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Asian/sushi Restaurant Greeter/hostess - On Call
In Food and Beverage, we sincerely embrace the communities and cultures where we live, appreciate the values diversity brings, actively learn from others, and in turn, warmly share our knowledge and experiences with colleagues and guests.
We highly value and show appreciation to individuals who strive to always do their best in all that they do, who are motivated by self-esteem and teamwork, and who treat others with fairness and respect. We require and admire integrity, honesty and accountability as measures of strength in our character and welcome and hold close those who live these qualities in their personal and professional lives.
Most importantly, we are a company that practices caring. Every one of our team members needs to feel that they are cared about and that they genuinely extend their own care to all,... be they guests or colleagues, strangers or friends. We believe that when you spend time, enjoying what you do, with people and in a place you care about, you will discover just how important you are in making a difference in the success of our livelihoods and in the health of our spirit.
Maryland Live Casino & Hotel Food and Beverage Team
CORE VALUES:
•
CLEAN - Must make the property shine and look impeccable while maintaining a neat, CLEAN and crisp personal appearance.
•
SAFE - Must make guests feel SAFE and comfortable through creating a worry-free, carefree experience.
•
FAST - Provide FAST and efficient service with accuracy. Meet service time requirements and anticipate guest needs.
•Responsible for cleaning and maintaining host area, steps, and entrance to restaurant.
•Other duties as assigned.
SCOPE AND MAIN PURPOSE OF JOB:
The Asian Restaurant Host(ess) is responsible for greeting and thanking restaurant guests with a positive, friendly attitude, ensuring guests are seated in a timely manner; Engaging guests throughout service and seeing that guests' needs are anticipated and communicated.
Requirements EDUCATIONAL REQUIREMENTS
•A high school diploma or equivalent GED is preferred.
•One (1) year of restaurant experience preferred.
KNOWLEDGE, SKILLS, AND ABILITIES
•Knowledge of Asiatic Culture Service Etiquette, including Japanese preferred. Excellent Product knowledge of Asiatic foods/beverages and preparation preferred.
•Ability to communicate in English and fluently in another Asian language (Chinese, Japanese, Korean, Vietnamese or other...)
•Knowledge of or ability to learn sanitation laws and health regulations.
•Ability to communicate and perform assigned duties under pressure.
•Ability to perform in an interruptive environment.
WORKING CONDITIONS:
•Loud, high-energy 24-hour work environment.
•Standing and walking for duration of shift.
•Repetitive lifting up to 25 pounds.
•Bending, stooping and squatting.
•Due to the casino environment, frequent exposure to "adult" language.
•Flexible working schedule to include weekends and holidays.
•Working in a team environment.
•You will work in an environment where smoking is allowed.
•You will be exposed to alcohol, smoke, bright lights, and noise.
•You will need to walk up and down stairs to the Casino floor.
•You will work in 24/7 high energy casinos with over 300,000 square feet of gaming, hotel and entertainment space.
Job Info
Hanover, MD
Live! Casino - Hotel
Posted on:
12/08/2018
Position Available: Immediately
Work Permit:
Applicants who do not already have legal permission to work in the location of this job will not be considered.
Management Position: No
Description
Job DescriptionDescription
About Us and our F&B Service Culture
In Food and Beverage, we sincerely embrace the communities and cultures where we live, appreciate the values diversity brings, actively learn from others, and in turn, warmly share our knowledge and experiences with colleagues and guests.
We highly value and show appreciation to individuals who strive to always do their best in all that they do, who are motivated by self-esteem and teamwork, and who treat others with fairness and respect. We require and admire integrity, honesty and accountability as measures of strength in our character and welcome and hold close those who live these qualities in their personal and professional lives.
Most importantly, we are a company that practices caring. Every one of our team members needs to feel that they are cared about and that they genuinely extend their own care to all,... be they guests or colleagues, strangers or friends. We believe that when you spend time, enjoying what you do, with people and in a place you care about, you will discover just how important you are in making a difference in the success of our livelihoods and in the health of our spirit.
Maryland Live Casino & Hotel Food and Beverage Team
CORE VALUES:
•
CLEAN - Must make the property shine and look impeccable while maintaining a neat, CLEAN and crisp personal appearance.
•
SAFE - Must make guests feel SAFE and comfortable through creating a worry-free, carefree experience.
•
FAST - Provide FAST and efficient service with accuracy. Meet service time requirements and anticipate guest needs.
•Responsible for cleaning and maintaining host area, steps, and entrance to restaurant.
•Other duties as assigned.
SCOPE AND MAIN PURPOSE OF JOB:
The Asian Restaurant Host(ess) is responsible for greeting and thanking restaurant guests with a positive, friendly attitude, ensuring guests are seated in a timely manner; Engaging guests throughout service and seeing that guests' needs are anticipated and communicated.
Requirements EDUCATIONAL REQUIREMENTS
•A high school diploma or equivalent GED is preferred.
•One (1) year of restaurant experience preferred.
KNOWLEDGE, SKILLS, AND ABILITIES
•Knowledge of Asiatic Culture Service Etiquette, including Japanese preferred. Excellent Product knowledge of Asiatic foods/beverages and preparation preferred.
•Ability to communicate in English and fluently in another Asian language (Chinese, Japanese, Korean, Vietnamese or other...)
•Knowledge of or ability to learn sanitation laws and health regulations.
•Ability to communicate and perform assigned duties under pressure.
•Ability to perform in an interruptive environment.
WORKING CONDITIONS:
•Loud, high-energy 24-hour work environment.
•Standing and walking for duration of shift.
•Repetitive lifting up to 25 pounds.
•Bending, stooping and squatting.
•Due to the casino environment, frequent exposure to "adult" language.
•Flexible working schedule to include weekends and holidays.
•Working in a team environment.
•You will work in an environment where smoking is allowed.
•You will be exposed to alcohol, smoke, bright lights, and noise.
•You will need to walk up and down stairs to the Casino floor.
•You will work in 24/7 high energy casinos with over 300,000 square feet of gaming, hotel and entertainment space. |
{% load staticfiles %}
<!-- Static navbar -->
<nav class="navbar navbar-default navbar-static-top">
<div class="container">
<div class="navbar-header">
<button type="button" class="navbar-toggle collapsed" data-toggle="collapse" data-target="#navbar" aria-expanded="false" aria-controls="navbar">
<span class="sr-only">Toggle navigation</span>
<span class="icon-bar"></span>
<span class="icon-bar"></span>
<span class="icon-bar"></span>
</button>
<a class="navbar-brand" href="{% url 'home' %}">eCommerce 2</a>
</div>
<div id="navbar" class="navbar-collapse collapse">
<ul class="nav navbar-nav">
<li><a href="{% url 'home' %}">Home</a></li>
<!-- <li><a href="{% url 'about' %}">About</a></li>
<li><a href="{% url 'contact' %}">Contact</a></li> -->
<form class="navbar-form navbar-left" method="GET" role="search" action='{% url "products" %}'>
<div class="form-group">
<input type="text" class="form-control" placeholder="Search" name="q" value='{{ request.GET.q }}'>
</div>
<!-- <button type="submit" class="btn btn-default">Submit</button> -->
</form>
</ul>
<ul class="nav navbar-nav navbar-right">
<li>
<a href='{% url "cart" %}'><i class="fa fa-shopping-cart fa-navbar-cart"></i> <span id="cart-count-badge" class='badge'>{{ request.session.cart_item_count }}</span></a>
</li>
{% if request.user.is_authenticated %}
<li><a href="{% url 'orders' %}">Orders</a></li>
<li><a href="{% url 'auth_logout' %}">Logout</a></li>
{% else %}
<li><a href="{% url 'registration_register' %}">Register</a></li>
{% if not request.user.is_authenticated and not "/accounts/login" in request.get_full_path %}
<li class="dropdown">
<a href="#" class="dropdown-toggle" data-toggle="dropdown" role="button" aria-expanded="false">Login <span class="caret"></span></a>
<ul class="dropdown-menu" role="menu">
<form class='navbar-form' method='POST' action='{% url "auth_login" %}'>{% csrf_token %}
<div class='form-group'>
<input type='text' class='form-control' name='username' placeholder='Username' />
</div>
<div class='form-group'>
<input type='password' class='form-control' name='password' placeholder='Password' />
</div>
<button type='submit' class='btn btn-default btn-block'>Login</button>
</form>
<p class='text-center'><a href='{% url "auth_password_reset" %}'>Forgot password</a>?</p>
<!-- <li><a href="#">Action</a></li>
<li><a href="#">Another action</a></li>
<li><a href="#">Something else here</a></li>
<li class="divider"></li>
<li class="dropdown-header">Nav header</li>
<li><a href="#">Separated link</a></li>
<li><a href="#">One more separated link</a></li> -->
</ul>
</li>
{% endif %}
{% endif %}
</ul>
</div><!--/.nav-collapse -->
</div>
</nav> |
Devil’s Peak Brewing Company
Born as a response to the one-dimensional beer culture in South Africa, The Devil’s Peak Brewing Company (DPBC) produces brews unlike any other beer you’re likely to find locally. The designs reflect the intricacy of the different beer styles and the brewer’s passion to create beers with unique character and depth. Craftsmanship, idiosyncratic style and the underlying belief of ‘truth to material’ – where the insides inform the outside – equals 4 distinctive styles with 4 equally standout designs. Recently released, the designs are already winning public attention at beer appreciation festivals and on Facebook and twitter. |
Q:
Moving position of CMV Home Button?
I'm relatively new at working with the CMV (and .css).
What's the easiest way to change the Home button's position?
To clarify, I'm attempting to move the position of the button itself down about 100px.
A:
This really should be done via css in a css file. However the CMV html template for the map controls is a bit dated and so not as easy to reposition as the positioning is done directly in the template. I will add updating this approach as a to-do item for the next version of the cmv core application
In the meantime, you can accomplish what you want by changing line 13 in the template. To move it down 100px, change top:94px to top:194px. This will also move down the locate button if you are using that.
|
This bus driver was clearly reminiscing his childhood afternoons spent watching The Magic School Bus on the telly. He also clearly cannot differentiate between cartoons and the reality: buses can’t fly.
Give a man a fish and he can eat for a day. Give a man a keyboard and an internet connection and he will get very angry in comments sections. This #StopTheKnot video has riled people both here at home and abroad. |
SAP HANA XS Advanced - Modules within Node.js SAP HANA applications
Details
, but you certainly aren’t limited to only the functionality provided by XSJS and XSODATA. You have access to the full programming model of Node.js as well. In this section you will learn how to extend your existing Node.js module in the SAP Web IDE for SAP HANA.
You will learn about how to create and use reusable code in the form of Node.js modules. You will use package.json to define dependencies to these modules which make the installation of them quite easy. You will use one of the most popular modules – express - which helps with the setup the handling of the request and response object. You will use express to handle multiple HTTP handlers in the same service by using routes.
You will learn about the fundamentals of the asynchronous nature of Node.js You will also see how this asynchronous capability allows for non-blocking input and output. This technique is one of the basic things that makes Node.js development different from other JavaScript development and also creates one of the reasons for its growing popularity. You will see how these techniques are applied to common operations like HTTP web service calls or even SAP HANA database access. You will also see how to create language translatable text strings and HANA database queries from Node.js
Your final part of this section will demonstrate the ease at which you can tap into the powerful web sockets capabilities of Node.js You will use web sockets to build a simple chat application. Any message sent from the SAPUI5 client side application will be propagated by the server to all listening clients.
Step 1: Require modules
Return to the Node.js module and the server.js source file. You currently have enough Node.js logic to start up the XSJS compatibility and that’s it. You would like to extend server.js to also handle some purely Node.js code as well. At the same time we still want to support our existing XSJS and XSODATA services from this server.js.
Begin by adding two new Node.js module requirements toward the beginning of the file. One is for the http built-in module and the other is the popular open source module express. One of the most common tasks in Node.js is acting as a web server and handling http requests/responses. Express is a module that wraps and extends the low level http library and provides many additional services.
If you look into the code, you will see a script called initialize is called, and is supposed to be found in the current application directory. Under js, create a directory called utils and a file named initialize.js.
The server.js script is calling two function modules, initExpress and initXSJS. You will define them here as reusable functions using module.exports as in the code below. Copy it into the initialize.js script.
Before you add the initialization for XSJS so that the previous modules can also be executed, take a look at the code you have just pasted.First, the @sap/logging module is used to write log and trace files. By calling the external node module, passport, you are enforcing the UAA authentication you defined earlier. How? By using a JWT (JSON web token) strategy, a middleware function passes the request to the actual route which, in this case, is the xsuaa service you defined at the very beginning using the XS client. You are then injecting an instance of sap-hdb-ext, called xsHDBConn. This module is designed to expose pooled database connections as a Node.js middleware.
Step 2: Add XSJS initialization
You still want to configure the XSJS module but no longer want it to listen directly on your service port. Instead you will start your own instance of express add a route handler for /node which you can code your own Node.js logic. Then you will add the XSJS as a root route handler. Here is the ending piece of code for initialize.js.
Back in server.js, after calling the initialization scripts, you are calling a router. You will now create a router that will accommodate the different Express route requests. You will mimic part of the folder structure suggested in the Express module documentation and create a folder called router with a file called index.js in it.
Routes determine how your application will respond to different requests based on the URI and HTTP method. In the following code, a call aiming at the path node like https://app_URL:port/node will be handled by a script called myNode:.
You are now responding to a GET request to the relative root location of the file (in this case, router). However, in the index.js file inside the router folder, you set the path to /node in the URL. You need to map this route in the xs-app.json file in the HTML5 module.
Step 5: Add the route in the web module
Because it is the web module that is exposed for calls, you need to add the new /node route to the file xs-app.json:
Back in step 1 you added the modules express and passport using the require function. You now need to add them manually in the package.json file with the rest of the dependencies for the js module.
Step 6: Build and Run
You can now run the js module.
After it has built and started running, you can run the web module. This will open a new tab with the index.html. Test the original .xsjs service is running properly after the paths were changed with the route handler. Replace html with xsjs: |
Plan A for the Cardinals offseason was to raise the ceiling with some superstar players. Once Plan A signed with the Red Sox and Cubs, respectively, it became clear that Plan B was to raise the floor.
As a result, the Cardinals Opening Day bench was a significant upgrade over recent years. Couple that with the positional flexibility from corner guys like Brandon Moss and now Matt Holliday, and what you have is the makings of a more dynamic lineup, which can be shuffled and arranged on a daily basis to maximize it's productivity.
...and that hasn't exactly been Mike Matheny's strong suit.
During his four years at the helm, Matheny has made it clear he likes his guys to have specific roles. He likes each position to have a "starter." He likes defined roles in the bullpen. When circumstances upset those roles, for instance - and I'm just picking an example at random here - in the 2014 NLCS, the shit tends to hit the fan.
Flexibility is hard to quantify, and MLB rosters can take a variety of shapes. One season, you might have eight healthy guys that are clearly the best at their position. Another year, injuries could force the manager to make moves. But comparing the four years of the Matheny Administration to the final four seasons of the La Russa reign, the numbers bear out the assumption that Matheny prefers a more static lineup.
During that span, on average, Matheny has used just 83 different lineups per season. La Russa averaged 100. And if you replace 2010 - a real outlier year for La Russa - with 2007, he averaged 108 lineups per season.
In 2013, Matheny used only 64 different lineups, and relied on his most-preferred lineup 31 times. If you omit the outlier season for La Russa, he never used fewer than 98 lineups, as many as 125, and sent the same lineup out onto the field just nine times.
Whether or not the Matheny Way or the La Russa Way is somehow better, this year's roster seems designed to be more dynamic.
In acquiring Jedd Gyorko, the team clearly sought a right-handed bat who could complement (and maybe even platoon) with Kolten Wong, not to mention a Major League Quality hitter who could give Johnny Peralta the occasional day off. The flexibility of Moss and now Holliday to play corner outfield or first base gives them a number of ways to juggle those spots.
The Red Baron wrote a great piece a couple weeks back about how the Cardinals could take advantage of platoon splits this season. And on Opening Day, they pretty much executed the plan, with Holliday moving to first to get an extra righty in the lineup in the form of Pham.
Of course, an inning-and-a-half later, Tommy pulled his Pham and Holliday was back in left field. So that may be the last we see of Holliday-to-first platoon against lefties. But then with Jon Niese on the mound last night, Wong hit the bench and Gyorko lived up to his hype as a lefty masher by gyrking one into the seats in left.
So even in this tiny, two-game sample, we're seeing some hints of creativity from Matheny and a willingness to use that deep bench to exploit platoon advantages. That's encouraging, although we've seen him dabble with players and lineups in the past only to soon revert back to starting Mike's Guys™ every day.
It will take some time to get a sense whether Matheny's tendencies as a manager are shifting to best suit the makeup of the 2016 Cardinals. Are guys getting enough rest? Are players being rotated in a way to better exploit platoon advantages, optimize defense, etc.? These aren't conclusions we can make after a game or two, but patterns that emerge over the long season.
Let's meet back here in a couple months and see how it's going. |
[Ultrastructural observation on the spermiogenesis of Pagumogonimus skrjabini].
The spermiogenesis of Pagumogonimus skrjabini was observed on the ultrastructural level by transmission electron microscopy. The development and the morphological characteristics of spermatids and sperm were studied. The Golgi comples was located near the anterior region of the spermatid nucleus and developed into an acrosomal cap of the sperm. Accordingly, an acrosomal structure of P. skrjabini was recognized in this study. The mature sperm of P.s. was composed of head and tail. The head part was filled with dense nucleus and the nucleoplasma extended into the anterior end of the median part of the sperm tail. A conjunction part was seen between the head and the tail. The tail was composed of a median part and a terminal part. At the beginning of median part of the tail, two axial filaments were found on each side. Each axial filament was composed of two central filaments and nine pairs of cuter filaments. The two central filaments were surrounded by fibrous sheath, both of which were connected with spoke-like structures. Mitochondria were arranged in a line in the ventral portion of the central filaments. At the terminal part of the tail, two axial filaments lay closely together. |
---
abstract: 'We provide a full characterisation of the large-maturity forward implied volatility smile in the Heston model. Although the leading decay is provided by a fairly classical large deviations behaviour, the algebraic expansion providing the higher-order terms highly depends on the parameters, and different powers of the maturity come into play. As a by-product of the analysis we provide new implied volatility asymptotics, both in the forward case and in the spot case, as well as extended SVI-type formulae. The proofs are based on extensions and refinements of sharp large deviations theory, in particular in cases where standard convexity arguments fail.'
address: 'Department of Mathematics, Imperial College London'
author:
- Antoine Jacquier and Patrick Roome
title: 'Large-Maturity Regimes of the Heston Forward Smile'
---
[^1]
Introduction
============
Consider an asset price process $\left({\mathrm{e}}^{X_t}\right)_{t\geq0}$ with $X_0=0$, paying no dividend, defined on a complete filtered probability space $(\Omega,\mathcal{F},(\mathcal{F}_t)_{t\geq0},{\mathbb{P}})$ with a given risk-neutral measure ${\mathbb{P}}$, and assume that interest rates are zero. In the Black-Scholes-Merton (BSM) model, the dynamics of the logarithm of the asset price are given by ${\mathrm{d}}X_t=-\frac{1}{2}\sigma^2 {\mathrm{d}}t +\sigma {\mathrm{d}}W_t$, where $\sigma>0$ represents the instantaneous volatility and $W$ is a standard Brownian motion. The no-arbitrage price of the call option at time zero is then given by the famous BSM formula [@BS73; @M73]: $C_{\textrm{BS}}(\tau,k,\sigma) := {\mathbb{E}}\left({\mathrm{e}}^{X_\tau}-{\mathrm{e}}^k\right)_+
={\mathcal{N}}\left(d_+\right)-{\mathrm{e}}^k{\mathcal{N}}\left(d_-\right)$, with $d_\pm:=-\frac{k}{\sigma\sqrt{\tau}}\pm\frac{1}{2}\sigma\sqrt{\tau}$, where ${\mathcal{N}}$ is the standard normal distribution function. For a given market price $C^{\textrm{obs}}(\tau,k)$ of the option at strike ${\mathrm{e}}^k$ and maturity $\tau$, the spot implied volatility $\sigma_{\tau}(k)$ is the unique solution to the equation $C^{\textrm{obs}}(\tau,k)=C_{\textrm{BS}}(\tau,k,\sigma_{\tau}(k))$.
For any $t,\tau>0$ and $k\in{\mathbb{R}}$, we define as in [@B04; @VL] a forward-start option with forward-start date $t$, maturity $\tau$ and strike ${\mathrm{e}}^k$ as a European option with payoff $\left({\mathrm{e}}^{X_{\tau}^{(t)}}-{\mathrm{e}}^k\right)^+$ where $\label{eq:XtTauDef}X_{\tau}^{(t)}:=X_{t+\tau}-X_t$ pathwise. By the stationary increment property, its value is simply $C_{\textrm{BS}}(\tau,k,\sigma)$ in the BSM model. For a given market price $C^{\textrm{obs}}(t,\tau,k)$ of the option at strike ${\mathrm{e}}^k$, forward-start date $t$ and maturity $\tau$, the forward implied volatility smile $\sigma_{t,\tau}(k)$ is then defined (see also [@B04]) as the unique solution to $C^{\textrm{obs}}(t,\tau,k)=C_{\textrm{BS}}(\tau,k,\sigma_{t,\tau}(k))$. The forward smile is a generalisation of the spot implied volatility smile, and the two are equal when $t=0$.
The literature on implied volatility asymptotics is extensive and has drawn upon a wide range of mathematical techniques. Small-maturity asymptotics have received wide attention using heat kernel expansion results [@Benarous]. More recently, they have been studied using PDE methods [@BBF; @Hagan; @Pascucci], large deviations [@DFJV; @FJSmall], saddlepoint methods [@FJL11], Malliavin calculus [@Gobet; @KT04] and differential geometry [@GHLOW; @PHL]. Roger Lee [@Lee] was the first to study extreme strike asymptotics, and further works on this have been carried out by Benaim and Friz [@BF1; @BF2] and in [@Guli; @GuliMass; @GS; @FGGS; @DFJV; @Marco]. Large-maturity asymptotics have only been studied in [@Tehr; @FJ09; @JMKR; @JM12; @FJM10] using large deviations and saddlepoint methods. Fouque et al. [@Fouque] have also successfully introduced perturbation techniques in order to study slow and fast mean-reverting stochastic volatility models. Models with jumps (including Lévy processes), studied in the above references for large maturities and extreme strikes, ‘explode’ in small time, in a precise sense investigated in [@Alos; @AndLipton; @Tank; @Nutz; @MijTankov; @FL].
On the other hand the literature on asymptotics of forward-start options and the forward smile is sparse. Glasserman and Wu [@GW10] use different notions of forward volatilities to assess their predictive values in determining future option prices and future implied volatility. Keller-Ressel [@KR] studies the forward smile asymptotic when the forward-start date $t$ becomes large ($\tau$ fixed). Bompis [@Bo13] produces an expansion for the forward smile in local volatility models with bounded diffusion coefficient. In [@JR12] the authors compute small and large-maturity asymptotics for the forward smile in a general class of models (including stochastic volatility and time-changed exponential Lévy models) where the forward characteristic function satisfies certain properties (in particular essential smoothness of the re-scaled limit). In [@JR13] the authors prove that for fixed $t>0$ the Heston forward smile explodes as $\tau$ tends to zero. Finally, empirical results on the forward smile have been carried out by practitioners in Balland [@Ba], Bergomi [@B04], Bühler [@B02] and Gatheral [@G06].
Under some conditions on the parameters, it was shown in [@JR12] that the smooth behaviour of the pointwise limit $\lim_{\tau\uparrow\infty}\tau^{-1}\log{\mathbb{E}}({\mathrm{e}}^{u X_{\tau}^{(t)}})$ yielded an asymptotic behaviour for the forward smile as $
\sigma_{t,\tau}^2(k\tau)=v_0^{\infty}(k)+v_1^{\infty}(k,t)\tau^{-1}+\mathcal{O}(\tau^{-2}),
$ where $v_0^{\infty}(\cdot)$ and $v_1^{\infty}(\cdot,t)$ are continuous functions on ${\mathbb{R}}$. In particular for $t=0$ (spot smiles), they recovered the result in [@FJ09] (also under some restrictions on the parameters). Interestingly, the limiting large-maturity forward smile $v_0^{\infty}$ does not depend on the forward-start date $t$. A number of practitioners (see eg. Balland[@Ba]) have made the natural conjecture that the large-maturity forward smile should be the same as the large-maturity spot smile. The result above rigorously shows us that this indeed holds if and only if the Heston correlation is close enough to zero.
It is natural to ask what happens when the parameter restrictions are violated. We identify a number of regimes depending on the correlation and derive asymptotics in each regime. The main results (Theorems \[theorem:largematasympcalls\] and \[theorem:HestonLargeMatFwdSmileNonSteep\]) state the following, as $\tau$ tends to infinity: $$\begin{aligned}
{\mathbb{E}}\left({\mathrm{e}}^{X^{(t)}_{\tau}}-{\mathrm{e}}^{k\tau}\right)^+
& =
{\mathcal{I}}\left(k,\tau,V'(0),V'(1),{1\hspace{-2.1mm}{1}}_{\{\kappa<\rho \xi\}}\right) +
\frac{\phi(k,t)}{\tau^{\alpha}}{\mathrm{e}}^{-\tau\left(V ^*(k)-k \right)+\psi(k,t)\tau^\gamma}
\left(1+\mathcal{O}\left(\tau^{-\beta}\right)\right),\\
\sigma_{t,\tau}^2(k\tau)
& =
v_0^{\infty}(k,t)+v_1^{\infty}(k,t)\tau^{-\lambda}+{\mathcal{R}}(\tau,\lambda),\end{aligned}$$ for any $k\in{\mathbb{R}}$, where ${\mathcal{I}}$ is some indicator function related to the intrinsic value of the option price, and $\alpha$, $\gamma$, $\lambda$ strictly positive constants, depending on the level of the correlation. The remainder ${\mathcal{R}}$ decays to zero as $\tau$ tends to infinity. If $t=0$ (spot smiles) we recover and extend the results in [@FJ09].
The paper is structured as follows. In Section \[sec:largematreg\] we introduce the different large-maturity regimes for the Heston model, which will drive the asymptotic behaviour of forward option prices and forward implied volatilities. In Section \[sec:largematregfsaymp\] we derive large-maturity forward-start option asymptotics in each regime and in Section \[sec:largematregfsmaymp\] we translate these results into forward smile asymptotics, including extended SVI-type formulae (Section \[sec:SVIExtended\]). Section \[sec:nonsteepnumerics\] provides numerics supporting the asymptotics developed in the paper and Section \[sec:proofsnonsteep\] gathers the proofs of the main results.
**Notations:** ${\mathbb{E}}$ shall always denote expectation under a risk-neutral measure ${\mathbb{P}}$ given a priori. We shall refer to the standard (as opposed to the forward) implied volatility as the spot smile and denote it $\sigma_\tau$. The forward implied volatility will be denoted $\sigma_{t,\tau}$ as above and we let ${\mathbb{R}}^*:={\mathbb{R}}\setminus\{0\}$ and ${\mathbb{R}}^*_+:=(0,\infty)$. For a sequence of sets $(\mathcal{D}_{ \varepsilon})_{\varepsilon>0}$ in ${\mathbb{R}}$, we may, for convenience, use the notation $\lim_{\varepsilon \downarrow 0}\mathcal{D}_{ \varepsilon}$, by which we mean the following (whenever both sides are equal): $
\liminf_{\varepsilon \downarrow 0}\mathcal{D}_{ \varepsilon}
:= \bigcup_{\varepsilon>0}\bigcap_{s\leq\varepsilon}\mathcal{D}_{s}
= \bigcap_{\varepsilon>0}\bigcup_{s\leq\varepsilon}\mathcal{D}_{s}
=:\limsup_{\varepsilon\downarrow 0}\mathcal{D}_{\varepsilon}.
$ Finally, for a given set $A\subset{\mathbb{R}}$, we let $A^o$ and $\overline{A}$ denote its interior and closure (in ${\mathbb{R}}$), $\Re(z)$ and $\Im(z)$ the real and imaginary parts of a complex number $z$, and ${\mathrm{sgn}}(x) = 1$ if $x\geq 0$ and $-1$ otherwise.
Large-maturity regimes {#sec:largematreg}
======================
In this section we introduce the large-maturity regimes that will be used throughout the paper. Each regime is determined by the Heston correlation and yields fundamentally different asymptotic behaviours for large-maturity forward-start options and the corresponding forward smile. This is due to the distinct behaviour of the moment explosions of the forward price process $(X_{\tau}^{(t)})_{\tau>0}$ in each regime . In the Heston model, the (log) stock price process is the unique strong solution to the following SDEs: $$\label{eq:Heston}
\begin{array}{rll}
{\mathrm{d}}X_t & = \displaystyle -\frac{1}{2}V_t{\mathrm{d}}t+ \sqrt{V_t}{\mathrm{d}}W_t, \quad & X_0=0,\\
{\mathrm{d}}V_t & = \kappa\left(\theta-V_t\right){\mathrm{d}}t+\xi\sqrt{V_t}{\mathrm{d}}Z_t, \quad & V_0=v>0,\\
{\mathrm{d}}\left\langle W,Z\right\rangle_t & = \rho {\mathrm{d}}t,
\end{array}$$ with $\kappa>0$, $\xi>0$, $\theta>0$ and $|\rho|<1$ and $(W_t)_{t\geq0}$ and $(Z_t)_{t\geq0}$ are two standard Brownian motions. We also introduce the notation $\mu:=2\kappa\theta/\xi^2$. The Feller SDE for the variance process has a unique strong solution by the Yamada-Watanabe conditions [@KS97 Proposition 2.13, page 291]). The $X$ process is a stochastic integral of $V$ and is therefore well-defined. The Feller condition, $2\kappa\theta\geq\xi^2$ (or $\mu\geq 1$), ensures that the origin is unattainable. Otherwise the origin is regular (hence attainable) and strongly reflecting (see [@KT81 Chapter 15]). We however do not require the Feller condition in our analysis since we work with the forward moment generating function (mgf) of $X$. Define the real numbers $\rho_-$ and $\rho_+$ by $$\label{eq:RhoPM}
\rho_{\pm} := \frac{{\mathrm{e}}^{-2 \kappa t} \left(\xi({\mathrm{e}}^{2\kappa t}-1) \pm({\mathrm{e}}^{\kappa t}+1) \sqrt{16 \kappa ^2 {\mathrm{e}}^{2 \kappa t}+\xi ^2 (1-{\mathrm{e}}^{\kappa t})^2}\right)}{8 \kappa },$$ and note that $-1\leq \rho_-<0<\rho_+$ with $\rho_\pm=\pm 1$ if and only if $t=0$. We now define the large-maturity regimes: $$\label{eq:Regimes}
\left.
\begin{array}{llr}
{\mathfrak{R}}_1: & \textit{Good correlation regime:} & \rho_{-}\leq\rho\leq\min(\rho_{+},\kappa/\xi); \\
{\mathfrak{R}}_2: & \textit{Asymmetric negative correlation regime:} & -1<\rho<\rho_{-}\text{ and }t>0;\\
{\mathfrak{R}}_3: & \textit{Asymmetric positive correlation regime:} & \rho_{+}<\rho<1\text{ and }t>0;\\
& {\mathfrak{R}}_{3a}: & \rho\leq \kappa/\xi;\\
& {\mathfrak{R}}_{3b}: & \rho>\kappa/\xi;\\
{\mathfrak{R}}_4: & \textit{Large correlation regime:} &
\kappa/\xi<\rho\leq\min(\rho_{+},1).
\end{array}
\right.$$ In the standard case $t=0$, ${\mathfrak{R}}_1$ corresponds to $\kappa\geq \rho\xi$ and ${\mathfrak{R}}_4$ is its complement. We now define the following quantities: $$\label{eq:DefUpmU*pm}
u_{\pm}:=\frac{\xi-2\kappa\rho\pm\eta}{2\xi(1-\rho^2)}
\qquad \text{and}\qquad
u_{\pm}^{*}:=\frac{\psi\pm\nu}{2 \xi ({\mathrm{e}}^{\kappa t}-1)},$$ with $$\label{eq:etanu}
\eta:=\sqrt{\xi^2(1-\rho^2)+(2\kappa-\rho\xi)^2},\qquad
\nu:=\sqrt{\psi^2-16\kappa^2{\mathrm{e}}^{\kappa t}}
\qquad \text{and}\qquad
\psi:=\xi ({\mathrm{e}}^{\kappa t}-1)-4\kappa\rho{\mathrm{e}}^{\kappa t},$$ as well as the interval $\mathcal{D}_\infty\subset{\mathbb{R}}$ by
\[eq:DInfinityLargeMaturity\]
${\mathfrak{R}}_1$ ${\mathfrak{R}}_2$ ${\mathfrak{R}}_{3a}$ ${\mathfrak{R}}_{3b}$ ${\mathfrak{R}}_4$
---------------------- -------------------- -------------------- ----------------------- ----------------------- -------------------- --
$\mathcal{D}_\infty$ $[u_{-},u_{+}]$ $[u_{-},u^*_{+})$ $(u^*_{-},u_{+}]$ $(u^*_{-},1]$ $(u_{-},1]$
Note that for $t>0$, $u_{+}>u_{+}^*>1$ if $\rho\leq\rho_{-}$ and $u_{-}<u_{-}^*<0$ if $\rho\geq\rho_{+}$ Also $\nu$ defined in is a well-defined real number for all $\rho\in [-1,\rho_-]\cup[\rho_+,1]$. Furthermore we always have $u_{-}<0$ and $u_{+}\geq 1$ with $u_{+}=1$ if and only if $\rho=\kappa/\xi$. We define the real-valued functions $V$ and $H$ from $\mathcal{D}_{\infty}$ to ${\mathbb{R}}$ by $$\label{eq:VandH}
V(u) := \frac{\mu}{2}\left(\kappa-\rho\xi u-d(u)\right)
\qquad\text{and}\qquad
H(u) :=
\frac{V(u)v {\mathrm{e}}^{-\kappa t}}{\kappa\theta -2 \beta _t V(u)}
-\mu\log\left(\frac{\kappa\theta-2 \beta _t V(u)}{\kappa\theta \left(1-\gamma \left(u\right)\right)}\right),$$ with $d$, $\beta_t$ and $\gamma$ defined in . It is clear (see also [@FJ09] and [@JM12]) that the function $V$ is infinitely differentiable, strictly convex and essentially smooth on the open interval $\left(u_{-},u_{+}\right)$ and that $V(0)=0$. Furthermore $V(1)=0$ if and only if $\rho\leq\kappa/\xi$. For any $k\in{\mathbb{R}}$ the (saddlepoint) equation $V'(u^*(k))=k$ has a unique solution $u^*(k)\in(u_-,u_+)$: $$\begin{aligned}
\label{eq:u*}
u^*(k):=\frac{\xi-2\kappa\rho+\left(\kappa\theta\rho+k\xi\right)\eta\left(k^2\xi^2+2k\kappa\theta\rho\xi+\kappa^2\theta^2\right)^{-1/2}}{2\xi \left(1-\rho^2\right)}.\end{aligned}$$ Further let $V^*:{\mathbb{R}}\to{\mathbb{R}}_+$ denote the Fenchel-Legendre transform of $V$: $$\label{eq:VStarDefinition}
V^*(k):=\sup_{u\in\mathcal{D}_{\infty}}\left\{uk-V(u)\right\},
\qquad\text{for all }k\in{\mathbb{R}}.$$ The following lemma characterises $V^*$ and can be proved using straightforward calculus. As we will see in Section \[sec:LDP\], the function $V^*$ can be interpreted as a large deviations rate function for our problem.
\[lemma:V\*Characterisation\] Define the function $W(k,u)\equiv u k-V(u)$ for any $(k,u)\in{\mathbb{R}}\times [u_-,u_+]$. Then
- ${\mathfrak{R}}_1$: $V^*(k) \equiv W(k,u^*(k))$ on ${\mathbb{R}}$;
- ${\mathfrak{R}}_2$: $V^*(k) \equiv W(k,u^*(k))$ on $(-\infty, V'(u_+^*)]$ and $V^*(k) \equiv W(k,u^*_+)$ on $(V'(u^*_{+}), +\infty)$;
- ${\mathfrak{R}}_{3a}$: $V^*(k) \equiv W(k,u^*_-)$ on $(-\infty, V'(u^*_{-}))$ and $V^*(k) \equiv W(k,u^*(k))$ on $[ V'(u_-^*), +\infty)$;
- ${\mathfrak{R}}_{3b}$: $$V^*(k) \equiv
\left\{
\begin{array}{ll}
W(k,u^*_-), \quad & \text{on } (-\infty, V'(u^*_-)),\\
W(k,u^*(k)), \quad & \text{on } [V'(u^*_-), V'(1)],\\
W(k,1), \quad & \text{on } (V'(1), +\infty);
\end{array}
\right.$$
- ${\mathfrak{R}}_4$: $V^*(k) \equiv W(k,u^*(k))$ on $(-\infty, V'(1)]$ and $V^*(k) \equiv W(k,1)$ on $(V'(1),+\infty)$.
Forward-start option asymptotics {#sec:largematregfsaymp}
================================
In order to specify the forward-start option asymptotics we need to introduce some functions and constants. As outlined in Theorem \[theorem:largematasympcalls\], each of them is defined in a specific regime and strike region where it is well defined and real valued. In the formulae below, $\gamma$, $\beta_t$ are defined in , $u^*_{\pm}$ in and $V$ in . $$\label{eq:a}
\left\{
\begin{array}{rlrl}
a_1^{\pm}(k) & := \displaystyle \mp \frac{2|k-V'(u_{\pm}^*)|}{\zeta^2_{\pm}(k)},
& a_2^{\pm}(k) & := \displaystyle \frac{\mu{\mathrm{e}}^{-\kappa t}}{16\beta _t^2}
\frac{\xi ^2 v V''(u_{\pm}^*)-8 \beta _t^2 {\mathrm{e}}^{\kappa t} V'(u_{\pm}^*) \left(k-V'(u_{\pm}^*)\right)}{V'(u_{\pm}^*)\left(k-V'(u_{\pm}^*)\right)^2}, \\
\widetilde{a}_1^{\pm} & := \displaystyle \mp\left|\frac{{\mathrm{e}}^{-\kappa t}\kappa\theta v}{4 V'(u_{\pm}^*)V''(u_{\pm}^*)\beta_t^2}\right|^{1/3},
& \widetilde{a}_2^{\pm} & :=\displaystyle -\frac{(\kappa\theta{\mathrm{e}}^{-\kappa t})^{2/3}}{12 \xi ^2 v^{1/3}\beta _t^{4/3} } \frac{16 V'(u_{\pm}^*) V''(u_{\pm}^*) \beta _t^2 {\mathrm{e}}^{\kappa t}+\xi ^2 v V'''(u_{\pm}^*)}{2^{1/3} |V'(u_{\pm}^*)|^{2/3} V''(u_{\pm}^*)^{5/3}},
\end{array}
\right.$$ where $$\label{eq:variance}
\zeta^2_{\pm}(k)
:= \displaystyle 4 \beta _t \left(\frac{V'(u_{\pm}^*)(k-V'(u_{\pm}^*))^3}{\kappa\theta v {\mathrm{e}}^{-\kappa t}}\right)^{1/2};
$$ $$\label{eq:e}
\left\{
\begin{array}{rlrl}
e_0^{\pm}(k) & := \displaystyle -2\beta_t a_1^{\pm}(k)V'(u_{\pm}^*),
& e_1^{\pm}(k) & :=-\beta_t \left[V''(u_{\pm}^*)a_1^{\pm}(k)^2+2V'(u_{\pm}^*) a_2^{\pm}(k)\right], \\
\widetilde{e}_0^{\pm} & := \displaystyle -2\beta_t \widetilde{a}_1^{\pm}V'(u_{\pm}^*),
& \widetilde{e}_1^{\pm} & := -\beta_t \left[V''(u_{\pm}^*)(\widetilde{a}_1^{\pm})^2+2V'(u_{\pm}^*) \widetilde{a}_2^{\pm}\right],
\end{array}
\right.$$ $$\label{eq:c0}
\left\{
\begin{array}{ll}
c_0^{\pm}(k) := \displaystyle -2a_1^{\pm}(k)\left(k-V'(u_{\pm}^*)\right),
\quad
c_2^{\pm}(k) :=\displaystyle \left(\frac{\kappa\theta\left(1-\gamma(u_{\pm}^*) \right) }{e_0^{\pm}(k)}\right)^{\mu}, & \\
c_1^{\pm}(k) := \displaystyle v {\mathrm{e}}^{-\kappa t} \left(\frac{a_1^{\pm}(k) V'(u_{\pm}^*)}{e_0^{\pm}(k)}-\frac{\kappa\theta e_1^{\pm}(k)}{2 e_0^{\pm}(k)^2 \beta _t}\right)-a_2^{\pm}(k) \left(k-V'(u_{\pm}^*)\right)+\frac{1}{2}
a_1^{\pm}(k)^2 V''(u_{\pm}^*),&
\end{array}
\right.$$
$$\label{eq:c0bound}
\left\{
\begin{array}{ll}
\widetilde{c}_0^{\pm} := \displaystyle \frac{3}{2} (\widetilde{a}_1^{\pm})^2 V''(u_{\pm}^*),
\quad
\widetilde{c}_2^{\pm} := \left(\frac{\kappa\theta\left(1-\gamma(u_{\pm}^*) \right)}{\widetilde{e}_0^{\pm}}\right)^{\mu},
\quad
g_0 := \frac{v {\mathrm{e}}^{-\kappa t} V(1)}{\kappa\theta -2 \beta _t V(1)}, & \\
\widetilde{c}_1^{\pm} := \displaystyle v {\mathrm{e}}^{-\kappa t} \left(\frac{\widetilde{a}_1^{\pm} V'(u_{\pm}^*)}{\widetilde{e}_0^{\pm}}-\frac{\kappa\theta\widetilde{e}_1^{\pm}}{2 (\widetilde{e}_0^{\pm})^2 \beta _t}\right)+\widetilde{a}_1^{\pm}\widetilde{a}_2^{\pm} V''(u_{\pm}^*) +\frac{(\widetilde{a}_1^{\pm})^3 V'''(u_{\pm}^*)}{6},&
\end{array}
\right.$$
$$\label{eq:phi0largetime}
\phi_0(k)
:= \displaystyle \frac{1}{\sqrt{2\pi V''(u^*(k))}}
\left\{
\begin{array}{ll}
\displaystyle \frac{\exp\left({H(u^*(k))}\right)}{u^*(k) (u^*(k)-1)},
& \text{if }k\in{\mathbb{R}}\setminus\{V'(0),V'(1)\},\\
\displaystyle \left(-1-{\mathrm{sgn}}(k)\left(\frac{V'''(u^*(k))}{6 V''(u^*(k))}-H'(u^*(k))\right)\right),
& \text{if }k\in\{V'(0),V'(1)\}.
