointed head of Consumer Financial Protection Bureau (CFPB).
CFPB payday rule hits the Federal Register.
Mark Warner's payday lending bill.
Pavaso partners with eOriginal.
Head to head comparison of Elevate Credit, competitors.
How blockchain can serve the underserved.
Coinbase intros platform for hedge fund investors.
Is AQR at the vanguard of alternative investing?
Why bitcoin and real estate investing go hand in hand.
How Affirm makes money.
Lending is all about recovery.
Financial advisors published false information online.
The Visa debit card for cryptocurrencies.
Glint to offer multi-currency card for spending gold.
Komodo to bring dICO to market with Monaize.
China Citic, Baidu launch direct bank.
How consumer lending in China is booming. AT: "This is a very interesting read."
Last week's top news.
Flender to expand into eastern Europe, Spain.
Interview with Charles Egly of Younited Credit. AT: "Lend Academy podcast."
Allied Irish Banks invest 30M Euro in Transfermate.
Funding roundup.
Douugh partners with Choice Financial.
How startups simplify digital lending.
How fintechs assist MSMEs.
Aegon Life searching for fintechs to partner with.
Five fintech platforms that fund honeymoons.
Crowd Genie looking to raise 31.6M in ICO.
Are crowdfunding, P2P lending good business financing options?
The Global Trade Connectivity Network.
Indonesia to fix fintech rules.
PayPal to sell $ 6 billion in consumer loans to Synchrony Financial (TechCrunch), Rated: AAA
Orchard Platform Initiates Analysis: Severe Weather's Effect on U.S. Unsecured Consumer Lending Industry (Crowdfund Insider), Rated: AAA
Why Cleveland Fed should retract its online lending study (American Banker), Rated: AAA
Cleveland Federal Reserve Pulls Document Critical of Peer to Peer Lending as Critics Question Research Methodology (Crowdfund Insider), Rated: AAA
GS Marcus Deep-Dive (PeerIQ), Rated: AAA
Is LendingClub Making a Permanent Shift to Higher Quality Borrowers? (Lend Academy), Rated: AAA
Are Banks and Credit Unions Prepared for a New Mobile Era? (The Financial Brand), Rated: AAA
This $ 700 Billion Industry Has Been Untouched By Tech, Until PeerStreet Changed Everything (Forbes), Rated: A
Mulvaney as CFPB head? Five things to know (American Banker), Rated: A
CFPB final payday/auto title/high-rate installment loan rule published in Federal Register (The National Law Review), Rated: A
The controversy around Mark Warner's payday lending bill, explained (The Week), Rated: A
Pavaso Forms Partnership With eOriginal (National Mortgage Professional), Rated: A
Elevate Credit (ELVT) and Its Peers Head to Head Comparison (Dispatch Tribunal), Rated: A
How Blockchain Technology Can Serve the Have-nots (Wharton), Rated: A
Coinbase is going after big hedge fund money with its new cryptocurrency security platform (Business Insider), Rated: A
AQR: The Vanguard of Alternative Investing? (Morningstar), Rated: A
RealEstateInvestingProfits.com Explain Why Bitcoin and Real Estate Investing are Joining Forces (PRUnderground), Rated: A
How does Affirm make money? (Vator.tv), Rated: B
In lending its all about the recovery (AltFi), Rated: A
Financial advisers found publishing false credentials online (Which?), Rated: A
UK FinTech to Launch a Bitcoin Visa Debit Card with Support for Altcoins (Cryptocoins News), Rated: A
Fintech startup Glint de-cloaks to offer a multi-currency account and card that supports spending gold (TechCrunch), Rated: B
Komodo (KMD) bringing first dICO to Market with Monaize (Oracle Times), Rated: B
China Citic, Baidu launch direct bank in fintech push (Reuters), Rated: AAA
Young, Carefree and Unsecured (Bloomberg), Rated: AAA
WeiyangX Fintech Review (Crowdfund Insider), Rated: A
Flender looking at expansion into eastern Europe and Spain (The Business Post), Rated: AAA
Charles Egly of Younited Credit (Lend Academy), Rated: A
Allied Irish Banks invests €30m in payments fintech firm Transfermate (City A.M.), Rated: A
Fintech funding round-up (Banking Technology), Rated: A
Fintech start-up Douugh scores partnership with US mutual bank Choice (Financial Review), Rated: AAA
Fintech startups simplify digital money lending (Sunday Guardian Live), Rated: AAA
How fintech startups are assisting MSMEs, the biggest contributors to the Indian economy (YourStory), Rated: A
In digital drive, Aegon Life looking for fintech partnerships (Zeebiz), Rated: B
Five fintech platforms which will make all your honeymoon dreams come true (Business-Standard), Rated: B
P2P lender Crowd Genie targets raising up to $ 31.6m via ICO (Deal Street Asia), Rated: A
Are crowdfunding and P2P lending good options for business financing? (e27), Rated: A
OJK Will Fix The Rules Regarding Fintech (Gatra News), Rated: B
PayPal announced today it has agreed to sell $5.8 billion in consumer credit receivables to Synchrony Financial, in an expanded relationship between the companies. The deal also includes Synchrony's acquisition of $1 billion in participation interests in PayPal receivables held by certain investors and a chartered financial institution, the company said.
As a result of today's deal, the two companies will expand their partnership by making Synchrony Bank the exclusive issuer of the PayPal Credit online consumer financing program available to PayPal customers in the U.S. for the next 10 years, replacing Comenity.
In a recent blog, Orchard Platform posted initial research how much Hurricanes Harvey and Irma affected the U.S. unsecured consumer lending industry. According to Orchard Platform, approximately 91% outstanding loans in Florida were in designated FEMA disaster areas including metropolitan areas Tampa, Orlando, Miami and Jacksonville.
"The population of loans in the areas affected by Harvey experienced a 3x increase from July 2017 to September 2017," according to Orchard Platform Credit Analytics Manager Nicholas Del Zingaro. "All consumer unsecured loans in Texas experienced a 170 bps increase in Current to 30 Roll Rate over the same period. Irma made landfall on September 10th, but the Florida and Irma designated areas within Florida and Southern Georgia already show signs of distress, with the Current to 30 Rate increasing from 1.5% to 2.5%. The total population had an uptick of 30 bps from August to September."
Source: Crowdfund Insider
The Marketplace Lending Association is calling upon the Federal Reserve Bank of Cleveland to temporarily retract and revise its report on online lending due to what we see as serious flaws in the authors' reliance on certain underlying data.
In our view, this paper — "The Taste of Peer-to-Peer Loans" — and its accompanying materials show that a lack of precision and understanding of subject matter can result in significant inaccuracies. The report's authors presented findings that seemed to reflect issues with the P-to-P industry, but they actually relied on data from a much broader category of loans. The result was a misleading and brutally critical report about the P-to-P industry that was actually based in part on data from more traditional loans.
Earlier this month Crowdfund Insider posted a research report published by the Cleveland Federal Reserve that was highly critical of the peer to peer lending industry (marketplace lending) in the US. The report, authored by Yuliya Demyanyk at the Cleveland Federal Reserve, Elena Loutskina at the University of Virginia, and Daniel Kolliner at the University of Maryland, has since disappeared from the Federal Reserve site.
Marcus, was launched in October 2016 amidst mixed perceptions from market participants. One-year later, however, Marcus has achieved its $2 Bn origination objective – making it the fastest growing lending platform that PeerIQ tracks.
GS Marcus expects to originate $13 Bn over three years – the exact amount that Wells Fargo consumer balances have shrunk over the last twelve months as detailed in the PeerIQ Lending Earnings Insights report. GS expects to grow revenue from the Marcus platform to over $1Bn by capturing roughly 6% of the $250Bn unsecured consumer loan market:
Source: PeerIQ
GS CFO Marty Chavez notes that Marcus has an aggressive ~3.5% ROA objective. By comparison, Discover's ROA is currently ~2.4% and has only achieved a quarterly 3.5% ROA once in the last ten years.
Although the statistics look similar, each lender is measuring loss-rates somewhat differently:
Lending Club and Prosper cumulative loss rates on 36-month prime term loans are ~12% – as estimated by ratings agencies during a base case (not thru cycle) scenario.
GS projects thru-the-cycle annual credit losses of 4.0%. Therefore, GS is betting that it will outperform on losses thru-the-cycle.
Discover's 3.2% loss-rate is a realized statistic from the most recent 10-Q.
Discover management notes that loss rates are re-normalizing to higher levels. Indeed, Discover's loss rate was 2.1% two year ago in 3Q 2015 and management expects losses will continue to re-normalizing going forward.
We believe a comparable thru the cycle loss-rate for Discover would meet or exceed 4%. By way of comparison, the Discover loan portfolio experienced a peak charge-off rate during the financial crisis of ~7%. (and continued to deliver a positive ROA).
* GS estimate of 4%, Lending Club and Prosper based on 3-year ratings agencies cum. Loss estimate of 12%. Discover based on 3Q-10Q realized
Source: PeerIQ, GS Investor Presentation, Public Filings, Bloomberg.
A recent post on the Lend Academy Forum spurred a discussion about the potential future of LendingClub, particularly as it relates to the types of borrowers they serve. While we don't have insight into what LendingClub's plans are, there are several things that have happened over the last two years that help us hypothesize that LendingClub's strategy may be shifting.
LendingClub recently sent an email titled "How LendingClub Notes May Help You Generate Long-Term Wealth". In it, they tout returns in the 4-6% range, a far cry from the returns some investors saw in LendingClub's early days. The 4-6% range they present is footnoted, clarifying that this includes only grades A-C.
After I began writing this article LendingClub coincidentally announced in their recent earnings call that loan grades F and G would no longer be available to investors These loans have an average interest rate of 24.16% on LendingClub's platform. Moving forward, the loans will be brought in house as part of a test portfolio for LendingClub.
You can clearly see the expansion of C grade loans, which has increased to 36.09% of total originations in 2017, the most ever.
C grade loans currently make up just shy of 50% of 60 month loans.
After years of strong mobile growth being driven by younger demographic segments, the majority of recent, more modest growth can be attributed to the 55 and older generation. In fact, consumers in the 55+age group have a three-year compound annual growth rate (CAGR) of nearly 8% compared to only 2% for the 18 to 34 segment, according to a study from Deloitte.
As in 2016, close to 90% of consumers viewed their phone within an hour of waking up, with roughly 80% doing the same within an hour of going to sleep.
Interestingly, the Deloitte research found that over 70% of younger demographic groups believe they are using their phones too much and are looking for ways to limit dependence. Alternatively, only 13% of consumers over 55 had the same concerns.
Source: The Financial Brand
When consumers were asked about the way they communicated on mobile phones, all options increased in 2017, including text messaging (91%), voice calls (86%), email (81%), social messaging (72%) and video calls (30%). The increase in voice calls reversed a four-year decline.
The survey found a significant growth in use of mPayments in 2017, albeit against a rather low base number. According to Deloitte, consumers who said they made an in-store mobile payment with a smartphone or other device in 2017 reached 29%, which is a 50% increase over 2016. Those who used mPayments weekly also increased by 50% in 2017, (from 8% to 12%).
The PeerStreet platform lets accredited private investors access the huge market of real estate loans, backed by big data and advanced underwriting to identify loans that can give consistent returns.
Brett Crosby, Co-Founder and COO of PeerStreet, has extensive experience in analytics from his time working at Googleas Director of Product Marketing.
What did you do before this?
I was the co-founder of a company called Urchin, which was early in the web analytics space. We were acquired by Google in 2005, and turned Urchin into Google Analytics. I stayed at Google for 10 years, building and launching Google Analytics, launching mobile ads, local ads, the go-to market on social initiatives at Google, and Google Drive. After that, I was running global growth on Chrome, Gmail Docs and Drive.
If President Trump taps Office of Management and Budget Director Mick Mulvaney as interim head of the Consumer Financial Protection Bureau, as is widely expected, he will be a sea change from outgoing head Richard Cordray.
Mulvaney, a former congressman from South Carolina, was a fierce critic of the bureau when in Congress and he sat on the Financial Services Committee.
The CFPB's final payday loan rule was published in today's Federal Register. Lenders covered by the rule include nonbank entities as well as banks and credit unions. In addition to payday loans, the rule covers auto title loans, deposit advance products, and certain high-rate installment and open-end loans. For a summary of the rule, see our legal alert.
At issue is the different ways that states try to handle payday lenders. Some states try to crack down on them with caps on interest rates. But other states are more lenient. And the situation is further complicated by big national banks, which operate under federal law and only have to comply with interest rate caps in the state they're chartered in.
That loophole enables national banks to engage in "rent-a-charter" schemes. Since these banks aren't subject to an interest rate cap (or are subject to a more lenient one), they can issue a predatory loan, then immediately sell that loan to a smaller payday lender barred by state law from issuing it on its own.
Pavaso Inc. has announced that it has selected eOriginal to support lenders in the digital mortgage process. Specifically, Pavaso will utilize eOriginal's electronic promissory note (eNote) and electronic vaulting (eVault) services.
Net Margins Return on Equity Return on Assets
Elevate Credit 0.13% 1.40% 0.14%
Elevate Credit Competitors -27.28% -16.64% -8.05%
51.8% of Elevate Credit shares are held by institutional investors. Comparatively, 40.9% of shares of all "Professional Information Services – NEC" companies are held by institutional investors.
Gross Revenue NetIncome Price/Earnings Ratio
Elevate Credit $580.44 million -$22.37 million 357.00
Elevate Credit Competitors $242.33 million -$13.85 million 84.22
Some 2.7 billion people worldwide today have zero access to capital. Despite lacking any credit history or verifiable economic identity, these so-called unbanked or under-banked individuals can now access global capital markets with a $10 Android phone, thanks to blockchain-based economic identity platforms like BanQu or Humaniq that create a unique hash of verifiable authenticity — similar to a social security number — from a simple retina scan or selfie. The total market opportunity this group represents is a staggering $380 billion, according to a recent report.
On Thursday, Coinbase, the San Francisco-based cryptocurrency exchange, announced a new platform that might quell the anxieties of big money investors looking to invest in crypto. The platform, called Coinbase Custody, was built specifically to meet the needs of such investors, including hedge funds and family offices, according to a Medium post by Coinbase CEO Brian Armstrong.
Just last week, an unidentified user accidentally deleted the code library required to use recently created digital wallets within Parity, a popular digital-wallet provider, and cryptocurrencies have long been associated with the chasms of the deep, dark web.
The service will charge users a $100,000 startup fee. Armstrong said there will also be a monthly fee based on assets.
Because index funds pose little competitive threat, expense ratios for the leading alternative funds are far higher than elsewhere in the industry. These days, the vast majority of conventional fund sales go into funds that have expense ratios of less than 0.60%–usually much less. With alternatives, on the other hand, a 1% expense ratio is considered low-cost. Most of the larger funds have expense ratios approaching 1.5%, which would doom them were they not alternatives.
Given all these differences, it's not surprising that, for alternatives, industry leadership is upside down. The giants are absent. Among them, Vanguard, BlackRock, Fidelity, Capital Research, and T. Rowe Price run a grand total of $7 billion in alternative mutual funds. In contrast, the management firm AQR controls $29 billion.
Source: Morningstar Direct
RealEstateInvestingProfits.com, a strategic consulting and real estate investing educational platform that is responsible for a combined 1,000 closings and nearly a 100m in total sales volume focused on wholesale/flips with their partners and affiliates, are making a compelling argument, suggesting that real estate and Bitcoin should be natural partners and doing their best to open eyes to possibilities in this area.
With the Real Estate Market Size growing from $7.1 trillion in 2015 to $7.4 trillion in 2016, currency movements effectively reduced the size of the global real estate investing market by approximately 2.3% in the dollar (USD) terms according to MSCI Research, the question many are asking – Can Bitcoin be a positive disruptor to save on third-party fees and high transactions exchanges from lending?
Affirm makes it easy to repay the loan, send out email and SMS text messages to remind the customer of upcoming payments. Users can pay theur Affirm bills online, by debit card or ACH transfer, and sign up for autopayment.
The company makes money the same way that a credit card does: by charging interest of between 10 percent and 30 percent.
There's a problem with shadow banking and alternative finance. It's called what to do when bad stuff happens.
Whereas most consumer lenders will struggle to provide 5%, platforms such as Funding Circle, Assetz and ThinCats can easily provide a net yield in excess of 5%, even after allowing for losses.
This deeper understanding of the lending process and defaults is all for the good but I think it raises a much more critical issue, especially relevant for SME lending – how do lenders cope with problem borrowers?
But Funding Circle also has a sweet spot in lending tens and hundreds of thousands of pounds which means they tend to avoid lending large sums in the £500k to £50m bracket (in fact nothing towards to the top end).
Which? Money analysed 43 advice firms which are listed on on Unbiased.co.uk – a comparison service that allows you to find a financial adviser – which stated they employed certified financial planners. These are advisers who hold a specific certification from the Chartered Institute of Securities and Investment (CISI).
Some 63% of them (27 firms), however, did not actually employ any such advisers.
Seven out of 24 firms (29%) were also falsely claiming to be accredited by the Society of Later Life Advisers (SOLLA), and 14 out of 72 firms (19%) claimed to have advisers with chartered financial planner status, despite not employing anyone who was, in fact, chartered.
A London-based fintech startup is planning on launching a prepaid Visa debit card, giving users the option to spend a range of cryptocurrencies across the U.K.
On Tuesday, the London Block Exchange (LBX) launched, headed by an 18-year Credit Suisse veteran.
The cryptocurrencies include bitcoin, ethereum, ripple, litecoin and monero.
London-based Glint has been pretty stealthy about what it planned to offer, despite several funding rounds and a vague description that it wanted to a create new "global currency" based on gold. Well, today the fintech startup is finally de-cloaking with a staggered launch of its multi-currency account, app and card that does indeed let you store your money in gold and convert it back to fiat currency at the point of payment.
Komodo (KMD) has a much smaller market cap at $233 million. Komodo promises to be a block chain interoperable network to allow transactions across coins to help financial institutions bring banking to freelancers, small business owners, and other underserved customers accept and send payments.
Monaize is now teaming up with Komodo for the first dICO to facilitate financial transactions using cryptocurrency.
China Citic Bank Corp (601998.SS) and search engine giant Baidu Inc (BIDU.O) launched on Saturday a direct banking joint venture, dubbed AiBank, to capitalize on China's rapidly growing fintech sector.
Consumer lending is booming in China, thanks to a less thrifty younger generation who have cast off the save-at-all-costs mentality of their parents.
China's unsecured consumer loans amounted to just 9 percent of gross domestic product in the first nine months of this year, compared with 15 percent in the U.S., according to consultants Oliver Wyman. The educated 18- to 36-year-old borrowers LexinFintech targets tend to be ignored by banks, even though their job prospects mean that they're unlikely to default.
Auto financing, meanwhile, has exploded to account for more than a third of car purchases last year from 8 percent in 2011, according to CLSA Ltd. data.
Peer-to-peer lender PPDAI Group Inc., which listed in New York last week, also said that its rates exceeded 36 percent once fees are included. The company's shares are trading below their offer price.
People's Bank of China, China's central bank, has made plans to launch a united platform by the end of 2017 for collecting personal credit information and assessing people's credit ratings.
The new platform is expected to cover data from non-traditional market participants, especially Fintech industry (e.g. peer to peer lending), which will complement the existing credit data mechanism, increase supervision over non-traditional financial sectors and effectively reduce systematic risks.
Third-party credit service agencies may also become shareholders in the new platform with a ratio of 8% respectively.
On November 16th, Jianpu Technology Inc. announced it would be listed on the NYSE. Goldman Sachs, Morgan Stanley and JP Morgan are the bookrunners for the deal.
On November 16th, Xiamen Financial Affairs Office released the first P2P lending firms fillings in China. However, what drew the media's attention more in the fillings is a firm called Jing Dong Xu Hang Online Lending Information & Intermediary Service Ltd. This company is a wholly owned secondary subsidiary of JD Finance.
Flender, a peer-to-peer lending start-up backed by Ding founder and Esat Digifone co-founder Mark Roden, is eyeing up an expansion to eastern Europe and Spain after it launches in Britain next year.
Younited Credit is the largest online consumer lender in Continental Europe having crossed €500 million in total loans issued earlier this year.
In this podcast you will learn:
What the banking environment is like in France.
The long road they took to get a banking license.
The typical borrowers coming to Younited Credit today.
The terms of the loans they are offering today in France.
The kinds of investors on their platform today.
How they structure their investor offerings.
The three different ways they make money.
How their insurance product works.
The yield to investors of their different offerings.
Who Charles sees as their competitors.
How they are expanding their business to Italy and Spain.
Some of the large investors they are working with today.
Their approach to technology and underwriting.
Some of the alternative data they are using to feed into their algorithms.
How Brexit has impacted their business.
How they are using the €40 million they raised recently.
Where they are at with regards to profitability.
Allied Irish Banks (AIB) has invested €30m in (£27m) in business-to-business international payments start-up Transfermate, it will announce today.
The investment could value Transfermate at between €250m to €300m, sources said.
As reported in May, peer-to-peer lending start-up Flender was seeking to get €1 million in funding and is targeting a UK launch after getting full authorisation from the Financial Conduct Authority.
Over in Israel, Tipigo Ventures, which offers an artificial intelligence (AI) powered wealth management platform, has raised $1 million in seed funding. The firm says this puts its valuation at $10 million.
Kuants, an algorithmic trading platform, will "co work" and "co live" at IA's start-up academy during a three-month long acceleration programme.
Staying in India, Sumeru Enterprise Tiger Business Solutions, a Bengaluru-based banking software start-up, has raised $900,000 from unnamed investors in India and the US.
Sydney-based fintech start-up Douugh has scored a partnership with US mutual bank Choice Financial, as it readies to launch its smart banking personal assistant, Sophie.
As part of the open banking partnership, Douugh will launch an integrated bank account and debit card with the bank, giving it the ability to accept deposits. Choice Financial has also invested in Douugh, as part of a $2.5 million seed round.
Amongst those startups who have been simplifying digital lending are Rubique, InCred, ZestMoney, Qbera, Loan Singh etc.
Working as a medium for a customer and financial institutions, data analytics performed on hundreds of data points on Rubique's platform assess the creditworthiness of the customers (loan origination qualification), bringing predictability by giving them eligible offers to choose from.
Using real-time processing is also part and parcel for Zest Money, based in Bangalore since 2015, whose USP is its simple digital process, fast approval time and flexible products, with the benefit of multiple options to pay EMIs.
Using cutting-edge technology and proprietary credit underwriting algorithms, based on alternative data sources, Loan Singh enables frictionless lending to creditworthy and underserved borrower segments. It mainly provides personal loans (for salaried individuals), Professional Certification Loans (for students pursuing skill development and certification programmes), and Small Ticket Unsecured Personal and Consumer Loans (through third-party associations).
Others like InCred, founded in January 2017 and based out of Mumbai, focus on giving credit to those customers who have traditionally been underserved by large banks and NBFCs.
While contributing eight percent to the nation's total GDP, micro, small and medium enterprises (MSMEs) also provide for 40 percent of the total export. Producing over 10,000 different types of products, these small-scale ventures are also responsible for 45 percent of the entire manufacturing output.
To begin with, last year's demonetisation drive has propelled the digital onboarding of a number of MSMEs.
However, given a favourable environment and a slew of radical changes, there is still a huge credit deficit that is still unmet for the sector. This is exactly where the number of mushrooming fintech startups step in. To disrupt the status quo and level the playing field, a number of fintech lenders are supporting these small-scale ventures. The fact was made evident by McKinsey, claiming that nearly 75 percent of the emerging fintech lenders are helping MSMEs with lending, payment systems, retail banking, wealth management and more.
Aegon Life Insurance is exploring partnerships with fintech firms to expand customer base through a digital push of selling policies online, a top company official said.
Here is a list of five fintech platforms, which will make all your honeymoon dreams come true:
Faircent.com
Singapore-based P2P lending solutions provider Crowd Genie plans to conduct an initial coin offering (ICO) of its CGCOIN currency, aiming to raise up to ETHB100,000 ($31.69 million).
For the better part of a decade, banks have relaxed their lending conditions, too, so small businesses are finding more success with being approved for a loan.
But banks are not the only ones providing funding to small businesses, as they are actually reluctant to lend money to such enterprises in some jurisdictions. In China, for example, state-owned banks are not too fond of lending to individuals and small businesses. However, here P2P lending is a booming market, with around 2,200 p2p lenders and a market valued at US$100 billion.
Not all small businesses have the capability to launch their own ICOs, nor build their own blockchains over Ethereum, however. For this purpose, a startup called Starbase will empower any business or individual to crowdfund using cryptocurrencies and tokens without building their own network.
It is based on this point of view that MAS and the Hong Kong Monetary Authority decided to collaborate on a Blockchain-based cross-border trade finance platform. The platform, which is called Global Trade Connectivity Network (GTCN), is an open-sourced Blockchain platform and will be launched at the start of 2019.
The authority noted that as of September this year, 24 P2P lending companies consisting of 16 local companies and 8 foreign companies have been registered and licensed in OJK. Meanwhile 31 P2P lending companies are in the process of registration.
Author Allen TaylorPosted on November 20, 2017 Categories Aegon Life, affirm, alternative investing, AQR, Bankbazaar, Banks, Bitcoin, Blockchain, business financing, CFPB, China Citic, Coinbase, consumer lending, Credit Unions, CreditMantri, Crowd Genie, cryptocurrencies, Daily News Digest, digital lending, Douugh, elevate credit, eOriginal, faircent, Featured, FED, financial advisors, fintech, Flender, Glint, ICO, Indonesia, Komodo, Kuants, lendingclub, Loantap.in, Marcus, mobile banking, Monaize, News, online banks, Online Lending, Orchard Platform, p2p lending, Pavaso, payday lending, payday lending rule, paypal, PeerIQ, PeerStreet, real estate investing, Rubique, SMEs, Tipigo Ventures, trade financing, Transfermate, Visa, Younited Credit
News Comments Today's main news: Kabbage secures $200M credit facility from Credit Suisse for AI-based lending expansion. Consumer Financial Protection Bureau (CFPB) files suit against Think Finance. Royal Bank of Scotland to launch robo under NatWest brand. Hexindai names Citi depository bank for American Depository Receipt Program. ICICI Bank, Paytm partner on short-term credit. Today's main analysis: PeerIQ Lending […]
Today's main news: Kabbage secures $200M credit facility from Credit Suisse for AI-based lending expansion. Consumer Financial Protection Bureau (CFPB) files suit against Think Finance. Royal Bank of Scotland to launch robo under NatWest brand. Hexindai names Citi depository bank for American Depository Receipt Program. ICICI Bank, Paytm partner on short-term credit.
Today's main analysis: PeerIQ Lending Earnings Insight Report. P2P lending in India.
Today's thought-provoking articles: Borrowers bungle credit card debt consolidation. LendingTree's holiday shopping survey results. The best personal loans of 2017. The best bad credit loans of 2017. The best small business loans of 2017. Beyond the bank.
Kabbage gets $200M credit facility. AT: "Congratulations! Another great achievement from Kabbage, one of the leading small business lenders in the alternative space. They're planning to use the funds for expansion of AI-based loans."
DBRS assigns ratings to Kabbage Asset Funding 2017-A-LLC.
Are borrowers bungling credit card debt consolidation? AT: "The focus is on the Cleveland Fed study recently published, but this is a good read."
Fed flags online lending.
PeerIQ Lending Earnings Insight Report. AT: "A must-read."
Marcus is booming.
What LendingTree's holiday shopping survey suggests about consumers. AT: "Judging from this survey, the economy is doing well–parents are planning to spend more for Christmas this year."
LendingTree breaks 52-week high.
Sharestates launches one-click closing.
Why people ignore bitcoin tax laws.
The best personal loans of 2017. AT: "Consumer-facing, but a must-read for personal lenders with comparisons."
The best bad credit loans of 2017. AT: "Consumer-facing, but a must-read for bad-credit lenders with comparisons."
The best small business loans 2017. AT: "Consumer-facing, but a must-read for small business lenders with comparisons."
Square letting users buy and sell bitcoin through cash mobile app.
YieldStreet surpasses $200M in originations. AT: "Congratulations."
Goji receives $15M funding.
CFPB aims for Think Finance, accuses service company of deception. AT: "This is huge. Regulation is no longer aims at lenders but at service companies that provide the underlying architecture."
CFPB has done a good job.
CFPB needs a Cordray-like watchdog.
Opportunities, challenges for digital lending.
Traditional banks should help govern fintech.
Office of the Comptroller of the Currency takes tamer stance.
Alternative investing fees matter.
Royal Bank of Scotland to launch robo.
Beyond the bank. AT: "The interesting thing is that there is still speculation that online lenders will become white label tech providers to banks. Personally, I don't think this will define the industry."
Big 3 feature Women in Fintech powerlist.
Victory Park Capital takes majority ownership of Borro.
How LendInvest's buy-to-let offer compares to other lenders.
Crowdproperty pitches to raise 600K GBP through equity crowdfunding.
Monzo planning an IPO.
UK fintech sets new funding record.
Millennials say advisors are inaccessible, expensive.
Cardiff to host new network of tech hubs.
Hexindai names Citi a depository bank.
Unified credit rating system expected soon.
Fintech companies dominate top 10 global firms.
WeShareBonds raises 12M Euro.
P2P lending in India disrupting traditional finance. AT: "With comparisons to other nations."
Taurus Coin opens for business.
Sharesies considering a robo platform.
ICICI Bank, Paytm partner on short-term credit.
Narrow banking: An idea whose time has come.
Reserve Bank of India planning p2p lending norm clarifications.
Active.ai raises over $8M.
Canadian companies make top 50 global fintech companies list.
Kabbage gets $ 200M from Credit Suisse to expand its AI-based business loans (TechCrunch), Rated: AAA
DBRS Assigns Ratings to Kabbage Asset Funding 2017-A LLC (DBRS), Rated: A
Study suggests many borrowers bungle credit card debt consolidation (Credible), Rated: AAA
Fed flags online lending as subprime redux, but market hits back (Global Capital), Rated: A
PeerIQ Lending Earnings Insights Report (PeerIQ), Rated: AAA
Goldman Sachs' lending platform is booming (Business Insider), Rated: A
LendingTree Holiday Shopping Survey Suggests Bigger Budgets, Selfless Spending and Mobile Shopping Among Parents this Holiday Season (Business Insider), Rated: AAA
LendingTree Inc. (TREE) Breaks into New 52-Week High on November 16 Session (Equities.com), Rated: A
Sharestates, America's Private lender, Launches Their One Click Closing Option (PR Newswire), Rated: A
Bitcoin Tax Laws Are A Nightmare So People Ignore Them (International Business Times), Rated: A
The Best Personal Loans of 2017 (U.S. News), Rated: AAA
The Best Bad Credit Loans of 2017 (U.S. News), Rated: AAA
The Best Small Business Loans of 2017 (U.S. News), Rated: AAA
Square Now Letting Some Users Buy & Sell Bitcoin Through Cash Mobile App (Crowdfund Insider), Rated: A
YieldStreet Surpasses $ 200M in Originations, Bolsters Leadership Team and Launches New Website amid Period of High Growth (BusinessWire), Rated: A
$ 15 Million Investment Round Fuels Accelerated Growth at Goji (PR Newswire), Rated: A
CFPB Guns for Think Finance. Files Suit Alleging Consumer Deception in Repaying Loans Not Legally Owned (Crowdfund Insider), Rated: AAA
Richard Cordray's CFPB Has Done Its Job Well (Bloomberg), Rated: A
We need a watchdog at Consumer Financial Protection Bureau (Washington Post), Rated: A
Five opportunities and challenges in digital lending (American Banker), Rated: A
Traditional banks should help govern fintech (Reuters), Rated: A
Under Trump, Banking Watchdog Trades Its Bite for a Tamer Stance (The New York Times), Rated: A
Buffett's 'Million-Dollar Bet' shows how much fees matter (Herald Tribune), Rated: A
Royal Bank of Scotland to launch robo-advice under NatWest brand (Financial Times), Rated: AAA
Beyond the bank (Prospect Magazine), Rated: AAA
RateSetter, Funding Circle, Zopa feature in Women in Fintech Powerlist (P2P Finance News), Rated: A
VPC takes majority stake in online lender Borro (P2P Finance News), Rated: A
How does LendInvest's buy-to-let offering stack up to P2P rivals? (P2P Finance News), Rated: A
Crowdproperty Pitches to Raise 600K GBP through Equity Crowdfunding (P2P-Banking), Rated: A
Digital bank Monzo sizes up IPO (AltFi), Rated: A
UK FinTech Beats Brexit Blues with New Funding Record (Digit), Rated: A
Millennials say advisers are inaccessible and too expensive (Financial Times), Rated: A
Cardiff will host one of the first a new network of tech hubs being set up in a £21m investment (Wales Online), Rate: B
P2P Lender Hexindai Names Citi As Depositary Bank for American Depositary Receipt Program (Crowdfund Insider), Rated: AAA
Unified credit rating system expected soon, say experts (China Daily), Rated: A
China Fintech Companies Dominate Top-10 List of Global Innovators (China Money Network), Rated: B
Online Lender WeShareBonds Raises €12 Million in Mission to Help Finance French SMEs (Crowdfund Insider), Rated: A
Peer to Peer (P2P) Lending in India: A positive disruption to traditional financing, albeit cautious approach required (CARE Ratings), Rated: AAA
TAURUS COIN OPENS FOR BUSINESS GLOBALLY (EIN News), Rated: A
Sharesies to look at robo-advice once regulator rules on exemptions (The National Business Review), Rated: A
ICICI Bank and Paytm partner for short term credit for users (Medianama), Rated: AAA
Narrow banking is an idea whose time has come (livemint), Rated: A
RBI likely to issue clarifications on P2P lending norms soon (Money Control), Rated: B
Active.Ai raises over $8 million in Series-A round (India Times), Rated: A
WEALTHSIMPLE, LEAGUE AMONG KPMG'S TOP 50 FINTECH COMPANIES (Betakit), Rated: B
After picking up $250 million in equity funding from Softbank earlier this year, the small business loans and finance company Kabbage — which uses only algorithms and machine learning (no humans) to determine an applicant's eligibility — is announcing another big infusion of money. The company is picking up $200 million from Credit Suisse in a revolving credit facility that it will use for loans.
Specifically, Kathryn Petralia, who is the COO and co-founded the company with Rob Frohwein, said the funding will help the company increase the number of loans it can make to larger companies in the US. The average size of those loans will grow to "north of $200,000," she said.
DBRS, Inc. (DBRS) assigned ratings to the following classes of loans extended by a group of lenders to Kabbage Asset Funding 2017-A, LLC (the Facility):
— Up to $148,150,000 of Class A Loans rated A (sf)
— Up to $24,868,000 of Class B Loans rated BBB (sf)
The Facility is a warehouse financing arranged for the benefit of Kabbage, Inc. (Kabbage) to support originations of small business loan receivables. Kabbage acts as servicer for the Facility.
Thanks in part to the rise of fintech companies that make loans online, more than 16 million Americans now have personal loans — an increase of 64 percent in the last five years.
Payoff — a personal lender that specializes in helping consumers tackle credit card debt — says its internal data shows borrowers who paid off at least $5,000 in credit card balances between August 2016 and January 2017 saw a 40-point increase in their FICO score within four months.
Cleveland Fed study: a cautionary tale
One year after taking out a P2P loan, borrowers had credit scores that were 16 points lower than those of the non-P2P borrowers they were matched to, on average — an impact that persisted for four years, the authors said in their working paper.
The study found no evidence that P2P lenders are providing access to credit for "underbanked" consumers — borrowers taking out P2P loans were obtain other credit from traditional banks at rates similar to other consumers.
But a number of companies on the list, including Lightstream, BestEgg, LendingPoint, Earnest, and RocketLoans weren't around in 2012, when the most recent loans studied by the Cleveland Fed researchers were made. Several others — including Avant, CommonBond, Pave, and Upstart — were just getting off the ground at the time.
The growth in fintech lending has been a driver in overall personal loan growth, with 16.1 million consumers owing $106 billion in personal loan debt as of June 30, 2017. That's up from the 9.8 million borrowers who owed $45 billion in personal loan debt in mid-2012.
ABS participants, speaking with GlobalCapital this week, hit back at the report, written by Yuliya Demyanyk, senior research economist, Daniel Kolliner, research analyst, both of the Cleveland Fed; and Elena Loutskina a professor of business administration at the University of Virginia's Darden School of Business, and contributing author at the Cleveland Fed.
The authors challenge the belief that peer-to-peer (P2P) loans have expanded credit to borrowers with limited access to debt since the financial crisis.
A central argument of the Fed's report was that sector has not done much to expand access to debt for borrowers with low credit scores.
In securitization, subprime consumer ABS has thinned since the crisis. For the five major credit card issuers, the years 2008-2016 saw revolving credit available to US borrowers with a Fico score of less than 660 reduced by approximately $142bn, according to data published by online lender Elevate earlier this year.
Even portfolios backing recent marketplace loan ABS have a weighted average Fico score above 680, the level that defines so-called 'near prime' credits, despite the notable dip in collateral quality. Marlette, for example, had an average score of 705 for its most recent transaction which was priced in October, while Prosper had a weighted average Fico score of 709, according to data from Kroll Bond Rating Agency.
We are pleased to release our inaugural Lending Earnings Insights report.
Below are some of the main themes that we explore in this tracker:
Large banks continue to retrench. Wells Fargo's loan portfolio is down $13 Bn YOY. Loss reserves are down at all major banks except at GS due to the ramp-up in their consumer lending portfolio. Goldman Sachs expects lending initiatives to add $2 Bn in revenue in the coming years. GS loan loss reserve increased 50% and GS had the highest improvement in ROE across its peer group.
Credit re-normalization trend continues remains a recurring theme across all major lending groups. Overall, loss-rates on recent vintages are increasing versus prior recent vintages, although performance remains stronger than pre-crisis levels. Card issuers are increasing loan loss reserves at a higher rate than loan growth, indicating expectations of higher losses going forward. Discover and American Express increased loan loss provisions ~50% although loan growth is at 9% and 14% respectively.
Consumer installment lenders do not anticipate an increase in loss rates, after having recalibrated loss expectations and increased reserves in 2016. OneMain had the smallest increase in loss reserves at 4%.
Consumers now have access to greater supply of credit and credit demand continues to grow. Consumer average debt-to-incomes are below pre-crisis levels.
Several lenders cited the shifting competitive landscape and the role of technology in driving innovation and risk management.
Where are we in the credit cycle?
Other banks may be tempted to emulate Goldman Sachs' move into lending, but they should proceed with caution. Although consumer credit demand in the US
LendingTree, the nation's leading online loan marketplace, recently conducted its Holiday Shopping Survey among 1,050 Americans aged 25 to 55 with at least one child. The results show that people generally expect to give more than they expect to receive, and although only 55 percent of respondents have a set budget this holiday season, 76 percent plan to spend the same amount or more on holiday shopping compared to last year.
According to the survey, the average holiday shopping budget across all age groups was $943, although 45 percent of respondents say they don't have a set budget for holiday shopping this year. LendingTree's 2016 holiday survey found that 56 percent of respondents planned to shop for the holidays without a pre-set budget.
Additionally, 29 percent say they plan to spend more on holiday shopping in 2017 than they did in the 2016 holiday season.
Source: LendingTree
Parents are setting a low bar for their children's gift giving abilities in 2017, with 68 percent of parents expecting to receive no gifts from or on behalf of their kids.
Most parents (80 percent) plan to spend at least $100 per child this year while 37 percent of parents plan to spend at least $250 per child. Although 62 percent of parents say they try to spend the same amount on each child, younger children have a slight advantage with 12 percent of parents admitting to spending more on younger children and only 6 percent of parents use a child's behavior to dictate how much money is spent on their gifts.
A debit card is the primary form of payment for holiday shopping for 46 percent of respondents, as well as the primary form of payment for across all groups. Second to debit cards, 29 percent designated cash as their primary form of payment, and only 21 percent are primarily credit card users – although credit cards are considered more secure than cash or debit cards. A recent CompareCards by LendingTree survey found that 66 percent of Americans think debit cards are as safe or safer than credit cards for payments, when in fact debit cards don't offer the same consumer protections as credit cards.
Respondents expect to do 50 percent of their shopping online and 34 percent of their shopping on their mobile phone. Millennials (age 35 and under) expect to do 40 percent of their shopping on their phone, the largest of any other age group.
Shares of LendingTree Inc. (TREE) broke into a new 52-week high yesterday, hitting a peak of $281.80. Shares closed at $279.40 after opening at $271.75 for a move of 2.83%. The company now has a market cap of $3.34 billion.
Sharestates, an online real estate investment marketplace, announced today the launch of its new One Click Closing tool, a feature that will allow return borrowers to visit a page where they can upload all the details and documents required for a new loan, allowing for a seamless transfer of closing date information without further communications. The launch of this new tool coincides with the company's overarching goal of providing borrowers with a streamlined funding process, while providing them the opportunity to solely focus on identifying viable real estate investment opportunities.
A survey of 564 American bitcoin users by the online loan marketplaceLendEDU, revealed more than 35 percent didn't plan to report bitcoin-related gains or losses on their tax returns. On average, respondents said the current fiat value of their bitcoin holdings were $2,930.85, although that will probably continue to rise along with bitcoin's market price.
Some of the most common requirements for a personal loan are:
Minimum credit score: Most lenders require that you have at least fair or good credit when applying for a personal loan. Each lender sets its own cutoff for what it considers to be excellent, good, fair or bad credit. In general, fair credit is a FICO score between 580 to 669 and good credit is a score between 670 to 739. Most companies require a score of at least 600, but some have greater requirements. A higher score will increase your ability to be approved, and the higher your score, the lower interest rate you'll qualify for too.
Clean credit history: Lenders don't like to see defaults, collections or bankruptcies. If you have one or more of these on your credit report, you might not be approved for a personal loan. If you're approved, you may have to pay an exorbitant interest rate.
Stable employment: A lender needs to know that if it lends you money, you'll have the means to repay it over time. Without a stable job, you could miss payments or default on the loan. Proof of employment validates your loan application.
Proof of identification: Lenders usually need to see proof of identification, such as a copy of your driver's license or passport, before approving your loan. Identity theft is common and they want to prevent thieves from taking out loans under another person's credit.
Choosing a Personal Loan Company
There are two types of lenders you can choose from: banks and peer-to-peer lenders. Banks offering personal loans include SoFi and LightStream and peer-to-peer lenders include Upstart, LendingClub, Prosper and Peerform.
Marketplace-based lenders usually have less strict credit score requirements than their bank-based counterparts. For example, LendingClub and Peerform only require a FICO score of 600 while bank-based companies such as SoFi and Payoff have minimum FICO scores of 660.
Every lender has a minimum and maximum loan amount. For example, SoFi will lend up to $100,000 while Payoff lends up to $35,000. If you need to borrow $45,000, then only look at lenders who offer that amount or more.
Best Personal Loan Companies of 2017
Best for very good credit, low APR and no origination fees: LightStream
Best for very good credit, low APR, no origination fees and a range of offerings: SoFi
Best for very good credit and low APR with merit-based qualifications: Earnest
Best marketplace for fair to good credit with merit-based qualifications: Upstart
Best bank for fair to good credit with merit-based qualifications: LendingPoint
Best for fair to good credit with a co-signer option: LendingClub
Bad credit usually is a FICO score below 640. FICO is the main scoring system for consumer credit, with credit score rangesdefined as:
Exceptional (800 to 850)
Very Good (740 to 799)
Good (670 to 739)
Fair (580 to 669)
Very Poor (300 to 579)
Payday Loans Versus Personal Loans
Payday Loans Personal Loans for Bad Credit
Lenders Online, brick-and-mortar Online, brick-and-mortar
Loan Amounts Typically less than $500 $1,000 to $50,000
Loan Terms Two to four weeks One to five years
Interest Rates 200 to 400 percent APR 36 percent APR or less
Some alternative payday loan companies market themselves as more socially responsible than traditional payday lenders because they offer better terms. They also want to help consumers rebuild their shaky credit and make payments on time. For instance, LendUp provides financial education and rewards existing borrowers who repay their loans to be eligible for loans at larger amounts and lower rates. Fig Loans only charges fees to cover the costs of the loan.
Choosing a Bad Credit Lender
Consumers should evaluate lenders based on the following criteria:
Type of lending company
Credit history and general qualifications
Co-signer option
Additional eligibility qualifications
Interest rates and types
Repayment options
According to the National Small Business Association, 69 percent of small businesses used financing in 2016, including loans, credit cards, venture capital and crowdfunding. The remaining 31 percent were not able to obtain adequate financing.
According to data from the U.S. Small Business Administration, small business bank loans totaled nearly $600 billion in 2015. At the same time, lending from alternative sources such as finance companies and peer-to-peer, or P2P, marketplace lenders amounted to $593 billion.
There are two categories of alternative lenders, direct and peer-to-peer lenders:
1. Direct lenders: Direct lenders are finance companies that fund your loan with capital other than a bank and without a middleman such as a broker, investment bank or private equity firm. Some direct lenders, such as LiftFund, offer SBA loans. Typically, small to midsize businesses borrow from direct lenders.
2. Peer-to-peer lenders: Online peer-to-peer lending directly connects you with investors who usually have a diversified loan portfolio made up of small portions of loans. A loan is often divided among several investors.
Choosing a Small Business Loan
Minimum years in business
Minimum annual revenue
Best Small Business Loans of 2017
Best for very small businesses: Kabbage
Best for borrowers with low credit scores: OnDeck
Best for new businesses: Accion
Best for low APR: LendingClub
Best for invoice financing: Fundbox
Financial service company Square is reportedly now testing out bitcoin on some of its Cash mobile app users. The new feature will allow users to buy and sell the cryptocurrency through the app.
YieldStreet, the alternative investment platform that is working to change the way wealth is created, announced that it has surpassed $200 million in originations and has added two new executives to its leadership team: Volfi Mizrahi as Managing Director of Originations and Ivor Wolk as General Counsel. Their additions come on the heels of the appointment of Hrishi Dixit as CTO earlier this year.
Goji announced a $15 million investment round led by Hudson Structured Capital Management Ltd., doing business as HSCM Bermuda.
The Consumer Financial Protection Bureau (CFPB) has filed suit in federal court against Think Finance, a Fintech that leverages its technology to power online lending platforms. The CFPB says the suit was filed for its "role in deceiving consumers into repaying loans that were not legally owed."
The CFPB alleges that Think Finance illegally collects on loans that are void under state laws governing interest rate caps or the licensing of lenders.
The actual filing states:
"From 2011 through at least 2015, Defendant has performed critical functions for three separate lending businesses owned by Native American Tribes: (1) Great Plains Lending, LLC (Great Plains); (2) MobiLoans, LLC (MobiLoans); and (3) Plain Green, LLC (Plain Green) (collectively, the Tribal lenders). Defendant is therefore a "service provider" under CFPA. 12 U.S.C. § 5481(226)."
Read the filing here.
As recently as a decade ago, the U.S. had no single regulator tasked with looking out for the interests of consumers in financial markets. Fragmented oversight allowed all kinds of bad behavior to fall through the cracks. Mortgage brokers hid the true terms of loans in piles of nearly indecipherable documents. Banks changed the order of transactions to extract the maximum overdraft fees from poor customers. Payday lenders offered products designed to trap people in an unending cycle of debt.
Cordray has accomplished a lot. The CFPB designed new, simpler mortgage-loan disclosures. It shed light on banks' overdraftpractices. It created the first federal rules to make payday lending less predatory. It gave the public reams of valuable information, such as a database that allows consumers to compare credit-card agreements. Its practice of publishing complaints pushed financial institutions to be more responsive. Its investigation of Wells Fargo brought national attention to the fake-accounts issue.
Some of its practices (in particular, preferringdiscretionary enforcement over explicit rule-making) are less than ideal and ought to be revisited; in other areas (such as auto lendingand credit reporting) its authority should be expanded.
Cordray's departure gives President Trump an opportunity to appoint a new leader, and I'm concerned that this will derail the watchdog agency's consumer-first mission.
In Cordray's parting statement, he wrote that the agency has recovered $12 billion for nearly 30 million consumers.
Despite the rapid growth of online and mobile lending in recent years, many banks are still just getting started.
Traditional lenders should demand that online financial companies protect consumer privacy and money interests, Federal Reserve Governor Lael Brainard said on Thursday.
Banks often pay tech companies for the information they gather on borrowers. For that reason, Brainard said, those lenders can set high standards in consumer protection and privacy.
The regulator, the Office of the Comptroller of the Currency, which oversees the nation's biggest banks, has made it easier for Wall Street to offer high-interest, payday-style loans. It has softened a policy for punishing banks suspected of discriminatory lending. And it has clashed with another federal regulator that pushed to give consumers greater power to sue financial institutions.
One category of alternative investments includes hedge and private equity funds. I am not in favor of such investments, especially for the average investor, and I am not alone.
Typically, hedge fund fees are 2 percent plus an incentive, or "carry," of 20 percent of the profits. The result is billions of dollars for the managers and far less for clients.
In 2015, Buffett lagged his hedge fund rival for the first time since 2008, gaining 1.4 percent versus Protégé's 1.7 percent. However, 2016 saw Buffett gain 11.9 percent to Protégé's 0.9 percent. At the end of 2016, Buffett's index fund gained 7.1 percent per year, or $854,000 in total, compared with 2.2 percent per year, or $220,000, for Protégé.
Royal Bank of Scotland is launching robo-advice for more than 5m customers as banks return to the investment market after a string of fines and as regulators attempt to plug the UK's wide "advice gap".
The state-backed lender is claiming to be the first in the UK to launch automated online investment advice when it opens on Monday under its NatWest brand. The service is designed for the majority of customers and for people with as little as £500 to invest as a lump sum.
The process, which costs £10 plus fees for the investment, is aimed at customers who lack the confidence to invest alone but do not wish to pay higher charges for full-blown financial advice, such as tax and inheritance planning.
Five high street banks remain responsible for more than 80 percent of business lending, prompting calls for greater market choice both from challenger banks and from emerging FinTech (financial technology) firms. However, the introduction of new platforms and new players raises questions of integration, innovation and regulation.
Launched seven years ago, Funding Circle has since hosted 70,000 lenders and is responsible for 2 percent of total UK business lending. While going head-to-head with the banks it is worth noting that Funding Circle is also growing the market.
So what next for financial services? Will the disruptors get disrupted? Possibly. One attendee suggested that FinTech firms would soon be providing 'white label' versions of their services to traditional banks that would then offer those services to customers. Others suggested that the likes of Amazon and Google would provide the biggest future threat not just to banks but to today's FinTech leaders.
FEMALE employees at RateSetter, Funding Circle, Zopa and Landbay have all made this year's Women in Fintech Powerlist, which celebrates the achievements and talent of women across the sector.
RateSetter's entrants are: Alexa McAlister, head of partnerships; Angela Yotov, head of legal; Joanna Wright, chief risk officer; Katie Brown, corporate counsel; Laurence Perrin, head of compliance; Lucy Bott, head of customer operations; and Maud Holma, finance counsel.
Women from Funding Circle who made the list are: Alysha Randall, global finance director; Lisa Jacobs, chief strategy officer; Lucy Vernall, global general counsel and global head of compliance; Maria Weaver, chief people officer; Panni Morshedi, managing director of international; Swati Lay, chief information security officer; and Vittoria Reimers, VP loan operations.
VICTORY Park Capital Specialty Lending (VPC) has continued its expansion into balance sheet lending by upping its investment in online secured lender Borro.
The alternative finance-focused investment trust now has the largest stake in the firm, which provides loans secured on luxury assets, owning around 49 per cent.
LendInvest said it will be offering buy-to-let loans through intermediaries ranging between £50,000 and £5m, with rates starting at 3.69 per cent depending on whether borrowers take a two, three or five-year fixed rate.
In comparison, Landbay facilitates fixed rate and tracker buy-to-let loans of between £70,000 and £500,000 from 3.55 per cent.
And fellow P2P lender LandlordInvest offers loans of between £30,000 and £300,000 from five per cent.
UK p2p lending marketplace Crowdproperty is currently pitching on Seedrs to raise 600K GBP from the crowd at a pre-money valuation of 5.9M GBP.
In an advertisement for a new Chief Financial Officer, Monzo has let on about its plans for an IPO within three to four years time.
The posting, published today on Monzo's job site, says:
"Alongside the CEO, you'll be heavily involved in future capital raising – pitching to investors and negotiating the best terms. In the next 3-4 years, it's likely you'll be responsible for taking the company through an IPO."
The UK fintech sector has enjoyed a record year of investment in 2017 with more than £2 billion invested across 182 deals, according to research body FinTech.Global.
Additionally, the compound annual growth rate (CAGR) of 10.7% experienced by firms valued below £75 million between 2014 and 2016 was supplemented with a further investment of £1.2 billion for this bracket in 2017. As for companies valued above the £75 million figure, a further £877.1 million has been committed across eight deals.
Investment in the UK's top-10 fintech firms accounted for nearly 46.7% of the total investment between Q1 and Q3 2017. The largest of these deals went to Gryphon Insurance in June, valued at £179.6 million. Of the top 10 deals, four went to lending firms – Prodigy Finance, 1stStop Group, Neyber and Funding Circle, valued at £457.5 million. Of the remaining six deals, two went to challenger banks Tandem and Atom, two went to enterprise software companies Options Technology and Darktrace, and the final two went to insurtech companyies Gryphon Insurance and Revolut.
Millennials view financial advice as an industry of "exclusivity, inaccessibility and high cost," according to research carried out by the financial advice trade body.
The poll of 178 millennials found 78 per cent of them believed they could only receive advice if they had investible wealth in excess of £50,000 but a significant number wanted advice when they had £10,000 or less.
Online-only options appealed to just 12 per cent of those surveyed.
A new tech hub will be set up in Cardiff as part of a UK-wide network of regional hubs under a £21m investment announced by the UK Government.
On Wednesday, Chinese peer-to-peer (P2P) lending platform Hexindai announced it has appointed Citi's Issuer Services business as the depositary bank for its American Depositary Receipt (ADR) program. According to the online lender, the program was established through an initial public offering of its American Depositary Shares (ADSs), priced at $10 per ADS, which raised approximately $50 million.
A unified platform for collecting personal financial information and assessing people's credit ratings is being planned, and it is expected to be launched soon as a part of the central bank's regulatory framework, experts told China Daily.
It will complement the existing credit center of the People's Bank of China, the nation's central bank.
Compared with the ranking last year, two more Chinese firms were added to the top 50 list. Online marketplace lending company Dianrong and credit card and online financial service firm U51.com, or 51Xinyongka, rose to top 50. Among the 100 fintech companies, lending and payments focused companies lead in terms of sectors.
WeShareBonds, an AMF-registered crowd lending platform, has raised €12 million to continue financing French SMEs. The new funding includes both the closing of WeShareBonds' second credit fund (Prêtons aux PME 2018) to finance French SMEs for €10 million and an equity increase of €2 million to finance platform growth.
The overall size of the NBFC sector in India has grown significantly during the last few years with increasing share of NBFC total assets to bank total assets (approximately around 15 per cent of the total banking assets).
P2P Lending in UK: In UK, the P2P market has seen active retail investor participation. The outstanding loan book in the UK industry is approximately around £2.9 billion (~Rs.25,000 crore) as on Q3-2017 as compared to £0.75 billion (~Rs.6,500 crore) as on Q4-20142 . Based on the outstanding loan book as on Q3-2017, the key players in the segment are Funding Circle, Zopa, FolktoFolk, Ratesetter and ThinCats capturing majority of the market.
P2P Lending in USA: P2P industry in USA is around $20 billion (~Rs.1.3 trillion) in 2016 up from $18 billion in 20153 . P2P lending in the USA has seen active participation of institutional investors (approximately 70 per cent of the total investor volumes) lending to borrowers through the platforms. In the USA, three dominant players capture majority of the market which include Lending Club, Prosper and Sofi.
P2P Lending in China: Globally, China has the largest market size of P2P lending which started in 2006. As of January 2017, there were total 2388 P2P platforms in China4with trading volume in 2015 touching $67 billion (~Rs.4.4 trillion) which is ten times that of UK and four times bigger than USA. However unlike USA and UK, the China P2P market is dominated by large number of small and medium size firms.
P2P Lending in India: Globally P2P lending has been in existence for more than ten years; however, it has been evolving in India in the last couple of years. Given the recent RBI guidelines, companies will now need to obtain NBFC–P2P license and will come under the purview of the regulator. There are more than 50 P2P online platforms operating in India. I-Lend, LenDenClub, Faircent, Lendbox, i2iFunding, Monexo, India Money Mart, Rupaiya Exchange are some of the leading P2P platforms operating in India. Currently, some of the leading P2P platforms claim to disburse loans amounting to ~Rs.1 to 2 crore a month. Outstanding loans under P2P model is estimated to have reached ~Rs.50-60 crore.
Source: CARE Ratings
United Kingdom's leading peer to peer cryptocurrency lender Taurus Coin today launched its online lending marketplace with USD 150M lending capital, bringing the world's fastest growing form of lending to the South-East Asia, Gulf, Africa, Europe, and India investors.
Start-up fund distribution platform Sharesies will develop a plan to provide personalised, automated financial advice, known as robo-advice, once there is more clarity about regulatory exemptions.
Last month the Financial Markets Authority decided to grant an exemption to enable the provision of robo-advice services under the current financial advice regime and said it aims to finalise the exemption and be open for applications in early 2018.
ICICI Bank and Paytm have partnered to offer short-term credit to users. The credit given to customers will be interest-free for 45 days and the bank says that it will give loans up to Rs 20,000. Initially, this will be allowed for select ICICI Bank customers who are on Paytm and will be extended to other bank customers as well who use Paytm.
Once the credit limit is set up for a customer, a consolidated bill is generated on the first day of the next month, which has to be paid by the 15th day of the same month. Customers can use their Paytm Wallet, debit card or internet banking of any bank for an easy repayment of their dues.
In August, the bank launched product called Instant Credit Card where certain pre-approved customers of the bank will be able to get a virtual credit card much before the physical card is delivered to them. A physical card will be sent to the customer's address in 5-7 days.
Meanwhile, the bank's rival, HDFC Bank said that it would start offering a virtual credit card for customers through its PayZapp wallet, as indicated by this Financial Express report. HDFC Bank has the maximum number of credit cards in circulation with 9.03 million. Meanwhile, ICICI Bank has 4.34 million credit cards.
Paytm's rival in the wallet space, MobiKwik has partnered with Bajaj Finance to to bring in credit facilities to a wallet business.
In April this year, PayU India will be investing $50 million in its product LazyPay over the next few years. The credit facility could extend for amounts from Rs 3,000 and even up to Rs 10,000, depending upon customer behaviour.
For the recently concluded festival season, e-commerce player Flipkart started to offer EMIs on debit cards on high-value purchases.
The most heartening takeaway from last week's public sector bank executive jamboree was the discussion around differentiated lending structures. The ThinkShop (earlier editions were called Gyan Sangams) suggested that large banks focus on corporate lending, while smaller lenders focus on retail loans or specific geographies.
Taking differentiated lending to its logical end, the time has come to consider converting the worst performers among state-owned lenders to narrow banks, which won't lend at all.
Narrow banks are safe banks. By not lending, and using their deposits to buy government bonds, they carry virtually no credit risk. There is no danger of non-performing loans and frequent injections of equity capital that has to be funded by taxpayers. For the Reserve Bank of India (RBI) too, supervision gets easier. There is no need for deposit insurance.
The Reserve Bank of India is soon likely to issue clarifications on the guidelines for peer-to-peer (P2P) lending platforms relating to the lending limits, trusteeship and other operational norms.
Active.Ai raises over million in Series-A round (India Times), Rated: A
Singapore-based fintech platform Active-.Ai has raised $8.25 million in a series-A round, which was led by Vertex Ventures, Creditease Holdings and Dream Incubator. Existing investors Kalaari Capital and IDG Ventures also participated.
Toronto-based Wealthsimple, League, and SecureKey were among the company recognized in the top 50 category. In the emerging stars category, Toronto-based Borrowell, Wave, and Sensibill were recognized.
The list was put together by FinTech investment firm H2 Ventures and KPMG FinTech.
Author Allen TaylorPosted on November 17, 2017 Categories Active.ai, alternative investing, Artificial Intelligence, bad credit loans, Banks, Bitcoin, buy-to-let, CFPB, Citi, Credit, credit card debt, CrowdProperty, Daily News Digest, DBRS, Equity Crowdfunding, Featured, FED, fintech, fintech charter, Goji, Hexindai, ICICI Bank, IPO, kabbage, kpmg, lawsuits, LendingTree, LendInvest, Marcus, Millennials, mobile apps, Monzo, narrow banking, NatWest, News, OCC, Online Lending, p2p India, p2p lending, Paytm, PeerIQ, personal loans, Regulation, Reserve Bank of India, robo-advice, Royal Bank of Scotland, Sharesies, Sharestates, short-term credit, small business loans, Square, Taurus Coin, Think Finance, Victory Park Capital, WeShareBonds, women in FinTech, YieldStreet
News Comments Today's main news: PayPal launches P2P funding platform.True Accord lands $22M in funding.Lendable hits 100M GBP lending milestone.P2P Global Investments fund sees huge reduction in U.S. consumer loan exposure.Yirendai's Q3 results.Klarna, PPRO partner on credit payment across Europe. Today's main analysis: The latest trends in consumer credit.The corporate bond market suffers indigestion. Today's […]
Today's main news: PayPal launches P2P funding platform.True Accord lands $22M in funding.Lendable hits 100M GBP lending milestone.P2P Global Investments fund sees huge reduction in U.S. consumer loan exposure.Yirendai's Q3 results.Klarna, PPRO partner on credit payment across Europe.
Today's main analysis: The latest trends in consumer credit.The corporate bond market suffers indigestion.
Today's thought-provoking articles: Marcus is winning the personal loans arms race.Why customer acquisition is difficult for financial startups.
TransUnion report on latest consumer credit trends. AT: "This is a must-read report concerning credit card, auto, mortgage, and personal loan trends."
Corporate bond market suffers a huge bout of indigestion.
Marcus is winning the personal loans arms race. AT:"We must ask why Marcus seems to be winning. Are the leading alternative lenders losing market share to Marcus's attack on fees? Is it simply because they've got the backing of Goldman Sachs. For sure, there does seem to be a competitive play going on, but is that the total picture?"
TrueAccord receives $22M in Series B funding. AT: "This is interesting because the product is machine learning applied to debt collections. Congratulations."
Why customer acquisition is difficult for financial startups. AT: "Customer acquisition costs are important for long-term profitability. A company can take a short-term loss in exchange for the long-term benefit, but it's a risky proposition. Interesting that Wealthfront reduces its marketing budget every year."
CleanCapital closes $3.7M funding round.
196 million consumers have access to a variety of revolving lines of credit.
A tale of two Fed studies. AT: "Online lenders up in arms over the recent Cleveland Fed study should remember that the study is based on different data than a previous Fed study that was favorable. In truth, one cannot base a conclusion on the study of a single online lender. I think the real picture is what lies beyond both Fed studies."
Critics call it predatory lending.
Kill the airline voucher while you're at it.
Consumer Financial Protection Bureau requests information on free access to credit scores.
A podcast discussing the Cleveland Fed study on P2P lending.
Lend360 personal experience: A new era in online lending.
2017 holiday budgets by city.
Magilla Loans ranked as top intermediary by National Real Estate Investor.
Varde Partners acquires CreditShop.
Lendable hits 100M GBP lending milestone. AT: "Congratulations."
P2P Global Investments fund sees large reduction in U.S. consumer loan exposure.
Millennial borrowers' demands underline need for digital lending.
Fintech needs a meaningful investment fund.
1 in 20 nurses have taken out a payday loan to pay bills.
Yirendai Q3 results.
Lexin Fintech Holdings is the latest play on tech.
Hui Ying reports Q3 results.
PPRO, Klarna partner on credit payment methods.
Banco BNI Europa, EDEBEX partner on Portuguese SMEs.
CreditGate24 expands into Germany.
Fintech puts power into customer hands.
Financial services and the generation game.
PayPal launches P2P funding platform. AT: "How long will it be before PayPal moves into equity crowdfunding?"
Cash in the digital world.
Capital markets transformation demands evolutionary approach.
The effective cost of credit.
How FinMomenta plans to change how financial loans are disbursed.
Canadians to use blockchain for digital IDs.
Dubai International Financial Center unveils $100M fund for fintech startup investment.
New Q3 2017 TransUnion Industry Insights Report reveals latest consumer credit trends (TransUnion Email), Rated: AAA
Corporate bond market suffers bout of indigestion (Morningstar), Rated: AAA
Goldman Sachs' Marcus is winning the personal loans arms race (Tearsheet), Rated: AAA
TrueAccord Nabs $ 22M Series B To Bring Machine Learning To Debt Collections (Forbes), Rated: AAA
Why customer acquisition is so difficult for financial startups (Tearsheet), Rated: AAA
Solar Finance Pioneer CleanCapital Closes 3.7M Investment Round to Help Investors Tap Solar Market (EIN Presswire), Rated: A
As Black Friday Nears, a Record 196 Million Consumers Now Have Access to Various Forms of Credit Cards and Other Revolving Lines of Credit (TransUnion Email), Rated: A
Online Lending And A Tale Of Two Fed Studies (PYMNTS), Rated: A
Fintech critics call it predatory lending (CBS News), Rated: A
While Killing The Check, Kill The (Airline) Voucher, Too (PYMNTS), Rated: A
Real Estate Fintech Firm Unison Announces 2018 Expansion Plans (Crowdfund Insider), Rated: A
CFPB requests information on free access to credit scores (American Banker), Rated: A
Dave Wilson's Chart of the Day for Nov. 14 (Bloomberg), Rated: A
Lend360: A New Era in Online Lending (Crowdfund Insider), Rated: A
2017 Holiday Budgets by City (WalletHub), Rated: A
National Real Estate Investor Ranks Magilla Loans a Top Financial Intermediary with $ 1B in Commercial Loans (PR Newswire), Rated: B
Värde Partners Acquires CreditShop (PR Newswire), Rated: B
Lendable hits £100m lending landmark (P2P Finance News), Rated: AAA
P2P Global Investments fund sees large reduction in US consumer loan exposure (AltFi), Rated: AAA
Demands of millennial borrowers underline need for digitisation in lending (AltFi), Rated: A
Fintech sector needs a meaningful investment fund says challenger bank boss (Yorkshire Post), Rated: A
More than one in 20 nurses have taken out a payday loan to pay bills, survey claims (ChronicleLive), Rated: A
Yirendai Reports Third Quarter 2017 Financial Results (PR Newswire), Rated: AAA
Have phone will borrow: Here's the latest tech play on China's spendthrift youth (SCMP), Rated: A
Hui Ying Financial Holdings Corp. Reports Unaudited Third Quarter 2017 Financial Results (PR Newswire), Rated: A
PPRO, Klarna team up for credit payment methods across Europe (The Paypers), Rated: AAA
Banco BNI Europa and Belgian Fintech EDEBEX celebrate a partnership to support Portuguese SMEs (BNI Europa Email), Rated: A
Swiss Fintech Goes Germany (FiNews), Rated: A
Fintech putting power in hands of the customer (Belfast Telegraph), Rated: A
A Digital Future: Financial Services and the Generation Game (EIU.com), Rated: B
PayPal launches P2P funding platform (Business Insider), Rated: AAA
Bringing Cash Into The Digital World – On A Global Stage (PYMNTS), Rated: A
TRANSFORMATION IN CAPITAL MARKETS DEMANDS EVOLUTIONARY APPROACH, FIND CELENT AND FINASTRA (Global Banking and Finance), Rated: A
David Chaston reviews the effective cost of credit, being interest plus standard fees, of taking out a payday loan (Interest.co.nz), Rated: A
How FinMomenta plans to change the way financial loans are disbursed (Money Control), Rated: A
Forget iris scans, Canadians to use Blockchain for digital IDs (Information Management), Rated: A
DIFC unveils $ 100 million fintech investment fund (The National), Rated: B
With the holiday shopping season officially kicking off during Black Friday next week, TransUnion's (NYSE: TRU) just released
Credit Card Lending Metric Q3 2017 Q3 2016 Q3 2015 Q3 2014
Number of Credit Card Loans
Borrower-Level Delinquency Rate (90+ DPD)
Average Debt Per Borrower $5,483 $5,323 $5,229 $5,251
Prior Quarter Originations* 15.5 million 17.6 million 15.3 million
Average New Account Credit Lines*
*Note: Originations are viewed one quarter in arrears to account for reporting lag.
Q3 2017 Credit Card Loan Performance by Age Group
Age/Variable 90+ DPD Annual Pct. Change Average Loan Balances Per Consumer Annual Pct. Change
Gen Z (1995 – present) 2.55% 15.5% $1,101 28.5%
Millennials (1980-1994) 2.48% 5.6% $4,028 12.0%
Gen X (1965-1979) 2.10% 7.7% $6,997 4.9%
Baby Boomers (1946-1964) 1.11% 8.8% $6,351 0.8%
Silent (Until 1945) 0.74% 10.2% $3,928 0.2%
Q3 2017 Auto Loan Trends
Auto Lending Metric Q3 2017 Q3 2016 Q3 2015 Q3 2014
Number of Auto Loans
Average Debt Per Borrower $18,567 $18,361 $17,946 $17,351
Prior Quarter Originations* 7.1 million 7.3 million 7.2 million 6.8 million
of New Auto Loans*
Mortgage Delinquency Rates Continue Extended Decline
Q3 2017 Mortgage Loan Trends
Mortgage Lending Metric Q3 2017 Q3 2016 Q3 2015 Q3 2014
Number of Mortgage Loans
Average Debt Per Borrower $199,417 $193,489 $189,428 $186,577
of New Mortgage Loans*
Personal Loan Balances Reach All-Time High as Delinquency Rates Decline
Q3 2017 Unsecured Personal Loan Trends
Personal Loan Metric Q3 2017 Q3 2016 Q3 2015 Q3 2014
Total Balances $112 billion $100 billion $83 billion $66 billion
Number of Unsecured Personal Loans
3.53%%
Average Balance of New Unsecured Personal Loans*
The corporate bond market suffered a bout of indigestion last week. Between absorbing a healthy amount of new issues and profit-taking from early year-end window-dressing, corporate credit spreads widened, albeit from levels that are still near multiyear lows. The average spread of the Morningstar Corporate Bond Index (our proxy for the investment-grade bond market) widened 5 basis points to +104. In the high-yield market, the BofA Merrill Lynch High Yield Master Index widened 24 basis points to end the week at +376.
Source: Morningstar
Energy Companies' Credit Quality Expected to Continue to Improve;
2018 Oil Forecast $55-$60
A confluence of global events recently drove the crude oil futures price curve into backwardation, a condition in which a commodity's market price today (or spot price) is higher than the price for further-out month contracts. As of this writing, the spot price for West Texas Intermediate crude is $56.90/barrel and the December 2018 contract is priced at $55.70/barrel. Typically, the oil market trades in contango, which is the opposite of backwardation. In contango, a commodity's spot price is below the price for further-out month contracts.
See the full Morningstar report here.
Marcus by Goldman Sachs said it was going to lend $2 billion to customers by the end of this year. As of late Monday, it had already done that.
Lending Club has reported losses exceeding $200 million over the last six quarters; Prosper has lost $210 million since the start of 2016, despite various cost-cutting measures, and lost its unicorn status. Even OnDeck Capital, which focuses on small businesses, is struggling to become profitable, having reported losses over eight consecutive quarters.
Goldman sees a $13 billion lending opportunity with Marcus over three years, CFO Marty Chavez said Tuesday in remarks at the Bank of America Merrill Lynch Future of Financials Conference.
online lending fees
San Francisco based TrueAccord, announced today that is has closed $22M in additional funding led by Arbor Ventures, with participation from existing and new investors. The Series B funding follows a period of sustained and rapid growth for the company.
Customer acquisition is expensive. For a large bank it could cost between $1,500 and $2,000 to acquire a retail banking customer, according to Ciaran Rogers, director of marketing at StratiFi, an early stage startup that helps advisors manage portfolio risk. In credit cards the cost could be in the hundreds, not thousands — according to David True, a partner at PayGility Advisors. An expensive customer could be as high as about $800, which would include the cost of teasers and bonus loyalty points. At startups it could be between $5 to about $300 for one customer. Fintechs want to spend less money on that — Wealthfront, for example, decreases its marketing budget year after year.
Partnerships with bigger brands have been one way to bring that cost down. For example, Canada's fifth largest bank, CIBC, is reportedly in talks with robo-adviser Wealthsimple over a referral deal in which the bank would send some of its customers to the digital investment startup.
At MoneyLion, the cost of customer acquisition is about $5 or less, said chief marketing officer Tim Hong. MoneyLion launched in 2013 and now touts about 1.5 million customers. Earlier this year, Luvleen Sidhu, president and chief strategy officer of the all-digital BankMobile, said it spends about $10 to acquire an account.
CleanCapital, an online marketplace for clean energy investing, announced today that it closed its Series A with a total investment of $3.7 million. This investment came through 50 investors to include FinTech and cleantech leaders as well as SeedInvest's Selections Fund in this latest round.
Over the past five years, the solar market grew an average rate of 72% per year, fueled by regulatory stability and reduced install costs.
With the holiday shopping season officially kicking off during Black Friday next week, TransUnion's (NYSE: TRU) just released Q3 2017 Industry Insights Report found that 195.9 million consumers now have access to revolving credit such as bank-issued and private label credit cards. According to the report, this is the highest level of revolving credit access since TransUnion began measuring the variable and is greater than the 192.6 million consumers who had access to such credit products in Q3 2016.
TransUnion's analysis found that average private label card originations in the holiday season (defined as November and December) for 2016 was 148% of the average originations for the January through October timeframe. This is tracking in line with recent rises observed in 2015 (156%) and 2014 (164%).
Other Consumer Credit Headlines from the Industry Insights Report:
Total Credit Balances Rise despite Slowdown in New Credit Card Accounts
Auto Loan Market Shifting Toward Less Risky Consumers
In 2010, digital lenders originated $249 million in unsecured personal loans, and by 2016 that number had grown ninety-fold.
Cleveland's Dark Outlook
That detail alone isn't necessarily bad news – after all, having more debt doesn't necessarily mean the online lending customers are doing worse. But paired with other data, the news looks pretty grim. According to the Cleveland Fed survey, the online lending customers also showed lower credit scores on average, more delinquent debt and more total debt outstanding.
The findings further suggest that in some cases, the three- to five-year installment loans of up to $30,000 to $40,000 often offered by online lending sites are not being used for their intended purpose of consolidating credit card debt into a single, lower-interest loan. Instead, customers were using those loans to rack up more debt and maxing out the cards they used to pay off the loans.
Philly, Chicago And A Very Different Result
The earlier report did note that outcomes varied depending on the specific borrower profile and their precise lending requirements. However, because of the expanded and more inclusive credit ranking criteria, consumers who might otherwise be "credit invisible" or appear to have a sub-prime score are able to get a more complete evaluation that considers a wider array of factors.
The lack of regulatory clarity raises concerns, they said, over whether customers are treated fairly, have "equal access to credit, and receive offers that can be easily compared and understood," suggesting that alt lenders need to compete on a level playing field with their regulated bank counterparts.
Why The Discrepancy?
The Cleveland Fed study examined data from TransUnion for consumers who had been identified as having taken out "online bank-based loans." That includes a much wider set of businesses and lenders than is technically defined by the more traditional online lenders.
The Philly-Chicago study focused entirely on data from Lending Club, a marketplace lender.
The Cleveland Fed study goes even further. It claims that P2P lending is a ticking time bomb in which loose lending and cascading defaults could lead to another crash like the one the US suffered in 2008 when the subprime lending bust took down major banks and insurers, disabling the housing market for years.
Astrada points specifically to the high interest rates that prospective borrowers with poor credit could have to pay. Many websites offering P2P loans advertise 5 percent loans with terms of one to five years. This may look good to those who would like to roll their credit card debt of 25 percent into a P2P loan.
But the reality is that for many borrowers, the interest rate is much higher. He emphasizes that one up and coming P2P lender about to go public claims that its average loan portfolio across its business model is nearly 150 percent.
In the seventh installment of the "Kill the Check" series, PYMNTS' Karen Webster sat down with Ingo Money CEO Drew Edwards to get a sense of how airlines can use push payments to quell misfires and compensate passengers for their troubles when things go awry.
The digital nature of push payments also helps airlines as they can better control when, where and how such monies are spent. In addition, Edwards noted, push payments are instantly reconciled with the airline's own accounting functions as they are being used.
On Monday, home ownership investment platform Unison announced 2018 expansion plans.
The platform reported that in 2017 alone it has expanded into five additional states including Illinois, New York, Arizona, New Jersey and Pennsylvania, bringing its total footprint to twelve states plus Washington D.C.
In addition to announcing 2018 expansion plans, Unison also revealed multiple promotions and additions to its management team. These are the following:
Jim Riccitelli assumed the role of President focusing on facilitating Unison's rapid expansion and supporting Unison's trademark focus on consumer education and financial literacy.
Bill Walker and Brian Elbogen, former Managing Directors, have been promoted to Chief Revenue Officer and Chief Strategy Officer
Laura Wensley has been brought on as Director of Finance
Rayan Rafay has been promoted to Chief Operating Officer of Unison's investment management business
John Arens, who is General Counsel at Unison, has taken on additional responsibilities as Managing Director of Business Operations
Heather Phillips has joined as Associate General Counsel
The Consumer Financial Protection Bureau is seeking more information about consumers' experience with free access to credit scores.
In two separate notices published in the Federal Register on Monday, the CFPB said it wants more data on which companies consumers are using to obtain their free scores. The bureau also said it is updating a public list of companies that offer free access to a credit score.
GUESTS: David Wilson Stocks Editor Bloomberg News Discussing his Chart of the Day "Here's a 'secret' about tech stocks from Rich Bernstein."
Julie Verhage Reporter Bloomberg Editorial Discussing the Cleveland Federal Reserve Bank slamming the peer-to-peer lending business, calling it predatory and asking for more regulation.
I can sense that most of the folks attending the conference that got hurt by Operation Chokepoint feels a bit vindicated by the latest roll back of many new proposals laid out by the CFPB, Consumer Finance Protection Bureau. Just recently the United State senate overruled the CFPB's arbitration rule. The overruling by CFPB and essentially a no-confidence vote happened about a week after Lend360 concluded in Dallas, Texas.
I attended Dan Quan's "The Regulator's View of Fintech" session on the last day of the conference. Dan manages the small dollar lender desk at the Consumer Financial Protection Bureau, at his panel, Dama Brown from FTC and Shamoil Shipchandler from SEC all spoke about an era of collaboration with lenders. The tones from all three regulatory representative is vastly different than that of five years ago where mass regulation and penalties were the topic of discussion.
I feel like the industry has finally evolved from a cat and mouse game with the regulators to a more collaborative marketing participants as this industry continues to mature.
The National Retail Federation predicts the average per-person tab this holiday season will reach $967, up nearly 3.4 percent since 2016.
Americans are on track to end 2017 with more than $60 billion in additional credit-card balances, according to WalletHub's projections. That figure puts us perilously close to the nearly $1 trillion grand total recorded at the height of the Great Recession.
Holiday Budget by City
Overall Rank*
Holiday Budget
1 Naperville, IL $2,381
2 Sugar Land, TX $2,368
3 Bellevue, WA $2,367
4 Sunnyvale, CA $2,360
5 Carmel, IN $2,330
6 Milpitas, CA $2,262
7 League City, TX $2,225
8 Maple Grove, MN $2,221
9 Allen, TX $2,163
10 Columbia, MD $2,032
In order to determine the cities with the biggest holiday budgets, WalletHub's analysts compared 570 cities across five key metrics: 1) Income, 2) Age, 3) Debt-to-Income Ratio, 4) Monthly Income-to-Monthly Expenses Ratio and 5) Savings-to-Monthly Expenses Ratio.
Magilla Loans, a search engine for loans which connects borrowers to banks without requesting personal information, has been recognized by National Real Estate Investor (NREI), a leading authority on trends in the commercial real estate market, as one of the 2017 Top Financial Intermediaries for commercial real estate loans arranged within the last calendar year. The ranking identifies Magilla Loans as a reliable and efficient service which satisfies the needs of commercial real estate developers and executives.
Read our featured analysis on Magilla Loans.
CreditShop and Värde Partners today announced that Värde will acquire Austin-based CreditShop. CreditShop is a specialty finance company focused on providing consumer friendly credit products and personal loans to prime and near-prime consumers. CreditShop is the 25th largest MasterCard and Visa credit card issuer in the United States.
In March 2017, CreditShop acquired a $1.6 billion MasterCard credit card portfolio from Barclaycard. The company expects to launch its own credit card products in 2018.
LENDABLE has announced that it has hit the £100m cumulative lending milestone in less than four years since launch.
The peer-to-peer consumer lender said on Monday that it is the third UK consumer lender after Zopa and RateSetter to reach this milestone and that it reached it in the fastest time.
The £798m P2P Global Investments fund has entered into an agreement to sell a significant proportion of its exposure to US consumer loans.
The transaction represents a reduction of £36.9m net exposure or 4.56 per cent of the fund's net asset value (NAV) and £167.1m in gross exposure.
A new report from Equiniti finds that 30 per cent of consumers aged 18-25 have borrowed more than £1,000 over the past year. This equates to approximately 2 million people, according to estimates: the highest proportion of any generational group.
The report draws on data from a survey of 2,001 UK consumers in August 2017. 32 per cent were classified millennials, 34 per cent generation-x and 34 per cent baby boomers. 52 per cent were women, 48 per cent were men.
Since 2015, borrowing (of over £1,000) has increased by 17 per cent among millennials, 9 per cent for generation-x and just 1 per cent for baby boomers.
Speaking at a fintech summit in Leeds, Mr Letts also questioned whether the challenger banks were radically different from mainstream banking, positing that some had simply "put new clothes on the emperor".
"On other the other side are what I call the 'neobanks', people coming in with much hurrah and hysteria and telling everyone that the big banks are finished and that they are going to take over the world .
"If you set up a fund, from Government, that invested in fintechs and you had a billion pound fund where do you think businesses will come to? It is very simple."
The Royal College of Nursing's (RCN) workforce survey found that 6% of nurses have been forced to take out one of the high interest rate loans in the last year to meet their daily bills and living expenses.
Meanwhile, one in four has borrowed money from friends, family or their bank, 23% have taken on an additional paid job and half did overtime to cover their bills and expenses, according to the poll of 7,720 nurses from across the UK.
Yirendai Ltd. (NYSE: YRD) ("Yirendai" or the "Company") today announced its unaudited financial results for the quarter ended September 30, 2017
For Three Months Ended
in RMB million
YoY
Amount of Loans Facilitated
Total Net Revenue
Total Fees Billed (non-GAAP)
Adjusted EBITDA(1) (non-GAAP)
Adjusted Net Income (2) (non-GAAP)
In the third quarter of 2017, Yirendai facilitated RMB 12,185.4 million (US$1,831.5 million) of loans to 192,725 qualified individual borrowers through its online marketplace, representing a year-over-year growth of 117%; 75.7% of the borrowers were acquired from online channels; 57.2% of the loan volume was originated from online channels and nearly 100% of the online volume was facilitated through mobile.
In the third quarter of 2017, Yirendai facilitated 214,967 investors with total investment amount of RMB 13,510.0 million(US$2,030.6 million), 100% of which was facilitated through its online platform and 92% of which was facilitated through its mobile application.
For the third quarter of 2017, total net revenue was RMB 1,513.9 million (US$227.5 million), an increase of 28% from the previous quarter and 73% year-over-year; net income was RMB 303.0 million (US$45.5 million), and increase of 13% from the previous quarter and a decrease of 12% year-over-year. The decrease of net income is mainly because that, in the third quarter of 2016, the Company recognized a tax credit of RMB 151.7 million because one of its subsidiaries became qualified as a software enterprise which makes it eligible for an exemption of enterprise income tax for 2015 and 2016. Excluding the impact of the tax credit, adjusted net income in the third quarter of 2016 was RMB 192.6 million.
An online lender targeting spendthrift 24 to 36 year olds is the latest fintech firm from China to bet on the willingness of Chinese youth to go into debt for the newest smartphone – and on the willingness of US investors to bid up its shares.
Shenzhen-based Lexin Fintech Holdings, which operates an online e-commerce platform offering instalment shopping, is following in the footsteps of Chinese microcredit providers Qudian and Hexindai which raised US$900 million and US$50 million in their US IPOs in October and November, respectively.
They were also out in force for the recent Singles' Day shopping festival, which saw sales on Alibaba's e-commerce platforms reach 168 billion yuan (US$25.3 billion). During the first hour of the 24 hour shopping spree the number and value of orders on the Fenqile platform rose three and six times respectively compared with the same period last year.
Hui Ying Financial Holdings Corp. (OTCQB: SFHD) ("Hui Ying" or the "Company"), a leading online financial credit facility solution provider servicing Small-to-Medium Enterprises ("SMEs") and individual borrowers in China, today announced its financial results for the three and nine months ended September 30, 2017.
Third Quarter 2017 Highlights
For the Three Months Ended September 30,
($ millions, except per share data)
Loan origination service fee
Loan repayment management fee
Financing income from entrusted loans
Other income (expenses)
$ (0.24)
EPS – diluted
Total loans facilitated through our platform increased by 73.3% to RMB 2.6 billion for the third quarter of 2017, from RMB 1.5 billion for the same period of last year, as China's online peer-to-peer lending platform industry continued to grow significantly during the third quarter, coupled with the increased marketing campaign, promotion activities on our platform as well as increased brand awareness of our online marketplace.
Total revenues more than doubled to $14.28 million for the third quarter of 2017 from $7.07 million for the same period of last year, as a result of increase in loans facilitated through our platform and the contribution from the newly launched entrusted loan business. Revenues from loan origination service fee, loan repayment management fee and financing income from entrusted loans were $8.39 million, $5.25 million and $0.64 million, respectively, for the third quarter of 2017 compared to $5.11 million, $1.97 million and nil, respectively, for the same period of last year.
Net income was $4.76 million, or $0.06 per diluted share, for the third quarter of 2017, compared to $1.46 million, or $0.02 per diluted share, for the same period of last year.
PPRO Group and Klarna have announced an agreement aimed at enabling PSPs to offer credit-based payments through PPRO`s payment hub to European merchants.
The partnership will be marketed to PPRO's payment service providers customer base and will provide access to Klarna's services and consumers across Sweden, Norway, Finland, Denmark, the Netherlands, Germany, Austria, and the UK.
Banco BNI Europa and Edebex have announced today the celebration of a new partnership for immediate availability of an online platform for the purchase and sale of invoices to Portuguese companies with cash requirements, offering an innovative alternative to financial credit and traditional factoring.
CreditGate24 is opening its first branch outside Switzerland in Berlin, seeking to develop the German market for digital financing and investment, the company said in a statement today.
A number of areas have been established or significantly impacted by FinTech; Peer-to-Peer (P2P) lending, mobile payments, and instant payment notifications, to name a few.
The ongoing bank branch closures across Ireland and the UK demonstrates the changing climate.
With a reduction in branches, banks are investing heavily in technology to reduce costs, to improve their customer experience and to increase customer self-service in an attempt to ward off the threat of FinTech start-ups.
The blockchain is another example of FinTech and one which has been a hot topic across multiple industries for a number of years.
A digital future: financial services and the generation game is a report sponsored by Banco Santander for presentation at the Tenth Santander International Banking Conference, written by The Economist Intelligence Unit.
It assesses how people's expectations of their financial services providers are changing and how technology must be deployed to meet them. The report is based on extensive desk research and in-depth interviews, conducted in August-October 2017 with 14 representatives of financial institutions and companies.
PayPal has launched Money Pools, a service that allows its users to create fundraising pages where their contacts can contribute money for a shared item or event, like a group gift or trip,
In the latest Data Drivers installment, Steve Villegas, vice president of Partner Management at PPRO Group, told PYMNTS' Karen Webster that "alternative payments are going to drive the future of eCommerce." But between the promise and the reality, some connectivity is on order, bringing consumers payment options – and merchants toward better conversion rates when it comes to online commerce.
Data Point One: 17.6 Percent
This is the percentage of credit card penetration worldwide – a lot of cards, but not a lot of penetration on a global stage.
Alternative payments may capture 50 percent of transactions this year, globally speaking, he said.
As has been widely reported, Alibaba grabbed as much as $25 billion in sales to 225 countries and regions. Roughly 90 percent of transactions were completed on mobile devices.
Data Point Two: Three Billion
This is the number of people estimated worldwide to be without a bank account – and yet, armed with mobile devices, can be brought into the world of digital transactions and can participate fully in the global economy.
Data Point Three: 38 Percent
This is the average rate of eCommerce growth of the 11 fastest growing countries globally. That far outpaces the 12 percent a year eCommerce growth seen in the U.S.
China provides a stark reminder of the explosive boost to eCommerce, at 64 percent year over year. Other areas that have high eCommerce growth rates include Indonesia and Malaysia. Growth is high both in bank payment-related transactions and with eWallet. Russia is also showing growth, Villegas stated.
New research from Celent (commissioned by Finastra) which examines the future transformation of capital markets, identifies six key drivers of change over the next five years to 2022:
Digitalization of the trade and client lifecycle
The Fintech revolution
The need to integrate with an evolving ecosystem
The trend for banks to focus on core capabilities and outsourcing of non-core functions
Advances in big data, machine learning and data analytics
The rise of open APIs and micro-services in helping banks deliver increased agility
The report, 'The Great Transformation in Capital Markets – Revolution to Evolution', examines the changes that have already taken place in capital markets since the 2008 crisis, the wave of big transformation projects undertaken since 2011-12 designed to optimize operations and reduce costs, and expected trends in the transformation journey over the next five years. It incorporates the findings of detailed discussions conducted with 17 tier one and two global capital markets institutions, predominantly in the US and Europe but also across Asia and Latin America.
In this table we have set out what each lender says you must repay for borrowing $500. (We targeted 30 days but not every lender offers that.) Then we calculated the effective annual interest rate for entering into that deal. This is different to the interest rate the lender discloses because we bundle up that interest rate with any set-up loan fees into an effective rate. But we haven't included any fees if you default; this analysis assumes the borrower meets all payments on time.
making one repayment (except as noted) Borrow Repay in (days) % daily % pa
in random order … $ $ # effective effective
Ferratum $500 $748 30 1.352% 493.4%
Save My Bacon $500 $828 30 1.696% 618.9%
Need Cash Today $500 $640 28 (4 wp) 1.458% 532.1%
Moola $500 $640 28 (4 wp) 1.458% 532.1%
Zebra Loans $500 $835 30 1.724% 629.3%
Payday Advance $500 $932 30 2.097% 765.6%
Payday Loan $500 $932 30 2.097% 765.6%
Can'twait $500 $731 30 1.274% 465.0%
Cash Relief $500 $748 30 1.352% 493.4%
Smart Cash $500 $691 30 1.084% 395.8%
Just Cash $500 $748 30 1.352% 493.4%
Little Loan Shop $500 $748 30 1.352% 493.4%
Seed Cash – 3 monthly pymts $500 $950 91 (3 mp) 1.128% 411.8%
Cash Burst – 2 monthly pymts $500 $1189 61 (2 mp) 2.029% 740.6%
Real Finance $500 $665 30 0.956% 348.9%
Easy Cash $500 $605 30 0.537% 232.5%
Cash till Payday $500 $647 30 1.700% 620.5%
Money Shop $500
Easy Financing $500
Instant Cash Online $500 $637 30 0.812% 296.3%
Ex-bankers Brahma Mahesh, Naveen Madgula, and a techie for 17 years at Hexaware – Praveen Krishnam founded FinMomenta last year, launching its loans platform Tachyloans in May.
The startup borrows from the emerging trend of servicing small-ticket loans online for individuals and SMEs.
"The loan approval process in banks is very subjective. It is dependent on a human perception of the loans officer. It kills the whole idea of credit scoring. That's the reason banks are able to service just about 2%-5% of the huge working class of about 60 crore population. Others just depend on money lenders. Banks don't touch these people because they don't have a credit history," says FinMomenta co-founder Brahma Mahesh.
The interest rates on Tachyloans range from 11.5% to 25% depending on the FinMomenta credit rating – the better the credit score, the lower the interest rate.
By the end of 2018, the company is targeting to service 1,500 loans and 34,000 loans in next 5 years, which will increase its loan portfolio to about Rs 500 crore.
Consumers will be able to sign up for new digital-identity system developed by SecureKey Technologies Inc. and underpinned by IBM Corp.'s blockchain technology in the first half of 2018. They'll be able to instantly prove who they are to banks, telecom providers and governments using apps on their phones and Windows devices, according to Greg Wolfond, chief executive officer of Toronto-based SecureKey.
Canada's six-largest lenders, including Toronto-Dominion Bank and Royal Bank of Canada invested C$30 million ($24 million) in the project.
The Dubai International Financial Center (DIFC) it has established a $100 million fund to invest in fintech start-ups, the latest move in the freezone's bid to position itself as the regional centre for the fast-growing and disruptive sector.
Author Allen TaylorPosted on November 15, 2017 Categories Banco BNI Europa, Blockchain, Capital Markets, CleanCapital, consumer credit, Consumer Financial Protection Bureau, consumer loans, Corporate Bonds, cost of credit, Credit, credit payment, credit scores, CreditGate24, CreditShop, Daily News Digest, digital ID, digital lending, EDEBEX, Featured, FED, FinMomenta, fintech, Goldman Sachs, Hui Ying, Klarna, Lend360, LexinFintech, Magilla Loans, Marcus, Millennials, News, p2p funding, p2p lending, P2PGI, Payday loans, paypal, personal loans, PPRO, predatory lending, revolving credit, SMEs, startups, Transunion, TrueAccord, Yirendai
Monday June 5 2017, Daily News Digest
In the June 1 issue of Lending-Times, we highlighted a TransUnion report on how to identify and fight online fraud. A typo suggested we didn't recommend report, however, we highly recommend it and you can download it here. News Comments Today's main news: After buying George Banco RateSetter will not lend to its customers. Amartha receives regulatory […]
In the June 1 issue of Lending-Times, we highlighted a TransUnion report on how to identify and fight online fraud. A typo suggested we didn't recommend report, however, we highly recommend it and you can download it here.
Today's main news: After buying George Banco RateSetter will not lend to its customers. Amartha receives regulatory approval from Indonesian Financial Services Authority.
Today's main analysis: Common Bond's securitization of student loans.
Today's thought-provoking articles: Impact of the latest adjustments to P2P/MPL. WeiyangX Fintech review. Three reasons for the boosting Cash Loan in China.
Latest PeerIQ Loan Performance Monitor. GP:" Common Bond's student loan securitzation. And very interesting data on Lending Club's delinquency. "
Global Debt Registry appoints Charlie Moore as president. GP :"Congratulations!". AT: "Congratulations."
Investing in real estate: Single-family or multi-family homes? AT: "A good case for investing in multi-family through crowdfunding platforms."
GTCR acquires Sage Payment Solutions.
Where incumbents are making investments in wealth tech. GP:" For instance, Goldman Sachs and JP Morgan Chase are co-investors in Motif, Northwestern Mutual and Citi Ventures are co-invested in Betterment, and UBS and Santander InnoVentures are co-investors in SigFig.". "AT: "Some of the world's largest financial institutions invest in wealth tech startups and robo-advisors."
AutoGravity car financing app now available across New Jersey. GP:"Car choosing and financing, all via a mobile app. Maybe we can apply this model to other markets."
Fed Governor: Data aggregators impact bank safety, soundness as part of fintech stack.
PeerStreet's Jessica Murray named HousingWire rising star.
Roostify names Sandeep Aji VP of products.
How to get a second chance with your bank.
RateSetter not to lend to George Banco's customers. GP:"Rate Setter bought the former wholesale lenders and decided not to lend to its customers anymore. Strange. We wonder what happened there. ".AT: "This is a reversal of their original intent."
P2P platforms facing hybrid dilemma. GP:"THE PEER-TO-PEER finance industry could be on its way to becoming a polarised market, where the biggest firms stick to their core P2P lending activities and the rest are forced to evolve into hybrid models."
Safety in banking.
Peer-to-peer lending promised 6%, but this man is red-faced and in the red. GP:"Diversificaiton is key, and Lending Club recommends at least 300 different notes per lender."
Assetz Capital Review.
Assetz Capital adds ex-bank specialist to regional director team.
From beach to boardroom.
WeiyangX Fintech Review.
Three primary reasons for the boosting Cash Loan in China.
Fintech is king of Lithuania's tech revolution.
MPL News Roundup from Lend Academy. AT: "A mix of new news and some already reported."
The impact of the latest adjustments in peer-to-peer/MPL. AT: "An excellent read."
Crowdfunding, millennial buyers and higher mortgage rule real estate in 2017.
Crowdsurfer adds Zopa data.
How robo-advice fees compare to multi-asset funds.
Trov connects with AXA Insurance & celebrates UK launch.
Fintech Australia announces new board of directors.
Monexo: Trying to fill the void.
Pinjam gears up for growth spurt this year.
Amartha receives regulatory approval in Indonesia.
Why your financial planner should be a robot.
In Singapore, fintech boom missing the tech.
Some options if you want advice from a robot.
Latest PeerIQ Loan Performance Monitor (PeerIQ), Rated: AAA
Global Debt Registry Appoints Charlie Moore as President (PR.com), Rated: A
Investing in Real Estate: Single Family Homes or Multi-Family? (Crowdfund Insider), Rated: A
GTCR Announces Acquisition of Sage Payment Solutions (Guru Focus), Rated: A
Where Incumbents Are Making Investments In Wealth Tech (CB Insights), Rated: A
AUTOGRAVITY CAR FINANCING APP NOW AVAILABLE ACROSS NEW JERSEY (AutoGravity), Rated: A
FRB Governor: Data Aggregators Impact Bank Safety, Soundness as Part of the "Fintech Stack" (JD Supra), Rated: A
PeerStreet's Jessica Murray Named One of HousingWire's 2017 Rising Stars (BusinessWire), Rated: B
Roostify Names Sandeep Aji as Vice President of Products (BusinessWire), Rated: B
How To Get A Second Chance With Your Bank (NASDAQ), Rated: B
RateSetter decides not to lend to George Banco's customers (P2P Finance News), Rated: AAA
P2P platforms facing hybrid dilemma (P2P Finance News), Rated: A
Safety in banking (SilverSeek), Rated: A
Peer-to-peer lending promised 6%, but I've been left red-faced and in the red (The Guardian), Rated: A
Assetz Capital Review – 30 Days Access Account with 4.75% Target Rate (P2P-Banking), Rated: A
Assetz Capital adds ex-bank specialist to regional director team (P2P Finance News), Rated: B
From beach to boardroom: Iced coffee king surfing to success (London Loves Business), Rated: B
WeiyangX Fintech Review (Crowdfund Insider), Rated: AAA
Three primary reasons for the boosting Cash Loan in China (Xing Ping She Email), Rated: AAA
Fintech is King of Lithuania's Tech Revolution (Red Herring), Rated: A
Marketplace Lending News Roundup – June 3 (Lend Academy), Rated: AAA
The impact of the latest adjustments in peer-to-peer/marketplace lending (AltFi), Rated: AAA
Crowdfunding, millennial buyers and higher mortgage rule real estate in 2017 (International Business Times), Rated: A
Crowdsurfer adds Zopa data (Finextra), Rated: A
How robo advice fees compare to multi-asset funds (AltFi), Rated: A
Insurtech App Trov Connects with AXA Insurance & Celebrates UK Launch (Crowdfund Insider), Rated: A
FinTech Australia Announces New Board of Directors (Crowdfund Insider), Rated: A
Monexo: Trying to fill the void (Free Press Journal), Rated: A
Fintech Firm Pinjam Gears Up for Growth Spurt This Year (Jakarta Globe), Rated: A
P2P Lender Amartha Receives Regulatory Approval from Indonesian Financial Services Authority (Crowdfund Insider), Rated: AAA
Why Your Financial Planner Should Be a Robot (Knowledge.insead.edu), Rated: A
In Singapore, fintech boom is missing the 'tech' (Southeast Asia Globe), Rated: B
Some options if you want advice from a robot (IOL), Rated: A
The US economy generated a seasonally adjusted 138 K jobs last month (vs. expectation of 185K) bringing the jobless rate to 4.2% and another step closer to full employment. As inflation risks emerge, the Fed is widely expected to increase rates another 25 bps at the upcoming FOMC meeting on June 14th.
On the regulatory front, the US House will vote on a bill sponsored by Jeb Hensarling (R-TX) to reform and repeal portions of the landmark Dodd-Frank financial reform bill.
On the securitization front, student lending originator CommonBond priced its $232 Mn private student loan ABS. Goldman Sachs was the structuring lead, and co-leads include Barclays and Citi. Also, AB Alert reports that Lending Club is preparing a multi-seller deal which includes collateral from multiple originators including potentially loans from Lending Club's own balance sheet. As PeerIQ noted in the summer of last year, we believe marketplace lenders that can offer whole loan investors a reliable path to liquidity and low-cost permanent financing can generate a competitive advantage.
PeerIQ is pleased to present the PeerIQ Loan Performance Monitor. The monitor tracks interest rates, delinquency, and charge-off rates for both platforms across vintages and grades.
Source: PeerIQ MPL Loan Performance Monitor June 2017
Global Debt Registry (GDR), the asset certainty company known for its loan validation expertise, today announced Charlie Moore, the firm's Chief Commercial Officer, has been named President as former Chairman and CEO Mark Parsells returns to his FinTech consultancy practice.
As President, Moore will be focused on the continued delivery of loan level diligence services to the investment community, leading the daily operations of the company. Moore previously led the firm's commercial operations including business development, partnerships and marketing and has over 20 years of experience building financial services technology businesses in the U.S. and Europe.
"Institutional multifamily" typically means dozens, if not hundreds, of distinct units within a single property, managed by a seasoned professional management firm. These properties have many different tenants, with a diversity of employment situations and lease structures. If one tenant leaves abruptly, many others will remain in place, and overall rental income will suffer only marginally. Multifamily managers can further mitigate vacancy risk by structuring leases to end on a rolling basis. Single family investments don't carry the same benefit – a tenant living in a single-family home constitutes 100% economic loss for as long as the property remains vacant.
Investing through online crowdfunding platforms gives individual investors the opportunity to invest in a small piece of large multifamily projects that are institutional grade and have passed the underwriting of well-established lenders and co-investors who often have decades and billions of dollars of investing under their belt. The same can't be said of most single-family investments.
While these benefits are most apparent for direct owners of (investors in) property, this benefit of multifamily investing should be passed along to individuals who co-invest via an online (crowdfunding) platform.
GTCR, a leading private equity firm, announced today that it has entered into a definitive agreement to acquire Sage Payment Solutions, Inc. ("SPS" or the "Company") for $260 million. SPS, headquartered in Reston, Virginia, is a leading provider of payment processing and merchant acquiring solutions in North America. GTCR is acquiring SPS from The Sage Group plc (LSE: SGE) ("Sage"), a global provider of integrated accounting, payroll and payment solutions headquartered in the UK. GTCR is partnering with SPS management to pursue organic growth initiatives and fund future acquisitions in the payment processing industry. To support this strategy, GTCR has committed up to $350 million of equity capital to the platform. The transaction is expected to close in the third quarter following receipt of regulatory approvals and other consents.
SPS provides credit card, ACH, check, gift and loyalty card processing services to small and medium-sized businesses ("SMBs") in the United States and Canada.
Deals to wealth tech startups hit a record of 30 investments in Q1'17 amid a number of new early-stage entrants globally. In particular, robo-advisors have been gaining prominence and taking on incumbents in nearly 20 countries around the world.
Since 2012, several banks and wealth management firms have made co-invests in wealth tech. For instance, Goldman Sachs and JP Morgan Chase are co-investors in Motif, Northwestern Mutual and Citi Ventures are co-invested in Betterment, and UBS and Santander InnoVentures are co-investors in SigFig.
Incumbents have made the most investments to companies that fall into our robo-advisors category, including Betterment, Motif, Personal Capital, WealthNavi, Folio, ForwardLane, and SigFig
Blackrock invested in Personal Capital, the second most well-funded wealth tech company with approximately $207M in funding.
LearnVest is the only featured company on our map to have exited.
AutoGravity, a FinTech pioneer revolutionizing car shopping and financing with the power of the smartphone, has unveiled an innovative mobile application to help car buyers in the Garden State finance any new or used car in minutes in just four easy steps.
With its unique platform, the AutoGravity app guides car buyers through an intuitive four-step process:
Choose a car – Select any make, model and trim of any new or used car.
Find a dealer – Choose from AutoGravity's proprietary national dealership database; geolocation helps quickly identify nearby dealers that sell the car the selected.
Search for financing – Car buyers can scan their driver's license and connect to social media to quickly pre-fill the finance application.
Select a lender – Receive up to four binding finance offers in minutes, then select a loan or lease offer and head to the dealership to complete the purchase.
In a recent speech at the Northwestern Kellogg Public-Private Interface Conference, Federal Reserve Board Governor Lael Brainard indicated that the relationships between banks and data aggregators within the "fintech stack" may present safety and soundness concerns that warrant oversight by the FRB (and perhaps other prudential regulators).
Governor Brainard indicated that banks will need to apply significant resources to update their data infrastructure to allow access to real-time data for third-party developers.
Governor Brainard explained that because banks are more tightly regulated than the average fintech company, consumer protection and safety and soundness considerations should supersede experimental innovation.
While some banks may elect to give access to data aggregators, Governor Brainard observed that other banks may be unwilling or unable to provide permissioned access to third parties due to fears about compliance with laws and regulations and the ability to monitor and control the use and access to data. She then noted that the Fed's supervisory role should focus on ensuring that financial institutions subject to its supervision operate safely and follow applicable law. At the same time, she stated that the Fed has "a strong interest in permitting socially beneficial innovations to flourish, while ensuring the risks that they may present are appropriately managed, consistent with the legal requirements."
PeerStreet, a marketplace for investing in real estate backed loans, is honored to announce that its VP of Strategy, Jessica Murray, has been named to HousingWire's 2017 Rising Stars list of young leaders to watch in the housing industry. HousingWire's 2017 Rising Stars list recognizes talent that demonstrate leadership and innovation, inspiring not only those within their company, but also their communities and the industry at large.
In her time at PeerStreet, Murray established the company voice through social media, content marketing, customer communications and placed media while serving as the Head of Communications. In her current role, Murray maintains many strategic and operational responsibilities, which also include managing PeerStreet's capital markets and hiring.
Roostify, a provider of automated mortgage transaction technology, today announced it has named Sandeep Aji as Vice President of Products. Aji will be responsible for overseeing the continued development of Roostify's mortgage technology platform – from enabling more API-driven capabilities to improving user experience for lenders and consumers.
Prior to Roostify, Aji was Co-Founder and CEO of Impartus, a cloud-based, SaaS platform for higher education.
Despite a higher cost of capital, an online loan may be necessary for a small business. The reason: There has been a continued downtrend in lending from banks to small businesses. "Together, 10 of the largest banks issuing small loans to business lent $44.7 billion in 2014, down 38% from a peak of $72.5 billion in 2006," reports The Wall Street Journal. Meanwhile, nonbank lenders have seized the opportunity and captured 26% market share up from 10%.
RATESETTER has announced that it has decided not to lend directly to George Banco's customers as there are "better uses of our development resources".
The 'big three' peer-to-peer lender bought a stake in the guarantor loan provider, which was a former wholesale lending partner, last month. It had also agreed to lend directly to its 10,000 customers, with George Banco acting as introducer.
The business and consumer lender will keep its equity stake in the firm and its co-founder Peter Behrens will remain as a non-executive director on George Banco's board.
THE PEER-TO-PEER finance industry could be on its way to becoming a polarised market, where the biggest firms stick to their core P2P lending activities and the rest are forced to evolve into hybrid models.
A wide range of industry onlookers have told Peer2Peer Finance News that it will be impossible for smaller firms to achieve profitability without either expanding into balance sheet lending, merging with direct lenders or morphing into a business model closer to that of a collective investment scheme.
"It's incredibly difficult to build a straightforward P2P business to the size where it becomes profitable," said Andy Davis, author of a report that pointed to hybrid models as an inevitable evolution in the sector.
"It's intrinsically more profitable to arrange and lend rather than only arrange. We're going to start seeing hybrid loans emerge."
P2P is ultimately just a subset of direct non-bank lending, he argued, but with different technology in place and different market access. When a direct lender sets up a P2P platform, its return on capital goes up exponentially and it can immediately recycle those returns to originate more lending.
"Hybrid lending from some providers will increasingly be the chosen solution. This is not an issue or a problem for investors in and of itself, " added 4th Way analyst Neil Faulkner.
It was probably with sound money and sound banking in mind that Goldmoney recently announced a tie-up with a British-based and regulated peer-to-peer lender, which enables owners of gold and silver bullion to use it as collateral to raise funds.ii The purpose of this article is to explain how honest banking worked before fractional reserve banking was devised. This is the logic behind the recently announced collaboration between Goldmoney and Lend & Borrow Trust Company Ltd.
On 23rd May, Goldmoney announced an investment and collaboration in and with the UK-based peer-to-peer lending platform, Lend & Borrow Trust Company Ltd. LBT is unique, being the only peer-to-peer facility in Western financial markets that allows businesses and individuals to use their investment-grade physical bullion as collateral against loans, without the loan obligations and collateral being comingled with other customer business.
At no time is LBT a principal in the transaction, so lenders and borrowers can agree an interest rate without having to take LBT's creditworthiness into account, based solely on physical gold or silver as collateral.
The logic of a collaboration between Goldmoney and LBT is obvious, in that it enables customers to raise finance using bullion. But there is an underlying sound-money logic as well. Between them, Goldmoney and LBT are the template for sound-money banking as it existed before fractional reserve banking became the standard banking model, after Britain's Bank Charter Act of 1844.
When James Patterson invested £1,000 in the peer-to-peer (P2P) lender Funding Circle back in 2015, his hope was that his money would grow a bit faster than the pitiful rates of interest offered by his bank. At the time, the relatively new lender was promising returns of 5-6% a year – 10 times more than his bank.
However, almost a year and a half on, his investment is now worth just £988 – a loss of £12. It's because one of the firms that 10% of his money was lent to defaulted, leaving his account £128 in the red – a sum that his other investments at the platform have struggled to make up.
It has delivered some impressive returns to savers in recent years but, Patterson says, after his experience, he will not be investing anymore.
James Meekings, co-founder and managing director of Funding Circle, says Patterson will be back in the black in the next couple of months as the firm expects to recover some of his losses which, in turn, will be passed on to him.
Recently I opened an account at p2p lending marketplace Assetz Capital to gain some first hand experiences. Assetz Capital offers secured business loans to small and medium British SMEs. I decided to start with the 30 days access account as it is mostly hands off and deposited a tiny amount, which was credited within an hour.
Assetz Capital has a minimum investment amount of 1 GBP. Assetz is open to international investors, but a UK bank account is required.
Assetz also offers a quick access account with 3.75% target rate, designed to provide immediate access to cash, in normal market conditions, for investors. Currently 19 million GBP are invested in this account. Further account types are the 'Great British Business Account' (GBBA) with 7% target rate, the 'Green Energy Income Account' (GEIA) with 7% target rate and the 'Manual Loan Investment Account' (MLIA) with 5.5% to 18% gross rate. See comparison of Assetz accounts. Assetz also features a secondary market without fees providing liquidity.
ASSETZ CAPITAL has hired former bank finance specialist Samantha Williamson to boost its team of regional relationship directors.
Williamson will supervise the peer-to-peer lending platform's activities in the South Manchester region.
She previously worked as business development manager at financial independent broker Positive Commercial Finance, helping firms grow through non-traditional finance avenues.
Prior to that, she served as senior real estate manager at Barclays and commercial lending manager at Santander, both posts located in the Manchester area.
Who's bankrolling you?
We are. And so is Funding Circle. The banks have been completely useless.
What advice would you give other entrepreneurs trying to secure that kind of finance?
It depends on how much you're looking for and how long you've been trading, but if you need money, I'd go crowd funding, 100 per cent. If you are well under the table with trading, I'd take a look at funding circle.
Search engine giant Baidu Inc. is to quit crowdfunding market and pay more attention on artificial intelligence"]. Users will not see the "Baidu Crowdfunding" channel when they log in their Baidu Finance account, but will still be able to check the crowdfunding history.
On May 25, China Rapid Finance Limited, a leading online consumer lending marketplace in China, reported its unaudited financial results for the quarter ended March 31, 2017.
Transaction and service fees
USD16.8 million
Consumption loans
USD6.7 million
Operating aspect:
Number of new borrowers added in the first quarter of 2017 was approximately 545,000. As of March 31, 2017, the Company had reached approximately 2 million unique borrowers on its marketplace since inception, and the total number of loans facilitated on the Company's platform grew to approximately 15 million.
Total loan volume facilitated on the Company's marketplace in the first quarter of 2017 increased to USD485 million, primarily driven by the rapid expansion of consumption loans, which accounted for USD405 million of the total loan volume.
Total number of consumption loans facilitated in the first quarter of 2017 was 4 million, while total number of maintenance loans facilitated was 6,000.
Ant Financial, the financial arm of Chinese e-commerce giant Alibaba, became the latest player in Hong Kong's competitive mobile payments market after it announced the launch of its mobile wallet for Hong Kong users on May 24.
On May 25, Ant Financial announced to launch the car insurance rating mechanism for the insurance industry to improve the risk management capability.
By Dr. Yang Li
From 2017, the number of cash loan companies have increased tremendously in China. Various kinds of cash loan firms have mushroomed, including CashBus, MagicCash, GoldBar of JingDong, Ants Borrow of Alibaba, WeiliDai of Tencent, etc. So far, there are already thousands of small cash loan platforms exists in China, and many of them have received fund financing from top VC investors such as Sequoia Capital, Innovation Works, and ZhenFund.
Why cash loan growth explosively in a short period? The following three reasons may explain.
Reason 1: The lower threshold of credit system by Big data method.
In the past, credit system was mainly referred to Central bank credit system, however, it could not cover most people. The information of vast majority of low-income, unregistered social groups have not been collected in the credit system, but they have extensive borrowing needs.
As big data technology developing fast these years, many data companies are growing rapidly, and they acquired data for business use. Owing to the big data credit system, cash loan platforms are able to evaluate the borrower's credit situation from multi-dimensions: traits of character, consumption habits, loan demand, repayment willingness, etc. In this way, the problem of information asymmetry between the investors and borrowers is eliminated, making the cash loan business prosper in the broadest social group.
Reason 2: Vertical specialization of cash loan industry provides more business opportunities
The division of the cash loan industry is now divided into receipt, audit, lending and collection, each process are served by independent and professional companies or teams. With the booming of cash loans, an ecological chain around the industry has been derived, including data processing companies, business consulting companies, law office specialized in collection, etc.
The vertical specialization of the industry made cash loan platforms extremely convenient in obtaining customers, audit management and collection, so that the platforms can save more costs and gain more business opportunities.
Reason 3: Changing of the public consumption concept stimulated loan demands
With the improvement of people's living standards and the popularization of deficit spending concept, the public consumption concept has been changing a lot.
Consumer demand is beginning to diversify. There are not only the need for food and clothing, but also spiritual needs of learning, fitness and travel, etc. And the consumers' attitudes are gradually transforming from rational consumption to perceptual advanced consumption. Spending "future money" at "the present" is becoming a common social spending habit, for example, more and more people choose to purchase cars, houses and 3C electronic products on installment. The growing advanced consumption has stimulated the loan demand across society.
Under Soviet rule Lithuania became known as a center for laser technology and bioscience, the latter of which now accounts for 1% of GDP and is growing at almost 25% annually. Last year businessmen, scientists and the government signed an agreement to make Lithuania the European hub for health and biotech innovation by 2020.
But it is fintech that has taken the strongest grip on the country's tech scene. TransferGo, WoraPay, Blender, Simplex and IBS are just a few of a small but growing clique of firms taking advantage of strong local talent, low wages and public pledges.
Lithuania is the only jurisdiction in the EU to have a special-purpose banking license, allowing the foundation of a bank with registered capital of just €1m ($1.1m).
Vaidas Adomauskas first imagined WoraPay, a payment platform, while waiting to pay for food at a restaurant. Now it is backed with almost $1m in funding and is headquartered in London–which many believe to be Europe's fintech capital.
Capitalizing on the Lithuanian fintech craze, Rise, the Barclays-backed Rise coworking franchise, opened a location in capital city Vilnius last year. It has 50 working spaces, an auditorium and conferencing facilities for entrepreneurs trying to get a foothold in financial tech.
Behind the Scenes at Orchard Platform, a Struggle to Innovate from The New York Times – The long and winding road of Orchard's plans for a secondary loan market.
Peer-to-peer lender RateSetter raises £13m, Woodford and Artemis lead from AltFi – In the UK RateSetter has closed another £13M equity round as they get closer to full FCA approval.
SoFi and JetBlue Help Customers Managing Student Loans Earn Reward Travel from PR Newswire – This week SoFi announced that you can earn up to 50,000 JetBlue reward points by refinancing a student loan.
Did someone cancel the fintech revolution? from Finextra – The promise of fintech has not yet been released says Accenture in a new report.
Kind of Blue from FinTech Junkie – The latest from Frank Rotman comparing startups to jazz and what to do when you hit a wrong note.
Are Small Business Borrowers Bank-Loyal to a Fault? from deBanked – Despite low approval rates banks are still the top choice for entrepreneurs looking for a loan.
The impact of the latest adjustments in peer-to-peer/marketplace lending from AltFi – Good summary of the latest developments at the big four marketplace lenders in the UK.
Funding Circle ditches property
Leading small business loans marketplace Funding Circle announced that it would be winding up its property-secured lending in April, with a view to stopping entirely by mid-2018.
AltFi Data's analytics engine shows that only one quarterly cohort of Funding Circle's property-backed lending resulted in any bad debt. This came in the third quarter of 2015. Bad debts for this cohort have reached 4.67 per cent – but it's important to note that recoveries may still be made, and that this is just one of 17 quarterly cohorts. The size of this cohort is about £34m.
RateSetter stops wholesale lending
AltFi Data told us in March that RateSetter had originated £273m loans to lending businesses, equating to 15.6 per cent of its £1.748bn cumulative lending total at the time.
The firm has now lent a little over £1.9bn in loans, of which 15.4 per cent are wholesale. As can be seen in the chart below, the overall trend is down.
More capacity at MarketInvoice?
MarketInvoice announced the launch of a new longer-term product (MarketInvoice Pro) in February. This allows businesses to draw an open funding line, secured against their outstanding invoices.
Well, since unveiling the new product in February, MarketInvoice has posted back-to-back monthly origination records (versus all previous months in its existence), with £42m in March and £35m in April.
But this isn't yet feeding through in terms of outstanding principal per month, which is hovering at around £25m per month, versus an all-time high of £35m.
Investors are falling over themselves for Zopa loans, but should they be?
The net returns delivered to Zopa investors has been fairly consistent at between 4.5 and 5.0 per cent for the past two and a half years. But the rate being paid by its borrowers is climbing.
Zopa's average gross interest rate has steadily increased from 5.3 per cent at the outset of 2015 to 8.4 per cent in April 2017. The reason for this is simply that a higher proportion of Zopa's loans are now being made to "riskier" borrowers. But the returns being offered by the platform haven't yet adjusted to reflect this.
Interestingly, real estate crowdfunding is not limited to the US market. It is actually one of the hottest trends in the overall global realty market today. Realty crowdfunding platforms are continuously being launched in the UAE, Asia and even Egypt. In fact, a leading Singapore-based realty crowdfunding platform recently raised around S$1 million (AU$0.98 million) in the first funding round for a company.
If crowdfunding is the signature trend of real estate in 2017, the rise of millennial home buyers is a close second. The oldest millennials are now in their mid-30s and are planning to have their own houses. Marriage is on the cards for most of them, further creating the urgency for a new home. Most jobs have been designed for the 25 to 34 age bracket, with wages happily rising. Overall, it is a highly favourable situation for millennials to think of a new house this year.
On the other hand, the recent Brexit fallout has had a major impact in the contemporary real estate scene. With UK realty currently going through an uncertain phase, the US real estate scene is fast hogging the limelight in the global property market. The Chinese market, too, is currently moving along a slow tide, which presents an advantage for US developers. The American commercial real estate is to benefit in particular, and speculations are on the rise about steady foreign investments in the country.
Zopa, the world's first and one of the largest peer-to-peer (P2P) lenders, has lent in excess of £2.3 billion to customers in the UK, and the addition of its data set will deepen Crowdsurfer's insight into the global alternative finance market.
Cambridge-based Crowdsurfer analyses data from more than 900 different alternative finance platforms, including equity, bonds, SME debt, P2P and more, and has mapped more than ten million transactions to provide the most in-depth take on global trends and patterns in alternative finance.
Will robo advice spark a price war? We crunched the numbers looking at how much platforms charge compared to a typical multi asset fund.
Vanguard, a U.S. based passive fund manager, is planning to sell its index funds directly to UK consumers, charging just 0.23 per cent annually. Previously, individuals had to invest in funds through an intermediary or, more recently, via robo advisor to get access to the company's funds.
Taking a look at the top UK robo advice platforms, we found that an investment of £10,000 would cost an average of £6.79 a month in both management, platform, and fund fees.
A £10,000 investment in 2015 held in the average fund in the IA's 20-60% Shares sector would have cost an average £9.83 per month for a mixed fund, while the average fund in the IA's 40-85% Shares sector would cost £10.25, according to data on the average ongoing charges figure from the Investment Association.
Putting this all together, for the average robo advice platform a £10,000 portfolio amounts to approximately 0.81 per cent fees, or £81 over a year. In comparison, the average multi-asset fund charges between 1.18 per cent and 1.23 per cent over the course of the year, with IA's 40-85% Shares sector the higher of the two. This amounts to £118-£123 on £10,000, or approximately £40 more than the average robo advice portfolio.
Pre-RDR fees would have cost £15.42 and £14.92 on a £10,000 investment.
Moneyfarm, a robo advisor based in the UK and Italy, stands out because it doesn't charge a management fee for any investments under £10,000, just the fund fee. Investments over £10,000 are charged 0.6 per cent.
Trov has launched its on-demand insurance platform in the UK, in partnership with AXA Insurance. Users are provided with a personalized quote and can quickly turn insurance on (or off) for an item without the need for any interaction with a traditional insurance agent.
FinTech Australia has announced the election of its new Board of Directors. The new Board is said to align with constitutional changes regarding gender diversity and representation from a broad number of states.
The new board members are:
David Ball – CEO and co-founder of HyperBank (Queensland representative)
Natalie Dinsdale – Director of Marketing at Tyro (NSW representative)
Luke Howes – Co-founder and CEO of Proviso (South Australian representative)
Lucy Liu – Chief Operating Officer of Airwallex (Victorian representative)
Alan Tsen – CEO of The Week in Bitcoin (Victorian representative)
Emma Weston – Co-founder and CEO of AgriDigital (NSW representative)
Our business structure focuses on P2P lending to three segments of borrowers – salaried individuals, practising professionals, and small and medium enterprise (SME). Right now, however, we focus only on the salaried individual segment.
Our typical borrower profile is salaried, aged 25-30 years, with an average salary of Rs 25, 000 for which the average borrowing works out upto Rs 1.50 lakh. Such a working population today is much more independent and amenable to migration. This in turn brings a lot of minor expenses and there such loans are very useful. Such borrowers are often under the under the banks' lending radar who offer them Rs 8-10 lakh loans to start with but we create options for them (based on the amount requirement). The borrowers that we target are usually digitally savvy and appreciates the benefits that we bring to the table.
What are the benefits offered to borrowers?
First and foremost is time-saving. The borrower is made known in a minute if the loan is available or not.
Another benefit of Monexo is that it is active in the entire activity chain of P2P lending – origination, screening, profile-grading, pricing of each application, disbursement, client servicing and lastly debt collection. This is right now a key differentiator among our contemporaries.
Describe the business structure and how it would attract lenders?
Our fees are taken out of repayments made to lender (2.5 per cent) based on their actual EMI receipts.
Borrowers are graded in categories from M1 down to M8. They get an automatic upgrade when they create a repayment track record. One key criteria is that debt should not be more than 60 per cent of the borrowers' income.
Our typical business process is approval of only 25 per cent of the applications submitted. This is because most of the 75 per cent are already defaulters somewhere.
Fintech company Gadai Pinjam Indonesia is gearing up for a growth spurt this year in a mission to expand the reach of financial services to unbanked small and medium enterprises.
The company, which provides pawnshop services and micro-loans through its online platform Pinjam.co.id, eyes to disburse between Rp 100 billion ($7.52 million) and Rp 200 billion this year, increasing up to 10 times its loan outstanding.
Pinjam will cooperate with state-owned post Pos Indonesia as well as some gold shops, to increase the number of outlets where their customers can pawn their goods. It plans to have more than 100 points in Jakarta by the end of this year.
Jakarta based Amartha (PT Amartha Micro Fintek) a peer to peer lending platform launched in 2010, is now officially registered with the Directorate of Institutional and Product IKNB (Financial Industry Non Bank) Financial Services Authority ( FSA).
Amartha said the approval by the financial regulators will boost public confidence in the platform and investing. Currently Amartha claims to have successfully financed over 34,000 micro businesses in parts of Indonesia to more than 10,000 registered investors, with total funds distributed to 87 billion rupiah (USD $6.5M).
In 2015 the Singapore-based bank, DBS, surveyed 600 local mothers in their 30s about retirement. The results were revealing. Three-quarters had not started planning for their retirement. Only 25 percent thought they would have sufficient funds to retire on. The average Singaporean household, headed by a 45-year-old, spends US$3,800 per month. However, 69 per cent believe they would be able to retire on less than US$2,200 a month, while 38 per cent believe it would be less than US$1,500.
According to a 2015 Nielsen survey, six out of 10 Singaporeans only start saving for their retirement once they reach 45. They believe they will just need to double their current savings to retire comfortably with peace of mind.
In China, the social pension is the primary source of retirement income. However, 43 percent of respondents in a survey conducted by the Society of Actuaries in 2016 believe the government or their company will cut their benefits in the future. With an estimated 329 million Chinese turning 65 by 2050, it is projected there will be a US$118 trillion pension deficit.
At 55, the average male has US$98,000 and female has US$85,000, bringing the total household retirement assets at around US$183,000. However the couple now has only 12 years until retirement.
Big banks are putting a lot of effort in to improve their customer experience. WeChat Pay might not be big right now, but Alibaba bought Lazada, so Alipay's coming. That's going to change a lot of things. Banks are trying hard to capture their customers' attention and build strong ties. Small-to-medium enterprise lending and security is big, particularly in Singapore. How do you protect your data? Singapore is a very strong private banking hub: a lot of money is parked here from very strange people. You don't want to have this information leaked, so the regulation techspace is being upgraded.
Jaco van Tonder, the director of advisory services at Investec Asset Management, says robo-advisers are useful to clients who cannot afford to pay for face-to-face professional financial advice.
Personal Finance looks at three offerings in the South African market:
1. Sygnia RoboAdvisor. The service was launched last year by listed asset manager Sygnia. Depending on your investment requirement and risk appetite, RoboAdvisor will expose you to unit trusts, exchange traded funds (ETFs), money market funds or cash.
• Minimum investment amounts: lump sum of R10 000 or a monthly debit order of R500.
• Management fees: 0.5% a year including VAT.
2. iTransactGo. This service is operated by Johannesburg-based exchange traded product investment platform iTransact. The company was established in 2010, and it launched its robo-adviser service last month.
• Minimum investment: lump sum of R5 000 or a minimum monthly contribution of R300.
• Investment term: there is a minimum term of one year.
• Management fees: not more than 1.14% a year including VAT, depending on the size of the investment.
3. Bizank. The company is independently owned and has appointed Anchor Capital as the asset manager of its robo-adviser. The robo-adviser, which was launched last year, creates a portfolio to meet your investment goal (for example, retirement or buying a house) based on your responses to its questions.
• Minimum investment: a lump sum of R10 000 or a monthly debit order of R1 000.
• Management fees: between 1% and 1.5% excluding VAT.
Author Allen TaylorPosted on June 5, 2017 Categories Amartha, Analysis, Assetz Capital, Banks, cash loan, Crowdsurfer, Daily News Digest, data aggregators, Featured, FED, financial planning, fintech, George Banco, Global Debt Registry, insurtech, Lend Academy, marketplace lending, Millennials, Monexo, mortgage, p2p lending, PeerIQ, PeerStreet, Pinjam, RateSetter, real estate crowdfunding, real estate investing, Robo-advisors, Roostify, Sage Payment Solutions, Trov, UK p2p, wealth tech, Zopa
More on Libor and that Japan connection
October 17 is going to be a big day for global USD money markets. It's the deadline by which prime money market reforms must adjust to floating NAV models, leaving only those funds investing in government securities able to offer par value protection. …
October 17 is going to be a big day for global USD money markets. It's the deadline by which prime money market reforms must adjust to floating NAV models, leaving only those funds investing in government securities able to offer par value protection. The likes of Zoltan Pozsar at Credit Suisse are expecting banks to lose a significant whack of unsecured bank funding as a result.
Continue reading: More on Libor and that Japan connection
Author Izabella KaminskaPosted on August 12, 2016 Categories BoJ, FED, Japanese banks, Libor, Money Market Funds, OBFR, Zoltan Pozsar
Benign neglect of NIMs in the US
Central bankers in Europe have been thinking a lot about The Death of Banks lately. Not so much in the US.
There's good reason for that, of course. Europe has been bleeding out banks with negative rates, so policy makers there have become painfully awa…
There's good reason for that, of course. Europe has been bleeding out banks with negative rates, so policy makers there have become painfully aware of the banks' role implementing monetary policy.
The US Federal Reserve, on the other hand, has been keeping banks alive with a steady drip of interest on excess reserves, or IOER, to control rates in a financial system awash with liquidity. The Fed's releasing a policy statement today (we understand if you forgot about that in the heli-frenzy before the BoJ on Friday).
Of course, keeping banks on life support with IOER doesn't help net interest margins. NIM is a key measure of bank profitability. It's also closely tied to the US yield curve -- which is unfortunate for banks, because that sucker has been positively steamrolled lately by the combination of low yields abroad, low inflation expectations and rising US policy rates.
Continue reading: Benign neglect of NIMs in the US
Author Alexandra ScaggsPosted on July 27, 2016 Categories Banking, FED, NIM | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 1,655 |
{"url":"https:\/\/gcdigitalliteracy.globalclassroom.us\/stratus\/course\/index.php?categoryid=1","text":"# Digital Literacy in the Workplace | $159 |$25 Special Offer!\n\nToday\u2019s organizations increasingly depend on technological applications to enhance productivity,\u00a0inter-organization collaboration, and knowledge management, all in order to gain a competitive advantage. Traditional and emerging technological applications facilitate accessing and sharing information, interoperability, and collaboration, and facilitate knowledge management in ways that were previously not possible. Consequently, many jobs require a working knowledge of computers and the Internet to perform basic functions. Additionally, as technology continues to evolve, more jobs will require employees to be digitally literate, and those who are digitally literate are more likely to be economically secure.\n\nBridging the economic and developmental divides is in large measure a matter of increasing digital literacy and access for people who have been left out of the information and communications technology revolution. Being digitally literate in the 21st century, however, involves more than being able to effectively and efficiently operate a computer and navigate the Web. As with traditional literacy, being digitally literate incorporates abilities such as being able to both encode and decode communications facilitated through various digital methods.\n\nSkills associated with digital literacy include finding, evaluating, and creating both information and communications transmitted through digital means. However, as with the changing nature of technology, the concept of \u201cdigital literacy\u201d is constantly evolving. While the root of being digitally literate remains relatively constant, devices such as tablet computers (iPads), smart phones, and emerging applications, such as Web 2.0 tools, are constantly redefining what it means to be digitally literate. This course addresses such issues.\n\nObjectives:\n\n1. Explore the fundamentals of computing, including basic terminology, file and folder\u00a0management, file types, and various productivity tools.\n2. Investigate issues related to navigating eLearning (online and blended) learning\u00a0environments, including exploring various tools of learning management systems.\n3. Define eLearning and examine characteristics of an effective eLearner and eLearning\u00a0environment.\n4. Investigate the \u201cnew digital divide.\u201d\n5. Examine similarities and differences between \u201cdigital natives\u201d and \u201cdigital immigrants.\u201d\n6. Define and apply the five characteristics of evaluating web resources when accessing\u00a0information on the Web.\n7. Explore dangers associated with using the Internet.\n8. Investigate and apply practices for minimizing dangers when using the Internet.\n9. Define the social web.\n10. Examine how social networking helps users establish a \"digital identity\" and how important\u00a0it is to protect your personal digital identity.\n11. Examine the features and functionality of various Web 2.0 tools and how they can be used in\u00a0various settings.\n12. Explore new hardware and software applications.\n13. Create and utilize an open source suite for multimedia productivity.\n14. Explore issues associated with creating effective digital communications.\n\nCourse Author:\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0Richard Hartshorne\nCourse Type:\nSelf-Paced\nCourse Duration:\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 5 modules\n\nReady to Register? There are 3 easy ways to enroll.\n\nIf you have any questions, please call Student Services at 877-257-2597.\n\n### Digital Literacy - Workplace Digital Literacy - Workplace\n\nToday\u2019s organizations increasingly depend on technological applications to enhance productivity, inter-organization collaboration, and knowledge management, all in order to gain a competitive advantage. Traditional and emerging technological applications facilitate accessing and sharing information, interoperability, and collaboration, and facilitate knowledge management in ways that were previously not possible. Consequently, many jobs require a working knowledge of computers and the Internet to perform basic functions. Additionally, as technology continues to evolve, more jobs will require employees to be digitally literate, and those who are digitally literate are more likely to be economically secure.","date":"2021-01-20 01:35:16","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\": 0.18353302776813507, \"perplexity\": 5010.237653523703}, \"config\": {\"markdown_headings\": true, \"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-2021-04\/segments\/1610703519843.24\/warc\/CC-MAIN-20210119232006-20210120022006-00467.warc.gz\"}"} | null | null |
#import "NSView.h"
@interface NSView (IBViewAdditions)
- (struct CGRect)IDE_IB_convertRectToScreen:(struct CGRect)arg1;
@end
| {
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<?php
namespace Drupal\Tests\content_moderation\Functional;
use Drupal\node\Entity\NodeType;
/**
* Tests permission access control around nodes.
*
* @group content_moderation
*/
class NodeAccessTest extends ModerationStateTestBase {
/**
* Modules to enable.
*
* @var array
*/
public static $modules = [
'content_moderation',
'block',
'block_content',
'node',
'node_access_test',
];
/**
* Permissions to grant admin user.
*
* @var array
*/
protected $permissions = [
'administer workflows',
'access administration pages',
'administer content types',
'administer nodes',
'view latest version',
'view any unpublished content',
'access content overview',
'use editorial transition create_new_draft',
'use editorial transition publish',
'bypass node access',
];
/**
* {@inheritdoc}
*/
protected function setUp() {
parent::setUp();
$this->drupalLogin($this->adminUser);
$this->createContentTypeFromUi('Moderated content', 'moderated_content', FALSE);
$this->grantUserPermissionToCreateContentOfType($this->adminUser, 'moderated_content');
// Add the private field to the node type.
node_access_test_add_field(NodeType::load('moderated_content'));
// Rebuild permissions because hook_node_grants() is implemented by the
// node_access_test_empty module.
node_access_rebuild();
}
/**
* Verifies that a non-admin user can still access the appropriate pages.
*/
public function testPageAccess() {
// Initially disable access grant records in
// node_access_test_node_access_records().
\Drupal::state()->set('node_access_test.private', TRUE);
$this->drupalLogin($this->adminUser);
// Access the node form before moderation is enabled, the publication state
// should now be visible.
$this->drupalGet('node/add/moderated_content');
$this->assertSession()->fieldExists('Published');
// Now enable the workflow.
$this->enableModerationThroughUi('moderated_content', 'editorial');
// Access that the status field is no longer visible.
$this->drupalGet('node/add/moderated_content');
$this->assertSession()->fieldNotExists('Published');
// Create a node to test with.
$this->drupalPostForm(NULL, [
'title[0][value]' => 'moderated content',
'moderation_state[0][state]' => 'draft',
], t('Save'));
$node = $this->getNodeByTitle('moderated content');
if (!$node) {
$this->fail('Test node was not saved correctly.');
}
$view_path = 'node/' . $node->id();
$edit_path = 'node/' . $node->id() . '/edit';
$latest_path = 'node/' . $node->id() . '/latest';
// Now make a new user and verify that the new user's access is correct.
$user = $this->createUser([
'use editorial transition create_new_draft',
'view latest version',
'view any unpublished content',
]);
$this->drupalLogin($user);
$this->drupalGet($edit_path);
$this->assertResponse(403);
$this->drupalGet($latest_path);
$this->assertResponse(403);
$this->drupalGet($view_path);
$this->assertResponse(200);
// Publish the node.
$this->drupalLogin($this->adminUser);
$this->drupalPostForm($edit_path, [
'moderation_state[0][state]' => 'published',
], t('Save'));
// Ensure access works correctly for anonymous users.
$this->drupalLogout();
$this->drupalGet($edit_path);
$this->assertResponse(403);
$this->drupalGet($latest_path);
$this->assertResponse(403);
$this->drupalGet($view_path);
$this->assertResponse(200);
// Create a pending revision for the 'Latest revision' tab.
$this->drupalLogin($this->adminUser);
$this->drupalPostForm($edit_path, [
'title[0][value]' => 'moderated content revised',
'moderation_state[0][state]' => 'draft',
], t('Save'));
$this->drupalLogin($user);
$this->drupalGet($edit_path);
$this->assertResponse(403);
$this->drupalGet($latest_path);
$this->assertResponse(200);
$this->drupalGet($view_path);
$this->assertResponse(200);
// Now make another user, who should not be able to see pending revisions.
$user = $this->createUser([
'use editorial transition create_new_draft',
]);
$this->drupalLogin($user);
$this->drupalGet($edit_path);
$this->assertResponse(403);
$this->drupalGet($latest_path);
$this->assertResponse(403);
$this->drupalGet($view_path);
$this->assertResponse(200);
// Now create a private node that the user is not granted access to by the
// node grants, but is granted access via hook_node_access().
// @see node_access_test_node_access
$node = $this->createNode([
'type' => 'moderated_content',
'private' => TRUE,
'uid' => $this->adminUser->id(),
]);
$user = $this->createUser([
'use editorial transition publish',
]);
$this->drupalLogin($user);
// Grant access to the node via node_access_test_node_access().
\Drupal::state()->set('node_access_test.allow_uid', $user->id());
$this->drupalGet($node->toUrl());
$this->assertResponse(200);
// Verify the moderation form is in place by publishing the node.
$this->drupalPostForm(NULL, [], t('Apply'));
$node = \Drupal::entityTypeManager()->getStorage('node')->loadUnchanged($node->id());
$this->assertEquals('published', $node->moderation_state->value);
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 9,299 |
{"url":"http:\/\/mathhelpforum.com\/trigonometry\/33456-verifying-trig-identities.html","text":"# Math Help - Verifying trig identities\n\n1. ## Verifying trig identities\n\nVerify the identity: 2cos4xsin2x=2sin3xcos3x-2cosxsinx.\n\nI know I have to use the product to sum identities but im still stuck.\nSo far I have, sin(3x+3x)+sin(3x-3x)-sin(x+x)-sin(x-x)\nWhat happens to the sin after you subtract 3x-3x and x-x?\n\n2. Originally Posted by kelsey3\nVerify the identity: 2cos4xsin2x=2sin3xcos3x-2cosxsinx.\n\nI know I have to use the product to sum identities but im still stuck.\nSo far I have, sin(3x+3x)+sin(3x-3x)-sin(x+x)-sin(x-x)\nWhat happens to the sin after you subtract 3x-3x and x-x?\n\n$\\underbrace{2\\sin(3x) \\cos(3x)}_{=\\sin(6x)} -2 \\sin(x) \\cos(x)$\n\n$\\underbrace{\\sin(6x)}_{\\sin(2x)\\cos(4x)+\\sin(4x)\\c os(2x)} -2\\sin(x)\\cos(x)$\n\n$\\sin(2x)\\cos(4x)+\\underbrace{\\sin(4x)}_{2\\sin(2x)\\ cos(2x)}\\cos(2x) -2\\sin(x)\\cos(x)$\n\n$\\sin(2x)\\cos(4x)+2\\sin(2x) \\underbrace{\\cos^{2}(2x)}_{\\frac{1+\\cos(4x)}{2}} -\\underbrace{2\\sin(x)\\cos(x)}_{\\sin(2x)}$\n\n$\\sin(2x)\\cos(4x)+\\sin(2x) +\\sin(2x)\\cos(4x)-\\sin(2x)$\n\n$2 \\sin(2x) \\cos(4x)$\n\nYeah!!\n\nIm curious if there is a faster way? anyone?\n\n3. Think Soroban got it in the other, double-posted thread.","date":"2015-04-26 09:53:49","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\": 6, \"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\": 0.8933001160621643, \"perplexity\": 2780.1221412881123}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2015-18\/segments\/1429246654264.98\/warc\/CC-MAIN-20150417045734-00208-ip-10-235-10-82.ec2.internal.warc.gz\"}"} | null | null |
{"url":"https:\/\/ask.sagemath.org\/questions\/42110\/revisions\/","text":"# Revision history [back]\n\n### Solve a differential equation using series expansions.\n\nGiven an ODE such as $$y''+x^2y'+y=0$$ Is it possible to get sage to display the solution in the from (at least the first few terms of the expansion) $$y=a_o\\left(c_0+c_1x+c_2x^2+\\dots\\right) + a_1\\left(d_0+d_1x+d_2x^2+\\dots\\right)$$\n\nmy attempts:\n\n### Solve a differential equation using series expansions.\n\nGiven an ODE such as $$y''+x^2y'+y=0$$ Is it possible to get sage to display the solution in the from (at least the first few terms of the expansion) $$y=a_o\\left(c_0+c_1x+c_2x^2+\\dots\\right) + a_1\\left(d_0+d_1x+d_2x^2+\\dots\\right)$$\n\nmy attempts:\n\nEDIT: I have made some progress, functional but it is not pretty. second attempt","date":"2021-09-21 05:53:32","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\": 0.7660297155380249, \"perplexity\": 411.3738909971202}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.3, \"absolute_threshold\": 10, \"end_threshold\": 5, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2021-39\/segments\/1631780057158.19\/warc\/CC-MAIN-20210921041059-20210921071059-00230.warc.gz\"}"} | null | null |
Marijuana Dispensaries Renew Push to Open in East Maricopa County
East Valley Tribune: "Tempe had more interest than any East Valley city last year when municipalities began sifting through applications for dispensary locations. Mesa has also seen renewed interest, while Gilbert's had a single inquiry. Chandler hasn't had any new interest, and a city official said it's unlikely a dispensary will find a place to open in that community despite the state's medical marijuana program going forward. The would-be dispensary owners flooded cities last year with applications, totaling more than 80 in the East Valley. Tempe was overwhelmed with about 50 applications, while Mesa fielded 35. Chandler and Gilbert had only a few applications per city. . . . Department of Health Services rules administrator Tom Salow said he doesn't expect the agency will issue all of the 126 licenses this year that are available. Tribal nations will likely block the 18 dispensaries that the geographical system would set aside for them."
By On the Net|2012-02-13T06:45:07-07:00February 13th, 2012|Stories & Articles, Zoning|Comments Off on Marijuana Dispensaries Renew Push to Open in East Maricopa County | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 76 |
Leptoromys és un gènere de rosegador extint de la família dels aplodòntids, que actualment només conté una espècie vivent, el castor de muntanya. Visqué a Nord-amèrica durant l'estatge Rupelià de l'època de l'Oligocè.
Referències
Aplodòntids
Esciüromorfs extints
Rosegadors de l'Oligocè | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 5,081 |
package io.lettuce.core.event.connection;
import java.net.SocketAddress;
import io.lettuce.core.internal.LettuceAssert;
/**
* @author Mark Paluch
* @since 3.4
*/
abstract class ConnectionEventSupport implements ConnectionEvent {
private final String redisUri;
private final String epId;
private final String channelId;
private final SocketAddress local;
private final SocketAddress remote;
ConnectionEventSupport(SocketAddress local, SocketAddress remote) {
this(null, null, null, local, remote);
}
ConnectionEventSupport(String redisUri, String epId, String channelId, SocketAddress local, SocketAddress remote) {
LettuceAssert.notNull(local, "Local must not be null");
LettuceAssert.notNull(remote, "Remote must not be null");
this.redisUri = redisUri;
this.epId = epId;
this.channelId = channelId;
this.local = local;
this.remote = remote;
}
/**
* Returns the local address.
*
* @return the local address
*/
public SocketAddress localAddress() {
return local;
}
/**
* Returns the remote address.
*
* @return the remote address
*/
public SocketAddress remoteAddress() {
return remote;
}
/**
* @return the underlying Redis URI.
*/
String getRedisUri() {
return redisUri;
}
/**
* @return endpoint identifier.
*/
String getEpId() {
return epId;
}
/**
* @return channel identifier.
*/
String getChannelId() {
return channelId;
}
@Override
public String toString() {
StringBuilder sb = new StringBuilder();
sb.append(getClass().getSimpleName());
sb.append(" [");
sb.append(local);
sb.append(" -> ").append(remote);
sb.append(']');
return sb.toString();
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 6,580 |
Do you have the guts to look at your gadget's guts?
Ever wondered how they arrange all the pieces of your smart phone inside such a small space? What about fixing the screen on your computer or phone? Well that's what Kyle Wiens, CEO of iFixit – a collaborative repair community and parts retailer, is here to talk about this week. Fixing electronics yourself is something that just about anybody can do, and will save you money as well as give yourself a bit of confidence. Not to mention, that you can feel good about helping the environment by helping eliminate waste.
Jump to this week's SHOW PAGE to find out more about teardowns, do-it-yourself electronic repair, and of course the week in review for technology news.
…Not in the philosophical sense, but in the genealogical one. If you are interested in finding out, then this week's show is for you. Chuck Roberts joins Tom to tell us all about how to use your computer to build a family tree. From general information on getting started to details about the best ways to organize what you find, this week's show will give you the best of the digital and analog when tackling such a task. Find out more at the [SHOW PAGE].
Before the interview, our Week in Review will start and end with energy savings in data centers, move on through with smart traffic lights on to something else creepy about Facebook that may have slipped past you and your account settings. And while we cover telecommunications, these fiber optics for the human nervous system from SMU and DARPA will look into the future of medical technology.
This week, we dive into The Future of Application Creation with David Milliron [MORE ABOUT THE SHOW] from Caspio.
We have details about a new charging station for electric vehicles planned for NYC along with a wireless way to weigh water. We'll you don't weigh water …but that sounded better than wireless monitoring. The City is installing units to help consumers understand how much water they are using. The idea is if you use N gallons of water in your shower today, you should try to use N-1 the next time you shower. It is also easier to conserve when you have metrics to follow (See also 20th Century manufacturing). Tune in this Sunday at 6pm EDT | 3pm MST to 1100 KFNX's live stream link at their site or head over to the show page to listen and download the show and get the links early next week.
We are lining up guest bloggers, guests for interviews and guests to submit 2-3 minute 'drops' on a technology, product or pain. If that is you or you know someone who fits the bill, email producer AT imitechtalk dot com or web AT imitechtalk dot com.
Tom attended the most recent CEA LineShow in NYC and brought back information on some very interesting technology products. 3D TVs, solar iPhone stations, how to dispose of your digital waste in NYC and much, much more await when you listen and check out all the links from the exhibitors we interviewed [LISTEN TO SHOW].
Hope you enjoyed the show. Check back later this week for info on our July 18 show.
One more thing speaking of CES – registration for the 2011 CES is now open. Start thinking about what you want to see and hear about from CES.
[Download/Listen] Doug Smith gave several interesting facts about a more "green" economy. Check the links to our Week In Review stories and let us know what you think.
Thanks for stopping by and come back on Thursday for a description of the show this next week!
We had a great show last night with Lori Grunin from CNET! If you haven't went to buy your new camera yet, Lori will be back this Sunday to tell us how to edit all those photos from our new digs.
Innovators, entrepreneurs, visionaries, and eco-designers will return to New York City to discuss the future of sustainability for the consumer electronics industry. Register now.
Five speakers have just been announced for the 2009 lineup!
The South by South West Interactive Festival features five days of exciting panel content and amazing parties. Attracting digital creatives as well as visionary technology entrepreneurs, the event celebrates the best minds and the brightest personalities of emerging technology. Whether you are a hard-core geek, a dedicated content creator, a new media entrepreneur, or just someone who likes being around an extremely creative community, SXSW Interactive is for you!
Attend the Interop Conference for a comprehensive, integrated view of technologies that will give your business a competitive edge. Learn how the recent surge of IT innovation can help you cut costs, get closer to your customers and increase revenue.
Don't forget we are on twitter – imitechtalk. Follow us for tech tips and reminders about upcoming shows. | {
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} | 92 |
Monday was a .NET day with one speech dedicated to the overview of Microsoft HoloLens – a world's first holographic computer and the most awaited device on AR/VR market. The participants got acquainted with its hardware and software platforms and had a chance to test Microsoft HoloLens in real life. The second speech focused on functional programming for .NET developers.
says Volodymyr Chyrva, Development Director at Sigma Software.
Sigma is a long-term business partner wherever information technology makes a difference. We are a global player with a Nordic base. We deliver the smartest solutions to support our customers' business aims. Sigma is owned by Danir AB and has about 3000 employees in eleven countries. Sigma operates in Ukraine since 2006 and the local team currently consists of over 700 IT professionals. | {
"redpajama_set_name": "RedPajamaC4"
} | 8,343 |
Q: VS code: java error en la clase principal He estado probando Java en netbeans y en VS code, también
En el primero no tengo problemas para hacer algún programa, en VS code, al momento de ejecutar un programa me lanza el siguiente error:
Error: no se ha encontrado o cargado la clase principal App1
Causado por: java.lang.NoClassDefFoundError: App1/App1 (wrong name: App1)
este es el código:
El archivo lo tengo en una carpeta que se llama App1
Este error solo me aparece si tuviera el archivo en una carpeta, si lo tengo en el área de trabajo no me lanza ningún error.
¿Qué debo de hacer?
A: Si estas en vs code, elimina el package de la línea numero 1, es muy probable que tu estructura de archivos y paquetes en el mismo folder de espacio de trabajo este mal estructurada.
te comparto este link para más información.
https://code.visualstudio.com/docs/java/java-tutorial
A: No estoy seguro, pero creo que Java es muy quisquilloso a la hora de estructurar tu aplicación. La estructura que sigue con Maven es muy característica. Tal vez el fichero .xml que ejecuta el programa intente buscarlo en el directorio src/main/java y no lo encuentre y al tener la clase main en una carpeta que se llama App1 te sale ese error.
En este enlace puedes encontrar información sobre la estructura de la que te hablo, pero seguramente la habrás visto ya en netbeans.
https://hop2croft.wordpress.com/2011/04/28/estructura-basica-de-un-proyecto-con-maven/
A: Encontramos que la ruta puede ser un problema. En MAc si tu proyecto esta en el escritorio, resultó que el nombre muy largo puede ser causa del problema.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 6,562 |
Authentic and Abundant Love: The Culture Project
Northeast Ohio Catholic Magazine
Peter and Andrew did not hesitate to drop their nets when Jesus called out to them, "Follow me, and I will make you fishers of men" (Mt 4:19). I always thought if I was in their shoes it would play out much differently. I would be too comfortable casting my net and would be too afraid to follow. But when I encountered the mission of The Culture Project, I was captivated by the call the Lord was extending to me. Though I was afraid, I did not hesitate and I decided to follow.
The Culture Project gives presentations on human dignity, sexual integrity, and virtuous living to young people in schools, parishes, communities, and at community events. I encountered them during my senior year of college and their missionary spirit, joy, and authentic nature moved me to drop my post-graduation plans and apply to the organization. Through my own experiences, I had witnessed the lies our culture tells young people about love and self-worth. By becoming a missionary with the organization, I knew I would be able to speak truth to these lies and help combat a toxic culture.
During my two years of service, I encountered young ladies fed up with what the world had to say about their worth and young men wanted to become men of virtue. But there were also many young people broken down by what the world had offered them. They were real, raw, and honest about what they had been facing, but they were not afraid to venture on a new, more fulfilling path. It is in these encounters that I knew my decision to leave behind my net behind and follow Christ was more than worth it.
Regarding the mission of The Culture Project, I sat down with our Founder and Executive Director Cristina Barba to share more about how the mission started and what it is trying to achieve:
What was your basis/reasoning for founding The Culture Project?
My friends and I had lived the ways of the culture and found it wanting. We were sick of what the world had to offer us. We found there was more for ourselves and our peers. We felt that we needed to do more. The hook-up culture had sold us short. The world had sold young people short. We thought that the truth needed to get out, and there was a power in a community of men and women who were practicing what they preached. We based the mission program on four pillars: formation, community, prayer and work.
What are some adversities or challenges that you are fighting against today?
Two of the biggest challenges our mission faces are: finding good men and technology. Finding the right young men who are ready and willing to be a part of this mission can be tough. Mission work often appeals to women, and a lot of guys interested in this type of work might be interested in pursuing religious life or priesthood. Some are ready to get married and start a family. We see a crisis in men who want to step up or pause their pursuits in order to serve this mission. The second is technology. Many of our culture's issues are perpetuated by technology. It pulls us further away from reality and the real world. We live in our culture of isolation, and our overuse of technology is a part of that. The youth that we are encountering are facing this every single day; especially in terms of the sexual, secular and negative self-worth messages that bombard them.
What is your favorite verse or quote that inspires the mission of The Culture Project?
Recently, I watched a documentary on Pope Saint John Paul II's visit to Poland. He is a great hero of mine and a patron of the work of The Culture Project. He said, "You are not who they say you are. Let me remind you of who you are." Youth today struggle with our identity in terms of who the culture tells us we are, especially when it comes to our sexuality. What we do at The Culture Project is to encounter the youth, put a pause on that thinking and say, "You are not who the world says you are." We have the opportunity to remind them that they are made in the image and likeness of God. This quote sums up what we are trying to achieve.
The Culture Project gives presentations on human dignity, sexual integrity, and virtuous living to young people in schools, parishes, and other settings. Their mission is to encounter youth and young adults, remind them of their dignity, and encourage them to be men and women of character. A team of five Culture Project missionaries will be serving the Diocese of Cleveland, Ohio beginning in October. To learn more or request a visit from the Cleveland team to your location, go to www.thecultureproject.org.
Alyssa Sanchez is a former missionary with The Culture Project and now works in the headquarters office as the Office and Mission Administrator.
This article was previously published in Radiant Magazine. | {
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} | 3,520 |
John K. Castle is a pioneer in private equity investing in addition to being an adventurer, philanthropist and volunteer.
About John Castle
Castle Harlan
Castle Connolly
Branford Castle
Education & Philanthropies
Happy Warrior Award
Adventures on Land and Water
Restoration of the JFK Winter White House
Partial List of Media Links & Videos
John K. Castle, Pioneer in Private Equity Investing, Presented "Happy Warrior Award" At 2017 Al Smith Dinner
John K. Castle, chairman and CEO of the New York private equity firm Castle Harlan Inc., is the 2017 Happy Warrior Award recipient from The Alfred E. Smith Memorial Foundation, which was presented Thursday at the 2017 Al Smith Dinner.
Castle shared speaking roles with keynote speaker Paul Ryan, Speaker of the U.S. House of Representatives, and acclaimed actress Patricia Heaton, the dinner's master of ceremonies. The foundation holds the annual dinner, which brings together political and business leaders and features good-natured humorous quips, to raise funds for needy children of the Archdiocese of New York, as well as to honor Smith, a four-term governor of New York and the 1928 Democratic Presidential candidate. The dinner raised more than $3.5 million.
Smith was known as "The Happy Warrior" for his ability to maintain his positive outlook while tackling pressing social issues. Each year the foundation carries on the tradition of the "Happy Warrior" by recognizing an individual who exemplifies the character and leadership of Smith. Castle, this year's recipient, helped build private equity investing into a major sector of investing and capital formation. He also is a leading philanthropist in health and education, and an adventurer.
Castle's business career is long and distinguished. He is chairman and chief executive officer of Castle Harlan, Inc., a private merchant bank, and chairman and chief executive officer of Branford Castle, Inc., a firm that makes long-term investments in small to medium-sized private companies. Castle also is an advisory director of the DuPont Corporation Investment Management Co. Castle has committed a substantial portion of his time and resources to public service in education, healthcare and religious organizations over the past 35 years, and a number of philanthropies. He is also an adventurer who has traveled the world.
Castle received his bachelor's degree from the Massachusetts Institute of Technology, his MBA as a Baker Scholar with High Distinction from Harvard, and he has been awarded four Honorary Doctorate degrees.
Copyright 2019 - johnkcastle.com | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 3,055 |
4 in 10 people are worried about multiculturalism. A thread on Reddit takes a more positive view.
Sam Woolfe
A recent poll has found that four in 10 people think multiculturalism threatens British culture. Following the survey, a thread began on Reddit, asking people what 'British culture' means to them. The answers take a more positive view of multiculturalism. They also show that whatever is 'British' is still very much present.
Defining British culture
Many people may view immigration and globalisation in a negative light. But the top comment on the Reddit threat takes a different view:
Being British is about driving a German car to an Irish pub for a Belgian beer, then getting a taxi home driven by a Pakistani man, grabbing an Indian curry or a Turkish kebab on the way, to sit on Swedish furniture and watch American shows on a Japanese TV.
Other comments highlight typical British characteristics, both good and bad:
Eating fish and chips in the rain.
British culture to me is self-deprecation and multiculturalism hasn't taken that away at all.
Sarcasm and shared moaning, funny turns of phrase and taking the piss out of your mates.
Complaining about the weather, whether it is too hot, too cold, too wet or too snowy.
Self-deprecating attitude towards oneself and the country as a whole while simultaneously maintaining a certain pride in that fact and the country regardless
Overriding air of doom and gloom
Half arsed approach to religion (only relevant for christenings, weddings and funerals)
Mostly eating foreign food and then having a roast on a Sunday
Respect for the Police, Fire service and NHS. (I think the police one is a very British thing)
We're a diverse nation
Although it's easy to point out very British traits, such as politeness and a stiff upper lip, British culture is still quite hard to define. Other Redditors remarked:
Being socially reserved and outwardly polite. Being overly reliant on alcohol as a social bond. Being self-deprecating about ourselves and our country.
I generally think we're a pretty diverse nation though, which is why trying to define what being British is seems such a fruitless endeavour. The answer will vary depending on where you go. I imagine people in London would give quite a different answer to people in Cornwall.
I think most people can see all cultures are complex mixes of different cultures that vary over time. It's complicated and detailed, very hard to define in simple terms.
I'd say that this multiculturalism is what has defined us in the past and will do in the future. I'm not sure if there is actually anything in our culture that is purely British, except maybe queuing and apologising for everything, regardless of if it's our fault or not.
This recent survey revealed that a large minority of people in the UK feel that people who move here from other countries are failing to integrate. Most, though, don't believe this is a problem. They believe workers from abroad are essential to the smooth running of the country. And in terms of 'British culture' disappearing, well, it doesn't look like sarcasm and moaning are disappearing anytime soon.
– Join us, so we can keep bringing you the news that matters.
Featured image via Beyond Bespoke
Someone asked Vince Cable if the LibDems were 'too nice'. Well stop laughing and let's look at their record.
Nearly £20k is raised in a 'Crowdfunder for Katie Hopkins'. But she's going to freak at the small print. | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 2,882 |
#include "coda.h"
typedef struct coda_tree_node_struct coda_tree_node;
struct coda_tree_node_struct
{
const coda_type *type;
int num_items;
void **item;
coda_tree_node *all_children; /* node that contains items that are applicable for all indices */
int num_indexed_children;
long *index; /* can be '-1' for attributes */
coda_tree_node **indexed_child;
};
coda_tree_node *coda_tree_node_new(const coda_type *type);
void coda_tree_node_delete(coda_tree_node *node, void (*free_item)(void *));
int coda_tree_node_add_item_for_path(coda_tree_node *node, const char *path, void *item, int leaf_only);
int coda_tree_node_get_item_for_cursor(coda_tree_node *node, coda_cursor *cursor, void **item);
| {
"redpajama_set_name": "RedPajamaGithub"
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{"url":"https:\/\/my-assignmentexpert.com\/2022\/03\/29\/%E6%95%B0%E5%AD%A6%E4%BB%A3%E5%86%99-%E9%9A%8F%E6%9C%BA%E8%BF%87%E7%A8%8B%E4%BB%A3%E8%80%83the-law-of-the-iterated-logarithm\/","text":"19th Ave New York, NY 95822, USA\n\n# \u6570\u5b66\u4ee3\u5199| \u968f\u673a\u8fc7\u7a0b\u4ee3\u8003|The law of the iterated logarithm\n\nA population starts with one individual at time $n=0: Z_{0}=1$.\n\nAfter one unit of time (at time $n=1$ ) the sole individual produces $Z_{1}$ identical clones of itself and dies. $Z_{1}$ is an $\\mathbb{N}_{0}$-valued random variable.\n\n(a) If $Z_{1}$ happens to be equal to 0 the population is dead and nothing happens at any future time $n \\geq 2$.\n\n(b) If $Z_{1}>0$, a unit of time later, each of $Z_{1}$ individuals gives birth to a random number of children and dies. The first one has $Z_{1,1}$ children, the second one $Z_{1,2}$ children, etc. The last, $Z_{1}^{\\text {th }}$ one, gives birth to $Z_{1, Z_{1}}$ children. We assume that the distribution of the number of children is the same for each individual in every generation and independent of either the number of individuals in the generation and of the number of children the others have. This distribution, shared by all $Z_{n, i}$ and $Z_{1}$, is called the offspring distribution. The total number of individuals in the second generation is now\n$$Z_{2}=\\sum_{k=1}^{Z_{1}} Z_{1, k}$$\n(c) The third, fourth, etc. generations are produced in the same way. If it ever happens that $Z_{n}=0$, for some $n$, then $Z_{m}=0$ for all $m \\geq n$ \u2013 the population is extinct. Otherwise,\n$$Z_{n+1}=\\sum_{k=1}^{Z_{n}} Z_{n, k}$$\n\nmy-assignmentexpert\u2122\u00a0\u968f\u673a\u8fc7\u7a0bStochastic Process\u4f5c\u4e1a\u4ee3\u5199\uff0c\u514d\u8d39\u63d0\u4ea4\u4f5c\u4e1a\u8981\u6c42\uff0c \u6ee1\u610f\u540e\u4ed8\u6b3e\uff0c\u6210\u7ee980\\%\u4ee5\u4e0b\u5168\u989d\u9000\u6b3e\uff0c\u5b89\u5168\u7701\u5fc3\u65e0\u987e\u8651\u3002\u4e13\u4e1a\u7855 \u535a\u5199\u624b\u56e2\u961f\uff0c\u6240\u6709\u8ba2\u5355\u53ef\u9760\u51c6\u65f6\uff0c\u4fdd\u8bc1\u00a0100%\u00a0\u539f\u521b\u3002my-assignmentexpert\u2122\uff0c \u6700\u9ad8\u8d28\u91cf\u7684\u968f\u673a\u8fc7\u7a0bStochastic Process\u4f5c\u4e1a\u4ee3\u5199\uff0c\u670d\u52a1\u8986\u76d6\u5317\u7f8e\u3001\u6b27\u6d32\u3001\u6fb3\u6d32\u7b49 \u56fd\u5bb6\u3002 \u5728\u4ee3\u5199\u4ef7\u683c\u65b9\u9762\uff0c\u8003\u8651\u5230\u540c\u5b66\u4eec\u7684\u7ecf\u6d4e\u6761\u4ef6\uff0c\u5728\u4fdd\u969c\u4ee3\u5199\u8d28\u91cf\u7684\u524d\u63d0\u4e0b\uff0c\u6211\u4eec\u4e3a\u5ba2\u6237\u63d0\u4f9b\u6700\u5408\u7406\u7684\u4ef7\u683c\u3002 \u7531\u4e8e\u7edf\u8ba1Statistics\u4f5c\u4e1a\u79cd\u7c7b\u5f88\u591a\uff0c\u540c\u65f6\u5176\u4e2d\u7684\u5927\u90e8\u5206\u4f5c\u4e1a\u5728\u5b57\u6570\u4e0a\u90fd\u6ca1\u6709\u5177\u4f53\u8981\u6c42\uff0c\u56e0\u6b64\u968f\u673a\u8fc7\u7a0bStochastic Process\u4f5c\u4e1a\u4ee3\u5199\u7684\u4ef7\u683c\u4e0d\u56fa\u5b9a\u3002\u901a\u5e38\u5728\u7ecf\u6d4e\u5b66\u4e13\u5bb6\u67e5\u770b\u5b8c\u4f5c\u4e1a\u8981\u6c42\u4e4b\u540e\u4f1a\u7ed9\u51fa\u62a5\u4ef7\u3002\u4f5c\u4e1a\u96be\u5ea6\u548c\u622a\u6b62\u65e5\u671f\u5bf9\u4ef7\u683c\u4e5f\u6709\u5f88\u5927\u7684\u5f71\u54cd\u3002\n\nmy-assignmentexpert\u2122 \u4e3a\u60a8\u7684\u7559\u5b66\u751f\u6daf\u4fdd\u9a7e\u62a4\u822a \u5728\u7edf\u8ba1Statistics\u4f5c\u4e1a\u4ee3\u5199\u65b9\u9762\u5df2\u7ecf\u6811\u7acb\u4e86\u81ea\u5df1\u7684\u53e3\u7891, \u4fdd\u8bc1\u9760\u8c31, \u9ad8\u8d28\u4e14\u539f\u521b\u7684\u7edf\u8ba1Statistics\u4ee3\u5199\u670d\u52a1\u3002\u6211\u4eec\u7684\u4e13\u5bb6\u5728\u968f\u673a\u8fc7\u7a0bStochastic Process\u4ee3\u5199\u65b9\u9762\u7ecf\u9a8c\u6781\u4e3a\u4e30\u5bcc\uff0c\u5404\u79cd\u968f\u673a\u8fc7\u7a0bStochastic Process\u76f8\u5173\u7684\u4f5c\u4e1a\u4e5f\u5c31\u7528\u4e0d\u7740 \u8bf4\u3002\n\n\u2022 \u65f6\u95f4\u5e8f\u5217\u5206\u6790Time-Series Analysis\n\u2022 \u9a6c\u5c14\u79d1\u592b\u8fc7\u7a0b Markov process\n\u2022 \u968f\u673a\u6700\u4f18\u63a7\u5236stochastic optimal control\n\u2022 \u7c92\u5b50\u6ee4\u6ce2 Particle Filter\n\u2022 \u91c7\u6837\u7406\u8bba sampling\u00a0theory\n\n## \u6570\u5b66\u4ee3\u5199| \u968f\u673a\u8fc7\u7a0b\u4ee3\u8003|growth rate\n\nLet $S_{n}=X_{1}+X_{2}+\\cdots+X_{n}$ be the partial sum of independent identically distributed variables, as usual, and suppose further that $\\mathbb{E}\\left(X_{i}\\right)=0$ and $\\operatorname{var}\\left(X_{i}\\right)=1$ for all $i$. To date, we have two results about the growth rate of $\\left{S_{n}\\right}$.\nLaw of large numbers: $\\frac{1}{n} S_{n} \\rightarrow 0$ a.s. and in mean square.\nCentral limit theorem: $\\frac{1}{\\sqrt{n}} S_{n} \\stackrel{\\mathrm{D}}{\\rightarrow} N(0,1)$\nThus the sequence $U_{n}=S_{n} \/ \\sqrt{n}$ enjoys a random fluctuation which is asymptotically regularly distributed. Apart from this long-term trend towards the normal distribution, the sequence $\\left{U_{n}\\right}$ may suffer some large but rare fluctuations. The law of the iterated logarithm is an extraordinary result which tells us exactly how large these fluctuations are. First note that,\n$$U=\\limsup {n \\rightarrow \\infty} \\frac{U{n}}{\\sqrt{2 \\log \\log n}}$$\n\n## \u6570\u5b66\u4ee3\u5199| \u968f\u673a\u8fc7\u7a0b\u4ee3\u8003|iterated logarithm\n\nTheorem. Law of the iterated logarithm. If $X_{1}, X_{2}, \\ldots$ are independent identically distributed random variables with mean 0 and variance 1 then\n$$\\mathbb{P}\\left(\\limsup {n \\rightarrow \\infty} \\frac{S{n}}{\\sqrt{2 n \\log \\log n}}=1\\right)=1$$\nThe proof is long and difficult and is omitted (but see the discussion in Billingsley (1995) or Laha and Rohatgi (1979)). The theorem amounts to the assertion that\n$$A_{n}=\\left{S_{n} \\geq c \\sqrt{2 n \\log \\log n}\\right}$$\noccurs for infinitely many values of $n$ if $c<1$ and for only finitely many values of $n$ if $c>1$, with probability 1 . It is an immediate corollary of (1) that\n$$\\mathbb{P}\\left(\\liminf {n \\rightarrow \\infty} \\frac{S{n}}{\\sqrt{2 n \\log \\log n}}=-1\\right)=1$$\njust apply (1) to the sequence $-X_{1},-X_{2}, \\ldots$\n\n## \u6570\u5b66\u4ee3\u5199| \u968f\u673a\u8fc7\u7a0b\u4ee3\u8003|GROWTH RATE\n\n$$U=\\limsup {n \\rightarrow \\infty} \\frac{U {n}}{\\sqrt{2 \\log \\log n}}$$\n\n## \u6570\u5b66\u4ee3\u5199| \u968f\u673a\u8fc7\u7a0b\u4ee3\u8003|ITERATED LOGARITHM\n\n$$\\mathbb{P}\\left(\\limsup {n \\rightarrow \\infty} \\frac{S {n}}{\\sqrt{2 n \\log \\log n}}=1\\\u53f3\uff09=1 \u5428H\u548cpr\u25cb\u25cbFs\u4e00\u25cbnG A_{n}=\\left{S_{n} \\geq c \\sqrt{2 n \\log \\log n}\\right} \u25cbCC \\mathbb{P}\\left(\\liminf {n \\rightarrow \\infty} \\frac{S {n}}{\\sqrt{2 n \\log \\log n}}=-1\\right)=1$$\n\n## Matlab\u4ee3\u5199\n\nMATLAB \u662f\u4e00\u79cd\u7528\u4e8e\u6280\u672f\u8ba1\u7b97\u7684\u9ad8\u6027\u80fd\u8bed\u8a00\u3002\u5b83\u5c06\u8ba1\u7b97\u3001\u53ef\u89c6\u5316\u548c\u7f16\u7a0b\u96c6\u6210\u5728\u4e00\u4e2a\u6613\u4e8e\u4f7f\u7528\u7684\u73af\u5883\u4e2d\uff0c\u5176\u4e2d\u95ee\u9898\u548c\u89e3\u51b3\u65b9\u6848\u4ee5\u719f\u6089\u7684\u6570\u5b66\u7b26\u53f7\u8868\u793a\u3002\u5178\u578b\u7528\u9014\u5305\u62ec\uff1a\u6570\u5b66\u548c\u8ba1\u7b97\u7b97\u6cd5\u5f00\u53d1\u5efa\u6a21\u3001\u4eff\u771f\u548c\u539f\u578b\u5236\u4f5c\u6570\u636e\u5206\u6790\u3001\u63a2\u7d22\u548c\u53ef\u89c6\u5316\u79d1\u5b66\u548c\u5de5\u7a0b\u56fe\u5f62\u5e94\u7528\u7a0b\u5e8f\u5f00\u53d1\uff0c\u5305\u62ec\u56fe\u5f62\u7528\u6237\u754c\u9762\u6784\u5efaMATLAB \u662f\u4e00\u4e2a\u4ea4\u4e92\u5f0f\u7cfb\u7edf\uff0c\u5176\u57fa\u672c\u6570\u636e\u5143\u7d20\u662f\u4e00\u4e2a\u4e0d\u9700\u8981\u7ef4\u5ea6\u7684\u6570\u7ec4\u3002\u8fd9\u4f7f\u60a8\u53ef\u4ee5\u89e3\u51b3\u8bb8\u591a\u6280\u672f\u8ba1\u7b97\u95ee\u9898\uff0c\u5c24\u5176\u662f\u90a3\u4e9b\u5177\u6709\u77e9\u9635\u548c\u5411\u91cf\u516c\u5f0f\u7684\u95ee\u9898\uff0c\u800c\u53ea\u9700\u7528 C \u6216 Fortran \u7b49\u6807\u91cf\u975e\u4ea4\u4e92\u5f0f\u8bed\u8a00\u7f16\u5199\u7a0b\u5e8f\u6240\u9700\u7684\u65f6\u95f4\u7684\u4e00\u5c0f\u90e8\u5206\u3002MATLAB \u540d\u79f0\u4ee3\u8868\u77e9\u9635\u5b9e\u9a8c\u5ba4\u3002MATLAB \u6700\u521d\u7684\u7f16\u5199\u76ee\u7684\u662f\u63d0\u4f9b\u5bf9\u7531 LINPACK \u548c EISPACK \u9879\u76ee\u5f00\u53d1\u7684\u77e9\u9635\u8f6f\u4ef6\u7684\u8f7b\u677e\u8bbf\u95ee\uff0c\u8fd9\u4e24\u4e2a\u9879\u76ee\u5171\u540c\u4ee3\u8868\u4e86\u77e9\u9635\u8ba1\u7b97\u8f6f\u4ef6\u7684\u6700\u65b0\u6280\u672f\u3002MATLAB \u7ecf\u8fc7\u591a\u5e74\u7684\u53d1\u5c55\uff0c\u5f97\u5230\u4e86\u8bb8\u591a\u7528\u6237\u7684\u6295\u5165\u3002\u5728\u5927\u5b66\u73af\u5883\u4e2d\uff0c\u5b83\u662f\u6570\u5b66\u3001\u5de5\u7a0b\u548c\u79d1\u5b66\u5165\u95e8\u548c\u9ad8\u7ea7\u8bfe\u7a0b\u7684\u6807\u51c6\u6559\u5b66\u5de5\u5177\u3002\u5728\u5de5\u4e1a\u9886\u57df\uff0cMATLAB \u662f\u9ad8\u6548\u7814\u7a76\u3001\u5f00\u53d1\u548c\u5206\u6790\u7684\u9996\u9009\u5de5\u5177\u3002MATLAB \u5177\u6709\u4e00\u7cfb\u5217\u79f0\u4e3a\u5de5\u5177\u7bb1\u7684\u7279\u5b9a\u4e8e\u5e94\u7528\u7a0b\u5e8f\u7684\u89e3\u51b3\u65b9\u6848\u3002\u5bf9\u4e8e\u5927\u591a\u6570 MATLAB \u7528\u6237\u6765\u8bf4\u975e\u5e38\u91cd\u8981\uff0c\u5de5\u5177\u7bb1\u5141\u8bb8\u60a8\u5b66\u4e60\u5e94\u7528\u4e13\u4e1a\u6280\u672f\u3002\u5de5\u5177\u7bb1\u662f MATLAB \u51fd\u6570\uff08M \u6587\u4ef6\uff09\u7684\u7efc\u5408\u96c6\u5408\uff0c\u53ef\u6269\u5c55 MATLAB \u73af\u5883\u4ee5\u89e3\u51b3\u7279\u5b9a\u7c7b\u522b\u7684\u95ee\u9898\u3002\u53ef\u7528\u5de5\u5177\u7bb1\u7684\u9886\u57df\u5305\u62ec\u4fe1\u53f7\u5904\u7406\u3001\u63a7\u5236\u7cfb\u7edf\u3001\u795e\u7ecf\u7f51\u7edc\u3001\u6a21\u7cca\u903b\u8f91\u3001\u5c0f\u6ce2\u3001\u4eff\u771f\u7b49\u3002","date":"2022-08-13 09:52:29","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\": 0.8238719701766968, \"perplexity\": 1001.177121682642}, \"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-2022-33\/segments\/1659882571911.5\/warc\/CC-MAIN-20220813081639-20220813111639-00702.warc.gz\"}"} | null | null |
\section{Introduction}
The directional signalling capabilities of base stations (BSs) that have multiple transmit antennas enable a variety of techniques~\cite{SymbollevelandMulticast} for simultaneously transmitting independent messages to multiple single-antenna receivers, including dirty paper coding~\cite{TheCapacityRegion}, vector perturbation precoding~\cite{Avectorperturbationtechnique2}, lattice reduction precoding~\cite{latticereductionaided}, Tomlinson-Harashima precoding~\cite{Precodinginmultiantenna}, rate splitting~\cite{RobustTransmissioninDownlink}, per-symbol beamforming~\cite{ConstructiveMultiuserInterference}, and conventional linear beamforming~\cite{ShiftingtheMIMO}. Of these signalling techniques, conventional linear beamforming has the simplest implementation and will be the focus of this paper. In particular, we will consider scenarios in which the users that have been scheduled for transmission specify the quality-of-service (QoS) that they expect to receive. In that setting, the BS designs the set of beamformers to ensure that the signal-to-interference-and-noise ratio (SINR) at each receiver meets the target level that is implicitly specified by that user's QoS requirements. When the BS has perfect knowledge of the channel to each user, the beamformers that minimize the total transmitted power required to achieve the SINR targets can be efficiently found \cite{Jointoptimal,Reference2,Solutionofthemultiuser,OptimalMultiuserTransmit}. However, in practice these channels are estimated and possibly predicted. In time division duplexing (TDD) systems the estimation is typically performed during the training phase on the uplink, whereas in frequency division duplexing (FDD) systems, each receiver estimates its channel and feeds back a quantized version of that estimate to the BS. Since the BS has only estimates of the users' channels, it can only estimate the receivers' SINRs. Those estimates are, quite naturally, uncertain and hence there is a possibility that a design performed using the estimated channels will fail to meet the SINR targets when the beamformers are implemented.
A prominent approach to designing a precoder that can control the consequent outage is to postulate a model for the uncertainty in the channel estimates and to seek designs that control the outage probability under that uncertainty model. In some cases the approach involves jointly designing the beamforming directions and the power allocated to these directions (e.g.,~\cite{Optimalpowercontrol,Probabilisticallyconstrained,OutageConstrained,LowComplexityRobustMISO}), while in other cases the beamforming directions are designed based on the channel estimates only, and the uncertainty model is incorporated into the design of the power loading; e.g., \cite{Coordinateupdate,Arobustmaximin,ATractableMethod}. Unfortunately, in most settings the outage constraint has proven to be intractable (an exception is the case in \cite{Coordinateupdate}), and hence the goal has been to develop computationally efficient algorithms that can manage the outage probability. One possible strategy for doing so is to seek ``safe" approximations of the robust optimization problem \cite{Reference4}. When such approximations result in a feasible design problem, the solution is guaranteed to satisfy the constraints of the original problem, but these approximations can be quite conservative; e.g., \cite{OutageConstrained,Probabilisticallyconstrained}. An alternative strategy is to develop approximations of the outage constraint that typically provide good performance, but might not necessarily guarantee that their solution is feasible for the original problem; e.g., \cite{LowComplexityRobustMISO,Optimalpowercontrol}. The approach taken in this paper falls into that class.
The development of the proposed offset-based approach begins with the rewriting of the SINR constraint as the non-negativity of a random variable. That random variable is a non-convex quadratic function of the uncertainties, in which the quadratic kernel is a quartic function of the beamformers. Then, we approximate the non-negativity constraint on the random variable by the constraint that its mean is larger than a given multiple of its standard deviation. For the case of Gaussian channel uncertainties, the mean and standard deviation are quadratic and quartic function of the beamformers, respectively. That fact enables the application of semidefinite relaxation techniques to obtain a convex formulation of (a relaxed version of) the approximated problem. While that design technique is quite effective, the computational cost of solving the convex conic program with semidefinite constraints is significant. By making a further approximation that is suitable for systems with reasonably small uncertainties, we obtain a design formulation for which the KKT optimality conditions have a simpler structure. That simpler structure facilitates the development of an approximate solution method that only requires the iterative evaluation of closed-form expressions. Further approximations reveal a connection with the low-complexity technique developed in~\cite{LowComplexityRobustMISO}.
An analysis of the computational cost of these precoder design techniques shows that it is the calculation of the beamforming directions that consumes most of the required computational resources, and that when these directions are defined in-advance, the computational load can be significantly reduced. Accordingly, we develop variants of our precoder design algorithms that perform power loading on a set of fixed beamforming directions. These algorithms have low computational costs, and provide performance that is close to that of the optimal power loading algorithm~\cite{Coordinateupdate}. Furthermore, for systems with a large number of antennas (i.e., ``massive MIMO") in which the channel hardens, we develop a variant of our power loading algorithm that has a computational cost that grows only linearly with the number of antennas.
In practice, the BS has limited power available for transmission, and it is possible that the power required to serve the scheduled users with the required outage probabilities may exceed that limit. In some of these scenarios, some users suffer from a weak channel, or from having their channels closely aligned with those of other users. When that happens, such users consume most of the power transmitted by the BS. This suggests opportunities to reschedule users. On the other hand, some users might be close to the BS and experiencing a relatively strong channel; a case that suggests opportunities for doing some sort of power saving. The proposed power loading algorithm provides an explicit relationship between the required outage probabilities and the consumed power, which allows us to address these issues. Using this explicit power-outage relationship we can reduce the required power when the resulting increases in the outage probabilities are tolerable, and we can identify users that consume excessive amounts of power.
The above-mentioned designs are ``fair" in the sense that they seek to provide each user with their specified outage probability. However, the proposed design techniques are quite flexible, and can accommodate other objectives, such as the sum of the outage probabilities. As we will demonstrate, such designs can improve the average performance of the users.
\section{System model}
We consider a scenario in which a BS that has $N_t$ antennas communicates with $K$ single-antenna users over a narrow-band channel. In the linear beamforming transmission case, the transmitted signal can be written as $\mathbf{x}= \sum_{k=1}^K\mathbf{w}_k s_k,$ where $s_k$ is the normalized data symbol intended for user $k$, and $\mathbf{w}_k$ is the associated beamformer vector. For later reference we let $\mathbf{u}_k =\mathbf{w}_k /\|\mathbf{w}_k \| $ denote the beamforming direction for user $k$, and let $\beta_k=\|\mathbf{w}_k \|^2$ denote the power allocated to that direction. Hence, $\mathbf{w}_k= \sqrt{\beta_k}\mathbf{u}_k $. The received signal at user $k$ is modelled as
\begin{equation}\label{rcvd_sig}
y_k= \mathbf{h}_k^H \mathbf{w}_k s_k + \textstyle\sum_{j \neq k}\mathbf{h}_k^H \mathbf{w}_j s_j + n_k,
\end{equation}
where $\mathbf{h}_k^H$ is the vector of complex channel gains between the antennas at the BS and user $k$, and $n_k$ is the additive zero-mean circular complex Gaussian noise at that user.