\end{array}
\right.$$
$$\left\{
\begin{array}{llrl}
\phi_\pm (k) & := \displaystyle \frac{c_2^\pm(k){\mathrm{e}}^{c_1^\pm(k)}}{\zeta_{\pm}(k)u^*_\pm(u^*_\pm-1)\sqrt{2\pi}},
& \widetilde{\phi}_\pm & := \displaystyle \frac{\tilde{c}_2^\pm{\mathrm{e}}^{\tilde{c}_1^\pm}}{u^*_\pm(u^*_\pm-1)\sqrt{6\pi V''(u^*_{\pm})}},\\
\phi_2 (k) & := \displaystyle \frac{-{\mathrm{e}}^{g_0}}{\Gamma(1+\mu)}
\left(\frac{2\mu (\kappa-\rho\xi)^2 (k-V'(1))}{\kappa\theta-2\beta_t V(1)}\right)^{\mu},
& \phi_1 & := \displaystyle \frac{-{\mathrm{e}}^{g_0}}{2\Gamma (1+\mu/2)}
\left(\frac{\mu(\kappa-\rho\xi)^2 \sqrt{2 V''(1) }}{\kappa\theta-2\beta_t V(1)}\right)^{\mu},
\end{array}
\right.$$
Since $u^*_-<0$ and $u^*_+>1$, we always have $V'(u^*_+)>0$ and $V'(u^*_-)<0$. Furthermore, $V''(u^*_{\pm})>0$ and one can show that $\gamma(u^*_{\pm})\neq1$; therefore all the functions and constants in , , , and are well-defined and real-valued. $\phi_0$ is well-defined since $V''(u^*(k))>0$ and $\phi_2$ and the constant $\phi_1$ are well-defined since $\kappa\theta-2\beta_t V(1)>0$. Finally define the following combinations and the function ${\mathcal{I}}:{\mathbb{R}}\times{\mathbb{R}}_{+}^*\times{\mathbb{R}}^3\to{\mathbb{R}}$ : $$\begin{array}{llllll}
{\mathcal{H}}_0:
& \alpha = \frac{1}{2}, & \beta = 1, & \gamma = 0, & \phi \equiv \phi_0, & \psi \equiv 0,
\\
\widetilde{{\mathcal{H}}}_\pm:
& \alpha = \frac{\mu}{3}-\frac{1}{2}, & \beta = \frac{1}{3}, & \gamma = \frac{1}{3},
& \phi \equiv \widetilde{\phi}_{\pm}, & \psi \equiv \tilde{c}_0^{\pm},
\\
{\mathcal{H}}_\pm:
& \alpha = \frac{\mu}{2}-\frac{3}{4}, & \beta = \frac{1}{2}, & \gamma = \frac{1}{2}, & \phi \equiv \phi_{\pm},
& \psi \equiv c_0^{\pm}, \\
{\mathcal{H}}_1:
& \alpha = -\frac{\mu}{2}, & \beta = \frac{1}{2}, & \gamma = 0, & \phi \equiv \phi_{1}, & \psi \equiv 0,
\\
{\mathcal{H}}_2:
& \alpha = -\mu, & \beta = 1, & \gamma = 0, & \phi \equiv \phi_{2}, & \psi \equiv 0,
\end{array}$$ $$\label{eq:intrinsiclargetimenonsteep}
{\mathcal{I}}(k,\tau,a,b,c)
:= \left(1-{\mathrm{e}}^{k \tau}\right){1\hspace{-2.1mm}{1}}_{\{k<a\}}+{1\hspace{-2.1mm}{1}}_{\{a<k<b\}}
+c{1\hspace{-2.1mm}{1}}_{\{b\leq k\}}
+\frac{1-c}{2}{1\hspace{-2.1mm}{1}}_{\{k=b\}}
+\left(1-\frac{1}{2}{\mathrm{e}}^{k \tau}\right){1\hspace{-2.1mm}{1}}_{\{k=a\}}.$$
We are now in a position to state the main result of the paper, namely an asymptotic expansion for forward-start option prices in all regimes for all (log) strikes on the real line. The proof is obtained using Lemma \[lem:nonsteeplem\] in conjunction with the asymptotics in Lemmas \[lemma:V\*Asymptotics\], \[lemma:FourierAsymptotics\], \[lem:v0v1F\] and \[lem:v0v1\].
\[theorem:largematasympcalls\] The following expansion holds for forward-start call options for all $k\in{\mathbb{R}}$ as $\tau$ tends to infinity: $${\mathbb{E}}\left({\mathrm{e}}^{X^{(t)}_{\tau}}-{\mathrm{e}}^{k\tau}\right)^+
=
{\mathcal{I}}\left(k,\tau,V'(0),V'(1),{1\hspace{-2.1mm}{1}}_{\{\kappa<\rho \xi\}}\right) +
\frac{\phi(k,t)}{\tau^{\alpha}}{\mathrm{e}}^{-\tau\left(V ^*(k)-k \right)+\psi(k,t)\tau^\gamma}
\left(1+\mathcal{O}\left(\tau^{-\beta}\right)\right),$$ where the functions $\phi$, $\psi$ and the constants $\alpha$, $\beta$ and $\gamma$ are given by the following combinations[^2]:
- ${\mathfrak{R}}_1$: ${\mathcal{H}}_0$ for $k\in{\mathbb{R}}$;
- ${\mathfrak{R}}_2$: ${\mathcal{H}}_0$ for $k\in(-\infty, V'(u_+^*))$; $\widetilde{{\mathcal{H}}}_+$ for $k=V'(u_+^*)$; ${\mathcal{H}}_+$ for $k\in(V'(u_+^*),+\infty)$;
- ${\mathfrak{R}}_{3a}$: ${\mathcal{H}}_-$ for $k\in(-\infty, V'(u_-^*))$; $\widetilde{{\mathcal{H}}}_-$ for $k=V'(u_-^*)$; ${\mathcal{H}}_0$ for $k\in(V'(u_-^*),+\infty)$;
- ${\mathfrak{R}}_{3b}$: ${\mathcal{H}}_-$ for $k\in(-\infty, V'(u_-^*))$; $\widetilde{{\mathcal{H}}}_-$ for $k=V'(u_-^*)$; ${\mathcal{H}}_0$ for $k\in(V'(u_-^*),V'(1))$; ${\mathcal{H}}_1$ at $k=V'(1)$; ${\mathcal{H}}_2$ for $k\in(V'(1),+\infty)$;
- ${\mathfrak{R}}_4$: ${\mathcal{H}}_0$ for $k\in(-\infty, V'(1))$; ${\mathcal{H}}_1$ for $k=V'(1)$; ${\mathcal{H}}_2$ for $k\in(V'(1),+\infty)$;
In order to highlight the symmetries appearing in the asymptotics, we shall at times identify an interval with the corresponding regime and combination in force. This slight abuse of notations should not however be harmful to the comprehension.
\[rem:largematfwdstartoption\]
(i) Under ${\mathfrak{R}}_1$, asymptotics for the large-maturity forward smile (for $k\in\mathbb{R}\setminus\{V'(0),V'(1)\}$) have been derived in [@JR12 Proposition 3.8].
(ii) For $t=0$, large-maturity asymptotics have been derived in [@FJL11] under ${\mathfrak{R}}_1$ and partially in [@JM12] under ${\mathfrak{R}}_4$.
(iii) All asymptotic expansions are given in closed form and can in principle be extended to arbitrary order.
(iv) When ${\mathcal{H}}_{\pm}$ and ${\mathcal{H}}_2$ are in force then $V^*(k)-k$ is linear in $k$ as opposed to being strictly convex as in ${\mathcal{H}}_0$.
(v) If $\rho\leq\kappa/\xi$ then $V^*(k)-k\geq0$ with equality if and only if $k=V'(1)$. If $\rho>\kappa/\xi$ then $V^*(k)-k\geq -V(1)>0$. Since $\gamma\in[0,1)$, the leading order decay term is given by ${\mathrm{e}}^{-\tau\left(V ^*(k)-k\right)}$.
(vi) Under ${\mathcal{H}}_2$ (which only occur when $\rho>\kappa/\xi$ for log-strikes strictly greater than $V'(1)$), forward-start call option prices decay to one as $\tau$ tends to infinity. This is fundamentally different than the large-strike behaviour in other regimes and in the BSM model, where call option prices decay to zero. This seemingly contradictory behaviour is explained as follows: as the maturity increases there is a positive effect on the price by an increase in the time value of the option and a negative effect on the price by increasing the strike of the forward-start call option. In standard regimes and for sufficiently large strikes the strike effect is more prominent than the time value effect in the large-maturity limit. Here, because of the large correlation, this effect is opposite: as the asset price increases, the volatility tends to increase driving the asset price to potentially higher levels. This gamma or time value effect outweighs the increase in the strike of the option.
(vii) In ${\mathfrak{R}}_4$, the decay rate $V^*(k)-k$ has a very different behaviour: the minimum achieved at $V'(1)$ is not zero and $V^*(k)-k$ is constant for $k\geq V'(1)$. There is limited information in the leading-order behaviour and important distinctions must therefore occur in higher-order terms. This is illustrated in Figures \[fig:LargeCorrelRegime10years\] and \[fig:LargeCorrelRegime20years\] where the first-order asymptotic is vastly superior to the leading order.
(viii) It is important to note that $u_{\pm}^*$ and $V^*$ depend on the forward-start date $t$ through and the regime choice. However, in the uncorrelated case $\rho=0$, ${\mathfrak{R}}_1$ always applies and $V^*$ does not depend on $t$. The non-stationarity of the forward smile over the spot smile (at leading order) depends critically on how far the correlation is away from zero.
In order to translate these results into forward smile asymptotics (in the next section), we require a similar expansion for the Black-Scholes model, where the log stock price process satisfies ${\mathrm{d}}X_t = -\frac{1}{2}\Sigma^2 {\mathrm{d}}t + \Sigma{\mathrm{d}}W_t$, with $\Sigma>0$. Define the functions $V^*_{\textrm{BS}}:{\mathbb{R}}\times{\mathbb{R}}^*_{+}\to{\mathbb{R}}$ and $\phi_{\textrm{BS}}:{\mathbb{R}}\times{\mathbb{R}}^*_{+}\times{\mathbb{R}}\to{\mathbb{R}}$ by $V^*_{\textrm{BS}}(k,a):=
\left(k+a/2\right)^2/(2 a)$ and $$\phi_{\textrm{BS}}(k,a,b) \equiv
\frac{4a^{3/2}}{(4k^2-a^2)\sqrt{2\pi}}\exp\left(b\left(\frac{k^2}{2a^2}-\frac{1}{8}\right)\right)
{1\hspace{-2.1mm}{1}}_{\{k\neq\pm a/2\}}
+\frac{b-2}{2\sqrt{2 a \pi}}
{1\hspace{-2.1mm}{1}}_{\{k=\pm a/2\}},$$ so that the following holds (see [@JR12 Corollary 2.11]):
\[Cor:BSOptionLargeTime\] Let $a>0$, $b\in{\mathbb{R}}$ and set $\Sigma^2:=a+b/\tau$ for $\tau$ large enough so that $a+b/\tau>0$. In the BSM model the following expansion then holds for any $k\in{\mathbb{R}}$ as $\tau$ tends to infinity (the function ${\mathcal{I}}$ is defined in): $${\mathbb{E}}\left({\mathrm{e}}^{X^{(t)}_{\tau}}-{\mathrm{e}}^{k\tau}\right)^+
=
{\mathcal{I}}\left(k,\tau,-\frac{a}{2},\frac{a}{2},0\right)
+
\frac{\phi_{\textrm{BS}}(k,a,b) }{\tau^{1/2}}{\mathrm{e}}^{-\tau\left(V^*_{\textrm{BS}}(k,a)-k\right)}\left(1+\mathcal{O}(\tau^{-1})\right).$$
Connection with large deviations {#sec:LDP}
--------------------------------
Although clear from Theorem \[theorem:largematasympcalls\], we have so far not mentioned the notion of large deviations at all. The leading-order decay of the option price as the maturity tends to infinity gives rise to estimates for large-time probabilities; more precisely, by formally differentiating both sides with respect to the log-strike, one can prove, following a completely analogous proof to [@JR13 Corollary 3.3], that $$-\lim_{\tau\uparrow\infty}\tau^{-1}\log{\mathbb{P}}\left(X^{(t)}_{\tau} \in B\right) = \inf_{z\in B} V^*(z),$$ for any Borel subset $B$ of the real line, namely that $(X^{(t)}_{\tau}/\tau)_{\tau>0}$ satisfies a large deviations principle under ${\mathbb{P}}$ with speed $\tau$ and good rate function $V^*$ as $\tau$ tends to infinity. We refer the reader to the excellent monograph [@DZ93] for more details on large deviations. The theorem actually states a much stronger result here since it provides higher-order estimates, coined ‘sharp large deviations’ in [@BR02] (see also [@Bercu Definition 1.1]). Now, classical methods to prove large deviations, when the the moment generating function is known rely on the Gärtner-Ellis theorem. In mathematical finance, one can consult for instance [@FJSmall], [@FJ09] or [@JMKR] for the small-and large-time behaviour of stochastic volatility models, and [@Pham] for an insightful overview. The Gärtner-Ellis theorem requires, in particular, the limiting logarithmic moment generating function $V$ to be steep at the boundaries of its effective domain. This is indeed the case in Regime ${\mathcal{R}}_1$, but fails to hold in other regimes. The standard proof of this theorem (as detailed in [@DZ93 Chapter 2, Theorem 2.3.6]) clearly holds in the open intervals of the real line where the function $V$ is strictly convex, encompassing basically all occurrences of ${\mathcal{H}}_0$. The other cases, when $V^*$ becomes linear, and the turning points $V'(0)$ and $V'(1)$, however have to be handled with care and solved case by case. Proving sharp large deviations essentially relies on finding a new probability measure under which a rescaled version of the original process converges weakly to some random variable (often Gaussian, but not always); in layman terms, under this new probability measure, the rare events / large deviations of the rescaled variable are not rare any longer. More precisely, fix some log-moneyness $k\in{\mathbb{R}}$; we determine a process $(Z_{\tau,k})_{\tau>0} := ((X^{(t)}_{\tau,k}-k\tau)/\tau^{\alpha})_{\tau>0}$, and a probability measure ${\mathbb{Q}}_{k,\tau,\alpha}$ via $$\frac{{\mathrm{d}}{\mathbb{Q}}_{k,\tau,\alpha}}{{\mathrm{d}}{\mathbb{P}}}:=\exp\left( u^*_{\tau}(k)X_{\tau}^{(t)}-\tau\Lambda^{(t)}_{\tau}(u^*_{\tau}(k))\right),$$ where $u^*_{\tau}(k)$ is the unique solution to the equation $\partial_{u}\Lambda^{(t)}_{\tau}(u_{\tau}^*(k))=k$, with $\Lambda^{(t)}_{\tau}$ denoting the (rescaled) logarithmic moment generating function of $X_\tau^{(t)}$ (See Section \[sec:FwdLimits\]). The characteristic function $\Phi_{\tau,k,\alpha}(u):={\mathbb{E}}^{{\mathbb{Q}}_{k,\tau,\alpha}}({\mathrm{e}}^{{\mathtt{i}}u Z_{\tau,k}})$ has some expansion as $\tau$ tends to infinity. Once this pair has been found, the final part of the proof is to express call prices (or probabilities) as inverse Fourier transforms of the characteristic function multiplied by some kernel, and to use the expansion of $\Phi_{\tau,k,\alpha}(u)$ to determine the desired asymptotics. The main technical issues, and where the different regimes come into play, arise in the properties of the asymptotic behaviour of $\Phi_{\tau,k,\alpha}(u)$ and $u^*_\tau(k)$ as $\tau$ tends to infinity (and on the value $\alpha$ one has to choose). More precise details about the main steps of the proofs are provided at the beginning of Section \[sec:proofsnonsteep\] and in Section \[sec:Genmethlargetime\].
Sharp large deviations, or more generally speaking, probabilistic asymptotic expansions, à la Bahadur-Rao [@Bahadur], can also be proved via other routes. In particular, the framework developed by Benarous [@Benarous] (and applied to the financial context in [@DFJV1; @DFJV]) is an extremely powerful tool to handle Laplace methods on Wiener space and heat kernel expansions. However, the singularity of the square root diffusion (in the SDE for the variance) at the origin falls outside the scope of this theory. Incidentally, Conforti, Deuschel and De Marco [@CDDM] recently proved a (sample path) large deviations principle for the square root diffusion, giving hope for an alternative proof to ours. As explained in [@JR13; @JR15], the forward-start framework on the couple $(X_\tau,V_\tau)_{\tau\geq 0}$, solution to , starting at $(0,v)$, can be seen as a standard option pricing problem on the forward couple $(X_{\tau}^{(t)}, V_{\tau}^{(t)})_{\tau\geq 0}$, solution to the same stochastic differential equation , albeit starting at the point $(0, V_t)$, namely with random initial variance. This additional layer of complexity arising from starting the SDE at a random starting point makes the application of the Benarous framework as well as the Conforti-Deuschel-De Marco result, a fascinating, yet challenging, exercise to consider.
Forward smile asymptotics
=========================
\[sec:largematregfsmaymp\]
We now translate the forward-start option asymptotics obtained above into asymptotics of the forward implied volatility smile. Let us first define the function $v_0^{\infty}:{\mathbb{R}}\times{\mathbb{R}}_{+}\to{\mathbb{R}}$ by $$\label{eq:v0nonsteep}
v_0^{\infty}(k,t) := 2 \left(2 V^*(k) -k+2 \mathcal{Z}(k) \sqrt{V^*(k)(V^*(k)-k)}\right),\quad\text{for all }k\in{\mathbb{R}}, t\in{\mathbb{R}}_+$$ with $\mathcal{Z}:{\mathbb{R}}\to\{-1,1\}$ defined by $
\mathcal{Z}(k) \equiv
{1\hspace{-2.1mm}{1}}_{\{k\in[V'(0),V'(1)]\}} +{\mathrm{sgn}}(\rho\xi-\kappa){1\hspace{-2.1mm}{1}}_{\{k>V'(1)\}}-{1\hspace{-2.1mm}{1}}_{\{k<V'(0)\}}
$ and $V^*$ given in Lemma \[lemma:V\*Characterisation\]. Define the following combinations: $$\begin{array}{llllll}
{\mathcal{P}}_0:
& \chi \equiv \chi_0, \quad & \eta \equiv 1, \quad & \lambda =1, \quad & {\mathcal{R}}(\tau,\lambda) = \mathcal{O}(\tau^{-2\lambda}),\\
\widetilde{{\mathcal{P}}}_\pm:
& \chi \equiv \widetilde{c}_0^{\pm}, \quad & \eta \equiv 1, \quad & \lambda = \frac{2}{3},
\quad & {\mathcal{R}}(\tau,\lambda) = o(\tau^{-\lambda}),\\
{\mathcal{P}}_\pm:
& \chi \equiv c_0^{\pm}, \quad & \eta \equiv 1, \quad & \lambda = \frac{1}{2},
\quad & {\mathcal{R}}(\tau,\lambda)=
\left\{
\begin{array}{ll}
\displaystyle o\left(\tau^{-\lambda}\right),
& \text{if } \mu\ne 1/2,\\
\displaystyle \mathcal{O}\left(\tau^{-2\lambda}\right),
& \text{if } \mu = 1/2,
\end{array}
\right.\\
{\mathcal{P}}_1:
& \chi \equiv0, \quad & \eta \equiv 0, \quad & \lambda = 0,
\quad & {\mathcal{R}}(\tau,\lambda)= o(1).
\end{array}$$ Here $c_0^{\pm}$ and $\widetilde{c}_0^{\pm}$ are given in and and $\chi_0:{\mathbb{R}}\setminus\{V'(0),V'(1)\}\to{\mathbb{R}}$ is defined by $$\label{eq:chi0largetime}
\chi_0(k,t) \equiv
H(u^*(k))+
\log \left(\frac{4 k^2-v_0^{\infty}(k,t)^2}{4 (u^*(k)-1) u^*(k) v_0^{\infty}(k,t)^{3/2} \sqrt{V''(u^*(k))}}\right),$$ with $V$ and $H$ given in and $u^*$ in . We now state the main result of the section, namely an expansion for the forward smile in all regimes and (log) strikes on the real line. The proof is given in Section \[sec:fwdsmileproofnonsteeplargemat\].
\[theorem:HestonLargeMatFwdSmileNonSteep\] The following expansion holds for the forward smile as $\tau$ tends to infinity: $$\sigma_{t,\tau}^2(k\tau)
=
v_0^{\infty}(k,t)+v_1^{\infty}(k,t)\tau^{-\lambda}+{\mathcal{R}}(\tau,\lambda),
\qquad\text{for any }k\in{\mathbb{R}},$$ where $v_1^{\infty}:{\mathbb{R}}\times{\mathbb{R}}_{+}\to{\mathbb{R}}$ is defined by $$v_1^{\infty}(k,t)
:=
\left\{
\begin{array}{ll}
\displaystyle\frac{8 v_0^{\infty}(k,t)^2}{4 k^2-v_0^{\infty}(k,t)^2}\chi(k,t),
\quad & \text{if }k\in{\mathbb{R}}\setminus\{V'(0),V'(1)\},\\
\displaystyle 2\eta(k)\left[1-\sqrt{\frac{v_0^{\infty}(k,t)}{V''(u^*(k))}}\left(1+{\mathrm{sgn}}(k)\left(\frac{V'''(u^*(k))}{6 V''(u^*(k))}-H'(u^*(k))\right)\right)\right],
\quad & \text{if }k\in\{V'(0),V'(1)\},
\end{array}
\right.$$ with the functions $\chi,\eta$, the remainder ${\mathcal{R}}$ and the constant $\lambda$ given by the following combinations[^3]:
- ${\mathfrak{R}}_1$: ${\mathcal{P}}_0$ for $k\in{\mathbb{R}}$;
- ${\mathfrak{R}}_2$: ${\mathcal{P}}_0$ for $k\in(-\infty, V'(u_+^*))$; $\widetilde{{\mathcal{P}}}_+$ for $k=V'(u_+^*)$; ${\mathcal{P}}_+$ for $k\in(V'(u_+^*),+\infty)$;
- ${\mathfrak{R}}_{3a}$: ${\mathcal{P}}_-$ for $k\in(-\infty, V'(u_-^*))$; $\widetilde{{\mathcal{P}}}_-$ for $k=V'(u_-^*)$; ${\mathcal{P}}_0$ for $k\in(V'(u_-^*),+\infty)$;
- ${\mathfrak{R}}_{3b}$: ${\mathcal{P}}_-$ for $k\in(-\infty, V'(u_-^*))$; $\widetilde{{\mathcal{P}}}_-$ for $k=V'(u_-^*)$; ${\mathcal{P}}_0$ for $k\in(V'(u_-^*),V'(1))$; ${\mathcal{P}}_1$ for $k\in[V'(1),+\infty)$;
- ${\mathfrak{R}}_4$: ${\mathcal{P}}_0$ for $k\in(-\infty, V'(1))$; ${\mathcal{P}}_1$ for $k\in[V'(1),+\infty)$.
\[rem:contlargemat\]
(i) In the standard spot case $t=0$, the large-maturity asymptotics of the implied volatility smile was derived in for ${\mathfrak{R}}_1$ only (i.e. assuming $\kappa> \rho\xi$). In the complementary case, ${\mathfrak{R}}_4$, the behaviour of the smile for large strikes become more degenerate, and one cannot specify higher-order asymptotics for $k \geq V'(1)$.
(ii) The zeroth-order term $v_0^{\infty}$ is continuous on ${\mathbb{R}}$ (see also section \[sec:SVIExtended\]), which is not necessarily true for higher-order terms. In ${\mathfrak{R}}_2$, ${\mathfrak{R}}_{3a}$ and ${\mathfrak{R}}_{3b}$, $v_1^{\infty}$ tends to either infinity or zero at the critical strikes $V'(u^*_{+})$ and $V'(u^*_{-})$ (this is discussed further in Section \[sec:nonsteepnumerics\]). In ${\mathfrak{R}}_1$, $v_1^{\infty}$ is continuous on the whole real line.
(iii) Straightforward computations show that $0<v_0^{\infty}(k)<2|k|$ for $k\in{\mathbb{R}}\setminus\left[V'(0), V'(1)\right]$, and $v_0^{\infty}(k)>2|k|$ for $k\in\left(V'(0), V'(1)\right)$, so that $v_1^\infty$ is well defined on ${\mathbb{R}}\setminus\{V'(0), V'(1)\}$. On $(-\infty, V'(u_-^*))\cup(V'(u_+^*),\infty)$, $c_0^{\pm}>0$, so that in Regimes ${\mathfrak{R}}_2$ on $(V'(u^*_{+}), \infty)$ and in ${\mathfrak{R}}_{3b}, {\mathfrak{R}}_{3b}$ on $(-\infty, V'(u_-^*))$, $v_{1}^{\infty}$ is always a positive adjustment to the zero-order term $v_0^{\infty}$; see Figure \[fig:asymeffect\] for an example of this ’convexity effect’.
Theorem \[theorem:HestonLargeMatFwdSmileNonSteep\] displays varying levels of degeneration for high-order forward smile asymptotics. In ${\mathfrak{R}}_1$ one can in principle obtain arbitrarily high-order asymptotics. In ${\mathfrak{R}}_2$, ${\mathfrak{R}}_{3a}$ and ${\mathfrak{R}}_{3b}$ one can only specify the forward smile to arbitrary order if $\mu=1/2$. If this is not the case then we can only specify the forward smile to first order. Now the dynamics of the Heston volatility $\sigma_t:=\sqrt{V_t}$ is given by $
{\mathrm{d}}\sigma_t=\left(\frac{2\mu -1}{8\sigma_t}\xi^2-\frac{\kappa \sigma_t}{2}\right){\mathrm{d}}t +\frac{\xi}{2} {\mathrm{d}}W_t,
$ with $\sigma_0=\sqrt{v}$. If $\mu=1/2$ then the volatility becomes Gaussian, which this corresponds to a specific case of the Schöbel-Zhu stochastic volatility model. So as the Heston volatility dynamics deviate from Gaussian volatility dynamics a certain degeneracy occurs such that one cannot specify high order forward smile asymptotics. Interestingly, a similar degeneracy occurs in [@JR13] for exploding small-maturity Heston forward smile asymptotics and in [@DFJV] when studying the tail probability of the stock price. As proved in [@DFJV], the square-root behaviour of the variance process induces some singularity and hence a fundamentally different behaviour when $\mu \ne 1/2$. In ${\mathfrak{R}}_2$, ${\mathfrak{R}}_{3a}$ and ${\mathfrak{R}}_{3b}$ at the boundary points $V'(u^*_{\pm})$ one cannot specify the forward smile beyond first order for any parameter configurations. This could be because these asymptotic regimes are extreme in the sense that they are transition points between standard and degenerate behaviours and therefore difficult to match with BSM forward volatility. Finally in ${\mathfrak{R}}_{3b}$ and ${\mathfrak{R}}_4$ for $k>V'(1)$ we obtain the most extreme behaviour, in the sense that one cannot specify the forward smile beyond zeroth order. This is however not that surprising since the large correlation regime has fundamentally different behaviour to the BSM model (see also Remark \[rem:largematfwdstartoption\](iii)).
SVI-type limits {#sec:SVIExtended}
---------------
The so-called ’Stochastic Volatility Inspired’ (SVI) parametrisation of the spot implied volatility smile was proposed in [@G04]. As proved in [@GJ11], under the assumption $\kappa>\rho\xi$, the SVI parametrisation turn out to be the true large-maturity limit for the Heston (spot) smile. We now extend these results to the large-maturity forward implied volatility smile. Define the following extended SVI parametrisation $$\sigma^2_{{\mathrm{SVI}}}(k,a,b,r,m,s,i_0,i_1,i_2):=a+b\left(r(k-m)+i_0 \sqrt{i_1(k-m)^2+i_2(k-m)+i_0 s^2}\right),$$ for all $k\in{\mathbb{R}}$ and the constants $$\left\{
\begin{array}{rl}
\omega_1 & := \displaystyle\frac{2\mu}{1-\rho^2}
\left(\sqrt{\left(2\kappa+\xi^2-\rho\xi\right)^2+\xi^2\left(1-\rho^2\right)}-\left(2\kappa+\xi^2-\rho\xi\right)\right),
\qquad
\omega_2:=\frac{\xi}{\kappa\theta},\\
a_{\pm} & := \displaystyle\frac{\kappa\theta }{2 \left(u^*_{\pm}-1\right) u^*_{\pm} \beta _t},
\quad
b_{\pm} := 4 \sqrt{\left(u^*_{\pm}-1\right) u^*_{\pm}},
\quad
r_{\pm} := \frac{2(2 u^*_{\pm}-1)}{b_{\pm}},
\quad m_{\pm}:=\left(u^*_{\pm}-\frac{1}{2}\right)a_{\pm},\\
\widetilde{a} & := -2 \widetilde{m},
\quad
\widetilde{b} := 4 \sqrt{-\widetilde{m}},
\quad
\widetilde{r} := \frac{1}{2\sqrt{-\widetilde{m}}},
\quad
\widetilde{m}:=\mu(\kappa-\rho\xi),
\end{array}
\right.$$ where $u^*_{\pm}$ is defined in and $\beta_t$ in . Define the following combinations: $$\begin{array}{lllllllll}
{\mathcal{S}}_0:
& a=\frac{\omega_1(1-\rho)^2}{2}, & b=\frac{\omega_1\omega_2}{2}, & r=\rho, & m=-\frac{\rho}{\omega_2}, & s=\frac{\sqrt{1-\rho^2}}{\omega_2}, & i_0=1, & i_1=1, & i_2=0,
\\
{\mathcal{S}}_\pm:
& a=a_{\pm}, & b=b_{\pm}, & r=r_{\pm}, & m=m_{\pm}, & s=\frac{1}{8}a_{\pm}, & i_0=-1, & i_1=1, & i_2=0,
\\
{\mathcal{S}}_1:
& a=\widetilde{a}, & b=\widetilde{b}, & r=\widetilde{r}, & m=\widetilde{m}, & s=0, & i_0=1, & i_1=0, & i_2=1.
\end{array}$$ The proof of the following result follows from simple manipulations of the zeroth-order forward smile in Theorem \[theorem:HestonLargeMatFwdSmileNonSteep\] using the characterisation of $V^*$ in Lemma \[lemma:V\*Characterisation\].
\[cor:SVINonSteep\] The pointwise continuous limit $\lim_{\tau\uparrow\infty}\sigma^2_{t,\tau}(k\tau)=\sigma^2_{{\mathrm{SVI}}}(k,a,b,r,m,s,i_0,i_1,i_2)$ exists for $k\in\mathbb{R}$ with constants $a,b,r,m,s,i_0,i_1$ and $i_2$ given by[^4]:
- ${\mathfrak{R}}_1$: ${\mathcal{S}}_0$ for $k\in{\mathbb{R}}$;
- ${\mathfrak{R}}_2$: ${\mathcal{S}}_0$ for $k\in(-\infty, V'(u_+^*))$; ${\mathcal{S}}_+$ for $k\in[V'(u_+^*),+\infty)$;
- ${\mathfrak{R}}_{3a}$: ${\mathcal{S}}_-$ for $k\in(-\infty, V'(u_-^*)]$; ${\mathcal{S}}_0$ for $k\in(V'(u_-^*),+\infty)$;
- ${\mathfrak{R}}_{3b}$: ${\mathcal{S}}_-$ for $k\in(-\infty, V'(u_-^*)]$; ${\mathcal{S}}_0$ for $k\in(V'(u_-^*),V'(1))$; ${\mathcal{S}}_1$ for $k\in[V'(1),+\infty)$;
- ${\mathfrak{R}}_4$: ${\mathcal{S}}_0$ for $k\in(-\infty, V'(1))$; ${\mathcal{S}}_1$ for $k\in[V'(1),+\infty)$.
Financial Interpretation of the large-maturity regimes {#sec:finintuition}
======================================================
The large-maturity regimes in were identified with specific properties of the limiting forward logarithmic moment generating function. Each regime uncovers fundamental properties of the large-maturity forward smile, some of which having been empirically observed by practitioners. These regimes are not merely mathematical curiosities, but their studies reveal particular behaviours and oddities of the model. An intuitive question is how different the large-maturity forward smile and the large-maturity spot smile are. This is a metric that a trader would have a view on and can be analysed using historical data. Because of the ergodic properties of the variance process, at first sight it seems natural to conjecture that the large-maturity spot and forward smiles should be the same at leading order. More specifically, if $\sigma_{\tau}^{(t)}(k)$ denotes the Black-Scholes implied volatility observed at time $t$, i.e. the unique positive solution to the equation ${\mathbb{E}}\left[({\mathrm{e}}^{X_{t+\tau}-X_t}-{\mathrm{e}}^{k})^+|\mathcal{F}_t\right]=C_{\textrm{BS}}(\tau,k,\sigma_{\tau}^{(t)}(k))$, then by definition the forward implied volatility solves the equation $C_{\textrm{BS}}(\tau,k,\sigma_{t,\tau}(k))=\mathbb{E}[C_{\textrm{BS}}(\tau,k,\sigma_{\tau}^{(t)}(k))]$. If we suppose that $\lim_{\tau\uparrow\infty}\sigma_{\tau}^{(t)}(\tau k)=\sigma^{\infty}(k)$, where the function $\sigma^{\infty}$ is independent of $t$ (this is the case in Heston — it does not depend on $V_t$) then it seems reasonable to suppose that $C_{\textrm{BS}}(\tau,k,\sigma_{t,\tau}(k\tau))\approx C_{\textrm{BS}}(\tau,k,\sigma^{\infty}(k))$ and hence that $\sigma_{t,\tau}(k\tau)\approx \sigma^{\infty}(k) \approx \sigma_{\tau}(k\tau)$. It is therefore natural to conjecture (see for example [@Ba]) that the limiting forward smile $\lim_{\tau\uparrow\infty}\sigma_{t,\tau}(k\tau)$ is the same as the limiting spot smile $\lim_{\tau\uparrow\infty}\sigma_{\tau}(k\tau)$. Theorem \[theorem:HestonLargeMatFwdSmileNonSteep\] shows us that this only holds under the good correlation regime ${\mathfrak{R}}_1$, i.e. for correlations ‘close’ to zero. Deviations of the correlation from zero therefore effect how different the large-maturity forward smile is to the large-maturity spot smile.
Consider now the practically relevant (on Equity markets) case of large negative correlation (${\mathcal{R}}_2$). In Figure \[fig:asymeffect\] we compare the two limiting smiles using the zero-order asymptotics in Corollary \[cor:SVINonSteep\] when $\rho<\rho_{-}$. At the critical log-strike $V'(u^*_{+})$, the forward smile becomes more convex than the corresponding spot smile. Interestingly this asymmetric feature has been empirically observed by practitioners [@B04] and is a fundamental property of the model—not only for large-maturities. Quoting Bergomi [@B04] from an empirical analysis: “...the increased convexity (of the forward smile) with respect to today’s smile is larger for $k>0$ than for $k<0$...this is specific to the Heston model.”
![ Here $t=0.5, \tau=2, v=\theta=0.1, \kappa=2, \xi=1, \rho=-0.9$, so that ${\mathfrak{R}}_2$ applies. Circles correspond to the spot smile $K\mapsto\sigma_{\tau}(\log K)$ and squares to the forward smile $K\mapsto\sigma_{t,\tau}(\log K)$ using the zeroth-order asymptotics in Corollary \[cor:SVINonSteep\]. Here $\rho_{-}\approx-0.63$ and ${\mathrm{e}}^{2V'(u^*_{+})}\approx 1.41$.[]{data-label="fig:asymeffect"}](FwdvsSpotAsym.eps)
It is natural to wonder about the origin of this effect. Consider a standard European option with large strike $k>0$. A large number of sample paths of the stock price approach ${\mathrm{e}}^{k}$, but, because of the negative correlation, the corresponding variance tends to be low (the so-called ‘leverage effect’). For a delta-hedged long position this is exactly where we want the variance to be the highest (maximum gamma and vega). Hence there is a tendency for the (spot) implied volatility to be downward sloping for high strikes. On the other hand, consider a forward-start option with large strike $k>0$. Suppose that the variance is large at the forward-start date, $t$. Because of the negative correlation, the stock price will tend to be low here. But this is irrelevant since the stock price is always re-normalised to $1$ at this point. Hence there will be a greater number of paths where the re-normalised stock $S_u/S_{t}$ for $t\leq u\leq t+\tau$ is close to ${\mathrm{e}}^{k}$ and the variance is high relative to the (spot) case discussed above. The relative nature of this effect induces this ‘convexity effect’. When there is large positive correlation $\rho>\rho_{+}>0$ (${\mathcal{R}}_3$), then the large-maturity forward smile is more convex then the large-maturity spot smile for *low* strikes, $k<0$. This is the ‘mirror image’ effect of ${\mathcal{R}}_2$ and follows from similar intuition to above.
When $\rho>\kappa/\xi$ (${\mathcal{R}}_{3b}$ and ${\mathcal{R}}_4$) there is a transition point for large strikes where the smile is upward sloping and possibly concave (See Figures \[fig:LargeCorrelRegime10years\] and \[fig:LargeCorrelRegime20years\]). It is important to note that this effect materialises for both the large-maturity spot and forward smile and is due to the fact that paths where the stock price are high will tend to be accompanied by periods of very high variance because of the positive correlation.
The intuitive arguments given above for each regime are not specific to Heston. A natural conjecture is that all stochastic volatility models where the variance process has a stationary distribution will exhibit similar large-maturity regimes. However, the location of the transition points and the magnitude of the ‘convexity corrections’ may be quite different and model specific.
Numerics {#sec:nonsteepnumerics}
========
We first compare the true Heston forward smile and the asymptotics developed in the paper. We calculate forward-start option prices using the inverse Fourier transform representation in [@L04F Theorem 5.1] and a global adaptive Gauss-Kronrod quadrature scheme. We then compute the forward smile $\sigma_{t,\tau}$ with a simple root-finding algorithm. In Figure \[fig:HestLargeMatComp\] we compare the true forward smile using Fourier inversion and the asymptotic in Theorem \[theorem:HestonLargeMatFwdSmileNonSteep\](i) for the good correlation regime, which was derived in [@JR12]. In Figure \[fig:5yearnonsteep\] we compare the true forward smile using Fourier inversion and the asymptotic in Theorem \[theorem:HestonLargeMatFwdSmileNonSteep\](ii) for the asymmetric negative correlation regime. Higher-order terms are computed using the theoretical results above; these can in principle be extended to higher order, but the formulae become rather cumbersome; numerically, these higher-order computations seem to add little value to the accuracy anyway. In Figure \[fig:AsymNegTrans\] we compare the asymptotic in Theorem \[theorem:HestonLargeMatFwdSmileNonSteep\](ii) for the transition strike $k=V'(u^*_{+})$. Results are all in line with expectations.