Under this model, if we let $\sigma_k^2$ denote the noise variance, then the SINR at user $k$ is
\begin{equation}\
\text{SINR}_k= \frac{| \mathbf{h}_k^H \mathbf{w}_k|^2}{\sum_{j \neq k} | \mathbf{h}_k^H \mathbf{w}_j|^2 + \sigma_k^2}.
\end{equation}
The design of a set of beamformers $\{ \mathbf{w}_k\}_{k=1}^K$ so that the SINRs satisfy specified target values (i.e., $\text{SINR}_k\geq \gamma_k$) requires the knowledge of the channel vectors $\{\mathbf{h}_k\}_{k=1}^K$. However, the BS has only estimates of $\{\mathbf{h}_k\}_{k=1}^K$, and hence its estimates of the SINRs at the receivers are uncertain. Accordingly, we will incorporate the channel uncertainty model into the design process. In particular, we will consider systems in which the uncertainty can be modelled using the simple additive model,
\begin{equation}\label{uncertainty}
\mathbf{h}_k= \mathbf{h}_{e_k} +\mathbf{e}_k,
\end{equation}
where $\mathbf{h}_{e_k}$ is the BS's estimate of the channel to user $k$, and the uncertainty in that estimate is characterized by the distribution of the elements of $\mathbf{e}_k$.
In this paper, we will focus on scenarios in which $\mathbf{e}_k$ can be modelled as a circular complex Gaussian random variable with mean $\mathbf{m}_k$ and covariance $\mathbf{C}_k$; i.e., $\mathbf{e}_k \backsim \mathcal{CN} (\mathbf{m}_k, \mathbf{C}_k)$. One scenario in which that model is applicable is that of a TDD scheme operating in a slow fading environment, in which the BS estimates the channel on the uplink using a linear estimator and exploits channel reciprocity. When the channel gains are uncorrelated and the BS employs the best linear unbiased estimator (BLUE), $\mathbf{e}_k \backsim \mathcal{CN} (0, \sigma_{e_k}^2\mathbf{I})$, and we will pay particular attention to that case. (Robust beamforming schemes for uncertainty models tailored to the FDD case were developed in \cite{LowComplexityRobustMISO}.)
Now if we let $\delta_k$ denote the maximum tolerable outage probability for user $k$, the generic joint beamforming and power loading problem can be written as
\begin{subequations}\label{outage_min}
\begin{align}
\min_{\substack{\mathbf{w}_k}} \quad &\textstyle\sum_{k=1}^K \mathbf{w}_k^H \mathbf{w}_k \\
\text{subject to} \quad & \text{Prob}(\text{SINR}_k \geq \gamma_k)\geq 1- \delta_k, \quad \forall k. \label{sinr5}
\end{align}
\end{subequations}
This problem is hard to solve due to the intractable probabilistic outage constraint in \eqref{sinr5} even when the uncertainty is Gaussian~\cite{Optimalpowercontrol,Probabilisticallyconstrained,OutageConstrained}.
In order to resolve that intractability, a variety of approximations of the problem in \eqref{outage_min} by problems that are tractable have been proposed \cite{Optimalpowercontrol,Probabilisticallyconstrained,OutageConstrained,LowComplexityRobustMISO}.
In many cases, the class of approximations that is considered is restricted to the class of ``safe'' approximations~\cite{Reference4}.
Such approximations are structured so that they guarantee that any solution of the approximate problem is feasible for the original problem
in \eqref{outage_min}. However, in the downlink beamforming application, such approximations can be quite conservative, in the sense that the
feasible set of the approximate problem is significantly smaller than that of the original problem; cf. \eqref{outage_min}. That can result in
instances of the approximate problem being infeasible when the original problem has a solution, or in beamformer designs that consume significantly more power than necessary. The approximation that we will develop below is not structurally constrained in this way, but it typically performs well in practice. Furthermore, its simple form provides considerable flexibility in its application, and facilitates the development of highly-efficient algorithms.
\section{Principles of the offset-based approach}\label{sect3}
The derivation of the proposed approximation of the outage probability begins by rewriting $\text{SINR}_k \geq \gamma_k$ as $\mathbf{h}_k^H \mathbf{Q}_k \mathbf{h}_k - \sigma_k^2 \geq 0$, where
\begin{equation}
\begin{aligned}
\mathbf{Q}_k &= \mathbf{w}_k \mathbf{w}_k^H/\gamma_k-\textstyle\sum_{j \neq k} \mathbf{w}_j \mathbf{w}_j^H \\
&= \beta_k \mathbf{u}_k \mathbf{u}_k^H/\gamma_k-\textstyle\sum_{j \neq k} \beta_j \mathbf{u}_j \mathbf{u}_j^H.
\end{aligned}
\end{equation}
That is, the probability that $\text{SINR}_k \geq \gamma_k$ is the same as the probability that the term $\mathbf{h}_k^H \mathbf{Q}_k \mathbf{h}_k - \sigma_k^2$ is non-negative. Under the additive uncertainty model in \eqref{uncertainty}, we observe that $\mathbf{h}_k^H \mathbf{Q}_k \mathbf{h}_k - \sigma_k^2$ is an indefinite quadratic function of the uncertainty, $\mathbf{e}_k$. In particular, we can formulate the SINR constraint as follows
\begin{equation}\label{SINR_reformulation}
f_k(\mathbf{e}_k)=\mathbf{h}_{e_k}^H \mathbf{Q}_k \mathbf{h}_{e_k} + 2 \text{Re}(\mathbf{e}_k^H \mathbf{Q}_k \mathbf{h}_{e_k} ) + \mathbf{e}_k^H \mathbf{Q}_k \mathbf{e}_k - \sigma_k^2 \geq 0.
\end{equation}
The key observation that underlies the offset approximation is that for uncertainties $\mathbf{e}_k$ that are reasonably concentrated, if we design the beamforming vectors so that the mean value of $f_k(\mathbf{e}_k)$, denoted by $\mu_{f_k}$, is a significant multiple of its standard deviation, denoted by $\sigma_{f_k}$, then that user will achieve a low outage probability. If we let $r_k$ denote that multiple for the $k$th user, then the resulting approximation of the SINR constraint, $\text{Prob}(\text{SINR}_k \geq \gamma_k)\geq 1- \delta_k$, can be written as
\begin{equation}\label{offset_constr}
\mu_{f_k} \geq r_k \sigma_{f_k}.
\end{equation}
In order to develop an intuitive rationale for that approximation for the outage probability, we observe that when $\mathbf{e}_k$ in \eqref{uncertainty} is Gaussian, $f_k(\mathbf{e}_k)$ has a generalized chi-square distribution \cite{OntheDistributionof}. We also observe that the term that complicates the calculation of the relevant tail probability (i.e., Prob ($f_k(\mathbf{e}_k)<0$)) is the indefinite quadratic term $\mathbf{e}_k^H \mathbf{Q}_k \mathbf{e}_k$ in \eqref{SINR_reformulation}. To have reasonable outage performance, the norm of the channel uncertainty $\mathbf{e}_k$ in \eqref{uncertainty} should be relatively small compared to the norm of the channel; cf., \cite{MIMObroadcast}. In that case, the constant and linear terms in \eqref{SINR_reformulation} will tend to dominate the quadratic term. Furthermore, the distribution of $ \mathbf{e}_k^H \mathbf{Q}_k \mathbf{e}_k$ is ``bell shaped" since $\mathbf{Q}_k$ generically has one positive and $K-1$ negative eigenvalues. Now if we approximate the quadratic term $\mathbf{e}_k^H \mathbf{Q}_k \mathbf{e}_k$ by a Gaussian term of the same mean and variance, then the distribution of $f_k(\mathbf{e}_k)$ becomes Gaussian and the constraint in \eqref{offset_constr} provides precise control over the tail probability.
In other words, the constraint in \eqref{offset_constr} provides precise control of the tail probability of the Gaussian approximation of $f_k(\mathbf{e}_k)$. These insights, and the guidance that they provide on the choice of $r_k$, are discussed in more detail in Appendix~\ref{r_value_sel}.
To be able to use the offset approximation in \eqref{offset_constr} in a low-complexity design algorithm, we need to obtain expressions for $\mu_{f_k}$ and $ \sigma_{f_k}$ in terms of the design variables $\mathbf{w}_k= \sqrt{\beta_k}\mathbf{u}_k$. As shown in Appendix \ref{mean_var_der}, when $\mathbf{e}_k \backsim \mathcal{CN} (\mathbf{m}_k, \mathbf{C}_k)$
\begin{subequations}\label{mean_eqn_c}
\begin{align}
\mu_{f_k}& = \mathbb{E} \{f_k(\mathbf{e}_k)\} \nonumber \\
&= (\mathbf{h}_{e_k}+\mathbf{m}_k)^H \mathbf{Q}_k (\mathbf{h}_{e_k}+\mathbf{m}_k) - \sigma_k^2 + \mathbf{w}_k^H \mathbf{C}_k \mathbf{w}_k /\gamma_k \nonumber\\
& \qquad -\sum_{j \neq k} \mathbf{w}_j^H \mathbf{C}_k \mathbf{w}_j^H, \\
\sigma_{f_k}^2 & = \text{var} \{f_k(\mathbf{e}_k)\} \nonumber \\
&= 2 (\mathbf{h}_{e_k}+\mathbf{m}_k)^H \mathbf{C}_k^{1/2} \mathbf{Q}_{k}^2 \mathbf{C}_k^{1/2} (\mathbf{h}_{e_k}+\mathbf{m}_k) \nonumber \\
&\qquad +\text{tr} (\mathbf{C}_k^{1/2} \mathbf{Q}_{k} \mathbf{C}_k^{1/2} )^2.
\end{align}
\end{subequations}
From the perspective of beamformer design, an important observation is that $\mu_{f_k}$ is a non-convex quadratic function of the beamformers $\{\mathbf{w}_k\}_{k=1}^K$, but for fixed beamforming directions $\{\mathbf{u}_k\}_{k=1}^K$ it is a linear function of the power loading $\{\beta_k\}_{k=1}^K$. The variance $\sigma_{f_k}^2$ is a quartic function of the beamformers, and for fixed directions is a non-convex quadratic function of the power loading. In scenarios in which the model $\mathbf{e}_k \backsim \mathcal{CN} (0, \sigma_{e_k}^2\mathbf{I})$ is appropriate, these expressions simplify to
\begin{subequations}
\begin{align}
\mu_{f_k} & = \mathbf{h}_{e_k}^H \mathbf{Q}_k \mathbf{h}_{e_k}- \sigma_k^2 + \sigma_{e_k}^2 \Bigl(\beta_k /\gamma_k - \sum_{j \neq k}\beta_j \Bigr). \label{mean_eqn} \\
\sigma_{f_k}^2 & =2 \sigma_{e_k}^2 \mathbf{h}_{e_k}^H \mathbf{Q}_{k}^2 \mathbf{h}_{e_k} +\sigma_{e_k}^4 \text{tr} (\mathbf{Q}_{k}^2). \label{var_rel}
\end{align}
\end{subequations}
We will focus on this simplified case in the following sections.
\section{Offset-based robust beamforming}\label{mod_dir}
As discussed above, the robust beamforming problem in \eqref{outage_min} is fundamentally hard to solve due to the intractability of the probabilistic SINR outage constraint in \eqref{sinr5}. If we were to replace that constraint with its offset approximation, $\mu_{f_k} \geq r_k \sigma_{f_k}$, then the problem in \eqref{outage_min} can be closely approximated by
\begin{subequations}\label{outage_min2}
\begin{align}
\min_{\substack{\mathbf{w}_k}} \quad & \textstyle\sum_{k=1}^K \mathbf{w}_k^H \mathbf{w}_k \\
\text{subject to} \quad & \mu_{f_k} \geq r_k \sigma_{f_k}, \quad \forall k. \label{sinr3}
\end{align}
\end{subequations}
Some insights into the behaviour of solutions to \eqref{outage_min2} can be obtained by observing that when the values of $r_k$ are chosen to be the same, the beamforming vectors are designed so that users with a large SINR variance are provided with a larger SINR mean. To do so, those users with a lower SINR variance are not provided with as large mean SINR as they do not need the same protection against the uncertainty.
To develop an algorithm to obtain good solutions to \eqref{outage_min2}, we observe that in \eqref{sinr3} we have the term $\mu_{f_k}$ which is quadratic in $\mathbf{w}_k$, and we also have the term $ \sigma_{f_k}^2= 2 \sigma_{e_k}^2 \mathbf{h}_{e_k}^H \mathbf{Q}_k^2 \mathbf{h}_{e_k} +\sigma_{e_k}^4 \text{tr} (\mathbf{Q}_k^2) $, which includes the square of the matrix $ \mathbf{Q}_k$ and, accordingly, is quartic in $\mathbf{w}_k$. If we make the substitution $\mathbf{W}_k =\mathbf{w}_k \mathbf{w}_k^H$, then the functions in \eqref{sinr3} become linear and quadratic functions of $\mathbf{W}_k$ and the objective becomes linear. As such, the remaining difficulty in the reformulation of the problem is the set of rank-one constraints on $\mathbf{W}_k$. If we relax those constraints we obtain the following semidefinite relaxation of the problem in \eqref{outage_min2}
\begin{subequations}\label{outage_min4}
\begin{align}
\min_{\substack{\mathbf{W}_k, d_{1k}, d_{2k}}} \quad &\text{tr} \Bigl(\textstyle\sum_{k=1}^K \mathbf{W}_k \Bigr) \\
\text{s.t.} \quad & \mathbf{h}_{e_k}^H \mathbf{Q}_k \mathbf{h}_{e_k}- \sigma_k^2 + \sigma_{e_k}^2 \text{tr}(\mathbf{W}_k) /\gamma_k \nonumber \\
& \quad - \sigma_{e_k}^2 \text{tr} \Bigl(\textstyle\sum_{j\neq k} \mathbf{W}_j \Bigr) \geq r_k \| [d_{1k} \; d_{2k} ]\| , \\
& d_{1k} \geq \sqrt{2} \sigma_{e_k} \| \mathbf{h}_{e_k}^H \mathbf{Q}_k \|, \\
& d_{2k} \geq \sigma_{e_k}^2 \| \mathbf{Q}_k \|_F, \\
& \mathbf{W}_k \succeq \mathbf{0}, \quad \forall k,
\end{align}
\end{subequations}
where $\| \cdot \|_F$ represents the Frobenius norm of the matrix.
In this formulation, each SINR constraint in \eqref{sinr3} is replaced by three second order cone (SOC) constraints. Thus, the problem in \eqref{outage_min4} is a convex conic optimization problem and can be efficiently solved using interior point methods. Two refined implementations of those methods are easily accessible through the $\textsc{Matlab}$-based CVX tool \cite{cvx}. In our numerical experience, the rank of the optimal $\mathbf{W}_k$'s in \eqref{outage_min4} has always been one. When that occurs, the semidefinite relaxation is tight and the optimal beamformer vectors $\mathbf{w}_k$ can be directly obtained from the optimal matrices $\mathbf{W}_k$. This phenomenon has been established in some related beamforming problems \cite{Reference2,RobustSINR,UnravelingtheRankOne}, and has been observed numerically in a number of other downlink beamforming problems; e.g., \cite{OutageConstrained}.
\subsection{Low-complexity precoding algorithm}
Although the problem in \eqref{outage_min4} is convex, it contains $3K$ SOC constraints, plus the $K$ semidefinite constraints. As a result, solving \eqref{outage_min4} incurs a significant computational load even for a moderate number of antennas. In this section, we will first show how a mild approximation of the problem in \eqref{outage_min4} leads to an optimization problem with only $K$ SOC constraints. We will then use insights from the KKT conditions of that problem to show that it can be approximately solved using the iterative evaluation of a sequence of closed-form expressions.
The approximation is based on the observation, made above, that in practical downlink systems the uncertainty in the channel estimates must be small in order for the system to support reasonable rates \cite{MIMObroadcast}. In such scenarios, the term in \eqref{sinr3} containing $\sigma_{e_k}^4$ will typically be significantly smaller than the other term. Accordingly, $\sigma_{f_k}^2 \approx 2 \sigma_{e_k}^2 \mathbf{h}_{e_k}^H \mathbf{Q}_k^2 \mathbf{h}_{e_k}$ is a reasonable approximation.
Applying this approximation in the context of the problem in \eqref{outage_min2} we obtain the following approximation of \eqref{sinr3}
\begin{multline}\label{new_sinr_const}
\mathbf{h}_{e_k}^H \mathbf{Q}_k \mathbf{h}_{e_k}- \sigma_k^2 + \sigma_{e_k}^2 \mathbf{w}_k^H \mathbf{w}_k /\gamma_k - \sigma_{e_k}^2 \sum_{j \neq k} \mathbf{w}_j^H \mathbf{w}_j \\ \geq r_k \sqrt{2} \sigma_{e_k} \|\mathbf{h}_{e_k}^H \mathbf{Q}_k\|.
\end{multline}
The semidefinite relaxation of the resulting approximation of the problem in \eqref{outage_min2} can be written as
\begin{subequations}\label{outage_min5}
\begin{align}
\min_{\substack{\mathbf{W}_k}, \mathbf{d}_k} \quad &\text{tr} \Bigl(\textstyle\sum_{k=1}^K \mathbf{W}_k \Bigr) \\
\text{s.t.} \quad & \mathbf{h}_{e_k}^H \mathbf{Q}_k \mathbf{h}_{e_k}- \sigma_k^2 + \sigma_{e_k}^2 \text{tr}(\mathbf{W}_k) /\gamma_k \nonumber \\
& \quad - \sigma_{e_k}^2 \text{tr} \Bigl(\textstyle\sum_{j\neq k} \mathbf{W}_k \Bigr) \geq \| \mathbf{d}_k \|, \label{sinr4} \\
& \mathbf{d}_k = r _k \sqrt{2} \sigma_{e_k} \mathbf{Q}_k \mathbf{h}_{e_k}, \label{dconst} \\
& \mathbf{W}_k \succeq \mathbf{0}, \quad \forall i.
\end{align}
\end{subequations}
We note that the problem in \eqref{outage_min5} is over parameterized (the vectors $ \mathbf{d}_k$ are not needed), but this over parameterization will simplify the following analysis.
The problem in \eqref{outage_min5} is another convex conic program, but it has significantly fewer constraints than that in \eqref{outage_min4}; there are $K$ SOC constraints rather than the $3K$ in \eqref{outage_min4}. While it can be solved with less computational effort than \eqref{outage_min4}, the presence of the semidefinite constraints means that considerable effort is still required. To derive a more efficient algorithm, we examine the Lagrangian of \eqref{outage_min5}, assuming that the matrices $\mathbf{W}_k$ are of rank one. If we let $\nu_k$ denote the dual variable for the constraint in \eqref{sinr4}, and $\boldsymbol\psi_{f_k}$ denote the vector of dual variables for the equality constraint in \eqref{dconst}, the Lagrangian can be written as
\begin{multline}
\mathcal{L}(\mathbf{w}_k, \mathbf{d}_k, \nu_k,\boldsymbol\psi_{f_k})= \sum_{k=1}^{K} \mathbf{w}_k^H \mathbf{w}_k -\sum_{k=1}^{K}\nu_k \Bigl(\mathbf{h}_{e_k}^H \mathbf{Q}_k \mathbf{h}_{e_k} - \sigma_k^2 \\ +\sigma_{e_k}^2 \mathbf{w}_k^H \mathbf{w}_k /\gamma_k - \sigma_{e_k}^2 \sum_{j \neq k}\mathbf{w}_j^H \mathbf{w}_j - \| \mathbf{d}_k \| \Bigr) \\ -\sum_{k=1}^{K} \boldsymbol\psi_{f_k}^H (\mathbf{d}_k - r_k \sqrt{2} \sigma_{e_k} \mathbf{Q}_k \mathbf{h}_{e_k}).
\end{multline}
From the KKT conditions of the problem in \eqref{outage_min5}, we can deduce that
\begin{multline}\label{closed_form}
\mathbf{w}_k =\Biggl( \frac{\nu_k}{\gamma_k}\mathbf{h}_{e_k} \mathbf{h}_{e_k}^H-\sum_{j\neq k} \nu_j \mathbf{h}_{e_j} \mathbf{h}_{e_j}^H + \frac{\nu_k \sigma_{e_k}^2}{\gamma_k} \mathbf{I} -\sum_{j\neq k} \nu_j \sigma_{e_k}^2 \mathbf{I} \\ - \frac{ r_k \sqrt{2} \sigma_{e_k} }{\gamma_k} \text{Re} \{\boldsymbol\psi_{f_k} \mathbf{h}_{e_k}^H \} + \sum_{j\neq k} r_j \sqrt{2} \sigma_{e_k} \text{Re} \{\boldsymbol\psi_j \mathbf{h}_{e_j}^H \} \Biggr)\mathbf{w}_k,
\end{multline}
which is an eigen equation for the direction $\mathbf{u}_k$. Using a similar approach to the perfect CSI case \cite{OptimalMultiuserTransmit}, we can rearrange this equation to obtain the following fixed-point equation for $\nu_k$,
\begin{multline}\label{nu_mod2}
\nu_k^{-1} = \mathbf{h}_{e_k}^H \Biggl( \mathbf{I} + \sum_j \nu_j \mathbf{h}_{e_j} \mathbf{h}_{e_j}^H - \frac{\nu_k \sigma_{e_k}^2}{\gamma_k} \mathbf{I} +\sum_{j\neq k} \nu_j \sigma_{e_k}^2 \mathbf{I} \\ +\frac{ r_k \sqrt{2} \sigma_{e_k} }{\gamma_k} \text{Re} \{\boldsymbol\psi_{f_k} \mathbf{h}_{e_k}^H \} - \sum_{j\neq k} r_j \sqrt{2} \sigma_{e_k} \text{Re} \{\boldsymbol\psi_j \mathbf{h}_{e_j}^H \} \Biggr)^{-1} \\ \times \mathbf{h}_{e_k} \Bigl(1+\frac{1}{\gamma_k} \Bigr).
\end{multline}
The expressions in \eqref{closed_form} and \eqref{nu_mod2} share a similar structure to those obtained for the corresponding QoS problem in the case of perfect CSI at the BS \cite{OptimalMultiuserTransmit}, but the matrix components of each equation contain four additional terms that are dependent on the variance of the channel estimation error. To exploit this structure and obtain an efficient algorithm for good solutions to \eqref{outage_min5} we observe that if we were given $\{\boldsymbol\psi_{f_k}\}$, then we could solve the fixed-point equations in \eqref{nu_mod2} for $\{ \nu_k \}$, and then we could solve the eigen equations in \eqref{closed_form} for the beamforming directions $\{\mathbf{u}_k\}$. The solution could then be completed by performing the appropriate power loading, which will be explained in the following section. Therefore, if we could find a reasonable approximation for the vectors $\boldsymbol\psi_{f_k}$, we would obtain an iterative closed-form solution. To do so, we observe that the variable $\mathbf{d}_k$ in \eqref{dconst} appears in the Lagrangian in the term $ \nu_k \| \mathbf{d}_k \| -\boldsymbol\psi_{f_k}^H \mathbf{d}_k$.
Accordingly, from the stationarity component of the KKT conditions we have that $\| \boldsymbol\psi_{f_k} \| = \nu_k$ and that $\mathbf{d}_k$ and $\boldsymbol\psi_{f_k}$ are in the same direction; i.e., $\mathbf{d}_k/ \| \mathbf{d}_k\| = \boldsymbol\psi_{f_k} /\| \boldsymbol\psi_{f_k} \|$. Accordingly, we can write
\begin{equation}\label{psi}
\boldsymbol\psi_{f_k}= \nu_k \mathbf{d}_k/ \| \mathbf{d}_k\|.
\end{equation}
Since $\mathbf{d}_k = r _k \sqrt{2} \sigma_{e_k} \mathbf{Q}_k \mathbf{h}_{e_k}$, $\boldsymbol\psi_{f_k}$ explicitly depends on the beamforming directions, which have not yet been determined. However, we observe that if we substitute \eqref{psi} into \eqref{nu_mod2}, the terms involving $\mathbf{d}_k$ are multiplied by the standard deviation of the error, $\sigma_{e_k}$. As we have already argued in the derivation of the approximations that lead to \eqref{outage_min4}, $\sigma_{e_k}$ will be small in effective downlink beamforming schemes, and this suggests that reasonable initial approximations of the directions should yield a good approximation of $\{ \nu_k \}$, and hence a good set of beamforming directions. We suggest the use of the zero-forcing (ZF) directions \cite{Zeroforcingmethods} for the estimated channels, which we will denote by $\mathbf{u}_{z_k}$. When we use that initialization, the initial direction of $\mathbf{d}_k$ will be the same as $\mathbf{u}_{z_k}$, which allows us to rewrite the fixed-point equations in \eqref{nu_mod2} as
\begin{multline}\label{nu_mod}
\nu_k^{-1} = \mathbf{h}_{e_k}^H \Biggl( \mathbf{I} + \sum_j \nu_j \mathbf{h}_{e_j} \mathbf{h}_{e_j}^H - \frac{\nu_k \sigma_{e_k}^2}{\gamma_k} \mathbf{I} +\sum_{j\neq k} \nu_j \sigma_{e_k}^2 \mathbf{I} \\ +\frac{ r \sqrt{2} \sigma_{e_k} \nu_k }{\gamma_k} \text{Re} \{\mathbf{u}_{z_k} \mathbf{h}_{e_k}^H \} - \sum_{j\neq k} r \sqrt{2} \sigma_{e_k} \nu_j \text{Re} \{\mathbf{u}_{z_j} \mathbf{h}_{e_j}^H \} \Biggr)^{-1} \\ \times \mathbf{h}_{e_k} \Bigl(1+\frac{1}{\gamma_k} \Bigr).
\end{multline}
The derivations outlined above are summarized in the sequence of closed-form operations in Alg. \ref{Alg1}. While the initial approximation can be improved by using the beamformers obtained in step 4 to obtain a refined estimate of the direction of $\mathbf{d}_k$ and returning to step 2 of the algorithm, the simulation results in Section \ref{sec_sim} suggest that the one-shot approach taken in Alg. \ref{Alg1} produces a solution whose performance is quite close to that of the original offset-based design formulation in \eqref{outage_min4}. That suggests that in the scenarios that we have considered, the underlying approximations are working quite well.
\begin{algorithm}
\caption{Iterative closed-form beamformer design}
\label{Alg1}
\begin{algorithmic}[1]
\State Find the ZF directions $\{\mathbf{u}_{z_k}\}$.
\State Find each $\nu_k$ using \eqref{nu_mod}.
\State Find each $\mathbf{u}_k$ using the corresponding variant of \eqref{closed_form}.
\State Apply the power loading developed in Section \ref{per_user_Power_Loading_algorithm}.
\end{algorithmic}
\end{algorithm}
\subsection{Constant-offset algorithm \cite{LowComplexityRobustMISO}}\label{sec_org_offset_max}
As is apparent from the derivation in the previous section, one of the challenges that complicates the closed-form calculations is the quartic dependence of the variances $\sigma_{f_k}^2$ on the beamforming vectors $\mathbf{w}_k$. One way in which these complications can be reduced is to modify the offset approximation in \eqref{offset_constr} so that the mean, $\mu_{f_k}$, is constrained to be greater than a constant; i.e., the SINR constraint is replaced by $$ \mu_{f_k} \geq r_k. $$
If we make the approximation that the channel estimation errors are small enough that the third term on the right hand side of \eqref{mean_eqn} can be neglected, the semidefinite relaxation of the resulting approximation of \eqref{outage_min2} can be written as
\begin{subequations}\label{r_prob}
\begin{align}
\min_{\substack{\mathbf{W}_k}} \quad &\text{tr} \Bigl(\textstyle\sum_{k=1}^K \mathbf{W}_k \Bigr) \\
\text{s.t.} \quad & \mathbf{h}_{e_k}^H \mathbf{Q}_k \mathbf{h}_{e_k}- \sigma_k^2 \geq r_k, \label{sinr6}\\
& \mathbf{W}_k \succeq \mathbf{0}, \quad \forall k.
\end{align}
\end{subequations}
Interestingly, this problem arose previously in the context of a low-complexity solution to the robust beamforming design problem for FDD and TDD systems that use a zero-outage region approach, and the semidefinite relaxation was shown to be tight \cite{LowComplexityRobustMISO}. The zero-outage region approach provides robustness by requiring that the SINR constraints hold for all channels in a neighbourhood of the estimated channel.
The iterative closed-form solution to \eqref{r_prob} has a similar structure to that in Alg. \ref{Alg1}, but given the simpler structure of the problem, the Lagrange multipliers $\boldsymbol\psi_{f_k}$ disappear, and the expressions in \eqref{closed_form} and \eqref{nu_mod2} simplify to
\begin{equation}\label{closed_form2}
\mathbf{w}_k =\Biggl( \frac{\nu_k}{\gamma_k}\mathbf{h}_{e_k} \mathbf{h}_{e_k}^H-\sum_{j\neq k} \nu_j \mathbf{h}_{e_j} \mathbf{h}_{e_j}^H \Biggr)\mathbf{w}_k,
\end{equation}
\begin{equation}\label{nu}
\nu_k^{-1} = \mathbf{h}_{e_k}^H \Bigl(\mathbf{I}+\textstyle\sum_{j} \nu_j \mathbf{h}_{e_j} \mathbf{h}_{e_j}^H \Bigr)^{-1} \mathbf{h}_{e_k} \Bigl(1+\frac{1}{\gamma_k} \Bigr).
\end{equation}
After obtaining the beamforming directions from \eqref{nu} and \eqref{closed_form2}, the power loading in \cite{LowComplexityRobustMISO} is performed based on the fact that the constraints in \eqref{sinr6} are satisfied with equality at optimality. (If this were not the case for constraint $k$, then the power allocated to $\mathbf{w}_k$ could be reduced in a way that will still satisfy all the constraints and provide a lower objective value, contradicting the presumed optimality.) While doing so generates a solution to \eqref{r_prob}, significant performance gains can be obtained when the beamforming directions obtained from \eqref{closed_form2} are combined with the power loading algorithm presented in Section \ref{per_user_Power_Loading_algorithm}.
\subsection{Complexity analysis and further approximations}\label{sec_approxs}
The problems in \eqref{outage_min4} and \eqref{outage_min5} are convex optimization problems with SOC and semidefinite constraints. General purpose interior point methods for such problems require $\mathcal{O}(N_t^6)$ per iteration, which represents a significant computational load.
In contrast, the key computational steps in the iterative closed-form approximation, Alg. \ref{Alg1}, are those in \eqref{closed_form}, \eqref{nu_mod} and the calculation of the ZF directions that are used in the initialization. The ZF directions can be obtained in $\mathcal{O}(N_t^2 K)$ operations.
The computational cost of solving \eqref{nu_mod} is dominated by the matrix inversion required for each user and hence it grows as $\mathcal{O}(N_t^3 K)$. We can exploit the factorized matrix structure in \eqref{closed_form} which allows for an efficient use of the power iteration method. Therefore, the cost of step 3 grows as $\mathcal{O}(N_t K^2)$. We can see that it is the computation of the Lagrange multipliers \eqref{nu_mod} that requires most of the resources to compute the beamforming directions.
The constant-offset algorithm \cite{LowComplexityRobustMISO} that was reviewed in Section~\ref{sec_org_offset_max} does not require an initial set of directions and the expression for $\nu_k$ is significantly simpler. In particular, the matrix to be inverted is the same for each user, which reduces the number of computations required to $\mathcal{O}(N_t^3)$. Furthermore, additional approximations can be applied to avoid the matrix inversion all together.
When the channels are nearly orthogonal, as they tend to be in massive MISO channels that ``harden" as the number of antennas increases \cite{Multipleantennachannelhardening}, then if we let $\alpha_k= \|\mathbf{h}_{e_k}\|^2$, we can write $\textstyle\sum_{j} \nu_j \mathbf{h}_{e_j} \mathbf{h}_{e_j}^H$ in the form of an eigen decomposition $\textstyle\sum_{j} \nu_j \alpha_j \frac{\mathbf{h}_{e_j}}{\sqrt{\alpha_j} } \frac{\mathbf{h}_{e_j}^H}{\sqrt{\alpha_j}}$,
and hence,
$$\mathbf{h}_{e_k}^H \Bigl(\mathbf{I}+\textstyle\sum_{j} \nu_j \alpha_j \frac{\mathbf{h}_{e_j}}{\sqrt{\alpha_j} } \frac{\mathbf{h}_{e_j}^H}{\sqrt{\alpha_j}} \Bigr)^{-1} \mathbf{h}_{e_k} \approx \frac{\alpha_k}{1+\nu_k \alpha_k}.$$
Accordingly, we can approximate \eqref{nu} by $$\nu_k\approx \gamma_k/\alpha_k.$$
To find the channel norms $\alpha_k= \| \mathbf{h}_{e_k} \|^2$ we need only $\mathcal{O}(N_t)$ operations. Hence, that approximation enables us to compute all $\nu_k$s in only $\mathcal{O}(N_t K)$ operations.
\section{Offset-based robust power Loading}\label{per_user_Power_Loading_algorithm}
In this section, we will show how to apply the offset-based approach to the power loading problem that remains if the beamforming directions are chosen separately. Examples of choices for those directions include the maximum ratio transmission (MRT), zero-forcing (ZF), or regularized zero-forcing (RZF) directions, which are calculated from the estimated channels, or any of the directions generated by the previously described algorithms. Once the directions are chosen, we can rewrite the problem in \eqref{outage_min2} as
\begin{subequations}\label{outage_min6}
\begin{align}
\min_{\substack{\beta_k}} \ \quad & \sum_{k=1}^K \beta_k \\
\text{subject to} \quad & \mu_{f_k} \geq r_k \sigma_{f_k}, \quad \forall k, \label{sinr2}
\end{align}
\end{subequations}
where for fixed directions $\{\mathbf{u}_k\}$ the expressions for $\mu_{f_k}$ and $\sigma_{f_k}$ in \eqref{mean_eqn} and \eqref{var_rel} simplify to
\begin{subequations}\label{mean_eqn2}
\begin{align}
\mu_{f_k} & = |\mathbf{h}_{e_k}^H \mathbf{u}_k|^2 \beta_k/\gamma_k - \sum_{j \neq k} |\mathbf{h}_{e_k}^H \mathbf{u}_j|^2 \beta_j- \sigma_k^2 \nonumber\\
& \qquad +\sigma_{e_k}^2 \Bigl(\beta_k /\gamma_k - \sum_{j \neq k}\beta_j \Bigr). \\
\sigma_{f_k}^2 & =2 \sigma_{e_k}^2 \mathbf{h}_{e_k}^H \Bigl(\beta_k \mathbf{u}_k \mathbf{u}_k^H/\gamma_k-\sum_{j \neq k}\beta_j \mathbf{u}_j \mathbf{u}_j^H \Bigr)^2 \mathbf{h}_{e_k} \nonumber\\ &\qquad +\sigma_{e_k}^4 \text{tr} \Bigl(\beta_k \mathbf{u}_k \mathbf{u}_k^H/\gamma_k-\sum_{j \neq k}\beta_j \mathbf{u}_j \mathbf{u}_j^H \Bigr)^2. \label{var_rel2}
\end{align}
\end{subequations}
Since $\mu_{f_k} $ is linear in $\{\beta_k\}$ and $\sigma_{f_k}$ is a convex quadratic function of $\{\beta_k\}$, the problem in \eqref{outage_min6} can be rewritten as an SOC programming problem, and an optimal solution can be efficiently obtained using generic interior-point methods. However, to begin to develop a more efficient algorithm that exploits some of the specific features of the problem in \eqref{outage_min6}, we observe that at optimality the constraints in \eqref{sinr2} hold with equality. If this were not the case for constraint $k$, then $\beta_k $ could be reduced in a way that still satisfies the constraints and yet provides a lower objective value, which would contradict the presumed optimality. To use that observation, we note that if the variances $\sigma_{f_k}^2$ are fixed, then the set of equations $\{\mu_{k} =r_k \sigma_{f_k}\}$ yields $K$ linear equations in the $K$ design variables $\{\beta_k \}_{k=1}^K$. If we define $\boldsymbol{\beta}=[\beta_1, \beta_2,..., \beta_K]^T$, $\boldsymbol{\sigma}_f=[\sigma_{f_1}, \sigma_{f_2},..., \sigma_{f_K}]^T$, $\boldsymbol{\sigma}=[\sigma_{1}, \sigma_{2},..., \sigma_{K}]^T$, $\mathbf{r}=[r_1, r_2,...,r_k]^T$, and the matrix $\mathbf{A}$ such that $\mathbf{[A]}_{ii}= | \mathbf{h}_{e_i}^H {\mathbf{u}}_i |^2/\gamma_i+\sigma_{e_i}^2 /\gamma_i $, and $\mathbf{[A]}_{ij}= - | \mathbf{h}_{e_i}^H {\mathbf{u}}_j |^2 - \sigma_{e_i}^2$, $\forall i \neq j$, then the set of linear equations can be written as
\begin{equation}\label{A_eqn}
\mathbf{A} \boldsymbol{\beta} =\boldsymbol{\sigma}^2+ \boldsymbol\sigma_{f} \odot \mathbf{r},
\end{equation}
in which $\odot$ represents element-by-element multiplication. Once the values of $\{\beta_k \}$ have been found, we can update the value of $\boldsymbol{\sigma}_f$ using \eqref{var_rel2}. That suggests the iterative linearization algorithm for solving \eqref{outage_min6} that is summarized in Alg.~\ref{Alg2}.
\begin{algorithm}
\caption{The power loading algorithm}
\label{Alg2}
\begin{algorithmic}[1]
\State Initialize $\sigma_{f_k}=1$. Compute $\mathbf{A}$ and $\mathbf{A}^{-1}$.
\State Find $\boldsymbol{\beta}$ by solving the set of linear equations in \eqref{A_eqn}.
\State Update each $\sigma_{f_k}$ using \eqref{var_rel2}.
\State Return to 2 until a termination criterion is satisfied.
\end{algorithmic}
\end{algorithm}
By observing the dependence of $\boldsymbol{\sigma}_f$ on $\boldsymbol{\beta}$ in \eqref{var_rel2}, Alg. \ref{Alg2} can be written in the form of a fixed point technique by writing $\boldsymbol{\beta} =\mathbf{A}^{-1} \boldsymbol{\sigma}^2+ \mathbf{A}^{-1} (\boldsymbol\sigma_{f} \odot \mathbf{r})$. The eigenvalues of $\mathbf{A}^{-1}$ determine the convergence properties for these fixed-point equations. Since the matrix $\mathbf{A}$ typically has large diagonal values representing the signal powers, and lower values on the off-diagonal elements representing the interference powers, the eigen values of $\mathbf{A}^{-1}$ will typically be less than one. Our numerical experience not only confirms this observation, but also suggests that the number of iterations needed for near-optimal performance is very small. In terms of computational cost, the initialization step in Alg. \ref{Alg2} requires $\mathcal{O}(K^2 N_t)$ operations to compute $\mathbf{A}$ and $\mathcal{O}(K^3)$ operations to compute $\mathbf{A}^{-1}$. In each iteration the computational cost for step 2 is $\mathcal{O}(K^2)$ operations, and the cost of step 3 is $\mathcal{O}(K N_t^2)$ operations.
\subsection{Simplifying the SINR variance calculation}\label{simplified_var_subsect}
The above analysis shows that the only step in Alg. \ref{Alg2} whose computational cost grows faster than linearly in the number of antennas is the computation of $\sigma_{f_k}$. In massive MISO systems, the resulting computational load can be significant. To reduce the required computations, we observe that when the number of antennas is large and the channels are uncorrelated, the inner product between different channels will typically be relatively small. Since the beamforming directions will typically be closely aligned with the channel vectors, the inner product between different beamforming vectors will likely be small as well. This observation suggests removing the cross terms $\mathbf{u}_j^H \mathbf{u}_k, \forall j \neq k$ in \eqref{var_rel2}. That would yield the following approximations
\begin{equation}
\begin{aligned}
\mathbf{h}_{e_k}^H \mathbf{Q}_{k} \mathbf{Q}_{k} \mathbf{h}_{e_k} &= \mathbf{h}_{e_k}^H \Bigl(\beta_k \mathbf{u}_k \mathbf{u}_k^H/\gamma_k-\sum_{j \neq k}\beta_j \mathbf{u}_j \mathbf{u}_j^H \Bigr)^2 \mathbf{h}_{e_k} \\
&\approx |\mathbf{h}_{e_k}^H \mathbf{u}_k|^2 \beta_k^2/\gamma_k^2 + \sum_{j \neq k} |\mathbf{h}_{e_k}^H \mathbf{u}_j|^2 \beta_j^2,
\end{aligned}
\end{equation}
and
\begin{equation}
\begin{aligned}
\text{tr} (\mathbf{Q}_{k}^2)&= \text{tr} \Bigl(\beta_k \mathbf{u}_k \mathbf{u}_k^H/\gamma_k-\sum_{j \neq k}\beta_j \mathbf{u}_j \mathbf{u}_j^H \Bigr)^2 \\
&\approx \text{tr} \Bigl(\beta_k^2 \mathbf{u}_k \mathbf{u}_k^H \mathbf{u}_k \mathbf{u}_k^H /\gamma_k^2+\sum_{j \neq k}\beta_j^2 \mathbf{u}_j \mathbf{u}_j^H \mathbf{u}_j \mathbf{u}_j^H \Bigr) \\
&= \beta_k^2/\gamma_k^2+\sum_{j \neq k}\beta_j^2.
\end{aligned}
\end{equation}
The numerical results presented in Section~\ref{sec_sim} indicate that these approximations result in designs that are very close in performance to those obtained from the original formulations, even when the number of antennas is quite small. Furthermore, since the terms $|\mathbf{h}_{e_k}^H \mathbf{u}_j|^2$ are already computed in the initialization step that constructs the matrix $\mathbf{A}$, these approximations reduce the computational cost of updating $\boldsymbol\sigma_f$ in step 3 of Alg.~\ref{Alg2} from $\mathcal{O}(N_t^2 K)$ to $\mathcal{O}(K^2)$.
\subsection{User rescheduling}\label{userrescheduling}
One of the fundamental characteristics of the original outage constrained beamformer design problem in \eqref{outage_min2} is that for a certain set of channel estimates the problem may be infeasible. That is, there may be no set of beamformers that can satisfy the outage constraints. Furthermore, even when the problem is feasible, the solution may be impractical in the sense that the minimum transmission power required to satisfy the outage constraints may exceed the capability of the BS. The approximations of the original formulation in \eqref{outage_min4} and \eqref{outage_min5} retain these characteristics, and the power loading problem in \eqref{outage_min6} retains them, too. Fortunately, as we now explain, for systems in which each user specifies the same value for $r$, the structure of a closely-related power loading problem provides insights into which users should be rescheduled in order for the problem in \eqref{outage_min6} to be feasible, and for the solution of the problem to be within the capabilities of the BS. The auxiliary power loading problem that we will consider is that of maximizing a common offset coefficient subject to an explicit power constraint, namely
\begin{subequations}\label{outage_min7}
\begin{align}
\max_{\substack{\beta_k},r} \quad & r \\
\text{subject to} \quad & \textstyle\sum_{k=1}^K \beta_k \leq P_t, \label{pwr_cont} \\
\quad & \mu_{k} \geq r \sigma_{f_k}, \quad \forall k \label{sinr7},
\end{align}
\end{subequations}
where $P_t$ denotes the maximum transmission power of the BS. This problem is always feasible whenever all the estimated channels are different. (The value of $r$ can be decreased until all components of \eqref{sinr7} can be satisfied using a power loading that satisfies \eqref{pwr_cont}.) However, negative values and small positive values of $r$ correspond to cases with high probability of outage. The problem in \eqref{outage_min7} can be solved using an algorithm similar to that in Alg.~\ref{Alg2}. However, at the step analogous to step 2 of Alg.~\ref{Alg2}, we need an additional equation to determine the value for $r$. That equation arises from observing that the power constraint in \eqref{pwr_cont} holds with equality at optimality, and hence, from \eqref{A_eqn} and \eqref{pwr_cont} we have that
$$ r = \frac{P_t- \boldsymbol{1}^T \mathbf{A}^{-1} \boldsymbol\sigma^2}{ \boldsymbol{1}^T\mathbf{A}^{-1} \boldsymbol\sigma_{f}},$$
where $\boldsymbol{1}$ is the vector with all elements equal to one. This equation clearly demonstrates the relationship between the power budget and the robustness. More importantly, it shows that the users that correspond to the largest elements of $\mathbf{A}^{-1} \boldsymbol\sigma^2$ are the ones that play the biggest role in constraining the extent of robustness that can be obtained. That suggests that if the optimal value of $r$ in \eqref{outage_min7} is not large enough to provide the desired robustness level, one or more of those users corresponding to large values of $\mathbf{A}^{-1} \boldsymbol\sigma^2$ should be rescheduled. (We note that the use of good user selection algorithms, e.g., \cite{Ontheoptimalityofmultiantenna}, prior to the design of the beamforming directions will reduce the need to reschedule users, but the inherent capability of the proposed power loading algorithms to perform rescheduling provides significant performance gains when the initial user selection is imperfect.)
Once the optimal value of the auxiliary problem in \eqref{outage_min7} exceeds the desired value for $r$, the power minimization problem in \eqref{outage_min6} can be solved. Since the distribution of $f_k(\mathbf{e}_{k})$ is dominated by the Gaussian terms, values of $r$ in the range of 2 to 5 would be sufficient to obtain outage probabilities consistent with the expectations of contemporary applications; see Appendix~\ref{r_value_sel}.
\subsection{Average outage}\label{mod_algo}
The design formulations that we have considered up until this point have taken the form of minimization of the transmission power subject to (an approximation of) an outage constraint on each user for the current realizations of the channels. However, as we now illustrate, the proposed design approach is quite flexible and can accommodate other notions of outage.
Let us assume that we have the optimal power loading and offset coefficient for the problem in \eqref{outage_min7}, which provide all the users with essentially the same outage probability. We will denote those values by $\{\beta_k^\star\}_{k=1}^K$ and $r^\star$. Given this solution, the goal of this section is to perturb the value of the offset coefficient for each user so as to minimize the average outage probability over the users, and to adjust the power loading accordingly. To do so, we let $\delta_{r_k}$ denote the perturbation on the $k$th user's offset coefficient; i.e., $r_k=r^\star+\delta_{r_k}$.
As discussed in Section~\ref{sect3}, in typical operating scenarios the distribution of $f_k(\mathbf{e}_k)$ can be accurately approximated by a Gaussian distribution. In that case, the outage probability for a given value of the offset coefficient $r_k$ is simply the value of the complementary cumulative distribution function (CCDF) of the standardized normal distribution, $\mathcal{N} (0, 1)$, at the value of $r_k$. If we let $g(\cdot)$ denote the CDF of the standard normal distribution, the the problem of minimizing the outage probability becomes
\begin{subequations}
\begin{align}
\max_{\substack{\beta_k, \delta_{r_k}}} \quad & \textstyle\sum_{k=1}^K g(r^\star+\delta_{r_k}) \\
\text{s.t.} \quad & \textstyle\sum_{k=1}^K \beta_k = P_t,
\end{align}
\end{subequations}
where the condition $\textstyle\sum_{k=1}^K \beta_k = P_t$ ensures that the power used after perturbation will be the same as that used by the solution to \eqref{outage_min7}. That constraint can be shown to be equivalent to the linear constraint $\boldsymbol{1}^T \mathbf{A}^{-1} (\boldsymbol\sigma_{s} \odot \boldsymbol{\delta_{r}} )=0$, where $\boldsymbol{\delta_{r}}$ is the vector containing the scalars $\delta_{r_k}$. Furthermore, the CDF $g(\cdot)$ can be well approximated by a quadratic curve; see Fig.~\ref{fig1}.
\begin{figure}
\begin{center}
\epsfysize= 2.4in
\epsffile{fig2.eps}
\caption{The CDF of the standardized normal distribution, $\mathcal{N} (0, 1)$, denoted $g(r)$, and its least squares quadratic approximation over $r \in[ 1,3]$.
}\label{fig1}
\end{center}
\end{figure}
With this approximation in place, the problem can be stated as the following convex problem in $\boldsymbol\delta_{r}$
\begin{subequations}\label{pert_eqn}
\begin{align}
\max_{\substack{\boldsymbol{\delta_{r}}}} \quad & \textstyle\sum_{k=1}^K a_0(r^\star+\delta_{r_k})^2+ a_1 (r^\star+\delta_{r_k}) +a_2\\
\text{s.t.} \quad & \boldsymbol{1}^T \mathbf{A}^{-1} (\boldsymbol\sigma_{s} \odot \boldsymbol{\delta_{r}})=0,
\end{align}
\end{subequations}
where $a_0, a_1$ and $a_2$ are the coefficients of the quadratic approximation of $g(r)$.
If we let $\mathbf{b}= (\boldsymbol{1}^T \mathbf{A}^{-1}) \odot \boldsymbol\sigma_{s}$, then using an analysis of the KKT conditions of \eqref{pert_eqn}, we can derive the dual variable $\zeta$ of the equality constraint as
$$\zeta=\frac{-(2 a_0 r^\star +a_1) \mathbf{b}^H \boldsymbol{1}}{\mathbf{b}^H \mathbf{b}},$$
and the required $\boldsymbol{\delta_{r}}$ as
$$\boldsymbol{\delta_{r}}=\frac{-(2 a_0 r^\star + a_1)\boldsymbol{1} -\zeta \mathbf{b} }{2 a_0}.$$
Accordingly, whenever we have the optimal solution $\{\beta_k^\star\}_{k=1}^K$ and $r^\star$ of the problem in \eqref{outage_min7}, we can calculate $\zeta$, and the resulting perturbations of the offset coefficient $\boldsymbol{\delta_{r}}$. The modified offset coefficient vector $\mathbf{r}$ can be updated using $r_k=r^\star+\delta_{r_k}$. The power loading $\{\beta_k\}_{k=1}^K$ is then updated by using the linear equations arising from \eqref{sinr7} holding with equality; i.e., $\boldsymbol{\beta} =\mathbf{A}^{-1} \boldsymbol{\sigma}^2+ \mathbf{A}^{-1} (\boldsymbol\sigma_{f} \odot \mathbf{r})$.
\section{Simulation results}\label{sec_sim}
In this section, we will provide three sets of numerical results. First, we will provide simulation results that show the validity of the offset-based algorithms and compare the performance of the algorithms presented here to that of zero-outage region algorithms that obtain robustness by ensuring that outage does not occur for uncertainties that lie in a given region. Specifically we will compare with the sphere bounding (SB) algorithm presented in \cite{OutageConstrained}. Second, we will provide comparisons between the performance of the offset-based power loading algorithms proposed in Section \ref{per_user_Power_Loading_algorithm}, the optimal power loading algorithm in \cite{Coordinateupdate}, and the perturbation-based power loading algorithm that seeks to minimize the averaged outage, which was presented in Section~\ref{mod_algo}. In the third set of simulation results, we will demonstrate the performance gains that can be obtained by using the user rescheduling and the power saving described in Section~\ref{userrescheduling}. We will also show the validity of the low-complexity approximations presented in Section~\ref{sec_approxs}.
For the initial simulation setup, we will we consider a downlink system in which a BS serves three single-antenna users. We will assume that the BS has four antennas, and the three users are randomly distributed within a radius of 3.2km. The large scale fading is described by a path-loss exponent of 3.52 and log-normal shadow fading with 8dB standard deviation, and the small scale fading is modelled using the standard i.i.d. Rayleigh model. The channel estimation error is assumed to be zero-mean and Gaussian with covariance $\sigma_{e_k}^2 \mathbf{I}$. The receiver noise level is -90dBm, and the SINR target is set to 6dB. A simple channel-strength user selection technique is employed, where users are served only if $ 100\|\mathbf{h}_{e_k}\|^2/ \sigma_k^2 \geq \gamma_k$, where we consider 100 here as the implicit total power constraint.
Each of the algorithms that we consider involves a choice of a robustness measure. For the algorithms provided in this paper the robustness measure is the value of the offset coefficient $r_k$. For the sphere bounding algorithm in \cite{OutageConstrained} it is the size of the zero-outage region, and for the power loading algorithm in \cite{Coordinateupdate} it is directly the outage probability. To plot the performance curves, we randomly generate a set of channel realizations and provide the BS with estimates of those channels. Each algorithm is then used to design a set of beamformers that should provide the specified robustness. Using those beamformers we determine whether or not any user in the system with the actual channel realizations is in outage, and we calculate the corresponding transmission power. By repeating this experiment over thousands of channel realizations, we can plot the average outage probability over the users versus the average transmission power for the different algorithms when these algorithms provide a viable solution; by which we mean a solution that satisfies the constraints using a transmitted power that is less than 100. In fairness to all methods, the average is taken over those channel realizations for which all methods produce a viable solution.
In Fig.~\ref{sim1}, assuming $\sigma_{e_k}=0.1$, we plot the average outage probability versus the average total transmitted power for the proposed robust beamforming algorithms in \eqref{outage_min4}, \eqref{outage_min5}, Alg. \ref{Alg1}, and that of a system with the constant-offset directions described in Section.~\ref{sec_org_offset_max} and the suggested power loading in Section \ref{per_user_Power_Loading_algorithm}. As benchmarks, we plot the performance of the SB algorithm \cite{OutageConstrained}, and that of a system that employs the ZF directions combined with the power loading in Section \ref{per_user_Power_Loading_algorithm}. In Fig.~\ref{sim2}, we repeat the experiment for $\sigma_{e_k}=0.05$. We observe that the performance gap between the proposed algorithms becomes smaller when the error variance decreases, which justifies the validity of the approximations for small error size. We also note that the performance of the low-complexity robust beamforming algorithm in Alg. \ref{Alg1} is very close to that of the original formulation in \eqref{outage_min4}, and that both algorithms provide better performance than the SB algorithm (which incurs a significantly larger computational load). The relative performance of the ZF-based algorithm with the proposed power loading algorithm in Section~\ref{per_user_Power_Loading_algorithm} depends on the uncertainty size, where comparatively better performance results are obtained when the uncertainty size is larger. That observation means that while both the offset-based beamforming directions and power loading contribute to the excellent performance for small uncertainty size, as the uncertainty size increases the role of the offset-based power loading becomes more significant. The performance of the combination of the original constant-offset directions in Section~\ref{sec_org_offset_max} with the suggested power loading in Section~\ref{per_user_Power_Loading_algorithm} is not quite as good as that of the other offset-based approaches. However, decoupling the design of the beamforming directions and that of the power loading significantly reduces the computational cost (see Section~\ref{sec_approxs}), and greatly increases the flexibility of the design, as explained in Section~\ref{per_user_Power_Loading_algorithm}.
\begin{figure}
\begin{center}
\epsfysize= 2.8in
\epsffile{sim1.eps}
\caption{The average transmitted power against the outage probability for a system with 3 users, 4 BS antennas, $\gamma$ = 6dB, and $\sigma_{e_k}=0.1$.
}\label{sim1}
\end{center}
\end{figure}
\begin{figure}
\begin{center}
\epsfysize= 2.8in
\epsffile{sim2.eps}
\caption{The average transmitted power against the outage probability for a system with 3 users, 4 BS antennas, $\gamma$ = 6dB, and $\sigma_{e_k}=0.05$.
}\label{sim2}
\end{center}
\end{figure}
The second set of simulation results examines the performance gap between the proposed power loading algorithms in Section \ref{per_user_Power_Loading_algorithm}, and the power loading algorithm in \cite{Coordinateupdate} when the constant-offset directions are chosen; see Section.~\ref{sec_org_offset_max}.
In Fig.~\ref{sim3}, we plot the average outage probability versus the average transmitted power for the power loading algorithm in \cite{Coordinateupdate}, the power loading in Alg. \ref{Alg2}, and the modified power loading in Section~\ref{mod_algo}. (For the latter case, the quadratic approximation used in \eqref{pert_eqn} is the least-squares approximation in Fig.~\ref{fig1}.) While the algorithm in \cite{Coordinateupdate} is optimal in terms of the power required to achieve the specified outage probabilities for each user and for each channel realization, the proposed algorithms provide better average outage probability. This performance is achieved while requiring no more than five iterations in the power loading algorithm in Alg. \ref{Alg2}. As one would expect, the modified power loading algorithm in Section~\ref{mod_algo} provides an even lower average outage probability than that obtained by Alg. \ref{Alg2}.
\begin{figure}
\begin{center}
\epsfysize= 2.8in
\epsffile{sim3.eps}
\caption{The average transmitted power against the outage probability for a system with 3 users, 4 BS antennas, $\gamma$ = 6dB, and $\sigma_{e_k}=0.1$.
}\label{sim3}
\end{center}
\end{figure}
To assess the performance gains that result from the power control capabilities of the proposed power loading algorithms, we plot the outage probability of the problem in \eqref{outage_min7} with the constant-offset directions in Section~\ref{sec_org_offset_max} versus the number of antennas. In this case, we set the total power constraint $P_t=1$, and the number of users to six. We also plot the corresponding results when the approximations for obtaining the directions in Section~\ref{sec_approxs}, and those for obtaining the power loading in Section~\ref{simplified_var_subsect} are applied. In addition, we plot the performance of the proposed user rescheduling scheme (Alg. Sect. \ref{userrescheduling} (a)) and the user rescheduling when combined with the power saving (Alg. Sect. \ref{userrescheduling} (b)). We applied user rescheduling whenever the resulting offset $r$ in \eqref{outage_min7} is smaller than two, and the rescheduled user(s) are considered to be in outage. For the power saving algorithm we upper bound $r$ by 5. We observe from Fig.~\ref{sim4} that the proposed approximations provide almost the same outage performance over the whole range of antenna numbers. We also observe that the user rescheduling technique greatly enhances the outage performance, especially when the number of antennas is relatively low. (When the number of antennas is low, there is a greater probability of the channels not being sufficiently orthogonal.) Fig.~\ref{sim4} shows that the power saving algorithm (Alg. Sect.~\ref{userrescheduling} (b)) provides essentially the same performance as the regular algorithms, but significant power can be saved; the average actual transmitted powers used for that algorithm when the number of antennas are $[20,25,\cdots,60]$ are $[0.74, 0.71, 0.67, 0.65, 0.62, 0.59, 0.56, 0.54, 0.52]$ all of which are significantly smaller than the total power constraint $P_t=1$.
\begin{figure}
\begin{center}
\epsfysize= 2.8in
\epsffile{sim4.eps}
\caption{The outage probability versus the number of antennas for a system with 6 users, $\gamma$ = 6dB, and $\sigma_{e_k}=0.1$.
}\label{sim4}
\end{center}
\end{figure}
\section{Conclusion}
In this paper, a new offset-based approach is proposed for robust downlink beamforming. The approach is based on rewriting the SINR outage constraint as a non-negativity constraint on an indefinite quadratic function of the error in the base station's model of the channel. That non-negativity is then approximated by an offset-based constraint in which the mean of the function is required to be larger than a specific multiple of its standard deviation. This approach enabled the formulation of the robust beamforming design problem as a problem that can be transformed into a convex problem through the process of semidefinite relaxation (SDR). The computational complexity of the SDR problem can be further reduced when the uncertainty size is small, allowing for an iterative closed-form solution. When the beamforming directions are defined in advance, the offset-based approach generates a power loading algorithm that provides excellent performance and unique power control capabilities, while incurring only a small computational cost. The demonstrated performance gains, and the significant differences in computational cost exemplify the advantages of using the offset-based approach instead of the sphere bounding approach. Within the suite of algorithms generated by the offset based approach, the separation of the design into the constant-offset directions (Section~\ref{sec_org_offset_max}) and the proposed power loading (Section~\ref{per_user_Power_Loading_algorithm}) provides a compelling balance between performance, computational cost and design flexibility.
\appendices
\section{Choice of $r_k$} \label{r_value_sel}
From Cantelli's Inequality, which is sometimes referred to as the one-sided Chebyshev inequality, we know that for any random variable $\mathbf{X}$ with mean $\mu_x$ and variance $\sigma_x$,
$$\text{Prob}(\mathbf{X}-\mu_x \leq -r \sigma_x) \leq \frac{1}{1+r^2}.$$
Therefore, if we ensure that $\mu_x \geq r \sigma_x$ then $\text{Prob}(\mathbf{X} \leq 0) \leq \frac{1}{1+r^2}$. Accordingly, if we set $r_k=\sqrt{1/\delta_k-1}$, then the approximation in \eqref{offset_constr} is ``safe", in the sense that any solution to the corresponding problem in \eqref{outage_min2} is guaranteed to satisfy the original outage constraints in \eqref{outage_min}. However, for the distributions that typically arise in downlink beamforming Cantelli's Inequality is quite loose and the resulting beamformer design is quite conservative. Indeed, as we explained in Section~\ref{sect3}, for small uncertainties the distribution of $f_k(\mathbf{e}_k)$ is close to being Gaussian. If it were in fact Gaussian, then if the beamformers are designed such that $ \mu_{f_k} \geq r_k \sigma_{f_k}$ then the outage probability would be $Q(r_k)=\frac{1}{2} \text{erfc}(\frac{r_k}{\sqrt{2}})$, where $\text{erfc}(\cdot)$ is the complementary error function.
\section{Mean and variance derivations} \label{mean_var_der}
A Gaussian random variable $\mathbf{e}_k \backsim \mathcal{CN} (\mathbf{m}_k, \mathbf{C}_k)$ can be represented as $\mathbf{e}_k =\mathbf{m}_k + \mathbf{C}_k^{1/2} \hat{\mathbf{e}}_k$, where $\hat{\mathbf{e}}_k \backsim \mathcal{CN} (0, \mathbf{I})$. Using that representation we can write
\begin{equation}
\begin{aligned}
\mu_{f_k}& = \mathbb{E} \{f_k(\mathbf{e}_k)\} \\
&=(\mathbf{h}_{e_k}+\mathbf{m}_k)^H \mathbf{Q}_k (\mathbf{h}_{e_k}+\mathbf{m}_k) - \sigma_k^2 \\
&\quad +\mathbb{E} \{\hat{\mathbf{e}}_{k}^H \mathbf{C}_k^{1/2} \mathbf{Q}_{k}\mathbf{C}_k^{1/2} \hat{\mathbf{e}}_{k}\}\\
&= (\mathbf{h}_{e_k}+\mathbf{m}_k)^H \mathbf{Q}_k (\mathbf{h}_{e_k}+\mathbf{m}_k) - \sigma_k^2 + \mathbf{w}_k^H \mathbf{C}_k \mathbf{w}_k /\gamma_k \\
&\quad -\sum_{j \neq k} \mathbf{w}_j^H \mathbf{C}_k \mathbf{w}_j^H.
\end{aligned}
\end{equation}
The variance can be expressed as
\begin{equation}
\begin{aligned}
\sigma_{f_k}^2&=\text{var}\{f_k(\mathbf{e}_{k}) \} \\
&=\text{var}\{ 2 \text{Re}( \hat{\mathbf{e}}_{k}^H \mathbf{C}_k^{1/2} \mathbf{Q}_{k} (\mathbf{h}_{e_k}+\mathbf{m}_k )) \\
&\qquad + \hat{\mathbf{e}}_{k}^H \mathbf{C}_k^{1/2} \mathbf{Q}_{k}\mathbf{C}_k^{1/2} \hat{\mathbf{e}}_{k} \bigr\} \\
&=2 (\mathbf{h}_{e_k}+\mathbf{m}_k)^H \mathbf{C}_k^{1/2} \mathbf{Q}_{k}^2 \mathbf{C}_k^{1/2} (\mathbf{h}_{e_k}+\mathbf{m}_k) \\
&\qquad +\text{var}\{\hat{\mathbf{e}}_{k}^H \mathbf{C}_k^{1/2} \mathbf{Q}_{k}\mathbf{C}_k^{1/2} \hat{\mathbf{e}}_{k}\}+0^* \\
&=2 (\mathbf{h}_{e_k}+\mathbf{m}_k)^H \mathbf{C}_k^{1/2} \mathbf{Q}_{k}^2 \mathbf{C}_k^{1/2} (\mathbf{h}_{e_k}+\mathbf{m}_k) \\
&\qquad + \text{tr} (\mathbf{C}_k^{1/2} \mathbf{Q}_{k} \mathbf{C}_k^{1/2} )^2,
\end{aligned}
\end{equation}
where $\mathbf{[A]}_{ij}$ denotes the $(i,j)$th element of the matrix $\mathbf{A}$, and tr denotes the trace function. At the point marked with the asterisk we have used the fact that the expectation of the cross terms is equal to zero. This is true because $\mathbb{E} \{2 \text{Re}( \mathbf{e}_{k}^H \mathbf{Q}_{k} (\mathbf{h}_{e_k}+\mathbf{m}_k )) (\hat{\mathbf{e}}_{k}^H \mathbf{C}_k^{1/2} \mathbf{Q}_{k}\mathbf{C}_k^{1/2} \hat{\mathbf{e}}_{k}) \}$ consists of terms containing either similar or different components from the $\hat{\mathbf{e}}_{k} $ vector. Since $\hat{\mathbf{e}}_{k} $ has a zero mean, all terms with different indices will have a zero mean, while terms of similar indexes will take the form of a complex Gaussian raised to the power of three, which also has zero mean.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 9,973 |
Лой — примітивна вузька лопата, яка була поширена в Ірландії.
Лой (Loy) — місто в Ірландії, у графстві Тайрон.
Лой (Loy) — рід черевоногих молюсків.
Лой або Лойс () — особове ім'я.
Лой — прізвище.
Див. також
USS Loy (DE-160) — американський ескортний міноносець
Бухта Св. Лоя (St Loy's Cove) — бухта в Корнуолі (Велика Британія)
Лій — топлений жир рогатої худоби | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 5,406 |
Q: Change PyDev way of indenting function argument I'd like to use PEP8 accepted format:
# Hanging indents should add a level.
foo = long_function_name(
var_one, var_two,
var_three, var_four)
How to configure PyDev so when I open bracket and press Enter it moves caret to next line with increased indentation by one level? Currently after hitting Enter caret is set just below openning bracket.
Is it configurable at all?
A: You can configure that at:
Preferences > PyDev > editor > Typing > After '(' indent to its level (indents by tabs if unchecked).
A: You'll need to utilize \ but before you do that you need to put at least one after calling the function.
def superlongfunction(one,\
two):
return(one+two)
foo = superlongfunction(5,\
2)
Edit: That's what I'm used to do for visualization. I tried without \ and it worked, but you must have at least one right after calling the function.
foo = superlongfunction(5,
2)
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 7,262 |
\section{Introduction}
\subsection{Background}
\hskip 0.6cm
Let $G$ be a connected graph with vertex set $V(G)$ and edge set $E(G)$.
Let $d_{G}(u)$ be the degree of vertex $u$ in $G$.
The distance between vertex $u$ and vertex $v$ is denoted by $d_{G}(u,v)$.
If we replace each edge of the graph $G$ with a unit resistor and regard the graph $G$ as an electrical network $N$, then we define the effective resistance of vertex $u$ and vertex $v$ in the electrical network $N$ as the resistance distance between vertex $u$ and vertex $v$ in the graph $G$, and denoted by $\Omega_{G}(u,v)$.
In this paper, all notations and terminologies used but not defined can refer to Bondy and Murty \cite{bond2008}.
The Wiener index is one of the oldest and most studied topological index from application and theoretical viewpoints. As a extension of the Wiener index, The Kirchhoff index is an important measure which contains more information than the Wiener index and plays an essential role in the research of QSAR and QSPR.
The Wiener index \cite{wien1947} of graph $G$ is defined as
$W(G)=\sum\limits_{\{u,v\}\subseteq V(G)}d_{G}(u,v)$,
replacing distance with resistance distance in the definition of Wiener index, we can obtain the Kirchhoff index, which is defined as \cite{klra1993}
$$Kf(G)=\sum\limits_{\{u,v\}\subseteq V(G)}\Omega_{G}(u,v).$$
Some mathematical and physical interpretations of Kirchhoff index can be found in \cite{kldj1997,klzh1995}.
The extremal Kirchhoff index had been considered on unicyclic graphs \cite{yaji2008}, fully loaded unicyclic graphs \cite{gdch2009}, cacti \cite{whwa2010}, graphs with given cut edges \cite{deng2010}, graphs with a given vertex bipartiteness \cite{lipa2016}, random polyphenyl and spiro chains \cite{hkde2014},
linear hexagonal (cylinder) chain \cite{huli2020}, generalized phenylenes \cite{lilz2020,zhli2019}, M\"{o}bius/cylinder octagonal chain \cite{liwl2022},
linear phenylenes \cite{peli2017}, connected (molecular) graphs \cite{zhtr2009}, and so on.
Some molecular descriptors of polycyclic chains had been considered for many years.
Such as Wiener index \cite{chli2022,cayz2020}, Kirchhoff index \cite{maqi2022,suya2023,yakl2014,yasu2022,yawa2019,zhll2022}, Tutte polynomials \cite{chgu2019}, Merrified-Simmons index \cite{cayz2017}, Kekule structures \cite{tztd2019}, forcing spectrum \cite{zhji2021}, $k$-matching \cite{cazh2008}, Hosoya index \cite{qizh2012}, and so on.
Let $Q_{h}$ be the linear quadrilateral chain with $h$ squares and $S_{i}$ $(1\leq i\leq h)$ the $i$-th square of $Q_{h}$.
Then the $k$-polycyclic chain $P_{h}$ can be obtained from $Q_{h}$ by adding $k-4$ vertices to $S_{i}$ $(1\leq i\leq h)$ by adding $0$ (resp. $1,2,\cdots,k-4$) vertices to the top edge of $S_{i}$ $(1\leq i\leq h)$ and the remaining vertices to the bottom edge of $S_{i}$ $(1\leq i\leq h)$. In Figure \ref{fig-11}, either $D_{5}$ or $L_{5}$ is a special $P_{5}$, $Z_{6}$ is a special $P_{6}$.
For convenience, we suppose that we add $\lceil \frac{k-4}{2}\rceil$ vertices to the top edges of $S_{1}$ and $S_{h}$, $ \lfloor \frac{k-4}{2}\rfloor$ vertices to the bottom edges of $S_{1}$ and $S_{h}$,
and for the $S_{i+1}$ $(1\leq i\leq h-2)$, we give a number $w_{i}=0$ (resp. $1,2,\cdots,k-4$) to the $k$-polygon if the $k$-polygon is obtained by adding $w_{i}$ vertices to the top edge of $S_{i+1}$. Then we can use a $(h-2)$-vector $w=(w_{1},w_{2},\cdots,w_{h-2})$ to denote the $k$-polycyclic chain, where $w_{i}\in \{0,1,\cdots,k-4\}$.
Let $P_{h}(w)$ (or simply $P(w)$) be the $k$-polycyclic chain with $h$ $k$-polygons and $w=(w_{1},w_{2},\cdots,w_{h-2})$ be a $(h-2)$-tuple of $0,1,\cdots,k-4$.
The $k$-polycyclic chain $P(\underbrace{0,0,\cdots,0}_{h-2})$ or $P(\underbrace{k-4,k-4,\cdots,k-4}_{h-2})$ is called a helicene $k$-polycyclic chain, where $P(\underbrace{0,0,\cdots,0}_{h-2})\cong P(\underbrace{k-4,k-4,\cdots,k-4}_{h-2})$, and denoted by $D_{h}$.
If $k\geq 6$ is even, the $k$-polycyclic chain $P(\underbrace{\frac{k-4}{2},\frac{k-4}{2},\cdots,\frac{k-4}{2}}_{h-2})$ is called a linear $k$-polycyclic chain, and denoted by $L_{h}$.
If $k\geq 5$ is odd, then $k$-polycyclic chain $P(\underbrace{\lfloor\frac{k-4}{2}\rfloor,\lceil\frac{k-4}{2}\rceil,\lfloor\frac{k-4}{2}\rfloor,
\lceil\frac{k-4}{2}\rceil\cdots}_{h-2})$ or $P(\underbrace{\lceil\frac{k-4}{2}\rceil,\lfloor\frac{k-4}{2}\rfloor,\lceil\frac{k-4}{2}\rceil,
\lfloor\frac{k-4}{2}\rfloor\cdots}_{h-2})$ is called a zigzag chain, denoted by $Z_{h}$, where $P(\underbrace{\lfloor\frac{k-4}{2}\rfloor,\lceil\frac{k-4}{2}\rceil,\lfloor\frac{k-4}{2}\rfloor,
\lceil\frac{k-4}{2}\rceil\cdots}_{h-2})\cong P(\underbrace{\lceil\frac{k-4}{2}\rceil,\lfloor\frac{k-4}{2}\rfloor,\lceil\frac{k-4}{2}\rceil,
\lfloor\frac{k-4}{2}\rfloor\cdots}_{h-2})$. Figure \ref{fig-11} gives $D_{5}$ with $k=6$, $L_{5}$ with $k=6$ and $Z_{6}$ with $k=7$.
\begin{figure}[ht!]
\centering
\scalebox{.14}[.14]{\includegraphics{f-11.png}}
\caption{$D_{5}$ with $k=6$, $L_{5}$ with $k=6$ and $Z_{6}$ with $k=7$.}
\label{fig-11}
\end{figure}
\subsection{Main results}
\hskip 0.6cm
Our main results are shown as follows.
\begin{theorem}\label{t1-1}
Let $\mathcal{P}_{h}$ be the set of $k$-polycyclic chains with $h$ $k$-polygons \rm{($k\geq 5$)}. Then for any $G\in \mathcal{P}_{h}$, we have
$$Kf(G)\geq Kf(P(\underbrace{0,0,\cdots,0}_{h-2})),$$
with equality if and only if $G\cong D_{h}$.
\end{theorem}
\begin{theorem}\label{t1-2}
Let $\mathcal{P}_{h}$ be the set of $k$-polycyclic chains with $h$ $k$-polygons \rm{($k\geq 5$)}. Then for any $G\in \mathcal{P}_{h}$, we have
$$Kf(G)\leq Kf(P(\underbrace{\lfloor\frac{k-4}{2}\rfloor,\lceil\frac{k-4}{2}\rceil,
\lfloor\frac{k-4}{2}\rfloor,\lceil\frac{k-4}{2}\rceil,\cdots}_{h-2})),$$
with equality if and only if
$
G\cong
\begin{cases}
L_{h} &,\ if\ n\ is\ even\\[3mm]
Z_{h} &,\ if \ n\ is\ odd
\end{cases}
$.
\end{theorem}
Let $k=5,6,8$. Then by Theorems \ref{t1-1} and \ref{t1-2}, we have the following corollaries immediately, which is main results of \cite{suya2023,yakl2014,yasu2022,maqi2022}.
\begin{corollary}\label{c1-6}{\rm\cite{suya2023}}
Among all pentagonal chains with given the number of pentagons, the helicene pentagonal chain (resp. zigzag pentagonal chain) has the minimum (resp. maximum) Kirchhoff index.
\end{corollary}
\begin{corollary}\label{c1-3}{\rm\cite{yasu2022}}
Among all hexagonal chains with given the number of hexagons, the helicene hexagonal chain has the minimum Kirchhoff index.
\end{corollary}
\begin{corollary}\label{c1-4}{\rm\cite{yakl2014}}
Among all hexagonal chains with given the number of hexagons, the linear hexagonal chain has the maximum Kirchhoff index.
\end{corollary}
\begin{corollary}\label{c1-5}{\rm\cite{maqi2022}}
Among all octagonal chains with given the number of octagons, the helicene octagonal chain (resp. linear octagonal chain) has the minimum (resp. maximum) Kirchhoff index.
\end{corollary}
\subsection{Preliminaries}
\hskip 0.6cm
In the following, we introduce some important rules and transformations in an electrical network.
The first is the series connection rule and parallel connection rule.
\textbf{Parallel Connection Rule:}
If $h$ resistors are connected in parallel, then we replace them by a single resistor whose reciprocal of resistance is the sum of $h$ reciprocal of resistances (see Figure \ref{fig-12} (a)).
\textbf{Series Connection Rule:}
If $h$ resistors are connected in series, then we replace them by a single resistor whose resistance is the sum of $h$ resistances (see Figure \ref{fig-12} (b)).
\begin{figure}[ht!]
\centering
\scalebox{.18}[.18]{\includegraphics{f-12.png}}
\caption{Illustrations of parallel connection rule and series connection rule.}
\label{fig-12}
\end{figure}
Now we introduce a transformation between a resistor network $\Delta$ and a resistor network $Y$.
Let $N_{1}$, $N_{2}$ be resistor networks and $V^{*}\subseteq V(N_{1})\cap V(N_{2})$. Then we call $N_{1}$, $N_{2}$ are $V^{*}$-equivalent if $\Omega_{N_{1}}(u,v)=\Omega_{N_{2}}(u,v)$ for any $u,v\in V^{*}$. By $V^{*}$-equivalent, series and parallel connection rule, we have
\begin{proposition}\label{p13-3}{\rm\cite{kenn1899}}
Let $\Delta$ and $Y$ be two resistor networks (see Figure \ref{fig-13}). If $\Delta$ and $Y$ satisfy the following equations:
$$R_{io}=\frac{R_{ij}R_{ik}}{R_{ij}+R_{ik}+R_{jk}},\ R_{jo}=\frac{R_{ij}R_{jk}}{R_{ij}+R_{ik}+R_{jk}},\ R_{ko}=\frac{R_{ik}R_{jk}}{R_{ij}+R_{ik}+R_{jk}},$$
then $\Delta$ and $Y$ are $\{i,j,k\}$-equivalent.
\end{proposition}
\begin{definition}\label{d13-3}{\rm\cite{kenn1899}}{\rm($\mathbf{\Delta-Y}$ \textbf{Transformation})}
Let $\Delta$ and $Y$ be two resistor networks (see Figure \ref{fig-13}).
If $\Delta$ and $Y$ satisfy $\{i,j,k\}$-equivalent, then we can transform $\Delta$ to $Y$,
and call it a $\Delta-Y$ transformation.
\end{definition}
Clearly, a $\Delta-Y$ transformation is a technique that change a resistor network $\Delta$ to another equivalent resistor network $Y$.
\begin{figure}[ht!]
\centering
\scalebox{.15}[.15]{\includegraphics{f-13.png}}
\caption{The graph of $\Delta-Y$ Transformation.}
\label{fig-13}
\end{figure}
Now we introduce the definition of $S,T$-isomers in organic chemistry.
\begin{definition}\label{p13-4}{\rm\cite{pola1982}}{\rm($\mathbf{S,T}$-\textbf{isomers})}
Let $N_{1}$ and $N_{2}$ be two vertex-disjoint graphs, $u,v\in V(N_{1})$ and $u\neq v$, $x,y\in V(N_{2})$ and $x\neq y$. Let $S$ be the graph obtained from $N_{1}$ and $N_{2}$ by connecting $u$ with $x$, and $v$ with $y$, $T$ be the graph obtained from $N_{1}$ and $N_{2}$ by connecting $u$ with $y$, and $v$ with $x$. Then we call $S$ and $T$ are $S,T$-isomers.
\end{definition}
Figure \ref{fig-14} gives an illustration of $S,T$-isomers and a pair of hexagonal chains as $S,T$-isomers.
Let $\Omega_{G}(u)=\sum\limits_{v\in V(G)\setminus \{u\}}\Omega_{G}(u,v)$.
\begin{figure}[ht!]
\centering
\scalebox{.16}[.16]{\includegraphics{f-14.png}}
\caption{(a) $S,T$-isomers,\hspace{0.25cm} (b) A pair of hexagonal chains as $S,T$-isomers.}
\label{fig-14}
\end{figure}
\begin{lemma}\label{l13-5}{\rm\cite{yakl2014}}
Let $S,T,N_{1},N_{2},u,v,x,y$ be defined as in Figure \ref{fig-14}(a). Then
$$Kf(S)-Kf(T)=\frac{(\Omega_{N_{1}}(u)-\Omega_{N_{1}}(v))(\Omega_{N_{2}}(y)-\Omega_{N_{2}}(x))}
{\Omega_{N_{1}}(u,v)+\Omega_{N_{2}}(x,y)+2}.$$
\end{lemma}
Finally, we introduce some qualities about the resistance distance, especially, the triangular inequality.
\begin{lemma}\label{l13-6}{\rm\cite{klra1993}}
The resistance function on a graph is a distance function. Thus for any vertices $a,b,x \in V(G)$, we have
\rm{(i)} $\Omega_{G}(b,a)\geq 0$,
\rm{(ii)} $\Omega_{G}(a,b)=0$ if and only if $a=b$,
\rm{(iii)} $\Omega_{G}(a,b)=\Omega_{G}(b,a)$,
\rm{(iv)} $\Omega_{G}(a,x)+\Omega_{G}(x,b)\geq \Omega_{G}(a,b)$.
\end{lemma}
\section{Proof of Theorems \ref{t1-1} and \ref{t1-2}}
\hskip 0.6cm
Let $P(w)$ be a weighted $k$-polycyclic chain with $h$ $k$-polygons and $w=(w_{1},w_{2},\cdots,w_{h-2})$, where $0\leq w_{i}\leq k-4$ for $1\leq i\leq h-2$. Suppose the $k$-polygons in $k$-polycyclic chain are $C_{1},C_{2},\cdots,C_{h}$ in order. Let the top (resp. bottom) common vertices of $C_{i}$ and
$C_{i+1}$ are $a_{i}$ (resp. $b_{i}$) for $i=1,2,\cdots,h-1$.
\begin{figure}[ht!]
\centering
\scalebox{.16}[.16]{\includegraphics{f-21.png}}
\caption{Illustration of the transformation from $P(w)$ to $G_{h-1}$ in Lemma \ref{l2-1}.}
\label{fig-21}
\end{figure}
\begin{lemma}\label{l2-1}
Let $P(w)$ be a weighted $k$-polycyclic chain $($$k\geq 5$$)$ and the weight of edges in $C_{h}$ is $1$. Let $u,v\in V(C_{h})$ with $d_{P(w)}(u)=d_{P(w)}(v)=2$, $ua_{h-1}\in E(P(w))$ and $uv\in E(P(w))$. Then for any $z\in V(C_{1})\setminus \{a_{1},b_{1}\}$, we have $\Omega_{P(w)}(z,u)<\Omega_{P(w)}(z,v)$.
\end{lemma}
\begin{proof}
Firstly, we suppose that $z$ is the vertex adjacent to $a_{1}$ in $C_{1}$.
Now we show $\Omega_{P(w)}(z,u)<\Omega_{P(w)}(z,v)$.
Step 1: Replace the path with length $k-1$ from $z$ to $b_{1}$ (do not pass $a_{1}$) by an edge $zb_{1}$ with weight $k-1$, we can obtain the graph as Figure \ref{fig-21} (b).
Step 2: Translate the $\Delta$-network $za_{1}b_{1}$ to a $Y$-network with center $z_{1}$, we can obtain the graph $G_{1}$ as Figure \ref{fig-21} (c).
Step 3: Repeat steps $1$ and $2$, we can obtain the graph $G_{2}$ as Figure \ref{fig-21} (d).
$\vdots$
Step $h$: Repeat steps $1$ and $2$, we can finally obtain the graph $G_{h-1}$ as Figure \ref{fig-21} (e).
Let the weight of $z_{h-1}a_{h-1}$, $z_{h-1}b_{h-1}$ in $G_{h-1}$ be $\theta_{1}$ and $\theta_{2}$, respectively. Since the weight of $a_{h-1}b_{h-1}$ is $1$, by the Proposition \ref{p13-3}, it is easy to know that $0<\theta_{1}<1$ and $0<\theta_{2}<1$.
By the series connection rule and parallel connection rule, we have
$\frac{1}{\Omega_{G_{h-1}}(z_{h-1},u)}=\frac{1}{\theta_{1}+1}+\frac{1}{\theta_{2}+k-2}$,
$\frac{1}{\Omega_{G_{h-1}}(z_{h-1},v)}=\frac{1}{\theta_{1}+2}+\frac{1}{\theta_{2}+k-3}$,
and
$$\Omega_{P(w)}(z,u)=\Omega_{G_{h-1}}(z,u)=\Omega_{G_{h-1}}(z,z_{h-1})+\frac{(\theta_{1}+1)
(\theta_{2}+k-2)}{\theta_{1}+\theta_{2}+k-1},$$
$$\Omega_{P(w)}(z,v)=\Omega_{G_{h-1}}(z,v)=\Omega_{G_{h-1}}(z,z_{h-1})+\frac{(\theta_{1}+2)
(\theta_{2}+k-3)}{\theta_{1}+\theta_{2}+k-1}.$$
Thus
$$\Omega_{P(w)}(z,u)-\Omega_{P(w)}(z,v)=\frac{\theta_{1}-
\theta_{2}-k+4}{\theta_{1}+\theta_{2}+k-1}<0.$$
For any other vertex $z'\in V(C_{1})\setminus \{a_{1},b_{1}\}$, we can prove similarly.
This completes the proof.
\end{proof}
With the similar proof, we can strengthen the conclusion of Lemma \ref{l2-1}.
\begin{lemma}\label{l2-2}
Let $P(w)$ be a weighted $k$-polycyclic chain $($$k\geq 5$$)$ and the weight of edges in $C_{h}$ is $1$. Let $u,v\in V(C_{h})$ with $d_{P(w)}(u)=d_{P(w)}(v)=2$, $uv\in E(P(w))$, $d_{P(w)}(a_{h-1},u)<d_{P(w)}(a_{h-1},v)$ or $d_{P(w)}(b_{h-1},u)<d_{P(w)}(b_{h-1},v)$. Then for any $z\in V(C_{1})\setminus \{a_{1},b_{1}\}$, we have $\Omega_{P(w)}(z,u)<\Omega_{P(w)}(z,v)$.
\end{lemma}
\begin{lemma}\label{l2-3}
Let $P(w)$ be a $k$-polycyclic chain \rm{($k\geq 5$)}. Let $u,v\in V(C_{h})$ with $d_{P(w)}(u)=d_{P(w)}(v)=2$, $ua_{h-1}\in E(P(w))$ and $uv\in E(P(w))$. Then we have $\Omega_{P(w)}(u)<\Omega_{P(w)}(v)$.
\end{lemma}
\begin{proof}
We complete the proof by the following three cases.
{\bf Case 1}. $z\in V(C_{1})\setminus \{a_{1},b_{1}\}$.
By Lemma \ref{l2-1}, we have $\Omega_{P(w)}(z,u)<\Omega_{P(w)}(z,v)$.
{\bf Case 2}. $z\in V(C_{i})$ for $2\leq i\leq h-1$.
By the series connection rule and parallel connection rule, we can simply $P(w)$ to a weighted $k$-polycyclic chain which consists of $k$-polygons $C_{i},C_{i+1},\cdots,C_{h}$ such that the weight of edge $a_{i-1}b_{i-1}$ is less that $1$ and the weight of all other edges are $1$.
They by Lemma \ref{l2-1}, we have $\Omega_{P(w)}(z,u)<\Omega_{P(w)}(z,v)$.
{\bf Case 3}. $z\in V(C_{h})$.
By the series connection rule and parallel connection rule, we can simply $P(w)$ to a weighted $k$-polygons such that the weight of edge $a_{h-1}b_{h-1}$ is $r(<1)$ and the weight of all other edges are $1$. Then by
$\sum\limits_{z\in V(C_{h})}\Omega_{P(w)}(z,u)=\Omega_{P(w)}(b_{h-1},u)+\Omega_{P(w)}(a_{h-1},u)+\sum\limits_{z\in V(C_{h})\setminus\{a_{h-1},b_{h-1} \}}\Omega_{P(w)}(z,u)$
\hspace{0.10cm} and
$\sum\limits_{z\in V(C_{h})}\Omega_{P(w)}(z,v)=\Omega_{P(w)}(a_{h-1},v)+\Omega_{P(w)}(u,v)+\sum\limits_{z\in V(C_{h})\setminus\{a_{h-1},u \}}\Omega_{P(w)}(z,v)$, we have
$$\sum_{z\in V(C_{h})}\Omega_{P(w)}(z,u)=\frac{(r+1)(k-2)}{r+k-1}+\frac{r+k-2}{r+k-1}+\sum_{i=1}^{k-3}\frac{i(r+k-i-1)}{r+k-1}
,$$
$$\sum_{z\in V(C_{h})}\Omega_{P(w)}(z,v)=\frac{2(r+k-3)}{r+k-1}+\frac{r+k-2}{r+k-1}+\sum_{i=1}^{k-3}\frac{i(r+k-i-1)}{r+k-1}
.$$
Thus
$$\sum_{z\in V(C_{h})}\Omega_{P(w)}(z,u)-\sum_{z\in V(C_{h})}\Omega_{P(w)}(z,v)=\frac{(k-4)(r-1)}{r+k-1}<0.$$
This completes the proof.
\end{proof}
With the similar proof, we can strengthen the conclusion of Lemma \ref{l2-3}.
\begin{lemma}\label{l2-4}
Let $P(w)$ be a $k$-polycyclic chain $($$k\geq 5$$)$. Let $u,v\in V(C_{h})$ with $uv\in E(P(w))$, $d_{P(w)}(u)=d_{P(w)}(v)=2$, $d_{P(w)}(a_{h-1},u)<d_{P(w)}(a_{h-1},v)$ or $d_{P(w)}(b_{h-1},u)<d_{P(w)}(b_{h-1},v)$. Then we have $\Omega_{P(w)}(u)<\Omega_{P(w)}(v)$.
\end{lemma}
\begin{figure}[ht!]
\centering
\scalebox{.24}[.24]{\includegraphics{f-22.png}}
\caption{Illustration of $P(w)$ and $P(w^{*})$ of Lemma \ref{l2-5}.}
\label{fig-22}
\end{figure}
\begin{lemma}\label{l2-5}
Let $P(w)$ be a $k$-polycyclic chain $($$k\geq 6$$)$, where $w=(w_{1},w_{2},\cdots,w_{h-2})$ with $w_{i}=0$ or $k-4$ for some $i\in\{1,2,\cdots h-2\}$ and $0\leq w_{j}\leq k-4$ for any $j\in\{1,2,\cdots h-2\}\setminus \{i\}$. Let $w^{*}=(w_{1},\cdots,w_{i-1},t,k-4-w_{i+1},\cdots,k-4-w_{h-2})$, where $1\leq t\leq k-5$. Then $Kf(P(w^{*}))>Kf(P(w))$.
\end{lemma}
\begin{proof}
We only consider $w_{i}=k-4$. The case of $w_{i}=0$ is similar, we omit it.
Since $w_{i}=k-4$, we suppose the $k-2$ vertices on the top of the $(i+1)$-th polygons are
$u_{t},\cdots,u_{2},u_{1},u,x,x_{1},x_{2},\cdots,x_{k-t-4}$. The two vertices on the bottom of the $(i+1)$-th polygons are $v,y$ (see Figure \ref{fig-22}).
It is obvious that the two edges $ux,vy$ are the edge cut of $P(w)$.
Let $P(w^{*})$ be the graph obtained from $P(w)$ by deleting edges $ux,vy$ and adding edges $uy,vx$. Then $P(w)$ and $P(w^{*})$ are isomers.
Suppose that $N_{1}$ is the component of $P(w)\setminus \{ux,vy\}$ containing vertex $u,v$,
$N_{2}$ is the component of $P(w)\setminus \{ux,vy\}$ containing vertex $x,y$.
By Lemma \ref{l13-5}, we have
$$Kf(P(w))-Kf(P(w^{*}))=\frac{(\Omega_{N_{1}}(u)-\Omega_{N_{1}}(v))(\Omega_{N_{2}}(y)-\Omega_{N_{2}}(x))}
{\Omega_{N_{1}}(u,v)+\Omega_{N_{2}}(x,y)+2}.$$
If $z\in V(N_{1})\setminus \{u,v,u_{1},u_{2},\cdots,u_{t}\}$, then by the triangular inequality of resistance distance, we have
\begin{equation}\label{eq:21}
\Omega_{N_{1}}(v,z)\leq \Omega_{N_{1}}(v,u_{t})+\Omega_{N_{1}}(u_{t},z)<1+\Omega_{N_{1}}(u_{t},z)\leq t+\Omega_{N_{1}}(u_{t},z)=\Omega_{N_{1}}(u,z).
\end{equation}
If $z\in \{u_{1},u_{2},\cdots,u_{t}\}$, then by cut-vertex property of resistance distance, we have
\begin{equation}\label{eq:22}
\sum_{i=1}^{t}\Omega_{N_{1}}(v,u_{i})= t\cdot \Omega_{N_{1}}(v,u_{t})+\sum_{i=1}^{t-1}i<\sum_{i=1}^{t}i= \sum_{i=1}^{t}\Omega_{N_{1}}(u,u_{i}).
\end{equation}
Then by $\Omega_{N_{1}}(v,u)=\Omega_{N_{1}}(u,v)$, equations (\ref{eq:21}) and (\ref{eq:22}), we have
$$\Omega_{N_{1}}(v)=\sum_{z\in V(N_{1})}\Omega_{N_{1}}(v,z)<\sum_{z\in V(N_{1})}\Omega_{N_{1}}(u,z)=\Omega_{N_{1}}(u).$$
Now we show $\Omega_{N_{2}}(y)<\Omega_{N_{2}}(x)$. For convenience, we let $k-4-t=s$.
If $z\in V(N_{2})\setminus \{x,y,x_{1},x_{2},\cdots,x_{s}\}$, then by the triangular inequality of resistance distance of Lemma \ref{l13-6}, we have
\begin{equation}\label{eq:23}
\Omega_{N_{2}}(y,z)\leq \Omega_{N_{2}}(y,x_{s})+\Omega_{N_{2}}(x_{s},z)<1+\Omega_{N_{2}}(x_{s},z)\leq s+\Omega_{N_{2}}(x_{s},z) = \Omega_{N_{2}}(x,z).
\end{equation}
If $z\in \{x_{1},x_{2},\cdots,x_{s}\}$, then by cut-vertex property of resistance distance, we have
\begin{equation}\label{eq:24}
\sum_{i=1}^{s}\Omega_{N_{2}}(y,x_{i})= s\cdot \Omega_{N_{2}}(y,x_{s})+\sum_{i=1}^{s-1}i<\sum_{i=1}^{s}i= \sum_{i=1}^{s}\Omega_{N_{2}}(x,x_{i}).
\end{equation}
Then by $\Omega_{N_{2}}(y,x)=\Omega_{N_{2}}(x,y)$, equations (\ref{eq:23}) and (\ref{eq:24}), we have
$\Omega_{N_{2}}(y)<\Omega_{N_{2}}(x)$.
Thus $Kf(P(w^{*}))-Kf(P(w))>0$.
This completes the proof.
\end{proof}
\hskip 0.6cm
\textbf{Proof of Theorem \ref{t1-1}}.
Let $P(w)$ be the $k$-polycyclic chain with $h$ polygons and have the minimum Kirchhoff index.
Then by Lemma \ref{l2-5}, we have $w_{i}=0$ or $w_{i}=k-4$ $(k\geq 6)$ for any $i\in\{1,2,\cdots,h-2\}$. It is obvious that we also have $w_{i}=0$ or $w_{i}=1=k-4$ if $k=5$.
Suppose that $w_{1}=0$, next we show $w_{i}=0$ for $2\leq i\leq h-2$.
Otherwise, there exists $i(2\leq i\leq h-2)$ such that $w_{i-1}=0$ and $w_{i}=k-4$.
Suppose that the $k-2$ vertices on the top of the $(i+1)$-th polygons are
$u,x,x_{1},\cdots,x_{k-4}$, and the two vertices on the bottom of the $(i+1)$-th polygons are $v,y$.
It is obvious that the two edges $ux,vy$ are the edge cut of $P(w)$.
Let $P(w^{*})$ be the graph obtained from $P(w)$ by deleting edges $ux,vy$ and adding edges $uy,vx$. Then $P(w)$ and $P(w^{*})$ are isomers.
Suppose that $N_{1}$ is the component of $P(w)\setminus \{ux,vy\}$ containing vertex $u,v$,
$N_{2}$ is the component of $P(w)\setminus \{ux,vy\}$ containing vertex $x,y$.
By Lemma \ref{l13-5}, we have
$$Kf(P(w))-Kf(P(w^{*}))=\frac{(\Omega_{N_{1}}(u)-\Omega_{N_{1}}(v))(\Omega_{N_{2}}(y)-\Omega_{N_{2}}(x))}
{\Omega_{N_{1}}(u,v)+\Omega_{N_{2}}(x,y)+2}.$$
Since $w_{i-1}=0$, there are two vertices $a_{i-1}$ and $u$ in the top of $i$-th polygons, and $k-2$ vertices in the bottom of $i$-th polygons. By Lemma \ref{l2-3}, we have
$\Omega_{N_{1}}(u)<\Omega_{N_{1}}(v).$
We replace $s(=k-t-4)$ by $k-4$ in the proof of equations (\ref{eq:23}) and (\ref{eq:24}) of Lemma \ref{l2-5}, and we can show
$\Omega_{N_{2}}(y)<\Omega_{N_{2}}(x)$.
Then $Kf(P(w))-Kf(P(w^{*}))>0$. This is a contradiction with that $P(w)$ has the minimum Kirchhoff index. Thus $w_{i}=0$ for $1\leq i\leq h-2$.
This completes the proof.
\hfill $\blacksquare$
\begin{lemma}\label{l2-6}
Let $P(w)$ be a $k$-polycyclic chain $($$k\geq 6$$)$ with $h$ $k$-polygons, $w=(w_{1},w_{2},\cdots,$ $w_{h-2})$ where $0\leq w_{i}\leq k-4$ for $i\in\{1,2,\cdots,h-2\}$. If $w_{i}\geq \lceil \frac{k-4}{2}\rceil+1$ for some $i(1\leq i\leq h-2)$, we take $w^{*}=(w_{1},\cdots,w_{i-1},\lceil \frac{k-4}{2}\rceil,k-4-w_{i+1},\cdots,k-4-w_{h-2})$, then $Kf(P(w^{*}))>Kf(P(w))$.
\end{lemma}
\begin{proof}
Let $w_{i}=t\geq \lceil \frac{k-4}{2}\rceil+1$, the $t+2$ vertices on the top of the $(i+1)$-th polygons be
$u_{t-\lfloor \frac{2t-k+4}{2} \rfloor},\cdots,u_{2},u_{1},u,x,x_{1},x_{2},\cdots,x_{\lfloor \frac{2t-k+4}{2} \rfloor}$, and the $k-t-2$ vertices on the bottom of the $(i+1)$-th polygons be $v_{k-t-4},\cdots,v_{2},v_{1},v,y$, respectively.
It is obvious that the two edges $ux,vy$ are the edge cut of $P(w)$.
Let $P(w^{*})$ be the graph obtained from $P(w)$ by deleting edges $ux,vy$ and adding edges $uy,vx$. Then $P(w)$ and $P(w^{*})$ are isomers.
Suppose that $N_{1}$ is the component of $P(w)\setminus \{ux,vy\}$ containing vertex $u,v$,
$N_{2}$ is the component of $P(w)\setminus \{ux,vy\}$ containing vertex $x,y$.
By Lemma \ref{l13-5}, we have
$$Kf(P(w))-Kf(P(w^{*}))=\frac{(\Omega_{N_{1}}(u)-\Omega_{N_{1}}(v))(\Omega_{N_{2}}(y)-\Omega_{N_{2}}(x))}
{\Omega_{N_{1}}(u,v)+\Omega_{N_{2}}(x,y)+2}.$$
If $z\in V(N_{1})\setminus \{u,v,u_{1},u_{2},\cdots,u_{t-\lfloor \frac{2t-k+4}{2} \rfloor},v_{1},v_{2},\cdots,v_{k-t-4}\}$, then by the triangular inequality, cut-vertex property of resistance distance and $t\geq \lceil \frac{k-4}{2}\rceil+1$, we have
\begin{eqnarray*}
\Omega_{N_{1}}(v,z) & \leq & k-t-4+\Omega_{N_{1}}(v_{k-t-4},u_{t-\lfloor \frac{2t-k+4}{2} \rfloor})+\Omega_{N_{1}}(u_{t-\lfloor \frac{2t-k+4}{2} \rfloor},z)\\
& < & t-\lfloor \frac{2t-k+4}{2} \rfloor+\Omega_{N_{1}}(u_{t-\lfloor \frac{2t-k+4}{2} \rfloor},z)\\
& = & \Omega_{N_{1}}(u,z).
\end{eqnarray*}
If $z\in \{u_{1},u_{2},\cdots,u_{t-\lfloor \frac{2t-k+4}{2} \rfloor},v_{1},v_{2},\cdots,v_{k-t-4}\}$, then by cut-vertex property of resistance distance and $t\geq \lceil \frac{k-4}{2}\rceil+1$, we have
$$ \sum_{i=1}^{k-t-4}\Omega_{N_{1}}(v,v_{i})+\sum_{i=1}^{t-\lfloor \frac{2t-k+4}{2} \rfloor}\Omega_{N_{1}}(v,u_{i})<\sum_{i=1}^{k-t-4}\Omega_{N_{1}}(u,v_{i})+\sum_{i=1}^{t-\lfloor \frac{2t-k+4}{2} \rfloor}\Omega_{N_{1}}(u,u_{i}).$$
Thus
$$\Omega_{N_{1}}(v)=\sum_{z\in V(N_{1})}\Omega_{N_{1}}(v,z)<\sum_{z\in V(N_{1})}\Omega_{N_{1}}(u,z)=\Omega_{N_{1}}(u).$$
If $z\in V(N_{2})\setminus \{x,y,x_{1},x_{2},\cdots,x_{\lfloor \frac{2t-k+4}{2} \rfloor}\}$, then by the triangular inequality of resistance distance, we have
$$ \Omega_{N_{2}}(y,z)\leq \Omega_{N_{2}}(y,x_{\lfloor \frac{2t-k+4}{2} \rfloor})+\Omega_{N_{2}}(x_{\lfloor \frac{2t-k+4}{2} \rfloor},z)<1+\Omega_{N_{2}}(x_{\lfloor \frac{2t-k+4}{2} \rfloor},z)\leq \Omega_{N_{2}}(x,z).$$
If $z\in \{x_{1},x_{2},\cdots,x_{\lfloor \frac{2t-k+4}{2} \rfloor}\}$, then by cut-vertex property of resistance distance, similarly we have
$$ \sum_{i=1}^{\lfloor \frac{2t-k+4}{2} \rfloor}\Omega_{N_{2}}(y,x_{i})< \sum_{i=1}^{\lfloor \frac{2t-k+4}{2} \rfloor}\Omega_{N_{2}}(x,x_{i}).$$
Thus
$$\Omega_{N_{2}}(y)<\Omega_{N_{2}}(x).$$
Then $Kf(P(w^{*}))-Kf(P(w))>0$.
This completes the proof.
\end{proof}
Similar to the proof of Lemma \ref{l2-6}, we also have
\begin{lemma}\label{l2-7}
Let $P(w)$ be a $k$-polycyclic chain $($$k\geq 6$$)$ with $h$ $k$-polygons, $w=(w_{1},w_{2},\cdots,$ $w_{h-2})$ where $0\leq w_{i}\leq k-4$ for $i\in\{1,2,\cdots,h-2\}$. If $w_{i}\leq \lfloor \frac{k-4}{2}\rfloor-1$ for some $i(1\leq i\leq h-2)$, we take $w^{*}=(w_{1},\cdots,w_{i-1},\lfloor \frac{k-4}{2}\rfloor,k-4-w_{i+1},\cdots,k-4-w_{h-2})$, then $Kf(P(w^{*}))>Kf(P(w))$.
\end{lemma}
\textbf{Proof of Theorem \ref{t1-2}}.
Let $P(w)$ be the $k$-polycyclic chain with $h$ polygons and has the maximum Kirchhoff index.
Then by Lemmas \ref{l2-6} and \ref{l2-7}, we have $w_{i}=\lceil \frac{k-4}{2} \rceil$ or $w_{i}=\lfloor \frac{k-4}{2} \rfloor$ $(k\geq 6)$ for $1\leq i\leq h-2$. It is obvious that we also have $w_{i}=1=\lceil \frac{k-4}{2} \rceil$ or $w_{i}=0=\lfloor \frac{k-4}{2} \rfloor$ if $k=5$.
If $k$ is even, the conclusion holds. Next we only consider $k$ is odd, then $\lceil \frac{k-4}{2} \rceil>\lfloor \frac{k-4}{2} \rfloor$.
If there exists $i$ such that $w_{i-1}=w_{i}=\lceil \frac{k-4}{2} \rceil$ or $w_{i-1}=w_{i}=\lfloor \frac{k-4}{2} \rfloor$, we will obtain a contradiction.
We only consider $w_{i-1}=w_{i}=\lceil \frac{k-4}{2} \rceil$, the proof case of $w_{i-1}=w_{i}=\lfloor \frac{k-4}{2} \rfloor$ is similar.
Suppose the $\lceil \frac{k-4}{2} \rceil+2$ vertices on the top of the $(i+1)$-th polygons are $u,x,x_{1},\cdots,x_{\lceil \frac{k-4}{2} \rceil}$. The $\lfloor \frac{k-4}{2} \rfloor+2$ vertices on the bottom of the $(i+1)$-th polygons are $v,y,y_{1},\cdots,y_{\lfloor \frac{k-4}{2} \rfloor}$, respectively.
It is obvious that the two edges $ux,vy$ are the edge cut of $P(w)$.
Let $P(w^{*})$ be the graph obtained from $P(w)$ by deleting edges $ux,vy$ and adding edges $uy,vx$. Then $P(w)$ and $P(w^{*})$ are isomers.
Suppose that $N_{1}$ is the component of $P(w)\setminus \{ux,vy\}$ containing vertex $u,v$,
$N_{2}$ is the component of $P(w)\setminus \{ux,vy\}$ containing vertex $x,y$.
By Lemma \ref{l13-5}, we have
$$Kf(P(w))-Kf(P(w^{*}))=\frac{(\Omega_{N_{1}}(u)-\Omega_{N_{1}}(v))(\Omega_{N_{2}}(y)-\Omega_{N_{2}}(x))}
{\Omega_{N_{1}}(u,v)+\Omega_{N_{2}}(x,y)+2}.$$
Since $w_{i-1}=\lceil \frac{k-4}{2} \rceil>\lfloor \frac{k-4}{2} \rfloor$, then by Lemmas \ref{l2-2} and \ref{l2-4}, we have
$\Omega_{N_{1}}(u)>\Omega_{N_{1}}(v).$
Next we prove $\Omega_{N_{2}}(y)<\Omega_{N_{2}}(x).$
If $z\in V(N_{2})\setminus \{x,y,x_{1},x_{2},\cdots,x_{\lceil \frac{k-4}{2} \rceil},y_{1},y_{2},\cdots,y_{\lfloor \frac{k-4}{2} \rfloor}\}$, Note that the weight of edge $x_{\lceil \frac{k-4}{2} \rceil}y_{\lfloor \frac{k-4}{2} \rfloor} $ is less than $1$ and $\lceil \frac{k-4}{2} \rceil>\lfloor \frac{k-4}{2} \rfloor$. Then by the triangular inequality of resistance distance, we have
$ \Omega_{N_{2}}(y,z)\leq \lfloor \frac{k-4}{2} \rfloor+\Omega_{N_{2}}(y_{\lfloor \frac{k-4}{2} \rfloor},x_{\lceil \frac{k-4}{2} \rceil})+\Omega_{N_{2}}(x_{\lceil \frac{k-4}{2} \rceil},z)<\lceil \frac{k-4}{2} \rceil+\Omega_{N_{2}}(x_{\lceil \frac{k-4}{2} \rceil},z)= \Omega_{N_{2}}(x,z).$
If $z\in \{x_{1},x_{2},\cdots,x_{\lceil \frac{k-4}{2} \rceil},y_{1},y_{2},\cdots,y_{\lfloor \frac{k-4}{2} \rfloor}\}$. Note that the weight of edge $x_{\lceil \frac{k-4}{2} \rceil}y_{\lfloor \frac{k-4}{2} \rfloor} $ is less than $1$ and $\lceil \frac{k-4}{2} \rceil>\lfloor \frac{k-4}{2} \rfloor$. Then by cut-vertex property of resistance distance, similarly we have
$ \sum\limits_{i=1}^{\lceil \frac{k-4}{2} \rceil}\Omega_{N_{2}}(y,x_{i})+\sum\limits_{i=1}^{\lfloor \frac{k-4}{2} \rfloor}\Omega_{N_{2}}(y,y_{i})<\sum\limits_{i=1}^{\lceil \frac{k-4}{2} \rceil}\Omega_{N_{2}}(x,x_{i})+\sum\limits_{i=1}^{\lfloor \frac{k-4}{2} \rfloor}\Omega_{N_{2}}(x,y_{i}).$
Then
$\Omega_{N_{2}}(y)<\Omega_{N_{2}}(x)$.
Thus $Kf(P(w^{*}))-Kf(P(w))>0$. This is a contradiction with that $P(w)$ has the maximum Kirchhoff index.
This completes the proof.
\hfill $\blacksquare$
\section{Conclusions}
\hskip 0.6cm
In this paper, we completely solve the problem about the extremal $k$-polycyclic chains with respect to Kirchhoff index for $k\geq 5$, which extends the results of \cite{suya2023} for $k=5$, \cite{yakl2014,yasu2022} for $k=6$ and \cite{maqi2022} for $k=8$.
In addition to the Kirchhoff index, the Wiener index is also an important molecular descriptor.
Cao et al. \cite{cayz2020} determined the extremal Wiener indices in $k$-polycyclic chains with $k$ is even. Chen et al. \cite{chli2022} determined the expected values of Wiener indices in random $k$-polycyclic chains with $k$ is even.
Thus the problem of determining the extremal Wiener indices in $k$-polycyclic chains with $k$ is odd is still open. We intend to consider the above challenging problems in the future.
\vspace{4mm}
\noindent
{\bf Acknowledgements}\, This research is supported by the National Natural Science Foundation of China (Grant No. 11971180), the Guangdong Provincial Natural Science Foundation (Grant No. 2019A1515012052), the Characteristic Innovation Project of General Colleges and Universities in Guangdong Province (Grant No. 2022KTSCX225) and the Guangdong Education and Scientific Research Project (Grant No. 2021GXJK159).
\baselineskip=0.20in
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 5,727 |
\section{Introduction}
A large fraction of the world's water and energy resources are located in naturally fractured reservoirs within the earth's crust. Understanding the dynamics of such reservoirs in terms of flow, heat transport and fracture stability is crucial to successful application of engineered geothermal systems (also known as enhanced geothermal systems, EGS) for geothermal energy production. Reservoir development characteristics such as permeability creation and induced seismicity largely depend on the properties of preexisting fractures, porosity, permeability and fracture orientation within the local stress field. One of the primary driving mechanisms for permeability creation in EGS involves shear failure induced by fluid injection at high pressures \citep{barton1995,evans2005,hickman2010}. Along sections of the well that are free of natural fractures and in environments with low differential stress, tensile fractures may develop if the injection pressure exceeds the minimal principal stress. Shear and tensile fracture propagation and reactivation are not exclusive and might occur simultaneously during the stimulation of the reservoir \citep{mcclure2014}. Clearly, preexisting, critically stressed and optimally oriented fractures provide the most favorable conditions for enhancing permeability of EGS \citep{barton1995, combs2004, ghassemi2007}.\\
The basis for EGS are fractured reservoirs, which are usually geothermal plays of the ``hot dry rock'' type where the available water in the porous medium is considered negligible \citep{brown2012}. These conditions are found primarily in metamorphic or igneous terrains with low permeability and porosity, containing fractures and faults that provide the major pathways for fluid flow (e.g. Fenton Hill, Soultz, Basel, Cooper basin and Desert Peak \citep{kelkar2016,hooijkaas2006,haring2008,chen2009, hickman2010}). In geothermal energy systems, the fracture's surfaces serve as the main heat exchanger. Fractured reservoirs can be considered to consist of two distinct separate media, the fractures and the matrix, and different types of reservoirs can be defined that depend on their properties \citep{nelson2001}. In EGS, two cases typically prevail: 1) reservoirs with low porosity matrix for which both the permeability and the storage capacity of the rock mass are controlled by the fractures (cf. type 1 in \citep{nelson2001}) and 2) reservoirs with sufficient matrix porosity such that fluid storage is dominated by the matrix while the fractures contain only a small fraction of the fluid but control the permeability (cf. type 2 in \citep{nelson2001}).
Simulation of flow and transport through fractured porous media is challenging due to the high permeability contrast between the fractures and the surrounding rock matrix. However, accurate and efficient simulation of flow through a fracture network is crucial in order to understand, optimize and engineer reservoirs. Even after decades of research, this is still a very active research topic. Additionally, accurate estimations of the fracture stability are necessary in order to predict permeability evolution and forecast induced seismicity. Discrete fracture models (DFM) have been developed to address the computational problem of scales for fluid flow and heat transport. Various modeling frameworks for the simulation of fractured porous media in general and geothermal reservoirs in particular exist. Literature reviews of current modeling approaches in geothermal reservoirs and hot dry rock systems are presented by Willis-Richards and Wallroth, Sanyal et al. and O'Sullivan et al.\citep{willis1995,sanyal2000, osullivan2001}. Some better known open-source modeling frameworks related to this work include PFLOTRAN \citep{pflotran-user-ref}, OpenGeoSys \citep{kolditz2012} and DuMux \citep{flemisch2011}. Yet traditional conforming DFM, where the fractures are explicitly resolved by the numerical grid, suffer from computationally expensive pre-processing in the numerical grid generation and can encounter severe time step restrictions during the simulation when using explicit time-stepping and small cells around the fractures \citep{norbeck2014,sandve2012}. \\
An alternative approach uses the embedded discrete fracture models (EDFM), which treat fracture and matrix in two separate computational domains. The embedded fracture model was first introduced by Lee et al. for single phase problems and later extended to two-phase flow \citep{lee2001, li2008}. The embedded discrete fracture model is a promising technique in modeling the behavior of enhanced geothermal systems. Karvounis \citep{karvounis2013} employs EDFM and a statistical approach to better understand and possibly forecast seismicity induced seismicity by fluid injection during the stimulation phase of an EGS. Norbeck et al. additionally model fracture deformation by linear fracture mechanics \citep{norbeck2016}.
In this paper we present the to our best knowledge first open source implementation of an embedded discrete fracture model for single phase flow and heat transport with additional capabilities to determine fracture stability in fractured reservoirs. Slip tendency analysis is used in order to estimate fault reactivation potential in earthquake prone areas as well as fracture stability in geothermal reservoirs \citep[e.g.][]{morris1996, moeck2009}. Slip tendency is an indicator for the likelihood of slip. Using slip tendency, predictions on fracture instabilities during the hydraulic stimulation of a fractured reservoir are feasible without solving for the typically non-linear evolution of the stress equilibrium equation. THERMAID, an acronym for "Thermo-Hydraulic Energy Resource Modeling for Application and Development", is a fractured reservoir modeling framework implemented in \textit{MATLAB}, which can be used as a standalone simulation package for TH(m) cases in geothermal reservoirs or as a blue print for the re-implementation of the method e.g. in a high performance computing (HPC) framework. We coin the term TH(m) to indicate a coupled Thermo-Hydraulic code, and we use the lower case (m) to indicate simplified mechanics.\\
This paper is structured as follows. In the next section we present the methodology of the embedded discrete fracture model, and describe in detail the underlying theory of the fracture stability analysis. The implemented model is evaluated in the \nameref{sec1:results} section by comparing it with a widely used numerical model in several test cases. We conclude the paper by illustrating possible applications of the code using some examples and a discussion of the findings.
\section{Methodology}\label{sec1:method}
The conceptual idea of the EDFM is the distinct separation of a fractured reservoir into a fracture and a matrix domain. We introduce a transfer function to account for coupling effects between the two domains (cf. Figure \ref{fig1:edfm_domain}), so the fracture and matrix domains are computationally independent except for the transfer function. As the fractures are generally very thin and highly permeable compared to the surrounding matrix rock, the gradient of fluid pressure with the fracture normal to it is negligible. This allows for a lower dimensional representation of fractures (i.e. 1D objects within a 2D reservoir).
\begin{figure}[!htbp]
\centering
\includegraphics[width=0.55\linewidth]{1_domain_separation.pdf}
\caption{A fractured domain a) is separated in a uniform grid b) and a fracture grid c). The two resulting domains are coupled using the transfer function $\Psi^{fm}$ .}
\label{fig1:edfm_domain}
\end{figure}
\subsection{Conceptual model}
Numerical modeling of fractured reservoirs is not only challenging from the numerical and computational point of view, but also because it involves a variety of coupled thermal, hydraulic, mechanical and chemical (THMC) processes. THERMAID focuses on thermo-hydraulic processes and their coupling, with some additional mechanical processes considered within a simplified geomechanical model.
\begin{figure}[!b]
\centering
\includegraphics[width=0.99\linewidth]{2_ConceptualModelFig.pdf}
\caption{A conceptual model of a fractured domain with relevant thermo-hydro-mechanical processes. Underlined processes are included in THERMAID's current implementation. Processes in brackets are considered relevant but are currently not included mainly due to ambiguity in the simplified geomechanical model.}
\label{fig1:concept}
\end{figure}
Figure \ref{fig1:concept} shows the conceptual model of the most relevant thermal, hydraulic and mechanical processes in fractured reservoirs. The core processes implemented in THERMAID are fluid flow through pressure diffusion in the matrix and the fracture network and the accompanying heat transfer by advection and diffusion. Pressure and heat are also exchanged between the rock matrix and the fracture at the fracture walls by pressure and thermal diffusion and thermal advection, respectively.\\
Associated with the thermo-hydraulic processes, numerous thermo-mechanic or thermo-hydraulic processes could be activated, but only a limited amount and simplified processes are currently implemented in THERMAID. Poro-elastic and thermo-elastic deformation of the fractures and of the matrix is currently not implemented in the code. The fluid pressure in the fracture is, however, considered in the computation of fracture effective normal stress, which is an important parameter to evaluate fracture stability. If a fracture becomes unstable it will slip, and the associated dilation slip will increase the fracture transmissivity. The transmissivity increase is introduced in a simplified manner: if a fracture segment reaches the slip condition, its transmissivity is multiplied by a fixed permeability enhancement factor. In reality, slip on a fracture perturbs local the stress state, potentially affecting the stability of other fractures. This process is not implemented in the code because the direction and amount of slip is ambiguous. However, stress change induced by thermal changes is implemented, not yet in a fully coupled way, but instead computing the thermal stresses and superposing thermal stresses onto the ambient stresses.\\
In addition to the processes shown in Figure \ref{fig1:concept}, THERMAID properly accounts for gravity effects, internal pressure and heat sources, and the pressure- and temperature-dependence of fluid density and viscosity. The corresponding equations of state are given by \citep{sun2008} for density and \citep{alshemmeri2012} for the viscosity of water.
To summarize, THERMAID is a thermo-hydraulic code for fractured media that accounts for mechanical stability of the fractures, slip-induced transmissivity increase and thermally induced stresses. In the following, we introduce the governing equations for fluid flow and heat transport, couplings between fracture and matrix, and the implemented fracture stability analysis.
\subsection{Governing equations}
Flow in naturally fractured reservoirs is often described by the equations for nearly incompressible single-phase flow. We assume that the equations for nearly incompressible single-phase flow are valid in both matrix and the fractures. This simplification might not yield an adequate description of the flow in some fractured reservoirs where very large fracture apertures result in non-Darcian flow. The methodology presented here is however easily modifiable to extended Darcy flow models.
The pressure equation, derived from continuity and total mass balance equations for single-phase fluid flow, is:
\begin{equation}
\phi \left(\beta_f + \beta_r\right)\frac{\partial p}{\partial t} = \nabla \cdot \left[ \frac{k}{\mu}(\nabla p-\rho_fg)\right] + Q \label{eq1:pressure}
\end{equation}
where $\phi$ [-] is the porosity, $\rho_f$ $[\frac{kg}{m^3}]$ is the fluid density, $p$ [Pa] is the fluid pressure and $Q$ $[\frac{m^3}{s}]$ a source term. The compressibilities $\beta$ [Pa$^{-1}$] are denoted with the subscripts $f$ for fluid and $r$ for rock, respectively, $k$ $[m^2]$ is the permeability and $\mu$ [Pa$\cdot$s] the fluid viscosity. We consider only isotropic permeability $k$. Permeability is often linked to fracture aperture through the Cubic law, which has been shown to be useful in predicting fluid transport through fractured reservoirs and fractured porous media in general. However, it does not account for the roughness of the fracture or flow adjacent to the fracture walls due to the rock permeability. From the fluid pressure $p$, the fluid velocity is calculated using Darcy's law, i.e.
\begin{equation}
\mathbf{v} = - \frac{k}{\mu} \left[ \nabla p -\rho_f g\right]\label{eq1:darcy_v}
\end{equation}
The total mass balance equation derived above is separated into parts for the matrix and the fracture domains, i.e.
\begin{equation}
\phi^m\left( \beta_f + \beta_r\right)\frac{\partial p^m}{\partial t} = \nabla \cdot \left[ \frac{k^m}{\mu^m}(\nabla p^m-\rho_fg)\right] + \Psi^{mf} +Q^m \label{eq1:pressure_m}
\end{equation}
and
\begin{equation}
\phi^{f} \left(\beta_f + \beta_r\right)\frac{\partial p^f}{\partial t} = \nabla \cdot \left[ \frac{k^f}{\mu^f}(\nabla p^f-\rho_fg)\right] + \Psi^{fm} +Q^f
\end{equation}
where $\Psi^{mf}$ and $\Psi^{fm}$ are the flux transfer functions between the matrix and the fractures. Superscripts $m$ and $f$ denote matrix and fracture quantities respectively.
The heat transport equation is derived similarly to the pressure equation based on a balance of energy. We assume local thermal equilibrium so that $T=T_r =T_f$ where $T_r$ and $T_f$ are the temperatures of solid rock and fluid respectively. Taking average over an elemental volume we find
\begin{equation}
\overline{c_p\rho}\frac{\partial T}{\partial t} + {c_p}_f\rho_f\mathbf{v}\nabla T - \overline{\lambda}\nabla^2 T = \overline{q} \label{eq1:heat}
\end{equation}
where over-lined properties denote volume averaged mean values for the porous medium.
\begin{eqnarray}
\overline{c_p\rho} &= \phi ({c_{p}}_f\rho_f) + (1-\phi)({c_{p}}_r\rho_r) \label{eq1:heat_cp} \\
\overline{\lambda} &= \phi \lambda_{f} + (1-\phi)\lambda_{r} \label{eq1:heat_lambda}\\
\overline{q} &= \phi q_{f} + (1-\phi)q_{r} \label{eq1:heat_q}
\end{eqnarray}
In equations \ref{eq1:heat} to \ref{eq1:heat_q} the heat capacity $c_{p} \,[\frac{J}{kg K}]$, the thermal conductivity $\lambda \,[\frac{W}{m K}]$ and internal heat source $q \,[\frac{W}{m^3}]$ of solid rock and fluid have been introduced. The fluid velocity $\mathbf{v}$ used in the heat transport equation is the Darcy velocity given by equation \ref{eq1:darcy_v}.
The heat transport equation is separated into matrix and fracture parts according to the same procedure as for the fluid pressure equation
\begin{equation}
\overline{c_p\rho}^m\frac{\partial T^m}{\partial t} + ({c_p}_f\rho_f\mathbf{v})^m\nabla T^m - \overline{\lambda}^m\nabla^2 T^m = \overline{q}^m + \mathcal{X}^{mf}
\end{equation}
and
\begin{equation}
\overline{c_p\rho}^f\frac{\partial T^f}{\partial t} + ({c_p}_f\rho_f\mathbf{v})^f\nabla T^f - \overline{\lambda}^f\nabla^2 T^f = \overline{q}^f + \mathcal{X}^{fm}
\end{equation}
where $\mathcal{X}^{mf}$ and $\mathcal{X}^{fm}$ are the heat transfer functions between the damaged matrix and the fractures.
\subsection{Fracture matrix coupling}
To obtain a conservative set of equations, we apply a transfer function governing the mass and heat exchange between the two domains. The transfer function is treated as a source/sink term in the pressure and heat transport equations for damaged matrix and fracture, respectively, similar to classical well models \citep{peaceman1978}.\\
The transfer function for the pressure equation is defined as
\begin{equation}
\Psi^{fm} = CI \cdot \Xi \cdot (p^f-p^m)
\end{equation}
with $\Xi$ being the mean total mobility of the fluid, defined as the fraction of permeability and viscosity \citep{lee2001}. $CI$ is the connectivity index between matrix and fracture that is dependent on the numerical discretization (cf. next section). From the separated mass balance equations, it becomes immediately clear that the total flux between matrix and fracture has to be conserved:
\begin{equation}
\int \Psi^{mf} dV = - \int \Psi^{fm} dA
\end{equation}
The transfer function for the heat equation is similarly defined. However, as two heat transport mechanisms are present in the equation, the transfer function needs to account for both mechanisms. Thus, the transfer function is defined as:
\begin{equation}
\mathcal{X}^{fm} = {\mathcal{X}^{fm}}^{\nabla} + {\mathcal{X}^{fm}}^{\nabla^2}
\end{equation}
where the superscript $\nabla$ denotes the heat advection contribution and $\nabla^2$ denotes the heat conduction contribution. The heat conduction contribution ${\mathcal{X}^{fm}}^{\nabla^2}$ is derived using the same approach as in the pressure transfer function.
\begin{equation}
{\mathcal{X}^{fm}}^{\nabla^2} = CI \cdot \Lambda \cdot (T^f-T^m)
\end{equation}
Here, $\Lambda$ is the heat conductivity at the fracture-matrix interface which can be calculated as
\begin{equation}
\Lambda = \frac{2\cdot \lambda^f\cdot \lambda^m}{\lambda^f + \lambda^m} \label{eq1:Xi}
\end{equation}
using the definition of the averaged heat conductivity $\lambda$ given in equation \ref{eq1:heat_lambda}. The advection contribution $\chi_{fm}^{\nabla}$, on the other hand, explicitly shows the coupling to the pressure equation based on the Darcy velocity:
\begin{equation}
{\mathcal{X}^{fm}}^{\nabla} = \Upsilon \cdot \mathbf{v}^{fm} \label{eq1:Chi_advection}
\end{equation}
In equation \ref{eq1:Chi_advection} we introduce the fluid velocity $\mathbf{v}^{fm}$ and specific heat capacity $\Upsilon$ at the matrix-fracture interface. $\Upsilon$ is calculated analogous to equation \ref{eq1:Xi} and based on the averaged specific heat capacity given in equation \ref{eq1:heat_cp}. The fluid velocity at the matrix-fracture interface is defined as
\begin{equation}
\mathbf{v}^{fm}= - CI \cdot \Xi \cdot (\nabla p)^{fm} \label{eq1:v_fm}
\end{equation}
where $(\nabla p)^{fm}$ is the pressure gradient at the interface of matrix and fracture.
As discussed for the pressure transfer function also the heat transfer flux has to be conserved:
\begin{equation}
\int \mathcal{X}^{mf} dV = - \int \mathcal{X}^{fm} dA
\end{equation}
\subsection{Fracture stability}
Within THERMAID, a simplified analytical approach to fracture slip enables us to estimate fracture stability based on slip tendency analysis. Following Amonton's law for purely frictional fault reactivation
\begin{equation}
\tau = \mu_s \cdot \sigma_{\text{n}eff} \label{eq1:amonton}
\end{equation}
with $\tau$ as shear stress, $\sigma_{\text{n}eff}$ as effective normal stress ($\sigma_{\text{n}} -p$ and $p$ as fluid pressure), and $\mu_s$ as sliding friction coefficient \citep{byerlee1978}, slip tendency is the ratio of shear stress to effective normal stress on a surface \citep{morris1996}, i.e.
\begin{equation}
T_s = \frac{\tau}{\sigma_{\text{n}eff}}
\end{equation}
Fracture failure or slip is likely to occur if the shear stress to effective normal stress ratio equals or exceeds the frictional sliding resistance $\mu_s$. Thus we define the stability of a fracture as follows
\begin{equation}
T_s =
\begin{cases}
\frac{\tau}{\sigma_{\text{n}eff}} < \mu_s & \quad \text{(stable)}\\
\frac{\tau}{\sigma_{\text{n}eff}} \geq \mu_s & \quad \text{(unstable)}\\
\end{cases}
\end{equation}
Shear and effective normal stress acting on a given fracture depend on the orientation of the fracture plane within the effective principal stress field. If the effective principal stress field $(\vec{l},\vec{m},\vec{n})$ and the reference coordinate system $(\vec{x},\vec{y},\vec{z})$ of the simulation match, shear and normal stress can be calculated by simple expressions based on the dip angle of the fracture \citep[e.g.][]{miller2004}. Otherwise, stress transformations are needed in order to calculate the correct normal and shear stress in the fracture coordinate system $(\vec{u},\vec{v},\vec{w})$. The involved rotation matrices can be calculated if the orientations of the principal stresses are known \citep[e.g.][]{allmendinger2011}. Finally we compute the normal and shear stress on the fracture in the fracture coordinate system based on the 3D principal stresses. Note that in this calculation the influence of the intermediate principal stress component is taken into account despite the 2D model geometry.
An important addition to the effects of pore pressure and far field stresses for fracture stability is thermal stress. A body will change its shape and/or volume when exposed to a temperature change $\Delta T$. If the body's deformation is restricted, as it would be the case for a small volume inside a rock mass, the strain results in thermal stress.
\begin{equation}
\sigma_{T_{ij}} = \frac{E}{1-2\nu} \cdot \alpha \Delta T \delta_{ij} \label{eq1:sigma_t}
\end{equation}
where $\alpha$ is the coefficient of linear thermal expansion in $\frac{1}{K}$, $E$ is the Young's modulus ($Pa$) and $\nu$ the Poisson ratio (-). $\delta_{ij}$ is the Kronecker delta, which is 1 for identical indices $i$ and $j$, and 0 otherwise. The thermal stress is positive (relative compression) if the temperature difference is positive ($\Delta T > 0$), and if the temperature difference negative, the thermal stress is negative (relative tension).
In the following we assume that the thermal stress is independent of the fluid pressure and the in-situ stress state of the rock. Thus, the resulting stress can be obtained by superposition of the effective stress ($\sigma_{eff} = \sigma_{tot} - p$) and the thermal stress. We formulate the superposed effective stress as
\begin{equation}
\sigma_{eff} = \sigma_{tot} - p + \sigma_{T}
\end{equation}
which can be used in equation \ref{eq1:amonton} in order to account for thermal stress during the fracture stability analysis.\\
Clearly, other stress contributions as slip induced stresses and stresses induced by chemical reaction have to be considered in a general case. However, especially the estimation of slip induced stress changes is ambiguous as the amount of slip and slip direction for potentially failing fractures is not known a-priori unless the underlying equations for fracture slip are solved explicitly. Thus, for reasons of simplicity we restrict ourselves to only effective stress and thermally induced stress changes.
As fractures are reactivated they generally show an increase in aperture as the fracture surfaces are not smooth but have many asperities. Due to a strong aperture dependence of permeability \citep[e.g.][]{nemvcok2002}, where small changes in aperture result in very large changes in permeability, it can be assumed that unstable (or sliding) fractures undergo a stepwise change in fracture permeability \citep{millernur2000}. Here we adopt the most simple model
\begin{equation}
k^{f} = \begin{cases}
k^{f} & \quad \text{if } T_s < \mu_s \\
\gamma \cdot k^{f} & \quad \text{if } T_s \geq \mu_s \\
\end{cases}
\end{equation}
where $\gamma$ is an permeability enhancement factor.
This model successfully described the distribution of the induced seismicity in the Basel EGS site, and fluid-driven aftershock sequences \citep{miller2004,miller2015}.
\section{Implementation}\label{sec1:implementation}
We implemented the two-dimensional embedded discrete fracture method in MATLAB. Our implementation is based on the concepts used in \textit{MAFLOT}, an open source MATLAB flow solver \citep{kuenze2012}. As briefly discussed in the introduction, the matrix and fracture domains are discretized by regular Cartesian grids in 2D for the matrix and 1D for the fractures respectively (cf. Figure \ref{fig1:edfm_domain}).
\subsection{Numerical discretization in space}
Using a finite volume approach, we discretize the domain $\Omega$ as the integration over finite control volumes $\Omega_{ij}$ with $\Omega = \sum_{ij=1}^N \Omega_{ij}$. Using the Gauss theorem, the divergence integral over the volume can be rewritten as the surface integral normal to the boundary of the volume. Applied to a matrix grid cell on the right hand side (RHS) of the pressure equation \ref{eq1:pressure_m} this yields
\begin{equation}
\int_{\Omega_{ij}} \nabla\cdot \left(\frac{k}{\mu}\cdot\nabla p\right)^m + \Psi^{mf} + Q^m dV \Rightarrow \int_{\partial\Omega_{ij}}\left(\left(\frac{k}{\mu}\cdot\nabla p\right)^m +\Psi^{mf}\right)\cdot \mathbf{n} ds + \int_{\Omega_{ij}} Q^m dV
\end{equation}
Note that gravity is neglected here and in the remains of this section to better facilitate comprehension of the implementation. The pressure gradient over the cell boundary $\partial\Omega_{ij}$ is approximated by a two-point flux approximation that is second-order accurate in space. As the domain is generally heterogeneous in terms of rock properties, a harmonic averaging technique is used to calculate the appropriate values at the cell boundaries.
The discretization of the RHS of the temperature equation is analogous to the pressure equations and omitted here for brevity. It is worth noting, however, that the advection term must be treated with special care. In this EDFM implementation, we use an upwind method in the fractures in combination with a \textit{minmod-}flux limited QUICK scheme in the matrix \citep{courant1952,leonard1979,roe1986}.
\subsection{Connectivity index}
The connectivity index $CI$ between matrix and fracture is discretization-dependent, and defined based on the linear pressure distribution assumed within a grid cell intersected by a fracture \citep{hajibeygi2011}. It is defined as the length fraction $A_{ij,k}$ of fracture segment $k$ inside matrix cell $ij$ divided by the average distance $\langle d \rangle_{ij,k}$ between matrix cell $ij$ and fracture segment $k$.
\begin{equation}
CI_{ij,k} = \frac{A_{ij,k}}{\langle d \rangle_{ij,k}}
\end{equation}
The average distance $\langle d \rangle_{ij,k}$ can be calculated as
\begin{equation}
\langle d \rangle_{ij,k} = \frac{\int x_k(x')dx'}{V_{ij}} \label{eq1:d_mean}
\end{equation}
where $x_k$ is the distance from the fracture within the matrix cell and $V_{ij}$ the volume of the matrix cell. This allows a proper accounting for the reduced influence of a fracture segment on a matrix cell if the fracture segment does not cross the matrix cell through its center. In many cases equation \ref{eq1:d_mean} has to be evaluated by numerical integration. For rectangular grids however, there exists an analytical solution \citep{hajibeygi2011,pluimers2015}. For enhanced efficiency, the analytical expressions are used in our implementation.
\subsection{Fracture intersections}
Fracture intersections significantly impact flow dynamics in the reservoir. The additional fracture-fracture transmissivity can be calculated as
\begin{equation}
T_{i,j}= \frac{\alpha_i \cdot \alpha_j}{\alpha_i + \alpha_j} \quad \textrm{with } \alpha_i = \frac{b_i \Xi_i}{0.5\cdot \Delta x^f}
\end{equation}
where $b_f^i$ denotes the fracture aperture, $\Xi_i$ the total mobility and $\Delta x^f$ the numerical discretization spacing in the fracture \citep{karimi2003}.
\subsection{Time-discretization}
The time derivatives in equations \ref{eq1:pressure} and \ref{eq1:heat} are treated using the backward Euler method, which is an implicit time-discretization with local truncation error $\mathcal{O}(h^{2})$. The method is unconditionally stable theoretically allowing arbitrarily large time steps. In practice, when encountering non-linear behavior, such as the temperature- and pressure-dependent evolution of fluid density, issues with non-convergence might appear and place an indirect restriction on the time-step. Nonetheless, much larger time steps are allowed in the implemented method when compared to explicit schemes.
\subsection{Solution strategy}
We adapt a serial iterative scheme in order to accurately account for the coupling between the pressure and transport equations. In strongly coupled problems, multiple iterations must be used to capture any arising nonlinearities. In most cases, the flow and transport exhibit rather loose coupling in which only a few iterations are needed to converge to the solution. If on the other hand, fracture stability ceases and permeability enhancement in unstable fracture parts is used, the number of iterations might increase significantly and even limit the timestep.
\section{Results}\label{sec1:results}
We present the results of three benchmark experiments and an application experiment that provide insight into the capabilities of THERMAID and validate the implemented method. Fracture permeability and aperture are treated as independent from each other in the following. This allows simulating 'filled' fractures with relatively high aperture and comparably small permeability and allows fracture permeability estimates independent of Cubic law.
First we validate the implemented model with a simple flow problem independently. We then evaluate the coupled results of fluid flow and heat transport on a simple geometry and on a more realistic complex fracture network. The final numerical experiment is the application of the implemented approach to a field scale problem were we take advantage of the implemented fracture stability analysis in order to characterize the stimulated reservoir during injection of a geothermal reservoir.
\subsection{Validation of the pressure equation}
In order to validate the implementation of flow equations of the model, we use an analytical solution for the steady-state flow in a porous medium in the presence of a fracture \citep{strack1982,kolditz2012}. Figure \ref{fig1:setup_H} shows the benchmark geometry, a square with a length of 10 m with a 2m-long inclined fracture in the center of a square domain. The aperture of the fractures is fixed at $b = 0.05m$. Uniform flow is maintained by imposing a specific discharge $q_0$ from the left boundary into the domain. To compare numerical results with the analytical solution, pressures calculated by the analytical solution are used at the lateral boundaries, i.e. $p_{in} = 49646$ Pa and $p_{out} = -49646$ Pa (cf. Figure \ref{fig1:setup_H}). On the top and bottom a no-flow Neumann boundary is applied. The remaining material properties of the numerical model are shown in Table \ref{tab:param}.
\begin{table}[!hb]
\caption{Model parameters used for the inclined fracture solution.}
\vspace{-0.5cm}
\begin{center}
\begin{tabular}{|l|l|l|l|}
\hline
& Parameter & Value & Unit\\ \hline
$\alpha$ & Fracture angle & 45 & $^{\circ}$\\ %
$b_{max}$ & Maximum fracture aperture & 0.05 & m\\ %
$L $ & Fracture length & 2 & m \\ %
$k^m$ & Matrix permeability & $1\cdot 10^{-12}$ & m$^2$ \\ %
$k^f$ & Fracture permeability & $1\cdot 10^{-10}$ & m$^2$ \\ %
$\mu$ & Fluid viscosity & $1\cdot 10^{-3}$ & Pa s \\ %
$q_0$ & Specific discharge & $1\cdot 10^{-4}$ & m s$^{-1}$ \\ \hline
\end{tabular}
\label{tab:param}
\end{center}
\vfill
\centering
\includegraphics[width=0.55\linewidth]{setup_H.png}
\caption{Numerical setup to evaluate the performance of the flow equations solution. Incompressible fluids are assumed in this benchmark experiment. On the left boundary a constant pressure of 49646Pa is assumed. The right boundary is set to -49646Pa to enforce the specific discharge $q_0$. On the other boundaries a no-flow boundary condition is applied.}
\label{fig1:setup_H}
\end{table}
The pressure distribution obtained by THERMAID is shown in Figure \ref{fig1:bench_H}a. The lateral uniform flow is disturbed in the vicinity of the inclined fracture where the flow is faster than in the surrounding porous media. Figure \ref{fig1:bench_H}b shows the pressure profile along a diagonal line from the bottom-left to the top-right. The results show very good agreement between the numerical solution obtained by THERMAID and the analytical solution. We quantify the difference between our model with the reference by the 'normalized root mean squared error (NRSME)' as well as the 'normalized mean absolute error (NMAE)'.
\begin{align}
NRSME &= \frac{\sqrt{\frac{\sum_{i=1}^n \left(x_i - x_i^{ref}\right)^2}{n}}}{max(x_i^{ref}) - min(x_i^{ref})} \\
\nonumber \\
NMAE &= \frac{\frac{\sum_{i=1}^n |x_i - x_i^{ref}|}{n}}{max(x_i^{ref}) - min(x_i^{ref})}
\end{align}
We decided to use two measures of performance due a recent debate on both measures \citep[e.g.][]{willmott2005,chai2014}. Especially \citep{chai2014} suggest that a combination of measures is required to assess model performance. We observe errors of well below $1\%$ (NRSME:
$0.21\%$ and NMAE: $0.19\%$) that validate the implementation of the fluid flow equations.
\begin{figure}[htbp]
\centering
\subfloat[]{ \includegraphics[width=0.44\textwidth]{bench_H_matrix.png}}
\hfill
\subfloat[]{\includegraphics[width=0.46\textwidth]{bench_H.png}}
\caption{a) Pressure field computed for the flow field including a single inclined fracture. b) Comparison between simulated (continuous red curve) and analytical derived (empty black circles) pressure distribution along diagonal from bottom left to top right of the model}
\label{fig1:bench_H}
\end{figure}
\subsection{Validation of the heat transport equation}
We validate the coupled flow and heat transport equations using a benchmark geometry that consists of two perpendicular 5m-long fractures intersecting in the middle of a square domain (cf. Figure \ref{fig1:setup_TH2}). The aperture of the fractures is fixed at $b = 1mm$. The domain is 100m by 100m square domain with Dirichlet boundary conditions on the left and right sides. On the left a constant pressure of 10MPa is applied, whereas the right side is fixed to 0MPa. On the top and bottom a no-flow Neumann boundary is applied. The domain is initially at $T_0=180^{\circ}C$, which is a typical temperature for economic heat extraction in a geothermal reservoir. The inflow temperature at the left side of the domain is set to $T_{in}=50^{\circ}C$ (cf. Figure \ref{fig1:setup_TH2}). The material parameters for this benchmark were chosen realistically and are shown in Table \ref{tab1:param_1}. The benchmark's results are evaluated after 40 years of simulation. The matrix domain is discretized by 301x301 cells while the fractures are modeled by 304 fracture segments (152 each). The reference solution is computed by \textit{COMSOL} on a conforming discrete fracture network with a high resolution grid.
\begin{figure}[!ht]
\centering
\vspace{0.25cm}
\includegraphics[width=0.55\linewidth]{7_setup_TH2.png}
\caption{Numerical setup to evaluate the performance of the coupled flow and heat transport solution. A simple fracture geometry and incompressible fluids are used. On the left boundary a constant pressure of 10MPa at 50$^{\circ}$C is applied. The right boundary is set to 0Pa. On the other boundaries a no-flow boundary condition is applied. The interior has an initial temperature of 180$^{\circ}$C. All parameters for this model setup are shown in Table \ref{tab1:param_1}.}
\label{fig1:setup_TH2}
\end{figure}
\begin{table}[!htbp]
\caption{Properties used in the coupled flow and heat transport models. Superscripts: f - fracture, m - matrix. Subscripts: f - fluid, r - rock.}
\begin{center}
\begin{tabular}{|l|l|l|}
\hline
Permeability & $k^{f} = 1\cdot 10^{-11}m^2$ & $k^m = 10^{-16} m^2$\\ \hline
Porosity & $\phi^{f} = 0.3$ & $\phi^m = 0.3$\\ \hline
Density & $\rho_{f} = 1\cdot 10^{3}\frac{kg}{m^3}$ & $\rho_{r} = 2.5\cdot 10^{3}\frac{kg}{m^3}$\\ \hline
Viscosity & $\mu_{f} = 1\cdot 10^{-3}Pa\cdot s$ & \\ \hline
Specific heat & $c_{p_f} = 4000 \frac{J}{kg\cdot K}$ & $c_{p_r} = 1000 \frac{J}{kg\cdot K}$\\ \hline
Heat conductivity & $\lambda_f = 0.5\frac{W}{m\cdot K}$ & $\lambda_r = 2.0\frac{W}{m\cdot K}$\\ \hline
\end{tabular}
\label{tab1:param_1}
\end{center}
\end{table}
Figure \ref{fig1:TH2_fracture} shows the temperature in both fractures after 40 years of coupled flow and heat transport simulation. Additionally, Table \ref{tab1:TH2_error} shows the quantitative error analysis for the fracture temperatures. We observe a very good agreement between the temperature distribution in both fractures with the reference solution. The horizontal fracture presents changes in temperature over most its extent, which is in accordance with the principal flow direction. As the vertical fracture is not aligned with the flow, a rather homogeneous temperature decrease is observed to about $140^{\circ}C$ after 40 years. This is in good agreement with the matrix temperatures at the position of the fracture. Nevertheless, a significant change in temperature is observed close to the intersection of both fractures. Here the fracture-fracture interaction is clearly visible as both fractures show nearly identical temperatures at the intersection (cf. Figure \ref{fig1:TH2_fracture}). The quantitative error analysis shows differences between our solution and the reference of $\sim 0.8\%$ for the vertical fracture and $\sim 0.1\%$ for the horizontal fracture although the two error measures differ slightly (cf. Table \ref{tab1:TH2_error}).
\begin{figure}[!ht]
\centering
\includegraphics[width=0.92\linewidth]{11_TH2_fracture.png}
\caption{Fracture temperatures through vertical and horizontal fractures. For both fractures we see very good agreement between the implemented method and the reference solution.}
\label{fig1:TH2_fracture}
\end{figure}
\begin{table}[!hb]
\caption{NRSME and NMAE errors for the first coupled fluid flow and heat transport equation benchmark.}
\begin{center}
\begin{tabular}{|r|l|l|}
\hline
& $\mathbf{T_{vertical}}$ & $\mathbf{T_{horizontal}}$ \\ \hline
\textbf{NRMSE [\%]}& $0.89$ & $0.16$\\ \hline
\textbf{NMAE [\%]}& $0.71$ & $0.11$ \\ \hline
\end{tabular}
\label{tab1:TH2_error}
\end{center}
\end{table}
Ultimately, the benchmark shows that our model accurately solves the coupled flow and heat transport equations for this geometry. The simulated time-frame is consistent with the estimated lifetime of a typical enhanced geothermal reservoir and additionally shows that the implemented time-marching scheme is accurate for the problem at hand.
\subsection{Validation of the heat transport equation on a complex fracture network}
We evaluate the coupled flow and heat transport on a more complex fracture geometry. The geometry (Figure \ref{fig1:setup_TH}) consists of a total of 13 fractures within a square domain. Boundary and initial conditions are equal to the previous experiment. The fracture aperture is set to $b = 0.5mm$. The remaining parameters governing the heat transport are consistent with the benchmark in the last section and shown in Table \ref{tab1:param_1}. We evaluate the results after 40 years of simulation. The reference solution computed by \textit{COMSOL} contains 419'594 DOF. In this experiment we evaluate also grid dependence of the implemented model by comparing the results for different resolution simulations with the reference.
\begin{figure}[!htbp]
\centering
\includegraphics[width=0.5\linewidth]{8_setup_TH.png}
\caption{Numerical setup to evaluate the performance of the coupled flow and heat transport solution with a more realistic complex fracture geometry. Incompressible fluids are used. On the left boundary a constant pressure of 10MPa at 50$^{\circ}$C is applied. The right boundary is set to 0Pa. On the outer boundaries a no-flow boundary condition is applied. The interior has an initial temperature of 180$^{\circ}$C. All parameters for this model setup are shown in Table \ref{tab1:param_1}.}
\label{fig1:setup_TH}
\end{figure}
In the previous section we focused on the temperature distributions in the fractures. Here we take a closer look at the matrix temperature distributions. Figure \ref{fig1:TH_matrix}a shows the final pressure distribution for a matrix grid resolution of 301x301. The temperature distribution in the domain after 40 years of simulation is shown in \ref{fig1:TH_matrix}b. Both pressure and temperature fields show a heterogeneous distribution due to the influence of the fractures.
\begin{figure}[htbp]
\centering
\subfloat[]{\includegraphics[width=0.49\textwidth]{12_TH_pressure.png}}
\hfill
\subfloat[]{ \includegraphics[width=0.49\textwidth]{13_TH_temp.png}}
\caption{a) Pressure distribution for the complex fracture geometry. The heterogeneous pressure distributions shows the significant influence of the fractures. b) Final temperature distribution in the matrix for the complex fracture geometry. The heterogeneous pressure distributions leads to inhomogeneous fluid velocities, which is consequently shown in the temperature evolution.}
\label{fig1:TH_matrix}
\end{figure}
Figures \ref{fig1:TH_error}a and \ref{fig1:TH_error}b show the percental deviation of our solution from the reference for the matrix grid resolution of 301x301 of our model. The pressure solution shows only small errors with a NRMSE of $0.35\%$. In the lower third of the domain between 20m and 60m in $x$-direction, a region of elevated error ($\sim 1\%$) is present (cf. Figure \ref{fig1:TH_error}a). Bigger deviations are visible close to some fracture tips where typically on the high-pressure (inflow) side of the fracture our model overestimates the matrix pressure compared to the reference. The low-pressure (outflow) sides of the fractures show predominantly underestimations of pressure. Interestingly, fractures that exhibit error concentration around one of the tips, do not necessarily show the opposite error on the other side of the fracture. Maximum pressure deviations from the reference are below $\pm 5\%$. \\
The errors in the temperature distribution are generally larger than for the pressure. Figure \ref{fig1:TH_error}b shows the percentage error at the final stage of the simulation for a matrix grid discretization of 301x301. Compared with the error in the pressure solution, we find that our model seems to always overestimate the matrix temperature compared to the reference. The normalized RMS error for this resolution is $2.22\%$. We suspect that the elevated temperature deviations are caused by the relatively small error in the pressure solution. The small error in the pressure leads to comparably larger differences in flow velocities that are controlling heat advection. Thus, over a simulation of 40 years this error accumulates to the values observed here.
\begin{figure}[htbp]
\centering
\subfloat[]{\includegraphics[width=0.49\textwidth]{14_TH_err_p.png}}
\hfill
\subfloat[]{ \includegraphics[width=0.49\textwidth]{15_TH_err_T.png}}
\caption{Deviation from reference solution: \textbf{a)} Pressure. A region of elevated error is visible in the lower part of the domain. Significant deviations are also visible at some fracture tips. \textbf{b)} Temperature. Measured at the final stage of the simulation. The error in the pressure solution are reproduced in the temperature solution. The matrix temperatures are generally overestimated compared with the reference.}
\label{fig1:TH_error}
\end{figure}
We further want to investigate the influence of the resolution on the accuracy of the results, so we compare four different resolutions (101x101, 301x301, 501x501 and 1001x1001). We investigate the improvement of solution accuracy in the pressure and heat transport solutions by using the NRMSE and NMAE values compared to the high resolution solution obtained by \textit{COMSOL}. Table \ref{tab1:TH_error_resolution} shows both error measurements for all resolutions. We find a general improvement of the accuracy with an increase in resolution. For the temperature, this is a decrease in NRMSE from $4.6\%$ (101x101) to $1.2\%$ (1001x1001). The pressure error is consistently about one magnitude smaller, showing a decrease from $0.78\%$ (101x101) to $0.15\%$ (1001x1001). Overall we find a significant increase in accuracy with an increase in resolution. Nevertheless, the deviation is not changing significantly between 501x501 and 1001x1001 ($1.74\%$ vs $1.22\%$ in case of the temperature). This indicates a systematic difference between the reference solution and our method. There are multiple possible origins of this systematic error. Since we observe the systematic deviation also in the pressure, we think it is likely to be a difference in methodology concerning the fluid flow equation. These differences could include the treatment of fracture-fracture intersections, the definition of matrix-fracture interface permeability, and inherent numerical differences between finite element and finite volume methods. Nevertheless, we find very good agreement between the reference simulation and our implementation for large parts of the model. Even in regions of significantly elevated deviation, we find acceptable agreement with differences below $10\%$ between the two methods. The definitive source of the difference is currently not resolved but presents excellent future research opportunities.
\begin{table}
\caption{NRSME and NMAE errors for the second coupled fluid flow and heat transport equation benchmark in dependence of resolution.}
\begin{center}
\begin{tabular}{|r|l|l|l|l|}
\hline
& $\mathbf{p^m}$ & & $\mathbf{T^m}$ & \\ \hline
& \textbf{NRMSE [\%]}& \textbf{NMAE [\%]}& \textbf{NRMSE [\%]}& \textbf{NMAE [\%]} \\ \hline
\textbf{101 x 101} & 0.78 & 0.64 & 4.59 & 4.0 \\ \hline
\textbf{301 x 301} & 0.35 & 0.26 & 2.22 & 1.87 \\ \hline
\textbf{501 x 501} & 0.23 & 0.17 & 1.74 & 1.51 \\ \hline
\textbf{1001 x 1001} & 0.12 & 0.12 & 1.11 & 1.02 \\ \hline
\end{tabular}
\label{tab1:TH_error_resolution}
\end{center}
\end{table}
\subsection{Utilization of the fracture stability analysis}
We present the fracture stability analysis to show the influence of permeability enhancement and thermal stress on fracture stability. We model fluid injection into a complex fracture network with a range of fracture orientations. The geometry consists of a total of 196 fractures within a square domain (Figure \ref{fig1:setup_FSA}). The borehole is located in the middle of the domain with an open hole section of 6m. The fracture aperture in the reservoir is set to $b = 0.1mm$. The remaining parameters used in this section are shown in Table \ref{tab1:param_2}. The upper limit of the fractured reservoir domain is assumed to be at 5km depth. The injection pressure is held constant at 25MPa. The principal stresses are oriented as shown in Figure \ref{fig1:stress_systems}, which corresponds to a normal faulting regime. The magnitudes of the principal stresses are 125MPa, 107.5MPa and 81.25 MPa respectively, which corresponds to a relative stress ratio of $R=0.4$. The in-situ pore pressure is assumed to be hydrostatic ($\sim 50$MPa). The stress conditions roughly resemble the relative conditions at the Fenton Hill and Hijiori geothermal projects although both projects were situated above 4km depth \citep{xie2015,barton1988,oikawa2000}. We evaluate the results after 10 days of continuous fluid injection.
\begin{figure}[!htbp]
\centering
\includegraphics[width=0.65\linewidth]{16_setup_FSA.png}
\caption{Numerical setup to evaluate the fracture stability on a realistic complex fracture geometry. A constant injection pressure of 25MPa is applied in the borehole (blue line). On the outer boundaries a no-flow boundary condition is applied. All parameters for this model setup are shown in Table \ref{tab1:param_2}.}
\label{fig1:setup_FSA}
\end{figure}
\begin{figure}[!htbp]
\centering
\includegraphics[width=0.90\linewidth]{setup_FSA_stress.png}
\caption{Principal stress orientations in the fracture stability analysis. The principal stress field is aligned with a NED coordinate system. The orientation of the reference plane (simulation plane) within the principal stress field is shown as well as the resulting stress orientations in 2D view.}
\label{fig1:stress_systems}
\end{figure}
\begin{table}[!htb]
\caption{Properties used in the fracture stability analysis model. Superscripts: f - fracture, m - matrix. Subscripts: f - fluid, r - rock.}
\begin{center}
\begin{tabular}{|l|l|l|}
\hline
Permeability & $k^{f} = 1\cdot 10^{-12}m^2$ & $k^m = 10^{-18} m^2$\\[5pt] \hline
Porosity & $\phi^{f} = 0.9$ & $\phi^m = 0.1$\\[5pt] \hline
Density & $\rho_{f} = 1\cdot 10^{3}\frac{kg}{m^3}$ & $\rho_{r} = 2.5\cdot 10^{3}\frac{kg}{m^3}$\\[5pt] \hline
Viscosity & $\mu_{f} = 1\cdot 10^{-3}Pa\cdot s$ & \\[5pt] \hline
Specific heat & $c_{p_f} = 4000 \frac{J}{kg\cdot K}$ & $c_{p_r} = 1000 \frac{J}{kg\cdot K}$\\[5pt] \hline
Heat conductivity & $\lambda_f = 0.5\frac{W}{m\cdot K}$ & $\lambda_r = 2.0\frac{W}{m\cdot K}$\\[5pt] \hline
Thermal expansion. coeff.& $\alpha = 7.9\cdot 10^{-6} K^{-1}$ & \\[5pt] \hline
Shear modulus& $G = 29.0 GPa$ & \\[5pt] \hline
Poisson ratio & $\nu = 0.25$ & \\[5pt] \hline
\end{tabular}
\label{tab1:param_2}
\end{center}
\end{table}
\begin{figure}[!htbp]
\centering
\includegraphics[width=0.9\linewidth]{17_FSA_pressure_case1.png}
\caption{Matrix pressure in the reservoir after 10 days of injection. Due to the orientation of the fractures a preferential flow direction in the vertical direction is visible.}
\label{fig1:FSA_pressure_1}
\end{figure}
Figure \ref{fig1:FSA_pressure_1} shows the pressure distribution after 10 days of injection. Due to the orientation of the pre-existing fractures, a preferential flow direction in the vertical direction is visible. Slight pressure changes due to the injection are measured at distances up to 55m in the vertical and 35m in the horizontal directions from the injection point. The zone of 10MPa pressure changes extends roughly 10m around the borehole. Very high pressures $>20$MPa are concentrated in the direct vicinity of the injection.\\
The in-situ fracture stability is influenced by the additional injected fluid pressure. Figure \ref{fig1:FSA_1} shows the final normalized fracture slip tendency. Note that a normalized slip tendency value of 1 represents a fracture that is eligible for failure and slip. We observe a range of values in the reservoir based on the fractures' orientations. The average fracture stability is high with values well below the failure condition. However, closer to the injection the increased slip tendency due to the injection is visible. Zones with fluid overpressure of $>5$MPa show significant increase in slip tendency (yellow colors in the plot). The region with at least 10MPa additional fluid pressure is very close to or eligible for slip on the fracture.
\begin{figure}[!ht]
\centering
\includegraphics[width=0.9\linewidth]{18_FSA_case1.png}
\caption{Fracture stability in terms of normalized slip tendency in the reservoir after 10 days of injection. Values are normalized by the friction coefficient $\mu=0.6$. High values denote higher slip tendency. The general fracture stability is good at levels well below the instability condition. Stability reduces closer to the injection. Very close to the injection point high fluid pressures lead to unstable fracture segments (red). }
\label{fig1:FSA_1}
\end{figure}
\paragraph{Permeability enhancement}
In the previous section fracture segments eligible for slip did not have any feedback on the fluid pressure distribution. Here we investigate this feedback by introducing the stepwise permeability enhancement for failing fractures. The setup used is identical to the previous section except a 10-fold increase in permeability is assumed for failing fracture segments.\\
Figure \ref{fig1:FSA_pressure_2} shows the pressure distribution after 10 days of injection if permeability enhancement is considered. Although the general flow directions remain unchanged, the fluid pressure distribution shows significant differences in extent and magnitude.
\begin{figure}[!htb]
\centering
\includegraphics[width=0.9\linewidth]{19_FSA_pressure_case2.png}
\caption{Matrix pressure in the reservoir after 10 days of injection if permeability enhancement is used. Here a enhancement factor of $\gamma=10$ is used. Due to the constant injection pressure and increased permeability in unstable fracture segments, the high-pressure zone is increased drastically.}
\label{fig1:FSA_pressure_2}
\end{figure}
Fracture stability is changed drastically if permeability enhancement is considered. The in-situ fracture stability remains unchanged in the outer regions of the domain at very stable levels. On the other hand, most of the fractures within the overpressured regions show elevated slip tendency with fractures closer to the injection being eligible for slip. Compared to the previous simulation, 20-times more fracture segments are unstable and capable of slip. Failing fractures, which have increased in permeability allow fluid to propagate more easily. As we assume a constant pressure injection, the amount of injected fluid is increased significantly. In this way a much larger stimulated area is observed compared to the case without permeability enhancement.\\
Thermal stress has only a small influence during the relatively short injection period of 10 days in this simulation. The resulting thermal stress distribution is shown in Figure \ref{fig1:FSA_2} and shows thermal stresses concentrated at the borehole. As the fractures within the vicinity of significant thermal stress are eligible for slip also by the injection fluid pressure, no additional unstable fracture segments are observed. However, in a recent study investigating the role of thermal stress in a geothermal reservoir in detail we found that thermal stresses can facilitate slip on non-optimally oriented fractures, and this is especially important in long-term injection scenarios where the thermal stress changes become more significant with time \citep{jansen2017}. \\
\begin{figure}[!htb]
\centering
\subfloat[]{\includegraphics[width=0.49\linewidth]{20_FSA_case2.png}}
\hfill
\subfloat[]{\includegraphics[width=0.49\linewidth]{21_FSA_thermal_stress_case2.png}}
\caption{\textbf{a)} Fracture stability in terms of normalized slip tendency in the reservoir after 10 days of injection for the case of permeability enhancement. Values are normalized by the friction coefficient $\mu=0.6$. High values denote higher slip tendency. Stability drastically reduces closer to the injection as the high fluid pressure zone is much bigger if permeability enhancement is used (red). \textbf{b)} Thermal stress after 10 days of injection into the fracture network. The thermal stress is concentrated close to the injection well. The color-scale in the figure starts at 0.25MPa with the darkest blue. Everything below is neglected in the graphical representation and shown in the background color. Note that the absolute value of the thermal stress is shown and all thermal stress here is tensional.}
\label{fig1:FSA_2}
\end{figure}
The experiments presented here show the importance of fracture stability analysis. We showed that a stepwise permeability increase in potentially failing fracture segments has a major impact on the stimulated reservoir volume and allows fracture slip in larger parts of the domain. This emphasizes the importance of coupling thermo-hydraulic models with the mechanical changes during fracture slip.
\section{Conclusion}\label{sec1:conclusion}
We developed, implemented and validated a fractured reservoir modeling framework in \textit{MATLAB} for investigating coupled thermo-hydraulic problems including fracture stability analysis.
Our results show with high confidence that the accuracy of the implemented \textit{MATLAB} package are within the limits of commercial simulators for fractured reservoirs. Especially the results of the coupled flow and heat transport on a complex fracture network show the importance of discrete fractures in numerical analysis of fractured reservoirs. Both pressure and temperature distributions show heterogeneities due to fracture-matrix interactions. THERMAID presents easy access to the underlying implementation that enables rapid prototyping as well as detailed investigations of the embedded discrete fracture model and coupled processes in naturally fractured reservoirs. \\
As discussed earlier in the results, the deviations in the pressure solution could be caused by different treatment of fracture-fracture intersections or the definition of matrix-fracture interface permeability between the models. Currently there is no clear indication about which weighting to use at fracture-matrix interfaces, which presents an excellent future research opportunity for combined laboratory and numerical experiments. We assume that differences in the temperature solutions are caused by the deviations in the pressure solution that are magnified with time.
We showed that the embedded discrete fracture model is a viable alternative to the existing methods. As numerical discretization is simplified compared to conforming discrete fracture models, dynamic changes of the fracture network are possible without large numerical overhead. The extension of the embedded discrete fracture model to three dimensions has not been discussed so far in this article. Due to the relatively simple numerical discretization the extension to three dimensions is feasible. However, THERMAID is currently only developed in a 2D version. This is however not a limitation of the embedded fracture model but due limiting factors of the achievable computational performance in \textit{MATLAB}. Nevertheless, the approach taken in THERMAID could be efficiently re-implemented and extended to 3D in a high-performance computing environment. The embedded discrete fracture model is not necessarily restricted to regular grids and can be extended to general geometries. However, using regular grids can be advantageous for the application of massively parallel computation techniques to further increase computational efficiency and enable large scale, high resolution simulations.\\
Our results show the importance of including the mechanical behavior of fractures and the reservoir in thermo-hydraulic simulations. Although the deformation process during fracture slip was not explicitly taken into account, the assumed step-wise increase in fracture permeability during slip provides the necessary feedback for the pressure equation in order to capture the observed increase in injectivity during hydraulic stimulation. We propose that changes in permeability and aperture should be incorporated in all models that seek to fully understand the thermo-hydraulic evolution during fluid injection in fractured reservoirs. Although our model exhibits a very simplified view on the complex fracture mechanics, it still provides important insight into reservoir stimulation that helps in identifying some challenges and opportunities for future studies. More advanced models currently under development will consider both pre-existing fractures as in the present work, but also the generation of new fractures in response to the evolving stress state from both thermo-and hydraulic perturbations. Future models might also include fracture roughness and solve the full equilibrium equations to estimate aperture changes that influence permeability. Recently, progress in this direction has been made using boundary element methods, multi-point stress approximations (MPSA) and the novel extended finite volume method (XFVM) \citep{norbeck2016,ucar2016,deb2016}. However, these models are not yet as computationally efficient as to allow an adaption for THERMAID. Currently, induced seismicity can not be quantified in terms of magnitude because slip on the fracture is not computed. Moreover, fracture slip can occur in a seismic or aseismic manner, thus further complicating the assessment of induced seismicity. These are all areas that we are currently pursuing in order to extend and refine THERMAID's capabilities. \\
Although this paper focuses on the application of enhanced geothermal systems, other possible applications for THERMAID include seasonal thermal energy storage in fractured aquifers, and natural or anthropogenic fluid-driven earthquake sequences. Furthermore, a wide range of research questions related to fractured reservoirs and their properties can be addressed using THERMAID. Even beyond the current model capabilities, we expect further applications and research opportunities because the open source code will allow a community to evolve and contribute to this common platform. The open source distribution and GNU GPL v3.0 license enables the scientific community to use and modify THERMAID to their needs. The implementation in \textit{MATLAB} ensures that even novice programmers can easily understand the underling equations and their implementation and develop their own numerical models based on the examples provided with THERMAID. Simulation of coupled processes in fractured reservoirs is becoming increasingly important in today's research. With THERMAID we present an alternative starting point from which new insight can be gained into the complex coupled processes in fractured domains in the subsurface.
\section*{Acknowledgements}
We thank the Swiss National Fond (SNF), for the financial support through the grant 'NFP70:Energy Turnaround' under the project no. 153971.
\section*{Computer Code Availability}
\begin{itemize}
\item \textbf{Project name:} THERMAID
\item \textbf{Project home page:} https://github.com/gujans/THERMAID
\item \textbf{Referenced archived version DOI:} 10.5281/zenodo.1175829
\item \textbf{Operating system(s):} Platform independent
\item \textbf{Programming language:} MATLAB
\item \textbf{Other requirements:} built and tested with MATLAB R2015b
\item \textbf{Licence:} GNU GPL v3.0 or later
\item \textbf{Any restrictions to use by non-academics:} The terms of the GNU GPL v3.0 or later apply.
\end{itemize}
\section*{Competing interests}
The authors declare that they have no competing interests.
\section*{Funding}
This work was supported by the Swiss National Fond (SNF), for the financial support through the grant 'NFP70:Energy Turnaround' under the project no. 153971.
\section*{Bibliography}
\bibliographystyle{abbrvnat}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 2,516 |
More Praise for the First Edition of _Make Your Contacts Count_
"A must-read for anyone who wants to get a job or make a career change."
—Lisa Keathley, Career Development Center Director, School of Foreign Service, Georgetown University School of Foreign Service
"This book is the first tool I recommend to the people I coach."
—Linda Marks, Corning, Inc.
"Baber & Waymon show how to connect for the right reasons, ones that make others feel important and valued, rather than used."
—Bruce Nolan, Director, Treasury Executive Institute, U.S. Government
"One of the top books ever produced on the subject."
—Dinah Adkins, National Business Incubation Association
"Essential knowledge for any professional or businessperson."
—Jack Cole, Ph.D., The Johns Hopkins University
"A must-read to gain the competitive edge."
—Anne Kelly, CEO, Federal Consulting Group
"As the world becomes flatter, our relationships become even more important. Unless you live in a cave, insolated from the world, you need this book."
—Ane Powers, The White Hawk Group, LLC
# Make Your Contacts Count
**Networking Know-How for Business and Career Success**
Second Edition
Anne Baber
Lynne Waymon
American Management Association
New York • Atlanta • Brussels • Chicago • Mexico City • San Francisco
Shanghai • Tokyo • Toronto • Washington, D.C.
_Special discounts on bulk quantities of AMACOM books are available to corporations, professional associations, and other organizations. For details, contact Special Sales Department, AMACOM, a division of American Management Association, 1601 Broadway, New York, NY 10019.
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_Library of Congress Cataloging-in-Publication Data_
_Baber, Anne_
_Make your contacts count : networking know-how for business and career success / Anne Baber and Lynne Waymon.—2nd ed_.
_p. cm_.
_Includes index_.
_ISBN-13: 978-0-8144-7402-0_
_ISBN-10: 0-8144-7402-0_
_1. Career development. 2. Business networks. 3. Social networks. 4. Interpersonal relations. 5. Business etiquette. 6. Success in business. I. Waymon, Lynne. II. Title_.
_HF5381.B143 2007_
_650.1'3—dc22_ | _2006031977_
---|---
_© 2007 Anne Baber and Lynne Waymon.
All rights reserved.
Printed in the United States of America_.
_This publication may not be reproduced, stored in a retrieval system, or transmitted in whole or in part, in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior written permission of AMACOM, a division of American Management Association, 1601 Broadway, New York, NY 10019_.
_Printing number_
_10 9 8 7 6 5 4 3 2 1_
## Contents
**Preface: Get Ready for State-of-the-Art Networking**
The Time Is Right
Are You Ready?
The Contacts Count Networking System
**Part I: Survey Your Skills and Mindset**
**Chapter 1: Assess Your Skills**
Instructions
Observing the "Netiquette"
Assessing Your Comfort Level
Being Strategic
Meeting People
Using Networking Organizations
Making the Most of Events
Achieving Bottom-Line Results
Following Through
Check Your Results
Next Steps
**Chapter 2: Change Your Mindset**
Come In from the Cold
The Ten Biggest Misconceptions About Networking
Ten Turnoffs in the Language of Networking
You Say You're Shy?
Catch Your Critic
Convert Your Critic into Your Coach
Believe the Best About Yourself and Others
Bonus: Know Your Style
**Part II: Set Your Strategy**
**Chapter 3: Teach Trust**
Move from Taking to Trusting
Teach That You Can Be Trusted
The Trust Matrix
Avoid Manipulation
**Chapter 4: Develop Your Relationships**
Move Through the Six Stages
The Next Move Is Up to You
Rate Your Relationships
Have Questions About the A's?
**Chapter 5: Go with Your Goals**
Size Your Project to Match Your Goal
Check Out Your Choices
Assess Your Network
Plan Your Strategic Positioning Project
Bonus: Get Off to a Good Start
**Part III: Sharpen Your Skills**
**Chapter 6: Know the "Netiquette"**
Enter Enthusiastically
Brighten Up Your Body Language
ENGAGE Your Partner
Tune Up Your Tone of Voice
Consider Closeness
Watch What You Put in Your Mouth
Treat Touching as Taboo
Forego Flirting
Pay Your Way
Exchange Business Cards Effectively
Join Groups Comfortably
Bonus: Ten Tips on the Nuances of "Netiquette"
**Chapter 7: Avoid the Top Twenty Turn-Offs**
**Chapter 8: "Who Are You?"**
Why Remembering Names Is Hard
Learn Someone's Name
Teach Your Name
Try These Twenty Tips
Break Up Bunches of Introductions
Deal Skillfully with Forgotten Names
Give Yourself a Tagline
Yes, Mind Your Manners
The Introduction Rule: FIRST IS FOREMOST
**Chapter 9: "What Do You Do?"**
Why Most Answers Bomb
Make the Right Things Happen
Give It Your BEST
Be Interesting
Try These Tips
Read These Frequently Asked Questions
**Chapter 10: "What Are We Going to Talk About?"**
Listen for Your Cue
Use Success Stories to Tell What's New
Figure Out Your Agenda
Begin with the Right Side
What Do You Have To Give?
What Do You Want To Get?
Give and Get with Ease
Practice Agenda-Making
Go Public with Your Agenda
Exchange Something
**Chapter 11: Make Conversation Flow**
Listen Generously
Use Your EARS
How Listening Generously Pays Off
Be Seriously Curious
Tell Success Stories
Construct Your Story Carefully
Sample These Stories
People Want to Know...
**Chapter 12: End with the Future in Mind**
Prepare for the Next Time
Listen for the Bell
Eight Ways To Leave
A Ritual for Leave-Taking
Do You Have Questions?
**Chapter 13: Follow Through**
Focus on Follow Through
Figure Out Your Reasons to Reconnect
Face Your Fears
Fill in the Blanks on Your Calendar
The Five Goals of Follow Through
Freshen Up Your Relationships
Find the Way
Bonus: Five More Ingenious Ways to Fit In Follow Through
**Part IV: Select Your Settings**
**Chapter 14: Network at Work**
Got the Right Word?
Bank On the Benefits
Ten Ways to Get on Board Quickly
Assess Your Corporate Culture
How Strong Is Your Inside Network?
Map Out a Plan
Pair Up with Peers
Avoid Erroneous Assumptions
Overcome the Barriers
Bonus: After Organizational Earthquakes, Rebuild Your Network
**Chapter 15: Make It Rain Clients**
What People Think
Professionalize Your Practice Development
Make Conversations Count
What's One Conversation Worth?
Create Constellations
Cross-Sell Your Clients
Make Asking for Referrals a Ritual
**Chapter 16: (Net)Work from Home**
Tune In to the Trends
Conquer the Challenges
Link Up Your Life and Your Livelihood
Bonus: Create a Constellation
**Chapter 17: Make the Most of Your Memberships**
Size Your Network to Fit Your Needs
Link Up One-on-One
Access Anybody
Join Groups
Choose Groups Strategically
Understand the Hierarchy
Know the Group Before You Join
Orchestrate Who Knows You
The Twelve Biggest Mistakes Members Make
Jump Right In
**Chapter 18: Rev Up Referral Groups**
See How They Run
Shop Around
Check It Out
Don't Just Join, Join In
Start Small
Spice Up the Meetings
Start Your Own
**Chapter 19: Connect at Conventions**
Expand Your Expectations
Get Ready, Get Set: Before You Go
Show Up at the Conference
Follow Up After You Get Home
Later On, Get Re-inspired
Bonus: Plan Meetings That Get People Talking
**Chapter 20: Jump-Start Your Job Hunt**
Use The Contacts Count Networking System
Twenty-Five Tactics to Find a Job Fast
Bonus: Manage Your Strategy Support Group
**Index**
**About the Authors**
## Preface
Get Ready for State-of-the-Art Networking
Everybody has contacts. Not everybody has contacts that count. How about you?
As we've researched, written about, and spoken on networking, we've discovered that only a fraction of people intuitively know how to network. Everybody else can—and must—learn how.
### The Time Is Right
When we began researching our first book in 1987, networking had its niche as a job-hunting and career advancement tool. And women, wanting to break through the glass ceiling, saw how men in business helped each other and called it "the old-boys' network." Women's networking groups were springing up to encourage women to learn the skills and to help each other. Networking was seen as an extracurricular activity, not one sanctioned by or encouraged by employers unless, of course, it was for sales or business development.
Things have changed since then. In the last several years, networking—often under a pseudonym—has become a hot topic. Professors in business schools, senior executives in government, and heads of corporations all tout relationship building as the key to success.
You can hardly pick up a business school journal without seeing an article on networking. University research is exploring such topics as "social capital," "communities of practice," and "horizontal integration."
One of the competencies now required for entry into the Senior Executive Service—the U.S. government's top ranking jobs—is Building Coalitions/Communications. That competency is defined as the ability to explain, advocate, and express facts and ideas in a convincing manner and to negotiate with individuals and groups internally and externally. That competency also requires the ability to develop an expansive professional network with other organizations and to identify the internal and external politics that impact the work of the organization.
Corporate leaders are now seeing the business value of relationship building. The CEO of one Fortune 500 company recently announced an initiative with five elements. Four of those imperatives can only be accomplished as employees become proficient in the concepts and skills we cover in this book. What could be a louder call or clearer definition for networking than this imperative: "Establish enduring, inclusive relationships within (the company) and with our customers, employees, teammates, and community. Enable mutually beneficial partnerships that take full advantage of internal and external synergies. Understand the impact of personal behavior on others, and place a high priority on honesty and integrity."
So, just why is networking so important in today's business environment?
Networking is now _the essential professional competency_ for employees at all levels. They need to develop strategic networking skills and practices to excel at creating, cultivating, and capitalizing on the cross-functional relationships that get things done and affect the bottom line.
Networking is now _the most important tool_ for intelligence gathering. In business settings, such as conferences, trade shows, meetings, and even golf outings, people need leading-edge networking skills to find the latest information on resources, trends, and best practices.
Networking is now _the antidote_ to the coming brain drain as baby boomers retire. Experienced employees need networking expertise so they can pass on their valuable organizational and technical knowledge to newer, younger staff members.
Networking is now _the critical strategy_ for business development. Professionals and entrepreneurs need to know how to gain visibility and credibility in their target markets, and how to build and maintain relationships for long-term growth.
Networking is now _a must-have capability_ for professional association members. Members need networking skills to take advantage of great connections at professional association meetings and conferences and to bring back new ideas and practices into their places of business.
Networking is _the know-how_ for doing business in the U.S. International businesspeople and students need to get comfortable with and competent in the cultural ground rules for building relationships with Americans.
Networking is _the method_ for personnel retention because it creates feelings of inclusion and helps people from diverse backgrounds feel listened to and valued at work.
Networking remains _the primary technique_ that people use to find new jobs, change careers, or land on their feet after a layoff, merger, or reorganization. People who are looking for career advancement need practical networking strategies to become the natural and only choice in the job market.
### Are You Ready?
As you go through life, from graduation to the grave, you'll have thousands of encounters with people in business and social settings.
You know that networking is important. Yet, you may wonder exactly how to create, cultivate, and capitalize on networking relationships and opportunities.
You know you should get networking on your calendar. Yet, you may have trouble trying to fit it into your busy lifestyle.
You know you need other people's help to get your projects, ideas, and initiatives off the ground. Yet, you may be unsure about how to connect with key people to get the job done.
You know that visibility often results in promotability. Yet, you may feel uncomfortable and shy about how to raise your profile in a way that fits your organizational culture.
You know that networking is the way to expand your customer/client list. Yet, you may wish you knew more about planning networking Projects that give you top of the mind awareness with clients and referral sources.
You know you should join groups to network. Yet, you may be unclear about how to select the best ones and how to use them to advance your career and business goals.
You know it's valuable to go to professional meetings, trade shows, and conferences. Yet, you may wish you were better equipped to uncover resources, opportunities, and best practices at these venues.
You know that your executives are pushing teamwork, connectivity, horizontal integration, social capital, and the idea that business development is everybody's job. Yet, you may need to sharpen your interpersonal skills to be a player.
That's why we wrote this book: So that you can make networking an art...not an accident.
### The Contacts Count Networking System
This book, the second edition of our fifth book, presents, for the first time in print, The Contacts Count Networking System (see Figure P-1). Here, you will find step-by-step guides to all aspects of networking. Most important, our System will help you take your networking from scattershot to streamlined and strategic.
Part I of this book gives you the tools to Survey Your Skills and Mindset. Measuring your mastery of various networking skills will give you a baseline. As you complete the comprehensive fifty-question Self-Assessment, you'll be able to see which skills are solidly part of your repertoire, and which ones you'll need to work on. But your skill level is only one part of your starting base. It also makes sense for you to become more conscious of the beliefs you hold about networking so that you're not held back by misconceptions or outdated attitudes.
**FIGURE P-1. The Contacts Count System.**
Part II provides the underlying networking concepts you'll need to Set Your Strategy. After you know where you stand, you're ready for some big-picture concepts. We believe that the key to building relationships is trust. The Trust Matrix portrays the power of demonstrating your Character and Competence with everyone you meet. Once you understand how trust is created, you'll want to see what kinds of relationships are possible as you network. The Six-Stages Model provides a realistic picture of the variety of relationships that are present in anyone's network. Understanding the six stages—and knowing what to do and say at each stage—puts you in charge of the growth and development of your network. As you clarify your goals, you'll network more strategically to achieve them. And sizing your networking Projects to fit your goals will guarantee you get the most from your investment of time and money.
Part III helps you Sharpen Your Skills. Central to our System are the essential face-to-face networking skills that will take you comfortably and professionally from Hello to Goodbye. Because we have worked with people in almost all walks of life, in all kinds of businesses, at every level of the hierarchy, you can be sure that we've covered all the bases. Using focus groups, interviews, and research, we've developed the most complete encyclopedia of real-world, field-tested networking know-how. We've invented models and formulas, collected best practices and examples, created checklists and quizzes, and identified Frequently Asked Questions and the most common networking dilemmas. As you apply this comprehensive and practical spectrum of skills, you will be equipped to reach your goals.
Part IV guides you as you Select Your Settings. If you want to network at work, you'll get the tools you need to assess your corporate culture, identify key people and activities, and build cross-functional relationships that get things done, affect the bottom line, and advance your career. If you want to network in the world, you'll find the tools to help you choose and get the most out of a variety of venues. You'll maximize the effect of your participation in professional associations, referral groups, civic and community organizations, volunteer and networking activities, at conventions and trade shows, and in social situations.
When you're hunting for a job or want to change careers, we'll coach you to apply your networking skills in well-chosen arenas until you hear the words, "You're hired!"
Survey your skills and mindset, set your strategy, sharpen your skills, and select your settings. Use this system to Make Your Contacts Count.
Our best to you,
Anne Baber and Lynne Waymon
P.S.: Contact us at www.ContactsCount.com to tell us about your networking successes.
## PART I
## Survey Your Skills and Mindset
Start where you are. Check out your mastery of real-world skills. Then, explore your beliefs about networking.
Want to find out how you stack up as a networker? Wonder if you understand the subtleties, know the strategies, and are using state-of-the-art skills? To spotlight your strengths and weaknesses, complete the Self-Assessment in Chapter 1. As you look at your results, you'll be ready to set your priorities and decide which chapters to read first. Then, you'll be able to use your time—and this book—in the best way.
Because you see the marketplace value of networking, you'll want to get rid of any ideas that could hold you back. Misconceptions about networking abound. You can clear your mind of them and adopt attitudes that will get you ready to make great connections.
## CHAPTER 1
## Assess Your Skills
Taking the Self-Assessment in this chapter will give you an overview of your specific networking behaviors, attitudes, and strategies.
This exercise will help you.
* Test your current level of mastery of state-of-the-art networking behaviors and beliefs.
* Increase your awareness of the vast repertoire of skills and strategies available to you as you build business relationships.
* Remind yourself of some techniques that you know but don't use as much as you could.
* Pinpoint topics you want to focus on to increase your impact, professionalism, and comfort.
* Verify your increased competency when you take the quiz again, after you've made The Contacts Count Networking System a way of life.
### Instructions
As you go through the Self-Assessment, we want you to know how we define some of the terms we've used. Then, you'll need to know how to select your answers. Finally, after you are finished with the Self-Assessment, you'll need to know how to assess your mastery and decide what to do next.
### Defining Some Terms
_Networking Event:_ All those business, quasi-business, and social situations in which you have opportunities to develop valuable connections.
_Organization:_ Any group you join for the purpose of making business connections (professional association, Chamber of Commerce, alumni group, business referral group, board, etc.).
_Company:_ Who you work for (your firm, your agency, your sole proprietorship, etc.).
### Selecting Your Answers
Below, you'll find eight sections: Observing the "Netiquette," Assessing Your Comfort Level, Being Strategic, Meeting People, Using Networking Organizations, Making the Most of Events, Achieving Bottom-Line Results, and Following Through. Each section concentrates on a specific area of the networking experience. The statements in each section focus on what you believe about networking and what you do and say when you are networking. For each statement, check one of the following as your response:
Rarely | for 0 to 20 percent of the time
---|---
Sometimes | for 20 to 50 percent of the time
Frequently | for 50 to 80 percent of the time
Almost Always | for 80 to 100 percent of the time
With each section, you'll find a commentary that will help you in your self-assessment.
### Observing the "Netiquette"
Look back at your answers as you consider these comments.
If you've ever had an awkward moment as you engaged in a networking activity, you know how daunting it is to feel as if you don't know what to do. As you learn the skills and techniques—and the rationales behind them—you'll find that you'll rarely find yourself in a situation you can't handle with aplomb and confidence.
Handing out lots of business cards isn't networking. See Chapter 6 for the rest of the story.
Observing the "Netiquette"
I talk to discover reasons to hand out my business card.
Rarely _____ Sometimes _____ Frequently _____ Almost Always _____
As I talk with someone, I'm trying to figure out a reason to give him my business card and get his.
Rarely _____ Sometimes _____ Frequently _____ Almost Always _____
I sense when I can begin talking about what I can offer or what my company provides.
Rarely _____ Sometimes _____ Frequently _____ Almost Always _____
I'm comfortable joining a group of people who are already talking.
Rarely _____ Sometimes _____ Frequently _____ Almost Always _____
I consciously work at talking only about 50 percent of the time.
Rarely _____ Sometimes _____ Frequently _____ Almost Always _____
I find interesting ways to say thank you when someone gives me a resource or referral.
Rarely _____ Sometimes _____ Frequently _____ Almost Always _____
If a contact doesn't reciprocate, I skillfully and tactfully point out how she can help me.
Rarely _____ Sometimes _____ Frequently _____ Almost Always _____
Do you worry about seeming too pushy? Too passive? If you're too pushy, you'll turn people off. If you're too passive, you won't get much out of networking.
When you're approaching a group, are you mentally back at the eighth grade dance, wondering if people will snub you? If you know the steps for joining (not breaking into) a group, you'll be able to do it with ease. The process appears in Chapter 6.
Do you, out of nervousness, find yourself chattering away, dominating the conversation? Or do you have a hard time holding up your end of the conversation with Success Stories and important topics to talk about? Give and take is basic to networking. Besides, you have to listen to learn what your contact needs. Chapter 7 will help you avoid all the top 20 networking turn-offs.
Do you say, "Thanks!" in ways that make you memorable, yet are appropriate? Corporate cultures, for example, differ. Appropriate ways to say thank you in IBM are bound to be different from what's done at an ad agency.
Great connectors observe and learn the "netiquette" in particular organizations from the members of those organizations. You can always ask the advice of a mentor at work, when deciding how to say, "Thanks!" Or, you can watch the pros in your association to figure out how quickly it's appropriate to "talk business" with potential clients at the meetings.
Do you sometimes feel that you are the only one in the relationship who is giving? Do you know what to do about that?
### Assessing Your Comfort Level
Look back at your answers as you consider these comments.
Networking has emerged as a respected business and career skill. Why, then, does it sometimes feel uncomfortable?
Assessing Your Comfort Level
I feel professional and comfortable when I'm networking.
Rarely _____ Sometimes _____ Frequently _____ Almost Always _____
I'm energized and excited as I enter a room full of people.
Rarely _____ Sometimes _____ Frequently _____ Almost Always _____
Networking is something I want to do, not just something I have to do.
Rarely _____ Sometimes _____ Frequently _____ Almost Always _____
I can talk easily about my successes.
Rarely _____ Sometimes _____ Frequently _____ Almost Always _____
When I talk with people, I find out something of interest to me.
Rarely _____ Sometimes _____ Frequently _____ Almost Always _____
At networking events, I can think of plenty of meaningful topics to talk about.
Rarely _____ Sometimes _____ Frequently _____ Almost Always _____
Few families today sit down to a long Sunday dinner where Uncle Charlie tells stories and Grandma chimes in with reminiscences. Good conversational skills are learned. Few people are born with the gift of gab. But anybody can learn how to use conversation to build networking relationships.
Often, the "ground rules" for networking are unclear. Because it's a "hidden" career and business skill that you're just expected to know, many people are unsure about what's considered professional.
In some circles, networking is mistakenly equated with hot-dogging, tooting your own horn, or grandstanding. Some people say, "I shouldn't have to network. My good work should stand on its own without my having to promote myself." But who will know what you do well and what you need if you don't develop ease in talking about those things? For tips on constructing and telling Success Stories, see Chapter 11.
What happens in the conversation is that after you exchange names and after you ask "What do you do?" there is a pause. It's the pause that comes right before the conversation about the weather. Here's the rule: Never—and we mean never—talk about the weather or the ball scores. Instead, see Chapter 10 to learn how to carry around with you a pocketful of topics you really want to talk about, topics that will convince others of your expertise, build your credibility, teach others to trust you, lead you to resources, and assure that opportunities drop into your lap.
### Being Strategic
Look back at your answers as you consider these comments.
Are you surprised that "being strategic" didn't turn out to be one of your strong points? In this sped-up world, it's all too easy to run from activity to event—whatever's available this week will do—and then wonder why networking doesn't work!
So slow down! Make a long-range plan about which Arenas or settings you want to become known in and for what reason. Which organizations should you join? Test drive them before you plunk down the membership dues. There's a quiz that will help you make smart choices in Chapter 17. Want to do Olympic-level networking? Design a Project for yourself that will make you the natural and only choice when opportunity comes knocking.
Being Strategic
I have a long-range, strategic plan for my networking efforts in each organization I belong to.
Rarely _____ Sometimes _____ Frequently _____ Almost Always _____
I join organizations because of my strategic business/career development plan.
Rarely _____ Sometimes _____ Frequently _____ Almost Always _____
Before I go to an event, I think of specific resources/tips/trends I have to offer to the people I'm likely to see there.
Rarely _____ Sometimes _____ Frequently _____ Almost Always _____
I initiate at least one networking meeting (breakfast/lunch, etc.) a week.
Rarely _____ Sometimes _____ Frequently _____ Almost Always _____
I let people know the types of problems I can solve, so they refer exactly the right kinds of opportunities to me.
Rarely _____ Sometimes _____ Frequently _____ Almost Always _____
I'm comfortable telling my contacts what I want or need.
Rarely _____ Sometimes _____ Frequently _____ Almost Always _____
### In networking, strategy equals results.
Do you find yourself resisting being strategic? Do you think it's just too calculating to decide on a networking goal and go after it? Do you wish things would "just happen" without your orchestrating them? Tell yourself that managing your networking contacts is okay; manipulating is not. When you are aware of the difference, you'll feel more comfortable making a strategic networking plan. Tell yourself that planning for visibility and credibility is just like any other planning you do for your business or your career: It makes sense. You have limited hours and dollars to spend in the marketplace, and—without a plan—you'll sink down into aimless activity that doesn't amount to anything.
### Meeting People
Look back at your answers as you consider these comments.
Are you surprised that giving your job title isn't the right thing to do? It will be more valuable to you if people know your talent, not your title. As people meet you for the first time, they don't care (yet!) that you're with Smith, Jones, Miller, Barnes and Blarney or that you work for Verizon. To craft answers to the inevitable "What Meeting People do you do?" question that features your talents rather than your title, the name of your company, or your occupation, consult Chapter 9. Hint: Your answers should make it easy for people to talk with you, and should begin to teach people about your Character and Competence.
Meeting People
When someone asks, "What do you do?" I avoid giving my job title (e.g., executive vice president of administrative services).
Rarely _____ Sometimes _____ Frequently _____ Almost Always _____
I use several methods to learn people's names.
Rarely _____ Sometimes _____ Frequently _____ Almost Always _____
I've figured out a way to teach others my name and make it memorable.
Rarely _____ Sometimes _____ Frequently _____ Almost Always _____
When I've forgotten someone's name, I know how to retrieve it comfortably.
Rarely _____ Sometimes _____ Frequently _____ Almost Always _____
When someone asks, "What do you do?" I avoid saying, "I'm with..." and giving the name of the organization I work for.
Rarely _____ Sometimes _____ Frequently _____ Almost Always _____
When people ask what I do for a living, my answer paints a vivid picture.
Rarely _____ Sometimes _____ Frequently _____ Almost Always _____
When someone asks, "What do you do?" I avoid leading with my occupation or job category (e.g., purchasing agent, lawyer, systems analyst, architect.)
Rarely _____ Sometimes _____ Frequently _____ Almost Always _____
Have you given up on remembering names? Don't despair. In Chapter 8 you'll find three ways to remember somebody's name and three ways to make your own memorable. That's important too. You'll be pleased to know that there are several things you can do when you forget someone's name besides to say, "Oh, no. I've forgotten your name."
To be a successful networker, you'll have to shed the old meeting/greeting rituals we know so well and do so mindlessly. These rituals restrict, rather than enhance, your ability to build relationships.
### Using Networking Organizations
Using Networking Organizations
The first year I'm a member of an organization, I take an active role by serving on a committee or doing some job.
Rarely _____ Sometimes _____ Frequently _____ Almost Always _____
When I join an organization, I attend at least 75 percent of its events.
Rarely _____ Sometimes _____ Frequently _____ Almost Always _____
I know whether my company supports business development/networking with time and money.
Rarely _____ Sometimes _____ Frequently _____ Almost Always _____
I introduce myself to the leader or speaker when I go to an event.
Rarely _____ Sometimes _____ Frequently _____ Almost Always _____
At networking events, I avoid spending time with people from my own organization.
Rarely _____ Sometimes _____ Frequently _____ Almost Always _____
I know how to increase my visibility in any organization I belong to.
Rarely _____ Sometimes _____ Frequently _____ Almost Always _____
Look back at your answers as you consider these comments.
Are you making the most of your memberships? What are the worst mistakes members make? See Chapter 17. Too often, when it comes to joining organizations, people say, "I'm too busy!" Or "I'm too bashful!" Or "I'm too broke!" If you're in the midst of a job search or starting a business or professional practice, you may wonder if joining is worth the time and money. Well, not if you just join and hang around on the fringes. Not if you spend all your time sitting with and talking to co-workers you see every day. Not if you fail to find ways to exhibit your Character and Competence. Not if you miss opportunities to teach people to trust you.
Do you know how to connect at conventions? If you really want to get your money's worth, see Chapter 19.
Visibility is valuable. You'll find some great ideas about increasing your visibility at work in Chapter 14, and at networking venues in Chapter 17.
### Making the Most of Events
Look back at your answers as you consider these comments.
Have you ever left a networking event grumping, "I don't know why I come to these things. I don't get a thing out of them." If you learn the skills in this book, you'll never have that experience again.
### How do your skills stack up?
Do you want to meet the movers and shakers? Arrive early.
Do you sometimes feel "stuck" talking to the same person long after you've exhausted topics of mutual interest? Most people do. Did you ever say, in leave-taking, "I think I'll go freshen my drink," and head in the opposite direction from the bar? You need our easy LEAVE NOW process, described in Chapter 12, so that you know how to end conversations professionally and comfortably. Then you'll be able to move on and meet a dozen people in a two-hour event.
Confused about getting down to business at networking events? Learn how and when to start talking about your company or your product or service.
People want to do business with people they trust. Do your contacts trust you? Do you trust them? What's the key to developing trust? See Chapter 3 for some surprising thoughts on the topic.
Wonder why you should bother to introduce one of your contacts to another? Find out the benefits of becoming a great connector, someone known for bringing people together.
Making the Most of Events
I know how to end a conversation comfortably and professionally and move on to the next person.
Rarely _____ Sometimes _____ Frequently _____ Almost Always _____
I avoid "ho-hum conversations" about topics like the weather or the ball scores.
Rarely _____ Sometimes _____ Frequently _____ Almost Always _____
I find it easy to turn the conversation toward what I do or my company does.
Rarely _____ Sometimes _____ Frequently _____ Almost Always _____
At a typical networking event (two hours) I introduce myself to ten to twelve people.
Rarely _____ Sometimes _____ Frequently _____ Almost Always _____
What I say at networking events is consciously designed to teach people to trust me.
Rarely _____ Sometimes _____ Frequently _____ Almost Always _____
I arrive early at networking events.
Rarely _____ Sometimes _____ Frequently _____ Almost Always _____
When I'm listening to people, I try to think of someone they'd like to meet, and then I introduce them to each other.
Rarely _____ Sometimes _____ Frequently _____ Almost Always _____
### Achieving Bottom-Line Results
Look back at your answers as you consider these comments.
This is where the rubber meets the road. The questions in this section reveal whether your network is working.
How about networking at work? Have you detected a resistance to networking inside your corporation, government agency, or institution? Why? How can you decide if your organization (whether you're a sole proprietor or part of a huge corporation) values and supports relationship-building? If you don't know how to read the culture, look at the quiz in Chapter 14. What if networking became "the right thing to do" in your corporate culture? You could help make that happen.
Achieving Bottom-Line Results
After an event, I can name at least three valuable pieces of information I've learned from others.
Rarely _____ Sometimes _____ Frequently _____ Almost Always _____
At work, I hear stories about how people have developed business and enhanced their careers through networking.
Rarely _____ Sometimes _____ Frequently _____ Almost Always _____
I can cite examples of how my networking activities have paid off for my organization.
Rarely _____ Sometimes _____ Frequently _____ Almost Always _____
I can say exactly how my networking activities have paid off for my own career.
Rarely _____ Sometimes _____ Frequently _____ Almost Always _____
I can point out examples of assistance or resources I've given to my contacts.
Rarely _____ Sometimes _____ Frequently _____ Almost Always _____
When I need something, I know whom to call.
Rarely _____ Sometimes _____ Frequently _____ Almost Always _____
When my key contacts talk about me, I notice they can vividly and accurately describe what I do.
Rarely _____ Sometimes _____ Frequently _____ Almost Always _____
How about you personally? Is your network ready to tap into? Can you point to bottom-line results from specific contacts you've cultivated? When you need a resource, a referral, or an idea—or maybe just a pat on the back—do you know exactly who to call?
How vibrant are your networking relationships? Have you been able to develop a cadre of people who are out in the world promoting you and actively looking for ways to contribute to your success?
Have you ever listened to a contact describe you, your capabilities, your successes to someone else. What a revelation! Use every encounter to teach your contacts something else about you.
### Following Through
Following Through
When I get a business card, I put the information in my database.
Rarely _____ Sometimes _____ Frequently _____ Almost Always _____
I Follow Through to provide something I've promised within 3 to 5 days.
Rarely _____ Sometimes _____ Frequently _____ Almost Always _____
Soon after a networking event, I re-connect with two or three people I talked with.
Rarely _____ Sometimes _____ Frequently _____ Almost Always _____
After an event, I have several requests to fulfill from people I talked with.
Rarely _____ Sometimes _____ Frequently _____ Almost Always _____
I am able to fit staying in touch with key contacts into my daily/weekly/monthly routine.
Rarely _____ Sometimes _____ Frequently _____ Almost Always _____
Look back at your answers as you consider these comments.
Wouldn't it be great to Follow Through creatively and consistently? Sure, we recommend that you use an electronic database. But technology isn't the complete answer for keeping in touch. The real key to Following Through is to have something to follow up about. So, be sure that, in addition to talking about yourself, you also find out what's on the other person's Agenda. What are her challenges, interests, needs, enthusiasms, dilemmas? Then you can Follow Through based on her needs, not your need for a new client or your interest in getting a better job. The best relationships are built on giving. When you Listen Generously and focus on giving more than you get, reconnecting becomes a whole lot easier.
### Check Your Results
Throughout the Self-Assessment, the best answers are Frequently and Almost Always. Look at your array of answers. If most of your answers are Frequently and Almost Always, you probably have mastered the networking concepts behind the statements. As you go through this book, you will get a kick out of delving deeper into what makes networking work, and you'll pick up some new tips and techniques along the way. If most of your answers are Sometimes and Rarely, you will be saying "Aha!" a lot as you read this book. You'll discover new ideas, new concepts, and, of course, skill-building tips and techniques. Soon you'll be able to put the new ideas to work for you as you build your network.
### Next Steps
After you've checked your results in the eight skill areas, take a look at the big picture. Maybe you're doing well in Meeting People, but need to focus on Being Strategic. Or maybe you find that your Comfort Level is high, but you fall down on Following Through. Use this Self-Assessment to set your priorities. Should you go straight to a specific topic or chapter? By giving attention to specific areas, you'll see results quickly. For instance, if the Self-Assessment points out that you need more skill in Meeting People, then you'll want to pay special attention to Chapters , , and . If you find that you were stymied by a majority of the statements, you may decide to start at the beginning of this book and read straight through to the end.
Take out your calendar. Make a note to retake the Self-Assessment in six months. That way, you'll have time to read this book and to put many of the tips and techniques to work. Go ahead. Experiment. Making a small change or adding a new behavior can have a dramatic effect on your ability to connect with people. We predict you'll be pleasantly surprised at how easily you've made the Contacts Count approach to networking a way of life.
## CHAPTER 2
## Change Your Mindset
What is networking anyhow? It's a term that's been around since the 1960s, but many people still don't have a clear idea of its meaning.
_Networking is the deliberate process of exchanging information, resources, support, and access in such a way as to create mutually beneficial relationships for personal and professional success._
That's our official definition, but there's no denying that the term networking has, through the years, collected some negative connotations. What about your mindset? Does networking's chilly image make you freeze up? Could it be that you have some misconceptions about making connections? Do you cringe at the language of networking? Do you think of yourself as shy? Do the things you say to yourself about networking undermine your efforts? In addition to determining what skills you're good at and which ones need sharpening, you also must check out your attitudes. Decide to clear your mind of all the unproductive and debilitating notions that might have crept in. Thinking positively about networking will give you the motivation to leap right in and start learning to be the best that you can be.
### Come In from the Cold
What images prevail as people talk about networking? Brrr! People say:
* "He gave me the cold shoulder."
* "What can I say to break the ice?"
* "I hate to make cold calls."
* "I got cold feet when I thought about going to the meeting alone."
* "I just froze up."
It's hard to feel excited about making contact when your mind is full of images like this. If you think of other people—people you might network with—as cold and rejecting, it will be hard for you to enjoy the moment, exchange information, or explore future opportunities.
### The Ten Biggest Misconceptions About Networking
**1.** "I do a good job," says Brad, an engineer. "I shouldn't have to network at work. My work should stand for itself."
### There are no fast-food networks.
Brad's mistaken. Smart employees use networking to stay in touch with internal customers and suppliers. Their networks alert them to problems before they get out of hand and help them spot emerging needs. These employees break through bureaucratic bottlenecks. They use personal contacts to get things moving and speed things up. They build constituencies and gain support for projects and proposals. They collaborate and cooperate. They create ad-hoc, cross-functional, problem-solving teams. When somebody says, "I need it yesterday," they come though. If Brad networked at work, he could serve his organization better and develop his reputation as a person who can get things done—fast.
**2.** "I tried networking last Thursday," says Mel, a franchise owner. "It doesn't work."
To cultivate a bountiful network takes months—maybe even years. Mel has a microwave mentality. You can't zap a relationship for thirty seconds. Networking is a long-term process.
**3.** "Networking is fine for the junior folks who are still struggling to climb the ladder," says Richard, a vice president of human resources. "But I don't need to network anymore."
### You never outgrow the need to network.
No matter what your title, you never outgrow the need to network. Richard could be using networking to his company's—and his—advantage. He could assure his company remains competitive by networking with people who have similar jobs in other companies. Networking is the best way to "benchmark," to check out the best practices and compare yourself with those one rung above you on the corporate ladder. And, the higher up someone is in the hierarchy, the more vulnerable he is. Middle management and staff positions disappear daily. Richard better get cracking to create his "safety net."
**4.** "Networking is manipulative," says Teresa, who owns a nanny agency. "I don't like the idea of arm twisting someone to do something for me."
True. You can't exploit others and expect to build long-term relationships. The way to avoid manipulation, though, is to give more than you receive. And, when you want something, be up front and overt about it. Look at the following statements. Saying, "To build my business I give free workshops for young parents to show them how to find and manage a nanny. Do you know anyone who'd like an invitation?" will likely prompt a better response than a "hidden agenda" question, such as, "So, do you know anyone with kids under five?"
**5.** "Networking is just schmoozing," says Karla, a manager of administrative services. "It's boring and...uncomfortable."
If Karla's conversations skitter over the surface, she needs to find out how to sidestep those superficialities and get down to business. She can learn how to go beyond the chitchat into conversations that can help her solve problems, come up with new ideas, and access valuable resources.
**6.** "I'm not looking for a job right now," says Diana, a purchasing manager. "I don't need to network."
Absolutely the worst time to begin networking is when you need to—after you decide to change jobs or lose your job. If you expect to be job-hunting anytime in the future—and in today's economy, it could be sooner than you think—you should be networking now. The length of your job search depends on the strength of your network. By increasing her visibility among her peers and superiors, Diana can achieve top-of-the-mind awareness. When they think "dynamite purchasing manager," her name will pop up. Not only can networking protect her if her job goes away, it also can lead to new job opportunities—even when she isn't looking.
**7.** "Networking has never done a thing for me or my career," says Kyle, a director of corporate planning.
### A single conversation can change your life.
That's hard to believe. People can provide access to vital information, such as news of a job opening before it's advertised, insights on industry trends, early warnings of happenings that could impact your business, and even great ideas for a business of your own. Many people can put a bottom-line figure on the value of a single conversation. And sometimes, a single conversation can change your life.
**8.** "Sure," says Carmen, a photographer. "I know how to network. You just hand out a business card."
Handing out cards isn't networking. Most cards end up in the trash. To network, you must create real, human connections.
**9.** "I wasn't born with the gift of gab," says Morrie, a CPA. "I'll never be any good at networking."
### Networkers are made, not born
Only about 10 percent of the people we've interviewed say they come by their conversational skills naturally. The rest of us need networking know-how. Luckily, anyone—even introverts—can learn to network. Once Morrie makes up his mind, he can become better and better at connecting with people.
**10.** "Networking is a waste of time," says Henri, an attorney. "I leave networking events asking myself, 'Why did I come?'"
More than 85 percent of people who attend networking events tell us they haven't figured out what they want to achieve. If you aim at nothing, you'll hit it, as the saying goes. Henri needs to go with goals in mind. That way, he'll find what he's looking for.
### Ten Turnoffs in the Language of Networking
Do the words that people use when they are talking about networking make you cringe? Here's our list of the Top Ten Turnoffs and why we don't like them.
**1.** "Schmoozing." That word makes networking seem so slimy and insincere!
**2.** "30-Second Commercial." Sure you want to "sell yourself" to your contact, but this phrase implies too much of a hard sell.
**3.** "Pick Your Brains." It makes us think of vultures coming in for the kill. We wish people would say, "I'd like to get your thoughts about something."
**4.** "Work a Room." So depersonalizing and one-sided, this phrase sounds as if you intend to work people over and take all you can.
**5.** "Information Interview." You don't have to make a specific appointment to gather valuable information. Using state-of-the-art networking skills, you can make networking a way of life.
**6.** "Tricks of the Trade." Let's not imply anything that smacks of manipulation. There are no "tricks" in our networking System; only upfront, clear offers to be helpful to each other.
**7.** "Favor Bank." Doing things for others is the right thing; doing things for others just so they'll "owe" you one is the wrong reason to give. Give without strings, without expectations of getting—that's the way to create a network that works.
**8.** "Power Lunch." Yes, invite a powerful contact to lunch, but don't call it that. It sounds too much as if you value people just for their positions.
**9.** "Business Card Exchange." Exchanging cards without building trust is non-productive. When you leave a networking event with twenty or thirty cards, what do you do with them? Toss 'em into the trash! Instead, look for reasons to exchange cards. Be alert for ways to move your relationship beyond the networking event into the future. Broadcasting your business cards makes only "cardboard connections," not real connections.
**10.** "Important People." Don't you hate it when you are talking with someone and that person is looking over your shoulder trying to find someone better to talk with? Give your whole attention to the person you are with. Anyone can turn out to be a wonderful contact!
### You Say You're Shy?
Do you think of yourself as shy? You're not alone. The first studies, in 1972 at Stanford University's Shyness Clinic, found that 40 percent of all Americans labeled themselves shy. But, that figure, Clinic Director Dr. Lynne Henderson notes, has steadily increased. Now, as many as 50 percent of us say we're shy.
Philip Zimbardo, who founded the Clinic, blames this rapidly growing fear of being with people on a variety of phenomena—from ATMs to video games to TVs—that reduce day-to-day informal contact with others. And, he says, children don't see their parents relating in a natural, easy, friendly way often enough. Families are smaller and often too busy to spend time honing conversation skills at the dinner table.
Zimbardo defines shyness as reticence and self-consciousness, not just in stressful social situations, but over all. He found that shy people are less popular, find fewer friends, exhibit lower self-esteem, make less money, say their lives are boring, demonstrate fewer leadership skills, are more likely to be depressed, have less social support, and are more likely to be lonely. Shyness will cost you, say these experts. You won't be as successful as someone who has learned to network, make a good appearance, and socialize on the job. Whether you always feel shy or just feel shy in certain situations, you can learn to be more comfortable. "Shyness is not a disease," Henderson says. "It's a habit pattern that can be relearned."
Remember that many confident, easy-going networkers (including us!) once were shy and uncomfortable. They've just learned new behaviors.
Instead of letting shyness hold you back from networking, learn the skills in this book. You'll become more confident and comfortable and each success will show you the value of connecting.
### Catch Your Critic
You climb into your car to go to a networking event. You put your key in the ignition. You turn into the street...then, all of a sudden, you're there. You have no recollection of the route you took, the traffic you coped with, or the signs and houses and businesses you passed.
You've been on autopilot. But, when you think about it, you remember that your Critic—the voice in your head—has been haranguing you.
The voice makes it very hard—sometimes impossible—for you to connect easily with others. Notice how your Critic sabotages you.
### Bad reviews and bad previews are the Critic's stock in trade.
During introductions, the voice in your head yells at you. Just when the person you're talking with gives his name, the Critic says, "You never can remember people's names." Sure enough, while the Critic is yelling, the other person's name is blotted out.
In the middle of a conversation, the voice mutters, "You never can think of anything to talk about." And guess what...it's a self-fulfilling prophecy. You aren't able to think of anything to say.
After you've been talking with someone for several minutes, the voice harangues, "This person would rather be talking to someone more important." And, you fade out of the conversation, stammering something about needing to freshen your drink.
The Critic is bad news. Your brain believes what you tell it about yourself. The good news is you can transform your Critic into a helpful Coach.
If you notice what your Critic says and don't like it, you can reprogram that voice in your head to give you positive and supportive messages instead of negative and defeating ones.
### Convert Your Critic into Your Coach
Teach the voice in your head to say something helpful and supportive. Whenever your Critic makes you feel uncomfortable and incapable, develop encouraging statements that make you feel confident and strong. Most of your statements will probably focus on your new beliefs that networking is valuable and that you can learn to do it well.
Changing the way you talk to yourself about your ability to network and combining that new mindset with the specific skills in this book will help you become a better networker. Look at Figure 2-1 for some ideas on how to change that negative Critic into a supportive Coach.
### Believe the Best About Yourself and Others
Your beliefs about yourself and other people will support you to succeed at connecting. Take Paul, for example. He used to be apprehensive about entertaining out-of-town clients at dinner. But, to cope with his reluctance about meeting new people, he's gotten in the habit of giving himself a pep talk. "I say to myself, 'They've got kids and hobbies and hopes and dreams.' I think about all the things we have in common. If I prepare, I'm okay."
**FIGURE 2-1. Turning Your Critic into Your Coach.**
Paul discovered through experience what psychologists have verified by studying the conversational patterns of people meeting for the first time. These researchers found that if people meeting for the first time believe they have a lot in common, they act very much as if they are old friends. They pay attention to subtle conversational clues and match each other's progress through the conversation. If one brings up a lighter, more informal topic, the other responds with a light topic of his own. If one says something self-revealing, the other follows.
On the other hand, if the strangers are told they have nothing in common, conversation limps along, and both parties feel they haven't connected. This research reinforces the idea that your attitude toward others impacts your success as a networker.
Make up your mind. That will make it a lot easier for you to learn the skills required in today's marketplace.
### Bonus: Know Your Style
You don't have to change your personality to be good at networking. Ever taken one of those "communication styles" profiles? Each style has both its comfort zone and its challenge zone. The multitude of communication style assessments on the market today use various terms to help you understand your approach. Whatever assessment you have taken, you should be able to find your profile below.
**_If you like to head "straight for the finish line,"_** it will feel natural for you to set goals, ask for what you want or need, walk into a room full of strangers, and figure out innovative ways to find good contacts. It will be a challenge for you to slow down and appreciate a conversation partner who has a different style, to take the time to develop trust, to share "air time," and Listen Generously.
**_If you like to "keep a lot of balls in the air,"_** it will feel natural to you to collaborate, to give first and give generously, and to learn networking skills and systems so that you feel more comfortable. It will be a challenge for you to stick to your goals, to ask for what you want or need, to tell Success Stories, to hold up your end of the conversation, and to take your 50 percent of the "air time."
**_If you like "hanging out,"_** it will feel natural to you to mix with others, to be enthusiastic when you are talking, and to strike up conversations. It will be a challenge for you to clarify your goals. You'll need to guard against overpowering people, spending too much time talking without focusing on the result you want, and forgetting about your goals because you are enjoying the process of connecting.
**_If you prefer to "go it alone,"_** it will feel natural to you to plan carefully what you will do at a networking event, to make your Agenda (What to Give and Get), to get one-on-one with people, and to be creative in your approach. It will be a challenge for you to make yourself get out there, to learn to trust others, to demonstrate that you are trustworthy, and to share personal information and Success Stories.
**_If you like "taking things easy,"_** it will feel natural to you to Listen Generously to your partner, to feel that networking is cooperating not competing, to save time by targeting your networking and to set goals. It will be a challenge for you to respond quickly in conversations, to focus on your needs and wants, and to feel a sense of urgency to get things done.
**_If you like to "floor it,"_** it will feel natural to you to seek out a variety of people and organizations, to enjoy the excitement of making those initial contacts, and to have lots of irons in the fire. It will be a challenge for you to be strategic, to Follow Through to deepen your contacts, to focus and concentrate on selected contacts rather than continuing to seek new ones, to limit the number of organizations you are a member of, and to take the time to develop trust.
**_If you like to get "your ducks in a row,"_** it will feel natural to you to use The Contacts Count Networking System—to learn the steps to become a more skillful networker, to practice your skills, to appreciate the Give/Get concept of networking because it's fair, and to keep track of your contacts and networking activities. It will be a challenge for you to get started, to keep from worrying about being perfect, to quit being concerned about meeting the "right people."
**_If you like to "go with the flow,"_** it will feel natural to you to be upfront about what you want and need, to appreciate the Contacts Count System's non-manipulative approach, to be candid with contacts to move relationships forward, and to give up the tired, old rituals we use when meeting people. It will be a challenge for you to focus on the details of making and maintaining contact, to Follow Through effectively and systematically, and to develop relationships over the long haul.
You don't have to change who you are. Whatever your style, know that you can customize all the ideas in this book to enhance who you already are.
## PART II
## Set Your Strategy
The biggest mistake networkers make is not being strategic. To be strategic, you must understand and use two underlying concepts: How trust develops and how relationships develop. These unique Contacts Count ideas will provide a strong base for all your networking activities. Then you can create the goals and Projects that get you where you want to go.
Do you know that trust building is _the_ most important networking activity? Even if you realize its importance, you may not know how to do it. What do you know about trust? We break it down into its two essential components—Character and Competence—and show you how to demonstrate them in every encounter.
Do you have a binary approach to networking? Do you think people are either a part of—or not a part of—your network? Networking relationships are much more complex than that. You can plot each one of your contacts on the Six-Stages Model. Once you know where they are on this map, you'll immediately be able to see what you can do next to enhance those relationships. You'll understand your relationships and how to develop them in a new way that makes re-connecting and staying in touch easy—and more and more beneficial for both you and your contacts.
Want to get the biggest bang for your networking buck? Create the networking Project that will take you to your goal.
## CHAPTER 3
## Teach Trust
Everybody says, "People want to do business with people they trust."
Have you ever been at a networking event where someone came up to you and introduced himself saying, "Hi, I'm George. I sell long-term care insurance. Do you need any?"
It's this kind of "going for the jugular" that gives networking a bad name. You don't know or trust George enough to do business with him.
Trust is the outcome of several (our research indicates six to eight) conversations in which you provide examples of your trustworthiness and observe your contact's behavior and listen to what he says to determine if he can be trusted.
People teach people to trust them.
If Peter, a career coach, takes sloppy minutes at the committee meeting, or forgets to return JoAnn's phone call, or has a couple of typos in his marketing brochure, JoAnn probably won't hire him. Nor will she recommend him to her friends and acquaintances.
On the other hand, if he handles his committee responsibilities carefully, and promptly returns calls, and has professional looking marketing materials, then JoAnn will probably think of him when she decides to make a career change and needs some help.
After JoAnn becomes Peter's client, if she cancels sessions at the last minute, fails to follow through on her "homework" that will help him guide her job hunt, and looks as if she came straight from her workout at the health club, Peter will be reluctant to recommend her for a job he knows about.
If JoAnn arrives on time, dresses professionally, completes all of the assessments Peter has provided, and speaks positively about her expertise and her current employer, then Peter probably will pass along the job lead.
### Move from Taking to Trusting
As people begin to network, they typically focus on trying to get something for themselves. There's nothing wrong with wanting your efforts to bear fruit. But, that's only part of the story. Networking is not just about TAKING.
The TAKING mindset works when you happen to connect with somebody who has what you want or needs what you're offering.
George, the guy who sells long-term care insurance, is focusing on TAKING. He's only interested in you if you, at that very moment, in the middle of the Chamber of Commerce's "Business After Hours" event, let's say, happen to need long-term care insurance and are willing (remarkably!) to buy it from him. His approach is no more than a face-to-face cold call. Chances are he'll go home after the event and grump, "This networking stuff is a bunch of hooey!"
Charlene went to that same Chamber event thinking about a problem: how to ship a fragile antique desk to her daughter in London. When she asked Bart, whom she'd just met, what he did, he said, "I ship anything anywhere, especially valuable things. I just sent a baby grand to the Philippines." Bingo!
But if that's the only kind of networking you are ready to do, you are not going to be very satisfied. During that networking event that lasted two hours and involved more than 800 people, how many serendipitous meetings like Charlene and Bart's do you think happened? Probably not very many.
When you begin to realize that the point of networking is to exchange something of value, you've begun to think of networking as TRADING.
TRADING is exciting. It makes you feel as if your networking efforts are worthwhile. Clarice, who has her own training company, went to a networking breakfast and met the college-age son of a member. She was delighted to meet Howie, who has a window-washing business, because she needed that service immediately. She took Howie's card so she could call him and get an estimate. When Howie found out what Clarice did, he suggested she talk with his dad because he knew his father's company was looking for someone to write a training manual. That's a great trade. But it's a one-time trade. Clarice will need her windows washed only once in a blue moon. Howie's business may not last beyond the summer, and he may not know anyone else who might need her training services.
When people think of networking as TAKING or TRADING, what they get is Single-Sale Networking. Though Single-Sale Networking may result in instant gratification, it's time-consuming, and you miss out on the long-term benefits. Unfortunately, many people feel that when they have achieved a TRADE, they have reached the epitome of networking.
### Networking isn't about taking; it's about teaching.
Long-term networking relationships are built by TEACHING people what you need and what to count on you for, and by learning the same about them. When you meet someone, take the time to be interested in that person and his or her business. Put your antenna up for resources, ideas, tips, information, or access that you could give to that contact. Look for ways to become known to that person and to educate that person about yourself and your capabilities.
Remember the old line: "It's not WHAT you know, it's WHO you know?" That's only partly true. Sure, WHAT you know is important. It's your expertise, your knowledge, what you are paid for. WHO you know is important, too. Those are the people you call when you are looking for an idea, a resource, a referral.
But just as important as WHAT you know and WHO you know is WHO KNOWS YOU. Does Fred know you so well that, when something comes into his life, you pop into his head. And he says to himself, "Oh, I've got to send this to Sean." Who knows you that well?
The big networking challenge, then, is how to teach your contacts who you are and what you are looking for, so they can send good things your way. An equally big challenge is how to learn about your contacts and what they are looking for, so you can send good things their way. Make sure you spend as much time learning about your contact's business and life as you do teaching him or her who you are.
When you put your emphasis on developing TRUST, then relationships become mutually beneficial.
### Teach That You Can Be Trusted
Your contacts will begin to trust you as you teach them about your Character and Competence. To teach your contacts about your Character:
* Do what you say you will do.
* Meet deadlines.
* Go for the win/win solution.
* Treat everyone you meet fairly.
* Be unfailingly reliable.
* Speak well of people even when they are not present.
* Come from a position of abundance, not scarcity.
* Collaborate rather than compete.
* When something goes wrong, ostentatiously make it right or compensate generously for your failure.
* Go the extra mile.
* Respect other people's time and possessions.
To believe in your Character, your contacts must either see you in action (observe your behavior) or must hear about you (listen to stories you tell about yourself that provide vivid examples of your Character).
If you promise your contact you'll call her on Tuesday, do it. That's how you teach someone that you will do what you say you will do.
If you promise you'll come up with ten items for the public television fund-raising auction by December 5, provide a dozen. That's how you teach someone that you meet deadlines, are reliable and go the extra mile.
Suppose you want to teach someone that when something goes wrong, you'll do more than make it right. Tell about the time you inadvertently charged a customer more than you should have for a job and how, when you discovered the mistake, you not only called to apologize and to ask if he would like his money returned or credited to his account, but also sent him a basket of cookies as an additional apology.
Suppose you want to teach someone that you're a stickler for details. Talk about the newsletter you edit for your professional association. Tell how you go to great lengths to be sure every name is spelled correctly and all details in the articles are correct.
### Your contacts won't help you until they trust you.
Suppose you want to teach someone that you are innovative. Tell about your work on the program committee that's resulted in an award-winning line-up of programs for the association.
Suppose you want to teach someone that you are a good organizer. Tell about the time you compiled information to submit to national to enter the Chapter of the Year contest. (See Chapter 11 for more ideas on using stories to teach about yourself.)
To teach your contacts about your Competence, you will need to reveal that you:
* Have earned the proper credentials.
* Stay at the leading edge of your profession.
* Have won praise and awards from your peers.
* Take life-long learning seriously.
* Are cited as an expert in the trade press or in the mass media.
* Teach or mentor others.
* Consult with others to share your expertise.
* Write for publication or speak in public.
* Do the job right—the first time.
* Are happy to discuss your procedures and processes with clients and customers.
* Handle "the little stuff" with care.
* Follow through to be sure that your work meets or exceeds expectations.
Here are some suggestions about ways to show your Competence: Frame diplomas, accreditation certificates, and customer kudos and hang them on the wall of your office. Invite your contact to lunch and give her the "grand tour." Send contacts articles that quote you, newspaper clippings or conference programs that show you speaking. Tell stories (See Chapter 11.) about consulting with others. Protect the confidentiality of the organization or person you consulted with. Tell other stories about doing the job right and handling "the little stuff" with care. Provide your contact with a sample of your work or a tour of your work site.
To build a strong network, make sure your contacts know your capabilities and are confident in your ability to perform. It's unreasonable to expect that people who don't know you will be comfortable giving you referrals or suggesting you for jobs. They have no idea of your special areas of expertise and have not known you long enough to be sure you will come through.
Use a similar process to learn about your contact's Character and Competence: look for the same behaviors and ask for stories.
### The Trust Matrix
The Trust Matrix, shown in Figure 3-1, graphically depicts the process of developing trust. When you first meet someone, you probably have little knowledge of his Competence and Character. And your contact probably doesn't know how you would rate in those areas either. That's why most relationships begin in the lower left quadrant of The Trust Matrix, with Competence and Character still to be determined. You are Acquaintances.
If a contact has a "bad experience" with you—or if you have a "bad experience" with your contact—your relationship will derail into the upper left quadrant or the lower right quadrant.
If your relationship moves to the upper left quadrant, your contact trusts your Character, but questions your Competence. He thinks you are Admirable But Not Able. This situation can be remedied. It could be that you have just changed jobs or just graduated and are in your first job or have moved into a new career field. Think about how you can teach your contact that you are Able—or Competent—as well as Admirable. Tell stories as you talk or send out a newsletter or press release to let contacts know about your increasing expertise. Take a visible role in organizations, so contacts can experience your Competence first hand.
**FIGURE 3-1. The Trust Matrix.**
If your relationship moves to the lower right quadrant, your contact trusts your Competence, but questions your Character. Your contact believes you are Able But Not Admirable. Everybody makes mistakes that contribute to concerns about Character or Competence. Can you repair the mistake and your reputation? If you were late to an event, show over time that lateness was a one-time aberration, not a habit. You will have to demonstrate repeatedly that your character is Admirable.
If you have positive experiences with each other, your networking relationship will move to the top right quadrant. You will become Advocates or Allies. Advocates speak well of you and your business, refer qualified customers or clients to you and create opportunities for you. Allies are trusted advisors. Your trust in the confidentiality of the relationship and the value of the relationship is so high that you feel comfortable sharing frustrations and trade secrets, and celebrating successes. Developing that level of trust takes time.
It's also possible that one or both of you may have established a reputation in the community. Ideally, you want contacts to hear good things _about_ you before they hear _from_ you. If your good reputation precedes you, your relationship may start off in the upper right hand quadrant. If so, you've "jump started" the process and can begin your relationship with Character and Competence assumed. That doesn't mean that you can relax and forget about the trust question. As your relationship continues, continue to reaffirm both your Character and Competence.
### Avoid Manipulation
Networking is not about manipulating other people. If you are absolutely honest about what you want to establish—a mutually beneficial, long-term, trusting, business relationship—you will not be manipulating your contacts. It's only when you connive to get something for yourself through misdirection, subterfuge, or telling only part of the truth that you are being manipulative. Don't do it. (See Chapter 10 for a more detailed discussion of the dangers of manipulation.) To build trust, you must convince your contact of your Character, as well as your Competence. Nothing destroys relationships quicker than one party feeling manipulated by the other. Be totally upfront about your motives.
## CHAPTER 4
## Develop Your Relationships
Networks are always becoming. They are never complete or static. Networking relationships offer the possibility of growth through six stages of development. Understanding these stages will help you figure out what behaviors are professional—not too pushy and not too passive. You'll be able to assess your current network and decide where to put your energies to widen and strengthen it.
Make a list of ten people you know. Include a variety of people: co-workers; people you know well and people you have just met; clients, customers, or vendors; people from a professional association or community organization; people from your leisure life. Keep these people in mind as you read about the Six Stages.
### Move Through the Six Stages
Study the Six-Stages Model, shown in Figure 4-1. On the outside are all the people you run into, however casually. At the center are those few people with whom you have very trusting and long-term relationships.
**_Accidents_.** In your lifetime, you will bump into thousands of people. These casual, unplanned, random encounters are Accidents. They probably will never be repeated. They are one-time-only meetings. You are thrown together for some period of time. So you talk to each other—in line for tickets to the hit play, in the emergency room waiting area, when you are 14C and she is 14B on the plane. Any person you meet outside a common context is an Accident. Networking relationships sometimes grow out of Accidents if you can find a reason to stay in touch.
**FIGURE 4-1. The Six Stages Model.**
**_Acquaintances_.** People that you run into because of who you are and what you do are Acquaintances. They have something in common with you. An Acquaintance may be a person who is a friend of a friend. You meet at your neighbor's daughter's wedding, for instance. You _might_ see an Acquaintance again. Then again, you might not. There's enough of a connection there that, if you _had_ to, you could probably find an Acquaintance again.
When someone mentions your Acquaintance Ramon Sanchez to you, you're likely to say, "Ramon Sanchez? Sounds familiar. I think I've met him. Isn't he a lawyer or a CPA or something like that?" He'd probably be as vague about you. You may be able to recall an Acquaintance's name, but you haven't really begun to learn about each other.
**_Associates_.** People you come in contact with on some regular basis for some period of time are Associates. You are both part of the same system. You've both joined the alumni association, or the swim club, or a professional association, or you work for the same employer. Because you see each other every week or every month—or even every year—you have the chance to learn each other's names and reconnect often enough to learn a bit about each other. But unless both of you work at the relationship, it will never develop. You will continue to see each other, chat briefly, and part, without providing any assistance to each other.
If Ramon has become an Associate, you might describe him like this: "Ramon is a attorney. He's a member of our Chamber. I think he specializes in estates. I could find him in the membership directory."
**_Actors_.** People with whom you exchange valuable information, resources, or leads are Actors. Sometimes you are the giver in the exchange; sometimes you are the receiver. Whether you realize it or not, you are looking for two things in each other. If you each find these two things, you'll want more activity with each other; if you don't see these two things exhibited in their conversation and behavior, you won't pursue the relationship. What are these two things? Character and Competence. (See Chapter 3.)
You gather information about each other and have each other's phone numbers and e-mail addresses. You know enough about him, and he knows enough about you, to be useful to each other. Once you get into an exchange relationship, you are Actors.
If you needed to find someone to advise you on setting up a trust, you might say, "Oh, Ramon. I met him at the Chamber." Once you make an appointment with Ramon, you become Actors. You make note of his phone number and address. When you meet with him, you look for evidence of his Character and Competence and begin to demonstrate yours.
**_Advocates_.** People who believe in each other's Character and Competence are Advocates. You know that your Advocates will come through, and they know that you will help them. You have developed a high level of trust with each other. Your antenna is up for information and resources for these people. And they, likewise, feed you opportunities. You speak well of them.
When Ramon becomes an Advocate, you look for ways to assist him. If someone mentions that she is concerned about how well organized her Mother's financial situation is, you recommend Ramon, saying, "I worked with an excellent attorney, Ramon Sanchez, who helped my mother set up a trust. Why don't you give him a call? I know his first consultation is free. I recommend him highly. You can reach him at this number."
**_Allies_.** People who are experts on you, your business, your career, your needs, your aspirations, and your vision are Allies. They know where you've been and where you're headed—and they want to help you get there! They are your senior advisors, and you are theirs. Because you talk about core life and business issues, you have established confidentiality as a ground rule of your relationship. You both are so committed to your mutual success that you serve on each other's unofficial Board of Directors. These are the people you turn to for sage advice—on how to climb the corporate ladder, on whether it's time to open a branch office in Denver, on how to deal with a difficult client. Allies commiserate with you when the going gets rough and celebrate with you when success is sweet.
When Ramon becomes an Ally, you might call and say, "Hey, Ramon, your daughter's wedding was beautiful. By the way, I know of a board opening that might be a good career move for you. Do you want me to give your name to the chairman? Also, let's get together next week. I'm thinking of making some big changes, and I'd like your thoughts."
Go back to that list of ten people you made. Decide what stage you are at with each person on your list. Assuming that your list of ten is representative of all the people you know, what does it tell you about how you need to expand and develop your network? What does it tell you about appropriate next steps with these ten people? If you made an even broader list of people you know, would there be a good mix of people from a variety of Arenas? Would it include people from your workplace, your profession, your industry, your friendship and leisure-time circle, your family?
### The Next Move Is Up to You
As you look at the Six-Stages Model, you'll become clear about what you need to do if you'd like to have more of a relationship with someone. Look at Figure 4-2 for further ideas on transitioning from one Stage to the next.
If you and Jim are Associates, your Number One job is to Listen Generously, so you can find out what he needs. When you find something that you can give, you will just naturally move to the Actor stage of exchanging.
If you and Monica are Actors, your Number One job is to show her your Character and Competence through everything you do and say.
If you and Horatio are Advocates, your Number One job is to promote him and to keep your antenna up for opportunities and possibilities to pass along to him.
If you and Svetlana are Allies, your Number One job is to tell the truth, provide support for all she does, and help her in any way you can to succeed in both her business and personal life.
You'll probably have hundreds or thousands of Accidents, Acquaintances, and Associates in your lifetime. You'll enter into an Actor relationship with fewer people, and even fewer will become Advocates and Allies. Of course, you can't make anyone move to the next stage with you, but you can say and do things that will make it more likely that the relationship will grow.
**FIGURE 4-2. Next Steps.**
### Rate Your Relationships
Networking is a process of teaching and learning. Choose someone on your list of ten and use this quiz to figure out where you are with that person. When you answer "No," you'll have a clue about what you want to be sure to tell—and ask—the next time you meet.
**1**. Does she recognize my name instantly when I call?
**2**. Does she know me well enough to recognize me "out of context," at the store, in a new group?
**3**. Does she know my face and my name well enough to come up to me in a crowd and introduce me accurately to others?
**4**. Has she found a reason to have my phone number or e-mail in her system?
**5**. Does she know the name of my company or organization?
**6**. Can she accurately describe what I do?
**7**. Can she give vivid examples of what I do?
**8**. Does she know that I am good at what I do and can she cite reasons why my service, product, or skills are superior?
**9**. Does she know of some independent verification of my expertise such as an award, a certification, or a third party endorsement?
**10**. Does she regularly send me valuable information and respond to requests from me?
**11**. Does she know what kind of customers, clients, or job opportunities will appeal to me and does she send them my way?
**12.** Does she always speak well of me to other people and pass my name around?
**13**. Does she regularly refer qualified customers, clients, or job opportunities to me?
**14**. Does she consistently create opportunities to stay in touch with me?
**15**. Does she treat the business, career, and life issues we talk about with confidentiality and caring?
This quiz highlights our finding that it often takes six or eight contacts with someone before he or she knows who you are, has learned your marketplace niche, and begins to trust you. Once that trust is established, you might be in touch once a week or once a year, depending on the relationship.
In networking, the ball's always in your court. It's up to you to take the next step to cultivate the relationship.
### Have Questions About the A's?
Here are some of the questions people have about the Six Stages.
**Q:** | "I have two Advocates and they don't like each other. What can I do?"
---|---
**A:** | Say to each of them separately, "I know you two don't see eye-to-eye. That's a huge advantage to me because I get two distinct points of view. I wish you could bury the hatchet, but if not, I'll still value advice and input you both give me."
**Q:** |
"I've made some incredible contacts with Accidents. I got a $50,000 training contract because I said hello to a fellow passenger in the van from the airport to the hotel. Why bother to cultivate long-term relationships when that works so well?"
**A:** | Making a magical connection with someone you meet on the fly (no pun intended) is great fun. But, over the long haul, you'll be glad to have the mature and solid, mutually beneficial relationships with Advocates and Allies. You'll get more than business from them; you'll get support and opportunities that are tailor-made for you, because they know you so well.
**Q:** | "Can customers and clients become Advocates and Allies? I guess we became Actors when I worked for them but that was four years ago. We have no relationship now."
**A:** | Sure they can. When you add buying and selling into the relationships, they become more complex. If both parties are of high Character and Competence, and if both are candid and open, the relationship can flourish.
**Q:** | "I heard that people often get jobs from secondary contacts. That sounds like the Acquaintances stage would bring the most benefits."
**A:** | You're right. If you are job hunting, it would be wise to seek out those Acquaintances—friends of friends—because they move in different Arenas than you do. That means when you contact them you expand your network. You have another advantage with Acquaintances: they give you trust because they trust your mutual friend or contact.
**Q:** | "When I looked at my ten people versus the Six-Stages Model, all were Associates. What am I doing wrong?"
**A:** | This situation is typical for younger networkers who haven't had the time—or perhaps the need—to build a network to help them advance their careers. First, you may simply have been thinking of groups you belong to as you made your list of ten. That would have led you to write down mostly Associates. Second, you may need to do more than join organizations. You may need to work at your network to deepen those relationships.
**Q:** | "I thought of ten people, but my best contacts are my friends. Do my friends fit on that model? Or are they outside?"
**A:** | Again, this situation is typical of networkers early in their careers. They're still hanging around with college buddies or people they grew up with. It's great to think of friends as networking contacts. Have conversations with them about keeping your antennas up, so you can provide opportunities for each other. And certainly, when you become Allies with someone, they become friends because you have shared your personal and career goals. It's wise however, to cultivate contacts strategically—to specialize a bit—so that you meet people in your career field or profession.
**Q:** | "I have people that I think of as Actors because I've done something for them, but they never do anything for me. How can I not only move to the next stage with them, but also be sure that the relationship is mutually beneficial?"
**A:** | Think about these people—the networking moochers in your life—as you answer the questions on the Rate Your Relationships quiz. Perhaps you haven't taught them what you're looking for and how they might be helpful. Next time you meet, have a couple of Success Stories to tell them, so that they can better appreciate your abilities and amiability. If you continue to feel that they aren't reciprocating, have a conversation in which you recount some of the things you've done for them and say, "Here's how you could help me." If that doesn't work, cross them off your list. There are a lot of people out there who will be happy to reciprocate.
**Q:** | "There's an Associate of mine that I don't want to have a relationship with at all. He's not a person whose ethics I admire. But he keeps calling and wanting to get together. What should I do?"
**A:** | Be busy when he calls, very busy.
**Q:** | "I think I have all Advocates and Allies in my network."
**A:** | Check it out by having conversations with some of these people and asking them, for example, to give a vivid example of what you do. If they can't, tell them Success Stories. They'll be likely to remember those anecdotes. Ask yourself if you are suffering from Tired Network Syndrome. You may have been at it so long, that you do have a cadre of well-established relationships. On the other hand, do you have networking needs that they are unable to fulfill—new passions in your life that your current contacts are not connecting with? If so, deliberately set out to join some new Arenas and meet some people who can share your new interests.
**Q:** | "How much time does it take to develop and nurture a relationship with an Ally?"
**A:** | Lots. Because you are so in tune, you may be tempted to cut back on the time you spend with these valuable contacts. Be sure that you do set aside a regular time to get together, perhaps dinner once a month. Challenge yourself to find some way that you can contribute to their success every time you meet. Listen Generously, be Seriously Curious and swap Success Stories. Then you'll be able to profit from—and continue to enjoy—those relationships for years to come.
## CHAPTER 5
## Go with Your Goals
Networking is small talk with a purpose. What's yours? What's going on in your life that makes you ready to work on your network?
You may have a vague notion that "Things go better with networking." You're right. Networking is not just a career skill; it's a life skill. It's for new grads, for people at any stage of their careers—or even for retirees. If you've decided that you want to know how to network better just because it's the smart thing to do, leap right in.
You may already have a personal goal—one that has nothing to do with your career. Perhaps you've decided you want to find a significant volunteer activity, something you can do to make a difference. You may already have a career goal. Perhaps you want to take the certification program offered by your professional association. You may already have a business goal. Perhaps you want to create a partnership with another entrepreneur. You may already have a workplace goal. Your organization may have announced initiatives that make building relationships a priority. Any goal you have can be the impetus for a networking Project.
### Size Your Project to Match Your Goal
You are CEO, president, and chairman of the board of your network. You will customize it to fit your own needs. It won't be a carbon copy of anybody else's network. It will include a unique set of contacts developed for your unique set of reasons. You will construct your network and select networking activities to move toward your goal, whatever it is.
If your goal is small, it might require only a small networking Project—one that can be accomplished in less than six months and is made up of only a couple of activities. It will take only a limited number of networking contacts and a limited amount of money, time, and effort. Wanting to benchmark your department's processes and procedures, for example, you might put together a group of several people in your professional association who have similar jobs. As each person hosts one meeting and provides an overview of his department, you'll all get a better idea of how your workplaces measure up.
### Never underestimate the power of networking to enhance your life, both professionally and personally.
If your goal is medium-sized, it might require a larger networking Project—one that could take six months to a year, with a corresponding outlay of money, time, and effort. Say you want to fulfill your department's new objective: Raise the visibility of our bank with small business owners. You will create a networking Project that includes a variety of substantial activities. You might join the Chamber and immediately volunteer for a committee that is assessing Chamber programs that target small businesses. And you might put together—and get bank sponsorship for—a workshop for members of the Home-Based Business Association. Over time, you will be able to open up many opportunities to meet small businesses owners.
If your goal is very far-reaching, you'll need to create a large-scale networking effort. It might take several years and require commitment, perhaps even self-sacrifice, not to mention money and time! These big, long-term, life-changing efforts are Strategic Positioning Projects. They will include many kinds of networking activities, pursued with high intensity. They will position you to be the natural and only choice when opportunity knocks.
### Check Out Your Choices
Whatever the size of your networking effort, you'll come to many ChoicePoints along the way. These are opportunities to make strategic decisions about what you will say or do. As you make these decisions, your goal is a guidepost showing you the right direction. Think, for example how many ChoicePoints—strategic decisions—you'll deal with as you attend just one networking event. You'll decide:
* What you want to get out of the event.
* What information you are prepared to give
* Who to talk with
* How to make your name memorable
* What to talk about
* What stories to tell that show what kind of expertise
* Whose cards to get and keep
* When you want to leave a conversation
And that's only a partial list. Using all the tips, tactics, and tools in this book will help you manage the many ChoicePoints and make the right decisions, every time.
### Assess Your Network
Before you begin to set the parameters for your networking Project, look at what you are currently doing and already have in place. Then you'll be able to decide what your networking Project will look like, what your investment will be, and how you can build on your existing circles.
**_Tally your time._** Grab a piece of paper and a calculator, and tally up the time you currently devote to networking. Consult your calendar for the previous twelve months. Can you document how much of your own and your employer's time you have spent? In our workshops, we've heard people say everything from a low of 20 hours a year to a high of 600 hours a year (about 12 hours a week). Of course, people in some careers spend almost all of their time networking.
**_Calculate the cash._** Now, figure out how much money you (and your organization) spent on your networking last year. Include dues and other expenses for activities, such as associations, professional groups, referral groups, country clubs, conferences, business lunches, trade shows, sports boxes and tickets, etc. Include money out of your own pocket, as well as money spent by your organization. What's the grand total of your financial commitment? We've heard amounts that ranged from a measly $75 to a grand $75,000.
**_Determine your return._** Are you surprised to see how _little_ time and money you actually spend, given how important meeting new people and re-connecting with long-time contacts are to your goals? Or are you shocked to realize how _much_ time and money you spend and want a better return on your investment? Do you need to increase the effectiveness of your networking to get your money's worth?
**_Decide on your investment._** How much time and money should you ideally spend? This is an important decision to make as you tune into your goals and begin to think what size networking Projects you want to do. You can't expect to spend five minutes and five bucks and change your life through networking. On the other hand, you can easily overspend or waste your hard-earned dollars or overextend yourself and waste your valuable time, if you don't make strategic decisions about where and when to spend your time and money. The bigger your goal, the more you are going to have to put into your networking efforts to achieve it.
**_Add up your Arenas._** Your Arenas are the circles you are known in and the groups you belong to. To maximize your opportunities for building your network, we recommend that you take part in six different Arenas.
### Adele's Arenas
Adele is a lawyer who specializes in serving clients in the high-tech industry. She has two goals: to gain "key player" status in that industry for her firm and, since she's in charge of her firm's new hire program, to increase by 25 percent the number of women associates hired over the next three years. Figure 5-1 is a chart she made of her involvement in six different Arenas.
### Your Turn
Create a chart like the one in Figure 5-1 and list all of your Arenas. First, list the Arenas you're already a part of because of who you are and what you already do. These Arenas have no membership dues, but they are circles of people you know: "I'm a parent who knows the parents of the other kids in my daughter's ballet class" or "I know my neighbors in Post Oak Farms." Then add the organizations you've paid to become involved with.
**FIGURE 5-1. Adele's Arenas.**
Fill out the rest of your chart. Make a note about the nature of your participation—what Role you play. Be honest. If you only attended two events last year, put that down. Then, itemize the Benefits of each organization. Do the Benefits of belonging contribute to the networking Project you are putting together to reach your goal? Finally, fill in your Reasons for belonging to that organization. Again, you may find that an organization is not a good fit for the direction that you are trying to go. Use your chart to assess the value to you of the organizations you belong to.
**_Assess the benefits._** Are you getting what you want from these organizations? If not, are they the wrong ones for you or do you need to find more strategic ways to participate? Does being involved contribute to your goals or have you outgrown your need for and interest in the group? Don't start joining new Arenas if you have not taken advantage of the opportunities in groups you're already a part of. As you assess your Arenas, ask yourself, "Where have I developed the most profitable contacts in the past?"
As Adele looked over her involvements, she decided that although Alumni Association events were fun, she'd be more likely to find clients and women to hire in other places. She was satisfied with what she was doing in the Technology Council. She noticed that her reason for joining the Board of Trade—contact with CPA firms—was not at the top of her list of current goals, but remembered several new clients in recent months had come from the group. As she thought about Women in Technology, she decided to consult the membership directory to see just how many lawyers were members. She found only four. None were a good fit with her law firm. She decided to terminate her membership. As she thought about the Dingman Center, she realized that, having lectured there for three years, she had already become an accomplished speaker and needed to find new reasons for continuing to teach there. She felt the Bar Association was a must.
As Adele assessed her involvements, she saw how important it is to make sure that organizations you belong to are the right ones for your networking Project.
**_Find new Arenas._** When you decide it is time to choose a new Arena, ask yourself, "Whom would I like to meet, and where can I find that type of person?" Look at the guidelines in Chapter 17.
### Plan Your Strategic Positioning Project
Here's how to get started planning your big Project to reach your big goal. If your goal is smaller, just scale down the networking activities you undertake.
Go for quantity as you come up with networking options to pursue your Strategic Positioning Project. You can put together a group to help you brainstorm. Or glean ideas from stories in the business press, from _The Wall Street Journal_ to _Fast Company._ Or ask colleagues or mentors for ideas. Or borrow models or adapt ideas from what others have done. You'll find dozens of ideas throughout this book.
### Design a big-time, long-term Project.
Use your network within your organization, your community, your profession, or your industry to do some preliminary research about possible activities. Then, outline your networking options. Projects evolve. You'll constantly be tweaking your Project as the months go on.
As we interviewed hundreds of networkers, we noticed that their Projects had similar characteristics. Most of their big-time, long-term, networking efforts could pass at least four out of the following five tests.
### The Doorway Test
Ask yourself, "Who do I need to know? Who are my ideal customers, clients or employers? How do I want them to perceive me? Where will I find these people? Where do they spend time? How can I participate in their Arenas?"
Will your Strategic Positioning Project position you so that the people who can help you achieve your goal are streaming by? Find—or create—a "doorway." Put yourself in that doorway so you meet those people and they begin to know you.
Melinda, a partner in a CPA firm, had to look to find the "doorway" that would lead to her goal: a constant stream of women business owners as clients. She found someone in her network to propose her name for the Board of Directors of the Women's Business Center, whose mission is to guide and support women business owners. The Board position gave her credibility and visibility. Melinda also taught classes at the Center on financial matters for growing businesses, whether they were start-ups or pulling in revenues of a million or more. As woman business owners became familiar with Melinda's expertise, some selected her as their CPA.
Mitch, a 28-year-old attorney, specialized in wills and trusts. When he moved to Chicago to join a firm there, he knew no one. He'd be the first to tell you that he found his Project—and his "doorway"—by accident. A guy who loved ballroom dancing, he noticed a newspaper article about a tea dance at a downtown hotel. He went to the dance and enjoyed himself hugely. The second time he attended, it hit him: The senior citizens at the dance were potential clients. Sure enough, as he got acquainted with his dancing partners, they naturally asked about his work and enough of them eventually became clients that the partners sat up and took notice.
To pass The Doorway Test, make sure your Project puts you in your target's doors or brings them to yours.
### The All or Nothing Test
If people see you doing one thing well, they will assume that you are good at everything. If people see you doing one thing poorly, they will assume that you do nothing well. That's the All or Nothing Rule. Is your Strategic Positioning Project a vehicle for demonstrating your Character and Competence? Even if your networking activity has nothing to do with your exact area of career expertise, people just naturally make the leap that you must be good at your job—if you are Competent—as you perform your networking activity.
### Your Project will make you the natural and only choice.
Evan wanted to become known to government human resources professionals who could hire the trainers and consultants his firm placed. Over many years, he became active in The Training Officers Conference, a professional group that meets monthly and has an annual conference. He showcased his approach to training and development in a luncheon speech he gave for the group called "Change Reaction and the Power of New Ideas." He became known to others by working on the Awards Committee. He soaked up information on trends and challenges from the many professional development programs he attended and from the conversations he had with contacts. He looked for ways to support other members and funnel good information to them. He established his Character and Competence. His long-term commitment to the well-being of the group and the people he meets there has often made his firm the natural and only choice when training and consulting are needed.
Lee's goal is to look for a satisfying, part-time retirement career. As a tax attorney, he had developed a reputation for being able to explain hard-to-understand concepts. Through his network, he was asked to teach in an executive MBA program. Teaching was a natural for him. Student evaluations tipped Lee off that storytelling was his strong suit. Lee created a Project to investigate storytelling as his next career. He used his network to find a speech coach and to discover the best storytelling festivals and workshops to attend.
To pass The All or Nothing Test, be sure your Project gives you places to showcase your abilities. Then, perform brilliantly. That way, you'll get a reputation for doing everything well.
### The Bottom-Line Test
Can you arrange the time in your schedule and the money in your budget to support your Strategic Positioning Project? Will your networking efforts take you one step forward toward your goal?
Louisa is a financial planner whose long-term goal is to be invited to give a two-minute "financial tip" for women every day on CNN. To prepare for that opportunity, she found a networking contact to propose her as host for a monthly show for women to her local cable TV station. She's developed a circle of contacts at the station and is learning from them everything she'll need to know to be comfortable in the world of TV. Louisa estimates that the Project will cost her about $5,000 this year and take about three days a month. That's a huge investment, but she's confident that the experience she's gaining with her show will lead to a syndicated show and ultimately that call from CNN.
### Invest time and energy to build the net worth of your network.
Dan's goal as a human resources manager is to make an extraordinary contribution at work. He's in charge of the corporation's Employee Satisfaction Survey. After talking with several trusted colleagues, he decided that his Strategic Positioning Project should be to join The Horizon Group, made up of people from other Fortune 500 companies who have the same responsibility. He knew that membership would help him and his company learn state-of-the-art survey strategies. To prompt an invitation to this exclusive group, he networked with several Horizon Group members he knew. When he was invited to join, Dan then had to convince his boss that budgeting for the Horizon Group's weeklong annual conference and the $5,000 membership fee would be worth it. He made a detailed proposal that was accepted, even in a year when money was tight.
Three months after Dan attended his first meeting, he proved to his boss the value of his Horizon Group networking Project. He had access to survey questions that had been vetted by experts, had contacts to ask for advice on how to adapt the survey when his company acquired a German subsidiary, and he had new ideas about how to announce the results of the survey for maximum impact. Dan was nominated for an Employee-of-the-Year award and was satisfied that he and his SPP were making—and would be able to continue to make—a major contribution to his organization's success.
To pass The Bottom-Line Test, make sure you are spending your time and money in the best way to reach your goal. Put together a Strategic Positioning Project with a big impact.
### The Five-Year Test
Does your Strategic Positioning Project set the stage for the phone call you want to receive in three to five years?
Deirdre has had jobs of increasing responsibility in the high-tech industry. She accepted a job offer from a company she really wanted to work for even though it was a lateral move. Not content to languish too long in this mid-level marketing position, she designed a Strategic Positioning Project. Her goal? To become known to people in the corporate hierarchy who might tap her unused talents and advocate for her when openings come up. She made a list of the folks she especially wanted to know—and to know her. She used her job as program chair for her professional development association to take the first step. When her committee decided to do a program on Employee Retention Strategies, Deirdre invited one of the people on her list, her second-level manager, to be on the panel. Deirdre had several opportunities to talk with the manager before the program and to show her Character and Competence as she handled all the planning details. She even offered to drive her manager to and from the event because parking was difficult and because, of course, she knew they'd have more time to talk.
To pass The Five-Year Test, be sure every activity in your Project helps to create the outcome you want down the road.
### The Pig in Mud Test
Does your Project represent a magnificent blend of your unique personal and professional interests? Does it represent who you are, what you value, what you like to do, where you want to go, and what you do best? Will your Project make you as happy as a pig in mud? We are from Kansas, and we have seen pigs in mud. They wallow. They roll. They close their eyes in ecstasy. They are happy, content, comfortable, and want to be right where they are and no place else.
Morris, owner of a mortgage company, graduated from the university 15 years ago. An active alumni and avid supporter of the university's basketball team, he had a strong network of contacts in the university's alumni association. He created a Project that helped his alma mater, allowed him expand his business in a very natural way, and made him very happy. Here's how it happened. It came to his attention that the president of the university didn't have any discretionary funds to use for worthy projects that came up from time to time. So Morris offered to start the President's Club. Through networking, Morris assembled a group of generous donors who created a fund for the President to use on innovative projects that would improve the University's visibility and attract talented students. When students wanted to enter a robotics contest in Japan, they were able to buy their supplies and their airline tickets with money from the fund. Imagine how the President bragged about his students when they won the contest, and how grateful he was to Morris for creating the fund!
In his nationwide fundraising campaign, Morris found that alums often asked about his business and some wanted him to handle their mortgages. Morris expanded his business so that he could do mortgages in many states and took his business to a whole new level.
When Jeanne decided that she wanted to find the "capstone job" of her career, she made a list of thirty-eight people in the non-profit area. Her plan? Talk with these folks as a way to survey what's going on in philanthropy. She'd call and ask them to lunch "to explore what's next in our career field." She went into these conversations with two goals: connecting and learning. Notice that job-hunting was not her primary goal. She had a job that she could stay in until she found that ideal position. "I'm Irish. The best fun in the world to me is talking and brainstorming with people," she says. "And our conversations were always two sided—their career and my career." After thirty-two conversations—over almost a year—she got the call she wanted. When she picked up the phone, one of her contacts said, "There's a new job here that I want to tell you about. We had you in mind when we wrote the job description for this new position." It was perfect.
To pass The Pig in Mud Test, be sure your Project makes you happy!
### Bonus: Get Off to a Good Start
**1.** Get specific about your goal. Write it down in detail.
**2.** Determine what size networking Project your goal demands.
**3.** Decide how much time and money you are going to commit to your Project.
**4.** Check these figures against the time and money you spent during the past 12 months. See any difference? What are the implications as you plan your Project?
**5.** Analyze the chart you made of the Arenas you are currently involved with and decide if you want to continue your participation, make your participation more strategic, or discontinue your participation.
**6.** Think what you could do in your current Arenas to move toward your goal.
**7.** If you decide you need to find new Arenas, make some notes about what you will be looking for.
**8.** Do you need help conceptualizing your Project? Decide if you want to ask individuals (Who?) for suggestions, put together a brainstorming group (Who?), look for models (Where?). Read all of this book for ideas.
**9.** Think about what research you'll need to do to figure out the right networking activities for your Project.
**10.** Begin to keep a list of your networking options.
**11.** Start talking about your goal and networking Project. This will do two things: help you clarify your Project and help you gather good ideas.
**12.** Decide if your plans so far will help you pass The Doorway Test and put you in touch with the people you need to meet.
**13.** Figure out if your plans so far will help you pass The All or Nothing Test and allow you to show your Character and Competence.
**14.** Do all of your plans so far pass The Bottom-Line Test? Can you spend the necessary time and money? And do the activities you have in mind clearly take you toward your goal? Are there peripheral or extraneous activities that you could cut to streamline your Project?
**15.** Can you see how doing this Project will result in your eventually getting the phone call you want or creating the opportunity that will signal you've reached your goal? Does your Project pass The Five Test?
**16.** Does the thought of forging ahead with your Project make you happy? Does it pass The Pig in Mud Test and blend your personal and professional interests?
**17.** As you plan and implement your networking Project, are you ready to learn everything you can about networking from this book?
We encourage you to start right now. We'll be cheering you on every step of the way to your goal.
## PART III
## Sharpen Your Skills
You say you weren't born with the gift of gab? That's okay. Anyone can learn how to turn small talk into smart talk. Here you'll discover state-of-the-art skills and secrets nobody ever told you about—the rules and tools that make networking easy. Whether you think of yourself as out-going and gregarious, or shy and retiring, you'll be able to enter a roomful of strangers enthusiastically, comfortably, and professionally. Worried that you don't know the no-nos? Take a look at the Top Twenty Turn-Offs, things no savvy networker would ever be caught dead doing.
What really happens when you meet someone? Do you muddle through those oh-so-important first few minutes? We'll help you rid yourself of the worn-out rituals that don't work. And you'll find out exactly what to say and do to manage those three Million-Dollar Moments that happen over and over again every time you shake hands. You'll learn how to get the most out of every conversation from "Hello" to "Good-bye." Want to know what to do next to enlist the people you meet in your network? You'll discover many ways to reconnect, stay in touch, and follow up.
## CHAPTER 6
## Know the "Netiquette"
Do you wonder, "What's the right thing to do?" If you know the networking protocols, you'll never be at a loss. You'll be able to put your best foot forward, in every networking moment.
### Enter Enthusiastically
For many people, the hardest part of networking isn't talking at all; it's entering a room full of strangers. Your Critic may be saying, "Everybody knows each other but me. Everybody's better at this than I am. Everybody's looking at me." Nonsense. That's just not so. "I enter a room in an upbeat way with a smile," says Martha. "That way people want to talk with me. Sometimes people coming into a room look so serious and forbidding that they give signals that say, 'Don't talk to me!' I like to let my body language show that I'm glad I'm here."
That's good advice. Often, people spend a lot of time before an event fussing with their hair, their clothes, their makeup. Of course, you'll want to look your best. But be sure, also, to put on a smile and energize yourself. The way you relate to space sends a message. The pace of your entry and the amount of space you take up indicate your level of confidence. If you move slowly, edge into the room keeping your back to the wall, and make little or no eye contact with people, you will look uncomfortable. Relax. To energize yourself, listen to your Coach's encouragement. (For more about your Coach, see Chapter 2.) Talk to yourself. Use positive statements like:
* "This is going to be interesting."
* "I'm ready and eager to talk to people today."
* "I wonder what great ideas and opportunities I can discover as I talk with these people."
### Brighten Up Your Body Language
What messages are you sending through your body language? As you talk with someone, you either provide "rewards," through your positive responses, or "punishments," through your lack of response or negative responses. If you are providing rewards, people will enjoy talking with you. If you provide only punishments, they'll move on to someone else. Look at Figure 6-1 for non-verbal cues to know.
**FIGURE 6-1. Non-Verbal Cues.**
### ENGAGE Your Partner
Use your body language to reward and encourage your partner in conversation. The letters ENGAGE can help you remember how to give a positive message through your body language as you talk with people.
**E = Establishing Eye Contact.** If you're looking anywhere on your partner's face, she will feel that you are looking her in the eye. In our culture, we break eye contact every seven or eight seconds or so. Glance away and then back. It's more flattering to your partner to glance down to the side and then back at your partner's face rather than over her shoulder, as if you are looking for someone else to talk with.
**N = Nodding.** Nod to show that you're following and enjoying the conversation.
**G = Geniality.** Be cheerful and cordial. Your geniality tells your partner that you're having fun in the conversation. Smile. Do it often, but appropriately. Nervousness can lead to smiling at serious or even sad topics. Women who are concerned about being perceived as assertive should be careful to smile only when the occasion or topic warrants.
**A = Aiming Your Attention.** Let your body language acknowledge that your full attention is concentrated on your partner. Lean slightly forward.
**G = Gesturing Appropriately.** Use your hands to emphasize key words or concepts. Watch other people to expand your use of gestures. The people who are the most comfortable will use more expansive motions. But small gestures work well in close-up conversation, too.
**E = Easing Your Posture.** Stand comfortably, with your feet slightly apart and your back straight. Center your weight so that you don't sway or feel off-balance.
In a crowded, noisy room, you can create a "bubble" for you and your conversational partner with your eye contact, your gestures, and your forward-leaning posture.
### Tune Up Your Tone of Voice
Do you sound whiny? Tentative? Bored? Do you sound like a schoolmarm? A drill sergeant? A mouse? A judge? How do you want to sound? What tone of voice would support you as you present yourself?
Read a couple of children's books into a tape recorder. Try to really "be" the characters and create different voices for each of them. This exercise has several benefits: You widen your range of available "voices," you find out how you sound, and when you have finished analyzing your tape, you have a nice present to give to a child you know!
You also might tape yourself in various situations, such as at the office or at the dinner table. If you just turn on the recorder and let it run, you'll eventually forget about it and record your voice in its "natural" state.
Make your voice sound more confident and energetic. If your voice sounds too high, deepen it. Move it down the scale one whole tone. If you sound draggy and tired, speed up your delivery. Radio announcers read about 150 words a minute. Moving along a bit faster than that will make you more interesting to listen to. If you seem to be talking in a monotone, emphasize key words by moving up or down the scale. Making some words higher or lower than the rest avoids a flat delivery and allows your energy and enthusiasm to come through. Also, increase or decrease your volume to emphasize key words.
### Consider Closeness
Each culture has different "rules" about how close conversation partners should stand when they talk to one another. In the U.S., we stand about three feet apart to have social conversations. If the room is very noisy or if the conversation becomes more personal, we move in, closing the range to perhaps eighteen inches. If we move closer than that, we are usually having an intimate conversation. If your partners in conversation typically move away from you, you may be violating this "rule." On the other hand, if you talk from farther away or move away during a conversation, your partner may think of you as "stand-offish" or "distant."
Sometimes a person moves away from his conversation partner for other reasons. Television commercials for products like deodorants and mouthwash have created anxiety about being close to other people. The most sensible thing to do is to take reasonable precautions and then don't worry about these problems.
### Watch What You Put in Your Mouth
In business or social situations, neither smoking nor excessive drinking is acceptable behavior.
If the event involves eating, use your best manners. At a stand-up networking meeting, choose foods that are easy to eat, such as grapes, crackers, or bits of cheese. Steer clear of the chewy, dripping, garlic-laced, hard-to-eat items at the hors d'oeuvres table. If you worry about a piece of spinach attaching itself to your front tooth or about dripping shrimp sauce on your tie, don't eat.
If the etiquette of eating concerns you, read a book on manners, find a course on etiquette in your community or encourage your organization to offer one. There's no point in letting concerns about manners sabotage your ability to be at ease with people.
### Treat Touching as Taboo
Except for shaking hands, it's usually inappropriate in business settings to touch your partner. Two people of the same gender and "rank," however, may be comfortable touching during a conversation. In general, though, it's either the female or the more powerful person who initiates touching during a conversation. If a woman touches, it may be interpreted as flirting. If a person of higher "rank" touches, it may be an indication to his partner that he or she is pulling rank, trying to establish control, or even engaging in sexual harassment.
Unless you have established a special relationship with someone in which touching is acceptable, don't do it. And even if you have a warm relationship with someone—especially someone of the opposite sex—it's better to err on the side of formality in public.
### Forego Flirting
Marilyn says, "As a woman, I'm concerned that people will misinterpret my friendliness. I don't want them to think I'm flirting."
Jorge says, "I worry about the Dos and Don'ts of networking with women."
Those worries are unnecessary if you're clear about the difference between flirting and networking and if you "read" the unwritten ground rules in different organizational cultures. For example, two past presidents of one professional organization—in this case, a man and a woman who have known each other for nearly two decades—hug when they meet. In other groups, that behavior might be off-limits.
In business situations, you'll want to avoid a sexual come-on. If you know how to flirt, then you can figure out how not to. But just in case you do it unconsciously, take a look at some common flirting behaviors.
Men flirt by extending eye contact beyond the normal length of time (more than eight seconds or so), which signals attention and interest. A man also may indicate a romantic interest by inappropriate or extended touching, a handshake that turns into a hand holding, for example. Or he may sit closer than necessary. Or he may assume responsibility for a woman's comfort through excessively solicitous hovering. It is not appropriate in a business setting for a man to open doors, help remove a woman's coat or pull out her chair. Or rather, it's not appropriate if the behavior is unilateral. If either party helpfully opens a door for the other person, who is encumbered, that's fine. A man doesn't need to offer to carry packages or suitcases. He shouldn't use a diminutive nickname—saying Katie for Kathryn, for example. If a man introduces himself as William, you'll _never_ hear a woman say, "Hi, Billy." That's a put-down in business. So are overly enthusiastic comments about a woman's dress or hair or other personal compliments beyond, "You're looking well."
Women flirt using many of the same tactics. Eye contact, especially lowering the head and looking up through the lashes, can be flirtatious. So can touching in a proprietary manner, fingering a man's tie, for example, or brushing lint off his shoulder. Women also flirt by touching their own hair or twisting a curl. They might stand closer than normal. Inappropriate laughter is another clue. So is calling a man "honey" or "love." Women might "serve" men, offering to bring them coffee, for example. Most comments about a man's clothing or hair are also off-limits.
None of these behaviors is appropriate in a business context.
### Pay Your Way
By the way, always pay your fair share of the cost of networking. It's better to go "dutch treat" than to pay for a networking contact's meals, for example. Remember, you're trying to establish a mutually beneficial relationship. Ignore titles, and work on developing peer relationships, not superior-subordinate relationships. If one person always pays, the relationship also will become unequal. You can't buy a networking contact. Most relationships work best when each person pays his or her way, not only with money, but also with valuable information or referrals or resources.
### Exchange Business Cards Effectively
Handing a business card to someone does not constitute a networking relationship. Strangely enough, the biggest mistake people make with business cards is giving them out too freely, too soon. When that happens, your contact will go back to the office, look at the card, say to himself, "I wonder who this is?" and throw the card in the wastebasket.
### Handing out your business card makes only a "cardboard connection."
Challenge yourself not to give your card out until you've found some connection, some reason for exchanging names and phone numbers. Approach conversations asking yourself, "I wonder what she needs that I can provide? Let's see if I can figure it out," and "I know what I'm looking for today. Wonder if I can find someone who has the information I need?"
You have to work to make a conversation lead to the exchange of cards. When you hit upon a reason to trade cards, you have accomplished something very important. You have extended the relationship beyond the event at which you met.
Jewell mentioned her aerobics class to Tess, a woman she met at a seminar. Tess mentioned that she wanted to find a low-cost exercise program, so Jewell asked for her card and said she'd send some information. The next time Jewell went to her aerobics class, she picked up a catalog from Parks and Recreation, took it home, and mailed it to Tess.
When you find a conversational connection and need to exchange cards, take a moment right then and there—or as soon as you leave the event—to note on the back of the card anything you want to remember about the person, the conversation, or what you agreed to do: "Has twins." "Went to Duke University." "Wants to know more about the Chamber of Commerce networking night." "Needs speaker." If you have promised to do something, follow through.
The next time you see your contact, you don't have to start all over again. You can build on the information that you provided. You might update your contact on how you are using the information he sent you: "Thanks so much for sending me Tom's phone number. He sounds like just the speaker we need for our sales meeting. We plan to meet next week."
If someone gives you her card the moment you meet, don't just stuff it in your pocket. Read it carefully. Use it as a visual aid. Look at her name and learn it. Discuss her title. Notice the organization. Ask about it. If it's an engineering firm, ask what kinds of engineering projects it specializes in.
### Tickle yourself. Jot reminders on the backs of cards you collect, so that you can follow up and follow through.
Then give her your card and use it as an aid to teach her about you. "My card says Robert James Hensy, but everybody calls me 'R.J.' My office is about five blocks from yours at Tenth and Broad. I've listed some of the health care management services my firm provides on the back of my card. Our new facility is for patients with Alzheimer's. Did you see the story about it in the Kansas City Star last Sunday?"
### Join Groups Comfortably
In any room full of people, most people will be talking in groups. You certainly can look around to find someone else who is not attached to a group and make a beeline for that person. Barbara says, "When I feel nervous about joining a group, I form my own. I look for someone standing alone and start a conversation with that person."
Or you can join a group. In our workshops, people ask, "How can I break into a group?" We tease them a bit: "Well, first, you find a big sledgehammer...." We choose not to use the phrase "break into a group." Breaking in implies that you must force yourself on the group, a violent act; joining implies that the group was incomplete without you!
To join in, signal that you're committed to becoming part of the conversation. Gently but firmly touch the arm of one person. Almost always, the circle will open up to allow you to come on in. Don't be tentative; show commitment by making eye contact with the speaker or smiling at one of the listeners. Take a few seconds to listen. You can start participating any time you feel tuned in to what's going on. When the conversation slows, turn to a person next to you and introduce yourself. Often, someone else in the group will initiate introductions. If people in the group seem to be acquainted, ask, "How do you all know each other?" as a way to prompt introductions.
If someone quickly introduces everyone in the group to you, don't despair. Simply go back to each individual later and say, "Let's introduce ourselves again. It's hard to catch everyone's name in a group."
If joining a group is uncomfortable for you, analyze why. Are you remembering high school? Most of us have vivid memories of feeling excluded—even people who were members of an "ingroup." As grown-ups, we still carry some of those adolescent feelings around with us. Analyze what your Critic says when you think about joining a group. When you bring the Critic's comments out into the light of day and examine them, they usually will seem quite ridiculous and based on leftover angst from your teenage years:
* "They don't want to talk to me."
* "They are talking about me."
* "They don't want to include me."
* "They will laugh at me or tell me to go away."
When a new person joins your group, smile, nod, and make eye contact. When whoever is talking comes to a stopping point, fill the newcomer in. Say, "Jack was just telling us about his new job." Then look back at Jack so he can continue.
Sometimes people worry that they might be joining a private or intimate conversation. Trust your powers of observation. You will be able to tell when a private conversation is taking place. Here are some clues. People may be touching. There may be visible emotion. Voices may be very low or higher than normal. They may move closer to each other.
If you enter a conversation that's too intimate, or if you don't like the topic, you can leave comfortably. If the conversation is too personal, say, "Looks like I've interrupted something. I'll talk with you later." Or, "This feels like a private conversation. I'll catch you later." Or, if the topic is not something you want to talk about, say, "Hey, I'll talk with you later. It looks like you're really getting into this topic." You may find that one of the people who's involved in the intimate conversation or talking about the topic you're trying to avoid will regard you as a savior and welcome you into the group as an excuse to reduce the level of intimacy or change the topic.
If you're still unsure about how to present yourself with greater confidence, ask yourself, "How would I act if I had just 10 percent more confidence? Or 25 percent more? Or 50 percent more?" Then act that way.
### Bonus: Ten Tips on the Nuances of "Netiquette"
Whether you are at a networking event or a ball game, keep these tips in mind to be comfortable and professional as you make contact.
**1.** Be polite, positive and politic. Don't ask, "Has your boss calmed down any?" Ask, "How's morale in your office these days?" Don't ask, "Have the lay-offs finally stopped at your company?" Ask, "Are things back to normal at your company?"
**2.** Check in with acquaintances. Meeting new people may actually be easier than beginning a conversation with a person you see only rarely and know only slightly. Don't berate yourself for not remembering all the details of that person's life or work. And, it's easy to put your foot in your mouth inadvertently when you begin a conversation with someone you haven't talked with recently. Assume that the person's life has changed. It probably has.
"A man I'd worked with several years ago had transferred to another division of the company, "says Jerry." Seeing him again, I asked about his wife. He said, 'Oh, we've been divorced for two years.'" To avoid that kind of slip-up, ask general questions rather than specific ones. "We haven't talked for a while. Catch me up on what you're doing." "How's your year been?" "What's changed for you since we've talked?" "What's new in your life?" These kinds of questions allow the other person to bring you up to date, revealing as much as he wishes. The answers will guide your conversation.
**3.** Go for the relationship, not the contract. When you meet casually, and the conversation moves to a "let's do some business" level, set up a convenient time to call or place to meet to complete the transaction. As you continue to talk in the casual meeting, build your relationship with your contact. If you swap stories about trout fishing with a new acquaintance and develop a strong rapport, he's more likely to think of you when he needs an accountant than if you spend most of the conversation aggressively pushing your accounting services.
### Go for the relationship, not the contract.
**4.** Relationships that bounce back and forth from friendship to selling are tricky, no doubt about it. Be scrupulously honest about your intentions to keep the boundaries clear and avoid abusing a friend's trust.
Nora and Lee had known each other for fifteen years. Lee's job was eliminated, and she went into business for herself. Nora, the marketing director for a law firm, called Lee and asked her to recommend a computer software trainer. More than a year later, after the two had visited at several professional meetings, Lee called Nora and set up a lunch meeting, saying, "I want to hear about the condo you are building and catch up with your life. I also want to tell you about a series of seminars I just finished doing for employees at the bank. I think your organization might find these seminars useful." Lee made her Agenda clear as she was issuing the invitation. Their lunch conversation ranged from personal items to the seminar series, and Nora asked for additional information about the seminars so she could consider them for her employees.
**5.** Make judicial choices about what parts of your personal life you are willing to talk about in networking situations. Mae, an attorney with a mid-sized firm, is building a reputation for being a fascinating person to talk to—about a lot of things. Even though adoption law isn't her specialty, she has two adopted children, so she shares what she knows and then refers prospective parents to an attorney with many years of experience in that field. She also is a stand-up comic. She is careful about bringing her passion for comedy into the conversation because she doesn't want it to affect people's perception of her as a lawyer.
**6.** Notice people. You may think it's not proper to ask about his tan or her beautiful necklace. Why not? As children, we may have asked embarrassing questions about the obvious ("Why is Aunt Betty's tummy so fat?") and been shushed rather than told she was pregnant. As children, we may have gotten another message, a far deeper one: "Don't notice other people." In school, teachers said, "Keep your eyes on your own paper." Other adults may have told us, "It's not polite to make personal remarks." Comedian Lenny Bruce once said, "When you are eight years old, nothing is any of your business." Some of the rules we were taught as children no longer apply.
"Before a business meeting got underway," Meg remembers, "I joined a group of three or four people who were chatting. One of them, John, had his arm in a cast. I figured the rest of the group had already asked him, 'What happened?' So I decided not to ask again. As the meeting started, I asked Betty what happened to John's arm. She said, 'Oh, I don't know. Nobody asked about that.'"
**7.** Make talking to you easy, not hard. Don't put yourself down. You will discourage conversation if you say, "I'm just a secretary." Or, "I'm just a housewife." After one of those statements, your conversation partner is at a loss for what to talk with you about. If you are feeling like an underdog in a situation, be sure you are prepared to talk about something interesting. See Chapter 10 for ideas.
**8.** Edit the jargon from your conversation. If you say, "I'm the EXO for the DDG," you'll stop the conversation cold. If you are talking with people outside your company and occupation, be sure you translate any specialized language into terms that anyone can understand.
**9.** Be on the lookout for role models. When you find someone who handles situations in a way that you'd like to, observe how they do it. It's all right to copy someone else's manner. Do it consistently, and it will become part of your own style.
**10.** Just say "Hi!" What's the most successful opener for starting a conversation with somebody? There's nothing complicated about the answer. The word is "Hi!" Just "Hi!"
It's a "Hi!" that flashes a message in neon lights: "I feel great about meeting you, and I'd like to talk with you." That "Hi!" says, "I'm happy to be here, and I'm looking forward to getting to know you." It's inviting and energizing and relaxing at the same time. It's inviting and energizing because it signals that you're a person who is committed to helping this conversation move along. It's relaxing because it signals that you're a person who can take care of himself in a conversation. It's not the kind of "Hi" that sends the message, "I'm just saying this to be polite, and I hope you won't take this as a signal that we have to talk." What a difference! And the difference comes from the tone of voice and the body language. Practice the two kinds of "Hi!" Feel the difference between the two.
### Talk to strangers. Take every opportunity to meet someone new.
Joan remembers learning about the power of just saying "Hi". "Shortly after I was married I went downtown for lunch with a neighbor, whose name was Sue Jones," she says. "We went to the department store cafeteria. You could sit anywhere. We went over to a table and put our trays down. Sue looked at the two strangers sitting there and said, 'Hi, I'm Sue Jones.' I never would have introduced myself to strangers. I was so stunned, so impressed. So, I made a button in my brain and labeled it Sue Jones. After that, when I would go to various events, I would push my button and say, 'Hi, I'm Joan Martinson.' That helped me to begin talking—just pretending I was Sue. I still do it. I still, ten years later, have my Sue Jones button. When I'm uncomfortable being myself or when I'm feeling shy, I push the button, become Sue Jones, and I'm immediately comfortable reaching out to people."
## CHAPTER 7
## Avoid the Top Twenty Turn-Offs
Do some networkers drive you crazy or display an appalling lack of "netiquette?" It's easy to recognize other people's "sins," the things that turn you off. Of course you wouldn't commit any of these faux pas, would you? You'll be a more attractive conversational partner if you avoid these twenty turn-offs.
**1. _Don't tell all the details._** The classic definition of a bore is someone who, when you ask him how he is, tells you. The person who insists on telling everything will soon lose his audience. Ever notice how infuriating it is to listen to someone who wonders out loud, "Let's see, was it Tuesday or Wednesday?" Who cares? Get on with the story. Don't include everything; sketch in the broad outlines. Follow the tips in Chapter 11 for telling brisk, compelling Success Stories about your experiences.
**2. _Don't do monologues and interrupt others._** This bore never lets you get a word in edgewise. He trounces your comments with non-stop verbiage of his own. Hogging the airtime, he insists on having the first, the middle, and the last word. He especially likes to interrupt subordinates and women to show who's more powerful. He ignores questions and insists on directing the conversation himself. Encourage and invite others to participate. If they don't leap in, ask them a question and wait for their answer. See Chapter 11 for ideas on turning monologues into dialogues.
**3. _Don't interrogate people._** Persistence is usually counted as a virtue. However, you have to know when to stop pushing and probing. The interrogator doesn't. Long after a topic has run out of steam, the interrogator is still battering his conversational partner. Tone of voice has as much to do with interrogation as the wording of questions or comments. They are delivered in an accusatory tone, often belittling or demeaning to his partner. He says, "You should...." It's one of his favorite phrases. "Why don't you...?" runs a close second. "Surely you know that...." ranks right up there. The interrogator generally has strong feelings about the topic and pushes his partner for agreement. State your ideas but don't try to convert others to your way of thinking or push your opinions down their throats. If an interrogator puts you on the spot, say, "Why on earth would you ask me about that? I never talk about...." Or, "Are you comfortable talking about that? I'm not." Depersonalize the topic. Move the question from one that makes you uncomfortable to a topic you'd like to discuss.
**4. _Don't insist on one-upmanship._** The person who always has a better story than yours or a better deal to crow about is committing one-upmanship. She can never merely accept a comment or story; she has to top it with one of her own. These people use conversation to make themselves look wonderful—sometimes at the expense of others. If you've closed a $1 million deal, they've got a $5 million one to tell about. If you went skiing at Keystone, they went to the Alps. Don't play the "Can You Top This?" game.
**5. _Don't seek free advice._** This person wants something for nothing. He corners doctors to ask about his physical symptoms, lawyers to ask about planning his estate, computer consultants for detailed advice on updating his firewall. He abuses his partners by asking them questions in a networking situation that he should be asking in a more formal situation. Don't ask for free professional advice when you should be paying for it.
**6. _Don't hide._** Some people, not wanting to appear self-centered, never tell you anything about themselves. You have to pry it out of them. Or they downplay what they have done, leaving you to feel quite foolish when you discover, through additional questions or other means, what it is they are obliquely referring to. If you bring up a topic, be sure it's one that you feel comfortable discussing fully.
**7. _Don't be dogmatic._** These people have all the answers and merely want to make converts to their way of thinking. They want confirmation of what they've already decided. Compare them to listeners who keep an open mind to gain understanding and sometimes, as a result of talking with others, even change their point of view. People do expect to be comfortable as they network. They don't want to be harangued by someone who is trying to change their mind or force opinions down their throats.
**8. _Don't give unsolicited advice._** If you evaluate your own life, you reveal yourself—not a bad thing to do in conversation. But, if you evaluate others' lives, you may offend them. Know the difference. Never say, "Why don't you...?" Or even worse, "You should...." Or "You should have...." If you feel that your experience might be helpful to someone else, ask permission. Say, "Would you like to hear about what I did in that kind of situation?"
### People expect to be comfortable as they network.
**9. _Don't be a bigot._** Unfortunately, bigots come in many varieties. These are the people who make ethnic, religious, or sexual comments to put down others. Bigots insult various people in various ways. They stereotype, lumping people into groups and making comments about that group as if those comments apply equally to all. They generalize based on a single experience or a small number of experiences. Or they tell jokes that play off stereotypes. In any form, bigotry is highly offensive and exhibiting it discourages others from trusting you. If someone makes a bigoted remark, practice your assertive behavior. Force him to examine his prejudices. Call attention to them. A hearty executive grabbed a woman's hand at a meeting saying "You must be Marvin's secretary!" The woman calmly replied, "Why would you think I'm someone's secretary?" It may be necessary to keep the bigot's good will and to help him save face. If so, speak in a light tone of voice. Tact is the knack of making a point without making an enemy.
**10. _Don't whine._** Whiners never have anything good to say. They go on and on about their health (Dreadful!), the economy (Dreadful!), today's teenagers (Dreadful!). You know the type. What a downer! Don't allow yourself to be perceived as a whiner.
### Networking events are places to make plans to get together later.
**11. _Don't do hard sells._** When Benny goes to a networking event, he thinks he should get somebody's name on a contract right then and there. He walks up to people and says, "Hi, I'm Benny. I sell specialty advertising items. I could make your business card into a refrigerator magnet and have it ready for you by next week." Contrary to what many people think, a networking event is not a place to sell. It's a place to make contact with people so that you can arrange to meet them later to do business. Benny isn't focusing on building trust; he's putting all his energy into finding a customer. He needs to back off and cultivate the relationship. That's when he'll begin to attract clients.
**12. _Don't assume you will get paid._** Gina gave Carlotta the name of a potential client to call. Carlotta followed up and contacted the prospect. She got the contract and wrote Gina a note of thanks for the lead, saying that she would be on the alert for something to send Gina's way. Gina's response was to bill Carlotta for a 10 percent "finder's fee." Most networking implies reciprocity. It is okay to charge a finder's fee only if that agreement is made up front. Generally, expert networkers consider such arrangements to be self-defeating because they do not necessarily develop the relationship. They'd rather have someone looking out for an opportunity to give them. If, however, someone does something for you that results in business, and you know you'll never have an opportunity to pay her back, then you might consider paying a finder's fee or sending a gift. Thea wrote a magazine article quoting Dorothy, who lived in the Midwest, Dorothy got a call from someone who read the article. That person became a client of Dorothy's. In gratitude, Dorothy sent Thea a finder's fee.
**13. _Don't make unreasonable requests._** Be careful what you ask for. Una, a part-time professor, told her class of eleven graduate students to call and interview certain middle managers in her company. The next time one of the managers saw Una, he asked her, rather pointedly, to check with him before making that kind of assignment in the future because it was too time-consuming for him to talk to so many students during his busy workday. The professor apologized and was especially careful for several months to look for ways to be helpful to all the managers to repay them for the inconvenience she had thoughtlessly caused them.
**14. _Don't confuse contacts with friends._** "Networking, ugh!" says Yvonne. "I just want to be friends with people." You can develop friendships with networking contacts, but it takes time, something many business people don't have. It's possible to have many "warm" business contacts without turning them into friendships. Women are more likely than men to be confused by the invisible line that separates contacts from confidants. Don't worry about it. Be friendly with your contacts. If a friendship takes root, fine; if not, you still have a good contact.
**15. _Don't abuse people's trust._** Your work may deal with sensitive information or business intelligence. Don't pass it on without careful thought. As you tell stories to illustrate your Character and Competence, disguise real situations if they are sensitive. If you are a real estate agent who sold a house that was part of a divorce settlement, for instance, you would, of course, avoid using your clients' real names. Don't give out a contact's name or use one person's name to make contact with another person unless you ask permission first. The world is a small place, and word gets around if you violate someone's confidence. Always be trustworthy.
**16. _Don't be so eager to provide resources that you pass along names of people or organizations that you haven't thoroughly checked out._** Before you give your contact a name, be sure that person's or organization's performance will reflect well on you. Jon asked Charlie if he knew of a good foundation repair company. Jon, wanting to be helpful, told him the name of the contractor who was working on his neighbor's basement. Unfortunately, the neighbor, unbeknownst to Jon, had just filed suit against the contractor.
**17. _Don't burden others with inappropriate or intimate information._** You have to know someone very well before it's appropriate to discuss your daughter's divorce or how much you spent on your last vacation.
**18. _Don't expect to get without giving._** The absolutely worst thing you can do is to take repeatedly from a person without reciprocating by sending information, referrals, or opportunities her way.
### Anybody can learn to make small talk and to use small talk to create valuable networking relationships.
**19. _Don't refuse to play the game._** Some people are hard to talk to because they don't take "small talk" seriously. They haven't bothered to plan ahead, to think of good topics, to notice what's on their Agenda, or to collect experiences to talk about that reveal their interests, Character, and Competence. They soak up a lot, but don't contribute. They throw all the burden of coming up with things to talk about on their conversation partners' shoulders. And heaven forbid they should ask a question or demonstrate any interest in your life! You should be ready to fully participate in the conversation.
### Networking is a journey, not a destination. Enjoy the trip.
**20. _Don't forget to enjoy the journey._** It's easy to keep your eyes on the horizon—on your business goals—and fail to enjoy the ride. Sure, you're looking for business or career benefits, but relax. Take time to appreciate people's unique gifts beyond their usefulness to you at the moment. Allow yourself to enjoy encountering the unexpected. Just as you might drive around the bend and discover a wonderful view, a conversation may veer off, and you'll find you're talking about something you haven't even thought about for years or something that changes your view of the world. People who enjoy window-shopping or browsing through a bookstore or antique shop can get the same kick out of being with people. People take courses in art appreciation; networking can be a "course" in the appreciation of people and of life. Be ready to be delighted.
## CHAPTER 8
## "Who Are You?"
There are Three Big Questions that always come up in every networking encounter:
**1.** "Who are you?"
**2.** "What do you do?"
**3.** "What are we going to talk about?"
These questions are "Million-Dollar Moments." As you deal with them, you can either begin a relationship that might be worth who knows how much over a lifetime. Or you can muddle through these moments, relying on the worn-out rituals we all know so well for meeting and greeting. If you learn how to maximize these moments, you can make sure your business relationships start off in the best possible way.
This chapter will help you deal with the first question, "Who are you?" Learning names ranks as the Number One concern for people in our workshops: 97 percent of participants say they have trouble with this important skill. The good news is that you can learn to remember names and make yours memorable, too.
### Why Remembering Names Is Hard
Okay. You're right. When people meet, they rarely say to one another, "Who are you?" What exactly do they do?
One person takes the initiative and sticks out her hand, saying, "Hi, I'm Jennifer Allsgood."
Shaking her hand, the other person responds, "Rob Schafer. Nice to meet you." "You too," says Jennifer.
How long does that name exchange take, do you suppose? In our workshops, people are amazed when they count how many seconds an introduction takes—three or four! No wonder people have trouble remembering names! You are asking the impossible of yourself to think that you can learn someone's name—and teach that person your name—in only a few seconds. Notice, too, that if you think of this moment as teaching and learning, not just saying and hearing, you'll be recognizing this moment's true importance.
### The average name exchange takes less than five seconds. No wonder we can't remember people's names!
What's the rush?
People say, "I just whiz through the name thing so I can move on to the good stuff." But in networking, names are "the good stuff." It will be mighty difficult for you to initiate a relationship with someone if you don't know that person's name. So, slow down and linger longer over the name exchange.
### Learn Someone's Name
When someone says her name, do not immediately reply with your own. Instead, focus initially on learning hers.
Here are three things you can do. These ideas are so simple you may be tempted to dismiss them. Don't do that! Practice so that you can use them every time you meet someone. They work.
### To remember a name, use it immediately.
**1. _Repeat the first name_**. Say, "It's nice to meet you, Jennifer." You may think that you habitually do that, but our research indicates that less than 25 percent of people involved in introductions repeat the name. Train yourself to do it every time. Hang on to the name in order to introduce Jennifer to at least one other person at that event. Whether you make that introduction thirty minutes later or three hours later, Jennifer will appreciate that you bothered to remember her name. Notice that you are focusing only on the first name. That's fine. It's the tried and true principle of "divide and conquer." Learn the person's first name first.
**2. _Ask for the last name again or confirm it._** Say, "And your last name was...?" Or, "Tell me your last name again." Or, "And your last name is Allsgood?" The person will repeat her last name: "It's Allsgood." Or say, "Yes, it's Allsgood." Usually, people will say their last names very distinctly when you ask only for the surname. One problem with the old ritual is that people are so used to saying their names that they say them too quickly, crunching their first and last names together. When you ask specifically for the last name, your partner will say her last name clearly.
**3. _Ask a question or make a comment about the person's name._** Comment either on the first name or the last name. It's a chance for you to say the name again. Here are some suggestions.
* "Do you like to be called Jenny or Jennifer?"
* "Allsgood sounds like it might be an English name. Do you know where it came from?"
### Teach Your Name
Notice that you have not yet said your own name! Now is the time to do just that. Be ready to help someone learn your name. There are three things you can do.
**1. _Give 'em a double dip._** Say your first name twice. "I'm Rob, Rob...Shafer."
**2. _Separate and articulate._** Say your first and last name with a tiny pause in between and pronounce your last name crisply and distinctly. "I'm Rob, Rob (pause) Shafer."
**3. _Make your name memorable._** Say something about your name to help the person you're talking with remember it. Spelling your name is a good idea because a majority of us are visual learners. We learn best when we can see the letters in our mind's eye. To help you learn her name, Jennifer might have spelled Allsgood when you asked her for it again. Nancy Mann says, "It's Mann with two 'ns.' I'm the only woman who's a Mann who's in real estate in Kansas City."
There are several concerns people have as they begin to use this system:
* **_"It feels awkward at first."_** Yes, it does. We are used to playing the Name Game like Ping Pong: Your name to me; my name back to you. That's the ritual. You'll need to practice to be comfortable with the new system because the timing is different.
* **_"What should I do when people are wearing nametags?"_** Use the nametag as a visual aid. When Bob is learning Jennifer's name, he can say, looking at the nametag, "I see your name is Jennifer. Do you ever go by a nickname?" When he is introducing himself, Bob can say, "My nametag says 'Robert,' but I prefer 'Rob.'"
* **_"Sometimes the other person—who is still playing Ping Pong—will interrupt. What do I do then?"_** Go with the flow. Answer the other person's question. Then go back to ask something else or say something else about that person's name later. The point is to talk longer about the names.
* **_"Sometimes I'm introduced to someone and they say, 'I never can remember names.' How should I respond?"_** That's your cue to say, "You can remember mine. Here's how. It's Sherry Hunter. Sherry like the drink. You can remember Hunter because I hunt down computer problems and fix them."
### Try These Twenty Tips
**1.** Continue to use the other person's name as the conversation moves along. "Are you a new member, Fred?"
**2.** Look for a personal connection, perhaps someone else you know with the same name. Make the connection out loud. Tell your partner, "Hi, Adam. Good to meet you. Adam was my college roommate's name, so it will be easy for me to remember yours." Or, "Nice to meet you, Harriet. Wasn't your name mentioned as one of the new board members?"
**3.** Visualize a picture to help you remember the name. Associate the name with a picture in your mind. If you meet someone in a leadership position whose name is Arthur, visualize him as King Arthur with the Knights of the Roundtable. (Some people like this technique; others say it just confuses them. Use it if it's helpful.)
**4.** Ask the person to spell his or her name. "Is that Carl with a 'C' or a 'K'?" "Is that M-a-r-y or Merry as in Christmas?" If the person is wearing a nametag, you still may comment on the spelling, "I see that you spell Marsha with an 'S.'" Because most people are visual learners, seeing the letters of the name in your mind helps.
**5.** Ask how the person got her name. "Do you know why you were named Savannah? Were you named for the city?" We find that nearly half of our workshop participants can tell a story about how they got their names.
**6.** If you know something about the person, even though you've never met, mention it. Acknowledge his or her uniqueness. "I understand that this new orientation program was your idea, Kay."
**7.** If you notice that people often have trouble understanding your name, it may be because you are saying your name, not teaching it, and are running the two words together. This problem may be accentuated if your first name ends and your last name begins with a vowel. If your name is difficult for people to understand, separate the two names like this, "Hi, my first name is Marla; my last name is Anderson."
**8.** If your name comes from a culture less familiar to the people you are meeting, then you'll have to make a special effort to teach your name. Barbara Rodvani says, "Rodvani, think of a van going down a road, Rodvani." Ankur says, "It's like encore—take a bow."
**9.** If your last name is a hyphenated combination, say so. It's very difficult for people to understand a first name plus _two_ last names. "Hi, I'm Maureen, Maureen...James-Martin. James is my maiden name and Martin is my husband's last name."
**10.** Come up with several ways to help people remember your name. As you say your name, give a little extra information so that you have a chance to repeat your name for your partner. It can be as simple as saying, "Jack's a nickname for Jackson." Or tell people where your name came from. "Stanton was my grandfather's name. I like having his name because he encouraged me to start my business." "My first name is Andreal. My mother liked the name Andrea, but she wanted something unique, so she added an 'L.'"
**11.** Keep your energy level high—rev it up. Let your body language and tone of voice indicate that you're seriously trying to learn your partner's name and teach your name. People say that this is very flattering.
**12.** Always say the person's name again as you leave to reinforce your learning." It was good to meet you, Ronda."
**13.** Give yourself a realistic goal. At a networking event, for example, vow to really learn the names of five people before you leave.
**14.** Decide whether you want to teach your first name or your last name. If you want your contact to be able to find you in his industry directory or the phone book, concentrate on your last name. Your co-author, Anne Baber teaches both names this way: "Hi, I'm Anne, Anne Baber. It's Ann with an "e" and Babe with an "r."
**15.** Design a way to teach your name and what you do at the same time. Debbie, a new franchise owner does just that. She says, "Hi, I'm Debbie Danforth with Decorating Den. Just remember 'D' for Debbie and 'D' for Decorating Den."
**16.** If you don't like the association that people make when they hear your name, say something to redirect their attention. Even though the TV show is long gone Mindy found that people were always asking her, "Where's Mork?" So, she decided to say, "Hi, I'm Mindy, Mindy . . Jones. Mindy, like Lindy, but with an 'M.'"
**17.** Set up a positive association. Don't use a memory hook that links you with a negative impression. Annabel Lector used to say, "Hi, I'm Annabel Lector, like the killer in 'Silence of the Lambs.'" Now she says, "Hi, I'm Annabel Lector, like rector—an English clergyman."
**18.** If your name is very common, you can still make it memorable. Tom Smith says, "There are fourteen Tom Smith's in the phone book, but I'm the only one who's here tonight!" Joe Jones says, "I wish I were from Indiana so you could call me Indiana Jones. But I'm from Iowa, so call me Iowa Jones."
**19.** When you meet someone with a foreign-sounding name, don't assume that she is from another country. When Ying-Chie, a third-generation American, introduces herself, she often is asked, "Where are you from?" She replies, hiding her irritation, "San Francisco."
**20.** If your name is memorable or connects easily to some idea, you may become bored or annoyed with what people say about it. "People always say, 'Just like the bird,' when I say my name," complains Rick Robin. Find a way to use that connection. Rick, a realtor, might say, "Yes! And, as a realtor, I always find just the right nest for people!"
### Break Up Bunches of Introductions
You've joined a group of people, and one of them is quickly introducing each person to you. You're thinking, "I'll never remember all these names!" What to do?
Smile. Say hello as each person is introduced. After a while, when the group breaks up, go back to each individual and introduce yourself, one-on-one, using the new system.
### Deal Skillfully with Forgotten Names
Have you ever seen someone across the room and said to yourself, I know that person. What is her name?" This is not an age-related problem; it's a brain overload problem. Let's face it, you know hundreds of people—coworkers, customers, colleagues, cousins—so the expectation that you'll never forget a name is unrealistic.
Above all, avoid the following scenario.
You see someone whose name you think you should remember across the room. You make eye contact, then hang your head, shuffle over with a discouraged look on your face, limply put out your hand and apologetically announce, "Ooooh, no! I've forgotten your name." If the person wants to make you feel better, she'll say, "Oh, I've forgotten your name too," even if she remembers it!
This low-energy start has no place to go but down as you stand around mutually beating yourselves up with a duet of, "I'm so bad with names." "No, I'm much worse." After you commiserate about how dumb you are, you finally reintroduce yourselves, all the while protesting that you'll probably forget each other again.
What should you do? You have several options. Try one of these ways to re-connect, even if you can't remember the person's name:
Walk up to the person, stick out your hand and say, "I remember you, I'm Craig." You're banking on the ritual. The other person will most likely say his name back.
If you do recall the situation in which you met or a topic you discussed, refer to that. "I remember meeting you at the conference, and we talked about job opportunities in Denver. Tell me your name again!" Or, "As I remember, we talked about the seminar you'd just attended. I'm Todd Watson. "That way, you acknowledge that your prior meeting was memorable. Since you've offered your name, your partner will usually follow your cue and give his name.
Ask a friend to remind you of the forgotten name. "Jerry, I know I've met that guy over there with the red tie. Remind me of his name."
Don't worry about it. Hope that the person's name will occur to you as the conversation goes along. Often, as you begin talking, you'll remember the name.
### Give Yourself a Tagline
Often, we give Taglines about ourselves almost automatically. They are short "identifiers" that usually answer your partner's unspoken questions:
* "Why are you here?"
* "Who are you?"
* "How do you relate to me?"
Here are some examples.
**_Geography Tagline._** "Hi, I'm Lois. I'm in the office across from the elevator." Use "geography" to make a connection. I'm here because I work across the hall.
**_History Tagline._** "Hi, I'm Sue, Sue...Gost. We played on the company volleyball team together last summer. You've got a great serve." Use your history together to make a connection. You know each other through the volleyball team. You're also throwing in an acknowledgment of your teammate's expertise.
**_Relationship Tagline._** "Hi, I'm Melinda, Melinda...Sommers, Harry's secretary." Clarify your relationship to indicate the connection.
**_Title/Role Tagline._** "Hi, I'm George, George...Pope. I'm the editor of the company newsletter." Explain what your job is to create a connection.
**_Reason-for-Attending Tagline._** "Hi, I'm Lou, Lou...Logan. I'm new in town and interested in meeting people in healthcare." Use your reason for coming to the event to make a connection. That will signal to people that they can help you by introducing you to people in healthcare.
**_Refer-Me-Please Tagline._** You also can create a tagline that allows you to be passed along quickly to a networking partner who is more appropriate for your purposes. You can say, "I'm John Jones. I'm hoping to talk to some people who have used web conferencing and find out which vendors they like best. Do you know anyone who has done a webinar recently?" Chances are, if your first partner doesn't meet your need, she'll pass you along to someone more qualified. This is a shortcut that will help you find what—or who—you are looking for.
Use your Tagline to connect with your conversation partner by telling:
* Your location. "I have an office across from the graphics department."
* Your history. "Didn't we volunteer together last year at the homeless shelter?"
* Your relationship. "I work for the CEO."
* Your place in the organization. "I'm in the marketing department."
* Your purpose. "I'm hoping to meet people who have experience with ethics training."
* How you happened to be in the room. "I'm so glad I got to this meeting. I've admired the speaker from afar for years."
* Who you'd like to be passed along to "I'm looking for someone who's savvy about marketing on the Internet."
### Yes, Mind Your Manners
Life is more casual today and few people can quote chapter and verse on the protocol of introductions. But it will make you feel more confident to know the rules, so here they are.
**_Shaking Hands and Standing Up._** Anyone who is introduced to anyone else should offer to shake hands. Gender and age used to govern who made the first move. Those distinctions are obsolete today. Reaching out to shake hands should be almost simultaneous. It's proper to stand when introductions are being made unless you are seated in a restaurant or are in some other environment that makes standing difficult.
**_To Introduce Peers to Each Other._** Say either name first. It doesn't matter which one comes first. Use both first and last names and speak distinctly. "Jackie Arnold, this is Rob Baker." Give each some additional information about the other person, if you know them well enough. "Jackie, Rob is on the audit staff. Rob, Jackie is in human resources." Using their names several times will be helpful to them.
**_To Introduce a Superior to a Subordinate._** In today's workplace, we're moving away from focusing on these matters of rank. Nevertheless, to follow the rules, follow this pattern. Say the name of the superior first. "Mr. Brown (or Don, if you use the person's first name), I'd like you to meet Bob Davis. Bob, this is Don Brown." Again, it's helpful to all concerned if you can give some additional information—often just a title will do. "Don, Bob is in our legal department. Bob, Don has been division manager for as long as I've been with the company."
**_To Introduce a Customer._** When introducing customers to people in your business, treat the customer as the superior. Say the customer or client's name first to honor that relationship. "Mr. Smith, I'd like you to meet Mary Jones. Mary, this is Al Smith. We installed one of our systems in his business last week. Al, Mary heads our training staff."
**_To Introduce Women._** It used to be proper always to introduce a man to a woman – that means, you'd treat the woman as if she were the superior, saying her name first. That rule is obsolete and rank should prevail. Since so many people are confused about the rules, don't make any assumptions about the rank of a woman whose name is said first.
**_To Introduce Older People._** It also used to be proper to introduce a younger person to an older person, saying the name of the older person first. Again, today's protocol would be to ignore age.
**_To Introduce a Person With No Business Status._** When introducing someone who has no business status (such as your mother), say the name of the company person first if he or she outranks you. If the company person is a peer or of lower rank, say your mother's name first to honor her.
### The Introduction Rule: FIRST IS FOREMOST
The only rule you need to know about introductions: Say the name of the higher-ranking person or the person you want to honor first. First is foremost is the rule. That's all you need to remember.
## CHAPTER 9
## "What Do You Do?"
In New Zealand, they might ask it this way: "Wot do ye do fer a crust, mate?" In the U.S., it's the inevitable question, the second big question that always comes up in every networking encounter: "What do you do?"
How you answer this oft-asked question is crucial. It's the second "Million-Dollar Moment." But, as was the case with "Who are you?" all too often, the ritual answers we have learned so well, and use so effortlessly, get in the way of building relationships and finding out more about each other.
### Why Most Answers Bomb
When people ask, "What do you do?" do you respond with:
Your occupation, job type, or category? "I'm an attorney." That's CEMENT. The response falls like a dead weight—a block of cement—at the other person's feet. There may be thirty-seven other attorneys in the room. You just missed the chance to make yourself unique. Your conversation partner is likely to simply say: "Oh...nice." Unfortunately, that's the type of comment listeners will probably always make when you describe yourself in this way.
Your title? "I'm Assistant Information Systems Manager with the Northeast Division of Management Information Systems, a division of Integrated Information Management, Inc." That's FOG. Giving a title—especially a long, complicated, jargon-filled one—leaves you surrounded by a thick cloud of words. And your conversation partner is likely to say, "Oh...nice."
Your industry? "I'm in real estate." That's THE BLOB. That response puts you right into the middle of the great gray blob of the other twenty-three people your conversation partner knows who are also in real estate. You've missed your chance to tell about your special talents in the real estate industry that make you different from all the other twenty-three. And again, your conversation partner, not knowing what else to say, will probably reply with a polite "Oh...nice."
The name of the organization you work for? "I'm with Disney." That's THE FLAG. That response wraps you in the flag of the organization. You aren't going to be known for your talents and capabilities if you say that; your only identity will be as one of those "Disney people"—a dangerous situation if you ever are laid off.
What's the problem? These commonplace responses to "What do you do?" aren't conversation-builders; they're conversation-stoppers.
Your contact may have learned to deal with CEMENT and THE BLOB by asking questions: "What kind of law do you practice?" "Which one of the real estate companies are you with?" But you won't have made it easy for him to talk with you. And you will have missed the boat when it comes to teaching him anything about your capabilities and talents.
### Make the Right Things Happen
Your conversation partner has a TV screen in her head. Most people do. When you tell her about your work, there are two possibilities.
On the one hand, she may see nothing but static—a blizzard on the screen. That's what people see when you have responded to "What do you do?" with CEMENT, FOG, or THE BLOB—nothing. Giving the name of your organization, especially if it's well-known, as in THE FLAG, may feel good for the moment. And your conversation partner will picture your company. But you could get so much more mileage out of your answer. Ask yourself, "What do I want my contact to see on the TV screen in her head? What one thing do I want her to know about me?" When you come up with _that_ , you'll know what to say.
### Give It Your BEST
Use our formula—the BEST/TEST—to construct your answer. The first sentence of your reply tells the one thing of all your many talents and skills you do BEST. The second sentence gives a brief example and is a TESTimonial to your talents. It should briefly show how you saved the day, served the client, or solved the problem.
Use only ten or fifteen words in your first sentence to tell what you do BEST. Keep it snappy and jargon-free. Aim to be understood by a ten-year-old. Include exciting, colorful, vivid language.
Andy says, "I tried the two ways of answering 'What do you do?' at a recent conference when I wanted to get into conversation with the executive director of the President's Council on Physical Fitness and Sports. When I said, 'I'm with the Association of Pedestrian and Bicycle Professions,' I got a blank stare. I've figured out that, when people hear the association's name, they often think I either run the Tour de France or own a messenger service! But later that day I had a second chance. I tried again, saying, 'I work for an association that helps people find information and resources on how to build more walking and bicycling into their lives and their communities. We just gave a grant that created 300 more miles of trails in Colorado.' The look of interest on the executive director's face was all I needed to convince me that using the BEST/TEST formula to answer 'What do you do?" is much better."
Consider this example. Kathy used to say, "I'm a senior manager with the construction advisory practice of a professional services firm." That answer created a FOG in the listener's mind. But using the BEST/TEST formula Kathy now gets conversations going with this answer: "I help clients when their dream construction project turns into their worst nightmare—you know, when construction projects aren't delivered on time, on budget or to the right quality standards. I just mediated a conflict over construction of a power plant in Asia." Her new answer makes pictures appear on the TV screen in the listener's mind.
Terri used to describe herself as "a marketing consultant," a CEMENT answer. Now, she tells what she does best: "I help people get the word out about their products and services." She updates her TEST constantly to provide a vivid picture of her succeeding with clients: "Last week, I wrote a news release that got one of my clients, a CPA, on the front page of Tuesday's business section. He's had seven calls so far from prospective clients since the article appeared!" What do you know about Terri from this short anecdote?
She writes news releases that get results for clients. A CPA is her client. Using the BEST/TEST allows Terri to teach you about her Character and her Competence to build the trust that is necessary to establish an effective networking relationship.
Now, imagine that you run into a CPA who says to you, "I want to let women entrepreneurs know about my services for small businesses." Wouldn't Terri's name and expertise pop up in your mental Rolodex™? Assuming you've learned enough about her Character and Competence, wouldn't you mention Terri to this CPA?
By the way, we hope you won't call your answer to the "What do you do?" question an "Elevator Speech" or a "30-second commercial." Those labels devalue and diminish the very important trustbuilding and teaching process that goes on in this ritual. Your answer is not a "commercial." It is a carefully crafted couple of sentences that you will use to spark a conversation and begin to teach your contact about your Character and Competence.
### Be Interesting
Rather than eliciting the comment, "Oh...nice," when you tell what you do, aim for this comment: "Tell me more." When people asked Buford, "What do you do?" he used to give his title. It was so long that he had to stop and take a breath in the middle: "I'm director of student financial aid in the student affairs division at the University of Missouri (gasp!) Kansas City." And people said, "Oh...nice."
Then he came up with another way—a much more interesting way—to put it. He started saying, "I give away $32 million a year. One student we gave a four-year scholarship to just graduated with honors and came by the office to thank us." Did people want to hear more? You bet!
Lisa who's a program analyst at the Department of State, told us in a workshop that she wished she'd known the BEST/TEST formula when she was at a luncheon attended by then Secretary of State Colin Powell. "Powell sat down beside me, introduced himself and asked, 'What do you do?' All I could come up with was, "I work for Mr. Baker." That was such a poor conversation starter that Powell then turned to talk to someone else. I can't blame him! Here's the answer I wish I'd given: 'I design office spaces here at State. We just finished a rush job, doing 880 offices for six different bureaus.'"
Look at Figure 9-1 for some more examples of vivid ways people tell what they do.
**FIGURE 9-1. Transforming Your Answer.**
### Try These Tips
**_Say the right thing in your BEST/TEST._** Don't choose being interesting over teaching people what you _really_ want them to know about you. A pharmaceutical saleswoman got people's attention when she said, "I sell drugs." But after thinking it over, she decided that was not what she wanted to teach people. She now says, "I educate doctors about new drugs, so they can give their patients the most up-to-date information on prescriptions."
**_Tell your talent, not your title._** Titles tell what you are, not what you do. Instead, paint a picture in the other person's mind of you in action, you at your best.
**_Avoid acronyms and jargon._** When the person you're talking with is unfamiliar with your "insider lingo" they will feel put off.
**_Resist the ego trip._** If you work for a well-known or prestigious group, resist the urge to bask in that organization's glory. If you must say "I'm with Hallmark," or "I'm at the Department of State," be sure to also include a talent or an example. We guarantee that the name of your organization alone won't start the conversation you want. And worse yet, you just missed the chance to teach someone about your talents and successes.
**_Ask a question._** A variation on the BEST/TEST is to answer "What do you do?" with a question. David asks, "Has your bank ever put your money in somebody else's account?" Whether the answer is yes or no, he says, "I'm working with Federal Reserve banks nationwide to design a system so that never happens again."
### Read These Frequently Asked Questions
Here are some questions people ask about the BEST/TEST method.
**Q:** | "Is it _ever_ okay to tell my title and the name of the company I work for?"
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**A:** | Sure. You can say those things later in the conversation if you wish. But beware. It's a dangerous thing to fall in love with your title and your company. The pleasure you take in introducing yourself with your title and your company affiliation is an indication of your dependency on them for your self-image. In today's volatile economy, you could find yourself out of work tomorrow. (Find more resources at our website, www.FireProofYourCareer.com.)
It's far better to teach people about your abilities. For instance, you might want them to know that you are an outstanding trainer who knows how to convey complicated technical information. Your talents and your reputation will get you your next job, not your current title and company.
**Q:** | "What if I wear several hats?"
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**A:** | Prepare several different BEST/TESTs. Select the right one to use depending on whom you are talking to and what you want him to know about you. For instance, when one of your co-authors is talking to meeting planners, she might say, "I get people talking at conventions." (BEST) "I just gave the kickoff keynote on convention networking at the Healthcare Educators annual meeting." (TEST)
When one of us is talking to people in professional services, she might say, "I help lawyers get the most out of their professional memberships and turn contacts into clients." (BEST) "I just finished a four-part seminar for attorneys at McCann, Henry, & Wisecoff." (TEST)
When one of us is talking to people in the publishing industry, she might say, "I wrote the book on networking." (BEST) "A book club just bought 59,400 copies." (TEST)
**Q:** | "Isn't this bragging?"
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**A:** | Some people in our workshops say, "Oh, I could never say something like that. I'd feel like I was bragging." But are you? See the "What do you do?" question as an invitation to tell what you're excited about, working on, or proud of. A lot of it has to do with your delivery—so show you were as excited and thrilled as your client. Your body language and tone of voice can show you're excited about the results you bring for your internal or external clients, or the students you serve, or the association members you keep informed.
As the great American humorist Will Rogers once said, "If you done it, it ain't braggin'." If you're talking about your Mercedes Benz, your yacht and your house in Switzerland, that's bragging. But if you're talking about a project you poured your time, talent, and creativity into, that's not bragging.
Another way to make your answer more conversational is to start with a question. Judy does that sometimes. She says, "You know the U.S. government buys millions of dollars of products and services each year? Well, my clients come to me for advice on whether or not selling to the government is a good bet for them. I was so pleased yesterday when one client called and said, 'I think your advice just saved us about $200,000 and a lot of headaches.'"
When you're asked what you do, the best way to start a conversation is to be enthusiastic and specific about your accomplishments. How else will people learn what to count on you for, what you're good at, whom they should refer to you, and what opportunities they should send your way?
**Q:** | "How will I know when I have a good answer to the question, 'What do you do?'"
---|---
**A:** | Ask yourself these three questions:
**1**. Does my answer give a specific, positive picture of me succeeding, me doing what I want to be known for? Does it teach about my Character, my Competence? Does it show what I want to do more of?
**2**. Does my answer encourage people to say, "Tell me more?" Does it invite questions and conversation without being maddeningly mysterious? The real estate agent who merely says "I'm a miracle worker" is being too cagey. She needs to add "...for home buyers." Her TEST can further clarify her claim as she says, "I just found a house for a newly married couple who both use wheelchairs—and at a price they can afford in a neighborhood they love."
**3.** Do I deliver my answer in an excited, upbeat way, in a tone of voice that expresses my delight in serving my customers or solving the problems, rather than sounding full of myself?
**Q:** | "What should I do if the person I'm talking with gives me CEMENT (her job type), FOG (her title), or THE BLOB (her industry)?"
---|---
**A:** | Ask questions designed to draw out specific examples, learn about special expertise, or hear about unique projects. Ask:
* "What's a typical day like in your work?"
* "Tell me about a recent project you've been working on."
* "What have you been doing this week."
* "What's your favorite project these days at work?"
**Q:** | "I'm in a technical field. I have a Ph.D. I can't imagine being so folksy—especially when I'm with my peers and everyone is trying to one-up the other person."
---|---
**A:** | It's okay to use your title or the jargon of your profession if you are speaking to other people in the same specialty. But be sure to supplement that with a vivid TESTimonial, so that people have a clear idea of your expertise.
**Q:** | "I hate what I do. I'm just an office manager. It's so boring. I'm trying to change careers. What should I say?"
---|---
**A:** | If you don't like what you are doing, don't talk about it. Instead talk about the five percent of your job you do like or what you have done in the past or what you want to do in the future. Mary, who posed the question, now describes the part of her job she likes the most: "I'm an expert scheduler and organizer." (BEST) "Last year, when my company relocated, I managed the move. It was so exciting to see us go from an up-and-running office into thousands of boxes and back out again in record time and with a minimum of trauma." (TEST)
### Give contacts specific examples of projects, so that they can describe to others–accurately and vividly–what you do.
**Q:** | "Won't I need several answers depending on who I'm talking to and how well they know my kind of work?"
---|---
**A:** | Absolutely! We recommend you have four or five answers you're comfortable giving. One might be for an informal setting like a backyard barbecue or when you're at the swimming pool with your kids. One might be for use internally and designed to teach people in your organization how your work contributes to the bottom line. Another might be for when you're at a conference, designed to teach how your skills would apply in a new career arena you're thinking of moving to.
**Q:** | "How can I keep my answers to 'What do you do?' from sounding stilted or canned? People aren't used to hearing something this long or detailed."
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**A:** | True. A BEST/TEST answer will set you apart from the crowd. Say your answer with enthusiasm and as if you know the listener will be interested. Practice it until you know it as well as you know your own name. Be sure to keep your TEST short. Your answer is not an item on your resumé and shouldn't sound like "resumé-speak." Keep the language conversational and jargon-free so it flows off your tongue easily. Watch for people's reactions and modify your answers until you get the responses you want.
## CHAPTER 10
## "What Are We Going to Talk About?"
There's one conversation that everybody knows, word-for-word. You can hear it at networking events all across the U.S., from Tacoma, Washington, to Tampa, Florida. It goes like this:
_"Hi, how are you?"_
"Good. How are you?"
_"Not bad. What's new?"_
"Not much. What's new with you?"
_"Not much. Been real busy."_
"Me too. Good to see you."
_"You too. We'll have to get together sometime."_
"Great idea. I'll give you a call."
_"Well, bye. See you later."_
This is a conversation in search of a topic! Without one or more topics that _you_ want to talk about, you'll waste your time in purposeless chit-chat like that one-size-fits-all conversation.
It pays to be prepared to talk about topics you care about. Having an Agenda—a plan for your networking conversations—is vital.
### Listen for Your Cue
Two cues should prompt you to say to yourself, "Time to use my Agenda."
One cue is hearing, "How are you?" or "What's new?"
The other cue is the pause, as you and your conversation partner search for something to talk about. Often that pause comes just after you've finished talking about what you both do—and just before the conversation about the weather or the ball scores!
Do you wonder, "How can I direct the conversation to my business or to topics that are really important to me?" Here's how to manage the third "Million-Dollar Moment."
### Use Success Stories to Tell What's New
When somebody asks her, "What's new?" Ilsa says, "I'm training for a marathon. If I tell people that, then they'll ask me, the next time we meet, 'How's the marathon training going?' That keeps me motivated, and committed to my goal." Another reason for talking about the marathon is that she is looking for pledges for the charity run.
When somebody asks him, "What's new?" Sam says, "I've moved my business. My new location is right next to the Metro—and the rent is actually lower!" The reason he gives for choosing that topic is that he wants to show how easy it is to get to his graphic design business.
### When someone asks, "What's new?" tell a Success Story that shows you saving the day, solving the problem, or serving the customer.
Both Ilsa and Sam have good answers and good reasons for their answers to "What's new?"
But all too often, people reply, "Not much. What's new with you?" and sink into another one of those superficial conversations like the one at the beginning of this chapter.
The best reply to "What's new?" is to tell a Success Story. A Success Story is a short, punchy anecdote. It teaches your conversation partner what you do, what you're interested in (like Ilsa), how you serve customers or clients, something important about your business (like Sam) or what you or your firm have to offer. For more information on constructing your Success Stories, see Chapter 11.
### Figure Out Your Agenda
More than 85 percent of people we surveyed as they arrived at networking events hadn't figured out exactly why they'd come. They knew they wanted something, but hadn't figured out what! They hadn't thought about who they wanted to meet, what they wanted to find out, and how they were going to achieve their goals. They didn't know what they wanted to share, tell people about, pass along. Predictably, about the same percentage of people, surveyed on their way out, said they wished they'd gotten more out of the event.
To be an effective networker, have a clear purpose in mind before you begin talking with people. That purpose comes from knowing what's on your ever-changing Agenda. As you focus on your Agenda, you'll feel eager and excited about connecting with people.
Your networking Agenda is a mental or written list of what you have To Give and what you want To Get.
### Make an Agenda of things to talk about, and don't leave home without it.
Since we each have unique purposes, we each have unique Agendas.
Bill, a teacher, has a small real estate business on the side. He was elated when he finally found an accountant he trusted and enjoyed working with. In the weeks that followed, he enthusiastically recommended her when talking with other business owners. At the same time, Bill was looking for someone to work for him part-time, doing mailings and keeping his database up to date. He had something to _give_ in his conversations—his recommendation of an accountant; he had something he was trying to _get_ in his conversations—the name of a qualified person who might like to work part time. He had an Agenda.
Using this chapter, you'll learn how to construct your personal Agenda to focus on the things you want To Get and, equally important, the things you have To Give as you make connections with other people. Having an Agenda will energize and empower you so that you'll benefit from your networking. And using your Agenda will help you uncover the commonalities and needs that move the relationship from the Associates to the Actors Stage. You'll feel comfortable and capable of enjoying yourself, making contact, gathering information, and seeking out opportunities.
You may have dreaded networking situations in the past because you felt you didn't have anything to talk about. Actually, when it comes to topics, the problem is _not_ that there's _nothing_ to talk about. The problem is that there's _too much_ to talk about! Hundreds of topics a day come crashing in on you via newspaper, TV, radio, e-mail, junk mail, and the Internet. Often no one topic looks that much more interesting than any other. So it's hard, perhaps impossible, to select from all of the ideas racing around in your head. An Agenda simplifies the situation. And you automatically care about—and have energy for—the topics on your own Agenda list.
### Begin with the Right Side
There are two sides to your Agenda: what you want To Get and what you have To Give. Most people, as they think about networking, focus on what's in it for them. That's not the right place to begin. In fact, the second biggest mistake people make about networking (after not being strategic) is to think that it's about getting. It's not about getting. It's about giving.
### There are two sides to networking: giving and getting. You are only in control of one side—guess which one. It only makes sense to work on the side you control 100 percent.
Russell Simmons, cofounder of Def Jam Records, says the following in Business 2.0, an online magazine: "You get a lot of benefit from giving, not from taking. You have to fill a void, give people something that's meaningful and useful."
Giving—not taking—is the way to build your network. It's not just a nice thing to do. It's the smart thing to do.
Psychologists have discovered a quirk of human nature that we call The Reciprocity Principle. It goes like this: If you give somebody something, he will try to give you something back. It gets even better. If you give somebody something, he will insist on giving you _more_ than you gave him.
Doesn't that sound exactly like what you want to happen when you're networking? So, to plug into The Reciprocity Principle, give first, give freely.
You're actually in control of only half of the networking process—the giving part. Does it make sense to focus on the getting part—something you have little control over?
### The Reciprocity Principle: When you give people something, they will insist on giving you even more back.
Have lots to give and give generously. Be helpful to others. Often, you'll benefit from contact with someone whom you can't immediately—or perhaps ever—pay back. Believe in the great network in the sky—that if you give, you will get—somehow, somewhere, someday.
Five years ago, an executive gave Ellen some career advice. This year, when she received the Member of the Year award from her professional association, she mentioned the executive in her acceptance speech, thanked him, and told how she felt inspired to mentor others because of his help early in her career.
Giving positions you as a resource. As Donna, a senior sales representative, says, "Think of yourself as networking at a larger scale than giving information about your business. I might help a woman I know find a spot on a non-profit board, for instance, not because I know I'll get business from it, but rather because I want people to see me as the person you come to when you don't know where to find something. I want to be seen as someone who knows a lot of resources and people."
At first, it may seem that you are giving more than you are getting from your networking relationships. If so, you are networking the right way.
### What Do You Have to Give?
People often scratch their heads and say, "Give? I don't know what I have to give." To create your To Give list, think about your accomplishments, skills, enthusiasms, and resources.
Most of us really do have plenty to give—ideas, expertise, phone numbers, introductions to other people. The possibilities are endless. But, if you have a hard time figuring out exactly what you can offer, try using this formula.
Think to yourself, "Give MORE."
**_M = Methods_.** Can you make life easier for your contacts?
* "My expertise on how—and how not—to build a brick patio."
* "Information about how to run web conferences as an alternative to on-site training."
* "How to negotiate the best deal on severance if you've been laid off."
**_O = Opportunities._** Can you alert people to an opportunity?
* "An apartment to sublet for six months."
* "Rex kittens (extremely short-haired cats for people with allergies)."
* "A job opening at Allied Sciences."
**_R = Resources._** Can you offer someone or something?
* "The name and phone number of a great band for weddings and parties."
* "The name of my veterinarian who makes house calls."
* "A great article I just read on how people react to website design."
* "A place—my conference room—to hold meetings."
**_E = Enthusiasms._** Are you excited about something?
* "My professional association. Our programs for professional development are terrific!"
* "Taking jazz singing lessons from one of the best teachers in Washington, D.C."
As you become more aware of what you have to give to others, you'll always be able to narrow down the universe of topics to a list of things you want to talk about. The things you have to give automatically become topics. These topics connect you with the people you meet. They also let people know what to count on you for. Listen for or create moments to offer what's on your To Give list.
Being prepared to give means taking stock of your accomplishments, resources, skills, and enthusiasms. It means acknowledging that you are a unique and special human being with a contribution to make. If you wish you had more to give others, it may be a sign that you need to stock up. Do more. Experience more. Learn more. Risk more. Take a negotiation skills class. Learn Thai cooking. Take that vacation you've been talking about. Go ahead with the catering business you've dabbled in for so long. Anything you become enthusiastic about becomes something to share. Your enthusiasms are things you're so excited about that you'd talk to anybody, anywhere, anytime about them. When you live to the fullest, you'll just naturally have lots of resources, experiences, and opinions to give to others.
### Get ready to give. Before an event, list three resources, tips, or opportunities to tell people about.
Having things to give makes it easy for you to go from just associating with people to interacting and exchanging with them. When you Listen Generously and find resources and ideas to give, you automatically move from the Associate Stage into the Actor Stage with your contact. As you give, you provide evidence of your Character and Competence. You create trust so that your contacts want to send opportunities your way.
### What Do You Want to Get?
After you've thought through what you have to give, it's time to think about getting. The list of things you want to find, connect with, create, understand, learn, and know about also is endless. Look at your desk; look at your life. What problems are you trying to solve? What opportunities do you want to investigate? What are your upcoming challenges?
To help you jog your memory, think, "Get REAL." Notice the examples.
**_R = Results._** What outcome do you want?
* "Office furniture I can afford."
* "Training so I can get up to speed on my computer graphics software."
* "Tips on growing BIG tomatoes."
**_E = Expertise._** What do you want to know about?
* "A good, convenient day camp for my nine-year-old."
* "Tips on appearing on a TV talk show."
* "The best way to find good employees for my start-up."
**_A = Access._** Who or what do you need to find?
* "A part-time secretary with a background in the health field."
* "A publisher for my book."
* "A good caterer for our next sales meeting."
**_L = Leads._** Who do you need to meet?
* "People who are thinking of selling their homes and moving to retirement complexes."
* "A veterinarian to join my business referral group."
* "Someone who knows about careers in training and development."
* "An experienced emcee for the Chamber of Commerce Trade Show."
### Give and Get with Ease
In our workshops, we ask people to make lists of things they'd like To Give and To Get. When they realize they do have a lot to offer, they immediately feel more comfortable about making conversation with purpose and pizzazz. When they bring what they really need to the surface of their minds, they immediately feel more eager to network. We ask people to choose one item from each list—one thing they have To Give and one thing they'd like To Get—and write both items on stick-on nametags.
The people in the room become a living bulletin board, a human swap meet. They talk with each other about what they've written on their nametags. As they begin to discover each other, the energy level in the room heats up, the excitement builds, and the noise level rises. Figure 10-1 shows some of the things a group of our workshop participants wanted To Give and To Get.
You'll notice that in this Sample Agenda there are no exact matches between the have To Give and want To Get items. If you don't know of a classy restaurant in New York City, introduce your conversation partner to Phil, who used to live there. With an Agenda, you'll be able to find common interests with your conversation partners. What counts is that on any of these topics, the talk will be meaningful and useful—and therefore valuable—for someone. As in the rest of life, sometimes you'll give and sometimes you'll get. Clarify your Agenda and talk with people about the topics on it. Author and seminar leader Zig Ziglar says, "You can have everything in life you want—if you help enough other people get what they want."
**FIGURE 10-1. Sample Agenda Items.**
When you go public with your Agenda like this, networking becomes an exciting process of search and connection.
Now, you probably aren't going to actually write an item from your Agenda on your nametag the next time you go to a business or social event. But, you can prepare for any occasion by making a written list of items you have To Give and want To Get. Put the list in your pocket. You won't need to refer to it; you'll simply feel the confidence that comes from being prepared.
You can assume that the other people in the room—even if they don't realize it—also have Agendas. Discovering their Agendas—and following your own Agenda—will become a whole new approach to networking. "Chance," it is said, "favors the prepared mind." Preparing your Agenda is preparing your mind for making great connections.
Stan is vice president of sales for a burglar alarm company. He has twenty-four salespeople spread over a four-state area. Figure 10-2 is an Agenda he used at a networking event.
Did Stan find what he was looking for? At the end of the meeting, he had the names of two chiropractors, three businesses that agreed to donate items for the auction, and a Boston contact.
When someone asked him, "What's new?" Stan started talking about the Cajun cooking class he was taking. His conversation partner told Stan about a store that sells nothing but spices that he's going to investigate.
With an Agenda, you'll see results from your networking every time.
### Practice Agenda-Making
Think of an upcoming networking situation. Take a moment and list some things you'd like To Give in your conversations with others—resources, ideas, skills, experiences, talents, and enthusiasms, for example. Then list what you'd like To Get, find, connect with, know more about, and create in your life. Be as specific as possible. If you put "Happiness" on your want To Get list, for example, you'll be disappointed because no one can give that to you.
**FIGURE 10-2. Stan's Agenda.**
### Go Public with Your Agenda
The cardinal rule about anyone's networking Agenda is this: "If there's no mystery, there's no manipulation." Managing conversations is quite different from manipulating other people. Managing is okay; manipulating is not. Effective networking is based on saying what you want and making sure that you take every opportunity to contribute to the success of others by giving anything you can.
### In networking, be up front, be honest. If there's no mystery, there's no manipulation.
Were you brought up to believe that saying outright what you want is pushy, self-centered, and overbearing? Were you taught that you should not see people as opportunities? Many people were. Sometimes, as a kid, you might have had to toss out subtle clues or be indirect to get what you wanted. As an adult, it's best to be direct. Tell your contacts, straight out, what's on your Agenda. Your honesty about your purposes will increase your sense of competence and professionalism. With these new ground rules for networking, the people you meet become opportunities for you, and you become an opportunity for them.
Here's a surefire test to determine if your Agenda is manipulative. Ask yourself, "How would I feel if my Agenda were the headline on the front page of tomorrow morning's newspaper: _Joe Jackson Hunts Job in Healthcare?_ What if everybody knew what I wanted? What if they could see right through the subtle clues to what I actually have in mind? Would they feel good about me and my purpose? Would I?" If the answer is yes, your Agenda item is a good one to talk about.
Avoid asking for information that people normally are paid to provide. Don't describe a legal problem you're having and ask for advice from a lawyer at a networking event. Don't describe a problem you're having with your computer and ask for advice from a computer consultant you meet at a party. On the other hand, it makes sense to find out what kinds of cases a lawyer handles. That kind of information would be acceptable and might be valuable in the future, both to you and to the lawyer, whose name and specialty would then be on file in your mental Rolodex.™
### Be prepared to be spontaneous.
Get comfortable telling people how they might be helpful to you in the future. Imagine that you're in a networking situation and complete the following sentence: "I'd like to know you better because..."
You could say, "I'd like to know you better because I'd like to know more about what you do as a marketing manager," or "I'd like to stay in touch so we can share strategies about how to make our home-based businesses grow."
How would you feel about going public with the reason? Perhaps you'd like to know this person better because he could be in a position to hire you some day. Is there any benefit in keeping that Agenda hidden? What could be the benefits of sharing that reason with the person? Perhaps you'd like to know this person better because she could probably refer potential clients to you. Is there any reason you can think of that you shouldn't tell her that?
Sally, who owns a tutoring business that employs forty-two tutors, said to the principal of a private school, "I'd like to become known to you because I imagine people often ask you to recommend tutors for their kids."
Corrine, who has her own training company, had lunch with Diana, who is in the marketing department of a greeting card company. Corrine was aboveboard about her Agenda and said to Diana, "I hope that, when you need training, you'll think of me. I'd love to work with you on a project." A few days later, Corrine saw a column by Humorist Dave Barry making fun of an ad he'd seen for a service that would send cards to your friends and family for you after you were dead. She knew Diana would get a kick out of it, so she sent her a copy of the column. It's little things that build relationships.
### Exchange Something
If you still have negative feelings about accepting help from others or being beholden to others, focus your energy on making an exchange. The way to make a fair exchange is to offer something equally valuable. Give something back in the conversation.
One thing you _can_ give at any time is appreciation. Take the time to say "Thank you" to people who help you. Make your thanks prompt. Write a note that same day. In the note, be specific about what Jack did for you: "Thank you for giving me Omar's phone number." Tell Jack what you did with the information. That lets Jack know you thought it was important. "I have called Omar and set up an appointment for next Tuesday." Being specific does something else. It could be that Jack will see Omar between now and next Tuesday. Your note may help Jack to remember to mention you. Now, that's networking!
If you are uncomfortable with the idea of going after what you want, remind yourself that people are free to choose. You will certainly say "No" when someone asks for information or offers a service or product you don't want. Trust your conversation partner to say "No" if you offer something he doesn't want or ask for something he isn't comfortable giving.
If you are uncomfortable with the idea of "selling yourself," think of it as giving others the opportunity to take advantage of (in a positive way) your expertise, your talent, your training. You're a resource to them. Believe in yourself and promote yourself. If you offer a service or resource that no one wants right then, what have you lost? Nothing. What have you gained? Others may tuck that information away and use it later.
Build every relationship for the long term. Never assume that you can use and discard people. Harriet remembers: "I ran into a woman at the swimming pool. We'd taught first grade together years ago. We hadn't kept in touch, but we hadn't burned any bridges either. When I was looking for new clients, I remembered seeing her at the pool. She'd told me that her husband was a new manager and felt like he was in over his head. I sent her a brochure about my consulting services. A few weeks later, her husband called me for coaching on management skills."
The idea of the Agenda is a powerful one. It will help your networking be more pleasurable, purposeful, and profitable. Share it. Teach others about the idea of the Agenda, and you will increase your chances of getting what you want, but not at anyone else's expense.
A light bulb went on for one of our workshop participants. She said, "Oh, I get it. You've got to be prepared to be spontaneous!"
## CHAPTER 11
## Make Conversation Flow
When we ask people to describe what happens in a good conversation, here's what they say:
"There's lots of give and take."
"We move from one topic to the next easily."
"I feel comfortable and listened to."
"I learn a lot about the other person."
"I get a good picture of what she's like and what she's interested in."
"We find out what we have in common."
"Time seems to pass quickly."
"It's easy to shut out the distraction of lots of other conversations going on nearby."
"We talk a long time without running out of things to say."
"Things just flow."
So what do people do to make conversations flow so easily? They rely on three conversational skills to enjoy, explore, and exchange when they're talking with people. Good conversations happen when you Listen Generously, are Seriously Curious, and tell Success Stories from your own experience. Some of these ideas will help you begin a conversation with someone new; others can be used as you Follow Through with contacts you've known for some time. Put these tools to use in your next conversation.
### Listen Generously
Networking doesn't mean doing all the talking. The first thing good conversationalists do is give others a chance to talk. Listening Generously means hearing not only the words, but also the _needs_ of your conversation partner. As you explore a variety of topics, be alert for opportunities to offer a resource, an idea, an introduction, or just a word of encouragement.
Listening is _not_ just waiting for your turn to talk. Unfortunately, many people act that way in conversations—impatiently waiting _instead_ of listening. Listening is work. Don't think of it as a passive activity where you just nod every once in a while as you wait for your turn. Listening is active. To be a good listener, give your undivided attention and focus. You _speak_ at a rate of about 150 words a minute; but you can _think_ more than 500 words a minute. That's one reason you must train yourself to pay attention rather than allow your mind to wander off on tangents. Listening is a challenge. Networking venues are often noisy. You'll probably be trying to listen in a room where lots of other lively conversations are going on. People retain only a small fraction of what they hear. But if you use the tactics in this chapter, you should be able to do much better at the quiet side of networking.
### Use Your EARS
Let the EARS formula remind you to listen better.
**_E_ = _Encourage your partner._** Nod, indicate with your body language that you are following what he's saying.
**_A_ = _Acknowledge your partner._** Restate or sum up her point of view.
**_R_ = _Respond to your partner._** Comment, ask questions to get more information, provide information or answers.
**_S_ = _Save what's being said._** Mentally store important pieces of information for future reference.
### How Listening Generously Pays Off
You'll reap these five benefits as you listen attentively:
**1. _You'll stand out._** Giving true attention is so rare (especially at networking events, where people have a tendency to glance around the room to see who else is there) that you will make a positive impression. Ken says, "I create an imaginary bubble around me and the person I'm talking with. Six elephants could dance through the room, and I probably wouldn't notice." You can bet people remember talking with him.
**2. _You'll find out how to Follow Through._** Listen for what's on the other person's Agenda. Listen for his challenges, interests, and enthusiasms. Bill heard a need as James talked about moving from a downtown office to a home office. A few days later, Bill sent James an article about home office design. Bill isn't selling file cabinets. He's a computer coach who sees business value in becoming known by giving first.
### Listen for your contact's Agenda
Remember, it takes six to eight contacts with someone before you know each other well enough to have established a long-term business relationship. So listen for reasons to stay in touch.
**3. _You'll develop a reputation as a great connector._** Who would your conversation partner like to meet? To find out, listen. When Carla introduced herself as an interior designer who focuses on the senior citizen market, Mitzi immediately said, "I want to introduce you to Hank. He's an expert on marketing to the 50-plus generation." Listen for links, what people have in common. "You went to the University of Chicago? So did Dan. Let me take you over and introduce you." Or, "You're on the program committee for your women's network at work? Sherrie's active in her company's network. Would you like to meet her?"
When you become known as somebody who knows everybody, people will call you and ask you if you know someone who.... As you link people together, you give to them and plug into The Reciprocity Principle. They will try to give you something back.
**4. _You'll be able to bridge to what's on your Agenda._** Suppose you and your conversation partner are talking about the horrors of business travel. You'd like to bridge to your need to find a conference center for your sales meeting. Listen carefully and make the transition. "Sounds like you've clocked a lot of miles to far away places, Fred. You know, that reminds me. I'm looking for something close to home, and you might be able to help. I wonder if you know of any conference centers within about 75 miles of the city. I need to find a place for my June meeting of 200 salespeople."
**5. _You'll learn something._** There's an old saying: "A good listener is not only popular, but after a while, he knows something." As you listen, you'll increase your understanding and knowledge of the topic under discussion.
Carlos, an architect, was visiting friends in San Francisco. At a dinner party they gave, he met Bill, who is in the import/export business. Carlos couldn't imagine what they could possibly have in common, but he listened intently as Bill explained how he sought out the work of artists around the world. Several months later, one of Carlos's clients wanted an unusual work of art for the lobby of his new building. Carlos remembered Bill and enlisted his help in finding just what the client wanted.
### Be Seriously Curious
When we were four years old, we were curious about everything. Nothing escaped our interest. Everything got our undivided attention. But somewhere along the way, we have learned to look cool, as if we've seen it all, as if nothing surprises us. To be a great connector, re-connect with some of that four-year-old curiosity. Find a role model. Anyone under the age of five will do. Notice the questions he asks, the energy he has for finding out.
Seeing the fisherman bring his boat up onto the beach, four-year-old Matt ran up to have a look at the catch. He had dozens of questions: "Why is that one striped, but this one is spotted?" "Does that kind grow any bigger?" "Are all of these fish good to eat?" "Which one is poisonous?" "Do you like to touch them?" "Can they make noise under water?"
Asking good questions is the second tool you need to make conversation flow. Questions help you uncover a need or a commonality and move more quickly into the Actors Stage of relationship building, where you are actively exchanging information.
Often people ask questions that are too broad, too vague, and too ritualistic. They ask "What's new," or "Hi, Bob, what's going on?" One of the very best questions—because it narrows the scope—is "What have you been working on lately?"
Learn more about being Seriously Curious from these ten tips:
**1. _Organize some openers._** Do your brain a favor and, ahead of time, think of several openers. What could you say to start a conversation with the person sitting next to you at a workshop? How would you begin a conversation with someone at the hors d'oeuvres table? Here are some possibilities:
* "The title of this session, 'How To Stop the Brain Drain,' really grabbed my attention. Is your organization finding that a problem, too?"
* "Are you a first-timer like me or a long-time member?"
* "This speaker made a good point about career security. What do you think of her ideas?"
**2. _Ask about origins and history._** Asking about beginnings is a good way to hear about how people got where they are and to learn more about their Character and Competence. Ask:
* "How did the project begin?"
* "How did you meet your business partner?"
* "How did you get into marketing?"
* "How did you come up with this unusual packaging idea?"
**3. _Notice other people._** When you say, "I noticed you were on the edge of your seat during the speech," it's a compliment. When you take the time to notice people out loud, you'll find that the interaction deepens and the conversation becomes more personal. You've let the other person know that he is visible to you, that you are thinking about him. That's when a relationship begins. Here are some examples.
* "You seemed to really enjoy giving that presentation. Have you always felt comfortable talking in front of groups?"
* "I noticed your pin. It's very beautiful. Is there a story behind it?" (Often, people ask, "Where did you get it?" Avoid that question. It could sound envious or even predatory—as if you want to go right out and buy one just like it.)
* "I noticed you made sure everyone got a chance to give their ideas in the meeting. Does that come naturally or did you learn some techniques from the class the company offers?"
**4. _Appreciate other people._** When was the last time someone told you something they appreciated about you, for no reason, out of the blue? Maybe you were a little embarrassed, but wasn't it wonderful? Didn't it brighten your day and give you a special connection to that person? Your willingness to give appreciation to other people is a sign of your confidence and strength. As your capacity for gratitude grows, your ability to give grows. No phony baloney stuff here, please. Just ask yourself from time to time, when you're with people, "What do I appreciate about this person? What would feel good to acknowledge about this person?"
**5. _Take clichés seriously._** Listen for clichés and know how to handle them. When someone says, "How are you?" and you reply, "Fine," you've just completed a dead-end routine. Coping with these ritual conversations is one reason people hate to network. The question "How are you?" is too big, too open-ended. Here are some tactics that will help you move from a ritual conversation into one that goes somewhere:
If you're bored, bore in. Take the cliché one step further, explore it. Or make it more personal. Or make your response unexpected and playful.
If you ask, "How have you been?" and your partner replies, "Busy," ask:
* "What's a typical busy day like for you?"
* "What do you do on a busy weekend?"
* "If you decided tomorrow not to be busy any more, what would you quit doing?"
* "Do you remember times in your life that you haven't been as busy? Did you like it?"
**6. _Do something about the weather._** Everybody talks about the weather, but nobody does anything about it. How can you say something different or personal about it? How can you move the conversation to a more business-like topic? As with other ritual topics, if you are Seriously Curious, the weather can become interesting. If someone says, "Terrible weather we're having," then ask:
* "Have you ever lived anyplace where the weather's worse?"
* "I find the weather really affects my energy? Do you notice that too?"
If Dan says, "What a beautiful day!" you might ask:
* "What are you doing now that it's warm outside?"
* "Do ups and downs in the weather affect your sales much?"
Asking serious questions about a superficial topic turns your partner into an ordinary expert. Notice how these questions lead to more important topics. Your partner's answers provide clues to new topics to follow up on.
Don't forget your Agenda. Eleanor says, "My interest in talking about the weather was zero until I started planning for retirement. Now, I direct the weather conversation to my Agenda, saying, 'Yes, this humidity's a killer. It's recently hit me that when I retire, I can live anywhere on earth. Where do you suppose has the best weather?" Even boring weather conversations will come alive when you tie them to your real interests or needs.
**7. _Encourage dialogue._** Good conversation is a dialogue, not a monologue. Plan to talk only about 50 percent of the time, and you'll be remembered as someone who was interested as well as interesting. Notice people's body language—it often signals when they have something to add. Check out whether people are with you by saying things like, Have you ever experienced that?" or Is this something you're dealing with at work, too?" Look at Figure 11-1 for the difference.
**FIGURE 11-1. Going for Dialogue.**
**8. _Dig for gold._** Imagine that someone says to you, "My life is just crazy right now." That's a goldmine statement. If you dig deeper, you might find the mother lode. Surprisingly enough, most people respond with, "Oh, me too." They ignore the obvious question waiting to be asked, "What's going on that's crazy?" In one such exchange, Lee got a surprising answer: "I'm interviewing people for a new job we've created in our department." Lee applied for that job and was hired. Wasn't she glad she decided to go for the gold?
As Jean and Chuck talked before the board meeting started, he said, "What a week! I've never seen anything like it." Instead of responding with the usual cliché, "I know what you mean," Jean went for the goldmine, saying "Tell me more. What's going on in your life?" Chuck said, "We're trying to find a cat sitter to live in our house this summer." Jean suggested her boss's daughter, a college student who loved cats.
**9. _Interview people._** Imagine you're writing a magazine article about your conversation partner. Ask profile questions. These are the kinds of questions you may have seen in the American Express ads.
Put a few of your favorites from this list in your repertoire in case a conversation lags. Many of them are somewhat playful and likely to encourage a more personal response than some of the questions that are more conventional. Don't be afraid to try them.
* A typical day in your life?
* Personal philosophy?
* Business philosophy?
* Favorite anything: TV show, magazine, singer, song, performer, music, website, author, book, movie, actor, actress, meal, snack?
* Favorite gadget?
* Favorite thing to do on a Sunday afternoon?
* Personal hero?
* Motto?
* Dream vacation?
* What your Dad (Mom) always told you?
* Worst job?
* Biggest obstacle you had to overcome in your life?
* What people in high school thought you were like?
* What you wish you could stop doing?
* Someone you'd give anything to meet?
* Something you hope you never have to do?
* What you'd be doing if you weren't doing what you are doing?
* One thing you'd like to change about your work or business?
* Advice you'd give young people?
* What you've learned about life?
* Issues that matter to you? Action you take in support of those issues?
Answer the question yourself first if you feel that the question might be seen as intrusive. Obviously this kind of thing can be overdone; you'll want to be sensitive and appropriate. Use a playful tone of voice. It's fun to try profile questions on people you think you know well: your business partner, your teenager, your parents, your office mates, or even your boss.
**10. _Invite other people to talk._** Want people to tell you more? Prompt them to continue the conversation. Encourage your conversation partners. If you've been talking about executive coaching for employees, ask, "And, how about your company? Do you coach executives?"
Jim Collins, author of _Built to Last_ and _Good to Great_ , says: "If you want to have an interesting conversation, be interested. If you want to meet interesting people, be interested in the people you meet—their lives, their history, their story. Where are they from? How did they get there? What have they learned? By practicing the art of being interested, the majority of people become fascinating teachers; nearly everyone has an interesting story to tell."
### Tell Success Stories
There's a third skill that will make your conversations flow—storytelling.
### Use Success Stories to teach contacts about you.
At lunch with a client, on the flight to Denver with your boss, with the hiring manager as you wait for your job interview to begin, at the health club—wherever you are, your skill at telling stories will make people enjoy talking with you. Stories help the listener see you in action. Your anecdote puts a vivid picture in Joe's head so he remembers to refer you to his client. Your anecdote teaches Susan what you're good at or what you might be looking for so you can move to the Actor Stage of relationship building. Your example shows Mary more about your Character and Competence, so she feels confident recommending you for the promotion. To have memorable conversations, hone your ability to tell stories.
How good are you at telling a brief story, example, or anecdote that teaches people about you—your interests, your challenges, your talent? Don't be put off by the word _success._ Of course, we're not advising you to brag. That would detract from your Character and Competence. But if you construct your Success Story carefully, it can do a lot for you: It can give a vivid example of your expertise, enhance your credibility, teach people to trust you, and make people want to do business with you.
When someone at a networking event asked Carrie, who has her own PR business, "What's new," here's what she said. "I was really scrambling last week. I was in the middle of creating a brochure for Oak Tree Mall, and my office was flooded after that big rainstorm. So, I rented a computer, worked at home, and got the layout to the client on deadline, just as I had promised. Boy, was he happy!"
Here's what our workshop participants said Carrie's story taught them about her:
* "She won't let anything make her miss a deadline."
* "She's reliable."
* "She'll do what it takes to get the job done on time."
* "She's resourceful."
* "She doesn't give up."
* "She handles crises well."
* "She's doing work for a prestigious client, so she must be good."
* "You can trust her; she'll come through."
As you can see, you can get a lot of mileage out of a good Success Story: a reputation for going to heroic lengths to meet your deadlines, for delighting clients, for having a top business in town as a client. Best of all, it gives your contact a concrete picture of exactly what you do.
### Construct Your Story Carefully
Before you go to your next networking event, where you're sure to be asked the inevitable question, What's new," plan a couple of Success Stories. As you construct your story, use the letters in the word SUCCESS as your guide:
**_S_ = _Strategic._** Make sure your story fits your Agenda. Think about what you want people to know about you or your business, then build your story to teach that point.
**_U_ = _Unique._** Point out what makes you stand out from the crowd. If you're in real estate, for example, don't just say, "I've been selling lots of houses." That's expected. Give a specific and interesting example of a sale. "Last week, I found a home for a couple who both needed home offices. Both of them wanted first floor offices with outside access, lots of light, and great views. I found just the home, one that had two sunny rooms with French doors opening to a patio just off the driveway." This story teaches your conversation partner that you can find the unusual home.
**_C_ = _Clear._** Be sure you eliminate all the jargon of your profession.
**_C_ = _Concrete._** Give a couple of specific details to help your partner see a vivid picture. Those colorful words will stick in the other person's mind more easily than generalities. Notice that you can "see" the home the realtor was describing.
**_E_ = _Exciting._** Let your enthusiasm shine through. Use vivid language, an upbeat tone of voice, and a speedy, not draggy, delivery. Make it memorable.
**_S_ = _Short and Succinct._** Edit your story to a maximum of 6–10 sentences.
**_S_ = _Service-Oriented._** Be sure that your story teaches how well you served the internal or external client, solved the problem, saved the day.
It's a good idea, as you begin to sharpen your storytelling skills, to write out your stories. Then you can throw out the jargon, add vivid details, shorten the length to 6–10 sentences, refocus your stories to make your point or teach something about yourself. Plan Success Stories on several different topics, then use the one that seems most appropriate to the person you are talking with. Carry a notebook with you, so that you can capture ideas for stories. Develop one story a week until finding and telling anecdotes has become a conversational habit you feel confident about. As you are working on this skill, practice your stories on your family.
After you tell your story, ask your conversation partner a question that will elicit his or her story. Our favorite question comes from Ann, who asks, ' ' And what are you excited about these days?"
### Sample These Stories
### Karen's Story
I'm working on a project to teach mothers in prison to read to their kids when they come to visit. I work for the Prince Georges County Library System. I teamed up with people in other state agencies to get a grant to teach women inmates at the state prison storytelling skills to use with their pre-schoolers during family visits. I convinced several publishers to donate the books, and it was so much fun to teach the mothers. They really got into making up different voices for the characters in the story. The women loved reading to each other in the practice sessions almost as much as they loved reading to their own children. We had the party yesterday where all the kids came and each child got to take home a book to remember their time with their mom."
So what does Karen's story tell you about her? She knows how to work with others outside the library system. She's good at identifying needs and creating programs to meet them. She's an innovator, a problem-solver. Is her story memorable? Sure!
### Lynne's Story
"I just got back from teaching a three-day training course in Seattle. Boy, what a challenge! When I walked into the hotel conference room I noticed that one whole wall was windows overlooking the bay. 'Great room,' I thought to myself. But within the first hour of the class, I could see that my twelve students and I were mesmerized by the bay, longing to be out there. So that evening, I went to the cruise line that sends boats to Victoria and back and said, 'Could I bring twelve people on board tomorrow?' For only a little more money, we had our class on the water for the next two days. We met in a small conference room on board, I brought my flip chart, we worked very hard, and people were delighted to have breaks and lunchtime to stroll around the deck."
So what do you know about Lynne? She's a problem-solver and a leader. She's observant. She's resourceful. She's not willing to put up with an unworkable situation, and she'll go the extra mile to create a good learning environment for her students.
### Claire's Story
"I saw the most touching scene last night. I do mediation with couples who are working out custody and financial agreements. The couple I saw on Monday was very hostile and angry. So I just kept providing structure, helping them notice when they found agreement, and reminding them of their goal to stay out of court. The conversation was rocky for at least an hour and then something clicked with them. They left the office, and when I stood up to stretch my legs, I happened to look out the window. There they were, out in the parking lot, hugging. It wasn't a 'Let's get back together' hug. It was more of a 'We can work together for the good of each other and the kids' hug. I was so touched and I thought to myself, 'This is why I do this work.'"
So what did Claire teach you with her story? That she's persistent. That she succeeds even with the most difficult cases. That her work touches her heart.
### Allen's Story
"My company, a worldwide consulting firm, is so big that it's hard to know who the experts are. I sent out an e-mail and asked anyone who had a background or an interest in public health to respond. Surprisingly, more than seventy colleagues answered. We formed a Community of Practice. When any one of us is working on a client proposal in that area, it's easy to quickly identify people with specific knowledge. Not only can we tap into their expertise, but we also have an easier time staffing projects. We might ask a specialist to join the team to work with that client. Now I have experts that I can draw on for input when I'm doing client proposals or staffing work. And, of course, everyone else in the CoP knows more about my expertise."
So what do you know about Allen? He takes the initiative to improve the whole organization. He is determined to give clients the best his organization can offer.
### People Want to Know...
We have taught hundreds of people to tell stories. Here are some of the questions they ask us.
**1. _"What if I can't think of any stories?"_** Challenge yourself to notice the moments in your life that you'd like to tell others about. Look for experiences in your leisure and professional life that will show who you are. Listen carefully as others tell stories from their lives. Notice that, for the most part, they are talking about everyday events. Don't think you've got to have earthshaking stories—like about a time you rescued someone from a burning building or won a medal at the Olympics. Just look for times that brought out your best or would illustrate your Character and Competence. Keep that notebook with you so you can hang on to your ideas until you get a chance to write them out and edit them.
Ask yourself: "What would I like to teach people about me?" Look back at the goal you set as you read Chapter 5. Would it help you reach your goal if your contacts knew more about your Character and Competence? Do you want people to know that you're a stickler for details? That you're creative? That you're compassionate? That you can be tough when the going gets rough? That you know a lot about designing "green" buildings? Then look for anecdotes that give an example of those things. Figure 11-2 may give you some ideas on how to find your own stories.
**2. _"How can I get into my story? When do I tell it?"_** Look for a lull in the conversation. Or tell your story in response to "How are you?" or "What's new?" Think of a "transition sentence" that alerts the listener you're about to tell a story. Notice how Claire said, "I saw the most touching scene last night." That's her signal to others that she's going to tell an anecdote.
Here are some other good ideas for transition sentences.
To change the subject when there's a pause in the conversation, say, "I've been meaning to tell you about..." or "The most amazing thing happened last week..."
**FIGURE 11-2. Stories That Make the Point.**
To link your anecdote back to a previous conversation, say, "Remember when you told me about working with that client from China? I had a similar experience yesterday that I'd thought you'd be interested in."
To acknowledge your conversation partner's expertise, say, "I know you're a guy who loves the latest high-tech gadgets. Let me tell you about one I used that sure saved the day recently..."
To explore your contact's thoughts about an idea you have, say, "You know something interesting happened the other day, and I was curious about what you would think."
To update your contact on new talents or interests you've been developing say, "I had a "first" the other day in my life as a manager. Let me tell you about it."
**3. _"What are some tips for shaping my story and making it fun to listen to?"_** Relive the moment with relish. Craft your story so it paints a vivid picture. The best stories do several things. They reveal your interests, challenges, and talents so that people have an expanded idea of what to call on you for or what to send your way. And they are memorable enough that the listener could repeat them to others with some degree of accuracy. Ask yourself if you'd want to listen to your story. Practice it several times so you get to the point quickly.
Most stories have a "turnaround"—a moment when you had to do something, come up with a solution, solve a problem. Or a moment when you learned something about yourself or how the world works. Think of the childhood formula for a good story: "Once upon a time...Suddenly...Luckily...Happily ever after." If there's not a "Happily ever after," can you at least point to a lesson learned?
**4. _"Won't people think I'm "grandstanding" or "hot-dogging" if I tell a story?"_** No. Good conversationalists know how to pepper their conversation with brief, interesting vignettes about who they are and the experiences they've had. Most people won't be there to see your shining moments—how you captured your audience as you spoke at the conference even though the fire alarm went off in the middle of your presentation, or how you survived a camping trip with a dozen eight-year-olds. Think of your story as a gift to the conversation because it offers your conversational partner clues about what topics to bring up next, how to help you, or how to introduce you to others.
"A few years ago," a sports columnist wrote, "I followed Norm Stewart, Missouri's legendary basketball coach, out of a party. He was stopped by ten different people. He made every one of those people feel like the most special person in the world. His secret? He always had a good story to tell."
**5. _"What if I accidentally tell my story to the same person twice? Or what if someone overhears me telling the same story? That will be embarrassing!"_** Build a collection of stories. Make it a habit to notice moments in your business and leisure life that will make a good anecdote. Figure out how to tell it so that you teach the listener something new about your qualities and skills. Find and tell a new one every week until you have stockpiled enough that you can choose to tell the one that fits the situation or your conversation partner's interests. Jamie has lots of tales to tell that show how she juggles a part-time consulting business and her five-year-old triplets. But those might not be the best stories to tell to clients. She has other stories about getting her pilot's license and about competing on a Masters swimming team that she might tell at a business luncheon to teach about her qualities.
**6. _"What if everything I do is classified or I worry about client confidentiality?"_** To respect client confidentiality, disguise the particulars or combine several clients' experiences into one to make your story generic.
When Louis asked Cathleen, a CPA, "What's new with you," she said, "One of my clients was upset about a tax penalty for something that happened a couple of years ago. I wrote the most persuasive letter I could devise to the Department of Revenue about the situation. When they backed down and removed the penalty, my client was so relieved."
If your job is top secret, you can still tell stories from your personal life that highlight the qualities that make you special. Or you can genericize your stories from work so that they tell only what you are allowed to reveal.
There are three big conversational skills: generous listening, seriously curious questioning, and strategic storytelling. Master these and you'll be ready to talk to anybody.
## CHAPTER 12
## End with the Future in Mind
If someone asked you, "What's the most difficult moment in networking?" would you say "Ending the conversation"? Many people do. Introductions and meeting people are stressful, they'll tell you, but at least there's a routine: You shake hands and exchange names. On the other hand, there is no protocol for ending conversations and exiting can often seem awkward.
### Prepare for the Next Time
Your Critic may move into high gear when someone—even someone you've made a good connection with—ends a conversation with you. If you are the one doing the leaving, you may feel guilty because it seems as if you are rejecting or abandoning the other person. As a result of these feelings, people say, "I believe I'll freshen my drink," and walk away, not even bothering to head in the direction of the bar. Or they may simply say, with no intention of doing so, "I'll see you later." Or they may drift away when a third person enters the conversation.
To change your mindset about the final moments of a conversation, imagine that you'll continue your dialogue at some time in the future. Always assume that you will see your conversation partner again. Think, "I'm just beginning this relationship. It will be exciting to see it develop." Always prepare for the next time. Making a conscious closing will set the tone for your next meeting.
### Listen for the Bell
Tune in to the timetable. There's a bell that goes off in people's minds after a conversation has been going on for about five minutes. At networking events and at many quasi-business gatherings, such as cocktail parties or receptions, people have a vague notion that they should speak with as many people as possible. You will be able to tell from your conversation partner's body language when he is ready to move on. He will look away, gather his possessions, and perhaps even move farther away from you. Notice the bell in your head—your intuitive sense of when it's time to say good-bye.
### Eight Ways to Leave
Honesty is rare in the final moments of a conversation, but that's what works best. Be totally honest. Here are eight ways to leave a conversation gracefully and competently with your own integrity—and your contact's—intact.
**1. _Center on your Agenda._** Your Agenda will serve you well as you make conscious closings. Saying, "I want...I must...I need..." eliminates the feeling that you are abandoning your conversation partner. Shift the attention to where you are going and the purpose that is motivating you.
Here are some suggestions for closing a conversation by referring to your Agenda:
* "I'm going to circulate and welcome some of the new people."
* "I need to see three more people before I leave tonight."
* "I must speak to the membership chairman before he leaves."
* "I want to see if there are any other engineers (or people from my industry, or home-based business people) here."
**2. _Ask your contact for a referral._** To change conversation partners, ask your current partner for a referral to someone else in the room. Say:
* "I want to find other people who are working at home. Do you know anyone like that?"
* "Do you know anyone here who is involved with management training?"
* "I'm going to the annual meeting next month. Do you know anybody who went last year?"
* "Do you know of anyone who is thinking about moving to a new office this year? My company is expanding its office design services."
**3. _Take your contact along with you._** If you feel uncomfortable ending a conversation and walking away from someone, invite that person to go with you:
* "Let's see if we can find the registration booth."
* "Want a drink? I'm thirsty."
* "Would you like to come with me to talk with the new president? I want to ask her about next month's program."
**4. _Introduce your contact._** As you look around the room, you may see someone you want to introduce your conversation partner to. Don't think of this as a way to get rid of somebody. Instead, always think, Who do I know here that my contact might need to meet?"
An example: "Lenora, you mentioned you're going to Vancouver next month. I want to introduce you to Sam. He grew up there and could tell you all about the sights."
Or: "Tom, as soon as Bill arrives, I want to get you two together. Last month you said you were thinking of franchising your stores, and he's a franchise lawyer. I'll bet you two would have a lot to talk about."
**5. _Play concentration._** Remember that kids' game where you lay all the cards face down on the table? You turn over a ten of hearts, but you can't have it until you find a match. Your challenge is to remember after several turns where that ten of hearts is.
You can play Concentration in a room full of people, too. You meet Marjory, an interior designer, who specializes in helping seniors downsize and move to smaller quarters. A few minutes later, you talk with Cynthia, who says she's writing a book titled _Moving Mother._ You think to yourself, "I must introduce Cynthia to Marjorie. What a match!" You go out of your way to bring them together. Whether you stay with that conversation or not, they will remember you as a person who knows everybody.
**6. _Sum up and appreciate._** One of the most memorable ways to close is to sum up the conversation and show appreciation for your conversation partner. To do that, shake hands and acknowledge the conversation and its importance to you. You could even acknowledge the importance in your life of the relationship you have with your contact that perhaps goes way beyond this encounter. Find a specific quality in the other person or a moment in the conversation that you can genuinely express appreciation for:
* "If the other members of ASID are as enthusiastic as you are, I'm going to be very glad I joined."
* "Wonderful to see you and to hear about the trade show."
* "I'm so glad to know more about your department."
* "Thanks for telling me about your new marketing tactics. I'm looking forward to hearing how they are working next month."
**7. _Explain the next steps._** Finally, say what you will do next, or what you would like for your contact to do next, to continue the relationship. Many of these suggestions are reassuring to your contact because, in contrast to just melting away, you are being very specific. We call these Magnet Statements because they are designed to pull you back together at some point in the future. They provide the energy to continue the conversation and build the relationship. They signal interest. Let your sincerity shine through. Look the person in the eye. Ask the person for his or her card so you'll have the necessary information to re-connect. Jot a note to yourself on the back of the card while you are still with the person or soon after you part. Say what you will do or what the next step in your relationship will be:
* "I'm going to send you that article we talked about."
* "This idea really jelled for me when you explained it. I'd like to hear more when we get together."
* "I'll ask Jim to call you."
* "I'll see you at the next meeting."
* "I don't want to monopolize you this evening. Can we arrange to meet later?"
* "I hope we can do business after the holidays."
Or ask your contact to follow up:
* Give me a call next week, and we'll set up a time for me to tell you about my publishing experiences. I'm glad you asked me for advice. I'm always eager to help a fellow author. Here's my card."
**8. _Shake hands and leave._** After making these final statements, shake hands and leave quickly. No dilly-dallying. Use your body language to emphasize your purposeful leave-taking.
Remember watching a wonderful mini-series? Remember the good feelings of expectation you had when you saw the words "To Be Continued..." on the TV screen? That's how you want to leave your contact: Those words hanging in the air, setting the stage for the next episode in your relationship.
### A Ritual for Leave-Taking
To close a conversation easily, remember this LEAVE NOW formula:
**_L_** = Let go of your conversation partner after five minutes.
**_E_** = Explain what you must do. Be honest.
**_A_** = Act on your Agenda.
**_V_** = Volunteer a referral.
**_E_** = Exit easily to another conversation by taking your conversation partner with you.
**_N_** = Note what's gone on between you. Sum up the conversation and appreciate something your contact said or did.
**_O_** = Outline the next step for your contact.
_**W**_ = Walk. Shake hands and leave, purposefully.
### Do You Have Questions?
Here are some of the questions people have about endings.
**Q:** | "What if I'm talking with someone and we're interrupted?"
---|---
**A:** | Look for a way to reconnect before the meeting ends. Mark was listening intently to Susan talk about her expanding business when two other people joined their group and the conversation got sidetracked to another topic. Soon, the chair called the meeting to order. Susan and Mark ended up at different tables. Mark wanted to go on with their conversation because he figured she'd need his office design services sometime in the next year as she added more office space. Before he left the luncheon, he made a point of approaching Susan again, asking for her card, and offering to send her an article he'd written on office lighting.
**Q:** | "What if you really want to keep on talking?"
**A:** | Occasionally, you will find yourself in a conversation that's too good to leave. You want to keep talking with your contact even though the unwritten "rule" says, "Circulate!"
Hal began to talk with Marilyn, the speaker. After five minutes or so, he began to feel uncomfortable because she wasn't having an opportunity to visit with anyone else. Even though they both clearly wanted to continue their conversation, it seemed rude to do so. Finally Hal said, "I don't want to monopolize you. Let's plan to get together sometime in the next month. I'll call so we'll both have our calendars handy."
**Q:** | "I leave events with a pocket full of business cards. I know I should keep and organize the contact information. Got any tips?"
---|---
**A:** | You'll keep the contact information for some people because you like their energy or there's good chemistry between the two of you and you want to be on the lookout for ways to continue the relationship. Other cards serve as a reminder that you made a specific commitment that requires Follow Through. And you probably exchanged some cards because you uncovered a commonality or a need that invites further exploration. For instance, when Marilyn and Joe discovered they were both researching which career fairs to attend for their company's college recruitment activities, they had a natural reason to exchange cards.
Put data about people you want to keep track of in your contact management system. This software is a must for the serious net worker because it helps you use your contact information in a variety of ways. Before Mike's three-day business trip to Atlanta, he compiled a list of everyone he knows there, so he could decide whom to see. Rochelle called up a list of everyone she'd met last year at her association's annual meeting, so she could refresh her memory about these people before seeing them again this year.
## CHAPTER 13
## Follow Through
All too often, networkers spend lots of energy making initial contacts and then don't know how to cultivate them so that the relationships pay dividends down the road. Mike, Nancy, and Susan are typical of networkers everywhere. They're trying to figure out how to stay connected. Here's what they say.
"I go to networking events and meet a lot of people and then—nothing happens. What am I doing wrong?" asks Mike, a CPA.
"I talked to a coworker at last month's in-house training session. She's in a division I'd like to transfer to. I can't figure out what to do next. Soon, she'll forget who I am and what we talked about," says Nancy, a middle manager in a Fortune 100 company.
"Bob's company is similar to mine, and I'm sure I could learn a lot from him. Come to think of it, we're doing some advertising that he'd probably like to know about, but I'm not sure what the next step is in getting to know him," says Susan, a sales rep.
### Focus on Follow Through
If your networking isn't paying off, give more attention to following through. Follow Through is the act of carrying a motion to its natural completion. Follow Through insures that baseball players and golfers achieve maximum force on the ball. Follow Through insures that you as a networker achieve maximum impact.
Follow Through begins with a good conversation, one in which you Listen Generously and are Seriously Curious to find out what's on your conversation partner's Agenda. A meaty conversation will give you ideas. The best Follow Through is based on the other person's Agenda, not yours.
Ideally, you'll suggest another meeting during that first conversation with someone. You might say, "I'd like to talk with you more about that. How about if I call you on Monday to set up a time to get together?" Say, for example, "I'll give you a call next month so that we can get together for lunch." You'll want to set up a chain reaction of six to eight encounters to establish a networking relationship. You won't have to initiate every meeting. You know you'll see Tom at the next task force meeting, for example, and you'll reconnect with Clara at the board meeting.