In the large correlation regime ${\mathfrak{R}}_4$, we find it more accurate to use Theorem \[theorem:largematasympcalls\] and then numerically invert the price to get the corresponding forward smile (Figures \[fig:LargeCorrelRegime10years\] and \[fig:LargeCorrelRegime20years\]), rather than use the forward smile asymptotic in Theorem \[theorem:HestonLargeMatFwdSmileNonSteep\]. As explained in Remark \[rem:largematfwdstartoption\](iv) the leading-order accuracy of option prices in this regime is poor and higher-order terms embed important distinctions that need to be included. This also explains the poor accuracy of the forward smile asymptotic in Theorem \[theorem:HestonLargeMatFwdSmileNonSteep\] for the large correlation regime. As seen in the proof (Section \[sec:fwdsmileproofnonsteeplargemat\]), the leading-order behaviour of option prices is used to line up strike domains in the BSM and Heston model and then forward smile asymptotics are matched between the models. If the leading-order behaviour is poor, then regardless of the order of the forward smile asymptotic, there will always be a mismatch between the asymptotic forms and the forward smile asymptotic will be poor. Using the approach above bypasses this effect and is extremely accurate already at first order (Figures \[fig:LargeCorrelRegime10years\] and \[fig:LargeCorrelRegime20years\]).
In all but ${\mathfrak{R}}_1$, higher-order terms can approach zero or infinity as the strike approaches the critical values ($V'(u^*_{+})$ or $V'(1)$), separating the asymptotic regimes, and forward smile (and forward-start option price) asymptotics are not continuous there (apart from the zeroth-order term), see also Remark \[rem:contlargemat\](i). Numerically this implies that the asymptotic formula may break down for strikes in a region around the the critical strike. Similar features have been observed in [@JR13] where degenerate asymptotics were derived for the exploding small-maturity Heston forward smile.
Proof of Theorems \[theorem:largematasympcalls\] and \[theorem:HestonLargeMatFwdSmileNonSteep\] {#sec:proofsnonsteep}
===============================================================================================
This section is devoted to the proofs of the option price and implied volatility expansions in Theorems \[theorem:largematasympcalls\] and \[theorem:HestonLargeMatFwdSmileNonSteep\]. We first start (Section \[sec:FwdLimits\]) with some preliminary results of the behaviour of the moment generating function of the forward process $(X_\tau^{(t)})_{\tau>0}$, on which the proofs will rely. The remainder of the section is devoted to the different cases, as follows:
- Section \[sec:ProofConvex\] is the easy case, namely whenever the function $V^*$ in is strictly convex, corresponding to the behaviour ${\mathcal{H}}_0$, except at the points $V'(0)$ and $V'(1)$.
- In Section \[sec:Genmethlargetime\], we outline the general methodology we shall use in all other cases:
- Section \[sec:asymmetricproofs\] tackles the cases ${\mathcal{H}}_\pm$, $\widetilde{{\mathcal{H}}}_\pm$ and ${\mathcal{H}}_2$, corresponding to the function $V^*$ being linear;
- Section \[sec:V0V1limFT\] is devoted to the analysis at the points $V'(0)$ and $V'(1)$
- Section \[sec:fwdsmileproofnonsteeplargemat\] translates the expansions for the option price into expansions for the implied volatility.
Forward logarithmic moment generating function (lmgf) expansion and limiting domain {#sec:FwdLimits}
-----------------------------------------------------------------------------------
For any $t\geq 0$, $\tau>0$, define the re-normalised lmgf of $X_{\tau}^{(t)}$ and its effective domain $\mathcal{D}_{t,\tau}$ by $$\label{eq:MGFFwd}
\Lambda^{(t)}_\tau\left(u\right):=\tau^{-1}\log{\mathbb{E}}\left({\mathrm{e}}^{uX_{\tau}^{(t)}}\right),
\quad\text{for all }
u\in\mathcal{D}_{t,\tau}:=\{u\in{\mathbb{R}}:|\Lambda^{(t)}_\tau\left(u\right)|<\infty\}.$$ A straightforward application of the tower property for expectations yields: $$\label{eq:LambdaTau}
\tau\Lambda^{(t)}_\tau\left(u\right)
=
A\left(u,\tau\right)+\frac{B(u,\tau)v{\mathrm{e}}^{-\kappa t}}{1-2\beta_t B(u,\tau)}
-\mu\log\left(1-2\beta_t B\left(u,\tau\right)\right),
\quad\text{for all }u\in\mathcal{D}_{t,\tau},$$ where $$\begin{aligned}
A(u,\tau) & :=
\frac{\mu}{2}\left(\left(\kappa-\rho\xi u- d\left(u\right)\right)\tau-2\log\left(\frac{1-\gamma\left(u\right)\exp\left(-d\left(u\right)\tau\right)}{1-\gamma\left(u\right)}\right)\right),\nonumber\\
B(u,\tau) & := \frac{\kappa-\rho\xi u-d(u)}{\xi^2}\frac{1-\exp\left(-d\left(u\right)\tau\right)}{1-\gamma\left(u\right)\exp\left(-d\left(u\right)\tau\right)},\nonumber\\
d(u) & := \left(\left(\kappa-\rho\xi u\right)^2+u\left(1-u\right)\xi^2\right)^{1/2},
\qquad
\gamma(u) := \frac{\kappa-\rho\xi u-d\left(u\right)}{\kappa-\rho\xi u+d\left(u\right)},
\qquad
\beta_t := \frac{\xi^2}{4\kappa}\left(1-{\mathrm{e}}^{-\kappa t}\right).
\label{eq:DGammaBeta}\end{aligned}$$ The first step is to characterise the effective domain $\mathcal{D}_{t,\tau}$ for fixed $t\geq0$ as $\tau$ tends to infinity. Recall that the large-maturity regimes are defined in with $u_{\pm}$ and $u^*_{\pm}$ given in .
For fixed $t\geq 0$, $\mathcal{D}_{t,\tau}$ converges (in the set sense) to $\mathcal{D}_{\infty}$ defined in Table \[eq:DInfinityLargeMaturity\], as $\tau$ tends to infinity.
Recall the following facts from [@JR12 Lemma 5.11 and Proposition 5.12] and [@JM12 Proposition 2.3], with the convention that $u_{\pm}^{*}=\pm\infty$ when $t=0$:
(i) $ [0,1] \subset [u_{-},u_{+}] \cap(-\infty,u^*_{+})\subset\mathcal{D}_{t,\tau}$ for all $\tau>0$ if $\rho<0$;
(ii) $ [0,1] \subset [u_{-},u_{+}] \cap(u^*_{-},\infty)\subset\mathcal{D}_{t,\tau}$ for all $\tau>0$ if $0<\rho\leq \kappa/\xi$;
(iii) $ [0,1] \subset [u_{-},u_{+}] \subset\mathcal{D}_{t,\tau}$ for all $\tau>0$ if $\rho=0$;
(iv) $ [0,1] \subset [u_{-},1] \cap (u^*_{-},\infty) \subset\mathcal{D}_{t,\tau}$ for all $\tau>0$ if $\rho> \kappa/\xi$;
(v) $1<u^*_{+}<u_{+}$ if and only if $\rho\in (-1,\rho_{-})$ and $u_{-}<u_{-}^*<0$ if and only if $\rho\in (\rho_{+},1)$. We always have $\rho_{-}\in(-1,0)$ and $\rho_{+}>1/2$. In the latter case it is possible that $\rho_{+}\geq1$ in which case $u_{-}^*\leq u_{-}$.
Then for fixed $t\geq0$, the lemma follows directly from (i)-(iv) in combination with property (v).
The following lemma provides the asymptotic behaviour of $\Lambda_{\tau}^{(t)}$ as $\tau$ tends to infinity. The proof follows the same steps as [@JR12 Lemma 5.13], using the fact that the asset price process $({\mathrm{e}}^{X_t})_{t>0}$ is a true martingale [@AP07 Proposition 2.5], and is therefore omitted.
\[lem:fwdmgflargetauexpansion\] The following expansion holds for the forward lmgf $\Lambda^{(t)}_{\tau}$ defined in ($V$ and $H$ given in ): $$\Lambda^{(t)}_{\tau}(u)=
\left\{
\begin{array}{ll}
\displaystyle V(u)+\tau^{-1}H(u)\left(1+\mathcal{O}\left({\mathrm{e}}^{-d\left(u\right)\tau}\right)\right),
&\quad\text{for all }u\in\mathcal{D}_{\infty}^{o}\setminus\{1\}, \text{ as }\tau\text{ tends to infinity},\\
\displaystyle 0, &\quad\text{for }u=1\text{ and all } \tau>0.
\end{array}
\right.$$
\[rem:largematnonsteep\]
(i) When $\rho>\kappa/\xi$ (${\mathfrak{R}}_{3b}$ and ${\mathfrak{R}}_4$), we have $\lim_{u \uparrow 1}\Lambda^{(t)}_{\tau}(u)=V(1)\neq 0$, so that the limit is not continuous at the right boundary $u=1$. For $\rho\leq\kappa/\xi$ we always have $V(1)=H(1)=0$ and $1\in\mathcal{D}_{\infty}^{o}$.
(ii) For all $u\in\mathcal{D}_{\infty}^{o}$, $d(u)>0$, so that the remainder goes to zero exponentially fast as $\tau$ tends to infinity.
The strictly convex case {#sec:ProofConvex}
------------------------
Let $\overline{k}:=\sup_{a\in\mathcal{D}_{\infty}}V'(a)$ and $\underline{k}:=\inf_{a\in\mathcal{D}_{\infty}}V'(a)$. When $k\in(\underline{k},\overline{k})\setminus\{V'(0), V'(1)\}$, an analogous analysis to [@JR12 Theorem 2.4, Propositions 2.12 and 3.5], essentially based on the strict convexity of $V$ on $(\underline{k}, \overline{k})$, can be carried out and we immediately obtain the following results for forward-start option prices and forward implied volatilities (hence proving Theorems \[theorem:largematasympcalls\] and \[theorem:HestonLargeMatFwdSmileNonSteep\] when ${\mathcal{H}}_0$ holds):
\[lem:fwdstoptsteep\] The following expansions hold for all $k\in(\underline{k},\overline{k})\setminus\{V'(0), V'(1)\}$ as $\tau$ tends to infinity: $$\begin{aligned}
{\mathbb{E}}\left({\mathrm{e}}^{X^{(t)}_{\tau}}-{\mathrm{e}}^{k\tau}\right)^+
&=
{\mathcal{I}}\left(k,\tau,V'(0),V'(1),0\right) +
\frac{\phi_0(k,t)}{\tau^{1/2}}{\mathrm{e}}^{-\tau\left(V ^*(k)-k \right)}
\left(1+\mathcal{O}\left(\tau^{-1}\right)\right), \\
\sigma_{t,\tau}^2(k\tau)&=v_0^{\infty}(k,t)+\frac{8 v_0^{\infty}(k,t)^2}{4 k^2-v_0^{\infty}(k,t)^2}\chi_0(k,t)\tau^{-1}+\mathcal{O}(\tau^{-2}),\end{aligned}$$ with $V^*$ given in Lemma \[lemma:V\*Characterisation\], ${\mathcal{I}}$ and $\phi_0$ in and , $v_0^{\infty}$ in , $\chi_0$ in and $$\label{eq:steepstrikes}
(\underline{k},\overline{k}) =
\left\{
\begin{array}{ll}
{\mathbb{R}}, & \text{in }{\mathfrak{R}}_1,\\
(-\infty,V'(u^*_{+})), & \text{in }{\mathfrak{R}}_2,\\
(V'(u^*_{-}),+\infty), & \text{in }{\mathfrak{R}}_{3a},\\
(V'(u^*_{-}),V'(1)), & \text{in }{\mathfrak{R}}_{3b},\\
(-\infty,V'(1)), & \text{in }{\mathfrak{R}}_4.
\end{array}
\right.$$
We sketch here a quick outline of the proof. For any $k\in(\underline{k},\overline{k})$, the equation $V'(u^*(k))=k$ has a unique solution $u^*(k)$ by strict convexity arguments. Define the random variable $Z_{k,\tau}:=(X_{\tau}^{(t)} - k\tau)/\sqrt{\tau}$; using Fourier transform methods analogous to [@JR12 Theorem 2.4, Proposition 2.12]) the option price reads, for large enough $\tau$, $${\mathbb{E}}\left[{\mathrm{e}}^{X^{(t)}_{\tau}}-{\mathrm{e}}^{k\tau}\right]^+
=
{\mathcal{I}}\left(k,\tau,V'(0),V'(1),0\right) +
\frac{{\mathrm{e}}^{-\tau(k(u^*(k)-1)-V(u^*(k)))}{\mathrm{e}}^{H(u^*(k))}}{2\pi}
\int_{{\mathbb{R}}}\frac{\Phi_{\tau,k}(u)\sqrt{\tau}{\mathrm{d}}u}{[u-{\mathtt{i}}\sqrt{\tau}(u^*(k)-1)][u-{\mathtt{i}}\sqrt{\tau}u^*(k)]},$$ where $\Phi_{\tau,k}(u)\equiv{\mathbb{E}}^{\widetilde{{\mathbb{Q}}}_{k,\tau}}({\mathrm{e}}^{{\mathtt{i}}u Z_{k,\tau}})$ is the characteristic function of $Z_{k,\tau}$ under the new measure $\widetilde{{\mathbb{Q}}}_{k,\tau}$ defined by $\frac{{\mathrm{d}}\widetilde{{\mathbb{Q}}}_{k,\tau}}{{\mathrm{d}}{\mathbb{P}}}:=\exp\left( u^*(k)X_{\tau}^{(t)}-\tau \Lambda_{\tau}^{(t)}(u^*(k)\right)$. Using Lemma \[lem:fwdmgflargetauexpansion\], the proofs of the option price and the forward smile expansions are similar to those of [@JR12 Theorem 2.4 and Proposition 2.12] and [@JR12 Proposition 3.5]. The exact representation of the set $(\underline{k},\overline{k})$ follows from the definition of $\mathcal{D}_{\infty}$ in Table \[eq:DInfinityLargeMaturity\] and the properties of $V$.
Other cases: general methodology {#sec:Genmethlargetime}
--------------------------------
Suppose that $\overline{k}$ (defined in Section \[sec:ProofConvex\]) is finite with $V'(\overline{u})=\overline{k}$. We cannot define a change of measure (as in the proof of Lemma \[lem:fwdstoptsteep\]) by simply replacing $u^*(k)\equiv\overline{u}$ for $k\geq\overline{k}$ since the forward lmgf $\Lambda_{\tau}^{(t)}$ explodes at these points as $\tau$ tends to infinity (see Figure \[fig:mgfexplosion\]).
![ Regime ${\mathfrak{R}}_2$: Circles plot $u\mapsto V(u)$. Squares, diamonds and triangles plot $u\mapsto V(u)+H(u)/\tau$ with $t=1$ and $\tau=2,5,10$. Heston model parameters are $v=0.07$, $\theta=0.07$, $\rho=-0.8$, $\xi=0.65$ and $\kappa=1.5$. Also $\rho_{-}\approx -0.56$, $u^*_{+}\approx 9.72$ and $u_{+}\approx 14.12$. []{data-label="fig:mgfexplosion"}](LargeMatMGFBlowUp.eps)
One of the objectives of the analysis is to understand the explosion rate of the forward lmgf at these boundary points. The key observation is that just before infinity, the forward lmgf $\Lambda_{\tau}^{(t)}$ is still steep on $\mathcal{D}_{t,\tau}^{o}$, and an analogous measure change to the one above can be constructed. We therefore introduce the time-dependent change of measure $$\label{eq:MeasureChange}
\frac{{\mathrm{d}}{\mathbb{Q}}_{k,\tau}}{{\mathrm{d}}{\mathbb{P}}}:=\exp\left( u^*_{\tau}(k)X_{\tau}^{(t)}-\tau\Lambda^{(t)}_{\tau}(u^*_{\tau}(k))\right),$$ where $u^*_{\tau}(k)$ is the unique solution to the equation $\partial_{u}\Lambda^{(t)}_{\tau}(u_{\tau}^*(k))=k$ for $k\geq \overline{k}$. We shall also require that there exists $\tau_1>0$ such that $u^*_{\tau}(k)\in\mathcal{D}_{\infty}^{o}$ for all $\tau>\tau_1$ and $u^*_{\tau}\uparrow\overline{u}$; therefore Lemma \[lem:fwdmgflargetauexpansion\] holds, and we can ignore the exponential remainder ($d(u)>0$ for all $u\in\mathcal{D}_{\infty}^{o}$) so that the equation $\partial_{u}\Lambda^{(t)}_{\tau}(u_{\tau}^*(k))=k$ reduces to [^5] $$\label{eq:u^*tau}
V'\left(u^*_\tau(k)\right)+\tau^{-1}H'\left(u^*_\tau(k)\right)=k.$$ In the analysis below, we will also require $u^*_{\tau}(k)$ to solve and to converge to other points in the domain (not only boundary points). This will be required to derive asymptotics under ${\mathcal{H}}_0$ for the strikes $V'(0)$ and $V'(1)$, where there are no moment explosion issues but rather issues with the non-existence of the limiting Fourier transform (see Section \[sec:V0V1limFT\] for details). We therefore make the following assumption:
\[assump:ustartaueq\] There exists $\tau_1>0$ and a set $\mathcal{A}\subseteq{\mathbb{R}}$ such that for all $\tau>\tau_1$ and $k\in\mathcal{A}$, Equation admits a unique solution $u^*_\tau (k)$ on $\mathcal{D}_{\infty}^{o}$ satisfying $\lim_{\tau\uparrow\infty}u^*_\tau(k) = u^*_{\infty}\in\overline{\mathcal{D}_{\infty}}\cap (u_-,u_+)$.
Under this assumption $|\Lambda^{(t)}_{\tau}(u^*_\tau(k))|$ is finite for $\tau>\tau_1$ and $\mathcal{D}_{\infty}=\lim_{\tau\uparrow \infty} \{u\in{\mathbb{R}}:|\Lambda_{\tau}^{(t)}(u)|<\infty\}$. Also ${\mathrm{d}}{\mathbb{Q}}_{k,\tau}/{\mathrm{d}}{\mathbb{P}}$ is almost surely strictly positive and by definition ${\mathbb{E}}[{\mathrm{d}}{\mathbb{Q}}_{k,\tau}/{\mathrm{d}}{\mathbb{P}}]=1$. Therefore is a valid measure change for sufficiently large $\tau$ and all $k\in\mathcal{A}$.
Our next objective is to prove weak convergence of a rescaled version of the forward price process $(X_{\tau}^{(t)})_{\tau>0}$ under this new measure. To this end define the random variable $Z_{\tau,k,\alpha}:=(X_{\tau}^{(t)} - k\tau)/\tau^{\alpha}\label{eq:ztaukalpha}$ for $k\in\mathcal{A}$ and some $\alpha>0$, with characteristic function $\Phi_{\tau,k,\alpha}:{\mathbb{R}}\to\mathbb{C}$ under ${\mathbb{Q}}_{k,\tau}$: $$\label{eq:CharacNonSteep}
\Phi_{\tau,k,\alpha}(u):={\mathbb{E}}^{{\mathbb{Q}}_{k,\tau}}\left({\mathrm{e}}^{{\mathtt{i}}u Z_{\tau,k,\alpha}}\right).$$ Define now the functions $D:{\mathbb{R}}_{+}^*\times\mathcal{A}\to{\mathbb{R}}$ and $F:{\mathbb{R}}^*_{+}\times \mathcal{A}\times{\mathbb{R}}^*_{+}\to{\mathbb{R}}$ by $$\label{eq:DF}
D(\tau,k):=\exp\Big[{-\tau\Big(k (u^*_{\tau}(k)-1)-V(u^*_{\tau}(k))\Big)}+H(u^*_{\tau}(k))\Big], \quad
F(\tau,k,\alpha):= \frac{1}{2\pi}\int_{{\mathbb{R}}} \Phi_{\tau,k,\alpha}(u)\overline{C_{\tau,k,\alpha}(u)} {\mathrm{d}}u,$$ where $\overline{C_{\tau,k,\alpha}(u)}$ denotes the complex conjugate of $C_{\tau,k,\alpha}$ in , namely: $$\label{eq:CConj}
\overline{C_{\tau,k,\alpha}(u)} =\frac{\tau^{\alpha}}{(u-{\mathtt{i}}\tau^{\alpha}(u^*_{\tau}-1)(u-{\mathtt{i}}\tau^{\alpha}u^*_{\tau})}.$$ The main result here (proved in Appendix \[sec:prooflargematfourtransf\]) is an asymptotic representation for forward-start option prices:
\[lem:nonsteeplem\] Under Assumption \[assump:ustartaueq\], there exists $\beta>0$ such that for all $k\in\mathcal{A}$, as $\tau\uparrow\infty$: $$\begin{aligned}
{\mathbb{E}}\left({\mathrm{e}}^{X_{\tau}^{(t)}}-{\mathrm{e}}^{k\tau}\right)^{+}
= \begin{dcases*}
D(\tau,k) F(\tau,k,\alpha)\left(1+\mathcal{O}({\mathrm{e}}^{-\beta\tau})\right), & if $u_{\tau}^*(k)>1$,\\
(1-{\mathrm{e}}^{k\tau})+ D(\tau,k) F(\tau,k,\alpha)\left(1+\mathcal{O}({\mathrm{e}}^{-\beta\tau})\right), & if $u_{\tau}^*(k)<0$, \\
1+D(\tau,k) F(\tau,k,\alpha) \left(1+\mathcal{O}({\mathrm{e}}^{-\beta\tau})\right), & if $0<u_{\tau}^*(k)<1. $
\end{dcases*}\end{aligned}$$
We shall also need the following result on the behaviour of the characteristic function of $Z_{\tau, k, \alpha}$
\[lem:charactsimp\] Under Assumption \[assump:ustartaueq\] there exists $\beta>0$ such that for any $k\in\mathcal{A}$ as $\tau\uparrow\infty$: $$\Phi_{\tau,k,\alpha}(u) = \exp\Big(
-{\mathtt{i}}u k \tau^{1-\alpha} +
\tau\left(V\left({\mathtt{i}}u \tau^{-\alpha} + u^*_{\tau}\right) - V\left(u^*_{\tau}\right)\right)
+ H\left({\mathtt{i}}u \tau^{-\alpha} + u^*_{\tau}\right) - H\left( u^*_{\tau}\right)
\Big)
\left(1+\mathcal{O}({\mathrm{e}}^{-\beta\tau})\right),$$ where the remainder is uniform in $u$.
Fix $k\in\mathcal{A}$. Analogous arguments to Lemma \[lem:largetimesaddlefwdmgf\](iii) yield that $\Re d\left({\mathtt{i}}u \tau^{-\alpha}+a\right)>d(a)$ for any $a\in\mathcal{D}_{\infty}^o$. Assumption \[assump:ustartaueq\] implies that for all $\tau>\tau_1$, $\Re d\left({\mathtt{i}}u \tau^{-\alpha}+u_{\tau}^*(k)\right)>d(u_{\tau}^*(k))$. It also implies that $u^*_\infty < u_+$, and hence there exists $\delta>0$ and $\tau_2>0$ such that $u_{\tau}^*(k)<u_{+}-\delta$ for all $\tau>\tau_2$. Now, since $d$ is strictly positive and concave on $(u_{-},u_{+})$ and $d(u_{-})=d(u_{+})=0$, we obtain $d(u_{\tau}^*(k))>d(u_{+}-\delta)>0$. This implies that the quantities $\mathcal{O}\left(\exp\left[-d\left(\frac{{\mathtt{i}}u}{\tau^{\alpha}}+u_{\tau}^*(k)\right))\tau\right]\right)$ and $\mathcal{O}\left({\mathrm{e}}^{-d(u_{\tau}^*(k))\tau}\right)$ are all equal to $\mathcal{O}\left({\mathrm{e}}^{-d(u_{+}-\delta)\tau}\right)$ for all $k\in\mathcal{A}$. Using the definition of $Z_{\tau,k,\alpha}$, the change of measure and Lemmas \[lem:fwdmgflargetauexpansion\] and \[lem:largetimesaddlefwdmgf\], we can write $$\begin{aligned}
\log\Phi_{\tau,k,\alpha}(u)
&=\log {\mathbb{E}}^{{\mathbb{Q}}_{k,\tau}}\left[{\mathrm{e}}^{{\mathtt{i}}u Z_{\tau,k,\alpha}}\right]
=\log {\mathbb{E}}\left[
\exp\left(u^*_{\tau}X_{\tau}-\tau\Lambda_{\tau}^{(t)}\left(u^*_{\tau}\right)
+ \frac{{\mathtt{i}}u}{\tau^{\alpha}}\left(X_{\tau} - k\tau\right)\right)\right] \\
&=-{\mathtt{i}}u k \tau^{1-\alpha} +\tau\left(\Lambda_{\tau}^{(t)}\left({\mathtt{i}}u /\tau^{\alpha} + u^*_{\tau}\right)-\Lambda_{\tau}^{(t)}\left( u^*_{\tau}\right)\right) \\
&=-\frac{{\mathtt{i}}u k}{\tau^{\alpha-1}}
+\tau\left[V\left(\frac{{\mathtt{i}}u}{\tau^{\alpha}} + u^*_{\tau}\right) - V(u^*_{\tau})\right]
+ H\left(\frac{{\mathtt{i}}u}{\tau^{\alpha}} + u^*_{\tau}\right)-H\left( u^*_{\tau}\right)
+ \mathcal{O}\left[{\mathrm{e}}^{-d\left({\mathtt{i}}u \tau^{-\alpha}+u_{\tau}^*\right)\tau}\right]
- \mathcal{O}\left({\mathrm{e}}^{-d(u_{\tau}^*)}\tau\right)\\
&= -{\mathtt{i}}u k \tau^{1-\alpha} +\tau\left(V\left({\mathtt{i}}u/ \tau^{\alpha} + u^*_{\tau}\right)-V\left( u^*_{\tau}\right)\right)+H\left({\mathtt{i}}u /\tau^{\alpha} + u^*_{\tau}\right)-H\left( u^*_{\tau}\right)
+\mathcal{O}\left({\mathrm{e}}^{-d(u_{+}-\delta)\tau}\right).\end{aligned}$$ Since $d(u_{+}-\delta)>0$ the remainder tends to zero exponentially fast as $\tau$ tends to infinity. The uniformity of the remainder follows from tedious, yet non-technical, computations showing that the absolute value of the difference between $\log\Phi_{\tau,k,\alpha}(u)$ and its approximation is bounded by a constant independent of $u$ as $\tau$ tends to infinity.
Asymptotics in the case of extreme limiting moment explosions {#sec:asymmetricproofs}
-------------------------------------------------------------
We consider now the cases ${\mathcal{H}}_\pm$, $\widetilde{{\mathcal{H}}}_\pm$ and ${\mathcal{H}}_2$, corresponding to the limiting lmgf $V$ being linear.
\[lemma:u\^\*tau\] Assumption \[assump:ustartaueq\] is verified in the following cases:
(i) ${\mathfrak{R}}_2$ with $\mathcal{A}=[V'(u^*_{+}),\infty)$ and $u_{\infty}^*=u_{+}^*$;
(ii) ${\mathfrak{R}}_{3a}$ and ${\mathfrak{R}}_{3b}$ with $\mathcal{A}=(-\infty,V'(u^*_{-})]$ and $u_{\infty}^*=u_{-}^*$.
(iii) ${\mathfrak{R}}_{3b}$ and ${\mathfrak{R}}_4$ with $\mathcal{A}=(V'(1),\infty]$ and $u_{\infty}^*=1$.
Consider Case (i) and re-write as $H'(u^*_{\tau}(k))/\tau=k-V'(u^*_{\tau}(k))$. Let $k\geq V'(u_{+}^*)$; since $V$ is strictly convex on $(u_{-},u_{+})$, we have $H'(u^*_{\tau}(k))/\tau=k-V'(u^*_{\tau}(k))\geq V'(u_{+}^*)-V'(u^*_{\tau}(k))>0$. We now show that $H'$ has the necessary properties to prove the lemma. The following statements can be proven in a tedious yet straightforward manner (Figure \[fig:HPrimeLargeMatAsymmetric\] provides a visual help):
(i) On $(0,u^*_{+})$ there exists a unique $\bar{u}\in(0,1)$ such that $H'(\bar{u})=0$;
(ii) $H':(\bar{u},u^*_{+})\to{\mathbb{R}}$ is strictly increasing and tends to infinity at $u^*_+$.
Therefore (i) and (ii) imply that a unique solution to exists satisfying the conditions of the lemma with $u^*_{\tau}(k)\in(\bar{u},u^*_{+})$. The function $H'$ is strictly positive on $(\bar{u},u^*_{+})$, and hence for large enough $\tau$, $u^*_{\tau}(k)$ is strictly increasing and bounded above by $u^*_{+}$, and therefore converges to a limit $L\in [\bar{u},u^*_{+}]$. If $L\in [\bar{u},u^*_{+})$, then the continuity of $V'$ and $H'$ and the strict convexity of $V$ implies that $\lim_{\tau\uparrow \infty}V'(u^*_{\tau}(k))+H'(u^*_{\tau}(k))/\tau=V'(L)<V'(u^*_{+})\leq k$, which is a contradiction. Therefore $L=u^*_{+}$, which proves Case (i). Cases (ii) and (iii) are analogous, and the lemma follows.
In the following lemma we derive an asymptotic expansion for $u^*_{\tau}(k)$. This key result will allow us to derive asymptotics for the characteristic function $\Phi_{\tau,k,\alpha}$ as well as other auxiliary quantities needed in the analysis.
\[lemma:a-dynamics\] The following expansions hold for $u^*_\tau(k)$ as $\tau$ tends to infinity:
(i) In Regimes ${\mathfrak{R}}_2$, ${\mathfrak{R}}_{3a}$ and ${\mathfrak{R}}_{3b}$,
(a) under ${\mathcal{H}}_{\pm}$: $
u^*_\tau(k) = u_\pm^*+a_1^\pm(k)\tau^{-1/2} + a_2^\pm(k)\tau^{-1}
+ \mathcal{O}\left(\tau^{-3/2}\right);
$
(b) under $\widetilde{{\mathcal{H}}}_{\pm}$: $
u^*_\tau(k) = u_\pm^* + \widetilde{a}_1^\pm \tau^{-1/3} + \widetilde{a}_2^\pm \tau^{-2/3}
+\mathcal{O}\left(\tau^{-1}\right);
$
(ii) In Regimes ${\mathfrak{R}}_{3b}$ and ${\mathfrak{R}}_4$,
(a) For $k>V'(1)$: $u^*_\tau(k)=1-\frac{\mu}{(k-V'(1))}\tau^{-1}+\mathcal{O}(\tau^{-2})$;
(b) For $k=V'(1)$: $u^*_\tau(k) = 1-\tau^{-1/2}\sqrt{\frac{\mu}{V''(1)}}+\mathcal{O}\left(\tau^{-1}\right)$,
with $a_1^{\pm}$, $a_2^{\pm}$ and $a_3^{\pm}$ defined in and $u_{\pm}^*$ in .
Consider Regime ${\mathfrak{R}}_2$ when ${\mathcal{H}}_+$ is in force, i.e. $k>V'(u_+^*)$, and fix such a $k$. Existence and uniqueness was proved in Lemma \[lemma:u\^\*tau\] and so we assume the result as an ansatz. This implies the following asymptotics as $\tau$ tends to infinity: $$\label{eq:VGammaAsymp}
\left\{
\begin{array}{rl}
V(u^*_\tau(k)) & = \displaystyle V(u^*_+) + \frac{a_1V'(u^*_{+})}{\sqrt{\tau}}
+ \left(\frac{a_1^2 V''(u^*_{+})}{2} + a_2V'(u^*_{+})\right)\frac{1}{\tau}
+\mathcal{O}\left(\frac{1}{\tau^{3/2}}\right), \\
V'(u^*_\tau(k)) & = \displaystyle V'(u^*_{+}) + \frac{a_1V''(u^*_{+})}{\sqrt{\tau}}
+ \left(\frac{a_1^2 V'''(u^*_{+})}{2} + a_2V''(u^*_{+})\right)\frac{1}{\tau}
+\mathcal{O}\left(\frac{1}{\tau^{3/2}}\right), \\
\gamma(u^*_\tau(k)) &
= \displaystyle \gamma(u^*_{+}) + \frac{a_1\gamma'(u^*_{+})}{\sqrt{\tau}}
+ \left(\frac{a_1^2\gamma''(u^*_{+})}{2} + a_2\gamma'(u^*_{+})\right)\frac{1}{\tau}
+\mathcal{O}\left(\frac{1}{\tau^{3/2}}\right),\\
\gamma'(u^*_{\tau}(k))
& = \displaystyle \gamma'(u^*_{+}) + \frac{a_1\gamma''(u^*_{+})}{\sqrt{\tau}}
+ \left(\frac{a_1^2\gamma'''(u^*_{+})}{2}+a_2 \gamma''(u^*_{+})\right)\frac{1}{\tau}
+\mathcal{O}\left(\frac{1}{\tau^{3/2}}\right).
\end{array}
\right.$$ We substitute this into and solve at each order. At the $\tau^{-1/2}$ order we obtain $
a_1^{+}(k)=\pm\frac{{\mathrm{e}}^{-\kappa t/2}}{2 \beta _t} \sqrt{\frac{\kappa\theta v}{V'(u_+^*) \left(k-V'(u_+^*)\right)}},
$ which is well-defined since $k-V'(u_+^*)>0$ and $V'(u_+^*)>0$. We choose the negative root since we require $u^*_\tau\in(0,u^*_{+})\subset\mathcal{D}_{\infty}^o$ for $\tau$ large enough. In a tedious yet straightforward manner we continue the procedure and iteratively solve at each order (the next equation is linear in $a_2$) to derive the asymptotic expansion in the lemma. The other cases follow from analogous arguments.
We now derive asymptotic expansions for $\Phi_{\tau,k,\alpha}$. The expansions will be used in the next section to derive asymptotics for the function $F$ in .
\[lemma:CharactNonSteepExp\] The following expansions hold as $\tau$ tends to infinity:
(i) In Regimes ${\mathfrak{R}}_2$, ${\mathfrak{R}}_{3a}$ and ${\mathfrak{R}}_{3b}$,
(a) under ${\mathcal{H}}_{\pm}$: $
\Phi_{\tau,k,3/4}(u) = {\mathrm{e}}^{-\zeta^2_\pm(k)u^2/2}\left(1+\max(1,u^s)\mathcal{O}\left(\tau^{-1/4}\right)\right);
$
(b) under $\widetilde{{\mathcal{H}}}_{\pm}$: $
\Phi_{\tau,k,1/2}(u) =
{\mathrm{e}}^{-3 V''\left(u_\pm^*\right)u^2/2}\left(1+\max(1,u^s)\mathcal{O}\left(\tau^{-1/6}\right)\right);
$
(ii) In Regimes ${\mathfrak{R}}_{3b}$ and ${\mathfrak{R}}_4$,
(a) For $k>V'(1)$: $
\Phi_{\tau,k,1}\left(u\right) = \exp\left(-{\mathtt{i}}u (k-V'(1) ) - \frac{u^2 V''(1)}{2\tau}\right)\left(1-{\mathtt{i}}u \frac{(k-V'(1))}{\mu}\right)^{-\mu}
(1+\max(1,u^s)\mathcal{O}(\tau^{-1}));
$
(b) For $k=V'(1)$: $
\Phi_{\tau,k,1/2}\left(u\right) = \exp\left(-{\mathtt{i}}u \sqrt{\mu V''(1)} - \frac{u^2 V''(1)}{2}\right)\left(1-{\mathtt{i}}u \sqrt{\frac{V''(1) }{\mu}}\right)^{-\mu}
(1+\max(1,u^s)\mathcal{O}(\tau^{-1/2})),
$
for some integer $s$ different from one line to the other. Recall that $\Phi_{\tau,k,\alpha}$ is defined in and $\zeta^2_{\pm}$ in . Furthermore, as $\tau$ tends to infinity the remainders in (i) and (ii)(b) are uniform in $u$ for $|u|<\tau^{1/6}$ and the remainder in (ii)(a) is uniform in $u$ for $|u|<\tau^{2/3}$.
\[rem:limitprops\]
(i) In Case (i)(a), $Z_{\tau,k,3/4}$ converges weakly to a centred Gaussian with variance $\zeta^2_{\pm}(k)$ when ${\mathcal{H}}_\pm$ holds.
(ii) In Case (i)(b), $Z_{\tau,k,1/2}$ converges weakly a centred Gaussian with variance $3V''(u_{+})$ when $\widetilde{{\mathcal{H}}}_\pm$ holds.
(iii) In Case(ii)(a), $Z_{\tau,k,1}$ converges weakly to the zero-mean random variable $\Xi-\gamma$, where $\gamma:=k-V'(1)$ and $\Xi$ is a Gamma random variable with shape parameter $\mu$ and scale parameter $\beta:=(k-V'(1))/\mu$. Lemma \[lem:Gammaasymplargecorrel\] implies that the limiting characteristic function satisfies $\int_{-\infty}^{\infty}\left(1-{\mathtt{i}}u\beta\right)^{-\mu}{\mathrm{e}}^{-V''(1)u^2/(2\tau)}u^j{\mathrm{d}}u=\mathcal{O}(1)$ for any $j\in\mathbb{N}\cup\{0\}$.
(iv) In Case(ii)(b), $Z_{\tau,k,1/2}$ converges weakly to the zero-mean random variable $\Psi+\Xi$, where $\Psi$ is Gaussian with mean $-\sqrt{\mu V''(1)}$ and variance $V''(1)$ and $\Xi$ is Gamma-distributed with shape $\mu$ and scale $\sqrt{V''(1)/\mu}$.
We now prove Case (i)(a) in Regime ${\mathfrak{R}}_2$, as the proofs in all other cases are similar. In the forthcoming analysis we will be interested in the asymptotics of the function $e_{\tau}$ defined by $$\label{eq:etau}
e_{\tau}(k) \equiv \sqrt{\tau}\left(\kappa\theta -2 \beta _t V(u^*_\tau(k))\right).$$ Under ${\mathfrak{R}}_2$, in Case (i)(a), $\left(\kappa\theta -2 \beta _t V(u^*_\tau)\right)$ tends to zero as $\tau$ tends to infinity, so that it is not immediately clear what happens to $e_{\tau}$ for large $\tau$. But the asymptotic behaviour of $V(u^*_\tau)$ in and the definition yield the following result:
\[lemma:e-dynamics\] Assume ${\mathfrak{R}}_2$ and ${\mathcal{H}}_+$. Then the expansion $e_{\tau}(k) = e_0^+(k) + e_1^+(k)\tau^{-1/2}+\mathcal{O}\left(\tau^{-1}\right)$ holds as $\tau$ tends to infinity, with $e_0$ and $e_1$ defined in and $u_{\pm}^*$ in .
Consider Regime ${\mathfrak{R}}_2$ when ${\mathcal{H}}_+$ is in force, i.e. $k>V'(u_+^*)$, and fix such a $k$, and for ease of notation drop the superscripts and $k$-dependence. Lemma \[lem:charactsimp\] yields $$\log\Phi_{\tau,k}(u) = -{\mathtt{i}}u k \tau^{1/4} +\tau\left(V\left(\frac{{\mathtt{i}}u}{ \tau^{3/4}} + u^*_{\tau}\right)-V\left( u^*_{\tau}\right)\right)+H\left(\frac{{\mathtt{i}}u }{\tau^{3/4}} + u^*_{\tau}\right)-H\left( u^*_{\tau}\right)
+\mathcal{O}(\tau^{-1/4}).
\label{eq:phiztaunonsteepderiv}$$ Using Lemma \[lemma:a-dynamics\], we have the Taylor expansion (similar to ) $$\begin{aligned}
\label{eq:Vustarttaui}
V\left(u^*_{\tau}+{\mathtt{i}}u/\tau^{3/4}\right)=\frac{\kappa\theta}{2\beta_t}+\frac{a_1V'}{\sqrt{\tau}}+\frac{{\mathtt{i}}u V'}{\tau^{3/4}}+\left(\frac{V''a_1^2}{2}+V' a_2\right)\frac{1}{\tau}+\frac{{\mathtt{i}}u a_1 V''}{\tau^{5/4}}+\mathcal{O}\left(\frac{1}{\tau^{3/2}}\right),\end{aligned}$$ as $\tau$ tends to infinity, where $V$, $V'$ and $V''$ are evaluated at $u^*_{+}$. Using we further have $$\begin{aligned}
V\left(u^*_{\tau}+{\mathtt{i}}u/\tau^{3/4}\right)-V\left(u^*_{\tau}\right)
& = \frac{{\mathtt{i}}u V'(u^*_{+})}{\tau^{3/4}}+\frac{{\mathtt{i}}u a_1 V''(u^*_{+})}{\tau^{5/4}}+\mathcal{O}\left(\frac{1}{\tau^{3/2}}\right),
\label{eq:Vustarttauidiff}\\
\gamma\left(u^*_{\tau}+{\mathtt{i}}u/\tau^{3/4}\right)
& = \gamma(u^*_{+})+\frac{a_1\gamma'(u^*_{+})}{\sqrt{\tau}}+\frac{{\mathtt{i}}u \gamma'(u^*_{+})}{\tau^{3/4}}+\mathcal{O}\left(\frac{1}{\tau}\right).
\label{eq:Gammaustartauuidiff}\end{aligned}$$ We now study the behaviour of $H\left({\mathtt{i}}u /\tau^{3/4} + u^*_{\tau}\right)$, where $H$ is defined in . Using Lemma \[lemma:e-dynamics\] and the expansion for large $\tau$, we first note that $$\begin{aligned}
\label{eq:den}
e_{\tau} - 2 \beta _t \sqrt{\tau} \left[V(u^*_{\tau}+\frac{{\mathtt{i}}u}{\tau^{3/4}})-V(u^*_{\tau})\right]
& = e_0-\frac{2\beta_t{\mathtt{i}}u V'}{\tau^{1/4}}+\frac{e_1}{\sqrt{\tau}}
-\frac{2\beta_t {\mathtt{i}}u a_1 V''}{\tau^{3/4}}+\mathcal{O}\left(\frac{1}{\tau}\right),\end{aligned}$$ with $e_{\tau}$ defined in . Together with , this implies $$\begin{aligned}
\label{eq:HFirstTerm1}
\frac{v {\mathrm{e}}^{-\kappa t}V(u^*_{\tau}+{\mathtt{i}}u/ \tau^{3/4})}{\kappa\theta -2 \beta _t V(u^*_{\tau}+{\mathtt{i}}u/ \tau^{3/4})}
& = \frac{\sqrt{\tau}v {\mathrm{e}}^{-\kappa t} V(u^*_{\tau}+{\mathtt{i}}u/ \tau^{3/4})}{e_{\tau}-2 \beta _t \sqrt{\tau} \left(V(u^*_{\tau}+{\mathtt{i}}u/ \tau^{3/4})-V(u^*_{\tau})\right)}\nonumber\\
& = \frac{\kappa\theta v {\mathrm{e}}^{-\kappa t}\sqrt{\tau}}{2 e_0 \beta _t}+\frac{{\mathtt{i}}\kappa\theta u v e^{-\kappa t} V' \tau^{1/4}}{e_0^2 }+v {\mathrm{e}}^{-\kappa t} \left(\frac{a_1 V'}{e_0}-\frac{e_1 \kappa\theta }{2 e_0^2 \beta _t}\right) -\frac{\zeta_{+}^2u^2}{2}+\mathcal{O}\left(\frac{1}{\tau^{1/4}}\right),\end{aligned}$$ with $\zeta_{+}$ defined in . Substituting $e_0$ in into the second term in we find $$\label{eq:simplify}
\frac{{\mathtt{i}}\kappa\theta u v e^{-\kappa t} V' }{e_0^2 }={\mathtt{i}}u \left(k-V'\right).$$ Following a similar procedure using $e_\tau$ we establish for large $\tau$ that $$\begin{aligned}
\label{eq:HFirstTerm2}
&\frac{v {\mathrm{e}}^{-\kappa t}V(u^*_{\tau})}{\kappa\theta -2 \beta _t V(u^*_{\tau})}=\frac{\kappa\theta v {\mathrm{e}}^{-\kappa t}\sqrt{\tau}}{2 e_0 \beta _t}+v {\mathrm{e}}^{-\kappa t} \left(\frac{a_1 V'}{e_0}-\frac{e_1 \kappa\theta }{2 e_0^2 \beta _t}\right) +\mathcal{O}\left(\frac{1}{\sqrt{\tau}}\right),\end{aligned}$$ and combining , and we find that $$\label{eq:HFirstTerm12}
\frac{V(u^*_{\tau}+{\mathtt{i}}u/ \tau^{3/4})v {\mathrm{e}}^{-\kappa t}}{\kappa\theta -2 \beta _t V(u^*_{\tau}+{\mathtt{i}}u/ \tau^{3/4})}
- \frac{V(u^*_{\tau})v {\mathrm{e}}^{-\kappa t}}{\kappa\theta -2 \beta _t V(u^*_{\tau})}
={\mathtt{i}}u \left(k-V'\right)\tau^{1/4}-\frac{\zeta_+^2u^2}{2} +\mathcal{O}\left(\frac{1}{\tau^{1/4}}\right).$$ We now analyse the second term of $\exp(H({\mathtt{i}}u /\tau^{3/4} + u^*_{\tau})-H(u^*_{\tau}))$. We first re-write this term as $$\begin{aligned}
&\exp\left(-\mu\log \left(\frac{\kappa\theta-2 \beta _t V(u^*_{\tau}+{\mathtt{i}}u/ \tau^{3/4}) }{\kappa\theta \left(1-\gamma \left(u^*_{\tau}+{\mathtt{i}}u/ \tau^{3/4}\right)\right)}\right)
+\mu\log \left(\frac{\kappa\theta-2 \beta _t V(u^*_{\tau}) }{\kappa\theta \left(1-\gamma \left(u^*_{\tau}\right)\right)}\right)\right) \\ \nonumber
&=\left(\left(\frac{\kappa\theta-2 \beta _t V(u^*_{\tau}+{\mathtt{i}}u/ \tau^{3/4})}{\kappa\theta-2 \beta _t V(u^*_{\tau})}\right)\left(\frac{1-\gamma \left(u^*_{\tau}+{\mathtt{i}}u/ \tau^{3/4}\right)}{1-\gamma \left(u^*_{\tau}\right)}\right)^{-1}\right)^{-\mu},\end{aligned}$$ and deal with each of the multiplicative terms separately. For the first term we re-write it as $$\begin{aligned}
\frac{\kappa\theta-2 \beta _t V(u^*_{\tau}+{\mathtt{i}}u/ \tau^{3/4})}{\kappa\theta-2 \beta _t V(u^*_{\tau})}=\frac{e_{\tau}-2 \beta _t \sqrt{\tau} \left(V(u^*_{\tau}+{\mathtt{i}}u/ \tau^{3/4})-V(u^*_{\tau})\right)}{e_\tau},\end{aligned}$$ and then we use the asymptotics of $e_{\tau}$ in \[lemma:e-dynamics\] and equation to find that as $\tau$ tends to infinity, $$\begin{aligned}
\frac{\kappa\theta-2 \beta _t V(u^*_{\tau}+{\mathtt{i}}u/ \tau^{3/4})}{\kappa\theta-2 \beta _t V(u^*_{\tau})}=1+ \mathcal{O}\left(\frac{1}{\tau^{1/4}}\right).\end{aligned}$$ For the second term we use the asymptotics in and to find that for large $\tau$ $$\left(\frac{1-\gamma \left(u^*_{\tau}+{\mathtt{i}}u/ \tau^{3/4}\right)}{1-\gamma \left(u^*_{\tau}\right)}\right)^{-1}=\left(\frac{1-\left(\gamma+a_1\gamma'/\sqrt{\tau}+{\mathtt{i}}u \gamma'/\tau^{3/4}+\mathcal{O}(1/\tau)\right)}{1-\left(\gamma+a_1\gamma'/\sqrt{\tau}+\mathcal{O}(1/\tau)\right)}\right)^{-1}
=1+\mathcal{O}(1/\tau^{3/4}).$$ It then follows that for the second term of $\exp(H({\mathtt{i}}u /\tau^{3/4} + u^*_{\tau})-H(u^*_{\tau}))$ that for large $\tau$ we have $$\label{eq:HSecTerm12}
\exp\left(-\mu\log \left(\frac{\kappa\theta-2 \beta _t V(u^*_{\tau}+{\mathtt{i}}u/ \psi_\tau) }{\kappa\theta \left(1-\gamma \left(u^*_{\tau}+{\mathtt{i}}u/ \psi_\tau\right)\right)}\right)
+\mu\log \left(\frac{\kappa\theta-2 \beta _t V(u^*_{\tau}) }{\kappa\theta \left(1-\gamma \left(u^*_{\tau}\right)\right)}\right)\right) = 1+\mathcal{O}\left(\frac{1}{\tau^{1/4}}\right).$$ Further as $\tau$ tends to infinity, the equality implies $$\begin{aligned}
\label{eq:Vustarttauidifftau}
\tau\left(V(u^*_{\tau}+{\mathtt{i}}u/ \tau^{3/4})-V(u^*_{\tau})\right)
={\mathtt{i}}u V'(u^*)\tau^{1/4}+\mathcal{O}(\tau^{-1/4}).\end{aligned}$$ Combining , and into completes the proof. The proof of the uniformity of the remainders and the existence of the integer $s$ follow the same lines as the proof of [@BR02 Lemmas 7.1, 7.2].
In order to derive complete asymptotic expansions we still need to derive expansions for $D$ and $F$ in . This is the purpose of this section. We first derive an expansion for $D$ which gives the leading-order decay of large-maturity out-of-the-money options:
\[lemma:V\*Asymptotics\] The following expansions hold as $\tau$ tends to infinity:
(i) In Regimes ${\mathfrak{R}}_2$, ${\mathfrak{R}}_{3a}$ and ${\mathfrak{R}}_{3b}$,
(a) under ${\mathcal{H}}_\pm$: $
D(\tau,k) =
\exp\left(-\tau (V^*(k)-k)+\sqrt{\tau}c_0^{\pm}(k)+c_1^{\pm}(k)\right)
\tau^{\mu/2}c_2^{\pm}(k)(1+\mathcal{O}(\tau^{-1/2}));
$
(b) under $\widetilde{{\mathcal{H}}}_\pm$: $
D(\tau,k) =
\exp\left(-\tau (V^*(k)-k)+\tau^{1/3}c_0^\pm + c_1^{\pm}\right)\tau^{\mu/3}c_2^{\pm}
(1+\mathcal{O}(\tau^{-1/3}));
$
(ii) In Regimes ${\mathfrak{R}}_{3b}$ and ${\mathfrak{R}}_4$,
(a) For $k>V'(1)$: $
D(\tau,k)=
\exp\left(-\tau (V^*(k)-k)+\mu+g_0\right)
\left(\frac{2(k-V'(1))(\kappa-\rho\xi )^2}{ \left(\kappa\theta -2 V(1) \beta _t\right)}\right)^{\mu}
\tau^{\mu}
(1+\mathcal{O}(\tau^{-1}));
$
(b) For $k=V'(1)$: $
D(\tau,k)=
\exp\left(-\tau (V^*(k)-k)+\mu/2+g_0\right)
\left(\frac{2(\kappa-\rho\xi )^2 \sqrt{V''(1) \mu}}{\left(\kappa\theta -2 V(1) \beta _t\right)}\right)^{\mu}
\tau^{\mu/2}(1+\mathcal{O}(\tau^{-1/2})).
$
where $c_0$, $c_1$ and $c_2$ in , $g_0$ in and $V^*$ is characterised explicitly in Lemma \[lemma:V\*Characterisation\].
Consider Regime ${\mathfrak{R}}_2$ in Case(i)(a) (namely when ${\mathcal{H}}_+$ holds), and again for ease of notation drop the superscripts and $k$-dependence. We now use Lemma \[lemma:a-dynamics\] and to write for large $\tau$: $$\begin{aligned}
\label{eq:step1asymp}
{\mathrm{e}}^{-\tau\left(ku^*_{\tau}-V(u^*_{\tau})\right)}
& = \exp\left[-\tau(k u^*_{+}-V(u^*_{+}))-\sqrt{\tau}a_1(k-V')+r_0-a_2 k +\mathcal{O}(\tau^{-1/2})\right] \\ \nonumber
& = {\mathrm{e}}^{-\tau V^*(k)-\sqrt{\tau}a_1(k-V')+r_0-a_2 k}\left[1+\mathcal{O}(\tau^{-1/2})\right],\end{aligned}$$ with $r_0 := \frac{1}{2}V''a_1^2 + V' a_2$ and where we have used the characterisation of $V^*$ given in Lemma \[lemma:V\*Characterisation\]. We now study the asymptotics of $H(u^*_\tau)$. Using the definition of $e_\tau$ in we write $$\label{eq:step2asymp}
{\mathrm{e}}^{{H(u^*_{\tau})}}
= \exp\left(\frac{V(u^*_{\tau})v {\mathrm{e}}^{-\kappa t}}{\kappa\theta -2 \beta _t V(u^*_{\tau})}
\right)
\left[\frac{\kappa\theta-2 \beta _t V(u^*_{\tau})}{\kappa\theta \left(1-\gamma\left(u^*_{\tau}\right)\right)}\right]^{-\mu}
= \tau^{\frac{\mu}{2}}
\exp\left(\frac{V(u^*_{\tau})v {\mathrm{e}}^{-\kappa t}}{\kappa\theta -2 \beta _t V(u^*_{\tau})}
\right)
\left[\frac{e_{\tau}}{\kappa\theta \left(1-\gamma \left(u^*_{\tau}\right)\right)}\right]^{-\mu},$$ and deal with each of these terms in turn. Now by we have, as $\tau$ tends to infinity, $$\begin{aligned}
\label{eq:step3asymp}
&\frac{v {\mathrm{e}}^{-\kappa t}V(u^*_{\tau})}{\kappa\theta -2 \beta _t V(u^*_{\tau})}=\frac{\kappa\theta v {\mathrm{e}}^{-\kappa t}\sqrt{\tau}}{2 e_0 \beta _t}+v {\mathrm{e}}^{-\kappa t} \left(\frac{a_1 V'}{e_0}-\frac{e_1 \kappa\theta}{2 e_0^2 \beta _t}\right) +\mathcal{O}\left(\frac{1}{\sqrt{\tau}}\right).\end{aligned}$$ Using the asymptotics of $e_{\tau}$ given in Lemma \[lemma:e-dynamics\] and those of $\gamma$ in we find $$\begin{aligned}
\label{eq:step4asymp}
\left(\frac{e_{\tau}}{\kappa\theta\left(1-\gamma \left(u^*_{\tau}\right)\right)}\right)^{-\mu}
= \left(\frac{e_{0}+e_1/\sqrt{\tau}+\mathcal{O}\left(1/\tau\right)}{\kappa\theta \left(1-\gamma\right)+\kappa\theta a_1\gamma'/\sqrt{\tau}+\mathcal{O}\left(1/\tau\right)}\right)^{-\mu}
=
\left(\frac{\kappa\theta\left(1-\gamma \right)}{e_0}\right)^{\mu}
\left(1+\mathcal{O}\left(\frac{1}{\sqrt{\tau}}\right)\right).\end{aligned}$$ Using the definition of $e_0$ in , note the simplification $-a_1(k-V')+\frac{\kappa\theta v {\mathrm{e}}^{-\kappa t}}{2 e_0 \beta _t}=-2 a_1 (k-V')$. Combining this, , , and we find that $$D(\tau,k):=
{\mathrm{e}}^{-\tau\left(k(u^*_{\tau}-1)-\Lambda_{\tau}^{(t)}\left(u^*_{\tau}\right)\right)}
=
\exp\left(-\tau (V^*(k)-k)+\sqrt{\tau}c_0^{+}+c_1^{+}\right)\tau^{\mu/2}c_2^{+}
(1+\mathcal{O}(\tau^{-1/2})),$$ with $c_0^+$, $c_1^+$ and $c_2^+$ in . All other cases follows in an analogous fashion and this completes the proof.
In Lemma \[lemma:FourierAsymptotics\] below we provide asymptotic expansions for the function $F$ in . However, we first need the following technical result, the proof of which can be found in [@BR02 Lemma 7.3]. Let $p$ denote the density of a Gamma random variable with shape $\lambda$ and scale $\nu$, and $\widehat{p}$ the corresponding characteristic function: $$\label{eq:gammadensitycharact}
p(x) \equiv \frac{1}{\Gamma(\lambda)\nu^{\lambda}}x^{\lambda-1}{\mathrm{e}}^{-x/\nu}{1\hspace{-2.1mm}{1}}_{\{x>0\}},
\qquad \widehat{p}(u) \equiv (1-{\mathtt{i}}\nu u)^{-\lambda}.$$
\[lem:Gammaasymplargecorrel\] The following expansion holds as $\tau$ tends to infinity: $$\int_{{\mathbb{R}}} \exp\left({-{\mathtt{i}}\gamma u - \frac{\sigma^2u^2}{2\tau} }\right)u^{\beta} \widehat{p}(\gamma u){\mathrm{d}}u
=\sum_{r=0}^{q}\frac{2\pi \sigma^{2r}}{{\mathtt{i}}^{\beta} \gamma^{2r+\beta+1}2^{r}r! \tau^{r}}p^{(2r+\beta)}(1)+\mathcal{O}\left(\frac{1}{\tau^{q+1}}\right),$$ with $\gamma,\nu,\lambda\in{\mathbb{R}}^*_{+}$, $\beta\in\mathbb{N}\cup\{0\}$, $q\in\mathbb{N}$ and $p^{(n)}$ denoting the $n$-th derivative of the Gamma density $p$.
\[lemma:FourierAsymptotics\] The following expansions hold as $\tau$ tends to infinity (with $\zeta_{\pm}$ in and $u^*_{\pm}$ in ):
(i) In Regimes ${\mathfrak{R}}_2$, ${\mathfrak{R}}_{3a}$ and ${\mathfrak{R}}_{3b}$,
(a) under ${\mathcal{H}}_\pm$: $
F(\tau,k,3/4) =
\frac{\tau^{-3/4}}{\zeta_\pm(k) u^*_+(u^*_{\pm}-1)\sqrt{2\pi}}(1+\mathcal{O}(\tau^{-1/2}))
$;
(b) under $\widetilde{{\mathcal{H}}}_\pm$: $
F(\tau,k,1/2) =
\frac{\tau^{-1/2}}{u^*_{\pm}(u^*_{\pm}-1)\sqrt{6\pi V''(u^*_{\pm})}}(1+\mathcal{O}(\tau^{-1/3}))
$;
(ii) In Regimes ${\mathfrak{R}}_{3b}$ and ${\mathfrak{R}}_4$,
(a) For $k>V'(1)$: $F(\tau,k,1)
=-\frac{{\mathrm{e}}^{-\mu}\mu^{\mu}}{\Gamma(1+\mu)}(1+\mathcal{O}(\tau^{-1}))$;
(b) For $k=V'(1)$: $F(\tau,k,1/2)
=-\frac{{\mathrm{e}}^{-\mu/2}(\mu/2)^{\mu/2}}{2\Gamma(1+\mu/2)}(1+\mathcal{O}(\tau^{-1/2}))$.
Again, we only consider here Regime ${\mathfrak{R}}_2$ under ${\mathcal{H}}_+$ in Case (i)(a). Using the asymptotics of $u^*_{\tau}$ given in Lemma \[lemma:a-dynamics\], we can Taylor expand for large $\tau$ to obtain $
\overline{C(\tau,k,3/4)}
=\frac{\tau^{-3/4}}{(u_+^*-1) u_+^*}(1+\mathcal{O}(\tau^{-1/2})),
$ where the remainder $\mathcal{O}(\tau^{-1/2})$ is uniform in $u$ as soon as $u=\mathcal{O}(\tau^{3/4})$. Combining this with the characteristic function asymptotics in Lemma \[lemma:CharactNonSteepExp\] we find that for large $\tau$, $
F(\tau,k,3/4)
=\frac{1}{\tau ^{3/4} \left(u_+^*-1\right) u_+^*}\int_{{\mathbb{R}}}\exp\left(-\frac{\zeta_+^2(k)u^2}{2}\right)(1+\mathcal{O}(\tau^{-1/4})){\mathrm{d}}u.
$ Using Lemma \[lem:expsmalllargetime\], there exists $\beta>0$ such that as $\tau$ tends to infinity we can write this integral as $$\begin{aligned}
\int_{-\infty}^{\infty}\exp\left(-\frac{\zeta_+^2(k)u^2}{2}\right)\left(1+\mathcal{O}(\tau^{-1/4})\right){\mathrm{d}}u
&=\int_{-\tau^{3/4}}^{\tau^{3/4}}\exp\left(-\frac{\zeta_+^2(k)u^2}{2}\right)
\left(1+\mathcal{O}(\tau^{-1/4})\right){\mathrm{d}}u
+\mathcal{O}({\mathrm{e}}^{-\beta \tau}) \\
&=\int_{-\tau^{3/4}}^{\tau^{3/4}}\exp\left(-\frac{\zeta_+^2(k)u^2}{2}\right){\mathrm{d}}u
\left(1+\mathcal{O}(\tau^{-1/4})\right)
+\mathcal{O}({\mathrm{e}}^{-\beta \tau}) \\
&=\int_{{\mathbb{R}}}\exp\left(-\frac{\zeta_+^2(k)u^2}{2}\right){\mathrm{d}}u \left(1+\mathcal{O}(\tau^{-1/4})\right)
= \frac{\sqrt{2\pi}}{|\zeta(k)|}\left(1+\mathcal{O}(\tau^{-1/4})\right).\end{aligned}$$ The second line follows from Lemma \[lemma:CharactNonSteepExp\] and in the third line we have used that the tail estimate for the Gaussian integral is exponentially small and absorbed this into the remainder $\mathcal{O}(\tau^{-1/4})$. By extending the analysis to higher order the $\mathcal{O}(\tau^{-1/4})$ term is actually zero and the next non-trivial term is $\mathcal{O}(\tau^{-1/2})$. For brevity we omit the analysis and we give the remainder as $\mathcal{O}(\tau^{-1/2})$ in the lemma. Case (i)(b) follows from analogous arguments to above and we now move onto Case (ii)(a). Using the asymptotics of $u^*_{\tau}$ in Lemma \[lemma:a-dynamics\] we have $
\overline{C(\tau,k,1)}
= -\left(\frac{\mu}{\nu(k)} - {\mathtt{i}}u\right)^{-1}+\mathcal{O}(\tau^{-1})
= \frac{-\nu(k)}{\mu}\left(1-\frac{{\mathtt{i}}u \nu(k)}{\mu}\right)^{-1}+\mathcal{O}(\tau^{-1}),
$ where we set $\nu(k):=k-V'(1)$ and the remainder $\mathcal{O}(\tau^{-1})$ is uniform in $u$ as soon as $u=\mathcal{O}(\tau)$. Using the characteristic function asymptotics in Lemma \[lemma:CharactNonSteepExp\] and Lemma \[lem:expsmalllargetime\], there exists $\beta>0$ such that as $\tau$ tends to infinity: $$\begin{aligned}
F(\tau,k,1)
& =\frac{-\nu}{2\pi \mu}\int_{-\tau}^{\tau} \exp\left(-{\mathtt{i}}u \nu - \frac{u^2 V''(1)}{2 \tau}\right)
\left(1-\frac{{\mathtt{i}}u \nu}{\mu}\right)^{-\mu}
\left[\left(1-\frac{{\mathtt{i}}u \nu}{\mu}\right)^{-1}+\mathcal{O}\left(\tau^{-1}\right)\right] {\mathrm{d}}u
+ \mathcal{O}({\mathrm{e}}^{-\beta \tau}) \nonumber \\
& = \frac{-\nu}{2\pi \mu}\int_{-\tau}^{\tau} \exp\left(-{\mathtt{i}}u \nu - \frac{u^2 V''(1)}{2 \tau}\right)
\left(1-\frac{{\mathtt{i}}u \nu}{\mu}\right)^{-1-\mu} {\mathrm{d}}u \left[1+\mathcal{O}\left(\tau^{-1}\right)\right]+\mathcal{O}({\mathrm{e}}^{-\beta \tau}).\label{eq:largecorrelextend}\end{aligned}$$ The second line follows from Lemma \[lemma:CharactNonSteepExp\] and Remark \[rem:limitprops\](iii). Further we note that $$\begin{aligned}
\left|\int_{|u|>\tau} \exp\left(-{\mathtt{i}}u \nu - \frac{u^2 V''(1)}{2 \tau}\right)
\left(1-\frac{{\mathtt{i}}u \nu}{\mu}\right)^{-1-\mu} {\mathrm{d}}u \right|
&\leq \tau \int_{|z|>1} {\mathrm{e}}^{-\frac{1}{2}\tau z^2 V''(1)}
\left(1+\frac{ z^2 \tau^2 \nu^2}{\mu^2}\right)^{-1-\mu} {\mathrm{d}}z \\
&\leq \tau \int_{|z|>1} {\mathrm{e}}^{-\frac{1}{2}\tau z^2 V''(1)} {\mathrm{d}}z
=\mathcal{O}\left({\mathrm{e}}^{-\Delta\tau}\right), \end{aligned}$$ for some $\Delta>0$ as $\tau$ tends to infinity. Combining this with we can write $$\begin{aligned}
F(\tau,k,1)
& = \frac{-\nu}{2\pi \mu}\int_{-\infty}^{\infty} \exp\left(-{\mathtt{i}}u \nu - \frac{u^2 V''(1)}{2 \tau}\right)
\left(1-\frac{{\mathtt{i}}u \nu}{\mu}\right)^{-1-\mu} {\mathrm{d}}u \left[1+\mathcal{O}\left(\tau^{-1}\right)\right],\\
& = \left(-\frac{{\mathrm{e}}^{-\mu}\mu^{\mu}}{\Gamma(1+\mu)}
+\mathcal{O}\left(\tau^{-1}\right)\right)\left[1+\mathcal{O}\left(\tau^{-1}\right)\right],\end{aligned}$$ where we have absorbed the exponential remainder into $\mathcal{O}(\tau^{-1})$, and where the second line follows from Lemma \[lem:Gammaasymplargecorrel\]. We now prove (ii)(b). Using the asymptotics of $u^*_{\tau}$ for large $\tau$ in Lemma \[lemma:a-dynamics\], we obtain $\overline{C(\tau,k,1/2)} = \frac{1}{a_1(1+{\mathtt{i}}u/a_1)}+\mathcal{O}(\tau^{-1/2})$, with $a_1=-\sqrt{\frac{\mu}{V''(1)}}$ and where the remainder $\mathcal{O}(\tau^{-1/2})$ is uniform in $u$ as soon as $u=\mathcal{O}(\tau^{1/2})$. Using the characteristic function asymptotics in Lemma \[lemma:CharactNonSteepExp\] and analogous arguments as above we have the following expansion for large $\tau$: $$F(\tau,k,1/2)
=\frac{1}{2\pi a_1}
\int_{{\mathbb{R}}} \frac{\exp\left({\mathtt{i}}u a_1 V''(1) -\frac{1}{2}u^2 V''(1)\right)}{(1+{\mathtt{i}}u /a_1)^{1+\mu}}{\mathrm{d}}u \left(1+\mathcal{O}\left(\tau^{-1/2}\right)\right).$$ Let $n$ and $\widehat{n}$ denote the Gaussian density and characteristic function with zero mean and variance $V''(1)$. Using , we have $$\int_{{\mathbb{R}}}{\mathrm{e}}^{-{\mathtt{i}}\omega u}\widehat{n}(u)\widehat{p}(u){\mathrm{d}}u
=2\pi \mathcal{F}^{-1}(\widehat{n}(u)\widehat{p}(u))(\omega)
=2\pi \mathcal{F}^{-1}(\mathcal{F}(n * p))
=2 \pi \int_{0}^{\infty} n(\omega-y) p(y) {\mathrm{d}}y,$$ so that $$\frac{1}{2\pi a_1}\int_{{\mathbb{R}}} \frac{\exp\left({\mathtt{i}}u a_1 V''(1) - \frac{1}{2}u^2 V''(1)\right)}{(1+{\mathtt{i}}u /a_1)^{1+\mu}}{\mathrm{d}}u
=\frac{1}{a_1}\int_{0}^{\infty}n(-a_1 V''(1) - y) p(y) {\mathrm{d}}y.$$ This integral can now be computed in closed form and the result follows after simplification using the definition of $a_1$ and the duplication formula for the Gamma function.
Asymptotics in the case of non-existence of the limiting Fourier transform {#sec:V0V1limFT}
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In this section, we are interested in the cases where $k\in\{V'(0),V'(1)\}$ whenever ${\mathcal{H}}_0$ is in force, which corresponds to all the regimes except ${\mathcal{R}}_{3b}$ and ${\mathcal{R}}_4$ at $V'(1)$. In these cases, the limiting Fourier transform is undefined at these points. We show here however that the methodology of Section \[sec:Genmethlargetime\] can still be applied, and we start by verifying Assumption \[assump:ustartaueq\]. The following quantity will be of primary importance: $$\label{eq:Upsilon}
\Upsilon(a) := 1 + \frac{a \rho\xi}{\kappa-\rho\xi}{\mathrm{e}}^{\kappa t},$$ for $a\in\{0,1\}$, and it is straightforward to check that $\Upsilon$ is well defined whenever ${\mathcal{H}}_0$ is in force.
\[lem:v0v1ustarexist\] Let $a\in\{0,1\}$ and assume that $v \ne \theta\Upsilon(a)$. Then, whenever ${\mathcal{H}}_0$ holds, Assumption \[assump:ustartaueq\] is satisfied with $\mathcal{A}=\{V'(a)\}$ and $u_{\infty}^*=a$. Additionally, if $v<\theta\Upsilon(a)$, then there exists $\tau_1^*>0$ such that $u_{\tau}^*(k)<0$ if $a=0$ and $u_{\tau}^*(k)>1$ if $a=1$ for all $\tau>\tau_1^*$, and if $v>\theta\Upsilon(a)$, then there exists $\tau_1^*>0$ such that $u_{\tau}^*(k)\in (0,1)$ for all $\tau>\tau_1^*$;
Recall that the function $H$ is defined in . We first prove the lemma in the case $a=0$, in which case $\Upsilon(0) = 1$. Note that $H'(0)>0(<0)$ if and only if $v/\theta<1(>1)$ and $H'(0)=0$ if and only if $v=\theta$. Now let $k=V'(0)$ and $v<\theta$ and consider the equation $H'(u)/\tau=V'(0)-V'(u)$. Since $H'$ is continuous $H'$ is strictly positive in some neighbourhood of zero. In order for the right-hand side to be positive we require our solution to be in $(-\delta_0,0)$ for some $\delta>0$ since $V$ is strictly convex. So let $\delta_1\in(-\delta_0,0)$. With the right-hand side locked at $V'(0)-V'(\delta_1)>0$ we then adjust $\tau$ accordingly so that $H'(\delta_1)/\tau_1=V'(0)-V'(\delta_1)$. We then set $u_{\tau_1}=\delta_1$. It is clear that for $\tau>\tau_1$ there always exists a unique solution to this equation and furthermore $u^*_{\tau}$ is strictly increasing and bounded above by zero. The limit has to be zero otherwise the continuity of $V'$ and $H'$ implies $\lim_{\tau\uparrow\infty}V'(u^*_{\tau})+H'(u^*_{\tau})/\tau=V'(\lim_{\tau\uparrow\infty}u^*_{\tau})<V'(0)$, a contradiction. A similar analysis holds for $v>\theta$ and in this case $u^*_{\tau}$ converges to zero from above. When $v=\theta$ then $u^*_{\tau}=0$ for all $\tau>0$ (i.e. it is a fixed point). Analogous arguments hold for $k=V'(1)$: $H'(1)>0(<0)$ if and only if $v/\theta > \Upsilon(1)$ ($<\Upsilon(1)$) and $H'(1)=0$ if and only if $v/\theta=\Upsilon(1)$. If $v/\theta>\Upsilon(1)$ ($<\Upsilon(1)$) then $u_\tau^*$ converges to $1$ from below (above) and when $v/\theta = \Upsilon(1)$, $u_{\tau}^*=1$ for all $\tau>0$.
We now provide expansions for $u^*_{\tau}$ and the characteristic function $\Phi_{\tau,k,1/2}$. Define the following quantities: $$\label{eq:v0v1apam}
\alpha_0:=
\frac{2{\mathrm{e}}^{-\kappa t}(v-\theta)\kappa}{\theta((2\kappa-\xi)^2+4\kappa\xi(1-\rho^2))}
,\quad
\alpha_1:=
\frac{2{\mathrm{e}}^{-\kappa t}(\kappa-\rho\xi)^2}{\kappa\theta ((2\kappa-\xi)^2+4\kappa\xi(1-\rho^2))}
(\theta\Upsilon(1) - v).$$ The proofs are analogous to Lemma \[lemma:a-dynamics\] and \[lemma:CharactNonSteepExp\] and omitted. Note that the asymptotics are in agreement with the properties of $u^*_{\tau}(k)$ in Lemma \[lem:v0v1ustarexist\].
\[lem:v0v1\] Let $a\in\{0,1\}$ and assume that $v \ne \theta\Upsilon(a)$. When $k=V'(a)$, the following expansions hold as $\tau$ tends to infinity (for some integer $s$): $$\begin{aligned}
u^*_\tau(k) & = a + \alpha_a\tau^{-1} + \mathcal{O}\left(\tau^{-2}\right),\qquad
D(\tau,k) = {\mathrm{e}}^{\tau V'(a)(1-a)}\left(1+\mathcal{O}\left(\tau^{-1}\right)\right),\\
\Phi_{\tau,k,1/2}(u) & = {\mathrm{e}}^{-\frac{1}{2}u^2V''(a)}
\left(1+\left({\mathtt{i}}\alpha_a u V''(a) - \frac{{\mathtt{i}}u^3 V'''(a)}{6}+{\mathtt{i}}u H'(a)\right)\tau^{-1/2}+\max(1,u^s)\mathcal{O}(\tau^{-1})\right).\end{aligned}$$
We now define the following functions from ${\mathbb{R}}^*\times\{0,1\}$ to ${\mathbb{R}}$ and then provide expansions for $F$ in : $$\label{eq:varpiauxil}
\left\{
\begin{array}{ll}
\varpi_1(q,a) & :=
{\mathrm{e}}^{q^2 V''(a)/2}\pi \left[2 {\mathcal{N}}(q\sqrt{V''(a)})-1-{\mathrm{sgn}}(q)\right],\\
\varpi_2(q,a) & :=
-\sqrt{\frac{2\pi}{V''(a)}}+{\mathrm{e}}^{q^2 V''(a)/2}\pi q \left[1+{\mathrm{sgn}}(q)-2 {\mathcal{N}}\left(q\sqrt{V''(a)}\right)\right],\\
\varpi_3(q,a) & :=
\frac{\sqrt{2\pi}(q^2V''(a)-1)}{(V''(a))^{3/2}}
- 2\pi q^2 |q| \exp\left(\frac{q^2V''(a)}{2}\right){\mathcal{N}}\left(-|q|\sqrt{V''(a)}\right),\\
\varpi(q,a) & :=
\frac{\varpi_1(q)}{2\pi}+\frac{1}{2\pi \sqrt{\tau}}\left((a_1V''(a)+H'(a))\varpi_2(q,a)
+\frac{V'''(a) \varpi_3(q,a)}{6}\right).
\end{array}
\right.$$
\[lem:v0v1F\] Let $a\in\{0,1\}$ and assume that $v \ne \theta\Upsilon(a)$. Then the following expansions hold as $\tau$ tends to infinity (with $a_0$ given in ): $$F\left(\tau,V'(a),1/2\right) =
\frac{{1\hspace{-2.1mm}{1}}_{\{a=1\}}{\mathrm{sgn}}(\alpha_1) - {1\hspace{-2.1mm}{1}}_{\{a=0\}}{\mathrm{sgn}}(\alpha_0)}{2}
- \frac{1}{\sqrt{2\pi \tau V''(a)}}
\left[1 + {\mathrm{sgn}}(a)\left(\frac{V'''(a)}{6V''(a)}-H'(a)\right)\right]\left[1+\mathcal{O}\left(\frac{1}{\tau}\right)\right].$$
Consider the case $a=0$. Set $P(u):={\mathtt{i}}\alpha_0 u V''(0)-{\mathtt{i}}u^3 V'''(0)/6+{\mathtt{i}}u H'(0)$ and note that $\overline{C(u,\tau,1/2)}:=\frac{1}{\left(-{\mathtt{i}}u-u^*_{\tau}\sqrt{\tau}\right)} - \frac{1}{\left(-{\mathtt{i}}u-u^*_{\tau}\sqrt{\tau}+\sqrt{\tau} \right)}$. Using Lemma \[lem:v0v1\] and the definition of $F$ in : $$\label{eq:v0v1Fint}
F(\tau,V'(0),1/2) =
\frac{1}{2\pi} \int_{{\mathbb{R}}}{\mathrm{e}}^{-V''(0) u^2/2} \overline{C(u,\tau)} (1+P(u)\tau^{-1/2}+\mathcal{O}(\tau^{-1})) {\mathrm{d}}u.$$ We cannot now simply Taylor expand $\overline{C(u,\tau,1/2)}$ for small $\tau$ and integrate term by term since in the limit $\overline{C(u,\tau,1/2)}$ is not $L^1$. This was the reason for introducing the time dependent term $u^*_{\tau}(V'(0))$ so that the Fourier transform exists for any $\tau>0$. Indeed, we easily see that $\overline{C(u,\tau,1/2)}=- {\mathtt{i}}/u + \mathcal{O}(\tau^{-1/2})$. We therefore integrate these terms directly and then compute the asymptotics as $\tau$ tends to infinity. Note first that since $|\overline{C(u,\tau,1/2)}|=\mathcal{O}(1)$, then $\overline{C(u,\tau,1/2)} (1+P(u)\tau^{-1/2}+\mathcal{O}(\tau^{-1}))
=\overline{C(u,\tau,1/2)}(1+P(u)\tau^{-1/2})+\mathcal{O}(\tau^{-1})$. Further for any $q\neq0$, $\int_{{\mathbb{R}}}{\mathrm{e}}^{-V'' (0)u^2/2}\frac{1}{-{\mathtt{i}}u - q} {\mathrm{d}}u =\varpi_1(q,0)$, $
\int_{{\mathbb{R}}}{\mathrm{e}}^{-V''(0) u^2/2}\frac{{\mathtt{i}}u}{-{\mathtt{i}}u - q} {\mathrm{d}}u =\varpi_2(q,0)$ and $\int_{{\mathbb{R}}}{\mathrm{e}}^{-V''(0) u^2/2}\frac{{\mathtt{i}}u^3}{-{\mathtt{i}}u - q} {\mathrm{d}}u =\varpi_3(q,0)$. Now using the definition of $\varpi$ in and exchanging the integrals and the asymptotic (an analogous justification to the proof of Lemma \[lemma:FourierAsymptotics\](i)) in we obtain $$F(\tau,V'(0),1/2)
=\varpi\left(u^*_{\tau}\sqrt{\tau},0\right) - \varpi\left((u^*_{\tau}-1)\sqrt{\tau},0\right)
+ \mathcal{O}\left(\tau^{-1}\right).$$ Using Lemma \[lem:v0v1\] and asymptotics of the cumulative normal distribution function we compute: $$\begin{aligned}
\varpi\left(u^*_{\tau}\sqrt{\tau},0\right) & = \varpi
\left(\alpha_0\tau^{-1/2}+\mathcal{O}\left(\tau^{-3/2}\right),0\right)
=-\frac{{\mathrm{sgn}}(\alpha_0)}{2}
-\frac{6H'(0)V''(0)-V'''(0)}{6\sqrt{2\pi}(V''(0))^{3/2}\sqrt{\tau}}
+\mathcal{O}\left(\tau^{-1}\right),\\
\varpi((u^*_{\tau}-1)\sqrt{\tau},0) & =
\varpi\left(-\sqrt{\tau} + \alpha_0 \tau^{-1/2}+\mathcal{O}\left(\tau^{-3/2}\right),0\right)
=\frac{1}{\sqrt{2\pi V''(0) \tau}}
+\mathcal{O}\left(\tau^{-1}\right).\end{aligned}$$ The case $a=1$ is analogous using $\varpi(\cdot, 1)$ and the lemma follows.
Consider ${\mathfrak{R}}_{3b}$ and ${\mathfrak{R}}_4$ with $k=V'(1)$ in Section \[sec:asymmetricproofs\]. Here also $u^*_{\tau}(k)$ tends to $1$ and it is natural to wonder why we did not encounter the same issues with the limiting Fourier transform as we did in the present section. The reason this was not a concern was that the speed of convergence ($\tau^{-1/2}$) of $u^*_{\tau}$ to $1$ was the same as that of the random variable $Z_{\tau,k,1/2}$ to its limiting value. Intuitively the lack of steepness of the limiting lmgf was more important than any issues with the limiting Fourier transform. In the present section steepness is not a concern, but again in the limit the Fourier transform is not defined. This becomes the dominant effect since $u^*_{\tau}(k)$ converges to $1$ at a rate of $\tau^{-1}$ while the re-scaled random variable $Z_{\tau,k,1/2}$ converges to its limit at the rate $\tau^{-1/2}$.
Forward smile asymptotics: Theorem \[theorem:HestonLargeMatFwdSmileNonSteep\] {#sec:fwdsmileproofnonsteeplargemat}
-----------------------------------------------------------------------------
The general machinery to translate option price asymptotics into implied volatility asymptotics has been fully developed by Gao and Lee [@GL11]. We simply outline the main steps here. There are two main steps to determine forward smile asymptotics: (i) choose the correct root for the zeroth-order term in order to line up the domains (and hence functional forms) in Theorem \[theorem:HestonLargeMatFwdSmileNonSteep\] and Corollary \[Cor:BSOptionLargeTime\]; (ii) match the asymptotics.