### Figure Out Your Reasons to Reconnect
Why _do_ you want to get back in touch? Here are three great reasons.
**1. _Chemistry._** You like the person and you can imagine because of what he does (chairs the diversity task force, for instance), who he knows (people in Seattle where you plan to move next year), the experiences he's had (worked in London), that it will be mutually beneficial if you have the time to teach and learn and explore together.
**2. _Commitment._** You had a rich, Agenda-based conversation that requires some specific next step. You promised to provide a phone number or website or piece of information.
**3. _Commonality._** You found you have something in common or uncovered a need that begs to be explored.
### Face Your Fears
Do you feel uncomfortable about taking that first step to reestablish a dialogue—especially if quite a bit of time has elapsed since you met with your contact?
Rob handles that problem like this. He calls and asks, "Isn't it about time for our annual lunch?"
Maybe it hasn't been a year, but you're still afraid that the person you'd like to re-connect with won't remember you. You could ask a colleague or friend to re-introduce you.
You have a good reason to reconnect. You'd like to build a relationship because you think the person would be useful to you and perhaps you could help him, too. But that probably feels like far too much to ask for initially. People tell us they'd feel more comfortable if they had "an excuse" for calling or setting up another encounter. But when you think of follow up as Follow Through, you don't need an "excuse." Follow Through becomes a legitimate, integral, natural _completion_ of the process, not an add-on or afterthought.
Reconnecting does become more comfortable when you reopen a conversation by reminding your contact of something you have in common. These ideas will help your contact remember you.
**_Refer back to when and how you met._** "We met in that computer course a couple of weeks ago. Are you using the software they suggested? I've figured out some tricks I'll be glad to show you."
**_Refer to a common need._** "Since we're both starting businesses, I was interested in what you said about looking for office space. I'm working on that, too. How about getting together to talk about strategy?"
**_Refer to proximity._** "We work near each other; let's get together for lunch." Or, "We live in the same neighborhood, let's meet at the deli for supper next week." Or, "We sat at the same table at the Chamber of Commerce dinner. I'd like to know more about the sales training program you mentioned."
**_Refer to a common background._** "I noticed that we both went to the University of Florida. I got a flyer saying there's going to be an alumni get-together to watch the game next week. Do you want to go?" Or, "Don't you have a degree in English, too? I'd be very interested to know how you made the transition to PR. How about coffee later this week?"
**_Appreciate your contact's contributions._** "You're doing a great job heading up the program committee. I did that for the Des Moines chapter, and I know what a big job it is. We developed a great checklist for planning any event. If you think your committee members could use it, I'll drop it by your office."
**_Refer to a common acquaintance._** "You know Burt, don't you? When I talked with him, he suggested we get together. I'm heading up the fundraiser for the hospital. Burt said you did one last year and might be able to give me some pointers. How about breakfast next week?"
**_Refer to time or money savers._** "I heard you say you're feeling overwhelmed with paperwork. I was too, so I hired an office organizer. I'd be happy to share some of her tips with you. They helped me completely overhaul my office. Would you like to come over and see what I did?"
### Fill in the Blanks on Your Calendar
Set a goal for the number of networking calls and meetings you want to do every week. When you are planning your week, pull out your calendar and schedule your networking. Don't just think about it, do it.
When people decide to create an active network of 50 to 250 or more contacts, they worry about how much time it will take. It takes less time than you might think.
### Dean and Marta's Story
Dean and Marta met at a luncheon meeting of their professional group. As entrepreneurs, they were always looking for work. They shared many common interests and friends. They were very clear with each other about their goals. He said, "I'm a career coach for lawyers who want to reassess their career options." She said, "I teach executives how to handle high-stakes press conferences." Their networking is practically effortless because they both know what the other has to give and wants to find. As you look at their interactions over a period of one year, you can see that Dean and Marta spent only about three hours networking with each other during that time.
February: Marta sent Dean a news article telling about an upcoming, one-week course for lawyers on career changing issues. Dean followed up and was invited to be a guest speaker.
April: Dean called Marta with the name of a law firm looking for a motivational speaker. Marta passed the lead on to a speaker who had referred an executive client to her.
May: Marta and Dean had lunch, updated each other on recent successes and challenges, and enjoyed each other's company.
September: Marta referred her lawyer cousin to Dean for career counseling.
December: Dean and Marta chatted at a holiday party. Marta told Dean she was looking for clients in Europe. Dean introduced Marta to a lawyer he knew who had recently returned after spending a year in London.
It's not the amount of time, but the quality of the interaction that counts in networking.
### The Five Goals of Follow Through
What are you trying to achieve as you nurture networking relationships? Take another look at the Rate Your Relationships quiz in Chapter 4. Notice how relationships develop. Aim for these five goals as you stay in touch.
Teach your contacts:
**1.** Your name and how to reach you easily.
**2.** Exactly what you do.
**3.** To have faith in your ability to serve or supply them—or people they refer to you—expertly.
**4.** What kinds of clients, customers or job opportunities you are seeking and what you can refer to them.
**5.** What kind of information and opportunities you are looking for.
As you reach these goals with your contacts, you will begin to—and continue to—reap the benefits of networking.
### Freshen Up Your Relationships
A businessperson said to us, "I realized that I don't make enough phone calls—general how-are-you, was-just-wondering-how-things-are-going, stay-in-touch type calls—to people in my network. I think I'm afraid the call is going to take too much time—too much of my time, too much of their time. Part of the problem is I'm not sure how to end the call. Any suggestions or thoughts on how I can call more and stress less?"
Before you call, think of three things to say to your contact. You speak at about the rate of 150 words per minute so even in a three-minute call you can say quite a bit. Remember networking is about teaching, so every conversation, whether face-to-face or by phone, is a chance to teach. Here are three ideas for your phone call Agenda.
**1.** Come up with a bit of information that verifies that you know who your contact is and what he is interested in. "You know, I was thinking about you the other night when we had dinner at a Korean restaurant. I was remembering the stories you told about trying all the delicacies when you were living in Seoul. I wasn't quite that brave."
**2.** Have something to give. "You'd mentioned your job was requiring more and more in the way of negotiating skills. I just heard that there's a speaker on that very topic at the Wharton Alumni Club Tuesday night—would you like to be my guest?"
**3.** Think of a Success Story that teaches your contact something about your interests and your expertise. "My job has taken an interesting turn: I'm not only writing the speeches for the executives, I'm now coaching them on their presentation skills—something that requires a great deal of tact!"
When you get your contact on the phone, ask, "Is this a good time to talk for a couple of minutes?"
As you chat, listen for new information about what might be on his or her Agenda so you can respond generously, either by giving some useful information immediately or sending some information later. Or listen for a topic to talk about when you next meet. A good question to get your contact to talk is to ask, "What have you been working on lately?"
To end the call, talk about the next step for your relationship. If this is a person you want to see on a regular basis, build that idea into your comment. "Let's have our quarterly lunch in late September."
### Find the Way
As you decide how you will Follow Through, here are some of the things you'll need to consider. Do you want face time or can you use the written word or electronic communication? Do you want to become visible to only one person or many people at the same time? How much time can you afford to spend? What's your budget? As you choose your method, take these tactical tradeoffs into account. Balance the time or money each method takes with its potential for building the relationship. Then decide which way is right for you, with a particular contact, in a particular situation. Follow Through ideas come in all flavors. Choose from among these thirty ideas—one for every day in the month.
**1. _Share a cab._** Split the cab fare as you go to a meeting or event.
**2. _Park and walk._** Park your car in a new spot in the company parking lot every day and chat with a different person as you walk into the building.
**3. _Lend a book._** Deliver a book or CD you have enjoyed to a contact. As you visit, ask about projects that person is working on and be ready to tell about your latest successes and challenges.
When Mike met Charles at a Rotary International luncheon, Charles said, "I'm on the library's waiting list for that business bestseller." Mike asked for his business card and said, "I have a copy. I'll give you a call and bring it over to you."
**4. _Forward the freebies._** Provide access to events, people, and resources – dinner with a visiting author, your library of training DVDs, a sneak preview of a movie, tickets to a sports event, or speech.
**5. _Pull up a chair._** At a meeting or event, plan to sit next to someone you'd like to know better. Call your contact before the event and say, "Hey, we haven't had a chance to talk for a while. Let's sit together at the luncheon and catch up."
**6. _Have a bunch to lunch._** Ask a few people you'd like to know better to lunch. Pick your lunch bunch carefully so that the benefits of their becoming better acquainted with you and with each other are obvious. Marcella, who owns a small advertising agency, frequently invites a mix of clients and potential clients to a catered lunch in her conference room. "They seem to enjoy meeting each other. Often, the stories my current clients tell to my potential clients 'sell' them on using my services."
**7. _Tip the talkers._** Before a meeting begins, chat with the speaker or emcee. Let that person know what interests you about the topic and your experience with it. Presenters appreciate knowing more about their audiences. A mention of you from the podium acts almost like an endorsement and certainly gives you more visibility and credibility.
Fred, the owner of a franchise sign shop, gave other attendees a ready-made way to Follow Through with him after the presentation. He showed up early at a workshop on marketing and talked to the speaker. When she asked about his work, he said, "I make signs and banners for all kinds of businesses. I also do a complete range of signs that comply with Americans With Disabilities Act regulations." When a workshop participant asked about the ADA signage regulations, the speaker said, "Fred's company has done a lot of that. Fred, stand up, so people will be able to find you during our coffee break."
**8. _Find someone to thank._** Late Friday afternoon, when not much else is going on, look back over your week and find five people to thank. Karishma sent a funny card to Elena, a coworker who'd tutored her on the new software. Mike shot an e-mail to Bill thanking him for a referral. Jane ordered a gift basket of coffee and specialty chocolates for the three people who supported her the most during her successful job search. Mary made a phone call to Stan, appreciating his advice on companies who would reliably handle the office move she was managing.
**9. _Host a meeting._** Want to show some influential people where your business is located and give them a clear image of what you do? Offer to have the committee or board meeting at your place. To make your business real to attendees, give a quick guided tour. Talk informally about awards on the wall, new equipment, new capabilities, and various services you provide.
**10. _Extend an invitation._** Want to see someone more frequently? Encourage your contact to visit and perhaps join an organization you already belong to.
**11. _Speak out._** Speak to the local chapter of an association. Provide a news release to your local newspaper or business publication about the program. Send that same news release to contacts you think might be interested in the topic and ask them to let you know if they will attend. Be sure to say hello, and Follow Through with a note saying, "Nice to see you."
**12. _Throw a party._** Invite contacts to your place of business to give people a better idea of what you do. If you work at home, team up to find an interesting place for your open house. Select your co-host carefully. Look for someone with whom you might have customers in common.
Artist Carol works at home, so she teamed up with frame shop owner, Kari, to showcase both businesses with an after hours wine and cheese party. She and Kari invited both past and potential customers to view Carol's drawings and to see Kari's frames.
**13. _Add food._** You can extend almost any activity by adding a meal or a cup of coffee at the end. That gives you time to talk. After the soccer practice, plan a picnic. After the training session at work, go out for a drink or coffee.
Sue had met several women at the health club, but they didn't usually have time for extended conversation. That's when she came up with the Breakfast Bunch. The group meets for breakfast one Saturday a month after they exercise.
**14. _Honor the volunteers._** Have you enlisted volunteers for your favorite charity? Bring them together.
Wendie's final project for her graduate degree was designed to encourage eleven-year-old girls to think about careers in science and to provide role models for them. She invited her contacts—professional women in scientific areas—to help with the project. After the project was over, she invited the women to a "dutch treat" celebration brunch at a local restaurant to introduce them to each other.
**15. _Drop by._** Turner wrote a letter to his franchise training director recommending Gloria, a speaker, for the next training conference and sent her a copy. Rather than calling to say thanks, Gloria dropped by Turner's store. She said, "Your letter was wonderful. I appreciated it so much. I will follow up with the training director. Do you have time to give me the grand tour of your store and to tell me about your products and services? I want to understand exactly what you and your fellow franchisees do before I make a proposal to the training director. And want to be able to recommend you to anyone I run into who needs printing."
**16. _Face it._** When face-to-face contact isn't feasible, send your face. Have a note card that fits into a business envelope printed up with your photo on it.
**17. _Send a postcard._** Out of town on business or vacation? Take some addresses with you. Buy a handful of postcards, or before you go, have postcards printed with your picture, your logo, a saying that makes people think of your service or product, or some interesting facts about your industry. Write a note confirming a future meeting with your contact.
When Ron (a professional speaker and humorist) travels for business, he always takes a stack of the postcards he had specially designed so he can mail notes to prospects and clients. Ron's postcard shows him standing in front of a huge hotel ballroom full of rows and rows of empty seats. The caption under Ron's big smile says, "Wish you were here!"
**18. _Notice publicity._** Peruse the newspaper watching for publicity about any of your contacts. Clip articles and send them with sticky notes. Or cut out your contact's advertisements and send with a note telling what made the ad leap off the page and grab your attention.
**19. _Get feedback._** Ask your contact to review something you've written. Tom writes a short column each month for his professional association magazine. As a supplier of services that members often need, he finds it an excellent way to establish his credibility and name recognition in his specialized marketplace. About three weeks before his deadline, he sends his column to a couple of prospects, clients, or referral sources and asks for their comments, suggestions, and a reality check. They are honored to be seen as a sounding board and often have good examples or suggestions.
**20. _Send the news._** If your business involves providing information—and who's doesn't?—produce a print or electronic newsletter or blog. Highlight your successes, new products and services. Show how clients or customers benefit. Help your contacts see how they could use your expertise. Quoting or featuring customers also enhances your credibility and testifies to your Character and Competence.
**21. _Provide a calendar._** Send your contact a calendar of events you'll be involved in or clients you will be working with. This idea works well for musicians, artists, trainers, consultants, craftspeople, speakers, and freelancers, for example.
**22. _Give a goodie._** Send your contact a bagel and cream cheese or a couple of donuts—or even a single, specialty tea or coffee bag—along with some information you'd like that person to take time to look at. Notice that this technique "creates" a coffee and bagel break in your contact's day for her to focus on your information.
**23. _Announce your news._** Get the word out about an achievement, a move, or a promotion. Send a news release, postcard, or note. This is a good way to teach contacts about your Competence.
**24. _Read all about it._** Send your contacts an article that mentions you as an expert. If you haven't been in the news recently, send an article that gives information on the kind of service or product you provide. That way, you can "piggyback" on an article in the news media, positioning yourself as an expert.
Jim's firm analyzes overhead costs for small and mid-sized businesses. When The Wall Street Journal featured an article on rising overhead costs for small businesses, he sent copies, with a personal note and his brochure to twenty potential clients.
**25. _Wish 'em a happy._** Send a card on an unusual holiday—Fourth of July, your birthday, Labor Day—to avoid having your message become just one of many at the end of the year. Or send birthday cards to contacts on their birthdays.
**26. _Be a winner._** Entering a professional association awards program takes time and effort, but if you win, it's worth it. All the reviewers who evaluate the entries learn of your expertise. Whether you win or not, send a thank you note to each of them.
**27. _Delegate responsibility._** Make a list of people you want to stay in touch with and have your assistant send a short, personal message, drafted by you, every couple of months.
**28. _Create a quiz._** Design a quiz to teach people about your product or service. Put the quiz on a wallet-sized card to give out to potential clients or distribute it by fax, mail, or e-mail.
Jeff, who owns a carpet store, created a quiz: "Do You Know How to Buy Carpet?" When he meets someone who is thinking about buying carpet, he says, "Give me your business card, and I'll send you my quiz. It will help you know what to look for as you make your decision. Of course, I hope you'll come by my store as you are shopping."
**29. _Give yourself a job._** Find a reason to interview your contact. Gene was on the planning committee for the next regional conference for his professional association. When he met Melissa, he asked her if he could interview her about what she'd like to see on the program. Andie was putting together a proposal for an employee survey. She asked if she could talk with Margi about "lessons learned" from the survey Margi's company had done last year.
**30. _Add to their library._** Give contacts a copy of a book you have written or a book that relates to the product or service you provide. One key idea in Following Through is to provide something to contacts that they will keep a long, long time, so that your name and phone number are available and visible for a long, long time. Giving a book accomplishes just that.
### Bonus: Five More Ingenious Ways to Fit In Follow Through
You can do networking on the run. Use bits and pieces of time effectively. Make multitasking a way of life. Notice that all of these ideas allow you to network as you are doing something you already have on your calendar or to-do list.
**1. _Piggyback._** Have coffee with a contact after the meeting or get to the event early so you can talk with the movers and shakers.
**2. _Enlist a volunteer._** Ask a contact to join you in a charitable activity like Habitat for Humanity, so that you see her more frequently.
**3. _Share a sandwich._** Rather than sitting with your usual group, ask someone in another department to have lunch with you in the company cafeteria.
**4. _Sweat together._** Ask a contact to join you for a walk or a bike ride.
**5. _Take to the skies._** Flying to a conference? Call a contact and arrange to sit next to each other on the plane.
## PART IV
## Select Your Settings
Where will you use your networking skills?
Inside your corporation, non-profit, government agency? Here are the tools you need to assess and navigate the complexity of your workplace culture. You'll get state-of-the-art tactics that will show you the best ways to create your network at work.
To develop business? You can professionalize your practice development. You'll find out how to make it rain clients.
As you work at home? You'll learn how to connect as you go it alone.
In various networking venues, such as professional associations or Chambers of Commerce? You'll find out how to select the best organizations for your purposes and how to make the most of your memberships.
In referral groups? You'll discover how to get the most out of relationships in your group.
At conventions? You'll see how to get the information, inspiration, and—most important—the interaction you came for.
As you job hunt? You'll be able to use the special networking skills most job-hunters don't know about to find the perfect job faster or leap from one career field to another.
Whatever the setting, you'll find cutting-edge ideas in these chapters that will help you advance your career as you put your portfolio of networking skills to work in the world.
## CHAPTER 14
## Network at Work
No matter where you work—a corporation, government agency, university, association, non-profit, or other kind of organization—networking is a pivotal professional competency. It's the BEST way to get the job done, make things work, improve the processes, and advance your career.
### Got the Right Word?
Recognize that in some organizations the word "networking" makes people uncomfortable. Don't be fooled. A lot of networking is no doubt going on, but under the alias of "relationship building," "teamwork," "collaboration," "social acumen," "connectivity," "social capital," 'horizontal integration," "inclusion," "collaborative knowledge networks," or "communities of practice." Check your own organizational initiatives for hints that relationship building is a corporate priority.
### Bank On the Benefits
Professors from the nation's top business schools (e.g., Harvard, Wharton, Kellogg) are writing articles touting the benefits and necessity of networking at work. Their research substantiates what we've found as we've worked to support all kinds of organizations as they teach employees how to network. Relationship building has become a corporate priority, and yet people are not sure how to go about making it part of the corporate culture. Interestingly, no one department or function has claimed "ownership" of networking in most organizations. No one area has "championed" or "sponsored" an organization-wide approach. The time is right for all the stakeholders to get together and coordinate their plans to advance networking as a core competency. The stakeholders might logically include marketing, business development, career development, mentoring programs, leadership and employee development programs, corporate communications, human resources, executive coaching, affinity groups, and diversity programs, etc. Who in your organization recognizes and teaches networking as a skill-set everybody needs?
The benefits of networking outweigh the efforts. Persuade the powers that be that networking is vital to the health of your organization. And clarify networking's benefits to your organization and to you.
Networking at work helps you to:
* **_Keep getting the big picture._** Things change fast. Use your network to keep up with what's going on. What percent of sales in your organization are from products or services that didn't exist five years ago? Employees have to stay on their toes, just to know what products their company makes. To remind yourself of the rate of change in your workplace, take a look at the last few issues of your company's annual report.
* **_Bolster the bottom line._** Understand that your job depends on the success of the organization. Look around for ways to link up efforts to produce income. Sherman facilitated internal strategic planning sessions at the bank. He realized that offering help with strategic planning to non-profits might attract them to bring their business to the bank. He asked himself, "What other things might these organizations need?" He enlisted people from several other departments to provide clients and prospects with a package of services.
* **_Venture into the white spaces._** Look at the organization chart. What do you see? Boxes linked by some vertical lines that indicate the chain of command? Now, look between the boxes. What do you see? White space? In most organizations, that white space is unexplored territory for networking. That's where you'll find the unmet needs, the undiscovered problems, the opportunities, and the connections that will enhance your career and allow you to contribute more to the organization's success.
An enterprising government employee in Fairfax County, VA, made a list of all languages spoken by employees in various departments so they could serve their increasingly diverse customers better. Her willingness to venture out into the white space helped her showcase her skills and get a better job. It was a win for the county, a win for the citizens, and a win for her.
* **_Uncork bureaucratic bottlenecks._** If you create temporary project teams to tackle problems and launch initiatives, you'll make a name for yourself. Increase collaboration with other departments.
Patricia used her internal network to change an operational policy that was causing difficulties for people in several departments every quarter. She talked with a key contact in the operations department to find out the history and exact intent of the current policy. Her contact suggested that she talk with two managers who had strong feelings about the policy. She interviewed those managers to determine their concerns. She researched and drafted her policy amendment. Then she consulted with two of her peers to see if they had any additional information she should consider. She also wanted to gain their support for her proposal. One of them told her about some recent legal developments that she was unaware of.
Without her network, Patricia might have created an unacceptable proposal. With her network, she was able to gather the information and feedback she needed, while building support among key people for the amendment. Her policy change was accepted.
Any time you are working on something that will affect people outside your own department, take the time to "field test" your idea. That way, you'll build support for the idea because you've included others in the process. "No surprises" is a cardinal rule of corporate life. Pre-testing ideas prevents surprises.
* **_Expand your knowledge base._** Figure out what resources you need and put together a network made up of people representing many different interests and areas of expertise. If you introduce your contacts to each other, you can encourage information and skill sharing among all the members of the group. As you network, you expose yourself to new ideas and ways of doing things. This "cross pollination" almost always benefits the organization.
* **_Create your safety net._** Take responsibility for your own career self-management. Network to increase your visibility within your organization so that opportunities find you! Explore options in case your job goes away. In these days of rightsizing and restructuring, it's smart to keep your ears open. Ask yourself, "What skills do I have that could be used in other areas of the organization?" Figure out how to showcase those skills. What can you do so that others become aware of your capabilities?
Maria offered to manage the 10-K run for a local charity. Sue noticed how much the community sponsors liked working with her and how well-organized she was. When a job opened up in Sue's department, she thought of Maria.
Gary wanted to move from a technical area into training. When line managers were invited to teach a career management course, Gary volunteered. He was able to brush up on training skills as he took the train-the-trainer course and to showcase his teaching skills as he presented the career course. When the training department was looking for a technical trainer, Gary's name came up and he was able to make the switch.
* **_Access inside information._** Through networking you can get information before it is public. And that information comes with the evaluation and insight your contact adds—something you'll never get from an official announcement.
* **_Develop a power base._** In organizations, power comes from being an information broker—someone who can stimulate collaboration among many different groups and make things happen. This power has nothing to do with your place on the organization chart.
* **_Round up talent._** Put together a circle of contacts with diverse skills that broaden your ability to get things done and insure the success of your projects and initiatives.
### Ten Ways to Get on Board Quickly
Got a new job? It may take you as many as six months to feel that you are in control of your job. When your orientation takes that long, your organization loses the ability to tap into your creativity, knowledge, new perspectives, expertise, industry contacts, and fresh ideas.
What can you do to leap into the saddle ASAP? Here are ten ideas. Notice how many of them deal with building an internal network! You'll learn how to do that in the rest of this chapter.
**1.** Recognize that you, in a new situation, will need to notice the cultural ground rules and be aware of some of the organizational history before leaping into action.
**2.** Know that most organizations think that providing you with information is the key to helping you become productive quickly.
**3.** Get clear that your best strategy is to build relationships, not gulp information. The more connected you feel, the more you'll feel satisfied and committed to your new job.
**4.** Often introductions to others in an organization aren't strategic and are done during a quick walk down a hallway. Ask your boss, "Who do I need to get to know?" Say, "When you introduce me, I know you'll be telling about my background. It will help me out if you'll also fill me in on the other person's roles and projects. That way, it will be easier for me to go back later and delve into things I need to get up to speed on."
**5.** Ask for or select on your own a "buddy of the week" for at least your first month on the job. This will give you someone to ask questions of.
**6.** Ask questions and engage in conversations in which you explore your co-workers' and subordinates' abilities, skills, and knowledge. Talk to people about their roles and responsibilities.
**7.** Take advantage of your newbie "halo." When you begin, you have a window of time in which people expect you to be a bit different. Even if networking is not the norm in your organization, you can use this time to get out of your cubicle and meet as many people as you can.
**8.** Ask for assignments that bring you in contact with others, not stand-alone projects.
**9.** Jump over to other groups and find out how your group and theirs are connected.
**10.** Identify the in-house experts and resource people. Ask everyone you talk with, "Who else should I get to know?" When the same names keep popping up, you will have found the key influencers. Call and arrange to meet. Ask your boss to contact these folks in advance of your call so you are never "calling cold."
### Assess Your Corporate Culture
If you've decided that you need to work on your network, begin by assessing your corporate culture. Is your organization network-friendly? To determine how supportive your workplace is, ask yourself these questions:
Do corporate initiatives mention relationship building in any way? If so, there is recognition at the top that building "social capital" is valuable.
Is training offered? You can suggest networking workshops or suggest that the skills be embedded in existing leadership or employee development training.
Are you encouraged to belong to professional associations and to attend both monthly meetings and conferences?
Are you encouraged to volunteer in the community, serve on boards, etc.?
Is it easy—and expected—for you to collaborate with people in other departments—to venture out into the white space on the organizational chart?
How much money will your organization spend on professional association dues and conferences for you? Collateral expenses, such as travel, lodging, etc.? Is anyone tracking whether the organization is getting its money's worth?
Are networking activities/goals included in your annual performance plan?
Are you rewarded when your networking contributes to the success of the organization?
Recognize that in some organizations, networking violates the cultural ground rules. If that's your assessment, talk with your boss and your colleagues about the reasons for networking inside your organization. Use ideas from this chapter to convince them that networking at work pays off—for the organization and for the individual.
Some forward-thinking organizations are deliberately working to create a more collaborative culture by setting up mentoring programs, encouraging the formation of communities of practice, sponsoring women's networks and other affinity groups, and providing ways for people to interview others to discuss lateral moves and opportunities for upward mobility.
Even if you've decided that your organization's culture isn't very network-friendly, you'll still find ideas in this chapter that will work for you. Focus not on self-serving objectives, but on serving internal and external customers, streamlining internal processes, getting the job done, and impacting the bottom line.
### How Strong Is Your Inside Network?
Use this quiz to rate the strength of your current inside network.
**1.** Do you know people at all levels of the organization? Do they know your name and what you do?
**2.** Do you know all the people whose work intersects yours in any way?
**3.** Do you know people who have jobs you might like to have some day?
**4.** Are you involved in any cross-functional efforts or interdepartmental activities (e.g., temporary assignments, committees, task forces, special projects, volunteer activities)?
**5.** Are you plugged into the grapevine? Do you find out what's up before your boss tells you?
**6.** Do you take every opportunity to meet face-to-face to define and discuss complex problems, shifting priorities, areas of responsibility?
**7.** Do you know and talk with others about tools to get the job done today and trends that will impact your job in the future?
**8.** Do you have effective internal channels through which to send information?
**9.** When you see a problem that involves people from various areas, do you take the initiative to bring people together to solve it?
**10.** Do you drop by to see people—even when you don't need anything?
Could you say "Yes" to most of those questions? If not, make building your inside network a priority.
### Map Out a Plan
Draw a map of your contacts. On a big piece of paper, draw a circle and put your name in it. Then draw circles with the names of everyone at work that you interact with. Add circles with the names of everyone you think you should interact with. If you don't know the name of one of these people, use his or her title or describe the type of job he or she has. For example, you may decide you want to know the editor of the corporate newsletter or someone in the IT department who can troubleshoot problems. You can find out those people's names later. Who else do you want or need to know in order to solve problems, contribute to the bottom line, uncork bottlenecks, and create career security? This map is your current and potential network at work.
On your map, rate each relationship:
Write **_E_** next to the name of any person with whom you believe you have a positive, mutually beneficial relationship.
Write **_R_** next to the name of any person with whom you believe you have a negative, unpleasant, or unproductive relationship.
Write _**S**_ next to the name of any person with whom you have no relationship or a neutral relationship.
(STOP: DO NOT READ ANY FURTHER UNTIL YOU HAVE MADE YOUR MAP!)
You may be amazed as you take a few minutes and analyze your map.
Where is your circle? At the center of the page? At the top? How big is your circle compared to the other circles? Do they vary in size? Did you connect all the other circles to your own with lines? Did you arrange the other people's circles in any way? You may gain some insights about the "distance" you perceive between you and other people by this analysis. What else can you discover about your network? Do you have a lot of **_S_** ratings? What does that tell you? Are there lots of circles without names?
### ENHANCE, START, or REPAIR your relationships.
Next create your strategic plan to increase the breadth of your network, the strength of your network, and the effectiveness of your network.
Using your map, make a list of everyone who has a **_E_** by his or her name. Begin with this group. The next time you get together with one of these people, take the opportunity to mention the way you have worked together—the problems you've solved, the processes you've improved, the projects you've collaborated on. Remind the person of milestones in the history of your relationship—how you met, what you have done for each other, your successes—even your failures. Say how much you appreciate that person. Let him know you want to continue your relationship and to **ENHANCE** it. Verbalize your trust in that person and your reliance on that person's expertise. Ask for his or her help and offer your help. You don't have to be specific. Say, "I know I can count on you to brainstorm with me when I need a good idea. You know you can always call on me for help." Use this meeting to confirm your relationship. Resolve to consistently be on the lookout for ways to contribute to that person's success and to stay in touch on a regular basis—even when you don't need anything. You might want to suggest that you have lunch once a month, for example.
Next, make a list of everyone who has a **S** by his or her name. Choose someone on the list and come up with a concrete reason to **START** getting together. You might say, for example, "I'd like to work with you to streamline this process. I'd like to explore how we could cut several days out of the processing time." Or, if you are thinking about your career, you might say, "Long-term, I'd like to make a move from my staff position to one that has more impact on the bottom line. I'd like to know how you did it." Remember that to get, you must give. As you become acquainted, listen for that person's workplace goals and challenges so that you can contribute. Listen Generously so that you can give more than you receive. Take every opportunity to be helpful. Understand that you will need to meet, face-to-face, with the person six to eight times to build a mutually trusting and beneficial relationship. Trust is built on an appreciation of the other person's Character and Competence. Put those meetings on your calendar over a period of six months or so. Supplement those meetings with e-mail and phone calls.
Finally, ask yourself what's going on with the relationships you rated **_R_.** If one of these relationships involves a person you absolutely must work effectively with, make it priority to **REPAIR** that situation. "But," you may be saying to yourself, "You don't know Joe—he's impossible!" Or, "Nobody could work with Kayla!" It's true. Sometimes you must deal with difficult people. And sometimes, sticky situations and shrinking resources can make even the nicest people hard to get along with. If that's the case, see the situation as an opportunity to develop your own influencing skills. Get a copy of _The Empowered Manager: Positive Political Skills at Work_ , by Peter Block. Learn his system for cultivating Allies and for limiting the negative impact of Adversaries, Opponents, Bedfellows and Fence-Sitters. Block would suggest meeting with the problem person and saying something like this: "In the past, we've had conflicts. I'd like to change that, so that when we work together it's positive and productive. Let's talk about how we can overcome the past and start fresh." Suggest "ground rules" for how you'll work together in the future. You might say, "When you see a problem, come to me first, rather than talking to Sue and Bill and Wang." Or, "If your people are running behind by more than thirty-six hours, let me know so we can adjust things here." Over time, you may or may not be able to improve the relationship or even get the project back on track, but your sense of confidence and your reputation as a relationship builder will grow. If you have tried to change the nature of your interactions without success, give it up as a lost cause and spend your time and energy connecting with someone who is more amenable to working with you.
### Pair Up with Peers
Besides networking with the people your job intersects with, it's valuable to exchange information with your peers, people at your own level throughout your organization. Jerry, who worked in the corporate planning department, happened to sit down at the same lunch table with Marcia, who was the corporate speechwriter. He told her about his involvement with the school district where he lived. She said she'd just written a speech for the chairman of the board on education and knew that he was very interested in finding a way to support projects like the one Jerry was working on. She suggested that Jerry talk with the chairman about corporate funding for the project. Jerry did and was tapped to head up the corporation's efforts with educational institutions.
Your peers can provide support for you outside your own work group. They can give you information that is vital to your career. And they can increase your visibility in the organization.
### Avoid Erroneous Assumptions
Know the rules. Here are eight erroneous assumptions to avoid as you network at work.
**1.** "People I work with are automatically part of my network." Not true. You must create and nurture the relationships.
**2.** "Everyone is an equally good networking contact." Not true. Seek out the experts, the influencers, and people who will give back. As you talk with people in your network, agree to respond quickly to their requests.
**3.** "It's his job to give me information. I shouldn't have to 'make nice' to get it." Not true. You'll get better help faster when you are obviously willing to help. Listen Generously to your contact. Does that person need something you can supply? If you can't discover anything, ask, "How can I help you?"
**4.** "Since we work at the same organization, I can access anyone." Not true. The "best" contacts are busy people. Use referrals, references, and introductions by a third party. And become known for the people you connect.
**5.** "I don't have to prepare like I would for an external contact, since we're colleagues." Not true. Don't waste your contact's time. Before you ask for something, do as much study or research as possible. That will provide you with some basic information. Jot down what you've found as you've tried to solve the problem or find the out-of-the-ordinary information. Note any blind alleys you discovered. Form your question(s) carefully. Make your quest interesting and intriguing for your contact. Link your question or need to something of interest to your contact. That way, you're not just asking for a handout. Is there a payoff for him or her? Try to find one. If you can't, be sure you volunteer to be helpful.
**6.** "My request is so important that my contact will drop everything to answer it yesterday." Not true. Make sure you give your contact enough time. If you need something, don't procrastinate. Ask early, before you are desperate.
**7.** "When I receive the information, the interaction is over." Not true. Get back in touch to tell your contact "the rest of the story," and what use you made of what he or she gave you.
**8.** "I said, 'Thanks.' That should be enough." Not true. Size your "Thank you" to match the size of the favor. Send a handwritten note. Take your contact out to lunch. Send a funny card. Write a note to your contact's boss. Take every opportunity to give credit publicly.
### Overcome the Barriers
Yes, there are barriers in most organizations. Old hierarchies and ways of doing things linger on, even after executive pronouncements that give employees "permission" to network. And you may have some beliefs that make networking challenging for you. Here are some questions—and answers—that may help.
**Q:** | "I do a good job. Shouldn't my work stand on its own? Do I really have to 'promote' myself?"
---|---
**A:** | In an ideal world, your work would be noticed and appreciated. But in the real world, you must make your good work visible. It's often true that people who are promoted most often and get the biggest salary increases are not necessarily the most technically competent, but are those who are willing to make their competence visible.
**Q:** | "I don't think networking is 'the thing to do' here. But I can see the benefits. Any ideas for a first step?"
**A:** | Talk with your coworkers in the department and, when you are sure you have their support, enlist your boss in a low-risk, high-payoff activity, like a brown-bag lunch with another department. When the corporate communications department at a telecommunications company invited the human resources department to lunch, it was the beginning of a rich collaboration. As people got to know each other, they integrated their strategic planning so that an HR request for the production of a training calendar was on the corporate communications department's annual schedule. If you collaborate, you can negotiate to even out the workload, so all the projects don't hit at the same time.
**Q:** | "I'm okay networking with peers, but I freeze up when I have to talk to the higher-ups. How can I be more comfortable?"
**A:** | Get in touch with your Agenda. Is it something you would not want as a headline in tomorrow's newspaper? "I want to snow this person. I want him to like me better than my colleague, so that I'll get a promotion." That's a hidden Agenda. There's a small, but significant difference in an upfront Agenda you could legitimately make conversation from. Say, "I want to clearly demonstrate in this conversation the level of my expertise about this subject to my boss," or "I want my fundraising skills to be visible, so I'm a natural candidate for the new venture."
**Q:** | "Won't talking with higher ups be seen as bootlicking or grandstanding?"
**A:** | If you sincerely and openly network with everyone in your organization—your work group, your subordinates, and your peers, not just people above you in the organization—you will feel more comfortable. That kind of openness will become your way of operating in the organizational environment. Remember that people above you are often eager for "news," especially good news, from the people like you who are on the front lines.
**Q:** | "What if I run into an executive and he asks me, 'What's going on in your area?'"
**A:** | Always have a Success Story to tell about yourself or your team. See Chapter 11 for how to create Success Stories.
**Q:** | "I'm new. I don't know anybody. Where should I start?"
**A:** | Take a tip from Susan, who says, "When I joined the company, I made up my own orientation program. Every day in the cafeteria, I found someone sitting alone and asked, 'May I join you?' then I asked 'What do you do here?' I was in charge of library and information services, and, in order to serve the organization well, I decided I needed to know a key person in each department. Every time I hire people I tell them, 'Your job is to represent the library, so get to know people.' I taught each of them my 'cafeteria method.' Now, I tell each of the 26 people I supervise to develop their own contact circle, 'Your group will be different from mine, and that way we'll know what the people we serve need—sometimes even before they know they need it.' Networking internally helps us form cross-functional teams and gather the intelligence that makes our department a catalyst for change."
### Bonus: After Organizational Earthquakes, Rebuild Your Network
Reorganizations, mergers, acquisitions, and layoffs can destroy the relationships you've carefully built within your organization. You may feel like hunkering down and hiding. Don't! Reconstruct your network, using these ideas:
* Acknowledge that "the good old days" are gone forever, then—quickly—commit yourself wholeheartedly to the new enterprise.
* Focus up. Let your boss know you are ready to help to create the new order.
* Bring your subordinates on board. Meet one-on-one for lunch. Or get together after work. Create a new sense of camaraderie and adventure. Recommit to doing a good job. Look forward, not backward.
* Reach out to peers you relate with on the job—anyone whose job intersects yours.
* Escape your cubicle and build new bridges with related departments, functions, and divisions.
* Volunteer for cross-departmental teams and activities.
* Spend time shoring up relationships with customers, clients, suppliers, and vendors. (If they believe your company is in chaos, they may defect. Be positive. Also remember, these networking contacts could provide your next job.)
* Find ways to relate to competitors, perhaps through industry associations. Always be upbeat and positive about what's been going on in your organization. Boost your company's reputation, every chance you get. Remember, your next job could come from your current competitor.
## CHAPTER 15
## Make It Rain Clients
If you're a lawyer, CPA, or doctor, what's the word that's most abhorrent to you? How about "sales?" When you hear it, do you grimace? Flinch? Cringe? Blanch? The word "marketing" is only slightly less upsetting. So, you call it something else—"rainmaking."
Whatever you call it, you probably know that, in today's marketplace, even professionals like you have to sell. You must bring clients in the door, or you will find yourself on the way out!
"But," you plead, "Isn't there a _professional_ way to do it?"
Law firms need "finders, minders, and grinders." Minders are people who handle clients; grinders grind out the work; and finders are rainmakers. All three are valuable, but in most firms everyone must market.
People who take active roles in professional associations find that they are a comfortable place to sell. The loss in billable hours is more than offset by these benefits: credibility for their firms; visibility that enhances their firms' reputations; referrals from other professionals; access to key people in business, government, and legal circles; and, of course, rainmaking.
You can come up with other comfortable ways to sell. Successful rainmakers know how to create networks from which referrals flow. In this chapter, you'll find out what people in professional services _really_ think about networking. You'll discover new ways to develop your practice. You'll learn how to strategize and set up a professional Relationship Management Program for your firm. And you'll get advice on how to create the most powerful networking process of all—the Constellation.
### What People Think
In a group of CPAs that included twenty-five-year veterans as well as those new to the profession, people voiced these concerns about networking:
* "I have so little free time I'm reluctant to spend it doing something I dislike. How can I get over my reluctance and fit networking into my already-too-busy life?"
* "Don't you have to spend a lot of time at this before you get any rewards?"
* "Do I begin talking about business or do I try small talk first?"
* "I tend to seek out people I know rather than new faces. How can I find ways to split my time and meet new people?"
* "Not only do I want to remember other people's names, I also want them to remember mine. How can I make that happen?"
* "How subtle or direct should I be about marketing my services when I meet someone for the first time?"
* "How do I get to the point of making reciprocal referrals?"
* "Can I afford to go to networking events or will I miss too much billable time?"
* "How can I follow up to develop my practice?"
* "How can I tactfully use social and business occasions to 'sell' my services?"
* "How can I find the 'movers and shakers' at a meeting?"
Most of these concerns aren't unique to professionals like you. But there are three things that make networking especially hard for you: your lack of time, your ambivalence about accepting client development as part of your role, and your conviction that you shouldn't have to sell yourself. Here's how to streamline the process and get more comfortable at it.
### Professionalize Your Practice Development
Take a strategic view. Set up a Relationship Management Program to provide a structured, customized blueprint for your activities. Almost anyone can initiate this planning session. The benefits of careful planning include:
* Constant referrals to qualified clients
* Additional business from current clients
* Access to inside information on business trends and resources
* Higher visibility in your field and in the community
To make networking part of your overall business plan, assess and target, initiate and connect, track and measure, and renew and reassess.
**_Assess and target._** Where have your current clients come from? Assess your current client base to determine how you found each other. Is there a pattern you can build on? Where are your potential clients? Which organizations and circles of influence will logically be your best targets?
Kent's architecture firm specializes in designing schools. He and his partners, therefore, should focus their networking efforts on organizations that serve school administrators. Although it makes sense to use this tactic, the number of businesspeople who actually seek out and become active in organizations that serve their clients is surprisingly small. Put your energy where your potential clients are.
### You can do more than hope that your networking efforts will pay off. You can make it happen.
Which organizations are members of your firm currently affiliated with? Can they justify their memberships or did they just join without thinking about who they need to meet? Can they explain exactly how those memberships put them in touch with the right people? How much time are they spending each month? What is the cost of belonging to each organization? Can they document the return on their investment? Use the chart in Chapter 5 to assess various Arenas.
Does it make more sense to target individuals rather than join groups? If so, which individuals would provide the largest numbers of solid leads and referrals? How can you find these people?
What amount of time can and should people be spending on client development? If the primary goal is billable hours, how can the firm encourage and reward people for taking the time to develop networking relationships?
**_Initiate and connect._** Once you have determined your targets and identified organizations and circles that you should be more active in, how are you going to deploy the members of your firm?
Pair up organizations and colleagues whose backgrounds or interests match. Set goals for the next six months and create networking Projects to achieve them. Use ideas in this book to get up to speed quickly and to become active and visible.
**_Track and measure._** Devise a system for tracking leads and quantifying your efforts. Marshall was surprised to discover that twelve of fifteen new clients for his advertising firm came from one client services team. What were they doing? They had adopted a structured, pre-planned method for always asking for referrals.
**_Renew and reassess._** Determine how you will stay in touch with prospects, referral sources, and past clients. How will you update them on new services and remind them of your expertise? What contact management system will you use? How long will you stay active in an organization if you are not getting any referrals through your contacts there?
Consider these questions as you plan:
**1.** What strategic goals and plans in our organization make Relationship Management a priority now?
**2.** Which relationships have brought us the best information and referrals in the past? What has been the pattern of development in those relationships?
**3.** What kinds of relationships do we need to stay informed in our field today? Who do we already know? Who do we need to know? Where will we meet these people?
**4.** What contacts do we already have who can provide introductions that will establish maximum credibility?
**5.** What behaviors tend to build trust with key contacts? What is the best protocol for initiating relationships? How can we best exhibit our Competence and Character to contacts?
**6.** What skills do we need to cultivate business connections? How are we going to train people over time? Do we need a formal training program to help people recognize the many ChoicePoints in their networking activities? Should we establish a mentoring program?
**7.** What do people have to offer individually, as they establish mutually beneficial business connections? Is each individual prepared to "sell" his or her own specialty and to explain why that service is superior?
**8.** Are people prepared to "cross sell," to pass prospects and referrals to others in the firm? What process do we have or could we develop to keep people informed about each other's areas of expertise?
**9.** What are the plans of each individual in our organization for making contacts in a variety of Arenas?
**10.** What systems are in place to support members of our firm as they go forward? Are there ways to reward people for their efforts? Are we sending a conflicting message when we emphasize billable hours and also encourage networking?
### Make Conversations Count
Here are some tips to help you get the most from your networking activities.
Often, people ask questions of professionals at networking events that are requests for professional guidance. These folks may not mean to ask for a free consultation; they may just be trying to make conversation with you on a topic you might find interesting. You'll want to develop a clear, but clever way to encourage your conversation partner to make an appointment.
Work on your answer to the question, "What do you do?" When you reply with your profession ("I'm an attorney"), you make your conversation partner do all the work. Instead, come up with a BEST/TEST answer (see Chapter 9) that gets the conversation going. Rose doesn't even mention that she's a CPA in her answer. She says, "We work with passionate entrepreneurs, helping them grow their businesses." Or, "We help people increase their net worth. We showed a client that formed as an LLC how they are overpaying their income tax." Be sure your example highlights a part of your practice that you want to expand and not something you are lukewarm about doing.
### Networking can help you turn contacts into clients.
To let people know about your specialty, have interesting information that you are comfortable giving away. Eldon works with businesspeople when they want to move from sole proprietorships to corporations. He has amassed a lot of useful information for business owners who are feeling overwhelmed as their businesses expand overnight, and often suggests that they join an executive roundtable group, so they can get advice from other entrepreneurs.
Listen with an ear for problems. When Conrad mentioned his father's illness, Kira introduced him to a counselor who dealt with succession planning for family businesses. The counselor later sent the family back to Kira, who handled the legal issues.
Don't define your role too narrowly. Marcie used to say, "I am a business analyst and management consultant, who specializes in setting up flex-time programs." She realized that potential clients often didn't appreciate how much impact flex-time programs could have on employee turnover. So, now she says, "I help companies retain employees." She finds that talking about the problem, rather than the solution brings her more opportunities.
Tell Success Stories. (See Chapter 11.) One way to memorably describe your problem-solving capabilities is to tell stories. Respecting client confidentiality, you can disguise the particulars and tell how you solved a problem, or saved someone money, or increased productivity or profits. Make your stories brief and dynamic.
When someone asked Mark, a CPA, "What's new?" he said, "I just saved a client, whose business almost tanked last year, a ton of money by convincing him to move from his rented office back home."
### What's One Conversation Worth?
Often, people who market their services can determine the value of a single conversation. Joyce, a financial consultant, met a man who asked her, "What would you do if you had $600,000 to invest?" Her answer obviously impressed him because he did ask her to manage his portfolio. She figures she made $18,000 from that one conversation.
When Chuck, who teaches problem-solving in corporations, took his seat on an airplane, he was delighted to find himself next to a person who had just been promoted to head up the creativity department at a large chemical company. Unfortunately, that man was exhausted, and Chuck was able to have only a short conversation with him before the man fell asleep and slept through the entire flight. However, as Chuck boarded the rental car van, another man caught up with him. "Excuse me for eavesdropping," he said, "but I was sitting behind you and I'm very interested in the kind of creativity training you were talking about." Chuck has provided $48,000 worth of seminars for that roundabout contact.
### Create Constellations
Your strongest referrals can come from other professionals with whom you share clients. Put together your own referral group—your Constellation—by forming alliances with top-notch professionals in other fields who have access to your market. One group is made up of a lawyer, a CPA, an interior designer who concentrates on the senior citizen market, and a person who runs estate sales. When a widow decides to move into a retirement apartment, the estate sale person can refer her to the interior designer. Either the estate sale person or the interior designer can refer her to the CPA, who can refer her to the lawyer. The referral cycle can begin with any member of the group so that a client for one of the members soon becomes a client for all of the members. (See Chapter 16 for more on creating a Constellation and Chapter 18 for more on referral groups.)
George, a Chicago CPA, has been developing Constellations for many years. "The key to business is obtaining new clients," he says. "The problem is to do it professionally. Networking is the solution."
As he helps other professionals grow their practices, he reaps the benefits. He cultivates relationships with bankers, lawyers, insurance agents, and stockbrokers—all people who can refer business his way. "Once I have identified potential candidates to link up with, allocating the time to get to know them is the biggest challenge." He goes at it systematically, setting up a series of meetings. He meets with people over lunch or dinner, visits them in their offices, and invites them to his. "I don't go and say, 'I need you to give me new business.' I say 'Here's what I can do for you and, hopefully, you'll be in position to pay back the favor.'
### Create customer-common alliances with other businesspeople. Refer your customers to your allies, and get referrals in return.
"I do a selling job on myself and a fact-finding job on the other person," he says. He finds out exactly what the person does so he can qualify contacts for him and refer the right clients. "I make sure if a contact needs a banker, I send him to the right one. You need to know specialists in various areas, but I cultivate only a handful of 'partners.' I want to keep them happy. I couldn't keep eighteen bankers happy."
Actually, George doesn't _send_ any clients to his 'partners.' His trademark is that he _hand-delivers_ the clients. He calls the banker, makes an appointment, and personally takes his client to the banker's office.
He's trained all seventy-five professionals in his firm to use the same practice development method. When he recruits on campuses for new accountants, he makes sure they are marketing-oriented. As new grads join the firm, George trains them to build relationships with people in their own age bracket, creating networks that will grow in influence and power through the years as their careers advance.
### Cross-Sell Your Clients
Can you sell more services to existing clients? The cost of getting new business far exceeds the cost of getting more business from clients you already serve. That's why internal referrals are so important in professional services firms.
Conrad, a CPA, teaches the client services teams in his firm to handle their engagements so that they listen for other organizational problems his firm might help to solve. His teams are trained to pick up on any needs they hear about. "You say the staff needs training on this new accounting software? Our training department can do that for you. I'll set up a meeting." "You say the CEO is having a business planning retreat for all the top staff? Our management consultants can facilitate your strategic planning session. Let me call Kathryn, and we'll get our people ready to make a proposal to your CEO." Conrad's people cross-sell the firm's capabilities constantly.
### Make Asking for Referrals a Ritual
If you don't get referrals, perhaps you haven't established a routine for asking for them. Make asking an integral part of what you do with every client. You'd never fail to mention billing procedures, would you? Asking for referrals should be even more central.
Bill Cates, dubbed the "Champion of Referral Selling," suggests "foreshadowing" as a way to let the client know that you'll eventually ask. Early in the relationship, often even before someone has become a client, it's possible to "foreshadow" with comments like, "Since my business is built on referrals..." or "Sam was referred to me by Janna at Kidder, Wilson, and Smith."
Teach everyone in your firm the following eight steps so that they know how to ask clients or their business contacts for referrals.
**1. _Recall your track record._** Encourage your client or contact to remember what you did for him. Ask, "What do you particularly appreciate about the way we worked with you, handled your project, managed your engagement, pursued your case?"
**2. _Remind clients that you count on referrals._** Encourage clients to become part of your referral system. Say, "You probably remember my saying that about 60 percent of our business comes from referrals from satisfied clients like you."
**3. _Review their circles._** Help clients or contacts think of people who might be ready for your services. Say, "I know you're on the board of the Country Club. Are there others on the board who are also ready to do some serious financial planning?" Or, "I was so pleased when you said your partners complimented you about the new addition we designed for your house. Are any of them ready to look at some photos of other work we've done, and talk with me about what they'd like to do to update their homes?" Or, "Which new members of the Chamber of Commerce do you recommend I contact to let them know about our firm's accounting services?"
**4. _Receive specifics._** Ask for any specific information that will help make your first contact successful. "Let me be sure I've got Lisa's last name spelled correctly. Does she have a cell phone?" "What is a good time to reach Paul? At home or at work?" "Which of our services do you think Martina would find most interesting right now?" "Why do you think Ron would be interested at this time?"
**5. _Raise the possibility of success._** Encourage clients and contacts to pave the way. Say, "It would be really helpful to me if you'd give Mary a call to let her know I'll be getting in touch. Naturally, I'd appreciate your telling her how satisfied you were with my work." Or, Would you be willing to send Sacha a note telling her I'm going to call?" Or, "How about if the three of us get together for breakfast Friday? That way you can introduce us, and Fred can hear firsthand about what we've done for you."
**6. _Rapidly make contact._** If your client gives you a referral, follow up quickly. If you don't, it could be embarrassing to you and your client.
**7. _Recount the results._** Get back to your client with your thanks. Give appreciation, whether the contact resulted in business or not. Closing the loop and letting your client know what happened is just common courtesy. In some types of referral relationships, you may have agreed to pay a referral fee—a percentage of the dollar amount of the business. If you made that kind of agreement, be sure the check goes out quickly.
**8. _Reciprocate._** Return the favor. Be on the lookout for ways you can help your client succeed. Provide resources, information, or referrals to the client or contact who gave you the referral.
Go ahead, make it rain.
## CHAPTER 16
## (Net)Work from Home
Are you part of the home-based business revolution—the growing segment of the working population attracted by the 30-second commute, the flexible hours, the tax breaks, the idea of being your own boss, and a sky's-the-limit income potential? The _Wall Street Journal_ reports that there are now 30 million home-based businesses, and that 8,000 new ones are starting up each week.
### Tune In to the Trends
Three trends are responsible for the surge in stay-at-home workers. First, mergers and corporate downsizing continue to "free up the futures" of many talented, experienced people. Second, technologies necessary to set up an office at home are widely available, relatively inexpensive, getting easier for people to use. Third, the "sandwich generation" has a need for more flexible schedules as they juggle child care and elder care.
According to _Success Magazine_ , the top ten home-based businesses include business consulting and services, computer services and programming, financial consulting and services, marketing, advertising and public relations, medical practices and services, graphics and visual arts, security, real estate, writing, and independent sales. And, of course, there are many franchises that work well as home-based businesses, such as Computer Tots, Decorating Den, and Molly Maids. Some types of franchises start off as home-based businesses, then later experience such growth that they move to commercial space.
### Conquer the Challenges
Two challenges arise for people who are (net)working from home.
If you're someone who thrived on the camaraderie of belonging to a large organization and having lots of coworkers to kibitz with, working solo may seem lonely. You may spend too much time attending events to get the contact you crave. If that's you, make a plan for your networking activity. Make good decisions about which events and which groups will give you the best returns. Plan your calendar accordingly.
On the other hand, if you like working alone, you might get too comfortable "cocooning." Do you hate to put on that suit and head out to the luncheon meeting? Are you too busy to volunteer for the hospital fundraiser? Are you so tied to your terminal that you're reluctant to meet new people and to reconnect with people you know? Don't forget that business success depends on networking for new clients or customers and on gathering the latest business intelligence so you stay at the cutting edge.
Do you have a goal that is attainable through networking? Review Chapter 5. Decide why you are networking. Create your Project. Then, finding the time to network will make sense. Schedule your networking just like you schedule your other business activities. Use every social and business event as a time to explore trends that will affect your business (that new zoning law), find resources (that space to hold client focus group meetings), and tell people about your successes (that appointment to the governor's small business advisory committee). Proactively look for and give leads and referrals.
Successful home-based businesspeople use networking to find customers, suppliers, distributors, lenders, investors, joint-venture partners, other kinds of partners, and mentors. In the process, you will strike up valuable connections that will fuel your business growth for years to come.
When Jeff wanted to explore the idea of franchising his business, he thought of Al, a lawyer he had grown to trust and respect as he worked on committees at the Board of Trade. From her contacts at the Northern Virginia Technology Council, Sau Ching learned how to prepare a package to send to potential investors when she was ready to expand her business importing dolls. She might have followed guidelines in a book or from an Internet article, but her chance to talk over the fine points with other businesspeople who'd been through the venture capital process saved her precious time and ensured her success.
Many entrepreneurs find themselves networking as they go about their day-to-day tasks. Think of how often you reach out to shake hands and say hello. Now, think of what would happen if a good percentage of those encounters generated sales leads or productive new ideas. Sound far-fetched? It shouldn't—not if you're as prepared to take advantage of chance meetings as you are of regular business appointments.
### People want to do business with people they trust.
Lee's business was making costumes. She needed a source for unusual trimmings. She found one, sitting next to her on a flight to Louisville.
Remember, you're not networking until the people you meet know your name, understand what type of customers or clients you want, trust your Character and Competence, and believe you'll reward them for their efforts on your behalf.
In this chapter, you'll find tips for making networking from home work for you. Do refer to other chapters for additional ideas before you finalize your networking business plan.
### Link Up Your Life and Your Livelihood
_Make linking your life._ Instead of thinking of networking as something else to put on your already crowded to-do list, see it as life itself. Everywhere you go, from back-yard barbecues to trade shows, you meet people. Be sure that you can answer the questions, "What do you do?" and "What's new?" in ways that teach people about your service or product. Be prepared with up-to-date examples of your recent successes, projects, and clients. The more you tell, the more you sell.
Linda owns Bed and Biscuit, an inn for pets. She always has a new story that reminds people of the tender loving care she lavishes on her animal guests. "I'm boarding a llama this week," she reports, "and she's so happy when I groom her that she stands very still and breathes loudly through her nose." Or, "Mattie, the German Shepherd, had puppies with me while her owners were on vacation. They were so happy with our newborn care that they mentioned us in their Christmas letter!"
**_Join—or start—a referral group._** A referral group or Constellation can bring you plenty of business. Follow the guidelines in the Bonus at the end of this chapter. Emily, of Discovery Toys, started her own group of six women who all live in her neighborhood. The strength of their group is its diversity. In addition to Emily, the group includes a musician who plays for private parties, a graphic artist, a nanny-finder, an image consultant, and an interior designer. They used each other's services and bought each other's products to get acquainted. The classical guitarist had a session with the image consultant and came up with a new look that got her noticed and added flair to her performance. The nanny-finder hosted a party for all her nannies at which Emily showed her toys.
### Create customer-common Constellations.
Bart, an insurance agent, teamed up with an attorney and a financial advisor. Reva, owner of Hire a Handywoman, a home fix-it service, belongs to a group that has thirty-eight members, all in different businesses. She can track 20 percent of her income to that group—during her first year of membership.
**_Spiff up your image._** Act and appear in public as if you had an office downtown in the high rent district. Remember, you may meet people at the grocery store or library or copy center. So, wear business dress or at least business casual when you go out, not your oldest, grubbiest sweat suit—even if that's your favorite work outfit! Always have your business cards with you, even on a run to the post office.
Angie was in line at the post office waiting to charge up her postage meter when she ran into Loren, an independent sales rep doing the same errand. He sells women's handbags; Angie markets a local craftsman's line of purses and briefcases. A year later, they still laugh about how they met when Loren commented on her leather backpack. Loren now sells Angie's line.
**_Choose to stand out._** Select three to six Arenas and become visible in them. Pick one, such as a local home-based business association, for the business support you need from peers. Choose two populated by your clients or by people who can refer work to you. Susan, a caterer, became active in The Association of Wedding Professionals. Jaime, a remodeler, joined a real estate group, figuring those people might provide some good referrals to home-owners who needed to have a few things done before putting their houses on the market. Use your hobbies or interests to make contact. Evan, a computer consultant, counts seven new clients this year from contacts he made at his health club. Keep your family up-to-date about exactly what you are doing and what kinds of resources or clients you want to find.
### Make sure people hear about you before they hear from you.
Make sure people hear about you before they hear from you. Imagine meeting a prospect at a networking event who says, "Didn't I see an article by you in last month's Chamber newsletter?" Or, "I heard you interviewed on the Women's Resource Network radio show several weeks ago." Getting visible in organizations and in your community allows you to make a name for yourself and enhance your credibility with people who count—even before you meet.
**_Pick your not-so-prime-times to network._** Everybody's got high and low energy times and busy and not so busy times. Go with the flow. Mari's mornings are hectic, so she avoids breakfast meetings. Don is a morning person so he schedules meetings early in the day. Many home-based business people comment that they shy away from lunch events which can eat up three or even four hours in the middle of the day. Analyze your biorhythms, your work flow, and your family obligations and plan accordingly.
**_Accomplish your Agenda._** Decide before you leave the house what you want to accomplish. Create your Agenda every day and focus on it. Here are some sample Agenda items:
* "Meet at least two people on the Board of Directors and offer to be on a committee to raise my visibility in the organization."
* "Find out which temporary help agency others have used and liked."
* "Meet a divorce lawyer who might eventually send tax work my way."
* "Identify businesses large enough to hold catered events several times a year and invite them to my Tasting Party."
* "Meet people whose parents are considering moving into a retirement complex."
**_Reconnect and Follow Through._** When you return to your office from a networking event, go through any business cards you received and decide how to reconnect. Do this immediately, before you get caught up in the tasks on your desk. Ideally, you'll use one of the many contact management software packages on the market to keep track of people and stay in touch. In addition to putting phone numbers and addresses at your fingertips, your system will also alert you to Follow Through (e.g., "Call Fred March 1 about the annual dinner") and will make creating mailing labels a cinch.
**_Promote your business creatively._** Since you don't have a store front and don't have a big sign, what can you do to draw attention to your business? Gina, a watercolorist, teamed up with Gary, who owns a frame shop in a busy mall. They hosted a Saturday "sidewalk art show" to display his frames and her paintings. Candy's stuffed animal business, Bunny Rabbit Babies, hopped in the spring time, but hibernated in the fall and winter. She donated fifteen bunnies to the playroom in the children's wing of the hospital, which led to a contract with the hospital gift shop. That exposure led to another contract with a hotel gift shop. Those two contracts tripled her business in one year.
**_Holler "Help!"—and get it._** With only the cat and the philodendron to talk to, you may feel you are home alone. The truth is that, all across town, others just like you are wondering, "Am I on the right track?" "Do I have what it takes?" "How can I grow my business?" "Who can I talk to when I get down in the dumps?" The two best sources of support are other home-based business people like yourself and hired experts.
Get the support and encouragement you need by networking with others who work from home. Lynnette, who is in business for herself and by herself, already belonged to several networking groups of people who do training and development. But she wanted to talk over her business strategy and get support from people who were outside her own profession. So, she and three other entrepreneurs started The Presidents' Group. They meet once a month for an hour or so in a restaurant. First, each person has a fifteen-minute turn. Everybody gets a chance to tell about a recent accomplishment. For people who work alone, celebrating success with others is important. "At first, we were shy about sharing our accomplishments. I guess we all grew up being told not to brag," Lynnette says. "But we found that when we took credit for our successes, it was easier to tackle the problems." Next, people take turns asking for help or feedback from the group on one issue or challenge. This problem can be anything from "How can I market my services so that I have more work in December?" to "How do you like this design for my new business card?" In the group, Lynnette says, "The friendships deepened as our business savvy increased." Four years later, the group is still meeting.
Use your network to hire experts. Ask around for an accountant who specializes in home-based businesses, or for a graphic artist who can give you the look you want in your next brochure. You can't do it all yourself. Often, those professionals you hire will turn out to be some of your best referral sources. After all, they know exactly what you do. And they know that, when your business grows, so will theirs.
### Bonus: Create a Constellation
Some of the most mutually beneficial contacts are the people who share customers. Put together your own group—a Constellation of people who have customers in common with you. Form alliances with top-notch professionals in other fields who have access to your market. For example, a financial advisor invited an insurance broker, a realtor, a CPA, and a lawyer to be part of her group. A client for one of the members often becomes a client for all of the members.
Here's how to start your Constellation:
Notice what other businesses your clients use most often.
Identify five or six potential partners.
Spend time with them individually. Teach your Character and Competence and learn about theirs. You must be able to trust that they will treat your valuable clients well before you invite them to join your Constellation.
Find out exactly what your potential partners do so you can qualify clients for them and refer the right ones to them. Teach them exactly what kinds of clients you are looking for, so they can do the same for you.
Whenever possible, plan a face-to-face meeting to introduce your Constellation member to the client you're referring. When that's not possible, do all you can to warm up your colleague's first call to the client.
Meet regularly so that you can keep up-to-date on new services and new successes. A fourth Friday lunch works well for many groups.
Sandra, a marketing consultant, created The Marketing Consortium. "Think of it as a six-leaf clover," she suggests, "with me in the middle and six related companies on the leaves." These companies do media buying, sales promotions, direct marketing, sales training, new business development, and strategic planning. Sandra's company provides creative support with ads and brochures. She always asks her clients if they need anything her "leaves" could provide. "I don't get a percentage; I just expect business in return."
Constellations are one very targeted way to create your network. You don't need to know lots of people; you need to know the right people well.
## CHAPTER 17
## Make the Most of Your Memberships
There are two ways to create your network: link up with people individually and join organizations. Some of your best networking contacts will come from one-on-one relationships—your next door neighbor, your accountant, your cousin in Tulsa. You find these people, or they find you, outside any formal organization. You'll also benefit from becoming involved with several groups if you select them carefully, keeping in mind the time and money you have to spend and your career and business goals.
### Size Your Network to Fit Your Needs
There is no magic number for the size of your network. If fifty people think of you when a certain kind of expertise is needed, speak well of you to their colleagues, and consider you a source of advice and information, you may have an excellent network—depending on your current situation and goals.
If, however, you fear being laid off, want to start your own business someday, have a business that could profit from having more customers or clients, or are facing a major life change (such as moving across the country, or leaving the military), then you need to think bigger.
We recommend that you become active in six networking Arenas. That may sound like a lot, but consider that you are probably already a member of a family, a religious organization, a professional or trade group, a leisure-time activity (hobby group, health club, etc), and you may have "kid" connections. Those groups add up to five Arenas. Think of all of those venues as networking opportunities.
### Link Up One-on-One
Your personal network will include people from many different sources. You may forge strong business alliances with:
* Coworkers, past and current.
* Bosses, past and current.
* Neighbors.
* Past college professors or continuing education instructors.
* Relatives.
* People who provide services to you (your doctor, dentist, office supply store proprietor, etc.).
Customize your one-on-one network by being on the lookout for chance encounters. See every accidental meeting as an opportunity to explore new ideas, find out about new marketplaces, or just enjoy the serendipitous connections that life sends your way.
Rita, a CPA, stops every morning for coffee. She made it a point to learn the names of the people who serve her and chat about work—theirs and hers. One day, as she was waiting in line, the clerk asked the customer in front of her, "How are you doing today?" The man said, "Terrible! I've got to find a new accountant. Mine moved to Chicago." The clerk pointed at Rita and said, "There's one right behind you! Rita, meet Alonzo."
### Access Anybody
Regardless of whoever you need to know, somebody you know most likely knows somebody who knows them. This small-world phenomenon was studied in-depth by the late Stanley Milgram, a psychologist. What he discovered proves that it's possible for you to make contact with just about anyone you wish—because you have friends who have friends who have friends.
Milgram wondered if it would be possible for a package to be passed from a specific, randomly selected person in Omaha to a specific, randomly selected person in Boston using "friends" (people you know by their first names) as the conduits. He discovered that not only was it possible, but it took surprisingly few people—typically only five or six—for two individuals half a continent apart to make contact through a chain of acquaintances.
As Milgram explained it, if you know just fifty people on a first-name basis, and they know fifty people, you have access to 2,500 contacts. If that group each knows fifty people, you could potentially reach 125,000 people. And if they each know fifty people you could reach more than six million contacts.
When Annette, who lives in Kansas City, was attending the international conference of her professional association in Montreal, she went to a reception honoring international delegates. In the corner of the room sat a woman wearing a sari. Because her sister Maureen, who lives in Washington, D.C., had just adopted a baby from India, Annette went over and introduced herself. She said, "I have a new nephew who was born in India. "The woman introduced herself, saying, "I'm Cerena from Bombay." Amazingly, Cerena's mother had been Annette's new nephew's foster mother in India.
You may hear people talk about various networking "generations." Your "first generation" network includes the people you know directly. Your "second generation" network includes the people known by the people you know. The "third generation" includes their contacts. And the "fourth generation"—the six-plus million pool—includes their contacts. It is possible to relate to "fourth generation" contacts, but only if you've been passed along by people who have established a great deal of trust.
Michael, who works for a social justice non-profit organization, points out the importance of asking people you know for the resources you need. "What I need is often two or three links down the chain. So I put out the word. Networking at the beginning of the project for almost any initiative I put together is one way to insure that it will turn out to be effective with our constituents."
Of course, the range and quality of your contacts will dramatically affect your ability to use the small-world phenomenon to find resources and opportunities. Great networkers know people who know people who know people in a variety of Arenas—organizations, subcultures, marketplaces, groups, and niches.
Take MaryLou in Philadelphia, for instance. Her new boss Shelia moved to town from Seattle. Shelia's hobby is collecting antique inkwells. "How obscure," thinks MaryLou, who knows nothing about inkwells or antiques. But, using her network, she discovers that a friend of a friend is immersed in the antique subculture of Pennsylvania. MaryLou is able to connect Shelia with the people who know the best shows, the best appraisers, and the best dealers in Pennsylvania. Shelia is thrilled and sees that MaryLou is resourceful and creative.
The possibilities are limitless. But your time and money are not. So that's why great networkers don't just count on one-on-one connections. They join groups to put them in contact with large numbers of people.
### Join Groups
Joining organizations is the best way to build relationships with a multitude of people and expand your personal and professional network by creating instant Associates. For any given interest, job type, industry, or business, you'll have many possible groups to choose from.
Jon is an architect in Easton, Maryland. He specializes in designing hospitals and has a personal and professional interest in landscaping. He _could_ join the American Institute of Architects, and any of its many special interest groups. He _could_ join the Board of Trade, and the American Institute of Landscape Architects, and the local Rotary Club, and the American Association of Hospital Administrators, and the Chesapeake Healthcare Association. Then, there's the group that's restoring plant life along the Chesapeake Bay. Or how about the Lion's Club? Or he could join the alumni group for his alma mater, Boston University. Or how about a referral group? Jon has many ChoicePoints. You probably do too. Jon needs a process to narrow down the choices and find the best groups for him. So do you.
### Choose Groups Strategically
Use the list that follows as you make strategic choices about which Settings to focus on. Not all groups are equally useful for networking. Your choice will depend on your goals and on the characteristics of the group.
These Arenas are arranged from the most highly structured and intentional networking groups at the top of the list, to the most "accidental" and serendipitous at the bottom. The groups at the top focus on bringing people together to do business; the groups farther down the list have other goals and networking becomes a sideline. Groups at the top will actually teach you how to network and outline appropriate behaviors. In groups farther down the list, the ground rules are foggier, so the more skilled at networking you are, the more successful you'll be.
### Understand the Hierarchy
**_Customer Common Groups._** These groups (Constellations) are made up of businesses that have customers in common. Owners of businesses that beautify and maintain the home, for example, an interior designer, a real estate agent, a home remodeler, a lawn care professional, and a chimney sweep might band together to refer work to each other. To start your own group, see the Bonus: Create a Constellation at the end of Chapter 16.
**_Special Purpose Networks._** Some networks are created with one purpose in mind. In one Midwest city, for example, entrepreneurs started a special network to attract venture financing. Job hunting support groups are another example.
**_Business Referral Groups._** Small or home-based business people, sales professionals, or people in professional services and others benefit from these groups. The groups' missions are tightly focused on getting business and generating referrals from each other. Only one business in each "category" may join. For example, members will include only one florist, one electrician, one accountant. At meetings, members learn about each other's products and services. A commitment to attend and generate leads is essential to the success of the group. See Chapter 18 for a detailed look at these groups.)
**_Networking Organizations._** These groups often have the word "networking" in their names. They may have other goals, such as professional development for members. But they will focus on providing opportunities to build relationships. To help people get acquainted, these organizations may offer special interest groups, such as a book club, an investment group or a business owners group. One women's networking organization, The Central Exchange in Kansas City, has as its motto: "The thing that sets us apart is the people we bring together."
### Make strategic choices about where to network.
**_Professional and Trade Associations._** Whatever your job type, whatever your industry or profession, there is at least one professional association for you, if not several. Ask experienced people in your chosen field to recommend which one would be right for you. Watch the business section of your newspaper for meeting announcements. Check the Encyclopedia of Associations at your local library, or go to a group's website for membership information and the name of a local contact.
**_Industry-Specific Organizations._** These organizations put you in touch with people in other companies. An aviation association, for example, brings together people from all of the carriers, as well as related businesses. Because they face similar problems, these people can be great resources for each other. Visibility in one of these groups may help your upward mobility, since there is usually a lot of opportunity for job movement among similar organizations. Within these industry-specific groups, there often are subgroups for people with various kinds of jobs—a purchasing group, for example. These subgroups provide access to your peers and leaders across the industry.
**_Workplace Task Forces/Committees._** Don't forget opportunities to network at work. Get out of your cubicle and mix with others by serving on the Run for Fun Committee, or The United Way planning team, or the Diversity Task Force. These are good ways to increase your visibility at work and to hear about other areas of your organization that might need your skills.
**_Chambers of Commerce._** Whether you're self-employed or work for an organization, your local Chamber of Commerce will welcome you. Although this group's mission focuses on civic improvement, economic development, and legislative efforts to favor business, at Chamber meetings, you'll come in contact with people from a wide variety of workplaces with a wide variety of interests. Networking is certainly a big part of the picture.
**_Civic and Service Organizations._** These groups include such organizations as Rotary International, the Lions Club, and many others. They focus on service to the community and civic improvement. The relaxed, informal conversations you have there help others trust you and, long-term, can lead to job opportunities, new customers, and access to all kinds of resources.
**_Volunteer Groups._** When Simon agreed to help build new play equipment in the community park, little did he know hammering nails with Martin would lead to a five-year contract to videotape every corporate presentation made at Martin's company. Volunteering is a way to blend a passion for giving back to the community with the chance to establish long-term business relationships as you demonstrate your Character and Competence.
**_Hobby/Health/Sports Activities._** Some of Pat's first customers when he started his home-based graphic design business were the people he'd met singing in a barbershop quartet. As you enjoy leisure-time activities, remember to teach others about your skills and talents and Listen Generously for how you can contribute to the quality of their personal and professional lives.
Sondra and Marilyn both showed up at the health club at 6 a.m. on Mondays, Wednesdays, and Fridays. They could have just continued to exchange pleasantries, but instead used their exercise time to explore ideas about "life after the corporation." Two years, and lots of miles on the treadmill later, they quit their jobs and started a painting and wallpapering business.
**_Alumni Groups._** A special kind of camaraderie grows out of having attended the same school. Alumni clubs put you in touch with people of all ages and walks of life. Although these clubs focus on promoting the school, raising funds, or supporting the teams, networking is an important part of the mix.
**_Religious Organizations._** While business may not be the first thing you talk about at your church, synagogue, or mosque, it's undeniably true that being active in a religious community does establish relationships from which businesses and careers may eventually grow. Bob and George got to know each other so well at choir practice, that when George was asked to open up a new division in Milwaukee, Bob introduced him to his brother there.
**_Kid Connections._** Your son plays on a soccer team or a basketball team or takes swimming lessons. How many times have you waited impatiently for the coach to end practice when you could have been developing your relationships with other parents? Kudos go to Amy whose sidelines conversations with Amera resulted in Amy providing some management training for Amera's organization—even though their daughters were on rival teams!
**_Seatmates._** There you are, in an airplane for three hours, elbow to elbow with your seatmate. Sure, you might want to read or nap, but remember that a lot of travelers make business contacts with people they meet on airplanes. On a trip to Chicago, Bob sat next to David, a sales rep for a box manufacturing company. Bob told David he was looking for a heart-shaped box for his company's new specialty food product. David sent him the specs the next day and got the contract.
**_Wild Cards._** Networking with people whose perspective is completely different from yours broadens your horizon in unexpected ways. As you seek out contacts with people you seem to have nothing in common, each conversation becomes an adventure. Assume that everyone you meet is important. These wild card contacts can be winners.
Any place people are is a networking opportunity...if you have the know-how.
### Know the Group Before You Join
To find the right networking groups, associations, or organizations for you, check the Internet, the phone book, your local library, or the business pages of your newspaper. Ask other people in your profession what organizations they benefit from the most. Ask customers and clients what groups they belong to. There may be an "associate member" category for suppliers to their industries.
Once you've identified a few organizations, remember that you are about to place a very talented person—you—in a key position, so look before you leap. Attend a couple of meetings as a guest. Talk to new members and board members. Read several issues of the newsletter. Scan the membership directory. Before you write your check and commit your time, assess the organization's value to you by answering the following questions:
**1.** How many members are there? The bigger the better for networking, but it may be easier to move into leadership positions or gain visibility in other ways in smaller groups.
**2.** Can I get excited about the group's mission? Will its activities help me reach my networking goals?
**3.** Are people in the group likely to need my product or service or to refer business to me? Are people in the group likely to provide valuable resources or information?
**4.** What do people say about the group? What's its reputation in the profession or community?
**5.** What opportunities will the group offer me to associate with my peers? With stars in the field?
**6.** Does the group set a good networking culture by encouraging people to introduce themselves and talk to each other about important business and career Agendas?
**7.** Does the group have special activities to help newcomers feel welcome and meet people?
**8.** How easy is it to participate? How quickly could I move into a leadership role that would give me visibility and career experience?
**9.** Do the leaders seem genuinely excited about their participation or are they playing the "somebody has to do it" game?
**10.** Are the programs cutting edge? Do the topics and speakers provide valuable professional growth?
**11.** What would my time commitment be? Can I make that commitment for at least one year?
**12.** What exactly could I contribute to this group in order to become visible?
### Orchestrate Who Knows You
Joining a group doesn't mean you join anybody's network—or that they join yours. Your membership gives you a place to develop relationships with your fellow members, who are your Associates. Great networkers work on becoming visible and valuable, and as a consequence, memorable. Sure, what you know is important. And who you know is important. But focus your energy on expanding the number of people who know YOU.
Use the organizations you join to:
* Demonstrate your skills and expertise.
* Discover new career directions or make a job change.
* Gain recognition for your accomplishments and successes.
* Find new resources and best practices.
* Establish your reputation with people you might want to network with.
Take a high-profile role in organizations you belong to. Write an article for the newsletter. Provide a program. Staff the registration table at the monthly meeting. Get elected to the board of directors. Set up a job bank if your group doesn't have one. Enter your work in the annual awards program—an excellent way to become known for your abilities. Demonstrate your speaking skills, your budgetary wizardry, your organizing expertise, your leadership prowess.
When people see you in action in an organization, they make up their minds about your Character and Competence—even if they've never met you. If you do a great job as treasurer, people will assume that you are an excellent IT manager or an outstanding salesperson. Conversely, if you've promised to do something, but don't come through, people will assume that you are not a competent attorney or public relations practitioner. It's the All or Nothing Rule: If you do one thing well, people will assume you do everything well.
The strength and expanse of your network depends on how many people know you so well that when resources or opportunities drop into their lives, you pop into their minds as the person to call.
### The Twelve Biggest Mistakes Members Make
**1.** They join, but don't go. They show up so sporadically that they can't reap the many benefits of membership.
**2.** They appear, but don't interact. They eat another olive, listen to the speaker, and leave.
**3.** They skip the networking portion, arriving just in time for the meal. They duck out just as the speaker finishes. Then they wonder why networking doesn't work for them.
**4.** They talk and sit with people they already know.
**5.** They make no effort to be visible; instead, they try to blend into the crowd.
**6.** They wait for others to make the first moves.
**7.** They think handing out business cards is networking.
**8.** They give up too soon. They hop from one organization to another, never giving themselves or others time to establish relationships.
**9.** They have "non-conversations." ("Hi, how are you?" "Not bad. How are you?" "Not bad. What's new?") They never get around to productive conversations.
**10.** They arrive without an Agenda. They come without any idea of what they have To Give or what they want To Get.
**11.** They are unaware of "netiquette" within the group. They violate "good networking" protocols.
**12.** They forget that the best way to show Character and Competence is to contribute time and energy.
### Jump Right In
You don't have to wait to be elected to the board. You don't even have to sign up with a committee. Here are ten ways to jumpstart your participation in any group and to instantly begin to get involved.
### After you become a member, the important work of creating relationships begins.
**1.** **_Come with a purpose._** More than 85 percent of people we surveyed confessed that, when they attend networking events, they have no specific purpose in mind, nothing they want to find or connect with, or learn. Before you go, decide what you want To Get—and what you have To Give. See Chapter 10 for step-by-step instructions on setting your Agenda.
**2.** **_Plan ahead._** What can you do before a networking event to make sure you meet the people you need to meet? Call the administrator and ask for a list of attendees, so you know who is coming, or go to the group's website. You may find photos of board members, or a list of people who have won awards, or mini-profiles of members. This information can be used as conversation starters.
**3.** **_Show up._** Arrive early and stay late. Be in the moment. Clear your mind. Set aside thoughts of work piling up back at the office. Give quality time to your contacts.
**4.** **_Act like a host, not a guest._** You are president of your own network, even when you are attending an organization's event! Take responsibility for the success of the meeting. Greet newcomers, even if you aren't yet an old-hand yourself. As you make others comfortable, you'll feel more comfortable too.
**5.** **_Give yourself a job._** Look around and find a way to be helpful. Pitch in at the registration desk. Pass out programs. Doing something will give you a reason for starting conversations.
**6.** **_Introduce yourself to the leaders._** Seek out the president, membership chair, or program chair. Ask questions about the organization. Tell them why you have joined or are thinking about joining.
**7.** **_Talk and sit with people you don't know._** Nearly 75 percent of people we surveyed admitted that they end up sitting next to the people they came with. If you are going to the event with people from your own office, agree beforehand that you won't sit together.
**8.** **_Link up with your competitors._** Often people avoid talking with people who are in the same business or profession. But they can be excellent contacts, and may even refer business to you, eventually. Be ready to offer some information or a resource to your competitor to start a positive interaction.
**9.** **_Help others connect._** Introduce people to each other and build your reputation as an expert networker. Say, "Oh, Sarah—I just met Ona, who has also just started her own business. Let me introduce you to her."
**10.** **_Show off your wares or your services._** Provide a demonstration or a sample. Contribute a door prize. Do a display. Speak on a panel. Gayle, who sells fine leather products, carries a briefcase that shows off her wares.
## CHAPTER 18
## Rev Up Referral Groups
What if, all over town, there were people who knew your business so well and who were so invested in your success that they consistently referred just the right kind of clients and resources to you? What an impact that could have on your business!
Of course, you're probably already getting some referrals from past customers and contacts. But what if you made networking for referrals a major part of your overall client development strategy?
In this chapter, you'll find out how referral groups work, how to pick one or start one, how to make yours succeed, how to plan activities to help members get to know each other's business, and how to do the right things—and avoid the pitfalls—so that you reap the benefits.
### See How They Run
It's 7 a.m. Members gulp coffee and gobble bagels. By 7:15, they're getting down to business. First, they take turns introducing themselves and their businesses. Next, three members present ten-minute briefings on their products or services. The travel agent drapes a lei around the shoulders of her business suit and tells about her new Hawaiian package tour. A handyman describes how he saved the day for a homeowner who frantically called him from work after a neighbor noticed an ominous stream of water gushing forth under the garage door. A stockbroker talks about the trends he sees and why international stocks are a growing part of most portfolios.
After the briefings, members—who have already been to the bank with the income from business referred to them _by_ other members or business _from_ other members—jump up to say thanks. People also give thanks for resources, business advice, and valuable contacts. New referrals are exchanged. By 8:15, members are heading for their offices or businesses.
We tracked the progress of one group of seventy people over a one-year period. The dollars reported ranged from a low of $3,100 worth of referred business in one week to more than $94,000. (OK, there's a Lexus dealer in the group who got the lion's share of that week's business!)
"I got 25 percent of my business through the network last year," says Bob, a commercial photographer. His group is a nonprofit organization in Virginia that grew out of several local business and community groups. Participants are required to provide a total of at least twenty leads a year for others in the organization, which meets every other week. "That puts everyone in the group on your sales force," Bob says.
Many referral clubs are run as profit-making businesses. You pay an entry fee and monthly charges to participate. Look on the Internet under "Business Referral Clubs" to find out more about charges, benefits, and requirements.
In local chapters of Ali Lasen's Leads Clubs, business owners, sales people, managers, and professionals give brief presentations and exchange leads. The Clubs offer workshops on networking, a free newsletter, and ways to advertise your business.
Business Network International (BNI) provides a structured and supportive system of giving and receiving business. The founder, Ivan Misner, claims that some participants have added as many as fifty new clients in the first two years! BNI points out that you might spend as much as $2,000 for a one-time newspaper ad. But with a small membership fee, a local BNI club can turn thirty or forty other club members into your own sales force. Their statistics assert that the average member gives about forty-five leads per year. BNI has chapters in many countries around the world, from Malaysia to Sweden.
One of the members is Vincent, a builder in Van Nuys, California. Leads from a real estate broker in his chapter, who recommended him for two remodeling jobs, brought him $90,000 worth of work. The added business is great, says Vincent, but more important to the long-term growth of his firm are the networking skills he has gained. "I'm a better networker now just because I spend time doing it," he says.
PowerCore, based in Atlanta, has forty-two referral clubs with 717 members in the Atlanta metro area. Founded by Wendy Kinney, PowerCore's process gets results. A unique internal mentoring program called PowerLinks helps new members get connected. Power-Partners and coaches also are active in helping new members get up to speed fast. Kinney offers special workshops, free to members, to help them develop referral networking skills and self-promotion skills.
Three types of businesses benefit the most from PowerCore referral groups, reports Kinney: highly competitive businesses (such as real estate, insurance, remodeling); service or information-based businesses (including Web designers, CPAs, dentists, attorneys, and printers); and businesses that are so new or unique that potential customers don't even know they exist (including virtual assistants, professional organizers, and home stagers).
Conrad, a former Naval officer turned real estate agent, attributes more than $60,000 in commissions to referrals from his PowerCore Team in his second year of membership. Lucy, who sells nutritional supplements finds that the continued contact at weekly PowerCore meetings helps her break through the negative image some people have of network marketing companies like the one she's involved with. Terry, who owns a franchise business, raves that in his first year with the group, 10 percent of all his business came through PowerCore. In his second year of involvement that figure leaped to 21 percent. Robert, who's new in the insurance business, booked more than 30 percent of his business through his Team in his first year. Tracy, an interior designer, reports that half of her new clients in the last six months have come from her Team referrals.
### Shop Around
Shop around until you find a group that meets your needs. To find groups, ask people in your network. Sometimes, groups are sponsored by Chambers of Commerce or government small business development agencies. Others are the brainchildren of individual business people. One very successful club in Maryland was started by a bridal consultant and a stockbroker. Watch your newspaper for meeting notices. Check with your library to see if it has a list of groups. Call the Chamber of Commerce. Inquire at local business support centers run by universities or community colleges, or ask the Small Business Administration.
### Find the referral group that's right for you.
New groups are springing up and membership in established groups is booming. The variety of formats and systems is amazing. The cost of membership in a club varies widely. Some groups charge an initial entry fee. Others just charge per meeting. Some charge enough to fund joint advertising projects or a newsletter to help members get to know each other. Non-profit groups may charge only for refreshments. Others include the price of breakfast or lunch. Some charge only when you show up. Others require that you pay three to six months in advance. Some groups meet in a restaurant or hotel. Others find meeting space at the office or store of a member to keep costs down. Find a model that works for you.
One group in Arizona has 146 members; other groups swear that a membership of twelve to fifteen people is optimal. Most groups allow only one of each kind of business to participate. That means, for instance, that one travel agent isn't competing against another travel agent in the group. Other groups allow anyone to join. Some groups carefully track and report on every lead given and taken. Other groups are much more informal and still get results.
The Arizona group teaches its members to say, "I'm a resource for anything you need." Members hand out each other's cards. Even though the group is very large, smaller Success Teams, made up of four to six people, get together each week for lunch. They know that learning more about how to help each other is a prerequisite for success.
Most referral groups publish some kind of membership roster, brochure, or flyer for members to give to their clients whenever appropriate. One Maryland group publishes a newsletter that introduces both members and prospects to the businesses. That group has also tried some joint advertising, using the theme, "Do business with people you trust."
Can your business profit from a referral group? Some types of businesses tend to achieve quicker results—referrals—than others. Caterers, gift services, bridal consultants, travel agents, printers, and florists are examples of businesses that do well. Accountants, lawyers, architects, doctors, and financial planners must be more patient to make their memberships pay off.
The amount of trust it takes to turn over your financial future to someone is much greater than the trust it takes to turn over the printing of your next business card. Remember too, that the dollar amounts of the referrals vary greatly. Dave, a real estate agent, passed along a remodeling job to Chuck that eventually netted $62,000. LeeAnn, a computer consultant, gave Jerry a lead for his advertising specialties business that resulted in an order for twenty T-shirts, with a $300 value.
The most effective groups recognize that three ingredients build strong referral groups:
**1. _The Quality of Members._** Choose people who are known for their Character and Competence.
**2. _The Quality of the Referrals._** Give leads that are qualified and preferably come with a personal introduction from you.
**3. _The Quality of the Members' Networking Skills._** Teach members how to cultivate relationships and pro-actively look for business opportunities for each other.
### Check It Out
Investigate groups before you join. Find out about:
**1. _Membership and Other Fees._** Low-cost groups may be as effective as high-cost groups.
**2. _Attendance Rules._** Some groups ask members who miss a certain number of meetings to give up their seat to someone else in that business category.
**3. _Expectations About How Many Leads a Member Must Provide._** How much pressure is there to produce? Sometimes the quality of the leads diminishes when people are hounded for quantity.
**4. _Time Commitments._** In addition to the scheduled meetings, how much additional time is required?
**5. _Leadership Responsibilities._** Will you be expected to serve on committees?
**6. _Entrance Requirements._** Are there rules about the size of your business or how long you have been in business?
**7. _Categories of Businesses in the Group._** Is the group noncompetitive, with only one member per category?
**8. _What Members Say About the Group's Value._** What's the turnover?
**9. _The Group's Reputation in the Business Community._** What do people think of the group?
**10. _The Group's Track Record._** How long has the group has been in existence? What bottom-line value do members place on their participation?
### Don't Just Join, Join In
Once you sign up, take your share of responsibility for making the group an effective referral source for you. Here are some guidelines for participating:
* Be there. If you don't attend—every time—you won't reap the benefits.
* Give it time. Referral group leaders say it takes a year for your business to begin getting the number and kind of referrals that make a big difference in your bottom line. Advertising gurus say prospects must hear or see your message nine times before they become customers. And, since prospects aren't really paying attention two-thirds of the time, it takes twenty-seven exposures to make nine impressions. If your group meets every other week, that's a year's worth of "exposures" before your message will sink in.
* Create sound bites. As you introduce yourself week after week, focus on the different aspects of your business and attach a one or two-sentence Success Story to give a vivid example.
Don, VP of sales and marketing for a temporary agency, never introduces himself with his title. Instead, he says things like, "We screen our temporaries—screen like we're panning for gold. Last week, a customer called me and said, 'We made our project deadline, thanks to the incredible people you sent us,'" or "Do you need an extra pair of hands? My agency can send you people who are experts on any of eighteen different software packages," or "We're the people-power people." Sometimes, Don holds up a paper doll he bought at a party supply store. As he talks, he pulls that one paper doll out into a string of twenty-five, and says, "Whatever you're short of, we have someone with that skill who can be there within 24 hours. Yesterday, I filled a request for a bilingual receptionist in less than three hours."
* Help the group grow. Bring prospective members whose businesses represent unfilled categories. If your group doesn't have a photographer, find one who'd be interested in joining and whose expertise and business practices would reflect well on you and the other members. Be alert for prospective members whose customers might need your services also. If you build decks, find a landscaper. The homeowner who is interested in putting in a rock garden might also want a new deck.
* Demonstrate your Character and Competence. In everything you do and say, show people they can trust you. Then, they won't hesitate to refer you to one of their customers or friends.
* Get together in ever-changing groups of three or four outside the referral group meeting to learn more about how you can help each other get business.
* Tell people how they can help you and who your ideal clients are.
* Listen Generously so that you know what kinds of information or leads to give. The Reciprocity Principle _does_ work. If you give, you will receive.
* Ask others in the group for feedback about how you introduce yourself and how you describe your business. Others will help you refine what you say to make it as effective as possible.
* Encourage the group to provide books or training programs or hire networking specialists to do workshops to enhance member's skills.
* Practice all of the networking skills in this book and teach them to others.
### Start Small
To speed your success, do one-month blitzes. Put together Success Teams of four or five members, and use these ideas to strengthen your relationships quickly. Next month, create new teams. Here are twelve ideas for Success Teams:
**1.** Meet several times. Eat lunch or breakfast together.
**2.** Give each teammate ten of your business cards to hand out to potential clients or customers.
**3.** "Test" each other to make sure _each one of you_ can describe—accurately and vividly—exactly what your teammates have to offer.
**4.** Visit each other's places of business.
**5.** Ask each member of the team to describe in detail his ideal customer (or an actual customer) so that you know who each person is looking for.
**6.** Ask each teammate to tell you about a current business or personal challenge and do everything in your power to assist him or her in coming up with a solution.
**7.** Designate a spokesperson to share your team's experience with the rest of the members at a regular meeting of the referral group.
**8.** Talk about where you might find or run across people to refer your teammates to. Keep the idea of referrals at the top of your mind.
**9.** Ask each teammate to tell stories about satisfied customers or clients. Why exactly, were they so happy with the team member's product or service?
**10.** Provide a special incentive for teammates to try your product or service and to experience how good you are!
**11.** Brainstorm with your teammates how else you might be able to help each other.
**12.** Teach these activities to others in the referral group.
### Spice Up the Meetings
After several months, Jan's group was getting stale. The steering committee decided it was time to change the meeting format and wake people up. They decided to start off every meeting with three, five-minute one-on-one's. In pairs, members were asked to discuss questions that would nudge them out of "Ho-hum" and into "No kidding!"
Here are the questions they used over several months:
How did you get started doing what you do? Why are you in the business you're in?
Where is your place of business? Is it easily accessible? Can you describe how to get there from here? Is it the best place for your business to be? Pros and cons?
What's your problem/concern/challenge today or this week? Brainstorm possible resources or solutions with your partner.
What's your unique capability? What do you do that most others in your line of work don't do? If you don't have a unique capability, what could it be?
Describe a recent satisfied customer. How do you know the person was satisfied?
If you had a smart person to help you all day today, what would you have that person do?
What one thing gets you down? What do you do to get "up" again?
What's the one reason for your success so far? What have you done right?
What's the best mistake you ever made—the one that you learned the most from?
What would you like to quit doing in your business?
What would you like to start doing in your business?
If money were no obstacle, what do you need to improve your business?
What other two member's businesses are most compatible with yours? For example, which businesses target the same customers?
What did you learn in school that is not true?
What did your mom/dad tell you that is true?
What's your best marketing tool?
What would you like a prospective customer to hear about your business?
What motivates you?
What do you get the biggest kick out of?
When is your slowest time of the year? What do you do about that?
### Start Your Own
Some groups struggle along. Perhaps too many of the members are start-ups. Perhaps turnover is high because people have unrealistic expectations about the amount of business they'll get—and how quickly. Some people don't understand the need for trust and expect instant referrals. Perhaps the group is _supposed_ to be a referral group, but spends a lot of its time listening to programs about topics irrelevant to members' business growth. (One group's newsletter listed the following programs: What's New at the Zoo, Afghanistan Today, and The Responsibility of the Media in a Democratic Society!) Perhaps people just plain don't have the networking skills it takes to be good referrers. If you are involved with a struggling group, don't assume the _idea_ doesn't work; assume that the _group_ doesn't work. One solution is to start your own.
There are pros and cons to starting your own referral group. The biggest plus is that you can handpick members, rather than link up with an already established group of people. Another benefit of starting your own group is that you'll have lots of input to design a meeting format that gives you maximum interaction. If you put together a small steering committee, there will be several of you to handle the finances, find the meeting place, and set ground rules—all important (but time-consuming) activities.
The design of some groups shows real strategic genius. The president of a security systems company carefully teaches his salespeople, who are scattered throughout the country, how to start their own referral groups. He shows them how to select the core group, so that members are from businesses that serve customers who will probably also need burglar alarms and fire safety equipment.
Remember, whether you join a group or start your own, each group has a personality, a reputation, a networking culture. Make yours upbeat, generous, friendly, and professional.
## CHAPTER 19
## Connect at Conventions
What's so hard about going to a convention? You send in your registration, buy your plane ticket, pack your suitcase, and go. Right?
Well, that's the way _most_ people go to conventions. But what you actually gain from a conference depends on how you interact with the other attendees. You could read a book or professional journal and get virtually the same information you'll receive from attending a convention. The difference, though, is obvious. A conference brings people face-to-face. If you don't make effective contact with the other people at the convention, you'll go home feeling vaguely dissatisfied.
Amazingly, there are shelves and shelves of books in the library devoted to _planning_ meetings and conventions. But there are no books on how to be an effective participant. Use the ideas in this chapter for what to do before, during, and after the conference. You can make conventions valuable business experiences that are worth your time and effort.
### Expand Your Expectations
Contacts Count surveys show that attendees and their bosses expect three things from the conference experience: information, inspiration, and interaction.
The meeting planner designs the program to accomplish the first two. Without taking much initiative, you'll soak up information and inspiration as you attend the sessions and listen to the speakers. But generating the one-on-one connections that enrich and expand that knowledge and motivation is, for the most part, up to you. It's your responsibility to create the individual interactions that make the time and money spent worthwhile.
Bringing people together face-to-face is expensive. In the business community, there are periodic efforts to replace meetings with video or Web teleconferencing. On the other hand, there are strong arguments for convening in person. Your organization will be more eager to pay for you to attend conventions, if you can bring back bottom-line benefits. Use the tips in the rest of this chapter for planning ahead, connecting at the conference, following up when you're back at work, and justifying the expense.
### Get Ready, Get Set, Before You Go
Take these steps to get ready.
**Set your Agenda.** Make a written list of your To Gives and To Gets. By making an Agenda, you customize the conference so that it exactly meets your needs. (See Chapter 10 for specific instructions on making Agendas.)
On the To Give side, jot down things to share with the people you meet:
* New resources you've discovered.
* Special expertise you've developed.
* Problems you've solved.
* Successes you've had.
Even if you are a newcomer, you still have insights and enthusiasms you might offer.
If you are a veteran, make a note about information you can provide to those just coming into the field and about what will be on your Agenda when you talk with the other "old hands."
On the To Get side, go for the gold and make your list as long as possible. Jot down what you want to find:
* Answers to challenges you're facing.
* Solutions to problems you're dealing with.
* Resources you need to succeed.
* People you'd like to meet.
Martina is an independent consultant in human resources who focuses on pharmaceutical companies. Her convention Agenda included a variety of items, as shown in Figure 19-1.
Martina is ready for meaty conversations. When someone asks her, "How are you?" or "What's new?" she can turn to her Agenda for a topic. And if the person she's chatting with doesn't know about "overseas contracting," for example, Martina can ask, "Do you know anyone here I could talk to about that?" She'll probably be passed along to someone with that expertise.
**_Take along other people's Agendas._** To build your relationships with colleagues, your boss, your sales people, or your business partner, collect their concerns and hunt for answers for them at the meeting.
**FIGURE 19-1. Agenda Items.**
After you are clear about your Agenda, tell people. Business people often see conventions as "just a joy junket." Sell your attendance. Send a memo to key people in your organization telling them that you're going to the conference. Attach a copy of the program. Your colleagues may know some of the speakers and may be able to tell you, "Dr. Dud is a waste of time, but be sure to hear Ms. Up-and-Coming." Ask if there's anything you can do for the organization while you are in the convention's host city. Your memo also lets others in your company know you are serious about your professional development. Emphasize that your attendance is not just your annual trip to the Sunbelt, but an educational experience and a way to gather state-of-the-art strategies. Then report on the results when you return to work.
**_Choose your sessions in advance._** Select your sessions before you arrive at your hotel room. Advance planning while you're still in the office will allow you to shape your experience to your goals. Select sessions carefully. Focus on the knowledge you need and the skills you want to develop. Look for the right sessions that will force you to reevaluate, plan for the future, and expand your horizons. Carefully pick a wild card session on a topic you think you may never have a use for. Invariably, that's the one that will open new doors for you or shine new light on old problems.
Divvy up sessions among several people from your organization who are attending or decide to attend one key session as a group and have a "How-are-we-going-to-apply-this?" get-together immediately afterward.
Begin to think about some topics that particularly interest you and to formulate the questions you'd like to ask at the session before you go.
**_Design your own sessions._** Recognize that some of the best sessions are not listed in the conference brochure—they're set up by you! Arrange them _before_ you leave for the conference. Here are some ideas. Visit a branch office or corporate headquarters to increase your knowledge of the business or to visit an internal customer. Meet with a key prospect or customer in the city where your conference is being held. Contact a speaker before the conference begins to suggest getting together for breakfast or lunch.
Set up a meeting with a board member, a guru, an expert, a counterpart from a similar organization, or colleague you've lost touch with. If a twosome seems intimidating, invite several people (they don't have to know each other!) to go out to dinner.
Before the meeting, Arnetta called a couple of acquaintances who were attending and invited them to dinner. At the conference when she met someone new or saw a colleague, she said, "I'm getting together a bunch of people to eat dinner at a great Italian restaurant. If you'd like to come, meet in the lobby at 6:30." Arnetta's sense of adventure was contagious. Nine people showed up and enjoyed a leisurely dinner exploring connections and common challenges.
Plan an out-of-the-ordinary experience to stimulate your creativity. That experience might have a business payoff. At a convention in New Orleans, Don took a guided tour of jazz spots. Six months later, he was producing a video program for his company and searching for appropriate music. A blues tune he'd heard in one of the jazz joints popped into his mind. His video later won an award. In Orlando, while other conventioneers were at Disney World, Jenna visited a shelter for homeless women. She got excited about a program that encourages employees to donate clothes so the women from the shelter can dress appropriately for job interviews. When she started a similar program back home, she improved life for others and gained positive PR for her image consulting business.
**_Build in time to relax, unwind, exercise, and see the sights._** Seek out information about the city before you go. Look at the city's home page on the Internet. Also browse though travel magazines. How can you take advantage of the location of the conference? If you don't find ways to take advantage of the site, you might as well be staying at home. What new things do you want to see? Plan ahead to use recreation—or even regional cuisine—to stimulate your creativity.
**_Volunteer for a job at the conference._** Call ahead and offer to help out. Find the name of the chairperson of a committee on which you'd like to serve, and volunteer your talents. It's a rare group that can't use an extra pair of hands. You'll find it easy to make contact with people that way. You'll also gain professional visibility, mingle with the leaders, and build a nationwide network.
Or give _yourself_ a job to do at the conference. Before you leave home, arrange to bring back a report on some aspect of the conference to a person in your organization, to your company newspaper, to your city newspaper, or to your local professional group. Having a job to do will strengthen your Agenda, and you'll feel as if you have an even better reason to meet people, ask questions, and take notes.
### Show Up at the Conference
Be there! Set aside thoughts of the work stacking up back on your desk and the messages piling up in your e-mail inbox. Use these ten tips for making great convention connections.
**1. _Arrive at the convention early._** The important people—speakers, conference organizers, association leaders—are likely to arrive early for "pre-meeting meetings." Rub shoulders with the successful people. They are the ones with the connections. Find a mentor or a role model.
**2. _Wear a smile._** Make your body language say, "I'd be easy to talk to." React to visual clues. Comment on jewelry, a necktie, a T-shirt, a nametag listing a state you've traveled in. If you're in line to pick up theater tickets with someone who is wearing the same kind of convention badge you are, go with the obvious. Say, "Hi, I'm Jack, Jack Armstrong. I'm at the NCAC convention, too."
**3. _Volunteer (again!) to help._** Give out nametags, fill in for a panelist whose plane got fogged in, distribute hand-outs for the speaker. Participation leads to relationships.
**4. _Take advantage of the "meetings in the hallway."_** Make the most of all the informal and unstructured moments. Introduce yourself to someone sitting near you before the keynoter begins to speak. Strike up a conversation with someone in the hotel lobby or at a luncheon. Welcome a newcomer. Congratulate a new board member at the opening reception. If you meet just one person who can help you boost your sales, advance your career, land a new job, or solve a problem that's festering back on your desk, you'll call the convention a success.
**5. _Introduce yourself to speakers or panelists._** Welcome them before the program, and let them know why you chose their session. Often, they are eager for more information on who's in the audience, so if they aren't busy getting ready, talk to them. They may mention you in their presentation—instant visibility! Or talk to the speaker afterwards. Ask if he or she is free for coffee or lunch to continue the discussion. It's rare to be turned down. If people can't do what you ask, they may offer something even better in return. One speaker said to Robert, "I can't go to lunch right now, but why don't you join Wolf and me for supper." Wolf turned out to be CNN anchorman Wolf Blitzer! If you feel shy about issuing an invitation to a speaker, ask two or three other people with similar interests to lunch and then ask the speaker to join the group.
**6. _Participate in the sessions._** Ask a question. This does several things. It forces you to think actively rather than just sitting passively and taking it all in. When you speak, stand and talk loud enough to be heard. Introduce yourself and tell where you're from or what organization you're with. You'll be remembered because you have been seen. Your visibility makes it easier for people to come up to you after the session and start a conversation. If there is time, you might even announce that you'd like to talk to people who "have successfully used an executive search firm" or "have solved the problem of doing long-distance sales training when field offices are spread out"—whatever is on your Agenda. Your question may attract others who have the same interests. Also, listen carefully to other people's questions. Follow up after the session by getting with those people, commenting on their questions or asking more about their point of view. Those are instant conversation starters.
**7. _Sit with strangers._** At sessions and meals, don't sit with people you already know. Use that time to meet someone new. Tell yourself that there are no accidental meetings, and try to figure out what you and the other person have in common. Find out what others are looking for and help them connect with resources and contacts.
**8. _Look for excuses to introduce people to each other._** Listen for commonalties, then be a great connector. "Fred, I want you to meet Sam. You both are in charge of leadership development programs in your organizations." "Mary, I want you to meet Sunita. You both are program chairs for your chapters. I know you'll have lots to talk about!"
**9. _Consult the list._** You may receive a list of attendees in your registration packet. Use it to look for people you'd like to meet. Take every opportunity to start conversations. Welcome first-timers. Thank an association leader for his or her hard work. Say hello in the elevators. Meet people who have the kind of job you have now. Meet others who have the kind of job you think you'd like to have next. Meet people from your own geographical area. Discuss regional or state activities. You might find out about new activities you'd like to take part in; or if you're already active, you might enlist an enthusiastic new member for your regional, state, or local organization. Your Agenda will give you ready-made topics.
**10. _Be prepared to job hunt—even if you don't think you're looking for a job._** Update your resumé and take a dozen copies with you. Make sure your business cards are up to date. Put together a few samples of your work. If there is a placement service, sign up and set up interviews with prospective employers to practice your interviewing skills. Throw in a U.S. map. You may want to be able to locate Bigville on the map. Interviewing gives you an idea of your marketability and the going rate in other parts of the country for the kind of work you do. If you are in the process of hiring, interview people for your job opening. Even if you don't fill the job with a person you interviewed at the conference, you'll have a benchmark against which to measure the people you interview when you return home. You also may be able to pick up a job description for a job you are creating. Then you won't have to write one from scratch.
### Follow Up After You Get Home
Sit down at your computer with your notes, and compile a list of major ideas, resources, and contacts. Turn sketchy notes into action steps. Make a list of people to Follow Through with. Do it within a week. Did you promise to send your counterpart, a franchisee in Albuquerque, that interesting article on selling to baby boomers? Do it! Did you say you'd review someone's resumé and send it back? Do it!
Tell your boss about the conference. Then follow up with a memo. Pass along all of the exciting ideas you heard. Tell the boss who you talked to, what sessions you attended, what you learned, and why it's valuable to you and the organization.
It's a great opportunity to meet with the people in your company you asked for advice or who gave you a job to do. Translate what you learned into positive observations, suggestions, or plans for your company or organization.
Go ahead and write that article for your company newspaper. Your fellow employees might be fascinated to hear what the keynote speaker, filmmaker Ken Burns, said about leadership. Of course, if you're really gutsy, you will have gone up to him and asked him for any experiences he's had that would especially apply to your industry.
Take a tip from Beth. As she was sorting through all the cards she received at the national convention, she reread the notes she'd made on the backs. Then, she hit on another idea. She pulled out her association directory and highlighted in yellow all the folks she'd met. She tagged them with sticky notes—yellow for members and blue for vendors. She copied the notes from the backs of the cards into the directory and refreshed the name/face connection. Next year, when she goes to the convention, she'll pack the book or tear out the highlighted pages and take them with her.
### Later On, Get Re-Inspired
Finally, six months later, take out your notes and re-read them. Choose a rotten, rainy Monday morning for this exercise. You'll find that all the ideas and enthusiasm and inspiration you felt while you were at the convention come flooding back. That's what a convention is for: To give you ideas and to stimulate you. William James, the psychologist, was talking once about the time it takes for the unconscious to incubate ideas. He said, "We learn to swim in the winter and skate in the summer." You may find that ideas from the conference have now incubated and are ready to be hatched.
Also, at this six-month point, send notes to some of the people you talked with at the last convention. Ask them if they are going to be attending the next one. Keep in touch with your contacts. Put them on your holiday card list. That will make going to the next conference much easier: You'll be looking forward to seeing, not strangers, but your valuable business contacts.
Assess how well you did at focusing on your Agenda. Did you answer your questions? Did you solve the problems you took with you to the conference? Did you meet many of the people you wanted to meet? If you can say yes, then you have succeeded at the art of conventioneering.
### Bonus: Plan Meetings That Get People Talking
Chances are that sometime, somewhere, you'll be involved in planning a convention, a sales kick-off, a regional conference, or a meeting for a local group. The sessions may be fantastic, the food may be delicious, the hotel may rate five stars, but what brings people back are the connections they make at the meeting you are in charge of. Look at Figure 19-2 for some more tips.
Here are twelve ideas to increase the interaction.
**1. _Prepare attendees before the conference._** Put articles and tip sheets on "networking know-how" in your magazine, newsletter, and convention publicity/registration packets. Make the "netiquette" of making contact in this organization crystal clear. That will give attendees the confidence to go from casual conversations to great connections.
**2. _Make nametags novel._** Print first names BIG. As conversation starters, add special ribbons or colored stickers designating interests to help people find each other.
**3. _Give out several blank nametags._** Tell people to wear a new one each day and write on it something they're eager to talk with others about. Or pre-print nametags to say, "Talk to me about..." and ask attendees to write in a topic they want or need to talk about. Agenda-based nametags make opening conversations easier.
**FIGURE 19-2. Meetings That Get People Talking.**
**4. _Maximize the mix and mingle._** Include some short, structured, one-on-ones or threesomes to encourage mixing and meeting. Choose an energetic, well-known person to lead the session. Plan the questions carefully to fit the group's interests and culture.
**5. _Manage the music._** Use music to energize or entertain, but keep the volume low and don't let it compete with conversation. People will leave your expensive reception in droves if they can't hear each other!
**6. _Give a hand for hospitality._** Train leaders, staff, and volunteers how to introduce people to each other. Teach them how to encourage conversations at mixers and sessions. Invite people to go out to dinner or attend special events.
**7. _Give people a way to begin conversations._** As they come into a session, pose a provocative question or topic on the screen and invite people to chat about it until the session starts.
**8. _Spark up your speakers._** Remind speakers that attendees want to talk with each other, as well as listen to experts. Suggest ways for speakers to include some interaction in their sessions. Also invent ways for speakers to meet attendees. Encourage speakers to attend receptions and meals where attendees can talk informally with them.
**9. _Make the most of meals._** Table talk isn't easy when the room is noisy and the tables are set for eight to ten. Work with the hotel on noise control and request smaller tables of four to six whenever possible. Using a Table Talk Host, or simply a bold sign at each table, suggest a topic or issue people can discuss over lunch. Topics that generate a lot of energy are ethical issues, future trends, or what's going on in local chapters. On "free" nights, invite people to sign up to go out to dinner in small groups. Designate a restaurant, host/hostess, and meeting place. Announce a "Lunch Bunch" for first-timers, people interested in certification, or people who want to swap tips on doing research on the Internet, etc.
**10. _Schedule small success groups._** Create small group meetings or rap sessions around topics of interest. Give time for the groups to meet two or three times during the conference, then by e-mail during the year. Offer a "Meet the Pros" session. Attendees sign up in advance for 30-minute round table discussions with a "pro," who talks and answers questions about a pre-announced issue or topic.
**11. _Foster follow up._** Be bold about suggesting how people can follow up and stay in touch after the meeting—on the Internet, with conference calls, at regional or special interest group meetings.
**12. _Find out with a focus group._** Ask attendees, "How could we make this a more network-friendly event?" Interview them at a breakfast focus group or with an e-mail poll. Trust them to tell you what helps them get the interaction they came for.
## CHAPTER 20
## Jump-Start Your Job Hunt
If you're looking for a job, you probably opened to this chapter first. Good! We'll help you switch career fields, re-enter the job market after taking time off, get your first job after graduating, bounce back from a layoff, transition from military to civilian life, find new options and new directions, advance in your career field, or just find a better job faster—no matter what your current situation.
Whether your job change is voluntary or necessary, networking know-how is the key to your success. The length of your job search is inversely related to the strength of your network.
The stronger your network, the shorter your job search. The weaker your network, the longer your job search. Two-thirds of job-finders say they found their jobs through networking. As you apply The Contacts Count Networking System, you'll be able to create a circle of contacts that will give you "hot information" on openings; access to people you need to know; resources and services you must have to conduct your job search; and support to boost your spirits along the way. As you use the Twenty-Five Tactics to Find a Job Fast, you'll be confident that you are going about your job search in the most professional and efficient way.
### Use the Contacts Count Networking System
Read this chapter to see how the four parts of The Contacts Count Networking System can help you jumpstart your job search. Then start at the beginning of this book. Learn everything you need to know to be well equipped to use networking as your primary job-hunting tool.
**1. _Survey your skills and your mindset._** What do you really know about networking? Taking the Self-Assessment in Chapter 1 will help you see which skills you've mastered and which ones you need to bone up on. You'll want to become so skillful a networker that connecting—comfortably and professionally—becomes a way of life for you. If you lack relationship-building skills, job hunting becomes a stressful and artificial process of sending out resumés and giving out business cards to strangers. Imagine the difference when your contacts want to help you!
Then, using Chapter 2, take a hard look at the attitudes and beliefs you have about networking. Choose a mindset that will make you more comfortable reaching out to people, even though, as a job-seeker, you feel in dire need of contacts.
**2. _Set your strategy._** As you network to job hunt, there are two key concepts you must understand—how to teach people to trust you and how to intensify and deepen your relationships. See Chapters and . If you don't know and use these concepts, your contacts will be superficial and a lot less likely to provide valuable assistance.
Assessing the state of your current network—how much time and money you're spending and what groups you're already connected to—will help you set your goals. You'll find that, as you network, you will arrive at many ChoicePoints—opportunities to make strategic decisions about what you'll do or say. And you'll find that thinking of your job hunt as a Strategic Positioning Project will give you an edge. See Chapter 5.
**3. _Sharpen your skills._** Use your results on the Self-Assessment to guide you as you decide which skills to tackle first. But be aware that the skill chapters take you from Hello to Follow Though, so it makes sense to work your way through all of them. We guarantee that even the most sophisticated networker will discover plenty of new ideas. Need more help to get up to speed fast? Find a role model. Hire a coach. Enroll in a Contacts Count workshop or online session. Do whatever it takes, so your superior networking skills make it possible for you to feel relaxed and competent, whatever the situation.
**4. _Select your settings._** If you want to change jobs, you may need the information about networking inside your organization in Chapter 14. If you must find a job, you'll discover in Chapter 17 that not all networking venues are equal in the opportunities they can provide to a job hunter. You'll get help figuring out the best organizations to become active in—and how to up your visibility instantly. It's tempting, if you're out of work and paring expenses to the bone, to cut out membership dues. Don't do it. You'd be cutting off your lifeline to job banks that focus on your occupation, mentoring from the best and brightest in your field, professional development that you can add to your resumé, and access to fellow members who are predisposed to help you. Think about the benefits to you of attending a conference. Where else could you find that number of potential contacts? See Chapter 19 for ways to capitalize on your conference experience.
### Twenty-Five Tactics to Find a Job Fast
Use these tactics to succeed with your job search networking Project.
**1. _Talk to yourself._** This tactic will support you as you get ready to present yourself in a positive way. We know that it works and that it's essential. For twenty days, write for twenty minutes. Get in touch with your feelings—especially if you have been laid off. Pour out on the pages your challenges, frustrations, and fears, and get rid of your feelings of betrayal, and loss, and even guilt. Don't use this time to make plans and lists for your job hunt. As you jettison the junk in your head this way, you'll get unstuck and be ready to move on. If you don't let go of this stuff, it will drag you down.
**2. _Know who you are._** If your job is everything and you lose it, who are you? It's not fair or true, but in this society we often get hooked into thinking we are our job titles, and if you haven't got one, you're nobody. The truth is that your job is what you do, not who you are. Since several of your job changes will be involuntary, as the result of downsizings, mergers, or moves, you'll be wise to separate who you are from what you do. Vow now, as you job hunt, to develop a strong identity for yourself outside of your work—as a fundraiser for the Boy Scouts, for example, or as board member of a charitable organization. Start now as you are job hunting. Find at least one activity where you can excel and feel good about yourself and your talents. Be aware that good networking contacts often come from these extracurricular activities where you've had a chance to show what you can do.
**3. _Enlist your family's support._** As you begin networking to find a job, don't overlook your own family. It's a low stress, low risk place to begin. As you talk to your near and dear, go though your resumé and practice saying what you are looking for. Ask relatives about their circles of contacts. You might find that your brother-in-law's father can introduce you to a key person.
**4. _Update your business card._** Maybe you've been laid off or are reluctant to use the business card from your current organization because you want to develop a new career identity. So get a new one. Include all the standard identifiers on your new card, including your special career designations or certifications, and a few words about your skills. Carla used "Financial Manager" and a special e-mail address and a special website on which she posted her resumé and accomplishments. Bradley used "Computer Security Operations and Disaster Recovery," CISSP (his earned designation of Certified Information Systems Security Professional), and a cell phone number he got to handle all his job search calls.
Why not have several cards? They are easy to make on your computer or order at an office supply store. They help you teach people about your multiple job search identities. Laura was looking for advancement in her field as a marketer of new products. One card stressed her expertise as a professional manager of marketing teams; the other described her as a New Product Marketing Strategist.
**5. _Join or start a job-hunting strategy and support group._** Job-hunting is lonely. People who link up with others keep their spirits up and learn from each other. You can find these groups sponsored by religious organizations, adult education centers, or women's centers. If you can't find a group, start one. See the Bonus section at the end of this chapter for tips. At one meeting, the group helped Gary brainstorm ways to develop contacts in a certain company and think through his answer to the tricky job interview question: "How much are you willing to travel?" The group listened while Monica fumed about companies that granted her interviews, then never let her know when they postponed hiring or filled the job. The group role-played with Jack to help him put his best foot forward as he prepared for an interview in a field he wanted to switch to, but lacked a lot of direct experience in.
**6. _Choose twenty organizations._** To focus your job search, choose twenty organizations you could imagine working for. Make your choices based on the organizations' locations, benefits plans, missions, predicted growth, or just because you have friends who work there. Set up a manila folder for each company and begin to fill it with every bit of "intelligence" you can find. Look on these companies' websites, order their annual reports, scan the newspaper and trade journals for articles about them. Most important, make the goal of your networking to build relationships with people in these twenty companies—or people who know people in these companies. Look through your network to see whom you already know people who might provide access or have tips for you. You'll hear about a job opening before it is announced, and you'll feel like an insider in the job interview because you'll know the company's history, trends, goals, culture, and practices.
**7. _Select six Arenas._** Pull out the list of the Arenas you're already involved with. If you haven't yet made that list, see Chapter 5. Are you already active in six Arenas? Are they the right ones for your job search? If you are changing career fields, join an association that serves the job or industry you've targeted. Look over the hierarchy of networking opportunities in Chapter 17 to be sure you've got the right mix.
**8. _Raise your visibility._** Now that you've joined, the work of building relationships and making yourself visible begins. Go to every meeting. Take advantage of everything the group has to offer. Get involved so people can see your talents. Use the networking portion of the meetings to find the people you want to know better. Visit the group's website. There may be an online directory that will help you pick out people you want to contact. Talk with members about what they do and how they got their jobs. Ask for their advice on conducting your job search.
**9. _Pick fifty people._** You can't network with everybody, so choose fifty people for your job-hunt network. Start your list now. The more diverse your network, the better because, when you know people in a variety of Arenas and with a variety of backgrounds, they will hear about opportunities that you won't hear about. Your best job leads will come, not from who you know, but from who they know. Include those family members you talked with and people from the groups you belong to. If your list doesn't total fifty, think again. Include friends, neighbors, someone who serves with you on the condo Board of Directors, and selected alumni from your alma mater. Look for people who might know people in the twenty organizations you've targeted.
**10. _Relocate the "information interview."_** When coauthor Anne worked for a large corporation, she frequently got calls from people who were job hunting in Kansas City. They asked to talk with her about her job and the field of corporate communications. Anne's usual response was a tough one, but realistic. She'd say, "My company doesn't pay me to talk to you during my work day about my job or your job hunt." Then to soften the blow of that much candor—and because she did want to be helpful—she would invite the caller to attend her professional organization's monthly luncheon. She'd tell him she'd talk with him there and also make sure he met others in the field.
Be creative about when and where you do these interviews. Talk to people at a party, in your carpool, or at a convention. Don't assume that they can donate their company's time to talk with you.
**11. _Teach people to trust you._** Two things teach that you are trustworthy: What you say and what you do.
Janna knew she'd see Joe, who works at a company she'd like to interview with, at the neighborhood barbecue on Saturday. She decided he'd trust her Competence more if she gave him some third party evidence of her success and expertise. She found a way to mention being asked to speak on a panel of experts about Ethics in the Workplace. Remember the All or Nothing Rule: If you do one thing well, people will assume that you do everything well.
**12. _Observe the protocols._** Let your contact know your plans. If someone gives you information, tell the person what you intend to do with it. Don't jump the gun. Allow your contact to determine the timing of what happens next or when you will follow through. "I casually mentioned a job opportunity I knew about to an unemployed acquaintance," says one executive. "I had planned, if she was interested, to phone my contact and arrange a meeting. Our telephone conversation was interrupted and when I called her back fifteen minutes later, she already had called my contact about the position, using me as a reference. I was very embarrassed. That job had been offered to me, and I had not yet refused it."
Pay your way. When you call a contact and ask that person to meet you for breakfast or lunch or a drink, the least you can do is offer to pay. Be sure also that you schedule the meeting at a location that's as convenient as possible for your contact. Dress appropriately for the meeting. Treat every meeting with every contact as a job interview. Remember that you are teaching your contacts who you are. Help them to see you in the position to which you aspire!
**13. _Prepare an answer for "what do you do?"_** Come up with several ways to answer this question. Avoid the mysterious "I'm in transition." Each answer should reflect a different talent you have, direction you'd like to go, or setting you'd like to work in. Each way should be vivid enough to make you memorable.
Marly, a veterinarian at the U.S. Department of Agriculture, used to give this answer: "I decide how many camels come into the U.S.. I just got back from Oman, and there are 300 on the way." But she realized she was becoming known as "The Camel Lady."
She wanted to leave her veterinary career behind and parlay the negotiating and language skills she'd learned bargaining about camels into a new career negotiating international contracts and agreements. So she changed her answer to, "I help people come to agreement. Last week, after months of patient coaching, I got six people who speak three different languages to sign contracts for their companies to work together." What she doesn't tell you is that these six people were talking about camels! With her new answer, she taught her contacts to see her in a new light—and to think of her when they heard of opportunities that could use her negotiating and language skills.
Raj, who wants to change from hospital personnel administration to directing a customer service improvement program in a healthcare organization, says, "I'm studying how hospitals can better serve their clients. I've just interviewed four experts for a presentation I'm giving next week at the American College of Healthcare Executives conference." When Benita was asked, "What do you do?" she didn't have an organization, but she had an answer. "I just won Salesperson of the Year from my firm because I increased my sales by 12 percent in a crowded market." What she didn't say was that three days earlier she had lost her job because her company moved to Phoenix! It's not that these job-seekers want to keep their job search a secret. Not at all. It's just that they want to bring it up at the right time, in the right way, after they've shown their Character and Competence.
**14. _Broaden your outlook._** As you teach people what you're looking for, don't limit yourself to a specific title. Instead, aim to teach your contacts about your talents. Garth called Benjamin and said, "I'm looking for a position as a Chief Financial Officer." Benjamin told Garth he didn't know of any jobs for CFOs. A few days later, Benjamin heard about a fast-growing waste-management company that wanted to raise capital for expansion. He didn't think of passing that tip along to Garth because Garth hadn't let him know that he'd been very successful in his previous job at raising capital. Garth had made the mistake so many job-hunters make—restricting himself to one title.
### Give first: Give freely.
**15. _Give first._** As a job seeker, you may feel that you don't have anything to give. Not true! Before you go to a social or professional event, tune into the To Give side of your Agenda. As you talk with someone, Listen Generously and think, "Who do I know or what resources am I in touch with that my contact might benefit from?" When her husband's job transfer took Angela to San Antonio, she started her networking to find a job there from scratch. She met Lynn at the first American Business Women's Association meeting she attended. Lynn mentioned she was beginning her yearly search for the perfect summer camp for her eleven-year-old. Angela remembered an article she'd read reviewing local camps. She promised to send it to Lynn. What a good reason for exchanging cards! When Lynn called to say thanks, Angela told her more about the kind of job she was looking for, and Lynn gave her a lead.
**16. _Get exactly what you need._** Be prepared to ask for valuable information and resources. Fill out the To Get side of your Agenda before you go anywhere. Approach each situation with a list of resources and information you're looking for. Then ask directly for what you need. Are you looking for people who work for those twenty companies? Ask people at a networking event. Or look for other information and resources you need. Ask, "Have you run across any information recently on negotiating salaries?" Or, "I'd like to start a Strategy and Support group with other people who are changing careers. Know anybody who might be interested?" Or, "I came here hoping to meet people who work in some aspect of health research. Do you know anyone who does that? "See Chapter 10 for tips about how to update your Agenda and guide your conversations.
**17. _Ask stepping stone questions._** These are questions whose answers will get you moving in the right direction and that signal your creativity and diligence. It doesn't work to say to someone you've just met or who is an Associate or Actor in your network, "I'm looking for a job. Are there any openings at your company?" The problem is that you're asking your contact to go out on a limb for you before he knows much about you.
For Dana, who was looking for a job managing and editing association publications, a good Stepping Stone Question was, "Do you know anyone who's a member of the local chapter of the American Society of Association Executives here in town? I'd like to talk with someone in the group before I attend a meeting." Bret, who is transitioning to civilian life from a military career in procurement, doesn't ask for a job in purchasing. He asks, "Do you know of any companies that are buying heavy equipment or building new plants?" It's always good to ask for advice. An example: "If I wanted to work for your company, what are the top two qualities I'd have to have to get hired?" When you hear the answers, you can respond with stories that illustrate that you have those qualities.
**18. _Immerse yourself in the culture._** Want to look like an insider by the time you get to the job interview? Spend time with people who have the job you'd like to have. As you volunteer for committees, go to certification classes, or attend conventions, notice what they read, how they dress, what they talk about, and who their gurus are. Listen for special terms company employees use to describe what they do.
Kevin was winding down his career in the Marine Corps. He had enrolled in an MBA program at a university near the base. As he got to know his classmates, he gathered information about their organizations. He listened to Jonas talk about the business benefits of playing on the company softball team and asked Al about the appropriate garb for "dress-down" days at his company and personnel hiring practices for ex-military.
Look for someone who will take you under his wing, answer questions, look over your resumé, give you advice, and introduce you to others. A mentor might be a friend who's already in the field, a past college professor, or someone you meet in a professional association. Many groups make getting a mentor easy by offering a formal program, complete with training for mentor and mentee.
Of course you can read about IBM, if that's one of the twenty corporations you chose when you planned your strategy. But the best way to get the scoop on the latest happenings is from people, not print. Review your list of fifty people. Who works for IBM? Who knows someone who works for IBM? Which organizations do you belong to that might have members who work for IBM? Who in your alumni association works there? Then look for ways to get into conversation with these people to learn all you can. Strategically build your network of people who are active in the Arena you've targeted.
**19. _Work for free._** Offer to help someone. Mary Alice, a whiz with all kinds of graphics software packages, wanted to learn more about careers in corporate training, so she offered to create a brochure for the freelance trainer who lived next door. In the process, she picked up more knowledge of the field and demonstrated her expertise. Her relationship with the trainer eventually led to several introductions and a corporate job designing internal training materials. Mary Alice is now one step closer to her goal: making the transition into delivering training.
### Showcase your skills to the people who count.
**20. _Give yourself a job._** As a job-hunter, you may have more time, so take on high-visibility jobs others don't have time to do. After Bob was laid off, he attended a regional convention of his professional association that was taking place in his hometown and made himself useful and visible. He manned the registration desk so he could greet people, introduced a prominent speaker to the group of 400, and drove a past president of the association to the airport. Soon after, the past president called with a job lead for him.
**21. _Look for problems to solve._** Tom Jackson, prolific author on job-hunting techniques says, "A job is an opportunity to solve a problem—and there is no shortage of problems." Ask yourself, "What kinds of problems have I been most successful with? "When you describe what you do in terms of the problems you solve, you put a picture of success in people's minds. Listening for problems can give you ideas on how to describe the skills on your resumé in the exact language your prospective employer uses. When Linda heard Jake talk about his company's new initiative in "internal customer service, "that's what she called her expertise in the cover letter and resumé she sent to him. Talking about problems you solve might even result in an organization inventing a job just for you.
**22. _Don't make unreasonable, inappropriate requests._** Dan brought 100 copies of his resumé to the networking meeting, left some on the display table, and gave one to everybody he talked to, saying, "If you know anyone who needs my skills, ask them to call me." This is the kind of behavior that gives networking a bad name. Why would you take the chance of recommending Dan to anyone you know? You'd want to be sure of Dan's Character and Competence before telling your contacts and colleagues about him. His request was presumptuous.
**23. _Fill in the blank spot on your resumé._** Take a university course or get an advanced degree. Education builds your credibility and puts you in touch with contacts who can help you.
Martha Lee had been out of the job market for thirteen years. She began a degree program at a university that was designed for working professionals. She arranged for an internship with the organization one of her professors worked for and later—after she'd proved herself—was hired part-time. Networking with classmates provided her with additional leads and opportunities.
Dottie hadn't planned on taking time off from work. Then she had twin boys. To keep herself current, she worked on her master's degree while she was at home with the boys for three years. When she did decide to look for a job, there was no gap on her resumé. The time she had spent at home was "covered" by her pursuit of the advanced degree. She also networked with professors and classmates and quickly found a job.
**24. _Say thanks._** Let people know how much you appreciate their time, information, and support. Do it throughout your job hunt and again when you find a job. To stand out, send hand-written notes. Tell your contacts what steps you took to put their help to use or how that information led to additional resources. Jan says, "My thank you cards are every bit as important as my business cards. I want people who help me move my job search along to know that I noticed and valued their efforts on my behalf." After you land a job, send one of your new business cards to your contacts.
### Your network is your safety net. Make sure it's in place before you need it.
**25. _Network in your new job._** Once people get a job, they often make two mistakes. They think, "Whew! That's over. Now I can stop networking." Wrong! Your network must always be ready to act as a safety net. In today's uncertain economic climate—mergers, downsizing, reorganizations—you never know when you are really going to need your business and personal contacts. Or they think, "I'm so busy with my new responsibilities that I don't have time to network!" Sure, you want to get up to speed fast on the tasks you are expected to accomplish. But also spend time building relationships and setting up your network. Read Chapter 14 for tips on how to cultivate a wide circle of contacts, outside your department, as you start your new job. And go to www.FireProofYourCareer.com for more tips about how networking can protect your career. Always be eager to stay and prepared to go.
### Bonus: Manage Your Strategy Support Group
If you join or create a job search/career change strategy support group with others who are in transition, here's a process that works. With this group you'll commiserate, celebrate, brainstorm, problem-solve, and make agreements about completing the sometimes challenging tasks that will help you find a job.
Meet every week or every other week. Limit the group to no more than five people. That way, each person will get the time he or she needs. Choose these people carefully to be sure they are positive and motivated. The group works best if each of you is looking for work in a different job function and or setting.
Before the meeting:
* In your private notebook, list all your accomplishments since your last meeting.
* List issues or challenges you want to discuss with the group.
At the meeting:
* Take twenty minutes to tell your accomplishments and get help with your challenges. Set a timer and stick to the allotted time for each person.
* When it's your turn to have the group's attention, tell your accomplishments, then tell the group what kind of help you'd like:
_The Echo._ "I need to talk in order to figure out where I am on this. I'll talk, you listen and say back what you think I'm saying."
_The Dress Rehearsal._ "I'll briefly tell you the situation, then I'd like one of you to be the other person. I'll practice what I'm planning to say, then I'd like your feedback."
_The Huddle._ "I'd like help mapping out a step-by-step strategy or plan for doing_______."
_The Brainstorm._ "I'm too close and can't see the forest for the trees. Let's brainstorm all the possible and impossible things I could do regarding the situation I'll briefly describe to you."
_Ann Landers._ "I'll briefly tell you what's going on. Then you give me advice—tell me what you would do. I may not take your advice, but I'd like to hear what it is."
_Mount Vesuvius._ "I'd like to gripe and fume and blow my stack for three minutes, just to get it out of my system. (Set the timer!) Then I'll choose an option from above."
* Write down any agreements you make with your group, steps you'll take, or scripts you've come up with in your notebook.
* Set the time and place for your next meeting. Consider that a missed or postponed meeting may mean there's something you'd rather not deal with, then choose to go toward the issue rather than burying it.
* Do an energy check at the end of each meeting. Let each person say, "On a scale of 1 (low) to 10 (high), after this meeting I feel like a______." Then tell why and what you need to feel good about going forward toward more career security or your new job. Thank people for their attention and advice.
## Index
The index that appeared in the print version of this title does not match the pages in your eBook. Please use the search function on your eReading device to search for terms of interest. For your reference, the terms that appear in the print index are listed below.
Accidents, as relationships
achieving bottom line results, self-assessment
Acquaintances, as relationships
Actors, as relationships
Advocates, as relationships
Agenda
construction of
creating a list
discovering other people's
for ending conversations
To Get list
To Give list
giving _vs._ getting
going public with
sample items
Allies, as relationships
All or Nothing Test, The, for Strategic Positioning Projects
alumni groups
appropriate behavior, _see_ "netiquette"
Arenas
assessment of
defined
finding new
for focusing job search
for home-based workers
sample chart
Associates, as relationships
Barry, Dave
behavior, appropriate, _see_ "netiquette"
being approachable
being positive
being strategic
_vs._ manipulating
self-assessment
BEST/TEST method
formula
frequently asked questions about
tips for responses
body language
ENGAGE formula for
non-verbal cues
Bottom-Line Test, The, for Strategic Positioning Projects
Bruce, Lenny
business card exchange
effective procedure for
_vs._ making real connections
Business Network International (BNI)
business referral groups
Cates, Bill
Chambers of Commerce
Character
role in developing trust
teaching others about your
ChoicePoints
civic organizations
clients
cross-selling of
generating referrals from
respecting confidentiality of
closeness, "rules" for
Coach, inner
comfort level, self-assessment
communication styles, types of
Competence
role in developing trust
teaching others about your
Constellations, _see also_ groups; referral groups
creation of
of home-based workers
contacts
_vs._ friends
introducing
and networking "generations"
Contacts Count Networking System described
for jumpstarting job search
conventions
do's and don'ts
expectations from
follow up after
sample Agenda items
selecting sessions to attend
setting Agendas for
tips for networking at
volunteering for jobs at
conversational skills
asking profile questions
being interested
encouraging dialogue
learning
Listening Generously
saying "Hi!" as opener
Seriously Curious questioning
storytelling
strategic storytelling
turn-offs to avoid
conversations
good
joining in
LEAVE NOW formula
Magnet Statements
positive attitude during
putting a value on
"rules" for closeness
timetable for ending
tips for ending
topics for, _see_ Agenda
value of
corporate culture
assessment of
learning about
Critic, inner
and anxiety about joining a group
conversion into positive Coach
customer common groups
dialogue
encouragement of
_vs._ monologue
Doorway Test, The, for Strategic Positioning Projects
EARS formula
ENGAGE formula, for body language
ENHANCE-START-REPAIR rating system
eye contact
Five-Year Test, The, for Strategic Positioning Projects
flirting, as inappropriate behavior
Follow Through
goals for
key role in networking
reasons for reconnecting
self-assessment
through proactive listening
tips for
friends, _vs._ contacts
geniality
gestures
getting
_vs._ giving
giving in return
REAL formula for
sample items
giving
_vs._ getting
MORE formula for
sample items
when job hunting
goals
for Follow Through
networking effort required for achieving
types of
goal setting
to increase Arenas
for networking Projects
for relationships
for Strategic Positioning Projects
weekly calendar for
groups, _see also_ Constellations; referral groups
assessing the value of
choosing strategically
hierarchy of
joining of
jumpstarting participation in
mistakes by members of
reasons for joining
hard sell, avoiding
Henderson, Lynne
hobby groups
home-based businesses, types of
industry-specific organizations
information interviews
internal networking, _see_ networking at work
introductions, _see also_ names etiquette of
remembering names during
use of taglines
Jackson, Tom
James, William
job hunting
Contacts Count Networking System
and information interview
managing strategy support group for
networking as key to
observing protocols for
as problem-solving opportunity
tactics for
Kinney, Wendy
Lasen, Ali
Leads Clubs
leave-taking
frequently asked questions about
LEAVE NOW formula
tips for
Listening Generously
benefits of
EARS formula for
as key to good conversation
Magnet Statements
making the most of events, self-assessment
manipulation, avoidance of
meeting people, self-assessment
membership in groups, _see_ groups
mergers, rebuilding network after
Milgram, Stanley
"Million-Dollar Moments"
monologues
avoiding
_vs._ dialogues
multitasking
names, _see also_ introductions; taglines
dealing with forgotten
difficulty of remembering
making yours memorable
technique for remembering
tips for remembering
"netiquette"
for body language
for eating
for entering a room
for flirting
how to learn
for joining groups
for paying your own way
self-assessment
tips for
for touching
turn-offs to avoid
networking, _see also_ Follow Through
avoiding manipulation
avoiding postponement of
defined
for developing a reputation
dynamic nature of
"generations"
getting beyond the chitchat
give/get concept
_vs._ grandstanding
from home, _see_ networking from home
importance in business environment
as life skill
as a long-term process
major misconceptions about
as more than TAKING
need for goals
need for reciprocity
recent emphasis on
and relationships, _see_ relationships
to remain competitive
resistance to within organizations
setting goals for
single-sale
as TEACHING
through multitasking
tips for
tips for professionals
as TRADING
turnoffs in language of
at work, _see_ networking at work
networking at work
assumptions to avoid
benefits of
as corporate priority
ENHANCE-START-REPAIR map for
in a new job
overcoming barriers to
rating the inside network
networking efforts
according to size of goal
ChoicePoints for
Strategic Positioning Projects
networking events
ChoicePoints for
defined
networking from home
best times for
challenges of
creative promotion
and Follow Through
linking work and life
selecting key Arenas
networking organizations
self-assessment
networking Projects
major, _see_ Strategic Positioning Projects
planning of
setting specific goals for
networks
one-on-one
size considerations in
nodding
non-verbal cues
organizations, _see also_ groups
defined
for focusing job search
peers, pairing up with
personal information, disclosure of
Pig in Mud Test, The, for Strategic Positioning Projects
posture
Powell, Colin
PowerCore referral clubs
professional organizations
questions
digging for gold
personal profile
Seriously Curious, _see_ Seriously Curious questioning
stepping stone
"What do you do?", _see_ "What do you do?" questions
Reciprocity Principle, The
reconnecting
reasons for
tips for
referral groups, _see also_ groups
characteristics of
checking out
finding the right
guidelines for participating in
key ingredients of
starting your own
referrals
asking for
creating a Constellation
eight steps for generating
Relationship Management Program
assessment of contacts
benefits of
setting up of
relationships
Accidents as
Acquaintances as
Actors as
Advocates as
Allies as
Associates as
building, _see_ networking
Following Through on
friendships _vs._ selling
for the long term
management of
rating of
six stages of
transitioning from stage to stage
religious organizations
reorganizations, rebuilding networks after
Rogers, Will
role models, need for
seatmates
self-assessments
checking results
Seriously Curious questioning
as key to good conversation
tips for
shyness
defined
learning to overcome
pervasiveness of
Shyness Clinic (Stanford University)
Simmons, Russell
single-sale networking
Six-Stages Model
questions and answers about
transitioning from stage to stage
special purpose networks
storytelling
constructing a story
conversational skills for
frequently asked questions about
SUCCESS formula for
Strategic Positioning Projects
for demonstrating Character and Competence
planning of
tests for
SUCCESS formula, for storytelling
success stories
frequently asked questions about
that make a point
Success Teams, suggestions for
taglines
for making a connection
purpose of
samples
teaching others
about who you are
about your Character
about your Competence
touching, appropriate behavior for
trust, developing
importance of
role of character and competence
Trust Matrix, The
visibility, increasing of
voice, finding the right tone
volunteer groups
"What do you do?" questions
answers when job hunting
BEST/TEST method for answering
choosing the right answer
common responses
making responses interesting
tips for responses
transforming your responses
work, relationship building at, _see_ networking at work
workplace committees
Ziglar, Zig
Zimbardo, Philip
## About the Authors
Contacts Count is the premier training company for face-to-face, business and career networking, offering a wide variety of learning opportunities in the U.S. and Canada. We (Anne Baber and Lynne Waymon) are the founders and principals. For the past seventeen years, we've helped clients realize the many strategic applications for networking and have worked with them to increase their employee and member expertise. We also are currently expanding our cadre of outstanding Certified Contacts Count Presenters.
Networking is the pivotal professional capability for today's businessperson. We recognize the commitment and practice it takes to learn the skills in this book. We want to help you put more profit, purpose, and pleasure into all of your business and career relationships.
If you are a member of an association or networking organization, pass our contact information along to the person or committee responsible for programming. Our keynotes and workshops, always rated "Outstanding," will help your members put these leading-edge networking skills to use immediately.
If you work for an organization—from a high-tech firm to a university to a defense contractor—we can help you foster a positive networking culture of inclusiveness and inquiry, both within the organization and in outside relationships. Our consulting and training services help organizations create strategic plans to build cross-functional relationships that get the job done, boost profits, and expand influence in the marketplace.
### Our Products and Services
1. Presentations: Keynotes, Workshops, Training Courses, Webinars, Teleseminars ("Networking Know-How: The Contents Court System for Savvy Professionals & Smart Companies")
2. Skill-Building Materials: Books, E-books, CDs, articles for reprint, Guide ( _Ten Networking Activities for Events at Associations, Alumni Groups, & Corporations_), _The Fireproof Your Career Toolkit_
3. Coaching and Consulting for individuals and organizations
4. In-house Train-the-Trainer Programs to certify employees to deliver our programs internally
5. In-house Licensure, allowing organizations to use our copyrighted materials
6. Training for individuals who wish to become Certified Presenters
### Our Corporate, Government and Association Experience
* Leadership Programs for Fortune 500 Companies
* Special Interest and Affinity Group Workshops at Fortune 500 Companies
* Skill-Building Training for Non-Profits, Corporations, and Associations.
* Keynotes and Workshops at Professional Association Conferences
* Career/Management Development Programs for Corporations, Government Agencies, and Associations
* Training for Universities, MBA Programs, Government Agencies, Corporations, Keynotes and Workshops for Alumni Groups, Chambers of Commerce, Business Expos
* Webinars and Tele-seminars for Associations, Corporations, and Career Development Firms
### Recent Clients
Lockheed Martin; Deloitte Financial Advisory Services; National Geographic Society; Georgetown University; Marquette University; Corning, Inc.; U.S. Departments of State, Commerce, Agriculture, Navy; BoozAllen; National Association of Home Builders; American Council of Engineering Companies; Public Relations Society of America; Tri-State Women Entrepreneur's Expo; National Association of Female Executives; Medical Librarians Association; Bank of America; Dupont, Inc.; First Horizon Bank; Heery, Inc.
### Our Websites
www.ContactsCount.com
Visit and sign up for our free Contacts Count e-newsletter.
www.FireProofYourCareer.com
### Our Contact Information
Anne Baber
13433 W. 80th Terrace
Lenexa, KS 66215
Phone: 1-913-492-6873
E-mail: ABaber@ContactsCount.com
Lynne Waymon
1400 East West Hwy., Suite 1228
Silver Spring, MD 20910
Phone: 1-301-589-8633
FAX: 1-301-589-8639
E-mail: LWaymon@ContactsCount.com
| {
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} | 2,510 |
Sainte-Léocadie (in catalano Santa Llocaia) è un comune francese di 143 abitanti situato nel dipartimento dei Pirenei Orientali nella regione dell'Occitania.
Geografia fisica
Fa parte della regione storica nota come Cerdagna.
Storia
Simboli
Nello stemma è raffigurata santa Leocadia, patrona del comune, e sullo sfondo i pali d'Aragona.
Società
Evoluzione demografica
Note
Altri progetti
Collegamenti esterni
Sainte--Leocadie | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 9,665 |
Peel the Papaya, removed the inner part, seeds and chop them into medium sized pieces as shown in the picture below.
Grind around 3 cups of fresh grated coconut into smooth paste by adding little amount of water. Squeeze the ground coconut, by placing it in a clean cotton cloth.
After this process, add 2 cups of water to the squeezed coconut, mix well and squeeze it again. Add this thin coconut milk, chopped papaya in a vessel and cook. Once its cooked, add jaggary, pinch of salt and bring them to boil. Keep this in low flame for about 5 mins.
Take little amount of thin coconut milk or milk in a bowl, add rice flour / batter and mix them well. Add this to the vessel containing cooked papaya and bring them to boil. Add thick coconut milk and bring them to boil. Add mashed cardamom seeds, cloves and keep the payasa in low flame for about 1-2 mins. | {
"redpajama_set_name": "RedPajamaC4"
} | 6,181 |
{"url":"https:\/\/meta.stackoverflow.com\/questions\/336840\/question-not-seeking-debugging-help-closed-for-seeking-debugging-help-without-mc","text":"# Question not seeking debugging help closed for seeking debugging help without MCVE?\n\nSee this question.\n\nIt was closed by five voters for the \"seeking debugging help\" reason. That was obviously ridiculous, since it was not seeking debugging help to start with.\n\nI flagged this, with the following note:\n\nThis question is not seeking debugging help, so how can it be closed for that reason? It's a perfectly valid question. close-voters think the OP should have provided an attempt, then they should down-vote for not showing enough research effort, not close-vote. If they think it's a duplicate--and there are definitely related questions on the site--then they should find them and vote to close as a duplicate.\n\nThe moderator responded:\n\ndeclined - I'm guessing they seized on the MCVE aspect here. If you feel it should be reopened, you are free to vote for that.\n\nI don't understand. How can it be OK for them to \"seize\" on the MCVE aspect when the very premise of this close reason--that it is seeking debugging help--does not apply here? People can downvote all they want, but I don't see why it's OK to close for wrong reasons. It's a manifestation of the phenomenon we have all seen where people \"double downvote\" by giving a made-up close reason.\n\n\u2022 The moderator isn't saying that it's OK. He's just surmising what the close voters' thought process was, and suggesting that you cast a normal reopen vote when you disagree with a close reason, instead of flagging for a moderator. \u2013\u00a0BoltClock Oct 24 '16 at 16:36\n\u2022 @BoltClock So maybe I'm misunderstanding flags and\/or the role of the moderator. Is that idea that obviously flawed\/wrong close votes are not something to flag, or for moderators to deal with? Sure I could vote to re-open, but that's pretty close to doing nothing, given the chances of four other people to come along and also vote to re-open. \u2013\u00a0user663031 Oct 24 '16 at 16:49\n\u2022 Vote to reopen first, and if nothing seems to be coming out of it, flag for a moderator. \u2013\u00a0BoltClock Oct 24 '16 at 16:53\n\nLike we have nothing better to do with our votes ...\n\nYou're right. That question shouldn't have been closed as NO MCVE. It could have been closed as Too Broad or even as a duplicate of this, and that has answers with jQuery so it must be a good one.\n\nDo note that not only 5 voters choose to close that question, also 3 others decided to leave it closed. So already 8 users didn't think that question was worth keeping. The down votes made the question eligible for the Roomba which happened yesterday.\n\nNow it can be confusing for OP's when their post is closed and a wrong reason is given. In this case I can see why also the MCVE one was applicable as for the code they showed it wasn't clear what did work and what didn't or even if it did run. In that sense pointing the OP to the MCVE close reason can be helpful, where too broad probably didn't help. And yes, down voting such lack of effort posts is a wise thing to do.\n\nSo by casting an un-delete vote and re-open vote I helped a bit so we can do The Right Thing\u2122 which seems to be down voting, close voting and delete voting. instead we made a train-wreck out of it :(\n\n\u2022 Let's also hope that The Right Thing\u2122 includes not deleting it. It's hovering at 2 delete votes right now and I get this sinking feeling that someone's going to tip it over the scale very soon. \u2013\u00a0Makoto Oct 24 '16 at 17:51\n\u2022 @Makoto Why do we need that question on the site? There are duplicates already, it shows no effort or research, and is basically someone asking us to do their work for us. If you'd like more examples of that kind of question, I'm sure anyone in the javascript tag can come up with more than a few for you... \u2013\u00a0Heretic Monkey Oct 24 '16 at 18:01\n\u2022 @MikeMcCaughan: The whole point of bringing it back is to point to the duplicate. There's no other value besides that; at a bare minimum, the OP will have a reference to go off of rather than nothing useful. I'd rather see it closed as a duplicate which may actually have a chance of helping the OP than have it deleted or removed from the site outright. \u2013\u00a0Makoto Oct 24 '16 at 18:02\n\u2022 @Makoto But the question would not make a good duplicate \"signpost\", because it was a code dump with barely a sentence of explanation. The OP can search the internet for themselves; searching on the title of the question returns How can I merge properties of two JavaScript objects dynamically? as the first hit, which would get them where they need to go. \u2013\u00a0Heretic Monkey Oct 24 '16 at 18:10\n\u2022 Confused about why it would be called a \"code dump\", or a reference would be made to the code they showed. No code was showed, or dumped. The OP didn't know where to start, which is not an unreasonable place to be. The point is that the 5 voters chose to close for invalid, unreasonable reasons. You are giving too much credence to folks with itchy fingers on the closevote button. There is no abstract notion of \"worth keeping\". There are downvotes, which are completely up to the discretion of the voter, and then there are valid, justified close votes. \u2013\u00a0user663031 Oct 24 '16 at 18:59\n\u2022 @torazaburo sure and those valid, justified close votes just happened. Let's move on. \u2013\u00a0rene Oct 24 '16 at 19:05\n\u2022 As a quick-note, even if we forget all the close votes and (un)delete votes and downvotes there after this Meta question, that particular SO question, even closed as duplicate, would have still been roombad in roughly 2 weeks. And this is not saying the close reason was appropriate, it certainly wasn't, and duplicate clearly would have been better. But since we're discussing this particular question... it actually wouldn't have changed anything. To discuss the appropriateness of close votes in a general manner, it would be better to make a separate post. \u2013\u00a0Tunaki Oct 24 '16 at 19:08\n\u2022 And yes the question was roombad before... so after closing it as duplicate, it would have been roombad a second time, if this is even a thing... \u2013\u00a0Tunaki Oct 24 '16 at 19:11\n\u2022 @torazaburo: \"The point is that the 5 voters chose to close for invalid, unreasonable reasons.\" No, the point is that a question which deserved to be closed was closed. Was it closed with the right close reason? Probably not. But that's far less important than the fact that a question which ought to be closed has been closed. \u2013\u00a0Nicol Bolas Oct 24 '16 at 19:13\n\u2022 @NicolBolas Umm, what was the right close reason? \u2013\u00a0user663031 Oct 24 '16 at 19:13\n\u2022 @rene The entire point of my post, as I would have thought would have been clear, is that those close votes were not valid or justified, by any interpretation. Essentially, you are saying that anyone can close anything for any reason at any time in the face of any set of facts to the contrary. If that is so, let's just get rid of all the close reasons and replace them with a single one saying \"I don't like this question and think it should be closed\". \u2013\u00a0user663031 Oct 24 '16 at 19:14\n\u2022 @torazaburo Actually, what you were told was that it's not worth the effort to undelete a question, reopen it, just so that you can then close it for a different reason and re-delete it. Nobody has told you that it's fine to close any quesiton for any reason, they've just told you that when a close-worthy question is closed for a wrong, or less applicable, reason, it's not really worth fixing, all the more so when it's deleted. \u2013\u00a0Servy Oct 24 '16 at 19:21\n\u2022 @torazaburo: \"Umm, what was the right close reason?\" Well, right now it is closed as a duplicate. So let's go with that. \u2013\u00a0Nicol Bolas Oct 24 '16 at 19:49\n\u2022 @torazaburo If you had spent half as much energy improving the question to a point where it wasn't such crap as you have defending your position, we wouldn't be having this discussion. \u2013\u00a0Heretic Monkey Oct 25 '16 at 15:03\n\u2022 @MikeMcCaughan This was never about the particular question. It was about people using random, inapplicable close reasons to close vote question they don't like. Actually, it was already answered early in the thread: vote to re-open, then possibly raise a flag if that doesn't work. \u2013\u00a0user663031 Oct 25 '16 at 18:44\n\nI believe \"no MCVE\" is \"be nice\" close reason as any other action on that post would presume lack of effort on OP part.\n\nIf I assume user made honest effort to research the question (like searched for \"JavaScript combine two objects\" or even \"JavaScript extend\"), than at that point it is reasonable to expect they need help with implementation and not to explain the same thing over and over again. And here you have \"no MCVE\" as close reason from me.\n\nAlternative - downvote and vote to close and claim OP did not bother to make any effort... While likely true it may not fit to \"be nice\" policy.\n\n\u2022 You seem to think that one first decides to close on some abstract grounds, and then goes rummaging around to find a close reason that fits some criteria such as \"being nice\". But actually, one should close vote if there's a close reason that actually fits, and not close vote if there's not. In this case, to repeat myself, the \"debugging\" close reason does not fit. Instead, the OP does not know where to begin. He doesn't even know the terms \"merge\" or \"deep merge\". If that level of ignorance, or failure to do research, is worthy of downvoting, then fine, downvote away. \u2013\u00a0user663031 Oct 25 '16 at 3:30\n\nHere is a screenshot of the question so it can be discussed now that it is deleted: https:\/\/i.stack.imgur.com\/Y6WUQ.png\n\nI agree with you that the question is on-topic, so I voted to reopen it. Your flag got declined though because you were asking a moderator to do something that you could do by yourself: to vote to reopen. He probably thought that the flag was unneccesary.\n\n\u2022 So you want this undeleted and then re-opened which seems to be a broad question, assuming there isn't a dupe? \u2013\u00a0rene Oct 24 '16 at 16:55\n\u2022 @rene sure, why not? Merging two objects seems a task that could be useful to wide audience. \u2013\u00a0user000001 Oct 24 '16 at 16:56\n\u2022 It is utter crap but I casted an undelete vote to see how many wondeful answers that will get \u2013\u00a0rene Oct 24 '16 at 16:57\n\u2022 It is undeleted now, reopen vote casted \u2013\u00a0rene Oct 24 '16 at 17:02\n\u2022 And it is deleted again \u2013\u00a0psubsee2003 Oct 24 '16 at 18:11\n\u2022 @psubsee2003 It's been reopened and is waiting on more undelete vote before it will be a live, open question \u2013\u00a0TylerH Oct 24 '16 at 18:21","date":"2019-10-16 02:21:09","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 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\": 0.3109360635280609, \"perplexity\": 1187.9294231475656}, \"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-2019-43\/segments\/1570986661296.12\/warc\/CC-MAIN-20191016014439-20191016041939-00509.warc.gz\"}"} | null | null |
download all files
## Requirements
PHP >= 5.6.4 ,
PHP Curl extension
## Install The Dependencies
now type this line on your console
```
composer install
```
```
php artisan key:generate
```
## migrate
```
php artisan migrate
```
## Add App key
go to streamlab website http://streamlab.io/
then open application get the `key` and `token` add them to `config/stream_lab.php`
```php
return[
'app_id'=>'',
'token'=>''
];
```
## Create Cahnnel
make sure you create public `channel` to your application call `geo`
## Start
```
php artisan serve
```
and enjoy
| {
"redpajama_set_name": "RedPajamaGithub"
} | 3,015 |
\section{introduction}
{\it Quantum walka} are dynamical systems related to quantum physics.
Many researchers study this subject in a number of frameworks.
They commonly use a pair of a Hilbert space $\Hil$ and a unitary operator $U$ on $\Hil$. The Hilbert space is associated to some space $X$. The space $\Hil$ is given by $\ell_2(X)$, $L^2(X)$, or their amplification.
There are two families of quantum walks. One is the family of discrete-time quantum walks.
These walks give unitary representations $(U^t)_{t \in \Z}$ of the integer group $\Z$.
We can regard the integer $t$ as the number of steps of some procedure.
The other is the family of continuous-time quantum walks.
Such a walk gives a unitary representation $(\exp(i t H))_{t \in \R}$ of the real group $\R$.
We can regard the real number $t$ as the flow of time.
For a discrete-time quantum walk $(U^t)_{t \in \Z}$, does there exist a {\it good} continuous-time quantum walk $(\exp(i t H))_{t \in \R}$ satisfying $\exp(i X) = U$?
A related open problem is proposed in \cite{Ambainis}.
For every unitary operator $U$, there exists a self-adjoint operator $H$ such that $\exp(i H) = U$. However, $X$ is not necessarily a {\it good} operator.
The unitary operator $\exp(i t H)$ ignores the base space $X$. Namely, there might exist unit vectors $\xi$ and $\eta$ such that the support of $\eta$ in $X$ is distant from that of $\xi$, and that the transition probability $|\langle \exp(i t H) \xi, \eta \rangle|^2$ is not small.
This means that the dynamical system by $(\exp(i t H))_{t \in \R}$ moves unit vectors too fast.
Therefore, we eliminate such a pathological walk and concentrate on walks satisfying some regularity. In this paper, we consider three kinds of regularity called {\it uniformity}, {\it smoothness}, or {\it analyticity} for operators on $\Hil$. Uniformity is the weakest, and analyticity is the strongest.
To the best of the author's knowledge,
all the known examples of one-dimensional quantum walks are analytic.
In this paper, we consider the case that the space $X$ is the integer group $\Z$, the local degree of freedom of $\Hil$ is finite, and $(U^t)_{t \in \Z}$ is a discrete-time homogeneous {\it analytic} quantum walk.
We determine whether $(U^t)_{t \in \Z}$ is realized by a continuous-time {\it uniform} quantum walk in Theorem \ref{theorem: continuous-time}.
To show this theorem, in Subsection \ref{subsection: uniform intertwiner}, we determine
the space of uniform intertwining operator between two one-dimensional homogeneous analytic quantum walks.
Before stating the main theorem (Theorem \ref{theorem: continuous-time}), we need to clarify the definition of one-dimensional quantum walks and regularity for operators on $\Hil$ in Section \ref{section: one-dimensional QW} (Definitions \ref{definition: one-dimensional QW}, \ref{definition: regularity}, \ref{definition: regularity for QW}).
Many results in Section \ref{section: one-dimensional QW} can be applied to general one-dimensional quantum walks, which are not necessarily of finite degree of freedom.
We also propose a new equivalence relation between one-dimensional quantum walks called {\it similarity}.
This new notion allows us to treat quantum walks in a flexible manner.
Similar walks have the same asymptotic behavior (Theorem \ref{theorem: similarity and limit distribution}).
For the argument in this paper,
we need a structure theorem on one-dimensional homogeneous quantum walks in \cite{SaigoSako}.
A concise abstract of the paper \cite{SaigoSako} is described in Subsection \ref{subsection: review of SS}.
Readers who wants to concretely understand the contents of this paper are recommended to read examples, skipping lemmas and propositions.
Among several examples, Example \ref{example: 4-state Grover walk} and Example \ref{example: 3-state Grover walk} introduce the $4$-state Grover walk and the $3$-state Grover walk.
Example \ref{example: 3-state Grover walk is continuous-time} shows that the $3$-state Grover walk can be realized by a continuous-time quantum walk, while
Example \ref{example: 4-state Grover walk is not continuous-time} shows that the $4$-state Grover walk can not.
\section{Definitions and basic properties of $1$-dimensional quantum walks}
\label{section: one-dimensional QW}
We construct a general framework for one-dimensional quantum walks as follows.
\begin{definition}\label{definition: one-dimensional QW}
One-dimensional {\rm discrete-time} quantum walk is a triplet $(\mathcal{H}$, $(U^{t})_{t \in \Z}$, $D)$ of
\begin{itemize}
\item
a Hilbert space $\mathcal{H}$,
\item
a unitary representation $(U^t)_{t \in \Z}$ of $\Z$ on $\mathcal{H}$,
\item
and a self-adjoint operator $D$. (In most cases, $D$ is unbounded.)
\end{itemize}
We call $U = U^1$ the generator of the quantum walk.
\end{definition}
\begin{definition}
One-dimensional {\rm continuous-time} quantum walk is a triplet $(\mathcal{H}$, $(U^{(t)})_{t \in \R}$, $D)$ of
\begin{itemize}
\item
a Hilbert space $\mathcal{H}$,
\item
a one-parameter group $(U^{(t)})_{t \in \R}$ of unitary operators on $\mathcal{H}$ which is continuous with respect to the strong operator topology,
\item
and a self-adjoint operator $D$. (In most cases, $D$ is unbounded.)
\end{itemize}
The self-adjoint operator $\lim_{t \to 0} \frac{U^{(t)} - 1}{it}$ is called the generator of the quantum walk.
\end{definition}
For the rest of this paper, we concentrate on one-dimensional quantum walks, so we simply call them {\it quantum walk}.
In most of preceding research, quantum walks are regarded as a dynamical system on some geometric space.
To fit the quantum walks defined above, we have only to define the operator $D$ as the observable of position on a one-dimensional space.
However, the above definition allows more flexible interpretations of quantum walks.
The self-adjoint operator $D$ can be other observables such as the momentum operator.
\subsection{Regularity on quantum walks}
By physical requirement,
we often assume a kind of regularity for operators such as {\it uniformity}, {\it smoothness},
or {\it analyticity}.
Note that for a map $f$ from the real line or a complex domain to a Banach space $B$, we can define differentiability on $f$ using the limit $\lim_{\Delta x \to 0} \frac{f(x + \Delta x) - f(x)}{\Delta x}$ in norm.
\begin{definition}\label{definition: regularity}
Let $D$ be a self-adjoint operator on $\Hil$ and let $U$ be a bounded operator on $\Hil$.
\begin{itemize}
\item
The operator $U$ is said to be {\rm uniform} with respect to $D$, if
the map
\[\R \ni k \mapsto e^{i k D} U e^{-i k D} \in \mathcal{B}(\mathcal{H})\]
is continuous with respect to the norm topology.
This condition implies that $k \mapsto e^{i k D} U e^{-i k D}$ is uniformly continuous.
\item
The operator $U$ is said to be {\rm smooth} or {\rm in the ${\rm C}^\infty$-class} with respect to $D$, if
the map
\[\R \ni k \mapsto e^{i k D} U e^{-i k D} \in \mathcal{B}(\mathcal{H})\]
is a smooth mapping with respect to the variable $k \in \R$.
\item
The operator $U$ is said to be {\rm analytic} with respect to $D$, if
there exists a holomorphic extension of the map
\[\R \ni k \mapsto e^{i k D} U e^{-i k D} \in \mathcal{B}(\mathcal{H})\]
defined on a domain of the form $\{\kappa \in \C | -\delta < {\rm Im}(\kappa) < \delta \}$.
\end{itemize}
\end{definition}
These conditions related to transition probability in quantum mechanics.
Consider the case that the spectrum of $D$ stands for position of some particle and that $U$ corresponds to some dynamical system.
Let $E(\cdot)$ be the spectral measure of $D$.
Let $\xi$ and $\eta$ be unit vectors in $\Hil$.
Suppose that the support of the measure $\langle E(\cdot) \xi, \xi \rangle$ is distant from that of $\langle E(\cdot) \eta, \eta \rangle$.
The conditions on regularity of $U$ mean that the matrix coefficient $\langle U \xi, \eta \rangle$ is small, if the support of $\xi$ with respect to the spectral decomposition of $D$ is distant from that of $\eta$.
See \cite[Definition 3.1]{SaigoSako} and \cite[Lemma 4.1]{SaigoSako}.
See also Proposition \ref{proposition: converging to an atom}.
Among the three conditions, uniformity is the weakest, and analyticity is the strongest.
The space of all the uniform operators forms a C$^*$-algebra.
The space of all the smooth operators forms a $*$-subalgebra.
The space of all the analytic operators also forms a $*$-subalgebra.
The main subject of this paper is {\it uniform} intertwiner between two homogeneous discrete-time {\it analytic} quantum walks.
If the operator $U$ is smooth with respect to $D$, the $m$-th derivative of $k \mapsto e^{i k D} U e^{-i k D}$ is given by the commutator
$i^m e^{ik D} [D, [D, \cdots [D, U] \cdots]] e^{-ik D}$. We put the commutator $[\cdot, \cdot]$ $n$-times here.
In particular, $[D, [D, \cdots [D, U] \cdots]]$ is a bounded operator.
This is a consequence of the following lemma.
\begin{lemma}\label{lemma: smoothness}
Let $V \colon \Hil_1 \to \Hil_2$ be a bounded operator.
Let $D_1$ be a self-adjoint operator on $\Hil_1$ and let
$D_2$ be a self-adjoint operator on $\Hil_2$.
Suppose that $\R \ni k \mapsto e^{ik D_2} V e^{-ik D_1} \in \mathcal{B}(\Hil_2 \leftarrow \Hil_1)$ is differentiable in the operator norm topology.
Then $V$ is a map from the domain of $D_1$ to that of $D_2$,
and
$D_2 V - V D_1 \colon {\rm dom}(D_1) \to \Hil_2$ is bounded with respect to the norm of $\Hil_1$.
The derivative of $k \mapsto e^{ik D_2} V e^{-ik D_1}$ is $i e^{ik D_2} (D_2 V - V D_1)e^{-ik D_1}$
\end{lemma}
\begin{proof}
Denote by $W$ the limit
\[W = \lim_{k \to 0} \dfrac{e^{ik D_2} V e^{-ik D_1} - V}{ik}\]
in the norm topology.
Let $\xi$ be an element of the domain of $D_1$.
Then we have
\begin{eqnarray*}
\frac{e^{ik D_2} - 1}{ik} V \xi
=
\frac{e^{ik D_2} V e^{-ik D_1} - V}{ik} \xi
-
e^{ik D_2} V \frac{ e^{-ik D_1} - 1}{ik} \xi.
\end{eqnarray*}
As $k$ tends to $0$, the first term converges to $W \xi$.
The norm of $e^{ik D_2}$ is uniformly bounded and $e^{ik D_2}$ converges to $1$ in strong operator topology.
The vector $\frac{ e^{-ik D_1} - 1}{ik} \xi$ converges to $- D_1 \xi$ in norm.
Therefore, the vector $\frac{e^{ik D_2} - 1}{ik} V \xi$ converges to
$W \xi + V D_1 \xi$. It follows that
$V\xi \in {\rm dom}D_2$, $D_2 V \xi = W \xi + V D_1 \xi$.
We calculate the derivative as follows:
\begin{eqnarray*}
& &
\lim_{\Delta k \to 0}
\frac{
e^{i (k + \Delta k) D_2} V e^{- i (k + \Delta k) D_1}
-
e^{i k D_2} V e^{- i k D_1}}
{\Delta k}
\\
&=&
i e^{i k D_2}
\lim_{\Delta k \to 0}
\frac{
e^{i \Delta k D_2} V e^{- i \Delta k D_1} - V}
{i \Delta k}
e^{- i k D_1}
\\
&=&
i e^{i k D_2}
W
e^{- i k D_1}.
\end{eqnarray*}
\end{proof}
\begin{definition}\label{definition: regularity for QW}
A discrete-time or continuous-time quantum walk $\left( \mathcal{H}, \left( U^{(t)} \right), D \right)$ is said to be
{\rm analytic} ({\rm smooth}, or {\rm uniform}), if for every $t$, $U^{(t)}$ is analytic (smooth, or uniform, respectively) with respect to $D$.
\end{definition}
\begin{remark}
In the case of continuous-time quantum walks, the author is not so sure about the above definition. A definition might be given by the relation between $D$ and the generator of the one-parameter unitary group $(U^{(t)})_{t \in \R}$, and would be stronger than our condition.
To state the main result of this paper,
we choose the weaker condition as above.
\end{remark}
\subsection{Basic examples of quantum walks}
The following are examples of quantum walks. The examples \ref{example: constant quantum walk} to \ref{example: the inverse Fourier transform} are analytic. We also define several notations, which are often used for the rest of this paper.
\begin{example}[Constant quantum walk]
\label{example: constant quantum walk}
Let $\alpha$ be a complex number whose absolute value is $1$.
The triplet $(\Hil, (\alpha^t)_{t \in \Z}, D)$ is a discrete-time quantum walk.
\end{example}
\begin{example}
[Discrete-time free quantum walk]
\label{example: discrete free QW}
Let $r$ be a positive real number.
Let $D_r$ be the diagonal operator on $\ell_2 (r \Z)$ defined by $D_r(\delta_x) = x \delta_x, x \in r \Z$.
Denote by $S_r$ the bilateral shift $\delta_x \mapsto \delta_{x+r}, x \in r \Z$.
Then $(\ell_2(r \Z), (S_r^t)_{t \in \Z}, D_r)$ is a one-dimensional discrete-time quantum walk.
We call $(\ell_2(r \Z), (S_r^t)_{t \in \Z}, D_r)$ {\it the discrete-time free quantum walk}.
The map
\[\R \ni k \mapsto \exp(i k D_r) S_r \exp(- i k D_r) \in \B(\ell_2(r \Z))\]
is given by
$\exp(i k D_r) S_r \exp(- i k D_r) = \exp(i k r) S_r$.
This can be extended to a holomorphic map defined on the complex plane $\C$.
The positive number $r$ is often defined by $1$.
\end{example}
\begin{example}
[Continuous-time free quantum walk]
Let $D$ be the multiplication operator on $L^2 (\R)$ given by the function $x \mapsto x$ on $\R$.
Let $X$ be the differential operator $- \frac{d}{i dx}$ on $L^2 (\R)$.
The one-parameter unitary group $(\exp(it X))_{t \in \R}$ generated by $X$ is the translation operator given by
\[[\exp(it X)(\xi)](x) = \xi(x - t), \quad \xi \in L^2(\R), x \in \R.\]
Then $\left( L^2(\R), (\exp(i t X))_{t \in \R}, D_1 \right)$ is a continuous-time quantum walk.
We call $( L^2(\R)$, $(\exp(it X))_{t \in \R}$, $D_1 )$ {\it the continuous-time free quantum walk}.
\end{example}
\begin{example}
[$(2 \times 2)$-matrix]
Let
$
u_x =
\left(
\begin{array}{cc}
a_x & b_x\\
c_x & d_x
\end{array}
\right), x \in \Z
$
be a two-sided infinite sequence of complex unitary matrices.
Define a unitary operator $U$ on $\ell_2(\Z) \otimes \C^2$ as follows:
\begin{eqnarray*}
U (\delta_x \otimes \delta_1) &=& a_x \delta_{x - 1} \otimes \delta_1 + c_x \delta_{x + 1} \otimes \delta_2,\\
U (\delta_x \otimes \delta_2) &=& b_x \delta_{x - 1} \otimes \delta_1 + d_x \delta_{x + 1} \otimes \delta_2, \quad x \in \Z.
\end{eqnarray*}
Let $D$ be the diagonal operator on $\ell_2(\Z)$ defined in Example \ref{example: discrete free QW}. (In the present case, $r$ is $1$).
Then $(\ell_2(\Z) \otimes \C^2, (U^t)_{t \in \Z}, D \otimes {\rm id})$ is a discrete-time quantum walk.
\end{example}
\begin{example}
[Homogeneous $(2 \times 2)$-quantum walk]
\label{example: 2by2 homogeneous}
In the previous example, consider the case that
$u_x$ is a constant sequence
$u_x =
\left(
\begin{array}{cc}
a & b\\
c & d
\end{array}
\right)$.
Then the unitary operator $U$ is given by
$U = \left(
\begin{array}{cc}
a S_1^{-1} & b S_1^{-1}\\
c S_1 & d S_1
\end{array}
\right)$.
The triplet
$(\ell_2(\Z) \otimes \C^2, (U^t), D_1 \otimes {\rm id})$ is a homogeneous discrete-time quantum walk.
\end{example}
\begin{example}\label{example: the inverse Fourier transform}
Denote by $\T$ the set of all the complex numbers
whose absolute values are $1$.
The dual group of $r \Z$ is given by
$\T_{2 \pi r^{-1}} = \R /(2 \pi r^{-1} \Z)$ via the coupling
\[r \Z \times \T_{2 \pi r^{-1}} \ni (x, k + 2 \pi r^{-1} \Z) \mapsto \exp(i x k) \in \T.\]
We distinguish $\T_{2 \pi r^{-1}}$ from $\T$ in this paper.
The subscript $2 \pi r^{-1}$ is equal to the length of the torus $\T_{2 \pi r^{-1}}$.
We denote by $c_x$ the character on $\T_{2 \pi r^{-1}}$ defined by $x \in r \Z$.
Denote by $\mathcal{F}_r \colon L^2(\T_{2 \pi r^{-1}}) \to \ell_2(r \Z)$
the Fourier transform given by $c_x \mapsto \delta_x$.
The inverse Fourier transform $\widehat{D_r} = \F_r^{-1} D_r \F_r$ of $D_r$ in Example \ref{example: discrete free QW} is
\[ \left[ \widehat{D_r}(\xi) \right] \left( e^{ik} \right)
=
\dfrac{1}{i}\frac{d \xi}{dk} \left( e^{ik} \right),
\quad \xi \in {\rm C}^\infty(\T_{2 \pi r^{-1}}),
k + 2 \pi r^{-1} \Z \in \T_{2 \pi r^{-1}}.\]
We simply denote by $D_r$ the inverse Fourier transform $\widehat{D_r}$.
Here $D_r$ stands for the differential operator $\frac{d}{i d k}$.
The inverse Fourier transform $\F_r^{-1} S_r \F_r$ of the bilateral shift $S_r$ in Example \ref{example: discrete free QW} is the multiplication operator $M[c_r]$.
The inverse Fourier transform of the discrete-time free quantum walk in Example \ref{example: discrete free QW} is
$(L^2(\T_{2 \pi r^{-1}}), (M[c_r]^t)_{r \in \Z}, D_r)$.
This is also a quantum walk.
\end{example}
\begin{example}\label{example: the inverse Fourier transform of 2 by 2}
The inverse Fourier transform $\widehat{U} = (\F_1^{-1} \otimes {\rm id}) U (\F_1 \otimes {\rm id})$ of $U$ in Example \ref{example: 2by2 homogeneous} is
\[\widehat{U} = \left(
\begin{array}{cc}
a M[c_1]^{-1} & b M[c_1]^{-1}\\
c M[c_1] & d M[c_1]
\end{array}
\right).\]
Here $M[c_1]\colon L^2(\T_{2 \pi}) \to L^2(\T_{2 \pi})$ is the multiplication operator given by $c_1$.
The triplet
$\left( L^2(\T_{2 \pi}) \otimes \C^2, \left( \widehat{U}^t \right)_{t \in \Z}, D_1 \otimes {\rm id} \right)$ is also a quantum walk.
\end{example}
\begin{example}
[Quantum walk by a multiplication operator]
\label{example: QW by multiplication operator}
Let $\lambda \colon \T_{2 \pi r^{-1}} \to \T$ be a Borel function.
Denote by $M[\lambda] \colon L^2(\T_{2 \pi r^{-1}}) \to L^2(\T_{2 \pi r^{-1}})$ the multiplication operator given by $\lambda$.
The triplet $\left( L^2(\T_{2 \pi r^{-1}}), (M[\lambda]^t)_{t \in \Z}, D_r = \frac{d}{i d k} \right)$ is a discrete-time quantum walk.
This type of quantum walks will often appear in this paper.
The walk is analytic (smooth, or uniform), if $\lambda$ analytic (smooth, or continuous, respectively).
\end{example}
\begin{example}[Direct sum]
Let $\left(\Hil_1, \left(U_1^{(t)}\right), D_1\right)$ and $\left(\Hil_2, \left(U_2^{(t)}\right), D_2\right)$ be two continuous-time or discrete-time quantum walks.
Then $(\Hil_1 \oplus \Hil_2$, $(U_1^{(t)} \oplus U_2^{(t)} )$, $D_1 \oplus D_2 )$ is also a quantum walk.
We call it {\it the direct sum quantum walk}.
\end{example}
\begin{example}[Amplification]
Let $m$ be a natural number.
Let $\left( \Hil, \left( U^{(t)} \right), D \right)$ be a quantum walk.
Then $\left(\Hil \otimes \C^n, \left(U^{(t)} \otimes {\rm id}\right), D \otimes {\rm id}\right)$ is also a quantum walk. We call it {\it the amplification quantum walk}.
\end{example}
Analyticity, smoothness, and uniformity are preserved under direct sum and amplification.
\subsection{Intertwiners between two quantum walks and their regularity}
\begin{definition}
Let $(\mathcal{H}_1, (U_1^{t})_{t \in \Z}, D_1)$ and $(\mathcal{H}_2, (U_2^{t})_{t \in \Z}, D_2)$
be two one-dimensional discrete-time quantum walks.
A bounded operator $X \colon \Hil_1 \to \Hil_2$ is called an {\rm intertwiner} between them, if it satisfies $X U_1 = U_2 X$.
\end{definition}
An intertwiner between $(\Hil_1, (U^{t})_{t \in \Z}, D)$ and itself is nothing other than an operator $X \in \mathcal{B}(\Hil)$ which commutes with $U$.
\begin{definition}
If
the mapping $\R \ni k \mapsto e^{i k D_2} X e^{-ik D_1} \in \mathcal{B}(\Hil_2 \leftarrow \Hil_1)$ is continuous (or smooth),
the intertwiner $X$ is said to be {\rm uniform} (or {\rm smooth}) with respect to $D_1$ and $D_2$.
If there exists a holomorphic extension of the map
\[\R \ni k \mapsto e^{i k D_2} X e^{-i k D_1} \in \mathcal{B}(\mathcal{H})\]
defined on a domain of the form
$\{\kappa \in \C | -\delta < {\rm Im}(\kappa) < \delta \}$,
$X$ is said to be {\rm analytic}.
\end{definition}
The bounded operator $X \colon \Hil_1 \to \Hil_2$ defines an operator
\[\widetilde{X} =
\left(
\begin{array}{cc}
0 & 0\\
X & 0\\
\end{array}
\right)
\colon \Hil_1 \oplus \Hil_2
\to
\Hil_1 \oplus \Hil_2 .
\]
The operator $X$ is an intertwiner between $U_1$ and $U_2$, if and only if $\widetilde{X}$ commutes with $U_1 \oplus U_2$.
The intertwiner $X$ is uniform (smooth, or analytic) with respect to $D_1$ and $D_2$,
if and only if $\widetilde X$ is uniform (smooth, or analytic, respectively) with respect to $D_1 \oplus D_2$.
Let $(\mathcal{H}_3, (U_3^{t})_{t \in \Z}, D_3)$ be another quantum walk.
If $X_1 \colon \Hil_1 \to \Hil_2$ is an intertwiner between $U_1$ and $U_2$, and if
$X_2 \colon \Hil_2 \to \Hil_3$ is an intertwiner between $U_2$ and $U_3$,
then
If $X_2 X_1 \colon \Hil_1 \to \Hil_3$ is an intertwiner between $U_1$ and $U_3$.
If $X_1$ is uniform with respect to $D_1$ and $D_2$, and if $X_2$ is uniform with respect to $D_2$ and $D_3$, then
$X_2 X_1$ is uniform with respect to $D_1$ and $D_3$.
Smoothness and analyticity are also preserved under this composition procedure.
To determine whether there exists a non-zero uniform intertwiner between given two homogeneous quantum walks, we use the following as a key tool.
\begin{proposition}\label{proposition: converging to an atom}
Let $r(1)$ and $r(2)$ be positive real numbers.
Let $V$ be a bounded operator from $\ell_2(r(1) \Z)$ to $\ell_2(r(2) \Z)$.
Let $D_{r(1)}$ be the diagonal operator $\delta_y \mapsto y \delta_y, y \in r(1) \Z$ on $\ell_2 (r(1) \Z)$.
Let $D_{r(2)}$ be the diagonal operator $\delta_x \mapsto x \delta_x, x \in r(2) \Z$ on $\ell_2 (r(2) \Z)$.
Suppose that
$V$ is uniform with respect to $D_{r(1)}$ and $D_{r(2)}$.
For $y \in r(1) \mathbb{N}$, define a probability measure $p_y$ on $\mathbb{R}$ by
\[ p_y = \sum_{x \in r(2) \Z} \left| \left\langle V \delta_y, \delta_x \right\rangle_{\ell_2(r(2) \Z)} \right|^2 \delta_{x / y},\]
where $\delta_{y / x}$ stands for the point mass at $y / x \in \R$.
Then for every positive number $\delta$, we have
\[\lim_{y \to \infty} p_y((-\infty, 1 - \delta] \cup [1 + \delta, \infty)) = 0\]
\end{proposition}
\begin{proof}
For every real number $k$, the matrix coefficient of $\exp (i k D_{r(2)}) V \exp (-ik D_{r(1)})$ at $(x, y)$ is
\[\left\langle \exp (i k D_{r(2)}) V \exp (-ik D_{r(1)}) \delta_y, \delta_x \right\rangle_{\ell_2(r(2) \Z)}
= \exp(ik (x - y)) \left\langle V \delta_y, \delta_x \right\rangle_{\ell_2(r(2) \Z)}.
\]
For a positive real number $\sigma$, define an operator $V_\sigma$ by
\[V_\sigma = \int_{-\infty}^\infty \exp \left( -\frac{k^2}{2 \sigma^2} \right) \exp (i k D_{r(2)}) V \exp (-ik D_{r(1)}) \dfrac{dk}{\sqrt{2 \pi} \sigma}.\]
Note that the operator norm of $V_\sigma$ is no more than that of $V$.
We also note that as $\sigma$ tends to $0$, $V_\sigma$ converges to $V$ in the operator norm topology.