We illustrate this with a few cases from Theorem \[theorem:HestonLargeMatFwdSmileNonSteep\]. Consider ${\mathfrak{R}}_{3b}$ and ${\mathfrak{R}}_4$ with $k>V'(1)$. We have asymptotics for forward-start call option prices for $k>V'(1)$ in Theorem \[theorem:HestonLargeMatFwdSmileNonSteep\]. The only BSM regime in Corollary \[Cor:BSOptionLargeTime\] where this holds is where $k\in(-\Sigma^2/2,\Sigma^2/2)$. We now substitute our asymptotics for $\Sigma$ and at leading order we have the requirement: $k>V'(1)$ implies that $k\in(-v_0(k)/2,v_0(k)/2)$. We then need to check that this holds only for the correct root $v_0$ used in the theorem. Note that we only use the leading order condition here since if $k\in(-v_0(k)/2,v_0(k)/2)$ then there will always exist a $\tau_1>0$ such that $k\in(-v_0(k)/2+o(1),v_0(k)/2+o(1))$, for $\tau>\tau_1$. Suppose now that we choose the root not as given in Theorem \[theorem:HestonLargeMatFwdSmileNonSteep\]. Then for the upper bound we get the condition $kV(1)>0$. Since $V(1)<0$ we require $V'(1)<0$ and then this only holds for $V'(1)<k<0$. This already contradicts $k>V'(1)$ but let’s continue since it may be true for a more limited range of $k$. The lower bound gives the condition $(k-V(1))k>0$. But the upper bound implied that we needed $V'(1)<k<0$ and so further $k<V'(1)$. Therefore $V'(1)<k<V(1)$ but this can never hold since simple computations show that $V'(1)>V(1)$. Now let’s choose the root according to the theorem. For the upper bound we get the condition $
-\sqrt{(V^*-k)^2+k(V^*(k)-k)}< V^*(k)-k = -V(1)>0
$ and this is always true. For the lower bound we get the condition $-\sqrt{(V^*-k)^2+k(V^*(k)-k)}< V^*(k)=k-V(1)$ and this is always true for $k>V'(1)$ since $V'(1)>V(1)$. This shows that we have chosen the correct root for the zeroth-order term and we then simply match asymptotics for higher order terms.
As a second example consider ${\mathfrak{R}}_2$ and $k>V'(u_{+}^*)$ in Theorem \[theorem:HestonLargeMatFwdSmileNonSteep\]. Substituting the ansatz $\sigma_{t,\tau}^2(k\tau)=v_0^{\infty}(k)+v_{1}^{\infty}(k,t)\tau^{-1/2}
+ v_{2}^{\infty}(k,t)\tau^{-1} + \mathcal{O}(\tau^{-3/2})$ into the BSM asymptotics for forward-start call options in Corollary \[Cor:BSOptionLargeTime\], we find $${\mathbb{E}}\left({\mathrm{e}}^{X_{\tau}^{(t)}}-{\mathrm{e}}^{k\tau}\right)^+
= \exp\left(-\alpha_0^{\infty} \tau +\alpha_1^{\infty} \sqrt{\tau }+\alpha_2^{\infty}\right)
\frac{4v_0^{3/2} }{\sqrt{2 \pi\tau} \left(4k^2-v_0^2\right)}\left(1+\mathcal{O}\left(\tau^{-1/2}\right)\right),$$ where $\alpha_0^{\infty}:= \frac{k^2}{2v_0^{\infty}}-\frac{k}{2}+\frac{v_0^{\infty}}{8}$, and $\alpha_1^{\infty}:= v_1^{\infty} \frac{4 k^2-v_0^2 }{8 v_0^2}$ and $\alpha_2^{\infty}$ is a constant, the exact value does not matter here. We now equate orders with Theorem \[theorem:largematasympcalls\]. At the zeroth order we get two solutions and since $V'(u^*_{+})>V(1)$, we choose the negative root such that matches the domains in Corollary \[Cor:BSOptionLargeTime\] and Theorem \[theorem:largematasympcalls\] for large $\tau$ (using similar arguments as above). At the first order we solve for $v_{1}^{\infty}$. But now at the second order, we can only solve for higher order terms if $\mu=1/2$ due to the term $\tau^{\mu/2-3/4} = \tau^{-1/2}$ in the forward-start option asymptotics in Theorem \[theorem:largematasympcalls\]. All other cases follow analogously.
Proof of Lemma \[lem:nonsteeplem\] {#sec:prooflargematfourtransf}
==================================
Define the function $C_{\tau,k,\alpha}:{\mathbb{R}}\to\mathbb{C}$ by $$\label{eq:Cdeflargetime}
C_{\tau,k,\alpha}(u):=\frac{\tau^{\alpha}}{(u+{\mathtt{i}}\tau^{\alpha}(u^*_{\tau}-1)(u+{\mathtt{i}}\tau^{\alpha}u^*_{\tau})},$$ with its conjugate given in .
\[lem:L1lem\] There exists $\tau^*_0>0$ such that $\int_{{\mathbb{R}}}|\Phi_{\tau,k,\alpha}(u) \overline{C_{\tau,k,\alpha}(u)}| {\mathrm{d}}u
<\infty$ for all $\tau>\tau^*_0$, $k\in\mathcal{A}$, $u^*_{\tau}(k)\not\in\{0,1\}$.
We compute: $$\begin{aligned}
\nonumber
\int_{{\mathbb{R}}}\left| \Phi_{\tau,k}(u) \overline{C_{\tau,k,\alpha}(u)} \right| {\mathrm{d}}u
&=
\int_{|u|\leq \tau^{\alpha}}\left| \Phi_{\tau,k,\alpha}(u) \overline{C_{\tau,k,\alpha}(u)} \right| {\mathrm{d}}u
+\int_{|u|>\tau^{\alpha}}\left| \Phi_{\tau,k,\alpha}(u) \overline{C_{\tau,k,\alpha}(u)} \right| {\mathrm{d}}u \\ \label{eq:finiteineq}
&\leq \frac{2\tau^{-\alpha}}{|u^*_{\tau}(k)(u_{\tau}^*(k)-1)|} \int_{|u|\leq \tau^{\alpha}}\left| \Phi_{\tau,k,\alpha}(u) \right| {\mathrm{d}}u
+\int_{|u|>1}\frac{{\mathrm{d}}u}{u^2},\end{aligned}$$ where the inequality follows from the simple bounds $$\left|\overline{C_{\tau,k,\alpha}(u)} \right|\leq \frac{\tau^{-2\alpha}}{|u^*_{\tau}(k)(u_{\tau}^*(k)-1)|},\text{ for all }|u| \leq \tau^{\alpha},
\qquad
\left|\overline{C_{\tau,k,\alpha}(u)} \right|\leq \frac{\tau^{\alpha}}{u^2}
\qquad
\text{and}\qquad
|\Phi_{\tau,k,\alpha}|\leq1.$$ Finally is finite since $u^*_{\tau}(k)\neq 1$, $u^*_{\tau}(k)\neq 0$.
We denote the convolution of two functions $f,h\in L^1({\mathbb{R}})$ by $(f\ast g)(x):=\int_{{\mathbb{R}}}f(x-y)g(y) {\mathrm{d}}y$, and recall that $(f\ast g)\in L^1({\mathbb{R}})$. For $f\in L^1({\mathbb{R}})$, we denote its Fourier transform by $(\mathcal{F}f)(u):=\int_{{\mathbb{R}}}{\mathrm{e}}^{{\mathtt{i}}u x}f(x) {\mathrm{d}}x$ and the inverse Fourier transform by $
(\mathcal{F}^{-1}h)(x):=\frac{1}{2\pi}\int_{{\mathbb{R}}}{\mathrm{e}}^{-{\mathtt{i}}u x}h(u) {\mathrm{d}}u.
$ For $j=1,2,3$, define the functions $g_j:{\mathbb{R}}_+^2\to{\mathbb{R}}_+$ by $$g_j(x,y):=\left\{
\begin{array}{ll}
(x-y)^+, \quad & \text{if } j=1, \\
(y-x)^+,\quad & \text{if } j=2,\\
\min(x,y),\quad & \text{if } j=3.
\end{array} \right.$$ and define $\widetilde{g}_j:{\mathbb{R}}\to{\mathbb{R}}_+$ by $\label{eq:gjtilde}
\widetilde{g}_j(z) := \exp\left(-u^*_{\tau}(k) z \tau^{\alpha}\right)g_j({\mathrm{e}}^{z \tau^{\alpha}},1)$. Recall the ${\mathbb{Q}}_{k,\tau}$-measure defined in and the random variable $Z_{k,\tau,\alpha}$ defined on page . We now have the following result:
\[lem:optpricerepnonsteep\] There exists $\tau^*_1>0$ such that for all $k\in\mathcal{A}$ and $\tau>\tau^*_1$: $$\begin{aligned}
~\label{eq:Parslargetime}
{\mathbb{E}}^{{\mathbb{Q}}_{k,\tau}}\left[\widetilde{g}_j(Z_{k,\tau,\alpha})\right]
= \begin{dcases*}
\frac{1}{2\pi}\int_{{\mathbb{R}}} \Phi_{\tau,k,\alpha}(u)\overline{C_{\tau,k,\alpha}(u)} {\mathrm{d}}u, & if $j=1,u_{\tau}^*(k)>1$,\\
\frac{1}{2\pi}\int_{{\mathbb{R}}}\Phi_{\tau,k,\alpha}(u)\overline{C_{\tau,k,\alpha}(u)} {\mathrm{d}}u, & if $j=2, u_{\tau}^*(k)<0$, \\
-\frac{1}{2\pi}\int_{{\mathbb{R}}}\Phi_{\tau,k,\alpha}(u)\overline{C_{\tau,k,\alpha}(u)} {\mathrm{d}}u, & if $j=3, 0<u_{\tau}^*(k)<1. $
\end{dcases*}\end{aligned}$$
Assuming (for now) that $\widetilde{g}_j\in L^1({\mathbb{R}})$, we have for any $u\in{\mathbb{R}}$, $
\left(\mathcal{F}\widetilde{g}_{j}\right)(u)
:=\int_{{\mathbb{R}}}\widetilde{g}_j(z){\mathrm{e}}^{{\mathtt{i}}u z}{\mathrm{d}}z,
$ for $j=1,2,3$. For $j=1$ we can write $$\int_0^{\infty}{\mathrm{e}}^{-u^*_{\tau}z\tau^{\alpha}}\left({\mathrm{e}}^{z\tau^{\alpha}}-1\right){\mathrm{e}}^{{\mathtt{i}}u z}{\mathrm{d}}z
=\left[\frac{{\mathrm{e}}^{z\left({\mathtt{i}}u-u^*_{\tau}\tau^{\alpha}+\tau^{\alpha}\right)}}
{\left({\mathtt{i}}u-u^*_{\tau}\tau^{\alpha}+\tau^{\alpha}\right)}\right]_{0}^{\infty}
-\left[\frac{{\mathrm{e}}^{z\left({\mathtt{i}}u-u^*_{\tau}\tau^{\alpha}\right)}}
{\left({\mathtt{i}}u-u^*_{\tau}\tau^{\alpha}\right)}\right]_{0}^{\infty}
=C_{\tau,k,\alpha}(u),$$ which is valid for $u^*_{\tau}(k)>1$ with $C_{\tau,k,\alpha}$ in . For $j=2$ we can write $$\int_{-\infty}^0{\mathrm{e}}^{-u^*_{\tau}z\tau^{\alpha}}\left(1-{\mathrm{e}}^{z\tau^{\alpha}}\right){\mathrm{e}}^{{\mathtt{i}}u z}{\mathrm{d}}z
=\left[\frac{{\mathrm{e}}^{z\left({\mathtt{i}}u-u^*_{\tau}\tau^{\alpha}\right)}}
{\left({\mathtt{i}}u-u^*_{\tau}\tau^{\alpha}\right)}\right]_{-\infty}^{0}-\left[\frac{{\mathrm{e}}^{z\left({\mathtt{i}}u-u^*_{\tau}\tau^{\alpha}+\tau^{\alpha}\right)}}
{\left({\mathtt{i}}u-u^*_{\tau}\tau^{\alpha}+\tau^{\alpha}\right)}\right]_{-\infty}^{0}
=C_{\tau,k,\alpha}(u),$$ which is valid for $u_{\tau}^*(k)<0$. Finally, for $j=3$ we have $$\begin{aligned}
\int_{{\mathbb{R}}}{\mathrm{e}}^{-u^*_{\tau}z\tau^{\alpha}}\left({\mathrm{e}}^{z\tau^{\alpha}}\wedge1\right){\mathrm{e}}^{{\mathtt{i}}u z}{\mathrm{d}}z
&=\int_{-\infty}^{0}{\mathrm{e}}^{-u^*_{\tau}z\tau^{\alpha}}{\mathrm{e}}^{z\tau^{\alpha}}{\mathrm{e}}^{{\mathtt{i}}u z}{\mathrm{d}}z
+\int_{0}^{\infty}{\mathrm{e}}^{-u^*_{\tau}z\tau^{\alpha}}{\mathrm{e}}^{{\mathtt{i}}u z}{\mathrm{d}}z \\
&=\left[\frac{{\mathrm{e}}^{z\left({\mathtt{i}}u-u^*_{\tau}\tau^{\alpha}+\tau^{\alpha}\right)}}
{\left({\mathtt{i}}u-u^*_{\tau}\tau^{\alpha}+\tau^{\alpha}\right)}\right]_{-\infty}^{0}
+\left[\frac{{\mathrm{e}}^{z\left({\mathtt{i}}u-u^*_{\tau}\tau^{\alpha}\right)}}
{\left({\mathtt{i}}u-u^*_{\tau}\tau^{\alpha}\right)}\right]_{0}^{\infty}
=-C_{\tau,k,\alpha}(u),\end{aligned}$$ which is valid for $0<u^*_{\tau}(k)<1$. From the definition of the ${\mathbb{Q}}_{k,\tau}$-measure in and the random variable $Z_{k,\tau,\alpha}$ on page we have $${\mathbb{E}}^{{\mathbb{Q}}_{k,\tau}}\left[\widetilde{g}_j(Z_{\tau,k,\alpha})\right]
=\int_{{\mathbb{R}}}q_j(k\tau^{1-\alpha}-y)p(y) {\mathrm{d}}y = (q_j\ast p)(k\tau^{1-\alpha}),$$ with $q_j(z)\equiv\widetilde{g}_j(-z)$ and $p$ denoting the density of $X_{\tau}^{(t)} \tau^{-\alpha}$. On the strips of regularity derived above we know there exists $\tau_0>0$ such that $q_j\in L^1({\mathbb{R}})$ for $\tau>\tau_0$. Since $p$ is a density, $p\in L^1({\mathbb{R}})$, and therefore $$\label{eq:conv}
\mathcal{F}(q_j\ast p)(u)=\mathcal{F}q_j(u) \mathcal{F}p(u).$$ We note that $\mathcal{F}q_j(u)\equiv\mathcal{F}\widetilde{g}_j(-u)\equiv\overline{\mathcal{F}\widetilde{g}_j(u)}$ and hence $$\label{eq:simpconv}
\mathcal{F}q_j(u) \mathcal{F}p(u)
\equiv{\mathrm{e}}^{{\mathtt{i}}u k\tau^{1-\alpha}}\Phi_{\tau,k,\alpha}(u) \overline{C_{\tau,k,\alpha}(u)}.$$ Thus by Lemma \[lem:L1lem\] there exists $\tau_1>0$ such that $\mathcal{F}q_j \mathcal{F}p\in L^1({\mathbb{R}})$ for $\tau>\tau_1$. By the inversion theorem [@R87 Theorem 9.11] this then implies from and that for $\tau>\max(\tau_0,\tau_1)$: $$\begin{aligned}
{\mathbb{E}}^{{\mathbb{Q}}_{k,\varepsilon}}\left[\widetilde{g}_j(Z_{\tau,k,\alpha})\right]
&= (q_j\ast p)(k\tau^{1-\alpha})
=\mathcal{F}^{-1}\left(\mathcal{F}q_j(u) \mathcal{F}p(u)\right)(k\tau^{1-\alpha}) \\
&=\frac{1}{2\pi}\int_{{\mathbb{R}}} {\mathrm{e}}^{-{\mathtt{i}}u k\tau^{1-\alpha}}\mathcal{F}q_j(u) \mathcal{F}p(u) {\mathrm{d}}u
= \frac{1}{2\pi}\int_{{\mathbb{R}}} \Phi_{\tau,k,\alpha}(u) \overline{C_{\tau,k,\alpha}(u)} {\mathrm{d}}u .\end{aligned}$$
We now move onto the proof of Lemma \[lem:nonsteeplem\]. We use our time-dependent change of measure defined in to write our forward-start option price for $j=1,2,3$ as $${\mathbb{E}}\left(g_j({\mathrm{e}}^{X_{\tau}^{(t)}},{\mathrm{e}}^{k\tau})\right)
= {\mathrm{e}}^{-\tau\left[ku^*_{\tau}(k)-\Lambda^{(t)}_{\tau}\left(u^*_{\tau}(k)\right)\right]}{\mathrm{e}}^{k\tau}{\mathbb{E}}^{{\mathbb{Q}}_{k,\tau}}\left[\widetilde{g}_j(Z_{\tau,k,\alpha})\right],$$ with $Z_{\tau,k,\alpha}$ defined on page . We now apply Lemma \[lem:optpricerepnonsteep\] and then convert to forward-start call option prices using Put-Call parity and that in the Heston model $({\mathrm{e}}^{X_t})_{t\geq0}$ is a true martingale [@AP07 Proposition 2.5]. Finally the expansion for $\exp\left({-\tau\left(k (u^*_{\tau}(k)-1)-\Lambda^{(t)}_{\tau}(u^*_{\tau}(k))\right)}\right)$ follows from Lemma \[lem:fwdmgflargetauexpansion\].
Tail Estimates
==============
\[lem:expsmalllargetime\] There exists $\beta>0$ such that the following tail estimate holds for all $k \in \mathcal{A}$ and $u^*_{\tau}(k)\not\in\{0,1\}$ as $\tau$ tends to infinity: $
\left| \int_{|u|>\tau^{\alpha}} \Phi_{\tau,k,\alpha}(u) \overline{C_{\tau,k,\alpha}(u)} {\mathrm{d}}u \right| = \mathcal{O}({\mathrm{e}}^{-\beta \tau}).
$
By the definition of $\Phi_{\tau,k,\alpha}$ in we have $
|\Phi_{\tau,k,\alpha}(z \tau^{\alpha})|=\exp\left(\tau(\Re [\Lambda_{\tau}^{(t)}({\mathtt{i}}z+u^*_\tau)]-\Lambda_{\tau}^{(t)}(u^*_\tau))\right).
$ For $|z|>1$ we have the simple estimate $
\left| \overline{C_{\tau,k,\alpha}(z\tau^{\alpha})} \right|
\leq \tau^{-\alpha}/z^2,
$ and therefore $$\left| \int_{|u|>\tau^{\alpha}} \Phi_{\tau,k,\alpha}(u) \overline{C_{\tau,k,\alpha}(u)} {\mathrm{d}}u \right|
\leq
\tau^{\alpha}\int_{|z|>1}\left| \Phi_{\tau}(z \tau^{\alpha}) \right| \left| \overline{C_{\tau,k,\alpha}(z \tau^{\alpha})} \right| {\mathrm{d}}z
\leq
\int_{|z|>1}{\mathrm{e}}^{\tau(\Re [\Lambda_{\tau}^{(t)}({\mathtt{i}}z+u^*_\tau)]-\Lambda_{\tau}^{(t)}(u^*_\tau))} \frac{{\mathrm{d}}z}{z^2},$$ for all $\tau>0$. We deal with the case $z>1$. Analogous arguments hold for the case $z<-1$. Lemma \[lem:largetimesaddlefwdmgf\](i) implies that there exists $\tau_1$ such that for $\tau>\tau_1$: $$\int_{z>1}{\mathrm{e}}^{\tau(\Re [\Lambda_{\tau}^{(t)}({\mathtt{i}}z+u^*_\tau)]-\Lambda_{\tau}^{(t)}(u^*_\tau))} \frac{{\mathrm{d}}z}{z^2}
\leq
{\mathrm{e}}^{\tau(\Re [V({\mathtt{i}}+u^*_\tau)]-V(u^*_\tau))+\mathcal{O}(1)} \int_{z>1} \frac{{\mathrm{d}}z}{z^2}.$$ Using Lemma \[lem:largetimesaddlefwdmgf\](ii) we compute $$\Re \Lambda_{\tau}^{(t)}({\mathtt{i}}+u^*_\tau)-\Lambda_{\tau}^{(t)}(u^*_\tau)
=\Re V({\mathtt{i}}+u^*_\tau)-V(u^*_\tau)
+(\Re H({\mathtt{i}}+u^*_\tau)-H(u^*_\tau))/\tau
+\mathcal{O}(\tau^{-n}),$$ for any $n>0$. Now using that $V$ and $H$ are continuous and Assumption \[assump:ustartaueq\] we have that $
\Re V({\mathtt{i}}+u^*_\tau)-V(u^*_\tau)=\Re V({\mathtt{i}}+u_{\infty})-V(u_{\infty})+o(1)
$ and $
\Re H({\mathtt{i}}+u^*_\tau)-H(u^*_\tau)=\Re H({\mathtt{i}}+u_{\infty})-H(u_{\infty})+o(1),
$ as $\tau$ tends to infinity. Lemma \[lem:largetimesaddlefwdmgf\](iii) implies that $\Re V({\mathtt{i}}+u_{\infty})-V(u_{\infty})<0$ and the lemma follows.
\[lem:largetimesaddlefwdmgf\]
(i) The expansion $ \exp(\Lambda_{\tau}^{(t)}({\mathtt{i}}z+u^*_{\tau}))= \exp(V({\mathtt{i}}z+u^*_{\tau})+ H({\mathtt{i}}z+u^*_{\tau})\tau^{-1})\mathcal{R}(\tau)$ holds as $\tau$ tends to infinity where $\mathcal{R}(\tau)={\mathrm{e}}^{\mathcal{O}({\mathrm{e}}^{-\beta\tau})}$ for some $\beta>0$ and $\mathcal{R}$ is uniform in $z$.
(ii) There exists $\tau_1^*$ such that $\Re \Lambda_{\tau}^{(t)}({\mathtt{i}}z+u^*_{\tau})\leq \Re \Lambda_{\tau}^{(t)}({\mathtt{i}}{\mathrm{sgn}}(z)+u^*_{\tau})$ for all $z>|1|$ and $\tau>\tau_1^*$.
(iii) For all $a\in\mathcal{D}_{\infty}^{o}$ the function ${\mathbb{R}}\ni z \mapsto \Re V({\mathtt{i}}z+a)$ has a unique maximum at zero.
<!-- -->
(i) The proof of the expansion follows from Assumption \[assump:ustartaueq\] and analogous steps to the proofs of Lemma \[lem:fwdmgflargetauexpansion\] and Lemma \[lem:charactsimp\]. The proof of uniformity of the remainder $\exp\left( \Lambda_{\tau}^{(t)}({\mathtt{i}}z + a) -V({\mathtt{i}}z + a)-H({\mathtt{i}}z + a)\tau^{-1}\right)$ in $z$ involves tedious but straightforward computations and is omitted for brevity. See Figure \[fig:hestlargetimetail\](a) for a visual illustration.
(ii) Assumption \[assump:ustartaueq\] implies that there exists $\tau_1^*$ such that $u_{\tau}^*\in\mathcal{D}_{\infty}^{o}$ for all $\tau>\tau_1^*$. So we need only show that for all $\tau>0$ and $a\in\mathcal{D}_{t,\tau}^{o}$: $\Re \Lambda_{\tau}^{(t)}({\mathtt{i}}z+a)\leq \Re \Lambda_{\tau}^{(t)}({\mathtt{i}}{\mathrm{sgn}}(z)+a)$ for all $z>|1|$. The proof of this result involves tedious but straightforward computations and is omitted for brevity. See Figure \[fig:hestlargetimetail\](b) for a visual illustration.
(iii) The proof of (iii) is straightforward and follows the same steps as [@JR12 Appendix C]. We omit it for brevity.
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J.D. Deuschel, P.K. Friz, A. Jacquier and and S. Violante. Marginal density expansions for diffusions and stochastic volatility, Part I: Theoretical foundations. *Communications on Pure and Applied Mathematics*, [67]{}(1): 40-82, 2014.
J.D. Deuschel, P.K. Friz, A. Jacquier and and S. Violante. Marginal density expansions for diffusions and stochastic volatility, Part II: Applications. *Communications on Pure and Applied Mathematics*, [67]{}(2): 321-350, 2014.
J. Figueroa-López, R. Gong and C. Houdré. Small-time expansions of the distributions, densities, and option prices of stochastic volatility models with Lévy jumps. *Stochastic Processes and their Applications*, [122]{}: 1808-1839, 2012.
D. Florens-Landais and H. Pham. Large deviations in estimation of Ornstein-Uhlenbeck model. *J.Appl. Prob.*, [36]{}: 60-77, 1999.
M. Forde and A. Jacquier. Small-time asymptotics for implied volatility under the Heston model. *International Journal of Theoretical and Applied Finance*, [12]{}(6), 861-876, 2009.
M. Forde and A. Jacquier. The large-maturity smile for the Heston model. *Finance and Stochastics*, [15]{}(4): 755-780, 2011.
M. Forde, A. Jacquier and R. Lee. The small-time smile and term structure of implied volatility under the Heston model. *SIAM Journal of Financial Mathematics*, [3]{}(1): 690-708, 2012..
M. Forde, A. Jacquier and A. Mijatović. Asymptotic formulae for implied volatility in the Heston model. *Proceedings of the Royal Society A*, [466]{}(2124): 3593-3620, 2010.
J.P. Fouque, G. Papanicolaou, R. Sircar and K. Solna. Multiscale Stochastic Volatility for Equity, Interest Rate, and Credit Derivatives. CUP, 2011.
P. Friz, S. Gerhold, A. Gulisashvili and S. Sturm. Refined implied volatility expansions in the Heston model. *Quantitative Finance*, [11]{} (8): 1151-1164, 2011.
K. Gao and R. Lee. Asymptotics of Implied Volatility to Arbitrary Order. *Finance and Stochastics*, [18]{}(2): 349-392, 2014.
J. Gatheral. A parsimonious arbitrage-free implied volatility parameterization with application to the valuation of volatility derivatives. [madrid2004.pdf](faculty.
baruch.cuny.edu/jgatheral/madrid2004.pdf), 2004.
J. Gatheral. The Volatility Surface: A Practitioner’s Guide. John Wiley & Sons, 2006.
J. Gatheral, E.P Hsu, P. Laurence, C. Ouyang, T-H. Wong. Asymptotics of implied volatility in local volatility models. *Mathematical Finance*, [22]{}: 591-620, 2012.
J. Gatheral and A. Jacquier. Convergence of Heston to SVI. *Quantitative Finance*, [11]{}(8): 1129-1132, 2011.
P. Glasserman and Q. Wu. Forward and Future Implied Volatility. *Internat. Journ. of Theor. and App. Fin.*, [14]{}(3), 2011.
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A. Gulisashvili. Asymptotic formulas with error estimates for call pricing functions and the implied volatility at extreme strikes. *SIAM Journal on Financial Mathematics*, [1]{}: 609-641, 2010.
A. Gulisashvili. Left-wing asymptotics of the implied volatility in the presence of atoms. *International Journal of Theoretical and Applied Finance*, [18]{}(2), 2015.
A. Gulisashvili and E. Stein. Asymptotic Behavior of the Stock Price Distribution Density and Implied Volatility in Stochastic Volatility Models. *Applied Mathematics & Optimization*, [61]{} (3): 287-315, 2010.
P. Hagan and D. Woodward. Equivalent Black volatilities. *Applied Mathematical Finance*, [6]{}: 147-159, 1999.
P. Henry-Labordère. Analysis, geometry and modeling in finance. Chapman and Hill/CRC, 2008.
S. Heston. A closed-form solution for options with stochastic volatility with applications to bond and currency options. *The Review of Financial Studies*, [6]{}(2): 327-342, 1993.
A. Jacquier and A. Mijatović. Large deviations for the extended Heston model: the large-time case. *Asia-Pacific Financial Markets*, [21]{}(3): 263-280, 2014.
A. Jacquier, M. Keller-Ressel and A. Mijatović. Implied volatility asymptotics of affine stochastic volatility models with jumps. *Stochastics*, [85]{}(2): 321-345, 2013.
A. Jacquier and P. Roome. Asymptotics of forward implied volatility. *SIAM J. on Financial Mathematics*, [6]{}(1): 307-351, 2015.
A. Jacquier and P. Roome. The small-maturity Heston forward smile. *SIAM J. on Financial Mathematics*, [4]{}(1): 831-856, 2013.
A. Jacquier and P. Roome. Black-Scholes in a CEV random environment: a new approach to smile modelling. Preprint available at [arXiv:1503.08082](http://arxiv.org/abs/1503.08082), July 2015.
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S. Karlin and H. Taylor. A Second Course in Stochastic Processes. Academic Press, 1981.
M. Keller-Ressel. Moment Explosions and Long-Term Behavior of Affine Stochastic Volatility Models. *Mathematical Finance*, [21]{}(1): 73-98, 2011.
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R.W. Lee. Option Pricing by Transform Methods: Extensions, Unification and Error Control. *Journal of Computational Finance*, [7]{}(3): 51-86, 2004.
R.W. Lee. The Moment Formula for Implied Volatility at Extreme Strikes. *Mathematical Finance*, [14]{}(3): 469-480, 2004.
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A. Mijatović and P. Tankov. A new look at short-term implied volatility in asset price models with jumps. Forthcoming in *Mathematical Finance*, 2013.
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S. Pagliarani, A. Pascucci and C. Riga. Adjoint expansions in local Lévy models. *SIAM Journal on Financial Mathematics*, [4]{}(1): 265-296, 2013.
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P. Tankov. Pricing and hedging in exponential Lévy models: review of recent results. *Paris-Princeton Lecture Notes in Mathematical Finance*, Springer, 2010.
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D. Williams. Probability With Martingales. CUP, 1991.
[^1]: AJ acknowledges financial support from the EPSRC First Grant EP/M008436/1. The authors would also like to thank an anonymous referee for useful comments.
[^2]: whenever ${\mathcal{H}}_0$ is in force, the case $k=V'(a)$ is excluded if $v = \theta \Upsilon(a)$, with $\Upsilon$ defined in , for $a\in\{0,1\}$.
[^3]: whenever ${\mathcal{P}}_0$ is in force, the case $k=V'(a)$ is excluded if $v = \theta \Upsilon(a)$, with $\Upsilon$ defined in , for $a\in\{0,1\}$.
[^4]: whenever ${\mathcal{S}}_0$ is in force, the case $k=V'(a)$ is excluded if $v = \theta \Upsilon(a)$, with $\Upsilon$ defined in , for $a\in\{0,1\}$.
[^5]: A similar analysis can be conducted even if $u^*_{\tau}(k)$ is not eventually in the interior of the limiting domain, but then one will need to use the full lmgf (not just the expansion) in .
|
READ ALSO:
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NEW DELHI: Over 25,000 people died and 3,882 were injured in railway accidents in 2014 , a government report has said.The National Crime Records Bureau (NCRB) report said 28,360 cases of ' railway accidents ' were reported during the year, showing a decrease of 9.2% compared to 2013 (31,236 cases).The majority of railway accidents (61.6%) was due to fall from trains or collision with people on the tracks (17,480 out of 28,360 cases), the report said.Maharashtra reported the maximum such cases, accounting for 42.5% of total cases of fall from train or collision of trains with people. A railway official said most such cases were reported from suburban services in the state and railways was running a campaign to educate people not to cross the tracks.Of the total 25,006 deaths in railway accidents, 14,391 persons died due to either fall from trains or collision of trains with people on the tracks, accounting for 57.6% of total deaths in railway accidents.A total of 469 cases of railway accidents occurred due to mechanical defects like poor design, track faults, bridge/tunnel faults during 2014. In Andhra Pradesh, 385 persons died in railway accidents due to mechanical defects.Sabotage by extremists/terrorists/others caused 13 and five railways accidents in Madhya Pradesh and Uttar Pradesh respectively that led to loss of 18 lives.It was found that around 60 accidents were reported due to fault of drivers that killed 67 people.The NCRB said 2,547 cases of 'railway crossing accidents' were reported which led to deaths of 2,575 people and injury to 126 people. Telangana (1,061 cases) reported the maximum number of railway crossing accidents, accounting for 41.7% of the total.The report also said that 836 deaths took place at unmanned railway crossings and 78.2% of such incidents were reported in Assam alone during 2014.Interestingly, most railway accidents (4,966 out of 28,360) were reported between 6 am and 9 am, accounting for 17.5% of the total. Between 9 am and 12 noon, 16.9% of railway accidents were reported.Maharashtra reported the maximum accidents between 3 pm and 6 pm and between 6 pm and 9pm, accounting for 23.9% (11,902 cases) and 28.7% (13,927 cases) respectively. |
GC-MS analysis of the lipophilic principles of Echinacea purpurea and evaluation of cucumber mosaic cucumovirus infection.
An analytical GC-MS method based on nonpolar fused silica capillary column was developed to analyze the lipophilic constituents, mainly alkamides, from the root extracts of Echinacea purpurea (L.) Moench. In particular, the proposed method was applied to evaluate the phytochemical impacts of cucumber mosaic cucumovirus (CMV) infection on the plant's lipophilic marker phytochemicals. Methanolic (70% v/v) extracts, obtained from root materials by ultrasonic treatments, were subjected to liquid-liquid extraction with n-hexane-ethyl acetate (1:1 v/v) to recover the lipophilic, volatile to semivolatile, principles. Seventeen components, including the 11 alkamides known to E. purpurea roots, were identified in the GC-MS traces of the analyzed fractions and efficiently separated in a turnaround time of 25 min. CMV infection was found to be responsible for significant variations in the relative compositions of the major constituents, in particular germacrene D, Dodeca-2E, 4E, 8Z, 10Z(E)-tetraenoic acid isobutylamide cis/trans isomers, Undeca-2Z, 4E-diene-8, 10-diynoic acid isobutylamide and Dodeca-2E, 4Z-diene-8, 10-diynoic acid isobutylamide. |
Annual Admitted Student Day 2008
Annual Admitted Student Day a Success
Ringling College of Art and Design hosted its annual Admitted Student Day on April 19, 2008. This annual event, which is intended for prospective fall 2008 students who have already applied to Ringling College, offers students a surplus of information about the Ringling campus and community, degree programs, resident halls, student life, and much more. The events of the day were well received by both parents and students, with one student quoted as saying, “It was wonderful, well organized and extremely informative (fun too!!). The curriculum….seems perfect for me! Plus my Dad feels a lot more comfortable about paying for Ringling.”
From another student, Jose’ Yulnar Diaz Amaro, this was a significant day indeed– a day he has been looking forward to for a long time. Amara began his higher education studying programming and computer systems in Mexico, but he has always had a dream of working in animation and the film industry, a dream that the road of programming was not going to lead him down. For Amara, it has not been easy; he first had to learn English before he could even begin his studies in animation. But, over the last 5 years, since living in the US and working on his own art, he has accomplished this task and is now ready to tackle the Computer Animation program here at Ringling, starting this fall. Amara says, “I am very excited to be at Ringling. When I came here and was accepted into Animation, a student told me that it will help to know programming. I am excited about that because I already have a knowledge of programming – this is a dream come true.”
During this year’s festivities, students and parents participated in campus tours, a Dance Revolution contest, bungee bull riding, 1-on-1 discussions with Ringling faculty and staff, a BBQ with live music, and presentations by Dean of Students, Dr. Tammy Walsh, and Phyllis Schaen, Director of Career Services; They also enjoyed an Island Festival (complete with funnel cakes and snow cones), tours of the academic program buildings, meeting instructors, and chatting with current Ringling students.
This year’s Admitted Student Day attendance totaled 509, with 191 of those being potential students. 12 of those students have already confirmed their attendance, paid their fees for fall semester, and will be entering Ringling as freshmen in the fall of 2008! |
Q:
models not loading on phalcon multi module
I'm trying to implement phalcon multi module with namespace. normally its working. but models not loading from its location(/apps/models/). if I paste all of my models file into controller dir then its working. It should load from models dir. how could i solve this problem.
[Front Module]
$loader->registerNamespaces(
array(
'Multiple\Frontend\Controllers' => '../apps/frontend/controllers/',
'Multiple\Frontend\Models' => '../apps/frontend/models/',
));
[Blogs Model]
namespace Multiple\Frontend\Controllers;
use Phalcon\Mvc\Model;
class Blogs extends Model{}
i also try "namespace Multiple\Frontend\Models;" but not success.
getting error like:
Fatal error: Uncaught Error: Class 'Multiple\Frontend\Controllers\News' not found in C:\xampp\htdocs\pm\apps\frontend\controllers\IndexController.php:38 Stack trace: #0 [internal function]:
i have my dispatcher like:
public function registerServices(DiInterface $di)
{
# Registering a dispatcher
$di->set('dispatcher', function () {
$dispatcher = new Dispatcher();
$dispatcher->setDefaultNamespace("Multiple\Frontend\Controllers");
return $dispatcher;
});
i think the error : "Error: Class 'Multiple\Frontend\Controllers\Blogs' not found" for this cause default namespace is frontend\controller. how to solve it? please
A:
I think you need to add one extra line in your controller like...
namespace Multiple\Frontend\Controllers;
use Phalcon\Mvc\Controller;
use Multiple\Frontend\Models\Blogs as Blogs; //** This line should Add **//
class IndexController extends Controller
{
public function indexAction()
{}
}
|
Original Delicatessen Seeds
Based in Spain, Original Delicatessen have spent two decades researching and developing unique strains unlike any others available on the market. Rich in one-of-a-kind flavours and aromas, Original Delicatessen hope to pave the way for other seedbanks to divert from the norm and use new concepts in their products.
You can buy Original Delicatessen's aromatic, delicious strains directly from Seedsman now. |
Chart of the Week: China Exports
The government in Beijing has been aggressive in using money policy to damp down property speculation as a way to steer China’s hot economy away from a meltdown.
Avoiding dangerous bubbles makes long-term sense, but there have been short-term costs: the benchmark Shanghai Composite Index is down more than 20 percent year-to-date. It fell 5 percent on Monday alone.
The chart to the right shows the growth rate over the past eight years for China’s imports and exports (lines), along with the nation’s trade balance (vertical bars). After a huge contraction in 2008-09 during the global recession, both imports and exports have bounced back strongly.
In April, imports (which include raw materials for manufacturing) were up 51 percent year over year and exports were up 30 percent.
Four straight months of strong year-over-year recovery in China’s exports was likely a key factor considered by the government when it imposed anti-property speculation policies last month.
But the sovereign debt crisis in the eurozone is another key factor. Europe is China’s largest trading partner, and the debt crisis has led to a significant devaluation of the euro against the Chinese yuan.
The yuan, which is pegged to the U.S. dollar, is up 14 percent against the euro in just the past four months. The stronger yuan makes Chinese-made products more expensive in the eurozone, and this hurts exporters.
The three-month trend of China’s imports, a leading indicator of future exports, has already headed down. Should a meaningful export deceleration occur in the intermediate term, Chinese authorities may reverse policies to protect economic growth.
To get more insights and perspective from the U.S. Global Investors investment team, subscribe to our weekly Investor Alert.
The Shanghai Composite Index (SSE) is an index of all stocks that trade on the Shanghai Stock Exchange. |
Month: October 2016
Craig Nybo
Format: Paperback
Language: 1
Format: PDF / Kindle / ePub
Size: 14.77 MB
Downloadable formats: PDF
There's an amazing scene about cloning in this one! So we don’t worry too much about genetically modified crops because crops were given to us to benefit humankind. Such reasoning will be unmasked as irrational, and ironically corporate executives reading Playboy and anti-porn radical feminists will make strange bedfellows in jointly unmasking the irrationalism. The site gives them the information they need to make healthy lifestyle choices.
A.A. Jordan
Format: Paperback
Language: 1
Format: PDF / Kindle / ePub
Size: 13.71 MB
Downloadable formats: PDF
But we could, over time, if we were careful, took our time and paid attention to ethical concerns we might be able to: Cure or prevent genetic diseases such as cystic fibrosis, hemophilia, sickle-cell, and Down's Syndrome. You’ve paid that gargantuan tuition to be taught and not to self-educate, right? Benford, Gregory & Eklund, Gordon If the Stars Are Gods. 1977, Berkley. Advanced reproductive technologies, cell and chromosome manipulation, genetics, and genomics are the major fields that pave the way for human genetic engineering.
The kinds of genetic engineering that could be done with today's technology are rather limited, and scientists should expect the particular problem of higher intelligence to be especially tricky. This online news resource highlights scientific discoveries accomplished with the help of NSF support. Hence some will contend that urban mythology is the expression of real suffering and a protest against real suffering, the sigh of the oppressed creature, the heart of a heartless world, and the soul of soulless conditions, the opium of the people.
KG Stutts
Format: Paperback
Language: 1
Format: PDF / Kindle / ePub
Size: 14.58 MB
Downloadable formats: PDF
The scene in which Vincent’s parents are told that he will not be allowed to go to school with other children (because the school cannot afford the skyrocketing insurance rates required to cover an in-valid child) includes a close-up of the gate shutting in Vincent’s face; the only other object visible in the shot is young Vincent’s hand clutching at the closed gate. At which point it joined the Uncanny X-Men. By Amy Worden, Inquirer Harrisburg Bureau HARRISBURG - A state senator thinks that Pennsylvania ought to do what no other state, including left-leaning California, has done: require that food products be labeled to show whether they contain genetically engineered ingredients.
Michael Cordy
Format: Paperback
Language: 1
Format: PDF / Kindle / ePub
Size: 14.68 MB
Downloadable formats: PDF
Both doctors split their time between Boston and the KwaZulu-Natal Province in South Africa, which has among the highest HIV/AIDS prevalence rates in the world. Campbell Memorial Award for best science-fiction novel have been selected, announced Christopher McKitterick, Director of the Gunn Center for the Study of Science Fiction. C4L is an interdisciplinary early childhood curriculum (preK) that focuses on science and math while incorporating literacy connections along the way. S. experiment even implanted human embryos, left over from in-vitro fertilization labs, in artificial wombs, but had to stop after six days to comply with regulations.
Or both are revivified in the worlds charged with microelectronic and biotechnological politics. A neural network consists of layers of processing units called nodes joined by directional links: one input layer, one output layer, and zero or more hidden layers in between. A new regulation concerning wall materials goes into effect next year. Available from Curriki, the nonprofit global community for K–12 educators seeking open learning resources, the project supports Next Generation Science Standards and Engineering for Physics, as well as Common Core State Standards for collaboration and communication.
Endi Webb
Format: Paperback
Language: 1
Format: PDF / Kindle / ePub
Size: 9.85 MB
Downloadable formats: PDF
But people who argue against the possibility of thinking computers are usually those who do not have a strong understanding of science and technology, or those who claim that human intelligence requires some kind of heavenly essence. In fact, one does not have to go to fiction to find examples of societies and movements that have used genetics as a form of social control. The Qur'an states (76.1-3): Has there not been over man a long period of time when he was a nothing not even mentioned?
For excellent reasons, most Marxisms see domination best and have trouble understanding what can only look like false consciousness and people's complicity in their own domination in late capitalism. Burroughs, the often-controversial author, is perhaps best known as the author of Naked Lunch and numerous other novels, including Junkie, Nova Express, the Cut-Up Trilogy, and Cities of The Red Night. It is obvious that in any complex interdependent system (i.e., a nonlinear system ), the alteration or removal of one part will cause a ripple effect of changes throughout; thus GAs naturally incorporate polygeny and pleiotropy. "In the genetic algorithm literature, parameter interaction is called epistasis (a biological term for gene interaction).
Piers Anthony
Format: Paperback
Language: 1
Format: PDF / Kindle / ePub
Size: 13.01 MB
Downloadable formats: PDF
Designed as a accompaniment to the PBS Kids television series SciGirls (which focuses on citizen science this season), students can create a food web to model an ecosystem (All Tangled Up); observe and identify neighborhood birds (Bird Is the Word); explore cloud characteristics (Cloud Clues); create a field guide (Out and About); look for phenomenal phenology in the community (Season Seeking), and identify frog calls (Wetland Band). The Spartans and the Nazis are classic examples of eugenic societies, but the process has history all over the world.
K. Makansi
Format: Paperback
Language: 1
Format: PDF / Kindle / ePub
Size: 12.50 MB
Downloadable formats: PDF
Certainly, few disciplines have within them so much potential for both good and evil. Despite the fact that this is rather like trying to present the international tobacco industry as a humanitarian organisation devoted to the health and welfare of mankind, most journalists and observers swallowed this absolutely outrageous lie without a murmur of protest. Students can reinforce what they learned through online connect-the-dot activities and word searches. |
Click on the image above to download a moderately sized image in
JPEG format (possibly reduced in size from original)
Original Caption Released with Image:
Context image
All this week, the THEMIS Image of the Day has been following on the real Mars the path taken by fictional astronaut Mark Watney, stranded on the Red Planet in the book and movie, The Martian.
Generally smooth and rolling terrain covers most of this portion of Schiaparelli Crater's floor. Because the impact that made Schiaparelli occurred billions of years ago, nature has had ample time to leave lava and sediments in the crater and to erode them. The ridge in the image's southern end is part of an eroded crater rim, one of many such smaller impact craters that have collected on Schiaparelli's floor since it formed. (This image was taken as part of a study for the Mars Student Imaging Project by a high-school science class.)
Here astronaut Mark Watney's great overland trek reaches its end. He arrives safely at the Mars Ascent Vehicle (MAV), which was sent in advance for the next Mars mission crew. The rocket will get him off the ground and into Mars orbit, where he can be picked up by a rescue ship coming from Earth.
NASA's Jet Propulsion Laboratory manages the 2001 Mars Odyssey mission for NASA's Science Mission Directorate, Washington, D.C. The Thermal Emission Imaging System (THEMIS) was developed by Arizona State University, Tempe, in collaboration with Raytheon Santa Barbara Remote Sensing. The THEMIS investigation is led by Dr. Philip Christensen at Arizona State University. Lockheed Martin Astronautics, Denver, is the prime contractor for the Odyssey project, and developed and built the orbiter. Mission operations are conducted jointly from Lockheed Martin and from JPL, a division of the California Institute of Technology in Pasadena. |
Q:
JFrame background color won't change?
So, I'm trying to display a simple JFrame but I'm unable to change the background color? Did a few searches and they all suggest to use useContentPane which I have.
import java.awt.*;
import javax.swing.*;
public class Login {
public static void main(String[] args) {
createWindow();
}
private static void createWindow() {
JFrame frame = new JFrame("Login System");
frame.getContentPane().setBackground(Color.darkGray);
frame.setSize(350, 350);
frame.setDefaultCloseOperation(JFrame.EXIT_ON_CLOSE);
JTextField username = new JTextField();
frame.add(username);
frame.setVisible(true);
}
}
The JFrame size works but the background does not and the TextField does not, pretty new to this. Any ideas? All help is appreciated.
A:
The code works fine. The problem is that your text field takes up the entire frame. Try:
//frame.add(username);
frame.add(username, BorderLayout.NORTH);
|
Our View: Replacement regret
Thursday , September 27, 2012 - 10:12 AM
Editorial Board
Any one of the millions of hard-working, talented individuals who have had to deal with the refrain, "you can be easily replaced," whether from an obtuse boss or a clueless observer, can take some satisfaction in the fiasco that has roiled the National Football League due to the use of replacement referees.
The best referees are locked out, so the first three weeks have featured replacements. Games have been longer, indecision has become a major feature in the games, and every week more NFL coaches, accustomed to competence, blow their stacks. On Monday night, the situation jumped the shark when, on a last-second pass, officials ruled a clear interception as a touchdown due to the receiver having a portion of his hand on the ball, which was firmly clasped in the arms of the defender.
It hardly bears mentioning that the receiver committed an easy-to-spot infraction of offensive pass interference, not caught by the replacement refs.
Now, we don’t care on these pages who wins NFL football games. Nor are we taking a stand on the financial dispute between the NFL and its regular referees that have led to the lockout — although we’re not surprised that Monday night’s fiasco led to reports Wednesday afternoon that this dispute may finally have ended.
The risible replacement referees currently contaminating the NFL serve as a reminder that many jobs are hard, and professionals with years of experience and training often deserve far more respect than backseat critics allow them.
There are a lot of tough jobs out there that tend to be thankless, and result in more complaints than compliments. When we see those same jobs attempted by lesser-qualified individuals, it’s often a revelation of how fortunate we are to have qualified people in tough jobs. |
The present invention relates to defensive weapons and more particularly to a spring whip which can be conveniently carried by a user and can be easily placed in its whipping position.
With the increased amount of crime, and especially attacks on individuals, there is a great need for defensive weapons which can be used to inflict a limited amount of pain so as to act as a deterrent. The weapon should be of a type which can be conveniently carried by an individual, and at the same time be available for immediate operation should an emergency situation arise. Numerous such defensive weapons are currently available. However, most of them require a great amount of time to place into operation. Such time delay can frequently result in harm to the individual before he has an opportunity to even assemble the defensive weapon. Other prior art devices are extremely dangerous and provide hazards to the individual carrying them and are therefore generally avoided. Still other devices are only available for summoning aid by sounding alarms, but do not provide an immediate weapon which can inflict pain and ward off an attacker.
A useful self-defense weapon has been described in my U.S. Pat. No. 3,554,546. In that patent there is described a spring whip which can be utilized as a defensive weapon, and which is formed of interconnected lengths of springs, wherein the selected diameter of the springs provide a compact, telescoped arrangement which attributes to the convenience of carrying the device. The conventional spring construction which consists of a succession of helical turns permits interconnecting of the spring lengths in their extended operative position by merely providing variations in the diameters of cooperating helical turns which produce a wedging engagement between adjacent spring lengths. The lengths of springs are stored in a housing which also serves as a hand grip. In order to facilitate the movement of the springs from their storage to their projected position extending from the housing, a number of weights are movably disposed within the hollow interior of one of the springs and are confined within that spring by bending respective opposite ends of the spring. As the spring whip is projected from the housing, the plural weights move within that spring and roll toward the remote end of the spring to aid in the projection of the springs into their extended position.
While such spring whips have been found quite useful, numerous problems have presented themselves with such devices to detract from their most efficient operation. For example, the lengths of spring did not have an arrangement for restraining their movement within the housing, and accordingly there was a tendency for the springs to loosen from the housing and accidentally move into their extended position. Furthermore, the movement of the weights provided an awkward arrangement in the projection of the lengths of springs, since they had a tendency to roll and move within the spring and continuously provided a source of annoying noise and disturbance. Furthermore, because the ends of the smallest spring were bent to retain the plural weights, the remote end of the housing had a sharp pointed edge, which had a tendency to cut and harm the individual carrying the weapon even when the weapon was not being used.
These and other various problems provided an inconvenience and shortcoming to the spring whip described in the aforementioned patent. |
Q:
Set a div's height to the remaining height in the page
I've found multiple answers on this question, but they all do not seem to work, especially the CSS way. I've moved to the jQuery method, however, it doesn't work neither.
I've got two divs at the moment: A header and a content div. The header has a height of 120px, I'd like the content div to be located below the header and have the remaining height of the screen. I've found multiple answers, I've tried to implent those methods, however, it still didn't work. So that's why I'm asking this question.
Here is my current code:
<script src="//ajax.googleapis.com/ajax/libs/jquery/1.10.2/jquery.min.js"></script>
<script>
var one = $('#header').height(),
two = parseInt($(window).height() - one);
$('#border').height(two)
</script>
<div class="header" id="one">hi</div>
<div class="border" id="two">hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br><div>
CSS:
body {
margin: 0;
background-image:url(../images/bg.png);
background-repeat:repeat;
color: #FFFFFF;
height: 100%;
}
.header
{
position: absolute;
left: 0px;
top: 0px;
width: 100%;
height: 120px;
background-color: black;
box-shadow: 0px 0px 40px #000000;
}
.border
{
position: absolute;
left: 25%;
z-index:0;
top: 120px;
width: 50%;
background-color: white;
color: black;
padding: 5px;
overflow-y:scroll;
}
Thanks!
A:
From my knowledge you can achieve this by
pure css using position:absolute.
and with the help of jQuery
I've solve your problem by both ways check out below solutions
I've checked and find some bugs in your code
1. in your html code you forget to close your <div id="two">.
*2. In your css code there you applied height:100% to your class which are reflecting.*
3. In your jQuery code you are trying to set height but with wrong selectors. That's why your code is not working
so i've fixed it in my case you have to change in your css code, html code and jquery code
check this fiddle http://jsfiddle.net/yNg95/2/
here is Html
<div class="header" id="one">hi</div>
<div class="border" id="two">
hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>hi<br>
</div>
here is css code
*{margin:0; padding:0;}
body {
background-image:url(../images/bg.png);
background-repeat:repeat;
color: #FFFFFF;
overflow:hidden;
}
.header{
position: absolute;
left: 0;
top: 0;
right:0;
height: 120px;
background-color: black;
box-shadow: 0px 0px 40px #000000;
}
.border{
position: absolute;
left: 25%;
z-index:0;
width: 50%;
top:120px;
/*height: 100%;*/
background-color: white;
color: black;
padding: 5px;
overflow-y:scroll;
}
here is jquery code
var one, two;
one = $('.header').outerHeight(true);
two = $(window).height() - one;
console.log(one, two)
$('#two').css({height : two})
here is the fiddle for another option only by pure css
http://jsfiddle.net/sarfarazdesigner/neR4U/
|
Joe Sands
Joe Sands is a dad of six. He's a flaming atheist of the "I don't give a shit" type. He has friends across the spectrum of worldviews and loves to listen, ingest, then tell everyone that his way is always the correct way. Coffee is his kryptonite.
Really. It’s just his body. In fact, I bet you have thousands of photographs that you can look at. Photographs of your husband alive. Smiling. Crying. Yelling in white hot anger at the dog who just pooped on his new shoes. Laughing at something the grandkids did. Sure, you don’t get to touch the creepy dead skin, holding the hand of a lifeless pile of cells, preservation chemicals sloshing throughout the body cavities. But burned his soul? Seriously?
You didn’t get to say your last goodbye. You didn’t get to hold him again one last time. Our daughters didn’t get to, our families didn’t get to, and we were very close.
Yes you did. The last time he was alive. Saying goodbye to a dead guy is…I don’t know.
The family ended up burying Tony’s remains. His ashes were placed in a coffin, which is about a quarter mile from their family farm.
As you would have, his lifeless body. He’s in the ground now. It’s just taking up more space than it otherwise would have. Seriously, a gravestone is just a place to come pay homage to the dead, if you’re into that sort of thing. You don’t even need one grain of the ex-living to be buried there.
They want to be compensated for the mistake but said they don’t have a specific dollar figure to refer to at this point, and so far, no court dates have been set.
I can’t even. And they’ll get money too.
Hey Mayo and the funeral home: Do a better job next time. Don’t mix up the body. Apparently people care.
Update 4/5/2016 10:05 AM CST:
My bride read this and commented, proving I am a callous old codge. Her statement is actually really good:
“Yes you did. The last time he was alive. Saying goodbye to a dead guy is…I don’t know”
The last time they said goodbye was never supposed to be a permanent goodbye. Saying goodbye to the body is extremely healing because it allows you to no longer look to that body as what you have of that person but instead to the memories. When I said “goodbye” to Grandpa’s body in January it gave me a chance to memorize his face so that when I have memories of him I can picture his face smiling at me with his plaid flannel shirt on because that is the way I saw him last and I memorized that.
Saying goodbye to a dead guy can be extremely healing and helps the memories bring comfort for many.
Those of you who have been reading my words since the inception of Incongruous Circumspection, those of you who encouraged me to begin writing my thoughts on the virtual page, those of you who have, over the years, shared with me your stories of abuse, allowing me to expose those who hurt you so deeply, those of you who have worked your ass off, trying to get me to realize the error of my atheist ways, yes….even you, welcome. New readers, hello!
As has been my tradition, I will continue to expose my dear Mama and her lovely narcissistic letters to me and my family. I will still be accepting stories of abuse, whether it be sexual, spiritual, mental, physical, abuses of power, or any other sort of abuse that ends up hurting my readers, and needs to be exposed.
I will be writing personal letters to commenters who try to win me back to the bosom of Christ. I’ve checked. His bosom is rock hard and full of dust-caked sweat. It’s really not that great. Also, his blood tastes like blood, not wine. And, news flash, the last time I tried to get a carpet stain out of my pearly white Berber, it didn’t leave the rug as white as snow. Sorry.
I’ll be doing what I do best, as well – writing puff pieces about the chaos of my own home. Every time my kids yell at me and tell me I’m an idiot, you will know. Of course, I’ll spice it up to make me look like the good guy and they, gooder. My bride, as ambitious, beautiful, brilliant, and successful as she is, will grace these pages as often as she allows. I love that woman and you will see it.
I hate those who hurt others. They will not be spared my ire. I love those who care for others. They will be praised here.
And I like craft beer (for sipping) and good coffee. I add cream to bad coffee. That is, all bad coffee, except my father-in-law’s. He makes the most vile coffee this side of the Mississippi, and yet I made the social faux pas of declaring it “the best damn coffee, this side of the Mississippi,” when I was but a stupid newlywed. I’ve been paying for that compliment ever since, having not the heart or the ovaries to tell him the truth. Then again, after 15-years, maybe I kinda like it in a self-flagellation sort of way. |
Cameron has around 100 ‘sink estates’ in mind for bulldozing or other ‘improvements’ in places like Rochdale, Wandsworth and Tottenham. He thinks getting rid of the ‘bleak’ towerblocks could prevent gang culture, drug abuse and poverty.
He claimed decades of neglect were behind the riots that swept Britain in 2011 as three out of four rioters came from sink estates. |
Money
Money is an integral element in the Grand Theft Auto series, with its importance varying game by game. It is a statistic primarily represented by a counter on the player's HUD as the amount of money in hand. Missions are often emphasized as a reliable source of income, but the player may resort to other means of obtaining money in the game.
In early GTA games, money is emphasized as the key to unlocking new areas in the game, but it may also be used in various other activities. The formula was dramatically modified after Grand Theft Auto III, when money was only important for specific missions as the completion of missions unlocks new area instead; the former was removed entirely after Grand Theft Auto: San Andreas. Since GTA III, the primary use of money is to purchase of items and services, such as Health, Body Armor, clothing, and Weapons. Money is also needed for respraying and/or repairing vehicles at a spray shop.
Contents
2D Universe
For progression into new cities or areas and storyline in Grand Theft Auto 1 and Grand Theft Auto 2, money must be earned to the point a certain amount is fulfilled, often in millions.
Money in the two games are relatively easy to obtain. Acts of crimes, murder and traffic violations often award players with scores, giving the player small quantities of cash. Missions, however, grant players larger amounts of money, in addition to score multipliers that increases the aforementioned monetary award from street crimes by one fold for each mission. This formula, assuming the player continues to successfully complete missions, will result in the player obtaining progressively larger amounts of money until a certain amount is reached and the player may progress to the next city or area.
The use of money for other purposes was explored in GTA 2. With the ability to save games, the player must have a certain amount of money in hand to enter save points (comically represented by a "Jesus Saves" evangelical place of worship which demands donations in order for the player to "save" his "soul"). The game also offers several drive-in shops where the player may remove their wanted level, upgrade their vehicle with equipment, or install bombs, all at a cost.
3D Universe
GTA III
Money in GTA III.
In Grand Theft Auto III, the money system was completely refashioned. While certain street crimes still award players with small amounts of money, the score multiplier is removed, and pedestrians, except emergency personnel, drop cash onto the street upon death. Missions still provide substantial amounts of money, but sub-missions, which debuted in GTA III, serve as an additional source of income, awarding the player with increasingly more money as the sub-missions progress.
Money in GTA III is assigned a secondary role in game progression for specific missions only, when the player is required to pay 8-Ball large sums of money to construct a bomb in "Bomb Da Base Act II", and when the player must pay a large ransom to secure Maria Latore's freedom, who is kidnapped by Catalina and the Colombian Cartel, in "The Exchange". Outside missions, money remains important in the purchase of weapons, respraying of vehicles and the installation of car bombs. Sessions with prostitutes, another addition in the game, also incur a cost to the player, depending on how long the player requires her services.
GTA Vice City
In Grand Theft Auto: Vice City, instant monetary awards for street crimes are largely eliminated (saved the destruction of helicopters and hijacking taxis, which was later removed in GTA San Andreas), leaving missions, sub-missions and dropped pedestrian cash, and robbing stores (in addition to the destruction of parking meters in Downtown) as the only visible sources of income. The average amount of money awarded to the player and cost of items were also divided by 10 (i.e. the use of Pay 'n' Spray costs $100 in GTA Vice City, compared to $1,000 in GTA III). The game also reduces the number of missions where large sums of money was needed; only one such mission remains, "Keep Your Friends Close".
Maintaining the relevance of money in GTA, the player is offered the possibility of purchasing properties and businesses at varying costs. Upon completion of missions or sub-missions for one of said businesses, the business will begin amassing a certain amount of money each day, which the player may pick up at their own leisure. As a joke, the player can earn $50 "Good citizen" bonus by beating criminals chased by police (but without use of any firearms).
GTA San Andreas
Money in GTA San Andreas.
While the money system is largely unchanged from the last installment, Grand Theft Auto: San Andreas expanded on the number of options to earn money and spend it, by introducing a variety of new sub-missions, establishments where players may purchase food or clothes, vehicle customization and gambling. Monetary pickups in gang turf are present, and, like GTA Vice City, properties may still be purchased and produce income of their own. In this game it is also possible to be in debt if the player had lost a great amount of money in casino gaming. If this happens the henchmen of the casino's owner will go after CJ to kill him a bit later.
GTA Advance
The player gets large sums of cash for missions like in GTA III, however there are no more needs for the money aside from weapons.
GTA Liberty City Stories
The money system works just like the previous installments from the GTA III Era. Aside from a mission which requires the player to have enough money to pay for some explosives, the only other thing the player can spend it on is weapons, prostitutes, ferries, Pay 'n' Spray and bombs for cars.
GTA Vice City Stories
Money's importance increases by a bit and aside from the previous purposes. The player can again purchase properties and build the business assets from them into whatever type they please. Money can now be gained easily through the new addition of the empire building and instead of picking up the money from each property, the player gets it through a pager message at 16:00 (4 PM) each day.
HD Universe
GTA IV
The core of the money system is unchanged in Grand Theft Auto IV. However, profitable sub-missions, which were sources of income since GTA III, are reduced to Brucie Kibbutz's Exotic Exports, The Fixer's Assassinations, and Stevie's Car Thefts. But after completing all 30 of Stevie's car theft missions, the player can bring any vehicle they want to his garage for extra cash, at varying prices depending on the model and condition. The game also allows the player to open cash registers for small amount of cash (robbing the business), and blowing up a Securicar armored truck (scattering money on the street for the player to pick up). Like weapons, money now lies realistically on the ground with a yellow-green glow to catch the player's eye instead of floating in mid-air.
As in GTA San Andreas, the importance of money for the purchase of food and clothing is reintroduced. Outings with friends or girlfriends also require substantial amounts of money when going for a drink, eating or bowling. Players are also given the option of simply giving money away to street musicians (for health) and tramps.
Money is also the unit of measure of rank in GTA IV's multiplayer. The more money the player has, the higher their rank: |
[Usefulness of endovascular treatment for delayed massive epistaxis following endoscopic endonasal transsphenoidal surgery: a case report].
We report here a case of massive nasal bleeding from the sphenopalatine artery three weeks after endonasal transsphenoidal surgery. This 66-year-old male suffered from massive nasal bleeding with the status of hypovolemic shock. Under general anesthesia, an emergent angiography revealed an extravasation from the sphenopalatine artery. Trans-arterial embolization using coil and n-butyl-cyanoacrylate (NBCA) was performed following the diagnostic angiography. Complete occlusion of the injured artery was achieved. The patient showed good recovery from general anesthesia. Delayed nasal bleeding after endonasal transsphenoidal surgery is a rare but important complication. The sphenopalatine artery and its branch are located in the hidden inferior lateral corner of the sphenoid sinus and may be injured during enlargement of the sphenoid opening. When massive delayed nasal bleeding follows transsphenoidal surgery and damage of the internal carotid artery has been ruled out, endovascular treatment of the external carotid artery should be considered. |
Introduction {#Sec1}
============
Cells live in a state of constant environmental flux and must reliably receive, decode, integrate and transmit information from extracellular signals such that response is appropriate.^[@CR1]--[@CR4]^ Dysregulation of signal transduction networks leads to inappropriate transmission of signaling information, which may ultimately lead to pathologies such as cancer. Therefore, a central problem in systems biology has been to untangle how quantitative information of cellular signals is encoded and decoded. In general cells respond to one or more properties of a stimulus, such as its identity, strength, rate of change, duration and even its temporal profile.^[@CR5]--[@CR11]^ There are extensive studies on the dose-response curves to reveal how cells respond differentially to a signal with different strength. In comparison, how cells respond to the temporal code of signals is less studies, and its physiological relevance receives much attention recently since most extracellular signals exist only transiently and cellular responses show dependence on signal duration.^[@CR12]--[@CR16]^
Transforming growth factor-β (TGF-β) is a secreted protein that regulates both transient and persistent cellular processes such as proliferation, morphogenesis, homeostasis, differentiation, and the epithelial-to-mesenchymal transition (EMT).^[@CR17]--[@CR21]^ Because it plays essential roles in developmental and normal physiological process, and its dysregulation is related to cancer, fibrosis, inflammation, Alzheimer's disease and many other diseases, the TGF-β signaling pathway has been probed extensively as a potential pharmaceutical target.^[@CR22],[@CR23]^ Several quantitative studies have expanded our knowledge on how the TGF-β-SMAD signaling pathway transmits the duration and strength information of the signal. ^[@CR24]--[@CR28]^
TGF-β can activate both SMAD-dependent and multiple SMAD-independent pathways, which then converge onto some downstream signaling elements. It is unclear how cells transmit and integrate information of the TGF-β signaling distributed among these parallel pathways. Addressing this question requires studies beyond the TGF-β/SMAD axis as in earlier work, where quantifying SMAD proteins serves as the fundamental readout.^[@CR24]--[@CR26]^ Here, we focused on expression of SNAIL1, which is such a downstream target and plays a key role in regulating a number of subsequent processes. Our results confirmed that crosstalk between the SMAD-dependent and independent pathways is key for cells to decode and transmit temporal and contextual information from TGF-β. We posit that the mechanism may be a central mechanistic signal transduction process as many signaling pathways share the network structure.
Results {#Sec2}
=======
Network analysis reveals a highly connected TGF-β signaling network {#Sec3}
-------------------------------------------------------------------
Through integrating the existing literature, we reconstructed an intertwined TGF-β-SNAIL1 network formed with SMAD-dependent and SMAD-independent pathways (Supplementary Fig. [S1](#MOESM1){ref-type="media"}). For further studies we then identified a coarse-grained network composed of a list of key molecular species (Fig. [1](#Fig1){ref-type="fig"}, and Supporting text for details). Along the canonical SMAD pathway, TGF-β leads to phosphorylation of SMAD2 and/or SMAD3 (pSMAD2/3), followed by nuclear entry after recruiting SMAD4 and forming the complex. The complex acts as a direct transcription factor for multiple downstream genes including SNAIL1 and I-SMAD.^[@CR24],[@CR29]^ I-SMAD functions as an inhibitor of pSMAD2/3, thus closes a negative feedback loop. TGF-β also activates GLI1, a key component of the Hedgehog pathway, both through transcriptional activation by pSMAD2/3, and through suppressing the enzymatic activity of glycogen synthase kinase 3 (GSK3). The latter is constitutively active on facilitating GLI1 and SNAIL1 protein degradation in untreated epithelial cells,^[@CR30],[@CR31]^ thus suppressing GSK3 is expected to lead to GLI1 and SNAIL1 protein accumulation. Other non-SMAD signaling pathways, such as MAPK, ERK, et al. may also impact on SNAIL1 expression but to a less extent.^[@CR29],[@CR32]^ We represented them as "others" in the model without further explicit treatment within the period of TGF-β treatment studied here. Therefore, the network integrates multiple feed-forward loops that converge at the regulation of SNAIL1 transcription. In the following sections, we will examine the functional roles of individual pathways in the network using several human cell lines.Fig. 1TGF-β induced signaling crosstalk network converges to SNAIL1. Reconstructed literature-based pathway crosstalk for TGF-β induced SNAIL1 expression. The node "others" refer remaining SNAIL1 activation pathways that have minor contributions to the time window under study and thus are not explicitly treated
The canonical TGF-β/SMAD pathway initializes a transient wave of SNAIL1 expression {#Sec4}
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First we examined the TGF-β/SMAD/SNAIL1 pathway (Fig. [2a](#Fig2){ref-type="fig"}) by treating human MCF10A cells with recombinant human TGF-β1, and performing multicolor immunofluorescence (IF) using antibodies directed against pSMAD2/3, SNAIL1. As expected from the pSMAD/I-SMAD negative feedback loop, pSMAD2/3 proteins accumulated in the nucleus transiently, peaking at around 12 h after TGF-β1 treatment, followed by a decrease by 24 h (Fig. [2b, c](#Fig2){ref-type="fig"}). We confirmed the transient pSMAD2/3 dynamics by sampling 1100--2600 cells at each time point (Fig. [2c](#Fig2){ref-type="fig"}). The result is also consistent with reports in the literature.^[@CR24],[@CR26],[@CR33]^ Nuclear SNAIL1 concentration rose concurrently with pSMAD2/3 (Fig. [2b, c](#Fig2){ref-type="fig"}), then there was a transient dip at 24 h, followed by another increase then a persistant elevation for one week.^[@CR34]^Fig. 2The SMAD proteins induce the first wave of SNAIL1. **a** Canonical SMAD-dependent pathway for TGF-β activation of SNAIL1 highlighted from the network in Fig. [1](#Fig1){ref-type="fig"}. **b** Two-color immunofluorescence (IF) images of pSMAD2/3 and SNAIL1 of MCF10A cells induced by 4 ng/ml TGF-β1 at various time points. The scale bar is 10 μm and is the same for other IF images in this paper. **c** Distributions of nuclear pSMAD2/3 and SNAIL1 concentrations quantified from the IF images. Red vertical lines indicate the mean value of the distributions at time 0, and blue vertical lines represent that at 12 h (for pSMAD2/3) or at 48 h (for SNAIL1), respectively. The number marked in each figure panel is the number of randomly selected cells used for the analysis. Throughout the paper we report fold changes of concentration and amount relative to the mean basal value of the corresponding quantity. **d** Effects of early (added together with TGF-β) and late (48 h after adding TGF-β) pSMAD inhibition on the *SNAIL1* mRNA level in MCF10A cells. **e** Thorough parameter space search confirmed that with the model in panel **a** one can fit the pSMAD2/3 dynamics, but not the two-wave SNAIL1 dynamics. The experimental data are shown as violin plots with the medians given by black bars. Solid curves are computational results with parameter sets sampled from the Monte Carlo search, and the red curves are the best-fit results. **f** Fold change of *SNAIL1* mRNA levels in MCF7 and A549 cells measured with quantitative RT-PCR after TGF-β1 treatment. **g** Fold change of *SNAIL1* mRNA levels measured with quantitative RT-PCR at 72 h after TGF-β1 (T) treatment. For early inhibition (*T* + I) the inhibitor was added at the time of starting TGF-β1 treatment. For late inhibition (*T*−/+I) the inhibitor was added 48 h (for MCF7) and 24 h (for A549) after starting TGF-β1 treatment, respectively. The inhibition results were compared to the TGF-β treatment (T) result at the same time point
Next, we investigated the function of phosphorylated SMAD2/3 on promoting snail1 transcription during TGF-β treatment. In addition to adding TGF-β, we treated MCF10A cells with an inhibitor LY2109761, which prevents SMAD2/3 phosphorylation through inhibiting TGF-β receptor kinase activity (Fig. [2d](#Fig2){ref-type="fig"}). Without the inhibitor, the *SNAIL1* mRNA showed the two-wave dynamics consistent to that of the protein. When the inhibitor was added concurrently with TGF-β treatment, the *SNAIL1* mRNA level was reduced to \~9% of that of the control experiment (without inhibitor) by day 3. This result is consistent with previous observation that directly blocking SMAD2/3 phosphorylation or pSMAD activation at the early stage of TGF-β treatment depletes snail1 expression significantly (1--4), and indicates that indeed pSMAD2/3 are required for SNAIL1 initial activation. However, the *SNAIL1* mRNA level remained \~70% when the inhibitor was added 48 h after initiation of TGF-β treatment (when nuclear pSMAD2/3 concentration has dropped to a minimum). Furthermore we constructed a mathematical model that contains only the TGF-β/SMAD/SNAIL1 pathway, and performed a thorough parameter space search using a multi-configuration Monte Carlo algorithm (Supplementary Fig. [S2](#MOESM1){ref-type="media"}). The search revealed regions of the parameter space that quantitatively reproduced the transient pSMAD2/3 dynamics, but not the two-wave dynamics of SNAIL1 expression (Fig. [2e](#Fig2){ref-type="fig"}). This computational result further confirmed that pSMAD2/3 is less essential for the second wave of SNAIL1.
Furthermore, this SNAIL1 dynamics is not cell type specific as equivalent two-wave dynamics were seen for *SNAIL1* mRNA in MCF7 and A549 cells (Fig. [2f](#Fig2){ref-type="fig"}). Similar to that of MCF10A, it is more effective on inhibiting *SNAIL1* mRNA by adding LY2109761 together with TGF-β than later (Fig. [2g](#Fig2){ref-type="fig"}). The impact of SMAD phosphorylation inhibitor on A549 is less than that on MCF10A or MCF7 at either early or late inhibition, which could be due to the higher level of EMT-related factors in A549.^[@CR35]^ In total, these results reveal that pSMAD2/3 is essential for the early phase of SNAIL1 activation, but is less important for the secondary phase elevation and persistence of SNAIL1 expression/localization.
GLI1 contributes to activating the second wave of SNAIL1 {#Sec5}
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The regulatory network suggests that GLI1 may be responsible for the second wave of SNAIL1 (Fig. [3a](#Fig3){ref-type="fig"}). To test this hypothesis, we performed microscopy studies of SNAIL1-GLI1 using MCF10A cells. The distribution of SNAIL1 found in this study (Supplementary Fig. [S3a](#MOESM1){ref-type="media"}) was consistent with those from the pSMAD2/3-SNAIL1 studies. Elevated and sustained expression of GLI1 under TGF-β treatment (Fig. [3b, c](#Fig3){ref-type="fig"}) was clearly evident. More interestingly GLI1 also showed an unexpected multi-phasic dynamic. Around 8 h after TGF-β treatment, cytosolic GLI1 concentration started to increase. At 12 h when SMAD activities decreased toward basal levels there was a clear accumulation of GLI1 in the nucleus, which continued to increase through day 2. Notably, at this time point cells expressing a high level of nuclear SNAIL1 consistently showed high nuclear GLI1 concentrations (Supplementary Fig. [S3a](#MOESM1){ref-type="media"}). Expanding the mathematical model of the network to Fig. [2a](#Fig2){ref-type="fig"} also reproduced the temporal dynamics of pSAMD2/3 and SNAIL1 (Supplementary Fig. [S3b](#MOESM1){ref-type="media"}), supporting the role of GLI1 as the activator of the second wave of SNAIL1.Fig. 3GLI1 is a major contributor to activate the second wave of SNAIL1 expression. **a** TGF-β activates the GLI1/SNAIL1 module partly through pSMAD2/3. **b** IF images on protein levels of GLI1 (in the free form). Red and blue vertical lines indicate the mean values of the distributions at time 0 and at 48 h, respectively. **c** Distributions of nuclear GLI1 concentrations quantified from the IF images. **d** Experimental validation of the results for early (added together with TGF-β) GLI1 inhibition on the *SNAIL1* mRNA level in MCF10A cells. **e** Experimental validation of the results for late (48 h after adding TGF-β) GLI1 inhibition on the *SNAIL1* mRNA level in MCF10A cells. **f** Fold change of *GLI1* mRNA levels measured with quantitative RT-PCR at different time points after combined TGF-β1 treatment in MCF7 or A549 cells. **g** Fold change of *SNAIL1* mRNA levels measured with quantitative RT-PCR at 72 h after combined TGF-β1 and GLI1 inhibitor GANT61 treatment in MCF7 or A549 cells. For early inhibition (*T* + I) the inhibitor was added at the time of starting TGF-β1 treatment. For late inhibition (*T*−/+I) the inhibitor was added 48 h (for MCF7) and 24 h (for A549) after starting TGF-β1 treatment, respectively. TGF-β treatment group (T) is shown as a positive control
If GLI1 is mainly involved only in the later maintenance of SNAIL1 expression, it is reasonable to predict that inhibiting GLI1 activity, either at the onset of or at some subsequent time after TGF-β treatment, would have minimal effect on the pSMAD2/3 induced initial wave of SNAIL1 expression. However, GLI1 inhibition would severely reduce the second wave of SNAIL1 expression. Indeed this was what observed experimentally. When GLI1 inhibitor GANT61 was added together with TGF-β at the beginning of the experiment, the *SNAIL1* mRNA level was reduced to be 55% (at 12 and 24 h), 12% (at 48 h) and 7% (at 72 h) compared to that without inhibition at the corresponding time points (Fig. [3d](#Fig3){ref-type="fig"}). In another experiment adding the inhibitor 48 h after TGF-β treatment also reduced the mRNA level measured at 72 h to be 25% (Fig. [3e](#Fig3){ref-type="fig"}). These results are qualitatively different from those with the SMAD inhibitor (Fig. [2d](#Fig2){ref-type="fig"}).
To confirm that GLI1 activation is not restricted to the MCF10A cell line, we also examined MCF7 and A549 cells with TGF-β treatment. We observed similar increased and sustained GLI1 expression, albeit with initial slight downregulation before 12 h, possibly due to cell line specific activation of some GLI1 inhibition pathways (Fig. [3f](#Fig3){ref-type="fig"}). Furthermore, early and late GLI1 inhibition lead to a reduction of the *SNAIL1* mRNA level to be 13 and 22% for MCF7 cells, and to a less extent of 57 and 66% for A549 cells, respectively (Fig. [3g](#Fig3){ref-type="fig"}). Additionally, increased GLI1 expression after TGF-β treatment has been found for multiple liver cancer cell lines.^[@CR36]^ In toto these results support the role of GLI1 as a signaling relay from pSMAD2/3 to SNAIL1.
GSK3 in a phosphorylation form with augmented enzymatic activity accumulates at endoplasmic reticulum and Golgi apparatus {#Sec6}
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Next, we hypothesized that GSK3 is fundamental to the observed multi-phasic GLI1 dynamic (Fig. [3b](#Fig3){ref-type="fig"}). Most published studies suggest that GSK3 is constitutively active in untreated cells, facilitating degradation of SNAIL1 and GLI1; TGF-β treatment leads to GSK3 phosphorylation and inactivation, which leads to an accumulation of SNAIL1 and GLI1.^[@CR37],[@CR38]^
Initially we tested whether the above mechanism is sufficient to explain the multi-phasic GLI1 dynamics. We treated MCF10A cells in the absence of TGF-β with a GSK3 activity inhibitor. Given the above mechanism, one should expect the GSK3 inhibitor to promote both GLI1 and SNAIL1. In our experiment, SNAIL1 did increase in the nucleus and even more in the cytoplasm due to inhibition of GSK3-dependent SNAIL1 degradation, but there was no noticeable change in GLI1 expression in either nucleus or cytoplasm (Fig. [4a](#Fig4){ref-type="fig"}), suggesting additional signaling mechanisms may be involved.Fig. 4TGF-β induced temporal switch between active and inhibitive phosphorylation forms of GSK3 proteins. **a** IF images showed that inhibiting GSK3 enzymatic activity alone increased SNAIL1 accumulation but did not recapitulate TGF-β induced GLI1 nuclear translocation. **b** Quantification of the IF images of MCF10A cells at different time points after TGF-β treatment. Red vertical lines indicate the mean value of the distributions at time 0, and blue vertical lines represent that at 8 h (for GSK3^AA^) or at 12 h (for GSK3^D^), respectively. **c** IF images showing GSK3^AA^ localization at the endoplasmic reticulum center (ERC)
Besides the inhibitory serine phosphorylation (S21 in GSK-3α and S9 in GSK-3β), previous studies showed that tyrosine (Y279 in GSK-3α and Y216 in GSK-3β) phosphorylation leads to augmented enzymatic activity of GSK3.^[@CR39]^ As a convenience when discussing the three forms of GSK3, we refer the enzymatically active unphosphorylated form and the more active tyrosine phosphorylated form as "GSK3^A^" and "GSK3^AA^", respectively, and the inactive serine phosphorylation form as "GSK3^D^". Also we reserve "GSK3" for the total GSK3. As expected, microscopy studies showed an increased concentration of GSK3^D^ peaking around 12 h after TGF-β treatment (Fig. [4b](#Fig4){ref-type="fig"}, Supplementary Fig. [S4a](#MOESM1){ref-type="media"}). Large cell-to-cell variations in the concentration of GSK3^D^ were observed, however, the abundance of cytosolic and nuclear GSK3^D^ were essentially equivalent (the expression ratio was close to one) for cells without TGF-β treatment (Supplementary Fig. [S4b](#MOESM1){ref-type="media"}). This observation corroborates earlier report that the serine phosphorylation does not affect GSK3 nuclear location.^[@CR40]^ TGF-β treatment led to transient deviation of this ratio from equivalence, reflecting additional active and dynamic regulation of GSK3 including covalent modification, location and protein stability. Specifically prior to inhibitory serine phosphorylation we observed transient GSK3^AA^ accumulation in the perinuclear region peaking at 8 h (Fig. [4b](#Fig4){ref-type="fig"}, Supplementary Fig. [S4a](#MOESM1){ref-type="media"}). Close examination of higher magnification confocal images revealed that the GSK3^AA^ formed clusters in the endoplasmic reticulum (ER) and Golgi apparatus, but not associated with actin filaments (Fig. [4c](#Fig4){ref-type="fig"}, Supplementary Movie [1](#MOESM4){ref-type="media"} & [2](#MOESM5){ref-type="media"}). Given that a function of active GSK3 is to modify target proteins post-translationally, our observation suggests an unreported role for GSK3^AA^ accumulating at the ER and Golgi apparatus as to modify newly synthesized proteins before their release to the cytosol. Specifically previous studies showed that in mammalian cells a scaffold protein SUFU binds to GLI to form an inhibitory complex; SUFU phosphorylation by GSK3β prevents the complex formation, and exposes the GLI1 nuclear localization sequence.^[@CR41]^ This mechanism explains the observed increase of free GLI1 in the cytosol followed by nuclear translocation (Fig. [3b](#Fig3){ref-type="fig"}). Since the two phosphorylation forms, GSK3^AA^ and GSK3^D^, coexist within single cells at defined time points, we performed co-immunoprecipitation and found that the probability of having the two GSK3 phosphorylation forms in one molecule was rare (Supplementary Fig. [S4c](#MOESM1){ref-type="media"}).