The matrix coefficient of $V_\sigma$ is
\begin{eqnarray*}
\left\langle V_\sigma \delta_y, \delta_x \right\rangle_{\ell_2(r(2) \Z)}
&=& \int_{-\infty}^\infty \exp \left( -\frac{k^2}{2 \sigma^2} \right) \exp(ik (x - y)) \dfrac{dk}{\sqrt{2 \pi} \sigma} \left\langle V \delta_y, \delta_x \right\rangle_{\ell_2(r(2) \Z)}\\
&=&
\exp\left( -\frac{(x - y)^2}{2} \sigma^2 \right)
\left\langle V \delta_y, \delta_x \right\rangle_{\ell_2(r(2) \Z)}.
\end{eqnarray*}
Take arbitrary (small) positive real numbers $\delta$ and $\epsilon$.
There exists a (small) positive real number $\sigma$ such that
$\| V - V_\sigma \| < \epsilon$.
For such $\epsilon$ and $\sigma$,
there exists a positive number $K$ such that for every $y \in r(1) \Z$,
\[\sum_{x \in r(2) \Z, |x - y| \ge K} \exp\left( - {(x - y)^2} \sigma^2 \right) < \epsilon.\]
We consider the case that $y \in r(1) \Z$ is larger than $K / \delta$.
Then
\begin{eqnarray*}
p_y(\R \setminus (1 - \delta, 1 + \delta))
&=&
\sum_{x \in r(2) \Z, |x / y - 1| \ge \delta} \left| \left\langle V \delta_y, \delta_x \right\rangle_{\ell_2(r(2) \Z)} \right|^2.
\end{eqnarray*}
Since the inequality $|x / y - 1| \ge \delta$ implies $|x - y| \ge |y| \delta \ge K$,
we have
\begin{eqnarray*}
p_y(\R \setminus (1 - \delta, 1 + \delta))
&\le&
\sum_{x \in r(2) \Z, |x - y| \ge K} \left| \left\langle V \delta_y, \delta_x \right\rangle_{\ell_2(r(2) \Z)} \right|^2.
\end{eqnarray*}
We further obtain the following inequalities
\begin{eqnarray*}
& &p_y(\R \setminus (1 - \delta, 1 + \delta))\\
&\le&
\sum_{x \in r(2) \Z, |x - y| \ge K}
\left| \left\langle V_\sigma \delta_y, \delta_x \right\rangle
+ \left\langle (V - V_\sigma) \delta_y, \delta_x \right\rangle \right|^2\\
&\le&
2 \sum_{x \in r(2) \Z, |x - y| \ge K}
\left| \left\langle V_\sigma \delta_y, \delta_x \right\rangle \right|^2
+
2 \sum_{x \in r(2) \Z, |x - y| \ge K}
\left| \left\langle (V - V_\sigma) \delta_y, \delta_x \right\rangle \right|^2\\
&\le&
2 \sum_{x \in r(2) \Z, |x - y| \ge K}
\exp\left( - {(x - y)^2} \sigma^2 \right) \left| \left\langle V \delta_y, \delta_x \right\rangle \right|^2
+
2 \left\| (V - V_\sigma) \delta_y \right\|^2.
\end{eqnarray*}
By the assumptions on $\sigma$ and $K$, we have
\begin{eqnarray*}
p_y(\R \setminus (1 - \delta, 1 + \delta))
\le
2 \|V\| \epsilon
+
2 \|V - V_\sigma\|^2
\le
2 \|V\| \epsilon
+
2 \epsilon^2.
\end{eqnarray*}
It follows that the positive measure $p_y$ tends to $0$ on $\R \setminus (1 - \delta, 1 + \delta)$.
\end{proof}
\begin{remark}\label{remark: uniform Roe algebra}
A bounded operator $V \colon \ell_2 \Z \to \ell_2 \Z$ is continuous with respect to the standard diagonal operator $D_1 \colon \delta_x \to x \delta_x, x \in \Z$, if and only if $V$ is an element of the uniform Roe algebra ${\rm C}^*_{\rm u} (\Z)$ defined in \cite[Subsection 4.5]{Roe}.
We can easily prove it, using $V_\sigma$ introduced in the above proof.
In this paper, we regard ${\rm C}^*_{\rm u} (\Z)$ as the space of operators on $\ell_2(\Z)$ which are uniform with respect to $D_1$.
\end{remark}
\subsection{Similarity between discrete-time quantum walks}
\begin{definition}
\label{definition: similarity}
Discrete-time quantum walks $(\mathcal{H}_1, (U_1^{t})_{t \in \Z}, D_1)$ and $(\mathcal{H}_2$, $(U_2^{t})_{t \in \Z}$, $D_2)$ are said to be {\rm similar}, if
there exists a unitary operator $V \colon \Hil_1 \to \Hil_2$ which is a {\it smooth} intertwiner between $(\mathcal{H}_1, (U_1^{t})_{t \in \Z}, D_1)$ and $(\mathcal{H}_2, (U_2^{t})_{t \in \Z}, D_2)$.
If $V$ maps the domain of $D_1$ to that of $D_2$ and $D_2 V = V D_1$ holds, or equivalently, if the mapping $k \mapsto \exp(i k D_2) V \exp(-ik D_1)$ is constant, then these walks are said to be {\rm unitary equivalent}.
\end{definition}
If $(\mathcal{H}_1, (U_1^{t})_{t \in \Z}, D_1)$
and $(\mathcal{H}_2, (U_2^{t})_{t \in \Z}, D_2)$ are similar,
and if one of them is smooth, then
the other is also smooth.
Similarity is an equivalence relation on smooth quantum walks.
\begin{example}
Example \ref{example: discrete free QW} and Example
\ref{example: the inverse Fourier transform} are unitary equivalent.
Example \ref{example: 2by2 homogeneous} and Example \ref{example: the inverse Fourier transform of 2 by 2} are also unitary equivalent.
The Fourier transform is a smooth intertwining operator.
\end{example}
\begin{example}\label{example: replacing D}
Let $(\Hil, (U^{t})_{t \in \Z}, D_1)$ be a smooth quantum walk.
Let $D_2$ be a self-adjoint operator on $\Hil$.
If the mapping $\R \ni k \mapsto e^{ik D_2} e^{- i k D_1}$ is smooth, then the
quantum walks $(\Hil, (U^{t})_{t \in \Z}, D_1)$ and $(\Hil, (U^{t})_{t \in \Z}, D_2)$ are similar.
Indeed, the identity operator gives a smooth intertwining operator between them.
\end{example}
\begin{example}
Let $(\Hil, (U^{t})_{t \in \Z}, D)$ be a smooth quantum walk.
Let $V$ be a unitary operator on $\Hil$. If $V$ is smooth with respect to $D$, then
quantum walks $(\Hil, (U^{t})_{t \in \Z}, D)$ and $(\Hil, (V U^{t} V^{-1})_{t \in \Z}, D)$ are similar.
Indeed, the unitary operator $V$ gives a smooth intertwining operator between them.
\end{example}
Similarity is compatible with direct sum and with amplification.
\subsection{Asymptotic behavior of quantum walks}
Let $(\Hil, (U^t)_{t \in \Z}, D)$ be a discrete-time smooth quantum walk.
Fix a unit vector $\xi$ in $\Hil$.
We often call $\xi$ an initial unit vector of the quantum walk.
Let $E(\cdot)$ be the spectral projection of $D$.
Recall that $E$ maps Borel subsets of $\R$ to orthogonal projection in $\mathcal{B}(\Hil)$.
For every $t \in \mathbb{N}$, we have a probability measure on $\R$ defined by
$\langle E(\cdot) U^t \xi, U^t \xi \rangle$. We pay attention on the push-forward $p_t$ of the measure under the mapping $\R \ni x \mapsto x / t \in \R$.
The measure $p_t$ is given by
\[p_t(\Omega) = \left\langle E(t \Omega) U^t \xi, U^t \xi \right\rangle, \quad {\rm a\ Borel\ subset\ } \Omega \subset \R. \]
The vector $\xi \in \Hil$ is said to be {\it smooth} with respect to $D$, if $\xi \in \bigcap_{m \in \N} \mathrm{dom} (D^m)$.
We often assume that
the quantum walk $(\Hil, (U^t)_{t \in \Z}, D)$ is smooth and that the initial unit vector
$\xi$ is smooth.
It is not hard to see that for every integer $t$, $U^t \xi$ is also smooth with respect to $D$.
\begin{lemma}\label{lemma: moment}
The $m$-th moment of $p_t$ is
$\displaystyle \int_{v \in \R} v^m \cdot p_t(dv)
=
\left\langle \dfrac{1}{t^m} D^m U^t \xi, U^t \xi \right\rangle.$
\end{lemma}
\begin{proof}
For every $t \in \N$,
we calculate the $m$-th moment of $p_t$ as follows:
\begin{eqnarray*}
\int_{v \in \R} v^m \cdot p_t(dv)
&=&
\int_{v \in \R} v^m \cdot \langle E(t dv) U^t \xi, U^t \xi \rangle \\
&=&
\int_{x \in \R} \dfrac{1}{t^m} x^m \cdot \langle E(dx) U^t \xi, U^t \xi \rangle.
\end{eqnarray*}
This is nothing other than the right hand side of the lemma.
\end{proof}
\begin{definition}
If the weak limit of $p_t$ exists, it is called the {\rm limit distribution} of the quantum walk $(\Hil, (U^t)_{t \in \Z}, D)$ with respect to the vector $\xi$.
\end{definition}
\begin{lemma}\label{lemma: upper bound for moments}
Let $(\Hil, (U^t)_{t \in \Z}, D)$ be a discrete-time {\rm smooth} quantum walk.
Let $\xi \in \Hil$ be a unit vector.
Assume that $\xi$ is smooth with respect to $D$.
Then we have
\begin{eqnarray*}
\limsup_t \left| \int_{v \in \R} v^m \cdot p_t(dv) \right|
\le \|[D, U]\|^m.
\end{eqnarray*}
\end{lemma}
\begin{proof}
For a while, we fix $m$ and consider the case that $t$ is large.
By Lemma \ref{lemma: smoothness}, we can define a sequence of bounded operators $u_0, u_1, u_2, \cdots$ as follows:
\[u_0 = U, \quad u_1 = [D, u_0], \quad u_2 = [D, u_1], \cdots.\]
By smoothness of $\xi$, we can also define a sequence of vectors $\xi_0, \xi_1, \xi_2, \cdots \in \Hil$ by
\[\xi_t = D^t \xi, \quad t \in \N.\]
For smooth operators and smooth vectors, the following Leibniz rule holds:
\begin{eqnarray*}
D X \eta &=& X' \eta + X \eta', \quad {\rm where }\ X' = [D, X], \eta' = D \eta,
\\
\left[D, X Y\right] &=& X' Y + X Y', \quad {\rm where } \ X' = [D, X], Y' = [D, Y].
\end{eqnarray*}
By the Leibniz rule, the vector $D^m U^t \xi$ can be expressed as follows:
\begin{eqnarray}\label{equation: Leibniz rule}
D^m U^t \xi = \sum_{s \in I} u_{\sharp s^{-1}(t)} u_{\sharp s^{-1}(t - 1)} \cdots u_{\sharp s^{-1}(1)} \xi_{\sharp s^{-1}(0)}.
\end{eqnarray}
In this formula,
\begin{itemize}
\item
$s$ is an element of the index set
\[ I = \left\{ s \colon \{1, 2, \cdots, m\} \to \{0, 1, \cdots, t\} \ | \ \rm{a\ map}\right\},\]
\item
$s^{-1} (j)$ is the inverse image of $\{j\} \subset \{0, 1, \cdots, t\}$ under the mapping $s$.
\item
$\sharp s^{-1} (j)$ is the number of elements of the inverse image.
\end{itemize}
Define a subset $I_0$ of $I$ as follows:
\[ I_0 = \left\{ s \colon \{1, 2, \cdots, m\} \to \{0, 1, \cdots, t\} \in I \ | \
s^{-1}(0) = \emptyset, s {\rm\ is\ injective}
\right\}.\]
It is not hard to see that if $t$ is large, almost all the elements of $I$ are in $I_0$.
More precisely, $\lim_{t \to \infty} \sharp I_0 / \sharp I$ is $1$.
For $s \in I$, the norm of the term $u_{\sharp s^{-1}(t)}$ $u_{\sharp s^{-1}(t - 1)}$ $\cdots$ $u_{\sharp s^{-1}(1)}$ $\xi_{\sharp s^{-1}(0)}$ in the equation (\ref{equation: Leibniz rule})
is bounded by
\[\max\{1 = \|u_0\|, \|u_1\|, \cdots, \|u_m\| \}^m \max\{1 = \|\xi\|, \|\xi_1\|, \cdots, \|\xi_m\|\}.\]
It follows that
\begin{eqnarray*}
\limsup_t \left\| \dfrac{1}{t^m} D^m U^t \xi \right\|
&\le&
\limsup_t
\dfrac{1}{\sharp I}
\sum_{s \in I} \left\| u_{\sharp s^{-1}(t)} u_{\sharp s^{-1}(t - 1)} \cdots u_{\sharp s^{-1}(1)} \xi_{\sharp s^{-1}(0)} \right\|\\
&=&
\limsup_t
\dfrac{1}{\sharp I}
\sum_{s \in I_0} \left\| u_{\sharp s^{-1}(t)} u_{\sharp s^{-1}(t - 1)} \cdots u_{\sharp s^{-1}(1)} \xi_{\sharp s^{-1}(0)} \right\|\\
&\le&
\limsup_t
\dfrac{\sharp I_0}{\sharp I}
\|u_1\|^m
\\
&=&
\|[D, U]\|^m
\end{eqnarray*}
Combining with the equation in Lemma \ref{lemma: moment}, for every positive integer $m$, we have
\begin{eqnarray*}
\limsup_t \left| \int_{v \in \R} v^m \cdot p_t(dv) \right|
=
\limsup_t \left| \left\langle \dfrac{1}{t^m} D^m U^t \xi, U^t \xi \right\rangle \right|
\le
\|[D, U]\|^m.
\end{eqnarray*}
\end{proof}
\begin{proposition}\label{proposition: compact support}
Let $(\Hil, (U^t)_{t \in \Z}, D)$ be a discrete-time smooth quantum walk.
Let $\xi \in \Hil$ be a smooth unit vector.
For every $L \in \R$ larger than $\|[D, U]\|$,
we have
\[\lim_{t \to \infty} p_t((-\infty, -L] \cup [L, \infty)) = 0.\]
\end{proposition}
\begin{proof}
By Lemma \ref{lemma: upper bound for moments},
for every positive integer $m$, we obtain the following:
\begin{eqnarray*}
\limsup_t p_t ((-\infty, -L] \cup [L, \infty))
&\le&
\limsup_t \dfrac{1}{L^{2m}} \int_{v \in (-\infty, -L] \cup [L, \infty)} v^{2m} \cdot p_t (dv)\\
&\le&
\limsup_t \dfrac{1}{L^{2m}} \int_{v \in \R} v^{2m} \cdot p_t (d v)\\
&\le&
\dfrac{ \|[D, U]\|^{2m}}{L^{2m}}.
\end{eqnarray*}
Since the positive integer $m$ is arbitrary, the conclusion follows.
\end{proof}
\begin{corollary}
Let $(\Hil, (U^t)_{t \in \Z}, D)$ be a discrete-time smooth quantum walk.
Let $\xi \in \Hil$ be a smooth unit vector.
If the limit distribution exists, then its support is compact.
\end{corollary}
\begin{proposition}\label{proposition: two kinds of convergence}
Let $(\Hil, (U^t)_{t \in \Z}, D)$ be a discrete-time smooth quantum walk.
Let $\xi \in \Hil$ be a smooth unit vector.
Let $p_\infty$ be a Borel measure on $\R$.
The sequence of the measures $p_t$ weakly converges to $p_\infty$, if and only if it converges to $p_\infty$ in law.
\end{proposition}
`If part' of this proposition is a consequence of the general theory like \cite[Theorem 4.5.5]{Chung}.
We give a proof to make the argument self-contained.
\begin{proof}
If $p_t$ converges to $p_\infty$ in law,
then the support of $p_\infty$ is included in $[- \|[D, U]\|$, $\|[D, U]\|]$,
by Lemma \ref{lemma: upper bound for moments}.
If $p_t$ weakly converges to $p_\infty$,
then the support of $p_\infty$ is included in $[- \|[D, U]\|$, $\|[D, U]\|]$,
by Proposition \ref{proposition: compact support}.
Let $\epsilon$ be an arbitrary positive real number less than $1$.
Take a real number $L$ larger than $\|[D, U]\|$.
For a bounded continuous function $f$ on $\R$,
and for a polynomial function $g$ on $\R$ satisfying
\begin{eqnarray}\label{equation: f and g}
|g(v) - f(v)| < \epsilon, \quad v \in [- L, L],
\end{eqnarray}
we have
\begin{eqnarray}\label{equation: p infty}
\left| \int_{v \in \R} f(v) p_\infty(dv) - \int_{v \in \R} g(v) p_\infty(dv) \right| < \epsilon.
\end{eqnarray}
Since the function $f$ is bounded and $g$ is polynomial, there exists a positive integer $m$
such that
\begin{itemize}
\item
for $v \in ( - \infty, - L] \cup [L, \infty)$,
$| f(v) - g(v) | < \left( \dfrac{v}{L} \right)^{2m}$, and
\item
$\left( \dfrac{\|[D, U]\|}{L} \right)^{2m} < \epsilon$.
\end{itemize}
For such a natural number $m$, we have
\begin{eqnarray*}
& &
\left| \int_{v \in \R} f(v) p_t(dv) - \int_{v \in \R} g(v) p_t(dv) \right|\\
&\le&
\int_{-L}^L | f(v) - g(v) | p_t(dv) + \int_{v \in ( - \infty, - L] \cup [L, \infty)} | f(v) - g(v) | p_t(dv) \\
&\le&
\epsilon + \int_{v \in \R} \left( \dfrac{v}{L} \right)^{2m} p_t(dv).
\end{eqnarray*}
By Lemma \ref{lemma: upper bound for moments},
if $t$ is large enough,
then we have
\begin{eqnarray}\label{equation: p t}
\left| \int_{v \in \R} f(v) p_t(dv) - \int_{v \in \R} g(v) p_t(dv) \right|
\le
\epsilon + \left( \dfrac{\|[D, U]\|}{L} \right)^{2m} + \epsilon < 3 \epsilon.
\end{eqnarray}
Suppose
that $p_t$ converges to $p_\infty$ in law.
Take an arbitrary bounded continuous function $f$ on $\R$.
There exists a polynomial function $g$ satisfying the inequality
(\ref{equation: f and g}), by the approximation theorem of Weierstrass.
The inequality (\ref{equation: p infty}) follows.
If $t$ is large enough,
we obtain the inequality (\ref{equation: p t}).
By the definition of convergence in law, if $t$ is large enough,
\begin{eqnarray}\label{equation: g}
\left| \int_{v \in \R} g(v) p_t(dv) - \int_{v \in \R} g(v) p_\infty(dv) \right| < \epsilon.
\end{eqnarray}
The inequalities (\ref{equation: p infty}), (\ref{equation: p t}), and (\ref{equation: g})
implies
that if $t$ is large enough,
\begin{eqnarray*}
\left| \int_{v \in \R} f(v) p_t(dv) - \int_{v \in \R} f(v) p_\infty(dv) \right|
< 5 \epsilon.
\end{eqnarray*}
We conclude that $p_t$ weakly converges to $p_\infty$.
Conversely suppose that $p_t$ weakly converges to $p_\infty$.
Take an arbitrary polynomial function $g$.
Then there exists a bounded continuous function $f$ on $\R$ satisfying the inequality
(\ref{equation: f and g}).
Then we have the inequality (\ref{equation: p infty}).
If $t$ is large enough,
we obtain the inequality (\ref{equation: p t}).
By the definition of weak convergence, if $t$ is large enough,
\begin{eqnarray}\label{equation: f}
\left| \int_{v \in \R} f(v) p_t(dv) - \int_{v \in \R} f(v) p_\infty(dv) \right| < \epsilon.
\end{eqnarray}
The inequalities (\ref{equation: p infty}), (\ref{equation: p t}), and (\ref{equation: f})
implies
that if $t$ is large enough,
\begin{eqnarray*}
\left| \int_{v \in \R} g(v) p_t(dv) - \int_{v \in \R} g(v) p_\infty(dv) \right|
< 5 \epsilon.
\end{eqnarray*}
We conclude that $p_t$ converges to $p_\infty$ in law.
\end{proof}
When we discuss the limit distribution,
we can freely replace the original quantum walk with similar one.
\begin{theorem}\label{theorem: similarity and limit distribution}
Assume that two smooth quantum walks $(\mathcal{H}_1, (U_1^{t})_{t \in \Z}, D_1)$ and $(\mathcal{H}_2, (U_2^{t})_{t \in \Z}, D_2)$ are similar.
Let $V \colon \Hil_1 \to \Hil_2$ be a unitary operator which gives similarity between the quantum walks. Let $\xi$ be a unit vector in $\Hil_1$ which is smooth with respect to $D_1$.
Then the vector $V \xi$ is smooth with respect to $D_2$. The quantum walk $(\mathcal{H}_2, (U_2^{t})_{t \in \Z}, D_2)$ has limit distribution with respect to $V \xi$, if and only if $(\mathcal{H}_1, (U_1^{t})_{t \in \Z}, D_1)$ has limit distribution with respect to $\xi$. In this case, these limit distributions coincide.
\end{theorem}
\begin{proof}
By Lemma \ref{lemma: smoothness}, $V$ maps $\xi$ in $\mathrm{dom} (D_1^m)$ to an element of $\mathrm{dom} (D_2^m)$.
Let $p_{1, t}$ be the $t$-th probability measure of $(\mathcal{H}_1, (U_1^{t})_{t \in \Z}, D_1)$ with respect to $\xi$.
Let $p_{2, t}$ be the $t$-th probability measure of $(\mathcal{H}_2, (U_2^{t})_{t \in \Z}, D_2)$ with respect to $V \xi$.
By Lemma \ref{lemma: moment}, the $m$-th moment of $p_{1, t}$ is given by
\[\left\langle \dfrac{1}{t^m} D_1^m U_1^t \xi, U_1^t \xi \right\rangle.\]
The $m$-th moment of $p_{2, t}$ is given by
\[
\left\langle \dfrac{1}{t^m} D_1^m U_2^t V \xi, U_2^t V \xi \right\rangle
=
\left\langle \dfrac{1}{t^m} D_2^m V U_1^t \xi, V U_1^t \xi \right\rangle.
\]
By Proposition \ref{proposition: two kinds of convergence}, it suffices to show that for every $m$, as $t$ tends to infinity, the difference of these moments converges to $0$.
Define sequences of bounded operators $\{v_j \colon \Hil_1 \to \Hil_2\}$ and $\{u_j \colon \Hil_1 \to \Hil_1\}$ by
\begin{eqnarray*}
v_0 = V, & & v_j = D_2 v_{j -1} - v_{j -1} D_1,\\
u_0 = U_1, & & u_j = D_1 u_{j -1} - u_{j -1} D_1.
\end{eqnarray*}
By Lemma \ref{lemma: smoothness}, these operators are bounded.
Define vectors $\{\xi_j\} \in \Hil_1$ by
\begin{eqnarray*}
\xi_0 = \xi, \quad \xi_j = D_1 \xi_{j -1}.
\end{eqnarray*}
By the Leibniz rule, the vector $D_2^m V U_1^t \xi$ can be expressed as follows:
\[D_2^m V U_1^t \xi = \sum_{s \in J} v_{\sharp s^{-1}(t + 1)} u_{\sharp s^{-1}(t)} u_{\sharp s^{-1}(t - 1)} \cdots u_{\sharp s^{-1}(1)} \xi_{\sharp s^{-1}(0)}. \]
In the formula,
\begin{itemize}
\item
$s$ is an element of the index set
\[ J = \left\{ s \colon \{1, 2, \cdots, m\} \to \{0, 1, \cdots, t, t + 1\} \ | \ \rm{a\ map}\right\},\]
\item
$s^{-1} (j)$ is the inverse image of $\{j\} \subset \{0, 1, \cdots, t, t + 1\}$ under the mapping $s$.
\item
$\sharp s^{-1} (j)$ is the number of elements of the inverse image.
\end{itemize}
Define a subset $J_0$ of $J$ as follows:
\[ J_0 = \left\{ s \colon \{1, 2, \cdots, m\} \to \{0, 1, \cdots, t, t + 1\} \in J \ | \
s^{-1}(t+ 1) = \emptyset \right\}.\]
If $t$ is large, the number of elements in the coset $J \setminus J_0$ is much smaller than $t^m$.
That is $\lim_{t \to \infty} (\sharp J - \sharp J _0)/ t^m = 1$.
For $s \in J$, the norm of the term
\[v_{\sharp s^{-1}(t + 1)} u_{\sharp s^{-1}(t)} u_{\sharp s^{-1}(t - 1)} \cdots u_{\sharp s^{-1}(1)} \xi_{\sharp s^{-1}(0)}\] in $D_2^m V U_1^t \xi$
is bounded by
\[\max_{0 \le j \le m} \|v_j\| \left( \max_{0 \le j \le m} \|u_j\|\right)^m \max_{0 \le j \le m} \|\xi_j\|.\]
It follows that
\begin{eqnarray*}
\lim_{t \to \infty}
\left\| \dfrac{1}{t^m} D_2^m V U_1^t \xi
-
\dfrac{1}{t^m} \sum_{s \in J_0} v_{\sharp s^{-1}(t + 1)} u_{\sharp s^{-1}(t)} u_{\sharp s^{-1}(t - 1)} \cdots u_{\sharp s^{-1}(1)} \xi_{\sharp s^{-1}(0)}
\right\|
=
0.
\end{eqnarray*}
The second term in the limit is nothing other than the following vector:
\begin{eqnarray*}
\dfrac{1}{t^m} V \sum_{s \in J_0} u_{\sharp s^{-1}(t)} u_{\sharp s^{-1}(t - 1)} \cdots u_{\sharp s^{-1}(1)} \xi_{\sharp s^{-1}(0)}
=
\dfrac{1}{t^m} V D_1^m U_1^t \xi.
\end{eqnarray*}
It follows that the $m$-th moment
$
\left\langle \dfrac{1}{t^m} D_2^m V U_1^t \xi, V U_1^t \xi \right\rangle
$
of $p_{2, t}$
is asymptotically identical to
\[
\left\langle \dfrac{1}{t^m} V D_1^m U_1^t \xi, V U_1^t \xi \right\rangle
=
\left\langle \dfrac{1}{t^m} D_1^m U_1^t \xi, U_1^t \xi \right\rangle.
\]
This is nothing other than the $m$-th moment of $p_{1, t}$.
\end{proof}
\subsection{Homogeneous quantum walks}
\label{subsection: homogeneous}
We propose the following axiom on one-dimensional homogeneous quantum walks.
\begin{definition}\label{definition: homogeneous}
The quadruple $U = (\Hil, (U^t)_{t \in \Z}, D, S)$ is called a discrete-time {\rm homogeneous} quantum walk, if the following conditions hold:
\begin{itemize}
\item
The triple $(\Hil, (U^t)_t, D)$ is a quantum walk.
\item
$S$ is a unitary operator on $\Hil$.
\item
$U S = S U$.
\item
$S$ preserves the domain of $D$.
\item
$S^{-1} D S - D$ is a positive constant operator $r \cdot {\rm id}$.
\item
The spectral projection of $D$ corresponding to $[0, r) \subset \R$ has finite rank.
\end{itemize}
An operator $X$ on $\Hil$ is said to be homogeneous, if $X$ commutes with $S$.
It is said to be essentially homogeneous, if there exists a natural number $N$ such that the operator $X$ commutes with $S^N$.
Regularity (uniformity, smoothness, or analyticity) for an operator on $\Hil$ is determined by $D$ as in Definition $\ref{definition: regularity}$.
Two discrete-time homogeneous quantum walks are said to be similar, if
two triplets of quantum walks are similar, and if the forth entries are unitary equivalent via the intertwiner which gives similarity.
If we need to consider continuous-time homogeneous quantum walk, replace $(U^t)_{t \in \Z}$ with $(U^{(t)})_{t \in \R}$.
\end{definition}
Let $\Hil_0$ denote the spectral subspace of $D$ corresponding to $[0, r) \subset \R$.
Denote by $n$ the dimension of $\Hil_0$.
The natural number $n$ is called {\it the degree of freedom}.
The Hilbert space $\Hil$ is decomposed as follows:
\[\Hil = \overline{\cdots \oplus S^{-1} \Hil_0 \oplus \Hil_0 \oplus S \Hil_0 \oplus \cdots}.\]
Identifying $\Hil_0$ with $\C^n$,
we can easily show that $(\Hil, (U^t)_t, D)$ is similar to a quantum walk $(\ell_2(\Z) \otimes \C^n, (U^t)_t, D_1 \otimes {\rm id})$ and that $S$ is identified with the bilateral shift $S_1 \otimes {\rm id}$.
We may fix the original homogeneous quantum walk $U$ as $U = (\ell_2(\Z) \otimes \C^n, (U^t)_t, D_1 \otimes {\rm id}, S_1 \otimes {\rm id})$.
For the rest of this paper, we always assume that $U$ is of the form
\[(\ell_2(\Z) \otimes \C^n, (U^t)_{t \in \Z}, D_1 \otimes {\rm id}, S_1 \otimes {\rm id})\]
and that the generator $U$ is analytic with respect to $D_1 \otimes {\rm id}$.
\section{Structure theorems on intertwiners and commutant}
In this section,
we demonstrate the way to determine the space of {\it uniform} intertwining operators between discrete-time homogeneous {\it analytic} quantum walks.
As a corollary, the algebra of uniform operators which commute with a given walk is determined.
For the first half, we review the structure theorem in \cite{SaigoSako}.
We need to fix the notations related to Fourier analysis.
\subsection{Fourier analysis}\label{subsection: Fourier analysis}
We consider the group $r \Z$ generated by a positive real number $r$ and its dual.
All the characters of $r \Z$ are of the form
\[\widehat \chi_k (x) = \exp(i k x), \quad k \in \left[ 0, \dfrac{2 \pi}{r} \right), x \in r \Z. \]
We identify the dual group $\{\widehat{\chi}_k \ |\ 0 \le k < 2 \pi r^{-1} \}$ with
$\R / (2 \pi r^{-1} \Z)$.
We often denote by $\T_{2 \pi r^{-1}}$ the dual group $\R / (2 \pi r^{-1} \Z)$.
The subscript $2 \pi r^{-1}$ is equal to the length of the torus.
We introduce the counting measure on $r \Z$. The scalar multiple $\dfrac{r}{2 \pi} dk$ of the Lebesgue measure $dk$ defines the Haar measure on $\T_{2 \pi r^{-1}}$.
For $x \in r \Z$, the character $c_x$ on $\T_{2 \pi r^{-1}}$ is defined as
\[c_x(k) = \exp(i k x), \quad k \in \T_{2 \pi r^{-1}}.\]
The Fourier transform $\F_r \colon L^2(\T_{2 \pi r^{-1}}) \to \ell_2(r \Z)$ maps
$c_x$ to the definition function $\delta_x$ of $\{x\} \subset r \Z$.
\subsection{Model quantum walk}
Let $\lambda \colon \T_{2 \pi r^{-1}} \to \T = \{z \in \C \ |\ |z| = 1\}$ be an analytic function.
The function $\lambda$ defines the multiplication operator $M[\lambda] \colon
L^2(\T_{2 \pi r^{-1}}) \to L^2(\T_{2 \pi r^{-1}})$.
The triplet $(\ell_2(r \Z), (U_\lambda^{t})_{t \in \Z}, D_r)$
of
\begin{itemize}
\item
The Hilbert space $\ell_2(r \Z)$ of the square summable functions on $r \Z$,
\item
The Fourier transform $U_\lambda^t = \F_r M[\lambda]^t \F_r^{-1} \colon \ell_2(r \Z) \to \ell_2(r \Z)$,
\item
The diagonal operator $D_r$ given by $D_r(\delta_x) = x \delta_x, x \in r \Z$.
\end{itemize}
is called a {\it model quantum walk}.
Here we note that the inverse Fourier transform $(\F_r^{-1} \ell_2(r \Z), (\F_r^{-1} U_\lambda^{t}\F_r)_{t \in \Z}, \F_r^{-1} D\F_r)$ of the model quantum walk is identical to
\[\left( L^2(\T_{2 \pi r^{-1}}), (M[\lambda]^t)_{t \in \Z}, \dfrac{d}{i dk} \right).\]
The model quantum walk was first introduced in \cite{SaigoSako}. The formulation in \cite{SaigoSako} is different from that in this paper. However, the difference is not crucial.
In \cite{SaigoSako}, the parameter $r$ was a reciprocal $r = d^{-1}$ of a natural number $d$.
A quantum walk
\[\left( \ell_2 (\Z) \otimes \C^d, \left(\F_{d^{-1}} M[\lambda]^t \F_{d^{-1}}^{-1} \right)_{t \in \Z}, D_1 \otimes {\rm id} \right)\]
was called a model quantum walk in \cite{SaigoSako}.
The Hilbert space $\ell_2 (\Z) \otimes \C^d$ can be identified with $\ell_2(d^{-1} \Z)$ by
\[\delta_s \otimes \delta_k \mapsto \delta_{k d^{-1} + s}, \quad s \in \Z, k \in \{1, 2, \cdots, d\}.\]
As explained in Example \ref{example: replacing D},
the self-adjoint operator $D_1 \otimes {\rm id}$ can be replaced with $D_{d^{-1}}$.
It turns out that the model quantum walk in \cite{SaigoSako} is similar to
\[\left( \ell_2 (d^{-1} \Z), \left(\F_{d^{-1}} M[\lambda]^t \F_{d^{-1}}^{-1} \right)_{t \in \Z}, D_{d^{-1}} \right).\]
This is a special case of model quantum walks defined in this paper.
\subsection{Review of a structure theorem in \cite{SaigoSako}}
\label{subsection: review of SS}
We have already obtained a structure theorem for discrete-time homogeneous analytic quantum walks.
We review here the main result in \cite{SaigoSako} and adjust the notations for the argument of this paper.
Let $U = (\ell_2(\Z) \otimes \C^n, (U^t)_{t \in \Z}, D_1 \otimes {\rm id}, S_1 \otimes {\rm id})$ be an arbitrary homogeneous analytic quantum walk.
The inverse Fourier transform of $U$ is
\[\widehat{U} =
\left(
L^2(\T_{2 \pi}) \otimes \C^n,
((\F_1 \otimes {\rm id})^{-1} U^t (\F_1 \otimes {\rm id}))_{t \in \Z},
\frac{d}{i d k} \otimes {\rm id}
\right).
\]
The generator $\widehat{U}$ is a unitary element of $C(\T_{2 \pi}) \otimes M_n(\C)$, the space of $(n \times n)$-matrices whose entries are multiplication operators given by analytic functions on $\T_{2 \pi}$.
For every $k \in \T_{2 \pi}$, $\widehat{U}$ gives an $(n \times n)$-unitary matrix $\widehat{U}(k)$.
The unitary matrix provides a decomposition of $\C^n$ into eigenspaces.
By analyticity of the entries of $\widehat{U}$, we obtain not only analytic functions $\lambda(k)$ of eigenvalues of $\widehat{U}(k)$, but also analytic sections of eigenvectors whose fibers make orthonormal bases of $\C^n$.
We need to keep in mind that the eigenvalue functions $\lambda(k)$ are not necessarily single-valued.
To describe the multi-valued eigenvalue functions,
we make use of the torus $\T_{2 \pi d} = \R / (2 \pi d \Z)$,
and define a covering map $p_d \colon \T_{2 \pi d} \to \T_{2 \pi} = \R / (2 \pi \Z)$ by the standard quotient.
We obtain
\begin{itemize}
\item
natural numbers $d(1), d(2), \cdots, d(\nu)$ whose sum is $n$,
\item
analytic maps $\lambda_\iota \colon \T_{2 \pi d(\iota)} \to \T$, $(\iota = 1, 2, \cdots, \nu)$
\end{itemize}
such that for every $k \in \T_{2 \pi}$, the set of the eigenvalues of $\widehat{U}(k)$ is
\[\bigcup_{\iota = 1}^\nu \left\{ \left. \lambda_\iota \left(\widetilde{k} \right) \ \right|\ \widetilde{k} \in \T_{2 \pi d(\iota)}, p_{d(\iota)}\left(\widetilde{k}\right) = k\right\}.\]
Corresponding to this description of eigenvalues, an analytic sections ${\bf v}_\iota \left( \widetilde{k} \right)$ of eigenvectors do exist.
These sections naturally define a unitary operator
\[V \colon \bigoplus_{\iota = 1}^\nu L^2(\T_{2 \pi d(\iota)}) \to L^2(\T_{2 \pi}\to \C^n) = L^2(\T_{2 \pi}) \otimes \C^n\]
by the formula
\[[V (\xi_\iota)]\left( k \right) =
\sum_{\widetilde{k},\ p_\iota\left( \widetilde{k} \right) = k}
\xi_\iota \left( \widetilde{k} \right)
{\bf v}_\iota \left( \widetilde{k} \right),
\quad \xi_\iota \in L^2(\T_{2 \pi d(\iota)}),
k \in \T_{2 \pi}.\]
By analyticity of the sections of eigenvectors,
$V$ is analytic with respect to
\[\frac{d}{i d k} \oplus \frac{d}{i d k} \oplus \cdots \oplus \frac{d}{i d k}\ (\nu {\rm-times}) {\rm \ and \ } \frac{d}{i d k} \otimes {\rm id}.\]
The analytic unitary operator $V$ gives similarity between $\widehat{U}$
and the direct sum
\[\bigoplus_{\iota = 1}^\nu
\left(
L^2(\T_{2 \pi d(\iota)}),
(M[\lambda_\iota]^t)_{t \in \Z},
\frac{d}{i d k}
\right).
\]
Applying Fourier transform, we conclude that $U$ is similar to a direct sum of model quantum walks.
\begin{example}[$4$-state Grover walk]
\label{example: 4-state Grover walk}
Consider the following unitary operator on $\ell_2(\Z) \otimes \C^4 \cong \ell_2(\Z)^4$:
\[U =
\frac{1}{2}
\left(
\begin{array}{ccccc}
S_1^{-3} & 0 & 0 & 0 \\
0 & S_1^{-1} & 0 & 0 \\
0 & 0 & S_1 & 0 \\
0 & 0 & 0 & S_1^{3}
\end{array}
\right)
\left(
\begin{array}{ccccc}
- 1 & 1 & 1 & 1 \\
1 & - 1 & 1 & 1 \\
1 & 1 & - 1 & 1 \\
1 & 1 & 1 & - 1 \\
\end{array}
\right).
\]
We concretely calculate the eigenvalue function of $U$ and identify the decomposition into model quantum walks.
The inverse Fourier transform of $U$ is
\[\widehat{U}(k) =
\frac{1}{2}
\left(
\begin{array}{ccccc}
e^{-3ik} & 0 & 0 & 0 \\
0 & e^{-ik} & 0 & 0 \\
0 & 0 & e^{ik} & 0 \\
0 & 0 & 0 & e^{3ik}
\end{array}
\right)
\left(
\begin{array}{ccccc}
- 1 & 1 & 1 & 1 \\
1 & - 1 & 1 & 1 \\
1 & 1 & - 1 & 1 \\
1 & 1 & 1 & - 1 \\
\end{array}
\right),
\quad k \in \T_{2 \pi}.
\]
The characteristic polynomial is
\begin{eqnarray*}
& & \det \left( \lambda - \widehat{U}(k) \right)\\
&=& \lambda^4 + \frac{e^{3ik} + e^{ik} + e^{-ik} + e^{-3ik}}{2} \lambda^3
- \frac{e^{3ik} + e^{ik} + e^{-ik} + e^{-3ik}}{2} \lambda -1\\
&=& (\lambda - 1) (\lambda + 1)
\left\{ \lambda^2 + (\cos 3k + \cos k) \lambda + 1\right\}.
\end{eqnarray*}
We obtain two constant eigenvalue functions $\lambda_1(k) = 1$ and $\lambda_2(k) = -1$.
We focus on the roots of the last factor.
The roots are given by
\begin{eqnarray*}
\lambda_3(k) &=&
- \dfrac{\cos k + \cos 3k}{2} - i \sin k \sqrt{1 + 4 \cos^4 k},
\\
\lambda_4(k) &=&
- \dfrac{\cos k + \cos 3k}{2} + i \sin k \sqrt{1 + 4 \cos^4 k}.
\end{eqnarray*}
Thus we obtain four single-valued analytic eigenvalue functions
$\lambda_1(k) = 1$, $\lambda_2(k) = -1$, $\lambda_3(k)$, $\lambda_4(k)$,
satisfying
\[\det \left( \lambda - \widehat{U}(k) \right) = \prod_{j = 1}^4 (\lambda - \lambda_j(k)).\]
By the structure theorem, it turns out that
the $4$-state Grover walk
$\left(
\ell_2(\Z) \otimes \C^4,
(U^t),
D_1 \otimes {\rm id}
\right)
$
is similar to the direct sum:
\begin{eqnarray*}
\left(
L^2(\T_{2 \pi}) \otimes \C^4,
(1 \oplus (-1)^t \oplus M[\lambda_3]^t \oplus M [\lambda_4]^t)_{t \in \Z},
\left( \frac{d}{id k} \right)^{\oplus 4}
\right)
\end{eqnarray*}
Similarity is given by a composite of the inverse Fourier transform and a unitary operator $\widehat{V} = \left( \widehat{V}(k) \right)_{k \in \T_{2 \pi}}$ acting on $L^2(\T_{2 \pi}) \otimes \C^4$.
The unitary $\widehat{V}(k) \in M_4(\C)$ is given by an analytic decomposition into eigenvectors and therefore analytic with respect to $\left( \frac{d}{i dk} \right)^{\oplus 4}$.
\end{example}
\begin{example}[$3$-state Grover walk]
\label{example: 3-state Grover walk}
Consider the following unitary operator on $\ell_2(\Z) \otimes \C^3 \cong \ell_2(\Z)^3$:
\[U =
\frac{1}{3}
\left(
\begin{array}{ccccc}
S_1^{-1} & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & S_1 \\
\end{array}
\right)
\left(
\begin{array}{ccccc}
- 1 & 2 & 2 \\
2 & - 1 & 2 \\
2 & 2 & - 1 \\
\end{array}
\right).
\]
We concretely calculate the eigenvalue function of $U$ and identify the decomposition into model quantum walks.
The inverse Fourier transform of $U$ is
\[\widehat{U}(k) =
\frac{1}{3}
\left(
\begin{array}{ccccc}
e^{-ik} & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & e^{ik} \\
\end{array}
\right)
\left(
\begin{array}{ccccc}
- 1 & 2 & 2 \\
2 & - 1 & 2 \\
2 & 2 & - 1
\end{array}
\right),
\quad k \in \T_{2 \pi}.
\]
The characteristic polynomial is
\begin{eqnarray*}
& & \det \left( \lambda - \widehat{U}(k) \right)\\
&=& \lambda^3
+ \frac{e^{ik} + 1 + e^{-ik}}{3} \lambda^2
- \frac{e^{ik} + 1 + e^{-ik}}{3} \lambda
1\\
&=& (\lambda - 1)
\left( \lambda^2 + \frac{4 + 2 \cos k}{3} \lambda + 1\right).
\end{eqnarray*}
Here we obtain an eigenvalue function $\lambda_1(k) = 1$.
We focus on the roots of the last factor.
For $k \in [0, 2 \pi)$, the roots are given by
\begin{eqnarray*}
\lambda_2(k) &=&
- \dfrac{2 + \cos k}{3} - \frac{1}{3} i \sin \frac{k}{2} \sqrt{10 + 2 \cos k}
\end{eqnarray*}
and
$\lambda_2 (k + 2 \pi) = \overline{\lambda_2(k)}$.
Thus we obtain one single-valued analytic eigenvalue function $\lambda_1(k) = 1$ and
one multi-valued analytic eigenvalue function
\[k \mapsto \{\lambda_2(k), \lambda_2(k + 2 \pi)\}.\]
By the structure theorem,
it turns out that the direct sum
\begin{eqnarray*}
\left(
L^2(\T_{2 \pi}) \oplus L^2(\T_{4 \pi}),
(1 \oplus M[\lambda_2]^t)_{t \in \Z},
\frac{d}{i dk} \oplus \frac{d}{i dk}
\right)
\end{eqnarray*}
is similar to
the $3$-state Grover walk
$\left(
\ell_2(\Z) \otimes \C^3,
(U^t),
D_1 \otimes {\rm id}
\right)
$.
Inui, Konno, and Segawa in \cite{InuiKonnoSegawa} observed that the limit distribution of the $3$-state Grover walk is localized around $0 \in \R$.
The above decomposition gives another proof of their result, since the walk contains a constant quantum walk as a direct summand.
Similarity is given by a composite of
a unitary operator
\[\widehat{V} \colon L^2(\T_{2 \pi}) \oplus L^2(\T_{4 \pi}) \to L^2(\T_{2 \pi}) \otimes \C^3\]
and the Fourier transform
\[\F_1 \otimes {\rm id} \colon L^2(\T_{2 \pi}) \otimes \C^3 \to \ell_2(\Z) \otimes \C^3.\]
The unitary $\widehat{V}(k)$ is given by an analytic decomposition into eigenvectors of $\widehat{U}(k)$ corresponding to the eigenvalues $\{1, \lambda_2(k), \lambda_2(k + 2 \pi)\}$.
The unitary $\widehat{V}$ is analytic with respect to $\frac{d}{i dk}$.
\end{example}
Here we propose a problem.
When is a homogeneous analytic quantum walk $U$ realized by a continuous-time uniform quantum walk which is {\it not necessarily homogeneous}?
Theorem \ref{theorem: continuous-time} gives an answer.
Some model quantum walk can be decomposed into a direct sum of model quantum walks.
To analyze such a case, we need to decompose $U$ further.
\subsection{Prime model quantum walks}
Let $\lambda$ a periodic analytic map $\lambda \colon \R \to \T$ on $\R$
which is not constant.
For a positive period $2 \pi r^{-1}$ of $\lambda$, $\lambda$ gives an analytic map $\lambda \colon \T_{2 \pi r^{-1}} \to \T$.
We can construct model quantum walks from $\lambda$.
Since the period $2 \pi r^{-1}$ of $\lambda$ is not unique, the model quantum walk
$(\ell_2(r \Z), (U_\lambda^{t})_{t \in \Z}, D)$ is not uniquely determined by $\lambda$.
But the possible model quantum walks are closely related to that of the minimal period.
We note that the dual group of $(r / m) \Z$ is $\T_{2 \pi m r^{-1}} = \R / (2 \pi m r^{-1} \Z)$.
The length of $\T_{2 \pi m r^{-1}}$ is $m$-times longer than that of $\T_{2 \pi r^{-1}}$.
\begin{proposition}\label{proposition: decomposition into prime walks}
Let $\lambda$ be a periodic analytic map $\lambda \colon \R \to \T$ on $\R$ which is not a constant.
Let $2 \pi r^{-1}$ be the minimal period of $\lambda$.
Let $m$ be a natural number.
The model quantum walk
$\left( \ell_2\left( \dfrac{r}{m} \Z \right), (U_\lambda^{t})_{t \in \Z}, D_{r / m} \right)$
is similar to the direct sum
\[\left( \ell_2\left( r \Z \right), (U_\lambda^{t})_{t \in \Z}, D_r \right)^{\oplus m}\]
of the model quantum walks given by the minimal period.
\end{proposition}
\begin{remark}
For the definition of similarity, see Definition \ref{definition: similarity}.
Similarity as homogeneous quantum walks defined in Definition \ref
{definition: homogeneous}
does not necessarily hold.
\end{remark}
\begin{proof}
It suffices to show that
\[\left( L^2\left( \R / (2 \pi m r^{-1} \Z )\right), (M[\lambda]^t)_{t \in \Z}, \frac{d}{i d k} \right)\]
is similar to
\[\left(L^2\left( \R / (2 \pi r^{-1} \Z )\right), (M[\lambda]^t)_{t \in \Z}, \frac{d}{i d k} \right)^{\oplus m}\]
Recall that for $x \in (r / m)\Z$, $c_x(k) = \exp(i k x)$ defines a character of $\T_{2 \pi m r^{-1}} = \R / (2 \pi m r^{-1} \Z )$ and that
$\{c_x \ |\ x \in (r / m)\Z\}$ is an orthonormal basis of $L^2 (\T_{2 \pi m r^{-1}})$.
Since the minimal period of $k$ is $2 \pi r^{-1}$,
we can express the analytic map $\lambda$ by
\[\lambda(k) = \sum_{x \in r \Z} \alpha_x c_x.\]
Since $\lambda$ is analytic, as $|x| \to \infty$, $|\alpha_x|$ rapidly decreases.
Therefore the infinite sum
\[M[\lambda] = \sum_{x \in r \Z} \alpha_x M[c_x] \in \B(L^2(\T_{2 \pi m r^{-1}}))\]
converges in the operator norm topology.
By the relation $M[c_x](c_y) = c_{x + y}$,
for every $z \in \{0, r/m, 2r/m, \cdots, (m - 1) r/m\}$,
\[\Hil_z := \mathrm{\overline{span}} \{c_{x + z} \ |\ x \in r\Z\}\]
is invariant under the action of $M[\lambda]$ and under $\exp(i k \frac{d}{i dk})$.
The action of $M[\lambda]$ on $\Hil_z$ is unitary equivalent to that on $\Hil_0$.
The operator $\frac{d}{idk}$ on $\Hil_z$ corresponds to the sum of a constant operator and $\frac{d}{idk}$ on $\Hil_0$.
The Hilbert space $\Hil_0 = \mathrm{\overline{span}} \{c_{x} \ |\ x \in r\Z\}$ is naturally identified with $L^2(\T_{2 \pi r^{-1}})$.
It follows that
\[\bigoplus_z \left(\Hil_z, (M[\lambda]^t)_{t \in \Z}, \frac{d}{i d k} \right)\]
and
\[\left(L^2\left( \R / (2 \pi r^{-1} \Z )\right), (M[\lambda]^t)_{t \in \Z}, \frac{d}{i d k} \right)^{\oplus m}\]
are similar.
The former walk is unitary equivalent to the original quantum walk.
\end{proof}
\begin{example}
Consider the quantum walk $(\ell_2(\Z) \otimes \C^3, (U^t)_{t \in \Z}, D_1 \otimes \mathrm{id})$ generated by
$U =
\left(
\begin{array}{ccc}
0 & S_1 & 0 \\
0 & 0 & S_1 \\
1 & 0 & 0
\end{array}
\right)$.
The inverse Fourier transform is
$\widehat{U}(k) =
\left(
\begin{array}{ccc}
0 & e^{ik} & 0 \\
0 & 0 & e^{ik} \\
1 & 0 & 0
\end{array}
\right), k \in \T_{2 \pi}$.
The characteristic polynomial is $\lambda^3 - e^{2 i k}$.
The eigenvalue function is a multi-valued function
\[k \mapsto \{\lambda_1(k), \lambda_1(k + 2 \pi), \lambda_1(k + 4 \pi)\}\]
given by
\[c_{2/3} (k) = \exp(2 i k/3), \quad k \in 6 \pi.\]
By the structure theorem in \cite{SaigoSako},
the original quantum walk is similar to
\[\left( L^2(\T_{6 \pi}), \left( M[c_{2/3}]^t \right)_{t \in \Z}, \frac{d}{i d k} \right).\]
The function $\lambda_1 \colon \T_{6 \pi} \to \T$ has a non-trivial period $3 \pi$.
As in Proposition \ref{proposition: decomposition into prime walks},
This is similar to the following direct sum:
\[\left( L^2(\T_{3 \pi}), \left( M[c_{2/3}]^t \right)_{t \in \Z}, \frac{d}{i d k} \right)^{\oplus 2}.\]
This is unitary equivalent to the direct sum of two prime model quantum walks
\[\left( \ell_2 \left( (2 / 3) \Z \right), \left( M[c_{2/3}]^t \right)_{t \in \Z}, D_{2 / 3} \right)^{\oplus 2}.\]
Note that the original quantum walk $U$ and this direct sum are similar in the category of quantum walks, but not similar in the category of homogeneous quantum walks (Subsection \ref{subsection: homogeneous}).
\end{example}
\begin{definition}
A model quantum walk $\left( \ell_2\left( r \Z \right), (U_\lambda^{t})_{t \in \Z}, D \right)$
is said to be {\rm prime}, if the analytic map $\lambda \colon \T_{2 \pi r^{-1}} \to \T$ has no period other than $0$.
\end{definition}
Thus we have the following structure theorem.
\begin{theorem}\label{theorem: structure theorem}
Let $U = (\ell_2(\Z) \otimes \C^n, (U^t)_{t \in \Z}, D_1 \otimes {\rm id})$ be an arbitrary one-dimensional discrete-time homogeneous analytic quantum walk.
Then there exist
\begin{itemize}
\item
non-negative integers $l$, $m$,
\item
rational numbers $r(j)$, $(j \in \{1, \cdots, l\})$,
\item
prime model quantum walks
$\left( \ell_2(r(j) \Z), \left( U_{\lambda(j)}^t \right)_{t \in \Z}, D_{r(j)} \right)$, $(j \in \{1, \cdots, l\})$,
\item
complex numbers $\alpha(k)$, $(k \in \{1, \cdots, m\})$ whose absolute values are $1$,
\end{itemize}
satisfying
\begin{itemize}
\item
that the given analytic walk
$U$
is similar to the direct sum
\[
\bigoplus_{j = 1}^l
\left( \ell_2(r(j) \Z), \left( U_{\lambda(j)}^t \right)_{t \in \Z}, D_{r(j)} \right)
\oplus
\bigoplus_{k = 1}^m
\left( \ell_2(\Z), \left( \alpha(k)^t \right)_{t \in \Z}, D_1 \right)
\]
(The integers $l$ and $m$ can be zero. In the case that $l = 0$, erase the first half. In the case that $m = 0$, erase the second half.)
\item
and that the degree of freedom $n$ is equal to
$m + \sum_{j = 1}^l
r(j)^{-1}.$
\end{itemize}
\end{theorem}
\begin{proof}
As we explained in Subsection \ref{subsection: review of SS},
$U$ is similar to the direct sum of model quantum walks
\[
\bigoplus_{\iota = 1}^\nu
\left( \ell_2(r(\iota) \Z), \left( U_{\lambda(\iota)}^t \right)_{t \in \Z}, D_{r(\iota)} \right).
\]
The positive numbers $r(\iota)$ are reciprocals of natural numbers $d(\iota)$.
Consider the case that the analytic function $\lambda(\iota) \colon \T_{2 \pi d(\iota)} \to \T$ is not constant.
If it is not prime, we can further decompose the
the model quantum walk
into prime model quantum walks.
In such a case, $r(\iota)$ becomes larger, but the sum of reciprocals is preserved (see Proposition \ref{proposition: decomposition into prime walks}).
Each prime model quantum walk becomes a direct summand of the first half.
If the analytic function $\lambda(\iota) \colon \T_{2 \pi d(\iota)} \to \T$ is a constant function $\alpha(\iota)$,
then the corresponding model quantum walk is decomposed as follows:
\[\left( \ell_2(r(\iota) \Z), \left( \alpha(\iota)^t \right)_{t \in \Z}, D_{r(\iota)} \right)
\cong
\left( \ell_2(\Z), \left( \alpha(\iota)^t \right)_{t \in \Z}, D_1 \right)^{\oplus r(\iota)^{-1}}.
\]
These direct summands becomes direct summands of the second half.
\end{proof}
Examples
\ref{example: 4-state Grover walk}
and
\ref{example: 3-state Grover walk}
give decompositions into constant quantum walks and prime model quantum walks.
\subsection{Uniform intertwiner between two walks}
\label{subsection: uniform intertwiner}
For a while, we make use of two pairs of dual groups $(r(1)\Z, \T_{l(1)})$ and $(r(2)\Z, \T_{l(2)})$.
As explained in Subsection \ref{subsection: Fourier analysis},
for $\iota = 1, 2$, the length $l(\iota)$ of the torus $\T_{l(\iota)}$ is equal to $2 \pi r(\iota)^{-1}$.
The following lemma is the most important technical ingredient of this paper.
\begin{lemma}\label{lemma: diffeomorphism on T}
Let $\psi \colon \T_{l(2)} \to \T_{l(1)}$ be a diffeomorphism.
Let $f$ be a bounded Borel function $\T_{l(2)}$ whose support is not null.
Let $\widehat{V}$ be the composite $M[f] \circ \psi^*$ of
\begin{itemize}
\item
the pull back $\psi^* \colon L^2(\T_{l(1)}) \to L^2(\T_{l(2)})$ of $\psi$ and
\item
the multiplication operator $M[f] \colon L^2(\T_{l(2)}) \to L^2(\T_{l(2)})$.
\end{itemize}
If $\widehat{V} \in \mathcal{B}(L^2(\T_{l(2)}) \leftarrow L^2(\T_{l(1)}))$ is uniform with respect to the differential operators $\frac{d}{i d k}$ on $\T_{l(1)}$ and $\frac{d}{i d k}$ on $\T_{l(2)}$,
then on every interval contained in ${\rm supp} f$, $\psi(k) - k$ is constant.
\end{lemma}
\begin{proof}
For $y \in r(1) \Z$, let $c_y$ denote the function on $\T_{l(1)} = \R / (r(1) \Z)$ given by $c_y(k) = \exp(i k y)$.
For $x \in r(2) \Z$, let $c_x$ denote the function on $\T_{l(2)} = \R / (r(2) \Z)$ given by $c_x(k) = \exp(i k x)$.
For $x \in r(2) \Z$ and $y \in r(1) \Z$, the matrix coefficient $V_{x, y}$ of the Fourier transform of $\widehat{V}$ is given by
\begin{eqnarray*}
V_{x, y} = \left\langle M[f] \psi^* c_y, c_x \right\rangle_{L^2 \left( \T_{l(2)} \right)}
= \int_{0}^{l(2)} \exp(-i k x) f \left( k \right) \exp(i \psi \left(k \right) y)
\cdot \dfrac{dk}{l(2)}.
\end{eqnarray*}
Define a function $\Psi \colon \T_{l(2)} \to \T = \{z \in \C | \ |z| = 1\}$ by
$\Psi(k) = \exp(i \psi(k) r(1))$.
The above quantity is equal to
\begin{eqnarray}\label{equation: a new quantum walk}
V_{x, y} &=& \left\langle M \left[\Psi^{y / r(1)} \right] f, c_x \right\rangle_{L^2 \left( \T_{l(2)} \right)}.
\end{eqnarray}
Motivated by the above formula, we consider the homogeneous smooth quantum walk $(L^2(\T_{l(2)}), (M[\Psi^{y / r(1)}])_{y \in r(1) \Z}, \frac{d}{i d k})$ and the initial vector $f \in L^2(\T_{l(2)})$.
For the rest of this proof, consider the case that $y \in r(1) \Z$ is large, and we regard the integer $y / r(1)$ as time. Define an integer $t(y)$ by $y / r(1)$.
The Fourier coefficients $(V_{x, y})_{x} \in \ell_2(r(2) \Z)$ of $M[\Psi^{t(y)}] f \in L^2 (\T_{l(2)})$ gives a measure on $r(2) \Z$.
Denote by $p_y$ the push-forward measure along the mapping $r(2) \Z \ni x \to x / y \in \R$.
More precisely, $p_y$ is the sum $\sum_{x \in r(2) \Z} |V_{x, y}|^2 \delta_{x / y}$ of point masses
at $\{x/y \ |\ x \in r(2) \Z\}$.
We first consider the case that $f \colon \T_{l(2)} \to \C$ is smooth.
Denote by $D = \frac{d}{i dk}$ the differential operator acting on $L^2 (\T_{l(2)})$.
By Lemma \ref{lemma: moment},
the $m$-th moment of $p_y$ is identical to
\begin{eqnarray*}
& &
\left\langle \left(\dfrac{D}{y}\right)^m M\left[\Psi^{t(y)}\right] f, M\left[\Psi^{- t(y)}\right] f \right\rangle_{L^2 \left( \T_{l(2)} \right)}\\
&=&
\left\langle \left( M\left[\Psi^{- t(y)}\right] \dfrac{D}{y} M\left[\Psi^{t(y)}\right] \right)^m f, f \right\rangle_{L^2 \left( \T_{l(2)} \right)}\\
&=&
\left\langle \left( M\left[\Psi^{- t(y)} \dfrac{t(y)}{i y} \Psi' \Psi^{t(y) -1}\right] + \dfrac{D}{y} \right)^m f, f \right\rangle_{L^2 \left( \T_{l(2)} \right)}\\
&=&
\left\langle \left( M\left[\frac{\Psi'}{i r(1) \Psi}\right] + \dfrac{D}{y} \right)^m f, f \right\rangle_{L^2 \left( \T_{l(2)} \right)}.
\end{eqnarray*}
The function $\Psi' / (i r(1) \Psi)$ is equal to $\psi'$.
As $y \to \infty$, the moment of $p_y$ tends to
\[\left\langle M[\psi']^m f, f \right\rangle_{L^2 \left( \T_{l(2)} \right)}
= \int_0 ^{l(2)} \psi'(k)^m |f(k)|^2 \cdot \dfrac{dk}{l(2)}.
\]
This implies that $p_y$ converges in law to the push-forward of the measure $|f(k)|^2 \dfrac{dk}{l(2)}$ along the mapping $\psi' \colon \T_{l(2)} \to \R$.
It follows that $p_y$ weakly converges to the push-forward measure
(Proposition \ref{proposition: two kinds of convergence}).
The vector $f$ is not a unit vector, but the argument in Proposition \ref{proposition: two kinds of convergence} is valid.
Let us go back to the general case.
Suppose that $g \colon \T_{l(2)} \to \C$ is a general bounded Borel function.
Denote by $\widehat{W}$ the operator $M \left[g \right] \circ \psi^*$.
The matrix coefficients $W_{x, y} = \left\langle \widehat{W} c_y, c_x \right\rangle_{L^2(\T_{l(2)})}$ of the Fourier transform of $\widehat{W}$ are given by
\begin{eqnarray*}
W_{x, y} &=& \left\langle M \left[\Psi^{t(y)} \right] g, c_x \right\rangle_{L^2(\T_{l(2)})}.
\end{eqnarray*}
Let $q_y$ be the probability measure $\sum_{x \in r(2) \Z} \left|W_{x, y} \right|^2 \delta_{x / y}$.
Define a constant $C$ by
$\left\| g \right\|_{L^2 (\T_{l(2)})}$.
For an arbitrary positive number $\epsilon$, there exist a smooth function $f \colon \T_{l(2)} \to \C$ satisfying
$\left\|f - g \right\|_{L^2(\T_{l(2)})} < \epsilon$,
$\left\|f \right\|_{L^2(\T_{l(2)})} \le C$.
We denote by $\| \cdot \|_{{\rm cb}^*}$ the norm of linear functionals on the Banach space of bounded continuous functions on $\R$.
By the Cauchy–-Schwarz inequality, we have
\begin{eqnarray*}
\left\| p_y - q_y \right\|_{{\rm cb}^*}
&\le&
\left\| \left(|V_{x, y}|^2 \right)_{x} - \left( \left| W_{x, y} \right|^2 \right)_{x} \right\|_{\ell_1}\\
&\le&
\left\| (V_{x, y})_{x} + \left(W_{x, y} \right)_{x} \right\|_{\ell_2}
\left\| (V_{x, y})_{x} - \left(W_{x, y} \right)_{x} \right\|_{\ell_2}.
\end{eqnarray*}
By the Plancherel theorem, and by the equation $|\Psi(k)| = 1$, we have
\begin{eqnarray*}
& &
\left\| p_y - q_y \right\|_{{\rm cb}^*}\\
&\le&
\left\| M\left[\Psi^{t(y)} \right] f + M\left[\Psi^{t(y)} \right] g \right\|_{L^2(\T_{l(2)})}
\left\| M\left[\Psi^{t(y)} \right] f - M\left[\Psi^{t(y)} \right] g \right\|_{L^2(\T_{l(2)})}\\
&=&
\left\| f + g \right\|_{L^2(\T_{l(2)})}
\left\| f - g \right\|_{L^2(\T_{l(2)})}
\\
&\le&
2C \epsilon.
\end{eqnarray*}
By the Cauchy–-Schwarz inequality, we have
\begin{eqnarray*}
& &
\left\| \psi'_* \left( |f(k)|^2 \dfrac{dk}{l(2)} \right)
- \psi'_* \left( \left| g(k) \right|^2 \dfrac{dk}{l(2)} \right)
\right\|_{{\rm cb}^*}\\
&\le&
\left\| |f(k)|^2 \dfrac{dk}{l(2)}
-
\left| g(k) \right|^2 \dfrac{dk}{l(2)}
\right\|_{{\rm cb}^* (\T_{l(2)})}\\
&=&
\left\| |f|^2
-
\left| g \right|^2 \right\|_{L^1(\T_{l(2)})}\\
&\le&
\left\| f + g \right\|_{L^2(\T_{l(2)})}
\left\| f - g \right\|_{L^2(\T_{l(2)})}\\
&\le&
2C \epsilon.
\end{eqnarray*}
Because $p_y$ weakly converges to $\psi'_* \left( |f(k)|^2 \dfrac{dk}{l(2)} \right)$,
$\widetilde{p}_y$ weakly converges to $\psi'_* \left( \left| g(k) \right|^2 \dfrac{dk}{l(2)} \right)$.
By Proposition \ref{proposition: converging to an atom},
since the Fourier transform $W$ of $\widehat{W}$ is uniform with respect to the diagonal operators $D_{r(1)}$ and $D_{r(2)}$,
the weak limit of $q_y$ has to be concentrated on $1$.
It follows that $\psi'$ is the constant function $1$ on the support of $g$.
\end{proof}
\begin{lemma}\label{lemma: diffeomorphism between intervals}
Let $\widehat{V}$ be a bounded operator from $L^2(\T_{l(1)})$ to $L^2(\T_{l(2)})$.
Let $I(1)$ be an open interval of $\T_{l(1)}$ and
let $I(2)$ be an open interval of $\T_{l(2)}$.
Let $\psi \colon I(2) \to I(1)$ be a diffeomorphism.
Suppose that the derivatives $\psi'$ and $\left( \psi^{-1} \right)'$ are bounded.
Let $f \colon I(2) \to \C$ be a bounded Borel non-zero function.
Suppose that the restriction $\widehat{V} |_{L^2(I(1))}$ is identical to the composition operator $M[f] \circ \psi^*$ of
\begin{itemize}
\item
the pull back $\psi^* \colon L^2(I(1)) \to L^2(I(2))$ of $\psi$ and
\item
the multiplication operator $M[f] \colon L^2(I(2)) \to L^2(I(2)) \subset L^2(\mathbb{T})$.
\end{itemize}
If $\widehat{V}$ is uniform with respect to the differential operators $\frac{d}{i dk}$ on $L^2(\T_{l(1)})$ and $\frac{d}{i dk}$ on $L^2(\T_{l(2)})$,
then on every interval contained in ${\rm supp} f \cap I(2)$, $\psi(k) - k$ is constant.
\end{lemma}
\begin{proof}
Let $g$ be an arbitrary smooth function on $\T_{l(1)}$ such that the support ${\rm supp} (g)$ is a compact subset of $I(1)$.
The multiplication operator $M[g]$ maps $L^2(\T_{l(1)})$ to $L^2({\rm supp} (g)) \subset L^2(I(1))$.
Since $M[g]$ is uniform with respect to $\frac{d}{id k}$, $\widehat{V} M[g]$ is uniform with respect to $\frac{d}{id k}$. The operator $\widehat{V} M[g]$ expressed as follows:
\[\widehat{V} M[g]
= M[f] \circ \psi^* \circ M[g].\]
Choose a diffeomorphism $\phi \colon \T_{l(2)} \to \T_{l(1)}$ which is identical to $\psi$ on $\psi^{-1} ({\rm supp} (g))$.
Then we have
\[\widehat{V} M[g]
= M[f] \circ \phi^* \circ M[g]
= M[f \cdot (g \circ \phi)] \circ \phi^*.\]
By Lemma \ref{lemma: diffeomorphism on T},
$\phi$ is rotation on every interval included in ${\rm supp}f \cap {\rm supp}(g \circ \phi)$, and therefore, $\psi$ is rotation on every interval included in ${\rm supp}f \cap {\rm supp}(g \circ \psi)$.
It follows that for every interval included in ${\rm supp} f \cap I(2)$, the map $\psi$ is rotation.
\end{proof}
\begin{proposition}\label{proposition: uniform intertwiner}
Let $\lambda_1 \colon \T_{l(1)} \to \T$ and $\lambda_2 \colon \T_{l(2)} \to \T$ be analytic maps.
Assume that $\lambda_1$ and $\lambda_2$ do not have period.
Let $\left( \ell_2 \left( r(1) \Z \right), (U_1^{t})_{t \in \Z}, D_{r(1)} \right)$ and $( \ell_2\left( r(2) \Z \right)$, $(U_2^{t})_{t \in \Z}$, $D_{r(2)} )$ be the prime model quantum walks given by $\lambda_1$ and $\lambda_2$.
Assume that there exists a non-zero uniform intertwiner between them. Then $l(1)$ is equal to $l := l(2)$, and therefore $r(1)$ is equal to $r := r(2)$.
There exists (unique) $\alpha \in \T_l$ such that
\[\lambda_2 (k) = \lambda_1(k + \alpha), \quad k \in \T_l.\]
The set of all the uniform intertwiners
\[\left\{ V \colon \ell_2\left( r \Z \right) \to \ell_2\left( r \Z \right) \ \left| \
V U_1 = U_2 V,
k \mapsto e^{ik D_r} V e^{-i k D_r} {\rm \ is\ continuous} \right. \right\}\]
is equal to
$\left\{\left. \F_r M[\rho] \F_r^{-1} \circ \exp( i \alpha D_r) \ \right|\ \rho \colon \T_l \to \C {\rm\ continuous} \right\}$.
\end{proposition}
In the proof, for a Borel subset $B \subset \T_{l(\iota)}$, we denote by $1_B$ the definition function of $B$. Note that the multiplication operator $M[1_B]$ is the orthogonal projection $L^2(\T_{l(\iota)}) \to L^2(B)$.
\begin{proof}
Let $\widehat{V} \colon L^2(\T_{l(1)}) \to L^2(\T_{l(2)})$ be a non-zero uniform intertwiner between
\[\left( L^2(\T_{l(1)}), (M[\lambda_1]^t), \frac{d}{i d k} \right), \quad \left( L^2(\T_{l(2)}), (M[\lambda_2]^t), \frac{d}{i d k} \right).\]
For a Borel subset $J \subset \T$, the spectral projection $E_1(J)$ of $M[\lambda_1]$
is the orthogonal projection
\[E_1(J) \colon L^2(\T_{l(1)}) \to L^2( \lambda_1^{-1}(J) ) \subset L^2(\T_{l(1)}).\]
The spectral projection $E_2(J)$ of $M[\lambda_2]$
is the orthogonal projection
\[E_2(J) \colon L^2(\T_{l(2)}) \to L^2( \lambda_2^{-1}(J) ) \subset L^2(\T_{l(2)}).\]
The equation $\widehat{V} M[\lambda_1] = M[\lambda_2] \widehat{V}$ implies
$\widehat{V} E_1(J) = E_2(J) \widehat{V}$.
Since $\lambda_1$ and $\lambda_2$ are not constant function,
by the identity theorem of analytic functions,
the inverse images of a singleton in $\T$ with respect to $\lambda_1$ and $\lambda_2$
is at most finite.
Therefore, the operators $M[\lambda_1]$ and $M[\lambda_2]$ do not have point spectrum.
We also note that the number of critical values of $\lambda_1$ and $\lambda_2$ is finite.
It follows that there exists an open interval $J \subset \T$ satisfying the following:
\begin{itemize}
\item
The interval $J$ does not contain the critical values of $\lambda_1$ nor
those of $\lambda_2$.
\item
The operator $V E_1(J)$ is not zero. (Therefore $E_2(J) V$ is not zero.)
\end{itemize}
For $\iota = 1, 2$, $\lambda_\iota^{-1}(J)$ consists of
finitely many open intervals.
Note that the restriction of $\lambda_\iota$ on each connected component of $\lambda_\iota^{-1}(J)$ is diffeomorphism onto $J$.
Choose connected components $I(\iota) \subset \lambda_\iota^{-1}(J)$ such that
$M[1_{I(2)}] \widehat{V} M[1_{I(1)}] \neq 0$.
There exist smooth functions $g_\iota$ on $\T_{l(\iota)}$ such that the support of $g_\iota$ is included in $I(\iota)$ and \[M[g_2] \widehat{V} M[g_1] = M[g_2] M[1_{I(2)}] \widehat{V} M[1_{I(1)}] M[g_1] \neq 0.\]
The operator $M[g_2] \widehat{V} M[g_1]$ is also a uniform intertwiner between $M[\lambda_1]$ and $M[\lambda_2]$.
Replace $\widehat{V}$ with $M[g_2] \widehat{V} M[g_1]$.
Thus we have a non-zero uniform intertwiner $\widehat{V}$ between $M[\lambda_1]$ and $M[\lambda_2]$ and intervals $I(\iota) \subset \T_{l(\iota)}$, $J \subset \T$ satisfying the following
\begin{itemize}
\item
$\lambda_\iota | _{I(\iota)}$ are diffeomorphisms onto $J$,
\item
There exist closed intervals $K(\iota) \subset I(\iota)$ such that
$\widehat{V} = M[1_{K(2)}] \widehat{V} M[1_{K(1)}]$.
\item
$\widehat{V}$ is uniform with respect to the differential operators $\frac{d}{i d k}$ on $\T_{l(1)}$ and $\T_{l(2)}$.
\end{itemize}
Denote by $\psi$ the diffeomorphism $(\lambda_1 | _{I(1)})^{-1} \circ \lambda_2 | _{I(2)} \colon I(2) \to I(1)$.
Note that $\psi'$ and $(\psi^{-1})'$ are bounded on $K(2)$ and on $K(1)$.
Using a bounded Borel function $g$ on $J$, we can express an arbitrary bounded Borel function on $I(1)$ as $(g \circ \lambda_1) 1_{I(1)}$.
The image of $(g \circ \lambda_1) 1_{I(1)}$ through $\widehat{V}$ is
\begin{eqnarray*}
\widehat{V}((g \circ \lambda_1) 1_{I(1)})
=
\widehat{V} \circ M[g \circ \lambda_1] (1_{I(1)}).
\end{eqnarray*}
Since the operator $M[g \circ \lambda_1]$ is equal to the functional calculus
$\int_{t \in \T} g(t) E_1(d t)$ of $M[\lambda_1]$, we have
\begin{eqnarray*}
\widehat{V}((g \circ \lambda_1) 1_{I(1)})
=
\widehat{V} \circ \left(\int_{t \in \T} g(t) E_1(d t) \right) (1_{I(1)}).
\end{eqnarray*}
Define $f \in L^2(\T_{l(2)})$ by $\widehat{V}(1_{I(1)})$.
Because $\widehat{V}$ is an intertwiner between $M[\lambda_1]$ and $M[\lambda_2]$,
we have
\begin{eqnarray*}
\widehat{V}((g \circ \lambda_1) 1_{I(1)})
&=&
\left(\int_{t \in \T} g(t) E_2(d t) \right) \circ \widehat{V} (1_{I(1)})\\
&=&
\left(\int_{t \in \T} g(t) E_2(d t) \right) (f)\\
&=&
M[g \circ \lambda_2] (f)\\
&=&
M[f] ((g \circ \lambda_2) 1_{I(2)}).
\end{eqnarray*}
The function $(g \circ \lambda_2) 1_{I(2)}$ is equal to $((g \circ \lambda_1) 1_{I(1)}) \circ \psi$.
It follows that $\widehat{V} = M[f] \circ \psi^*$.
The function $f = \widehat{V}(1_{I(1)})$ is continuous.
Indeed, we can express $f$ as $\widehat{V}(g_3)$, using a continuous function $g_3$ on $I(1)$ such that $\mathrm{supp}(g_3)$ is included in $I(1)$ and that $g_3(k) = 1$ for $k \in K(1)$.
Since $\widehat V$ is uniform and $g_3$ is continuous, $f = \widehat{V}(1_{I(1)})$ is a continuous on $T_{l(2)}$.
Since $\widehat{V} = M[f] \circ
\psi^*$ is not zero, ${\rm supp} f$ has to contain an open interval.
By Lemma \ref{lemma: diffeomorphism between intervals},
the mapping $\psi |_{{\rm supp} f}$ is given by rotation on the open interval.
By the identity theorem of analytic functions, $l(1) = l(2)$ and
there exists $\alpha \in \R$ such that
$\lambda_2(k) = \lambda_1(k + \alpha)$ for every $k \in \T_{l}$.
Note that $\alpha$ is uniquely determined only by $\lambda_1$ and $\lambda_2$, because $\lambda_2$ does not have period.
Define $l$ by $l(1)$ and $r$ by $2 \pi / l$.
To identify the set of all the uniform intertwiners, take an arbitrary uniform intertwiner $\widehat{V}$ between $M[\lambda_1], M[\lambda_2]$.
There exists finite open intervals
\[J(1), J(2), \cdots, J(\nu) \subset \T_l\]
such that the union $\cup_{\sigma} J(\sigma)$ is the complement of the set of critical values of $\lambda_1$.
It follows that the union $\cup_{\sigma} (J(\sigma) - \alpha)$ is the complement of the set of critical values of $\lambda_2$.
We note that if $\tau \neq \sigma$, then the intertwiner
$M[1_{J(\tau) - \alpha}] \widehat{V} M[1_{J(\sigma)}]$ is zero.
Indeed,
there exist no open intervals $I(2) \subset J(\tau) - \alpha$ and $I(1) \subset J(\sigma)$ such that $\psi = (\lambda_1 | _{I(1)})^{-1} \circ \lambda_2 | _{I(2)} \colon I(2) \to I(1)$ is well-defined and that $\psi$ is rotation.
By the contrapositive of the last paragraph, the intertwiner $M[1_{J(\tau) - \alpha}] \widehat{V} M[1_{J(\sigma)}]$ is zero.
Thus we can express $\widehat{V}$ as follows:
\[\widehat{V} = \sum_{\sigma = 1}^\nu \widehat{V} M[1_{J(\sigma)}]
= \sum_{\sigma = 1}^\nu M[1_{J(\sigma) - \alpha}] \widehat{V} M[1_{J(\sigma)}].\]
For the corner $M[1_{J(\sigma) - \alpha}] \widehat{V} M[1_{J(\sigma)}]$ of $\widehat{V}$,
there exists a Borel function $f_\sigma \colon (J(\sigma) - \alpha) \to \C$ such that
\[M[1_{J(\sigma) - \alpha}] \widehat{V} M[1_{J(\sigma)}] = M[f_\sigma] \circ \phi_\alpha^*,\]
where $\phi_\alpha$ is the rotation by $\alpha \in \T_{2 \pi r^{-1}}$.
Define a Borel function $f$ on $\T_l$, combining $f_\sigma$. We obtain that
\[\widehat{V} = M[f] \circ \phi_\alpha^*.\]
Because $\widehat{V}$ is uniform, $f \colon \T_l \to \C$ has to be continuous.
Therefore the set of uniform intertwiners is included in
\[\{M[f] \circ \phi_\alpha^* \ |\ f \colon \T_l \to \C {\rm\ continuous.}\}\]
We can easily show the converse inclusion by direct computation.
The set of Fourier transforms of these operators is nothing other than the set in the proposition.
\end{proof}
Now we are ready to identify the set of uniform intertwiners between given two homogeneous analytic quantum walks.
Theorem \ref{theorem: structure theorem} means that every homogeneous analytic quantum walk is a direct sum of finitely many prime model quantum walks and constant quantum walks.
It suffices to identify the set of uniform intertwiners between these building blocks $U_1$, $U_2$, $\cdots$.
\paragraph{\bf Case 1}
First consider the case that
$U_1$ is a constant quantum walk
$(\ell_2(\Z), (\alpha^t)_t, D_1)$.
and
that
$U_2$ is a prime model quantum walk
$\left( \ell_2(r \Z), (U_{\lambda}^t)_t, D_{r} \right)$.
Let $V$ be an intertwiner between $U_1$ and $U_2$.
then we have
\[U_\lambda V = V \alpha = \alpha V.\]
Because $U_\lambda$ has no eigenvector other than the zero vector,
$V$ has to be $0$.
\paragraph{\bf Case 2}
Consider the case that
$U_1$ and $U_2$ are constant quantum walks.
We express them as follows
\[
U_1 = \left( \ell_2(\Z), (\alpha^t)_t, D_1 \right),
\quad
U_2 = \left( \ell_2(\Z), (\beta^t)_t, D_1 \right).
\]
If $\alpha \neq \beta$, then there exists no non-zero intertwiner between them.
If $\alpha = \beta$, then every operator is an intertwiner between them.
The collection of the uniform operators is the uniform Roe algebra ${\rm C}^*_{\rm u}(\Z)$. See Remark \ref{remark: uniform Roe algebra}.
\paragraph{\bf Case 3}
Consider the case that $U_1$ and $U_2$ are prime model quantum walks.
We express the quantum walks as follows
\[
U_1 = \left( \ell_2(r(1) \Z), (U_{\lambda_1}^t)_t, D_{r(1)} \right),
\quad
U_2 = \left( \ell_2(r(2) \Z), (U_{\lambda_2}^t)_t, D_{r(2)} \right),
\]
By Proposition \ref{proposition: uniform intertwiner},
if $r: = r(1) = r(2)$, and
if there exists $\alpha \in [0, 2 \pi r^{-1})$ such that $\lambda_2(k) = \lambda_1(k + \alpha)$,
then the set of intertwiners is
\[\left\{\left. \F_r M[\rho] \F_r^{-1} \exp (i\alpha D_r) \ \right|\ \rho \colon \T_{2 \pi r^{-1}} \to \C {\rm\ continuous} \right\}.\]
By Proposition \ref{proposition: uniform intertwiner},
if $r(1) \neq r(2)$, or if there does not exist $\alpha \in \T_{2 \pi r^{-1}}$ such that $\lambda_2(k) = \lambda_1(k + \alpha)$,
then there exists no non-zero uniform intertwiner between $U_1$ and $U_2$.
\begin{example}\label{example: two direct summands of 4-state Grover walk}
Consider the quantum walk generated by
\[W =
\frac{1}{2}
\left(
\begin{array}{cc}
- S_1^3 - S_1 & S_1 - S_1^{-1}\\
S_1 - S_1^{-1} & - S_1^{-1} - S_1^{-3}
\end{array}
\right) \in \B(\ell_2(\Z) \otimes \C^2).
\]
The characteristic polynomial of the inverse Fourier transform $\widehat{W}(k)$
is
\[\lambda^2 + (\cos 3k + \cos k) \lambda + 1.\]
The roots are given by $\lambda_3(k)$ and $\lambda_4(k)$ defined in Example \ref{example: 4-state Grover walk}.
The quantum walk is similar to
\[\left( L^2(\T_{2 \pi})^2, M[\lambda_3] \oplus M[\lambda_4], \dfrac{d}{i dk} \oplus \dfrac{d}{i dk} \right).\]
Let $V_U$ be the intertwining unitary operator from the $4$-state Grover walk $U$ to
\[1 \oplus (-1) \oplus M[\lambda_3] \oplus M[\lambda_4] \in \B(L^2(\T_{2 \pi})^4).\]
Let $V_W$ be the intertwining unitary operator from $W$ to
\[M[\lambda_3] \oplus M[\lambda_4] \in \B(L^2(\T_{2 \pi})^2).\]
Because $\lambda_4$ is not obtained by translation of $\lambda_3$,
the space of uniform operators intertwining $U$ and $W$ is
\[
V_W^{-1}
\left(
\begin{array}{ccccc}
0 & 0 & C(\T_{2 \pi}) & 0\\
0 & 0 & 0 & C(\T_{2 \pi})\\
\end{array}
\right)
V_U,
\]
where $C(\T_{2 \pi})$ is the space of multiplication operators given by continuous functions on the torus.
The operators in the middle map $(L^2(\T_{2 \pi}))^4$ to $(L^2(\T_{2 \pi}))^2$.
To identify $V_U$ and $V_W$, we only have to identify the eigenspace decomposition of $\widehat{U}(k)$ and $\widehat{W}(k)$.
The calculation is possible, but complicated. We omit identifying them.
\end{example}
\begin{example}\label{example: a direct summand of 3-state Grover walk}
Consider the quantum walk generated by
\[W =
\frac{1}{3}
\left(
\begin{array}{cc}
-2 - S_1 & \sqrt{2}i (S_1^{-1} - 1)\\
\sqrt{2}i (S_1 - 1) & - 2 - S_1^{-1}
\end{array}
\right) \in \B(\ell_2(\Z) \otimes \C^2).
\]
The characteristic polynomial of the inverse Fourier transform $\widehat{W}(k)$
is
\[\lambda^2 + \dfrac{4 + 2 \cos k}{3} \lambda + 1.\]
The roots are given by $\lambda_2(k)$ and $\lambda_2(k + 2 \pi)$ defined in Example \ref{example: 3-state Grover walk}.
The quantum walk is similar to
\[\left( L^2(\T_\pi), M[\lambda_2], \dfrac{d}{i dk} \right).\]
Let $V_U$ be the intertwining unitary operator from the $3$-state Grover walk $U$ to
\[1 \oplus M[\lambda_2] \in \B(L^2(\T_{2 \pi}) \oplus L^2(\T_{4 \pi})).\]
Let $V_W$ be the intertwining unitary operator from $W$ to
\[M[\lambda_2] \in \B(L^2(\T_{4 \pi})).\]
The space of uniform operators intertwining $U$ and $W$ is
\[
V_W^{-1}
\left(
0 \quad M(C(\T_{4 \pi}))
\right)
V_U.
\]
The operators in the middle map $L^2(\T_{2 \pi}) \oplus L^2(\T_{4 \pi})$ to $L^2(\T_{4 \pi})$.
\end{example}
In the above two examples, we see the cases that there exist non-zero intertwiners.
However, in many cases, there exists no non-zero uniform intertwiner.
For example, there exists no non-zero uniform intertwiner between the walk in Example \ref{example: two direct summands of 4-state Grover walk} and that in Example \ref{example: a direct summand of 3-state Grover walk}.
To show absence of a non-zero intertwiner, we need some systematic way of proof like the contrapositive of Proposition \ref{proposition: uniform intertwiner}, while
for existence of non-zero intertwiner, we might find intertwiners by some chance.
\begin{example}
Let $\rho$ be a positive real number less than $1$.
Consider the quantum walks $U_\rho$ on $\ell_2(\Z) \otimes \C^2$ generated by
$U_\rho = \left(
\begin{array}{cc}
\rho S_1^{-1} & \sqrt{1 - \rho^2} S_1^{-1}\\
\sqrt{1 - \rho^2} S_1 & \rho S_1
\end{array}
\right)$.
The eigenvalue functions of the inverse Fourier transform $\widehat{U_\rho}$ are
\[
\lambda_{\rho, +} = \rho \cos k + i \sqrt{1 - \rho^2 \cos^2 k},
\quad
\lambda_{\rho, +} = \rho \cos k - i \sqrt{1 - \rho^2 \cos^2 k}.
\]
If $\rho(1), \rho(2) \in (0, 1)$ and if $\rho(1) \neq \rho(2)$, then there exists no $\alpha \in \T_{2 \pi}$ satisfying $\lambda_{\rho(2), \pm}(k) = \lambda_{\rho(1), \pm}(k + \alpha)$.
It follows that there exists no uniform intertwiner between $U_{\rho(1)}$ and $U_{\rho(2)}$.
\end{example}
\subsection{Uniform commutant of a homogeneous analytic quantum walk}
For a quantum walk $(\Hil, (U^t)_{t \in \Z}, D)$,
we call the algebra
\[\left\{ V \in \B(\Hil) \left|
V U = U V,
k \mapsto \exp(i k D) V \exp(- i k D) {\rm \ is\ continuous} \right. \right\}\]
the {\it uniform commutant} of $U$.
The following are conclusions of Proposition \ref{proposition: uniform intertwiner}.
\begin{corollary}
The uniform commutant of
a prime model quantum walk
$( \ell_2( r \Z )$, $( U_\lambda^{t} )_{t \in \Z}$, $D_r )$
is identical to
\[\left\{\left. \F_r M[\rho] \F_r^{-1} \ \right|\ \rho \colon \T_{2 \pi r^{-1}} \to \C {\rm\ continuous} \right\}.\]
\end{corollary}
\begin{proof}
By the definition of a prime model quantum walk,
$\lambda$ has no rotational symmetry.
\end{proof}
The following is the motivation of the definition of {\it prime} model quantum walk.
\begin{corollary}
No prime model quantum walk is similar to
a direct sum of two (not necessarily homogeneous) one-dimensional uniform quantum walks.
\end{corollary}
\begin{proof}
For every prime model quantum walk,
the uniform commutant is
\[\left\{\left. \F_r M[\rho] \F_r^{-1} \ \right|\ \rho \colon \T_{2 \pi r^{-1}} \to \C {\rm\ continuous} \right\} \subset \B(\ell_2(r \Z)).\]
The set of all the orthogonal projections in this algebra is $\{0, {\rm id}\}$.
If the walk were a direct sum of two quantum walks, the set would contain a non-trivial projection.
\end{proof}
\begin{remark}
In \cite{SaigoSako}, the notion of indecomposable quantum walk is defined.
The condition of indecomposable model quantum walk is weaker than that of primeness.
Primeness means that the walk can not be decomposable in the category of quantum walks,
while indecomposability means that the walk can not be decomposable in the category of {\it homogeneous} quantum walks.
\end{remark}
Using Theorem \ref{theorem: structure theorem},
we identify the structure of the uniform commutant
of the discrete-time homogeneous analytic quantum walk
$(\ell_2(\Z) \otimes \C^n, (U^t)_{t \in \Z}, D \otimes {\rm id})$.
\begin{proposition}\label{proposition: uniform commutant}
Let $(\ell_2(\Z) \otimes \C^n, (U^t)_{t \in \Z}, D_1 \otimes {\rm id})$ be an arbitrary one-dimensional discrete-time homogeneous analytic quantum walk.
The uniform commutant
\[
\left\{
V \colon
\ell_2(\Z) \otimes \C^n \to \ell_2(\Z) \otimes \C^n \ |\
V U = U V, k \mapsto e^{ik D} V e^{-ikD} {\rm \ is\ continuous}
\right\}
\]
is isomorphic to the algebra of the following form
\[
\bigoplus_{j = 1}^l
\left(
C (\T_{r(j)}) \otimes M_{\mu(j)}(\C)
\right)
\oplus
\bigoplus_{k = 1}^m
\left(
{\rm C}^{*}_{\rm u} (\Z) \otimes M_{\nu(k)}(\C)
\right)
.\]
(The integers $l$ and $m$ can be zero. In such a case, erase the corresponding direct summand.)
The operator $U$ itself is located at the element of the form
\[
\bigoplus_{j = 1}^l
\left(
M[\lambda_j] \otimes {\rm id}_{\mu(j)}
\right)
\oplus
\bigoplus_{k = 1}^m
\left(
\alpha(k) \otimes {\rm id}_{\nu(k)}
\right).
\]
The structure of smoothness and analyticity is given by the self-adjoint operator
\[
\bigoplus_{j = 1}^l
\left(
\frac{d}{i d k} \otimes {\rm id}_{\mu(j)}
\right)
\oplus
\bigoplus_{k = 1}^m
\left(
D_1 \otimes {\rm id}_{\nu(k)}
\right)
.\]
\end{proposition}
\begin{remark}
The non-negative integers $l$ and $m$ in Proposition \ref{proposition: uniform commutant} can be different from those in Theorem
\ref{theorem: structure theorem}.
\end{remark}
\begin{proof}
Recall that $U$ can be decomposed into prime model quantum walks and constant quantum walks.
Let $U_1$ and $U_2$ be two direct summand of $U$.
By the argument in Cases {\bf 1}, {\bf 2}, {\bf 3} in the previous subsection,
if there exists a non-zero uniform intertwiner between $U_1$ and $U_2$ ,
then there exists a uniform unitary operator which intertwines them.
Therefore, existence of non-zero uniform intertwiner defines an equivalence relation between
direct summand of $U$.
Let $\{U_1, U_2, \cdots, U_\nu\}$ be such an equivalence class.
By the above argument, they are all constant quantum walks, or they are all prime model quantum walks.
Consider the case that
$U_1, U_2, \cdots, U_\nu$ are the constant walks.
By {\bf Case 2} in the previous subsection, these are identical.
Express them as $\left( \ell_2(\Z), (\alpha^t)_t, D_1 \right)$.
Combining the set of uniform intertwiners, we obtain the algebra
${\rm C}^*_{\rm u}(\Z) \otimes M_\nu(\C)$.
Consider the case that
$U_1, U_2, \cdots, U_\nu$ are prime quantum walks.
By {\bf Case 3} in the previous subsection, corresponding analytic functions
\[\lambda_1, \cdots, \lambda_\nu \colon \T_{2 \pi r^{-1}} \to \T\]
are mutually translations of each other.
There exist $\alpha_1, \cdots, \alpha_\nu \in \T_{2 \pi r^{-1}}$ such that
$\lambda_j(k) = \lambda_1 (k + \alpha_j)$.
Then we have $\lambda_j(k) = \lambda_l(k + \alpha_j - \alpha_l)$.
By {\bf Case 3} in the previous subsection,
the set of uniform intertwiners from $U_l$ to $U_j$ is
\[\left\{\left. \F_r M[\rho] \F_r^{-1} \exp (i (\alpha_j - \alpha_l) D_r) \ \right|\ \rho \colon \T_{2 \pi r^{-1}} \to \C {\rm\ continuous} \right\}.\]
In the case of $l = k$,
$U_l$ is located at $\F_r M[\lambda_l] \F_r^{-1}$.
The inverse Fourier transform is
\[\left\{\left.
\exp \left(\alpha_j \dfrac{d}{d k} \right)
M[\rho]
\exp \left(\alpha_l \dfrac{d}{dk} \right)^{-1}
\ \right|\
\rho \colon \T_{2 \pi r^{-1}} \to \C {\rm\ continuous} \right\}.\]
Note that the operator $\exp \left(\alpha_j \frac{d}{d k} \right) \in \B(L^2(\T_{2 \pi r^{-1}}))$
is the translation operator by $\alpha_j \in \T_{2 \pi r^{-1}}$ and that it is a normalizer of the space of multiplication operators $C(\T_{2 \pi r^{-1}})$.
Also note that $\exp \left(\alpha_j \frac{d}{d k} \right)$ commutes with the differential operator $\frac{d}{i d k}$.
Combining all the intertwiners, we conclude that
the set of uniform intertwiners between $U_1 \oplus \cdots \oplus U_\nu$
and itself
is isomorphic to
$C(\T_{2 \pi r^{-1}}) \otimes M_\nu(\C)$.
\end{proof}
\section{Realization by a continuous-time uniform quantum walk}
\begin{lemma}\label{lemma: realization by one-parameter}
Let $\nu$ be a natural number.
Let $r$ be a positive real number.
Let $\lambda \colon \T_{2 \pi r^{-1}} \to \T$ be a continuous map.
There exists a one-parameter group $(U^{(t)})_{t \in \R}$ of unitary operators in
$C (\T_{2 \pi r^{-1}}) \otimes M_{\nu}(\C)$ satisfying
\[U^{(1)} = M[\lambda] \otimes {\rm id}_{\nu} ,\]
if and only if the winding number of $\lambda$ is zero.
\end{lemma}
\begin{proof}
Suppose that the winding number of $\lambda$ is zero.
Then there exists a continuous function $h \colon \T_{2 \pi r^{-1}} \to \R$ such that $\exp(i h) = \lambda$.
The one-parameter unitary group
\[U^{(t)} = M[\exp(i t h)] \otimes {\rm id}_\nu \in C (\T_{2 \pi r^{-1}}) \otimes M_{\nu}(\C)\]
satisfies $U^{(1)} = M[\lambda] \otimes {\rm id}_{\nu}$.
Conversely suppose that there exists a one-parameter unitary group $U^{(t)} \in C (\T_{2 \pi r^{-1}}) \otimes M_{\nu}(\C)$ which satisfies $U^{(1)} = M[\lambda] \otimes {\rm id}_{\nu}$.
We make use of $C (\T_{2 \pi r^{-1}})$-valued determinant
\[\det \colon C (\T_{2 \pi r^{-1}}) \otimes M_{\nu}(\C) \to C (\T_{2 \pi r^{-1}}).\]
Since the map $\det$ is multiplicative, $\det U^{(t)}$ is a unitary element of $C (\T_{2 \pi r^{-1}})$.
The winding numbers $\{w(\det U^{(t)})\}_{t \in \R}$ define a group homomorphism from $\R$ to $\Z$.
It follows that $w(\det U^{(t)}) = 0$ for every $t \in \R$.
Therefore we have
\begin{eqnarray*}
\nu w(\lambda) = w(\lambda^\nu) = w(\det U^{(1)}) = 0
\end{eqnarray*}
and $w(\lambda) = 0$.
\end{proof}
\begin{theorem}\label{theorem: continuous-time}
Let $(\ell_2(\Z) \otimes \C^n, (U^t)_{t \in \Z}, D_1 \otimes {\rm id})$ be an arbitrary one-dimensional discrete-time homogeneous analytic quantum walk.
Let $\lambda_1, \lambda_2, \cdots$ be the eigenvalue functions of $U$
introduced in Subsection \ref{subsection: review of SS}.
Then the following conditions are equivalent
\begin{enumerate}
\item
There exists a one-dimensional continuous-time {\rm uniform} quantum walk $(\ell_2(\Z) \otimes \C^n, (U^{(t)})_{t \in \R}, D_1 \otimes {\rm id})$ such that
$U^{(1)} = U$.
\item
There exists a one-dimensional continuous-time {\rm homogeneous} and {\rm analytic} quantum walk $(\ell_2(\Z) \otimes \C^n, (U^{(t)})_{t \in \R}, D \otimes {\rm id})$ such that
$U^{(1)} = U$.
\item
All the winding numbers of $\lambda_1, \lambda_2, \cdots $ are zero.
\end{enumerate}
\end{theorem}
The first item looks much weaker than the second item,
but the following proof will show that both are equivalent to the third item.
\begin{remark}
We may further weaken the first condition.
We can eliminate the assumption that $(U^{(t)})_{t \in \R}$ is continuous with respect to the strong operator topology.
\end{remark}
\begin{proof}
The easier half of
Theorem 5.14 in \cite{SaigoSako} shows that the third condition implies the second one.
It suffices to show that the first condition implies the third one.
Suppose that there exists a one-parameter group $U^{(t)}$ of uniform unitary operator on $\ell_2(\Z) \otimes \C^n$ such that $U^{(1)} = U$.
Note that for every $t$, $U^{(t)}$ commutes with $U$.
We use the algebra
\[
\bigoplus_{j = 1}^l
\left(
C (\T_{r(j)}) \otimes M_{\mu(j)}(\C)
\right)
\oplus
\bigoplus_{k = 1}^m
\left(
{\rm C}^{*}_{\rm u} (\Z) \otimes M_{\nu(k)}(\C)
\right)
\]
in Proposition \ref{proposition: uniform commutant}.
Existence of $U^{(t)}$ means that
\[
\widehat{U} = \bigoplus_{j = 1}^l
\left(
M[\lambda_j] \otimes {\rm id}_{\mu(j)}
\right)
\oplus
\bigoplus_{k = 1}^m
\left(
\alpha(k) \otimes {\rm id}_{\nu(k)}
\right).
\]
can be realized by a one-parameter group of unitary operators in the algebra.
The winding numbers of constant functions $\alpha(k)$ are $0$, so the latter summand does not have to do with our problem.
We can concentrate on the operator
\[
M[\lambda_j] \otimes {\rm id}_{\mu(j)}
\in
C (\T_{r(j)}) \otimes M_{\mu(j)}(\C).
\]
By Lemma \ref{lemma: realization by one-parameter},
if this is realized by a one-parameter unitary group inside $C (\T_{r(j)}) \otimes M_{\mu(j)}(\C)$,
then the winding number of $\lambda_j$ is $0$.
\end{proof}
\begin{example}
Let $r$ be a real number greater than $0$ and less than $1$.
Let us consider the quantum walk
\[
U =
\left(
\begin{array}{cc}
r S^{-1} & - \sqrt{1 - r^2} S^{-1} \\
\sqrt{1 - r^2} S & r S
\end{array}
\right).\]
acting on $\ell_2(\mathbb{Z}) \otimes \mathbb{C}^2$.
The weak limit theorem for this walk has been shown in \cite{KonnoJMSJ}.
The characteristic polynomial of the inverse Fourier transform $\widehat{U}(k)$ is
\[f(k; z) = \lambda^2 - r \left( e^{ik} + e^{-ik} \right) \lambda + 1.\]
We express $z$ by $e^{i \theta}$.
The roots are
\begin{eqnarray*}
\lambda_1(k) &=& r \cos k + i \sqrt{1 - r^2 \cos^2 k},\\
\lambda_2(k) &=& r \cos k - i \sqrt{1 - r^2 \cos^2 k}.
\end{eqnarray*}
They are single-valued functions.
The winding numbers are both zero.
By Theorem \ref{theorem: continuous-time},
This can be realized by a continuous-time quantum walk.\qed
\end{example}
\begin{example}\label{example: 3-state Grover walk is continuous-time}
The $3$-state Grover walk in Example \ref{example: 3-state Grover walk} can be realized
by a continuous-time analytic quantum walk.
We have obtained the constant eigenvalue function $\lambda_1(k) = 1$ and a multi-valued analytic eigenvalue function $\lambda_2$.
The winding number of $\lambda_2 \colon \R / (4 \pi \Z) \to \T$ is zero.
For the same reason, the quantum walk in Example \ref{example: a direct summand of 3-state Grover walk} can be realized by a continuous-time quantum walk.
\end{example}
Even if a homogeneous analytic quantum walk $(U^t)_{t \in \Z}$ is realized by a continuous-time quantum walk $(U^{(t)})_{t \in \R}$, the walk $(U^{(t)})_{t \in \R}$ is not necessarily homogeneous.
\begin{example}
Let $\beta$ be an element of $\T_{2 \pi} = \R / (2 \pi \Z)$.
Assume that for every integer $x \in \Z$, $x \beta \in \T_{2 \pi}$ is not zero.
Let $\lambda \colon \T_{2 \pi} \to \T$ be an analytic map without period.
Assume that the winding number of $\lambda$ is zero.
Choose an analytic map $g \colon \T_{2 \pi} \to \R$ satisfying $\exp(i g) = \lambda$.
Define $\rho \colon \T_{2 \pi} \to \T$ and $h \colon \T_{2 \pi} \to \R$ by $\rho(k) = \lambda(k + \beta)$ and $h(k) = g(k + \beta)$.
Consider the direct sum of model quantum walks
\[
(\ell_2(\Z), (\F_1 M[\lambda]^t \F_1^{-1})_{t \in \Z}, D_1)
\oplus
(\ell_2(\Z), (\F_1 M[\rho]^t \F_1^{-1})_{t \in \Z}, D_1).
\]
This is realized by the continuous-time homogeneous quantum walk
\[
U^{(t)} =
\left(
\begin{array}{cc}
\F_1 M[\exp(it g)] \F_1^{-1} & 0\\
0 & \F_1 M[\exp(it h)] \F_1^{-1}\\
\end{array}
\right).
\]
We also consider the one-parameter family of unitary
\[
V^{(t)} =
\left(
\begin{array}{cc}
\cos 2 \pi t & - \sin 2 \pi t \cdot \exp( - i \beta D_1)\\
\sin 2 \pi t \cdot \exp( i \beta D_1) & \cos 2 \pi t\\
\end{array}
\right).
\]
Because the inverse Fourier transform $\exp\left( \beta \frac{d}{dk} \right)$ of the operator $\exp( i \beta D_1)$ is the translation operator on $L^2(\T_{2 \pi})$ by $- \beta \in \T_{2 \pi}$, $V^{(t)}$ commutes with $U^{(t)}$.
It follows that $V^{(t)} U^{(t)}$ is also a one-parameter group of unitary operators and realizes the given quantum walk $\F_1 M[\lambda]^t \F_1^{-1} \oplus \F_1 M[\rho]^t \F_1^{-1}$.
If $t$ is not an element of $\frac{1}{2}\Z$, then $V^{(t)} U^{(t)}$ is not homogeneous.
Moreover, it is not even virtually homogeneous in the sense of Definition \ref{definition: homogeneous}.
\end{example}
Theorem \ref{theorem: continuous-time} provides a powerful way to show that given quantum walk is not a restriction of continuous-time quantum walk.
If one wants to show that given quantum walk can be realized by a continuous-time quantum walk, ad hoc way might be useful, because we might be able to find concrete description.
However, if one wants to show that it can not be realized by a continuous-time quantum walk, ad hoc way can not be useful, and we need some systematic procedure.
The following Corollary gives a sufficient condition for such non-existence.
\begin{corollary}\label{corollary: non-existence}
Let $(\ell_2(\Z) \otimes \C^n, (U^t)_{t \in \Z}, D_1 \otimes {\rm id})$ be an arbitrary one-dimensional discrete-time homogeneous analytic quantum walk.
Denote by $\widehat{U} \in C(\T_{2 \pi}) \otimes M_n(\C)$ the inverse Fourier transform of $U$.
If the winding number of $\det \widehat{U} \colon \T_{2 \pi} \to \T$ is not zero, then
there exists no one-dimensional continuous-time uniform quantum walk $(\ell_2(\Z) \otimes \C^n, (U^{(t)})_{t \in \R}, D_1 \otimes {\rm id})$ such that
$U^{(1)} = U$.
\end{corollary}
\begin{proof}
Let $\lambda_1, \cdots, \lambda_l$ be eigenvalue functions of $U$ introduced in Subsection \ref{subsection: review of SS}.
Whichever the eigenvalue functions are single-valued or multi-valued,
the winding number of $\det \widehat{U}$ is the sum of the winding numbers of $\lambda_1, \cdots, \lambda_l$.
If the winding number of $\det \widehat{U} \colon \T_{2 \pi} \to \T$ is not zero, there exists an eigenvalue function whose winding number is not zero.
\end{proof}
\begin{example}
we consider a quantum walk defined by
\[U
=
\left(
\begin{array}{cc}
r S_1 & - b S_1 \\
\overline{b} & r
\end{array}
\right), \quad r \in \mathbb{R}, b \in \mathbb{C}, r^2 + |b|^2 = 1.\]
acting on $\ell_2(\mathbb{Z}) \otimes \mathbb{C}^2$.
Determinant of the Fourier dual is
\[
\det \widehat{U} (k) = \det
\left(
\begin{array}{cc}
r e^{ik} & - b e^{ik} \\
\overline{b} & r
\end{array}
\right)
= e^{ik}.
\]
The winding number of $\det \widehat{U} \colon \T_{2 \pi} \to \T$ is one.
By Corollary \ref{corollary: non-existence}, $U$ can not be realized by a continuous-time uniform quantum walk.
\end{example}
The converse of Corollary \ref{corollary: non-existence} does not hold true.
\begin{example}\label{example: 4-state Grover walk is not continuous-time}
We prove that the $4$-state Grover walk in Example \ref{example: 4-state Grover walk} can not be realized
by a continuous-time uniform quantum walk.
Determinant of the Fourier dual $\widehat{U}$ is a constant function, so we can not use
Corollary \ref{corollary: non-existence}.
We obtain four single-valued analytic eigenvalue functions
\[\lambda_1(k) = 1, \quad \lambda_2(k) = -1,
\quad \lambda_3(k), \quad \lambda_4(k)\]
in Example \ref{example: 4-state Grover walk}.
The winding numbers are
\[w(\lambda_1) = 0, \quad w(\lambda_2) = 0,
\quad w(\lambda_3) = 1, \quad w(\lambda_4) = -1.\]
By Theorem \ref{theorem: continuous-time}, the quantum walk given by $U$ can not be realized by a continuous-time analytic (not necessarily homogeneous) quantum walk.
By the same reason, the quantum walk in Example \ref{example: two direct summands of 4-state Grover walk} can not be realized by a continuous-time quantum walk.
\end{example}
\bibliographystyle{amsalpha}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 8,223 |
{"url":"http:\/\/math.stackexchange.com\/questions\/129618\/why-is-n-tau-m-m-tau-a-continuous-local-martingale-if-m-and-n","text":"# Why is $N^\\tau ( M - M^\\tau )$ a continuous local martingale if $M$ and $N$ are?\n\nWorking through my stochastic calculus script, I encountered the following identity, for which no proof is given: $\\langle M, N^\\tau \\rangle = \\langle M, N \\rangle^\\tau$, if $M, N$ are continuous local martingales, null at 0.\n\nI know that (by uniqueness of the bracket) it suffices to show the following:\n\nIf both $M,N$ are continuous local martingales and $\\tau$ is a stopping time, then $N^\\tau ( M - M^\\tau )$ is again a continuous local martigale.\n\nBut why is this true?\n\nI know how to proof this with the properties of the stochastic integral. But since the above identity is used in our script during the construction of the stochastic integral, I would like to prove it directly. Does anybody know how this can be done?\n\nTrying to prove it, I started like this:\n\n$M,N$ continuous local martingales $\\ \\Rightarrow \\$ there are localizing sequences $\\tau_n, \\sigma_n$ such that $M^{\\tau_n}$ and $N^{\\sigma_n}$ are bounded martingales. By the stopping theorem, $N^{\\tau_n \\wedge \\sigma_n \\wedge \\tau}$ and $M^{\\tau_n \\wedge \\sigma_n \\wedge \\tau}$ are bounded martingales, too.\n\nNow I would like to proof that $N^{\\tau_n \\wedge \\sigma_n \\wedge \\tau} (M^{\\tau_n \\wedge \\sigma_n } - M^{\\tau_n \\wedge \\sigma_n \\wedge \\tau})$ is a martingale, since this would then imply that $N^\\tau ( M - M^\\tau )$ is a continuous local martigale (with localizing sequence $\\tau_n \\wedge \\sigma_n$).\n\nThanks a lot for your help! Regards, Si\n\n-\nThe fact that $N$ is a local martingale is irrelevant. Any continuous adapted process will work. \u2013\u00a0 George Lowther Apr 9 '12 at 15:21\n\nLet $X^{(n)}=N^{\\kappa_n}\\cdot(M^{\\varrho_n}-M^{\\kappa_n})$, where $\\varrho_n=\\tau_n\\wedge\\sigma_n$ and $\\kappa_n=\\varrho_n\\wedge\\tau$ hence $\\kappa_n\\leqslant\\varrho_n$. Then, $\\mathrm dX^{(n)}_t=[\\kappa_n\\leqslant t\\leqslant\\varrho_n]\\cdot N_{\\kappa_n}\\cdot\\mathrm dM_t=[\\kappa_n\\leqslant t\\leqslant\\varrho_n]\\cdot N^{\\kappa_n}_t\\cdot\\mathrm dM_t^{\\tau_n}$ and $M^{\\tau_n}$ is a martingale hence $X^{(n)}$ is a martingale.","date":"2014-08-22 04:17:59","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\": 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\": 0.9467167258262634, \"perplexity\": 89.19172179456022}, \"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-2014-35\/segments\/1408500822560.65\/warc\/CC-MAIN-20140820021342-00192-ip-10-180-136-8.ec2.internal.warc.gz\"}"} | null | null |
Palm Treo: The reviews are upbeat; Can a device turn Palm around?
Palm may have a hit on its hands with the Treo Pro judging from the recent spate of reviews as NDAs ended on Wednesday. The larger question is whether this pricey device can give the company a boost in the bottom line.
Wednesday. The larger question is whether this pricey device can give the company a boost in the bottom line.
The Palm Treo Pro is targeted to the enterprise with its black finish and white key highlights. I was hoping the device would have that great soft-touch rubberized matte casing since I find that helps me grip the device and looks more like a business device. The Treo Pro has a glossy black covering and is a major fingerprint magnet with the back looking much like the black iPhone 3G. The device is thinner than all other Treo devices which is a great move by Palm since I think most Treo devices are too thick and chunky. The Treo Pro has a Centro rounded look to it with a similar QWERTY keyboard.... I am considering the Treo Pro since I do enjoy using Windows Mobile devices and the great thing about the Treo Pro is that you can use it like a non-touch screen device yet get all the power and functionality of a touch screen device when you need it. The excellent GPS signal acquisition, long battery life, Palm touches like the ringer and WiFi buttons, and standard 3.5mm headset jack are compelling to me.
In terms of design, performance and non-OS features, this is the best hardware Palm has ever made. Though my personal dislike of WinMo has me wishing it ran the PalmOS, the truth is that this is a phone for the corporate crowd, and WinMo 6.1 can do things and reach audiences the PalmOS never could. That it is aimed towards a business crowd also justifies the $550 unlocked price tag. People in the corporate world travel to other countries, and need to switch SIM cards quickly and effortlessly. The feature that deserves the most mention is the redesign—because it's beautiful.
The big question: Can a $549 unlocked device that isn't bundled with a carrier really turn a company around?
Palm has its design chops back--the Centro and now Treo Pro have established a look for Palm to roll out to other devices.
Palm's average selling prices will increase in the quarter since it was way too reliant on the $99 Centro for growth.
The Treo Pro is benefiting from low expectations. If Palm can consistently over deliver on promises it will be a more viable turnaround story.
The Treo Pro is expensive and now may not be time to play that card.
The Treo Pro is yet another Windows Mobile device.
Palm needs a revolution not an evolution with its devices to restore its lost mojo.
In other words, it's a jump ball when it comes to Palm prognostications. Analysts, however, are skewing to the upbeat side.
Pacific Crest analyst James Faucette said in a research note last week that the Treo 800w is topping Sprint and Palm's initial expectations. Meanwhile, the Treo Pro will launch in Europe with Vodafone, Telstra and other carriers. In the U.S. AT&T is the favorite to pick up the Treo Pro. Faucette is predicting shipments of 40,000 more Treo units in the fiscal second quarter due to the Treo Pro launch.
Faucette also noted that management has been adding the right people--it recently hired Jeff Devine, formerly Nokia's operations VP, to run Palm's operations, which have been a sore spot in recent quarters. Toss in Palm's new OS, code-named Nova, in the first half and the company could be on the turnaround path.
JP Morgan analyst Paul Coster reckons that the Treo Pro could bring Palm closer to a second half profit as average selling prices increase.
It is unclear to us why Palm announced that the Treo Pro would only be available as an unlocked version in the U.S. this autumn. We assume this means AT&T has either refused to carry the product or wants to sell its existing inventory of Treo 750s before committing to a launch date. It is also possible that the two sides may be still debating about price, or that Palm may be waiting to see where the price point of the Bold before committing to a price at AT&T. Ultimately, we still presume that AT&T will end up carrying the Treo Pro given its impressive specs relative to its current portfolio of WM devices, and AT&T's apparent willingness to carry just about any smartphone offered to satisfy its focus on offering the subscribers the broadest selection of handsets.
Nevertheless, McCourt argues that the Treo Pro will at least hold its own in the race to grab enterprise customers. That fact puts Palm on much better footing than it had 90 days ago (click to enlarge chart via McCourt).
Palm is expected to post a loss of 12 cents a share on revenue of $368.8 million for the second quarter. | {
"redpajama_set_name": "RedPajamaC4"
} | 1,347 |
\section{\boldmath Microscopic model for the superconducting channel of {YB\MakeLowercase{a}$_2$C\MakeLowercase{u}$_3$O$_{7-\delta}$} in the mixed state}
\subsection{Electronic structure}
YBa$_2$Cu$_3$O$_{7-\delta}$ (Y123) has two CuO$_2$ planes in the unit cell, CuO chains running along the $b$ axis, and a small orthorhombic distortion with inequivalent $a$ and $b$ axes ($b>a$) \cite{Jorgensen-1990-s}. We ignore the CuO chains and the distortion, absent in other cuprates, and therefore irrelevant for high-$T_c$ superconductivity. Some effects of the chains on the vortex-core spectra have been studied in Ref.~\onlinecite{Atkinson-2009-s}. We furthermore ignore the bilayer splitting for simplicity and represent the CuO$_2$ layers as a one-band tight-binding model on a perfect square lattice with parameter $a=3.85$~\AA. We have also performed calculations for a two-band system including bilayer splitting, and found only inessential quantitative differences in the vortex cores. The one-band model is more convenient for large-scale simulations. We use the tight-binding parameters $t_1=-281$~meV, $t_2=139$~meV, and $t_3=-44$~meV determined by photoemission in Ref.~\onlinecite{Schabel-1998a-s, *Schabel-1998b-s} for the first, second, and third neighbor hopping, respectively, ignoring $t_4$ and $t_5$ for simplicity. The chemical potential is set to $\mu=-356$~meV for an electron density $n=0.84$, corresponding to optimally hole-doped Y123 with 0.16 hole per unit cell. The dispersion relation measured from the chemical potential is $\xi_{\vec{k}}=2t_1[\cos(k_xa)+\cos(k_ya)]+4t_2\cos(k_xa)\cos(k_ya)+2t_3[\cos(2k_xa)+\cos(2k_ya)]-\mu$. The average group velocity on the Fermi surface is $\langle v_{\mathrm{F}}\rangle=4.11\times10^7$~cm/s. The Fermi surface is shown in Fig.~\ref{fig:uniform}(a). Due to a van Hove singularity at $-376$~meV, the DOS has a negative slope in the low-energy region, with more weight for the occupied states [Fig.~\ref{fig:uniform}(b)]. We set the amplitude of the $d$-wave order parameter to $\Delta_0=19$~meV. In the uniform superconductor, the gap $\Delta_{\vec{k}}=(\Delta_0/2)[\cos(k_xa)-\cos(k_ya)]$ has its maximum at the point $(\pi/a,0.74/a)$ of the Fermi surface, giving coherence peaks at $\pm17$~meV [Fig.~\ref{fig:uniform}(c)]. This amplitude and symmetry of the order parameter follow self-consistently from the Bogoliubov-de Gennes equations for an (instantaneous) attractive interaction $V=-247$~meV on nearest-neighbor bonds.
\begin{figure}[b]
\includegraphics[width=\columnwidth]{figS1}
\caption{\label{fig:uniform}
(a) Fermi surface and (b) dispersion relation. The green curve in (b) is the normal-state DOS with a van Hove singularity at $-376$~meV. (c) Zero-field tunneling conductance of Y123 at 0.4~Kelvin (solid blue, left scale, from Ref.~\onlinecite{Bruer-2016-s}) and its decomposition in superconducting (solid green) and non-superconducting (red) channels. The solid green curve (right scale) is the BCS DOS calculated with the dispersion shown in (b). The finite zero-energy DOS in the gap is due to the finite energy resolution of the calculation ($\approx 3$~meV). The dashed green curve is the corresponding normal-state DOS. The value $A=0.25$~eV~nS is adjusted such that the non-superconducting channel has no coherence peaks at $\pm17$~meV. The dashed blue curve is the sum of the red and dashed green ones.
}
\end{figure}
Figure~\ref{fig:uniform}(c) shows the base-temperature zero-field tunneling spectrum of Y123 \cite{Bruer-2016-s} and a possible decomposition in two channels. The superconducting channel (SC) is calculated with the tight-binding model, and the non-superconducting channel (NSC) is the difference. The relative weight of the two contributions is adjusted in such a way that the NSC has no structure---peak or dip---at the edges of the superconducting gap. The resulting NSC has an overall positive slope. The latter is sensitive to the choice of hopping parameters in the SC, and is therefore not a robust feature of the analysis. This slope is irrelevant for our study of vortex cores focusing on energies $\lesssim50$~meV. In contrast, the dips near $\pm50$~meV, the peaks near $\pm30$~meV, and the subgap peaks near $\pm5$~meV are robust properties of the spectrum measured by STS in regions where superconductivity is suppressed \cite{Bruer-2016-s}. The dashed blue line shows the spectrum expected in such a region where the superconducting gap is closed, assuming that the relative weight of the two channels remains unchanged. This is fully consistent with the spectra measured in non-superconducting regions \cite{Bruer-2016-s}. A more elaborate modeling was introduced in Ref.~\onlinecite{Bruer-2016-s}, that involved bilayer splitting as well as an interaction with the spin fluctuations. The main effect of these additional ingredients is to assign (part of) the dips at $\pm50$~meV to the SC rather than the NSC. As these energies are not our main concern and these sophistications are impractical in view of large-scale vortex calculations, here we disregard them.
\subsection{Isolated vortex, self-consistent solution}
We solve the Bogoliubov-de Gennes equations self-consistently with a single vortex at the origin using the method described in Ref.~\onlinecite{Berthod-2016-s}. The reader is referred to Ref.~\onlinecite{Berthod-2016-s} for all practical details, while here we only give the elements specific to the present calculation. As our Hamiltonian extends up to third neighbors, it spreads the wave function on the lattice with a diamond shape. We therefore consider a finite system with diamond shape and linear size $M$, having $1+2M(1+M)$ lattice sites. We use $M=200$ (80\,401 sites) for calculating the self-consistent order parameter and $M=500$ (501\,001 sites) for calculating the local density of states (LDOS). The order parameter requires a smaller system because the coherence length imposes a spatial cutoff. With $M=200$, a Chebyshev expansion order $N=6000$, and termination using the Jackson kernel \cite{Berthod-2016-s}, the calculation retrieves the exact order parameter within 0.1\%. For the LDOS, the spatial cutoff would be set by the mean free path, which is infinite in our model. With $M=500$ and an expansion order $N=2000$, we reach an energy resolution of roughly $3$~meV without perturbations associated with the system's boundaries.
The self-consistent order parameter is plotted in Fig.~\ref{fig:vortex}(a). The quantity $|\Delta(\vec{r})|$ is defined as the sum of the order-parameter modulus on the four bonds surrounding the site $\vec{r}$. It is well fitted by the isotropic ansatz $\Delta(r)=\Delta_0/[1+\xi_0/r\exp(-r/\xi_1)]$ with $\xi_0=17a$ and $\xi_1=29.5a$. The difference between the exact and approximate data is negative along the $x$ and $y$ directions and positive along the diagonals. The self-consistent order parameter indeed displays a small in-plane anisotropy unlike the ansatz, and relaxes faster to its asymptotic value along the diagonals than along the lattice axes, as already found in similar calculations \cite{Ichioka-1996-s, Berthod-2016-s}. A ``core size'' $\xi_c$ may be defined by the condition $\Delta(\xi_c)=\Delta_0/2$, yielding $\xi_c=11.5a=4.4$~nm. This agrees very well with the BCS expression of the coherence length $\xi=\hbar v_{\mathrm{F}}/(\pi\Delta_0)=4.5$~nm if the Fermi-surface average of the velocity is substituted for $v_{\mathrm{F}}$.
\begin{figure}[tb]
\includegraphics[width=0.96\columnwidth]{figS2}
\caption{\label{fig:vortex}
(a) Modulus of the self-consistent order parameter for an isolated vortex (blue) on each site of the tight-binding lattice (black). The difference between the self-consistent and ansatz solutions (see text) is shown in orange. The white disk indicates the core size, defined as the distance at which the order parameter is $\Delta_0/2$. Colored balls mark the sites where the LDOS is plotted along (b) the antinodal direction, (c) the nodal direction, and (d) at a fixed distance as a function of angle. The LDOS curves are shifted vertically in (b), (c), and (d) and the color encodes the distance to the vortex center. Note that the curves show the full LDOS $N(\vec{r},E)$ without subtraction. The dashed curves in (d) are calculated using the isotropic ansatz for the order parameter. The insets in (b) and (c) show the spatial distribution of the LDOS (low, white to high, black) at the indicated energies, with the red circle of radius $\xi_c$ indicating the core size.
}
\end{figure}
The LDOS plotted in Figs.~\ref{fig:vortex}(b), \ref{fig:vortex}(c), and \ref{fig:vortex}(d) displays a zero-energy peak that is maximum at the vortex center and decays differently in all directions. The in-plane anisotropy of the LDOS is not a consequence of the in-plane anisotropy of the order parameter, as illustrated in Fig.~\ref{fig:vortex}(d), where it is seen that the typical angular dependence of the LDOS remains unchanged if an isotropic order parameter is used. It is also not a consequence of the $d$-wave gap symmetry: repeating the calculation for an $s$-wave gap leads to the same anisotropic LDOS with an un-split peak along the antinodal directions and a split peak along the nodal ones. Note that the LDOS peak in Fig.~\ref{fig:vortex} is a genuine continuum, not the superposition of densely packed discrete core levels as in $s$-wave superconductors \cite{Franz-1998b-s, Berthod-2016-s}. In fact, the LDOS anisotropy relates to the dispersion and Fermi-surface anisotropies, which tend to favor low-energy LDOS structures in the directions normal to the Fermi surface. The two traces in Figs.~\ref{fig:vortex}(b) and \ref{fig:vortex}(c) span different distances from the core; Figure~\ref{fig:fig1}(a) of the main text allows one to compare these traces along the same distance. The specific signature of the core for an isolated vortex [Fig.~\ref{fig:fig1}(b) of the main text] is obtained by subtracting the zero-field spectrum (green curve in Fig.~\ref{fig:uniform}) from the vortex LDOS.
\subsection{Ideal vortex lattices, self-consistent solutions}
\begin{figure}[t]
\includegraphics[width=\columnwidth]{figS3}
\caption{\label{fig:lattice-10}
(a) Modulus of the self-consistent order parameter (blue) and difference between the self-consistent and ansatz solutions (orange) for a square vortex lattice with inter-vortex distance $d$ oriented along the principal directions of the microscopic lattice. Two LDOS traces are shown along (b) the line connecting next-nearest neighbor vortices and (c) the line connecting nearest-neighbor vortices. The colors encode the position with respect to the cores as indicated by the balls in (a). The dashed curves, only half of which are shown for clarity, are obtained using the ansatz order parameter instead of the self-consistent one. (d) Spatial distribution of the LDOS at two energies; the red circles of radius $\xi_c$ show the vortex cores.
}
\end{figure}
\begin{figure}[t]
\includegraphics[width=\columnwidth]{figS4}
\caption{\label{fig:lattice-11}
Same as Fig.~\ref{fig:lattice-10} for a square vortex lattice with inter-vortex distance $d=38\sqrt{2}a$ oriented along the diagonals of the microscopic lattice. Note that the microscopic lattice is rotated by 45$^{\circ}$ in the three-dimensional plot (a), as compared to Fig.~\ref{fig:lattice-10}(a). In all LDOS maps of (d) and Fig.~\ref{fig:lattice-10}(d), however, the microscopic lattice directions correspond to the horizontal and vertical directions of the maps.
}
\end{figure}
The self-consistent solution for an ideal square vortex lattice with inter-vortex distance $d=54a$ is displayed in Fig.~\ref{fig:lattice-10}(a). This corresponds to a density of 19 vortices on $90\times 90$~nm$^2$ as observed in the experiment (Fig.~\ref{fig:fig2} of the main text). The vortex lattice is aligned with the microscopic lattice, the nearest-neighbor vortices being found along the $x$ and $y$ directions. The order parameter is maximum in the center of the squares formed by four nearest-neighbor vortices and has saddle points with $|\Delta(\vec{r})|=12$~meV on the lines joining them, leading to a significant spatial modulation. Note that the solution has been constrained to have a maximum gap of 19~meV like in zero field for simplicity---and also because no measurable reduction of the gap size is observed experimentally at this field---requiring a slight increase of the interaction to $V=-260$~meV. $|\Delta(\vec{r})|$ has a rounded shape in the cores, which is captured by the ansatz (generalized for vortex lattices \cite{Berthod-2016-s}) with an increased value $\xi_0=115a$ and a reduced value $\xi_1=8a$ with respect to the isolated vortex. The core size defined as $\Delta(\xi_c)=\Delta_0/2$ is slightly increased to $\xi_c=15.9a=6$~nm compared with the isolated vortex, consistently with previous studies in the quantum regime \cite{Berthod-2016-s}.
The corresponding LDOS is shown in Figs.~\ref{fig:lattice-10}(b), \ref{fig:lattice-10}(c), and \ref{fig:lattice-10}(d). Although the relative difference between the exact and ansatz solutions is 12\% at maximum, the LDOS curves calculated with both order parameters are almost undistinguishable. This is the justification for using the non-self-consistent ansatz when studying the LDOS in disordered vortex configurations, for which a full self-consistent calculation is impractical. The zero-energy LDOS peak is considerably suppressed and broadened in the core with respect to the isolated vortex. We have checked that the vortex-lattice calculation correctly reproduces the isolated-vortex limit as the distance $d$ is increased: both spectra differ by less than 5\% for $d\gtrsim 170a$ ($B\lesssim 0.5$~T). At the field considered, however, both the spectral and spatial signatures of the vortex cores differ markedly from those of the isolated vortex seen in Fig.~\ref{fig:vortex}. We have also verified that the suppression of the zero-energy LDOS peak is not due to the increased value of $\xi_c$ and more rounded order parameter in the core: the spectra shown in Fig.~\ref{fig:lattice-10} remain qualitatively unchanged if we use the ansatz order parameter with the values of $\xi_0$ and $\xi_1$ corresponding to the isolated vortex. The reason for a suppressed vortex-core peak can be understood by comparing the low-energy LDOS in Figs.~\ref{fig:vortex}(b) and \ref{fig:lattice-10}(d). Because for the isolated vortex the LDOS extends farther along the (10) direction than along the (11) direction, in the vortex lattice the wavefunctions in different cores strongly overlap and the core states get delocalized. This overlap is suppressed when the vortex lattice is not precisely aligned with the (10) direction and/or the vortex positions are disordered, such that the zero-energy LDOS peak is restored in these situations (see below).
For comparison, we show in Fig.~\ref{fig:lattice-11} the self-consistent order parameter and LDOS for a square vortex lattice rotated 45$^{\circ}$ with respect to the tight-binding lattice with an inter-vortex distance $d=38\sqrt{2}a$, which corresponds to the same field as in Fig.~\ref{fig:lattice-10}. There are significant differences between the two vortex-lattice orientations (hereafter I and II), both in the self-consistent order parameter and in the LDOS. While in I the gap has saddle points between nearest-neighbor vortices and maxima between next-nearest-neighbor ones, the situation is reversed in II with the gap maxima between nearest-neighbor vortices. It appears that the order parameter doesn't move rigidly with the vortex lattice: when rotating the vortex-lattice orientation from I to II, the cores move but the gap maxima and saddle points stay in place. The shape of the core is also quite different in I and II, where a best fit to the ansatz gives $\xi_0=9.9a$, $\xi_1=15a$, and a core size $\xi_c=6.4a=2.5$~nm smaller than in zero field. There is more structure in the case II, because each saddle point is in fact replaced by two saddle points separated by a local minimum. This explains the larger discrepancy between the ansatz and the exact solution, which reaches 32\% for II at the local minima. As a result, the differences between the LDOS calculated with the ansatz and self-consistent order parameters are slightly larger in II than in I. These differences remain nevertheless small compared with the qualitative differences between the LDOS in I and II: the zero-energy peak is neither strongly suppressed nor split in II as it is in I; at the position of the local minimum between two saddle points in II, the LDOS has a double peak at zero energy, while at the saddle point in I the LDOS is gapped. We see in Fig.~\ref{fig:lattice-11}(d) that the low-energy LDOS is much more localized in the cores compared with I, which highlights the much weaker wavefunction hybridization along the (10) directions in case II.
The data in Figs.~\ref{fig:vortex}, \ref{fig:lattice-10}, and \ref{fig:lattice-11} demonstrate that the vortex-core LDOS is not only a function of field, but also and more importantly a function of the positions of neighboring vortices. Depending on where the neighbors are, the zero-energy LDOS peak may be sharp or not, split or not, etc. Experimentally, one therefore expects variability in the measured vortex-core spectra when the vortex positions are disordered. These figures also show that, in order to investigate theoretically this variability, it is sufficient to work with the ansatz order parameter, whose only inputs are the vortex positions and the values of $\xi_0$ and $\xi_1$.
\section{\boldmath LDOS calculations for disordered vortex lattices }
\subsection{Disordered vortex configurations}
\begin{figure}[b]
\includegraphics[width=\columnwidth]{figS5}
\caption{\label{fig:disorder}
(a) Typical disordered vortex configuration. The central square represents the STS field of view of $90\times90~\mathrm{nm}^2$, where the vortices are located as observed in the experiment. The square is rotated by 5.7$^{\circ}$ with respect to the crystal axes. The dotted red square indicates the system size used for calculating the LDOS at the point marked by a cross; as the cross moves, the dotted square moves with it. (b) Isotropic structure factor showing the absence of orientational order in the generated vortex positions. (c) Angular average of the structure factor. The power-law behavior for $k<2\pi/d$ is shown in red.
}
\end{figure}
\begin{figure*}[tb]
\includegraphics[width=0.7\textwidth]{figS6}
\hfill\parbox[b]{0.25\textwidth}{
\caption{\label{fig:optimization}
For each configuration of the vortices outside the STS field of view, the theoretical spectral traces calculated along the paths $\alpha'$--$\delta'$ shown in (a) are compared with the corresponding experimental traces along $\alpha$--$\delta$ shown in (b).
}
\vspace{5.4cm}}
\end{figure*}
The high sensitivity of the theoretical LDOS to vortex positions prompts us, for a meaningful comparison with the STS experiment, to use in the calculation the vortex positions as they appear under the STM. Three difficulties arise: (i) the LDOS maxima which are accessible experimentally may not sit exactly on the phase singularity points where the order parameter vanishes, due to a polarization of the LDOS by asymmetric supercurrents \cite{Berthod-2013a-s, Berthod-2016-s}; (ii) we cannot disregard vortices that are outside the STS field of view although we don't know their positions; and (iii) due to lack of atomic resolution on the Y123 surface, the orientation of the microscopic lattice is only known approximately via the macroscopic twin boundaries. We ignore (i), expected to be a small effect, and locate the vortices inside the STS field of view at the positions of largest $dI/dV$ contrast [see Fig.~\ref{fig:fig4}(a) of the main text]. Outside the field of view, we generate vortex positions with the same density as inside, which corresponds to a field of 4.85~T. The positions are chosen at random, however, with a hard-core repulsion constraining the inter-vortex distance to be at least $d_0=41a$. This value was selected to be as large as possible: for larger values the random generation process would be stuck, not able to fit in the required number of vortices. The resulting vortex configurations show some degree of order, similar to what is seen in the STS field of view. An example is shown in Fig.~\ref{fig:disorder}. The structure factor $S(\vec{k})=\mathscr{N}^{-1}\left|\sum_ne^{-i\vec{k}\cdot\vec{R}_n}\right|^2$, where $\vec{R}_n$ are the positions of the $\mathscr{N}$ vortices, is isotropic indicating no orientational order. The angular average $\bar{S}(k)=\mathscr{N}^{-1}\sum_{nm}J_0(k|\vec{R}_n-\vec{R}_m|)$, where $J_0$ is the Bessel function, shows oscillations of wavevector $2\pi/d_0$ due to the hard-core repulsion. Furthermore, a power-law suppression of $\bar{S}(k)$ is observed for $k<2\pi/d$, where $d=54a$ is the inter-vortex distance in the equivalent ordered square lattice. Such power law is reminiscent of hyperuniformity, i.e., a type of order characterized by the suppression of density fluctuations at long wavelengths \cite{Torquato-2003-s}, where $\bar{S}(k)\sim k^{2-\eta}$ with $0<\eta\leqslant 2$ in two dimensions. It is likely that in reality the vortices outside the field of view present more order than the configurations generated by our procedure \cite{Maggio-Aprile-1995-s}, but the experimentally available vortex positions are not sufficient for inferring such an order. While certain characteristics of the vortex ordering outside the field of view may influence the LDOS inside, we do not expect this to change any of the conclusions we draw from our analysis.
\subsection{Search for a good configuration of vortices}
\begin{figure*}[tb]
\includegraphics[width=0.7\textwidth]{figS7}
\hfill\parbox[b]{0.25\textwidth}{
\caption{\label{fig:statistics}
(a), (b), and (c) display three distributions of the vortices outside the STS field of view (black dots), and the vortices at fixed positions within the field of view (white dots). The corresponding spectral traces along the paths $\alpha'$--$\delta'$ of Fig.~\ref{fig:optimization} are plotted on the right.
}
\vspace{8.35cm}}
\end{figure*}
\begin{figure*}[tb]
\includegraphics[width=1\textwidth]{figS8}
\caption{\label{fig:swave}
LDOS calculated in the configuration of Fig.~\ref{fig:disorder}(a) for a superconducting gap of (a) $d_{x^2-y^2}$ and (b) $s$ symmetry. Apart from the gap symmetry, all model parameters are identical in (a) and (b). The LDOS is shown at various energies in different panels. The rightmost panels show the ratio as calculated in Fig.~\ref{fig:optimization}(a), but with the colormap extending from the minimum to the maximum of the data, hence a slightly different contrast.
}
\end{figure*}
We have generated 600 disordered vortex configurations for various orientations of the microscopic lattice with respect to the STS field of view. For each configuration, we generate the order parameter using the ansatz and the values of $\xi_0$ and $\xi_1$ corresponding to Fig.~\ref{fig:lattice-10}, and we calculate the LDOS along the four paths displayed in Fig.~\ref{fig:optimization}(a), that correspond to paths n\textsuperscript{o}6 and n\textsuperscript{o}8 in Fig.~\ref{fig:fig4} of the main text. These four traces share a common point [the cross in Fig.~\ref{fig:disorder}(a)]: we use the LDOS at this point as the reference spectrum and subtract it from the calculated LDOS along the four paths. The same procedure applied to the experimental data [Fig.~\ref{fig:optimization}(b)], allows us to compute a sum of squared differences as the figure of merit for each vortex configuration. We find that the agreement between measurement and simulation is systematically better when the four paths are close to a nodal direction. The best compromise is reached if the STS field of view is rotated by 5.7$^{\circ}$ relative to the microscopic lattice as shown in Fig.~\ref{fig:disorder}(a). This points to a tendency for the nearest-neighbor vortices to align along the crystal axes, and justifies a posteriori our use of the values of $\xi_0$ and $\xi_1$ corresponding to that orientation rather than that of Fig.~\ref{fig:lattice-11}. We emphasize again that the precise values of $\xi_0$ and $\xi_1$ play very little role in the LDOS. Figure~\ref{fig:disorder}(a) is the best among the 600 configurations; the four traces are compared with the experimental ones in Fig.~\ref{fig:optimization}. Using this configuration, we calculate the LDOS in the whole STS field of view with 1~nm resolution. For comparing with Fig.~\ref{fig:fig4}(a) of the main text, we compute $dI/dV$ by adding to our theoretical LDOS the non-superconducting channel using the formula quoted in Fig.~\ref{fig:uniform}, we evaluate the ratio between the calculated $dI/dV$ at 5 and 17~meV, and thus obtain the map and traces shown in Fig.~\ref{fig:optimization}(a) and Fig.~\ref{fig:fig4}(b) of the main text.
Figure~\ref{fig:statistics} displays three vortex configurations different from the best one shown in Fig.~\ref{fig:disorder}(a) and \ref{fig:optimization}, and the corresponding theoretical spectral traces. These configurations agree reasonably with experiment as well, with a figure of merit within the best 10\% out of the 600 considered. One notices, in particular, four spectroscopically very different cores with configuration (a), a reinforcement of the LDOS at intermediate distance in (b), trace $\alpha'$, similar to what is seen in Fig.~\ref{fig:fig2}(c) of the main text, and split spectra at the center of vortices $\alpha'$ and $\delta'$ for configuration (c), as seen in the experiment [Fig.~\ref{fig:optimization}(b)].
\subsection{Mixed-state LDOS and gap symmetry}
It is tempting to search signatures of the $d_{x^2-y^2}$ symmetry of the order parameter in the LDOS around vortices. In the semiclassical regime $k_{\mathrm{F}}\xi\gg1$, the vortices are slowly varying perturbations of the order parameter compared with atomic distances, and their Fourier components are mostly at low momenta. Therefore, the vortices provide only small momentum transfers and the interaction of the Bogoliubov quasiparticles with the vortex lattice is dominated by forward scattering. In that limit, the details of the Fermi surface are irrelevant and the only source of spatial LDOS anisotropy---apart from the vortex lattice itself---is indeed the order-parameter symmetry. In the quantum regime $k_{\mathrm{F}}\xi\sim1$ relevant for Y123, however, the vortex lattice scatters Bogoliubov quasiparticles with large momentum transfers of the order of $k_{\mathrm{F}}$ and the LDOS is therefore sensitive to the anisotropy of the Fermi surface. This situation generically leads to the development of LDOS structures along the directions normal to the Fermi surface, a fact well known from studies of impurity scattering \cite{Weismann-2009-s}. In Y123, the Fermi surface segments are mainly oriented along the crystallographic directions [see Fig.~\ref{fig:uniform}(a)], such that one expects structure in the vortex-lattice LDOS along the (10) and (01) lattice directions, irrespective of the order-parameter symmetry. This is confirmed by our numerical results shown in Fig.~\ref{fig:swave}: for both $d$- and $s$-wave symmetries, the LDOS structures are aligned with the microscopic lattice at all energies. No structure is observed along the (11) and equivalent directions, which are the directions of the gap nodes in reciprocal space. Thus the expectation that the LDOS would ``leak'' out of the vortices along the directions of the gap nodes is not confirmed. There are nevertheless differences between the LDOS calculated for $d_{x^2-y^2}$ and $s$ symmetries. The vortex states are more localized in the $s$-wave case, leading to a better contrast, especially at low energy. However, according to these simulations, an unambiguous determination of the order-parameter symmetry based on experimental LDOS maps around vortices appears to be hopeless.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 3,706 |
CMS to release doc payment data annually
by Zack Budryk |
Feb 2, 2015 11:27am
The federal government will publicly release Medicare physician payment data every year, according to the Wall Street Journal.
Last year, the Centers for Medicare & Medicaid Services (CMS) released a year's worth of physician payment data in the wake of a court ruling that threw out a 1979 injunction against the release of such information. However, physicians' groups objected to the decision, saying the data painted an incomplete picture, as the numbers indicated revenue, not net earnings. For example, according to the WSJ, the data indicated some of the strongest earnings among ophthalmologists, but doesn't state that much of the information included major expenditures such as drugs used to treat macular degeneration.
The American Medical Association (AMA) later called for CMS to contextualize the data and clarify that "higher payments from Medicare typically represent higher expenses."
In addition, the data released last year did not include any patient information, which makes it difficult to draw conclusions about outcomes or care quality, according to the WSJ. CMS drew further criticism over what detractors called unnecessarily complex formatting and gaps in the data, FierceHealthFinance previously reported.
The AMA has urged CMS to improve the publication process before releasing any more data, arguing that 2014's data release was inaccurate and enabled "sensationalist" coverage, the WSJ reported. CMS has not stated whether it will alter or expand the next data release.
Despite the industry criticism, Gail Wilensky, Ph.D., who headed Medicare under President George H.W. Bush, argued continuing to publish the data could provide a better perspective on spending and care patterns, according to the article.
"As imperfect as the data may be," she said, "its continued release makes it in the physicians' interest to make it better--more useful and accurate--instead of just fighting its release."
- read the WSJ article (subscription may be required)
New CVS Caremark program nixes cost-sharing for diabetes drugs
CVS Caremark's new RxZERO design takes aim at members' out-of-pocket costs for diabetes medications.
by Paige Minemyer Jan 29, 2020 12:59pm
Anthem's Medicare Advantage enrollment grew by 20% in 2019
Anthem beat revenue estimates for the fourth quarter of 2019 thanks to the accelerated launch of its PBM IngenioRx.
by Robert King Jan 29, 2020 7:26am
Lyft partnering with CommonSpirit Health in California, Arizona
Lyft is teaming up with CommonSpirit Health and LogistiCare Circulation to roll out transportation services to patients in California and Arizona.
by Heather Landi Jan 29, 2020 10:58am
Zack Budryk | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 3,518 |
Geoff Ogilvy wins the 106th U.S. Open
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Tiger Woods completed one of the best ball-striking weeks of his career to retain his British Open title with an emotional two-shot victory over... Read more
Tiger Woods wins his 12th major at the 88th PGA Championship
The PGA Championship returned to Medinah for the first time since 1999 when Tiger Woods captured his first PGA Championship. At the longest course... Read more
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Angel Cabrera wins the 107th U.S. Open
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Padraig Harrington wins the 136th Open Championship
Pádraig Harrington defeated Sergio García in a playoff to take the title. 156 players participated in this year's Championship. The purse was... Read more
Tiger Woods wins his fourth PGA Championship
Tiger Woods survived a few anxious moments on the back nine to clinch his fourth U.S. PGA Championship and 13th major title with a two-shot victory... Read more
Trevor Immelman wins the 72nd Masters
Trevor Immelman won with a score of -8 (280) and became the first winner since Raymond Floyd in 1976 to hold the lead at the end of each of the... Read more
Tiger Woods, in clear agony after his recent left knee operation, completed a staggering U.S.Open victory on Monday by defeating fellow-American... Read more
Padraig Harrington defends his Open Championship at the 137th British Open
Pádraig Harrington successfully defended his Open Championship title for his second Open title. Harrington shot four under on the back nine to pull... Read more
Padraig Harrington wins the 90th PGA Championship
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Angel Cabrera wins the 73rd Masters
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Lucas Glover wins the 2009 U.S. Open
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Stewart Cink wins the 138th Open Championship
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Y.E. Yang wins the 91st PGA Championship
Prior to 2009 the last major played at Hazeltine was the 2002 PGA Championship, the 84th PGA Championship, which was won by Rich Beem. The 2009... Read more
Tiger Woods is Hospitalized After Crashing his Cadillac Escalade
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Tiger Woods Publicly Apologizes About Alleged Affair
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Taiwanese Animated Recreation of Tiger Woods Crash Becomes Internet Hit
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Tiger Woods Alleged Mistress, Rachel Uchitel, Cancels Press Conference
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Tiger Woods' Mother-In-Law Rushed to Hospital
Tiger Woods' mother-in-law was rushed early this morning from the golfer's home to the same Florida hospital where he was taken after a car... Read more | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 4,694 |
{"url":"https:\/\/stats.stackexchange.com\/questions\/411210\/difference-between-two-dimensions-sampled-from-dirichlet-distribution","text":"# Difference between two dimensions sampled from Dirichlet distribution\n\nSay I'm doing Bayesian inference on a Dirichlet-Multinomial model:\n\n$$x \\in [1,2,3]; \\\\ x \\sim Multinomial(p_1, p_2, p_3); \\\\ p_1, p_2, p_3 \\sim Dirichlet(\\alpha_1, \\alpha_2, \\alpha_3); \\\\ \\alpha_n = \\alpha_{prior} + \\sum (x=n)$$\n\nI'm interested in whether $$p_1 > p_2$$. I can sample from the posterior for this parameter directly (in R):\n\nalphas = c(10, 12, 5)\nsamp.from.dirichlet = MCMCpack::rdirichlet(100000, alphas)\ndelta.dirichlet = samp.from.dirichlet[,2] - samp.from.dirichlet[,1]\n# Prob. that p2 > p1\nmean(delta.dirichlet > 0) # > 0.66683\n\n\nIs there a closed-form way to calculate this probability?","date":"2021-06-18 08:11:21","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\": 2, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.8133094906806946, \"perplexity\": 3304.4773039105967}, \"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-2021-25\/segments\/1623487635920.39\/warc\/CC-MAIN-20210618073932-20210618103932-00445.warc.gz\"}"} | null | null |
import contextlib
import types
import unittest
import sys
import os
from queue import Queue
from typing import Tuple, List, Dict, Any
from unittest import mock
from mock import MagicMock
from BlockServer.component_switcher.component_switcher import ComponentSwitcher
from BlockServer.core.macros import MACROS, PVPREFIX_MACRO
sys.path.append(os.path.abspath(os.path.join(os.path.dirname(__file__), '..')))
class MockChannelAccess(object):
MONITORS: List[Tuple[str, types.FunctionType]] = []
@staticmethod
def add_monitor(pv: str, callback_func: types.FunctionType):
MockChannelAccess.MONITORS.append((pv, callback_func))
class MockConfigListManager(object):
def __init__(self):
self.active_config_name = "active"
self.configs = ["active", "inactive1", "inactive2"]
self.components = ["comp1", "comp2"]
self.loaded_configs = {} # config/comp name : returned config
def get_configs(self):
return [{"name": conf_name} for conf_name in self.configs]
def get_components(self):
return [{"name": comp_name} for comp_name in self.components]
def load_config(self, name, *_, **__):
if name in self.loaded_configs:
return self.loaded_configs[name]
else:
return MagicMock()
def update(self, *args, **kwargs):
pass
class MockComponentSwitcherFileManager(object):
def __init__(self):
self.config = None
def read_config(self) -> List[Dict[str, Any]]:
return self.config
class TestComponentSwitcher(unittest.TestCase):
def setUp(self) -> None:
MockChannelAccess.MONITORS = []
self.config_list = MockConfigListManager()
self.write_queue = Queue()
self.reload_func = MagicMock()
self.file_manager = MockComponentSwitcherFileManager()
self.file_manager.config = [
{
"pv": "first",
"is_local": True,
"value_to_component_map": {
"A": "comp1",
"B": "comp2",
}
}
]
self.component_switcher = ComponentSwitcher(
config_list=self.config_list,
blockserver_write_queue=self.write_queue,
reload_current_config_func=self.reload_func,
file_manager=self.file_manager,
channel_access_class=MockChannelAccess,
)
def test_GIVEN_empty_config_file_WHEN_call_add_monitors_THEN_no_monitors_added(self):
self.file_manager.config = []
self.component_switcher.create_monitors()
self.assertEqual(len(MockChannelAccess.MONITORS), 0)
def test_GIVEN_2_pvs_in_config_file_WHEN_call_add_monitors_THEN_2_monitors_added(self):
self.file_manager.config = [
{
"pv": "first",
"is_local": False,
"value_to_component_map": {}
},
{
"pv": "second",
"is_local": False,
"value_to_component_map": {}
},
]
self.component_switcher.create_monitors()
self.assertEqual(len(MockChannelAccess.MONITORS), 2)
self.assertEqual(MockChannelAccess.MONITORS[0][0], "first")
self.assertEqual(MockChannelAccess.MONITORS[1][0], "second")
@mock.patch.dict("BlockServer.core.macros.MACROS", {PVPREFIX_MACRO: "some_prefix:"})
def test_GIVEN_local_pv_monitored_THEN_monitored_pv_has_local_prefix_appended(self):
self.file_manager.config = [
{
"pv": "first",
"is_local": True,
"value_to_component_map": {}
}
]
self.component_switcher.create_monitors()
self.assertEqual(MockChannelAccess.MONITORS[0][0], "some_prefix:first")
@mock.patch.dict("BlockServer.core.macros.MACROS", {PVPREFIX_MACRO: "some_prefix:"})
def test_GIVEN_non_local_pv_monitored_THEN_monitored_pv_does_not_have_local_prefix_appended(self):
self.file_manager.config = [
{
"pv": "first",
"is_local": False,
"value_to_component_map": {}
}
]
self.component_switcher.create_monitors()
self.assertEqual(MockChannelAccess.MONITORS[0][0], "first")
def test_GIVEN_monitor_is_triggered_THEN_action_gets_appended_to_bs_write_queue(self):
self.component_switcher.create_monitors()
self.assertEqual(len(MockChannelAccess.MONITORS), 1)
self.assertEqual(self.write_queue.qsize(), 0)
# Fire the monitor with value A
MockChannelAccess.MONITORS[0][1]("A", 0, 0)
self.assertEqual(self.write_queue.qsize(), 1)
func, args, status = self.write_queue.get()
self.assertEqual(func, self.component_switcher._edit_all_configurations)
# Component 2 should be removed, components 1 should be added as our monitor got value A
self.assertEqual(args, ({"comp2"}, {"comp1"}))
def test_GIVEN_monitor_is_triggered_with_non_zero_stat_THEN_action_is_ignored(self):
self.component_switcher.create_monitors()
self.assertEqual(len(MockChannelAccess.MONITORS), 1)
self.assertEqual(self.write_queue.qsize(), 0)
# Fire the monitor with value A and stat=1
MockChannelAccess.MONITORS[0][1]("A", 1, 0)
self.assertEqual(self.write_queue.qsize(), 0)
def test_GIVEN_monitor_is_triggered_with_non_zero_sevr_THEN_action_is_ignored(self):
self.component_switcher.create_monitors()
self.assertEqual(len(MockChannelAccess.MONITORS), 1)
self.assertEqual(self.write_queue.qsize(), 0)
# Fire the monitor with value A and sevr=1
MockChannelAccess.MONITORS[0][1]("A", 0, 1)
self.assertEqual(self.write_queue.qsize(), 0)
def test_GIVEN_monitor_is_triggered_with_an_unknown_value_THEN_action_does_not_get_appended_to_bs_write_queue(self):
self.component_switcher.create_monitors()
self.assertEqual(len(MockChannelAccess.MONITORS), 1)
self.assertEqual(self.write_queue.qsize(), 0)
# Fire the monitor with a fake (invalid) value
MockChannelAccess.MONITORS[0][1]("C", 0, 0)
self.assertEqual(self.write_queue.qsize(), 0)
def test_GIVEN_active_config_is_not_in_config_list_THEN_get_valueerror(self):
self.config_list.active_config_name = "invalid"
with self.assertRaises(ValueError):
self.component_switcher._edit_all_configurations(set(), set())
def test_GIVEN_component_to_be_removed_doesnt_exist_THEN_get_valueerror(self):
with self.assertRaises(ValueError):
self.component_switcher._edit_all_configurations({"nonexistent"}, set())
def test_GIVEN_component_to_be_added_doesnt_exist_THEN_get_valueerror(self):
with self.assertRaises(ValueError):
self.component_switcher._edit_all_configurations(set(), {"nonexistent"})
def test_GIVEN_no_components_to_be_added_or_removed_THEN_current_config_not_reloaded(self):
self.component_switcher._edit_all_configurations(set(), set())
self.assertFalse(self.reload_func.called)
def test_GIVEN_components_to_added_or_removed_THEN_current_config_reloaded(self):
self.component_switcher._edit_all_configurations({"comp1"}, {"comp2"})
self.assertTrue(self.reload_func.called)
def test_GIVEN_active_config_already_in_correct_state_THEN_not_saved_again(self):
mock_conf = MagicMock()
mock_conf.get_component_names.return_value = ["comp1"]
self.config_list.loaded_configs = {"active": mock_conf}
self.component_switcher._edit_all_configurations(
components_to_be_added={"comp1"}, components_to_be_removed={"comp2"})
self.assertFalse(mock_conf.save_inactive.called)
self.assertFalse(self.reload_func.called)
def test_GIVEN_active_config_not_in_correct_state_THEN_edited_and_saved(self):
mock_conf = MagicMock()
mock_conf.get_component_names.return_value = ["comp1"]
self.config_list.loaded_configs = {"active": mock_conf}
self.component_switcher._edit_all_configurations(
components_to_be_added={"comp2"}, components_to_be_removed={"comp1"})
self.assertTrue(mock_conf.remove_comp.called_with("comp1"))
self.assertTrue(mock_conf.add_comp.called_with("comp2"))
self.assertTrue(mock_conf.save_inactive.called)
self.assertTrue(self.reload_func.called)
| {
"redpajama_set_name": "RedPajamaGithub"
} | 8,168 |
\section{Introduction}
\setcounter{equation}{0}
\setcounter{equation}{0}
\noindent In 2011, Verlinde \cite{verlinde} conjectured a link between gravity and an entropic force. Such conjecture was
proved correct recently cite{p1}, in a phase-space, statistical mechanics' context.\vskip 3mm
\noindent \color{red} Here we will confront two viewpoints
\begin{itemize}
\item Verlinde's elegant thought experiment, based in black-hole related assumptions, that leads to a Newtonian $r^{-2}$ radial dependence for the entropic force,
with
\item A quantum statistical mechanics treatment of a Fermi (Bose) gas that, in its classical limit, leads to the same radial dependence.
\end{itemize}
\noindent \normalcolor Verlinde suggests in his thought experiment that gravitation should emerge as a result of information about the positions of material particles, connecting a gravity's thermal treatment to 't Hooft's holographic principle. Accordingly, gravitation is to be regarded as an emergent phenomenon. The idea generated immense attention. See for
instance \cite{times,libro}. A very nice overview regarding the statistical mechanics of gravitation is to be encountered in Padmanabhan's work \cite{india}, and references therein.\vskip 2mm
\noindent This conjecture originated works in cosmology, the dark energy hypothesis, cosmological acceleration, cosmological inflation, and loop quantum gravity. The associated literature is extensive \cite{libro}. A relevant input is due to Guseo \cite{guseo}, who demonstrated that the local entropy function, related to a
logistic distribution, is a catenary and vice versa, an invariance interpreted through Verlinde's conjecture regarding gravity as an entropic force. \cite{guseo} puts forward a new interpretation of the local entropy in a system.
\vskip 2mm
\noindent This paper does not deal with any of these issues, though. Considering that we proved Verlinde's conjecture in a classical context \cite{p1}, we wish here to continue a discussion initiated
in \cite{DOI} with regards to the quantal bosons/fermions scenario. In \cite{DOI} we used Boltzmann-Gibbs (BG) entropy.
Here we wish to undertake a Tsallis-treatment. Why? Because distinctive advantages will be accrued in this way.
\color{red} It turns out that Tsallis entropy is a potential for the entropic force for $q=4/3$ (see proof in \cite{p1} and also in the Appendix), which is not the BG case.
This makes Tsallis' entropy the natural information measure to link to gravitation. This should be natural enough, since it is well known that BG is the natural entropy for systems with short-range interaction, while Tsallis' is the one appropriate to long-range interactions \cite{tsallis}. \normalcolor \vskip 2mm
\noindent We base our considerations on Chapters 6 and 7 of \cite{lemons}, to which the reader is referred for details. \color{red} Only the microcanonical ensemble is used in this book, and thus here. \normalcolor It is assumed
that each fermion or boson possesses an average energy $E/ N$.
Such average energy approximation produces results that, while
approximate, describe important features of the ideal Fermi (Bose) gas \cite{lemons}. In fact, most of the book is devoted to this excellent approximation, that allows one to appeal to the micro-canonical ensemble. This entails that the entropy is the logarithm of the multiplicity $\Omega$, according to the celebrated Boltzmann-formula.
\subsection{Our goal}
\noindent \color{red} The present effort intends to contribute to the current debate/discussion regarding
Verlinde's proposal, based on a thought-experiment, for an alternative (entropic) interpretation of gravity. A theory of quantum gravitation does not yet exist. What do we want to achieve here then? We can not expect to obtain en emerging entropic force that will yield classical gravitation in the quantum domain. What we wish to ascertain is whether the classical limit of our quantum statistical mechanics' Verlinde-treatment does yield Newton's gravitation in such limit. We will prove that such is the case. Thus, we contrast Verlinde's thought experiment with a rigorous statistical mechanics' argumentation. Such is the logic of the present effort. \normalcolor
\section{Entropic force for bosons in the microcanonical ensemble}
\subsection{Quantum entropic force}
\setcounter{equation}{0}
We start by reminding the reader of the q-logarithm notion, defined according to \cite{tsallis} as
\begin{equation}
\label{eq2.1}
\ln_qx=\frac {x^{1-q}-1} {1-q}.
\end{equation}
\color{blue} Tsallis entropy is defined. for a given set of micro-states labeled by $i$, whose probability is $P_i$, as \cite{tsallis}
\begin{equation} S_q= -\sum_i\, P_i^q \, \ln_q P_i, \end{equation} with $q$ any real number.
An important portion of the immense Tsallis' literature \cite{tsallis} is devoted to ascertaining which is the appropriate value of $q$ in variegated scenarios. In our present environment we will see below that $q=4/3$. \normalcolor
\vskip 3mm \noindent We will use it to compute the multiplicity $\Omega$ for a Bose gas in the micro-canonical ensemble following Boltzmann's logarithmic prescription in a Tsallis-environment. Following \cite{lemons}, in a system of free bosons for which the number of accessible single-particle states is given by $n$, $\Omega$ can be thought of as the number of ways of distributing $N$ particles and $n-1$ ''partitions'' separating them \cite{lemons}. It then reads \cite{lemons}
\begin{equation}
\label{eq2.2}
\Omega(E,V,N)=\frac {(N+n-1)!} {N!(n-1)!}=\frac {\Gamma(N+n)} {\Gamma(N+1)\Gamma(n)},
\end{equation}
where the energy $E$, the volume $V$, and the number of bosons $N$ are the extensive variables of the problem at hand. In terms of these variables one has \cite{lemons}
\begin{equation}
\label{Nn}
\frac {N} {n}=\frac {N} {V}\left(\frac {N} {E}\right)^{\frac {3} {2}}
\left(\frac {3h^2} {4\pi em}\right)^{\frac {3} {2}},
\end{equation}
an important relation that we will often employ below. $m$ stands for the gas' particles' mass. $e$ is Euler's number, and $h$ Planck's constant.
The classical limit is attained for $N/n \ll 1$ \cite{lemons}. Remember also that the $\Gamma$ function for large values of $z$ can be approximated by
\begin{equation}
\label{eq2.4}
\Gamma(z)\approx\sqrt{2\pi}z^{z-\frac {1} {2}} e^{-z},
\end{equation}
which we can use for
$N>>1$, $n>>1$ to obtain
\begin{equation}
\label{eq2.5}
\Omega(E,V,N)\approx\frac {(N+n)^{N+n}} {\sqrt{2\pi}N^Nn^n},
\end{equation}
so that the micro-canonical Tsallis' entropy becomes ($k_B$ is Boltzmann's constant)
\begin{equation}
\label{eq2.6}
{\cal S}_q=Nk_B\ln_q\Omega(E,V,N)^{\frac {1} {N}}
\end{equation}
\color{red} For $q\rightarrow 1$ one has
\begin{equation}
\label{eq1}
{\cal S}=Nk_B\ln\Omega(E,V,N)^{\frac {1} {N}}=
k_B\ln\Omega(E,V,N).
\end{equation}
Thus, for $q\rightarrow 1$ Tsallis's microcanonical entropy in terms of the multiplicity becomes Boltzmann's celebrated one. \normalcolor
\noindent Now, \color{blue} it was seen in \cite{p1} that the gravitational interaction can be extracted,
out of the infinite family of different Tsallis' entropies associated to all possible $q-$values, only for $q=\frac{4}{3}$.
\color{red} For other $q-$values the gradient of $S_q$ becomes proportional to $1/r^{\nu}$ with $\nu \ne 2$ \cite{p1}. See Appendix for details. \color{blue} Thus, we are heuristically forced to select $q=4/3$.
Note that, for each value of $q$, Tsallis has introduced a different statistical mechanics.
For example, for $q=1$ we obtain the orthodox statistical mechanics of Boltzmann-Gibbs. What Tsallis did
it is not to define just a new (single) realization of statistical mechanics, but a new infinite set of different statistical mechanics' realizations \cite{tsallis}.
\normalcolor
Accordingly,
\begin{equation}
\label{eq2.7}
{\cal S}_{\frac {4} {3}}=3Nk_B(1-\Omega^{-\frac {1} {3N}}),
\end{equation}
and using again (\ref{Nn}) we write
\begin{equation}
\label{eq2.8}
n=V\left(\frac {E} {N}\right)^{\frac {3} {2}}
\left(\frac {4\pi em} {3h^2}\right)^{\frac {3} {2}}.
\end{equation}
It is important to realize that here we find that
\begin{equation} n \propto V.\label{realize}\end{equation}
It is seen possible at this point to cast $\Omega$ in the fashion
\begin{equation}
\label{eq2.9}
\Omega=\frac {e^\gamma} {\sqrt{2\pi}N^N},
\end{equation}
with
\[\gamma=\left[N+V\left(\frac {E} {N}\right)^{\frac {3} {2}}
\left(\frac {4\pi em} {3h^2}\right)^{\frac {3} {2}}\right]
\ln\left[N+V\left(\frac {E} {N}\right)^{\frac {3} {2}}
\left(\frac {4\pi em} {3h^2}\right)^{\frac {3} {2}}\right]-\]
\begin{equation}
\label{eq2.10}
\left[V\left(\frac {E} {N}\right)^{\frac {3} {2}}
\left(\frac {4\pi em} {3h^2}\right)^{\frac {3} {2}}\right]
\ln\left[V\left(\frac {E} {N}\right)^{\frac {3} {2}}
\left(\frac {4\pi em} {3h^2}\right)^{\frac {3} {2}}\right],
\end{equation}
or
\begin{equation} \gamma= (N+n) \ln{(N+n)} - n\ln{n}. \label{gama}\end{equation}
\vskip 3mm \noindent The entropy's gradient becomes then
\begin{equation}
\label{eq2.11}
\vec{\nabla}{\cal S}_{\frac {4} {3}}=k_B\Omega^{-\frac {1} {3N}}
\vec{\nabla}\gamma
\end{equation}
Now, since $V=\frac {4} {3}\pi r^3$ this entails
\begin{equation}
\label{eq2.12}
\frac{\partial{\cal S}_{\frac {4} {3}}} {\partial r}=k_B\Omega^{-\frac {1} {3N}}
\frac {\partial\gamma} {\partial r},
\end{equation}
since ${\cal S}_{\frac {4} {3}}$ depends just upon $r$. Thus,
\begin{equation}
\label{eq2.13}
\frac{\partial{\cal S}_{\frac {4} {3}}} {\partial r}=k_B
4\pi r^2\Omega^{-\frac {1} {3N}}
\frac {\partial\gamma} {\partial V}.
\end{equation}
Taking into account that
\[\frac {\partial\gamma} {\partial V}=\left(\frac {E} {N}\right)^{\frac {3} {2}}
\left(\frac {4\pi em} {3h^2}\right)^{\frac {3} {2}}\left\{
\ln\left[N+V\left(\frac {E} {N}\right)^{\frac {3} {2}}
\left(\frac {4\pi em} {3h^2}\right)^{\frac {3} {2}}\right]-\right.\]
\begin{equation}
\label{eq2.14}
\left.\ln\left[V\left(\frac {E} {N}\right)^{\frac {3} {2}}
\left(\frac {4\pi em} {3h^2}\right)^{\frac {3} {2}}\right]\right\},
\end{equation}
and remembering Verlinde's definition for the entropic force \cite{verlinde} we find
\begin{equation}
\label{eq2.15}
{\vec {F}}_e=-\lambda k_BT\vec{\nabla}{\cal S}_{\frac {4} {3}},
\end{equation}
or the nice result
\begin{equation}
\label{eq2.16}
F_e=-\lambda k_BT\frac{\partial{\cal S}_{\frac {4} {3}}} {\partial r},
\end{equation}
that by appeal to (\ref{eq2.14}) yields
\[F_e=-\frac {\lambda} {\beta} 4\pi r^2\Omega^{-\frac {1} {3N}}
\left(\frac {E} {N}\right)^{\frac {3} {2}}
\left(\frac {4\pi em} {3h^2}\right)^{\frac {3} {2}}\left\{
\ln\left[N+V\left(\frac {E} {N}\right)^{\frac {3} {2}}
\left(\frac {4\pi em} {3h^2}\right)^{\frac {3} {2}}\right]-\right.\]
\begin{equation}
\label{eq2.17}
\left.\ln\left[V\left(\frac {E} {N}\right)^{\frac {3} {2}}
\left(\frac {4\pi em} {3h^2}\right)^{\frac {3} {2}}\right]\right\},
\end{equation}
or
\[F_e=-\frac {\lambda} {\beta} 4\pi r^2\Omega^{-\frac {1} {3N}}
\left(\frac {E} {N}\right)^{\frac {3} {2}}
\left(\frac {4\pi em} {3h^2}\right)^{\frac {3} {2}}\left\{
\ln\left[N+\frac {4\pi r^3} {3}\left(\frac {E} {N}\right)^{\frac {3} {2}}
\left(\frac {4\pi em} {3h^2}\right)^{\frac {3} {2}}\right]-\right.\]
\begin{equation}
\label{eq2.18}
\left.\ln\left[\frac {4\pi r^3} {3}\left(\frac {E} {N}\right)^{\frac {3} {2}}
\left(\frac {4\pi em} {3h^2}\right)^{\frac {3} {2}}\right]\right\},
\end{equation}
that can also be cast as
\begin{equation}
\label{eq2.19}
F_e=-\frac {\lambda} {\beta} 4\pi r^2\Omega^{-\frac {1} {3N}}
\left(\frac {E} {N}\right)^{\frac {3} {2}}
\left(\frac {4\pi em} {3h^2}\right)^{\frac {3} {2}}
\ln\left[1+\frac {N} {V}\left(\frac {N} {E}\right)^{\frac {3} {2}}
\left(\frac {3h^2} {4\pi em}\right)^{\frac {3} {2}}\right].
\end{equation}
Note that $F_e$ does not diverge at the origin but vanishes there. However, this happens at distances to the origin of the order of one hundredth of the Planck-length. No practical consequences can be detected, though.
Minding (\ref{Nn}) we also have
\begin{equation}
\label{newFe}
F_e=-\frac {\lambda} {\beta} 4\pi r^2\Omega^{-\frac {1} {3N}}
\left(\frac {E} {N}\right)^{\frac {3} {2}}
\left(\frac {4\pi em} {3h^2}\right)^{\frac {3} {2}}
\ln\left[1+ N/n
\right].
\end{equation}
\color{red} Notice also that the entropic force vanishes at zero temperature and diverges when $T \rightarrow \infty$. This putatively happened at the Big-Bang. There we have $r=0$ as well, so that the behavior of $F_e$ is complicated. However, this does not matter because at these limits quantum gravity, unknown today, reigns.\vskip 3mm
\noindent \color{red} A word of caution is necessary here. Since a theory of quantum gravity does not exist yet, we must not naively think that these equations for $F_e$ can be taken at face value. What is really of interest here is just the classical limit of $F_e$, that we are going to discuss below. \normalcolor
\subsection{Bose's entropic force in the classical limit $N/n \ll 1$}
\noindent The idea is to judiciously employ (\ref{eq2.19}) and (\ref{newFe}) in this limit. From (\ref{Nn}), i.e.,
$\frac {N} {n}=\frac {N} {V}\left(\frac {N} {E}\right)^{\frac {3} {2}}
\left(\frac {3h^2} {4\pi em}\right)^{\frac {3} {2}},$ plus $V=4\pi r^3/3$, one sees that
\begin{equation} \label{2lim} N/n \ll 1 \,\, \rightarrow \,\, r >> 1. \end{equation}
From (\ref{Nn}) we also ascertain that we can replace the logarithm by its argument minus unity in (\ref{newFe}), that then becomes
\begin{equation} F_e=-\frac {\lambda} {\beta} 4\pi r^2\Omega^{-\frac {1} {3N}}
\left(\frac {E} {N}\right)^{\frac {3} {2}}
\left(\frac {4\pi em} {3h^2}\right)^{\frac {3} {2}} \left[\frac {N} {V}\left(\frac {N} {E}\right)^{\frac {3} {2}}
\left(\frac {3h^2} {4\pi em}\right)^{\frac {3} {2}}\right] \end{equation}
and
\begin{equation}
\label{eq2.20}
F_e=-\frac {\lambda} {\beta} 4\pi r^2\Omega^{-\frac {1} {3N}}
\frac {N} {V}.
\end{equation}
Now, according to (\ref{realize}) we have $ n \propto V$
and (\ref{eq2.5}) entails that in our limit we have
\begin{equation} \Omega \propto V^N. \end{equation}. Thus,
\begin{equation} \Omega^{-\frac {1} {3N}} \propto (1/V^{1/3}), \end{equation}.
so that $F_e$ becomes
\begin{equation}
\label{eq2.22}
F_e \propto - {r^2},
\end{equation}
where the proportionality constant is assumed to include Newton's gravitation constant $G$. This
proves (in statistical mechanics' fashion), for free bosons, Verlinde's second conjecture: in the classical limit, the corresponding entropic force decreases as $1/r^2$, like Newton's gravitation \cite{p1}. \color{red} This dependence of $F_e$ with $r$ is all what Verlinde actually proved in \cite{verlinde}, in a Beckenstein-like thought-experiment. There he {\bf assumes} the number of bits $N$ contained in an appropriate Beckenstein enfolding screen can be cast as $N=Ac^3/G\hbar$, with $A$ the screen's area.
Actually, this is, a priori, Verlinde's definition of $G$, that later will turn out to be gravitation's constant. Summing up, neither in Verlinde's derivation nor in ours we see $G$ emerging from first principles. It is introduced in an ad hoc fashion, ''by hand''.
\normalcolor
\section{Entropic force for fermions}
\subsection{Quantum entropic force for fermions}
\setcounter{equation}{0}
Here we deal with $N$ fermions and $n$ micro-states that can be occupied by just one fermion. We have a multiplicity $\Omega$ given by \cite{lemons}
\begin{equation}
\label{eq3.1}
\Omega=\frac {n!} {(n-N)!N!}=
\frac {\Gamma(n+1)} {\Gamma(n-N+1)\Gamma(N+1)}.
\end{equation}
For $N>>1$ and $n>>1$ one is allowed to write
\begin{equation}
\label{eq3.2}
\Omega=\frac {e} {\sqrt{2\pi}}\frac {(n+1)^{n+1}}
{(n-N+1)^{n-N+1}(N+1)^{N+1}},
\end{equation}
that can be recast as
\begin{equation}
\label{eq3.3}
\Omega=\frac {e} {\sqrt{2\pi}} e^\gamma,
\end{equation}
where $\gamma$ is
\begin{equation}
\label{eq3.4}
\gamma=(n+1)\ln(n+1)+(N-n-1)\ln(n+1-N)-(N+1)\ln(N+1).
\end{equation}
The derivative of $\gamma$ with respect to $V$ is
\[\frac {\partial\gamma} {\partial V}=\left(\frac {E} {N}\right)^{\frac {3} {2}}
\left(\frac {4\pi em} {3h^2}\right)^{\frac {3} {2}}\left\{
\ln\left[1+V\left(\frac {E} {N}\right)^{\frac {3} {2}}
\left(\frac {4\pi em} {3h^2}\right)^{\frac {3} {2}}\right]-\right.\]
\begin{equation}
\label{eq3.5}
\left.\ln\left[1+V\left(\frac {E} {N}\right)^{\frac {3} {2}}
\left(\frac {4\pi em} {3h^2}\right)^{\frac {3} {2}}-N\right]\right\}.
\end{equation}
Retracing now here the boson-steps of the preceding Section we find
\[F_e=-\frac {\lambda} {\beta} 4\pi r^2\Omega^{-\frac {1} {3N}}
\left(\frac {E} {N}\right)^{\frac {3} {2}}
\left(\frac {4\pi em} {3h^2}\right)^{\frac {3} {2}}\left\{
\ln\left[1+V\left(\frac {E} {N}\right)^{\frac {3} {2}}
\left(\frac {4\pi em} {3h^2}\right)^{\frac {3} {2}}\right]-\right.\]
\begin{equation}
\label{eq3.6}
\left.\ln\left[1+V\left(\frac {E} {N}\right)^{\frac {3} {2}}
\left(\frac {4\pi em} {3h^2}\right)^{\frac {3} {2}}-N\right]\right\}.
\end{equation}
\color{red} Comparing (\ref{eq2.19}) for bosons with (\ref{eq3.6}) for fermions we see that they are not identical This does not matter, though, since a theory of quantum gravity is not available yet and we should not take the above cited equations at face value. What matters are their classical limits and they do coincide, of course.\normalcolor Finally, we can also cast (\ref{eq3.6}) as
\[F_e=-\frac {\lambda} {\beta} 4\pi r^2\Omega^{-\frac {1} {3N}}
\left(\frac {E} {N}\right)^{\frac {3} {2}}
\left(\frac {4\pi em} {3h^2}\right)^{\frac {3} {2}}\left\{
\ln\left[1+\frac {4\pi r^3} {3}\left(\frac {E} {N}\right)^{\frac {3} {2}}
\left(\frac {4\pi em} {3h^2}\right)^{\frac {3} {2}}\right]-\right.\]
\begin{equation}
\label{eq3.7}
\left.\ln\left[1+\frac {4\pi r^3} {3}\left(\frac {E} {N}\right)^{\frac {3} {2}}
\left(\frac {4\pi em} {3h^2}\right)^{\frac {3} {2}}-N\right]\right\}.
\end{equation}
Notice that one has, according to the last equation,
\begin{equation}
\label{eq3.8}
1+\frac {4\pi r^3} {3}\left(\frac {E} {N}\right)^{\frac {3} {2}}
\left(\frac {4\pi em} {3h^2}\right)^{\frac {3} {2}}-N>0,
\end{equation}
so that there is a lower bound for $r$ since Eq. (\ref{eq3.8}) entails
\begin{equation}
\label{eq3.9}
r>\left[\frac {3(N-1)} {4\pi}\right]^{\frac {1} {3}}
\left(\frac {N} {E}\right)^{\frac {1} {2}}
\left(\frac {3h^2} {4\pi em}\right)^{\frac {1} {2}}.
\end{equation}
\noindent \color{red} Selecting $m=$ uranium's mass, $N$=500, $v=0.1 c$ ($c$=speed of light) we obtain:
$r=2.4 10^{-21} m$.
This might perhaps suggest a kind of space-quantization?. Notice also that the entropic force vanishes at zero temperature.
if we select $\lambda$ independent of $T$.\normalcolor
\subsection{Entropic force in the classical limit $N/n \ll 1$}
First of all we realize that (\ref{2lim}) holds in this situation too.
We approximate things now for the classical limit, starting with (\ref{eq3.7}), in the fashion
\[F_e=-\frac {\lambda} {\beta} 4\pi r^2\Omega^{-\frac {1} {3N}}
\left(\frac {E} {N}\right)^{\frac {3} {2}}
\left(\frac {4\pi em} {3h^2}\right)^{\frac {3} {2}}\left\{
\ln\left[V\left(\frac {E} {N}\right)^{\frac {3} {2}}
\left(\frac {4\pi em} {3h^2}\right)^{\frac {3} {2}}\right]-\right.\]
\begin{equation}
\label{eq3.10}
\left.\ln\left[V\left(\frac {E} {N}\right)^{\frac {3} {2}}
\left(\frac {4\pi em} {3h^2}\right)^{\frac {3} {2}}-N\right]\right\},
\end{equation}
or
\begin{equation}
\label{eq3.11}
F_e=\frac {\lambda} {\beta} 4\pi r^2\Omega^{-\frac {1} {3N}}
\left(\frac {E} {N}\right)^{\frac {3} {2}}
\left(\frac {4\pi em} {3h^2}\right)^{\frac {3} {2}} \ln\left[1-\frac {N} {V}\left(\frac {N} {E}\right)^{\frac {3} {2}}
\left(\frac {3h^2} {4\pi em}\right)^{\frac {3} {2}}\right],
\end{equation}
that also reads
\begin{equation} F_e= \frac {\lambda} {\beta} 4\pi r^2\Omega^{-\frac {1} {3N}}
\left(\frac {E} {N}\right)^{\frac {3} {2}}
\left(\frac {4\pi em} {3h^2}\right)^{\frac {3} {2}} \ln{[1- N/n]}.
\label{agrego}\end{equation} Expanding now the logarithm we arrive at
\begin{equation}
\label{eq3.12}
F_e=\frac {\lambda} {\beta} 4\pi r^2\Omega^{-\frac {1} {3N}}
\left(\frac {E} {N}\right)^{\frac {3} {2}}
\left(\frac {4\pi em} {3h^2}\right)^{\frac {3} {2}}
\left[-\frac {N} {V}\left(\frac {N} {E}\right)^{\frac {3} {2}}
\left(\frac {3h^2} {4\pi em}\right)^{\frac {3} {2}}\right],
\end{equation}
or, for the entropic force in the classical limit (CL)
\begin{equation}
\label{eq3.13}
F_e(CL) =-(N/V)\frac {\lambda} {\beta} 4\pi r^2\Omega^{-\frac {1} {3N}}.
\end{equation}
\color{red} It is of the essence now to ascertain the behavior of $\Omega$ in the classical limit. Thus, we focus attention upon $\gamma$.
For this, we go back to (\ref{gama}) at this point and realize, that in the classical limit, it reduces to \normalcolor
\begin{equation} \gamma \approx N \ln{(n)}.\end{equation}
Thus, according to (\ref{eq3.3}),
\begin{equation} \Omega \propto \exp{\gamma}= n^N.\end{equation}
Further, we have
\begin{equation} n \propto V\end{equation}, so that \begin{equation} \Omega \propto V^N.\end{equation} Accordingly
\begin{equation} \Omega^{-\frac {1} {3N}} \propto V^{-\frac {1} {3}},\label{fomega}\end{equation}
which finally yields, for the entropic force in the classical limit
\begin{equation}
\label{eq3.14}
F_e \propto -\frac{1}{r^2}.
\end{equation}
We have encountered a similar expression for the emergent entropic force in the classical
limit similar to that obtained for bosons.
\section{Conclusions}
\noindent Verlinde conjectured in 2011 that gravitation, instead of being an elementary force, is an emergent entropic one. This rather surprising conjecture had 3280 downloads and 717 cites in ArXiv! In a phase-space classical context it was actually proved true in \cite{p1}.
\vskip 3mm
\noindent Here we asked for its workings in a quantum scenario and it was proved again, this time in the classical limit of the quantum treatment. The relevant question is now: what kind of entropy yields gravitation as an entropic force?
\vskip 3mm
\noindent We responded in this work that such an entropy is Tsallis' one for $q=4/3$ \color{blue} (an heuristically choice of $q$),
\normalcolor both for fermions and for bosons.
\vskip 3mm
\noindent Remark that, for fermions, we have found a lower bound for the distance to the origin $r$.