Contrary to our observation that TGF-β regulates GSK3^AA^ dynamics, other studies posit that GSK3^AA^ is not regulated by external cues.^[@CR42]^ To resolve this paradox, we measured the relative amount of different GSK3 forms through silver staining (Fig. S4d). Among the three forms, the overall percentage of GSK3^D^ increased from a basal level of 37--65% at 12 h after TGF-β treatment. In contrast, only a small fraction of GSK3 molecules assumed the GSK3^AA^ form and its overall abundance was stable over time (from \~10% basal level to \~13% at 8 h then back to \~10% at 12 h after TGF-β treatment). Essentially GSK3^AA^ did not change in abundance but did change in localizations (homing to the ER and Golgi apparatus) to form a high local concentration, which imbue an important role in TGF-β signaling.
A temporal and compartment switch from active to inhibitory GSK3 phosphorylation smoothens the SMAD-GLI1 relay and reduces cell-to-cell heterogeneity on GLI1 activation {#Sec7}
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Based on the above results, we constructed an expanded network for TGF-β induced SNAIL1 expression (Fig. [5a](#Fig5){ref-type="fig"}), which integrates a role for GSK3 and its temporal change of enzymatic activities in the cytosol and nucleus (Supplementary Fig. [S4e](#MOESM1){ref-type="media"}). The model reproduces the multiphasic dynamics of GLI1 as well as that of pSMAD2/3 and SNAIL1 (Supplementary Fig. [S5a](#MOESM1){ref-type="media"}).Fig. 5The GSK3 phosphorylation switch smoothens the SMAD-GLI1 relay. **a** Proposed expanded network for TGF-β induced SNAIL1 expression. **b** Left: Schematic of a generic positive feedback loop network. Also shown in green is an additional reservoir of the molecules in inactive form (*X*~I~) that can convert quickly into the active form (*X*) upon stimulation. Right: The response time *t*~R~ is sensitive to the initial concentration, (*X*)~0~ vs. (*X*)~0~ + Δ(*X*)~0~. The inlet figure shows the dependence of Δ*t*~R~ on (*X*)~0~ with Δ(*X*)~0~ fixed. **c** Box plots of GSK3 inhibition experimental data. **d** Scattered plots of GSK3 inhibition experimental data. Red points are the center of the scattered plots and each ellipse encloses 97.5% of the data points. Both were drawn with the R package, *car::data.ellipse*. **e** Computational simulation of SMAD, SNAIL1 and GLI1 behavior with (solid line) or without (dotted line) initial boosting in cells with high basal GLI1 level (left panels) or low basal GLI1 (right panels)
To understand the function of the early nuclear accumulation of GLI1 induced by GSK3^AA^, it is important to recognize that GLI1 has a positive-feedback loop, and this network motif (Fig. [5b](#Fig5){ref-type="fig"}, left panel without the part in green) has characteristic sigmoidal shaped temporal dynamics, with the substrate concentration increasing slowly at first then accelerating with time until it approaches saturation (Fig. [5b](#Fig5){ref-type="fig"}, right panel, red curve). The response time, t~R~, defined as the time taken to reach a target concentration value (*X*)~R~, is highly sensitive to initial substrate concentration (*X*)~0~: in fact a slight increase in the initial concentration, Δ(*X*), can significantly shorten the response time (Fig. [5b](#Fig5){ref-type="fig"}, right panel, blue curve). In contrast, one can accelerate the response time with an expanded network (Fig. [5b](#Fig5){ref-type="fig"} including the green part) that the signal triggers a fast conversion of the substrate from a preformed inhibitory form (*X*~I~) to active form *X*, effectively a boost of (*X*)~0~ to (*X*)~0~ + Δ(*X*). For a fixed Δ(*X*) a greater acceleration is seen in cells with lower initial concentrations (Fig. [5b](#Fig5){ref-type="fig"}, right panel, inlet figure). Consequently despite variations of their initial concentration (*X*)~0~, most cells within a population can reach (*X*)~R~ by a targeted time point *t*~T~ in a series of temporally regulated events such as cell differentiation and immune response. Indeed, many examples of this modified feedback loop motif exist. Figure [S5b](#MOESM1){ref-type="media"} gives some examples involving members of intrinsically disordered proteins and inhibitors of DNA binding proteins, β-catenin and the STING motif for immune responses. In the present scenario the accelerated GLI1 dynamic ensures sufficient accumulation of GLI1 before nuclear pSMAD2/3 level decreases, essentially analogous to a relay race when the first runner can only release the baton after the second runner has grabbed it. Later when the GLI1 and SNAIL1 concentrations start to increase, the GSK3^A^→GSK3^D^ conversion became necessary to reduce the rates of their degradation catalyzed by active GSK3. Interestingly, this conversion takes place concurrently with maximal concentration of nuclear pSMAD2/3, which activates GLI1 and SNAIL1 transcription. Furthermore, the small initial concentration boost does not affect another major function of the positive feedback loop, which is to robustly buffer temporal and strength fluctuations of signals (Supplementary Fig. [S5c](#MOESM1){ref-type="media"}).^[@CR43]^
To test the functional roles of GSK3 suggested above, we performed a series of GSK3 activity inhibition experiments. First, we pretreated MCF10A cells with GSK3 inhibitor SB216763, washed out the inhibitor then added TGF-β1 (Supplementary Fig. [S5d](#MOESM1){ref-type="media"}). We predicted that the treatment would slow down GLI1 nuclear accumulation, and at later times decrease the overall increase of GLI1 and SNAIL1 compared to cells without GSK3 inhibitor. Indeed this was observed (Fig. [5c](#Fig5){ref-type="fig"}, TGF-β+/− GSK3_I). More interestingly, the scatter plots (Fig. [5d](#Fig5){ref-type="fig"}) show the distributions with and without the inhibitor are similar in cells with high GLI1, but in the presence of the inhibitor there is a population of non-responsive cells with low GLI1 and SNAIL1. This observation is consistent with model predictions that the GSK3-induced boost of initial GLI1 concentration leads to acceleration in the GLI1 and SNAIL1 dynamics, and this boost is more evident for cells with lower level of initial nuclear GLI1 (Fig. [5e](#Fig5){ref-type="fig"}). In a separate experiment (Supplementary Fig. [S5e](#MOESM1){ref-type="media"}), we did not wash out GSK3 inhibitor while adding TGF-β. In this case the inhibitor had opposite effects on GLI1 and SNAIL1 protein concentrations: it slowed down the initial release and translocation of GLI1 needed to accelerate the GLI1 accumulation, but also decreased GLI1 and SNAIL1 degradation that becomes pre-eminent when the proteins were present at high levels. Compared to the samples grown in the absence of the GSK3 inhibitor, we also observed slower and more scattered GLI1 nuclear accumulation and SNAIL1 increase on day 2, but by day 3 the overall levels of GLI1 and SNAIL1 were actually higher than the case without the inhibitor (Fig. [5d](#Fig5){ref-type="fig"}, TGF-β + GSK3_I).
The SMAD-GLI1 relay increases the network information capacity and leads to differential response to TGF-β duration {#Sec8}
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Our results show that TGF-β1 signaling is effected through pSMAD2/3 directly with fast pulsed dynamics concurrently with a relay through GLI1 which has a much slower dynamics. The signaling ported by these two channels converges on SNAIL1 with a resultant two-wave expression pattern. To further dissect the potential functional interactions between these two pathways, we performed mathematical modeling and predicted that the two distinct dynamics allows cells to respond to TGF-β differentially depending on stimulus duration (Fig. [6a](#Fig6){ref-type="fig"}). Short pulses of TGF-β only activate pSMAD2/3 and the first wave of transient SNAIL1 expression. When the signal duration is longer than a defined threshold value, activation of GLI1 will lead to the observed second wave of SNAIL1 expression. We confirmed the predictions with MCF10A cells (Fig. [6b](#Fig6){ref-type="fig"}). Both TGF-β1 pulses with duration of 2 and 8 h activated pSMAD2/3 and the first wave of SNAIL1 expressions. However, only the 8-h but not the 2-h pulse activated sustained GLI1 and the second wave of SNAIL1 expression, similar to those with continuous TGF-β1 treatment.Fig. 6The TGF-β-SNAIL1 network permits detection of TGF-β duration and differential responses. **a** Model predictions that the network generates one or two waves of SNAIL1 depending on TGF-β duration. The red line overlaid on the heatmap is a sampling time of the short-time TGF-β induction. The green line represents the long-time TGF-β treatment. **b** Single cell protein concentrations quantified from IF images of cells under pulsed and continuous TGF-β treatments. The solid lines divide the space into coarse-grained states with respect to the corresponding mean values without TGF-β treatments (=1). **c** Schematics of how cells encode information of TGF-β duration through a temporally ordered state space
Clearly cellular responses have different temporal profiles depending on the TGF-β duration, and one can use the information theory to quantify their information content.^[@CR9],[@CR10]^ In this study we utilized a more intuitive understanding of network function from an information encoding viewpoint. Consider the pSMAD complex, which has three coarse-grained states, high (H), medium (M), and low (L), and each of GLI1 and SNAIL1 has two states, H and L (Fig. [6c](#Fig6){ref-type="fig"}). Then one can use three 4-element states, (L, L; L, L), (H, L; L, L), (H, M; L, H) to roughly describe the case without TGF-β and the 2 and 8 h pulse results in Fig. [6b](#Fig6){ref-type="fig"}, where each number in a state represents in the order the 12 and 48 h concentrations of pSMAD2/3 and GLI1, respectively. The three states are part of a temporally ordered state space, and encode information of TGF-β duration roughly as not detectable, short, and long. The same information is encoded by the SNAIL1 dynamics as (L, L), (H, L), and (H, H), reflecting SNAIL1 as an information integrator of the two converging pathways.
Further modeling suggests that components in the network function cooperatively to encode the TGF-β information (Supplementary Fig. [S6b](#MOESM1){ref-type="media"}). Increasing or decreasing the nuclear GSK3 enzymatic activity tunes the system to generate the second SNAIL1 wave with a higher or lower threshold of TGF-β duration, respectively, while changing the cytosol GSK3 enzymatic activity has the opposite effect. Upregulation of GLI1, or downregulation of I-SMAD, both of which have been observed in various cancer cells, also decrease the threshold for generating the second SNAIL1 wave. Therefore cells of different types can share the same network structure, but fine-tune their context-dependent responses by varying some dynamic parameters, and for a specific type of cells dysregulation of any of the signaling network components may lead to misinterpretation of the quantitative information of TGF-β signal.
We have shown that the SNAIL1 dynamics is TGF-β1 duration dependent. To further confirm that cells respond differentially to TGF-β1 with different duration, we measured the mRNA levels of another four genes, all of which respond to TGF-β1 (Supplementary Fig. [S6c](#MOESM1){ref-type="media"}).^[@CR44],[@CR45]^ Gene *FN1* codes for the cell motility related protein fibronectin. Its expression is activated even by the 2-h TGF-β1 pulse, and increases with longer TGF-β1. Gene *CTGF*, whose product is an extracellular matrix protein and related to cell motility, is activated at similar extent by both 2 and 8-h TGF-β1 pulses, and its expression level increases by additional 14-folds with continuous TGF-β1 treatment. Expressions of genes *MMP2 and CLDN4*, coding proteins related to mesenchymal extracellular hallmark and cell migration, increase only slightly (less than two folds) with either 2 or 8 h TGF-β1 pulse, compared to the more significant change under continuous TGF-β1 treatment. Therefore, these downstream genes also show differential expression patterns depending on TGF-β1 duration, and cells activate different response programs correspondingly.
Discussion {#Sec9}
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TGF-β is a multifunctional cytokine that can induce a plethora of different and mutually exclusive cellular responses. A significant open question is how cells interpret various features of the signal and make the cell fate decision. TGF-β can activate a number of pathways interconnected with multiple crosstalk points. Our studies reveal that this interconnection is essential such that components of the network can function coordinately and appropriately to interpret the temporal (time and duration) information from TGF-β.
pSMADs are major inducers for the first wave of SNAIL1 expression {#Sec10}
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The two-wave dynamic of TGF-β-induced SNAIL1 expression has been observed in several cellular systems,^[@CR46],[@CR47]^ supporting the underlying relay mechanism discovered in this work. The first wave is fundamentally induced by pSMAD2/3, as evidenced from our SMAD inhibition experiments, and similarity between the dynamics of pSMAD2/3 and the first wave of SNAIL1. SNAIL1 may act as cofactor of pSMADs to induce other early response genes.^[@CR48]^ At later times the nuclear concentrations of pSMAD2/3 decrease though continue to contribute to SNAIL1 activation at a lower level.
GLI1 is a signaling hub for multiple pathways and temporal checkpoint for activating second-wave of sustained SNAIL1 expression {#Sec11}
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GLI protein has been traditionally attributed to the canonical Hedgehog pathway. Here, we show that TGF-β induction of GLI1 relays the signal to induce SNAIL1. Consistent with the present study, Dennler et al. reported SMAD3-dependent induction of the GLI family by TGF-β both in multiple cultured cell lines, and in transgenic mice overexpressing TGF-β^[@CR30]^ Many other signals such as PGF, EGF can also activate the GLI family, and a GLI code has been proposed to integrate input from different pathways and lead to context-dependent differential responses.^[@CR49]^ Our results confirmed this role of GLI1 as an intermediate information integrator and transmitter, and suggest that TGF-β must act above a threshold value of duration to activate the second wave of SNAIL1. This temporal checkpoint prevents spurious SNAIL1 activation and subsequent major cellular fate changes.
GSK3 fine-tunes the threshold of the GLI1 checkpoint and synchronizes responses of a population of cells {#Sec12}
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The functional switch from pSMAD2/3 to GLI1 relays information from TGF-β signaling beyond the initial induction of SNAIL1, and this relay is facilitated by a second relay from the active to the inactive phosphorylation form of GSK3 proteins. Active regulation of the abundance and nuclear location of GSK3^AA^ form has been observed in neurons.^[@CR50]^ In contrast to these earlier reports we observed an accumulation of GSK3^AA^ in the ER and Golgi apparatus. Mechanistically this may be caused by redistribution of cytosolic GSK3^AA^, or a simple accumulation of de novo synthesized and phosphorylated GSK3 proteins. The overall consequence is an increase in local GSK3 enzymatic activity, which forms part of the GSK3 switch that smooths the pSMAD2/3-GLI1 transition and the duration threshold of TGF-β pulse that generates the second wave of SNAIL1.
This seemingly simple process, which accelerates the response time through transient and minor increases in the initial concentration of a molecular species subject to positive feedback control, may have profound biological functions. Positive feedback loops are ubiquitous in cellular regulation, with a major function to filter both the strength and temporal fluctuations of stimulating signals and to prevent inadvertent cell fate change. This network, however, may have an inherently slow response time, and the response is highly sensitive to the initial concentration of the substrate that lead to large cell-to-cell variation of temporal dynamics. This variation and slow dynamic may be problematic for processes such as neural crest formation and wound healing where precise and synchronized temporal control is crucial for generating collective responses of multiple cells. The expanded network shown in Fig. [5b](#Fig5){ref-type="fig"} allows transient increase of the initial substrate concentration, and solves the seemingly incompatible requirements for the simple positive feedback motif on robustness against fluctuations as well as fast and synchronized responses. It assures that despite a possible broad distribution of basal expression levels of the protein, cells are activated within a designated period of time at the presence of persistent activation signal, without sacrificing the filtering function of the feedback loop.
Cells use TOSS formed by a composite network to increase information transfer capacity {#Sec13}
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Cells constantly encounter TGF-β signals with different strengths and duration, and must respond accordingly. It is well documented that biological networks reliably transmit information about the extracellular environment despite intrinsic and extrinsic noise in a subtle and functional way. However, quantitative analyses using information theory reveal that the dynamic of each individual readout is quite coarse with one or few bits.^[@CR9],[@CR10]^ This is a paradox. However, our results suggest that cells use multiple readouts to generate a TOSS with an expanded capacity to encode signal information and generate a far more subtle response system. For example, the SMAD motif has a refractory period due to the negative feedback loop and thus can accurately encode the duration information of TGF-β only within a limited temporal range. The GLI1 motif encodes information of longer TGF-β duration, which then saturates. This TOSS may be further expanded, such that the SNAIL1 motif itself possibly encodes information of longer TGF-β duration and relays to other transcription factors such as TWIST and ZEB, and leads to stepwise transition from the epithelial to the mesenchymal phenotype depending on the TGF-β duration.^[@CR34]^ Therefore although each motif has limited information coding capacity, a combination of motifs can code and transmit detailed signaling information. This is analogous to the design of a computer composed of many binary logic gates.
As with other signaling process, TGF-β signaling is context dependent, and the dynamic and regulatory network vary between cell types.^[@CR25],[@CR51]^ For the three cell lines we examined our results identify GLI1 as a major relaying factor for the TGF-β signaling. The inhibition experiments show that other possible peripheral links have minor contributions to SNAIL1 activation, while their weights may grow at time later than we examined. Consequently the present work has focused on the early event of TGF-β activation of SNAIL1, which is within 72 h for MCF10A cells. Nevertheless, the relay mechanism and the corresponding network structure identified here can be general for transmitting quantitative information of TGF-β and other signals. It is typical that an extracellular signal is transmitted through a canonical pathway with negative feedbacks and multiple non-canonical pathways, and these pathways crosstalk at multiple points, and Supplementary Fig. [S6d](#MOESM1){ref-type="media"} gives some examples including IL-12, DNA double strand breaking, and LPS. Therefore, the mechanism revealed in this work is likely beyond TGF-β signaling.
Network temporal dynamics is a key for effective pharmaceutical intervention {#Sec14}
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Upregulation of GLI1, and GSK3 and the responsive SMAD family has been reported in pathological tissues of fibrosis^[@CR52]^ and cancer,^[@CR49]^ and all three have been considered as potential drug targets. The present study emphasizes that in cell signaling timing is fundamental for function. The same network structure may generate drastically different dynamics with different parameters, as observed for different cell types. Consequently, effective biomedical intervention needs to take into account the network dynamics. We have already demonstrated that adding the inhibitors at different stages of TGF-β induction can be either effective vs. not effective on reducing SNAIL1 (by inhibiting pSMAD2/3), both effective (by inhibiting GLI1), and even opposite (by inhibiting GSK3). Actually, one may even exploit this dynamic specificity for precisely targeting certain group of cells while reducing undesired side effects on other cell types.
In summary through integrated quantitative measurements and mathematical modeling we provided a mechanistic explanation for how cells read TGF-β duration. Several uncovered specific mechanisms, such as expanding information transmission capacity through signal relaying, and reducing response times of positive feedback loops by increasing initial protein concentrations, may be general design principles for signal transduction.
Methods {#Sec15}
=======
Cell culture {#Sec16}
------------
MCF10A cells were purchased from the American Type Culture Collection (ATCC) and were cultured in DMEM/F12 (1:1) medium (Gibco) with 5% horse serum (Gibco), 100 μg/ml of human epidermal growth factor (PeproTech), 10 mg/ml of insulin (Sigma), 10 mg/ml of hydrocortisone (Sigma), 0.5 mg/ml of cholera toxin (Sigma), and 1× penicillin-streptomycin (Gibco). MCF7 cells were purchased from ATCC and cultured in EMEM medium (Gibco) with 10% FBS (Gibco), 10 mg/ml of insulin, and 1× penicillin-streptomycin. A549 cells were purchased from ATCC and were cultured in F12 medium (Corning) with 10% FBS and 1× penicillin-streptomycin. All cells were incubated at 37 °C with 5% CO~2~.
TGF-β induce and inhibitor treatment {#Sec17}
------------------------------------
Cells for TGF-β induction and inhibitor treatment were seeded at \~60--70% confluence without serum starvation. For TGF-β treatment, 4 ng/ml human recombinant TGF-β1 (Cell signaling) was added into culture medium directly. For inhibition experiment, 4 μM of LY2109761 (Selleckchem), 20 μM of GANT61 (Selleckchem), and 10 μM of SB216763 (Selleckchem) were used to inhibit SMAD, GLI, and GSK3, respectively. The medium was changed every day during treatment to keep the reagent concentration constantly. For reproducibility, we used cells within 10th--15th generations, same patches of reagents, serum, and tried to perform each group of experiments (e.g., those in Fig. [2c](#Fig2){ref-type="fig"}) together.
Immunofluorescence microscopy and data analysis {#Sec18}
-----------------------------------------------
Cells were seeded on four-well glass-bottom petri dishes at \~60% confluence overnight and treated with reagents (TGF-β1 and/or inhibitors). Three independent experiments were performed in every treatment. At designated time points, cells were harvested and stained with specific antibodies following procedure modified from the protocols at the Center of Biological Imaging (CBI) in the University of Pittsburgh. In general, cells were washed with DPBS for 5 min for three times followed by 4% formaldehyde fixation for 10 min at room temperature. Cells were then washed three times with PBS for 5 min every time. PBS with 0.1% TritonX-100 (PBS_Triton) was used for penetration. BSA of 2% in PBST was used for blocking before staining with antibodies. The first antibodies, anti-pSMAD2/3 (Santa Cruz), anit-SNAIL1 (Cell signaling), anti-GLI1 (Santa Cruz) were diluted by PBST with 1% BSA. Samples were incubated with the first antibodies at 4 °C overnight. Then cells were washed three times with 10 min for each before being incubated with the secondary antibodies, anti-mouse Alexa Fluor 647 (Abcam), anti-rabbit Alexa Fluor 647 (Abcam), anti-gaot Alexa Fluor 647 (Abcam), anti-mouse Alexa Fluor 555 (Abcam), anti-rabbit Alexa Fluor 555 (Abcam), anti-gaot Alexa Fluor 555 (Abcam), anti-mouse Alexa Fluor 488 (Abcam), anti-rabbit Alexa Fluor 488 (Abcam), anti-gaot Alexa Fluor 488 (Abcam), for 1 h at room temperature. After antibody incubation, cells were washed with PBS_Triton for 5 min and stained with DAPI (Fisher) for 10 min at room temperature. Cells were washed three times with PBS_Triton for 5 min and stored in PBS for imaging.
Photos were taken with Nikon A1 confocal microscopy at CBI. The microscope was controlled by the build-in software, Nikon NIS Elementary. All photos, except the photo for GSK3^AA^ subcellular localization, were taken with plan fluor 40× DIC M/N2 oil objective with 0.75 numerical aperture and 0.72 mm working distance. The scan field were chosen randomly all over the glass-bottom area. For identifying the GSK3^AA^ subcellular localization, plan apo λ 100× oil objective with 1.45 NA and 0.13 mm WD was used. The 3D model of GSK3^AA^ overlapped with ERC and DAPI were reconstructed from 25 of *z*-stack images in 11.6 μm and videos were produced also by NIS Element software. To minimize photobleaching, an object field was firstly chosen by fast scan, then the photos were taken at 2014 × 2014 pixel or 4096 × 4096 pixel resolution, for generation large data or for photo presentation, respectively.
CellProfiller was used for cell segregation and initial imaging analysis as what described in Carpenter et al.^[@CR53]^
### Image correction {#Sec19}
To keep identical background through all images, background correction was performed before further image processing. For each image fluorescent intensities in space without cells were used as local background. Photos that have obviously uneven illumination and background fluorescence were removed from further processing. Otherwise the mean background fluorescent intensity was obtained through averaging over the whole image, and was deducted uniformly from the image.
### Image segmentation {#Sec20}
Cell number and position were determined by nuclear recognition with DAPI. The global strategy was used to identify the nuclear shape, and the Otsu algorithm was used for further calculation. Clumped objectives were identified by shape and divided by intensity. Next, using the shrank nuclear shape as seed, cell shape was identified by the Watershed algorithm. For identifying the clusters of GSK3^AA^ formed around a nucleus, the nuclear shape was shrank manually by 3 pixels and used as a new seed to grown the boundary with the watershed method until reaching background intensity level. All parameters were optimized through an iterative process of automatic segmentation and manual inspection.
### Image quantification {#Sec21}
Averaged fluorescence density and integrated fluorescence intensity were calculated automatically with CellProfiler. The amount of the GSK3^AA^ form was quantified as the sum of intensities of pixels belonging to the cluster formed around a nucleus. Concentrations of all other proteins were quantified by the average pixel intensity within the nucleus or cytosol region of a cell. Next, the quantified results were examined manually, and those cells with either cell area, nuclear area, or fluorescent intensity beyond five folds of the 95% confidence range of samples from a given treatment were discarded, which account for less than 1% of the cells analyzed. Immunofluorescence data were further processed and plots were generated using customized R codes and Matlab codes.
Quantitative PCR {#Sec22}
----------------
Cells were seeded in 12-well plastic bottom cell culture plates and treated as described above. Three parallel experiments were performed in every treatment. Total RNA was isolated with the TRIZOL RNA isolation kit (Fisher), and mRNA was reversely transcribed with the RNAscript II kit (ABI). The stem-loop method was used for microRNA reverse transcription. The qPCR system was prepared with the SYBR green qPCR kit with designed primers (Supplementary table [1](#MOESM3){ref-type="media"}) and performed on StepOnePlus real-time PCR (ABI).
Immunoprecipitation and silver staining {#Sec23}
---------------------------------------
Immunoprecipitation was performed with SureBeads magnetic beads (Bio-Rad) following a protocol modified from the one provided by the manufacture. We washed beads with PBS with 0.1% Tween 20 (PBS_Tween) for three times, then harvested cells by RIPA (Thermo) with proteinase and phosphatase inhibitor (Roche). Samples were pre-cleaned with 100 μl of suspended Protein G per 450 μl of lysis mixture. Antibodies targeting GSK3 (Cell Signaling), GSK3^AA^ (Santa Cruz), and GSK3^D^ (Santa Cruz) were added into every 100 μl of bead mixture respectively. The mixture was rotated at 4 °C for 3 h. Beads that were conjugated with antibodies were washed with PBS_Tween. An amount of 100 μl of pre-cleaned lysis buffer was added into conjugated beads and rotated at 4 °C overnight. Targeted proteins were eluded from beads by incubating with 40 μl 1× Laemmli buffer with SDS at 70 °C for 10 min. For the samples an amount of 5 μl was used for western blot assay, and an amount of 30 μl was loaded for SDS-PAGE (Bio-Rad) and followed by silver staining (Fisher).
Network reconstruction and coarse-graining {#Sec24}
------------------------------------------
The full network from TGF-β1 to SNAIL1 (Supplementary Fig. [S1](#MOESM1){ref-type="media"}) was generated with IPA (Qiagen®). Specifically, all downstream regulators of TGF-β1 and upstream regulators of SNAIL1 in human, mice and rat were searched and added to the network. Then, direct or indirect relationships between every pair of regulators were searched and added to the network. After obtaining the whole network, regulators that have been reported to be activated later than SNAIL1 were removed. Examination of the network reveals that the network can be further organized into three groups: the TGF-β-SMAD-SNAIL canonical pathway, the TGF-β-GSK3-β-catenin pathway that has the most number of links, and others. We further noticed that GLI1 is a central connector of TGF-β, SMAD, GSK3 and SNAIL1. We performed western blot and IF studies on β-CATENIN and found that neither its concentration nor its location changes significantly on day 3, therefore we removed β-CATENIN from the network. In addition, previous studies report that the SMAD-GLI axis plays important role in TGF-β induced EMT.^[@CR31]^ Therefore we further grouped the network as the SMAD module, the GLI module, and the GSK3 module, as well as the remaining ones that we referred as "Others", and reached the network shown in Fig. [2a](#Fig2){ref-type="fig"}. Those molecular species not explicitly specified in Fig. [2a](#Fig2){ref-type="fig"} either have their effects implicitly included in the links, for example the link from TGF-β to GSK3, or are included in the links of "Others". This treatment is justified since our various inhibition experiments indeed showed that the three factors we identified affect SNAIL1 expression the most. These "other" species may contribute to snail1 activation at a time later than what considered in this work. Therefore we emphasize the network in Fig. [2a](#Fig2){ref-type="fig"} is valid only within the time window we examined, i.e., within 3 days after TGF-β1 treatment for MCF10A cells.
Data and code availability {#Sec25}
--------------------------
The data and customized Matlab codes for simulation is available in following link: <https://figshare.com/s/3d045c7da0db4fb4cd0f>.
Electronic supplementary material
=================================
{#Sec27}
Supplementary figures Supplementary information Supplementary tables Movie S1: Subcellular localization of GSK3AA (red) Movie S2: Subcellular localization of GSK3AA (red) overlaid with ERC 623 (green) and DAPI (blue, nuclear area)
**Publisher\'s note:** Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
These authors contributed equally: Jingyu Zhang and Xiao-Jun Tian.
Electronic supplementary material
=================================
**Supplementary information** accompanies the paper on the *npj Systems Biology and Applications* website (10.1038/s41540-018-0060-5).
This work was supported by the National Science Foundation \[DMS-1462049 to JX\], and the Pennsylvania Department of Health (SAP 4100062224). We would like to acknowledge the NIH supported microscopy resources in the Center for Biologic Imaging at University of Pittsburgh, specifically the confocal microscope supported by grant number 1S10OD019973-01.
J.Z., X.T., and J.X. initiated the project and designed the experiments; Y.C. helped on experimental design; J.Z performed the experiments; W.W. and S.W. helped on data analysis; X.T., J.Z., and J.X. constructed the mathematical models. J.Z., X.T. and J.X. wrote the manuscript with input from S.W.
Competing interests {#FPar1}
===================
The authors declare no competing interests.
|
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“I know that my first review of this matcha wasn’t the best, but that somehow hasn’t stopped me from nearly finishing the small pouch I ordered. Tonight was the best cup I’ve had so far. It was...”
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I know that my first review of this matcha wasn’t the best, but that somehow hasn’t stopped me from nearly finishing the small pouch I ordered. Tonight was the best cup I’ve had so far. It was sweet, creamy.. nice and frothy. I actually really enjoyed this.
This makes me think that maybe I didn’t give the matcha a fair shot. Still, I’m increasing the rating!
I really enjoy the red matcha option because it lacks caffeine.. but feels a bit more special than just popping a bag of rooibos into a cup. There is no question that there is that rooibos flavor. It’s smoother than some others I’ve had and I don’t detect any of that medicinal taste. If you don’t like rooibos, I am not sure you would like red matcha… no matter how strong you get your flavoring. For me, the rooibos is absolutely evident! If you’re kind of in between liking/disliking it.. I say that you should give it a try! The addition of a flavor might be enough to sway you. :)
I wish they offered a 50/50 blend with the green or white, I sometimes really like rooibos but often don’t… it depends on the blend, really. But with the review certificate program it can’t hurt to try! |
Some data storage systems include complex arrangements of storage disk arrays, configuration management interfaces, and storage processors. A system administrator faces many choices in making adjustments to the configuration of a data storage system in response to changing conditions, many of them resulting in suboptimal performance. Along these lines, the system administrator may seek advice with regard to provisioning storage resources while running certain applications. Such advice typically follows application best practices that convert storage provisioning needs into a specification of storage resources such as storage type, RAID type, de-duplication, etc.
A conventional approach to automating storage resource provisioning employs software that allows an administrator to provision storage resources based on predefined provisioning best-practices algorithms. Such software can employ a scripting language, such as the well-known Lua scripting language, that expresses the application best practices in procedural logic. The administrator may account for changes in application best practices by arranging for adjustments to the software to reflect those changes. |
Q:
kineticjs user input to change text size
I got 90% of the way through making a canvas for a mug designer with user image upload and text styling (colour, weight, font style etc..) but I couldn't get my head around a bug in my code.
now I'm using kinetic.js to make it and have a text field with keyup and want to link it to a user input to resize text, I have worked with canvas but am new to kinetics so an easy way to implement this would be helpful.
HTML:
<input type="text" id="textBox" placeholder="Your Text">
<br>Text Size:
<input type="range" id="textSize" min="4" max="100" step="1" value="30" />
js:
var text = new Kinetic.Text({
x: 20,
y: 30,
text: '',
fontSize: '30',
fontFamily: 'Calibri',
fill: 'black',
draggable: true
});
document.getElementById("textBox").addEventListener("keyup", function () {
text.setText(this.value);
layer.draw();
}, true);
var stage = new Kinetic.Stage({
container: 'container',
width: 375,
height: 200
});
var layer = new Kinetic.Layer();
layer.add(text);
stage.add(layer);
here's the full fiddle
I also need to have the "onLoad" image (of Yoda) to be inserted with the "imageLoader", so help on that would be good too. (will post as a separate question if necessary).
Any tips appreciated, thanks in advance
A:
There are two decent options I can think of: adjusting the font size and adjusting the scale of the text element. Very similar in either case.
Font size:
document.getElementById("textSize").addEventListener("change", function() {
var size = this.value;
text.fontSize(size);
layer.draw();
}, true);
Fiddle: http://jsfiddle.net/pDW34/8/
Scale:
document.getElementById("textSize").addEventListener("change", function() {
var scale = this.value/100*4;
text.scale({x: scale, y: scale});
layer.draw();
}, true);
Fiddle: http://jsfiddle.net/pDW34/9/
|
Q:
How to pass three parameter in url for retrieving data using json parser in android?
I have two spinner and date picker in the first class in the 1st spinner I have selected location the related stock point flow on the 2nd spinner and selected date, and three string object sent to next class using put string,get string and added to url item,item1,date. and run the application select location, stockpoint, date then press get table button it will force close. I debug it the object is sent and add successfully when cursor come to JSONObject json1 = jParser.getJSONFromUrl(url); it will close . when I comment url and replace predefined url it will run.. I cant identify the problem
public void torun() {
Bundle b=this.getIntent().getExtras();
String items=b.getString("item");
String items1=b.getString("item1");
String dates=b.getString("date");
// String url0 = "http://10.0.2.2:51382/RestServiceImpl.svc/json/?Location=ArihantWanarpet&GROUP=ArihantShowroom&asondate=2013-2-24";
String url = "http://10.0.2.2:51382/RestServiceImpl.svc/jsons/?Location="+items+"&GROUP="+items1+"&asondate="+dates;
// Hashmap for ListView
ArrayList<HashMap<String, String>> contactList = new ArrayList<HashMap<String, String>>();
// Creating JSON Parser instance
JSONParser jParser = new JSONParser();
// getting JSON string from URL
try {
JSONObject json1 = jParser.getJSONFromUrl(url);
// Getting Array of Contacts
JSONArray list = json1.getJSONArray(TAG_JSONDataResult);
// looping through All Contacts
for(int i = 0; i < list.length(); i++){
JSONObject c = list.getJSONObject(i);
// Storing each json item in variable
String GRPCODE = c.getString(TAG_GRPCODE);
String GRPNAME = c.getString(TAG_GRPNAME);
String QTY = c.getString(TAG_QNT);
String BUDGET = c.getString(TAG_BUDGET);
String STOCK = c.getString(TAG_STOCK);
String DIFF = c.getString(TAG_DIFF);
String DIFF_P = c.getString(TAG_DIFF_P);
String EQTY = c.getString(TAG_EQTY);
String EQTY_P = c.getString(TAG_EQTY_P);
String EBUDGET = c.getString(TAG_EBUDGET);
String ESTK = c.getString(TAG_ESTK);
String ESTK_P = c.getString(TAG_ESTK_P);
String EDIFF = c.getString(TAG_EDIFF);
String EDIFF_P = c.getString(TAG_EDIFF_P);
String DQTY = c.getString(TAG_DQTY);
String DQTY_P = c.getString(TAG_DQTY_P);
String DBUDGET = c.getString(TAG_DBUDGET);
String DSTK = c.getString(TAG_DSTK);
String DSTK_P = c.getString(TAG_DSTK_P);
String DDIFF = c.getString(TAG_DDIFF);
String DDIFF_P = c.getString(TAG_DDIFF_P);
String PQTY = c.getString(TAG_PQTY);
String PQTY_P = c.getString(TAG_PQTY_P);
String PBUDGET = c.getString(TAG_PBUDGET);
String PSTK = c.getString(TAG_PSTK);
String PSTK_P = c.getString(TAG_PSTK_P);
String PDIFF = c.getString(TAG_PDIFF);
String PDIFF_P = c.getString(TAG_PDIFF_P);
// creating new HashMap
HashMap<String, String> map = new HashMap<String, String>();
// adding each child node to HashMap key => value
map.put(TAG_GRPCODE, GRPCODE);
map.put(TAG_GRPNAME, GRPNAME);
map.put(TAG_QNT, QTY);
map.put(TAG_BUDGET, BUDGET);
map.put(TAG_STOCK, STOCK);
map.put(TAG_DIFF, DIFF);
map.put(TAG_DIFF_P, DIFF_P);
map.put(TAG_EQTY, EQTY);
map.put(TAG_EQTY_P, EQTY_P);
map.put(TAG_EBUDGET, EBUDGET);
map.put(TAG_ESTK, ESTK);
map.put(TAG_ESTK_P, ESTK_P);
map.put(TAG_EDIFF, EDIFF);
map.put(TAG_EDIFF_P, EDIFF_P);
map.put(TAG_DQTY, DQTY);
map.put(TAG_DQTY_P, DQTY_P);
map.put(TAG_DBUDGET, DBUDGET);
map.put(TAG_DSTK, DSTK);
map.put(TAG_DSTK_P, DSTK_P);
map.put(TAG_DDIFF, DDIFF);
map.put(TAG_DDIFF_P, DDIFF_P);
map.put(TAG_PQTY, PQTY);
map.put(TAG_PQTY_P, PQTY_P);
map.put(TAG_PBUDGET, PBUDGET);
map.put(TAG_PSTK, PSTK);
map.put(TAG_PSTK_P, PSTK_P);
map.put(TAG_PDIFF, PDIFF);
map.put(TAG_PDIFF_P, PDIFF_P);
//map.put(TAG_PHONE_MOBILE, mobile);
// adding HashList to ArrayList
contactList.add(map);
}
} catch (JSONException e) {
e.printStackTrace();
}
/**
* Updating parsed JSON data into ListView
* */
ListAdapter adapter = new SimpleAdapter(this, contactList,
R.layout.list_item,
new String[] { TAG_GRPCODE, TAG_GRPNAME, TAG_QNT, TAG_BUDGET, TAG_STOCK, TAG_DIFF, TAG_DIFF_P, TAG_EQTY, TAG_EQTY_P, TAG_EBUDGET, TAG_ESTK,
TAG_ESTK_P, TAG_EDIFF, TAG_EDIFF_P, TAG_DQTY, TAG_DQTY_P, TAG_DBUDGET, TAG_DSTK, TAG_DSTK_P, TAG_DDIFF, TAG_DDIFF_P, TAG_PQTY,
TAG_PQTY_P, TAG_PBUDGET, TAG_PSTK, TAG_PSTK_P, TAG_PDIFF, TAG_PDIFF_P },
new int[] {
R.id.l1, R.id.l2, R.id.l3, R.id.l4, R.id.l5, R.id.l6, R.id.l7, R.id.l8, R.id.l9, R.id.l10, R.id.l11, R.id.l12, R.id.l13, R.id.l14,
R.id.l15, R.id.l16, R.id.l17, R.id.l18, R.id.l19, R.id.l20, R.id.l21, R.id.l22, R.id.l23, R.id.l24, R.id.l25, R.id.l26, R.id.l27, R.id.l28, });
setListAdapter(adapter);
// selecting single ListView item
ListView lv = getListView();
// Launching new screen on Selecting Single ListItem
}
JSONParser class
public class JSONParser {
static InputStream is = null;
static JSONObject jObj = null;
static String json = "";
// constructor
public JSONParser() {
}
public JSONObject getJSONFromUrl(String url) {
// Making HTTP request
try {
// defaultHttpClient
DefaultHttpClient httpClient = new DefaultHttpClient();
HttpGet httpPost = new HttpGet(url);
HttpResponse httpResponse = httpClient.execute(httpPost);
HttpEntity httpEntity = httpResponse.getEntity();
is = httpEntity.getContent();
} catch (UnsupportedEncodingException e) {
e.printStackTrace();
} catch (ClientProtocolException e) {
e.printStackTrace();
} catch (IOException e) {
e.printStackTrace();
}
try {
BufferedReader reader = new BufferedReader(new InputStreamReader(
is, "iso-8859-1"), 8);
StringBuilder sb = new StringBuilder();
String line = null;
while ((line = reader.readLine()) != null) {
sb.append(line + "\n");
}
is.close();
json = sb.toString();
} catch (Exception e) {
Log.e("Buffer Error", "Error converting result " + e.toString());
}
// try parse the string to a JSON object
try {
jObj = new JSONObject(json);
} catch (JSONException e) {
Log.e("JSON Parser", "Error parsing data " + e.toString());
}
// return JSON String
return jObj;
}
}
logcat
07-03 07:25:21.177 632-632/com.android.exchange E/StrictMode: null
android.app.ServiceConnectionLeaked: Service com.android.exchange.ExchangeService has leaked ServiceConnection com.android.emailcommon.service.ServiceProxy$ProxyConnection@40d33ec0 that was originally bound here
at android.app.LoadedApk$ServiceDispatcher.<init>(LoadedApk.java:969)
at android.app.LoadedApk.getServiceDispatcher(LoadedApk.java:863)
at android.app.ContextImpl.bindService(ContextImpl.java:1418)
at android.app.ContextImpl.bindService(ContextImpl.java:1407)
at android.content.ContextWrapper.bindService(ContextWrapper.java:473)
at com.android.emailcommon.service.ServiceProxy.setTask(ServiceProxy.java:157)
at com.android.emailcommon.service.ServiceProxy.setTask(ServiceProxy.java:145)
at com.android.emailcommon.service.ServiceProxy.test(ServiceProxy.java:191)
at com.android.exchange.ExchangeService$7.run(ExchangeService.java:1850)
at com.android.emailcommon.utility.Utility$2.doInBackground(Utility.java:551)
at com.android.emailcommon.utility.Utility$2.doInBackground(Utility.java:549)
at android.os.AsyncTask$2.call(AsyncTask.java:287)
at java.util.concurrent.FutureTask.run(FutureTask.java:234)
at java.util.concurrent.ThreadPoolExecutor.runWorker(ThreadPoolExecutor.java:1080)
at java.util.concurrent.ThreadPoolExecutor$Worker.run(ThreadPoolExecutor.java:573)
at java.lang.Thread.run(Thread.java:856)
latest logcat error
07-03 07:47:04.487 1877-1877/com.androidhive.innovate E/AndroidRuntime: FATAL EXCEPTION: main
java.lang.RuntimeException: Unable to start activity ComponentInfo{com.androidhive.innovate/com.androidhive.innovate.AndroidJSONParsingActivity}: java.lang.IllegalArgumentException: Illegal character in query at index 49: http://10.0.2.2:51382/RestServiceImpl.svc/jsons/? Location=ArihantWanarpet&GROUP=ArihantShowroom&asondate=2013-7-3
at android.app.ActivityThread.performLaunchActivity(ActivityThread.java:2180)
at android.app.ActivityThread.handleLaunchActivity(ActivityThread.java:2230)
at android.app.ActivityThread.access$600(ActivityThread.java:141)
at android.app.ActivityThread$H.handleMessage(ActivityThread.java:1234)
at android.os.Handler.dispatchMessage(Handler.java:99)
at android.os.Looper.loop(Looper.java:137)
at android.app.ActivityThread.main(ActivityThread.java:5041)
at java.lang.reflect.Method.invokeNative(Native Method)
at java.lang.reflect.Method.invoke(Method.java:511)
at com.android.internal.os.ZygoteInit$MethodAndArgsCaller.run(ZygoteInit.java:793)
at com.android.internal.os.ZygoteInit.main(ZygoteInit.java:560)
at dalvik.system.NativeStart.main(Native Method)
Caused by: java.lang.IllegalArgumentException: Illegal character in query at index 49: http://10.0.2.2:51382/RestServiceImpl.svc/jsons/? Location=ArihantWanarpet&GROUP=ArihantShowroom&asondate=2013-7-3
at java.net.URI.create(URI.java:727)
at org.apache.http.client.methods.HttpGet.<init>(HttpGet.java:75)
at com.androidhive.innovate.JSONParser.getJSONFromUrl(JSONParser.java:36)
at com.androidhive.innovate.AndroidJSONParsingActivity.torun(AndroidJSONParsingActivity.java:105)
at com.androidhive.innovate.AndroidJSONParsingActivity.onCreate(AndroidJSONParsingActivity.java:64)
at android.app.Activity.performCreate(Activity.java:5104)
at android.app.Instrumentation.callActivityOnCreate(Instrumentation.java:1080)
at android.app.ActivityThread.performLaunchActivity(ActivityThread.java:2144)
... 11 more
A:
Ok....So finally you have to change your string of date to SimpleDateformat and then add that string to URL:
SimpleDateFormat dateFormat = new SimpleDateFormat(
"yyyy-MM-dd");
Date myDate = null;
try {
myDate = dateFormat.parse(dates);
} catch (ParseException e) {
e.printStackTrace();
}
SimpleDateFormat timeFormat = new SimpleDateFormat("yyyy-MM-dd");
String finalDate = timeFormat.format(myDate);
and then add to:
String myUrl;
myUrl = String.format("http://10.0.2.2:51382/RestServiceImpl.svc/jsons/? Location=%s&GROUP=%s&asondate=%s",items,items1,finalDate );
|
Friday, January 25, 2013
Save Our Streets Crown Heights is delighted to share with you an evaluation of S.O.S. that reports that shooting rates have decreased as a result of our efforts. The report, "Testing a Public Health Approach to Gun Violence," conducted by the Center for Court Innovation, took place over 29 months and shows that comparison neighborhoods had 20% higher rates of gun violence than our neighborhood did.