\newpage
| {
"redpajama_set_name": "RedPajamaArXiv"
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{"url":"https:\/\/en.wikipedia.org\/wiki\/Local_system","text":"# Local system\n\nIn mathematics, local coefficients is an idea from algebraic topology, a kind of half-way stage between homology theory or cohomology theory with coefficients in the usual sense, in a fixed abelian group A, and general sheaf cohomology which, roughly speaking, allows coefficients to vary from point to point in a topological space X. Such a concept was introduced by Norman Steenrod.\n\n## Definition\n\nA local system of modules on space X is\n\n1. A locally constant sheaf of modules ${\\displaystyle {\\mathcal {L}}}$ on X. That is, every point has an open neighborhood ${\\displaystyle U}$ such that ${\\displaystyle {\\mathcal {L}}|_{U}}$ is a constant sheaf.\n\nIf X is path-connected, this the same as\n\n2. A homomorphism ${\\displaystyle \\rho :\\pi _{1}(X,x)\\to {\\text{End}}_{R}(M)}$ (${\\displaystyle M={\\mathcal {L}}_{x}}$ in the above).\n\nAnother (stronger, nonequivalent) definition generalising 2, and working for nonconnected X, is\n\n3. A functor\n${\\displaystyle {\\mathcal {L}}:\\Pi _{1}(X)\\to {\\textbf {Mod}}(R)}$\nfrom the fundamental groupoid of ${\\displaystyle X}$ to the category of modules over a commutative ring ${\\displaystyle R}$. Typically ${\\displaystyle R=\\mathbb {Q} ,\\mathbb {R} ,\\mathbb {C} }$. What this is saying is that at every point ${\\displaystyle x\\in X}$ we should assign a module\u00a0:${\\displaystyle M}$ with a representations of ${\\displaystyle \\pi _{1}(X,x)\\to {\\text{Aut}}_{R}(M)}$ such that these representations are compatible with change of basepoint ${\\displaystyle x\\to y}$ for the fundamental group.\n\nHere's a proof that 1 and 2 are the same if X is path-connected.\n\n\u2022 Take ${\\displaystyle {\\mathcal {L}}}$ as in 1 and a loop ${\\displaystyle \\gamma }$ at x. It's easy so show that any (definition 1)-local system on ${\\displaystyle [0,1]}$ is constant. For instance, ${\\displaystyle \\gamma ^{*}{\\mathcal {L}}}$. So we get an isomorphism ${\\displaystyle (\\gamma ^{*}{\\mathcal {L}})_{0}\\simeq \\Gamma ([0,1],{\\mathcal {L}})\\simeq (\\gamma ^{*}{\\mathcal {L}})_{0}}$. But ${\\displaystyle \\gamma }$ gives an isomorphism between both sides and ${\\displaystyle {\\mathcal {L}}_{x}}$, whence an endomorphism of ${\\displaystyle {\\mathcal {L}}_{x}}$.\n\u2022 Take homomorphism ${\\displaystyle \\rho :\\pi _{1}(X,x)\\to {\\text{End}}_{R}(M)}$. Consider the constant sheaf ${\\displaystyle {\\underline {M}}}$ on the universal cover ${\\displaystyle {\\widetilde {X}}}$ of ${\\displaystyle X}$ with cover ${\\displaystyle \\pi :{\\widetilde {X}}\\to X}$ , and let ${\\displaystyle {\\mathcal {L}}}$ be the deck-transform-\u03c1-equivariant part:\n${\\displaystyle {\\mathcal {L}}_{U}=\\left\\{{\\text{sections }}s\\in {\\underline {M}}_{\\pi ^{-1}(U)}{\\text{ with }}\\theta \\circ s=\\rho (\\theta )s{\\text{ for all }}\\theta \\in {\\text{ Deck}}({\\widetilde {X}}\/X)=\\pi _{1}(X,x)\\right\\}}$\n\nThe proof shows that (for X path-connected) another equivalent definition of a local system is\n\n4. A sheaf ${\\displaystyle {\\mathcal {L}}}$ whose pullback ${\\displaystyle \\pi ^{*}{\\mathcal {L}}}$ by the universal cover ${\\displaystyle \\pi :{\\widetilde {X}}\\to X}$ is the constant sheaf.\n\nCall the map ${\\displaystyle \\pi _{1}(X,x)\\to {\\text{End}}_{R}(M)}$ the monodromy representation of the local system.\n\n## Examples\n\n\u2022 Constant sheaves. For instance, ${\\displaystyle {\\underline {\\mathbb {Q} }}_{X}}$. This is a useful tool for computing cohomology since the sheaf cohomology\n${\\displaystyle H^{k}(X,{\\underline {\\mathbb {Q} }}_{X})\\cong H_{\\text{sing}}^{k}(X,\\mathbb {Q} )}$\nis isomorphic to the singular cohomology of ${\\displaystyle X}$.\n\u2022 ${\\displaystyle X=\\mathbb {R} ^{2}\\setminus \\{0,0\\}}$. Since ${\\displaystyle \\pi _{1}(\\mathbb {R} ^{2}\\setminus \\{0,0\\})=\\mathbb {Z} }$, there are ${\\displaystyle S^{1}}$-many linear systems on X, the ${\\displaystyle \\theta \\in \\mathbb {R} }$ one given by monodromy representation\n${\\displaystyle \\pi _{x}(X)=\\mathbb {Z} \\to {\\text{Aut}}_{\\mathbb {C} }(\\mathbb {C} )}$ by sending ${\\displaystyle n\\mapsto e^{in\\theta }}$\n\u2022 Horizontal sections of vector bundles with a flat connection. If ${\\displaystyle E\\to X}$ is a vector bundle with flat connection ${\\displaystyle \\nabla }$, then\n${\\displaystyle E_{U}^{\\nabla }=\\left\\{{\\text{sections }}s\\in \\Gamma (U,E){\\text{ which are horizontal: }}\\nabla s=0\\right\\}}$\nis a local system.\nFor instance, take ${\\displaystyle X=\\mathbb {C} \\setminus 0}$ and ${\\displaystyle E=X\\times \\mathbb {C} ^{n}}$ the trivial bundle. Sections of E are n-tuples of functions on X, so ${\\displaystyle \\nabla _{0}(f_{1},...,f_{n})=(df_{1},...,df_{n})}$ defines a flat connection on E, as does ${\\displaystyle \\nabla (f_{1},...,f_{n})=(df_{1},...,df_{n})-\\Theta (f_{1},...,f_{n})}$ for any matrix of one-forms ${\\displaystyle \\Theta }$ on X. The horizontal sections are then\n${\\displaystyle E_{U}^{\\nabla }=\\left\\{(f_{1},...,f_{n})\\in E_{U}:(df_{1},...,df_{n})=\\Theta (f_{1},...,f_{n})\\right\\}}$\ni.e. the solutions to the linear differential equation ${\\displaystyle df_{i}=\\sum \\Theta _{ij}f_{j}}$.\nIf ${\\displaystyle \\Theta }$ extends to a one-form on ${\\displaystyle \\mathbb {C} }$ the above will also define a local system on ${\\displaystyle \\mathbb {C} }$, so will be trivial since ${\\displaystyle \\pi _{1}(\\mathbb {C} )=0}$. So to give an interesting example, choose one with a pole at 0:\n${\\displaystyle \\Theta ={\\begin{pmatrix}0&dx\/x\\\\dx&e^{x}dx\\end{pmatrix}}}$\nin which case for ${\\displaystyle \\nabla =d+\\Theta }$,\n${\\displaystyle E_{U}^{\\nabla }=\\left\\{f_{1},f_{2}:U\\to \\mathbb {C} \\ \\ {\\text{ with }}f'_{1}=f_{2}\/x\\ \\ f_{2}'=f_{1}+e^{x}f_{2}\\right\\}}$\n\u2022 An n-sheeted covering map ${\\displaystyle X\\to Y}$ is a local system with sections locally the set ${\\displaystyle \\{1,...,n\\}}$. Similarly, a fibre bundle with discrete fibre is a local system, because each path lifts uniquely to a given lift of its basepoint. (The definition adjusts to include set-valued local systems in the obvious way).\n\u2022 A local system of k-vector spaces on X is the same as a k-linear representation of the group ${\\displaystyle \\pi _{1}(X,x)}$.\n\u2022 If X is a variety, local systems are the same thing as D-modules that are in addition coherent as O-modules.\n\nIf the connection is not flat, parallel transporting a fibre around a contractible loop at x may give a nontrivial automorphism of the fibre at the base point x, so there is no chance to define a locally constant sheaf this way.\n\nThe Gauss-Manin connection is a very interesting example of a connection, whose horizontal sections occur in the study of variation of Hodge structures.\n\n## Generalization\n\nLocal systems have a mild generalization to constructible sheaves. A constructible sheaf on a locally path connected topological space ${\\displaystyle X}$ is a sheaf ${\\displaystyle {\\mathcal {L}}}$ such that there exists a stratification of\n\n${\\displaystyle X=\\coprod X_{\\lambda }}$\n\nwhere ${\\displaystyle {\\mathcal {L}}|_{X_{\\lambda }}}$ is a local system. These are typically found by taking the cohomology of the derived pushforward for some continuous map ${\\displaystyle f:X\\to Y}$. For example, if we look at the complex points of the morphism\n\n${\\displaystyle f:X={\\text{Proj}}\\left({\\frac {\\mathbb {C} [s,t][x,y,z]}{(stf(x,y,z))}}\\right)\\to {\\text{Spec}}(\\mathbb {C} [s,t])}$\n\nthen the fibers over\n\n${\\displaystyle \\mathbb {A} _{s,t}^{2}-\\mathbb {V} (st)}$\n\nare the smooth plane curve given by ${\\displaystyle f}$, but the fibers over ${\\displaystyle \\mathbb {V} }$ are ${\\displaystyle \\mathbb {P} ^{2}}$. If we take the derived pushforward ${\\displaystyle \\mathbf {R} f_{!}({\\underline {\\mathbb {Q} }}_{X})}$ then we get a constructible sheaf. Over ${\\displaystyle \\mathbb {V} }$ we have the local systems\n\n{\\displaystyle {\\begin{aligned}\\mathbf {R} ^{0}f_{!}({\\underline {\\mathbb {Q} }}_{X})|_{\\mathbb {V} (st)}&={\\underline {\\mathbb {Q} }}_{\\mathbb {V} (st)}\\\\\\mathbf {R} ^{2}f_{!}({\\underline {\\mathbb {Q} }}_{X})|_{\\mathbb {V} (st)}&={\\underline {\\mathbb {Q} }}_{\\mathbb {V} (st)}\\\\\\mathbf {R} ^{4}f_{!}({\\underline {\\mathbb {Q} }}_{X})|_{\\mathbb {V} (st)}&={\\underline {\\mathbb {Q} }}_{\\mathbb {V} (st)}\\\\\\mathbf {R} ^{k}f_{!}({\\underline {\\mathbb {Q} }}_{X})|_{\\mathbb {V} (st)}&={\\underline {0}}_{\\mathbb {V} (st)}{\\text{ otherwise}}\\end{aligned}}}\n\nwhile over ${\\displaystyle \\mathbb {A} _{s,t}^{2}-\\mathbb {V} (st)}$ we have the local systems\n\n{\\displaystyle {\\begin{aligned}\\mathbf {R} ^{0}f_{!}({\\underline {\\mathbb {Q} }}_{X})|_{\\mathbb {A} _{s,t}^{2}-\\mathbb {V} (st)}&={\\underline {\\mathbb {Q} }}_{\\mathbb {A} _{s,t}^{2}-\\mathbb {V} (st)}\\\\\\mathbf {R} ^{1}f_{!}({\\underline {\\mathbb {Q} }}_{X})|_{\\mathbb {A} _{s,t}^{2}-\\mathbb {V} (st)}&={\\underline {\\mathbb {Q} }}_{\\mathbb {A} _{s,t}^{2}-\\mathbb {V} (st)}^{\\oplus 2g}\\\\\\mathbf {R} ^{2}f_{!}({\\underline {\\mathbb {Q} }}_{X})|_{\\mathbb {A} _{s,t}^{2}-\\mathbb {V} (st)}&={\\underline {\\mathbb {Q} }}_{\\mathbb {A} _{s,t}^{2}-\\mathbb {V} (st)}\\\\\\mathbf {R} ^{k}f_{!}({\\underline {\\mathbb {Q} }}_{X})|_{\\mathbb {A} _{s,t}^{2}-\\mathbb {V} (st)}&={\\underline {0}}_{\\mathbb {A} _{s,t}^{2}-\\mathbb {V} (st)}{\\text{ otherwise}}\\end{aligned}}}\n\nwhere ${\\displaystyle g}$ is the genus of the plane curve (which is ${\\displaystyle g=({\\text{deg}}(f)-1)({\\text{deg}}(f)-2)\/2}$).\n\n## Applications\n\nThe cohomology with local coefficients in the module corresponding to the orientation covering can be used to formulate Poincar\u00e9 duality for non-orientable manifolds: see Twisted Poincar\u00e9 duality.","date":"2018-07-18 23:22:15","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 78, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.9817197918891907, \"perplexity\": 840.153039504319}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2018-30\/segments\/1531676590329.62\/warc\/CC-MAIN-20180718213135-20180718233135-00459.warc.gz\"}"} | null | null |
Q: Apache Ignite cache miss handler Currently we are using guava for caching few db entities. I am evaluating distribute apache ignite to replace guava. In guava have semantics get-if-absent using CacheLoader. How can achieve same functionality with ignite.
A: Got the answer through this great article https://dzone.com/articles/apache-ignite-how-to-read-data-from-persistent-sto . If there is a cache miss we can fetch it from db through cache store as below
public class PersonStore implements CacheStore<Long, Person> {
@SpringResource(resourceName = "dataSource")
private DataSource dataSource;
// This method is called whenever IgniteCache.loadCache() method is called.
@Override
public void loadCache(IgniteBiInClosure<Long, Person> clo, @Nullable Object... objects) throws CacheLoaderException {
System.out.println(">> Loading cache from store...");
try (Connection conn = dataSource.getConnection()) {
try (PreparedStatement st = conn.prepareStatement("select * from PERSON")) {
try (ResultSet rs = st.executeQuery()) {
while (rs.next()) {
Person person = new Person(rs.getLong(1), rs.getLong(2), rs.getString(3), rs.getInt(4));
clo.apply(person.getId(), person);
}
}
}
}
catch (SQLException e) {
throw new CacheLoaderException("Failed to load values from cache store.", e);
}
}
// This method is called whenever IgniteCache.get() method is called.
@Override
public Person load(Long key) throws CacheLoaderException {
System.out.println(">> Loading person from store...");
try (Connection conn = dataSource.getConnection()) {
try (PreparedStatement st = conn.prepareStatement("select * from PERSON where id = ?")) {
st.setString(1, key.toString());
ResultSet rs = st.executeQuery();
return rs.next() ? new Person(rs.getLong(1), rs.getLong(2), rs.getString(3), rs.getInt(4)) : null;
}
}
catch (SQLException e) {
throw new CacheLoaderException("Failed to load values from cache store.", e);
}
}
// Other CacheStore method implementations.
…
}
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 4,633 |
\section{Introduction}\label{s:intro}
The tenth paper in the series of the discovery of isotopes, the discovery of the barium isotopes is discussed. Previously, the discoveries of cerium \cite{Gin09}, arsenic \cite{Sho09a}, gold \cite{Sch09a}, tungsten \cite{Fri09}, krypton \cite{Hei09}, einsteinium \cite{Bur09}, iron \cite{Sch09b}, vanadium \cite{Sho09b}, and silver \cite{Sch09c} isotopes were discussed. The purpose of this series is to document and summarize the discovery of the isotopes. Guidelines for assigning credit for discovery are (1) clear identification, either through decay-curves and relationships to other known isotopes, particle or $\gamma$-ray spectra, or unique mass and Z-identification, and (2) publication of the discovery in a refereed journal. The authors and year of the first publication, the laboratory where the isotopes were produced as well as the production and identification methods are discussed. When appropriate, references to conference proceedings, internal reports, and theses are included. When a discovery included a half-life measurement the measured value is compared to the currently adopted value taken from the NUBASE evaluation \cite{Aud03} which is based on the ENSDF database \cite{ENS08}. In cases where the reported half-life differed significantly from the adopted half-life (up to approximately a factor of two), we searched the subsequent literature for indications that the measurement was erroneous. If that was not the case we credited the authors with the discovery in spite of the inaccurate half-life.
\section{Discovery of $^{114-151}$Ba}
Thirty-eight barium isotopes from A = $114-151$ have been discovered so far; these include seven stable, 18 proton-rich and 13 neutron-rich isotopes. According to the HFB-14 model \cite{Gor07}, $^{181}$Ba should be the last particle-stable odd-even neutron-rich nucleus, with the even-even barium isotopes reaching up to $^{188}$Ba. Along the proton dripline two more isotopes are predicted to be stable and it is estimated that five additional nuclei beyond the proton dripline could live long enough to be observed \cite{Tho04}. Thus, about 42 isotopes have yet to be discovered and almost 50\% of all possible barium isotopes have been produced and identified so far.
Figure \ref{f:year} summarizes the year of first discovery for all barium isotopes identified by the method of discovery. The range of isotopes predicted to exist is indicated on the right side of the figure. The radioactive barium isotopes were produced using heavy-ion fusion evaporation (FE), light-particle reactions (LP), neutron-capture reactions (NC), spontaneous fission (SF), neutron induced fission (NF), charged-particle induced fission (CPF), and projectile fragmentation or fission (PF). The stable isotopes were identified using mass spectroscopy (MS). Heavy ions are all nuclei with an atomic mass larger than A = 4 \cite{Gru77}. Light particles also include neutrons produced by accelerators. In the following paragraphs, the discovery of each barium isotope is discussed in detail.
\begin{figure}
\centering
\includegraphics[width=12cm]{barium-year.pdf}
\caption{Barium isotopes as a function of time when they were discovered. The different production methods are indicated. The solid black squares on the right hand side of the plot are isotopes predicted to be bound by the HFB-14 model. On the proton-rich side the light blue squares correspond to unbound isotopes predicted to have lifetimes larger than $\sim 10^{-9}$~s.}
\label{f:year}
\end{figure}
\subsection*{$^{114}$Ba}\vspace{-.85cm}
In 1995 Guglielmetti \textit{et al.} announced in \textit{Identification of the new isotope $^{114}$Ba and search for its $\alpha$ and cluster radioactivity} the discovery of $^{114}$Ba \cite{Gug95a}. $^{114}$Ba was produced at the Gesellschaft f\"ur Schwerionenforschung Unilac via the fusion evaporation reaction $^{58}$Ni($^{58}$Ni,2n)$^{114}$Ba at 4.2~MeV/u and identified using an on-line mass separator. ``With $\Delta$E-E telescopes we measured the total ($\beta$-decay) half-life to be T$_\beta$ = 0.43$^+{0.30}_{-0.15}$~s and the partial $\alpha$-decay half-life to be T$_\alpha \ge$ 1.2 $\times$ 10$^2$~s (1 MeV $le$ E$_\alpha \le$ 4 MeV) for $^{114}$Ba.'' This half-life agrees with the currently accepted value of 0.53(23)~s.
\subsection*{$^{115,116}$Ba}\vspace{-.85cm}
In the paper \textit{Decay studies of the neutron-deficient isotopes $^{114-118}$Ba} Janas \textit{et al.} reported the first observation of $^{115}$Ba and $^{116}$Ba in 1997 \cite{Jan97}. A 4.9 MeV/u $^{58}$Ni beam was accelerated by the linear accelerator UNILAC at GSI and bombarded enriched $^{58}$Ni and $^{60}$Ni targets. $^{115}$Ba and $^{116}$Ba were identified by measuring the energy and time of $\beta$-delayed protons following on-line mass separation. ``The least-square fit yielded $T_{1/2} = 0.45 \pm 0.05$ s for the decay half-life of $^{115}$Ba and a lower limit of 15\% for $b_{\beta p}$.'' In addition, $\beta$-delayed X-rays were detected for $^{116}$Ba. ``The time characteristics of the Cs KX-rays intensity, analyzed under the assumption of a single decay component, yielded $T_{1/2} = 1.3 \pm 0.2$ s for the half-life of $^{116}$Ba.'' These observed half-lives are currently the only measured values for $^{115}$Ba and $^{116}$Ba. The same group had previously mentioned the observation of these isotopes in a conference proceeding \cite{Gug95b}.
\subsection*{$^{117}$Ba}\vspace{-.85cm}
Bogdanov \textit{et al.} reported the discovery of $^{117}$Ba in 1977 in their article \textit{New Neutron-Deficient Isotopes of Barium and Rare-Earth Elements} \cite{Bog77}. An enriched $^{92}$Mo target was bombarded with an 180-190~MeV sulfur beam produced by the JINR Laboratory of Nuclear Reactions U-300 Heavy Ion Cyclotron and $^{117}$Ba was produced in the fusion evaporation reaction $^{92}$Mo($^{32}$S,2p5n). The isotope was separated with the BEMS-2 on-line ion source and identified by its delayed proton emission. ``As the proton emission of the A = 117 isobar, it is unambiguously related to $^{117}$Ba, since the reaction leading to the formation of $^{117}$Cs was energetically impossible in our experiments.'' The measured half-life of 1.9(2)~s is consistent with the accepted value of 1.75(7)~s.
\subsection*{$^{118}$Ba}\vspace{-.85cm}
In the 1997 paper \textit{Decay studies of the neutron-deficient isotopes $^{114-118}$Ba} Janas \textit{et al.} reported the first observation of $^{118}$Ba \cite{Jan97}. A 4.9 MeV/u $^{58}$Ni beam was accelerated by the linear accelerator UNILAC at GSI and bombarded enriched $^{58}$Ni and $^{60}$Ni targets on a $^{63}$Cu backing. $^{118}$Ba was produced in the fusion-evaporation reaction $^{63}$Cu($^{58}$Ni,1p2n) and identified by measuring $\beta$-delayed X-rays and $\gamma$-rays following on-line mass separation. ``From the time characteristics of the cesium KX-rays intensity the $^{118}$Ba half-life of $T_{1/2} = 5.2 \pm 0.2$~s was determined under the assumption of a single decay component.'' This observed half-life is currently the only measured value for $^{118}$Ba. The same group had previously mentioned the observation of this isotope in a conference proceeding \cite{Gug95b}.
\subsection*{$^{119}$Ba}\vspace{-.85cm}
In 1974 Bogdanov \textit{et al.} observed $^{119}$Ba, which they reported in the article \textit{Delayed-proton emitter $^{119}$Ba} \cite{Bog74}. An enriched $^{106}$Cd target was bombarded with an 85~MeV oxygen beam produced by JINR Laboratory of Nuclear Reactions U-300 Heavy Ion Cyclotron. $^{119}$Ba was produced in the fusion-evaporation reaction $^{106}$Cd($^{16}$O,3n). The delayed proton spectrum was measured with a telescope of a thin proportional-counter and a surface-barrier detector. ``The observed activity was due mainly to two radiators with half-lives T$_{1/2} = 5.0 \pm 0.6$ sec, with an excitation function peaking at E$_{^{16}O}$ = 85 MeV, and T$_{1/2} = 15.0 \pm 1.0 $ sec with its maximum yield at a higher energy... Thus, the most probable identification of the 5-second emitter is $^{119}$Ba.'' The extracted half-life agrees with the currently accepted value of 5.4(3)~s.
\subsection*{$^{120}$Ba}\vspace{-.85cm}
In 1974 Conrad \textit{et al.} reported the observation of $^{120}$Ba in their article \textit{Quasi-Rotational Bands in Neutron Deficient Doubly Even Ba Isotopes} \cite{Con74}. $^{120}$Ba was produced in the fusion evaporation reaction $^{106}$Cd($^{16}$O,2n) by bombarding cadmium with a 66~MeV oxygen beam provided by the MP Tandem of the Max-Planck-Institut f\"{u}r Kernphysik in Heidelberg, Germany. The isotope was identified by charged-particle-, neutron-, and gamma-gamma coincidence measurements: ``To identify $^{120}$Ba, which has 18 neutrons less than the most abundant barium isotope, neutron-gamma coincidences had to be applied in addition to charged particle and gamma-gamma coincidence measurements. The upper limit for the lifetime of the ground state of $^{120}$Ba is 90~sec.'' The first three $\gamma$-transitions in $^{120}$Ba were measured. The upper limit of 90~seconds is consistent with the accepted half-life value of 24(2)~s.
\subsection*{$^{121}$Ba}\vspace{-.85cm}
In the 1975 paper \textit{New Delayed-Proton Emitters $^{119}$Ba, $^{121}$Ba and $^{116}$Cs} Bogdanov \textit{et al.} reported the discovery of $^{121}$Ba \cite{Bog75}. The U-300 cyclotron of the Nuclear Reactions Laboratory at Dubna accelerated a $^{32}$S beam to a maximum energy of 190~MeV. ``The isotope $^{121}$Ba was observed in bombardment of niobium with sulfur ions in the reaction $^{93}$Nb($^{32}$S,p3n)$^{121}$Ba and with substantially greater yield in the reaction $^{92}$Mo($^{32}$S,2pn). The half-life is 29.7$\pm$1.5~sec.'' $^{121}$Ba was identified with the BEMS-2 mass separator. The observed half-life is currently the only available measurement. A month later the authors submitted the results to a different journal where they were published first \cite{Kar74}.
\subsection*{$^{122}$Ba}\vspace{-.85cm}
In 1974 Conrad \textit{et al.} reported the observation of $^{122}$Ba in their article \textit{Quasi-Rotational Bands in Neutron Deficient Doubly Even Ba Isotopes} \cite{Con74}. $^{122}$Ba was produced in the fusion evaporation reaction $^{108}$Cd($^{16}$O,2n) by bombarding cadmium with a 66~MeV oxygen beam provided by the MP Tandem of the Max-Planck-Institut f\"{u}r Kernphysik in Heidelberg, Germany. The isotope was identified by gamma-gamma coincidence measurements: ``Up to now a level scheme of $^{122}$Ba has not been published. For the lifetime of the ground state a value between 2.5 and 5 sec has been suggested \cite{DAu67}. From our data the partial level scheme shown in fig. 4 was obtained.'' The first six $\gamma$-transitions of the ground state band in $^{122}$Ba were measured.
\subsection*{$^{123}$Ba}\vspace{-.85cm}
Preiss and Strudler reported discovery of $^{123}$Ba in 1962 in their article \textit{New Neutron Deficient Barium Isotopes} \cite{Pre62}. $^{123}$Ba was produced via the fusion-evaporation reactions $^{113}$In($^{16}$O,p5n), $^{115}$In($^{16}$O,p7n), $^{113}$In($^{14}$N,4n), $^{115}$In($^{14}$N,6n), natural Sn($^{16}$O,$\alpha$xn)$^{123}$Ba and natural Sn($^{12}$C,xn)$^{123}$Ba; the beams were produced by the Yale University Heavy Ion Accelerator and had a maximum energy of 10.5~MeV/nucleon. $^{123}$Ba was identified measuring characteristic X-ray spectra following chemical separation. ``Mass assignments for the new Ba activities were based on the parent daughter genetics using Cs half-lives and $\gamma$-ray energies previously reported and/or found in the present study. The proposed half-lives and mass assignments are: $^{123}$Ba, 2$\pm$0.5 min; $^{125}$Ba, 6.5$\pm$0.5~min; and $^{127}$Ba, 10.0$\pm$0.5~min.'' The observed half-life for $^{123}$Ba is consistent with the currently accepted value of 2.7(4)~m.
\subsection*{$^{124}$Ba}\vspace{-.85cm}
$^{124}$Ba was first observed in 1967 by Clarkson \textit{et al.} as reported in \textit{Collective Excitations in Neutron-Deficient Barium, Xenon, and Cerium Isotopes} \cite{Cla67}. $^{124}$Ba was produced by the reactions $^{115}$In($^{14}$N,5n) at 84~MeV and $^{116}$Sn($^{12}$C,4n) at 80~MeV where the ions were accelerated with the Berkeley heavy-ion linear accelerator~(HILAC). Gamma-ray spectra were measured with a lithium-drifted germanium counter. ``Since $^{124}$Ba was produced by two reactions with different targets and projectiles, which both give the same transitions, this mass assignment is likewise considered to be quite certain.'' The first two transitions of the ground-state band were correctly identified.
\subsection*{$^{125}$Ba}\vspace{-.85cm}
$^{125}$Ba was identified correctly for the first time in 1968 by D'Auria \textit{et al.} in the article \textit{Deformation in the Light Ba Isotopes: Isomeric States of Ba$^{125}$ and Ba$^{127}$} \cite{DAu68}. $^{125}$Ba was produced in the fusion-evaporation reaction $^{115}$In($^{14}$N,4n) at Yale University. The isotope was chemically separated and $\beta$- $\gamma$- and X-ray spectra were measured. ``Previously unobserved and unassigned $\gamma$ rays resulting from the decay of Ba$^{125}$ were detected at 56$\pm$3, 76$\pm$2, 84$\pm$2, and 141$\pm$2 keV, decaying with a composite half-life of 3.0$\pm$0.5 min.'' This half-life agrees with the currently accepted value of 3.5(4)~m.
D'Auria \textit{et al.} interpreted this state as the high-spin ground state in addition to the presence of an 8(1)~m isomeric excited state. The existence of the isomeric state has not been confirmed. Preiss and Strudler had previously reported a half-life of 6.5(5)~m for $^{125}$Ba \cite{Pre62}. However, because this value is almost a factor of two larger than the accepted value and closer to the claimed isomeric state, it is likely that the state had been misidentified.
\subsection*{$^{126}$Ba}\vspace{-.85cm}
In the 1954 article \textit{New Chain Barium-126--Cesium-126} Kalkstein \textit{et al.} announced the discovery of $^{126}$Ba \cite{Kal54}. Indium oxide was bombarded with a nitrogen beam produced by the Berkeley Crocker 60~inch cyclotron with a maximum energy of 140~MeV. $^{126}$Ba was produced in the fusion-evaporation reaction $^{115}$In($^{14}$N,3n) and identified using a scintillation spectrometer, a scintillation coincidence spectrometer, and a time-of-flight mass spectrograph. Element assignments and genetic relations have been verified chemically, and the mass number assigned with the isotope separator. ``$^{126}$Ba decays principally by orbital electron capture with a half-life of 96.5$\pm$2.0~minutes.'' This half-life is consistent with the currently accepted value of 100(2)~m.
\subsection*{$^{127}$Ba}\vspace{-.85cm}
In 1952 Linder and Osborne reported the discovery of $^{127}$Ba in \textit{The Nuclides Ba$^{127}$, Ba$^{128}$ and Cs$^{128}$} \cite{Lin52}. A cesium nitrate target was bombarded with 190~MeV deuterons at Livermore. $^{127}$Ba was chemically separated and its activity measured with an end-window argon-alcohol-filled counter. ``A barium isotope of 12-minute half-life was found whose radiations were not directly characterized. However, positron emission is probable since electromagnetic radiation seemed to comprise no more than five percent of the total activity detectable on an end-window counter. By four rapid chemical separations made at ten-minute intervals a cesium activity was obtained from the barium whose half-life and radiation characteristics agree with those reported for Cs$^{127}$. Furthermore, the yield of this nuclide diminished roughly by a factor of two in each of the four successive separations. The 12-minute barium activity is thus Ba$^{127}$.'' The observed half-life agrees with the currently adopted value of 12.7(4)~m.
\subsection*{$^{128}$Ba}\vspace{-.85cm}
Fink and Templeton identified $^{128}$Ba in 1950 as described in \textit{Radioactive Isotopes of Barium} \cite{Fin50}. Cesium chloride was bombarded with 85~MeV protons in the 184~inch Berkeley cyclotron. The induced barium activities were chemically separated and the half-life was measured with a Geiger counter: ``The 2.4-day period is not $^{127}$Ba, otherwise it would produce 5.5-hour $^{127}$Cs as a daughter. The most probable assignment is $^{128}$Ba from the p,6n reaction, but this assignment lacks direct proof.'' The measured half-life of 2.4~d agrees with the presently accepted value of 2.43(5)~d. Thomas and Wiig had measured a 2.4(1)~d half-life in barium but they were only able to determine that the mass number was smaller than 129 \cite{Tho50}.
\subsection*{$^{129}$Ba}\vspace{-.85cm}
In 1950 the discovery of $^{129}$Ba was reported first by Thomas and Wiig in \textit{On Neutron Deficient Isotopes of Barium} \cite{Tho50}.
250-MeV protons accelerated by the Rochester 130-inch cyclotron were used to bombard spectroscopically pure cesium chloride. The half-life of the chemically separated barium fraction was measured by deflecting positrons into a counter tube with a permanent magnet. ``Parent-daughter separations performed more than 24 hours after the bombardment failed to show any $^{129}$Cs activity whereas earlier milkings did show the activity. This led to the conclusion that the 1.8-hour barium is $^{129}$Ba.'' The measured half-life of 1.8(2)~h is close to the currently accepted value of 2.23(11)~h. Less than a month later Fink and Templeton submitted their half-life measurement of 2.0(1)~h for $^{129}$Ba \cite{Fin50} which was published in the same issue of Physical Review immediately following the article by Thomas and Wiig.
\subsection*{$^{130}$Ba}\vspace{-.85cm}
In the 1936 paper \textit{The Isotopic Constitution of Barium and Cerium} Dempster reported the first observation of $^{130}$Ba \cite{Dem36}. The mass spectra were measured at the Ryerson Physical Laboratory at the University of Chicago: ``I have photographed several mass spectra of the barium ions formed in a high frequency spark between two barium electrodes, which show two still fainter isotopes at 130 and 132.''
\subsection*{$^{131}$Ba}\vspace{-.85cm}
$^{131}$Ba was first observed by Katcoff in 1945 and reported in Plutonium Project Records \cite{Kat45}. The results were subsequently published in a refereed journal in the article \textit{New Barium and Cesium Isotopes: 12.0d Ba$^{131}$, 10.2 Cs$^{131}$, and Long-Lived Ba$^{133}$} \cite{Kat47}. $^{131}$Ba was produced by neutron irradiation of BaCO$_{3}$ in the Clinton Pile of Argonne National Laboratory and separated via fractional precipitation experiments. ``The Ba$^{131}$ isotope decays predominantly by orbital electron capture with a half-life of 12.0 days, emitting gamma-radiations of about 0.26 Mev, 0.5 Mev, and roughly 1.2 Mev.'' This half-life agrees with the presently adopted value of 11.50(6)~d. Yu \textit{et al.} had submitted their observation of a 11.7(3)~d half-life \cite{Yu47} for $^{131}$Ba six months before Katcoff. Although they do not reference the Plutonium Project Records by Katcoff they utilized material made available by the Manhattan Project \cite{Man46}. Thus we still credit Katcoff with the first observation of $^{131}$Ba.
\subsection*{$^{132}$Ba}\vspace{-.85cm}
In the 1936 paper \textit{The Isotopic Constitution of Barium and Cerium} Dempster reported the first observation of $^{132}$Ba \cite{Dem36}. The mass spectra were measured at the Ryerson Physical Laboratory at the University of Chicago: ``I have photographed several mass spectra of the barium ions formed in a high frequency spark between two barium electrodes, which show two still fainter isotopes at 130 and 132.''
\subsection*{$^{133}$Ba}\vspace{-.85cm}
Cork and Smith reported the first observation of $^{133}$Ba in their 1941 article \textit{Radioactive Isotopes of Barium from Cesium} \cite{Cor41}. $^{133}$Ba was produced by bombarding cesium with 9.5 MeV deuterons at the University of Michigan. Following chemical separation the isotope was identified with a magnetic beta-spectrometer and by absorption measurements. The observed 40.0(5)~h half-life was identified as an excited state and could be ascribed to either $^{133}$Ba or $^{134}$Ba. ``However, it is known that the Rochester group have found this same activity by bombarding cesium with protons. In their bombardment it is possible to produce barium by the (P,N) and (P,$\gamma$) reactions. Although the latter process it known to occur, the former is much more probable and favors the assignment to Ba$^{133}$.'' The observed half-life of this isomeric state agrees with the present value of 38.9(1)~h.
\subsection*{$^{134}$Ba}\vspace{-.85cm}
In 1936 Blewett and Sampson reported the discovery of $^{134}$Ba in \textit{Isotopic Constitution of Strontium, Barium, and Indium} \cite{Ble36}. The mass spectrographic study of barium was performed at Princeton University by heating barium oxide from a tungsten filament. ``The curves for barium showed a peak at mass 134 making up 1.8 percent of the total emission...[which] lead us to believe that this is due to a new isotope of barium.''
\subsection*{$^{135-137}$Ba}\vspace{-.85cm}
$^{135}$Ba, $^{136}$Ba, and $^{137}$Ba were discovered by Aston as reported in his 1932 article \textit{The Isotopic Constitution and Atomic Weights of Caesium, Strontium, Lithium, Rubidium, Barium, Scandium and Thallium} \cite{Ast32}. The mass spectra were measured at the Cavendish Laboratory in Cambridge, UK: ``The production of sufficiently intense rays from barium salts is a matter of great difficulty, but after many attempts an anode containing the chloride mixed with a little iodide yielded mass-spectra showing beyond any doubt the presence of three new isotopes, 135, 136, 137.''
\subsection*{$^{138}$Ba}\vspace{-.85cm}
In 1925 Aston reported the first observation of $^{138}$Ba in \textit{The Mass Spectra of Chemical Elements, Part VI. Accelerated Anode Rays Continued} \cite{Ast25}. The mass spectra were measured at the Cavendish Laboratory in Cambridge, UK: ``In these experiments the anode consisted of a mixture of barium chloride and lithium bromide. Schumannized plates were used and the line $^{138}$Ba was obtained of very great intensity.''
\subsection*{$^{139}$Ba}\vspace{-.85cm}
Pool \textit{et al.} published the first identification of $^{139}$Ba in \textit{A Survey of Radioactivity Produced by High Energy Neutron Bombardment} in 1937 \cite{Poo37a}. Neutrons with energies up to 20 MeV, produced by bombarding lithium with 6.3 MeV deuterons at the University of Michigan, were used to irradiate many stable elements. In the summary table the observed half-life of 85~m was assigned to $^{139}$Ba. This assignment is supported by a previously published contribution to a conference: ``Barium becomes strongly radioactive with a half-life period of 85.6 min. The $\beta$-particles have the negative sign. Chemical analysis shows that the activity is most probably due to Ba$^{139}$.'' \cite{Poo37b}. This half-life agrees with the currently accepted value of 83.1(3)~m. Amaldi \textit{et al.} had reported a 80~m period in barium in 1935; however, no mass assignment was made \cite{Ama35}.
\subsection*{$^{140}$Ba}\vspace{-.85cm}
In the 1939 paper \textit{Nachweis der Entstehung aktiver Bariumisotope aus Uran und Thorium durch Neutronenbestrahlung; Nachweis weiterer aktiver Bruchst\"ucke bei der Uranspaltung} Hahn and Strassmann identified $^{140}$Ba for the first time at Berlin-Dahlem in Germany \cite{Hah39}. $^{140}$Ba was produced by irradiating Uranium with neutrons from a Ra-Be-neutron source. Decay curves were measured following chemical separation. A previously reported 300~h activity originally labeled as ``Ra IV'' \cite{Hah38,Hah39a} now identified as the fission product ``Ba IV'' was again observed. Based on the measured half-life of the daughter activity it was tentatively assigned to $^{140}$Ba: ``Was die anderen Barium isotope aus dem Uran anbelangt, so l\"a\ss t sich f\"ur das Ba IV vielleicht die Hypothese machen, da\ss\ es die Muttersubstanz des in der Literatur beschriebenen Radiolanthans von 31-46 Stunden Halbwertszeit mit dem vermutlichen Atomgewicht 140 ist.'' (Concerning the other from uranium produced barium isotopes, it is hypothesized that Ba IV may be the parent of the radioactive lanthanum which was reported with a half-life of 31-46 hours with the probable atomic weight of 140.) Hahn and Strassmann did not officially assign the 300~h activity to $^{140}$Ba in subsequent papers \cite{Hah39b,Hah39c,Hah40a,Hah40b}, although they confirmed the relationship of the activity to $^{140}$La \cite{Hah40b}. Subsequently this assignment was specifically made by other authors \cite{Bor43,Wei43} and it was generally adopted in the 1944 Table of Isotopes \cite{Sea44}. The final proof was given by mass-spectroscopic measurements in 1947 \cite{Hay48}.
\subsection*{$^{141}$Ba}\vspace{-.85cm}
Katcoff established the identification of $^{141}$Ba in 1945 in the Plutonium Project Record \textit{Radiations from 3.7h La$^{141}$} \cite{Kat45b}.
Uranyl nitrate was irradiated with slow neutrons produced with the Chicago cyclotron and the A=141 mass chain of the $^{141}$Ba-$^{141}$La-$^{141}$Ce relationship was established: ``About 75 min was then allowed for 3.7h La$^{141}$ to grow into the solution from its 18m Ba$^{141}$ parent...The $\beta$-decay curve shows a long-lived component (probably the 28d Ce$^{141}$ daughter of 3.7h La$^{141}$) and small amounts of 30h and 1.5h components; but the 3.7h component greatly predominates.'' The currently accepted value for the half-life of $^{141}$Ba is 18.27(7)~m. Hahn and Strassmann had originally reported this half-life in barium for the first time \cite{Hah42} modifying a previous observation of a single 14~m component \cite{Hah39} into two components of 18~m and 6~m. Hahn and Strassmann also observed the relationship of the 18~m half-life with a 3.5~h component in lanthanum. However, no specific mass assignment was made. In another paper in the Plutonium Project Record Goldstein mentioned the established relationship of the mass chain \cite{Gol44} referring to the work by Ballou and Burgus which was not included in the published record \cite{Bal43}.
\subsection*{$^{142}$Ba}\vspace{-.85cm}
The first accurate identification of $^{142}$Ba was published by Schuman \textit{et al.} in 1959 in the article \textit{Decay of Short-Lived Barium and Lanthanum Fission Products} \cite{Sch59}. Enriched $^{235}$U was irradiated in an MTR pneumatic rabbit facility of the Atomic Energy Division of the Phillips Petroleum Company at Idaho Falls, Idaho. Fission products were chemically separated and the $\beta$- and $\gamma$- decays of the fragments were measured. ``The gamma-ray spectra of the barium samples showed photopeaks decaying with two half-lives, 11 min for Ba$^{142}$ and 18 min for Ba$^{141}$ and in addition the lanthanum daughter photopeaks growing in.'' The reported half-life of 11(1)~m agrees with the presently accepted value of 10.6(2)~m. The 6~m half-life reported by Hahn and Strassmann in 1942 \cite{Hah42} was only tentatively assigned to $^{142}$Ba as late as the 1958 edition of the Table of Isotopes \cite{Str58} (Classification D: Element certain and mass number not well established). Mal\'y \textit{et al.} reported a value of 5.9~m for $^{142}$Ba in 1958 \cite{Mal58}. Due to the large discrepancy of this half-life with the correct value, we credit Schuman \textit{et al.} with the first correct identification of $^{142}$Ba.
\subsection*{$^{143}$Ba}\vspace{-.85cm}
Wahl \textit{et al.} reported the first identification of $^{143}$Ba in 1962 in the article \textit{Nuclear-Charge Distribution in Low-Energy Fission} \cite{Wah62}. $^{143}$Ba was produced from $^{235}$U fission induced by thermal neutrons. The neutrons were produced from reactions of 10 MeV deuterons accelerated by the Washington University cyclotron on a beryllium target. The half-life of $^{143}$Ba was measured by timed separations of its daughters. ``The half-life value of Ba$^{143}$ obtained was (12.0$\pm$1.2) sec.'' This value is consistent with the presently accepted value of 11.5(2)~s. Hahn and Strassmann had speculated about the existence of a short-lived barium isotope ($<$ 0.5~m) \cite{Hah42} which was tentatively assigned to $^{143}$Ba by the Plutonium Project Records \cite{PPR51}.
\subsection*{$^{144}$Ba}\vspace{-.85cm}
Amarel \textit{et al.} observed $^{144}$Ba in 1967 as reported in their article \textit{Half Life Determination of Some Short-Lived Isotopes of Rb, Sr, Cs, Ba, La and Identification of $^{93,94,95,96}$Rb as Delayed Neutron Precursors by On-Line Mass-Spectrometry} \cite{Ama67}. $^{144}$Ba was produced by fission of $^{238}$U induced by 150 MeV protons from the Orsay synchrocyclotron. Isotopes were identified with a Nier-type mass spectrometer and half-lives were determined by $\beta$ counting. The measured half-life for $^{144}$Ba was listed in the main table with 11.4(25)~s which is consistent with the currently adopted value of 11.5(2)~s.
\subsection*{$^{145}$Ba}\vspace{-.85cm}
Grapengiesser \textit{et al.} reported the observation of $^{145}$Ba in \textit{Survey of short-lived fission products obtained using the isotope-separator-on-line facility at Studsvik} in 1974 \cite{Gra74}. $^{145}$Ba was produced and identified at the OSIRIS isotope-separator online facility at the Studsvik Neutron Research Laboratory in Nyk\"oping, Sweden. In the long table of experimental half-lives of many different isotopes the half-life of $^{145}$Ba is quoted as 4.2(5)~s. This value agrees with the currently adopted value of 4.31(16)~s. The previous observation of $\gamma$-ray transitions attributed to $^{145}$Ba were not sufficiently accurate and not based on firm mass assignment \cite{Hop71}.
\subsection*{$^{146}$Ba}\vspace{-.85cm}
The paper \textit{Ground-State Bands in Neutron-Rich Even Te, Xe, Ba, Ce, Nd, and Sm Isotopes Produced in the Fission of $^{252}$Cf} published in 1970 reported the first identification of $^{150}$Ce by Wilhelmy {\it et al.} at Berkeley \cite{Wil70}. They measured $\gamma$-spectra following spontaneous fission of $^{252}$Cf and observed the first two $\gamma$-ray transitions of the $^{146}$Ba ground state band. They did not mention the first observation of $^{146}$Ba: ``The data, which in some of the cases can be correlated with previously reported results...'' implies that some of the cases were new observations.
\subsection*{$^{147}$Ba}\vspace{-.85cm}
Wohn \textit{et al.} reported the discovery of $^{147}$Ba in 1978 in their article \textit{Identification of $^{147}$Cs and Half-Life Determinations for Cs and Ba Isotopes with A=144-147 and Rb and Sr Isotopes with A=96-98} \cite{Woh78}. $^{147}$Ba was produced and identified by neutron induced fission of $^{235}$U at the On-line Separator f\"{u}r Thermisch Ionisierbare Spaltprodukte (OSTIS) facility of the Institut Laue-Langevin in Grenoble, France. ``Half-life determinations of Rb and Cs fission products available at an on-line mass separator have been made for several neutron-rich Rb, Sr, Cs, and Ba isotopes using both $\beta$-multiscale and $\gamma$-multispectra measurements. The half-lives and rms uncertainties (in sec) are...$^{147}$Ba, 0.70(6).'' The observed value for the half-life is close to the accepted value of 0.893(1)~s. Wohn \textit{et al.} were aware of a previous work for $^{147}$Ba published in a conference proceeding \cite{Ami76}. This work was submitted by Engler \textit{et al.} to a refereed journal \cite{Eng79} seven months later than Wohn \textit{et al.}. Engler \textit{et al.} claimed the first observation of $^{147}$Ba although they quote the work by Wohn \textit{et al.}.
\subsection*{$^{148}$Ba}\vspace{-.85cm}
Engler \textit{et al.} observed $^{148}$Ba for the first time as reported in the 1979 article \textit{Half-Life Measurements of Rb, Sr, Y, Cs, Ba, La and Ce Isotopes with A=91-98 and A=142-149} \cite{Eng79}. A $^{235}$U target was exposed to thermal neutrons at the Soreq Nuclear Research Centre in Yavne, Israel. $^{148}$Ba was identified with the Soreq-On-Line-Isotope-Separator (SOLIS). ``The isotopes $^{147,148}$Ba and $^{149}$La were identified for the first time and their half-lives measured. The values obtained, in seconds, are 0.72$\pm$0.07 for $^{147}$Ba, 0.47$\pm$0.20 for $^{148}$Ba and 1.2$\pm$0.4 for $^{149}$La.'' The half-life for $^{148}$Ba is consistent with the accepted value of 0.612(17)~s.
\subsection*{$^{149}$Ba}\vspace{-.85cm}
In the 1993 article \textit{Delayed-neutron branching ratios of precursors in the fission product region} Rudstam \textit{et al.} reported the observation of $^{149}$Ba \cite{Rud93}. $^{149}$Ba was produced and identified at the OSIRIS isotope-separator online facility at the Studsvik Neutron Research Laboratory in Nyk\"oping, Sweden. In the large table of delayed-neutron branching ratios and half-lives the half-life of $^{149}$Ba is quoted as 0.324(18)~s. This value agrees with the currently adopted value of 344(7)~s. Warner and Reeder had reported a half-life measurement for $^{149}$Ba seven years earlier in a conference proceeding \cite{War86}.
\subsection*{$^{150-151}$Ba}\vspace{-.85cm}
Bernas {\it{et al.}} discovered $^{150}$Ba and $^{151}$Ba in 1994 at GSI, Germany, as reported in {\it{Projectile Fission at Relativistic Velocities: A Novel and Powerful Source of Neutron-Rich Isotopes Well Suited for In-Flight Isotopic Separation}} \cite{Ber94}. The isotopes were produced using projectile fission of $^{238}$U at 750 MeV/nucleon on a lead target. ``Forward emitted fragments from $^{80}$Zn up to $^{155}$Ce were analyzed with the Fragment Separator (FRS) and unambiguously identified by their energy-loss and time-of-flight.'' The experiment yielded 13 individual counts of $^{151}$Ba. As shown in Figure 3 of the article many more counts of $^{150}$Ba were recorded though not explicitly mentioned in the text since Mach \textit{et al.} had reported the discovery of $^{150}$Ba in a conference abstract \cite{Mac87}. However, since this observation was never published in a refereed journal we credit Bernas {\it et al.} with the discovery of $^{150}$Ba.
\section{Summary}
The discovery of the barium isotopes has been cataloged and the methods of their discovery discussed. Many of the barium isotopes have a long and interesting history. The discoveries of $^{140}$Ba and $^{141}$Ba were directly linked to the discovery of fission. The half-lives of four isotopes ($^{128}$Ba, $^{139}$Ba, $^{142}$Ba, and $^{143}$Ba) were first measured without accurate mass assignments and two measurements were initially wrong ($^{125}$Ba and $^{142}$Ba). $^{129}$Ba and $^{147}$Ba were identified essentially simultaneously by two groups independently. It is also interesting to note that $^{120}$Ba, $^{122}$Ba, $^{124}$Ba, and $^{146}$Ba were first observed in $\gamma$-ray spectroscopy studies. Finally, the discovery of $^{149}$Ba and $^{150}$Ba had been reported in conference proceedings seven years prior to a publication in refereed journals.
\ack
This work was supported by the National Science Foundation under grants No. PHY06-06007 (NSCL) and PHY07-54541 (REU). MH was supported by NSF grant PHY05-55445. JQG acknowledges the support of the Professorial Assistantship Program of the Honors College at Michigan State University.
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{"url":"https:\/\/stats.stackexchange.com\/questions\/511880\/root-mean-square-error-when-having-multiple-prediction-horizons","text":"# Root-mean-square error when having multiple prediction horizons\n\nI have a basic question about the root-mean-square error (RMSE). I have a prediction using an ARIMA model. I predicted a time series and use a rolling-horizon approach with overlapping or non-overlapping prediction horizons.\n\nFor simplicity let's just say we have not overlapping prediction horizons and I calculate forecasted values from a time series with 30 entries having a prediction horizon of 10. This means that I have 2 iterations:\n\n1. Learning data: ts [0:10], prediction ts [11:20]\n2. Learning data: ts [0:20], prediction ts[21:30]\n\nNow I want to calculate the RMSE. My question is whether I should calculate the RMSE of each prediction iteration which leads to 2 different RMSE or whether I should store all predicted values and calculate only 1 RMSE for the whole time series? Further I would like to know if the total RMSE in this case would be equal to the average RMSE of the individual prediction iterations? I implemented a function i R to calculate both the local RMSE of the single prediction iterations and the global RMSE of the whole time series. It turned out that the total RMSE is not the average of the 2 local RMSE. As I would definitely not rule out that I made a mistake in the calculation, I would like to know from you whether the total RMSE should be the average of the local RMSE?\n\nHere is my R code for calculating both the total RMSE and the RMSE of the single prediction iterations.\n\n#This function calculates different error measures for a predicted timeSeries by using the automated\n#Arima model in a rolling horizon approach with overlapping or non-overlapping prediction horizons\n\nerrorCalculationArima <-\nfunction(timeSeries, predictionHorizon, overlappingPredictionHorizons) {\n\nlibrary(forecast)\nhelpIterationCounter=0\ncurrentTimeSlot=0\nsumOfSquarredErrors=0\ntotalSumOfSquarredErrors =0\n\n# Set the next time slots\nif(overlappingPredictionHorizons==TRUE) {\nstepToNextTimeSlot=1\nnumberOfAllPredictions = predictionHorizon* (length(timeSeries)-predictionHorizon)\n}\n\nif(overlappingPredictionHorizons==FALSE) {\nstepToNextTimeSlot = predictionHorizon\nnumberOfAllPredictions = (length(timeSeries)-predictionHorizon)\n}\n\npredictedValues_list <- vector(mode = \"numeric\", length = numberOfAllPredictions)\nactualValues_list <- vector(mode = \"numeric\", length = numberOfAllPredictions)\nhelpCounterCurrentPositionInTheArrayOfAllValues =1\n\nwhile(currentTimeSlot + predictionHorizon<length(timeSeries)){\ncurrentTimeSlot=currentTimeSlot+stepToNextTimeSlot\n\n#fit autoArimaModel for all values until the currentTimeSlot\nautoArima_model<-auto.arima(ts(timeSeries[(0):currentTimeSlot]))\n\n#forecast values using the autoArima model\nforecastedVal_autoArima_model <- forecast(autoArima_model,h=predictionHorizon)\nmessage<-paste(\"Forecast Number \", (helpIterationCounter+1), \", Beginning time slot: \", currentTimeSlot)\nprint(message)\n\n#Calculate sum of squarred errors for the prediction horizons\nsumOfSquarredErrors = 0\nfor (i in 1:predictionHorizon) {\npredictedvalue = forecastedVal_autoArima_model$mean[i] actualValue = timeSeries [currentTimeSlot + i] sumOfSquarredErrors = sumOfSquarredErrors + (predictedvalue - actualValue)^2 string_actualValue<-paste(\"Actual Value: \" , actualValue) string_predictedvalue<-paste(\"Predicted Value: \" , predictedvalue) print(string_actualValue) print(string_predictedvalue) #Fill in the predicted and actual values to the lists containing every value predictedValues_list[helpCounterCurrentPositionInTheArrayOfAllValues]=predictedvalue actualValues_list [helpCounterCurrentPositionInTheArrayOfAllValues]=actualValue helpCounterCurrentPositionInTheArrayOfAllValues = helpCounterCurrentPositionInTheArrayOfAllValues + 1 } #Calculate and print RMSE of the current predictionHorizon message_RMSE <- paste(\"RMSE: \", sqrt(sumOfSquarredErrors\/predictionHorizon)) print(message_RMSE) helpIterationCounter = helpIterationCounter + 1 writeLines(\"\\n\") } #Calculate total RMSE of all data totalSumOfSquarredErrors=0 for(i in 1:length(predictedValues_list)) { totalSumOfSquarredErrors = totalSumOfSquarredErrors + (predictedValues_list[i] - actualValues_list[i])^2 } message_TotalRMSE <- paste(\"Total RMSE for the whole time series: \", sqrt(totalSumOfSquarredErrors\/length(predictedValues_list))) print(message_TotalRMSE) } Can anyone say something about these issues? I'd appreciate every comment. What do you usually do when evaluating a prediction of time series? Do you calculate the total RMSE or the average of the different perdiction interations? Does nobody have an idea about that? Have you never used the RMSE to evaluate prediction results? ## 2 Answers Hello while I can not show you my code, as it is confidential due to company legal agreement, I developed a Genetic Time Series fitting VAR in levels, VECM, and VAR in differences and doing a kmeans cluster analysis afterwards for narrowing down solution space for the Genetic Algorithm, as you can run my tool with an overall search grid for all features and parameters, and then use insights from the kmean to adjust the params of every single feature. I worked 4-5 months on it and I can say that I swallowed a lot of theory and coding strategy e.g. syntactic sugar and how to save chromosome solutions of the GA, which hold the solutions for all VAR submodels. So I have some experience. Maybe it would be better to go to chat if you want to know more in detail. I have non overlapping windows, as i do a rolling forecast with a window-size only of one that rolls forward 52 weeks but one week at a time. So we have a 52 weeks forecast which is originally a 1 week fit, as the shortest window guarantees the best fit. This is our strategy to mimic a one year forecast. Imho, the RMSE is limited in its answers. For business implications you'll need R\u00b2 and MAPE\/MAE, with train\/test split, as the RMSE is useless for other stakeholders. I myself would stick also to the MASE of Rob Hyndman, as it can also tell you the fit of a time series that can drop to zero (intermittent demand time series). And the MASE tells you if your forecasting method is better than the most naive forecast there is, in other words, this week equals last week. I can not tell you anything about your RMSE questions, but it belongs to the point of view. I mostly work in the area of media\/ads\/ad awareness kpi sales, and no one would understand rmse. So I develop models, that everyone can get into. Update As the OP asked of a more general matter, I updated my question: Dear Peter, that is an excellent question and indeed it depends more on the nature of your data points. And I'll give you an excerpt from my forecasting tool in the shape of an output matrix to answer this question. First lets get some terminology: -when im talking about window size, i mean reevaluating data or making the training set bigger at each iteration. The iteration is determined by the forecast period. e.g. 52 weeks measn 52 iterations of window size 1 -when im talking about nahead i mean how many windows you want to forecast at once despite your refitting. it is possible to forecast e.g. 2 windows although you are only incorporating information of only window1. -when we are observing every iteration we normally speak of APE! Absolute Prediction Error, MAPE Only if there are more than two observations to have the mean of them. That means you get RMSE\/MAPe in dependence of window size (also knowns as crossvalidation of time series https:\/\/otexts.com\/fpp2\/accuracy.html) and forecasting period. The red dots are the windows. For example, have a look at the output of my VAR. I want to forecast a period of 52 data points. So you can imagine tha this matrix has a format of 52 rows. I have a forecasting window of 2, that means I'm forecasting from my actual point of time that is t - 1 (some would say its t, because I want to forecast now, but that doesnt matter) . So I want to forecast t. However with a window of 2 n.ahead I also forecast t + 1. First, we have a look at that matter before dealing with the cross validation. In theory, the best forecast we can get, is the shortest, thus: If your data is of 99 time periods or of 1000 a forecast of only t would result in better results in theory: Also look here, why we are normally not doing something on the whole time series: https:\/\/www.youtube.com\/watch?v=kgBDQ3baESw But as you can see from my data an n.ahead t + 1 could be sometimes better at V6 and V7. Because the time series at V4 thinks we may get a rising value due to 8696 which is not the case (8094), resulting in bad predictions for V5 but better predictions for V6, since we have a two window prediction, the V7 prediction for V6 is better since we now incorporate the falling trend of 8094(V5, original) in the train set of (results in 7870 which is better for the falling trend of 7371. The quintessence is, that it is dependent of the datapoints. I always talked about incorporating, because in theory I still revalidate my model at every prediction with a window size of 1. Although sometimes a 2 window size reevalutaion would be better. But I'm talking about a big automation process, It would be easy to implement and to look at which is the better forecast for a varying of windows. Now lets have a look at your questions: My question is whether I should calculate the MAPE\/MAE\/MSE of each prediction iteration which leads to 2 different RMSE or whether I should store all predicted values and calculate only 1 MAPE\/MAE\/MSE for the whole time series? The question for me is more about do you want to forecast the 2 periods at once or step by step. Because this mainly influence the MAPE APE as you can see from my data. You can do both, and more: But it should not end up like in the video, thus: 1.) You can calculate the difference between observed and predicted value in every iteration just as in the matrix. Then you can calculate a MAPE over your whole prediction\/observations. This results in a forecast, refitted or reevaluated at a window size of 1!!!!! and a forecasting period of 2 periods, since you have two values. Although you might get 2 RMSE APE at the first and second observation we ignore the first, it is incorporated in the overall fit. 2.) You can also forecast 2 weeks ahead with a window size of 1 that would result in an average forecast window as you can do something like this: But that would also result in one RMSE or MAPE. As you can see from my data, I have RMSE at every point, but I'm interested in the overall fit, when using a specific window size (1) and forecasting period (1 or 2). So when you are giving one RMSE you always tell the window size, and n.ahead (or t + n ) at which the data was measured imho. If your n.ahead and window size is greater than 1 e.g. 2 Than I would tell both RMSE. Further I would like to know if the total MAPE\/MAE\/MSE in this case would be equal to the average MAPE\/MAE\/MSE of the individual prediction iterations? And from what we have seen, you can derive that the avg should normally not be the same. See Image: If there is still something I got wrong, please tell me, because I invested nearly over an hour to get all together ^^ Update 2 I'm answering the quesions of the OP as it seems I had not not thoroughly reasoned my answer (which is correct), thus Np Peter I'm here to clarify everything. 1. I have a forecast of 2 yes. Atm always. That means Im forecasting two weeks at once from my specific point of time. Looking at the matrix that means: I'm in week 2 and forecast week 2 (V2) and the next week (V3). I have only 2 because one method in the vars package in R only allows a forecast horizon of at least 2 when using dummy variables. If I could, I would use 1, thus forecast V2. However, as you can see from my matrix this is no measuring problem. Because although I forecast two weeks, I always ignore the latter. That leads to a forecast of 1. So yes Im using 2 but I would advise to use a forecasting week of 1, for simplicity so far. 2.) to understand the size of my data you have to remember window size is not the forecasting period of 2. Window size comes from cross validation in time series and it means, window size is the size of the sliding window, thus if my data consist of 152 e.g. I would start at 100 weeks training data and forecast week nr. 101 and 102 (forecast 2), then with window size == 1 in the next iteration I have 101 training data with window size == 1, I forecast 102 and 103, and so on. Thus I got 2 APEs, but I only use one (However it would be possible to use both and I would get a MAPE of two weeks forecast 2). Although it seems I have overlapping windows, I only use the actual forecast of the curent week. 1. V1 to V7 are forecasted Units on Week 1 to 7, the rows also reflects weeks. The reds are rrors. Above the reds are real values of units not the forecasted units. normally I would have a diagonal line like in correlation analysis. That would by the way reflect window size = 1 (reenlarging training set by 1) and forecasting = 1. But as I have forecast = 2 I have a double sized diagonal line. Red value in line \"n.ahead = t\" is the error of the forecasted Unit value in the actual week. (Week 2 forecast 10884-8709 Actual value for week2) = -2175. But If I'm in week 1 (V1) and want not only to forecast week1 the line with \"n.ahead = t\" but also n.ahead t + 1 thus week 2, I would look at the error of -2016 = 8709-10725. It means I can forecast the week of 2 when I'm still in Week 1. The first one with -2175 was I already forecasted week one, now i want to forecast week 2, because I'm actually at week2. SO the answer to your question is just a pure YES! 1. There are differences in calculations as you can see below in calculating MAPE and APE. I would say in the end it is different but it wont matter that much as you get always different results as you can see in the last image. Or to be concrete when you do individual forecasts APE you would end up with 24.98%, 2.34 and 4.71, these are you APE at every prediction the Mape is near 11. Due to your rolling window and forecast your APE can change dramatically between every week. But in terms of MAPEs it wouldn matter so much. As we average all of it. But it is still never the same value. 2. You are right as I would have normally taken 1, as it is the best short term.But I use 2 because its a technical limitation of a package i circumvent that only with the matrix, thus that may be the main root of confusin i induced when talking about n forecast of 2. But as you already know i can refit my train data every week but I could forecast 2 or 3, 4 weeks in advance if I want. If this is useful or not depends on the data and the information. as you can see a data point at time point t may has good information for time point t + 1 but not for t + 2 or vice versa. So you have to be really aware of what you are doing. 3. APE(t): APE(t+1)[ MAPE (t) You find all the values in the matrix. 1. The rolling window mentionend in 1) and 2) has to be calculated in a loop. R wont enlarge your training set. But the RMSE is in the forcast library by rob hyndman namely this function: forecast::accuracy( as.ts(train.set), test.set ) \u2022 How do you judge$R^2\\$ quality in your work?\n\u2013\u00a0Dave\nMar 12, 2021 at 19:45\n\u2022 The term quality is a little bit wide-opened ... but,.we are not judging a model alone by R\u00b2, we first judge by MAPE\/MSE, which represents our loss, as R\u00b2 increases as a side effect, although there may be solutions with better R\u00b2 but higher loss, MASE is then accomplished on the final model output to be sure, the model is better than a naive forecast, (some time series may drop to zero). Afterwards we check kpis that are derived from business, e.g., the GA offers a vast amount of solutions, we try to narrow down the space to these solutions where media vars show a specific return on investment Mar 13, 2021 at 11:57\n\u2022 Thanks Patrick for your answer. Basically my question is not only related to RMSE but a general one regarding forecasts. Let's say I calculate the MAPE\/MAE\/MSE or whatever (my code can be adjusted to that). Mar 15, 2021 at 9:19\n\u2022 Usually when having time series you have different prediction iterations when you want to do a short-term forecast. Now I want to calculate the MAPE\/MAE\/MSE. My question is whether I should calculate the MAPE\/MAE\/MSE of each prediction iteration which leads to 2 different RMSE or whether I should store all predicted values and calculate only 1 MAPE\/MAE\/MSE for the whole time series? Further I would like to know if the total MAPE\/MAE\/MSE in this case would be equal to the average MAPE\/MAE\/MSE of the individual prediction iterations? Mar 15, 2021 at 9:20\n\u2022 I appreciate it. If you still have questions feel free to contact me over chat. Mar 16, 2021 at 11:46\n\nAveraging errors over different forecast horizons makes no sense in my opinion.\n\nSuppose, you produced a set of dynamic forecasts $$\\hat y(t+h|I_t)$$ where $$t$$ is the last actual observation used for a forecast, $$I_t$$ - information set and $$h$$ is the forecast horizon. In this case, especially for ARIMA type of the model, the error variance depends on the time horizon $$Var[\\hat y(t+h_1|I_t)-y(t+h_1)]\\ne Var[\\hat y(t+h_2|I_t)-y(t+h_2)]$$\n\nThus, I think you need to study and present MSE as a curve over the forecast horizons $$h$$: $$MSE(h)=1\/T\\sum_t[\\hat y(t+h|I_t)-y(t+h)]^2$$ You may be better at forecasting long horizons rather than short ones, for instance.\n\nThe procedure can be visualized as a table with rows representing $$t$$ - the last actual used for a forecast, and columns representing $$t+h$$ - the time period for which forecast is produced. In this case the forecast horizons $$h$$ are diagonals of the upper triagonal matrix. You average along these diagonals. You'll have as many MSE(h) as you have diagonals.\n\n\u2022 Thanks Aksakal for your answer. Would you advice me to calculate the error when having a sliding window of the whole time series and not build the average over all prediction iterations? So I should just seperately store the predictions of every iteration [t1,..., t10], [t11,...,t20] and then compare them to the real values at the end of the whole prediction to evaluate it? This is only possible when having non-overlapping prediction horizons. So if the answer is yes, I would wonder what to do with overlapping prediction horizons. Here I would have 10 different forecasted values e.g. for t10. Mar 15, 2021 at 17:56","date":"2022-05-16 21:28:30","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 1, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 10, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.6996235847473145, \"perplexity\": 1020.4060101004554}, \"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-20220516204516-20220516234516-00676.warc.gz\"}"} | null | null |
NYPD officer convicted of sexually abusing young girl in Brooklyn
NEW YORK (WABC) -- An NYPD officer has been convicted for the repeated abuse of a young girl in Brooklyn for nearly three years, starting when she was 10, authorities said Wednesday.
"This is a disturbing case where a young child was sexually abused and repeatedly violated by a family friend-now a police officer-whom she should have been able to trust," said Brooklyn District Attorney Ken Thompson. "The victim had the courage to speak out against the abuser and we will now see that he is punished."
Jacob Sabbagh, 34, of Midwood, Brooklyn, was convicted of second-degree course of sexual conduct against a child.
He faces up to seven years in prison when sentenced in September.
Prosecutors say Sabbagh, who joined the police force in 2009, was a family friend of the victim and would sleep over at her Flatbush home approximately every month.
The investigation revealed that on each of these visits, between June 2005 and March 2008, he repeatedly groped and fondled the victim and on occasion forced her to reciprocate.
The abuse began when the victim was 10 and ended when she turned 13, said Thompson.
The victim and her family moved out of the country and, when she was 16, he told her mother about the past abuse.
The case was reported to authorities in the country where they lived and later referred to the District Attorney's Victim Services Unit.
In controlled telephone conversations between the victim and the defendant, prosecutors said Sabbagh was heard admitting to the sexual abuse and apologizing for his actions, according to evidence presented at trial.
flatbushnypdsex abusechild sex assault | {
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Q: Instance Member Cannot Be Used on Type when pushing UIViewController I am programmatically setting up ViewControllers (no storyboard).
I want to pass data to the next VC, and while I know how to do that with a segue and a storyboard, I can't figure out how to do it purely programmatically.
I get the error "Instance Member Cannot Be Used on Type..."
// Create Next View Controller Variable
let nextViewController = CarbonCalculatorResultsViewController()
// Pass data to next view controller. There is already a variable in that file: var userInformation: UserInformation?
CarbonCalculatorResultsViewController.userInformation = userInformation
// Push next View Controller
self.navigationController?.pushViewController(nextViewController, animated: true)
Do I need to instantiate the next VC before I can pass data? That's what this answer seems to talk about yet I don't have a Storyboard. Thanks!
A: Step 1: Setup your destination class
In CarbonCalculatorResultsViewController class, declare a var to receive data like so:
class CarbonCalculatorResultsViewController: UIViewController {
var foo: String? {
didSet {
// What you'd like to do with the data received
print(foo ?? "")
}
}
ovevride func viewDidLoad() {
//
}
}
Step 2: Prepare data in your source class
let nextViewController = CarbonCalculatorResultsViewController()
// You have access of the variable in CarbonCalculatorResultsViewController
nextViewController.foo = <data_you_want_to_pass>
// Push next View Controller
self.navigationController?.pushViewController(nextViewController, animated: true)
Then, every time the CarbonCalculatorResultsViewController comes alive, the didSet{} of foo would be called.
A: You should pass the variable on the Object, not on the Class
Replace: CarbonCalculatorResultsViewController.userInformation = userInformation
With:
nextViewController.userInformation = userInformation
Note:
CarbonCalculatorResultsViewController is the Class.
nextViewController is the Object.
Your full code should look like this:
// Create Next View Controller Variable
let nextViewController = CarbonCalculatorResultsViewController()
// Pass data to next view controller. There is already a variable in that file: var userInformation: UserInformation?
nextViewController.userInformation = userInformation
// Push next View Controller
self.navigationController?.pushViewController(nextViewController, animated: true)
A: The current sample code (above) is setting a value to a static variable (owned by the CarbonCalculatorResultsViewController.Type.
What I believe you want to implement is the following instead:
// Create Next View Controller Variable
let nextViewController = CarbonCalculatorResultsViewController()
// Pass data to next view controller. There is already a variable in that file: var userInformation: UserInformation?
nextViewController.userInformation = userInformation
// Push next View Controller
self.navigationController?.pushViewController(nextViewController, animated: true)
This sample code is setting a value to the instance variable userInformation on the type nextViewController.
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/ StarWars
Rogue One Prequel Series
Star Wars: Lucasfilm President Admits She Consults Clone Wars Creator Dave Filoni About All Projects
By Patrick Cavanaugh - November 7, 2019 04:03 pm EST
While his name might not be quite as synonymous with Star Wars as George Lucas is, Star Wars: The Clone Wars and Star Wars Rebels creator Dave Filoni has become one of the key figures in all storytelling at Lucasfilm, with president Kathleen Kennedy confirming that she often seeks his input when moving forward on a project. When speaking with Vanity Fair, the exec didn't specify just how involved in each project Filoni is, but she made it quite clear that, when shaping the future of the Star Wars saga, Filoni and his years of experience in the franchise provides invaluable input.
"There isn't a thing that we do in the storytelling space that I don't check with Dave," Kennedy confessed. "What I find about Dave is you don't just sit down and have a discussion about plot or review characters inside the Star Wars world. You end up having meaningful, thoughtful discussions about what it is we're trying to say inside the storytelling. He has a lot of empathy."
Filoni first entered the galaxy far, far away with the animated Clone Wars feature film, which paved the way for the TV series, on which he served as the supervising director. At the time, there were no plans for a new live-action Star Wars film, with that animated series offering some of the most compelling storytelling the saga had seen in any medium.
Despite that series earning a passionate following from fans, it was unceremoniously cancelled without being given a proper finale. Filoni then developed the animated Star Wars Rebels, which earned four seasons, before the swell of fan support led to the confirmation that a final season of The Clone Wars would be coming to Disney+.
Filoni has begun to expand his resume in the saga, as he directed an episode of the first live-action TV series, Star Wars: The Mandalorian. The director is also reportedly involved with directing Season Two of the series, though it's unclear to what degree he will be developing the new episodes.
The director and Lucas worked closely together on The Clone Wars, which began years before the studio was sold to Disney and Lucas exited the company. When asked by Vanity Fair what advice he offered to Filoni about the series, Lucas recalled, "To stay open-minded and realize that there's still a lot to learn. There's always something to learn."
Star Wars: The Mandalorian will premiere on Disney+ on November 12th and Star Wars: The Clone Wars returns sometime next year.
What do you think of Kennedy's remarks? Let us know in the comments below or hit up @TheWolfman on Twitter to talk all things Star Wars and horror!
Colin Trevorrow's Star Wars: Episode IX Script Included Kylo Ren vs Darth Vader Duel
Star Wars: The Rise of Skywalker Originally Had Luke Skywalker in a Bigger Role
Rey's Lightsaber From Star Wars: The Rise of Skywalker Is More Familiar Than You Thought
Star Wars: George Lucas Holds Baby Yoda in New Behind-the-Scenes Photo
Taika Waititi's Star Wars Movie Could Put Akira and Thor: Love and Thunder in Jeopardy
Star Wars: The Rise of Skywalker's Rey and Kylo Ren Duel Almost Looked Like Empire Strikes Back
Taika Waititi Being Recruited for Star Wars Movie | {
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layout: post
date: '2016-05-19'
title: "Alfred Sung by Dessy D650 Strapless Short Bridesmaid Dress"
category: Alfred Sung
tags: [Alfred Sung]
---
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Кристофер Джозеф Санде (; род. 10 февраля 1964) — кенийский боксёр, представитель средней весовой категории. Выступал за сборную Кении по боксу во второй половине 1980-х годов, бронзовый призёр летних Олимпийских игр в Сеуле. В период 1989—2001 годов боксировал на профессиональном уровне.
Биография
Крис Санде родился 10 февраля 1964 года.
Любительская карьера
Наибольшего успеха как боксёр добился в сезоне 1988 года, когда вошёл в основной состав кенийской национальной сборной и благодаря череде удачных выступлений удостоился права защищать честь страны на летних Олимпийских играх в Сеуле. В категории до 75 кг благополучно прошёл первых троих соперников по турнирной сетке (в том числе и Франко Ваньяму), тогда как в четвёртом поединке на стадии полуфиналов со счётом 0:5 потерпел поражение от восточногерманского боксёра Генри Маске и тем самым получил бронзовую олимпийскую медаль.
Профессиональная карьера
Вскоре по окончании сеульской Олимпиады Санде покинул расположение кенийской сборной и в октябре 1989 года успешно дебютировал на профессиональном уровне. Выступал преимущественно на территории США, большинство поединков выигрывал, но случались и проигрыши.
В апреле 1993 года отправился в Великобританию боксировать с валлийцем Никки Пайпером за титул интерконтинентального чемпиона во второй средней весовой категории по версии Всемирной боксёрской ассоциации (WBA), но проиграл техническим нокаутом во втором раунде.
Во второй половине 1990-х годов уже проигрывал практически все свои поединки, став заправским джорнименом. На его пути оказывались такие известные боксёры как Роберт Аллен, Крис Джонсон, Тим Литтлз, Луис Рамон Кампас, Мануэль Собраль, Алехандро Гарсия, однако ни у кого из них он выиграть не смог и в 2001 году после череды поражений завершил спортивную карьеру. В общей сложности провёл на профессиональном ринге 41 бой, из которых 19 выиграл (в том числе 7 досрочно), 19 проиграл, в двух случаях была зафиксирована ничья.
Примечания
Ссылки
Крис Санде — страница на сайте Международного олимпийского комитета
Боксёры Кении
Боксёры 1980-х годов
Боксёры 1990-х годов
Боксёры 2000-х годов
Боксёры средней весовой категории
Боксёры на летних Олимпийских играх 1988 года
Бронзовые призёры летних Олимпийских игр 1988 года | {
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Q: How do I output a variable from JavaScript to HTML enter image description here
1)Make up the design below
2)License type output in Bebas font-output via font-face
3)Display the selected license type and total amount
4)The format of the link in the Buy Now button is formed at your discretion
(How to execute an item highlighted in italics)
A: You need to have an HTML element (like a paragraph) where you want to display the amount.
If you give the element an id attribute with the value "my-paragraph", then your script can get the element like myParagraph = document.getElementById("my-paragraph").
Then you can change the content of the element like myParagraph.textContent = myAmount.
Begin your learning journey on MDN: https://developer.mozilla.org/en-US/docs/Web/API/Document_Object_Model/Introduction
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 3,338 |
Category Archives: Press
PRESS RELEASE – 11.09.18
By hambacherforst 09/11/2018 09/11/2018 Actions, Backgrounds, Deforestation, General, Occupations, Press, Repression, Standard
ON THE POSSIBILITY OF "SECURITY STRIPS", AND AN ALLEGED MOLOTOV ATTACK
The police forces of NRW, it seems, are using every excuse they can, in order to legitimize their destructive escalation in the 12.000 year old Hambacher Forst.
Information has reached us, that the police is considering cutting of so-called "security strips" (clearing meter-wide strips of underwood) on both sides of the L276, a.k.a. the Secu-Road, despite the formerly announced cutting-stop lasting until October 14th.
The background for their decision to cut hundreds of square-meters is an alleged attack on RWE cutting equipment, with rocks and a "molotov-cocktail", early morning September 10th.
The evidence on the alleged "molotov"-attack is however disputable.
A journalist which was present at the so-called "crime scene" shortly after explains:
"On the ground was some kind of liquid, but there were no glass-shards, no bottles, or any other containers to be seen. Usually when the police is taking pictures of a crime-scene, they don't remove evidence, but here was nothing to be found."
The extravagant reporting of the alleged attack, which led only to superficial damages on the RWE equipment, and no one wounded, serves only as a distraction in the debate, from who are the biggest perpetrators here: energy-giant RWE and the state of NRW, led by interior minister Herbert Reul (CDU).
In the row of violent attacks in and around the Hambach Forest occupation, it is clear which party ways the heaviest. Until now the police presence have led to several attacks on activists, supporters, as well as media workers, including the driver of a mobile kitchen being threatened with a gun, and a peaceful activist being brought to the hospital with a broken arm.
Responsible for this violence is the interior minister of the state of NRW, Herbert Reul (CDU), who uses the weapons of the state, to protect the interests of energy-giant RWE, and the coal lobby, while attacking the civil rights of free protest, free assembly and free movement.
Any and all attacks on RWE, and their state-sponsored protectors, is therefore also written in the context of self-defense, against RWEs destruction of nature, living-spaces, as well as the global climate, and against the brutal violence of the police, on civil protest and the whole climate justice movement.
Although not all parts of the movement for the preservation of the forest, and the immediate phase-out of coal agree to the militancy used by some autonomous activists, it is clear that the tactical diversity within the movement has strong value, and is not only legitimate, but necessary in the fight to protect the Hambach Forest.
Press release – 09/06/2018
By hambacherforst 09/07/2018 09/08/2018 Dates, Deforestation, Occupations, Press, Repression
Evictions going on, Day X was proclaimed
Today, the state police of NRW continued the destructive preparation(destroying of soil structures and barricades) of the forest for the upcoming evictions. The police stormed the forest at 8 am in order to cut walking ropes and bridges between trees and to isolate tree houses this way.
In the subsequent attempt to destroy the stilt construction "Simona", a hazel tree was felled by RWE workers in the middle of "Oaktown".
In reaction to the felling of the first tree, the Action Alliance proclaimed "Day X", the crossing of the so-called "red line".
Shortly after the proclamation of Day X, which was spread all over Germany and even the world, police left the forest. Although the soil structures all were destroyed, the activists now had the opportunity to move freely again and rebuild structures.
The proclamation of Day X marked the beginning of a new phase in the struggle for the 12,000 year old Hambach Forest. Hundreds of activists will join this struggle tomorrow, not least through the support of "Aktion Unterholz", which promised to launch a weekend of resistance to support the Hambach Forest in the form of mass actions of civil disobedience. They will break police lines and block RWE and police equipment.
Despite the high likelihood that the police will be present again tomorrow, there is a chance they will restrain themselves. In this case Aktion Unterholz will support the fight in other forms, by helping to rebuild barricades and soil structures.
At 3:00 pm, the various participating organizations of the Action Alliance held a press conference to communicate the situation to regional, national, as well as international media.
They criticize the violent action of the police forces, as well as the protection of the interests of the coal and capital giant RWE. They condemn the Interior Minister of NRW, Herbert Reul (CDU) for ordering these actions.
Furthermore Ende Gelände presented their new action plan for late October, which includes lock-ons this year, reflecting their response to the increasingly extreme situation in the struggle for the Hambach Forest.
While the 6 activists who were arrested on September 5 are back in freedom, the first trial day for the case "Kim Neuland" started in the Düren district court. The case concerns the arrest and 17-day detention of activist Maya (known as "Kim Neuland") on 17 December 2016, in connection with the resistance against the cutting of the Hambach Forest. The trial will continue on September 14th.
The repression against the climate-justice activists is going on, who are fighting for the preservation of the forest, and against the capitalist extraction of dirty lignite. But as long as the felling of the forest is not stopped forever and the fossil-fueled capitalist exploitation of the planet continues, the resistance will continues as well!
Important Information (Phone numbers etc.)
By hambacherforst 09/07/2018 11/03/2018 Dates, Forest Walks, Mobilization, Press, Repression
The EA (legal team) is available 24/7, +49177 1897053. Get in touch if you stick in a check or in custody, observe police actions or need legal advice. And always let us know when you're out again!
There are some persons in prison at the moment. More info.
Press conference, 09/06/2018
By hambacherforst 09/06/2018 10/04/2018 Dates, Mobilization, Network, Press, Repression
Invitation to the common press conference, to the beginning of the tree felling, today at 3 pm
Kerpen-Buir, September 06th, 2018. This morning RWE started felling trees in the Hambach Forest. In the early hours of the morning a large crowd of police entered the forest, at half past ten the first tree was felled. Thus, the RWE Group and the police, commanded by the NRW state government, create irrevocable facts even before the official cutting period. With the so called Day X, a nationwide mass mobilization begins today. Thousands of people will engage for the conservation of the forest during the days to come with demonstrations, sitting blocks and forest walks.
The local initiative Buirer für Buir, activists of the occupation in the Hambach Forest, Ende Gelände and the Aktion Unterholz invite today to a joint press conference. It will take place at 3:00 pm in the Protestant Gemeindehaus (parish hall), Bahnstraße 42 in 50170 Kerpen-Buir. Then we drive together to the Hambach Forest, where we are available for interviews.
Today's clearing work again sharpen the conflict over the Hambach Forest. The Federal Environment Ministry, the unions of police (GdP) and services (Ver.di), local residents and environmentalists demanded in recent weeks a coal moratorium and a deforesting stop. The Berlin coal commission threatens to fail, several members announced in the event of cuttings their exit from the panel. Nevertheless Mr. Reul, Minister of the Interior, ordered on behalf of the state government a large-scale police campaign, in order to enforce tree felling for RWE.
On Sunday, September 9, the initiative Buirer for Buir is calling for a demonstration in connection with a walk in the forest.
The occupiers of the forest are preparing to prevent the evacuation of over 60 tree houses during several weeks.
Ende Gelände announces blockades of coal infrastructure at the Hambach opencast mine on October 06 and October 25-29.
Aktion Unterholz will stop the destruction of the forest from tomorrow, Friday, 07-09-2018, with civil disobedience.
Representatives of the broad spectrum of protests comment on today's developments as follows: "The escalation of the conflict by RWE represents a deep cut in our quality of life for the village of Buir. In addition to the forest and our neighboring villages, we also lose our peace. The Hambach Forest, for us a symbol of a future-oriented society, now threatens to become a memorial to the destruction of our future. We feel left alone by those responsible in the federal and state governments – forgotten," says Andreas Büttgen of the local initiative Buirers for Buir.
"We defend the forest against RWE and Minister of the Interior Reul. In the trees we fight for climate justice and against capitalism. It will not be easy to get out of the forest," says activist Momo of the occupations in the Hambach Forest.
"With the evictions and the felling, RWE clearly transgresses a red line for the climate. It is a scandal that the state government protects this way corporate profits and not the climate. We call for the carbon phase out, as an immediate measure of global climate justice," said Karolina Drzewo, press officer of the alliance Ende Gelände.
"The destruction of the Hambach Forest is intolerable. We will therefore prevent the eviction and clearance of police and RWE from tomorrow on with actions of mass civil disobedience. By this form of action, we take our future into our own hands", says Anna Schönberg of Aktion Unterholz.
For background discussions, interviews or any questions, we are always happy to help.
Buirer für Buir: Andreas Büttgen, +49 173 5146141, info@nullbuirerfuerbuir.de
Waldbesetzung: Momo, press_hambachforest@nullriseup.net
Ende Gelände: Karolina Drzewo, +49 152 04560800, presse@nullende-gelaende.org, ende-gelaende.org
Aktion Unterholz: Anna Schönberg, +49 1775944678, presse_unterholz@nullriseup.net, aktion-unterholz.org
PRESS ANNOUNCEMENT 06.09.18
By hambacherforst 09/06/2018 09/07/2018 Actions, Blutbuchingen, Deforestation, Forest Walks, General, Oaktown, Occupations, Press, Solidarity
TREEHOUSES BEING EVICTED, TREES CUT, TODAY MARKS DAY X
The polices forces of NRW have currently begun the eviction of treehouses in the Hambach Forest occupation. They will now be violently removing activists and destroying the infrastructure of the occupation, in order to prepare for RWEs continued clearing of the 12.000 year old Hambach Forest.
In this way Herbert Reul (CDU), interior minister of the state of Northrhine-Westphalia, is actively attacking climate justice and democratic civil rights, in support of fossil-fuel giant RWE and the coal-lobby, clearly showing the corruption inherent in the capitalist system.
With the eviction of the first tree houses, the Hambach Forest occupation, alongside several action-alliances (including Ende Gelände, Aktion Unterholz, and Buirer Für Buir) are calling out: Day X is here, and a new phase of the struggle has begun.
In the next days, while RWE employees, with support from the hired soldiers of the German state, attempt to evict the occupation, hundreds of people will join the forest struggle, in order to protect the climate justice protest-camp, and there are many ways to take part. See the text "How do I become active?" for more information.
Leila, an activist from the occupied forest, says: "This is the time for action. RWE wants the protest silenced, the activists imprisoned, the forest cut, and the German state is spending millions of euros in order to support this wish. This capitalist corruption is NOT to be accepted, and we will continue struggling until the last tree is cut, and beyond!"
Tagged day x, deforestation, eviction, police, Solidarity, violence
Press announcement: 05.09.2018
By hambacherforst 09/05/2018 09/05/2018 Dates, General, Occupations, Press
Invitation to press conference on Day X
Since August 26th the Hambach Forest has been in the crosshair of a massive police action, and we have reason to believe that an eviction of the Hambach Forest occupation soon could be a reality.
While thousands right-extremists in Chemnitz are hunting refugees and migrants, the Hambach Forest will be subject to one of the largest police actions that the state of NRW has ever seen. The lines are being crossed, and the State and police are undermining civil rights and democracy, in the name of the capitalist system.
The Hambach Forest, a forest that has existed since the end of the last iceage and for the last 6 years has been occupied by activists, has by now become a symbol of resistance.
The protest points its finger at the people who put profit before sustainability and social justice.
We want to point out and dismantle the extraction of brown-coal (lignite), and its absurd destruction of landscapes, villages and the environment.
Day X: on the day where the first tree in the Hambach Forest falls, we, the action-alliances that fight for the preservation of the forest, will hold a press conference.
Leila, an activist from the occupation, says about the current situation:
"The state of NRW is right now directing its violence against the people protecting nature and fighting climate change. This is an attack on the whole movement for climate justice, and clearly shows the reactionary face of the capitalist state."
It is still unclear how the police will act on Day X, but we will keep you informed when the time comes.
Please direct press contact towards:
press_hambachforset@nullriseup.net
Further actors of the press conference:
Aktion Unterholz
Emil Freytag, 0177 5944676, presse_unterholz@nullriseup.net, aktion-unterholz.org
Ende Gelände!
Karolina Drzewo, 0152 04560800,presse@nullende-gelaende.org, ende-gelaende.org
Buirer für Buir,
Andreas Büttgen, 0173 5146141, andreas.buettgen@nullbuirerfuerbuir.de
Press release from the Hambach Forest – Meadow search and partial evacuation
By hambacherwald 08/28/2018 08/29/2018 Meadow Occupation, Photos, Press, Repression
Today, on 28/08/2018, there are partial evictions in the meadow camp of the Hambach Forest occupation as consequence of a search. The purpose of this search is to provide evidence of past actions and to seize items to carry out further actions.
As part of this mission, which began at 7:20 clock, there were over 30 sending-offs and also some arrests.
As a result of the five-day long police operation, the Hambach Forest has now been declared a danger area, ie. no people are being let in right now.
We strongly condemn the escalating actions of the police. The violence emanating from the police is in no way de-escalating, and the destruction of livelihood with the legitimation of confiscation of dangerous objects is unacceptable.
The Press Group
For further information: press_hambachforest@nullriseup.net
Barricade Eviction Press Statement
By hambachforest 06/30/2018 07/02/2018 Backgrounds, Press, Repression
On 28.06.2018, the Hambach Forest between Cologne and Aachen was cleared of barricades. Some people from the movement would like to make a press statement.
At 9:15 the police entered the forest. The contact officer had previously called and informed us that there will be a large-scale police operation to clear the access routes of RWE forces, under the protection of the police. Furthermore, it was announced that RWE forces will confiscate "dangerous items" if necessary. What then followed went far beyond that. There were transgressions not only of interpersonal but also of legal limits, so that in our view the action must once again be regarded as a targeted attempt to provocate and divide the movement. Once again, facts are distorted to justify the incidents to the public.
The press release of the Aachen police says:
"In addition to RWE's barricade clearances, the forest floor was also inspected off the paths. Employees of the responsible authorities (Kolpingstadt Kerpen, Gemeinde Merzenich, Rhein-Erft- Kreis) and the police jointly assessed all objects found as bulky waste. These included plastic tarpaulins, pallets, wood, metal poles and old furniture. On behalf of the responsible regulatory authorities, RWE transported a total of 140 m³ of refuse in over 17 trucks today alone".
We want to make a correction here:
Tearing down sleeping and storage areas (tents, constructions, etc.), declaring them as bulky waste and then carting them out of the forest is a farce that is unacceptable. In the course of "barricade clearance" in search of "dangerous objects", structures were deliberately destroyed and subsequently presented as "garbage" in the press report of the Aachen police.
Another quote from the press release of the Aachen police:
"In the interests of transparency, an attempt was also made to enter into dialogue with the residents of the Meadow, but talks were rejected.
It must be made clear that the police have illegally gained access to the legal meadow camp. The police justified the breach of the law with the "execution of danger". In a situation where police hundreds gain access to people's private rooms and then demand communication, the behaviour of the meadow dwellers is more than legitimate.
Aachen Police:
"But it must be clear to everyone that the forest must be freely and above all safely accessible to everyone and that if disruptions and crimes occur, the officials will continue to intervene consistently!"
The forest is accessible to all civilians at all times, in contrast to the statement of the Aachen police, we are happy about conversations and discussions. The barricades are usually portrayed as dangerous but they primarily serve to protect the forest, to make clearing more difficult, and to protect against arbitrary violence by the police and the security company Mundt commissioned by RWE.
To the topic why we don't want the police in the forest, there was already a detailed article on our blog.
Another situation:
"A puppy dog ran to the contact policeman during the evacuation work, who assured the activists by telephone that they could pick him up without police control. Here with it came with the collection of the puppy to a break of this promise and the activist was held over 90min by the police with handcuffs. And was only allowed to leave again after providing personal details."
Note: It is consensus in the Hambacher Forest not to maintain contact with the Police. People doing so were asked to leave. The "institution" of contact policeman has been established after years of public critique of Forestry Ministry and Law enforcement for not engaging forest activists in discussion. It has been put in place only after events in which forest activists have been seriously injured, including by a moving vehicle by RWE security and consists of calling Hambacher Forest published numbers 15 minutes as the police is already on the way, before their appearance in the forest. Contact cops are not just "good cop" PR tools but also intelligence assets in charge of contacts and extracting information from anybody willing to talk to them. They are not legally obligated to honor any promises they make and often will say anything towards a strategic end in sight. Like for example doing everything possible that RWE continues to profit by wrecking our Planet. So much for public safety!
Rainbow Action vs RWE Agenda of Climate Chaos
By hambachforest 05/07/2018 05/07/2018 Art/Culture, Deforestation, Press, Repression
Celebrating 40 years of resistance to Lignite Mining in Rhineland the previous Sunday a "Rainbow Action" took place in the forest. Starting with hundreds of bicyclist converging from both Cologne and Aachen bringing back a cheerful, colorful and busy feeling breaking the Forest monotony of constantly anticipating and dreading large police actions and evictions. Anticipating or denying which has turned into such an art and sport in Hambi that perhaps bets should be taken with the winner who gets the date of next police action right getting as a reward a police baton or helmet from past actions that can be found at ZAD or at many infoshops. Those not guessing right or not knowing would and will be hit with hundreds more batons, sprayed in the face with pepper spray during those more unwelcome visits that could happen today, tomorrow or day after and which more and more resemble large scale military maneuvers and exercises.
If the majority had any actual and not just simbolic say in aproving these actions not just everyone in those photos but most of the German and World's population would choose to have those same resources invested in sustainable and ecological energy transition, re-training, education and not in Climate Chaos imposing repression of present day reality of RWE's Energie-Reich of North Rhine Westphalia and of German and Global North's Illusion of Coal Exits and energie transitions, while thermally nuking the Global South and destroying last remaining millenarian habitats left in all of Europe.
In the forest the continual insanity i f lignite extraction and dumping of over 150 millions of tons of carbon into the atmosphere from the mine destroying Hambi alone and the waves of brutality, arrogance and injustice that perpetuate it, all of it is also beginning to resemble less and less repression as we have known it and more and more feudal hunts in which the local peasants(the police) would form lines and funnel wild animals into awaiting feudal lords, in this case RWE, that would dispatch them with a spear or a crossbow, in our case locking-up and taking out of action forest and climate activists for months and months in prisons in ever greater numbers, stripped of any illusion of a due process while the whole habitat of Millenarian Hambacher Forest is being killed and erased: as if it were a hunting trophy in the Fossil Fuels Planet-wide safari keeping the whole Earth in the cross-hairs and not letting go of the trigger. Slaughtering last of the last while going on an on about either about Cease Fire or about how dangerous those animals really are.
These cheerfull and colorful photos are our own distraction from the bleak, gray planet wide cloud of coal dust, its microparticles and carbon, concussion grenades and CS gas and in the Global South of live bullets that is beginning to engulf us and the rest of the planet like the dome of coal driven smog and smoke that now covers most of Chinese cities. None of which would not be possible without the present campaign of Green Scare Propaganda (latest Bild tabloid article with Hambi on front Page "Eco Terrorism out of Control" in which local police chief is quoted as saying that Hambi eviction would take two months) that projects a darker image of ecological activism. This image being more more conducive to keeping climate activists and occupiers of the forest in prison for months without transparent and just(if such thing ever existed in capitalist legal machine) legal process, keeping the environmental groups and experts out of coal exit talks as RWE would like and will get if the climate emergency it is causing will continue to cause more political polarization, more fear based politics and more corruptions and shortsightedness of those in power. Propaganda and lies that attempt to erode this support seen in those fotos, divide one group from another and finally limit the discussion on why so little is done to address this global emergency when so many resources are used instead to attack and imprison those that point out and fight against these systemic problems.
If there can be "No Jobs on a Dead Planet" then also all the pension funds, gifts, investments and positions that RWE is offering right and left to influence the status-quo "decision"-making and which it uses to enlarge its Climate-Change-Denying-Idiot-Bubble and to continue to get a free ride and to maintain their planetary criminality and terrorism(giving actual criminals and freedom fighter world over a horrible rap)then all those perks and kick-downs that keep the capitalist machine oiled up and greased up are all pretty meaningless as well. Breathing money can not be any more successful than breathing CO2 especially in a "Culture" that connects one to the other and finances and justifies its own suffocation.
As this year started with arctic melting and ice-cover break-up in mid-winter and continues with extreme weather all-time temperature records aready in April and May it will continue with some of the largest police actions in (Post?)Modern Germany putting in place and reinforcing a global disaster that is so extreme, complete, irreversible and unprecedented that it will be especially the children and descendants of those who with wealth, privilege and arrogance are attempting to isolate themselves from this reality wh o will be wanting the answers the most.
The people in the Forest and many of those in those photos will not fall in the category of those that have done nothing or who helped to destroy all hope for stable and livable planet. It has been said that the best time to plant a tree is 10 years ago and the next best time is now. This must be another thing that Fighting Climate Change and trees have in common.
Join us then here and everywhere to resist this toxic, waves of scorching heatwaves fueled by fossil fools reality that few would be willing to accept if it wasn't forced upon the population, and upon those in the forest and those on the barricades with tanks with helicopters, with platoons of cops with dogs and horses and then followed with criminalization and unjust non-transparent prison sentences. Forced by hierarchical patriarchal power structures that manage to convince those they co-opted and sucked into their sphere of influence that they do not have any individual choice, which might be the case but that does not absolve any of those involved from responsibility for their actions. Responsibility that they will have to face in the imminent future of food insecurity, climate destabilization and ecological collapse. Responsibility that they all will have to face especially from those closest to them they are pulling with them into violent and precarious reality in which the weather becomes our enemy and specie after specie vanishes at the fastest rate in the history of this planet, the gretest extinction not the 6th(chronological designation)of Life ever. Hence once again the importace of Propaganda in redirecting the attention from, how State and power structures are imposing a reality of terror, huger and violence on the rest of the planet to focus on activists that are atempting to stop this globally suicidal machine.
Einstein said that if bees vanish(rather prophetic statement as colony collapse disorder managed to wipe out most of domesticated bees already) we are bound to follow. It should take no Einstein to guess what the faith and effect of "business as usuall" mentality on this Civilization, on humanity in general, and especially of this Planet as a whole will be and already is if more than half of all species are killed-off, climate turns each year more and more into an unstable nightmare and such vital resources as clean water and air become more and more scarce.
You want to imagine the future? Imagine a Coal Power Plant's Smoke Stack that perpetually collapses on each being and each habitat that exists and that will ever exist on this Planet, and that keeps collapsing and crashing down on them all forever……
Ok a little admission of guilt for all of those who did not spot it: the previous sentence is a total rip off of Orwell's quote. He said that if you want to imagine the future imagine a Jack-boot that steps on a human face forever. With both police chief of Duren: Winesbach(sp?) who quadruples at least every estimate of activists in the forests in his frequent especially tabloid interviews and not any less surreal Rhine Energie board of advisors member and police minister of North Rhine Westphalia Herbert Reul, both in a big hurry to change their sneakers and office shoes for army boots with the new draconian police laws that they are attempting to put in place and which they have already been testing on Hambi prisoners for awhile with minimal public outcry outside of anarchists communities. Will all of that this quote like that of Einstein's warning about the bees is turning to be another self-actualized prediction. So until at least the army boot and munitions factories get buried by the collapsing-coal-power-plant-chimneys of climate-chaos get ready for one bumpy ride. Get a guitar and/or gas mask, crypto up, form a reliable affinity group, don't forget to tell all un-solidaric assholes to fuck-off and no matter what have fun. Poetry, music and art have always been a crucial part in the arsenal of protest and resistance…
On a finite planet not just all good things but also the bad ones must come to an end and how and when they end is decided especially by those who have not buried themselves in the virtual media and industry manufactured "realities" and who for every connection to life and solidarity that is severed forcefully and intentionally form at least two in their place, two that become much stronger than ever before as their importance finally is realized.
For the Earth, For the Climate, For Hambi!!
A Radio Report from the USA about Hambi and the Rhenish Lignite Mining
By hambacherforst 01/12/2018 01/14/2018 Press
This e-mail was sent to us:
I'm just now getting around to sending out this link to the story about Germany's coal problem, about which we spoke last fall. The audio version, which aired nationwide on 300 radio stations in the US on the BBC co-production The World, is embedded here:
https://www.pri.org/stories/2017-11-15/germany-talks-good-game-climate-its-still-stuck-coal
Make sure to click on the audio player to hear what actually went on the air, as it is different from the text story (and includes your voices).
A text version was also reprinted on msn.com online and in several US newspapers. Here's one:
http://www.post-gazette.com/powersource/policy-powersource/2017/11/16/Analysis-Germany-talks-a-good-game-on-climate-but-it-s-still-stuck-on-coal/stories/201711160124
Thank you all very much for your time, help and expertise, without which none of this would have been possible. I hope our paths will cross again in the future.
Until then, best regards," | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 8,143 |
He was Crucified
Mark 15:25-39, Philippians 2:3-11, Isaiah 53:1-12
Go with me to a strange town where you have never been before. You are invited by a friend to go to a certain building of unusual shape. You enter into a dimly lit building, and immediately your attention is riveted by something at the front of the auditorium. There, hanging from one of the rafters is a hangman's noose. You stand there stunned. "What have I gotten myself into this time?" you say to yourself.
As you stand there, people begin to file past you and take a seat facing the platform where that horrible noose dangles. As some of these people pass you, you notice that some of them carry books in their hands that have the symbol of the hangman's noose imprinted. You notice that others are wearing fine gold chains around their necks from which dangles a miniature hangman's noose.
You turn to your friend and ask, "What is that thing at the front of the auditorium there for? Why are people wearing miniature gallows around their own necks? Your friend replies, "Oh, it is the symbol of our religion." Immediately thoughts go racing through your mind. Incredible! Who are these people? A bunch of sadists? They must be sick. I must have walked into a Gothic horror story.
I. The Cross in Christian Worship
And yet we have something just as horrible as a hangman's noose. We display a cross in our sanctuaries. Some of us wear a cross as an article of jewelry. Go into church after church across our city, nation and world and there you will see the cross on altars, walls, spires, furnishings, books and in stained glass windows. Some churches are even shaped in the sign of the cruciform. A million churches with countless crosses.
It is obvious to us that the cross is not on a par with a hangman's noose. In fact the very reverse is true. The cross has become a thing of beauty. Listen to our songs:
In the Cross of Christ I glory, Towering o'er the wrecks of time
All the light of sacred story gathers round its head sublime.
On a hill far away stood an old rugged cross, the emblem of suffering and shame
But I love that old cross, Where the dearest and best, For a world of lost sinners was slain
The hymnody of the church has continued down the centuries to consider the cross a thing of transcendent glory.
2. The Cross in Jesus' Day
But people have not always thought that way about crosses. Go back in time 2,000 years to a world where the cross reigned as one of the most horrible instruments of torture and death.
It was a most simple thing. Anyone could make one. Just two beams of wood, nailed one across the other. But when it was placed in the ground and a person was hung on it, it became a perpendicular couch of horror. Its diabolic design ensured the death of its victim, but it also ensured a slow agonizing death. It was a way to torture the victim slowly, but guaranteeing death at the end. A strong man could hang there for days, painfully dying with the eastern sun blazing down on him, thirst consuming him, and hour by hour his strength slipping from him until death comes.
But nails through hands and feet kill no one. They just hurt. Breaking the legs of a person does not kill either. But here is the genius of the cross. A person dies of eventual suffocation. When the hands and arms cannot stand the pain, then the legs push up to give the hands relief. But then the pain moves to the feet, and when that becomes unbearable the legs give way and then the hands bear the pain, and up and down the body moves, trying to find relief. But finally when the legs cannot push, the hands bear the load and then the person cannot catch his breath, and finally succumbs to hyperventilation and then suffocation. When the prisoner is strong this can go on for hours and days, and so to speed up the process the legs could broken and then death comes quickly.
Upon such crosses the Roman Empire suspended its victims. On such crosses, thieves, murderers, traitors, and other undesirables, were hung between earth and sky. Historians tell us that while Jesus was just a boy in Nazareth, there had been an insurrection against Rome. The legions came pouring into the Galilee area, and before they left, a forest of crosses were left behind, holding hundreds of victims.
The cross was such a horrible implement of death that it was usually reserved for the worst of men and the Hebrew Scriptures shouted, "Cursed is anyone that is hanged on a tree." The cross was one of the worst way that humanity had ever devised for the execution of its fellow man. It was worse than a hangman's noose or an executioner's sword.
3. The Transformation of the Cross
Then what caused the metamorphosis of a thing of horror to the revered symbol of our faith? Because one man, who was different than all others, one day hung upon such a cross.
On the hill of Golgotha, the hill shaped like a skull, the alchemy of God's grace changed that piece of wood and that geometric configuration forever. And now we sing, "In the cross of Christ I glory"
At times the church has spoken as though the cross itself effected the miraculous change in human fortunes. Some thought that the cross was a supernatural relic with a magic that could elicit miracles? Charlatans sold splinters of the cross to pious pilgrim who were hoping for magic or miraculous cures. These pieces of the cross were supposed to contain wonderful powers. But it was superstition. It had no basis in fact. The cross upon which Jesus died was simply a piece of wood that had been used many times before and would be used many more times to kill Rome's enemies.
Crosses cannot save people. They could not then and cannot now. They were instruments of death, not life. And had there not been a certain man, on a certain day, who had been nailed to one of those crosses, the cross would simply be a grim reminder of our barbaric ways. No one would try to understand it, for it would have no meaning.
But it all changed because Jesus of Nazareth, the Son of God, was nailed to it. For Jesus changed every thing that he touched.
He touched the leper, the crippled, the blind, the deaf and the speechless, and they were changed.
He touched the demoniacs, and their lives were changed.
He touched the dead and they came to life.
He touched sinful men and women, and they were never the same again.
He touched a boy's lunch and made it a banquet of God's best bread.
He touched a goblet of wine and a loaf of homemade bread, and it became a sacrament to the millions that came later.
He touched 12 men of rather common skills, and transformed them into the founding members of the Christian Church..
At his birth he touched the stable and it became a place of worship.
And in his dying he took hold of wood and nails that had been used to crucify people, and transformed the cross into a symbol of human redemption! Everything Jesus touched he changed. Everything he touched he transformed and redeemed.
All of that says something wonderful to me.
There is no event of my life, no matter how terrible, no matter how crippling, that he cannot transform.
There is no event of my days, no matter how humdrum, to which he cannot add new meaning.
There is no event so traumatic, that he cannot infuse with a new glory.
God specializes in things called impossible. God can take the dull lead of tragedy, sufferings, limitations, and set backs, and by his grace transform them into gold that shines with a new glory.
He can take a job that seems like a waste of life's best energies, that feels like an albatross around the neck, and can turn it into an adventure.
He can take the task of the homemaker with its endless succession of preparing meals, cleaning dirty floors, the washing of stained dishes, and the never ending pile of laundry, and can make it as valued as the work of any other calling.
The transformation of an ugly cross tells me that he can alter the significance of any of the circumstances of my life.
But, it also speaks to me of something even more significant than the transformation of my circumstances. It tells me that he can take my life and transform me. The poet has said,
He can take my life
Debauched with sinning
Ground in the dust of earth and common things
Can grant me by his grace a new beginning
Can cause my soul to soar on heaven's wings.
The cross has always declared to humanity, how much God loves us. It is a welcome word in a world without love.
But its second message is just as crucial. Love is not enough. To be loved as I am is not enough. I want to be different than I am. I need to have my life changed so I can contribute to the health of my world. The transformation of the cross, underscores that truth. Let God touch our lives, and who can predict the outcome! | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 2,922 |
Janet Mills begins selling her budget to both ends of Maine's political spectrum
by Michael Shepherd February 11, 2019 February 11, 2019
Gov. Janet Mills gives her inaugural address in January 2019.
Good morning from Augusta. Janet Mills transitioned in earnest from governor-elect to Maine governor with the Friday release of her $8 billion two-year budget proposal, which she will pitch to lawmakers in a budget address on Monday night at the State House.
It was a busy, but relatively painless January for the new Democratic governor as her administration worked to fulfill campaign promises it could control by expanding Medicaid and plotting their first steps to address the opioid crisis. That changed after the budget proposal.
Mills committed to no tax increases while promising increased spending and the result was a budget spending nearly every dollar expected to be available to Maine through mid-2021 while taking lots of half-measures toward campaign goals. Lots of people are finding issue with that.
The budget would constitute a large spending hike while falling short of what lots of progressives wanted to see. The most noticeable thing about Mills' budget was the topline figure that crested $8 billion, an 11 percent hike over the baseline of the current budget. It is reliant on projections saying Maine will take in $7.9 billion and would balance only after a transfer from the current fiscal year. It doesn't raise taxes or touch reserves.
The reaction from minority Republicans in the Legislature was predictable. They said it was imprudent to spend that much, leaving the budget committee in the difficult position of cutting the budget during months of negotiations that should result in a final budget by spring. The Legislature always rewrites budgets, so what we see now is only a framework.
Democrats will need Republican votes to pass a budget — which takes two-thirds votes from both legislative chambers — but they control the Legislature and hold the keys in Augusta this year. Their leaders were mostly muted in statements, with Senate President Troy Jackson, D-Allagash, saying the proposal "marks the start of a long but important process."
But progressive interests were critical. Among the half-measures in the budget was a $126 million K-12 education spending increase that falls short of meeting a never-met, voter-approved threshold of 55 percent of essential local costs, though Mills would met a campaign pledge of raising minimum teacher salaries from $30,000 to $40,000.
The leader of the liberal Maine Center for Economic Policy, which issued a budget framework with an income tax hike on higher earners, said it "locks in" the tax policy of former Gov. Paul LePage. A proposed state employee headcount increase of more than 100 positions may not be enough for the Maine State Employees Association, the union for state workers.
Mills previewed her budget address on Friday by noting that the budget would land in a moderate position. On Friday, Mills noted that Republicans would call her proposal an example of "government spending run amok" and liberals would say "state government must spend more, more, more." She disagreed, calling her framework "fiscally responsible and pragmatic."
What she predicted happened. LePage told WGAN on Monday that "the difference between Janet Mills and a drunken sailor is that a drunken sailor spends his own money." He went on to criticize her for relying on what he believes to be overly optimistic revenue projections for the next two years and for ignoring the budget plan he left for her, which he said included tax cuts.
Democrats will be quieter about their criticisms, letting their leaders on the budget committee try to tailor the budget to satisfy progressives. The governor will have a chance to expand on things at 7 p.m.
Lawmakers have a full Monday at the State House ahead of Mills' State of the Budget address this evening. Seven legislative committees will meet today to consider 29 referred bills.
Members of the Criminal Justice and Public Safety Committee at 10 a.m. will hold a work session to possibly vote on whether to recommend LD 94 for passage in the Legislature. The bill, which aims to prohibit teachers from distributing obscene learning material to students without their consent, received mixed reviews from teachers, parents and lawmakers last week.
Other bills before committee today include LD 33, An Act to Establish a Thanksgiving Youth Turkey Hunting Season; LD 179, An Act to Change the Name of Columbus Day to Indigenous Peoples Day, and LD 150, An Act to Improve Attendance at Public Elementary Schools. Find the full schedule here.
— Maine schools are again wrestling with what should be required of students before they can graduate from high school. Last year, the Legislature rolled back requirements that school districts use "proficiency-based" learning systems in which students must show they've mastered the state's various academic expectations in math, English, science, social studies and four other subject areas to graduate. That leaves the state with a mix of different graduation requirements that vary from district to district. Approximately 25 percent of superintendents said they expected their districts to stick to proficiency-based diplomas, and 26 percent said they expected their districts to adopt a hybrid proficiency- and credit-based diploma. About 11 percent of superintendents said they could not yet accurately predict changes to their diploma criteria.
— Lawyers for a woman accused of killing her 10-year-old daughter last year allege that her husband tortured her and the child. In court documents filed Friday, lawyers for Sharon Carrillo describe alleged acts of sexual and physical abuse committed by Julio Carrillo against their client and her daughter, Marissa Kennedy. Prosecutors have charged Sharon and Julio Carrillo with murdering the girl, who was Julio's stepdaughter. Sharon's attorneys are seeking a separate trial and to have information she gave to investigators suppressed, in part because they say Sharon was a victim, not an accomplice, in months of abuse that resulted in the child's death.
— The number of Democrats who want to be president in 2021 just keeps growing. Two U.S. senators officially launched campaigns this past weekend and other Democrats are hinting that they will soon follow suit. Massachusetts Sen. Elizabeth Warren kicked off her campaign Saturday, and Minnesota Sen. Amy Klobuchar defied blizzard conditions to announce Sunday that she's running for president. Meanwhile, Colorado Sen. Michael Bennet hinted that he would soon join the crowded Democratic presidential nomination derby.
— The federal government seems on the verge of another shutdown. The Associated Press reports that a special congressional committee impaneled to break an impasse between President Donald Trump and Democratic leaders over funding for a southern border wall hit a snag in talks designed to end the stalemate. Without a deal by Friday, funding to keep many government services in place will run out. Trump continues to let his staff handle negotiations as he heads to a campaign rally today in El Paso.
Joy sparkler
Before Marie Kondo started sparking joy by throwing away other people's stuff, there was Scott Thistle.
Scott now works for the Portland Press Herald, But before he moved to the high-rent district of State House press row, he shared an office with us when he worked for the Sun Journal.
Without windows, the place lacks ambiance. Add rows of institutional gray filing cabinets filled and topped with two decades of piled paperwork and detritus accumulated by a parade of frumpy, grumpy State House reporters and you can see why our office had become, at best, a grotto of grimness. Joy was not an option.
So on what was shaping up to be a slow news day during the summer of 2013, Scott decided it was time to clean out the office. Moving with the speed and urgency of the ski rescue expert that he is, Scott — with me as his lumper — filled two Dumpsters with old reports, studies, notebooks and mail sent to people who had not worked there in decades. We kept the TV with built-in VHS player, but hundreds of pounds of other stuff disappeared.
By Kondo's standards, he sparked an inferno of joy, converting a dust mite heaven and catacombs filled with dead legislation and unopened mail addressed to people who retired last century into usable space.
For that, we thank him. But I'm not letting him near my house.
I am an affirmed packrat. Facebook reminds me that my wardrobe hasn't changed since before there was Facebook. Our attic is filled with scorecards from 1960s Red Sox games, LPs, moldy photo albums, old newspapers, toddler clothes my kids wore 25 years ago and keepsakes like the football-shaped "Happy NFL Easter" card a neighbor gave me in the early 1980s.
I'm not getting rid of any of them. And joy is overrated. Here is your soundtrack. — Robert Long
Tagged: Amy Klobuchar, Donald Trump, elizabeth warren, Janet Mills, Julio Carrillo, Marissa Kennedy, Michael Bennet, Paul LePage, Scott Thistle, Sharon Kennedy, Troy Jackson
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"redpajama_set_name": "RedPajamaCommonCrawl"
} | 7,338 |
\section{Introduction}
Many compact extragalactic radio sources show variations in their radio
flux density as a function of time. At high frequencies ($\nu \gg
1$~GHz) this variability is usually interpreted as being
intrinsic to the source (e.g.\ Qian et al 1995, although
see Kedziora-Chudczer et al 1997).
Variability at lower frequencies (e.g.\ Hunstead 1972;
Ghosh \& Rao 1992) is normally attributed to
refractive interstellar scintillation, in which the
intensity variations are caused by distortions of the wavefront by
electron density gradients in an intervening screen of material
(Shapirovskaya 1978; Rickett et al 1984). There is evidence that
the parameters of such variability depend on the
Galactic latitude of the source (Spangler et
al. 1989; Ghosh \& Rao 1992), suggesting that the material causing the
scintillation is in our own Galaxy.
In some sources, both intrinsic variability and scintillation may be
occurring at the same time (e.g. Mitchell et al 1994). Such sources
show large but uncorrelated variations at high and low
frequencies. At frequencies $\nu \approx 1$~GHz, one might expect both
effects to occur; however, variability in this region of the spectrum
is largely unexplored. The Molonglo Observatory Synthesis Telescope
(MOST; Mills 1981; Robertson 1991) operates at a frequency of
843~MHz, and is thus well placed to study this regime. For calibration
purposes, the MOST monitors the flux density of $\sim$10 compact
extragalactic sources every day. Thus the full record of MOST
calibrations, running from 1984 until the commencement of the Wide
Field Project in 1996 (Large et al 1994),
forms an ideal database with which to study
variability in this intermediate frequency range.
A preliminary analysis of three MOST calibrators was made by
Campbell-Wilson \& Hunstead (1994), hereafter Paper I. It was shown
that flux density measurements with a relative accuracy of 2\% could be
extracted from the database. Over the period from 1990.1 to 1993.7, the
source MRC~B0409--752 was shown to be stable, while MRC~B0537--441 and
MRC~B1921--293 were found to be highly variable. In this paper we now
report on all 55 calibrators used by the MOST, over a thirteen year period.
In Section~\ref{obs} we explain how we process the calibrator measurements
in order to produce light curves for each source, and then determine
whether a source is variable or not. In Section~\ref{results} we present
light curves for all 55 sources, plus structure functions for those
sources found to be variable. In Section~\ref{discuss} we discuss
some individual sources in our sample, and consider whether any of
the observed properties correlate with Galactic latitude.
\section{Observations and Data Analysis}
\label{obs}
\subsection{SCAN Measurements}
\label{obs_scan}
The MOST is an east-west synthesis telescope, consisting
of two cylindrical paraboloids of dimensions 778~m~$\times$~12~m.
Radio waves are received by a line feed system of 7744 circular
dipoles.
The telescope is steered by mechanical rotation of
the cylindrical paraboloids about their long axis,
and by phasing the feed elements along the arms.
In a single 12-hour synthesis, the MOST can produce
an image at a spatial resolution of $43''\times43''{\rm cosec}(|\delta|)$
and at a sensitivity of $\sim$1~mJy~beam$^{-1}$ (where
1 jansky~[Jy]~$=10^{-26}$~W~m$^{-2}$~Hz$^{-1}$).
Before and after each 12-hour synthesis, the MOST typically observes
$\sim5$ calibration sources in fan-beam
``SCAN'' mode in order to determine the gain
and pointing corrections for the telescope. These sources are
chosen from a list of 55 calibrators, 45 of which
were chosen from the Molonglo Reference Catalogue (MRC)
at 408~MHz (Large et al 1981), using as selection criteria that they have
declination $\delta < -30^{\circ}$, Galactic latitude
$|b| > 10^{\circ}$, angular sizes $<10''$ and flux densities
$S_{\rm{408\,MHz}}>4$~Jy
and $S_{\rm{843\,MHz}}>2.5$~Jy; further discussion is
given by Hunstead (1991). This list was later
supplemented by seven flat-spectrum ($S_{\rm{408\,MHz}}<4$~Jy) sources
from the work of Tzioumis (1987), plus three compact sources
for which $\delta > -30^{\circ}$. The full list of calibrators is
given in Paper I.
For each SCAN observation the calibrator source is tracked for two
minutes, after which the mean antenna response is compared with the
theoretical fan-beam response to a point source.
From 1994 to 1996, over 58\,000 such measurements were made.
In each case, parameters such as the goodness-of-fit of the response
and the pointing offset from the calibrator position are
recorded, along which an amplitude which is the product of the
instantaneous values of the source flux density, the intrinsic
telescope gain and local sensitivity factors. The main
factors are strong but well-determined functions of
meridian distance\footnote{Meridian
distance, MD, is related to hour-angle, $H$, by $\sin {\rm MD} \approx
\cos \delta \sin H$ --- see Robertson (1991).} (MD) and of ambient
temperature (which ranges from $-$10\ifmmode ^{\circ}\else $^{\circ}$\fi C to +40\ifmmode ^{\circ}\else $^{\circ}$\fi C during the
year); the variation of sensitivity with MD is shown in
Figure~\ref{fig_md}. After applying corrections for these two
factors, the telescope gain for each SCAN is derived by
comparing the corrected amplitude with the tabulated flux
density of the corresponding source (see Table~1 of Paper~I).
The residual scatter in the gain determined from steady sources
(defined in Section~\ref{obs_analysis}) is typically 2\% RMS; this is
the fundamental limit to the uncertainty of measurements made
using the SCAN database.
\subsection{Selection Criteria}
\label{obs_select}
Various selection criteria are applied to the SCAN database before
accepting measurements for further analysis:
\begin{itemize}
\item{} The uncertainties in the
MD gain curve increase towards
large MD, and observations made outside the MD range
$\pm50$\ifmmode ^{\circ}\else $^{\circ}$\fi\ are excluded;
\item{} Observations made during routine performance testing (characterised by
a large number of successive SCANs of the same source) are discounted,
except where the standard deviation in gain was less than 5\%. In such
cases the group is treated as a single measurement with a gain equal to the
average of the group;
\item{} a poor fit to the antenna response can
often indicate a confusing source or a telescope malfunction, and
such data are excluded;
\item{} extreme values of the relative gain (below 0.5 or above 1.5) are
assumed to be discrepant and are discarded.
\end{itemize}
Because calibration observations are made just before and after each
synthesis, the database is typically clustered into SCANs
closely spaced in time. We define a ``block'' as a group of at least
three valid observations made within the space of an hour. We initially
exclude observations of 15 of the 55 calibrators (see Table~1 of Paper~I),
because of: (i) a flat spectrum ($\alpha > -0.5$, $S \propto
\nu^{\alpha}$), (ii) suspected variability or (iii)
the presence of a confusing
source. By averaging the gains determined from each SCAN within a
block, a representative gain for the telescope at that particular epoch
can be determined. This is then applied to each individual
observation within the block to obtain a measurement of
flux density for that source.
Some of the resultant light curves have thousands of data points,
generally sampled at highly irregular intervals. Some light curves have
significant scatter; it is not clear whether this scatter is due to
unrecognised systematic errors in our flux density determination,
to true variability on time-scales shorter than the
typical sampling interval, or to the presence of confusing sources
in the field. In any case, we chose to bin each light-curve at
30 day intervals; the mean of all flux densities within a given bin
becomes a single point on a smoothed light curve, and the standard
deviation of the measurements in that bin becomes the error bar
associated with this measurement\footnote{In cases where there
is only one measurement in a particular 30 day interval, the error is
nominally assigned to be 5\% of the measured flux density.}. While
binning the data filters out any genuine variability on time-scales
less than a month, the irregular sampling intervals of the observations
and the inherent uncertainty in a single SCAN's flux density make the
MOST database less than ideal for studying such short-term behaviour.
\subsection{Analysis of Variability}
\label{obs_analysis}
In order to quantify
which sources are variable and which are steady,
we calculate the $\chi^2$ probability
that the flux has remained constant for a given source
(e.g.\ Kesteven et al 1976). We first calculate the quantity
\begin{equation}
x^2 = \sum_{i=1}^{n} (S_i - \tilde{S})^2/\sigma_i^2
\end{equation}
where $\tilde{S}$ is the weighted mean, given by
\begin{equation}
\tilde{S} = \frac{\sum_{i=1}^{n} (S_i/\sigma_i^2)}{\sum_{i=1}^{n}
(1/\sigma_i^2)},
\end{equation}
$S_i$ is the $i$th measurement of the flux density for a particular
source, $\sigma_i^2$ is the variance associated with each
30-day estimate of $S_i$, and $n$ is
the number of binned data points for that source.
For normally-distributed random errors, we expect $x^2$ to be
distributed as $\chi^2$ with $n-1$ degrees of freedom. For each
source, we can then calculate the probability, $P$, of exceeding $x^2$ by chance
for a random distribution.
A high value of $P$ indicates that a source has a steady flux density
over the available time period; we classify a source as
steady (S) if $P>0.01$, and undetermined (U) if $0.001 < P < 0.01$.
However the $\chi^2$ test cannot distinguish between
sources which are genuinely variable
and those which simply have a large scatter in their light curve; both
light curves result in a low value of $P$. We distinguish between
these possibilities by computing the structure function (e.g. Hughes et al
1992; Kaspi \& Stinebring 1992) for each
source for which $P<0.001$.
The mean is subtracted from the binned time series $S_t$,
and these data are then normalised
by dividing by the standard deviation. This yields a
new time series $F_t$, from which the structure function
\begin{equation}
\Sigma_\tau = \langle [ F_{t + \tau} - F_t]^2 \rangle
\end{equation}
can be calculated,
where $\tau$ is a parameter known as the lag.
If a light curve contains scatter but no true variability, then
the structure function will have the value
$\Sigma_\tau \approx 2$ for all values of $\tau$. But when a source is truly varying,
we expect the resulting structure function to consist of three regimes:
\begin{itemize}
\item{Noise regime:}
at small lags, $\Sigma_\tau$ is more or less constant.
\item{Structure regime:}
as $\tau$ increases, $\Sigma_\tau$
increases linearly (on a log-log plot).
\item{Saturation regime:}
at high lags, the structure function
turns over and oscillates around $\Sigma_\tau = 2$ (for our
normalisation). If there is a second, longer, time-scale
in the data, the structure function can enter another
linear regime at longer lags before again saturating.
\end{itemize}
If a source has $P<0.001$ but shows no clear structure in its structure
function, we classify it as undetermined (U). Only sources which
have both $P < 0.001$ and show structure are classified as variable (V). In
these cases, the structure function can also be used to obtain a
characteristic time scale, $\tau_V$, for variability; we define
$\tau_V$ to be equal to twice the lag at which the structure function
saturates. We expect a structure function to be sensitive only
to time scales longer than about 100 days (i.e. a few multiples of
the sampling interval of the binned data). Furthermore, caution
should be applied when interpreting structure at
large values of $\tau$, as only a few points make a contribution
to $\Sigma_\tau$ at these long lags (e.g. Hughes et al 1992).
\section{Results}
\label{results}
Approximately 28\,000 SCANs meet the selection criteria described in
Section~\ref{obs_select}, and around 22\,000 of these fall within valid
blocks. The resulting light curves for the 55 MOST calibrators are given
in Figure~\ref{fig_sources_1}.\footnote{The corresponding data tables
are available at http://www.physics.usyd.edu.au/astrop/scan/ .}
Using the criteria described in
Section~\ref{obs_analysis}, 18 sources are found to be variable, 19 are
found to have steady light curves, and the remaining 18 are
undetermined. Each source in Figure~\ref{fig_sources_1} is marked with
a V, S or U corresponding to its classification.
Structure functions for the 18 variable sources are shown in
Figure~\ref{fig_struc}; for each source we have estimated the time
scale for variability, $\tau_V$, as marked on each plot. However, we
note that some of these estimates are very approximate, as a result of
the sparse and/or irregular sampling of the light curves. For example,
for MRC~B2326--477 we have assigned $\tau_V = 400$~d, but one could
just as easily argue that $\tau_V = 2000$~d. Furthermore, there is
evidence that the structure functions for some sources, such as
MRC~B1740--517, enter another
linear regime beyond the point where they saturate. This suggests
that there are variations on time scales
longer than we can measure with these data.
Some properties of the 18 variable sources are summarised
in Table~\ref{tab_variables}.
\section{Discussion}
\label{discuss}
\subsection{Individual Sources}
We restrict our comments here to the 18 sources found to be variable.
Many of these sources have been observed in snapshot mode at 5 GHz
with the Australia Telescope Compact Array (ATCA, Burgess 1998), and
are also ATCA secondary phase calibrators.
\medskip
\noindent{\bf MRC B0208$-$512}: VLBI modelling shows a strong core
(Preston et al 1989), and a jet-like feature (Tingay et al 1996).
Detected as an X-ray source in the {\em ROSAT}\ All-Sky Survey (Brinkmann et
al.\ 1994) and as a $\gamma$-ray source in the {\em EGRET}\ survey (Bertsch et
al.\ 1993).
\noindent{\bf MRC B0537$-$441}: See Paper I.
\noindent{\bf MRC B0943$-$761}: Close $2\farcs8$ double at 5 GHz (Burgess
1998). Detected as an X-ray source in the {\em ROSAT}\ All-Sky Survey
(Brinkmann et al 1994).
\noindent{\bf MRC B1151$-$348}: Radio spectrum peaks at $\sim$200 MHz.
A VLBI image shows a 90~mas double structure (King et al 1993).
\noindent{\bf MRC B1215$-$457}: Compact steep-spectrum source with a
strong, slightly resolved VLBI core (Preston et al 1989).
\noindent{\bf MRC B1234$-$504}: Compact steep-spectrum source, with no
optical counterpart on the UK Schmidt sky survey but possible stellar
identification on a CCD image obtained at the Anglo-Australian
Telescope (AAT) (Burgess 1998).
\noindent{\bf MRC B1424$-$418}: Discordant flux densities measured at
Parkes point to the source being variable at~5 GHz (Burgess, priv
comm). VLBI modelling shows an unequal 23 mas double structure
(Preston et al 1989).
\noindent{\bf MRC B1458$-$391}: Compact steep-spectrum source in a
crowded optical field; optical ID based on an AAT CCD image (Burgess
1998).
\noindent{\bf MRC B1549$-$790}: VLBI image shows a curved structure,
possibly a core plus jet (Murphy et al 1993).
\noindent{\bf MRC B1610$-$771}: Quasar with a flat radio spectrum and
very steep optical spectrum (Hunstead \& Murdoch 1980). VLBI
observations (Preston et al 1989) show a strong core surrounded by a
50~mas halo.
\noindent{\bf MRC B1718$-$649}: The nearest GHz-peaked-spectrum
source, with a radio spectrum peaking near 3 GHz. VLBI imaging shows
two sub-parsec-scale components separated by $\sim$2~pc (Tingay et
al.\ 1997).
\noindent{\bf MRC B1740$-$517}: Crowded optical field; galaxy ID by di
Serego Alighieri et al (1994) is confirmed by an AAT CCD image
(Burgess 1998).
\noindent{\bf MRC B1827$-$360}: Compact ultra-steep-spectrum source
identified with a galaxy in a very crowded field.
\noindent{\bf MRC B1829$-$718}: Candidate source for defining the VLBI
astrometric reference frame (Ma et al 1998).
\noindent{\bf MRC B1854$-$663}: Compact steep-spectrum source
identified with a faint galaxy (Burgess 1998).
\noindent{\bf MRC B1921$-$293}: See Paper I.
\noindent{\bf MRC B2052$-$474}: Radio spectrum steep at low frequency,
but flattens at high frequency; core dominated at 5 GHz, possibly
triple (Burgess 1998). Detected as an X-ray source by the {\em ROSAT}\
All-Sky Survey (Brinkmann et al 1994).
\noindent{\bf MRC B2326$-$477}: Detected as an X-ray source in the
{\em ROSAT}\ All-Sky Survey (Brinkmann et al 1994). One of the set of
defining sources for the VLBI astrometric reference frame (Ma et al 1997).
\subsection{General Properties}
If the observed variability is a result of refractive scintillation in
the Galactic interstellar medium (ISM), then we expect some sort of
dependence of one of modulation index $m=\sigma/\bar{S}$, characteristic timescale
$\tau_V$ or their product, $m\, \tau_V$, on the Galactic latitude, $b$
(e.g. Spangler et al. 1989; Ghosh \& Rao 1992). However, apart from a
weak tendency for larger $\tau_V$ to occur at lower $|b|$,
there is no obvious correlation in our
data. This is not surprising given the large uncertainties in
$\tau_V$ arising from the irregular sampling of the light curves, and
the fact that there are few variable sources at high latitudes (14 of
the 18 variable sources have $10^{\circ} < |b| < 30^{\circ}$).
An alternative indicator of the effects of the Galactic ISM is to test
whether variable sources are more likely to be found at low
latitudes. We consider this possibility in Figure~\ref{fig_hist},
where we plot the ratio of variable sources ($N_V$) to variable plus
steady sources ($N_V + N_S$) in different latitude bins. While the
statistics are poor, there is a clear indication that sources are more
likely to be variable at low latitudes, as found for northern
hemisphere sources (Cawthorne \& Rickett 1985; Gregorini et al. 1986).
This is unlikely to be caused by selection effects associated with a
dependence of spectral index on Galactic latitude (cf. Cawthorne \&
Rickett 1985), since the main criterion for source selection was
angular size ($\theta < 10''$). Thus the extensive monitoring data
for the MOST calibrators provide good evidence that the variability
observed at 843 MHz arises from scintillation in the local ISM.
While spectral index was not considered in selecting the majority of
the sample, Table \ref{tab_variables} shows that two-thirds of the variables
have flat or inverted spectra ($\alpha > -0.5$), consistent with
source angular size being the main determinant of source variability.
Surprisingly, the remaining third of the variables fall in the class of
compact steep-spectrum (CSS) sources which are generally believed to
be young sources still contained within their host galaxies, and not
known to vary at high frequencies. The latter sources display a lower
level of variability, as measured by the modulation index $m$, and
in four of the six cases their V classification appears to be due to one-off
events lasting $\sim$1 year.
To investigate the variability properties of the MOST calibrator
sample as a whole, in Figure~\ref{m_alpha}
we have plotted $m$ versus $\alpha$
for all 55 sources. This Figure shows a clear trend
towards higher average modulation index as the radio spectrum
flattens, with a suggestion of an upper envelope. Perhaps the simplest
explanation for this behaviour in the unified model for
powerful extragalactic radio sources is to link $m$ and
$\alpha$ through the orientation of the radio axis to the line of
sight (e.g. Orr \& Browne 1982). We assume that the `core' of a
classical triple source is the only part with components small enough
in angular size to scintillate. If the core contribution dominates,
as a consequence of Doppler boosting in the flat spectrum sources,
even small fractional variations will be readily detected. However,
the same fractional variations in the core of a steep-spectrum,
lobe-dominated source will go undetected. We can therefore understand
the trends in Figure~\ref{m_alpha} in a qualitative sense, and it is
possible that a more detailed analysis may provide useful constraints
on radio source models.
\section{Conclusions}
55 sources used for calibration purposes by the
MOST at a frequency of 843~MHz have been
observed irregularly over a 13 year interval.
We have developed an algorithm to process these data and produce
a light curve for each source. Our analysis shows that 18 of these
sources can be considered variable. There is some suggestion that
these sources are distributed at lower Galactic latitudes than the 19
sources whose flux densities are unvarying. This suggests that
variability at 843~MHz on time scales of 1--10 years is predominantly
due to scintillation in the Galactic ISM rather than effects intrinsic
to the source. A possible correlation between modulation index and
spectral index can be explained qualitatively in terms of a variation
in the core fraction with orientation of the radio axis to the line
of sight.
\small
\section*{Acknowledgements}
We thank Ann Burgess for providing us with unpublished ATCA images of
several sources and for supplying us with her improved
meridian-distance gain curve. We also thank Duncan Campbell-Wilson,
Lawrence Cram, Jean-Pierre Macquart,
Gordon Robertson, Mark Walker and Taisheng Ye for useful discussions
and advice, and an anonymous referee for a careful
reading of the manuscript.
This research has made use of the NASA/IPAC Extragalactic
Database (NED), operated by JPL under contract with NASA. The MOST is
supported by grants from the Australian Research Council, the
University of Sydney Research Grants Committee, and the Science
Foundation for Physics within the University of Sydney. BMG
acknowledges the support of an Australian Postgraduate Award and of
NASA through Hubble Fellowship grant HF-01107.01-98A awarded by the
Space Telescope Science Institute, which is operated by the Association
of Universities for Research in Astronomy, Inc., for NASA under
contract NAS 5--26555.
\section*{References}
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\clearpage
\begin{figure}
\centerline{\psfig{file=md_curve.ps,width=15cm,angle=270}}
\caption{The relative gain of the MOST as a function of meridian
distance. The data shown correspond to the asymmetric gain curve
of Burgess (priv comm), with additional
empirical corrections of order 2\%.}
\label{fig_md}
\end{figure}
\begin{figure}
\centerline{\psfig{file=sources_1.ps,width=16cm,angle=270}}
\caption{Light curves for 55 MOST calibrators. Sources
are marked S, U or V, corresponding to whether their
time behaviour is steady, undetermined or variable
respectively. Each
abcissa ranges between
MJD -- 40\,000 $=$~5500 (1983 Jun) and 10400 (1996 Nov), while
ordinates run between
0.5 and 1.5 times the nominal flux density for each source (see Table~1
of Paper~I). Data have been binned into 30 day intervals --- the error bars
shown are the 1$\sigma$ standard deviation of the individual data points within
each interval, or are set at 5\% in cases where only one data point falls
in a given 30 day period.}
\label{fig_sources_1}
\end{figure}
\begin{figure}
\centerline{\psfig{file=sources_2.ps,width=16cm,angle=270}}
Figure~\ref{fig_sources_1} (cont.)
\end{figure}
\begin{figure}
\centerline{\psfig{file=sources_3.ps,width=16cm,angle=270}}
Figure~\ref{fig_sources_1} (cont.)
\end{figure}
\begin{figure}
\centerline{\psfig{file=sources_4.ps,width=16cm,angle=270}}
Figure~\ref{fig_sources_1} (cont.)
\end{figure}
\begin{figure}
\centerline{\psfig{file=struc_1.ps,width=18cm,angle=270}}
\caption{Structure functions for the 18 variable sources.
The dashed horizontal line corresponds to $\Sigma_\tau = 2$,
the value at which a pure sinusoid will saturate. The dashed
vertical line corresponds to the approximate lag at which
the structure function saturates (the time scale for variability,
$\tau_V$, is defined to be twice this value).}
\label{fig_struc}
\end{figure}
\begin{figure}
\centerline{\psfig{file=struc_2.ps,width=18cm,angle=270}}
Figure~\ref{fig_struc} (cont.)
\end{figure}
\begin{figure}
\centerline{\psfig{file=lat_hist.ps,height=10cm,angle=270}}
\caption{The fraction of variable sources as a function of Galactic latitude.
Horizontal error bars represent the width of each latitude bin, while
vertical error bars have been derived by computing the fraction of variable
sources which results when half the undetermined
sources in that bin are reclassified as either variable (upper limits)
or steady (lower limits).}
\label{fig_hist}
\end{figure}
\begin{figure}
\centerline{\psfig{file=m_alpha.ps,width=15.5cm,angle=270}}
\caption{Plot of modulation index ($m$) versus spectral index
($\alpha$) for all 55 MOST calibrator sources. Note the tendency for
$m$ to increase as the spectrum flattens.}
\label{m_alpha}
\end{figure}
\clearpage
\begin{table}
\caption{Properties of variable sources in our sample.}
\label{tab_variables}
\begin{tabular}{lrrrclrr} \\[-3mm] \hline
Source & \multicolumn{1}{c}{$b$} & \multicolumn{1}{c}{$\tau_V$} &
$m^a$ & Ident$^b$ & \multicolumn{1}{c}{$z$} & $\alpha^c$
& \multicolumn{1}{c}{LAS$^d$}\\
& (deg) & (days) & & & & & \multicolumn{1}{c}{($''$)} \\
\hline \\[-3mm]
MRC~B0208$-$512 & $-$61.8 & 2000 & 0.047 & Q & 1.003 & $-$0.23 & 6.0 \\
MRC~B0537$-$441 & $-$31.1 & 1500 & 0.139 & Q & 0.894 & +0.25 & \ldots \\
MRC~B0943$-$761 & $-$17.4 & 300 & 0.023 & g & \ldots & $-$0.79 & 2.8 \\
MRC~B1151$-$348 & +26.3 & 400 & 0.019 & Q & 0.258 & $-$0.49 & $<$2.9 \\
MRC~B1215$-$457 & +16.5 & 300 & 0.019 & Q & 0.529 & $-$0.59 & $<$1.9 \\
MRC~B1234$-$504 & +12.0 & 1200 & 0.060 & Q? & $\ldots$ & $-$0.82 & $<$1 \\
MRC~B1424$-$418 & +17.3 & 300 & 0.074 & Q & 1.52 & $-$0.47 & $<$2.3 \\
MRC~B1458$-$391 & +17.0 & 400 & 0.022 & g & $\ldots$ & $-$0.67 & $<$2.4 \\
MRC~B1549$-$790 & $-$19.5 & 700 & 0.050 & g & 0.15 & $-$0.29 & $<$1.0 \\
MRC~B1610$-$771 & $-$18.9 & 400 & 0.052 & Q & 1.71 & $-$0.13 & $<$1 \\
MRC~B1718$-$649 & $-$15.8 & 1400 & 0.060 & g & 0.013 & +0.21 & \ldots \\
MRC~B1740$-$517 & $-$11.5 & 2500 & 0.063 & g & $\ldots$ & $-$0.08 & $<$1 \\
MRC~B1827$-$360 & $-$11.8 & 400 & 0.016 & g & $\ldots$ & $-$1.12 & $<$1.5 \\
MRC~B1829$-$718 & $-$24.5 & 400 & 0.048 & g & $\ldots$ & $-$0.35 & \ldots \\
MRC~B1854$-$663 & $-$25.5 & 400 & 0.022 & g & \ldots & $-$0.86 & $<$1.0 \\
MRC~B1921$-$293 & $-$19.6 & 3000 & 0.191 & Q & 0.352 & +0.38 & \ldots \\
MRC~B2052$-$474 & $-$40.4 & 300 & 0.084 & Q & 1.489 & $-$0.34 & 3.9 \\
MRC~B2326$-$477 & $-$64.1 & 400 & 0.037 & Q & 1.489 & $-$0.15 & \ldots \\
\hline \\[-3mm]
\end{tabular}
$^a$ Modulation index, defined by $m = \sigma / \bar{S}$ \\
$^b$ Q = quasar ; g = galaxy\\
$^c$ Spectral index $\alpha$ (where $S\propto \nu^{\alpha}$)
between 408 and 2700 MHz (or 4850 MHz if 2700 MHz flux density not available)\\
$^d$ Largest angular size at 5 GHz, measured with the Australia Telescope
Compact Array (Burgess 1998)
\end{table}
\end{document}
| {
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Nature Trail Built with help of Beaver Manufacturing
August 29, 2015 /in Community Involvement, News /by madisonstudios
From the Newton Citizen
Wade Marbaugh, staff writer
MANSFIELD — Taking a break from the muggy heat on Tuesday, Andy Taylor, owner of ArtScape Lawn & Turf Professionals in Conyers, and his business partner Benny Evans proudly showed off a new nature trail that has sprung up in Mansfield over the past month.
"When we first started the project it was a big one," Taylor said. "We were clearing trees and pushing out mulch and finding stumps this deep we had to remove."
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The city of Mansfield will celebrate its new "Nature Trail and Educational Path" with a ribbon cutting on May 22 at 1 p.m.
That's well ahead of schedule — Mayor Jefferson Riley estimated the trail would be ready in June when he led a groundbreaking ceremony on April 10.
The trail is located a quarter of a mile east of Ga. Highway 11 on Highway 213. It runs between 213 and Mansfield Elementary on 15 acres of land donated to the city for this purpose by Needham.
"The Needham Family has long been dedicated to supporting Mansfield Elementary School and several years ago began an effort to preserve this remarkable tract of undeveloped land," Riley said.
Mansfield Elementary students will use the nature trail as an outdoor classroom for environmental education. It will also be available for the public to enjoy.
Lindsey, who also serves as chair of the Mansfield Tree Board and recently was recognized by a mayor's proclamation for her volunteerism, worked closely with Riley, City Council representatives and Mike Dubin, COO of Beaver Manufacturing, to finalize the design of the trail.
Pickens rendered the design, selected trees, cattails and numerous native plants for the landscape, which includes areas where students can sit on tree stumps — to be painted bright colors by Outback, according to owner David Wheeler — for outdoor lessons.
The city applied $75,000 of a lawsuit settlement toward the project, and Beaver Manufacturing donated $31,000.
Under the guidance of Beryl Budd, retired Georgia Forestry Commission chief ranger in Newton County, Riley and the Mansfield Tree Board secured a grant for partial clearing of invasive trees and plants in the wooded portion of the 15 acres.
ArtScape was named general contractor with a low bid of $106,000.
"This is a big deal for Mansfield and I intend to go far beyond what the proposal calls for and make it a showplace," Taylor said at the groundbreaking.
Taylor, a Social Circle resident, agreed to cut the connecting trail to the elementary school through the woods and build a bridge as his donation to the project.
Riley lauded the City Council for its support of the project.
"The City Council recognizes the importance of preserving this land and the benefits of using it as an outdoor classroom," he said.
"We turned our shared vision into a reality."
Residents are encouraged to support trail upkeep, new plantings, and the creation of educational tools such as trail markers, tables, and trail activity kits for teachers, as well as trail activity guides for users of the trail.
For information on how to support the project, call Mansfield City Hall 770-786-1660.
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Park, road to be named for Mansfield's Ed Needham Mansfield Elementary | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 1,355 |
RSpec.shared_examples "is indexable" do |private_test=true|
let(:resource_ids) do
created_instance_ids(api_resource_name).map(&:to_i)
end
let(:ips) do
defined?(index_params) ? index_params : {}
end
context 'for a normal user' do
before(:each) do
default_request scopes: scopes, user_id: authorized_user.id if authorized_user
end
def run_get
get :index, ips
end
it 'should return 200' do
run_get
expect(response.status).to eq 200
end
it "should have a specified number of items by default" do
run_get
expect(json_response[api_resource_name].length).to eq n_visible
end
if private_test
it 'should not include nonvisible resources' do
private_resource
run_get
expect(resource_ids).to_not include private_resource.id
end
end
context "with response" do
before { run_get }
it_behaves_like 'an api response'
it_behaves_like 'an indexable etag response'
end
end
if private_test
context 'when the authorized_user is an admin' do
let(:authorized_user) { create(:user, admin: true) }
before(:each) do
private_resource
default_request scopes: scopes, user_id: authorized_user.id if authorized_user
end
context 'when an admin param is set' do
it 'should include the non-visible resource' do
get :index, ips.merge(admin: 'literally_anything!')
expect(resource_ids).to include private_resource.id
end
end
context 'when no admin param is set' do
it 'should not include the non-visible resource' do
get :index, ips
expect(resource_ids).to_not include private_resource.id
end
end
end
end
end
| {
"redpajama_set_name": "RedPajamaGithub"
} | 7,499 |
{"url":"http:\/\/clay6.com\/qa\/38136\/roots-of-the-equation-12x-2-mx-5-0-are-in-the-ratio-3-2-then-m-equals","text":"# Roots of the equation $12x^2+mx+5=0$ are in the ratio $3 : 2$,then $m$ equals\n\n$\\begin{array}{1 1}(A)\\;\\large\\frac{5}{12}&(B)\\;\\large\\frac{1}{12}\\\\(C)\\;\\large\\frac{5\\sqrt{10}}{12}&(D)\\;5\\sqrt{10}\\end{array}$\n\nLet the roots be $\\alpha,\\beta$\n$\\large\\frac{\\alpha}{\\beta}=\\frac{3}{2}$\n$\\alpha+\\beta=-\\large\\frac{m}{12}$\n$\\alpha\\beta=\\large\\frac{5}{12}$\n$\\alpha=\\large\\frac{3}{2}$$\\beta\\Rightarrow\\large\\frac{3}{2}$$\\beta^2=\\large\\frac{5}{12}$\n$\\Rightarrow \\beta^2=\\large\\frac{10}{36}\\Rightarrow \\beta=-\\large\\frac{\\sqrt{10}}{6}$\n$\\alpha=\\large\\frac{3}{2}$$\\beta=-\\large\\frac{\\sqrt{10}}{4}$\n$m=-12(\\alpha+\\beta)$\n$\\;\\;\\;=12(\\large\\frac{\\sqrt{10}}{6}+\\frac{\\sqrt{10}}{4})$\n$\\;\\;\\;=5\\sqrt{10}$\nHence (D) is the correct answer.","date":"2017-12-13 18:31:24","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\": 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\": 0.9590393900871277, \"perplexity\": 426.679192012766}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2017-51\/segments\/1512948530668.28\/warc\/CC-MAIN-20171213182224-20171213202224-00673.warc.gz\"}"} | null | null |
Rajdowe Mistrzostwa Świata w roku 1978 były 6 sezonem Rajdowych Mistrzostwach Świata FIA. Sezon składał się z 19 rajdów. We wszystkich rajdach byli punktowani kierowcy, ale tylko w 11 z 19 były punktowane zespoły. Najlepszym kierowcom rajdowym w roku 1978 (FIA Cup for Rally Drivers) został fiński kierowca Markku Alén startujący samochodami Fiat 131 Abarth i Lancia Stratos HF, drugi był Francuz Jean-Pierre Nicolas, a trzeci Fin Hannu Mikkola. Tytuł konstruktorów wygrał Fiat przed Fordem i Oplem.
Kalendarz
Wyniki
Klasyfikacja zespołowa
W sezonie 1978 system punktacji producentów taki jak w roku 1977. Składał się on z dwóch grup punktacji, które do siebie dodawano.
Wpierw punkty dla producenta zdobywał najwyżej sklasyfikowany samochód danej marki według klucza:
Dodatkowe punkty były przyznawane dla najwyżej sklasyfikowanego samochodu danej marki za zajęcie miejsca od pierwszego do ósmego w swojej grupie, pod warunkiem, że dany zespół znalazł się w pierwszej dziesiątce w klasyfikacji generalnej, według klucza:
W tabeli poniżej punkty nie brane pod uwagę w końcowej klasyfikacji ujęto w nawiasach. W klasyfikacji końcowej liczyło się tylko osiem najlepszych wyników. Rajdy nie liczone ujęto w nawiasach.
Klasyfikacja kierowców FIA Cup for Rally Drivers
W tej klasyfikacji mającej wyłonić najlepszego zawodnika mistrzostw, punktowano sześć pierwszych pozycji w rajdzie na zasadzie:
Kierowcy o tytuł FIA Cup for Rally Drivers rywalizowali w dziewiętnastu rajdach: wszystkich jedenastu rajdach o mistrzostwo producentów (do ostatecznej klasyfikacji liczyły się tylko pięć najlepszych wyników), pięciu rajdach o najwyższym współczynniku, zaliczanych do Mistrzostw Europy w roku 1977 (liczone dwa najlepsze rajdy) oraz trzech rajdach wybranych przez FIA (tu liczył się tylko jeden najlepszy start).
Linki zewnętrzne
Wyniki sezonu na stronie eWRC.com
Przypisy
1978 w sportach motorowych | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 9,889 |
Q: Making a paint program using XNA Hello I am trying to make a paint program using XNA and I followed the guide found in here as much as possible: How to create Paint-like app with XNA?
And it worked all great so far however there is one issue: the drawn rectangles dont connect together to form a line. I am out of ideas and I would appreciate any help offered. Heres my code below. Please also take a look at the image attached to get a better understanding.
using Microsoft.Xna.Framework;
using Microsoft.Xna.Framework.Graphics;
using Microsoft.Xna.Framework.Input;
using System;
using System.Collections.Generic;
namespace ProfAnas
{
/// <summary>
/// This is the main type for your game.
/// </summary>
public class Game1 : Game
{
GraphicsDeviceManager graphics;
SpriteBatch spriteBatch;
Texture2D canvas;
Vector2 brushPos;
public Game1()
{
graphics = new GraphicsDeviceManager(this);
Content.RootDirectory = "Content";
graphics.PreferredBackBufferWidth = 1920; // set this value to the desired width of your window
graphics.PreferredBackBufferHeight = 1080; // set this value to the desired height of your window
graphics.ApplyChanges();
}
/// <summary>
/// Allows the game to perform any initialization it needs to before starting to run.
/// This is where it can query for any required services and load any non-graphic
/// related content. Calling base.Initialize will enumerate through any components
/// and initialize them as well.
/// </summary>
protected override void Initialize()
{
// TODO: Add your initialization logic here
base.Initialize();
Color[] pixel = new Color[1920 * 1080];
for (int i = 0; i < pixel.Length; i++)
{
pixel[i] = Color.White;
}
pixel[1919] = Color.Red;
spriteBatch = new SpriteBatch(GraphicsDevice);
canvas = new Texture2D(this.GraphicsDevice, 1920, 1080);
canvas.SetData<Color>(pixel);
}
/// <summary>
/// LoadContent will be called once per game and is the place to load
/// all of your content.
/// </summary>
protected override void LoadContent()
{
// Create a new SpriteBatch, which can be used to draw textures.
spriteBatch = new SpriteBatch(GraphicsDevice);
// TODO: use this.Content to load your game content here
}
/// <summary>
/// UnloadContent will be called once per game and is the place to unload
/// game-specific content.
/// </summary>
protected override void UnloadContent()
{
// TODO: Unload any non ContentManager content here
}
/// <summary>
/// Allows the game to run logic such as updating the world,
/// checking for collisions, gathering input, and playing audio.
/// </summary>
/// <param name="gameTime">Provides a snapshot of timing values.</param>
protected override void Update(GameTime gameTime)
{
MouseState state = Mouse.GetState();
Color[] pixel = new Color[1920 * 1080];
canvas.GetData<Color>(pixel);
if (state.LeftButton == ButtonState.Pressed)
{
brushPos.X = state.X;
brushPos.Y = state.Y;
double piOn4 = Math.PI / 4;
int xComponent;
int yComponent;
int screenWidth = 1920;
int screenHeight = 1080;
int regionXHalfed = (int)Math.Ceiling((10.0 * Math.Cos(piOn4) + 30.0 * Math.Cos(piOn4))/2);
int regionYHalfed = (int)Math.Ceiling((10.0 * Math.Sin(piOn4) + 30.0 * Math.Sin(piOn4))/2);
double angle;
double centerToBoundary;
double pixelToCenter;
List<int> boundedPixel = new List<int>();
for (int row=(state.Y-regionYHalfed); row < (state.Y + regionYHalfed); row++)
{
for (int column= (state.X - regionXHalfed); column < (state.X + regionXHalfed); column++)
{
xComponent=column - state.X+1;
yComponent=row - state.Y+1;
if (xComponent == 0)
{
pixelToCenter = Math.Sqrt((double)xComponent*xComponent + (double)yComponent*yComponent);
if (Math.Abs(pixelToCenter) <= 5*Math.Sqrt(2))
{
boundedPixel.Add(((row) * screenWidth + column) + 1);
}
continue;
}
angle=Math.Atan( (double)yComponent / (double)xComponent);
if (angle>= (piOn4 - Math.Atan(1.0/3)) && angle<= (piOn4 + Math.Atan(1.0/3)))
{
centerToBoundary = 15 / Math.Cos(angle - piOn4);
pixelToCenter= xComponent / Math.Cos(angle);
if( Math.Abs(pixelToCenter) <= Math.Abs(centerToBoundary))
{
boundedPixel.Add(((row)* screenWidth + column)+1);
}
}
if (angle >= (piOn4 + Math.Atan(1.0 / 3)) && angle <= Math.PI/2)
{
centerToBoundary = 5 / Math.Cos(angle + piOn4);
pixelToCenter = xComponent / Math.Cos(angle);
if (Math.Abs(pixelToCenter) <= Math.Abs(centerToBoundary))
{
boundedPixel.Add(((row) * screenWidth + column) + 1);
}
}
if (angle >= 0.0 && angle <= (piOn4 - Math.Atan(1.0 / 3)))
{
centerToBoundary = 5 / Math.Cos(angle + piOn4);
pixelToCenter = xComponent / Math.Cos(angle);
if (Math.Abs(pixelToCenter) <= Math.Abs(centerToBoundary))
{
boundedPixel.Add(((row) * screenWidth + column) + 1);
}
}
if (angle >= -Math.PI / 2 && angle <= 0.0)
{
centerToBoundary = 5 / Math.Cos(angle + piOn4);
pixelToCenter = xComponent / Math.Cos(angle);
if (Math.Abs(pixelToCenter) <= Math.Abs(centerToBoundary))
{
boundedPixel.Add(((row) * screenWidth + column) + 1);
}
}
}
}
foreach (int i in boundedPixel)
{
if(i>=0)
pixel[i] = Color.Red;
}
}
canvas.SetData<Color>(pixel);
// TODO: Add your update logic here
base.Update(gameTime);
}
/// <summary>
/// This is called when the game should draw itself.
/// </summary>
/// <param name="gameTime">Provides a snapshot of timing values.</param>
protected override void Draw(GameTime gameTime)
{
GraphicsDevice.Clear(Color.CornflowerBlue);
// TODO: Add your drawing code here
spriteBatch.Begin();
spriteBatch.Draw(canvas, new Vector2(0, 0));
spriteBatch.End();
base.Draw(gameTime);
}
}
}
Thank you :)
A: As @adv12 suggested, the Update method of XNA games isn't as fast as the one of MSPaint's, and that's why you'll never have a line if drawing pixel-by-pixel (or in your case rectangle-by-rectangle) if you move your mouse fast across the canvas.
The possible solution is to draw new rectangles when you release left mouse button, between the rectangles that were created when left mouse button was pressed. That will give you twice -1 the rectangles you are currently drawing, and you will have a line, no matter how fast the Update is called.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 2,882 |
namespace content {
class SharedCorsOriginAccessList;
// A URLLoaderFactory used for the file:// scheme used when Network Service is
// enabled.
// If a caller needs a request that has a fetch request mode other than
// "no-cors", this class should be used on the UI thread.
class CONTENT_EXPORT FileURLLoaderFactory
: public network::SelfDeletingURLLoaderFactory {
public:
// Returns mojo::PendingRemote to a newly constructed FileURLLoaderFactory.
// The factory is self-owned - it will delete itself once there are no more
// receivers (including the receiver associated with the returned
// mojo::PendingRemote and the receivers bound by the Clone method).
//
// |shared_cors_origin_access_list| can be nullptr if only "no-cors" requests
// will be made.
//
// Thread pool tasks posted by the constructed FileURLLoaderFactory use
// |task_priority|.
static mojo::PendingRemote<network::mojom::URLLoaderFactory> Create(
const base::FilePath& profile_path,
scoped_refptr<SharedCorsOriginAccessList> shared_cors_origin_access_list,
base::TaskPriority task_priority);
FileURLLoaderFactory(const FileURLLoaderFactory&) = delete;
FileURLLoaderFactory& operator=(const FileURLLoaderFactory&) = delete;
private:
FileURLLoaderFactory(
const base::FilePath& profile_path,
scoped_refptr<SharedCorsOriginAccessList> shared_cors_origin_access_list,
base::TaskPriority task_priority,
mojo::PendingReceiver<network::mojom::URLLoaderFactory> factory_receiver);
// network::mojom::URLLoaderFactory:
~FileURLLoaderFactory() override;
void CreateLoaderAndStart(
mojo::PendingReceiver<network::mojom::URLLoader> loader,
int32_t request_id,
uint32_t options,
const network::ResourceRequest& request,
mojo::PendingRemote<network::mojom::URLLoaderClient> client,
const net::MutableNetworkTrafficAnnotationTag& traffic_annotation)
override;
void CreateLoaderAndStartInternal(
const network::ResourceRequest request,
network::mojom::FetchResponseType response_type,
mojo::PendingReceiver<network::mojom::URLLoader> loader,
mojo::PendingRemote<network::mojom::URLLoaderClient> client);
const base::FilePath profile_path_;
const scoped_refptr<SharedCorsOriginAccessList>
shared_cors_origin_access_list_;
const scoped_refptr<base::SequencedTaskRunner> task_runner_;
};
} // namespace content
#endif // CONTENT_BROWSER_LOADER_FILE_URL_LOADER_FACTORY_H_
| {
"redpajama_set_name": "RedPajamaGithub"
} | 7,884 |
Monday 30 June 2014 4:08 pm
Isis reveals European expansion plan under new caliphate
By: Sarah Spickernell
Today, the extremist group that has taken over large swathes of northern Iraq declared the formation of a caliphate: an Islamic state led by a supreme political and religious leader.
It demanded that Muslims around the world swear allegiance to their ruler, Abu Bakr al-Baghdadi, who they claim has ultimate authority over all Muslims.
"The legality of all emirates, groups, states and organizations becomes null by the expansion of the caliph's authority and the arrival of its troops to their areas,' said Abu Mohammed al-Adnani, a spokesman for the group, in a video posted online. "Listen to your caliph and obey him. Support your state, which grows every day."
It also announced that it has changed its name to The Islamic State, having formally been known as The Islamic State of Iraq and the Levant (Isis).
The group are known for being prolific online, garnering much of their support via social media, and now a map showing their intended expansion plan is being shared widely by its followers.
Revealing the places it is looking to occupy by 2020, the map shows that it is not only North Africa and the Gulf region that the group has set its sights on, but also some parts of Europe, namely Spain and the Balkan states including Greece, Romania and Bulgaria.
The plan seems to be largely based on the borders of Muslim-led countries with the Austro-Hungarian Empire prior to the First World War: the group has previously said that it would like to reinstate the geographical boundaries present before the war.
Spain was a Muslim-led country for 700 years prior to the 15th century. | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 8,199 |
\section{Introduction}
\label{Introduction}
The AdS/CFT correspondence \cite{juanAdS,gkPol,witHolOne} (see
\cite{MAGOO} for a review) relates a quantum field theory in one
dimension to a theory in one higher dimension that includes gravity.
The primary example is ${\cal N}=4$ super-Yang-Mills theory, which is
related to the dimensional reduction of type~IIB string theory on
$S^5$ to the five-dimensional non-compact geometry $AdS_5$ (anti
de-Sitter space). The boundary of the Poincar\'e patch of $AdS_5$ is
simply Minkowski space, except that the metric on the boundary is only
specified up to a conformal transformation. This is OK because ${\cal
N}=4$ super-Yang-Mills theory is conformal, even at the quantum level.
The correspondence is usually studied in a strong coupling region for
the gauge theory, where it is far from classical, but the dual gravity
picture is classical in the sense that curvatures are small on the
Planck and string scales.
In \cite{RSalt} it was proposed that slices of $AdS_5$ could serve as
an alternative to compactification manifolds. It was shown that when
the near-boundary region of $AdS_5$ is cut away and the bulk spacetime
simply ends on a wall of constant extrinsic curvature (a horosphere of
$AdS_5$ to be precise), there is a normalizable graviton mode which
has zero mass in the four dimensions of the boundary. The metric of
$AdS_5$ is
\eqn{AdSMetric}{
ds_5^2 = e^{2r/L} (-dt^2 + d\vec{x}^2) + dr^2
}
where $\vec{x}$ is an ordinary 3-vector. This metric is a solution
to $R_{\mu\nu} = -{4 \over L^2} g_{\mu\nu}$. Hypersurfaces of constant
$r$ are horospheres. The part of the metric that is cut away in
\cite{RSalt} is $r>r_*$ for some given $r_*$.
In fact the proposal of \cite{RSalt} was to glue two identical copies
of the sliced anti-de Sitter space together along the $3+1$
dimensional boundary. However the four-dimensional graviton
is quite a general phenomenon, and there is a
large freedom in what one might have on the other side of the
horospherical boundary of a given copy of $AdS_5$. An illustration of
this can be found in \cite{HV}, where a single copy of $AdS_5$ is
obtained as part of a type~IIB string compactification on an
orientifold of $T^6$. The relevant point is that for a vacuum state
of the theory, the extrinsic curvature of the boundary should be
proportional to the induced metric:
\eqn{ThetaReq}{
\Theta_{ij} = -{1 \over L} g^{\rm (induced)}_{ij} \,.
}
Regarding the boundary as a positive tension $3+1$-dimensional brane
separating two sliced copies of $AdS_5$, this amounts to the statement
that the stress energy of the brane should respect $3+1$-dimensional
Poincar\'e invariance. The constant of proportionality, $1/L$, is
required so that there is a balance between the brane tension and the
bulk cosmological constant. More generally, a codimension one
boundary of a five-dimensional space with no excitations on it should
have the same property that $\Theta_{ij} = -{1 \over L} g^{\rm
(induced)}_{ij}$.
The authors of \cite{RSalt} termed their construction an ``alternative to
compactification,'' which seemed appropriate because one can travel an
infinite spatial distance into the five-dimensional bulk. Efforts such as
\cite{HV} to realize the construction in string theory, using coincident
D3-branes to produce the $AdS_5$ background, represent strong-coupling
extrapolations of perturbative string compactifications, where the massless
four-dimensional graviton is the usual zero-mode of the spin 2 closed
string state. In such a picture, the Kaluza-Klein gravitons of
\cite{RSalt} are interpreted as the strong-coupling description of open
string excitations on the D3-branes, in line with the extensive literature
on absorption by D3-branes \cite{IgorAbsorb,gktAbsorb,gkSchwing}.
It was suggested by J.~Maldacena \cite{juanPrivate} that this ``alternative
to compactification'' should properly be viewed in light of the AdS/CFT
correspondence as a coupling of gravity to whatever strongly coupled
conformal theory the $AdS_5$ geometry is dual to. This view was also taken
by H.~Verlinde in \cite{HV,VerlindeTalk}. A convincing statement of the
case was made by E.~Witten \cite{WittenComment} in response to
\cite{SundrumTalk,GiddingsTalk}.
From now on I will restrict myself to a minimal scenario where a single
copy of $AdS_5$ is cut off by an end-of-the-universe brane. Such objects
are well known in type~I$'$ string theory \cite{PolchinskiWitten} and in
Horava-Witten theory \cite{HoravaWitten}, so there is no problem of
principle in having a true end of the universe. However my comments
basically apply to any compactification geometry which involves the
near-horizon part of $AdS_5$.
The idea that the scenario of \cite{RSalt} is best viewed in the
context of AdS/CFT has not been universally embraced, perhaps partly
because it is hard to see what to do with it. (That difficulty is not
usually regarded as fatal, but it does seem to have held up progress
on the current issue). The goal of section~\ref{FRW} is to make the
idea seem more definite by using it to derive the radiation-dominated
Friedmann-Robertson-Walker (FRW) cosmology. The approach is to change
the bulk spacetime from $AdS_5$ to AdS-Schwarzschild, but not to
excite any matter on the cutoff brane. The Hawking temperature of the
AdS-Schwarzschild geometry, measured with respect to time on the
cutoff brane, can be interpreted as the temperature of the CFT which
the bulk spacetime is dual to. Readers determined to understand the
construction from a brane-world perspective may find it most useful to
think of the bulk as the background of near-extremal D3-branes.
However, AdS/CFT allows us to reinterpret the entire AdS-Schwarzschild
geometry as a manifestation of the dynamics of a four-dimensional
conformal field theory at finite temperature. With the CFT
interpretation in mind, it is easier to understand why the
radiation-dominated FRW geometry emerges: all CFT's have the same
equation of state up to numerical factors, so the FRW equations take
the same form as they do in the radiation-dominated era of our
universe.
The literature on brane-world cosmology is large and somewhat
scattered; indeed I became aware of some recent overlapping papers
only after this work was completed. Early work on supergravity domain
walls in $AdS_4$ has been reviewed in \cite{Cvetic:1993xe}.
Cosmological solutions to Horava-Witten theory compactified on a
Calabi-Yau manifold were investigated in \cite{Lukas:1998qs}; see also
\cite{Chamblin:1999ya} for subsequent development, and
\cite{Chamblin:1999ea} for an analogous treatment of type~I$'$. The
works
\cite{KrausCosm,Mukohyama:1999qx,Binetruy:1999hy,Kehagias:1999vr}
include constructions equivalent to equations
\eno{Foliate}-\eno{TauForm}, although the CFT interpretation was not
offered. Reference \cite{Shiromizu:1999wj} includes some formal developments
and presents equations equivalent to \eno{EinsteinBold}. Discussions
of cosmological constraints in \cite{Binetruy:1999hy,Flanagan:1999cu}
have some overlap with section~\ref{Estimate}. And there are a number
of papers \cite{Nihei:1999mt,Kaloper:1999sm,DFGK,Csaki:1999mp,Vollick}
that use similar brane-world techniques. A more extensive list of
references on brane-world cosmology can be found in
\cite{Csaki:1999mp}. The whole approach is rather different from the
older string cosmology literature; see \cite{Brustein} for references.
Section~\ref{General} consists of some remarks on the general
framework of AdS/CFT with a cutoff brane. A generalization of the
prescription \cite{gkPol,witHolOne} for computing Green's functions is
suggested at the level of effective field theory. The
four-dimensional Einstein action can be derived in this formalism,
with the result $G_4 = 2 G_5/L$. This relation obtains regardless of
the location of the cutoff brane in $AdS_5$. In standard AdS/CFT,
where there is no cutoff brane and hence no normalizable graviton, the
terms responsible for the Einstein action were removed using local
counter-terms \cite{hs}. Corrections to Newton's force law are
discussed from the CFT perspective,
but I avoid presenting details since the idea of the
calculation is not original to me.
Matter on the cutoff brane is incorporated naturally in the formalism.
Although I do not propose a definite model, the idea is to have
visible sector matter on the cutoff brane, somewhat as in certain
heterotic M-theory models \cite{OvrutLecture}. Excitations of that
matter would have to dominate over the solution in section~\ref{FRW}
at least for $z \mathrel{\raise2pt\hbox{$\mathop<\limits_{\hbox{\raise3pt\hbox{$\sim$}}}$}} 10^{10}$ in order for the cosmology to be realistic.
In section~\ref{Estimate}, I make some rough numerical estimates. One
is to check the cosmological effects of visible sector matter losing
its energy to the CFT. In AdS language this corresponds to absorption
of bulk gravitons by the horizon of the Poincar\'e patch, as in
\cite{IgorAbsorb,gktAbsorb,gkSchwing}. The rate of energy loss is
directly related to the deviations from Newton's force law. Another
is to estimate parameters of string theory that would permit the
deviations from Newton's force law to be observed experimentally,
assuming that $AdS_5$ emerges from a string theory construction. To
obtain deviations at the scale of even a nanometer (still three orders
of magnitude below the sensitivity of proposed experiments) an
extremely low string scale is required---approximately $1 \, {\rm
GeV}$. I make some speculative remarks regarding low string scales at the
end of section~\ref{Estimate}.
Throughout the paper, $\mu$ will denote a five-dimensional bulk
spacetime index and $i$ is a four-dimensional index. In cases where
precision is required, I will denote five-dimensional coordinates as
$x^\mu = (t,\vec{x},r)$ and four-dimensional coordinates as $\xi^i =
(\tau,\vec{\xi})$. Throughout the paper, $\ell_{\rm Pl}$ will denote
the four-dimensional Planck length: $\ell_{\rm Pl} = \sqrt{G_4}
\approx 1.6 \times 10^{-33} \, {\rm cm}$ in units where $\hbar = c =
1$.
\section{A cosmological solution}
\label{FRW}
Let us start purely from a four-dimensional point of view, and turn on
a finite temperature for the conformal field theory which is small in
Planck units. If we calculate the corresponding energy density
$\rho$, use the trivial equation of state $p = \rho/3$, and apply the
standard equation
\eqn{aEOM}{
\left( \dot{a} \over a \right)^2 = {8\pi G_4 \over 3} \rho \,,
}
what must result is the standard radiation-dominated FRW cosmology,
\eqn{FRWMetric}{
ds^2 = -d\tau^2 + a(\tau)^2 d\vec{x}^2 \,,
}
where $a(\tau)^2$ is linear in $\tau$.\footnote{There is probably no
obstacle in principle to extending the calculation to positive spatial
curvature. However for negative spatial curvature the
pathology observed in \cite{WittenYau} might emerge.} I will always
use $\tau$ for four-dimensional cosmological time; $t$ will be
reserved for the Poincar\'e time in $AdS_5$. The only difficulty is
finding the constant of proportionality in $\rho \propto T^4$: if the
conformal field theory is interacting, it could be a non-trivial
exercise even in the limit $T \ll 1/\ell_{\rm Pl}$ where gravity loops
shouldn't matter. However if the theory in question has an AdS dual
where the supergravity approximation is good,
then the study of black holes in AdS guarantees the relation
\eqn{RhoRel}{
\rho = {3 \pi^2 \over 2} c T^4 \,,
}
where $c$ is the coefficient for the trace anomaly in a normalization
where $c = (N^2-1)/4$ for ${\cal N}=4$ $SU(N)$ super-Yang-Mills
theory.\footnote{The calculation of \cite{hs} shows that in the limit
where classical gravity is applicable to the AdS black holes there is
in fact only one independent coefficient in the trace anomaly. This
and \RhoRel\ are non-trivial constraints on theories which can have
AdS duals.} In this normalization, a single Abelian photon has $c =
1/10$. There is a standard relation in AdS/CFT \cite{hs}
\eqn{cStandard}{
c = {\pi \over 8} {L^3 \over G_5} \,.
}
It should be remarked that for ${\cal N}=4$ gauge theory at weak
coupling, \RhoRel\ becomes $\rho = 2\pi^2 c T^4$. This is the $4/3$
problem, first noted in \cite{gkPeet,StromingerUnp}, and now
understood as being a result of strong interactions.
So far we have employed the AdS/CFT correspondence merely as a tool for
determining a detail of the strong-coupling thermodynamics. However the
calculation can be done entirely on the AdS side if we take seriously the
idea that the cutoff brane is no more nor less than a coupling of gravity
to the conformal field theory. It seems inevitable from a string theory
perspective that the gravity would be quantized, and that the detailed
``structure'' of the cutoff brane encodes the details of the quantum
gravity; but by taking $T \ll 1/\ell_{\rm Pl}$ we should be able to ignore
this issue. The cutoff brane, or ``Planck brane,'' controls gravity, while
the bulk of $AdS_5$ controls the conformal field theory. By assumption,
the Planck brane is not appreciably influenced by finite temperature, but
the conformal field theory is; so we should retain \ThetaReq, but change
the bulk background from $AdS_5$ to AdS-Schwarzschild. The metric of
AdS-Schwarzschild is
\eqn{AdSSch}{\eqalign{
ds_5^2 &= e^{2r/L} \left( -h(r) dt^2 + d\vec{x}^2 \right) +
{dr^2 \over h(r)} \cr
h(r) &= 1 - b^4 e^{-4r/L} \cr
b &= \pi L T_0 \,.
}}
Here $T_0$ is the Hawking temperature associated with the time
coordinate $t$. It is a constant parameter of the
AdS-Schwarzschild solution. The calculation will deal only with
the coordinate patch covered by $(t,\vec{x},r)$.
Given an orientable surface with unit normal $n_\mu$ (which specifies a
notion of outside and inside by the direction in which it points), the
extrinsic curvature can be defined as
$\Theta_{\mu\nu} = -(\delta_\mu^\lambda -
n_\mu n^\lambda) \nabla_\lambda n_\nu$.
We will follow \cite{BK} in taking $n_\mu$ to
be the outward unit normal, which points toward the true boundary of
$AdS_5$. There is a set of solutions to \ThetaReq\ which form a
foliation of the coordinate patch in question:
\eqn{Foliate}{
{t \over L} = {e^{r/L} \over b^2} +
{1 \over 4b} \sum_{k=1}^4 i^k \log (1-i^k b e^{-r/L}) +
{t_0 \over L} \,,
}
where $t_0$ is a constant of integration specifying a particular
leaf. All leaves have the same induced metric since they are related
by translation in $t$. Using \ThetaReq\ with the same constant of
proportionality, $1/L$, avoids a four-dimensional cosmological
constant---more about this later. It proves most convenient to
parametrize a particular leaf by $(r_*,\vec{x})$, where $r = r_*$ is a
solution of \Foliate\ for $r$ in terms of $t$. Then the induced
metric is
\eqn{InducedG}{
ds_{\rm (induced)}^2 =
-{e^{4r_*/L} \over b^4} dr_*^2 + e^{2r_*/L} d\vec{x}^2 \,.
}
If we define
\eqn{TauDef}{
\tau = L {e^{2r_*/L} \over 2 b^2}
}
then the metric \InducedG\ assumes the standard FRW form:
\eqn{TauForm}{\eqalign{
ds_{\rm (induced)}^2 &= -d\tau^2 + a(\tau)^2 d\vec{x}^2 \cr
a(\tau) &= b \sqrt{2 \tau \over L} \,.
}}
Thus we do indeed observe the linear $a(\tau)^2$ that we expected.
This behavior is strictly a consequence of conformal invariance: any
conformal field theory provides a source term for Einstein's equations
just like a bunch of massless photons.
We can be a little more quantitative and rederive the coefficient in
\RhoRel\ from the new perspective. In the late time limit, we can use
the relation
\eqn{GFiveFour}{
{1 \over G_4} = {1 \over G_5}
\int_{-\infty}^{r_*} dr \, e^{2 (r-r_*)/L}
= {L \over 2 G_5} \,.
}
Actually this relation comes from a Kaluza-Klein reduction of
five-dimensional gravity to four for a horospherical Planck brane in
pure $AdS_5$. It should be OK to leading order for a brane in an
asymptotically $AdS_5$ region of bulk spacetime, provided the brane is
only slightly curved on the scale $L$. Such a brane is locally like a
horosphere of $AdS_5$. It is perhaps more common in the literature to
see a modification of \GFiveFour\ that arises from removing the factor of
$e^{-2 r_*/L}$ from inside the integral.
The discrepancy is merely due to the fact that our four-dimensional
metric is precisely the induced metric, whereas more commonly the
four-dimensional metric on a horosphere is taken to be $e^{-2 r_*/L}
ds_{\rm (induced)}^2$. The form of \GFiveFour\ is forced on us by the
choice $ds_4^2 = ds_{\rm (induced)}^2$, which does seem the natural
one in the present context.
Combining the relation
\eqn{LateTimes}{
{1 \over 4\tau^2} = \left( {\dot{a} \over a} \right)^2 =
{8\pi G_4 \over 3} \rho
}
with \GFiveFour\ and \cStandard\ leads to
\eqn{RhoLate}{
\rho = {3 b^4 \over 16 \pi G_5 L e^{4 r_*/L}} =
{3\pi^2 \over 2} c (e^{-r_*/L} T_0)^4 \,.
}
Now, the temperature $T_0$ measured with respect to the time $t$ is
not the same as the temperature $T$ measured with respect to the time
$\tau$; rather,
\eqn{TTRel}{
T = {dt \over d\tau} T_0 = e^{-r_*/L} T_0 \,,
}
where we have used the relation $t = \sqrt{2L\tau}/b$. So \RhoLate\
is indeed identical to \RhoRel, coefficient and all.
The foregoing calculation is more than a formal manipulation: it is an
illustration that {\it string theory on $AdS_5$ is
identical to a 3+1-dimensional conformal field theory.}
We wanted our cosmology to be driven by the
conformal field theory dual to the bulk AdS geometry rather than by
anything on the Planck brane. So we left the Planck brane in its
ground state and made the bulk AdS geometry thermal by adding a black
hole horizon.
It is worth remarking that no stabilization mechanism was employed
because none was needed. From a brane-world point of view, the worst
has already happened: the negative tension brane of \cite{RShierarchy}
has retreated to infinity,\footnote{The reality of such a negative
tension object is something I am only provisionally willing to allow
for the sake of argument, since I am aware of no fully satisfactory
string theory construction of it as yet in an AdS background.} and the
delicate near-horizon cusp has been cut off by a finite temperature
horizon. The effect of that horizon is most transparent when viewed
in light of the AdS/CFT correspondence: it means that the
$3+1$-dimensional conformal field theory is at finite temperature.
In a generic bulk geometry, the retreating Planck brane would cause
the four-dimensional Newton constant to change. In this regard, an
asymptotically AdS space is very special: provided we use the induced
metric on the Planck brane (rather than some warping of it) as the
four-dimensional metric, \GFiveFour\ will hold asymptotically
when the Planck brane is moving in the asymptotically AdS region with
curvatures which are small compared to $L$.
That $ds_{\rm (induced)}^2$ turned out to be {\it exactly} the
radiation-dominated FRW metric should excite some suspicion.
Mightn't there be quantum gravity effects at
sufficiently early times which modify the picture? We derived the
agreement between \RhoRel\ and \RhoLate\ using a late-time relation,
\GFiveFour. It seems like a massive conspiracy that the physics of
early times would arrange for the cosmology to remain exactly
radiation-dominated FRW. There are limits to what we can assert about
physics at early times without specifying the nature of the Planck brane.
It seems inevitable however that \ThetaReq\ will receive corrections at
higher orders in derivatives.
Is this real cosmology? Not as it stands: nucleosynthesis would be
dramatically spoiled if the ``hidden CFT'' that $AdS_5$ represents had
any sizable effect on the radiation-dominated era of our universe.
However it is straightforward to extend the discussion by adding
matter to the brane, and its stress tensor, $T_{ij}^{\rm (matter)}$,
could take over from $T_{ij}^{\rm (CFT)}$ at late times. (If all we're
worried about is nucleosynthesis, then late times means $z \mathrel{\raise2pt\hbox{$\mathop<\limits_{\hbox{\raise3pt\hbox{$\sim$}}}$}}
10^{10}$). The AdS/CFT equation relevant to such a scenario is
\eqn{EinsteinBold}{\eqalign{
&G_{ij}^{\rm (induced)} - 8 \pi G_4 T_{ij}^{\rm (CFT)} -
8 \pi G_4 T_{ij}^{\rm (matter)} = \cr
&\qquad\quad
-{2 \over L} \left[ \Theta_{ij} - \left( \Theta +
{3 \over L} \right) g_{ij}^{\rm (induced)} \right] -
8 \pi G_4 T_{ij}^{\rm (matter)} \,.
}}
This equation is a rearrangement of a formula obtained in \cite{BK}
in the course of deriving quasi-local stress-energy tensors for
AdS/CFT in various dimensions.\footnote{Only I have changed the sign
of the Einstein tensor. This is necessary because of a difference
in sign conventions. The derivation in section~\ref{General} will
serve as a check that the signs in \EinsteinBold\ are consistent.} I
have used \GFiveFour\ to define $G_4$. It is necessary to check that
visible matter does not lose energy to the CFT fast enough to spoil
the cosmology. Since the CFT's couplings are essentially of
gravitational origin, this is perhaps plausible. An estimate will be
presented in section~\ref{Estimate}.
Actually, \EinsteinBold\ is only an approximate statement of translation,
via AdS/CFT, from purely four-dimensional quantities to quantities which
constrain how the Planck brane sits in the five-dimensional spacetime.
Equations with physical meaning arise from setting either side equal to
zero. In this section we considered a case where $T_{ij}^{\rm (brane)} =
0$; then setting the right hand side equal to zero amounts to requiring
\ThetaReq. Solving \ThetaReq\ gave us back radiation-dominated FRW
cosmology, which perhaps sounded surprising; but the identity
\EinsteinBold\ makes it inevitable, because if the right hand side
vanishes, so must the left hand side.
Suppose now we knew all about the matter on the brane, and discovered
that it generated a positive cosmological constant: $-8 \pi G_4
T_{ij}^{\rm (matter)} = \Lambda g_{ij}^{\rm (induced)}$ with $\Lambda
> 0$. Assuming the AdS part to be at zero temperature, we would then
reduce \EinsteinBold\ to
\eqn{EBSimp}{
G_{ij}^{\rm (induced)} + \Lambda g_{ij}^{\rm (induced)} =
-{2 \over L} \left[ \Theta_{ij} - \left( \Theta +
{3 \over L} + {L^2 \Lambda \over 2} \right)
g_{ij}^{\rm (induced)} \right] \,.
}
It is straightforward to show that setting the right hand side equal
to zero leads to a hypersurface in $AdS_5$ whose induced metric is
$dS_4$. Approximately this calculation has appeared elsewhere in the
literature, for instance
\cite{Nihei:1999mt,Kaloper:1999sm,DFGK,Vollick}. A direct analog in
lower dimensions was treated in \cite{Cvetic:1993xe}, where a fairly
general discussion of induced metrics on codimension one domain walls
in $AdS_4$ was also given.
\section{The general framework}
\label{General}
The equation \EinsteinBold\ is an approximate first variation of a
more general relation, which is the natural extension of the
prescriptions of \cite{gkPol,witHolOne}:
\eqn{DarkSide}{\eqalign{
S_{\rm eff}[\gamma_{ij},\psi] &=
S_{\rm 4d\,gravity}[\gamma_{ij}] +
S_{\rm 4d\,matter}[\gamma_{ij},\psi] +
W_{\rm CFT}[\gamma_{ij}] \cr\noalign{\vskip1.5\jot}
&= \mathop{\rm extremum}_{g_{ij}^{\rm (induced)} = \gamma_{ij}} \left(
S_{\rm bulk}[g_{\mu\nu}] + S_{\rm brane}[g_{ij}^{\rm (induced)},\psi]
\right) \,.
}}
The metric
$\gamma_{ij}$ is the metric on the Planck brane, and $\psi$ are the
extra matter fields which live on the Planck brane.
The first equation indicates a natural way of splitting up the
four-dimensional effective action into four-dimensional gravity, the
CFT, and the four-dimensional matter which comes from excitations on
the Planck brane. The second equation is the actual statement of
AdS/CFT, which in this case includes a ``brane reduction'' of
five-dimensional gravity to four dimensions, as envisaged in
\cite{RShierarchy,RSalt}. $W_{\rm CFT}$ is the generating functional
of connected Green's functions of the conformal field theory, with a cutoff
imposed at some energy scale $\Lambda$. See \cite{SusskindWitten} for
an early discussion of cutoffs in AdS/CFT.
One way to define the cutoff $\Lambda$ is as the energy of a fundamental
string stretched from the Planck brane all the way to the horizon of
$AdS_5$. This gives $\Lambda \sim L/\alpha'$ if we measure energies with
respect to a time $\tau$ on the Planck brane such that $\gamma_{\tau\tau} =
-1$. There can however be ambiguities in normalizing $\Lambda$, depending
on the physical question one is asking, as explained in
\cite{PeetPolchinski}. If $AdS_5$ is generated as the background of many
coincident D3-branes, then we can imagine peeling one of them off and
bringing it close to the Planck brane. A fundamental string stretched from
this D3-brane back to the main cluster has the interpretation of a massive
W-boson. In the supergravity description, this fundamental string
stretches to the horizon of $AdS_5$. Thus the relation $\Lambda \sim
L/\alpha'$ has a simple motivation in terms of a BPS quantity, namely the
mass for the heaviest W-boson which can be included in the effective theory
by Higgsing the CFT. A comprehensive discussion of interactions might
require a more precise specification of how a geometric cutoff in $AdS_5$
translates into a cutoff in the four-dimensional theory.
The extremum on the right hand side of \DarkSide\
is taken subject to the boundary condition that
the metric induced from $g_{\mu\nu}$ on the cutoff brane is
$\gamma_{ij}$. It is the saddle-point approximation to quantum
gravity in the bulk. If we wanted to do quantum gravity in some more
complete way (i.e.~string theory), we would make the replacement
\eqn{Quantize}{
{\rm extremum} \, S \to {1 \over i} \log \int [{\cal D} g] e^{i S} \,,
}
where $\int [{\cal D} g]$ represents path integration. (Path integration
in the sense of \Quantize\ would amount to closed string field theory---a
subject where our understanding is incomplete. We might however imagine some
other way of improving the saddle point approximation). In a real
string theory model, there would probably be many more bulk fields besides
the metric $g_{\mu\nu}$ that $S_{\rm bulk}$ would depend on, and they would
also have to have their boundary values specified in the extremum (or path
integral). $W_{\rm CFT}$ would depend on those boundary values, and there
would also be new terms added to $S_{\rm 4d\,gravity}$ for the dynamics of
the zero modes of the extra fields. How much of a problem all this extra
junk is depends on the couplings to the standard model fields. The
optimistic view is that such couplings are about as important in particle
physics contexts as the coupling of electrons and quarks to gravity. The
zero modes of extra bulk fields would modify long-distance four-dimensional
gravity if they remained massless, but any sort of confinement or mass
generation mechanism could prevent this problem.
Like all of AdS/CFT, \DarkSide\ is a claim to be substantiated rather
than an assumption. However, it is difficult to give a complete proof
because
$W_{\rm CFT}$ is a complicated non-local functional of $\gamma_{ij}$
whose exact form is independently accessible only through
a strong coupling QFT computation. If one takes the boundary to be
the true boundary of $AdS_5$, the evidence is compelling \cite{MAGOO}
that the extremum on the right hand side of \DarkSide\ does indeed
lead to the generating functional of connected Green's functions for a
CFT.\footnote{A conformal transformation on $\gamma_{ij}$ is needed
as the cutoff is removed to keep $\gamma_{ij}$ finite. However,
$W_{\rm CFT}$ without a cutoff only depends on the conformal class of
$\gamma_{ij}$.} Through the UV-IR relation we understand that cutting off a
portion of $AdS_5$ should change physics in the ultraviolet only.
Thus \DarkSide\ is true insofar as it is well-defined (that is, on the
level of an effective field theory on energy scales much lower than the
cutoff $\Lambda$) provided we can show that $S_{\rm
4d\,gravity} + S_{\rm 4d\,matter}$ emerges from the extremum on the
right hand side. That is what I will actually demonstrate concretely.
In the process I will derive \GFiveFour\ in a general setting, and
also check that the sign that seemed worrisome in \EinsteinBold\ is
OK.
The proof is piggy-backed on the calculations of \cite{hs}. In order
to keep the presentation self-contained, I will recapitulate parts of
that work. The setting is a foliation
of a five-dimensional Einstein manifold ${\cal M}$ (for instance,
$AdS_5$ or AdS-Schwarzschild), whose boundary has a metric in the
conformal class of a specified metric $\bar{g}_{(0)}$, and whose metric can
be written in the form \cite{FeffGraham}
\eqn{FeffMetric}{
ds_5^2 = g_{\mu\nu} dx^\mu dx^\nu = {1 \over 4 \rho^2} d\rho^2 +
{1 \over \rho} \bar{g}_{ij} d\xi^i d\xi^j \,.
}
In \FeffMetric\ and the following equations, $L$ has been set to
$1$. The metric $\bar{g}_{ij}$ can depend on $\rho$, but according to
\cite{hs,FeffGraham} it has an expansion
\eqn{gExpand}{
\bar{g} = \bar{g}_{(0)} + \rho \bar{g}_{(2)} +
\rho^2 \log\rho \, \bar{h}_{(4)} + \rho^2 \bar{g}_{(4)} + \ldots \,.
}
Here $\bar{g}_{(2)}$ and $\bar{h}_{(4)}$ are tensors constructed from
$\bar{g}_{(0)}$ using two and four derivatives, respectively.
The expansion breaks down after the logarithmic term, in the sense
that the $\bar{g}_{(n)}$ are no longer covariant tensors. Fortunately
the first two terms of \gExpand\ are all that we will need.
Explicitly,
\def{\buildrel \scriptscriptstyle{o} \over R}{}{{\buildrel \scriptscriptstyle{o} \over R}{}}
\def{\buildrel \scriptscriptstyle{o} \over a}{}{{\buildrel \scriptscriptstyle{o} \over a}{}}
\eqn{GTwoForm}{
\bar{g}^{(2)}_{ij} = {1 \over 2} \left( {\buildrel \scriptscriptstyle{o} \over R}{}_{ij} -
{1 \over 6} {\buildrel \scriptscriptstyle{o} \over R}{} \, \bar{g}^{(0)}_{ij} \right) \,,
}
where ${\buildrel \scriptscriptstyle{o} \over R}{}_{ij}$ is the Ricci tensor of $\bar{g}^{(0)}_{ij}$ and
${\buildrel \scriptscriptstyle{o} \over R}{}$ is the associated Ricci scalar.
The action under the extremum in \DarkSide\ is
\eqn{BulkAction}{\eqalign{
&S_{\rm bulk}[g_{\mu\nu}] + S_{\rm brane}[g_{ij}^{\rm (induced)},\psi]
= \cr
&\qquad\quad {1 \over 16 \pi G_5} \int_{\cal M} d^5 x \, \sqrt{g}
\left[ R + 20 \right] +
{1 \over 16 \pi G_5} \int_{\partial {\cal M}_\epsilon} d^4 \xi \,
\sqrt{g^{\rm (induced)}} \left[ -2 \Theta + \alpha \right] \,.
}}
We have located the cutoff brane on the hypersurface $\partial {\cal
M}_\epsilon$ defined by the equation $\rho = \epsilon$. We have also
defined
\eqn{AlphaDef}{
\alpha = \alpha_0 + {16 \pi G_5 \over \sqrt{g^{\rm (induced)}}}
{\cal L}_{\rm matter}(g_{ij}^{\rm (induced)},\psi) \,.
}
The constant $\alpha_0$ is what we will adjust to balance the tension
of the Planck brane against the bulk cosmological constant. An
imperfect adjustment would lead to the $dS_4$ induced metric, as
commented on after \EBSimp. Thus we are not claiming to make headway
on the cosmological constant problem; rather, we are pushing it into
the Planck brane. The extrinsic curvature term in \BulkAction\ is
necessary in order to have a well-defined variational principle.
Extremizing \BulkAction\ subject to $g_{ij}^{\rm (induced)} = \gamma_{ij}$
can be achieved by letting $ds_5^2$ have the form \FeffMetric\ and then
setting $\bar{g}_{ij} = \epsilon \gamma_{ij}$ at $\rho = \epsilon$
(this is at least true up to errors which will be subleading in a
derivative expansion). Then (cf.~(10) of \cite{hs})
\eqn{ExtremizedBulk}{
\mathop{\rm extremum}_{g_{ij}^{\rm (induced)} = \gamma_{ij}} \left(
S_{\rm bulk}[g_{\mu\nu}] + S_{\rm brane}[g_{ij}^{\rm (induced)},\psi]
\right) = {1 \over 16 \pi G_5} \int d^4 \xi \, {\cal L}
}
where
\eqn{EllDef}{\eqalign{
{\cal L} &= 4 \int_\epsilon {d\rho \over \rho^3} \sqrt{|\det\bar{g}|} +
\left[ {1 \over \rho^2} (-8 + 4 \rho \partial_\rho + \alpha)
\sqrt{\bar{g}} \right]_{\rho = \epsilon} \cr
&= \sqrt{|\det\bar{g}_{(0)}|} \left[
{\alpha - 6 \over \epsilon^2} +
{\alpha \over 2\epsilon} \mop{tr} \bar{g}_0^{-1} \bar{g}_2 -
\log \epsilon \, {\buildrel \scriptscriptstyle{o} \over a}{}_{(4)} + \hbox{(finite)} \right] \,.
}}
Here we have defined
\eqn{aZeroDef}{
{\buildrel \scriptscriptstyle{o} \over a}{}_{(4)} = -{1 \over 8} {\buildrel \scriptscriptstyle{o} \over R}{}^{ij} {\buildrel \scriptscriptstyle{o} \over R}{}_{ij} +
{1 \over 24} {\buildrel \scriptscriptstyle{o} \over R}{}^2 \,.
}
This quantity was identified in \cite{hs} as the conformal anomaly of
the CFT. The AdS/CFT prescription as detailed there is simply to
remove the terms that diverge as $\epsilon \to 0$ via local
counterterms. This is the only sensible course if the ultimate goal
is to take $\epsilon \to 0$ so that the cutoff boundary becomes the
true boundary. Instead we want to keep the cutoff boundary at a
finite, arbitrary $\epsilon$ and regard the induced metric
$\gamma_{ij}$ on $\partial {\cal M}_\epsilon$ as the Einstein metric
of the four-dimensional world. Rewriting \EllDef\ in terms of
$\gamma_{ij}$, one finds
\eqn{EllSimp}{
{\cal L} = \sqrt{|\det\gamma|} \left[
\alpha - 6 + {1 \over 2} R - \log\epsilon \, a_4 + \ldots \right] \,,
}
where now $R$ is the Ricci scalar of the metric $\gamma_{ij}$ and
$a_4$ is defined as in \aZeroDef, only using curvature tensors
pertaining to $\gamma_{ij}$ rather than to $\bar{g}^{(0)}_{ij}$. One
might fear that the logarithmic term in \gExpand\ would contribute to
the $\log\epsilon$ term in \EllSimp. It doesn't because $\mop{tr}
\bar{g}_{(0)}^{-1} \bar{h}_{(4)} = 0$.
Because powers of $\epsilon$ cancel in \EllSimp\ (and $\epsilon$ is
finite anyway) there is no longer an expansion parameter in \EllSimp.
The expansion can only be justified as a derivative expansion,
provided that the embedding of the cutoff brane in the
five-dimensional Einstein space involves only curvatures which are
slight on the length scale $L$. Combining \AlphaDef\ and
\ExtremizedBulk\ with \EllSimp, setting $\alpha_0 = 6$, and
repristinating powers of $L$, we find
\eqn{FinalDark}{\eqalign{
&\mathop{\rm extremum}_{g_{ij}^{\rm (induced)} = \gamma_{ij}}
\left( S_{\rm bulk}[g_{\mu\nu}] +
S_{\rm brane}[g_{ij}^{\rm (induced)},\psi] \right) = \cr
&\qquad\quad{}
{L \over 32 \pi G_5} \int d^4 \xi \, \sqrt{\gamma} R +
\int d^4 \xi \, \sqrt{\gamma}
{\cal L}_{\rm matter}(\gamma_{ij},\psi) +
W_{\rm CFT}[\gamma_{ij}]
}}
where $W_{\rm CFT}$ includes the $\log\epsilon$ term in \EllSimp\
plus all the other terms which we indicated with $\ldots\,$. We indeed
verify the relation $G_4 = 2 G_5/L$. Also, since the Ricci scalar
came in with the right sign in \FinalDark, the signs of \EinsteinBold\
are consistent. The calculation leading to \FinalDark\ is similar to
Kaluza-Klein reduction, the main difference being that the relation
$G_4 = 2 G_5/L$ does not involve the total length of the fifth
dimension (which could be infinite), but rather the curvature scale of
the five-dimensional geometry. This makes the current scenario rather
different from those of \cite{Dimopoulos}, where the circumference of
the extra dimensions does affect the four-dimensional Planck length.
Extremizing \BulkAction\ with respect to $g_{\mu\nu}$ without requiring
$g_{ij}^{\rm (induced)} = \gamma_{ij}$ would amount,
at leading order in
derivatives, to setting the right hand side of \EinsteinBold\ to zero, as
well as satisfying the bulk equations of motion.
Given some information regarding the structure of the Planck brane,
higher derivative corrections to \ThetaReq\ and to the right hand side
of \EinsteinBold\ would be accessible through a more meticulous treatment
of this unrestricted extremization problem. The trick of
\DarkSide\ is to extremize first with the induced metric held fixed
and then argue that the
extremization that remains to be carried out gives us the equations of
four-dimensional gravity (and brane matter if we want it),
plus something non-local
which we called $W_{\rm CFT}$. The claim that this something arises
equivalently by integrating out a CFT below a cutoff $\Lambda$
is the substance of AdS/CFT and
the basis for the suggestions in \cite{juanPrivate,HV,WittenComment}.
The argument \FeffMetric-\FinalDark\ stands in relation to the
observation \cite{RSalt} of a normalizable graviton approximately as
the derivation of the low-energy effective action of string theory via
beta-functions stands in relation to the calculation of the massless
string spectrum.
It should be possible to relate corrections to Einstein's equations and hence
Newton's force law directly to the anomaly term $a_4$, proceeding along
the lines of \cite{LiuTseytlin}. However it is more
transparent to follow the analysis of \cite{WittenComment}, where we
merely differentiate $W_{\rm CFT}[\gamma_{ij}]$ twice with respect to
$\gamma_{ij}$ to obtain the first correction to the graviton
propagator (see figure~\ref{figB}).
\begin{figure}
\centerline{\psfig{figure=figB.eps,width=5in}}
\caption{Contributions to the graviton propagator, following
\cite{WittenComment}: a) free graviton propagation; b) leading CFT
correction. The blob between the stress tensor insertions is intended
to denote the full $\langle TT \rangle_{\rm CFT}$
correlator.}\label{figB}
\end{figure}
The position space two-point function of the CFT stress tensor has
the form $\langle T(x) T(0) \rangle \sim c/x^8$. In momentum space
this is $\langle T(p) T(-p) \rangle \sim c \, p^4 \log p$. The corrected
graviton propagator is
\eqn{PropCorrect}{\eqalign{
G^{(2)}(p) &\sim {1 \over p^2} +
{1 \over p^2} \ell_{\rm Pl}
\left( c \, p^4 \log p \right) \ell_{\rm Pl} {1 \over p^2} \cr
G^{(2)}(x) &\sim {1 \over x^2} +
{c \, \ell_{\rm Pl}^2 \over x^4} \,,
}}
where the factors of the four-dimensional Planck length are vertex
factors for the coupling of the stress tensor to the graviton. The
altered propagator gives rise to deviations from Newton's $1/r^2$
force law, estimated already from the AdS side in \cite{RSalt}:
\eqn{ForceCorrect}{
F = {G m_1 m_2 \over r^2} \left( 1 +
a_1 {L^2 \over r^2} + \ldots \right) \,,
}
where $a_1$ is a dimensionless number on the order of unity.
In \ForceCorrect\ I have used \cStandard\
and \GFiveFour\ to combine $c \, \ell_{\rm Pl}^2$ into $\pi L^2/4$. I do
not claim any originality for the computation in \PropCorrect\ and
\ForceCorrect. The only further addition I would make to the recorded
comments in \cite{WittenComment} is that the coefficient of leading
correction is indeed computable from the CFT side: up to factors of order
unity it is $G_4$ times the central charge of the CFT.
Clearly, by differentiating \DarkSide\ and keeping track of all the
Lorentz structure we could obtain the corrected propagator in complete
detail and extract the exact value of $a_1$. I will refrain from
entering into this computation here because another group is pursuing
similar lines \cite{GKR}. It was important however to present the
general outline of the analysis because it will figure prominently in
the next section.
\section{Bounds and estimates}
\label{Estimate}
Note that \cStandard\ and \GFiveFour\ together imply that the central
charge is $c = {\pi \over 4} {L^2 \over \ell_{\rm Pl}^2}$, where as usual
$\ell_{\rm Pl}$ is the four-dimensional Planck length. For $AdS_5$
backgrounds arising from type~IIB geometries including D3-branes, $c \sim
N^2$ where $N$ is the number of D3-branes. So $N \sim L/\ell_{\rm Pl}$.
To be definite, let us suppose that $L$ is on the order of a micron.
Direct measurements of gravity already restrict $L \mathrel{\raise2pt\hbox{$\mathop<\limits_{\hbox{\raise3pt\hbox{$\sim$}}}$}} 1 \, {\rm mm}$,
and proposed experiments might probe Newton's force law to distances as
small as a micron. $L \sim
1 \, \mu{\rm m}$ means $N \sim 10^{29}$. This number seems on the
high side for a string compactification: something has to soak up all the
five-form flux. D3-brane charge is conserved, so it is true that if
we managed to set $N = 10^{29}$ through some arcane string theory
construction, we wouldn't worry about it wiggling. As disciples of
AdS/CFT we would also be relieved that five-dimensional quantum
gravity effects aren't an immediate problem. However, a large hidden
CFT is very dangerous in cosmology. Nucleosynthesis, for example,
would be spoiled if $\rho_{\rm CFT} \mathrel{\raise2pt\hbox{$\mathop>\limits_{\hbox{\raise3pt\hbox{$\sim$}}}$}} \rho_{\rm SM}$, where
$\rho_{\rm SM}$ is the energy density of the Standard Model fields.
Let us assume then that $\rho_{\rm CFT} \ll \rho_{\rm SM}$ around the
time of nucleosynthesis. Because the CFT has a large number of
degrees of freedom as compared to the Standard Model, this is possible
only if the CFT is much colder than Standard Model excitations. Suppose
that the Standard
Model and the CFT are to a good approximation decoupled. Then
$\rho_{\rm CFT}$ and $\rho_{\rm SM}$ decrease in fixed ratio during
the radiation-dominated era, up to factors of order unity associated
with freezing out the various massive fields of the Standard Model.
In the matter-dominated era, $\rho_{\rm CFT}$ and $\rho_{CBR}$ decrease
in fixed ratio. So we can guarantee that nucleosynthesis is unaffected by
the CFT if we demand $\rho_{\rm CFT} \ll
\rho_{\rm CBR}$ today. This translates roughly to $T_{\rm CFT} \mathrel{\raise2pt\hbox{$\mathop<\limits_{\hbox{\raise3pt\hbox{$\sim$}}}$}}
T_{\rm CBR}/c^{1/4} \approx 10^{-14} \, {\rm K}$ today if we want $L$
on the order of a micron.\footnote{We have used the AdS/CFT prediction
$\rho \sim cT^4$. Naively counting flat directions in ${\cal N}=4$
super-Yang-Mills theory suggests $\rho \sim \sqrt{c} T^4$. Even if
this were somehow true for a special CFT, it would only soften
\eno{NScaling} to $T_{\rm CFT} \mathrel{\raise2pt\hbox{$\mathop<\limits_{\hbox{\raise3pt\hbox{$\sim$}}}$}} T_{\rm CBR}/N^{1/4}$.} To
summarize,
\eqn{NScaling}{
c \sim N^2 \,, \qquad\ L \sim N \ell_{\rm Pl} \,, \qquad\
T_{\rm CFT} \mathrel{\raise2pt\hbox{$\mathop<\limits_{\hbox{\raise3pt\hbox{$\sim$}}}$}} T_{\rm CBR}/\sqrt{N} \,.
}
Suppose the CFT is cold enough at some early time to satisfy $\rho_{\rm
CFT} \ll \rho_{\rm SM}$.\footnote{Section~\ref{FRW} treated the opposite
limit. One should be able to use the equation \EinsteinBold\ with $T^{\rm
(CFT)}_{ij}=0$ to find a hypersurface in $AdS_5$ whose induced metric is
real-world cosmology. But this is only an equivalent means to find what we
already know by solving Einstein's equations. In this section we will
``cast down the ladder'' and work directly in four dimensions whenever
possible.} Cosmology at later times could still be spoiled if energy leaks
too quickly from visible matter into the CFT. The analogous problem in
theories with compact extra dimensions is cooling by emission of bulk
gravitons \cite{DimopoulosTwo}. To evaluate whether there is a problem in
our case, we must investigate the mechanisms of thermal equilibration
between the CFT and the other matter in the universe, operating on the
assumption that the CFT is very cold. Fortunately the tools are already
partly in hand. Standard Model particles can lose energy to the conformal
field theory through processes controlled by the graph in
figure~\ref{figA}a). The inclusive rates from these graphs are related to
the $1/r^4$ correction to Newton's law through the unitarity relation
illustrated in figure~\ref{figA}b).
\begin{figure}
\centerline{\psfig{figure=figA.eps,width=6in}}
\caption{a) Standard Model particles losing energy via graviton
exchange to CFT excitations. b) The inclusive rate is given by a
unitarity cut of the first correction to the graviton
propagator.}\label{figA}
\end{figure}
In particular, the inclusive rate goes as $\ell_{\rm Pl}^2 L^2$. By
dimensional analysis the contribution they make to the loss of
Standard Model energy density over time is
\eqn{RhoLoss}{
\left( {d\rho \over d\tau} \right)_{\rm lost} = -a_2
\ell_{\rm Pl}^2 L^2 T_{\rm SM}^9 \,,
}
where $a_2$ is a dimensionless number of order unity and $T_{\rm SM}$
is the temperature of Standard Model excitations. From an AdS point
of view, \RhoLoss\ is literally the rate at which energy density falls
across the horizon to be absorbed by the D3-branes. Three powers of
$T_{\rm SM}$ come from the absorption cross section
\cite{IgorAbsorb,gktAbsorb,gkSchwing}; also there are powers of
$T_{\rm SM}$ from the finite temperature kinematics of the Standard
Model particles. Energy density also decreases because of Hubble
expansion: in total,
\eqn{RhoEvolve}{
{d\rho_{\rm SM} \over d\tau} = -{\dot{a} \over a} T_{\rm SM}^4 -
\ell_{\rm Pl}^2 L^2 T_{\rm SM}^9 \,,
}
where we have dropped factors of order unity. One such factor is the
central charge of the Standard Model fields which are light compared to the
temperature at any given time. Approximately this same factor appears in
both terms on the right hand side of \RhoEvolve, so it doesn't matter much
for the relative size of the terms. However $a_2$ does matter, and it
should be computed if a more accurate estimate than the one presented here
is desired. We have also suppressed a term in \RhoEvolve\ for CFT energy
leaking back into visible fields, but that is OK since we are operating on
the assumption that the CFT is cold.
To determine whether the CFT is appreciably affecting the cosmology,
one should compare the two terms in \RhoEvolve. Their ratio is
\eqn{FMerit}{
\kappa = \ell_{\rm Pl}^2 L^2 T_{\rm SM}^5 H^{-1}
}
where $H^{-1} = a/\dot{a}$ is the inverse Hubble time (a function of
$\tau$). The CFT will not appreciably affect cosmology as long as
$\rho_{\rm CFT} \ll \rho_{\rm SM}$ and $\kappa$ is small. What
small means in this context depends on all the ``factors of order
unity'' that we have dropped. All these factors are calculable: once
we have \cStandard\ and \GFiveFour\ the rest is essentially
kinematics. In order to make some preliminary estimates I will assume
that the Hubble expansion term in \RhoEvolve\ dominates over the
energy loss term when $\kappa \ll 1$.
The state of the universe today does not lead to a dramatic bound on
$L$: for instance, estimating $\kappa$ for the rate of energy loss from the
CBR to the CFT gives
\eqn{LBoundToday}{
L^2 \ll {1 \over \ell_{\rm Pl} H_o^{-1} T_{\rm CBR}^5} \sim
10^{33} \, {\rm cm}^2 \,,
}
which is easily passed by any realistic theory. However the bound
tightens as one goes back in time. Tracing the matter-dominated
cosmology back to the time of last scatter at $z \sim 10^4$, one
obtains roughly
\eqn{LBoundLastScatter}{
L^2 \ll 10^{33} \, {\rm cm}^2
\left( {a_{\rm last\ scatter} \over a_o} \right)^5
{H_o^{-1} \over H_{\rm last\ scatter}^{-1}}
= 10^{33} \, {\rm cm}^2
\left( {a_{\rm last\ scatter} \over a_o} \right)^{7/2}
= 10^{19} \, {\rm cm}^2 \,,
}
still not meaningfully restrictive. Tracing the radiation-dominated
cosmology back to nucleosynthesis at $z \sim 10^{10}$, one obtains
\eqn{LBoundNucleosynthesis}{
L^2 \ll 10^{19} \, {\rm cm}^2
\left( {a_{\rm nucleosynthesis} \over a_{\rm last\ scatter}}
\right)^5
{H_{\rm last\ scatter}^{-1} \over H_{\rm nucleosynthesis}^{-1}}
= 10^{21} \, {\rm cm}^2
\left( {a_{\rm nucleosynthesis} \over a_{\rm last\ scatter}}
\right)^3
= 10 \, {\rm cm}^2 \,.
}
Still this bound is satisfied with four orders of magnitude to spare
(in $L$) if we suppose $L$ to be on the order of a micron. I
emphasize the extreme simple-mindedness of the estimates: all I have
done in \LBoundNucleosynthesis\ is to write
\eqn{ReallySimple}{
L^2 = \kappa \, {1 \over \ell_{\rm Pl}^2 T_{\rm SM}^5 H^{-1}}
\approx \kappa \, {1 \over
\ell_{\rm Pl}^2 T_{\rm CBR}^5 H_o^{-1}}
{1 \over z_{\rm last\ scatter}^{7/2}} \,
\left( z_{\rm last\ scatter} \over z_{\rm nucleosynthesis}
\right)^3 \,,
}
and then demand $\kappa \ll 1$. The powers of $z$ in
\LBoundLastScatter-\ReallySimple\ arise from the relations $H^{-1} \sim
a^{3/2}$ for the matter dominated cosmology and $H^{-1} \sim a^2$ for
the radiation-dominated cosmology. In view of the actual number
obtained in \LBoundNucleosynthesis, a more accurate estimate would be
desirable. One can also attempt to trace cosmology back to larger $z$
and tighten the bound on $L$ further, if one feels convinced that
$\kappa$ must still be small for $z > 10^{10}$.
An independent bound on $L$ could obtained by checking the effect on
supernovas of energy loss to the CFT, as in \cite{DimopoulosTwo}. The
energy scales here are on the order of $30 \, {\rm MeV}$, so a
slightly better bound than \LBoundNucleosynthesis\ might be expected.
There is yet another way to set a bound on $L$ if we assume that the
$AdS_5$ geometry comes from type~IIB string theory through some
Freund-Rubin ansatz or related compactification. In such
compactifications, the extra five dimensions have the same length
scale $L$ as $AdS_5$. Suppose $AdS_5 \times S^5$ is the relevant
geometry. Then the standard string theory relation $16 \pi G_{10} =
(2\pi)^7 g_s^2 \alpha'^4$ combined with \GFiveFour\ and $\mop{Vol} S^5 =
\pi^3 L^5$ leads us to
\eqn{LGalpha}{
L = g_s^{1/3} \left( 16 \pi^3 \alpha'^4 \over G_4
\right)^{1/6} \,.
}
Type~IIB theory has an S-duality symmetry which takes $g_s \to
1/g_s$. Thus we can assume that $g_s \leq 1$. A conventional value
of $\sqrt{\alpha'}$ would be only a few times the four-dimensional
Planck length, $\ell_{\rm Pl}$. This results in a bound on $L$ which
is also a few times $\ell_{\rm Pl}$. In order to make $L$ observably
big, we would have to make $\sqrt{\alpha'}$ big too. What is the
biggest $\sqrt{\alpha'}$ we could possibly imagine? In the old days
of string theory the answer would have been $\sqrt{\alpha'} \sim 1 \,
{\rm GeV}^{-1} \approx 0.2 \, {\rm fm}$: this is literally the Regge
slope of observed hadronic spectra. In recent literature
\cite{Antoniadis}, values of $\sqrt{\alpha'}$ as big as $1 {\rm
TeV}^{-1}$ have been regarded as acceptable. Plugging these numbers
into \LGalpha\ leads to
\eqn{ReggeBound}{\eqalign{
L \mathrel{\raise2pt\hbox{$\mathop<\limits_{\hbox{\raise3pt\hbox{$\sim$}}}$}} 10^{-7} \, {\rm cm} &\qquad
\hbox{for $\sqrt{\alpha'} \sim 1 {\rm GeV}^{-1}$} \cr
L \mathrel{\raise2pt\hbox{$\mathop<\limits_{\hbox{\raise3pt\hbox{$\sim$}}}$}} 10^{-11} \, {\rm cm} &\qquad
\hbox{for $\sqrt{\alpha'} \sim 1 {\rm TeV}^{-1}$} \,.
}}
The bad news is that deviations from Newton's force law on length
scales this small won't be detected any time soon. The good news is
that standard cosmology is no problem, as far back as nucleosynthesis
and further. If we assume that the radiation-dominated FRW solution
still pertains, we can estimate the redshift $z_*$ and the thermal
energies $T_*$ at which $\kappa = 1$. The result is
\eqn{SourEnergies}{\eqalign{
z_* \sim 10^{15} \,, \quad T_* \sim 100 \, {\rm GeV}
&\qquad \hbox{for $\sqrt{\alpha'} \sim 1 {\rm GeV}^{-1}$} \cr
z_* \sim 10^{18} \,, \quad T_* \sim 100 \, {\rm TeV}
&\qquad
\hbox{for $\sqrt{\alpha'} \sim 1 {\rm TeV}^{-1}$} \,.
}}
To get the numbers in \SourEnergies\ we have combined several
approximations and assumptions. The ``error in the exponent'' should
probably be taken to be about $\pm 2$. It is somewhat suggestive that
the values of $T_*$ we found are ``within errors'' of the boundary of
our direct knowledge of particle physics. If the string scale is at a
${\rm TeV}$, then physics changes sufficiently there that we can no
longer have any confidence that the radiation-dominated FRW cosmology
is relevant. Thus the second line of \SourEnergies\ only shows that
there are no cosmological problems as far back as we can trace the
theory. Strings at a ${\rm GeV}$ are a different matter, and we will
return to them shortly.
Although type~IIB string theory provides the best-understood vacua
involving $AdS_5$, it is conceivable that some other type of string
theory, even a non-critical string, could have an $AdS_5$ vacuum: see
for example \cite{PolWall}. For a non-critical string, \LGalpha\
would not be the right estimate, since some or all of the five compact
dimensions simply aren't there. Suppose the non-critical string lives
in $n$ dimensions, with $n \geq 5$. Assume also that it exhibits some
form of S-duality, so that the coupling cannot be parametrically
large. Then
\eqn{NonCritical}{
L \mathrel{\raise2pt\hbox{$\mathop<\limits_{\hbox{\raise3pt\hbox{$\sim$}}}$}} \left( \sqrt{\alpha'} \over
\ell_{\rm Pl} \right)^{\gamma} \sqrt{\alpha'}
}
where $\gamma = 2/(n-4)$. If we allow $n$ to range from $10$ to $5$,
the corresponding range of $\gamma$ is from $1/3$ to $2$. It is also
conceivable that some intersecting configuration of branes in critical
string theory could have an $AdS_5$ component in its near-horizon
geometry, and a different relation from \LGalpha\ could pertain if
some of the branes had more than $3+1$ worldvolume dimensions. I am
not currently aware of any completely well-defined, non-critical string theory
other than the $c \leq 1$ toys. Nor can I give a string theoretic example of
intersecting branes with an $AdS_5$ near-horizon geometry. Besides,
if the extra dimensions of the intersecting branes are larger than
$L$, then the salient physics of extra dimensions would be more along
the lines of \cite{Dimopoulos} than \cite{RSalt}. For the sake of a
concrete discussion, let us stick to \LGalpha, with \NonCritical\ as
a possible alternative.
Once we have ventured to set $\sqrt{\alpha'} \approx 1 \, {\rm GeV}^{-1}$,
the burning question is why all collider physics from a ${\rm GeV}$ up to a
${\rm TeV}$ isn't dramatically different. The simplest answer is rather
iconoclastic. It is that from a four-dimensional point of view, strings
are nothing more than QCD flux tubes. For energies well above $1 \, {\rm
GeV}$, but below the cutoff scale $\Lambda \sim L/\alpha'$, a better set of
variables is the particles of the Standard Model, plus a massless
propagating graviton. In the low-energy regime where strings are the good
variables, there is a massless graviton in the closed string spectrum. The
graviton must be present in a description of the theory at any scale: on
very general grounds \cite{WeinbergWitten} it is impossible for the graviton
to be a composite particle.\footnote{I thank M.~Strassler and R.~Sundrum
for discussions on this point and related issues.} The spectrum could also
include massless open strings if the Planck brane involves D-branes. The
gluons in the Standard Model lagrangian might be represented in this way at
low energies. Intuitively, the reason why a disk diagram with two gluon
boundary insertions and one bulk insertion of a graviton wouldn't couple
gluons to gravity on the scale of femtometers is that the wavefunction
overlap is small. This is the magic of extra dimensions (exploited
similarly in \cite{Dimopoulos}): \LGalpha\ is roughly a condition on how
big the extra dimensions have to be in the well-understood type~IIB string
theory examples to make low-energy strings consistent with gravity at the
four-dimensional Planck scale. The modified relation \NonCritical\ could
be pertinent for alternative models, as discussed in the previous
paragraph.
The view taken in the previous paragraph is distinct from those of
\cite{witHolTwo} or \cite{HV}. Wilson loops in AdS/CFT usually seek
out a location of large redshift in the bulk geometry in order to
lower their tension to the scale of confinement. The current scenario
has Wilson loops terminating on the Planck brane, and the relevant
geometry is the geometry near the Planck brane. I would not exclude
scenarios where a large redshift does exist near the Planck brane, and
the parameter entering into the Regge relation is a redshifted
$\alpha'$. If that is the way we think QCD strings are realized, then
once again the bound on $L$ is tighter than $L \mathrel{\raise2pt\hbox{$\mathop<\limits_{\hbox{\raise3pt\hbox{$\sim$}}}$}} 1 \, {\rm nm}$,
since the $\alpha'$ that enters \LGalpha\ is the
un-redshifted string tension.
Strings at a ${\rm GeV}$ seem like a natural apotheosis of the proposals of
\cite{Dimopoulos,Antoniadis}. We do not have to ``get rid'' of the
graviton if there are extra dimensions on the scale of a nanometer.
(Significantly smaller $L$ would work in a model where \NonCritical\
pertains). We do not have to worry about nucleosynthesis if the estimate
\LBoundNucleosynthesis\ bears out. But we do have to face some hard
questions. First, if $L \sim 1 \, {\rm nm}$, how do we manage to
accommodate $N = L/\ell_{\rm Pl} \approx 10^{26}$ D3-branes? Something has
to soak up all the Ramond-Ramond flux, and that sounds like an impossible
stretch for string compactifications (see for example \cite{HV}). $N$
comes out somewhat smaller if $g_s$ is small, or in models where
\NonCritical\ pertains with $n<10$. Second, string theory would have to
face up to hadron physics in the energy range between pions and partons.
Regge trajectories are as suggestive as they always were, but there is much
more to be explained. Processes where some or all of the final energy
winds up in CFT excitations are likely to be a problem. However the
relevant branching ratios typically depend on $L$ rather than $\alpha'$,
and amount to yet another way of setting an upper bound on $L$. Third,
strings could stretch from the Planck brane all the way into the $AdS_5$
bulk (to connect with a D3-brane if one wants to think in those terms) at
only a finite cost in energy. The mass of such a string is roughly
$L/\alpha'$, which comes out to be approximately $3000 \, {\rm TeV}$ if we
use $L \sim 1 \, {\rm nm}$. This is out of the range of colliders, but it
is nevertheless a dangerous number for any sort of loop computation because
these strings are so numerous: there are as many of them as there are
D3-branes. Their mass is bigger if \NonCritical\ applies:
$L/\alpha' \sim 10^{18 \gamma} \, {\rm GeV}$. Fourth and finally, if flux
tubes are long strings ending on the Planck brane, then what are quarks?
As observed in section~\ref{General},
there is a precise way of characterizing the strings stretched from the
Planck brane to the horizon of $AdS_5$: they are the massive $W$ bosons
associated with the separation of the Planck brane from the $L/\ell_{\rm
Pl}$ D3-branes that create the $AdS_5$ geometry. The scale of these masses
could be lowered, say to $30 \, {\rm TeV}$, if $L$ falls
sufficiently short of saturating the bound in the first line of
\ReggeBound. Or we could return to strings at a ${\rm TeV}$ and get
approximately the same $30 \, {\rm TeV}$ Higgs scale by saturating the
bound in the second line of \ReggeBound. Either way, we are left with a
version of \cite{VafaFrampton}, only with an enormous hidden sector gauge
group and strings at a ${\rm GeV}$ or a ${\rm TeV}$. In
\cite{VafaFrampton}, it seemed like coupling the CFT to gravity might
resurrect the hierarchy problem. This is less of a problem if the string
scale is smaller than or comparable to the scale of soft breakings of the
CFT: one may hope that stringy ``softness'' ameliorates the divergences of
gravity already at the string scale. There is no clear microscopic picture
of what the theory is without specifying the nature of the Planck brane.
However, the relation \LGalpha\ between Newton's coupling and other
low-energy quantities should not depend on the detailed properties of
the Planck brane.
In conclusion, insisting that $AdS_5$ has to come from string theory
provides a bound on $L$ which is sharper than we were able to obtain
from nucleosynthesis, and which appears to rule out experimental
observation of \ForceCorrect. There are two reasons why string theory
demands a small $L$. First, $L/\ell_{\rm Pl} \sim N$, where $N$ is
the number of units of Ramond-Ramond five-form flux. It is hard to
make this number really big in string compactifications. Second, $G_4
L^6 \mathrel{\raise2pt\hbox{$\mathop<\limits_{\hbox{\raise3pt\hbox{$\sim$}}}$}} \alpha'^4$, so we can only get big $L$ if we allow big
$\alpha'$. In an attempt to be maximally optimistic about the size of
$L$, we have reconsidered strings at a ${\rm GeV}$. Even this radical
step only gave us $L \mathrel{\raise2pt\hbox{$\mathop<\limits_{\hbox{\raise3pt\hbox{$\sim$}}}$}} 1 \, {\rm nm}$. If we make $\alpha'$ even
bigger, it only heightens the difficulties we encountered trying
to make sense of ${\rm GeV}$ strings. The strategies proposed in
AdS/CFT contexts to relate strings to QCD flux tubes generally have
the property that the fundamental $\alpha'$ is smaller than $1 \, {\rm
GeV}^{-2}$, implying a tighter bound on $L$.
\section{Discussion}
\label{Discussion}
The FRW cosmology found in section~\ref{FRW} is an interesting check
of the claim that the ``alternative to compactification'' proposed in
\cite{RSalt} is equivalent to a cutoff conformal field theory coupled to
four-dimensional gravity. However, as emphasized in
section~\ref{Estimate}, the CFT shouldn't make any sizeable
contribution to the actual cosmology of our universe at times later
than $z = 10^{10}$. Before that time, one is entitled to speculate
about the physical relevance of the solution of section~\ref{FRW}.
Suppose that the CFT and the visible sector matter on the Planck brane
were in thermal equilibrium at some early time. Assuming that the CFT
has a much larger central charge, we have $\rho_{\rm CFT} \gg
\rho_{\rm matter}$, and the solution found in section~\ref{FRW} should
approximately describe the cosmology. At late times one needs
$\rho_{\rm CFT} \ll \rho_{\rm matter}$. In parallel with
\cite{DimopoulosTwo}, we might imagine an inflationary scenario where
the inflaton lives on the Planck brane. Then reheating directly
affects only the visible sector, and if $\kappa$ is small by the
time of reheating there is substantially no thermal equilibration with
the CFT.
In a scenario with ${\rm GeV}$ strings, thermalization with the CFT
sets in significantly around $100 \, {\rm GeV}$ (although we must
recall that the estimates here were extremely crude). That alone
might lead us to rule this case out unless a reheating mechanism could
be proposed at a lower scale.
In known string compactifications, the number $N$ of D3-branes is
typically on the order of $10$. As many as $10^3$ D3-branes were
claimed to be attainable in certain orbifold examples \cite{HV}. If
we take this as a strict bound, then the relation $L/\ell_{\rm Pl}
\sim N$ puts our entire discussion at an inaccessibly small length
scale: $L \sim 10^{-30} \, {\rm cm}$ for $N = 10^3$. (As usual,
$\ell_{\rm Pl}$ is the four-dimensional Planck scale). The formalism
developed in section~\ref{General} could still be useful for
extracting a ``low-energy'' effective theory---``low-energy'' being
interpreted now as much less than $10^{16} \, {\rm GeV}$. Standard
inflation occurs around $10^{14} \, {\rm GeV}$, so it is possible one
might embed a ``conventional'' inflationary model in $AdS_5$ using the
$dS_4$ solution discussed after \EBSimp. The amusing aspect of such a
model is that there is a natural candidate for the pre-inflationary
universe: it is the radiation-dominated FRW solution found in
section~\ref{FRW}.
There are two solid conclusions to be drawn from the estimates of
section~\ref{Estimate}. First, our present understanding of
nucleosynthesis would not be threatened if deviations from Newton's
force law of the form \ForceCorrect\ were found. We already know that
such deviations cannot be present on scales much larger than a
millimeter, and this is enough to suppress the associated loss of
energy to the conformal field theory for $z$ as large as $10^{10}$.
Second, string theory as we understand it seems to forbid an $AdS_5$
space large enough to cause measurable deviations from Newton's force
law. Even if we are willing to take the string scale down to $1 \,
{\rm GeV}$ and regard strings as collective effects of QCD, $L$ still
can't be larger than $1 \, {\rm nm}$.
There is nothing sacred about an $AdS_5$ bulk spacetime: it has been
the focus of so much recent literature in part because it is simple.
Practically any string theory realization of $AdS_5$ will include
scalar fields, and if they have a non-trivial profile, large
deviations from $AdS_5$ are the generic behavior far from the
boundary. The literature on
renormalization group in AdS/CFT flows provides ample evidence of this
(see for example
\cite{GPPZfirst,DZ,gDil,kSfetsos,fgpwOne,GPPZone}).
Only a subset of these geometries can support
finite temperature, due to boundary conditions on the scalars at the
black hole horizon. A felicitous feature of $AdS_5$, which will not
be shared by generic ``RG flow'' geometries, is that the relation $G_4
= 2 G_5/L$ obtains no matter where the Planck brane is in the bulk
geometry. The formalism worked out in section~\ref{General} will
still retain its general features in a more generic bulk geometry, but
details will be rather different: for instance, it is no longer clear
that the induced metric on the cutoff brane will be the Einstein frame
metric.
A cutoff brane in a bulk geometry whose AdS/CFT dual is a quantum
field theory undergoing renormalization group flow corresponds to
gravity coupled to that same QFT. The proposal of \cite{RShierarchy}
is to put the Standard Model not on the cutoff brane, but rather on
some brane far from the boundary, where $g_{tt}$ is very small. There
are several ways that such a construction could be realized in string
theory. First, if the five-dimensional space-time ends at a finite
minimum of $g_{tt}$, then one can show that the end-of-the-world brane
must have negative tension. The best-understood constructions in the
current literature which admit negative tension end-of-the-world
branes are type~I$'$ string theory and certain Calabi-Yau
compactifications of Horava-Witten theory. At the classical level,
these constructions do not allow an $AdS_5$ bulk: there is always some
scalar that evolves across the five-dimensional bulk. There is not
yet compelling evidence that all scalars could be held fixed and an
$AdS_5$ bulk obtained. It would be possible to develop a formalism
similar to the one in section~\ref{General} for type~I$'$ or
Horava-Witten constructions, but it would have more the flavor of an
ordinary Kaluza-Klein reduction, where heavy fields are integrated out
and light fields are kept. The distinctive feature of \DarkSide\ is
that it enables us to obtain a non-local functional which summarizes
the dynamics of {\it infrared} degrees of freedom.
If there are no negative tension branes, the only option is for the
five-dimensional space-time to continue all the way to $g_{tt}=0$. If
there are scalars involved, the generic behavior is for curvatures to
become strong as $g_{tt} \to 0$. AdS/CFT has limited computational
power in such circumstances. The best hope is that string theory
provides a resolution of the strong curvatures. If visible sector
fields live on branes at strong curvatures, then we are not in a
position to say much about the physics. It is also conceivable
\cite{RandallLykken} that visible sector fields live on a probe brane
at small but nonzero $g_{tt}$. There are potential phenomenological
virtues to such a model, but it seems somewhat contrived.
String theory and string dualities have taught us that extra
dimensions are theoretically inexpensive. But the view of the fifth
dimension espoused in the current paper is not excessively literal:
rather than making the claim that there is actually a large extra
dimension of space waiting to be discovered, the statement is that an
extra dimension is a convenient way to describe collective phenomena
of a strongly coupled quantum field theory---in the present case, a
conformal field theory coupled to gravity. To make this seem more
definite, suppose measurements of gravity at a micron did after all
turn up deviations from Newton's law of the form \ForceCorrect. The
``AdS'' interpretation would be that gravitons are propagating in the
fifth dimension, while the ``CFT'' interpretation would be that a loop
of gauge bosons in a purely four-dimensional theory had contributed.
Which interpretation we prefer is a matter of ontology: if AdS/CFT is
right then they are absolutely indistinguishable on experimental
grounds. My current ontology isn't very happy either
with a CFT with $c \sim 10^{58}$ or with a fifth dimension with
curvatures on the scale of a micron. But it is in the subtle guises
of string duality and string compactification that I suspect extra
dimensions have the best chance of improving our understanding of the
physical world.
\section*{Acknowledgements}
Although it is tricky to assign credit in the absence of publications, I
should acknowledge the important contributions of J.~Maldacena and
E.~Witten. As far as I can ascertain, J.~Maldacena was the first to
enunciate the view that the Randall-Sundrum ``alternative to
compactification'' is nothing more nor less than gravity coupled to a
strongly interacting CFT; and as far as my personal knowledge extends,
E.~Witten was the first to suggest a definite calculation based on the
idea, namely a correction to Newton's law based on the two-point function
of the stress tensor of the CFT. I have also profited greatly from
conversations with H.~Verlinde. I thank the participants of the conference
``New Dimensions in String theory and Field theory'' at the ITP in Santa
Barbara for lively discussions---especially I.~Klebanov, M.~Gremm,
R.~Myers, V.~Periwal, B.~Ovrut, K.~\hbox{Skenderis}, S.~Giddings, S.~Elitzur,
E.~Witten, V.~Hubeny, L.~Bildsten, R.~Maimon, R.~Sundrum, M.~Strassler,
and G.~Horowitz.
I thank N.~Warner, K.~Pilch, D.~Freedman, O.~DeWolfe, A.~Karch,
E.~Silverstein, and S.~Kachru for earlier discussions, and particularly
D.~Gross for reading an early version of the manuscript and for useful
comments.
This research was supported by the Harvard Society of Fellows, and
also in part by the NSF under grant number PHY-98-02709, and by DOE
grant DE-FGO2-91ER40654. I thank the ITP at Santa Barbara for
hospitality while the work was carried out.
| {
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HomeCinemaMasalaSidra Iqbal: Brings Home the 1st GR8! Women Awards 2014
Sidra Iqbal: Brings Home the 1st GR8! Women Awards 2014
February 19, 2014 Aaliya Imtiaz Masala 0
Sidra was bestowed with the honor along with 16 other women from across the Middle East region who were recognized and celebrated at the event this year. Sidra is the first Pakistani to have ever been nominated for and awarded on this particular international award forum.
Sidra is a very popular TV host and a real life example of "beauty with brains" making her the perfect candidate to represent Pakistani women across the board on a global platform. Looking like a million dollars at the Dubai event in an outfit by ace designer Saniya Maskatiya, Sidra surely turned a lot of heads as she walked down the red carpet and later accepted her award on stage. Mr Ashish Bagga, Chief Executive Officer of the India Today group, the leading media group of India, presented the award to Sidra.
Speaking about receiving the award for her work, Sidra said "As a journalist in Pakistan we are faced with a fast changing political and security environment, sometimes so rapid in changing tunes that it leaves us gasping to understand the real nuances.
The assignments take me everywhere, from closed door media briefings at the Presidency, to scenes of utter devastation after a natural havoc, but the most trying of experiences has been to meet and interact with the courageous women who mourn the untimely deaths of their loved ones at the hands of extremist attacks. In the past few years more than 50,000 innocent civilians have lost their lives in these terrorist attacks in Pakistan.
This evil has struck very close to home too as friends and colleagues have lost their lives in this battle to uphold and safeguard our sensibilities. But it is not the battles that we have lost that haunt us the most, but the ones we never mustered the courage to fight. The GR8 Women Award in Journalism has rejuvenated my will and my spirits. May this forum of substance grow and flourish."
Bollywood stars Simi Garewal, Shriya Saran and Kangana Ranaut also received an award this year while past winners included big names such as Sushmita Sen and Karishma Kapoor. Other attendees at this year's event included leading Indian Television stars such as Rithvik Dhanjani and Barkha Dutt. The awards were aired live on Sony Entertainment.
The GR8 Women Awards have been organized for the past 3 years by the Indian Television Academy and serve to recognize women from all walks of life for their achievements.
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Pinterest's Power Players: Who You Need To Follow | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 8,223 |
The national flags of China and India in Mumbai. Photographer: Dhiraj Singh/Bloomberg
Imports From China Rebound In July Amid Increased Scrutiny
Pallavi Nahata@PallaviNahata
Sep 08 2020, 7:51 AM Sep 08 2020, 7:51 AM September 08 2020, 7:51 AM September 08 2020, 7:51 AM
Imports from China into India rebounded faster than overall shipments into the country in July, shows the fine print of trade data available until the month of July.The pick-up took the value of inbound shipments back to pre-Covid levels, the data shows. This, just as India announced one-off measures to restrict and monitor inbound imports from the neighboring nation amid military skirmishes.Merchandise imports from China contracted ...
Imports from China into India rebounded faster than overall shipments into the country in July, shows the fine print of trade data available until the month of July.
The pick-up took the value of inbound shipments back to pre-Covid levels, the data shows. This, just as India announced one-off measures to restrict and monitor inbound imports from the neighboring nation amid military skirmishes.
Merchandise imports from China contracted by 9.8% in July, compared to a contraction of 43.7% in June, according to data released by the Department of Commerce. The rebound was sharper than what was seen across India's aggregate imports, which contracted by 29.6% in July, after a drop of 48.5% the previous month. Imports from the United States and UAE, the second and third largest importers to India after China, contracted by about 30% in July, the data showed.
In value terms, imports from China rose to $5.58 billion in July, compared to $3.3 billion in June. The was the highest since January 2020 and Chinese imports accounted for nearly a fifth of India's total merchandise imports.
Exports to China, however fell marginally in July to $1.74 billion from $2.1 billion in the previous month. Consequently, India's trade deficit with China widened to $3.8 billion, the highest since January 2020.
With China's return to normalcy, foreign trade with China has recovered quicker than trade with other economies, hence reverting to previous trends, said Manoj Pant, director of the Indian Institute of Foreign Trade. Pant added that the trade policies changes announced in the aftermath of military tensions may not necessarily hurt merchandise imports from China. Instead, they are intended to function as a signalling mechanism, indicating that it may not longer be business as usual between the trade partners.
The measures are also intended to further build-up domestic capacity and draw in foreign direct investment.
Dharmakirti Joshi, chief economist at CRISIL, said that the change in some trade policies, along with the pandemic, have led to volatility in trade data.
An analysis of trade between China and India has thrown up confusing results. For instance, while there was a decline in India's trade deficit with China until June, when trade via Hong Kong was included, this result was a little less conclusive. There were also one-offs in the midst of the pandemic. For instance, China was importing steel from India, which was seen as an unusual move and was not expected to sustain.
However, over a longer period, there has been a trend of a narrowing trade deficit with China and this is likely to continue said Joshi. That said, by imposing more tariffs, India could end up with imports of more costly or less efficient items, said Joshi, adding that, a strategic change to produce more domestically will help lower the deficit going forward.
Also read: India's GDP Data Should 'Alarm', Says Raghuram Rajan | {
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Q: fs.readFileSync returning weird encoding I'm trying to use readFileSync to copy the contents of a Javascript file and then place it into an HTML file. However when I do so the outputted code is getting encoded and causing errors when the JS runs. For example = is getting encoded into =.
I've used different variations for the encoding like so
fs.readFileSync('dist/' + v + '.min.js')
fs.readFileSync('dist/' + v + '.min.js', 'utf8')
and the output doesn't change.
I'm using Webpack and the HtmlWebpackPlugin to read the file and serve it to a Handlebars template. When I log out the contents of the file it looks correct, so it's looking like Handlebars is mashing up the text for some reason.
Thanks for any help!
| {
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} | 1,472 |
{"url":"https:\/\/schoollearningcommons.info\/question\/5-the-product-of-roots-of-quadratic-equations-3-square-4-o-is-23199238-11\/","text":"## 5. The product of roots of quadratic equations 3x square-4x=o is \u200b\n\nQuestion\n\n5. The product of roots\n3x square-4x=o is\n\nin progress 0\n1 month 2021-08-18T06:21:59+00:00 1 Answer 0 views 0\n\nimaginary\n\nStep-by-step explanation:\n\nGiven equation is 3x\n\n2\n\n\u22124x+3=0\n\nTo find, the nature of the roots of the equation\n\nAn equation is said to have\n\n(i) two distinct and real roots if the discriminant b\n\n2\n\n\u22124ac>0\n\n(ii) equal real roots if b\n\n2\n\n\u22124ac=0\n\n(iii) no real roots or imaginary roots if b\n\n2\n\n\u22124ac<0\n\nIn the given equation a=3,b=\u22124,c=3\n\nHence the discriminant is (\u22124)\n\n2\n\n\u22124(3)(3)=16\u221236=\u221220<0\n\nTherefore the roots of the given equation are imaginary in nature.","date":"2021-09-19 07:38:25","metadata":"{\"extraction_info\": {\"found_math\": false, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.8174853324890137, \"perplexity\": 3304.24367167779}, \"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\/1631780056752.16\/warc\/CC-MAIN-20210919065755-20210919095755-00304.warc.gz\"}"} | null | null |
{"url":"http:\/\/www.thermospokenhere.com\/wp\/03_tsh\/C0250_3.01_work_ke_BODY\/work_ke_BODY.html","text":"THERMO Spoken Here! ~ J. Pohl \u00a9\u00a0 ( C0250~1\/15) ( C0350\u00a0-\u00a0 3.02 Work, PE: BODY)\n\n# 3.01 Work and Kinetic Energy: BODY\n\nNewton's 2'nd Law of Motion, a physical\/mathematical theory, tells the change of momentum as time proceeds provided the initial state of the BODY and the forces acting on the BODY for the duration are known. Since his death, scientists have tinkered with his Laws of Motion. Two new, ideas useful in understanding physical behaviors of a BODY have been determined. These new ideas are the constructs: Energy and Work. C0250_Work_KE_BODY\n\nHorizontal Kinetic Energy:\u00a0\u00a0\u00a0Force, velocity, momentum and displacement of a BODY are vectors. Vectors have three components in space; the math and calculus of vector equations is tedious. Although Newton used vectors, some want to use his ideas shortened in a way. HS physics education has popularized Newton by using the idea that space is three pieces, components... three scalar forms of the Laws of Motion. This writing will use the HS physics one-dimensional, scalar approach to establish (as an x-component consequence of Newton's Laws) kinetic energy.\n\nThe typical Earth-surface Cartesian space with it coordinates (0XYZ) divides naturally into planar spaces, OXY, OXZ and OYZ. Some pages ago, the x-component was extracted from his Newton's Laws of Motion. This is that equation; it applies in a horizontal coordinate (x or y) we will use \"x\" below:\n\n (1)1\n\nThe above is a scalar differential equation. Newton used calculus; we need calculus to proceed. A general differential displacement of the BODY is written as a vector: dS. The part of that dispalcement in the x-direction is the differential displacement: dx.\n\nWe will multiply Eqn (1) by dx. In so doing the idea \"motion by displacement dx,\" adds to the meaning of Eqn (1). Below we show the first step of equation change.\n\n (2)2\n\nSome algebra with differentials of calculus are in order. Beginning with the right side of (1), four steps are needed to arrive at our final form.\n\n (3)3\n\u2022 In expression (1) it is permissible to move the dt in the quotient to be beneath the dx resulting in form (2).\n\u2022 Form (2) becomes form (3) because dx\/dt is the definition of vx.\n\u2022 Next since mvx is a product, it can be expanded. See (4) above.\n\u2022 Finally since the mass of the body is constant, dm equals zero. Our result is (5).\n\nNow returning to Equation (1) we replace its left side term with (5) from Equation (3) to have (4a) below. Right of (4a) we insert notations of time dependence, \"(t)\" to terms that change. Note mass (BODY) having no \"(t),\" is constant. Equation form (4b).\n\n (4)4\n\nThis topic can be restricted to an \"x\" direction to be simpler but it still cannot be discussed without the calculus. Calculus at this level is not a 600-page everything-we-can-teach math text. It is just the sweet part. Just one or two differentials of physical importance. To understand, calculus and the first-order differential equation must be used. So!\n\nEquations (4a and b) are the same differential equation. Each represents the physical reality of horizontal motion as Newton came understand it and to represent it mathematically. Differentials of the above equation are creations of the mathematics, calculus. Differentials express change, as Newton would say, \"at the basis.\" Thus \"dvx\" means a very small (some say \"vanishingly small\") change of speed in (the x-direction). Right-of-equality (4) acts a force through a vanishingly small (thinking zero? Okay.) displacement. This is the \"talk\" of calculus.\n\nNewton's Laws of Motion: COMPONENT FORM\u00a0\u00a0The equation below expresses Newton's Laws of Motion. The equation is not mathematical; it is physical. It is not scalar; descriptions of motion in space require vectors. The equation is not vector algebraic. The equation is a first order differential equation with time as the independent variable. We often think that a specific amount of matter exists in space and time. Generally, every BODY exists in space with a time rate.\n\n (5)5\n\nA lesser, equally important, fact of any \"event\" of a BODY is its displacement - a vector. Displacement divided by the time required for that displacement is a vector called \"average velocity.\" We say no more about displacement except that we intend to multiply the above expression of momentum of a BODY by its displacement.\n\nFirst we specialize the above equation by writing it in component form. The sum of forces, \u03a3F, has two categories which are gravity and other forces experienced by the BODY. The others, \"pushes, contact, and pulls,\" act immediately at the system boundary and are called \"surface \" or \"applied\" forces. With these distinctions Newton's Laws of Motion become:\n\n (6)6\n\nThe above equation has three components, of course, but only two will require investigation. These are the Z-component and either the X or Y component. (they are similar). We begin by writing momentum of a Body, velocity of a Body and forces in component form:\n\n (7)7\n\nOnce the expansions of momentum of a BODY and the expanded expressions for force (3) are substituted into (2) and we obtain X-component and Z-component expressions:\n\n (8)8\n\nAbove left, is the scalar. Y-component (identical to the y-component) of Newton's Laws of Motion (First Law if sum of forces is zero and Second Law if the sum is not zero). These component consequences are what one might label \"horizontal to Earth\" (homogeneous mathematically) effects and results of motion with no forces - unchanged. The idea, \"zero applied and zero forces caused by horizontal motion force\" opens the realm... Uniform Motion.\n\nIn reality, nothing moves without effort (force sum not zero) and whatever moves changes position in time. The reality, movement (of a mass) and the time of that event... These facts are physically related. There is energy we know. As kids we are admonished, \"speed kills.\" We are cautioned \".It is bad luck to walk under a ladder.\" These are truths known even to youths, understood independent of textbooks, equations... not knowing Newton.\n\nWith scientific study and development quite a bit of terminology and nomenclature arises. Words of description assist learning but also impede it. In this section we introduce and study the simplest forms of energy and work. Work is one mechanism of change of energy. It is logical to call the equation we discuss here, the equation that relates energy change and work - the \"energy equation.\" But no sooner than we are familiar with the equation, we will begin to change it to a new equation more encompassing of energy, with new manners of work and a new idea - heat. We will want to call the next, new and improved equation the \"energy equation.\" But what then do we call the previous \"energy equation?\" The rule is that successive understandings represented in an equation include all terms of the previous understandings. The simplest energy consideration for an event is obtained from the last, super energy equation by reduction, by casting out inapplicable terms.\n\nNewton studied momentum differentially (very slow-motion camera), that is, at the limit of its \"very least\" change. No friction - that slow!\n\nMomentum, expressed at is basis, is, displacement, work, then kinetic and potential energy. To follow this path is to encounter momentum (a vector property) then displacement (also vector), then effect the multiplication vectorialy.\n\nKinetic energy, the initial energy form of matter, is a composite property. The answer to \"why does kinetic energy equal the mass times one half its speed squared.?\" is work. Work is the measure of energy; it prescribes the form of kinetic energy and others. Potential energy is a grand convenience that arises when Earth is included as part of the system. A thorough derivation of these energy forms requires serious vector mathematics.\n\nWork is a construct Work is not a property; it does not belong to a system. Work is mechanism, a where-with-all, or event, whereby energy of a body might changed. As importantly, work is the means whereby energy of a body is made quantitive.\n\nWhen work occurs, force (Newton's construct - a vector) acts at the system boundary and is displaced.\n\nThe body displaces (a vector entity) either in the direction of the vector force, in the opposite direction or any other direction. Work, being part force, part displacement (both vectors) is defined as the integral of their vector product.\n\n### ENERGY EQUATION - Mechanical Form\n\nA principal extension of Newton's Laws of Motion involved the scalar multiplication of the Laws of Motion by displacement of the system (body). The concept, the relation of system momentum with system energy was made (left of equality, immediately being kinetic energy. Right of\n\nSpecification of the system (and its essential aspects) occupies much of problem statements and solutions because every discussion of kinetic energy, potential energy, work, momentum... relates to a system. Below is presented the most elementary energy equation. The equation is a beginning; it will be expanded to serve more difficult physics, as we go.\n\nSufficient familiarity with kinetic and potential energy is assumed such that simple, ground-work considerations of energy analysis can be revisited. Once simple techniques of application are \"refreshed,\" a derivation be provided. Till then, elementary calculations regarding energy changes and work of simple systems will be addressed in a unified way by the beginning level energy equation in its increment form. This form is suited to events that are incremental in time, that is, have that aspects, \"start\" and \"stop.\"\n\n (9)9\n\nThe above equation is implicit. It relates energy change with the sum of works of an event. Summation signs remind us, \"... include all instances,\" thereof. The equation is written energy-explicit as:\n\n (10)10\n\nWe will carry the subscript, c.m., meaning \"in regard to the center of mass.\" Also the applicable velocities and elevations are those of the center of mass. The work W of the equation is all-inclusive and there are occasion when more than one work effect occur simultaneously. An unambiguous way to write work as it appears in energy equations is to precede it with the simple math symbol that means \"summation of.\" We will use the notation: \u03a3W.\n\nThermodynamics is not literature. Early goals of school are to mimic, to solve problems in the taught style to obtain some predicted number. The numbers of this writing are tainted by assumption and approximation. The problems are false and academic, admittedly. But their solutions serve as branches the only branches we have. once you understand the branches here, once you can climb to their ends, ... then you are as likely as anyone to see something new and understand it.\n\nThe next pages show calculations you might have seen in physics. But here there is a system approach. Refresh your understandings about mass, force, gravity, velocity, momentum and the energies, kinetic and potential.\n\nFOOTNOTES\n\n[1]\u00a0\u00a0While there are many cases of physical reality (and its event) for which a \"BODY\" system model is relevant, there are very many more where it is not. Physics exploits the BODY, seeing its mass to be constant, physics quickly casts rate of momentum change as the body mass times its \"rate of change of velocity\" ~ acceleration. The idea, acceleration, is not needed.\n\n## 3.01 Work and Kinetic Energy: BODY\n\nNewton's 2'nd Law of Motion (a mathematical expression) that tells the change of momentum in time provided the initial state of the BODY and the forces acting on that BODY for the duration are known. Since his death, scientists have tinkered with his Law of Motion. Two new ideas useful in understanding physical behaviors of a BODY have been determined. These new ideas are the constructs: Energy and Work.\n\nPremise presently unwritted!","date":"2018-02-18 23:55:51","metadata":"{\"extraction_info\": {\"found_math\": false, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.8415436148643494, \"perplexity\": 1202.699311952544}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2018-09\/segments\/1518891812293.35\/warc\/CC-MAIN-20180218232618-20180219012618-00611.warc.gz\"}"} | null | null |
\section{Introduction}
Managing inconsistencies in databases is a challenge that arises in several different contexts.
Data cleaning is the main approach towards managing inconsistent databases (see the survey \cite{DBLP:journals/ftdb/IlyasC15}). In data cleaning, clustering techniques and/or domain knowledge are used to
resolve violations of integrity constraints in a given inconsistent database, thus producing a single consistent database.
This approach, however, is often \emph{ad hoc}; for example, if a person has two different social security numbers in a database, which of the two should be kept?
The framework of database repairs and consistent query answering, introduced by Arenas, Bertossi, and Chomicki \cite{Arenas99}, is an alternative, and arguably more principled, approach to data cleaning.
In contrast to data cleaning,
the inconsistent database is left as is; instead, inconsistencies are handled at query time by considering all
possible repairs of the inconsistent database, where a {\em repair} of an inconsistent database $I$ is a consistent database $J$ that differs
from $I$ in a ``minimal" way.
The main algorithmic problem in this framework is to compute the
\emph{consistent answers} to a
query $q$ on a given database $I$, that is, the tuples that lie
in the intersection of the results of $q$ applied on each repair of $I$ (see the monograph \cite{Bertossi11}).
Computing the consistent answers to a query $q$ on $I$ can be computationally harder than evaluating $q$ on $I$, because an inconsistent database may have exponentially many repairs.
By now there is an extensive literature on the computational complexity of the consistent answers for different classes of constraints and queries \cite{CateFK12,Kolaitis12,Koutris17,Wijsen12,Wijsen13}.
For key constraints (the most common constraints) and for conjunctive queries (the most frequently asked
queries), the consistent answers appear to exhibit an intriguing trichotomy, namely, the consistent answers of every fixed conjunctive query under key constraints are either first-order rewritable (hence, polynomial-time computable), or are polynomial-time computable but not first-order rewritable, or are coNP-complete. So far, this trichotomy has been proved for self-join free conjunctive queries by Koutris and Wijsen \cite{Koutris16,Koutris17}.
Moreover, Koutris and Wijsen designed a quadratic algorithm that, given such a conjunctive query and a set of key constraints, determines the side of the trichotomy in which the consistent answers to the query fall. Prior to this work, Fuxman and Miller identified a class of conjunctive queries, called $C_\textit{forest}$, whose consistent answers are FO-rewritable \cite{Fuxman05,FuxmanM07}. Membership in $C_\textit{forest}$, however, is sufficient but not necessary condition for the FO-rewritability of the consistent answers.
Several academic prototype systems for consistent query answering have been developed \cite{Arenas03,Barcelo03,ChomickiH04,Fuxman05,FuxmanM05,Greco03,Kolaitis13,MannaRT11,MarileoB10}. In particular, the ConQuer system \cite{Fuxman05,FuxmanM05} is tailored to queries in the class $C_\textit{forest}$.
Other systems use logic programming \cite{Barcelo03,Greco03}, compact representations of repairs \cite{Chomicki04}, or reductions to solvers. Specifically, the system in \cite{MannaRT11} uses reductions to answer set programming, while the EQUIP system in \cite{Kolaitis13} uses reductions to binary integer programming and the subsequent deployment of CPLEX.
It is fair to say, however, no comprehensive and scalable system for consistent query answering exists at present; this state of affairs
has impeded the broader adoption of the framework of repairs and consistent answers as a principled alternative to data cleaning.
In this paper, we report on a SAT-based system for consistent query answering, which we call CAvSAT (Consistent Answers via \sat{}). The CAvSAT system leverages natural reductions from the complement of consistent query answering to \sat{} and to \wmaxsat{}. As such, it can handle the consistent answers to unions of conjunctive queries under \emph{denial} constraints, a broad class of integrity constraints that include functional dependencies (hence also key constraints) as a special case. CAvSAT is the first SAT-based system for consistent query answering.
We carried out a preliminary stand-alone evaluation of CAvSAT on both
synthetic and real-world databases. The first set of experiments involved the consistent answers of conjunctive queries under key constraints on synthetic databases in which each relation has up to one million tuples.
One of the \emph{a priori} unexpected findings is that, for conjunctive queries whose consistent answers are first-order rewritable, CAvSAT had comparable or even better performance to evaluating the first-order rewritings using a database engine, such as PostgreSQL.
The second set of experiments involved the consistent answers of (unions of) conjunctive queries under functional dependencies on restaurant inspection records in Chicago and New York with some of the relations exceeding 200000 tuples. The CAvSAT source code is available at the GitHub repository \url{https://github.com/uccross/cavsat} via a BSD-style license.
While much more work remains to be done, the experimental finding reported here provide evidence that a SAT-based approach can indeed give rise to a comprehensive and scalable system for consistent query answering.
\section{Basic Notions and Background}
\noindent{\bf Databases, Constraints, and Queries}~
A \textit{relational database schema} $\mathcal{R}$ is a finite collection of relation symbols, each with a fixed positive integer as its arity. The attributes of a relation symbol are names for its columns; attributes can also be identified by their positions, thus $Attr(R) = \{1, ..., n\}$ denotes the set of attributes of $R$.
An $\mathcal{R}$-\emph{database instance} or, simply, an $\mathcal{R}$-\emph{instance} is a collection $I$ of finite relations $R^I$, one for each relation symbol $R$ in $\mathcal R$. An expression of the form $R^I (a_1, ..., a_n)$ is a \textit{fact} of the instance $I$ if $(a_1, ..., a_n) \in R^I$.
Every $\mathcal R$-instance can be identified with the (finite) set of its facts. The \emph{active domain} of $I$ is the set of all values occurring in facts of $I$.
Relational database schemas are often accompanied by a set of integrity constraints that impose semantic restrictions on the allowable instances.
A \textit{functional dependency} (FD) $ \vec{x} \rightarrow \vec{y}$ on a relation symbol $R$ is an integrity constraint asserting that if two facts agree on the attributes in $\vec{x}$, then they must also agree on the attributes in $\vec{y}$. A \textit{key} is a minimal subset $\vec{x}$ of $Attr(R)$ such that the FD $ \vec{x} \rightarrow Attr(R)$ holds. In this case, the attributes in $\vec{x}$ are called \textit{key attributes} of $R$ and they are denoted by underlining their corresponding positions; thus, $R(\underline{A, B}, C)$ denotes that the attributes $A$ and $B$ form a key of $R$. Every functional dependency is expressible in first-order logic. For example, the key constraint $A,B \rightarrow C$ in $R(\underline{A, B}, C)$ is expressed by the first-order formula
$$\forall x, y, z, z' (R(x, y, z) \land R(x, y, z') \rightarrow z = z')$$
Functional dependencies are an important special case of \emph{denial constraints} (DCs), which are expressible by first-order formulas of the form
$$\forall x_1, ..., x_n \neg (\varphi(x_1, ..., x_n) \land \psi(x_1, ..., x_n)),$$
or, equivalently,
$$\forall x_1, ..., x_n (\varphi(x_1, ..., x_n) \rightarrow \neg \psi(x_1, ..., x_n)),$$
where $\varphi(x_1, ..., x_n)$ is a conjunction of atomic formulas and $\psi(x_1, ..., x_n)$ is a conjunction of expressions of the form $(x_i\; \mbox{op}\; x_j)$ with each $\mbox{op}$ a built-in predicate, such as $=, \neq, <, >, \leq, \geq$.
In words, a denial constraint prohibits a set of tuples that satisfy certain conditions from appearing together in a database instance.
Let $k$ be a positive integer. A \emph{$k$-ary query} on a relational database schema $\mathcal R$ is a function $q$ that takes an $\mathcal R$-instance $I$ as argument and returns a $k$-relation $q(I)$ on the active domain of $I$ as value. A \emph{boolean query} on $\mathcal R$ is a function that takes an $\mathcal R$-instance $I$ as argument and returns true or false as value. As is well known, first-order logic has been successfully used as a query language. In fact, it forms the core of SQL, the main commercial database query language.
A \textit{conjunctive query} is a first-order formula built using the relational symbols, conjunctions, and existential quantifiers. Thus, each conjunctive query is expressible by a first-order formula of the form
$$q(\vec{z}):= \exists \vec{w}\; (R_1(\vec{x_1}) \land ... \land R_m(\vec{x_m})),$$
where each $\vec{x_i}$ is a tuple consisting of variables and constants, $\vec{z}$ and $\vec{w}$ are tuples of variables, and the variables in $\vec{x_1}, ..., \vec{x_m}$ appear in exactly one of $\vec{z}$ and $\vec{w}$. Clearly, a conjunctive query with $k$ free variables $\vec{z}$ is a $k$-ary query,
while a conjunctive query with no free variables (i.e., all variables are existentially quantified) is a boolean query.
Conjunctive queries are also known as \emph{select-project-join} (SPJ) queries and are among the most frequently asked queries in databases. For example, the binary conjunctive query $q(s,t):= \exists c (\mathrm{Enrolls}(s,c) \land \mathrm{Teaches}(t,c))$ returns the set of all pairs $(s,t)$ such that student $s$ is enrolled in a course taught by teacher $t$, while the boolean conjunctive query $q():= \exists x, y, z (E(x,y)\land E(y,z)\land E(z,x))$ tests whether or not a graph with an edge relation $E$ contains a triangle.
\medskip
\noindent{\bf Repairs and Consistent Answers}~ Let $\mathcal{R}$ be a database schema and let $\Sigma$ be a set of integrity constraints on $\mathcal{R}$.
An $\mathcal R$-instance $I$ is \emph{consistent} if $I \models \Sigma$, that is, $I$ satisfies every constraint in $\Sigma$; otherwise, $I$ is \emph{inconsistent}.
A \emph{repair} of an inconsistent instance $I$ w.r.t. $\Sigma$ is a consistent instance $J$ that differs from $I$ in a ``minimal" way. Different notions of minimality give rise to different types of repairs (see \cite{Bertossi11} for a comprehensive survey). Here, we focus on \emph{subset repairs}, the most extensively studied type of repairs. An instance $J$ is a \emph{subset repair} of an instance $I$ if $J \subseteq I$ (where $I$ and $J$ are viewed as sets of facts), $J\models \Sigma$, and there exists no instance $J'$ such that $J'\models \Sigma$ and $J \subset J'\subset I$. From now on, by \emph{repair} we mean a subset repair.
Arenas, Bertossi, and Chomicki \cite{Arenas99} used repairs to give rigorous semantics to query answering on inconsistent databases. Specifically, assume that $q$ is a query, $I$ is an $\mathcal R$-instance, and $\vec{t}$ is a tuple of values. We say that $\vec{t}$ is a \emph{consistent answer} (also referred as a \emph{certain answer}) to $q$ on $I$ w.r.t. $\Sigma$ if $\vec{t} \in q(J)$, for every repair $J$ of $I$. We write $\cons{q, I, \Sigma}$ to denote the set of all \emph{consistent answers} to $q$ on $I$ w.r.t. $\Sigma$, i.e.,
$$\cons{q, I, \Sigma} = \bigcap\{q(J): \mbox{$J$ is a repair of $I$ w.r.t. $\Sigma$}\}.$$
If $\Sigma$ is a fixed set of integrity constraints and $q$ is a fixed query, then the main computational problem associated with the consistent answers is:
given an instance $I$, compute \cons{$q$, $I$, $\Sigma$}.
If $q$ is a boolean query, then computing the certain answers becomes the decision problem $\certainty{q,\Sigma}$: given an instance $I$,
is $q$ true on every repair $J$ of $I$ w.r.t.$ \Sigma$? When the constraints $\Sigma$ are understood from the context, we will write $\cons{q, I}$
and $\certainty{q}$, instead of $\cons{q, I,\Sigma}$
and $\certainty{q,\Sigma}$.
\medskip
\noindent{\bf Computational Complexity of Consistent Answers}
If $\Sigma$ is a fixed finite set of denial constraints and $q$ is a $k$-ary conjunctive query, where $k\geq 1$, then the following problem is in coNP: given an instance $I$ and a tuple $\vec{t}$, is $\vec{t}$ a certain answer to $q$ on $I$ w.r.t.\ $\Sigma$? This is so because to check that $\vec{t}$ is not a certain answer to $q$ on $I$ w.r.t.\ $\Sigma$, we guess a repair $J$ of $I$ and verify that $\vec{t}\not \in q(J)$ (note that $J$ is a subset of $I$, evaluating a fixed conjunctive query on a given database is a polynomial-time task, and testing if $J$ is a repair of $I$ w.r.t. denial constraints is a polynomial-time task as well). Similarly, if $q$ is a boolean conjunctive query, then the decision problem $\certainty{q,\Sigma}$ is in coNP.
Even for key constraints and boolean conjunctive queries, $\certainty{q,\Sigma}$ exhibits a variety of behaviors within coNP. Indeed, consider the queries
\begin{enumerate}
\item $\pathq():= \exists x, y, z\; R(\underline{x}, y) \land S(\underline{y}, z)$;
\item $\cycle():= \exists x, y\; R(\underline{x}, y) \land S(\underline{y}, x)$;
\item $\sink():= \exists x, y, z\; R(\underline{x}, z) \land S(\underline{y}, z)$.
\end{enumerate}
Fuxman and Miller \cite{FuxmanM07} showed that $\certainty{\pathq}$ is FO-rewritable, i.e., there is a first-order definable boolean query $q'$ such that $\cons{\pathq,\Sigma,I}=q'(I)$, for every instance $I$.
In fact, $q'$ is $\exists x, y, z\; R(\underline{x}, y) \land S(\underline{y}, z) \land \forall y'(R(\underline{x}, y') \rightarrow \exists z' S(\underline{y'}, z')).$
Wijsen \cite{WijsenR10} showed that $\certainty{\cycle}$ is in P, but it is not
FO-rewritable, while Fuxman and Miller \cite{FuxmanM07} showed that $\certainty{\sink}$ is coNP-complete via a reduction from the complement of
\textsc{Monotone 3-SAT}.
The preceding state of affairs sparked a series of investigations aiming to obtain classification results concerning the computational complexity of the consistent answers (e.g., see \cite{Grieco05,Kolaitis12,Lembo06,Wijsen09,WijsenR10}). The most definitive result to date is a \emph{trichotomy} theorem, established by Koutris and Wijsen \cite{Koutris15,Koutris16,Koutris17}, for boolean self-join free conjunctive queries, where a conjunctive query is \emph{self-join free} if no relation symbol occurs more than once in the query. This trichotomy theorem asserts that if $q$ is a self-join free conjunctive query with one key per relation symbol, then $\certainty{q}$ is FO-rewritable, or in P but not FO-rewritable, or coNP-complete. Moreover, there is a quadratic algorithm to decide, given such a query, which of the three cases of the trichotomy holds. It remains an open problem whether or not this trichotomy extends to arbitrary boolean conjunctive queries and to arbitrary functional dependencies or denial constraints.
\medskip
\section{Consistent Query Answering for Key Constraints}\label{sec:key-constraints}
In this section, we assume that $\mathcal{R}$ is a database schema and $\Sigma$ is a finite set of primary key constraints on $\mathcal R$, i.e., there is one key constraint per each relation of $\mathcal{R}$.
We first consider boolean conjunctive queries and, for each fixed boolean conjunctive query $q$,
we give a natural polynomial-time reduction from \certainty{$q$} to \unsat{}. We then extend this reduction to non-boolean conjunctive queries, so that for every fixed non-boolean conjunctive query $q$, the consistent answers to $q$ can be computed by iteratively solving \wmaxsat{} instances. In what follows, we heavily use the notions of \textit{key-equal groups} of facts and \textit{minimal witnesses} to a conjunctive query.
\begin{definition}\label{key-equal-group}\textbf{Key-Equal Group.} \emph{Let $I$ be an $\mathcal{R}$-instance. We say that two facts of a relation $R$ of $I$ are \textit{key-equal}, if they agree on the key attributes of $R$. A set $S$ of facts of $I$ is called a \textit{key-equal group} of facts if every two facts in $S$ are key-equal, and no fact in $S$ is key-equal to some fact in $I\backslash S$.}
\end{definition}
\begin{definition}\label{min-witness}\textbf{Minimal Witness.} \emph{Let $I$ be an $\mathcal{R}$-instance and let $S$ be a sub-instance of $I$. We say that $S$ is a \textit{minimal witness} to a conjunctive query $q$ on $I$, if $S \models q$, and for every proper subset $S'$ of $S$, we have that $S' \not\models q$.}
\end{definition}
For each relation $R$ of $I$, the key-equal groups of $R$ are computed by an SQL query that involves grouping the key attributes of $R$. Similarly, the set of minimal witnesses to a fixed conjunctive query $q$ on $I$ are computed efficiently as follows. A unique integer \textit{factID} is attached to each fact, by adding an attribute \textit{FactID} to each relation in $I$ that appears in $q$. Thus, a new instance $I'$ is built, where each relation $R'(\textit{FactID}, \vec{\underline{A}}, \vec{B})$ in $I'$ is obtained from a relation $R(\vec{\underline{A}}, \vec{B})$ in $I$. A new non-boolean query $q'$ is constructed, such that each atom $R'(\textit{factID}_R, \vec{\underline{x}}, \vec{y})$ in $q'$ is constructed from an atom $R(\vec{\underline{x}}, \vec{y})$ of $q$. The variables of $q'$ that correspond to the \textit{FactID} attributes are not existentially quantified. It is easy to see that each tuple (without duplicate \textit{factID}s) in $q'(I')$ is in 1-1 correspondence with a minimal witness to $q$ on $I$
\newpage
\noindent{\bf Boolean Conjunctive Queries}
Let $q$ be a fixed boolean conjunctive query over $\mathcal{R}$.
\begin{reduction}\label{reduction1}Given an $\mathcal R$-instance $I$, we construct a CNF-formula $\phi$ as follows.
For each fact $f_i$ of $I$, introduce a boolean variable $x_i$, $1\leq i\leq n$.
Let $\mathcal{G}$ be the set of key-equal groups of facts of $I$, and let $\mathcal{W}$ be the set of minimal witnesses to $q$ on $I$.
\begin{itemize}
\item For each $G_j \in \mathcal{G}$, construct the clause $\alpha_j = \underset{f_i \in G_j}{\lor} x_i$.
\item For each $W_j \in \mathcal{W}$, construct the clause $\beta_j = \underset{f_i \in W_j}{\lor} \neg x_i$.
\item Construct the boolean formula $\phi = \bigg(\overset{|\mathcal{G}|}{\underset{i=1}{\land}}\alpha_i \bigg) \land \bigg(\overset{|\mathcal{W}|}{\underset{j=1}{\land}}\beta_j$\bigg).
\end{itemize}
\end{reduction}
\preservecounter{proposition}
\movetoappendix{\subsection*{Proof of Proposition~\ref{prop1}}}
\copytoappendix{
\begin{proposition}\label{prop1}
Let $\phi$ be the CNF-formula constructed using Reduction \ref{reduction1}.
\begin{itemize}
\item The size of $\phi$ is polynomial in the size of $I$.
\item The formula $\phi$ is satisfiable if and only if \certainty{$q$, $\Sigma$} is false on $I$.
\end{itemize}
\end{proposition}
}
The proofs of all propositions are given in the Appendix.
\movetoappendix{
\begin{proof}
Let $n$ be the number of facts in $I$. There are exactly $n$ boolean variables used in $\phi$. Clearly, $|\mathcal{G}| \leq n$, therefore the number of $\alpha$-clauses is bounded above by $n$. Similarly, for each $G_j \in \mathcal{G}$, we have that $|G_j| \leq n$. Hence, the length of each $\alpha$-clause is also at most $n$. If $d$ is the number of atoms in $q$, then we have that $|\mathcal{W}| \leq n^d$; moreover, for every $W_j \in \mathcal{W}$, we have that $|W_j| \leq d$. Hence, the number of $\beta$-clauses in $\phi$ is at most $n^d$, and the length of each $\beta$-clause is bounded above by $d$. Since the query $q$ is not part of the input to the problem \certainty{$q$}, we have that $d$ is a fixed constant.
To prove the second part of the proposition, assume first that $\certainty{q}$ is false on $I$. Hence, there exists a repair $R$ of $I$ that falsifies $q$. Construct an assignment $\hat{a}$ to the variables in $\phi$ by setting $\hat{a}(x_i) = 1$ if and only if $f_i \in R$. Since exactly one fact from each key-equal group of $I$ is present in $R$, exactly one variable from each $\alpha$-clause is set to 1 in $\hat{a}$. Also, since $R \not\models q$, no minimal witness to $q$ is in $R$. Therefore, at least one variable from each $\beta$-clause is set to 0 in $\hat{a}$. Hence, $\hat{a}$ satisfies $\phi$. For the other direction, let $\hat{a}$ be a satisfying assignment to $\phi$. Since no two $\alpha$-clauses share a variable, we can construct a set $X$ of variables by arbitrarily choosing exactly one $x_i$ from each $\alpha$-clause, such that $\hat{a}(x_i) = 1$. Construct a set $R$ of facts of $I$, such that $f_i \in R$ if and only if $x_i \in X$. It is easy to see that $R$ contains exactly one fact from each key-equal group of $I$, and no minimal witness to $q$ on $I$ is present in $R$. Hence, $R$ is a repair of $I$ that falsifies $q$.
\end{proof}
}
\medskip
\noindent {\bf Non-boolean Conjunctive Queries}
Let $q$ be a fixed non-boolean query on $\mathcal R$, i.e., $q$ has one or more free variables. We extend Reduction \ref{reduction1} to Reduction \ref{reduction2}, so that one can reason about the certain answers to
$q$ on an $\mathcal R$-instance $I$ using
the satisfying assignments of the CNF-formula $\phi$ constructed via Reduction \ref{reduction2}.
\looseness = -1 We use the term \emph{potential answers} to refer to the answers to $q$ on $I$.
If $\vec{a}_l$ is such a potential answer, we write $q[\vec{a}_l]$ to denote the boolean conjunctive query obtained from $q$ by replacing the free variables in the body of $q$ by corresponding constants from $\vec{a}_l$.
\begin{reduction}\label{reduction2}\begin{sloppypar}
Given an $\mathcal R$-instance $I$, we construct a CNF-formula $\phi$ as follows.
For each fact $f_i$ of $I$, introduce a boolean variable $x_i$, $1\leq i\leq n$,
Let $\mathcal{G}$ be the set of key-equal groups of facts of $I$ and let $\mathcal{A}$ be the set of potential answer to $q$ on $I$. For each $\vec{a}_l \in \mathcal{A}$, let $\mathcal{W}^l$ denote the set of minimal witnesses to the boolean query $q[\vec{a}_l]$ on $I$. For each $\vec{a}_l \in \mathcal{A}$, introduce a boolean variable $p_1$,
$1\leq l \leq ..., |\mathcal{A}|$.
\begin{itemize}
\item For each $G_j \in \mathcal{G}$, construct the clause $\alpha_j = \underset{f_i \in G_j}{\lor} x_i$.
\item For each $\vec{a}_l \in \mathcal{A}$ and for each $W^l_j \in \mathcal{W}^l$, construct the clause ${\beta^l_j = \bigg(\underset{f_i \in W^l_j}{\lor} \neg x_i \bigg) \lor \neg p_l}$.
\item Construct the boolean formula $\phi = \bigg(\overset{|\mathcal{G}|}{\underset{i=1}{\land}}\alpha_i\bigg)\land\bigg(\overset{|\mathcal{A}|}{\underset{l=1}{\land}} \bigg( \overset{|\mathcal{W}^l|}{\underset{j=1}{\land}}\beta^l_j\bigg)\bigg)$.
\end{itemize}\end{sloppypar}
\end{reduction}
\preservecounter{proposition}
\movetoappendix{\subsection*{Proof of Proposition~\ref{prop2}}}
\copytoappendix{
\begin{proposition}\label{prop2}
Let $\phi$ be the CNF-formula constructed using Reduction \ref{reduction2}.
\begin{itemize}
\item The size of $\phi$ is polynomial in the size $I$.
\item There exists a satisfying assignment to $\phi$ in which a variable $p_l$ is set to 1 if and only if $\vec{a}_l \notin \cons{q, I}$.
\end{itemize}
\end{proposition}
}
\movetoappendix{
\begin{proof}Let $n$ be the number of facts in $I$. Let $m$ be the arity of the query $q$ and let $d$ and the number of atoms of $q$. Since an answer to $q$ is a set of $m$ facts, we have that $|\mathcal{A}| \leq n^m$. For each $l$, the number of witnesses in $\mathcal{W}^l$ is bounded by $n^d$. Therefore, there are at most $n^{m + d}$ $\beta$-clauses in $\phi$, each of length at most $d$. Since the query $q$ is not part of the input to \cons{$q$}, the quantities $m$ and $d$ can be treated are fixed constants. It follows directly from Proposition \ref{prop1} that both the number of $\alpha$-clauses and the length of each $\alpha$-clause in $\phi$ are bounded above by $n$.
To prove the second part of the proposition, assume first that $\vec{a}_l \notin \cons{q}$. Hence, there exists a repair $R$ of $I$, such that no minimal witness to $q[\vec{a}_l]$ is in $R$. Construct an assignment $\hat{a}$ to the variables in $\phi$ as follows. Set $\hat{a}(x_i) = 1$ if and only if $f_i \in R$. Set $\hat{a}(p_l) = 1$, and set $\hat{a}(p_j) = 0$ for all $j \neq l$. Since exactly one fact from each key-equal group of $I$ is in $R$, the assignment sets to 1 exactly one variable from each $\alpha$-clause. Since no minimal witness to $q[\vec{a}_l]$ is in $R$, at least one variable from each $\beta^l$-clause is set to 0 in $\hat{a}$, thus satisfying all $\beta^l$-clauses, even when $p_l$ is set to 1. All other $\beta$-clauses are satisfied trivially because of the assignment $\hat{a}(p_j) = 0$, for all $j \neq l$. In the other direction, let $\hat{a}$ be the satisfying assignment to $\phi$, such that $\hat{a}(p_l) = 1$. Since no two $\alpha$-clauses share a variable, we can construct a set $X$ of variables by arbitrarily choosing exactly one $x_i$ from each $\alpha$-clause, such that $\hat{a}(x_i) = 1$. Construct a set $R$ of facts of $I$, such that $f_i \in R$ if and only if $x_i \in X$. It is easy to see that exactly one fact from each key-equal group of $I$ is present in $R$. Since $\hat{a}(p_l) = 1$ and since all $\beta^l$-clauses are satisfied by $\hat{a}$, at least one fact from each minimal witness to $q[\vec{a}_l]$ is missing in $R$. Hence, $R$ must be a repair of $I$ such that $R \not\models q[\vec{a}_l]$.
\end{proof}
}
\begin{example}\label{example1}
Consider the flights information database in Table \ref{flights}. The database schema has three relations, namely, \textit{Airlines}, \textit{Tickets}, and \textit{Flights}; the key attributes of each relation are underlined.
This database is inconsistent, as the sets $\{f_1, f_3\}$ and $\{f_8, f_9\}$ of facts violate the key constraints of the relations \textit{Airlines} and \textit{Flights}, respectively.
Suppose we want to find out the codes of the flights that belong to an airline from Canada and fly to the airport OAK. This can be expressed by the unary conjunctive query $q(x):= \textit{Flights}(x,y,z,p,\text{`OAK'},q,r) \land \textit{Airlines}(z, \text{`Canada'})$.
\begin{table}
\caption{Flight information records.}\label{flights}
\begingroup
\renewcommand{\arraystretch}{1.1}
\setlength{\tabcolsep}{3pt}
\begin{minipage}[t]{0.4\textwidth}
\begin{tabular}[t]{|c||c|c|}
\hline
\multicolumn{3}{|c|}{\textit{Airlines}}\\\hline
\textit{Fact} & \underline{AIRLINE} & COUNTRY\\
\hline\hline
$f_1$ & Southwest & United States\\\hline
$f_2$ & Jazz Air & Canada\\\hline
$f_3$ & Southwest & Canada\\\hline
\end{tabular}
\end{minipage}
\begin{minipage}[t]{0.55\textwidth}
\hspace{-0.28in}
\begin{tabular}[t]{|c||c|c|c|c|}
\hline
\multicolumn{5}{|c|}{\textit{Tickets}}\\\hline
\textit{Fact} & \underline{PNR} & CODE & CLASS & FARE\\\hline\hline
$f_4$ & MJ9C8R & SWA 1568 & Economy & 430 USD\\\hline
$f_5$ & KLF88V & MI 471 & First & 914 USD\\\hline
$f_6$ & NJ5RT3 & SWA 1568 & First & 112 USD\\\hline
\end{tabular}
\end{minipage}\vspace{0.02in}
\begin{minipage}{\textwidth}
\begin{tabular}[t]{|c||c|c|c|c|c|c|c|}
\hline
\multicolumn{8}{|c|}{\textit{Flights}}\\\hline
\textit{Fact} & \underline{CODE} & \underline{DATE} & AIRLINE & FROM & TO & DEPARTURE & ARRIVAL\\
\hline\hline
$f_7$ & JZA 8329 & 01/29/19 & Jazz Air \ignore{Southwest} & GEG & OAK & 16:12 PST & 18:00 PST\\\hline
$f_8$ & SWA 1568 & 01/29/19 & Silkair & YYZ & YAM & 18:55 EST & \ignore{20:44}18:44 EST\\\hline
$f_9$ & SWA 1568\ignore{SWA 1959} & 01/29/19 & Southwest & LAX & OAK & 16:18 PST & 17:25 PST\\\hline
\end{tabular}
\end{minipage}
\endgroup
\end{table}
There are two potential answers to $q$, namely, `JZA 8329' and `SWA 1568', so we introduce their corresponding variables $p_1$ and $p_2$. Since the facts $f_1$ and $f_3$ form a key-equal group, we construct an $\alpha$-clause $(x_1 \lor x_3)$. Similarly, since the set $\{f_2, f_7\}$ of facts is a minimal witness to $q[\text{`JZA 8329'}]$, we construct the $\beta$-clause $(\neg x_2 \lor \neg x_7 \lor \neg p_1)$.
By continuing this way, we obtain the following CNF-formula $\phi$:\\
$(x_1\lor x_3)\land x_2 \land x_4 \land x_5 \land x_6 \land x_7 \land (x_8 \lor x_9) \land (\neg x_2 \lor \neg x_7 \lor \neg p_1) \land (\neg x_3 \lor \neg x_9 \lor \neg p_2)$.\\
Clauses $x_2$, $x_7$, and $(\neg x_2 \lor \neg x_7 \lor \neg p_1)$ force $p_1$ to take value 0 in each satisfying assignment of $\phi$, because the facts $f_2$ and $f_7$ appear in every repair of $I$, thus making `JZA 8329' a consistent answer to $q$. In contrast, there is a satisfying assignment of $\phi$ in which $p_2$ is set to 1, which implies that `SWA 1568' is not a consistent answer to $q$.
\end{example}
\medskip
\noindent{\bf Optimizing the reductions}\label{optimizations}
In real-life applications, a large part of the inconsistent database is consistent. For a boolean query $q$, if a minimal witness to $q$ is present in the consistent part of the database instance, then we can immediately conclude that \certainty{$q$, $I$, $\Sigma$} is true. This can be checked with simple SQL queries that involve grouping on the key attributes of each relation. Similarly, for non-boolean queries, the consistent answers coming from the witnesses that belong to the consistent part of the database can be computed efficiently using SQL queries. All additional consistent answers can then be found using the preceding reduction. In this case, we need to introduce boolean variables corresponding to only those facts that contribute to the additional potential answers. This significantly reduces the size of the CNF-formulas produced by Reductions \ref{reduction1} or \ref{reduction2}. This optimization has been used earlier in \cite{Kolaitis13}, where \cons{$q$, $I$, $\Sigma$} was reduced to an instance of binary integer programming.
\section{Consistent Query Answering Beyond Key Constraints}
In this section, we consider the broader class of denial constraints and the more expressive class of unions of conjunctive queries. Note that computing the consistent answers of unions of conjunctive queries under denial constraints is still in coNP, but the consistent answers of a union $Q := q_1\cup \ldots \cup q_k$ of conjunctive queries $q_1,\ldots, q_k$ is not, in general, equal to the union of the consistent answers of $q_1,\ldots,q_k$.
We give a polynomial-time reduction from \cons{$Q$, $I$, $\Sigma$} to \unsat{}, where $\Sigma$ is a fixed finite set of denial constraints and $Q$ is a fixed union of non-boolean conjunctive queries. The potential answers to $Q$ are treated in the same way as the potential answers to the conjunctive query $q$ in Reduction \ref{reduction2}; to this effect, we introduce a boolean variable for each potential answer.
The reduction we give here relies on the notions of \textit{minimal violations} and \textit{near-violations} to the set of denial constraints that we introduce next.
\begin{definition}\label{mv}
\textbf{Minimal violation.} \emph{Assume that $\Sigma$ is a set of denial constraints, $I$ is an $\mathcal{R}$-instance, and $S$ is a sub-instance of $I$. We say that $S$ is a \textit{minimal violation} to $\Sigma$, if $S \not\models \Sigma$ and for every set $S' \subset S$, we have that $S' \models \Sigma$.}
\end{definition}
\begin{definition}\label{nv}
\textbf{Near-violation.} \emph{Assume that $\Sigma$ is a set of denial constraints, $I$ is an $\mathcal{R}$-instance, $S$ is a sub-instance of $I$, and $f$ is a fact of $I$. We say that $S$ is a \textit{near-violation} w.r.t.\ $\Sigma$ and $f$, if $S \models \Sigma$ and $S \cup \{f\}$ is a minimal violation to $\Sigma$. As a special case, if $\{f\}$ itself is a minimal violation to $\Sigma$, then we say that there is exactly one near-violation w.r.t. $f$, and it is the singleton $\{f_{true}\}$, where $f_{true}$ is an auxiliary fact.}
\end{definition}
For a fixed finite set $\Sigma$ of denial constraints, the set of minimal violations to $\Sigma$ on a given database instance $I$ are computed as follows. The body of a denial constraint $d \in \Sigma$ is treated as a boolean conjunctive query $q_d$, possibly containing atomic formulas from $d$ that use built-in predicates such as $=$, $\neq$, $<$, $>$, $\leq$, and $\geq$, in addition to the relation symbols. The set of minimal witnesses to $q_d$ on $I$ is computed as described in Section \ref{sec:key-constraints}, which is also, precisely, the set of minimal violations to $d$. The union of the sets of minimal violations over all denial constraints in $\Sigma$ gives us the set of minimal violations to $\Sigma$. For each fact $f \in I$, the set of near-violations to $\Sigma$ w.r.t. $f$ can be obtained by removing $f$ from every minimal violation to $\Sigma$ that contains $f$.
Let $\mathcal{R}$ be a database schema, let $\Sigma$ be a fixed finite set of denial constraints on $\mathcal R$, and let $Q := q_1\cup \ldots \cup q_k$ be a union of conjunctive queries $q_1,\ldots,q_k$. Let $I$ be an $\mathcal{R}$-instance, and let $Q$ be the fixed union of $k$ non-boolean conjunctive queries $q_1, \ldots, q_k$.
\begin{reduction}\label{reduction3}\begin{sloppypar}
Given an $\mathcal R$-instance $I$, we construct a boolean formula $\phi'$ as follows.
\noindent Compute the following sets:
$\bullet$ $\mathcal{V}$: the set of minimal violations to $\Sigma$ on $I$.
$\bullet$ $\mathcal{N}^i$: the set of near-violations to $\Sigma$, on $I$, w.r.t. each fact $f_i \in I$.
$\bullet$ $\mathcal{A}$: the set of potential answers to $Q$ on $I$.
$\bullet$ $\mathcal{W}^l$: the set of all minimal witnesses to $Q[\vec{a}_l]$ on $I$, for each $\vec{a}_l \in \mathcal{A}$.
\noindent For each fact $f_i$ of $I$, introduce a boolean variable $x_i$, $1\leq i\leq n$. For the auxiliary fact $f_{true}$, introduce a constant $x_{true} = true$.
For each $N^i_j \in \mathcal{N}^i$, introduce a boolean variable $y^i_j$, and for each $\vec{a}_l \in \mathcal{A}$, introduce a boolean variable $p_l$.
\begin{enumerate}
\item For each $V_j \in \mathcal{V}$, construct a clause $\alpha_j = \underset{f_i \in V_j}{\lor} \neg x_i$.
\item For each $\vec{a}_l \in \mathcal{A}$ and for each $W^l_j \in \mathcal{W}^l$, construct a clause ${\beta^l_j = \bigg(\underset{f_i \in W^l_j}{\lor} \neg x_i\bigg) \lor \neg p_l}$.
\item For each $f_i \in I$, construct a clause $\gamma_i = x_i \lor \bigg(\underset{N^i_j \in \mathcal{N}^i}{\lor}y^i_j\bigg)$.
\item For each variable $y^i_j$, construct an expression $\theta^i_j = y^i_j \leftrightarrow \bigg(\underset{f_d \in N^i_j}{\land} x_d\bigg)$.
\item Construct the following boolean formula $\phi$:
$${\phi' = \bigg(\overset{|\mathcal{V}|}{\underset{i=1}{\land}}\alpha_i\bigg)\land\bigg(\overset{|\mathcal{A}|}{\underset{l=1}{\land}}\bigg(\overset{|\mathcal{W}^l|}{\underset{j=1}{\land}}\beta^l_j\bigg)\bigg) \land \bigg(\overset{|I|}{\underset{i=1}{\land}}\bigg(\Big(\overset{|\mathcal{N}^i|}{\underset{j=1}{\land}}\theta^i_j\Big)\land \gamma_i\bigg)\bigg)}$$
\end{enumerate}
\end{sloppypar}
\end{reduction}
\preservecounter{proposition}
\movetoappendix{\subsection*{Proof of Proposition~\ref{prop3}}}
\copytoappendix{
\begin{proposition}\label{prop3} Let $\phi'$ be the boolean formula constructed using Reduction \ref{reduction3}.
\begin{itemize}
\item The formula $\phi'$ can be transformed to an equivalent CNF-formula $\phi$ whose size is polynomial in the size of $I$.
\item There exists a satisfying assignment to $\phi'$ in which a variable $p_l$ is set to 1 if and only if $\vec{a}_l \not\in \cons{Q, I, \Sigma}$.
\end{itemize}
\end{proposition}
}
\movetoappendix{
\begin{proof}
Let $n$ be the number of facts of $I$. Let $d_1$ be the smallest number such that there exists no denial constraint in $\Sigma$ whose number of database atoms is bigger than $d_1$. Also, let $d_2$ be the smallest number such that there exists no conjunctive query in $Q$ whose number of database atoms is bigger than $d_2$. Since $\Sigma$ and $Q$ are not part of the input to \cons{$Q$}, the quantities $d_1$ and $d_2$ are fixed constants. We also have that $|\mathcal{V}| \leq n^{d_1}$, $|\mathcal{N}^i| \leq n^{d_1}$ for $1 \leq i \leq n$, $|\mathcal{A}| \leq n^{d_2}$, and $|\mathcal{W}^l| \leq n^{d_2}$ for $1 \leq l \leq |\mathcal{A}|$. The number of $x$-, $y$-, and $p$-variables in $\phi'$ is therefore bounded by $n$, $n^{d_1 + 1}$, and $n^{d_2}$, respectively. The formula $\phi'$ contains as many $\alpha$-clauses as $|\mathcal{V}|$, and none of the $\alpha$-clause's length exceeds $n$. Similarly, there are at most $n^{d_2}$ $\beta$-clauses, and none of their lengths exceeds $d_2+1$. The number of $\gamma$-clauses is precisely $n$, and each $\gamma$-clause is at most $n^{d_1+1} + 1$ literals long. There are as many $\theta$-expressions as there are $y$-variables. Every $\theta$-expression is of the form $y \leftrightarrow (x_1 \land ... \land x_d)$, where $d$ is a constant obtained from the number of facts in the corresponding near-violation. Each $\theta$-expression can be equivalently written in a constant number of CNF-clauses as $((\neg y \lor x_1) \land ... \land (\neg y \lor x_d)) \land (\neg x_1 \lor ... \lor \neg x_d \lor y)$, in which the length each clause is constant. This makes it possible to transform $\phi'$ into an equivalent CNF-formula $\phi$, whose size is polynomial in the size of $I$.
To prove the second part of the proposition, assume first that $\hat{a}$ is a satisfying assignment to the variables in $\phi'$ such that $\hat{a}(p_l) = 1$. Construct a database instance $R$ such that $f_i \in R$ if and only if $\hat{a}(x_i) = 1$. The $\alpha$-clauses make sure that no minimal violation to $\Sigma$ is present in $R$, meaning that $R$ is a consistent subset of $I$. The $\gamma$-clauses and the $\theta$-expressions encode the condition that, for every fact $f \in I$, either $f \in R$ or at least one near-violation w.r.t.\ $\Sigma$ and $f$ is in $R$. This condition makes sure that $R$ is indeed a repair of $I$. Since $\hat{a}(p_l) = 1$, the $\beta^l$-clauses ensure that at least one fact from each minimal witness to $Q[\vec{a}_l]$ is missing from $R$, meaning that $\vec{a}_l \not\in Q(R)$
In the other direction, given a repair $R$ that falsifies $Q[\vec{a}_l]$, build an assignment $\hat{a}$ as follows. Set $\hat{a}(x_i) = 1$ if and only if $f_i \in r$. Set $\hat{a}(p_l) = 1$, and set $\hat{a}(p_{l'}) = 0$ for all $l' \neq l$. Since $R \models \Sigma$, no minimal violation to $\Sigma$ is a subset of $R$, meaning that $\hat{a}$ satisfies all $\alpha$-clauses in $\phi'$. Also, for every fact $f\in I$, it must be the case that either $f \in R$ or at least one near-violation w.r.t. $\Sigma$ and $f$ is in $R$ (otherwise $R$ would not have been a repair of $I$). Therefore, all $\gamma$-clauses and $\theta$-expressions are also satisfied by the assignment $\hat{a}$. Since $R \not\models Q[\vec{a}_l]$, at least one fact from each minimal witness to $Q[\vec{a}_l]$ must be missing from $R$, meaning that there is at least one variable $x_i$ in each $\beta^l$-clause such that $\hat{a}(x_i) = 0$. Hence, all $\beta^l$-clauses are satisfied by $\hat{a}$, even when $\hat{a}(p_l) = 1$. All other $\beta$-clauses are satisfied trivially, since $\hat{a}(p_{l'}) = 0$, for all $l' \neq l$.
\end{proof}
}
\begin{example}\label{example2}
Consider the database instance from Table \ref{flights}. In addition to the three key constraints from Example \ref{example1}, suppose the schema now has two additional integrity constraints: \begin{enumerate*}[label=(\alph*)]
\item if a flight departs from YYZ, then its airline must be Jazz Air;\label{egd} and
\item for Southwest airlines, if two tickets have the same code, then the ticket with an economy class must have lower fare than the one with the first class.\label{denial}
\end{enumerate*}
These can be expressed as the following denial constraints:
\begin{align*}
\text{\ref{egd} }\forall x, y, z, w, p, q\; \neg &(\textit{Flights}(x, y, z, \text{`YYZ'}, w, p, q) \land z \neq \text{`Jazz Air'})\\
\text{\ref{denial} }\forall x, y, z, w, p, q\; \neg &(\textit{Flights}(x, y, \text{`Southwest'}, z, w, p, q)\land\textit{Tickets}(r, x, \text{`First'}, t)\\
&\land \textit{Tickets}(r', x, \text{`Economy'}, t') \land t \leq t')
\end{align*}
Let us say that we want to find the PNR numbers of the tickets booked with first class, or with Silkair airlines. This can be expressed as the union $Q:=q_1 \cup q_2$ of two unary conjunctive queries, where
\begin{align*}
q_1(x):=\; & \exists x, y, z\; \textit{Tickets}(x, y, \textit{`First'}, z)\\
q_2(x):=\; & \exists x, y, z, w, p, q, r, s, t\; \textit{Tickets}(x, y, z, w) \land \textit{Flights}(y, \textit{`Silkair'}, p, q, r, s, t)
\end{align*}
\begin{figure*}[!ht]
\caption{Minimal violations, minimal witnesses, and near-violations in Example \ref{example2}.}
\begin{varwidth}[t]{.4\textwidth}
Minimal violations to $\Sigma$:
\begin{itemize}
\item $\{f_1, f_3\}$, $\{f_8\}$, $\{f_4, f_6, f_9\}$
\end{itemize}
Minimal witnesses to $Q$:
\begin{itemize}
\item $\{f_5\}$, $\{f_6\}$, $\{f_4, f_8\}$
\end{itemize}
\end{varwidth
\hspace{4em
\begin{varwidth}[t]{.6\textwidth}
Near-violations to $\Sigma$:\vspace{0.1in}\\
\begin{varwidth}[t]{.5\textwidth}
\begin{itemize}
\item $f_1$ : $\{f_3\}$
\item $f_3$ : $\{f_1\}$
\item $f_4$ : $\{f_6, f_9\}$
\item $f_6$ : $\{f_4, f_9\}$
\end{itemize}
\end{varwidth}
\begin{varwidth}[t]{.5\textwidth}
\begin{itemize}
\item $f_8$ : $\{f_{true}\}$
\item $f_9$ : $\{f_4, f_6\}$
\item $f_2, f_5, f_7$ : None
\end{itemize}
\end{varwidth}
\end{varwidth}
\label{fig1}
\end{figure*}
\begin{figure*}[!ht]
\centering
\caption{The $\alpha$-, $\beta$-, $\gamma$-clauses, and the $\theta$-expressions in Example \ref{example2}.}
\vspace{-0.2in}
\begin{align*}
\alpha\text{-clauses: } & (\neg x_1 \lor \neg x_3), (\neg x_8), (\neg x_4 \lor \neg x_6 \lor \neg x_9)\\
\beta\text{-clauses: } &(\neg x_5 \lor \neg p_1), (\neg x_6 \lor \neg p_2), (\neg x_4 \lor \neg x_8 \lor \neg p_3)\\
\gamma\text{-clauses: } & (x_1 \lor y^1_1), (x_2),(x_3 \lor y^3_1),(x_4 \lor y^4_1),(x_5), (x_6 \lor y^6_1), (x_7), (x_8 \lor y^8_1), (x_9 \lor y^9_1)\\
\theta\text{-expressions: } & (y^1_1 \leftrightarrow x_3), (y^3_1 \leftrightarrow x_1), (y^4_1 \leftrightarrow (x_6 \land x_9)), (y^6_1 \leftrightarrow (x_4 \land x_9)), (y^8_1 \leftrightarrow x_{true}), \\
&(y^9_1 \leftrightarrow (x_4 \land x_6))
\end{align*}
\vspace{-0.2in}
\ignore{
\begin{align*}
\phi = & (\neg x_1 \lor \neg x_3) \land (\neg x_8) \land (\neg x_4 \lor \neg x_6 \lor \neg x_9) \land (\neg x_5 \lor \neg p_1) \land (\neg x_6 \lor \neg p_2)\\
& \land (\neg x_4 \lor \neg x_8 \lor \neg p_3) \land (x_1 \lor x_3) \land (x_2) \land (x_4 \lor y^4_1) \land (x_5) \land (x_6 \lor y^6_1) \land (x_7)\\
& \land (x_9 \lor y^9_1) \land (\neg y^4_1 \lor x_6) \land (\neg y^4_1 \lor x_9) \land (y^4_1 \lor x_6 \lor x_9) \land (\neg y^6_1 \lor x_4)\\
& \land (\neg y^6_1 \lor x_9) \land (y^6_1 \lor x_4 \lor x_9) \land (\neg y^9_1 \lor x_4) \land (\neg y^9_1 \lor x_6) \land (y^9_1 \lor x_4 \lor x_6)
\end{align*}}
\label{fig2}
\end{figure*}
The minimal witnesses to $Q$, the minimal violations to $\Sigma$, and the near-violations to $\Sigma$ w.r.t.\ each fact of the database are shown in Figure \ref{fig1}. With these, we construct the $\alpha$-, $\beta$-, $\gamma$-clauses, and the $\theta$-expressions of $\phi$, as shown in Figure \ref{fig2}.
Even though, for simplicity, it is not mentioned in Reduction \ref{reduction3}, we do the following optimization in practice: if $|N^i_j| = 1$, we do not introduce a variable $y^i_j$, but, we use the $x$-variable corresponding to the only fact in $N^i_j$. In each satisfying assignment to $\phi$, the variable $p_1$ must take the value 0. In contrast, this is not the case for $p_2$ and $p_3$. By Proposition \ref{prop3}, `KLF88V' is a consistent answer to $Q$, but `MJ9C8R' and `NJ5RT3' are not.
\end{example}
\section{Computing Consistent Answers via \wmaxsat{}}
By Proposition \ref{prop1}, the consistent answer to a boolean conjunctive query over a schema $\mathcal R$ with primary key constraints can be computed by solving the \unsat{} instance constructed in Reduction \ref{reduction1}. For non-boolean queries, however, in a CNF-formula $\phi$ constructed using Reduction \ref{reduction2} or \ref{reduction3}, one needs to identify each variable $p_l$ such that there exists at least one satisfying assignment to $\phi$ in which $p_l$ gets set to 1. By Proposition \ref{prop3}, the corresponding potential answers can then be discarded for being inconsistent. One way to do this is as follows. Add a clause $(p_1 \lor ... \lor p_{|A|})$ to $\phi$, and solve $\phi$ using a \sat{} solver. For each $p_l$ that gets set to 1 in the solution of $\phi$, remove the literal $p_l$ from $\phi$ and then solve $\phi$ again. Repeat this process until $\phi$ is no longer satisfiable. At the end of this iterative process, the potential answers corresponding to the $p$-variables that still occur positively in $\phi$ are precisely the consistent answers to $Q$ on $I$. This approach, however, requires many \sat{} instances to be solved when the number of potential answers is large. For this reason, we developed and tested a different method that uses solving \wmaxsat{} instances. The construction of these \wmaxsat{} instances is described in Reduction \ref{reduction4}.
\begin{reduction}\label{reduction4}
Let the setup be the same as that of Reduction \ref{reduction2} (or Reduction \ref{reduction3}).
\begin{enumerate}
\item Construct a CNF-formula $\phi$ using Reduction \ref{reduction2} (or Reduction \ref{reduction3}).
\item Make all clauses in $\phi$ hard.
\item For each $\vec{a}_l \in \mathcal{A}$, construct a unit $\epsilon$-clause $\epsilon_l = (p_l)$.
\item Make all $\epsilon$-clauses soft, and of equal weights.
\item Construct the WCNF-formula $\psi = \phi \land \bigg(\overset{|\mathcal{A}|}{\underset{l=1}{\land}}\epsilon_l\bigg)$.
\end{enumerate}
\end{reduction}
\vspace{-0.13in}
\begin{algorithm}[!ht]
\caption{Eliminating Inconsistent Potential Answers}\label{alg:eliminate}
\begin{algorithmic}[1]
\Procedure{EliminateWithMaxSAT}{$\psi, \mathcal{A}$}
\State \textbf{let} \textsc{Ans} $=$ \textbf{bool array}$[|\mathcal{A}|]$
\For {$l = 1$ to $|\mathcal{A}|$}
\State \textsc{Ans}[$l$] $\gets$ true
\EndFor\State \textbf{let bool} \textit{moreAnswers} $\gets$ true
\While{\textit{moreAnswers}}
\State \textit{moreAnswers} $\gets$ false
\State \textbf{let} \textit{opt} $\gets$ \maxsat{$(\psi)$}\Comment{Use \wmaxsat{} solver}
\For {$l=1$ to $|\mathcal{A}|$}
\If{\textit{opt}[$p_l$] = 1}
\State \textit{moreAnswers} $\gets$ true
\State \textsc{Ans}[$l$] $\gets$ false
\State Remove the unit clause $(p_l)$ from $\psi$
\State Remove all clauses containing the literal $\neg p_l$ from $\psi$
\State Add a new unit hard clause $(\neg p_l)$ to $\psi$
\EndIf
\EndFor
\EndWhile
\State \textbf{return} \textsc{Ans}
\EndProcedure
\end{algorithmic}
\end{algorithm}
The preceding Algorithm \ref{alg:eliminate} computes the consistent answers by iteratively solving \wmaxsat{} instances. It takes as inputs the instance $\psi$ constructed using Reduction \ref{reduction4} and the set $\mathcal A$ of potential answers. The idea is to eliminate, in each iteration, as many inconsistent answers from $\mathcal{A}$ as possible by solving $\psi$. After each iteration, $\psi$ is modified in such a way that additional inconsistent answers, if any, can be eliminated in subsequent iterations.
In Section \ref{experiments}, we carried out experiments in which it turned out that the number of iterations taken by Algorithm \ref{alg:eliminate} is less than 4, even when there is a large number of potential answers.
\movetoappendix{\subsection*{Proof of Proposition~\ref{prop4}}}
\movetoappendix{
We first state and prove Lemma \ref{lemma1}, that reasons about the satisfying assignments to the WCNF-formula $\psi$, constructed using Reduction \ref{reduction4}. This lemma is used in proving Proposition \ref{prop4}.
\begin{lemma}\label{lemma1}
Let $\phi'$ be the CNF-formula constructed in Step 1 of Reduction \ref{reduction4}, and $\psi_i$ be the \wmaxsat{} instance at the beginning of $i^{th}$ iteration of Algorithm \ref{alg:eliminate}. For all $i$, every optimal solution of $\psi_i$ satisfies all clauses in $\phi'$.
\end{lemma}
\begin{proof}
We prove Lemma \ref{lemma1} by induction on $i$. The CNF-formula $\phi'$ constructed in Step 1 of Reduction \ref{reduction4} can always be satisfied by setting all $x$-variables to 1, and all $p$-variables to 0. The clauses in $\phi$ being hard, $\epsilon$ being soft ensures that every optimal solution of $\psi_0$ satisfies all clauses in $\phi$. Assume that for some $i \geq 0$, every optimal solution to $\psi_i$ satisfies all clauses in $\phi$. At the end of iteration $i+1$, if \textit{moreAnswers} is true, the formula $\psi_{i+1}$ is constructed from $\psi_i$ by adding to $\psi_i$ the unit hard clauses $(\neg p_l)$. This forces every optimal solution of $\psi_{i+1}$ to satisfy all of these added clauses. Since no $p_l$ variable occurs positively in $\phi$, we have that, for all $i$, every optimal solution to $\psi_{i+1}$ still satisfies $\phi$.
\end{proof}
}
\copytoappendix{
\begin{proposition}\label{prop4} Algorithm \ref{alg:eliminate} returns an array \textsc{Ans} such that $\vec{a}_l \in \cons{Q, I}$ if and only if the entry \textsc{Ans[$l$]} is true.
\end{proposition}
}
\movetoappendix{
\begin{proof}
In one direction, for every $l$, if $\vec{a}_l \in \cons{Q, I}$, then, by Proposition \ref{prop3}, the variable $p_l$ takes value 0 in every assignment that satisfies $\phi$. By Lemma \ref{lemma1}, for every $i$, the optimal solution of $\psi_i$ also assigns value 0 to the variable $p_l$. As a result, Line 12 never gets executed, and the entry \textsc{Ans}[$l$] remains true.
For the other direction, we first prove that Algorithm \ref{alg:eliminate} always terminates. Observe that at the end of the $i^{th}$ iteration, for every $l$, a unit clause $(p_l)$ is present in $\psi_i$ if and only if \textsc{Ans}[$l$] is true. Hence, at the end of $i^{th}$ iteration, if \textit{moreAnswers} is true, then the optimal solution to $\psi_i$ must have assigned value 1 to at least one variable $p_l$ such that \textsc{Ans}[$l$] was previously true. Therefore, at the end of $i^{th}$ iteration, at least $i$ entries in \textsc{Ans} are false. It follows that the algorithm terminates after at most $|\mathcal {A}|$ iterations.
Since no clause in $\phi$ contains a positive literal $p_l$, the addition of a unit hard clause $(\neg p_l)$ to $\psi_i$ does not suppress any satisfying assignments to $\phi$ while finding the optimal solution to $\psi_{i+1}$. Therefore, in every iteration, the optimal solution of $\psi$ guarantees to satisfy the maximum number of $p_l$ variables for which the unit clause $(p_l)$ is still in $\psi$. As a result, Algorithm \ref{alg:eliminate} does not terminate until it marks the entries \textsc{Ans}[$l$] false, for all $l$, for which there exists a satisfying assignment to $\phi$ in which $p_l$ gets assigned to 1. In other words, by Proposition \ref{prop3}, for every inconsistent answer $\vec{a}_l$, the entry \textsc{Ans} gets marked as false.
\end{proof}
}
\section{Preliminary Experimental Results}\label{experiments}
\looseness = -1 We evaluated the performance of CAvSAT using two different scenarios. First, we experimented with large synthetically generated databases having primary key constraints. We implemented Reduction \ref{reduction2} without the optimization mentioned in Section \ref{optimizations}. We also implement Reduction \ref{reduction4} and Algorithm \ref{alg:eliminate}. We found out that for seven non-boolean FO-rewritable queries that were also used in \cite{Kolaitis13}, CAvSAT significantly outperformed the database evaluation of the FO-rewritings obtained using the algorithm from \cite{Koutris17}. We also implemented Reduction \ref{reduction2} with the optimization, and evaluated its performance on fourteen additional conjunctive queries whose consistent answers are coNP-complete or are in P but are not FO-rewitable. In the second scenario, we evaluated the performance of CAvSAT using Reduction \ref{reduction3} on a real-world database with functional dependencies.
The definitions of the queries used in the experiments, as well as the FO-rewritings of the first seven queries, can be found in the Appendix.
\medskip
\noindent{\bf Experimental Setup}
All experiments were carried out on a machine running on Intel Core i7 2.7 GHz, 64 bit Ubuntu 16.04, with 8GB of RAM. We used PostgreSQL 10.1 as an underlying DBMS, and MaxHS v3.0 solver \cite{Davies2011} for solving the \wmaxsat{} instances. Our system is implemented in Java 9.04.
\subsection{Synthetic Data Generation}
The synthetic data were generated in two phases: (a) generation of consistent data; (b) injection of inconsistency into consistent data. The parameters used to generate the data were the number of tuples per relation (\textit{rSize}), degree of inconsistency (\textit{inDeg}), and the size of each key-equal group (\textit{kSize}). \\
\textbf{Generating consistent data} Each relation in the consistent database was generated with the same number of tuples, so that injecting inconsistency with specified \textit{kSize} and \textit{inDeg} will make the total number of tuples in the relation equal to \textit{rSize}. For each query used in the experiment, the data was generated so the evaluation of the query on the consistent database results in a relation that has the size 15\% to 20\% of \textit{rSize}. The values of the third attribute in all of the ternary relations, were chosen from a uniform distribution in the range [1, \textit{rSize}/10]. This was done to simulate a reasonably large number of potential answers. The remaining attributes take values from randomly generated alphanumeric strings of length 10.\\
\textbf{Injecting inconsistency} In each relation, the inconsistency was injected by inserting new tuples to the consistent data, that share the values of the key attributes with some already existing tuples from the consistent data. The parameter \textit{inDeg} denotes in percentage the number of tuples per relation, that participate in a key violation. We conducted experiments with the varying values for \textit{inDeg}, ranging from 5\% to 15\%. The values of \textit{kSize} were uniformly distributed between 2 to 5. The non-key attributes of the newly injected tuples were uniform random alphanumeric strings of length 10.
\subsection{Experimental Results}
\noindent{\bf CAvSAT on FO-rewritable Queries }
In this set of experiments, we compare the performance of CAvSAT against the FO-rewritings of seven queries over the database with primary key constraints. For queries $q_1, \dots, q_7$, we computed the FO-rewritings using the algorithm of Koutris and Wijsen in \cite{Koutris17}. We refer to these FO-rewritings as \textit{KW-FO-rewritings}. Since the queries $q_1, \dots, q_4$ happen to be in the class $C_\textit{forest}$, we computed additional FO-rewritings for them using the algorithm implemented in the consistent query answering system ConQuer \cite{Fuxman05}; we refer to these rewritings as \textit{ConQuer-FO-rewritings}.
Each FO-rewriting was translated into SQL, and fed to PostgreSQL.
Table \ref{formula-size} shows the size of the WCNF-formulas produced by Reduction \ref{reduction4} without optimization (where Reduction \ref{reduction2} is used inside Reduction \ref{reduction4}), on these queries over the databases having one million tuples per relation. Figure \ref{unopt} shows the evaluation time of CAvSAT with these formulas using MaxHS v3.0 solver \cite{Davies2011}. The letter E denotes the time required for encoding the problem into a \wmaxsat{} instance, and the letter S denotes the time taken by Algorithm \ref{alg:eliminate}. The percentage adjacent to the letters S and E denotes the degree of inconsistency. Figure \ref{opt-vs-unopt} (left) shows the significant gain in performance due to the optimization, for databases of size one million tuples per relation and with 10\% inconsistency. This is not surprising; since 90\% of the data were consistent, it is expected that most of the consistent answers lie in the consistent part of the database. Table \ref{formula-size-optimized} shows the size of the WCNF-formulas produced by Reduction \ref{reduction4}, with the optimization in place.
Figure \ref{fo-vs-sat} (right) shows that for the queries $q_1, \dots, q_4$ in the class $C_\textit{forest}$, the performance of CAvSAT is slightly worse, but comparable, to their ConQuer-FO-rewritings. For all seven queries $q_1, \dots, q_7$, however, CAvSAT significantly outperformed their KW-FO-rewritings, as PostgreSQL hit the two hours timeout while evaluating each KW-FO-rewriting. In fact, this timeout was hit by all seven queries even for databases of size as small as 100K tuples per relation.
For $q_1, \dots, q_7$, the average number of iterations taken by Algorithm \ref{alg:eliminate} to eliminate all inconsistent potential answers was 2.85.
\begin{table}[ht]
\centering
\begin{varwidth}[t]{0.4\linewidth}
\caption{Size of CNF-formula}
\begin{center}
\renewcommand{\arraystretch}{1.1}
\setlength{\tabcolsep}{3pt}
\begin{tabular}[h]{|c|c|c|}\hline
\centering
Query & Variables & Clauses
\\\hline\hline
$q_1$ & 2.08M & 2.18M \\\hline
$q_2$ & 2.07M & 2.12M\\\hline
$q_3$ & 3.15M & 3.07M\\\hline
$q_4$ & 3.23M & 3.07M\\\hline
$q_5$ & 2.07M & 2.12M \\\hline
$q_6$ & 3.3M & 3.06M\\\hline
$q_7$ & 3.25M & 3.06M \\\hline
\end{tabular}\label{formula-size}
\end{center}
\end{varwidth}
\begin{minipage}[t]{0.59\linewidth}
\captionof{figure}{Evaluation time of CAvSAT without optimization, for 1M tuples/relation.}
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width=\textwidth,height=5cm,
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\label{unopt}
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\end{table}
\vspace{-4em}
\begin{figure*}[h]
\caption{Evaluation time of CAvSAT with and without optimization (left). Evaluation time of CAvSAT with optimization, in comparison with the KW-FO-rewriting and the ConQuer-FO-rewriting, for 1M tuples/relation with 10\% inconsistency (right).}
\begin{varwidth}[t]{.55\textwidth}
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\label{opt-vs-unopt}
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\hspace{1em
\begin{varwidth}[t]{.45\textwidth}
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\begin{axis}[
width=\textwidth,height=5cm,
xlabel=FO-rewritable queries,
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\label{fo-vs-sat}
\end{figure*}
\medskip
\noindent {\bf CAvSAT on Harder Queries}
In this set of experiments, we considered fourteen additional non-boolean conjunctive queries whose consistent answers are coNP-complete or in P but not FO-rewritable (Figure \ref{fo-queries} in the Appendix). Figure \ref{hard-queries} shows that the time required for the optimizing and then constructing the \wmaxsat{} instance using Reduction \ref{reduction4}, dominates over the time taken by Algorithm \ref{alg:eliminate}. The solver takes comparatively more time for the queries that have more free variables or more atoms. Table \ref{formula-size-optimized} shows the size of the CNF-formulas constructed by Reduction \ref{reduction4} (where Reduction \ref{reduction2} is used inside Reduction \ref{reduction4}) in this experiment. The average number of iterations taken by Algorithm \ref{alg:eliminate} to eliminate all inconsistent potential answers to a query was 3.2.
\vspace{-0.4in}
\begin{table}[ht]
\caption{The size of the CNF-formulas with optimization, for 1M tuples per relation.}
\renewcommand{\arraystretch}{1.1}
\setlength{\tabcolsep}{3pt}
\centering
\begin{tabular}[h]{|c|c|c||c|c|c||c|c|c|}\hline
Query & Variables & Clauses & Query & Variables & Clauses & Query & Variables & Clauses
\\\hline\hline
$q_1$ & 16.5K & 20.9K & $q_8$ & 16.6K & 16.8K & $q_{15}$ & 14.9K & 15K\\\hline
$q_2$ & 68.6K& 76.0K & $q_9$ & 58K & 57.7K& $q_{16}$ & 58.8K & 58.4K \\\hline
$q_3$ &31.9K &36.8K & $q_{10}$ & 31.3K & 36.6K & $q_{17}$ & 40.1K & 41.4K\\\hline
$q_4$ & 117.2K& 123.7K& $q_{11}$ & 105K & 118.1K & $q_{18}$ & 107.5K & 121.4K\\\hline
$q_5$ & 16.3K&20.6K& $q_{12}$ & 116.8K & 123.4K & $q_{19}$ & 114.4K & 120.7K \\\hline
$q_6$ & 32.8K & 33.2K & $q_{13}$ & 63.2K & 65.7K & $q_{20}$ & 53.4K & 63.7K\\\hline
$q_7$ &32.5K & 33.8K & $q_{14}$ & 53.9K & 59.2K & $q_{21}$ & 170K & 199K\\\hline
\end{tabular}\label{formula-size-optimized}
\end{table}
\vspace{-0.3in}
\begin{figure*}[h]
\caption{Evaluation time of CAvSAT for conjunctive queries with varying data complexity, with optimization, over the databases of size 1M tuples/relation.}
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width=\textwidth,height=4.5cm,
xlabel= Conjunctive queries with varying data complexity,
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\label{hard-queries}
\end{figure*}
\movetoappendix{
\begin{figure*}
\caption{Queries used in the experiments with synthetic data.}
\begin{itemize}
\item[] \textbf{FO-rewritable consistent answers:}
\item[] $q_1(z):=\; \exists x, y, v, w\; (R_1(\underline{x},y,z) \land R_2(\underline{y},v,w))$
\item[] $q_2(z,w):=\; \exists x, y, v\; (R_1(\underline{x},y,z) \land R_2(\underline{y}, v,w))$
\item[] $q_3(z):=\; \exists x, y, v, u, d\; (R_1(\underline{x},y,z) \land R_3(\underline{y},v) \land R_2(\underline{v},u,d))$
\item[] $q_4(z,d):=\; \exists x, y, v, u\; (R_1(\underline{x},y,z) \land R_3(\underline{y},v) \land R_2(\underline{v},u,d))$
\item[] $q_5(z):=\; \exists x, y, v, w\; (R_1(\underline{x},y,z) \land R_4(\underline{y,v},w))$
\item[] $q_6(z):=\; \exists x, y, x', w, d\; (R_1(\underline{x},y,z) \land R_2(\underline{x'},y,w) \land R_5(\underline{x},y,d))$
\item[] $q_7(z):=\; \exists x, y, w, d\; (R_1(\underline{x},y,z) \land R_2(\underline{y},x, w) \land R_5(\underline{x},y,d))$\\
\item[] \textbf{In P, but not FO-rewritable, consistent answers:}
\item[] $q_8(z,w):=\; \exists x, y\; (R_1(\underline{x},y,z) \land R_2(\underline{y},x,w))$
\item[] $q_9(z):=\; \exists x, y, w, u, d\; (R_1(\underline{x},y,z) \land R_2(\underline{y},x,w) \land R_4(\underline{y},u,d))$
\item[] $q_{10}(z,w,d):=\; \exists x, y, u\; (R_1(\underline{x},y,z) \land R_2(\underline{y},x,w) \land R_4(\underline{y},u,d))$
\item[] $q_{11}(z):=\; \exists x, y, w\; (R_1(\underline{x},y,z) \land R_2(\underline{y},x,w))$
\item[] $q_{12}(v, d):=\; \exists x, y, z, u\; (R_3(\underline{x},y) \land R_6(\underline{y},z) \land R_1(\underline{z},x, d) \land R_4(\underline{x, u},v))$
\item[] $q_{13}(v):=\; \exists x, y, z, u\; (R_3(\underline{x},y) \land R_6(\underline{y},z) \land R_7(\underline{z},x) \land R_4(\underline{x, u},v))$
\item[] $q_{14}(d):=\; \exists x, y, z, u\; (R_3(\underline{x},y) \land R_6(\underline{y},z) \land R_1(\underline{z},x,d) \land R_7(\underline{x},u))$
\\
\item[] \textbf{coNP-complete consistent answers:}
\item[] $q_{15}(z):=\; \exists x, y, x', w\; (R_1(\underline{x},y,z) \land R_2(\underline{x'},y,w))$
\item[] $q_{16} (z,w):=\; \exists x, y, x'\; (R_1(\underline{x},y,z) \land R_2(\underline{x'},y,w))$
\item[] $q_{17}(z):=\; \exists x, y, x', w, u, d\; (R_1(\underline{x},y,z) \land R_2(\underline{x'},y,w) \land R_4(\underline{y},u,d))$
\item[] $q_{18}(z,w):=\; \exists x, y, x', u, d\; (R_1(\underline{x},y,z) \land R_2(\underline{x'},y,w) \land R_4(\underline{y},u,d))$
\item[] $q_{19}(z,w,d):=\; \exists x, y, x', u\; (R_1(\underline{x},y,z) \land R_2(\underline{x'},y,w) \land R_4(\underline{y},u,d))$
\item[] $q_{20}(z):=\; \exists x, y, x', w, u, d, v\; (R_1(\underline{x},y,z) \land R_2(\underline{x'},y,w) \land R_4(\underline{y},u,d) \land R_3(\underline{u},v))$
\item[] $q_{21}(z,w):=\; \exists x, y, x', u, d, v\; (R_1(\underline{x},y,z) \land R_2(\underline{x'},y,w) \land R_4(\underline{y},u,d) \land R_3(\underline{u},v))$
\end{itemize}\label{fo-queries}
\end{figure*}
\begin{figure*}[!ht]
\caption{The KW-FO-rewritings of the consistent answers to queries $q_1$ to $q_7$.}
\begin{align*}
q_1(f^{(1)}):=\; &\exists s_1 \in R_1 (\forall r_1 \in R_1(\exists s_2 \in R_2(\forall r_2 \in R_2
\\ &(s_1.1 \neq r_1.1 \lor s_2.1 \neq r_2.1 \lor (r_1.2 = r_2.1 \land r_1.3 = f.1)))))\\
q_2(f^{(2)}):=\; &\exists s_1 \in R_1(\forall r_1 \in R_1(\exists s_2 \in R_2(\forall r_2 \in R_2
\\ &(s_1.1 \neq r_1.1 \lor s_2.1 \neq r_2.1 \lor (r_2.3 = f.1 \land r_1.2 = r_2.1 \land r_1.3 = f.2)))))\\
q_3(f^{(1)}):=\; &\exists s_1 \in R_1(\forall r_1 \in R_1(\exists s_2 \in R_3(\forall r_2 \in R_3(\exists s_3 \in R_2(\forall r_3 \in R_2
\\&((s_1.1 \neq r_1.1 \lor s_2.1 \neq r_2.1 \lor s_3.1 \neq r_3.1 \lor (r_2.2 = r_3.1 \land r_1.2 = r_2.1 \land r_1.3 = f.1))))))))\\
q_4(f^{(2)}):=\; &\exists s_1 \in R_1(\forall r_1 \in R_1(\exists s_2 \in R_3(\forall r_2 \in R_3(\exists s_3 \in R_2(\forall r_3 \in R_2
\\&(s_1.1 \neq r_1.1 \lor s_2.1 \neq r_2.1 \lor s_3.1 \neq r_3.1 \\&\lor(r_3.3 = f.1 \land r_2.2 = r_3.1 \land r_1.2 = r_2.1 \land r_1.3 = f.2)))))))\\
q_5(f^{(1)}):=\; &\exists s_1 \in R_1(\forall r_1 \in R_1(\exists s_2 \in R_4(\forall r_2 \in R_4
\\&(s_1.1 \neq r_1.1 \lor s_2.1 \neq r_2.1 \lor s_2.2 \neq r_2.2 \lor (r_1.2 = r_2.1 \land r_1.3 = f.1)))))\\
q_6(f^{(1)}):=\; &\exists s_1 \in R_2(\forall r_1 \in R_2(\exists s_2 \in R_5(\forall r_2 \in R_5(\exists s_3 \in R_1(\forall r_3 \in R_1
\\&(s_1.1 \neq r_1.1 \lor s_2.1 \neq r_2.1 \lor s_3.1 \neq r_3.1 \\&\lor (r_2.1 = r_3.1 \land r_1.2 = r_2.2 \land r_1.2 = r_3.2 \land r_3.3 = f.1)))))))\\
q_7(f^{(1)}):=\; &\exists s_1 \in R_2(\forall r_1 \in R_2(\exists s_2 \in R_5(\forall r_2 \in R_5(\exists s_3 \in R_1(\forall r_3 \in R_1
\\&(s_1.1 \neq r_1.1 \lor s_2.1 \neq r_2.1 \lor s_3.1 \neq r_3.1 \\&\lor (r_1.2 = r_2.1 \land r_1.2 = r_3.1 \land r_1.1 = r_2.2 \land r_1.1 = r_3.2 \land r_3.3 = f.1)))))))\\
\end{align*}
\vspace{-0.3in}\\
All variables range over tuples in relations; $f^{(i)}$ denotes a tuple with $i$ elements, $i = 1, 2$.
\label{fo-rewritings}
\end{figure*}
\begin{figure*}[!ht]
\caption{The SQL translations of the ConQuer-FO-rewritings of the consistent answers to queries $q_1$ to $q_4$.}
\begin{minipage}[t]{0.45\textwidth}
$q_1(R1\_3):$\\
Candidates as (\\
select distinct R1.3 as R1\_3,R1.1 as R1\_1\\
from R1, R2\\
where (R1.2 = R2.1))\\
Filter as (select R1\_1 \\
from Candidates C join R1 on C.R1\_1 = R1.1 \\
left outer join R2 on R1.2 = R2.1 \\
where (R2.1 is null) \\
union all select C.R1\_1 from Candidates C \\
group by C.R1\_1 \\
having count(*) $>$ 1)\\
select R1\_3 \\
from Candidates C \\
where not exists \\
(select * from Filter F where C.R1\_1 = F.R1\_1)\\
$q_3(R1\_3):$\\
Candidates as (\\
select distinct R1.3 as R1\_3, R1.1 as R1\_1\\
from R1, R3, R2\\
where (R3.2 = R2.1) and (R1.2 = R3.1))\\
Filter as (\\
select R1\_1 \\
from Candidates C join R1 on C.R1\_1 = R1.1 \\
left outer join R3 on R1.2 = R3.1 left outer join R2 on R3.2 = R2.1 \\
where (R3.1 is null OR R2.1 is null) \\
union all select C.R1\_1 from Candidates C \\
group by C.R1\_1 \\
having count(*) $>$ 1)\\
select R1\_3 \\
from Candidates C \\
where not exists \\
(select * from Filter F where C.R1\_1 = F.R1\_1)\\
\end{minipage}
\hspace{0.1\textwidth}
\begin{minipage}[t]{0.45\textwidth}
$q_2(R1\_3, R2\_3):$\\
Candidates as (\\
select distinct R1.3 as R1\_3, R2.3 as R2\_3,\\
R1.1 as R1\_1 from R1, R2\\
where (R1.2 = R2.1))\\
Filter as (select R1\_1 \\
from Candidates C join R1 on C.R1\_1 = R1.1 \\
left outer join R2 on R1.2 = R2.1 \\
where (R2.1 is null) \\
union all select C.R1\_1 from Candidates C \\
group by C.R1\_1 \\
having count(*) $>$ 1)\\
select R1\_3, R2\_3 \\
from Candidates C \\
where not exists \\
(select * from Filter F where C.R1\_1 = F.R1\_1)\\
$q_4(R1\_3, R2\_3):$\\
Candidates as (\\
select distinct R1.3 as R1\_3, R2.3 as R2\_3,R1.1 as R1\_1\\
from R1, R3, R2\\
where (R3.2 = R2.1) and (R1.2 = R3.1))\\
Filter as (select R1\_1 \\
from Candidates C join R1 on C.R1\_1 = R1.1 \\
left outer join R3 on R1.2 = R3.1 left outer join R2 on R3.2 = R2.1 \\
where (R3.1 is null or R2.1 is null) \\
union all select C.R1\_1 from candidates C \\
group by C.R1\_1 \\
having count(*) $>$ 1)\\
select R1\_3, R2\_3 \\
from Candidates C \\
where not exists \\
(select * from Filter F where C.R1\_1 = F.R1\_1)\\
\end{minipage}
\label{conquer-fo-rewritings}
\end{figure*}
}
\noindent{\bf Results on the Real-World Databases}
In this set of experiments, we evaluated the performance of CAvSAT using Reduction \ref{reduction3} on real-world data having key constraints on each relation, along with one functional dependency. The data used are about inspections of food establishments in New York and Chicago, and are taken from \cite{nydata} and \cite{chicagodata}. Part of this data have been previously used for evaluating data cleaning systems, such as HoloClean \cite{Rekatsinas17}. Since the structure of the schema or the constraints on the database were not specified by the source, we decomposed the data into four relations, and assumed reasonable key constraints for all relations and also one additional functional dependency, as shown in Table \ref{real-data-schema}. We evaluated the performance of Reduction \ref{reduction3} on six queries depicted in Figure \ref{dc-queries} in the Appendix. For example, query $Q_3$ returns the names of the restaurants, such that they are present in both New York and Chicago, and they were inspected on the same day. Figure \ref{real-queries} shows that the solver took the most amount of time to compute answers to this query. Not surprisingly, the evaluation time increases as the number of atoms or the number of free variables in the query grow. Table \ref{formula-size-realdata} shows the size of the CNF-formulas produced by Reduction \ref{reduction4} (where Reduction \ref{reduction3} is used inside Reduction \ref{reduction4}). No optimization was implemented in this set of experiments.
\movetoappendix{
\begin{figure*}[!ht]
\caption{Queries used on the real-world database.}
\begin{align*}
Q_1():=\; &\exists x, y, z, w, v, y', z', w', v' \; (\text{NY\_Rest}(x,y,z,w,v) \land \text{CH\_Rest}(x,y',z',w',v'))\\
Q_2(x):=\; &\exists y, z, w, v, y', z', w', v' \; (\text{NY\_Rest}(x,y,z,w,v) \land \text{CH\_Rest}(x,y',z',w',v'))\\
Q_3(x):=\; &\exists y, z, w, v, y', z', w', v', q, r, s, t, q', s', t' \; (\text{NY\_Rest}(x,y,z,w,v) \land \text{CH\_Rest}(x,y',z',w',v')\\ &\land \text{NY\_Insp}(y,q,r,s,t) \land \text{CH\_Insp}(y',q',r,s',t'))\\
Q_4(x, y):=\; &\exists z, w, v, q, r, s \; (\text{CH\_Rest}(x,y,z,w,v)\land \text{CH\_Insp}(y,q,r,s,\text{`Pass'}))\\
Q_5(x):=\; &\exists y, z, w, v, q, r, s \; (\text{CH\_Rest}(x,y,z,w,v)\land \text{CH\_Insp}(y,q,r,s,\text{`Fail'}))\; \cup\\
&\exists y, z, w, v, q, r, s \; (\text{NY\_Rest}(x,y,z,w,v)\land \text{NY\_Insp}(y,q,r,s,\text{`Fail'}))\\
Q_6(x, v):=\; &\exists y, z, w, y', z', w', v', q, r, s \; (\text{CH\_Rest}(x,y,z,w,v)\land \text{NY\_Rest}(x,y',z',w',v')\\&\land \text{NY\_Insp}(y',\text{`Not Critical'},q,r,s))\\
\end{align*}
\label{dc-queries}
\end{figure*}
}
\vspace{-0.3in}
\begin{table}[!ht]
\caption{The schema and the constraints of the real-world database.}
\centering
\begin{tabular}[t]{|l|l|}\hline
Relation & \# Tuples
\\\hline\hline
NY\_Insp (\underline{LicenseNo}, Risk, \underline{InspDate, InspType}, Result) & 229K \\\hline
NY\_Rest (Name, \underline{LicenseNo}, Cuisine, Address, Zip) & 26.5K\\\hline
CH\_Insp (\underline{LicenseNo}, Risk, \underline{InspDate, InspType}, Result) & 167K\\\hline
CH\_Rest (Name, \underline{LicenseNo}, Facility, Address, Zip) & 31.1K \\\hline
\end{tabular}
\vspace{0.08in}\\
\begin{tabular}[t]{|l|l|l|}\hline
Constraint & Type & Violations
\\\hline\hline
NY\_Insp (LicenseNo, InspDate, InspType $\rightarrow$ Risk, Result) & Key & 25.6\%\\\hline
NY\_Rest (LicenseNo $\rightarrow$ Name, Cuisine, Address, Zip) & Key & 0\%\\\hline
CH\_Insp (LicenseNo, InspDate, InspType $\rightarrow$ Risk, Result) & Key & 0.07\%\\\hline
CH\_Rest (LicenseNo $\rightarrow$ Name, Cuisine, Address, Zip) & Key & 5.86\%\\\hline
CH\_Rest (Name $\rightarrow$ Zip) & FD & 9.73\%\\\hline
\end{tabular}
\label{real-data-schema}
\end{table}
\vspace{-0.4in}
\begin{table}[!ht]
\begin{minipage}[t]{0.55\linewidth}
\captionof{figure}{Evaluation time of CAvSAT on real data.}
\centering
\vspace{0.12in}
\begin{tikzpicture}
\begin{axis}[ybar stacked,
width=\textwidth,height=4cm,
xlabel= Unions of conjunctive queries,
label style={font=\scriptsize},
xtick=data,
xticklabels={$Q_1$,$Q_2$,$Q_3$,$Q_4$,$Q_5$,$Q_6$},
legend style={at={(1,0.95)},font=\scriptsize,draw=none,fill=none},
legend cell align = {left},
bar width=6pt,
ymin=0, ymax=50,
enlargelimits=0.1,
y label style={at={(0.13,0.5)}},
x label style={at={(0.5,0.08)}},
xmin = 1,xmax=6,
ylabel=Eval. time (sec)]
\addplot[fill=coolblack!30!white,draw=none] coordinates {
(1, 11.216
(2,9.544)
(3,12.079
(4,9.035
(5,11.128
(6, 9.034
};\addlegendentry{Encoding time}
\addplot[fill=coolblack, draw=none] coordinates {
(1, 2.235)
(2,5.854)
(3,17.130)
(4,10.915)
(5,12.554)
(6, 7.437)
};\addlegendentry{Solving time}
\end{axis}
\end{tikzpicture}
\label{real-queries}
\end{minipage}
\hspace{2em}
\begin{varwidth}[t]{0.45\linewidth}
\caption{Size of the CNF-formula.}
\begin{center}
\renewcommand{\arraystretch}{1.1}
\setlength{\tabcolsep}{3pt}
\begin{tabular}[h]{|c|c|c|}\hline
Query & Variables & Clauses\\\hline\hline
$Q_1$ & 455.1K & 793.7K\\\hline
$Q_2$ & 456.5K& 794K \\\hline
$Q_3$ &455.1K &671.5K \\\hline
$Q_4$ & 476K & 861.5K\\\hline
$Q_5$ & 486.7K & 836.2K\\\hline
$Q_6$ & 455.5K & 1.12M\\\hline
\end{tabular}\label{formula-size-realdata}
\end{center}
\end{varwidth}
\end{table}
\newpage
\section{Concluding Remarks}
We designed and implemented CAvSAT, the first SAT-based system for consistent query answering. Our preliminary stand-alone evaluation shows that a SAT-based approach can give rise to a scalable system for consistent query answering. We note that, on queries with first-order rewritable consistent answers, CAvSAT had comparable or even better performance to evaluating the first-order rewritings using a database engine. This finding suggests a potential difference between theory and practice, since the study of first-order rewritability of the consistent answers was motivated from having an efficient evaluation of consistent answers using the database engine alone.
The next step in this investigation is to carry out an extensive comparative evaluation of CAvSAT with other systems for consistent query answering and, in particular, with systems that use reduction-based methods \cite{Kolaitis13,MannaRT11}.
\smallskip
\noindent{\bf Acknowledgments} Dixit is supported by the Center for Research in Open Source Software (CROSS) at UC Santa Cruz. Kolaitis is supported by NSF Grant IIS:1814152.
\clearpage
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 6,580 |
\section{Introduction}
The study of quantum many-body entanglement has provided many key insights into the structure of quantum states of matter. Low-energy states of quantum lattice systems obey typically the so-called ``area law" of the entanglement entropy \cite{Calabrese2009,Eisert2010,Cirac2012a}. As such, the area-law is a huge constraint on the classes of states that capture the relevant properties of matter at low energies. A more refined study has shown that, in fact, those states are captured by the so-called tensor network states, or simply ``tensor networks" \cite{Schuch2013b,Orus2013}. Such states obey naturally the area-law, and are at the basis of many theoretical and numerical developments in the study of quantum many-body systems and beyond \cite{Orus2014}. Examples of such states are, e.g., Matrix Product States (MPS)~\cite{Vidal2004a}, Projected Entangled Pair States (PEPS)~\cite{Verstraete2004b,Verstraete2006}, and the Multiscale Entanglement Renormalization Ansatz (MERA)~\cite{Vidal2007a}. These structures are, respectively, behind the so-called Density Matrix Renormalization Group algorithm (DMRG) for 1d systems~\cite{White1992}, PEPS-algorithms for 2d systems~\cite{Verstraete2008a}, and Entanglement Renormalization~\cite{Vidal2009a}.
The description of quantum many-body states in terms of tensor networks has several advantages. Apart from naturally obeying the area-law (and therefore capturing the correct expected entanglement behavior), TN states can also be manipulated efficiently (either exactly or approximately). Another advantage is the fact that both lattice and internal symmetries can be naturally incorporated. For instance, a description in terms of symmetric tensors~\cite{Singh2010a,Schuch2010a} can lead to important computational
advantages~\cite{Singh2010b,Singh2011,Singh2012,Weichselbaum2012,Singh2013},
and helps in the theoretical classification of phases of
matter~\cite{Rispler2015}. Moreover, gauge symmetries can also be naturally incorporated~\cite{Haegeman2015}, hence offering a natural framework to describe lattice gauge theories~\cite{Zohar2016a,Zohar2016b}.
In a seminal paper~\cite{Jiang2015}, S.~Jiang and Y.~Ran made the first attempt to organize PEPS into crude classes distinguished by short-range physics, related to the fractionalization of both on-site symmetries and space-group symmetries. In their work, the authors introduced (quite generally) the notion of projective symmetry group (PSG) for PEPS, enabling
to deal {\it a priori} with gauge equivalence between tensors. Using lattice quantum numbers, the authors predicted a number of district classes for spin-$\frac{1}{2}$ spin liquids on the Kagome lattice. More recently, a similar framework was applied to classify (trivial) \hbox{spin-$1$} PEPS on the square lattice~\cite{Lee2016}.
Our goal in this paper is to produce a simple classification scheme of all rank-5 SU(2)-spin rotational symmetric tensors. We characterize the tensors according to three criteria: i) the on-site physical spin-$S$, (ii) the local Hilbert space $V^{\otimes 4}$ of the four virtual (composite) spins attached to each site and (iii) the irreducible representations of the $C_{4v}$ point group of the square lattice. Using this scheme, we produce explicit expressions for all SU(2)-symmetric translationally and rotationally-invariant PEPS with bond dimension $D\leqslant 6$. As we shall see, one can recover all known SU(2)-symmetric PEPS on the square lattice as particular cases in our classification. Generically, to each of our symmetry class can be associated a $({\cal D}-1)$-dimensional manifold of spin liquids (potentially) preserving lattice symmetries and defined in terms of ${\cal D}$ independent tensors of a given bond dimension $D$. In addition, generic (low-dimensional) families of PEPS explicitly breaking particular point-group lattice symmetries (lattice nematics) and/or time reversal symmetry (chiral spin liquids~\cite{Kalmeyer1987,Schroeter2007}) can also be constructed. Finally, we apply this framework to search for new topological chiral spin liquids characterized by well-defined chiral edge modes, as revealed by their entanglement spectrum, and show how the symmetrization of a given double-layer PEPS leads to a chiral topological state with a gapless edge described by a SU(2)$_2$ Wess-Zumino-Witten (WZW) model~\cite{Witten1983}.
The paper is organized as follows: in Sec.~\ref{sec2} we elaborate on the specifics of our
classification~\footnote{The complete list of SU(2)-spin rotationally invariant PEPS with bond dimension $D\leqslant 6$ as well as the expressions of all the corresponding site tensors are provided in the Supplemental Material.}, show how many remarkable states of matter fit into it, explain the procedure to construct spin liquids, and point out the connection to previous work. In Sec.~\ref{sec3} we explain several attempts to obtain PEPS corresponding to higher-spin chiral topological quantum spin liquids. In particular, we show how a double-layer PEPS leads to a chiral topological state with a gapless edge described by a SU(2)$_2$ WZW model, which we characterize through its entanglement spectrum (ES). We also discuss how the PEPS tensor of such double-layer wavefunction can be expanded as a sum of ``fundamental" SU(2)-invariant PEPS tensors, which we also characterize. In Sec.~\ref{sec4} we wrap up our conclusions and outline several directions for future work. Finally, in Appendix~\ref{app1} we review the calculation of entanglement spectrum (ES) from 2d PEPS, and in Appendices~\ref{app2} and \ref{app3} we provide the coefficients for the PEPS tensors of several remarkable
and simple states, and of all tensors entering the
decomposition of the double-layer CSL, respectively.
\section{Classification of SU(2)-symmetric PEPS on the square lattice}
\label{sec2}
\subsection{Construction}
We consider here a square lattice (as shown in Fig.~\ref{FIG:lattice_tensor}(a)) with physical spin-$S$ site degrees of freedom. Hence, $d=2S+1$ basis states are assigned on each site. Here, we explicitly consider transitionally invariant states described by a PEPS built from a single tensor $A$, as the one shown in Fig.~\ref{FIG:lattice_tensor}(b). Each physical site has four virtual spins attached labelled on the figure by $u$, $l$, $d$ and $r$ along the up, left, down and right directions, respectively.
The virtual states $|v_\alpha\big>$ belong to some representation $V$ of SU(2) of total dimension $D$ which, generically,
is a direct sum $V=\oplus V_i$ of $\cal N$ irreducible representations (IRREP) $V_i$ of SU(2), each of partial dimension $2V_i+1$
and $D=\sum_{i=1}^{\cal N} (2V_i+1)$.
\begin{figure}
\begin{center}
\includegraphics[width=8cm,angle=0]{Figures/lattice_tensor}
\caption{[Color online] (a) Square lattice invariant under $C_{4v}$ point group symmetries (reflection axis are shown).
The generators of the point group are, e.g., the 90-degree rotation $R$, the reflection $R_x$ and
the inversion ${\cal I}=R_xR_y$. (b) A generic rank-5 PEPS tensor with one physical index $s$ and four virtual indices $u$, $l$, $d$ and $r$.}
\label{FIG:lattice_tensor}
\end{center}
\end{figure}
The corresponding translationally invariant PEPS~\cite{Cirac2012a} is obtained by assigning the same tensor $A$ on every physical site. Physically, the site tensor $A$ simply encodes a projector that maps the virtual space $V^{\otimes 4}$ onto all
$2S+1$ components of the physical spin-$S$. From the bond point of view, every pair of the nearest-neighbor (NN) virtual spins is projected to a block diagonal {\it virtual spin singlet state}. By construction the obtained wave function is a global spin singlet, i.e. invariant under SU(2) rotations. For a bi-partite lattice as the square lattice one can perform a simple spin rotation (by $\pi$ around the $Y$-spin axis)
on all sites of a given sublattice that transforms the virtual bond singlets into diagonal maximally entangled NN pair states
$|{\cal S}\big> =\sum_{\alpha=1}^D |v_\alpha v_\alpha \big>$. In this way the SU(2)-invariant PEPS becomes a simple
contraction of the tensor network of the $A$'s.
Our construction can also be easily generalized to states that are not SU(2)-singlets, i.e., have a total (average) spin component. For this, one could eventually build up tensors that transform under $S \neq 0$ IRREPS of SU(2) (sometimes dubbed "covariant" states). In this paper, however, we consider only "invariant" states under SU(2) -i.e. singlets-, which for simplicity we also call "symmetric".
The $2S+1$ components $A_s$ of a tensor $A$ encode the projectors $P_s : V^{\otimes 4}\rightarrow |s\big>$
onto the $|s\big>\equiv |S,S_z=s\big>$ physical state. Hence, the problem of enumerating all SU(2)-invariant PEPS
reduces to the finding of all (orthogonal) projectors that map
any virtual space $V^{\otimes 4}$ onto any spin-$S$ Hilbert space. Reversely, it amounts to enumerate
all $D^4$ orthogonal spin-$S$ wave functions that can be constructed out of the $(\oplus V_i)^{\otimes 4}$ basis
states.
In other words, we shall use the one-to-one correspondence between projectors and wave functions and
extract the tensor components from the wave functions.
One can write the physical state $|s\big>$ as
\begin{equation}
|s\big>=\sum_{\alpha_1,\alpha_2,\alpha_3,\alpha_4}
A_s(\alpha_1,\alpha_2,\alpha_3,\alpha_4) |\alpha_1,\alpha_2,\alpha_3,\alpha_4\big>,
\end{equation}
in terms of the $D^4$ virtual basis states
$|\alpha_1,\alpha_2,\alpha_3,\alpha_4\big>$ and of the tensor elements $A_s(\alpha_1,\alpha_2,\alpha_3,\alpha_4)$.
Since we can always find a set of $D^4$ orthogonal wave functions, the corresponding basic tensors fulfill the ``orthonormalisation'' property
\begin{eqnarray}
\sum_{\alpha_1,\alpha_2,\alpha_3,\alpha_4} [A_s(\alpha_1,\alpha_2,\alpha_3,\alpha_4)]^* B_{s'}(\alpha_1,\alpha_2,\alpha_3,\alpha_4)
\nonumber \\ =\delta_{s s'} \delta_{AB}\, ,
\end{eqnarray}
where the left-hand side defines some tensor inner product $A_s\cdot B_{s'}$.
We have performed such a program {\it analytically} using Mathematica for all possible virtual spaces with
$D\leqslant 6$. To reduce the cost of the computation, it is advantageous to use spin and (lattice) point group symmetries. First, it is convenient to decompose the virtual space $V^{\otimes 4}$ into all disconnected subspaces given
by the occupations $n_{\rm occ.}=\{n_1,\cdots, n_{\cal N}\}$ of the $\cal N$ spins $V_i$ ($\sum_{i=1}^{\cal N} n_i=4$), each subspace providing a different class of tensors. Secondly, we use the $S_z$ quantum number of the (physical) wave function. In fact, we start by computing all
$S_z=S$ wave functions (or equivalently all projectors onto the maximum $S_z=S$ subspace) and then apply the spin-lowering operator $S_-$ written in the virtual basis states. Simultaneously, we classify the various spin-$S$ wave functions according to
their point symmetry, i.e., according to the representations of the $C_{4v}$ point group: $A_1$ ($s$-wave),
$B_1$ ($d_{x^2-y^2}$-wave), $E$ (doubly-degenerate $p$-wave), $A_2$ ($g$-wave) and $B_2$ ($d_{xy}$-wave).
To accomplish such a purpose, we simply need to diagonalize, in the space spanned by the
$(\oplus V_i)^{\otimes 4}$ virtual basis
attached to a given site, simultaneously i) the total spin operator, ii) the 90-degree rotation operator $R$, iii) the reflection symmetry
$R_x$ operator, and iv) the inversion ${\cal I}=R_xR_y$ operator (see Fig.~\ref{FIG:lattice_tensor}(a)).
In practice, to perform this task efficiently, we have constructed in the $V^{\otimes 4}$ basis, the combined
(non-Hermitian) operator
\begin{equation}
{\cal O}_{\sigma,\sigma_z,\rho,\delta,\nu}=\sigma {\bf S}^2 + \sigma_z S_z + \rho R + \delta R_x + \nu D
\end{equation}
where the diagonal operator $D$ has specific diagonal elements characterizing each $n_{\rm occ.}$ sector.
The (real) coefficients $\sigma,\sigma_z,\rho,\delta,\nu$ are all chosen of very different magnitudes -- e.g. $10^{8},
10^{6}, 10^{4}, 10^{2}, 1$ -- in order to sort out the various eigenvalues (of order $1$).
Note that tensors belonging
to the $A_1$, $B_1$, $A_2$ and $B_2$ symmetry classes are purely real while the
$E$-symmetric tensors are intrinsically complex and come in complex conjugated pairs.
As a simple example, let us consider the case $V=\frac{1}{2}$ which contains
$2^4=16$ basis states, all in the unique $n_{\rm occ.}=\{4\}$ sector ($V$ here is a simple IRREP).
Diagonalizing ${\cal O}_{\sigma,\sigma_z,\rho,\delta,\nu}$ (omitting the $D$ part) gives 16 states (or tensor components) which can be grouped into
two singlets ($S=0$), three triplet ($S=1$) and one quintuplet ($S=2$), in agreement with the decomposition
$(\frac{1}{2})^{\rm \otimes 4}=(0\oplus 1)^{\rm \otimes 2}=2(0)\oplus 3(1)\oplus (2)$.
The $S=0$ outcomes correspond to ``classical'' TN, not considered afterwards. Inspection of the
eigenvalues associated to $R$ and $R_x$ reveals that one of the triplet states (or tensors) has $B_1$
symmetry, while the other two form a complex conjugate pair of $E$ symmetry. On the other hand,
the unique $S=2$ tensor is fully symmetric ($A_1$ IRREP) and corresponds to the spin-$2$ AKLT state
(see below).
\begin{table*} [htb]
\begin{center}
\resizebox{2.08\columnwidth}{!}{%
\begin{tabular}{@{} ccccc@{}}
\hline
\hline
V $\backslash$ S & 1/2 & 1 & 3/2 & 2 \\
\hline
$\frac{1}{2}$& & $\bf B_1$ $E$ & & $\bf A_1$ \\
\hline
$\frac{1}{2}\oplus 0$ & & & & \\
$n_{\rm occ.}=\{1,3\}$ & $\bf A_1$[$\bf B_1$] $\bf E$ & & & \\
$n_{\rm occ.}=\{2,2\}$ & & $\bf A_1^{(a)}$ $\bf A_1^{(b)}$[$\bf B_1$] $E$ $B_2$& & \\
$n_{\rm occ.}=\{3,1\}$ & $\bf A_1$[$\bf B_1$] $E^{(a,b)}$ $\bf A_2$[$\bf B_2$]& &$A_1$[$B_1$] $E$ &\\
\hline
$1$ & & $B_1$ $E^{(a,b)}$ $A_2$& & $A_1^{(a)}$[$B_1$] $A_1^{(b)}$ $E$ $B_2$\\
\hline
$\frac{1}{2}\oplus 0\oplus 0$ & & & & \\
$n_{\rm occ.}=\{1,1,2\}$ & $A_1^{(a,b)}$[$B_1^{(a,b)}$] $E^{(a-c)}$ $A_2$[$B_2$]& & & \\
$n_{\rm occ.}=\{2,1,1\}$ & & $A_1^{(a,b)}$[$B_1^{(a,b)}$] $E^{(a-c)}$ $A_2$[$B_2$]& & \\
\hline
$\frac{1}{2}\oplus \frac{1}{2}$ & & & & \\
$n_{\rm occ.}=\{1,3\}$ & & $A_1^{(a,b)}$[$B_1^{(a,b)}$] $E^{(a-c)}$ $A_2$[$B_2$]& &$A_1$[$B_1$] $E$ \\
$n_{\rm occ.}=\{2,2\}$ & & $A_1^{(a,b)}$[$B_1^{(a,b)}$] $B_1^{(c)}$ $E^{(a-e)}$ $A_2^{(a)}$ $A_2^{(b)}$[$B_2$] &
& $A_1^{(a)}$ $A_1^{(b)}$[$B_1$] $E$ $B_2$ \\
\hline
$1\oplus 0$ & & & & \\
$n_{\rm occ.}=\{1,3\}$ & & $\bf A_1$[$\bf B_1$] $E$ & & \\
$n_{\rm occ.}=\{2,2\}$ & & $B_1$[$A_2$] $E^{(a,b)}$ & & $\bf A_1^{(a)}$[$\bf B_2$] $\bf A_1^{(b)}$[$\bf B_1$] $E$ \\
$n_{\rm occ.}=\{3,1\}$ & & $A_1^{(a,b)}$[$B_1^{(a,b)}$] $E^{(a-c)}$ $A_2$[$B_2$] & &$A_1$[$B_1$] $E$ \\
\hline
$\frac{3}{2}$& & $B_1^{(a)}$ $B_1^{(b)}$ $E^{(a-c)}$ $A_2$ & & $A_1^{(a-c)}$ $B_1$ $E^{(a)}$ $E^{(b)}$ $A_2$[$B_2^{(a)}$] $B_2^{(b)}$ \\
\hline
$\frac{1}{2}\oplus 0\oplus 0\oplus 0$ & & & & \\
$n_{\rm occ.}=\{1,1,1,1\}$ & $A_1^{(a-c)}$[$B_1^{(a-c)}$] $E^{(a-f)}$ $A_2^{(a,b)}$[$B_2^{(a,b)}$]& & & \\
\hline
$\frac{1}{2}\oplus \frac{1}{2}\oplus 0$ & & & & \\
$n_{\rm occ.}=\{1,2,1\}$ & $A_1^{(a-c)}$[$B_1^{(a-c)}$] $E^{(a-f)}$ $A_2^{(a-c)}$[$B_2^{(a-c)}$]& & $A_1^{(a,b)}$[$B_1^{(a,b)}$] $E^{(a-c)}$
$A_2$[$B_2$] & \\
$n_{\rm occ.}=\{1,1,2\}$ & & $A_1^{(a,b)}$[$B_1^{(a,b)}$] $E^{(a-c)}$ $A_2$[$B_2$] & & \\
\hline
$1\oplus 0\oplus 0$ & & & & \\
$n_{\rm occ.}=\{1,1,2\}$ & & $A_1^{(a,b)}$[$B_1^{(a,b)}$] $E^{(a-c)}$ $A_2$[$B_2$] & & \\
$n_{\rm occ.}=\{2,1,1\}$ & & $A_1$[$B_1$] $E^{(a-c)}$ $A_2^{(a,b)}$[$B_2^{(a,b)}$]& & $A_1^{(a,b)}$[$B_1^{(a,b)}$] $E^{(a-c)}$ $A_2$[$B_2$]\\
\hline
$1\oplus \frac{1}{2}$ & & & & \\
$n_{\rm occ.}=\{1,3\}$ & $A_1^{(a,b)}$[$B_1^{(a,b)}$] $E^{(a-c)}$ $A_2$[$B_2$] &
& $A_1^{(a,b)}$[$B_1^{(a,b)}$] $E^{(a-c)}$ $A_2$[$B_2$] & \\
$n_{\rm occ.}=\{2,2\}$ & & $A_1^{(a-c)}$[$B_1^{(a-c)}$] $A_1^{(d,e)}$ $E^{(a-e)}$ $A_2^{(a,b)}$[$B_2^{(a,b)}$] $B_2^{(c,d)}$ &
& $A_1^{(a,b)}$[$B_1^{(a,b)}$] $B_1^{(c)}$ $E^{(a-e)}$ $A_2$[$B_2$] $B_2$\\
$n_{\rm occ.}=\{3,1\}$ & $A_1^{(a,b)}$[$B_1^{(a,b)}$] $E^{(a-d)}$ $A_2^{(a,b)}$[$B_2^{(a,b)}$] &
& $A_1^{(a-c)}$[$B_1^{(a-c)}$] $E^{(a-e)}$ $A_2^{(a,b)}$[$B_2^{(a,b)}$] & \\
\hline
$\frac{3}{2}\oplus 0$ & & & &\\
$n_{\rm occ.}=\{1,3\}$ & & &$\bf A_1$[$\bf B_1$] $E$ & \\
$n_{\rm occ.}=\{2,2\}$ & & $A_1^{(a)}$ $A_1^{(b)}$[$B_1$] $E$ $B_2$& & $B_1$[$A_2$] $E^{(a,b)}$ \\
$n_{\rm occ.}=\{3,1\}$ & $A_1$[$B_1$] $E^{(a,b)}$ $A_2$[$B_2$]& & $A_1^{(a,b)}$[$B_1^{(a,b)}$] $E^{(a-d)}$ $A_2^{(a,b)}$[$B_2^{(a,b)}$] & \\
\hline
$2$ & & $B_1^{(a,b)}$[$A_2^{(a,b)}$] $E^{(a-d)}$ & & $A_1^{(a,b)}$[$B_1^{(a,b)}$] $A_1^{(c,d)}$ $E^{(a-c)}$ $A_2$[$B_2^{(a)}$] $B_2^{(b,c)}$\\
\hline
\hline
\end{tabular}
}
\caption{List of all SU(2)-symmetric basic rank-5 tensors for physical spin $\frac{1}{2}\leqslant S\leqslant 2$ and bond dimension $D\leqslant 5$.
The virtual (physical) spin degrees of freedom $V$ ($S$) is displayed vertically (horizontally).
Whenever the virtual spin is a direct sum $V=\oplus V_i$, we decompose the virtual space $V^{\otimes 4}$ into
all subspaces given by the occupations $n_{\rm occ.}$ of the spins $V_i$. Note that, when two tensor classes are related by
``color exchange'', we keep only one of them in the list (e.g. for $V=\frac{1}{2}\oplus 0\oplus 0$
$n_{\rm occ.}=\{1,2,1\}$ is omitted since it is the ``color conjugate" of $n_{\rm occ.}=\{1,1,2\}$). Tensors are labelled according to their $C_{4v}$ point symmetry, $A_1$ ($s$-wave), $B_1$ ($d_{x^2-y^2}$-wave),
$E$ (doubly-degenerate $p$-wave),
$A_2$ ($g$-wave), $B_2$ ($d_{xy}$-wave). Subscripts $(a)$, $(b)$, etc... are used to differentiate non-equivalent tensors of the same class. Gauge-equivalent tensors (i.e. giving rise to the same PEPS) are listed between brackets, next to their gauge-related partners. Tensors giving rise to simple known wave functions
are highlighted in boldface (see text for a description and Supplemental Material for expressions).
}
\label{Table:tensor_list}
\end{center}
\end{table*}
Using the method described above, we have generated all SU(2)-invariant tensors up to $D=6$.
A subset of the list of tensors with $D\leqslant 5$ and $\frac{1}{2}\leqslant S\leqslant 2$ is displayed in Table~\ref{Table:tensor_list},
while the complete list for $D\leqslant 6$ is given as Supplemental Material. The columns correspond to different physical spins (limited to $S\leqslant 2$).
Vertically, the tensors are classified according to the representation $V=\oplus V_i$ of the virtual spins.
Each line corresponds to a sub-class given by a disconnected set of (virtual) basis states
characterized by the occupation numbers $n_{\rm occ.}$ of the $\cal N$ spins $V_i$.
It should be mentioned that all the tensors produced in our systematic procedure do not lead
necessarily to different PEPS due to remaining gauge degrees of freedom.
Indeed, imposing SU(2) and point group symmetries does not completely fix the gauge and some
freedom remains. Typically, tensors which have identical non-zero tensor elements up to a sign
are gauge equivalent as can be checked case by case. Table~\ref{Table:tensor_list} shows explicitly
all gauge-equivalent tensors.
\begin{table}[htb]
\begin{center}
\resizebox{0.98\columnwidth}{!}{%
\begin{tabular}{@{} ccccc @{}}
\hline
\hline
V $\backslash$ S & 1/2 & 1 & 3/2 & 2 \\
\hline
& & & &\\
$\frac{1}{2}$& 0/0/0/0/0 & 0/1/0/0/1 & 0/0/0/0/0 & 1/0/0/0/0 \\
& & & & \\
\hline
& & & &\\
$\frac{1}{2}\oplus 0$ & 2/2/1/1/3 & 2/2/0/1/2 & 1/1/0/0/1 & 1/0/0/0/0 \\
& & & &\\
$1$ & 0/0/0/0/0 & 0/1/1/0/2 & 0/0/0/0/0 & 2/1/0/1/1 \\
& & & & \\
\hline
& & & &\\
$\frac{1}{2}\oplus 0\oplus 0$ & 8/8/4/4/12 & 6/5/1/3/6 & 2/2/0/0/2 & 1/0/0/0/0 \\
& & & &\\
$\frac{1}{2}\oplus \frac{1}{2}$ & 0/0/0/0/0 & 6/9/4/3/13 & 0/0/0/0/0 & 6/3/0/1/3 \\
& & & &\\
$1\oplus 0$ & 0/0/0/0/0 & 3/5/3/1/8 & 0/0/0/0/0 & 5/3/1/3/4 \\
& & & &\\
$\frac{3}{2}$& 0/0/0/0/0 & 0/2/1/0/3 & 0/0/0/0/0 & 3/1/1/2/2 \\
& & & &\\
\hline
& & & &\\
$\frac{1}{2}\oplus 0\oplus 0\oplus 0$ & 21/21/12/12/33 & 12/10/3/6/13 & 3/3/0/0/3 & 1/0/0/0/0 \\
& & & &\\
$\frac{1}{2}\oplus \frac{1}{2}\oplus 0$ & 10/10/8/8/18 & 12/13/5/6/18 & 6/6/2/2/8 & 6/3/0/1/3 \\
& & & &\\
$1\oplus 0\oplus 0$ & 0/0/0/0/0 & 11/14/9/6/23 & 0/0/0/0/0 & 10/7/3/6/10 \\
& & & &\\
$1\oplus \frac{1}{2}$ & 4/4/3/3/7 & 5/5/3/4/8 & 5/5/3/3/8 & 5/4/2/2/6 \\
& & & &\\
$\frac{3}{2}\oplus 0$ & 1/1/1/1/2 & 2/3/1/1/4 & 3/3/2/2/5 & 3/2/2/2/4 \\
& & & &\\
$2$ & 0/0/0/0/0 & 0/2/2/0/4 & 0/0/0/0/0 & 4/2/1/3/3 \\
& & & &\\
\hline
& & & &\\
$\frac{1}{2}\oplus 0\oplus 0\oplus 0\oplus 0$ & 44/44/28/28/72 & 20/17/6/10/23 & 4/4/0/0/4 & 1/0/0/0/0 \\
& & & &\\
$\frac{1}{2}\oplus \frac{1}{2}\oplus 0\oplus 0$ & 28/28/20/20/48 & 25/24/11/14/35 & 12/12/4/4/16 & 6/3/0/1/3 \\
& & & &\\
$\frac{1}{2}\oplus \frac{1}{2}\oplus \frac{1}{2}$ & 0/0/0/0/0 & 33/39/24/21/63 & 0/0/0/0/0 & 21/15/3/6/18 \\
& & & &\\
$1\oplus 0\oplus 0\oplus 0$ & 0/0/0/0/0 & 27/31/22/18/53 & 0/0/0/0/0 & 17/13/6/10/19 \\
& & & &\\
$1\oplus \frac{1}{2} \oplus 0$ & 11/11/8/8/19 & 13/13/8/9/21 & 11/11/7/7/18 & 10/8/4/5/12 \\
& & & &\\
$1\oplus 1$ & 0/0/0/0/0 & 9/13/13/9/26 & 0/0/0/0/0 & 19/15/7/11/22 \\
& & & &\\
$\frac{3}{2}\oplus 0\oplus 0$ & 2/2/2/2/4 & 6/6/2/3/8 & 10/10/6/6/16 & 4/4/5/4/9 \\
& & & &\\
$\frac{3}{2}\oplus \frac{1}{2}$ & 0/0/0/0/0 & 7/12/8/5/20 & 0/0/0/0/0 & 15/10/7/10/17 \\
& & & &\\
$2\oplus 0$ & 0/0/0/0/0 & 1/4/5/2/9 & 0/0/0/0/0 & 10/7/3/6/10 \\
& & & &\\
$\frac{5}{2}$ & 0/0/0/0/0 & 0/3/2/0/5 & 0/0/0/0/0 & 5/2/2/4/4 \\
& & & &\\
\hline
\end{tabular}
}
\caption{Sets of the numbers ${\cal D}_1/{\cal D}_2 / {\cal D}_3 / {\cal D}_4 / {\cal D}_5$ of basic tensors belonging to the five $A_1$, $B_1$, $A_2$, $B_2$ and (doubly degenerate) $E$ IRREP of $C_{4v}$, respectively, which can be combined
in each $SU(2)$ $(V,S)$ symmetry class to give rise to fully-symmetric spin-$S$ spin liquids, for $D\leqslant 6$ and $\frac{1}{2}\leqslant S\leqslant 2$. Note each class defined by a direct sum $V=\oplus V_i$ of $\cal N$ SU(2) IRREP
includes all basic tensors of all sub-classes defined by $n_{\rm occ.}=(n_1,n_2,\cdots n_{\cal N})$ with $0\leqslant n_i\leqslant 4$ and $\sum_i n_i=4$. For instance, the $V=\frac{1}{2}\oplus \frac{1}{2}\oplus 0$ ($D=5$) class includes the $V=\frac{1}{2}$ ($D=2$), $V=\frac{1}{2}\oplus 0$ ($D=3$) and
$V=\frac{1}{2}\oplus \frac{1}{2}$ ($D=4$) classes.
}
\label{Table:numbers}
\end{center}
\end{table}
\subsection{Remarkable PEPS in this scheme}
Some of the low-$D$ PEPS obtained using our systematic construction and
listed in Table~\ref{Table:tensor_list} have already been introduced
in recent literature and/or correspond to well-known states of matter. Let us briefly list a few of them below, together
with some simple generalizations (see tensor expressions in Appendix~\ref{app2}).
\subsubsection{The spin-$S$, $S$ even integer, AKLT states}
The simplest 2d Affleck-Kennedy-Lieb-Tasaki (AKLT)~\cite{Affleck1987} state is obtained by decomposing a physical spin-2 on each site into four virtual
spin-$\frac{1}{2}$ spins, pairing every NN virtual spins into singlet and, finally, applying the projector $P: \frac{1}{2}^{\otimes 4}\rightarrow 2$
onto the fully symmetric $S=2$ subspace on every site. This state is known to have short-range correlations, and is given by the $A_1$ tensor of the $V=\frac{1}{2}$ line / $S=2$ column of Table~\ref{Table:tensor_list}. This construction can be straightforwardly
extended to higher physical spins by projecting four virtual spin-$\frac{k}{2}$, $k\in \mathbb{N}^*$, onto the fully symmetric spin-$S$, $S=2k$, physical subspace. This is realized by the unique $V=\frac{k}{2}$, $S=2k$, $A_1$ tensor
of dimension $D=k+1$ (see Appendix~\ref{app2} for $k=1$ and Supplemental Material for $k=1,\dots,5$). The 2d spin-$S$ AKLT states can serve as useful examples of trivial (featureless) states or
states with symmetry protected topological order~\cite{Chen2011} for $k$ even integer ($S=4p$) or $k$ odd integer
($S=4p+2$), respectively.
\subsubsection{The spin-$S$, $S$ odd integer, featureless paramagnets}
Starting with four $V=\frac{1}{2}$ virtual spins attached to each site and paired up into NN singlets, as in the AKLT construction, but projecting them onto $S=1$ on-site physical spins gives rise to a spin-$1$ featureless paramagnet~\cite{Jian2016}, also with short-range correlations. This is given by the $V=\frac{1}{2}$, $S=1$, $B_1$ tensor of Table~\ref{Table:tensor_list}. This construction can be
straightforwardly generalized to higher physical spins by simply
projecting four virtual spin-$\frac{k}{2}$, $k\in{\mathbb N}^*$, onto spin-$S$, $S=2k-1$, physical subspace.
This is always realized by the unique $V=\frac{k}{2}$, $S=2k-1$, $B_1$ tensor
of dimension $D=k+1$ (see Appendix~\ref{app2} for $k=1$ and Supplemental Material for $k=1,\dots,5$). We believe such states with $S>1$ are also featureless paramagnets.
Note that, for all $k$, the $B_1$ tensor comes always in pair with a (complex) $E$ tensor which leads to a {\it real} wave function and
might also have interesting properties.
\subsubsection{The spin-$\frac{1}{2}$ Resonating Valence Bond (RVB) state}
The RVB state is a spin-1/2 spin liquid
defined by an equal-weight superposition of all nearest-neighbor (NN) singlet configurations~\cite{Anderson1973}.
It is exactly given by the $V=\frac{1}{2}\oplus 0$, $n_{\rm occ.}=\{1,3\}$, $A_1$ (named ${\cal A}_1^{(1)}$ from now on)
or $B_1$ tensors (named ${\cal B}_1^{(1)}$ from now on) of the $S=\frac{1}{2}$ column of
Table~\ref{Table:tensor_list}. The NN RVB state was shown to be a ${\mathbb Z}_2$ topological spin liquid on the Kagome lattice~\cite{Schuch2012,Poilblanc2012,Poilblanc2013a}. On the square lattice it exhibits an extended $U(1)$ gauge symmetry and is critical~\cite{Albuquerque2010,Poilblanc2012}
(see below for the discussion of gauge symmetry).
\subsubsection{The long-range spin-$\frac{1}{2}$ RVB state (LR RVB)}
The LR RVB state is obtained by assuming a distribution of longer-range singlet bonds beyond NN
(yet still connecting two different sublattices). It is obtained within the PEPS formalism
by linearly combining the $V=\frac{1}{2}\oplus 0$, $n_{\rm occ.}=\{3,1\}$, $A_1$ (named ${\cal A}_1^{(2)}$ from now on)
tensor with the previous ${\cal A}_1^{(1)}$ NN RVB tensor.
\begin{equation}
{\cal A}_{\rm LRRVB}=\lambda_1 {\cal A}_1^{(1)} + \lambda_2 {\cal A}_1^{(2)},
\end{equation}
with $\lambda_1,\lambda_2\in \mathbb{R}$.
Alternatively, one can use the gauge-equivalent $B_1$ tensors, named ${\cal B}_1^{(1)}$ and ${\cal B}_1^{(2)}$.
The singlet bond distribution is controlled by the relative (real)
weight between the two tensors. Such a spin liquid ansatz turned out to be an excellent variational state for the frustrated
$J_1$--$J_2$ antiferromagnetic Heisenberg model on the square lattice~\cite{Wang2013}. It is interesting to notice that
this ansatz is in fact the most general $D=3$ SU(2)-invariant PEPS.
\subsubsection{The NN fermionic spin-$\frac{1}{2}$ RVB (NN fRVB)}
The NN fermionic-RVB (fRVB) is defined as an equal-weight superposition of dimer coverings where each (centro-symmetric) dimer is
written in the fermionic representation. It can be re-written as a spin-$\frac{1}{2}$ NN RVB state where, e.g., vertical dimers are assigned a
complex factor $i$ providing a completely different sign structure~\cite{Kotliar1988} than the above NN RVB state. This (real) wave function is given by the unique
$S=\frac{1}{2}$, $V=\frac{1}{2}\oplus 0$, $n_{\rm occ.}=\{1,3\}$, complex E tensor~\cite{Poilblanc2014}.
\subsubsection{The generalized spin-$S$ NN RVB}
The spin-$1$ RVB state can be obtained by attaching a single $S=1$
virtual spin on every site (accompanied by 3 spin-$0$). All NN virtual spins $1/2$ are again paired up into singlets which resonate.
The $S=1$, $V=1\oplus 0$, $n_{\rm occ.}=\{1,3\}$ $A_1$ (or $B_1$) tensor corresponds exactly to such a spin liquid.
This scheme can be generalized to any \hbox{spin-$S$} NN RVB and is always described by a single $V=S\oplus 0$, $n_{\rm occ.}=\{1,3\}$ $A_1$ (or $B_1$) tensor. The cases corresponding to $S=1$ and $S=\frac{3}{2}$ are highlighted in Table~\ref{Table:tensor_list} and the corresponding tensors are given in Appendix~\ref{app2}
(see Supplemental Material for all physical spin $S$ up to $S=2$).
\subsubsection{The spin-$S$ Resonating AKLT Loop (RAL) state}
As shown by Li et al.~\cite{Li2014}, the spin-$1$ RAL state involves two virtual spin-$\frac{1}{2}$ and two virtual spin-$0$ attached to every site, i.e.,
the virtual subspace is $V=\frac{1}{2}\oplus 0$, $n_{\rm occ.}=\{2,2\}$. Physically, NN virtual spin-$1/2$ are paired up into singlets
(as in the RVB state) and all virtual spins are then projected locally onto physical spins $1$ to produce AKLT loops. The
two $S=1$, $V=\frac{1}{2}\oplus 0$,
$n_{\rm occ.}=\{2,2\}$, $A_1^{(a)}$ and $A_1^{(b)}$ tensors encode the two possible site configurations of the loops with
180-degree or 90-degree angles. The RAL state on the square lattice is critical since the dimer-dimer correlations decay as a power law~\cite{Li2014}. Since the 1d AKLT chain can be extended to higher physical spin (see above for the 2d case), it is easy to generalize the RAL to a gas of resonating spin-$S$ AKLT chains, for all $S$ integer. It is given by the only two $V=\frac{S}{2}\oplus 0$ ($D=S+2$), $n_{\rm occ.}=\{2,2\}$, $A_1^{(a)}$ and $A_1^{(b)}$ tensors. The cases corresponding to $S=1$ and $S=2$ are highlighted in Table~\ref{Table:tensor_list} and the corresponding $S=1$ tensors are given in Appendix~\ref{app2}
(see Supplemental Material for all integer physical spin $S$ up to $S=4$).
\subsubsection{The spin-$1/2$ chiral spin liquid (CSL)}
The CSL is obtained by linearly combining the two previous ${\cal A}_1^{(1)}$ and ${\cal A}_1^{(2)}$
(with real coefficients) and the $V=\frac{1}{2}\oplus 0$, $n_{\rm occ.}=\{3,1\}$, $A_2$ tensor (named ${\cal A}_2$ from now on)
with a pure-imaginary coefficient~\cite{Poilblanc2015,Poilblanc2016}. The PEPS obtained from the resulting tensor (see Fig.~\ref{FIG:single_double}(a)),
\begin{equation}
{\cal A}_{\rm chiral}=\lambda_1 {\cal A}_1^{(1)} + \lambda_2 {\cal A}_1^{(2)} + i \lambda_c {\cal A}_2,
\label{Eq:chiral_sl}
\end{equation}
with $\lambda_1,\lambda_2, \lambda_c\in \mathbb{R}$, breaks time-reversal symmetry (provided
$\lambda_2 \lambda_c\ne 0$) and transforms into its complex conjugate state under
any of the reflection symmetries of Fig.~\ref{FIG:lattice_tensor}(a). It exhibits clear SU(2)$_1$ edge modes although there are some evidence for critical (singlet) bulk correlations.
\begin{figure}[htbp]
\begin{center}
\includegraphics[width=0.95\columnwidth,angle=0]{Figures/single_double}
\caption{(a) The three tensors of the spin-1/2 CSL involving the three virtual states of the $\frac{1}{2}\oplus 0$ (spin) representation,
$|\uparrow\big>$ and $|\downarrow\big>$ on the full lines, and $|0\big>$ on the dotted lines. (b) The natural basis of virtual states of the double layer tensor $|\uparrow\uparrow\big>$, $|\uparrow\downarrow\big>$, $|\downarrow\uparrow\big>$ and $|\downarrow\downarrow\big>$
($\in \frac{1}{2}\otimes\frac{1}{2}$, spin-$\frac{1}{2}$ on top {\it and} bottom layers),
$|\uparrow 0\big>$, $|\downarrow 0\big>$, $|0\uparrow\big>$ and $|0\downarrow\big>$
($\in \frac{1}{2}\oplus\frac{1}{2}$, spin-$\frac{1}{2}$ on top {\it or} bottom layer),
and $|00\big>$ ($\in 0$) is transformed (by a simple unitary transformation) into symmetric/antisymmetric states w.r.t to the exchange of layers. }
\label{FIG:single_double}
\end{center}
\end{figure}
\subsection{Constructing spin liquids and beyond}
\subsubsection{Generic spin liquids}
From a few of the previous examples, we see that tensors can be linearly combined to give new interesting states. In fact,
adding $\cal D$ (real) tensors $T^{(a)}$ as $\sum_{a=1}^{\cal D} \lambda_a T^{(a)}$ (involving $\cal D$ real coefficients $\lambda_a$)
belonging to the same ``class", i.e., characterized by the
same physical ($S$) and virtual ($V$) degrees of freedom and by the same IRREP of the point group $C_{4v}$, will lead to a $({\cal D}-1)$-dimensional
family of completely symmetric spin liquids which (potentially) do not break any symmetry, neither SU(2) nor point group symmetries.
The numbers $\cal D$ of tensors which can be combined
in each SU(2) $(V,S)$ symmetry class to give rise to fully-symmetric spin-$S$ spin liquids are given in Table~\ref{Table:numbers} for $D\leqslant 6$ and $S\leqslant 2$. Note that the counting of tensors of a given bond dimension includes all those of smaller bond dimensions which can be combined in each class.
Note also that, for a given linear combination of tensors, there is {\it a priori} no guarantee that all correlations, remain short range in such a symmetric state and the absence of spontaneous
symmetry breaking in the thermodynamic limit should, in principle, always be verified. We observe that the typical dimensions of the
PEPS families do not grow too fast, from ${\cal D}\sim 3-10$ for $D=4$ up to ${\cal D}\sim 10-40$ for $D=6$.
Also, it is interesting to notice that a subset of the symmetry classes does not provide a variational representation of half-integer spin-$S$.
\subsubsection{Lattice nematics}
If the $\cal D$ $T^{(a)}$ tensors belong to (at least) two different IRREP of the point group (while still involving the same virtual
and physical degrees of freedom), the resulting PEPS will explicitly
break the point group symmetry. For example, combining $A_1$ and $B_1$ tensors, or $A_2$ and $B_2$ tensors, will
produce a nematic state where vertical and horizontal directions will become non-equivalent (e.g. observables will acquire different
mean values).
As a concrete example, let us consider the $V=1$, $S=2$, gauge-equivalent $A_1^{(a)}$ and $B_1$ tensors of Table~\ref{Table:tensor_list}.
It is likely that these tensors produce a paramagnet similar to the $S=2$ AKLT state, although with gapped edge states.
The linear combination $A_1^{(a)}+B_1$ ($A_1^{(a)}-B_1$) of the two tensors gives a product of decoupled vertical (horizontal) spin-$2$ AKLT {\it chains} times a collection of independent horizontal (vertical) NN dimers (constructed from pairs of all the remaining virtual spin-$1$ not involved in the chains). Any partial superposition
like $\cos{\theta}\, A_1^{(a)}+\sin{\theta}\, B_1$, $\theta\in ]0,\pi/2[$, will give a lattice nematic state interpolating between the isotropic paramagnet and the array of AKLT chains.
\subsubsection{Breaking SU(2)-symmetry down to U(1): spin nematics and N\'eel antiferromagnets}
By breaking the global SU(2)-spin rotation invariance down to U(1)
one can construct, within our framework, two enlarged families of anisotropic quantum magnets. If TR-symmetry and space group symmetry are independently conserved, one obtains
anisotropic spin nematics~\cite{Andreev1984} for which
the spin Z-axis becomes non-equivalent from the two equivalent X and Y spin axis. If the combination of TR
with a unit translation is conserved, one gets N\'eel-like quantum magnets with a finite staggered magnetization.
In order to achieve this goal, one should remember that our
PEPS are defined in a physical basis where all spins on the B-sublattice have been rotated by $\pi$.
Each $S_z$-component (over the $d=2S+1$ components) of a given tensor hence contributes to a finite amplitude of the (physical) staggered magnetization ${\tilde S}_z^{\rm stag}=S_z$ in the original un-rotated basis.
The procedure to construct anisotropic magnets is therefore simple~; (i) One sorts out all tensors according to the virtual space $V$ (as before) and to
some maximum value $S_{\rm max}$
of $|S_z|$, defining the physical Hilbert space $S_z\in [-S_{\rm max},S_{\rm max}]$, by merging classes of different SU(2)-spin $S$
with the same orbital symmetry (i.e. IRREP of $C_{4v}$); (ii) One groups all tensors into $S_z=\pm |S_z|$ pairs of tensors
$T_{\pm}^{(a)}$ (where $a$ labels the pairs)
related by the $S_z\leftrightarrow -S_z$ symmetry (for $S_z\ne 0$);
(iii) On then constructs the linear superposition of the (normalized) on-site tensors for each local $S_z=s|S_z|$ physical index as
$\sum_{a,s} \lambda_{a,s} T_{s}^{(a)}$.
Generically, such a family of PEPS exhibits a finite staggered magnetization (in the original un-rotated basis)
as in a N\'eel state unless we further impose $\lambda_{a,s}=\lambda_{a,-s}$
to construct nematic states.
Note that Long Range Order (LRO) in the XY (spin) plane may still spontaneously appear in some domains
of the parameter space, despite
the local $U(1)$ symmetry.
\subsubsection{Complex $E$ tensors, TR-symmetry breaking and chiral spin liquids}
It is important to notice that, if the $\cal D$ tensors belong to the $A_1$, $B_1$, $A_2$ or $B_2$ IRREP, the resulting tensor is real
and, hence, invariant under time-reversal (TR) symmetry. For complex coefficients, or when $E$ tensors are combined, the resulting state generically
breaks TR, except at fine-tuned parameter subsets. Although chiral spin liquids with protected chiral edge modes have to be searched in
these classes of PEPS, we believe they probably span a tiny fraction of the TR-symmetry breaking PEPS manifold. Note also that not all $E$ tensors
give complex PEPS. For example, combining the three $S=1/2$, $V=\frac{1}{2}\oplus 0$, complex $E$ tensors (one tensor with $n_{\rm occ.}=\{1,3\}$ and two tensors with $n_{\rm occ.}=\{3,1\}$) surprisingly gives a real (SU(2)-symmetric) wave function. We believe this PEPS family can be viewed physically as an extension of the NN fRVB (see above)
in terms of fRVB states with longer-range (fermionic) dimers.
Similarly, the two PEPS given by the unique $S=1$, $V=\frac{1}{2}\oplus 0$, $E$ tensor and by the unique $S=3/2$, $V=\frac{1}{2}\oplus 0$, $E$ tensor are also real. Hence, we believe that the spin-$\frac{1}{2}$, spin-$1$ and spin-$\frac{3}{2}$ PEPS originated
from the $V=\frac{1}{2}\oplus 0$, $E$ tensors can probably all be mapped to real fermionic PEPS (fPEPS).
\subsubsection{Gauge symmetry and topological order}
Whether or not a PEPS of a given family exhibits topological order is a rather subtle issue. The existence of a gauge symmetry, i.e., an Invariant Gauge Group (IGG), plays a crucial role and is often a necessary condition. The AKLT states and the featureless paramagnets above have simple ${\cal N}=1$ virtual spaces with a single spin species in the four directions, $n_{\rm occ.}=4$. This is connected to ${\rm IGG}=\mathbb{I}$, characteristic of topologically {\it trivial} states.
Gauge symmetry (like ${\mathbb Z}_2$ or U(1)) can also be present, depending of the different $n_{\rm occ.}$ sectors involved in the construction
of the variational manifold. For instance, the NN RVB state is defined by a unit tensor with $V=\frac{1}{2}\oplus 0$ and $n_{\rm occ.}=\{1,3\}$ which,
in practice, implies that one and only one singlet dimer is attached to every site~\cite{Schuch2012}. This local constraint implies that the {\it number} of dimers cut by a line winding around an infinite cylinder is conserved, hence providing an infinite number of topological sectors associated to
a U(1) gauge symmetry~\cite{Poilblanc2012}. On the other hand, the LR RVB state mixes $n_{\rm occ.}=\{1,3\}$
and $n_{\rm occ.}=\{3,1\}$ tensors so that, then, only the {\it parity} of the number of dimers cut by a circumference is conserved,
hence reducing the number of sectors to two and the gauge symmetry to $\mathbb{Z}_2$.
More generally, the gauge symmetry can usually be inferred from the set of numbers of occupation
$[n_{\rm occ.}^1,n_{\rm occ.}^2,\cdots, n_{\rm occ.}^{\cal D}]$
of the $\cal D$ superposed tensors. In general, one can always use a {\it minimal} global virtual basis
$V=\oplus V_i$, direct sum of $\cal N$ IRREP $V_i$ of SU(2), for which all the occupation
numbers $n_{\rm occ.}^j$ are given by sets of $\cal N$ numbers, i.e.
$n_{\rm occ.}^j=\{n_1^j,\cdots, n_{\cal N}^j\}$ and $\sum_i n_i^j=4, \forall j$. Subsequently, topological order
can be characterized from the symmetry~\cite{Schuch2010a}.
However, (i) an extended gauge symmetry can emerge in the thermodynamic limit, as e.g. the U(1) symmetry in the case of the
$\mathbb{Z}_2$ LR RVB
state; (ii) Reversely, a mechanism of ``confinement" can suppress the topological
order associated to the underlying gauge symmetry~\cite{Schuch2013a}. In any case, topological order can always be inferred from a thorough
investigation of the transfer operator~\cite{Schuch2013a}, and not only from the local symmetry of the tensor.
\subsection{Connection with previous work}
Based on the original framework introduced by S.~Jiang and Y.~Ran~\cite{Jiang2015} and the notion
of projective symmetry group, Lee and Han provided a classification of (trivial) spin-$1$ PEPS on the square
lattice~\cite{Lee2016}, in terms of lattice quantum numbers.
In our classification scheme, we have recovered their results which correspond to a subset of our $S=1$
tensors~: the $V=\frac{1}{2}$ ($D=2$), $B_1$ tensor, the $V=1$ ($D=3$), $B_1$ and $A_2$ tensors, the
$V=1\oplus 0$ ($D=4$), $B_1$, $n_{\rm occ.}=\{1,3\}$ (RVB state) and $n_{\rm occ.}=\{2,2\}$ (RAL state) tensors.
While each tensor has (emergent) $U(1)$ IGG symmetry, a linear combination of them eliminates this gauge symmetry and produces a trivial state.
Note that these authors did not report about neither the $S=1$, $E$ tensors nor the $S=1$, $V=1\oplus 0$ ($D=4$), $n_{\rm occ.}=\{3,1\}$ tensors enumerated in Table~\ref{Table:tensor_list}.
Combining, e.g., the $S=1$, $V=1\oplus 0$ ($D=4$), $n_{\rm occ.}=\{1,3\}$ and $n_{\rm occ.}=\{3,1\}$ tensors
preserves a $\mathbb{Z}_2$ IGG and may lead to a topological spin liquid.
\section{Application: search for higher-spin ($S>\frac{1}{2}$) CSL}
\label{sec3}
One of the applications of our classification is the systematic construction of CSL
beyond the physical $S=\frac{1}{2}$ and virtual $V=\frac{1}{2}\oplus 0$ IRREPs, already realized. Our goal is to construct a family of TR symmetry-breaking PEPS with linear dispersing chiral edge modes described by a CFT beyond SU(2)$_1$. To characterize the edge modes, we analyze the entanglement spectrum (ES) of the corresponding PEPS when wrapped around an infinite cylinder and splitted in two parts (left and right), as explained in Appendix \ref{app1}. We describe below several natural routes to find such CSL.
\subsection{The complex $E$ tensors }
One of the first natural candidates would be the above $E$ tensors which are intrinsically complex.
Therefore, their corresponding PEPS can potentially break TR symmetry and, hence, be relevant
topological CSL.
However, we have found that some of the simplest, small $D$, $E$ PEPS wave functions are purely real
as seen above (hence showing a perfectly momentum-symmetric entanglement spectrum),
probably due to the fact that they can be re-casted in the form of real fPEPS wave functions,
as it is the case for the fRVB state. In this respect, we found that the ES
of the NN fRVB state is completely degenerate, hence saturating the upper bound of the entanglement
measures (e.g., the entanglement entropy per unit length $-{\rm Tr} (\rho\ln{\rho})/N_v$).
In addition, all {\it complex} $E$ PEPS we have tested turned out to show no well-defined chiral edge modes, although breaking TR symmetry.
For example, this is the case of the PEPS associated to the $S=1$, $V=1$, $E^{(a)}$ tensor
(see Table~\ref{Table:tensor_list} and Supplemental Material)
whose ES is shown in Fig.~\ref{Fig:ES_E}.
In fact, it is not clear however whether any $E$ tensor could give rise to a topological CSL whenever $S>1/2$.
\begin{figure}
\begin{center}
\includegraphics[width=\columnwidth,angle=0]{Figures/ESChi20d3D3_Ea}
\caption{[Color online] Entanglement spectrum computed on a $N_v=10$ (infinite) cylinder (with $\chi=20$)
of the PEPS constructed from the $S=1$, $V=1$, $E^{(a)}$ tensor. Data is plotted vs momentum (modulus $\pi$) along cylinder circumference. Note the spectrum is not symmetric w.r.t. $k=\pi/2$, reflecting TR symmetry-breaking of the PEPS. States can be grouped into SU(2)-multiplets, and $(s)$ labels $2s+1$ degenerate states.
Note, the spectrum does not seem to fit a simple (chiral) CFT like SU(2)$_2$. }
\label{Fig:ES_E}
\end{center}
\end{figure}
\subsection{The $A_1$ + i $A_2$ PEPS}
Inspired by previous studies, another promising route to construct CSL is to consider tensors of mixed $A_1 + i A_2$ (point group) symmetry,
where both the real and imaginary components can be a sum of tensors belonging to any given $A_1$ and $A_2$ classes involving
identical virtual and physical degrees of freedom. This is a direct generalization of Eq.~(\ref{Eq:chiral_sl}).
This procedure guaranties that the PEPS -- if complex -- transforms into its (orthogonal) complex conjugate state under
any of the reflection symmetries of Fig.~\ref{FIG:lattice_tensor}(a), a necessary condition for a CSL.
However, this construction does not necessarily imply that the PEPS breaks TR symmetry and the wave function
can, in some cases, be purely real (up to a global phase). In addition,
even if the PEPS breaks TR symmetry, there is no guarantee that it exhibits
linear dispersing edge modes described by a CFT characteristic of a topological CSL.
Before describing a successful case (the double-layer CSL and PEPS connected to it), we will show
below an example of some failure.
\subsection{Naive spin-$S$ generalizations of the spin-$\frac{1}{2}$ CSL}
It is known that some families of critical (non-chiral) spin-$S$ chains~\cite{Takhtajan1982,Babujian1982} bears a low-energy description in terms of the SU(2)$_k$ Wess--Zumino--Witten models with levels $k=2S$. It is therefore tempting to speculate that
(chiral) SU(2)$_k$ edge modes can originate from $V_i=\frac{k}{2}$ virtual spins
effectively interacting on the edge. Indeed, the $S=\frac{1}{2}$ CSL discussed above, which bears SU(2)$_1$ edge modes,
involves $V=\frac{1}{2}\oplus 0$ virtual states. In fact, inspecting Table~\ref{Table:tensor_list}, one sees that the
construction for $S=\frac{1}{2}$ can be easily extended for any spin-$S$ assuming $V=S\oplus 0$~:
one can always combine the unique $n_{\rm occ}=\{1,3\}$ spin-$S$ RVB tensor (in bold case in Table~\ref{Table:tensor_list}) with both the $n_{\rm occ}=\{3,1\}$ $A_1$ tensors (with a real coefficient)
and the $n_{\rm occ}=\{3,1\}$ $A_2$ tensor(s)
(with a pure-imaginary coefficients). We have computed the ES at a few points of the 3-dimensional (4-dimensional)
family of PEPS corresponding to the case $S=1$ ($S=\frac{3}{2}$) which turned out to be always gapped, ruling out chiral topological order.
\begin{figure}
\begin{center}
\includegraphics[width=\columnwidth,angle=0]{Figures/ESChi14_double}
\caption{Same as Fig.~\protect\ref{Fig:ES_E} for the PEPS obtained from the double-layer tensor of Eq.~\protect\ref{Eq:double_tensor},
$N_v=6$ and $\chi=14$. Whenever possible, the multiplet content of the levels is shown, following notations of Table~\ref{Table:su2_2}. The dashed lines are guides to the eye emphasizing the linear dispersion of the modes. }
\label{Fig:ES_double}
\end{center}
\end{figure}
\begin{table}[htb]
\begin{center}
\resizebox{0.95\columnwidth}{!}{%
\begin{tabular}{@{} cccc @{}}
\hline \hline
$n \backslash j $ & 0 & $\frac{1}{2}$ & 1 \\
\hline
& & &\\
0 & (0) {\color{red} [1]}& ($\frac{1}{2}$) {\color{red} [2]} &(1) {\color{red} [3]}
\\
1 & (1) {\color{red} [3]} & ($\frac{1}{2}$)+($\frac{3}{2}$) {\color{red} [6] } & (0)+(1) {\color{red} [4]}
\\
2 & (0)+(1)+(2) {\color{red} [9]} & 2($\frac{1}{2}$)+2($\frac{3}{2}$) {\color{red} [12]} & (0)+2(1)+(2) {\color{red} [12]}
\\
3 & (0)+3(1)+(2) {\color{red} [15]} & 4($\frac{1}{2}$)+3($\frac{3}{2}$)+($\frac{5}{2}$) {\color{red} [26]} & 2(0)+3(1)+2(2) {\color{red} [21]}
\\
& & &\\
\hline
\hline
\end{tabular}
}
\caption{[Color online] Towers of states of the SU(2)$_2$ WZW model, in each of the three sectors characterized by
the primary fields $j=0,\frac{1}{2},1$ (listed in each column) and conformal weights $\frac{1}{4}j(j+1)$. Each line corresponds to a Virasoro
level indexed by $n$. For each sector and each level, the (quasi-) degenerate states can be grouped in terms of exact SU(2) multiplets
like $n_0 (0) + n_1 (1) +\cdots$ (meaning $n_0$ singlets, $n_1$ triplets, etc...). The (red) numbers in brackets
correspond to the total number of states in each group of levels.
}
\label{Table:su2_2}
\end{center}
\end{table}
\subsection{Double-layer CSL and SU(2)$_2$ edge states}
\begin{figure}
\begin{center}
\includegraphics[width=0.65\columnwidth,angle=0]{Figures/FigDoub}
\caption{[Color online] Diagram corresponding to Eq.~\ref{Eq:double_tensor}~: the two layers of tensors ${\cal A}_{\rm chiral}$ are symmetrized by the isommetry $S$, which projects the physical dimensions in the 3-dimensional spin-$1$ subspace of $\frac{1}{2} \otimes \frac{1}{2}$, giving rise to the tensor ${\cal A}_{\rm double}$}
\label{FIG:Doub}
\end{center}
\end{figure}
To go beyond the above naive construction, we shall follow here a strategy borrowed from the field of the
Fractional Quantum Hall States (FQHS)~\cite{Laughlin1983,Moore1991}.
Recently, using interpretation of FQHS as conformal blocks in certain rational conformal field theories~\cite{Moore1991}, MPS representations of FQHS were exploited~\cite{Estienne2013a,Estienne2013b}, providing unprecedented
numerical accuracy~\cite{Estienne2015}.
While the (Abelian) Laughlin state can be written as a simple MPS, the non-Abelian states can be constructed as multilayer fractional quantum Hall wave functions upon symmetrization over the layer index~\cite{Greiter2009,Repellin2015}.
Very recently, symmetrization of topologically ordered PEPS was shown to be a powerful method for constructing new
topological models~\cite{Gonzales2016}. Here, we will apply this procedure
by considering {\it two layers} of the CSL defined by Eq.~(\ref{Eq:chiral_sl}).
The double-layer tensor $[{\cal A}_{\rm chiral}]^{\otimes 2}$
is symmetrized w.r.t the $\frac{1}{2}\otimes\frac{1}{2}$ physical variables,
\begin{equation}
{\cal A}_{\rm double}={\cal S}[{\cal A}_{\rm chiral}\otimes {\cal A}_{\rm chiral}] \, ,
\label{Eq:double_tensor}
\end{equation}
hence projecting onto the $S=1$ physical state (see Fig.~\ref{FIG:Doub}).
More precisely, its $S_z=+1,0,-1$ (physical) components are given by,
\begin{eqnarray}
{\cal A}_{\rm double}\vert_1&={\cal A}_{\rm chiral}\vert_{\frac{1}{2}}\otimes {\cal A}_{\rm chiral}\vert_{\frac{1}{2}} , \nonumber \\
{\cal A}_{\rm double}\vert_0&=
\frac{1}{\sqrt{2}} \{{\cal A}_{\rm chiral}\vert_{\frac{1}{2}}\otimes {\cal A}_{\rm chiral}\vert_{-\frac{1}{2}} \nonumber \\
&+ {\cal A}_{\rm chiral}\vert_{-\frac{1}{2}}\otimes {\cal A}_{\rm chiral}\vert_{+\frac{1}{2}}\}, \\
{\cal A}_{\rm double}\vert_{-1}&={\cal A}_{\rm chiral}\vert_{-\frac{1}{2}}\otimes {\cal A}_{\rm chiral}\vert_{-\frac{1}{2}} , \nonumber
\end{eqnarray}
where the virtual variables of the double-layer tensor on the left hand side are given by the tensor product of the virtual variables of the
two single layer CSL tensors on the right hand side. For convenience, we then realize a (unitary) change of the $D=9$ virtual basis,
from the $[\frac{1}{2}\oplus 0]_{\rm top}\otimes[\frac{1}{2}\oplus 0]_{\rm bottom}$ natural basis to the ``symmetric" basis
$1\oplus\frac{1}{2}\oplus\frac{1}{2}\oplus 0_s\oplus 0$ described in Fig.~\ref{FIG:single_double}(b),
where the two spin-$\frac{1}{2}$-representations correspond now to spin-$\frac{1}{2}$ states symmetric and antisymmetric w.r.t. layer exchange, respectively, and the spin-$0_s$ and spin-$0$ representations contain the $\frac{1}{\sqrt{2}}(\uparrow\downarrow-\downarrow\uparrow)$ and $00$ singlets, respectively.
It can be seen easily (see later for details) that ${\cal A}_{\rm double}$ inherits from ${\cal A}_{\rm chiral}$
SU(2)-spin rotation symmetry and lattice $A_1+iA_2$ (orbital) symmetry. It is therefore expected to break TR symmetry while preserving all
lattice symmetries, a key property of chiral spin liquids.
We have computed the ES of the double-layer tensor for $\lambda_1=\lambda_2=\lambda_c(=1/\sqrt{3})$ on an infinite $N_v=6$ cylinder and results are shown in Fig.~\ref{Fig:ES_double}.
Linearly dispersing branches are clearly seen. Lot of resemblance with the chiral SU(2)$_2$ CFT spectrum is seen. The later shown in Table~\ref{Table:su2_2} contains
three sectors labeled by the primary fields $j=0,\frac{1}{2},1$. The lowest branch of the ES agrees perfectly with the
content of the $j=0$ sector. We also observe two almost degenerate branches with some energy offset, but with the same slope,
each compatible with the theoretical expectation for the
$j=1/2$ sector. It should be noted that, although an energy offset of $3/16$ of the (average) level spacing
is expected for the $j=1/2$ branch, a larger offset (by a factor $\sim 4$) is observed which could be plausibly attributed to finite
perimeter ($N_v$) and finite-$\chi$ effects in the ES calculation (see Appendix \ref{app1}).
\begin{figure}[htbp]
\begin{center}
\includegraphics[width=0.9\columnwidth,angle=0]{Figures/expansion}
\caption{
All combinations (up to global $\pm\pi/2$ or $\pi$-rotations) obtained by expending the ${\cal A}_{\rm chiral}$
tensors (according to Eq.~(\ref{Eq:chiral_sl}))
simultaneously in the top and bottom layers of the (symmetrized) double-layer CSL (left-most column),
leading to
orthonormal tensors (right-most column) belonging to different representations of the virtual states
(middle column).
}
\label{Fig:double_layer}
\end{center}
\end{figure}
\subsection{Decomposition in terms of elementary tensors}
The double-layer tensor involves $D=9$ virtual states, making hard the computation of the ES on larger cylinders and with
larger MPS dimension $\chi$ to permit a more definite assignment of the edge theory. In addition, it is not clear whether the
observation of exact SU(2)$_2$ edge modes requires some degree of ``fine tuning".
For these two reasons, it is a good idea to try to construct simpler (i.e., with lower bond dimension $D$) PEPS
which, potentially, could exhibit chiral edge modes.
Our strategy is here to ``break up'' the double-layer tensor ${\cal A}_{\rm double}$ into independent parts
defined by smaller virtual spaces $V$
but still exhibiting SU(2)-spin rotation and $A_1 + i A_2$-lattice symmetries. To do so, we expend the ${\cal A}_{\rm chiral}$
tensors according to Eq.~(\ref{Eq:chiral_sl}) simultaneously in the top and bottom layers.
Fig.~\ref{Fig:double_layer} shows all possible combinations depending on the relative orientation of the various contributions
in the two layers. In fact, it can be shown that each part leads to the sum of a few orthonormal SU(2)-spin symmetric real
$S$ and $G$ tensors, whose virtual states belong to a smaller representation, subset of the overall Hilbert space
$V=[\frac{1}{2}\oplus 0]^{\otimes 2}$.
The $S$-tensors and $G$-tensors belong to the $A_1$ (s-wave) and $A_2$ (g-wave) IRREP and appear with real and pure-imaginary coefficients,
respectively, so that each part of the decomposition
bears an overall $A_1+i A_2$ symmetry. More precisely, we can write
\begin{equation}
{\cal A}_{\rm double} = \sum_{\alpha} s_\alpha S_\alpha + \sum_{\beta} g_\beta G_\beta
\label{Eq:decomposition}
\end{equation}
with
\begin{align}
s_1 & = \frac{\lambda _1 \lambda _2}{\sqrt{2}} &
s_2 & = \frac{1}{2} \sqrt{3} \lambda _1^2 \nonumber\\
s_3 & = \frac{\lambda_1^2}{2} &
s_4^{(a)} &= \frac{7 \lambda _2^2-3 \lambda _c^2}{12 \sqrt{2}} \nonumber\\
s_4^{(b)} &= \frac{1}{12} \sqrt{\frac{5}{2}} \left(\lambda _2^2+3 \lambda _c^2\right) &
s_5^{(a)} &= \frac{\lambda _2^2}{3 \sqrt{2}} \nonumber\\
s_5^{(b)} &= \frac{1}{12} \left(-\lambda _2^2-3 \lambda _c^2\right) &
s_6^{(a)} &= \frac{5 \lambda _2^2+3 \lambda_c^2}{12 \sqrt{2}} \nonumber\\
s_6^{(b)} &= \frac{1}{12} \left(\lambda _2^2-3 \lambda _c^2\right)&
s_7^{(a)} &= -\frac{\lambda _2^2+9 \lambda_c^2}{12 \sqrt{3}} \nonumber\\
s_7^{(b)} &= \frac{1}{3} \sqrt{\frac{5}{3}} \lambda _2^2 &
s_7^{(c)} &= \frac{17 \lambda _2^2+15 \lambda _c^2}{36 \sqrt{2}}\nonumber\\
s_7^{(d)} &= -\frac{25 \lambda _2^2+3 \lambda_c^2}{18 \sqrt{13}} &
s_7^{(e)} &= -\frac{1}{3} \sqrt{\frac{5}{26}} \left(\lambda _2^2-3 \lambda _c^2\right) \nonumber\\
s_8^{(a)} &= \frac{9 \lambda _2^2-5 \lambda _c^2}{12 \sqrt{3}} &
s_8^{(b)} &= \frac{1}{3} \sqrt{\frac{7}{6}} \lambda _c^2 S_8^{(b)} \nonumber\\
s_9 & = \frac{\lambda _1 \lambda _2}{2} &
s_{10} & = \frac{1}{2} \sqrt{5} \lambda _1 \lambda _2 \nonumber\\
g_1 & = - \frac{i \lambda _1 \lambda _c}{\sqrt{2}} &
g_4 & = -\frac{i \lambda_2 \lambda _c}{\sqrt{3}} \nonumber\\
g_5 & = -\frac{i \lambda _2 \lambda_c}{\sqrt{6}} &
g_7 & = - i \sqrt{\frac{5}{6}} \lambda _2 \lambda _c\nonumber\\
g_8 & = i \sqrt{\frac{2}{3}} \lambda_2 \lambda _c &
g_9 & = -\frac{1}{2} i \lambda _1 \lambda _c \nonumber\\
g_{10} & = \frac{1}{2} i \sqrt{5} \lambda _1 \lambda _c &&\nonumber
\label{Eq:decomposition_coeff}
\end{align}
where the subscripts of the $S$ ($s$) and $G$ ($g$) tensors (coefficients) label the different virtual spin representations according
to Fig.~\ref{Fig:double_layer}. The exact expressions of all $S$ and $G$ spin-$1$ tensors are providing in Appendix B, written in the same
$D=9$ overall basis so that any linear combination of tensors can easily be performed.
\begin{figure*}
\begin{center}
\includegraphics[width=1.8\columnwidth,angle=0]{Figures/Fig5}
\caption{[Color online] ES for the PEPS obtained from various subsets of Eq.~(\protect\ref{Eq:decomposition}), and $N_v=8$, $\chi=20$~:
(a) $S_1$ and $G_1$ tensors ($\frac{1}{2}\oplus\frac{1}{2}$, $D=4$);
(b) $S_3$, $S_4$ and $G_4$ tensors ($1\oplus 0$, $D=4$);
(c) $S_1$, $S_6$ and $G_1$ tensors ($\frac{1}{2}\oplus\frac{1}{2}\oplus 0_s$, $D=5$);
(d) $S_3$, $S_4$, $S_5$, $G_4$ and $G_5$ tensors ($1\oplus 0\oplus 0_s$, $D=5$).}
\label{Fig:ES_partial}
\end{center}
\end{figure*}
\begin{figure*}[htbp]
\begin{center}
\includegraphics[width=1.8\columnwidth,angle=0]{Figures/Fig8}
\caption{Same as Fig.~\protect\ref{Fig:ES_partial} for other subsets of
Eq.~(\protect\ref{Eq:decomposition}), and $\chi=10$~:
(a) $N_v=8$, for $S_1$, $G_1$, $S_2$, $S_6$, $S_9$ and $G_9$
tensors ($\frac{1}{2}\oplus\frac{1}{2}\oplus 0\oplus 0_s$, $D=6$);
(b) $N_v=6$, for $S_1$, $G_1$, $S_7$ and $G_7$ tensors ($1\oplus\frac{1}{2}\oplus\frac{1}{2}$, $D=7$);
(c) $N_v=6$, for $S_1$, $S_6$, $S_7$, $G_1$, and $G_7$ ($1\oplus\frac{1}{2}\oplus\frac{1}{2}\oplus 0_s$, $D=8$);
(d) $N_v=6$, for $S_1$, $S_2$, $S_3$, $S_4$, $S_7$, $S_{10}$, $G_1$, $G_4$, and $G_7$ ($1\oplus\frac{1}{2}\oplus\frac{1}{2}\oplus 0$, $D=8$).
}
\label{Fig:ES_partial2}
\end{center}
\end{figure*}
We have applied the above decomposition (\ref{Eq:decomposition}) for the same choice of the parameters $\lambda_1=\lambda_2=\lambda_c(=1/\sqrt{3})$ and computed the ES of a few ad-hoc linear combinations of $S$ and $G$ tensors (to keep the $A_1+iA_2$
symmetry) of larger and larger bond dimension $D$.
Keeping only the $S_1$ and $G_1$ tensors, on one hand, or the $S_3$, $S_4$ and $G_4$ tensors on the other hand,
enables to restrict the bond dimension to the same $D=4$ small value, although the two PEPS involve completely different
virtual degrees of freedom, $\frac{1}{2}\oplus\frac{1}{2}$ in the first case and $1\oplus 0$ in the second case.
We have found that their corresponding ES shown in Fig.~\ref{Fig:ES_partial} (a) and Fig.~\ref{Fig:ES_partial}(b), respectively, seem
to be both gapped.
By adding, gradually, more virtual degrees of freedom to the previous cases, it is interesting to see whether the gap
in the ES closes. For instance, adding the $S_6$ tensors to the $S_1$ and $G_1$ tensors enlarge the
virtual space to $\frac{1}{2}\oplus\frac{1}{2}\oplus 0_s$ and $D=5$. However, as seen in Fig.~\ref{Fig:ES_partial}(c), the gap seems to persist.
Similarly, adding the $S_5$ and $G_5$ tensors to the $S_3$, $S_4$ and $G_4$ tensors enlarge the
virtual space to $1\oplus 0\oplus 0_s$ and $D=5$, but does not close the gap either (see Fig.~\ref{Fig:ES_partial}(d)).
Next, we have considered the virtual representations $V=\frac{1}{2}\oplus\frac{1}{2}\oplus 0\oplus 0_s$ ($D=6$)
involving the $S_1$, $G_1$, $S_2$, $S_6$, $S_9$ and $G_9$ tensors,
$V=1\oplus\frac{1}{2}\oplus\frac{1}{2}$ ($D=7$)
involving the $S_1$, $G_1$, $S_7$ and $G_7$ tensors, $V=1\oplus\frac{1}{2}\oplus\frac{1}{2}\oplus 0_s$ ($D=8$)
involving the $S_1$, $S_6$, $S_7$, $G_1$, and $G_7$ tensors and
$V=1\oplus\frac{1}{2}\oplus\frac{1}{2}\oplus 0$ ($D=8$) involving the $S_1$, $S_2$, $S_3$, $S_4$, $S_7$, $G_1$, $G_4$, and $G_7$ tensors. Results are compared in Fig.~\ref{Fig:ES_partial2}(a-d).
As the number of tensors and the bond dimension increase, one does not observe any systematic trend, rather
the ES changes in some erratic fashion. The later seems nevertheless to remain gapped,
although in Fig.~~\ref{Fig:ES_partial2}(b) the (pseudo-)energy scale becomes very small.
In any case, the ES remains very different from the chiral ES of the $D=9$ double-layer CSL
shown in Fig.~\ref{Fig:ES_double}. This is a clear indication that there is a large degree of fine tuning in the latter wave function.
In other words, just imposing SU(2)-spin rotation and $A_1+iA_2$ orbital symmetries, as it was done for
the single layer CSL, is not sufficient to obtain a CSL. One reason might be that for SU(2)$_2$ CFT,
(i) spin-1 degrees of freedom would be needed on the boundary but
(ii) spin-1 chains are generically in the gapped Haldane phase.
A gapless spectrum in a spin-1 chain requires fine tuning~\cite{Takhtajan1982,Babujian1982}.
\section{Conclusions and outlook}
\label{sec4}
In this paper we have elaborated a classification scheme of all rank-5 SU(2)-symmetric tensors according to the on-site physical spin $S$, the local Hilbert space of the bond degrees of freedom, and the irreducible representations of the $C_{4v}$ point group of the square lattice. We have shown how many remarkable (Mott insulating) states of matter fall naturally into this classification. More generally, we have explained how our scheme can be used to systematically construct
families of translationally invariant many-body singlet states, preserving or breaking discrete (point group)
lattice symmetries, spin liquids and (lattice) nematics, respectively. However, we bring here a few words of caution~: first, we should mention that LRO
(associated e.g. to spontaneous translation and/or SU(2) symmetry breaking) may still appear
in the thermodynamic limit in some parameter regions,
the PEPS being in that case a fully symmetric ``Shr\"odinger cat state''.
Note that the existence of LRO in our symmetric PEPS can only be diagnosed by a
thorough numerical investigation, e.g. inspecting the low-energy spectrum
of the transfer operator~\cite{Schuch2013a}.
Secondly, it is likely that
not all translationally invariant and spin-rotationally symmetric spin liquids (on the square lattice)
can be expressed in terms of a PEPS based on a single on-site SU(2)-symmetric tensor.
However, we believe our classification encompasses
a very large manifold of symmetric spin liquids. Spin liquids not generated by our classification
may include e.g. those requiring a two-site (gauge) unit cell such as the (translationally invariant) $\pi$-flux PEPS
of the PSG classification~\cite{Jiang2015} or those requiring a different type of virtual particles like
fermions, Majoranas, anyons, etc...
We have also used our construction to systematically search for higher-spin ($S>1/2$) topological chiral spin liquids. One of our constructions uses a symmetrization over a double-layer PEPS, showing gapless chiral edge modes corresponding to a non-Abelian SU(2)$_2$ Wess-Zumino-Witten model, which we have
determined via the analysis of its entanglement spectrum.
This family of CSL can be seen as a 2-dimensional manifold (spanned by the parameters $\lambda_2/\lambda_1$
and $\lambda_c/\lambda_1$) imbedded in a much larger PEPS family (characterized by arbitrary superpositions of
the $S$ and $G$ tensors). This suggests that, more generally, non-Abelian CSL live on (relatively small)
fine-tuned manifolds of large PEPS families. Also, we believe that our construction can be extended to $k$-layers,
then showing SU(2)$_k$ gapless chiral edge modes.
We envisage these results as the first step of a broader research program, concerning the search of quantum spin liquids with tensor networks. Further work will involve, for instance, using the tensors produced in our classification in order to produce \emph{ans\"atze} for the numerical simulation of the frustrated Heisenberg model on the square lattice (improving the results of Ref.~\cite{Wang2013}), extending the classification scheme to the Kagome lattice, and using it to propose variational wave functions to search for new spin liquids for the Kagome Heisenberg Antiferromagnet.
Another straightforward extension of our classification is to enlarge the physical space to more
degrees of freedom e.g. to include extra U(1)-charge degrees of freedom~\cite{Poilblanc2014} to describe hole
doping a Mott (spin liquid or antiferromagnetic) insulator, or more orbital degrees of freedom ($N>2$ with SU(N) symmetry).
To build wavefunctions with specified U(1)-charge, covariant tensors can be constructed.
Lastly, we note that our PEPS are, by construction, translationally invariant. However, it is also straightforward to
describe states breaking translation invariance by considering different tensors at every site of a given supercell.
All these problems, and more, will be addressed in future work along these lines.
\bigskip
\begin{acknowledgments}
R.O. acknowledges the Laboratoire de Physique Th\'eorique, C.N.R.S. and Universit\'e de Toulouse, for hosting him during the period in which this work was initiated, as well as the MOGON computer cluster of the University of Mainz for computational resources. D.P. is supported by the TNSTRONG ANR (French Research Council) grant (2016-2020).
D.P. thanks Nicolas Regnault and German Sierra for help in completing Table~\ref{Table:su2_2}.
\end{acknowledgments}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 8,089 |
The real Dumping Ground from Tracy Beaker is in Cardiff, and we visited it
The DG is one of the seven wonders of the world
Susannah Griffin
Almost fifteen years on, and we're still obsessed with Tracy Beaker. The box set's on iPlayer, we're still deeply concerned about Rio's Maroon Five CD, and the cast are all grown up. So I'm sure we all have nostalgic memories of the antics which went on in Stowey House, also known as the Dumping Ground.
For those who don't remember, it played host to many a 'bog off', and some of The Story of Tracy Beaker's most iconic scenes, such as the protest in the name of Beaker, where everybody gathered outside and sat on their luggage.
That time Tracy, in a pure Beaker diva move, ordered a bouncy castle for her own birthday.
Or when Bouncer and Ben's wrestling ended in a thumb war, with Louise dressed incredibly wavy in the background.
And who could forget all those times Tracy stormed out of Cam's car on the big driveway and back into the DG?
After discovering the Duke and Elaine tandem scene in Roath Park, our suspicions grew that more of the show could have been filmed in Cardiff. And when half of the accents in the Dumping Ground were Welsh – Lol and Bouncer, Crash and The Wellards to name a few – it was unsurprising that the Dumping Ground might be located in the Welsh capital.
It turns out the house still exists in all its glory down Station Road, Llanishen, Cardiff. Just an hour walk away from Cardiff Uni and a five minute train journey from Cathays.
It's unmistakeable.
The house was used for series two and three of The Story of Tracy Beaker, and was featured in the classic Tracy Beaker movie – the one where her Hollywood mum comes back. So seeing the house that made our childhoods did not disappoint.
When we visited the legendary house, we found that it is now a gated residential building, separated into apartments called 'The Hollies'. We'd heard rumours it was an old people's home, but it's safe to say these were not true. The care home is actually opposite on the other side of the road.
Claim to fame, outside the holy DG gates
Unfortunately, no one answered the intercom and so we were't able to see what the garden or the inside of the house is like now. Although the front drive still looks the same, and there's still a grassy hedge patch in the middle.
After the devastation of realising we wouldn't be able to see the ins and outs of the property, we took to Google Maps to see if it looked the same from all angles. From this, it seems that Duke's summerhouse – THE place to hang out – still exists.
I'm convinced that green shed is Duke's summerhouse
Tracy Beaker's bedroom window still exists too
The location of the house itself was different from the sheltered manor house you'd expect. It's on a busy main road, next to a school and is surrounded by other houses.
Nonetheless, the other houses down Station Road are also an impressive size, so it seems like the Dumping Ground is situated in quite an affluent area. No bin problems or Cathays damp lingering in these parts of Cardiff.
According to Right Move, the property is now worth £1,199,000 and rent for the whole property would cost £3,450 pcm. It's safe to say that's a bit more hefty than the standard Cathays prices.
Either way it's nice to know that this nostalgic gem is practically on our doorstep. I don't think the new residents would appreciate being told to 'bog off' though.
Climate change protests are taking place in Cardiff this week
Grace Withers
Protestors are calling it the Summer Uprising
Cardiff Uni kicked anti-vaxx student off healthcare course and paid them £9000 compensation
Lauren Ryan
The uni paid them £5,000 for the "distress" caused
Cardiff is home to one of the most colourful streets in Wales
It may be brighter than all of our future's combined
We spoke to the Cardiff second year who started her own business at university
Isobel McAllister & Anna Rees
That's more ambition than some of us will ever have
Netflix just removed this controversial scene from 13 Reasons Why season one
Hayley Soen
The episode aired two years ago
These are the biggest ever wins on ITV's The Chase
Only true legends make it to this hall of fame | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 3,353 |
{"url":"https:\/\/www.lesswrong.com\/posts\/wXxPmc9W6kPb6i7vj\/notes-on-logical-priors-from-the-miri-workshop","text":"# 30\n\n(This post contains a lot of math, and assumes some familiarity with the earlier LW posts on decision theory. Most of the ideas are by me and Paul Christiano, building on earlier ideas of Wei Dai.)\n\nThe September MIRI workshop has just finished. There were discussions of many topics, which will probably be written up by different people. In this post I'd like to describe a certain problem I brought up there and a sequence of ideas that followed. Eventually we found an idea that seems to work, and also it's interesting to tell the story of how each attempted idea led to the next one.\n\nThe problem is a variant of Wei Dai's A Problem About Bargaining and Logical Uncertainty. My particular variant is described in this comment, and was also discussed in this post.\u00a0Here's a short summary:\n\nIn Counterfactual Mugging with a logical coin, a \"stupid\" agent that can't compute the outcome of the coinflip should agree to pay, and a \"smart\" agent that considers the coinflip as obvious as 1=1 should refuse to pay. But if a stupid agent is asked to write a smart agent, it will want to write an agent that will agree to pay. Therefore the smart agent who refuses to pay is reflectively inconsistent in some sense. What's the right thing to do in this case?\n\nThinking about that problem, it seems like the right decision theory should refuse to calculate certain things, and instead behave updatelessly with regard to some sort of \"logical prior\" inherited from its creators, who didn't have enough power to calculate these things. In particular, it should agree to pay in a Counterfactual Mugging with a digit of pi, but still go ahead and calculate that digit of pi if offered a straight-up bet instead of a counterfactual one.\n\nWhat could such a decision theory look like? What kind of mathematical object is a \"logical prior\"? Perhaps it's a probability distribution over inconsistent theories that the creators didn't yet know to be inconsistent. How do we build such an object, what constraints should it satisfy, and how do we use it in a decision algorithm?\n\n#### Attempt 1\n\nSometime ago, Benja Fallenstein and Paul Christiano came up with a way to build a probability distribution over all consistent theories. We start with a single empty theory with weight 1, and then refine it by successively adding all possible axioms. Each new axiom A is either already settled by the current theory T (i.e. either A or\u00a0\u00acA is provable in T), in which case we do nothing; or A is independent of T, in which case we choose randomly between T+A and T+\u00acA. This process gives nonzero probabilities to all finitely axiomatized consistent theories, and leads to well-behaved conditional probabilities like P(PA+Con(PA)|PA). (I gloss over the distinction between finitely and recursively axiomatizable theories. Building a variant of this construction that works for recursively axiomatizable theories is left as an exercise to the reader.)\n\nAt the workshop we tried to use a similar construction, but with local consistency instead of global consistency. (Let's say that a \"locally consistent theory\" is a set of statements that has no short proof of inconsistency, for some reasonable meaning of \"short\".) We keep adding axioms as before, but only check that the theory so far is locally consistent. Hopefully this way we might end up with a nice distribution over all theories, including inconsistent ones.\n\nThe trouble with this approach is that for some axiom A and some locally consistent theory T, both T+A and T+\u00acA might be locally inconsistent, so we won't be able to continue. We can try to salvage the idea by somehow adjusting the probabilities retroactively, but then it seems hard to guarantee that the probability of each statement goes to a well-defined limit, it might end up oscillating. So we abandoned this idea.\n\n#### Attempt 2\n\nInstead of trying to build up a prior over possibly inconsistent theories, we can try assigning each theory a \"quality\" between 0 and 1. For example, we can take a probability distribution over all local consistency constraints that a theory must satisfy, and define the \"quality\" of a theory as the probability that it satisfies a randomly chosen constraint. For purposes of that definition, the theories have to be complete, but possibly inconsistent. Theories with quality 1 will be the consistent ones, because they satisfy all the constraints. Then we can define the quality of an individual statement as the maximum quality of any theory that includes that statement.\n\nThis all seems fine so far, but at the next step we run into trouble. In Counterfactual Mugging with a logical coin, Omega needs to evaluate a counterfactual statement like \"if a digit of pi was different, the agent would do such-and-such thing\". We can try to straightforwardly define \"A counterfactually implies B\" as true if the quality of \"A and B\" is equal to the quality of A. Intuitively that means that the most consistent theory that includes A also includes B.\n\nUnfortunately, that definition does not have very good properties. Both \"PA counterfactually implies Con(PA)\" and \"PA counterfactually implies \u00acCon(PA)\" will be defined as true, because the theories PA+Con(PA) and PA+\u00acCon(PA) are both consistent and have quality 1. That doesn't seem to agree with our intuitions about how logical counterfactuals should work.\n\n#### Some ideas about logical counterfactuals\n\nOur intuitions about decision theory seem to say that some logical counterfactuals are \"natural\" and others are \"spurious\". For example, if the agent does not in fact take a certain action, then there's a \"natural\" proof of how much utility it would have implied, and also many \"spurious\" proofs that prove all sorts of absurdities by first simulating the agent and then using the fact that the action was not taken. (See\u00a0this post\u00a0for more about spurious proofs.)\n\nNatural counterfactuals look more correct to us than spurious ones because their proofs don't rely on fully simulating the agent. Maybe we could try weighting logical counterfactuals based on their proof length, and see where that idea takes us?\n\nIn simple decision theory problems, we are interested in counterfactuals like \"A()=1 implies U()=1\", \"A()=1 implies U()=2\", and so on, where A is the agent program that returns an action, and U is the universe program that contains the agent and returns a utility value. (See\u00a0this post\u00a0or this post for more about this setting.) Naively assigning some probability P(L) to each counterfactual statement with proof length L doesn't seem to work, because the problem might be so complex that all relevant statements have long proofs. But if we took some set of counterfactuals assumed to be mutually exclusive, with proof lengths L1, L2, ..., then it might be reasonable to give them probabilities like P(L1)\/(P(L1)+P(L2)+...). That looks suspiciously similar to Bayes' theorem, so it seems natural to try and come up with some sort of probabilistic interpretation where this will be an actual application of Bayes' theorem. Thankfully, it's easy to find such an interpretation.\n\nLet's take a theory like PA, generate random proofs in it with probability depending on length, and then throw out the invalid ones. If we view each statement of the form \"X implies Y\" as an \"arrow\" from statement X to statement Y, then the above process of picking random valid proofs induces a probability distribution on all true arrows, including those where statement X is false. Now we can define the conditional probability P(U()=u|A()=a) as the probability that a random arrow ends at \"U()=u\", conditional on the event that the arrow starts from \"A()=a\" and ends at some statement of the form \"U()=...\". This way, all possible statements \"U()=u\" for a given \"A()=a\" have probabilities that sum to 1, and the true statement \"A()=a\" leads to a single statement \"U()=u\" with probability 1. These seem like good properties to have.\n\nMore generally, when people are uncertain about some math statement, they often seem to reason like this: \"I guess I'm likely to find a proof if I try this method. Hmm, I checked a couple obvious proofs and they didn't work. Guess I have to lower my probability of the statement.\" Such reasoning corresponds to Bayesian updating as you observe new proofs, which is similar to the above setting.\n\nNow, can we extend this idea to obtain a \"logical prior\" over inconsistent theories as well?\n\n### Attempt 3, successful\n\nThere's nothing stopping us from using the above construction with an inconsistent theory, because the properties that follow from having a coherent probability distribution over arrows should still hold. Let's assume that we care about theories T1,T2,... which are incomplete and possibly inconsistent. (An example of such a theory is PA+\"the billionth digit of pi is even\"). In each of these theories, we can construct the above distribution on arrows. Then we can add them up, using some weighting of the theories, and get an overall distribution on statements of the form \"A()=a implies U()=u\". It has the nice property that you only need to sample finitely many proofs to compute the conditional probabilities to any desired precision, so it's easy to write a decision algorithm that will do approximate utility maximization based on these probabilities. If we only care about a single theory, like PA, then the algorithm seems to give the same answers as our earlier algorithms for UDT.\n\nThere's still a remaining problem to fix. If we care equally about the theory where the billionth digit of pi is even and the one where it's odd, then our algorithm will accept a straight bet on that digit of pi, instead of calculating it and accepting conditionally. An approach to solving that problem can be found in Wei Dai's proposed \"UDT2\", which optimizes over the next program to run, instead of the value to return. We can use that idea here, ending up with an algorithm that I will call \"UDT1.5\", to avoid incrementing the major version number for something that might turn out to be wrong. To put it all together:\n\n\u2022 We care about some program U() that returns a utility value.\u00a0Also we care about some probability distribution over theories T1,T2,... that are incomplete and possibly inconsistent.\n\u2022 Our decision algorithm A() attempts to find a program B such that running it will approximately maximize the expected value of U(), using the probabilities defined below. Then A runs B() and returns the resulting value.\n\u2022 For each possible program B, the conditional probability P(U()=u|A runs B) is defined as the probability that a random valid proof sampled from a random Ti\u00a0proves that \"if A runs B, then U()=u\", conditional on it being a proof that \"if A runs B, then U()=something\".\n\n(The algorithm can easily be extended to the case where A receives an argument representing observational data. Namely, it should just pass the argument to B, which corresponds to Wei Dai's UDT1.1 idea of optimizing over the agent's input-output map.)\n\nGiven a suitable prior, the above algorithm solves my original variant of Counterfactual Mugging with stupid and smart agents, but still correctly decides to calculate the digit of pi if offered a straight bet instead of a counterfactual one. And even apart from that, it feels right to\u00a0make decisions using a weighted sum of theories T1,T2,... as well as a weighted sum of universes U1,U2,... composing U. I'm not sure if the right decision theory should eventually work like this, using \"Tegmark level 4\" over possible universe programs and \"Tegmark level 5\" over logically inconsistent theories, or perhaps we can define a single kind of uncertainty that generalizes both indexical and logical. In any case, that's as far as the workshop went, and it seems worthwhile to continue thinking about such questions.\n\nMany thanks to all participants, to MIRI which invited me, to Kevin and Sean who hosted me, and anyone else I might have missed!\n\n# 30\n\nNew Comment\n\nWait, what is Tegmark level 5?\n\nIt's a name we made up for mathematically impossible universes that we still care about because we haven't yet proved them to be mathematically impossible. That becomes relevant in problems like Counterfactual Mugging with a logical coin.\n\nWhat is a \"locally consistent theory\"?\n\nLet's say \"a set of statements that has no short proof of inconsistency\", for some reasonable meaning of \"short\".\n\nThat's what I guessed, but you said \"So we abandoned this idea,\" implying that the rest of the article was completely different, while the rest of the article was about still proof lengths, just smoothing thresholds into weights, so I became skeptical of the guess. I don't have any suggestions for how to talk about false starts and how they relate, but I think it might be useful for insight into my confusion. (Actually, in this particular case, I do have a suggestion, which is to use the term \"proof length\" much earlier.)\n\nThanks! Made a small edit to the post.\n\nYou lost me at part\n\nIn Counterfactual Mugging with a logical coin, a \"stupid\" agent that can't compute the outcome of the coinflip should agree to pay, and a \"smart\" agent that considers the coinflip as obvious as 1=1 should refuse to pay.\n\nThe problem is that, I see no reason why smart agent should refuse to pay. Both stupid and smart agent know it as logical certainty that they just lost. There's no meaningful difference between being smart and stupid in this case, that I can see. Both however like to be offered such bets, where logical coin is flipped, so they pay.\n\nI mean, we all agree that a \"smart\" agent, that refused to pay here, would receive $0 if Omega flipped logical coin of asking if 1st digit of pi was an odd number, while \"stupid\" agent would get$1,000,000.\n\nNote that there's no prior over Omega saying that it's equally likely to designate 1=1 or 1\u22601 as heads. There's only one Omega, and with that Omega you want to behave a certain way. And with the Omega that designates \"the trillionth digit of pi is even\" as heads, you want to behave differently.\n\nAnd with the Omega that designates \"the trillionth digit of pi is even\" as heads, you want to behave differently.\n\nSpecifically, you want to bet on 'heads'. The trillionth digit of pi is a two.\n\nI think we need to find a trickier logical uncertainty as a default example. There is a (mildly) interesting difference between logical uncertainties that we could easily look up or calculate like \"Is 1,033 a prime?\" or \"is the trillionth digit of pi even?\" and logical uncertainties that can not be plausibly looked up. Both types of uncertainty are sometimes relevant but often we want a 'logical coin' that isn't easily cheated.\n\nAfter asking about this on #LW irc channel, I take back my initial objection, but I still find this entire concept of logical uncertainty kinda suspicious.\n\nBasically, if I'm understanding this correctly, Omega is simulating an alternate reality which is exactly like ours, and where the only difference is that Omega says something like \"I just checked if 0=0, and turns out it's not. If it was, I would've given you moneyzzz(iff you would give me moneyzzz in this kind of situation), but now that 0!=0, I must ask you for $100.\" Then the agent notices, in that hypothetical situation, that actually 0=0, so actually Omega is lying, so he is in hypothetical, and thus he can freely give moneyzzz away to help to real you. Then, because some agents can't tell for all possible logical coins if they are lied to or not, they might have to pay real moneyzzz, while sufficiently intelligent agents might be able to cheat the system if they are able to notice if they are lied to about the state of the logical coin. I still don't understand why a stupid agent would want to make a smart AI that did pay. Also, there are many complications that restrict decisions of both smart and stupid agents, given argument I've given here, stupid agents still might prefer not paying, and smart agents might prefer paying, if they gain some kind of insght to how Omega chose these logical coins. Also, this logical coin problemacy seems to me like a not-too-special special class of Omega problems where some group of agents is able to detect if they are in counterfactuals Note that the agent is not necessarily able to detect that it's in a counterfactual, see Nesov's comment. Yes, those agents you termed \"stupid\" in your post, right? The smart ones too, I think. If you have a powerful calculator and you're in a counterfactual, the calculator will give you the wrong answer. Well, to be exact, your formulation of this problem has pretty much left this counterfactual entirely undefined. Naive approximation, that the world is just like ours, and Omega just lies in counterfactual, would not contain such weird calculators which give you wrong answers. If you want to complicate problem by saying that some specific class of agents have a special class of calculators that one would usually think to work in certain way, but actually they work in a different way, well, so be it. That's however just a free-floating parameter you have left unspecified and that, unless stated otherwise, should be assumed not to be the case. Hmm, no, I assumed that Omega would be using logical counterfactuals, which are pretty much the topic of the post. In logical counterfactuals, all calculators behave differently ;-) But judging from the number of people asking questions similar to yours, maybe it wasn't a very transparent assumption... I asked about these differences in my second post in this post tree, where I explained how I understood these counterfactuals to work. I explained as clearly as I could that, for example, calculators should work as they do in real world. I did this explaining in hopes of someone voicing disagreement if I had misunderstood how these logical counterfactuals work. However, modifying any calculator would mean that there can not be, in principle, any \"smart\" enough ai or agent that could detect it was in counterfactual. Our mental hardware that checks if logical coin should've been heads or tails is a calculator the same as any computer, and again, there does not seem to be any reason to assume Omega leaves some calculators unchanged while changes results of others. Unless, this thing is just assumed to happen, with some silently assumed cutaway point where calculators become so internal they are left unmodified. However, modifying any calculator Calculators are not modified, they are just interpreted differently, so that when trying to answer the question of what happens in a certain situation (containing certain calculators etc.) we get different answers depending on what the assumptions are. The situation is the same, but the (simplifying) assumptions about it are different, and so simplified inferences about it are different as well. In some cases simplification is unavoidable, so that dependence of conclusions on assumptions becomes an essential feature. My current understanding of logical counterfactuals is something like this: if the inconsistent formal theory PA+\"the trillionth digit of pi is odd\" has a short proof that the agent will take some action, which is much shorter than the proof in PA that the trillionth digit of pi is in fact even, then I say that the agent takes that action in that logical counterfactual. Note that this definition leads to only one possible counterfactual action, because two different counterfactual actions with short proofs would lead to a short proof by contradiction that the digit of pi is odd, which by assumption doesn't exist. Also note that the logical counterfactual affects all calculator-like things automatically, whether they are inside or outside the agent. That's an approximate definition that falls apart in edge cases, the post tries to make it slightly more exact. (Btw, I think it should be mentioned that a central piece of motivation for this \"logical counterfactuals\" thing is that it's probably the same construction that's needed to evaluate possible actions in normal cases, without any contrived coins, for an agent that knows its own program. So for example although a counterfactual scenario can't easily \"lead\" to two different actions, two different actions in that scenario can still be considered as possibly even more (easily shown to be) contradictory \"logical counterfactuals\" that include additional assumptions about what the action is.) Try as I might, I cannot find any reference to what's canonical way of building such counterfactual scenarios. Closest I could get was in http:\/\/lesswrong.com\/lw\/179\/counterfactual_mugging_and_logical_uncertainty\/ , where Vladimir Nesov seems to simply reduce logical uncertainty to ordinary uncertainty, but this does not seem to have anything to do with building formal theories and proving actions or any such thing. To me, it seems largely arbitrary how agent should do when faced with such a dilemma, all dependent on actually specifying what it means to test a logical counterfactual. If you don't specify what it means, whatever could happen as a result. I am not sure there is a clean story yet on logical counterfactuals. Speaking for myself only, I am not yet convinced logical counterfactuals are \"the right approach.\" Hi Ilya, I am not yet convinced logical counterfactuals are \"the right approach.\" Me neither. Have you seen my post about common mistakes? To me it seems more productive and more fun to explore the implications of an idea without worrying if it's the right approach. I like \"breadth first search\" or more precisely \"iterative deepening\" better than \"depth first search.\" (DFS is not guaranteed to find the optimal solution, after all!) But if a stupid agent is asked to write a smart agent, it will want to write an agent that will agree to pay. Wait, I'm afraid I'm already lost and this question seems so simple as to suggest I'm missing some important premise of the hypothetical scenario: Why would the stupid agent want this? Why wouldn't it want to write a smart agent that calculates the millionth digit and makes the winning choice? Restatement of what I understand about the problem: You offer me lots of money if the millionth digit of pi is even and a small loss of it is odd. I should take the bet since I can't calculate the answer and it might as well be random . You offer me lots of money if the millionth digit of pi is even and a small loss of it is odd, and the chance to build a calculator to calculate the answer. I should still take the bet, even if my calculator tells me that it's odd. If I'm rephrasing it correctly it, then why?! If you're given the chance to make a calculator to solve the problem, why wouldn't you use it? What you're describing is not Counterfactual Mugging, it's just a bet, and the right decision is indeed to use the calculator. The interesting feature of Counterfactual Mugging is that Omega is using counterfactual reasoning to figure out what you would have done if the coin had come out differently. You get the money only if you would have paid up in the counterfactual branch. In that case the right decision is to not use the calculator, I think. Though other people might have different intuitions, I'm sort of an outlier in how much I'm willing to follow UDT-ish reasoning. The setup is such that muggings and rewards are grouped in pairs, for each coin there is a reward and a mugging, and the decision in the mugging only affects the reward of that same coin. So even if you don't know where the coin comes from, or whether there are other coins with the same setup, or other coins where you don't have a calculator, your decision on a mugging for a particular coin doesn't affect them. If you can manage it, you should pay up only in counterfactuals, situations where you hypothetically observe Omega asserting an incorrect statement. Recognizing counterfactuals requires that the calculator can be trusted to be more accurate than Omega. If you trust the calculator, the algorithm is that if the calculator disagrees with Omega, you pay up, but if the calculator confirms Omega's correctness, you refuse to pay (so this confirmation of Omega's correctness translates into a different decision than just observing Omega's claim without checking it). Perhaps in the counterfactual where the logical coin is the opposite of what's true, the calculator should be assumed to also report the incorrect answer, so that its result will still agree with Omega's. In this case, the calculator provides no further evidence, there is no point in using it, and you should unconditionally pay up. Perhaps in the counterfactual where the logical coin is the opposite of what's true, the calculator should be assumed to also report the incorrect answer, so that its result will still agree with Omega's. In this case, the calculator provides no further evidence, there is no point in using it, and you should unconditionally pay up. Yeah, that's pretty much the assumption made in the post, which goes on to conclude (after a bunch of math) that you should indeed pay up unconditionally. I can't tell if there's any disagreement between us... [-][anonymous]10y0 The origin of the logical coin seems relevant if you can compute it. Even if you know which side is counterfactual according to a particular logical coin, you might still be uncertain about why (whether) this coin (puzzle) was selected and not another coin that might have a different answer. This uncertainty, if allowed by the boundaries of the game, would motivate still paying up where you know reward to be logically impossible (according to the particular coin\/puzzle), because it might still be possible according to other possible coins, that you can't rule out a priori. [This comment is no longer endorsed by its author]Reply [-][anonymous]10y0 It seems to me that if you have a calculator, you should pay up exactly when you are in a counterfactual (i.e. you hypothetically observe Omega asserting an incorrect statement about the logical coin), but refuse to pay up if the alternative (Omega paying you) is counterfactual (in this case, you know that the event of being paid won't be realized, assuming these are indeed the boundaries of the game). There doesn't appear to be a downside to this strategy, if you do have a calculator and are capable of not exploding in the counterfactual that you know to be counterfactual (according to whatever dynamic is used to \"predict\" you in the counterfactual). (Intuitively, a possible downside is that you might value situations that are contradictory, but I don't see how this would not be a semantic confusion, seeing a situation itself as contradictory as opposed to merely its description being contradictory, a model that might have to go through all of the motions for the real thing, but eventually get refuted.) [This comment is no longer endorsed by its author]Reply Hm, yeah, that sounds really odd. I think the reason is sounds so odd is: how the hell is Omega calculating what your answer would have been if 1=0? If what Omega is really calculating is what you would have done if you were merely told something equivalent to 1=0, then sure, paying up can make sense. It seems to me that the relevant difference between \"1=0\" and \"the billionth digit of pi is even\" is that the latter statement has a really long disproof, but there might be a much shorter proof of what the agent would do if that statement were true. Or at least I imagine Omega to be doing the same sort of proof-theoretic counterfactual reasoning that's described in the post. Though maybe there's some better formalization of Counterfactual Mugging with a logical coin that we haven't found... Even if you're cutting off Omega's proofs at some length, there are plenty of math problems that people can't do that are shorter than high-probability predictions that people will or won't pay up. Certainly when I imagine the problem, I imagine it in the form of predicting someone who's been told that the trillionth digit of pi is even and then paying out to that person depending on their counterfactual actions. Of course, that leads to odd situations when the agent being predicted can do the math problem, but Omega still says \"no bro, trust me, the trillionth digit of pi really is even.\" But an agent who can do the math will still give Omega the money because decision theory, so does it really matter? If you're proposing to treat Omega's words as just observational evidence that isn't connected to math and could turn out one way or the other with probability 50%, I suppose the existing formalizations of UDT already cover such problems. But how does the agent assign probability 50% to a particular math statement made by Omega? If it's more complicated than \"the trillionth digit of pi is even\", then the agent needs some sort of logical prior over inconsistent theories to calculate the probabilities, and needs to be smart enough to treat these probabilities updatelessly, which brings us back to the questions asked at the beginning of my post... Or maybe I'm missing something, can you specify your proposal in more detail? Well, I was thinking more in terms of a logical prior over single statements, see my favorite here. But yeah I guess I was missing the point of the problem. Also: suppose Omega comes up to you and says \"If 1=0 was true I would have given you billion dollars if and only if you would give me 100 dollars if 1=1 was true. 1=1 is true, so can you spare$100?\" Does this sound trustworthy? Frankly not, it feels like there's a principle of explosion problem that insists that Omega would have given you all possible amounts of money at once if 1=0 was true.\n\nA formulation that avoids the principle of explosion is \"I used some process that I cannot prove the outcome of to pick a digit of pi. If that digit of pi was odd I would have given you a billion dollars iff [etc].\"\n\nAre you saying that Omega won't even offer you the deal unless it used counter-factual reasoning to figure out what you'll do once it offers?\n\nSo if Omega has already offered you the deal and you know the coin came out against your favor, and you find you are physically capable of rejecting the deal, you should reject the deal. You've already fooled Omega into thinking you'll take the deal.\n\nIt's just that if you've successfully \"pre-committed\" to the extent that a 100% accurate Omega has predicted you will take the offer, you'll be physically incapable of not taking the offer. It's just like Newcombs problem.\n\n[This comment is no longer endorsed by its author]Reply\n\nAnd if that's true, it means that the problem we are facing is, how to make an algorithm that can't go back on its pre-commitments even after it gains the knowledge of how the bet came out.\n\n[This comment is no longer endorsed by its author]Reply\n\nRetraction was unintentional - I thought this was a duplicate comment and \"unretract\" isn't a thing.\n\nYou can delete and then re-post a retracted comment if it has no replies yet.\n\nAs a sort of metaphor\/intuition pump\/heuristic thingy to make sure I understood things right: Can you think of this as A living in platonic space\/tegmark 4 multiverse \"before\" your universe and having unlimited power, you importing B from it and using that repeatedly as your limited AI, and the set of axiom sets A cares about as a \"level 5 tegmark multiverse\" with each of the logics being it's own tegmark 4 multiverse?\n\nPerhaps not so useful in building such an AI, but I find these kind of more intuitive rough approximations useful when trying to reason about it using my own human brain.\n\nI'm not sure \"existence\" is the best intuition pump, maybe it's better to think in terms of \"caring\", like \"I care about what these programs would return\" and \"I care about what would happen if these logical facts were true\". There might well be only one existing program and only one set of true logical facts, but we care about many different ones, because we are uncertain.\n\nI already have an intuition setup where \"what I care about\" and \"what really exists\" are equivalent. Since, you know, there's nothing else \"exists\" could mean I can think of and \"what I care about\" is what it seems to be used like?\n\nMay or may not need an additional clause about things that exist to things that exist also existing, recursively.\n\nLet's say you \"care\" about some hypothetical if you'd be willing to pay a penny today unconditionally in order to prevent your loved ones from dying in that hypothetical. If we take some faraway digit of pi, you'll find that you \"care\" about both the hypothetical where it's even and the hypothetical where it's odd, even though you know in advance that one of those provably does not \"exist\". And if you only had a limited time to run a decision theory, you wouldn't want to run any decision theory that threw away these facts about your \"care\". That's one of the reasons why it seems more natural to me to use \"care\" rather than \"existence\" as the input for a decision theory.\n\nSounds like both of us could use either interpretation without any difference in conclusion, and just find different abstractions useful for thinking about it due to small differences in what other kinds of intuitions we've previously trained. Just a minor semantic hiccup and nothing more.\n\nHow does a logical coin work exactly? To come up with such a thing, wouldn't Omega first need to pick a particular formula? If the statement is about the nth digit of pi, then he needs to pick n. Was this n picked at random? What about the sign of the test itself? If not, how can you be sure that the logical coin is fair?\n\nThe approach outlined in the post assumes that \"fairness\" of the coin is determined by your initial state of logical uncertainty about which math statements are true, rather than indexical uncertainty about which particular Omega algorithm you're going to face. Though I agree that's a big assumption, because we still don't understand logical uncertainty very well.\n\nA priori I wouldn't trust Omega to be fair. I only know that he doesn't lie. If Omega also said that he chose the logical statement in some fair way, then that would assure me the logical coin is identical to a normal coin. He can do this either using real uncertainty, like rolling a die to pick from a set of statements where half of them are true. Or he could use logical uncertainty himself, by not calculating the digit of pi before deciding to make the bet, and having a prior that assigns 50% probability to either outcome.\n\nFor what it's worth, the post assumes that Omega decides to participate in the game unconditionally, its code doesn't have a branch saying it should play only if such-and-such conditions are met. I'm not sure if that answers your question.","date":"2023-04-02 08:50:53","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\": 0.6890373826026917, \"perplexity\": 750.9256924055519}, \"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\/1679296950422.77\/warc\/CC-MAIN-20230402074255-20230402104255-00406.warc.gz\"}"} | null | null |
\section{Introduction}
A \emph{geometric graph}~\cite{ttgg} is an embedding of a graph
$G=(V,E)$ in ${\bf R}^2$ so that each vertex $v$ is associated with a unique
point $p$ in ${\bf R}^2$ and each edge is ``drawn'' as a straight line segment
joining the points associated with its end vertices.
Moreover, the edges incident on each vertex $v$ are given in angular
order around $v$, so that faces in the embedding of
$G$ in ${\bf R}^2$ are well-defined (e.g., using the next-clockwise-edge ordering).
Thus, we use the same notation and terminology to refer to
$G$ and its embedding.
If the edges in $G$ have no crossings, then $G$ is said to be a
\emph{plane graph},
while graphs that admit realizations as plane graphs
are \emph{planar graphs}~\cite{dett-gd-99,eg-sbdgt-70}.
Geometric graphs are natural abstractions of the geometric and
connectivity relationships that arise in a number of applications,
including road networks, railroad
networks, and utility distribution grids,
as well as
sewer lines and the physical connections defining the
Internet.
An example road network is shown in Figure~\ref{fig-ny}.
\begin{figure}[htb]
\begin{center}
\includegraphics[height=3in]{map.png}
\end{center}
\caption{A portion of the road network
surrounding the location of SODA 2009.
This image is from
\textsf{http://wiki.openstreetmap.org/},
under the Creative Commons attribution-share alike license.
}
\label{fig-ny}
\end{figure}
\begin{figure*}[hbt]
\centering\includegraphics[width=4.5in]{planarization}
\caption{A geometric graph and its planarization.}
\label{fig:planarization}
\end{figure*}
Although planar graphs and their plane graph realizations
have been studied extensively (e.g., see~\cite{t-pg-93}),
real-world geometric graphs often contain edge crossings.
Recent experimental studies by the first two authors gives
empirical evidence that real-world road networks typically have
$\Theta(\sqrt{n})$ edge crossings, where $n$ is the number of
vertices~\cite{eg-snprntaal}.
Motivated by this real-world example, therefore, we are interested in
studying algorithms for connected geometric graphs that
have a sublinear number of edge crossings. However, we use a weaker restriction on the number of crossings than the bounds that our evidence suggests for road networks: here we are interested in $n$-vertex geometric graphs that
have at most $O(n/\log^{(c)} n)$ edge crossings, for some constant
$c$, where $\log^{(c)} n$ denotes the $c$-th iterated logarithm
function.
We refer to such geometric graphs as \emph{restrained} graphs.
Given an $n$-vertex geometric graph $G$,
the \emph{planarization}\footnote{Our use of this term differs
from its use in the graph drawing literature
(e.g., see~\cite{dett-gd-99}), where it refers to
the problem of removing a minimal number of
edges to make $G$ be planar.}
of $G$ is the
graph $G'$ that is defined by the arrangement of the edges in $G$.
That is, as shown in Figure~\ref{fig:planarization}, we place a vertex in $G'$ for every vertex and pairwise edge
crossing in $G$, and we create an edge in $G'$ for every maximal edge segment
from $G$ that connects exactly two vertices in $G'$. Likewise, we
preserve the (clockwise/counterclockwise) ordering
of edges around corresponding vertices
in $G$ and $G'$, and we assume that intersection vertices in $G'$
similarly have their edges given in rotational order.
Thus, $G'$ is a plane graph having $n+k$ vertices,
where $k$ is the number of pairwise edge crossings among the edges in $G$.
By well-known properties of planar graphs (e.g., see~\cite[Prop.~2.1.6]{mt-gos-01}),
this implies that $G'$ has at most $3n+3k-6$ edges,
which in turn implies that $G$ has at most $3n+k-6$ edges.
Therefore, by restricting our attention to connected
geometric graphs with a sublinear number of edge crossings, we are,
by implication, focusing on connected geometric graphs that have $O(n)$ edges in their planarizations.
As mentioned above, a wealth of
algorithms are known for planar graphs and plane graphs.
Indeed, many of these algorithms, for such problems as single-source
shortest paths and minimum spanning trees, run in $O(n)$ time.
Much less is known for non-planar geometric graphs, however, which
motivates our interest in such graphs in this paper.
Specifically, we are interested in the following problems for
connected, restrained geometric graphs:
\begin{itemize}
\item
The \emph{Voronoi diagram} problem,
which is also known as the \emph{post office} problem:
we are given a set $P$ of $k$ vertices in a geometric graph $G$
and asked to determine for every other vertex $v$ in $G$ the
vertex in $P$ that is closest to $v$ according to the graph metric.
\item
The \emph{single-source shortest path} problem: we are given a
vertex $s$ and a geometric graph $G$ and asked to find the shortest
paths from $s$ to every other vertex in $G$.
\item
The \emph{polygon planarization} problem: given a geometric
graph defining a non-simple polygon
$P$ having $n$ vertices, compute the arrangement of all the edges of
$P$, including vertices defined by the pairwise crossings
of the edges in $P$.
\end{itemize}
In all these cases, we desire comparison-based
algorithms that require no additional
assumptions regarding the distribution of edge weights, so that our algorithms
can apply to a wide variety of possible edge weights that may vary for different users, including
combinations of distance,
travel time, toll charges, and subjective scores rating safety
and scenic interest~\cite{Epp-SJC-03}.
\subsection{Previous Related Work}
In the algorithms community, there has
been considerable prior work on shortest
path algorithms for Euclidean graphs
(e.g., see~\cite{gh-cspasm-05,hsww-csuts-05,%
kp-sppls-06,ss-hhhes-05,sv-speg-86,zn-spaeu-98}),
which are geometric graphs where edges are weighted by the
lengths of the corresponding line segments.
This prior work takes a decidedly
different approach than we take in this paper, however, in that it focuses on using special properties of the edge weights
that do not hold in the comparison model,
whereas we study road networks as geometric graphs with a sublinear
number of edge crossings and we desire
linear-time algorithms that hold in the comparison model.
The specific problems for which we provide linear-time algorithms are
well known in the general algorithms and computational geometry
literatures.
For general graphs with $n$ vertices and $m$ edges,
excellent work can be found on
efficient algorithms in the comparison model,
including single-source
shortest paths~\cite{clrs-ia-01,gt-adfai-02,r-rrsss-97},
which can be found in $O(n\log n + m)$
time~\cite{ft-fhtui-87}, and
Voronoi diagrams~\cite{a-vdsfg-91,ak-vd-00},
whose graph-theoretic version
can be constructed in $O(n\log n + m)$ time~\cite{e-tgvda-00,m-afaas-88}.
None of these algorithms run in linear time, even for
planar graphs. Linear-time algorithms for planar graphs
are known for single-source shortest paths~\cite{hkrs-fspap-97},
but these unfortunately do not immediately
translate into linear-time algorithms for non-planar geometric graphs.
In addition, there are a number of efficient
shortest-path algorithms that make assumptions about
edge weights~\cite{g-saspp-93,gh-cspasm-05,m-ssspa-01,t-usspp-99};
hence, they are not applicable in the comparison model.
Chazelle~\cite{c-tsplt-91a} shows that any simple polygon can be triangulated
in $O(n)$ time and that this algorithm can be extended to
determine in $O(n)$ time, for any polygonal chain $P$,
whether or not $P$ contains a self-intersection.
In addition, Chazelle posed as an open problem whether or not one can
compute the arrangement of a non-simple polygon in $O(n+k)$ time,
where $k$ is the number of pairwise edge crossings.
Clarkson, Cole, and Tarjan~\cite{cct-rpatd-92,cct-erpat-92} answer this question in the
affirmative for polygons with a super-linear number of crossings, as
they give a randomized algorithm that solves this problem in
$O(n\log^* n + k)$ expected time.
There is, to our knowledge, no previous
algorithm that solves Chazelle's open
problem, however, for non-simple polygons with a sublinear
number of edge crossings.
\subsection{Our Results}
In this paper, we
provide the first linear-time algorithm for planarizing a non-planar
connected geometric graph having a number of pairwise edge crossings, $k$,
that is sublinear in the number of vertices, $n$,
by an iterated logarithmic factor.
Specifically, we provide a randomized algorithm for planarizing
geometric graphs in $O(n + k\log^{(c)} n)$ expected time, which is linear for
restrained geometric graphs.
Given such a planarization, we show how it can be used to help
construct an $O(\sqrt{n})$-separator decomposition of the original
graph in $O(n)$ time.
Furthermore, we discuss how such separator decompositions can then be
used to produce linear-time algorithms for a number of problems,
including Voronoi diagrams and single-source shortest paths.
We also show how our planarization algorithm can be used to solve
Chazelle's open problem of planarizing non-simple polygons in
expected linear time
for polygons having a number of pairwise edge crossings that is
sublinear in $n$ by an iterated logarithmic factor.
Thus, combining this result with the polygon planarization
algorithm of Clarkson, Cole, and Tarjan~\cite{cct-rpatd-92,cct-erpat-92} provides a method for
planarizing an $n$-vertex polygon with $k$ edge crossings in
optimal $O(n + k)$ expected time, for all
values of $k$ except those in the range $[n/\log^{(c)}n, n\log^* n]$.
Our result also implies that the convex hull of restrained non-simple
polygons can be
constructed in $O(n)$ expected time, which, to the best of our knowledge, was
also previously open.
Besides planar separator decompositions, which we discuss below,
another one of the techniques we use in this paper is a method for
constructing a $(1/r)$-cutting for the edges of a geometric graph,
$G$.
This is a proper triangulation\footnote{%
A \emph{proper triangulation} is a
connected planar geometric graph such that every
face is a triangle and every triangular face has exactly three
vertices on its boundary.},
$T$, of the interior of
the bounding box containing $G$ such
that any triangle $t$ in $T$ intersects at most $(1/r)n$
edges of $G$.
Using existing methods (e.g., see~\cite{a-gpia-91i,bs-ca-95,h-cpctp-00}), one can construct such
a $(1/r)$-cutting for $G$ in $O(n\log r + (r/n)k)$ time, where $n$ is
the number of vertices in $G$ and $k$ is the number of pairwise edge crossings. However, in our application such a bound would be nonlinear, as we require $r$ to be large.
We show, in Section~\ref{sec-cuttings}, that for connected geometric graphs such a cutting can be constructed in the faster expected time bound
$O(ns + (r/n)k)$, where $r\le n/\log^{(s)} n$.
\section{Separator Decompositions}
\begin{figure*}[htb]
\centering\includegraphics[width=4.5in]{trapezoidalization}
\caption{Trapezoidal decomposition of a sampled subset of input graph edges.}
\label{fig:trapezoidalization}
\end{figure*}
One of the main ingredients
we use in our algorithms is the existence of small separators in
certain graph families (e.g., see~\cite{lt-stpg-79,m-fsscs-86}).
Several of the algorithms in this paper are based on the use of separators: we use them both as part of our algorithm for finding cuttings of geometric graphs, and later, once the graph has been planarized. Hence, we briefly review these tools here.
Given a graph $G=(V,E)$, a
subset $W$ of $V$ is an \emph{$f(n)$-separator} if the removal of the
vertices in $W$ separates $G$ into two subgraphs $G_1$ and $G_2$,
each containing at most $\delta n$ vertices, for some constant
$0<\delta < 1$.
It is well known that planar graphs have $O(\sqrt{n})$-separators
with $\delta=2/3$, and that such separators can be constructed in
$O(n)$ time~\cite{lt-stpg-79}.
Such separators are typically used in divide-and-conquer algorithms,
which involve finding a separator, recursively solving the problem in
the two separated subgraphs, and then merging the solutions together.
If the merge and divide steps can be solved in $o(n)$ time, however,
it is useful to have the entire recursive separator decomposition
computed in advance; for otherwise there is no way to beat an
$O(n\log n)$ time bound.
Such a \emph{separator decomposition} defines a binary tree $B$,
such that the root of $B$ is associated with the $f(n)$-separator for
$G$ and the subtrees of this root are defined recursively for the
graphs $G_1$ and $G_2$, respectively.
Previous work on separators includes the seminal contribution of
Lipton and Tarjan~\cite{lt-stpg-79}, who show that $O(\sqrt{n})$-sized
separators exist for $n$-vertex planar graphs and these can be computed
in $O(n)$ time.
Goodrich~\cite{g-psppt-95} shows that recursive
$O(\sqrt{n})$-separator decompositions can be constructed for planar
graphs in $O(n)$ time.
A related concept is that of geometric separators, which use
geometric objects to define separators in graphs defined by systems of intersecting disks
(e.g., see~\cite{ar-dsa-93,mttv-gsfem-95,mttv-sspnn-97,st-dpps-96}).
Eppstein
{\it et al.}~\cite{emt-dltag-93} provide a linear-time construction
algorithm for geometric separators which translates into an $O(n\log n)$ recursive separator
decomposition algorithm.
Because restrained graphs are not planar, the result of Goodrich does not immediately apply. However, it can be applied once we have planarized the graph, and it can also be applied to planar structures formed from subsets of the graph, such as the one we describe in the next section.
\section{Trapezoidal Decomposition of a Sample}
Suppose we are given a geometric graph $G$ having $n$ vertices
and $k$ pairwise intersections among its edges.
In this section, we describe our algorithm for constructing a
trapezoidal decomposition of a random sample of the edges of $G$.
That is, given the sample of edges, we construct the arrangement of these edges together with a set of vertical line segments through each edge endpoint and crossing, where each such segment is maximal with respect to the property of not crossing any other sampled edge, as shown below.
(See Figure~\ref{fig:trapezoidalization}.)
Our method is parameterized by $s$ where $r\le n/\log^{(s)} n$, and the sample probability is inversely proportional to $\log^{(s)} n$. We will later
show how to refine this sample so that we can
produce a cutting and then a planarization of $G$.
This first step of
our algorithm is essentially the same as performing $s$ levels of the
Clarkson, Cole, and Tarjan algorithm, except that their
method is for polygonal chains, whereas ours is for geometric graphs. Thus, we describe it at a high level.
Our algorithm begins with a trivial trapezoidal decomposition $T_0$ containing a single trapezoid
that encloses all of $G$. Call this trapezoid $t$. Let $C(t)=E$ be the \emph{conflict list} for $t$, that is,
the set of edges from $G$ that intersect the interior of $t$.
Then, for $i=1$ to $s$, we perform the following computation.
\begin{enumerate}
\item
Find a random sample $S_i$ of size $n/\log^{(i)} n$,
of the edges in $G$,
and for each trapezoid $t$ in $T_{i-1}$, use the Bentley-Ottmann
algorithm~\cite{bo-arcgi-79} to
construct the trapezoidal decomposition of the arrangement of the segments in
$C(t)\cap S_i$. Once all these trapezoidal decompositions are constructed,
merge them together to create a single trapezoidal decomposition, $T_i$, for
the segments in $S_i$. To be consistent with Clarkson, Cole, and Tarjan, we choose the samples such that $S_1 \subset S_{2} \subset \cdots \subset S_s$.
\item
Perform a depth-first traversal of $G$, while keeping track of the trapezoids in
the trapezoidal decomposition that are intersected during the
walk, so as to determine, for each trapezoid $t$ in $T_i$, the set $C(t)$. Since the geometric graph is connected, we never have to restart the depth-first traversal from a node whose location we do not already know. We can therefore use the arrangement of the sampled line segments to keep track of the intersected trapezoids at each step of the traversal. Thus we eliminate the need for time-consuming point-location data structure lookups.
\end{enumerate}
Let $T=T_s$ be the resulting final trapezoidal decomposition we get
from this computation, and let $S=S_s$ be the final random sample.
Using the framework established by Clarkson and Shor~\cite{cs-arscg-89} for
randomized divide-and-conquer algorithms, such as this, we can show
that
\begin{equation}
\label{eq-1}
E\left( |T| \right) = O\left(r+\left(\frac{r}{n}\right)^2 k\right)
\end{equation}
and
\begin{equation}
\label{eq-2}
E\left( \sum_{t\in T} |C(t)| \right) =
O\left( n+\left( \frac{r}{n}\right) k\right).
\end{equation}
In particular,
Equation~(\ref{eq-1}) is from their Lemma~4.1
and
Equation~(\ref{eq-2}) follows from their Corollary~4.4. The number of steps in the depth-first traversal is proportional to the total size of the conflict lists of the input geometric graph with the trapezoidal decomposition, which as we have seen above is small. A step from one trapezoid to a horizontally adjacent trapezoid may be accomplished in constant time, but a single trapezoid may have a non-constant number of neighbors above and below it, causing steps in those directions to take longer. But as Clarkson, Cole, and Tarjan show, the sum over all trapezoids of the conflict list size of the trapezoid multiplied by its number of neighbors remains linear in expectation, and this sum bounds the time to step vertically from one trapezoid to another using a sequential search along the trapezoid boundary to find the neighboring trapezoid. Therefore, we have the following preliminary result:
\begin{lemma}
Given a connected geometric graph $G$ with $n$ edges and $k$ pairwise edge crossings, and a
parameter $s$, we can in expected
time $O(ns+(r/n)k)$ find a random sample of $r=O(n/\log^{(s)} n)$
edges from $G$, the trapezoidal decomposition induced by the sample,
and the set of edges of $G$ crossing each trapezoid of the sample.
\end{lemma}
\section{Cuttings}
\label{sec-cuttings}
At this stage we take a detour from the Clarkson, Cole, and Tarjan algorithm.
For each trapezoid $g$ in $T$, let $\alpha_g=|C(g)|r/n$.
That is, $\alpha_g$ is the degree of excess that the conflict list
for $g$ has beyond what we would like for a $(1/r)$-cutting.
For each trapezoid $t$ with
$\alpha_t>1$, we form a random sample, $R_t$, of $C(t)$ of size
$2b\alpha_t \log \alpha_t$, where $b$ is the constant $K_{\max}$ from
Corollary 4.4 of Clarkson-Shor~\cite{cs-arscg-89}.
We then form the trapezoidal decomposition, $T_t$ of
the arrangement of the segments in $R_t$ using
any quadratic-time line segment arrangement algorithm~\cite{am-dplr-91,bdsty-arsol-92,ce-oails-92,ejps-pcoas-91}.
Thus, by Corollary 4.4 from Clarkson-Shor~\cite{cs-arscg-89},
the maximum size of any conflict list
of a trapezoid in $T_t$ is expected to be
less than
\begin{eqnarray*}
\left(\frac{|C(t)|}{|R_t|}\right)\log |R_t| &=&
\left(\frac{n}{r}\right)
\left(\frac{1}{\log \alpha_t^2}\right)\log (2\alpha_t\log
\alpha_t) \\
&\le& \frac{n}{r},
\end{eqnarray*}
for $\alpha_t\ge 4$.
Thus, we can repeat the above algorithm an expected constant number
of times until we have this condition satisfied, which gives us one
of the crucial properties of a $(1/r)$-cutting: namely, that each
cell intersects at most $(n/r)$ edges of $G$.
In addition,
the number of new trapezoids created inside $t$, as well as the
running time for creating the trapezoidal diagram $T_t$, is certainly at most
$O(|R_t|^2)$, which is
$O(\alpha_t^2 \log^2 \alpha_t)$.
More importantly, we have the following:
\begin{lemma}
Given the above construction applied to each trapezoid $t$ in $T$,
then
\[
E\left( \sum_{t\in T} \alpha_t^2 \log^2 \alpha_t\right)
= O\left(r+\left(\frac{r}{n}\right)^2 k\right).
\]
\end{lemma}
\begin{proof}
Our proof is based on an application of
Theorem~3.6 from the Clarkson-Shor framework.
To apply this theorem, we bound
\[
E\left( \sum_{t\in T} \alpha_t^2 \log^2 \alpha_t\right)
\]
by bounding the term,
$\alpha_t^2 \log^2 \alpha_t$,
by
\[
W\left({{|C(t)|} \choose c }\right),
\]
where $W$ is a positive concave function on ${\bf R}^+$ and $c$ is a constant.
Here, for the sake of an upper bound, we take $c=3$ and we define
\[
W(x) = \left(\frac{x^{1/3}}{N}\right)^2 \log^2 \frac{x^{1/3}+N}{N} ,
\]
where $N=n/r$.
Finally, to apply Theorem~3.6 from \cite{cs-arscg-89},
we need to observe that the number of trapezoids in $T$ that have a conflict list size
at most $c$ is proportional to the number
of trapezoids in $T$ that have a conflict list size at least $0$, which is $|T|$.
To see this, note that we can extend the vertical edges of any trapezoid in
$T$ in at most $O(1)$ ways until it hits $i=1,2,3$ other edges of the
random sample, $S$,
at which point we can extend this trapezoid horizontally in $O(1)$
ways until we hit $3$ segments in total.
Therefore,
by Theorem~3.6 from \cite{cs-arscg-89},
\[
E\left( \sum_{t\in T} \alpha_t^2 \log^2 \alpha_t\right)
\]
is
\[
O\left(r+\left(\frac{r}{n}\right)^2 k\right).
\]
\end{proof}
Thus, our refined
trapezoidal decomposition, $T'$, will
have size proportional to $|T|$.
It is still not quite a $(1/r)$-cutting, however, as it is not a
proper triangulation. Indeed, some trapezoids may
have many more than $4$ vertices on their boundaries
(see Figure~\ref{fig: highdegtrap}).
\begin{figure}[hbt]
\centering\includegraphics[width=3in]{highdegtrap}
\caption{Many trapezoids may be adjacent to another trapezoid along its top or bottom edges.}
\label{fig: highdegtrap}
\end{figure}
To refine $T'$ into a proper triangulation, we borrow an idea
from the fractional cascading framework of Chazelle and
Guibas~\cite{cg-fc1ds-86} to first refine $T'$ into
a trapezoidal decomposition such that each trapezoid has $O(1)$
vertices on its boundary, while keeping the total number of trapezoids to be
$O(|T'|)$, which is expected to be
\[
O\left(r+\left(\frac{r}{n}\right)^2 k\right).
\]
By triangulating the interior of each
such trapezoid, we will get a $(1/r)$-cutting whose size is still
$O(|T'|)$.
(See Figure~\ref{fig:cascade}.)
\begin{figure}[hbt]
\centering\includegraphics[width=3in]{cascade.png}
\caption{The cascading of trapezoidal rays.}
\label{fig:cascade}
\end{figure}
Construct the graph-theoretic planar dual $U$ to $T'$, and note that
we can direct the edges of $U$ so as to define four directed-acyclic
graphs, which respectively define the partial orders ``below,''
``above,'' ``left-of,'' and ``right-of'' among the trapezoids.
Without loss of generality,
let us direct $U$ according to the ``below''
relation, perform a topological sort, and process the trapezoids of
$T'$ from top to bottom according to this ordering.
When processing a trapezoid, $t$, we assume inductively that we have
determined the ordered list of vertices $V_t=(v_1,v_2,\ldots,v_j)$ on
$t$'s upper edge, which are bottom vertices of trapezoids above $t$.
To process $t$ we choose every other vertex, $v_{2i}$,
in $V_t$ and extend a vertical segment from $v_{2i}$ to the bottom of
$t$ to split $t$ in two for each such $v_{2i}$.
Doing this for every other vertex in $V_t$, therefore, splits $t$ and
increases the number of trapezoids by $\lfloor |V_t|/2\rfloor$.
We then repeat this computation by considering the new set of
trapezoids according to the ``above'' relation, from bottom to top.
Next, we do a similar computation for the ``left-of'' and
``right-of'' relations (except that now we extend segments parallel
to the top or bottom edges of our trapezoid in a way that partitions
its interior into non-crossing trapezoids).
When we have completed this last scan of the trapezoids, we will have
created a trapezoidal decomposition such that each trapezoid has
$O(1)$ vertices on its edges.
More importantly, we also have the following:
\begin{lemma}
The total number of trapezoids created by the above refinement
process is $O(|T|)$, which has expected value $O(r+(r/n)^2k)$.
\end{lemma}
\begin{proof}
We have already established that
$E(|T|)$ is $O(r+(r/n)^2k)$ and that
$E(|T'|)$ is $O(E(|T|))$.
So we have yet to show that the number of new trapezoids created
during any of our splitting processes is $O(|T'|)$.
We do this by an accounting argument.
Without loss of generality
, consider the processing according to the ``below'' relation.
Assume, for the sake of our analysis, that, at the beginning
of our computation, we give each vertical edge in our trapezoidal
decomposition \$2 and we require every vertical edge at the end of
the process to have at least \$1.
When we extend a vertical ray from an even numbered vertex
$v_{2i}$ at the top of a trapezoid, $t$ we can assume inductively that
the vertical edge above $v_{2i}$ has \$2, as does the vertical edge
directly to the left of this edge (which hits $t$ at vertex
$v_{2i-1}$).
Let us take \$1 from this vertical edge and from the one that hits
$t$ at $v_{2i}$, which leaves \$1 at each of those edges,
and use the \$2 to pay for the new vertical edge that we then extend
through $t$.
Therefore, since the two vertical edges we just took money from will
not be processed again, we can process each trapezoid and pay for
every action, while keeping \$1 for each trapezoid in our refined
trapezoidal decomposition.
Repeating this accounting argument for the ``above,'' ``left-of,''
and ``right-of'' relations completes the proof.
\end{proof}
Given a trapezoidal diagram having $O(1)$ vertices on the boundary of
each trapezoid, and each trapezoid intersecting at most $(n/r)$ edges
of our geometric graph $G$ we can easily triangulate
each trapezoidal face in this diagram to turn it into a
$(1/r)$-cutting with a number of triangles that is proportional to
the number of trapezoids.
(See Figure~\ref{fig:trias}.)
\begin{figure}[hbt]
\centering\includegraphics[width=3in]{triangles.png}
\caption{The triangulation step.}
\label{fig:trias}
\end{figure}
Thus, putting all the pieces together, we get the following.
\begin{theorem}
\label{thm-cutting}
Given a connected geometric graph $G$ having $n$ vertices and $k$ pairwise edge
crossings, one can construct a $(1/r)$-cutting for
the edges of $G$ of expected size $O(r+(r/n)^2k)$ in expected time
$O(ns+(r/n)k)$, for $r\le n/\log^{(s)} n$.
\end{theorem}
Taking $s$ as a constant gives us such a $(1/r)$-cutting
of expected size $O(r+(r/n)^2k)$ in expected time
$O(n+(r/n)k)$, and taking $s=\log^* n$ gives us
a $(1/r)$-cutting
of the same expected size (but with a potentially larger $r$) in
expected time $O(n\log^* n + (r/n)k)$, for any $r\le n$.
Since, in our applications involving restrained geometric graphs,
$k$ is sublinear in $n$ by an iterated logarithmic factor, we will be
taking $s$ to be a constant.
\section{Planarization}
In this section, we describe how to planarize a connected
geometric graph $G$ having $n$ vertices and $k$ edge crossings.
We begin by using the method
of Theorem~\ref{thm-cutting} to construct
a $(1/r)$-cutting, $C$, of the edges of $G$ of expected size $O(r+(r/n)^2k)$
in expected time $O(n+(r/n)k)$, where $r = n/\log^{(c+1)} n$, for a
fixed constant $c\ge 1$.
We then do a depth-first search of $G$, keeping track of the
triangles we cross in $C$ as we go, to compute, for each triangle $t$
in $C$, the set, $C(t)$,
of at most $(n/r)$ edges of $G$ that intersect $t$.
This takes $O(|C|n/r)$ time, which has expectation $O(n+(r/n)k)$.
We then apply Goodrich's separator decomposition algorithm~\cite{g-psppt-95}
to construct an $O(\sqrt{|D|})$-separator decomposition of the
graph-theoretic dual, $D$, to $C$.
Rather than taking this decomposition all the way to the point where
we would have subgraphs of $D$ of constant size, however, we stop
when subgraphs have size $O(\log^2 (n/r))$; hence, have separators of
size $O(\log (n/r))$.
Since $C$ is a triangulation, $D$ has degree $3$; hence, any
vertex separator for $D$ of size $g$ also gives us an edge separator
for $D$ of size at most $3g$.
Moreover, each edge of $D$ corresponds to a triangle edge in $C$,
which in turn crosses at most $(n/r)$ edges of $G$.
For each separator $H$ in our decomposition,
therefore, we can sort the edges of $G$ that cross each boundary of
a triangle in the separator
in time $O((n/r)\log (n/r))$ time.
There are $O(|D|/\log^2 (n/r))$ nodes at this level of the separator
decomposition tree; hence, there are
$O(|D|/\log^2 (n/r))\times O(\log (n/r)) = O(|D|/\log (n/r))$
triangles involved.
Thus, the total time for all these sorts is
$O(|D|(n/r)) = O(n+k)$.
After performing all these sorts of edges on the boundaries
of triangles in our separators, we can imagine that we have used
these boundaries to cut $G$ into $O(|D|/\log (n/r))$ regions
(including each triangle in one of our separators), such
that the edges of $G$ intersecting each region boundary
are given in sorted order.
(See Figure~\ref{fig:regions}.)
\begin{figure}[hbt]
\centering\includegraphics[width=3in]{regions.png}
\caption{Illustrating the regions and their boundary edges.}
\label{fig:regions}
\end{figure}
The total size of each subgraph is $O((n/r)\log^2 (n/r))$.
Moreover, the boundaries of these regions form a planar subdivision.
Thus, we have just subdivided our geometric graph $G$ into
$O(|D|/\log (n/r))$ disjoint geometric graphs.
In other words,
all $k$ edge crossings in $G$ have been isolated
into these small subgraphs.
For each subgraph $G_i$, use Chazelle's algorithm~\cite{c-tsplt-91a} to test
if all the faces of $G_i$ are simple in $O(|G_i|)$ time.
If all the faces of $G_i$ are in fact simple, then $G_i$ clearly
contains no edge crossings.
Thus, we can identify each small subgraph in this partition that
contains an intersection in time $O(|C|(n/r) + |G|)$, which has
expectation $O(n+k)$.
Clearly,
there are at most $k$ such subgraphs that contain edge crossings.
We complete our planarization algorithm, therefore, by running the
Bentley-Ottmann algorithm~\cite{bo-arcgi-79}
for each subgraph of $G$ that is identified
as having at least one edge crossing.
The time for each such invocation of the Bentley-Ottmann algorithm is
$O((n/r)\log^3 (n/r) + k'\log (n/r))$, where $k'\ge 1$ is the
number of edge crossings found.
Summing this over $k$ regions implies that the total time needed to
complete the planarization of $G$
is $O(k(n/r)\log^3 (n/r))$.
Substituting for $r$, we see that this time is
$O(k \log^{(c+1)} n \log^3 \log^{(c+1)} n)$,
which is $O(k \log^{(c)} n )$.
Therefore, we have the following:
\begin{theorem}
\label{thm-planarization}
Suppose one is
given a connected geometric graph $G$ with $n$ vertices and
$k$ edge crossings,
together with a $(1/r)$-cutting of the edges of $G$ of size
$O(r+(r/n)^2k)$, for $r=n/\log^{(c+1)} n$.
Then one can construct a planarization of $G$ (and the
trapezoidal decomposition of the arrangement of $G$'s edges), in
time $O(n+k\log^{(c)} n)$.
\end{theorem}
Combining this result with Theorem~\ref{thm-cutting}, we
get the following corollary.
\begin{corollary}
\label{cor-planarization}
Given a connected geometric graph $G$ having $n$ vertices and $k$ pairwise edge
crossings,
one can construct a planarization of $G$ in
expected time $O(n+k\log^{(c)} n)$.
\end{corollary}
\section{Applications}
In this section, we provide a number of applications of the above algorithms.
\subsection{Separator Decompositions of Restrained Geometric Graphs}
The algorithms in this section are based on the use of separators.
As mentioned above,
the separator-decomposition algorithm of Goodrich~\cite{g-psppt-95}
applies only to planar graphs.
Nevertheless,
given the tool of geometric graph planarization,
we can adapt Goodrich's result to restrained geometric graphs in a fairly
straightforward manner.
Given a restrained
geometric graph $G$, we planarize it using the algorithm above, creating the
planar graph $G'$.
As observed above,
$G'$ has total size $O(n)$.
Thus, we can use the result of Goodrich~\cite{g-psppt-95}
to compute a recursive
$O(\sqrt{n})$-separator decomposition of $G'$ in $O(n)$ time.
We convert this separator decomposition into a
$O(\sqrt{n})$-separator for $G$ by the following transformation.
For each node $v$ in a separator $W$ of $G'$
at a node $w$ in the separator decomposition tree $B$,
we do the following:
\begin{itemize}
\item
If $v$ is also a vertex in $G$, then we
add $v$ to the separator for $G$ corresponding to $w$, provided $v$ is not
already a member of a separator associated with an ancestor of $w$.
\item
If $v$ is an intersection point in $G'$, between edges $(a,b)$ and $(c,d)$
in $G$, then we add each of $a$, $b$, $c$, and $d$
to the separator for $G$ corresponding to $w$, provided it is not
already a member of a separator associated with an ancestor of $w$.
\end{itemize}
This gives us the following:
\begin{theorem}
\label{thm-deterministic}
Suppose we are given an $n$-vertex geometric graph $G$
and its planarization, $G'$, which is of size $O(n)$.
Then we can construct a recursive $O(\sqrt{n})$-separator decomposition of
$G$ in $O(n)$ time, for $\delta=2/3$.
\end{theorem}
\subsection{Single-Source Shortest Paths and Voronoi Diagrams}
Given an $n$-vertex
bounded-degree graph $G$ and a recursive $O(\sqrt{n})$-separator
decomposition for $G$,
Henzinger {\it et al.}~\cite{hkrs-fspap-97} show that one can compute shortest
paths from a single source $s$ in $G$ to all other vertices in $G$ in $O(n)$
time.
Using the separator decomposition algorithms presented above, then,
we can show that
their algorithm applies to restrained geometric graphs, even ones that do not have
bounded degree, by a simple transformation that replaces high-degree
vertices with bounded-degree trees of zero-weight edges.
Suppose we are given $K$ distinguished vertices in an $n$-vertex
restrained geometric graph $G$ and we
wish to construct
the \emph{Voronoi diagram} of $G$, which is a labeling of each
vertex $v$ of $G$ with the name of the distinguished vertex closest
to $v$.
As before, by replacing high degree vertices with bounded-degree
trees of zero-weight edges we can assume without loss of generality
that $G$ has constant degree.
In this case, we construct a recursive $O(\sqrt{n})$-separator
decomposition of $G$ using one of the algorithms of the previous
section.
Let $B$ be the recursion tree and let us label each vertex $v$ in $G$
with the internal node $w$ in $B$ where $v$ is added to the separator
or with the leaf $w$ in $B$ corresponding to a set containing $v$ where
we stopped the recursion (because the set's size was below our
stopping threshold).
Given this labeling, we can trace out the subtree $B'$ of $B$
that consists of the union of paths from the root of $B$ to the
distinguished nodes in $G$ in $O(n)$ time.
Let us now assign each edge in $B'$ to have weight $0$ and let us add
$B'$ to $G$ to create a larger graph $G'$.
Note that if we add each internal node $v$ in $B'$ to the separator
associated with node $v$ in $B$, then we get a recursive
$O(\sqrt{n})$-separator decomposition for $G'$, for each separator in
the original decomposition increases by at most one vertex.
Thus, we can apply the algorithm of Henzinger {\it et al.}~\cite{hkrs-fspap-97}
to compute the shortest paths in $G'$ from the root of $B'$ to every
other vertex in $G'$ in $O(n)$ time.
Moreover, since the edges of $G'$ corresponding to edges of $B'$ have
weight $0$, this shortest path computation will give us the Voronoi
diagram for $G$.
Therefore, we have the following:
\begin{theorem}
\label{thm-vd}
Given a connected $n$-vertex restrained graph $G$, together with
its planarization,
one can compute shortest paths from any vertex
$s$ or the Voronoi diagram defined by any set of $K$ vertices in $G$
in $O(n)$ time.
\end{theorem}
Incidentally, the above approach also implies a linear-time Voronoi diagram
construction algorithm for planar graphs, which was not previously known.
\section{Conclusions and Future Work}
We have provided linear-time algorithms for a number of problems
on connected restrained geometric graphs, which includes real-world
road networks. Our results allow for linear-time trapezoidalization, triangulation, and planarization of geometric graphs except for the very narrow range of the number of crossings for which neither our algorithm nor the previous $O(n\log^* n+k)$ algorithm is linear.
In addition, our methods imply linear-time algorithms for other problems on
such graphs as well. For example, one can use our algorithm to planarize a
restrained non-simple polygon and then construct its convex hull in linear
time by computing the convex
hull of the outer face of our planarization (e.g., by an algorithm
from~\cite{gy-fchsp-83,l-fchsp-83}).
There are a number of interesting open problems and future research directions
raised by this paper, including:
\begin{itemize}
\item
Can one close the $\log^{(c)} n$ gap on values of $k$
that admit optimal
solutions to Chazelle's
open problem of
computing a trapezoidal decomposition of an $n$-vertex non-simple polygon in
$O(n+k)$ time, where $k$ is the number of its edge crossings?
\item Can we planarize restrained geometric graphs deterministically in linear time? Such a result would allow us to apply separator-based divide and conquer techniques for minimum spanning trees~\cite{EppGalIta-JCSS-96} to construct them in linear time for this family of graphs. Known linear-time minimum spanning tree algorithms for arbitrary graphs require randomization~\cite{kkt-rltamst-95}, and known deterministic algorithms for this problem are superlinear~\cite{c-amsta-00}, although deterministic linear-time algorithms are known for planar graphs and minor-closed graph families~\cite{ct-fmst-76,Epp-HCG-00,m-tltam-04}.
\end{itemize}
\subsection*{Acknowledgment}
We would like to thank Bernard Chazelle for several helpful
discussions regarding possible approaches to solving his open problem
involving non-simple polygons.
This research was supported in part by the National Science Foundation, under
grants 0724806, 0713046, and 0830403,
and the Office of Naval Research, under
MURI Award number N00014-08-1-1015.
A preliminary version of this paper appeared in the ACM-SIAM
Symposium on Discrete Algorithms (SODA) as~\cite{egs-ltagg-09}.
\ifFull
\bibliographystyle{abbrv}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 3,353 |
Q: For named vectors and matrices, does [[ ever use partial matching without passing the exact=FALSE argument? I've recently read this section of the R Language Definition and become very confused. It says:
For vectors and matrices the [[ forms are rarely used, although they have some slight semantic differences from the [ form (e.g. it drops any names or dimnames attribute, and that partial matching is used for character indices).
Am I misreading it? It seems to say that [[ allows partial matching (presumably by default) for named vectors and matrices. However, I know this to be obviously false:
> lett<-setNames(sample(26),paste0(letters,letters,letters))
> lett[["aaa"]]#No partial match
[1] 23
> lett[["a"]]#Error
Error in lett[["a"]] : subscript out of bounds
so what did the language definition mean? It could be talking about the exact=FALSE flag that you can set, but this seems a very strange way to do that. In fact, a later section makes the following claim, which my above code disproves, so I'm even more confused:
For [[ and $ partial matching is used if exact matching fails, so x$aa will match x$aabb if x does not contain a component named "aa" and "aabb" is the only name which has prefix "aa". For [[, partial matching can be controlled via the exact argument which defaults to NA indicating that partial matching is allowed, but should result in a warning when it occurs.
Is the language definition simply out of date?
A: Actually, I think the language definition is - at least partially - indeed out of date. The help page of help("[[") regarding the exact argument states
Controls possible partial matching of [[ when extracting by a character vector [...]. The default is no partial matching. Value NA allows partial matching but issues a warning when it occurs. Value FALSE allows partial matching without any warning.
Usage supports this claim:
x[[i, exact = TRUE]]
x[[i, j, ..., exact = TRUE]]
The following code proves these defaults, as well.
set.seed(1)
lsub <- letters[1:3]
lett <- setNames(lapply(sample(3), c), paste0(lsub, lsub, lsub))
lett
#> $aaa
#> [1] 1
#>
#> $bbb
#> [1] 3
#>
#> $ccc
#> [1] 2
# partial matching
lett$a
#> [1] 1
lett[["aa", exact = FALSE]]
#> [1] 1
# no partial matching
lett[["aa"]]
#> NULL
# partial matching with warning
lett[["aa", exact = NA]]
#> Warning in lett[["aa", exact = NA]]: partial match of 'aa' to 'aaa'
#> [1] 1
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 5,970 |
Eventi
25 aprile - Papa Leone III durante una processione viene catturato e ferito dai Romani
29 novembre - Papa Leone III, aiutato da Carlo Magno, ritorna a Roma.
Nati
Morti
Calendario
Altri progetti
099 | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 4,692 |
Q: Jest test write on textarea input I am trying to write some mock string to a textarea using Jest and then press Enter but it does not work for whatever reason. My code so far:
test('Should add a message', () => {
const element = wrapper.find('textarea');
element.instance().value = 'abc';
element.simulate('keypress', { key: 'Enter' });
const newWrapper = wrapper.find('user');
expect(newWrapper.length).toBe(1);
});
My component:
<textarea
onKeyUp={sendMessage}
placeholder='Type your message here and press enter to send...'
cols='30'
rows='5'
></textarea>
Just to be clear the textarea is definitely there as the following test passes:
test('Should have a textarea', () => {
const element = wrapper.find('textarea');
expect(element.length).toBe(1);
});
A: In your case, what you are looking for is 'keyUp', not 'keypress'. And instead of key, use keyCode. The keyCode for Enter is 13.
Like this:
element.simulate('keyUp', { keyCode: 13 });
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 4,333 |
<?php
/**
*
*
* @package symfony
* @subpackage widget
* @author Fabien Potencier <fabien.potencier@symfony-project.com>
* @version SVN: $Id: sfWidgetFormSchemaFormatterList.class.php 5995 2007-11-13 15:50:03Z fabien $
*/
class sfWidgetFormSchemaFormatterList extends sfWidgetFormSchemaFormatter {
protected
$rowFormat = "<li>\n %error%%label%\n %field%%help%\n%hidden_fields%</li>\n",
$errorRowFormat = "<li>\n%errors%</li>\n",
$helpFormat = '<br />%help%',
$decoratorFormat = "<ul>\n %content%</ul>";
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 7,566 |
Calicut Heroes got the better of Ahmedabad Defenders 4-1(15-14,11-15,15-11,15-9, 15-8) on the 12th day of the ongoing Pro Volleyball in Kochi to continue their winning run. Jerome Vinith was the top scorer for the Heroes with 17 points while Novica Bjelica scored the most for Ahmedabad.
It was an interesting start to the match as Ahmedabad nosed ahead with a 5-3 lead owing to two errors from Calicut and three points scored by Novica Bjelica and Victor Sysoev. But Calicut showed why they were table-toppers pulling back things together and taking the lead at 8-6 at the first Technical Time Out (TTO). At 12-10, Calicut looked on their way to win the set but Ahmedabad called for a Super Point and converted on Calicut's serve to level the equation. The last few points were hard fought and it looked like anyone's set till 14-14. It was finally a spike by Paul Lotman which won the set for Calicut 15-14.
In the second set, Ahmedabad yet again went into an early lead but as it was the case in the previous set Calicut came back to level it 5-5. Ahmedabad was determined not to fall behind and went into the TTO an 8-6 lead. Calicut called for a Super Point at 6-9 and converted to close the gap to 8-9. Ahmedabad extended the lead even further as they went to 13-10. At 14-10, Ahmedabad had their first set point but could only seal the deal at 15-11 when Vipul Kumar's spike went out. The match was now level at 1-1.
Winning the second set seemed to have given Ahmedabad the momentum as they led 5-2 displaying good attack and defence. Calicut was in no mood to let Ahmedabad run away with the lead and levelled the score at 7-7. Ahmedabad somehow entered the TTO with a point advantage. After the TTO it was a neck-to-neck battle as no team was able to widen the gap. Ahmedabad called for a Super Point at 10-11 but failed to convert as Lotman's spike was too hot to handle. Calicut won the set as Ajith Lal closed it 15-11 with an inch perfect spike. Calicut led 2-1 in the match.
As the fan support increased, Calicut raised their game taking an 8-1 lead with Lotman acing a Super Serve at 6-1. Ahmedabad seemed to have lost their way. The conversion of a Super Point by GR Vaishnav also seemed futile as the gap of points was too much to handle. Calicut won the set 15-9 that ensured an unbeaten end to the league stage for Calicut.
The last set wasn't much of a competition as Calicut won the same by a score line of 15-8 winning their Super Point and acing a Super Serve both converted by skipper Jerome Vinith. Though, Ahmedabad won their Super Point at 6-10 but they never looked like being in the game. Calicut won the match 4-1. | {
"redpajama_set_name": "RedPajamaC4"
} | 3,204 |
Save money and get free rental delivery!
Check out our packages and see what gear suits your style.
Hauling your skis through the airport and then into a cab or shuttle can be a pain. Plus there's the extra baggage fees airlines charge. Avoid these hassles by renting online. When you arrive, your skis will already be there!
Having the right gear can make a huge difference in how much you enjoy the slopes. When you rent from the SkiBig3 Adventure Hub, you'll get local expert advice from our ski technicians that will equip you for our Rocky Mountain weather!
You could rent your gear once you arrive in Banff, but you may not be able to find exactly what you want. By reserving online, you'll to get what you want and you'll save not only time, but also money!
Select from our extensive fleet of skis and boards.
* Prices are subject to 5% GST, all prices are per day.
* Damage protection does not cover theft or loss.
** Credit card imprint is mandatory, Amex, M/C, and Visa accepted Ask about hotel delivery and pick-up options.
We hope you won't need to cancel your order, but we understand sometimes things just don't go to plan. We highly suggest purchasing travel insurance to protect yourself against any unforseen events. If cancellation occurs 7 days or more prior to the first day of use, items will be fully refundable less a $20 administration fee per person, plus taxes. If cancellation occurs within 7 days or more prior to the first day of usr, items and services are non refundable. | {
"redpajama_set_name": "RedPajamaC4"
} | 3,777 |
Hawaii News | Top News
Washington fire sparked by fatal plane crash burns buildings
By Martha Bellisle
SEATTLE » A wildfire ignited by a deadly plane crash in a rugged area of northern Washington state chased hundreds of people from their homes Friday and burned 10 to 12 buildings, including residences, authorities said.
The Federal Emergency Management Agency planned to send funding to help combat the blaze charring remote, dry land near Oroville, a small town close to the Canadian border. More than 400 people were evacuated, and 660 homes were threatened as winds picked up, officials said.
The 4.7-square-mile fire also posed a risk to roads, bridges, power and gas lines, and several private businesses in a state struggling with drought, which has made the parched terrain combustible.
The Obama administration said wildfires have been so bad this season that the Forest Service will exhaust its firefighting budget next week and will again have to tap into other programs for more money.
Tory King, a customer service worker at the Princess Center grocery store in downtown Oroville, said smoke has filled the town.
"All we can see here is smoke," she said.
A Cessna 182 crashed and sparked the fire that spread to the Canadian border. A body was found in the aircraft Thursday after crews responding to the blaze discovered the wreckage.
Local authorities hoped to get close enough Friday to see if there were any other victims, Okanogan County Sheriff Frank Rogers said. Officials with the Federal Aviation Administration also went to the site to try to identify the aircraft and investigate the crash.
"The plane was destroyed in the fire, so there were no numbers left to get a positive ID on the aircraft," Rogers said. "It was so hot that we couldn't get an identification."
Officials expected high winds in the remote region to fan the flames, said Josie Williams, spokeswoman for the Washington Incident Management Team No. 2. Most of the state is under a red-flag warning, meaning the temperatures are high and the landscape is crispy dry, Department of Natural Resources spokeswoman Janet Pierce said.
An evacuation shelter has been set up at Oroville High School.
Man sentenced to life in girlfriend's shooting death
Hopes high for karate's inclusion for 2020 Tokyo Olympics
More Hawaii News
Loan drop-off pulls down Territorial Savings' profit
UH graduate assistants' fight to unionize reaches high court | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 1,803 |
https://ucsb-cmptgcs-1a.github.io
Jekyll-based website for course materials for CCS CS 1A shared across
instructors.
To test locally:
* One time setup:
* `git clone` the repo
* Install rvm (the Ruby version manager)
* Run `./setup.sh` to install correct ruby version, bundler version, and bundle the gems
* From then on, to test the site locally:
* Run `./jekyll.sh
* Point browser to localhost:4000
| {
"redpajama_set_name": "RedPajamaGithub"
} | 9,677 |
{"url":"https:\/\/www.advancedliving.com\/magnesium-breakthrough\/?amp","text":"Supplements Vitamins Magnesium Breakthrough: Review the BiOptimizers Supplement\n\n# Magnesium Breakthrough: Review the BiOptimizers Supplement\n\nMagnesium Breakthrough by BiOptimizers is a magnesium supplement with 7 different types including chelate, citrate, bisglycinate, malate, aspartate, taurate and orotate ingredients in one formula.\n\nMagnesium Breakthrough is a nutritional supplement that claims to rapidly transform stress, hormones, and performance.\n\nBy taking two capsules of Magnesium Breakthrough daily, you can purportedly use all 7 forms of magnesium to upgrade multiple areas of performance. It\u2019s like a superpowered magnesium supplement.\n\nDoes Magnesium Breakthrough really work? What\u2019s the difference between Magnesium Breakthrough and other magnesium supplements? Find out everything you need to know about Magnesium Breakthrough today in our review.\n\n## What is Magnesium Breakthrough?\n\nMagnesium Breakthrough is a magnesium supplement sold online through MagBreakthrough.com\n\nThe supplement claims to solve magnesium deficiency. According to the Magnesium Breakthrough website, magnesium deficiency leads to severe health problems. It increases stress on your body, and stress is linked to heart disease, inflammation, obesity, mental illness, and other issues.\n\nThe makers of Magnesium Breakthrough claim all of these major health issues are linked to deficiency in a single mineral: magnesium.\n\nYou may think you take a good dose of magnesium per day, However, even if you eat plenty of magnesium-rich foods, you may not be getting enough magnesium.\n\nMagnesium Breakthrough aims to solve that problem by giving you all 7 forms of magnesium in a single, convenient capsule.\n\n## The Story Behind Magnesium Breakthrough\n\nMagnesium Breakthrough is promoted online by a man named Wade Lightheart, who claims he nearly died from stress two years ago \u2013 so he started taking Magnesium Breakthrough to help.\n\nWade claims he was one stress level away from being hospitalized when he sat down with fellow BiOptimizers co-founder Matt Gallant. Matt hooked Wade to a brain scanning machine and found Wade had 25% of the brain electricity of a 75-year old man.\n\nWade was so stressed out that it was affecting his brain energy, physical energy, and other aspects of his life.\n\nThat\u2019s when Matt recommended taking more magnesium, which he describes as \u201cthe superhero nutrient.\u201d\n\nAll of us experience stress daily. However, some of us manage stress better than others. In many cases, people with magnesium deficiency manage stress more poorly than people who get enough magnesium.\n\nHere\u2019s how the Magnesium Breakthrough website explains it:\n\n\u201cSince this nutrient is responsible for 300-600 different biochemical reactions in the body (including metabolism)\u2026 when your levels are low, you struggle with sleep, energy, metabolism, pain and more.\u201d\n\nThe sales page claims 80% of Americans are deficient in magnesium and that 99% of Americans don\u2019t get the \u201coptimal dose\u201d of magnesium per day.\u201d\n\nWhen you don\u2019t get enough magnesium, you get stuck in a vicious cycle. Your body gets more stressed, which worsens your physical and mental health. As your physical and mental health gets worse, you become more stressed. This cycle repeats, eventually leading to a range of issues.\n\nIn fact, the makers of Magnesium Breakthrough claim magnesium deficiency and stress \u201ccan disrupt all your body\u2019s processes,\u201d increasing your risk of anxiety, depression, digestive problems, headaches, sleep problems, heart disease, and more.\n\nTo solve this issue, Wade experimented with different doses of magnesium. He started taking 5g of magnesium per day divided into four doses.\n\nWithin a couple of months, this high magnesium dose fixed Wade\u2019s stress and burnout issues.\n\nAfter emphasizing the importance of magnesium, the makers of Magnesium Breakthrough propose their supplement as a solution (although it\u2019s important to note Magnesium Breakthrough contains 500mg of total magnesium instead of 5,000mg).\n\n## How Does Magnesium Breakthrough Work?\n\nYou can find plenty of magnesium supplements online and in health stores. Most multivitamins contain magnesium, and many of us get enough magnesium through our diets. So what makes Magnesium Breakthrough different? Why is Magnesium Breakthrough the best magnesium supplement on the market?\n\nMagnesium Breakthrough purportedly \u201cdefeats stress at a cellular level,\u201d among other benefits. Magnesium is also the fourth most abundant mineral in the body and is needed for everything from muscle development to blood circulation.\n\nSome of the advertised benefits of magnesium include:\n\n\u2022 Maintains normal muscle and nerve function\n\u2022 Builds strong bones\n\u2022 Maintains normal hearth rhythm\n\u2022 Lowers cortisol levels\n\nOf course, all ordinary magnesium supplements provide similar benefits. Your body needs magnesium for multiple crucial body processes.\n\nWhat makes Magnesium Breakthrough from other supplements is that it contains 7 forms of magnesium, targeting stress and other issues from multiple angles.\n\nMost magnesium supplements contain just one type of magnesium. According to the makers of Magnesium Breakthrough, this is a problem:\n\n\u201cOne of the biggest misconceptions about magnesium is that you just \u201cneed more\u201d of it and you\u2019ll be healthy and optimized. But the TRUTH is, there are many different types of magnesium \u2014 and each plays a critical role in different functions in your body. Most \u201chealthy\u201d people only get 1-2 forms at best (much of the population is deficient in all forms) \u2014 but when you get all 7 major forms of magnesium, that\u2019s when the magic happens.\u201d\n\nBy giving you all 7 types of magnesium, Magnesium Breakthrough purportedly leads to powerful benefits.\n\n## Types of Magnesium in Magnesium Breakthrough\n\nThe average person gets 1 to 2 forms of magnesium in their diet. However, there are 7 forms of magnesium overall. Magnesium Breakthrough contains all 7 forms of magnesium, each of which is linked with different benefits.\n\nHere are the benefits of the types of magnesium in Magnesium Breakthrough, according to the Magnesium Breakthrough sales page:\n\nMagnesium Chelate: This form of magnesium is important for building muscle, recovering from exercise, and supporting overall health.\n\nMagnesium Citrate: This form helps with the effects of obesity, with one study finding it supported arterial stiffness in overweight individuals.\n\nMagnesium Bisglycinate: This form treats excess stomach acid, which is linked with upset stomach, heartburn, and acid indigestion.\n\nMagnesium Malate: Magnesium malate is one of the most bioavailable forms of magnesium. It could help with migraines, chronic pain, and depression. This is the type of magnesium in fruits that gives them a tart taste.\n\nMagnesium Aspartate: This form of magnesium support the connection between your brain and muscles, your cardiac rhythms, and the overall acid-alkaline balance in your body, among other effects.\n\nMagnesium Taurate: This type of magnesium is best for your heart, according to Magnesium Breakthrough.\n\nMagnesium Orotate: Magnesium orotate is helpful for the heart and particularly beneficial for athletes.\n\nBy getting the optimal dose of all of these forms of magnesium, you can upgrade \u201cvirtually every function in your body,\u201d according to the Magnesium Breakthrough website.\n\n## Magnesium Breakthrough Ingredients\n\nMagnesium Breakthrough lists its full ingredient label online upfront, making it easy to compare the supplement with other magnesium formulas.\n\nAlthough the supplement makes a big deal out of its magnesium dosage, Magnesium Breakthrough contains just 500mg of magnesium (120% of your Daily Value) in each 2 capsule serving. Even if you\u2019re getting all 7 types of magnesium, it\u2019s only slightly higher than the recommended daily dose of magnesium.\n\nOther ingredients in Magnesium Breakthrough include manganese (1mg, or 43% of your Daily Value) and vitamin B6 (2mg, or 118% DV).\n\nThe company tests Magnesium Breakthrough to verify that it\u2019s free of mercury, lead, arsenic, and fluoride. This sounds obvious \u2013 but mercury is a problem with magnesium supplements. Some shady magnesium supplements have strong levels of mercury, and it\u2019s something to check before buying a magnesium supplement.\n\n## Scientific Evidence for Magnesium Breakthrough\n\nThe makers of Magnesium Breakthrough claim 80% of Americans are deficient in magnesium, with an additional 99% of Americans lacking the \u201coptimal dose\u201d of magnesium.\n\nExperts recommend that adults get 250 to 350mg of magnesium per day, although women who are lactating need as much as 400mg of magnesium per day.\n\nMost studies suggest approximately half of Americans are deficient in magnesium, which is less than the number mentioned on the Magnesium Breakthrough sales page.\n\nMany adults get enough magnesium through food sources. Pumpkin seeds, chia seeds, almond, spinach, cashews, peanuts, and cereal, for example, are all rich with magnesium. Other common sources of magnesium include soy milk, black beans, potatoes, rice, yogurt, oatmeal, bananas, salmon, milk, raisings, avocado, and beef, among other sources.\n\nThe most common magnesium supplements use magnesium oxide, magnesium citrate, and magnesium chloride. Because of FDA requirements, manufacturers only need to list the total amount of elemental magnesium in a supplement \u2013 not the breakdown of different types of magnesium.\n\nStudies show that forms of magnesium work in different ways.\n\nSome forms of magnesium dissolve well in liquid, for example. Others pass through your stomach acid more easily, allowing more magnesium to enter your gut.\n\nSome studies have found that magnesium aspartate, magnesium citrate, magnesium lactate, and magnesium chloride are more bioavailable than magnesium oxide and magnesium sulfate.\n\nInstead of summarizing each benefit of magnesium, we\u2019ll link to this 2017 review study published in Scientifica. That study reviewed evidence on magnesium to analyze its importance in clinical healthcare. Researchers concluded that magnesium supplementation \u201cappears to be a safe, useful, and well-documented therapy for several medical conditions,\u201d including everything from headaches to diabetes to premenstrual syndrome, among other conditions.\n\nOne study found magnesium supplementation could lower the risk of stroke by 12%. Other studies have linked magnesium deficiency to cardiovascular disease, osteoporosis, and other issues.\n\nOverall, there\u2019s plenty of scientific evidence supporting the use of magnesium as a supplement, and there\u2019s evidence showing different forms of magnesium work in different ways. Although Magnesium Breakthrough has not completed any clinical trials or scientific studies on its own, the supplement seems to work as advertised to support its advertised benefits.\n\n## Magnesium Breakthrough Pricing\n\nMagnesium Breakthrough is priced at $40 per bottle, although the price drops as low as$30 or $33 per bottle when ordering multiple units. Here\u2019s how pricing breaks down: 1 Bottle:$39.95 + $7.75 Shipping 3 Bottles:$97 + Free US Shipping\n\n6 Bottles: \\$182 + Free US & Canada Shipping\n\nAs part of a recent promotion, BiOptimizers bundles 1 bottle of MassZymes with each 3 bottle package and 3 bottles of MassZymes with each 6 bottle package.\n\n## Magnesium Breakthrough Refund Policy\n\nMagnesium Breakthrough comes with a 365-day refund policy.\n\nYou can request a complete refund on your purchase within 365 days of buying it.\n\nBiOptimizers is a nutritional supplement company founded in 2004. The company offers a range of supplements targeting different health and wellness goals, including Masszymes, P3-OM, and Kapex, among others.\n\nBiOptimizers was founded by Wade Lightheart (President) and Matt Gallant (Chief Executive Officer).\n\nYou can contact BiOptimizers via the following:\n\nEmail Form: https:\/\/bioptimizers.com\/legal\/contact.php\n\nPhone: 1-800-719-2467\n\nMailing Address: 5470 Kietzke Lane, Suite 300, Reno, NV 89511\n\n## Final Word\n\nMagnesium Breakthrough is a magnesium supplement that contains 7 different types of magnesium. Because it contains all 7 types of magnesium, Magnesium Breakthrough claims to be superior to other magnesium supplements.\n\nOverall, Magnesium Breakthrough could support various health and wellness goals as advertised online, although it\u2019s unclear how much more effective different types of magnesium are compared to, say, a single source of magnesium.\n\nFortunately, BiOptimizers backs up its claims with a 365-day refund policy, giving you one full year to request a refund if you\u2019re unsatisfied with Magnesium Breakthrough for any reason.","date":"2021-04-23 02:18:47","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\": 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\": 0.3420054018497467, \"perplexity\": 8909.066698157982}, \"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\/1618039626288.96\/warc\/CC-MAIN-20210423011010-20210423041010-00303.warc.gz\"}"} | null | null |
Q: How to load a html to a variable in next.config.js I use standard nextron (Next + electron.js) to jump start my project (https://github.com/saltyshiomix/nextron)
In one particular case, I need to read a file template.html to variable (String). so I think I'll use fs.readFileSync()
When I use import fs from "fs", the compiler complains that fs is not available. (refer to this Module not found: Can't resolve 'fs' in Next.js application)
The accepted answer on that thread, recommends me to update the next.config.js file into:
/// I need this to load image from static files
const withImages = require('next-images')
module.exports = withImages({
webpack: (config, { isServer }) => {
// Fixes npm packages that depend on `fs` module
if (!isServer) {
config.node = {
fs: 'empty'
}
}
return config
}
})
But the solution doesn't work for me.
import fs from 'fs'
const generatePDF = () => {
let content = fs.readFileSync("./template.html")
}
What's wrong here?
A: I think the problem is that you are trying to use fs module (which is node.js native module) on the client side. The webpack change you've added tells webpack to ignore fs module for the client (or replace it with the stub), but you are still trying to use it.
One way to workaround this is to render your component on the server side only. Please read this documentation - https://nextjs.org/docs/advanced-features/react-18/server-components
Honestly, generally it is bad idea to try to read files/templates on the client side.
But if you really need to get that html on the client, I recommend to create an API which will read html on the server and return as a string. Then you can insert it into the DOM as unsafe html. This will not require massing around with native node modules.
Hope this helps.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 3,595 |
Policies & Donations
The Old Westbury Library Special Collections & Archives supports the mission and guiding principles of SUNY Old Westbury to foster academic excellence and stimulate a passion and commitment for learning by creating and disseminating knowledge, advancing student learning, and supporting research and teaching.
The Library Special Collections & Archives selectively acquire and preserve collections that document Old Westbury College including primary and secondary source materials dating from the founding of the college through the present. In accordance with archival principles and standards, we preserve, organize, and create access to the materials collected.
The Special Collections & Archives department offers reference services to the SUNY Old Westbury community, including administrators, faculty, staff, students, and alumni, as well as the general public.
The purpose of this document is to outline the criteria used by the Special Collections & Archives for the selection, retention, preservation and promotion of material in the college's archives and special collections. The document will be used as a tool to aid decision making in order to meet the archive's responsibilities to the University and to the learning and research community at large. The strategy outlining what will be retained and preserved will improve the archive's ability to further develop its collections.
The Special Collections & Archives will:
Acquire and preserve records of enduring value produced by University offices, faculty, students and alumni. Including Graduate student theses and faculty publications.
Arrange and describe those records and make them accessible in accordance with established archival standards
Serve as an information resource center to assist the University community with research projects.
Encourage and assist the University community to recognize and make use of its institutional heritage via exhibits, presentations, and published resources.
OWLibrarian Twitter feed
Tweets by @OWLibrarian
Next: Policies & Donations >>
Last Updated: Aug 11, 2020 6:11 PM
URL: https://libguides.oldwestbury.edu/specialcollections | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 9,587 |
Tyranosaur Brex ? Khan Calls for Rebrexit - Prime Minister Boris Johnson ? Well Boris the 'Russian' used to be London Mayor :)
Category: Jand
Brexit extinction ? Tyranosaurus Brex ?
London Mayor Khan calls for New Brexit Vote as Theresa May's Shaky Political Stance belies her famous Curtsies
Fmr FM Boris Eyes Downing Street
Boris Johnson, Eyes on Downing Street, Sets Political Fires
Theresa May 'irritated' by leadership speculation
Boris Johnson 'digging own political grave' says Guto Harri
Boris Johnson Quotes
My chances of being PM are about as good as the chances of finding Elvis on Mars, or my being reincarnated as an olive.
My friends, as I have discovered myself, there are no disasters, only opportunities. And, indeed, opportunities for fresh disasters.
The dreadful truth is that when people come to see their MP they have run out of better ideas.
www.tweeterest.co.uk
Prime Minister Khan ? Well Boris the 'Russian' used to be London Mayor :)
The question for Boris Johnson — former mayor of London, former British foreign secretary and current potential British prime minister — was simple:
What's the worst mistake you've ever made?
There were many possibilities to choose from. But Mr. Johnson looked at his interlocutor, Arthur C. Brooks, the institute's president, and developed the glint in his eye that usually means he is about to deploy a well-rehearsed bluster-and-deflect response.
"My strategy is to litter my career with so many decoy mistakes, nobody knows which one to attack," Mr. Johnson declared. "In the last few minutes I've probably said something that the British media will say is absolutely outrageous, though I don't know what it is."
What Mr. Johnson did not mention was the cloud of intrigue, both personal (he is about to get a divorce) and political (he is probably plotting against Prime Minister Theresa May), wafting around him as he made his way across the Atlantic.
Articulate, charismatic and virtually unembarrassable, Mr. Johnson is one of the most popular leaders in a Conservative Party riven by internal dissent — and one of the few British politicians who is instantly recognizable to a foreign audience.
Along with Nigel Farage — the deeply anti-Europe U.K. Independence Party politician who used to be seen as a bit of a joke in Britain but has been touted by President Trump as someone whom "many people" believed should be ambassador to the United States — Mr. Johnson is emerging as the sort of leader Mr. Trump likes.
During his visit to Britain this summer, the president declared that Mr. Johnson would "make a great prime minister" because, he said, "he's got what it takes."
And Mr. Johnson, who once called Mr. Trump "stupefyingly ignorant," "clearly out of his mind" and unfit to be president, has lately taken to praising him back.
"I have become more and more convinced that there is method to his madness," he was quoted as having said at a private meeting of Conservatives in June.
At home, Mr. Johnson is seen as a deeply ambitious opportunist who masks his seriousness of purpose with a well-polished air of befuddled dishevelment and humorous nonchalance.
Like many Trumpian Republicans, Mr. Johnson has lately been tacking right, employing (in his case) an increasingly populist tone on issues like immigration, multiculturalism and Brexit, as the difficult process of Britain's extrication from the European Union is called.
Moderate Conservatives regard him as stealthy and dangerous.
"The cheeky chap of 'Have I Got News for You?' has morphed into a snarling populist," the Conservative commentator Matthew d'Ancona wrote recently, referring to a satirical game show that Mr. Johnson occasionally appeared on earlier in his career. "We need to approach his ambitions with deadly seriousness."
None of that was mentioned in Washington, where Mr. Johnson, 54, was in town to accept this year's Irving Kristol Award, which honors people who have made "exceptional intellectual and practical contributions to improve government policy, social welfare, or political understanding." Previous recipients include Benjamin Netanyahu and Paul Ryan.
Answering questions onstage from Mr. Brooks — sample: "Tell us, what does the special relationship mean, in your view?" — he discussed Russia, Europe, Winston Churchill, the Roman Empire and how the best way to promote unity in a Britain divided by discord over Brexit would have been for England to beat France in the World Cup.
Discussing his political evolution, Mr. Johnson described how his encounters with "bourgeois affluent hypocritical left-wing students" at Oxford University proved so unpleasant that he underwent a political conversion, virtually on the spot.
Mr. Johnson joking with President Trump as Prime Minister Theresa May walked past at a meeting in Brussels in May. This summer, Mr. Trump declared that Mr. Johnson would "make a great prime minister."CreditMatt Dunham/Associated Press
"My right-wing feelings were triggered, to use a modern word, by my sense of outrage at their glutinous hypocrisy," he said, speaking of his classmates. | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 4,822 |
Q: Anchor executes different function that it is assigned to For some reason a href with class RemoveSpell after clicking executes on a.AddSpell again. You can test out the problem here http://89.69.172.125/cms2.0/
$(document).ready(function(){
championNumber = 1;
$('a#AddChampion').on('click',function(){
$('div#ChampionInput').append(
'<div class="Champion">\
<a href="#" class="Remove">Remove</a>\
<br>\
<input type="text" class="ChampionInput" name="champion[]" placeholder="Champion '+championNumber+'">\
<a href="#" class="AddSpell">Add Spell</a>\
<br>\
<div>');
championNumber++;
});
$('div#ChampionInput').on('click', 'a.Remove',function(){
$(this).parent('div.Champion').remove();
});
$('div#ChampionInput').on('click', 'a.AddSpell',function(){
$(this).append(
'<div class="Spell">\
<br>\
<input type="text">\
<a href="#" class="RemoveSpell">Remove Spell</a>\
</div>\
');
});
$('div#ChampionInput').on('click', 'a.RemoveSpell',function(){
});
});
A: You should not add the remove spell as a child of addSpell element, need to insert it after that
$(document).ready(function() {
championNumber = 1;
$('a#AddChampion').on('click', function() {
$('div#ChampionInput').append(
'<div class="Champion">\
<a href="#" class="Remove">Remove</a>\
<br>\
<input type="text" class="ChampionInput" name="champion[]" placeholder="Champion ' + championNumber + '">\
<a href="#" class="AddSpell">Add Spell</a>\
<br>\
<div>');
championNumber++;
});
$('div#ChampionInput').on('click', 'a.Remove', function() {
$(this).parent('div.Champion').remove();
});
$('div#ChampionInput').on('click', 'a.AddSpell', function() {
$(this).after(
'<div class="Spell">\
<br>\
<input type="text">\
<a href="#" class="RemoveSpell">Remove Spell</a>\
</div>\
');
});
$('div#ChampionInput').on('click', 'a.RemoveSpell', function() {
$(this).closest('.Spell').remove();
});
});
* {
margin: 0;
padding: 0;
text-decoration: none;
}
body {
margin: 0 auto;
width: 100%;
max-width: 900px;
}
input {
border: 1px solid #999;
}
.ChampionInput {
margin-bottom: 5px;
}
<script src="https://ajax.googleapis.com/ajax/libs/jquery/1.11.1/jquery.min.js"></script>
<a href="#" id="AddChampion">Add Champion</a>
<form name="second_form" id="second_form" method="POST">
<div id="ChampionInput">
</div>
<br>
<br>
<input type="submit" name="submit">
</form>
A: Several problems. Invalid html nesting and not recognizing that events that occur on a descendent of an element will also trigger event handlers on the parent.
Events propagate up the DOM tree. So when you nest removeSpell inside addSpell and click removeSpell the event will also trigger on addSpell
This is in fact how you are delegating events using on(). You are assigning the event handler to div#ChampionInput which is up the DOM tree from both of these elements. This delegation relies on the event propagation or "bubbling" up from the targets
You can't have <a> as a descendent of <a>.
| {
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} | 131 |
package com.google.ads.googleads.v10.errors;
public final class FieldErrorProto {
private FieldErrorProto() {}
public static void registerAllExtensions(
com.google.protobuf.ExtensionRegistryLite registry) {
}
public static void registerAllExtensions(
com.google.protobuf.ExtensionRegistry registry) {
registerAllExtensions(
(com.google.protobuf.ExtensionRegistryLite) registry);
}
static final com.google.protobuf.Descriptors.Descriptor
internal_static_google_ads_googleads_v10_errors_FieldErrorEnum_descriptor;
static final
com.google.protobuf.GeneratedMessageV3.FieldAccessorTable
internal_static_google_ads_googleads_v10_errors_FieldErrorEnum_fieldAccessorTable;
public static com.google.protobuf.Descriptors.FileDescriptor
getDescriptor() {
return descriptor;
}
private static com.google.protobuf.Descriptors.FileDescriptor
descriptor;
static {
java.lang.String[] descriptorData = {
"\n1google/ads/googleads/v10/errors/field_" +
"error.proto\022\037google.ads.googleads.v10.er" +
"rors\"\330\001\n\016FieldErrorEnum\"\305\001\n\nFieldError\022\017" +
"\n\013UNSPECIFIED\020\000\022\013\n\007UNKNOWN\020\001\022\014\n\010REQUIRED" +
"\020\002\022\023\n\017IMMUTABLE_FIELD\020\003\022\021\n\rINVALID_VALUE" +
"\020\004\022\027\n\023VALUE_MUST_BE_UNSET\020\005\022\032\n\026REQUIRED_" +
"NONEMPTY_LIST\020\006\022\033\n\027FIELD_CANNOT_BE_CLEAR" +
"ED\020\007\022\021\n\rBLOCKED_VALUE\020\tB\357\001\n#com.google.a" +
"ds.googleads.v10.errorsB\017FieldErrorProto" +
"P\001ZEgoogle.golang.org/genproto/googleapi" +
"s/ads/googleads/v10/errors;errors\242\002\003GAA\252" +
"\002\037Google.Ads.GoogleAds.V10.Errors\312\002\037Goog" +
"le\\Ads\\GoogleAds\\V10\\Errors\352\002#Google::Ad" +
"s::GoogleAds::V10::Errorsb\006proto3"
};
descriptor = com.google.protobuf.Descriptors.FileDescriptor
.internalBuildGeneratedFileFrom(descriptorData,
new com.google.protobuf.Descriptors.FileDescriptor[] {
});
internal_static_google_ads_googleads_v10_errors_FieldErrorEnum_descriptor =
getDescriptor().getMessageTypes().get(0);
internal_static_google_ads_googleads_v10_errors_FieldErrorEnum_fieldAccessorTable = new
com.google.protobuf.GeneratedMessageV3.FieldAccessorTable(
internal_static_google_ads_googleads_v10_errors_FieldErrorEnum_descriptor,
new java.lang.String[] { });
}
// @@protoc_insertion_point(outer_class_scope)
}
| {
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} | 9,452 |
L'Anarchie est un journal (hebdomadaire paraissant le jeudi) individualiste anarchiste fondé le par Albert Libertad et Anna Mahé.
Historique
Au siège du journal, rue de la République à Romainville (à l'époque rue de Bagnolet), les compagnons groupés autour de Libertad (André Lorulot qui reprend le journal en 1908, Henriette Maîtrejean en 1911, Émilie Lamotte, Mauricius...) combattaient « les vices, habitudes et préjugés », le tabac et l'alcool, le « culte de la charogne » et le conformisme résigné de celui qui vote, se rend à la caserne, se marie et travaille.
On trouve parmi les rédacteurs de L'Anarchie, Émile Bill sous la signature d'Hémyle Bill, Raymond Callemin, dit Raymond la Science, et Viktor Lvovitch Kibaltchich sous la signature Le Rétif le futur Victor Serge membres de la bande à Bonnot. Prônant d'abord l'illégalisme, Henriette (dite Rirette) Maîtrejean et Victor Kibaltchich changent cette politique éditoriale en 1911 lorsque l'ancienne équipe disparaît à la suite d'un cambriolage.
484 numéros paraîtront entre le et le .
Périodiques
22 numéros disponibles sur Gallica,
L'Anarchie, Paris,
L'Anarchie, Paris,
L'Anarchie, Paris,
L'Anarchie, Paris,
L'Anarchie, Paris, n°176, ,
L'Anarchie, Paris, n°177, , .
La collection complète de l'anarchie est disponible sur le site "Fragments d'histoire de la gauche radicale"
Notes et références
Notes
Références
Voir aussi
Bibliographie
Alain Accardo, Albert Libertad, Gaetano Manfredonia, Le Culte de la charogne, Marseille, Agone, coll. « Mémoires Sociales », 2006
Jean Maitron, Le Mouvement anarchiste en France, Gallimard, coll. « Tel », 1992
René Bianco, Répertoire des périodiques anarchistes de langue française : un siècle de presse anarchiste d'expression française, 1880-1983, Aix-Marseille, 1987 .
.
Article
Albert Libertad, Et que crève le vieux monde, , lire en ligne.
Audiovisuel
Anne Steiner, Les anarchistes, corpus individualistes, Dictionnaire biographique Maitron, Centre d'Histoire Sociale, , voir en ligne.
Articles connexes
Histoire de l'anarchisme
L'Anarchie, journal de l'ordre, premier journal anarchiste attesté, dont L'anarchie reprend le titre.
Anarchisme
Anarchisme insurrectionnaliste
Presse anarchiste
Mauricius
Libertaire-Plage
Louis Rimbault
Liens externes
Centre international de recherches sur l'anarchisme (Lausanne) : notice.
Gallica : texte intégral.
Bibliothek der Freien, Berlin, List of digitized anarchist periodicals.
Presse anarchiste en France
Presse anarchiste individualiste
Presse hebdomadaire disparue en France
Titre de presse créé en 1905
Titre de presse disparu en 1914 | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 7,923 |
/*
CSS Name: Black Minimalism Theme
Description: For Black Minimalism Theme
Author: NextCellent
Version: 1.9.22
This is a template stylesheet that can be used with NextGEN Gallery. I tested the
styles with a default theme Kubrick. Modify it when your theme struggle with it,
it's only a template design
*/
/* ----------- Album Styles Extend -------------*/
.ngg-albumoverview {
margin-top: 10px;
width: 100%;
clear:both;
display:block !important;
}
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/*height: 130px;*/
overflow:hidden;
padding: 5px;
margin-bottom: 5px;
border: 1px solid #cccccc;
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text-align: left;
font-weight: bold;
margin:0px;
padding:0px;
font-size: 1.4em;
margin-bottom: 10px;
}
.ngg-thumbnail {
float: left;
margin: 0pt !important;
margin-right: 12px !important;
}
.ngg-thumbnail img {
background-color:#FFFFFF;
border:1px solid #A9A9A9;
margin:4px 0px 4px 5px;
padding:4px;
position:relative;
}
.ngg-thumbnail img:hover {
background-color: #A9A9A9;
}
.ngg-description {
text-align: left;
}
/* ----------- Album Styles Compact -------------*/
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padding-right:6px !important;
margin:0px !important;
text-align:left;
width:120px;
display:inline-block;
vertical-align:top;
}
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background:transparent url(albumset.gif) no-repeat scroll 0%;
height:86px;
margin:0pt 0pt 6px !important;
padding:12px 0pt 0pt 7px !important;
width:120px;
}
.ngg-album-compactbox .Thumb {
border:1px solid #000000;
margin:0px !important;
padding:0px !important;
width:91px;
height:68px;
}
.ngg-album-compact h4 {
font-size:15px;
font-weight:bold;
margin-bottom:0px;
margin-top:0px;
width:110px;
}
.ngg-album-compact p {
font-size:11px;
margin-top:2px;
}
/* ----------- Gallery style -------------*/
.ngg-galleryoverview {
overflow: hidden;
margin-top: 10px;
width: 100%;
clear:both;
display:block !important;
}
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float: left;
width: 20%;
}
.ngg-gallery-thumbnail {
float: left;
background: url(shadowAlpha.png) no-repeat bottom right !important;
background: url(shadow.gif) no-repeat bottom right;
margin: 10px 0 0 10px !important;
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.ngg-gallery-thumbnail img {
margin: -6px 6px 6px -6px;
background-color:#FFFFFF;
border:1px solid #A9A9A9;
display:block;
padding:4px;
position:relative;
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background-color: #A9A9A9;
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/* Images description */
font-size:90%;
padding-left:5px;
display:block;
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font-size:0.9em !important;
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display:block !important;
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text-align:center;
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font-weight:bold;
margin:0pt 6px;
}
.ngg-navigation a.page-numbers,
.ngg-navigation a.next,
.ngg-navigation a.prev,
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.ngg-navigation span.next,
.ngg-navigation span.prev {
border:1px solid #660000;
margin-right:3px;
padding:3px 7px;
}
.ngg-navigation a.page-numbers:hover,
.ngg-navigation a.next:hover,
.ngg-navigation a.prev:hover,
.ngg-navigation span.page-numbers:hover,
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background-color: #660000;
color: #FFFFFF;
text-decoration: none;
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/* ----------- Image browser style -------------*/
.ngg-imagebrowser {
}
.ngg-imagebrowser h3 {
text-align:center;
}
.ngg-imagebrowser img {
border:1px solid #A9A9A9;
margin-top: 10px;
margin-bottom: 10px;
width: 100%;
display:block !important;
padding:5px;
}
.ngg-imagebrowser-nav {
padding:5px;
margin-left:10px;
}
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float:left;
border:1px solid #DDDDDD;
margin-right:3px;
padding:3px 7px;
}
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float:right;
border:1px solid #DDDDDD;
margin-right:3px;
padding:3px 7px;
}
.ngg-imagebrowser-nav .counter {
text-align:center;
font-size:0.9em !important;
}
.exif-data {
margin-left: auto !important;
margin-right: auto !important;
}
/* ----------- Slideshow -------------*/
.slideshow {
margin-left: auto;
margin-right: auto;
text-align:center;
outline: none;
}
.slideshowlink {
}
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.ngg-slideshow {
overflow:hidden;
position: relative;
margin-left: auto;
margin-right: auto;
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.rs-slideshow {
position: relative;
}
.rs-slideshow .slide-container {
height: 100%;
left: 0;
overflow: hidden;
position: absolute;
top: 0;
width: 100%;
}
.rs-slideshow .slide-container img {
position: absolute;
}
.rs-controls a,
.rs-index-list li {
float: left;
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float: left;
list-style: none;
width: auto;
margin: 0;
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/* Hide the slide data container */
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display: none;
}
/* ----------- Single picture -------------*/
.ngg-singlepic {
display:block;
padding:4px;
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.ngg-left {
float: left;
margin-right:10px;
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float: right;
margin-left:10px;
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padding:5px 0px 0px 0pt;
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margin: 0pt 2px;
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width: 50px;
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border: 1px solid #000000;
}
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content: '' !important;
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float:left;
margin:0 2px 0px 2px !important;
overflow:hidden;
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border:1px solid #CCCCCC;
display:block;
padding:2px;
}
.ngg-gallery-list li.selected a{
-moz-background-clip:border;
-moz-background-inline-policy:continuous;
-moz-background-origin:padding;
background:#000000 none repeat scroll 0 0;
}
.ngg-gallery-list li img {
height:40px;
width:40px;
}
li.ngg-next, li.ngg-prev {
height:40px;
width:40px;
font-size:3.5em;
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li.ngg-next a, li.ngg-prev a {
padding-top: 10px;
border: none;
text-decoration: none;
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Q: Keep getting segfault when trying to split a sentence into an array of arrays I've only been learning programming for the last 2 months and it's my first time using StackOverflow, so hopefully I didn't screw up formatting.
I've been stuck on this exercise for a few days.
Description:
Parameters: #1. The string to be split.
#2. The delimiter character.
Return value: The array of new strings resulting from the split.
NULL if the allocation fails.
External functions allowed: malloc, free
Allocates (with malloc(3)) and returns an array
of strings obtained by splitting 's' using the
character 'c' as a delimiter. The array must be
ended by a NULL pointer.
I know surely there's an easier way to do it, but I'm extremely fearful of searching the web because I feel like I'm cheating myself out of learning by trying.
What I decided to do was creating a counter function that checks how many words will be between the delimiter (aka size of first array), then I check the size of each word and store that in an array of ints that I use to create the length of each array to which the first array will point to. Then I fill each array.
The tester I use says I've segmentation faults in two of the 5 tests, even though the output from my function matches the supposed output of the test.
I've been really stuck and starting to feel really bad for being behind my friends who already turned in the entire project.
Hopefully someone can teach me what I'm doing wrong. Thank you for your time!
char **ft_split(char const *s, char c)
{
int i;
int size;
int *elem;
char **new;
size = counter(s, c);
elem = sizeofeachstring(s, c);
i = 0;
if (s == NULL)
return (NULL);
new = malloc(sizeof(char *) * size);
if (new == NULL)
return (NULL);
while (i < size)
{
if (i < size)
new[i] = (char *)malloc(sizeof(char) * (elem[i] + 1));
i++;
}
fillarrays(s, c, new);
return (new);
}
static int counter(char const *s, char c)
{
int count;
int i;
i = 0;
count = 0;
while (s[i])
{
while (s[i] == c)
i++;
if (s[i] != c && s[i])
count++;
while (s[i] != c && s[i])
i++;
}
return (count);
}
static int *sizeofeachstring(char const *s, char c)
{
int i;
int len;
int n;
int *elem;
int size;
i = 0;
len = 0;
n = 0;
size = counter(s, c);
elem = (int *)malloc(sizeof(int) * size);
while (s[i] != '\0')
{
if (s[i] != c)
len++;
if ((s[i] != c && s[i + 1] == c && len > 0))
{
elem[n] = len;
len = 0;
n++;
}
i++;
}
return (elem);
}
static char **fillarrays(char const *s, char c, char **new)
{
int i;
int j;
int k;
j = 0;
i = 0;
k = 0;
while (s[i] != '\0')
{
while (s[i] == c)
i++;
if (s[i] == '\0')
break ;
if (s[i] != c)
{
new[j][k++] = s[i++];
if (s[i] == c)
{
new[j][k] = '\0';
if (s[i] != '\0' || s[i + 1] != '\0')
{
j++;
k = 0;
}
}
}
}
return (new);
}
A: try to fill the **new with NULL after fill the words in array
char **ft_split(char const *s, char c)
{
...
new = malloc(sizeof(char *) * (size + 1));
...
}
static char **fillarrays(char const *s, char c, char **new)
{
...
new[j] = NULL;
return (new)
}
from the name of your function i think you are studying at one of 42Network schools
additional thing you need to protect the malloc that's a norm error
| {
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Low prices. Free Nationwide Shipping On Orders Over R450. Less Is More.
Free nationwide shipping in SA on all products. You won't ever wait longer than 5 days.
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"redpajama_set_name": "RedPajamaC4"
} | 9,866 |
{"url":"http:\/\/www.show-my-homework.com\/2012\/10\/number-of-combinations.html","text":"# Number of Combinations\n\nA combination lock has 3 dials. Each of the dials has numbers ranging from 0-4. How many different combinations are possible with the lock?\n\nThe number of different combinations is equal to the maximum number that can be formed with numbers 0 to 4 having 3 digits. This means that we are numbering the possible combinations in base 5 This number is\n\n$444 (base 5) = 4*5^2+4*5+4 = 124 (base 10)$ different combinations.","date":"2018-02-20 19:28:25","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\": 0.6206822395324707, \"perplexity\": 352.369431914133}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 5, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2018-09\/segments\/1518891813088.82\/warc\/CC-MAIN-20180220185145-20180220205145-00304.warc.gz\"}"} | null | null |
Q: Typescript Array push only show latest item I have a list get from my API, like as :
{
"content": [
{
"trainingClassId": 1,
"examCode": "my exam 1",
"address": {
"addressId": 1,
"addressValue": "abc 1213"
},
"description": "only test",
"classId": null,
"startDate": 1511110800000,
"endDate": 1513702800000,
"examDate": 1511542800000
},
{
"trainingClassId": 2,
"examCode": "my exam 2",
"address": {
"addressId": 1,
"addressValue": "abc 1213"
},
"description": "only test",
"classId": null,
"startDate": 1511110800000,
"endDate": 1513702800000,
"examDate": 1511542800000
}
],
"last": true,
"totalElements": 2,
"totalPages": 1,
"size": 20,
"number": 0,
"first": true,
"sort": null,
"numberOfElements": 2
}
I want to convert long to date, so I create 1 object to binding data:
export class myApp {
id: number;
classId: string;
trainingDate: string;
examDate: string;
}
on my ts:
listData = new Array();
app_unit:myApp= new myApp();
listApp:any[];
ngOnInit() {
this.getAllTrainingClass();
}
async getAllTrainingClass(): Promise<void>{
await this.traningClassService.getdata().then(data =>this.listApp = data);
for(let ls of this.listApp){
this.app_unit.classId= ls.classId;
this.app_unit.examDate=this.convertTimestampToDate(ls.examDate);
this.app_unit.trainingDate=this.convertTimestampToDate(ls.startDate) +'-'+this.convertTimestampToDate(ls.endDate) ;
this.listData.push(this.app_unit);
}
the console log of has 2 item, but it is the latest item, like as:
{0:myApp
id:2
classId:null
examDate:"05/01/2018"
trainingDate:"05/11/2017-31/12/2017"
},
{1:myApp
id:2
classId:null
examDate:"05/01/2018"
trainingDate:"05/11/2017-31/12/2017"}
please advice me.
A: The problem is because "this.app_unit" is pointing to the same memory location, you are inserting the same object into the array, all pointing to the same memory location.
Put,
var app_unit:myApp= new myApp();
inside the for loop.
Thus, a new instance of the object is created with each loop.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 1,315 |
\section{\large{R\lowercase{oundtrip condition of the metasurface laser}}}
\label{sec_roundtripcond}
For the metasurface laser to work, the roundtrip condition has to be satisfied \cite{Wen2021}: the oscillating state has to be the same after every roundtrip inside the cavity. This means that, at steady state, the intracavity polarization and orbital angular momentum (OAM) conversions imparted by the different optical elements need to balance out, allowing to return to the same state after one cycle of transformations. In this section we show how the roundtrip condition is fulfilled in our metasurface laser by computing with Jones calculus \cite{Jones1941} the polarization and OAM state change after every optical element.
Since we work with forward and backward paths inside a cavity it is important to establish the frame of reference in which the polarization and OAM states are defined. We adopt the convention that the observer always looks at the beam following the direction of propagation \cite{Pistoni1995}, e.g. right-circularly polarized (RCP) and diagonal (D) states will become, respectively, left-circularly polarized (LCP) and anti-diagonal (A) states after reflection from a flat mirror. Similarly, the topological charge $\ell$ of an OAM state will become $-\ell$ upon reflection. Note that neither the spin angular momentum (SAM) nor OAM are changing upon reflection from the flat mirror \cite{Mansuripur2011}, the change in circular polarization state and topological charge sign are only due to the flip of the point of view of the observer.
We start by illustrating different schemes producing vortex beams with uniform polarization. First, we consider an array of $q$-plates, which are reciprocal elements \cite{Wen2021} based on the Pancharatnam-Berry (or geometric) phase. The Jones matrix of a $q$-plate is
\begin{equation}
Q = \begin{pmatrix}
\mathrm{cos}(m\phi) & -\mathrm{sin}(m\phi) \\
-\mathrm{sin}(m\phi) & -\mathrm{cos}(m\phi)
\end{pmatrix}
\end{equation}
where $m$ is the topological charge and $\phi$ the azimuthal coordinate of the beam. Since the $q$-plate operates on circular polarization states, a polarizing beam splitter (PBS) and a quarter-wave plate (QWP) are needed in the cavity to enforce circular polarization (Fig. \ref{figS_qJroundtripschemes}a). However, since the combination of PBS and QWP constitutes \textit{per se} the poor man's isolator, these elements alone would block the backward path in the cavity. To overcome this problem a Faraday rotator (FR), which is a non-reciprocal element, is added between the PBS and QWP, satisfying the roundtrip condition. To block reflections from the metallic mask encircling the metasurface array, the sample is slightly tilted by a few degrees. The output of this laser from the Talbot section of the cavity is a vortex crystal with RCP and charge $-m$. By rotating the QWP by $90^{\circ}$ the output switches to LCP with charge $m$, as verified experimentally (Fig. 2b,c of the main text).
\begin{figure*}[t]
\centering
\includegraphics[width=1\textwidth]{figS_qJroundtripschemes.pdf}
\caption{Roundtrip schemes of vortex lasers based on (\textbf{a}) $q$-plate, and (\textbf{b}) $J$-plate arrays, and the corresponding vector vortex laser schemes (\textbf{c},\textbf{d}), resulting in a superposition of beams with opposite spin and helicity. After every optical element the polarization state and topological charge of the OAM mode are given. PBS, polarizing beam splitter; FR, Faraday rotator; QWP, quarter waveplate; D, diagonal; A, anti-diagonal; V, vertical; H, horizontal; L, left-circular; R, right-circular.}
\label{figS_qJroundtripschemes}
\end{figure*}
Next, we consider an array of $J$-plates designed to operate on a linear polarization basis \cite{Devlin2017J}. In this case the $J$-plate is purely based on the propagation phase. Its Jones matrix is
\begin{equation}
J = \begin{pmatrix}
e^{i m\phi} & 0 \\
0 & e^{i n\phi}
\end{pmatrix}
\end{equation}
where $m$ and $n$ are two arbitrary topological charges. As shown in Fig. \ref{figS_qJroundtripschemes}b, one can use a QWP between the $J$-plate array and the mirror to switch between orthogonal linear polarization states. In this way, the reflected beam impinging on the $J$-plate will see a different topological charge, thanks to the metasurface birefringence. As shown by Jones calculus, the roundtrip condition is satisfied only if $-n-m=0$, corresponding to the relation $n = -m$ in the designed topological charges. This also shows that the laser scheme would not work for a spiral phase plate (SPP), which corresponds to a polarization independent $J$-plate with $n = m$. We will discuss a different scheme suitable for SPPs further on.
\begin{figure*}[t]
\centering
\includegraphics[width=1\textwidth]{figS_SPPJQWroundtripschemes.pdf}
\caption{(\textbf{a-c}), Roundtrip schemes of vortex lasers based on spiral phase plate arrays, which require a double reflector. (\textbf{d}), Scheme based on a SP-QW metasurface integrating both a spiral phase plate and a quarter waveplate. In this case a metallic mask is not needed on the sample. After every optical element the polarization state and topological charge of the OAM mode are given. PBS, polarizing beam splitter; QWP, quarter waveplate; V, vertical; H, horizontal; R, right-circular.}
\label{figS_SPPJQWroundtripschemes}
\end{figure*}
To complete the $q$- and $J$-plate schemes, we illustrate others that can be used to generate vector vortex beams (VVB). As shown in Fig. \ref{figS_qJroundtripschemes}c,d, the idea here is to change the polarization basis of the state impinging on the previous metasurfaces, i.e. we now use a linear polarization state for the $q$-plate and a circular polarization state for the $J$-plate. This is achieved by removing the FR and the QWP in the case of the $q$-plate, and by replacing the FR with a QWP in the case of the $J$-plate. Again, in the case of the $J$-plate the roundtrip condition is satisfied for $n=-m$. While a small tilt of the metasurface sample is needed in the schemes of Fig. \ref{figS_qJroundtripschemes}a-c, the scheme in Fig. \ref{figS_qJroundtripschemes}d allows to suppress reflections from the metallic mask without tilting the sample, thanks to the integrated poor man's isolator. The output from these lasers is a vector vortex crystal consisting of a superposition of orthogonal polarization states with different topological charge sign, i.e. an array of VVBs.
Getting back to the SPP, lasing schemes are possible by incorporating a double reflector in the cavity, such as a hollow-roof mirror or a Porro prism. The simplest cavity schemes are shown in Fig. \ref{figS_SPPJQWroundtripschemes}a,b. The key idea is that the double reflection prevents a topological charge accumulation upon a double pass through the SPP. It is important to remark that the position of the double reflector determines the side of the cavity where the vortices exist. In the configurations of Fig. \ref{figS_SPPJQWroundtripschemes}a,b, the sample needs to be slightly tilted to suppress unwanted reflections from the metallic mask. This can be avoided using the scheme of Fig. \ref{figS_SPPJQWroundtripschemes}c, which comprises a poor man's isolator, followed by a second QWP enabling the roundtrip. This second QWP could be integrated with a SPP into a single metasurface, which we call a SP-QW plate inspired by its constituting elements. Its Jones matrix is
\begin{equation}
SPQW = \frac{e^{i m\phi}}{\sqrt{2}}\begin{pmatrix}
1 & -i \\
-i & 1
\end{pmatrix}
\end{equation}
The advantage of a SP-QW is that a metallic mask is not needed on the sample, simplifying the fabrication process: in fact, only the light passing through the metasurfaces sees the embedded QWP managing to make a roundtrip. We fabricated a SP-QW and verified experimentally that the scheme in Fig. \ref{figS_SPPJQWroundtripschemes}d works. However, all the schemes in Fig. \ref{figS_SPPJQWroundtripschemes} suffer from several downsides due to the use of the double reflector. First, the double reflector does not allow to outcouple light, thus a beam splitter needs to be added inside the cavity, increasing the losses. Second, the beam reflected from the double reflector is flipped with respect to the axis of symmetry of the reflector, making the alignment with the axis of symmetry of the array critical and thus increasing the difficulty in building the cavity. Third, any double reflector creates a loss in the reflected field at the line where the two reflecting surfaces meet \cite{Litvin2007}. In view of all these additional complications, the plane parallel resonators based on $q$- and $J$-plates shown in Fig. \ref{figS_qJroundtripschemes} are much more preferable in practice.
\begin{figure*}[t]
\centering
\includegraphics[width=1\textwidth]{figS_azirad.pdf}
\caption{Topological charge analysis of azimuthally and radially polarized vector vortex crystals. (\textbf{a,e}) Schematics of the cavity in the vector vortex beam configuration. Also shown is the topological charge analysis setup. PBS, polarizing beam splitter; HWP, half-wave plate; SLM, spatial light modulator; CCD, charge-coupled device (camera); H, horizontal; V, vertical; R, right; L, left---all refer to the polarization state. $m$ is the topological charge. (\textbf{b,c,f,g}) Topological charge spectra measured by projecting the crystal in either the horizontal or vertical polarization state, as indicated by the arrows on the beam profiles. Only one beam of the array is shown, being representative of the vortex crystal. (\textbf{d,h}) Topological charge spectra of the radially and azimuthally polarized vortex crystals obtained by averaging the projective measurements for the horizontal and vertical polarization states.}
\label{figS_azirad}
\end{figure*}
\begin{figure*}[t]
\centering
\includegraphics[width=1\textwidth]{figS_topologicalsolutions.pdf}
\caption{Phase and intensity images of crystals emitted from a metasurface laser with array charge $\ell_a=1$ as obtained from numerical Fox-Li simulations. (\textbf{a},\textbf{b}) Resonator with a $1\%$ adjustment in the length of the telescope section. (\textbf{c},\textbf{d}) Ideal self-imaging resonator without any cavity length adjustment. The topological solution changes from $(0~|~1)$ (top) to $(-1~|~0)$ (bottom).}
\label{figS_topologicalsolutions}
\end{figure*}
\section{\large{A\lowercase{zimuthally and radially polarized vector vortex crystals}}}
The topological charge spectra of the vector vortex crystals presented in Fig. 2d,e of the main text present contributions with zero topological charge at $\ell = 0$. These contributions originate from the finite spin-orbit conversion efficiency of the metasurfaces, which leave a fraction of the beam passing through the metasurfaces unconverted, in both polarization and OAM. When using a $q$-plate array in the scalar vortex crystal configuration (Fig. \ref{figS_qJroundtripschemes}a), this unconverted residue has a circular polarization state (either RCP or LCP) which is orthogonal to that of the vortex crystal (LCP or RCP, respectively), thus it can be polarization filtered outside the cavity, leaving no trace in the topological charge spectrum of the vortex crystal (Fig. 2b,c of the main text). On the other hand, when using a $q$-plate array in the vector vortex crystal configuration, the unconverted beam's residue has linear polarization and cannot be filtered out in the topological charge analysis.
This can be understood by examining, for instance, the characterization of radially polarized light. It is important to note that the SLM used for the charge decomposition can only modulate horizontally polarized light, due to the alignment of the constituent liquid crystals. Therefore, we carry out two charge decompositions to analyze a vector vortex beam, one per each state of the linear polarization basis. For this purpose we use a half-wave plate (HWP) before the SLM (Fig. \ref{figS_azirad}a) to project either horizontally or vertically polarized light onto the SLM (Fig. \ref{figS_azirad}b,c). As one can see, a contribution at $\ell = 0$ of around 7$\%$ (Fig. \ref{figS_azirad}b) is only present for the case of horizontal polarization. This is due to the unconverted residue of the beam impinging on the metasurface array, which is horizontally polarized and has zero OAM charge (Fig. \ref{figS_azirad}a). Analogously, in the case of azimuthally polarized light, the unconverted residue is vertically polarized (Fig. \ref{figS_azirad}e) and is present when the state is projected onto the vertical polarization basis with a value of approximately 9$\%$ (Fig. \ref{figS_azirad}g). By averaging the measurements performed in the horizontal and vertical polarization bases, we obtain the charge spectra of the vector vortex crystals (Fig. \ref{figS_azirad}d,h).
\begin{figure*}[t]
\centering
\includegraphics[width=1\textwidth]{figS_simdecomp.pdf}
\caption{Parallelized topological charge characterization tested in numerical simulations. (\textbf{a}) Intensity and (\textbf{b}) phase of an array of vortices with charge 1. (\textbf{c}) Topological charge spectrum obtained from simulations replicating the experimental setup for charge characterization. (\textbf{d}) Modulated arrays obtained for different decomposition holograms.}
\label{figS_simdecomp}
\end{figure*}
\section{\large{T\lowercase{uning the topological solutions: Laser simulations}}}
We showed in the main text, by means of experiments and calculations, that different topological solutions can exist in the metasurface laser and that these can be selected by adjusting the cavity length of the telescope with a $\Delta z$ shift with respect to the ideal $4f$ self-imaging value. The model used in the main text calculated the transmission efficiency of an array of vortices for a single roundtrip in either the telescope or Talbot section of the laser, where the metallic mask embedded in the metasurface sample acts as a spatial filter. In this section we complete the analysis by presenting the results of Fox-Li simulations, which start from a noise seed and are iterative, considering not one but many roundtrips in the laser.
Fig. \ref{figS_topologicalsolutions} shows the phase and intensity of the crystals emitted from the metasurface laser for a relative adjustment of the cavity length of $1\%$ (A,B) and $0\%$ (C,D). Using the notation defined in the main text, in the first case the solution is $(0~|~1)$, while in the second case this becomes $(-1~|~0)$, in agreement with the experiments. It may be counter-intuitive observing donut beams in Fig. \ref{figS_topologicalsolutions}d with a topological charge of zero. This is due to the fact that the metasurface operates a phase-only transformation \cite{piccardo2020arbitrary}: the vortex beams propagating in the telescope section lose their topological charge after passing through the metasurfaces, but the intensity nulls in their beam profiles are preserved.
\section{\large{N\lowercase{umerical simulations of parallelized topological charge characterization}}}
In this work we introduce a parallelized technique that allows the topological charge spectrum of each vortex in the entire array to be analyzed all at once, regardless of the size of the array. This has the benefit of being extremely time efficient, as opposed to serial techniques analyzing one vortex at a time. Moreover, the topological spectrum is provided with spatial resolution, giving more information on the array than an average characterization analyzing the far-field with a cylindrical lens \cite{Qiao2021}. The technique is described in the Methods section and used in the main text to analyze the experimental measurements of vortex crystals. In Fig. \ref{figS_simdecomp} we apply it to a set of synthetic data obtained from numerical simulations replicating the experimental setup. We use a 10$\times$10 array of Laguerre-Gaussian modes with $p=0$ and $\ell=1$ (Fig. \ref{figS_simdecomp}a,b). The calculated topological spectrum shows that all the power is contained in the mode with charge $\ell = 1$, as expected (Fig. \ref{figS_simdecomp}c). The modulated arrays obtained for different decomposition holograms show the same features of the experiments (cf. Fig. \ref{figS_simdecomp}d and Fig. 2f of the main text), confirming the validity of the approach.
\section{\large{C\lowercase{aptions for Movies}}}
\textbf{Movie 1}---Calculated intensity (left) and phase (right) of an array of $10\times10$ vortices of topological charge $\ell=1$ propagating in the Talbot section of the cavity. $z_T$, Talbot distance.
\medskip
\textbf{Movie 2}---Calculated intensity (left) and phase (right) of an array of $10\times10$ vortices of topological charge $\ell=5$ propagating in the Talbot section of the cavity. $z_T$, Talbot distance.
\medskip
\textbf{Movie 3}---Transient showing the healing process of a topological charge defect with $\ell = 2$ (marked by a square) embedded in a metasurface array with $\ell = 1$ (right). At every roundtrip of the laser we show the dynamically-evolving phase profile of the vortex crystal. The simulation starts from a noise seed. Initially the charge corresponding to the defect device is $\ell = 2$, but after approximately 16 roundtrips the defect is healed and the charge becomes $\ell = 1$, remaining stable for any following iteration (also beyond the ones shown in the movie).
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 1,746 |
Lake City, Fl — Lake City Medical Center is pleased to announce the launch of minimally invasive Robotic Surgery at the local 91-bed acute care facility. The new surgical robot system is designed to help the surgeons transcend the limitations of conventional surgery and to provide a minimally invasive option for a wide range of procedures.
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Lake City Medical Center's timeline for beginning Robotic Surgery at the facility is the beginning of April. The initial implementation of the surgical robot will be for General Surgery procedures including but not limited to cholecystectomy, appendectomy, hernia repair, and gynecology.
Patients that require surgery completed using this system, can look forward to shorter recovery times, minimized surgical exposure, smaller and more precise incisions. We are truly thrilled to be a leader in this field and look forward to continue bringing Lake City and North Florida minimally invasive surgical options. | {
"redpajama_set_name": "RedPajamaC4"
} | 4,884 |
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