To learn more about S.O.S., check out the "About Us" or "Videos" tab above. If you want to get involved, check out the "How You Can Help" tab.
To download the study, click here.To read the press release, click here.To listen to a podcast interview with the authors, click here.To read a Q&A with the authors, click here.To read a recent New York Times article profiling one of the Violence Interrupters, click here. To listen to a recent CNN radio piece on the S.O.S. team, click here.
Devastatingly, in the late evening of New Year's Day, two 17-year-olds were shot and severely wounded while sitting in their car on the corner of Troy Avenue and Park Place. Two days later, S.O.S. stood on that same corner alongside local youth and members of the clergy to show the community that shootings will not go unnoticed or be tolerated. S.O.S. Clergy Action Network (C.A.N.) leader Reverend Kevin Jones, joined by Bishop Billips and Reverend Mathew Burke, appealed for peace through the S.O.S. bullhorn. Reverend Jones recalled growing up in the neighborhood, and, looking at a dilapidated building, noted how it had changed. “This was my block. I used to shop at that supermarket,” Jones said. “I’m still here… and now together we have to work together to stop the gun violence, and make our community whole again.”
The crowd standing against gun violence that night had a youthful energy, as it was filled with Youth Organizers, local high school students involved in our program to generate youth leadership in the struggle against gun violence. Marlon Peterson, deputy director at the Crown Heights Community Mediation Center, gathered the youth organizers together and told them that, because in this instance of gun violence both the shooters and victims were young people, the neighborhood might come to fear and shun people of their age. "Your presence here tonight shows the community that there is another way, there is another path for young people here. Youth don't have to be the face of violence. You're showing them tonight that youth can be the face of peace."
S.O.S. Holiday Party and Toy Giveaway
S.O.S. spread the holiday spirit in Crown Heights on Friday, December 21st, when the Crown Heights Community Mediation Center (CHCMC) opened its doors to children and families from the community for a holiday party and toy giveaway. Children poured into the Center wearing bright holiday smiles and transformed the space into a festive party full of music, food and laughter. CHCMC and S.O.S. staff quickly adopted the mood, leading the children in interactive games, carol singing and learning activities about how to make the neighborhood a safe and friendly place. At the end of the party the children lined up excitedly to collect gift bags that the S.O.S. team was distributing. The children were delighted with their gifts, leaving the Mediation Center with hands full of new toys and hearts full of Christmas cheer. We thank the shoppers and merchants of Park Slope and the Park Slope Civic Council who generously donated the toys and gifts that made this event possible.
Monday, January 7, 2013
CNN Radio News profiled the S.O.S. team's efforts to reduce shootings and killings in our neighborhood. Congratulations to Outreach Worker Derick Scott, Program Manager Allen James, and the rest of the S.O.S. team whose hard work made this possible.
Confronting criminals slows murders in NYC
(CNN) – New York ended 2012 with a historic low murder rate – 414 killings in all. It's the lowest the figure has been since police started keeping track in 1963.
Mayor Michael Bloomberg gives much of the credit to a combination of police tactics and some of the toughest gun laws in the country.
However, there is much more going on than stringent policing.
[3:02] "If you stake out a piece of territory like SOS has, basically a two square mile grid, we can suppress shootings here. We can lower the number of shootings here and the outreach team here has done that,” said Alan James, program manager at Save Our Streets Crown Heights, a community based project fighting gun violence in one Brooklyn neighborhood.
Wednesday, January 2, 2013
This Friday, January 4th at 5:30 pm Harlem Mothers SAVE, a coalition of mothers fighting against gun violence, will be holding a candlelight vigil to call attention to the spate of recent shootings in Upper Manhattan, and to light candles in memory of the victims of the Sandy Hook Elementary School shooting. The vigil will be held at 2471 Frederick Douglass Blvd/132nd street, and will bring together members of Harlem Mothers SAVE, Elected Officials, African American and Latina women and other community activists to call for action to stop the violence and slaughter of our children.
About the organization: Harlem Mothers SAVE (Stop Another Violent End) was established in 2006 by Jackie Rowe Adams, who lost two sons to gun violence with the assistance of NY State Assemblyman Keith Wright. The non-profit organization focuses on activism, victim services and education. April Tyler, a Harlem-based organizer and former Democratic district leader will lead the protest.
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Outreach Workers
S.O.S. outreach workers engage with those who are most likely to shoot or be shot. Our goal is to stop gun violence before it happens, and prevent retaliatory acts when it does.
Community Mobilization
S.O.S. Crown Heights works with a wide range of community partners; organizing BBQs, concerts, sporting events, rallies, and other events intended to strengthen our community and bring an end to gun violence
Crown Heights Mediation Center
S.O.S. is a project of the Crown Heights Mediation Center. Click on the image to go to the CHCMC blog or call 718-773-6886 for more information.
Center for Court Innovation
The Crown Heights Community Mediaiton Center is a project of the Center for Court Innovation |
Friday 26 February 2010 19.04 EST
First published on Friday 26 February 2010 19.04 EST
Gladstone Small was in the England team that dismissed Pakistan for 74 and then lost to them in the final
I remember the day really well. The weather hadn't been very good in Adelaide, it had been raining through the night so the wicket was a bit spicy.
Pakistan were a very gifted team with Inzamam-ul-Haq and Javed Miandad in the side. They also had Wasim Akram, who launched himself on to the international stage in the tournament. We'd played quite poorly against Zimbabwe and they'd beaten us, so we were under pressure. But we knew they had also struggled in the tournament and that this was our chance to knock them out.
We won the toss and put them in and things went very well. Phil DeFreitas got Inzamam first ball. I ended up getting Ijaz Ahmed and Moin Khan, their wicketkeeper-batsman, and finished with 2 for 29 off 10 overs. It was a great feeling bowling them out for 74.
We thought we'd got all this time to get the runs, but the heavy rain came in between innings and the game was delayed again and again. The overs kept getting reduced. In the end we had 10 overs to get them and they had this stupid rule for the tournament that meant that we needed to get 62 runs from those 10 overs. It was ridiculous. It should have been that we needed to get 14 or something like that.
We tried to go for them anyway, but it rained again and it was declared a wash-out and we shared the points, one each. In the end that was enough for them to qualify for the semi-finals.
The rain rule had done Pakistan a favour, but it also did us a favour in the semi-final against South Africa, where it helped us get through to the final.
But we really hadn't thought that we'd end up playing Pakistan in the final because they weren't playing that well. We should have knocked them out and they would never have had the chance to be so great in the final.
As it happened Wasim bowled Ian Botham, Allan Lamb and Chris Lewis in the space of five or six overs coming round the wicket. It was incredible bowling and it won them the match.
I do look back and think, well, England still haven't won the World Cup and that was probably our best chance. We had a tremendous team then. A lot of talent, a lot of experience. We've never got close since.
Then what happened
England lost to Pakistan in the final. Small retired in 1999 having taken 1,314 wickets and played in the last England team to win the Ashes in Australia. He will be back there with ITC Sports following England's tour next winter. |
Friday, September 11, 2009
Why the hesitation to form a Royal Commission of Inquiry , YAB Prime Minister?
Prime Minister Datuk Seri Najib announced yesterday that the Cabinet has set up a special task force to look into the Port Klang Free Zone ( PKFZ ) project. The task force will be headed by Chief Secretary to the Government Tan Sri Mohd Sidek .
The question the people want to know is why has the Cabinet not set up a Royal Commission of Inquiry to investigate the project which has become the" scandals of all scandals "?
The scandal is not a new issue and to date so many questions remain unanswered. Neither has any one been prosecuted although a special task force appointed has completed and submitted its investigation. What is causing the Cabinet's hesitation in setting up a RCI to fully investigate this RM 12.5 bil scandal?
Gerakan president Tan Sri Dr Koh Tsu Koon had earlier said that a RCI should be set up . He should tell the people whether he did make such a proposal at yesterday's Cabinet meeting and if he did , why were Cabinet members not supportive of his proposal?
When columnist Baradan Kuppusamy had written in an article titled " Tee Keat takes the hard- but right course" that Tee Keat seemed to be fighting a lone battle for transparency and accountability, the Prime Minister's office had issued a statement to clarify that the government was not indifferent towards the PKFZ issue and that relevant authorities, including the Public Accounts Committee were already conducting their investigations.
I had said that such clarification would not convince the people that the government had taken serious and sufficient efforts to investigate the scandal. I am sure again that if a government survey is carried out to gauge public opinion on the setting up of the special task force headed by Tan Sri Mohd Sidek and a RCI , the result will be overwhelming support for the setting up of a RCI.
The Straits Times has reported that the findings of the special task force headed by senior Malaysian lawyer Vinayak Pradhan had identified former transport minister Tan Sri Chan Kong Choy as among people who had not carried out their duties with adequate care. New Straits Times also said that they read the yet to be released report.
Many Malaysians are now wondering whether it is true that the yet to released report has been leaked. Why is there no official reaction from the Government on this?
A special task force headed by lawyer Vinayak Pradhan was set up . Now a super task force headed by Tan Sri Mohd Sidek is formed. Malaysians are now joking that a super super task force headed by the Prime Minister himself may be formed when the super task force submits its findings to the Cabinet.
The joke simply shows that the Cabinet has failed to convince the people of its commitment to transparency and accountability on the PKFZ scandal . The Cabinet should know that the single most important issue which has baffled the Malaysian public and the most important answer that they want to know is this - why is the Cabinet hesitant in setting up a RCI to investigate the RM 12.5 bil scandal? |
Former acting Democratic National Committee chairwoman Donna Brazile voiced her agreement Tuesday with Hillary Clinton’s judgment that President Donald Trump was not legitimately elected in 2016.
MSNBC host Chris Matthews said it was "confounding" that Clinton would call Trump’s election illegitimate, but Brazile said Clinton is right to draw attention to Russian interference. Matthews pushed her to say whether she actually agrees with Clinton, and Brazile continued to express doubts about the election.
"Donald Trump cracked the blue wall," she said. "He was able to win votes in Pennsylvania, Michigan, and Wisconsin, but, yes, I do believe Secretary Clinton is making a very important point. The role of social media, this coordination between WikiLeaks and possibly the Trump campaign—we need to get to the bottom of it."
Matthews pressed her to say whether she agreed with Clinton, and Brazile answered in the affirmative.
"My personal view is that it was not a legitimate election," Brazile said.
Matthews expressed shock and asked if that meant the election does not count. Brazile answered by asserting the theory that George W. Bush’s election over Al Gore was rigged by the Supreme Court.
"Chris, remember, I [was] the campaign manager of Al Gore in 2000, where, as you well know, the Supreme Court decided," Brazile said, referring to the Court’s decision on the Florida recount. "This election will always have an asterisk by it, and that's why I think Donald Trump should take steps as president to clean up our system so that no more foreign governments, no foreign meddling occur in the future."
Matthews pushed back against the assertion of illegitimacy by saying that is a third-world term and Clinton should never have used it. Brazile replied that she is not interested in semantics but in the "headwind" Clinton faced.
"I agree on that point, but you're arguing semantics when I want to argue that there was some big issues, big problems, and she faced a significant headwind in 2016," Brazile said. "I think what she means when she talks about legitimacy, she's talking about the foreign interference and meddling and what I describe in my book, Hacks."
Brazile’s book made waves for its accusation that Clinton had control over the DNC that helped her win a difficult primary against Sen. Bernie Sanders (I., Vt.), even as Brazile backtracked about whether the primary was "rigged." |
Q:
Open a new function with PHP
I have a page in PHP and I would open a function with a click.
Function shows me a query result, but when I write this code it doesn't work
<div class="btn-group">
<button type="button" class="btn btn-primary dropdown-toggle" data-toggle="dropdown">
Frequenza <span class="caret"></span>
</button>
<ul class="dropdown-menu" role="menu">
<?php
$query_frequenza="SELECT DISTINCT FREQUENZA FROM Dettagli_macchina WHERE macchine_id='$macchine' and Email='$_SESSION[login_user]'";
$result=mysqli_query($conne,$query_frequenza);
while($row=mysqli_fetch_array($result)){
$frequenza=$row['FREQUENZA'];
echo"<li><a href='#?frequenza=$frequenza' onclick='showfiltro2()'>$frequenza</a></li>";
}
?>
</ul>
</div>
<script type = "text/javascript">
function showfiltro2() {
document.getElementById("filtro2").style.display = "block";
document.getElementById("filtro1").style.display = "none";
}
</script>
<div id = "filtro2" style="display:none">
<?php
$filtro2=$_GET['frequenza'];
$query="SELECT DISTINCT * FROM Dettagli_macchina WHERE macchine_id='$macchine' and Email='$_SESSION[login_user]' and FREQUENZA='$filtro2' ";
$result=mysqli_query($conne,$query);
echo 'Found '. mysqli_num_rows($result) .'results';
echo "<table><tr>";
while ($row = mysqli_fetch_array($result)) {
echo "<tr><td>";
echo $row['COMPONENTE'];
echo "</td>";
echo "<td>";
echo $row['DETTAGLIO ATTIVITA'];
echo "</td>";
echo "<td>";
echo $row['FREQUENZA'];
echo "</td>";
echo "<td>";
echo $row['DATA-PREVISTA'];
echo "</td>";
echo "</tr>";
}
echo"</tr></table>";
?>
</div>
A:
Your question stems from a misunderstanding of how PHP and HTML work, and how data flows between the two.
First off it's important to remember that PHP and HTML are two completely separate parts, which do not interact with each other outside of the "request->reply" chain.
This means that all of the PHP code gets executed on the server, before the client gets the output of this processing. The server (PHP) doesn't care about what kind of output it is, nor does it understand how to parse it; For all PHP knows, it's all simple text.
After the PHP code has been completely parsed, the client receives the resulting text. Then it notices that it can understand this text as HTML, and parses it as a web-page. At this point the PHP code doesn't exist in the code at all, and the web browser (client) doesn't know anything about it.
It is unfortunate that so many tutorials keep mixing PHP and HTML code like you've done above, as this further confuses the two and makes it look like they're inter-communicative. What I recommend is to move all of your PHP code above any HTML-code, and do all of the processing before sending anything to the browser.
Not only will this make it a lot easier to actually keep track of, and understand, what's happening and why; But it will also allow you to add more functionality to your code, without trying to break the laws of physics. (For example: Deciding that you don't want to show a form to the user after all, half-way through the generation of said form.)
All this means that you don't "open a function" with a click. You send a request to the server with said click, and then the PHP code checks the incoming data for some predetermined condition (GET-parameter, etc), and then calls the function of said condition is fulfilled.
Something like this, in other words:
// First off we should use PDO, as mysql_*() is deprecated and removed in PHP7.
$db = new PDO ($dsn);
// Using prepared statements here, to prevent SQL injections.
$stmt = $db->prepare ("SELECT DISTINCT FREQUENZA FROM Dettagli_macchina WHERE macchine_id=:machineID and Email=:email");
$data = array (':machineID' => $macchine, ':email' => $_SESSION['login_user']);
if (!$stmt->exec ($data)) {
// Something went wrong, handle it.
}
// Initialize a variable to hold the generated menu, and a template to use when creating it.
$menuOut = $searchOut = '';
$menuTemplate = "<li><a href='#?frequenza=%s' onclick='showfiltro2()'>%s</a></li>";
// Using prepared statements we can iterate through all of the results with foreach().
foreach ($stmt->fetchAll () as $row) {
// Using htmlspecialchars() and rawurlescape() to prevent against XSS, and other HTML-injection attacks/mistakes.
// Notice where and in what order I've used the different functions, as one protects the URL as well.
$menuOut .= sprintf ($menuTemplate, htmlspecialchars (rawurlencode ($row['FREQUENZA'])), htmlspecialchars ($row['FREQUENZA']));
}
// Since this is probably the "function" you want to execute with said click, this is where we check if it
// has been sent by the client.
if (!empty ($_GET['frequenza'])) {
// Here you want to check to see if the parameter is actually something you'd expect, and not some random(?) garbage.
$filtro2 = $_GET['frequenza'];
// Again, prepared statements as your code was open to SQL injections!
$query = "SELECT DISTINCT * FROM Dettagli_macchina WHERE macchine_id=:machineID and Email=:email and FREQUENZA=:frequency";;
$stmt = $db->prepare ($query);
$data = array (
':machineID' => $macchine,
':email' => $_SESSION['login_user'],
':frequency' => $filtro2);
if (!$res = $stmt->exec ($data)) {
// Somethign went wrong with the query, handle it.
}
// Initialize a variable to hold the output, and the template to use for generating it.
$searchOut = '<table>';
$searchTemplate = '<tr><td>%s</td><td>%s</td><td>%s</td><td>%s</td></tr>';
$count = 0;
foreach ($stmt->fetchAll () as $row) {
// Again, protection against XSS and other HTML-breaking mistakes.
$searchOut .= sprintf ($searchTemplate,
htmlspecialchars ($row['COMPONENTE']),
htmlspecialchars ($row['DETTAGLIO ATTIVITA']),
htmlspecialchars ($row['FREQUENZA']),
htmlspecialchars ($row['DATA-PREVISTA']));
}
$searchOut = "<p>Found {$count} results</p>{$searchOut}</table>";
}
?>
<div class="btn-group">
<button type="button" class="btn btn-primary dropdown-toggle" data-toggle="dropdown">
Frequenza <span class="caret"></span>
</button>
<ul class="dropdown-menu" role="menu">
<?php echo $menuOut; ?>
</ul>
</div>
<script type="text/javascript">
function showfiltro2() {
document.getElementById("filtro2").style.display = "block";
document.getElementById("filtro1").style.display = "none";
}
</script>
<div id="filtro2" style="display: none">
<?php echo $searchOut; ?>
</div>
I've added some comments to explain what and why I've done things, as well as changed over from the old(!), deprecated and obsolete mysql_*() functions to PDO.
You can read more about how to use PDO in the PHP manual
|
* After someone drugs and tried to rape Delinda the Monticito tries to track the guy down, with Ed claiming he'll kill him. After finding out there are multiple victims and catching him Ed and Danny drive him out to the desert, with Danny begging Ed not to kill him.-->I am not digging this guy's grave!** Ed relents, then forces the predator to strip before digging his own grave with his bare hands. But the ground's too hot and hard, and he begs Ed to stop. His response?-->Did that work with my daughter? Did that work with any of your victims?** Then with a [[PreMortumOneLiner "What happens in Vegas, stays in Vegas."]] he shoots the creep as Danny yells, thankfully the gun wasn't loaded. As the police arrive Ed addresses Danny's concerns.-->[[MagnificentBastard You didn't think I was gonna shoot the guy?]] |
Triple-functional core-shell structured upconversion luminescent nanoparticles covalently grafted with photosensitizer for luminescent, magnetic resonance imaging and photodynamic therapy in vitro.
Upconversion luminescent nanoparticles (UCNPs) have been widely used in many biochemical fields, due to their characteristic large anti-Stokes shifts, narrow emission bands, deep tissue penetration and minimal background interference. UCNPs-derived multifunctional materials that integrate the merits of UCNPs and other functional entities have also attracted extensive attention. Here in this paper we present a core-shell structured nanomaterial, namely, NaGdF(4):Yb,Er@CaF(2)@SiO(2)-PS, which is multifunctional in the fields of photodynamic therapy (PDT), magnetic resonance imaging (MRI) and fluorescence/luminescence imaging. The NaGdF(4):Yb,Er@CaF(2) nanophosphors (10 nm in diameter) were prepared via sequential thermolysis, and mesoporous silica was coated as shell layer, in which photosensitizer (PS, hematoporphyrin and silicon phthalocyanine dihydroxide) was covalently grafted. The silica shell improved the dispersibility of hydrophobic PS molecules in aqueous environments, and the covalent linkage stably anchored the PS molecules in the silica shell. Under excitation at 980 nm, the as-fabricated nanomaterial gave luminescence bands at 550 nm and 660 nm. One luminescent peak could be used for fluorescence imaging and the other was suitable for the absorption of PS to generate singlet oxygen for killing cancer cells. The PDT performance was investigated using a singlet oxygen indicator, and was investigated in vitro in HeLa cells using a fluorescent probe. Meanwhile, the nanomaterial displayed low dark cytotoxicity and near-infrared (NIR) image in HeLa cells. Further, benefiting from the paramagnetic Gd(3+) ions in the core, the nanomaterial could be used as a contrast agent for magnetic resonance imaging (MRI). Compared with the clinical commercial contrast agent Gd-DTPA, the as-fabricated nanomaterial showed a comparable longitudinal relaxivities value (r(1)) and similar imaging effect. |
1.. Introduction {#s1}
================
Regulation of cell division is of great relevance for eukaryotes. Cells must proliferate throughout ontogenesis, tissue renewal and remodelling, and to repair damaged areas during wound healing. Defective cell-cycle checkpoints are a common feature of cancer cells and the inactivation of cell cycle regulators decides the physiological or pathological fate of stem cells. Although there are a large number of studies on the molecular and biochemical mechanisms controlling the cell cycle, the bioelectrical modulation of cell-cycle progression is still poorly understood. K^+^ channels have been implicated in the control of cell-cycle progression both through their influence on the membrane potential and non-canonical, permeation-independent mechanisms.
2.. Checkpoints and transmembrane potential regulate cell-cycle progression {#s2}
===========================================================================
The process that produces two daughter cells from a mother cell has been divided into several phases, each with very characteristic functional properties. Cell division in eukaryotes starts with the G1 (*gap* 1) phase, which separates the previous cell division from the period of DNA synthesis (S-phase), where chromosome replication is accomplished. This is followed by the second gap (G2) and the mitotic (M) phase. After M phase, a cell can proceed to a new G1 phase or enter a quiescent state (termed G0) that can last for a very long time, even for the rest of the life of the cell in the case of end-differentiated cells. The correct progression of the cycle is guaranteed because the initiation of a late event is strictly dependent on the successful completion of the preceding step. In eukaryotic cells, for example, mitosis will not start until the completion of DNA synthesis. The interdependency of events is owing to a series of surveillance or control mechanisms termed checkpoints, which have evolved to minimize the production and propagation of genetic inaccuracies \[[@RSTB20130094C1],[@RSTB20130094C2]\]. The complex machinery of cell-cycle checkpoints includes in all cases a sensor supervising the completeness of a particular task and a response element triggering the next downstream event, which will be a process involved in the actual replication and segregation of the DNA. For instance, the downstream event at the onset of S phase is DNA synthesis, the downstream event at the onset of mitosis is the assembly of the spindle and the downstream event at the end of mitosis is chromosome segregation \[[@RSTB20130094C3],[@RSTB20130094C4]\]. Thus, checkpoints are constitutive feedback control pathways safeguarding key cell-cycle transitions G1/S, G2/M and exit from mitosis \[[@RSTB20130094C5]\]. The key components of the mechanisms coordinating the downstream events are cyclin/cyclin-dependent kinase (CDK) complexes, which need to be expressed in a timely fashion and/or activated to allow cell-cycle progression.
The transmembrane potential has been reported as a cellular bioelectric parameter that influences the progression through the cell cycle \[[@RSTB20130094C6]\]. The concept came from the early experimental observation of a correlation between the resting membrane potential and the degree of mitotic activity \[[@RSTB20130094C7]\]; forcing the membrane potential of Chinese hamster ovary cells to a fixed hyperpolarized value completely inhibited DNA synthesis measured as \[^3^H\]thymidine incorporation, while cycling was recovered upon release of the potential ([figure 1](#RSTB20130094F1){ref-type="fig"}). Cell types with a very hyperpolarized resting potential, such as muscle cells and neurons, typically show little or no mitotic activity. Inversely, it was reported in the early 1970s that ouabain-induced depolarization was followed by the initiation of DNA synthesis and subsequent mitosis in chick spinal cord neurons \[[@RSTB20130094C8],[@RSTB20130094C9]\]. Moreover, it has been shown that the membrane potential is not constant during progression through the cell cycle \[[@RSTB20130094C10],[@RSTB20130094C11]\]. For example, the distribution of membrane potentials in cells from the breast cancer cell line MCF-7 is multimodal. The frequency of events at each maximum can be shifted when experimental treatments change the distribution of cells among the different phases of the cell cycle. The results of these experiments showed a pattern of positive correlation where the membrane potential hyperpolarizes during the G1/S transition, there is a significant contribution of depolarized cells towards G0/G1 and an enrichment in hyperpolarized cells towards G2/M transition \[[@RSTB20130094C12]\]. Figure 1.Complete block of DNA synthesis, measures as \[^3^H\]thymidine incorporation in cells with fixed hyperpolarized membrane potential. Reproduced from \[[@RSTB20130094C7]\] with permission. Open circles, control; black circles, manipulation of membrane potential.
3.. K^+^ channels as important players in the cell cycle {#s3}
========================================================
If the membrane potential is not constant along the cell cycle, cell-cycle-dependent changes in membrane permeability are required ([figure 2](#RSTB20130094F2){ref-type="fig"}). Potassium conductance governs the resting membrane potential in both excitable and non-excitable cells. In contrast to an action potential fired by a neuron, the potential changes along the cell cycle are much slower, gradual and smaller, and can be intuitively explained by modifications in the conductance that sets the resting membrane potential. Proliferation was one of the first identified aspects of cell physiology where potassium channels play a crucial role. The early observation that wide-spectrum potassium channel blockers inhibit proliferation \[[@RSTB20130094C13]\] has been repeatedly confirmed in many tissues and cell types (reviewed e.g. in \[[@RSTB20130094C6]\]). Many different potassium channels show cell-cycle-dependent variations of expression or activity \[[@RSTB20130094C14]--[@RSTB20130094C17]\]. Figure 2.Schematic of the behaviour of the membrane potential along the cell cycle. Different potassium channels show variations of expression or activity through the cell cycle, thus shifting the membrane potential towards hyperpolarized values, close to the equilibrium potential for potassium, at the border between G1 and S-phases.
For instance, a large conductance, voltage-gated K^+^ channel is expressed in unfertilized mouse oocytes; in the first cell cycle of fertilized oocytes, the channel is active throughout M and G1 phases, and inactive during S and G2. Thus, changes in channel activity set the membrane potential along the cell cycle in the oocyte \[[@RSTB20130094C18]\]. Increasing evidence shows that voltage-gated potassium channels are required for proliferation and may also help to determine the final identity and morphology of the cell \[[@RSTB20130094C19]--[@RSTB20130094C22]\]. The results of experiments in lymphocytes where the inhibition of K^+^ channel activity induces a reversible cell-cycle arrest \[[@RSTB20130094C23],[@RSTB20130094C24]\] or experiments where potassium channel blockers inhibit Schwann cell proliferation in a dose-dependent manner \[[@RSTB20130094C22],[@RSTB20130094C25],[@RSTB20130094C26]\] have been replicated in many systems and by many approaches; data from those experiments have been compiled already in several reviews (e.g. \[[@RSTB20130094C27]--[@RSTB20130094C31]\]).
Direct evidence for a change in ion channel composition in G1 phase was obtained from embryonic retinal cells, which express mainly two membrane conductances, delayed rectifier (I~K~) and inward rectifier (I~Kir~) potassium currents \[[@RSTB20130094C32]\]. Daughters of the same parental cell examined during and after mitosis always expressed similar I~K~ and I~Kir~ densities. However, non-sibling cells showed quantitative and qualitative differences in I~K~ and I~Kir~ densities. The heterogeneity therefore arises *after* cells re-enter G1, because the density distribution of potassium channels at cytokinesis is shown to be symmetric in both daughter cells \[[@RSTB20130094C33]\].
The mechanisms controlling ion channel densities along the cell cycle appear to be manifold. For example, K^+^ channel activity in mouse oocytes is at least partly independent of the nuclear cell-cycle clock, because channel activity continues to cycle in bisected embryos in the anucleate as well as the nucleate fragments \[[@RSTB20130094C34]\]. This suggests the active contribution of the cytoplasmic cell-cycle clock, which may involve changes induced by surface contractions and deformations before the cleavage of daughter cells on the channel activity \[[@RSTB20130094C34],[@RSTB20130094C35]\]. Thus, potassium channels are proposed to be involved in the signal transduction elicited by cell-cycle checkpoints, and help to elicit cell responses in the cell-cycle machinery, integrating the nuclear clock and the cytoplasmic cell-cycle clock. Pointing towards this hypothesis, there have been reports where K^+^ channel blockers (TEA) and depolarizing agents (veratridine) inhibit cell proliferation in oligodendrocyte progenitors in cell culture and cerebellar tissue slices, inducing G1 arrest through accumulation of p27^kip1^ and p21^CIP1^, two CDK inhibitors known to regulate cell proliferation \[[@RSTB20130094C36],[@RSTB20130094C37]\].
4.. Importance of K^+^ channels relies on both ionic conduction and permeation-independent mechanisms {#s4}
=====================================================================================================
The participation of K^+^ channels in the control of cell cycle could be an early event in evolution. The pore structure and the selectivity filter have been conserved between the prokaryotic and eukaryotic K^+^ channels \[[@RSTB20130094C38]\], which suggests that they evolved very early. The importance of K^+^ channels in the cell-cycle progression can also be illustrated in plant cells, for which K^+^ is a major nutrient. BY-2 tobacco cells require an increase in the K^+^ concentration in order to re-enter the cell cycle. The elevated K^+^ concentration increases the turgor pressure, which is required for cell growth. This is achieved by the activity of the inward rectifier K^+^ uptake channels \[[@RSTB20130094C39]\]. By contrast, mitosis requires a transient decrease in turgor pressure owing to K^+^ efflux channels. In what could be a reminiscence of this function, the role of K^+^ channels in homoeostatic cell volume regulation is well established, and they play a role in cell volume changes along the cell cycle \[[@RSTB20130094C40],[@RSTB20130094C41]\]. For instance, in a subset of human medulloblastomas, a voltage-gated K^+^ channel (K~V~10.2) seems to be required for the completion of mitosis, because it participates in cell volume reduction prior to cytokinesis \[[@RSTB20130094C21]\].
K^+^ channels also provide the driving force required for Ca^2+^ to enter the cell by shifting the membrane potential towards negative values. Ca^2+^ is an important mediator of intracellular signals implicated in the control of proliferation among other crucial processes in cell physiology, and by keeping the membrane potential at hyperpolarized values, K^+^ channels ensure efficient Ca^2+^ entry into the cell \[[@RSTB20130094C42]--[@RSTB20130094C45]\]. Still, regardless of whether the potassium gradient is used to generate driving force for Ca^2+^ or to change the cell volume, we traditionally tend to define the potassium current as the only effector, and ignore possible additional actions of the ion channel molecule itself. If only K^+^ flow was required, essentially any potassium channel expressed at the right moment would be able to affect cell-cycle progression. Experimental observations using either siRNA knockdown or specific blockers, for example antibodies, have repeatedly shown, however, that a specific potassium channel can be important for proliferation (e.g. \[[@RSTB20130094C46]--[@RSTB20130094C50]\]). This would indicate a permeation-independent, non-canonical mechanism that could involve protein--protein interactions, dependent or independent of the conformational changes of the channel mediated by voltage. Non-canonical functions \[[@RSTB20130094C51]\] have been described for at least the *Drosophila eag* channel \[[@RSTB20130094C52]\], its mammalian orthologue K~V~10.1 \[[@RSTB20130094C53]\], K~V~1.3 \[[@RSTB20130094C54]\] and K~Ca~3.1 \[[@RSTB20130094C55]\], which are still able influence cell proliferation in the absence of K^+^ permeation. Moreover, an alternatively spliced form of *Drosophila eag* that lacks the transmembrane regions, and therefore is not even a bona fide potassium channel has also been reported to influence intracellular signalling and alter cell morphology in the background of PKA/PKC activation \[[@RSTB20130094C56]\].
In more general terms, the fact that more than 70 genes encode K^+^ channels suggests an exquisite distribution of functions among specific molecular entities, rather than a homogeneous function for all potassium channels. Along these lines, the variability of K^+^ channels is further increased by the formation of heteromultimers, the influence of accessory subunits and a large number of post-translational modifications, such as glycosylation \[[@RSTB20130094C57]\], phosphorylation \[[@RSTB20130094C58]\] and sumoylation \[[@RSTB20130094C59]\]. There is substantial evidence that several K^+^ channels play a role in cell cycle and proliferation by means of both permeation-related and unrelated mechanisms ([figure 3](#RSTB20130094F3){ref-type="fig"}). Below, we describe some of them in more detail. Figure 3.K^+^ channels influence cell-cycle progression through permeation-related and non-canonical mechanisms. The former include cell volume regulation, modulation of membrane potential and generation of driving for Ca^2+^, while the latter rely on protein--protein interactions. K^+^ channel expression or function can in turn be regulated by progression through the cell cycle.
5.. K~V~1.3 {#s5}
===========
K~V~1.3 (together with K~Ca~3.1) was probably the first case showing the involvement of K^+^ channels in cell proliferation \[[@RSTB20130094C13],[@RSTB20130094C60]\]. In a very early report on T lymphocytes, mitogenesis induced by phytohaemagglutinin caused K^+^ channels to open more rapidly and at more negative membrane potentials, suggesting that they may play a role in mitogenesis \[[@RSTB20130094C13]\]. K~V~1.3 blockade was shown to suppress T-cell activation and Ca^2+^ signalling in human T cells owing to membrane depolarization, resulting in a reduced driving force for Ca^2+^ entry and impairment of activation by agents inducing mitogenesis \[[@RSTB20130094C61],[@RSTB20130094C62]\]. K~V~1.3 can act in conjunction with K~Ca~3.1, which is a Ca^2+^-dependent K^+^ channel activated by Ca^2+^--calmodulin \[[@RSTB20130094C63]\]. K~V~1.3 and K~Ca~3.1 have been found to cluster at the immunological synapse following contact with an antigen-presenting cell \[[@RSTB20130094C60]\]. Together, K~V~1.3 and K~Ca~3.1 modulate calcium-dependent cellular processes in immune cells, such as T-cell activation and proliferation \[[@RSTB20130094C43],[@RSTB20130094C64]\]. K~Ca~3.1 has also been implicated in the control of cell proliferation in rat mesenchymal stem cells, vascular smooth muscle cells (VSMCs), hepatocellular carcinoma cells as well as endometrial and prostate cancer cells \[[@RSTB20130094C45],[@RSTB20130094C46],[@RSTB20130094C65]--[@RSTB20130094C68]\], although in glioma cells K~Ca~3.1 knockdown abolished the current but did not affect proliferation \[[@RSTB20130094C69]\]. As K~Ca~3.1 seems to play a crucial role in glioma cell migration \[[@RSTB20130094C70]--[@RSTB20130094C75]\], it might be difficult to dissect both properties and the results can depend very strongly on the methods used to determine proliferation.
K~V~1.3 has also been implicated in the control of the cell cycle in many other cell types, such active microglia cells \[[@RSTB20130094C76],[@RSTB20130094C77]\], proliferating oligodendrocyte progenitors during G1/S transition \[[@RSTB20130094C37]\] and macrophages \[[@RSTB20130094C78]--[@RSTB20130094C80]\]. In human endothelial cells, vascular endothelial growth factor induces a K~V~1.3-dependent hyperpolarization that results in an increased Ca^2+^ entry, which is responsible for the effects on proliferation \[[@RSTB20130094C81],[@RSTB20130094C82]\]. It has been shown that the contractile activity of VSMCs controlling blood flow changes during the course of several vascular disorders and the cells acquire a proliferative and migratory phenotype \[[@RSTB20130094C83]\]. K~V~1.3 functional expression is associated with the proliferative phenotype, because the blockade of the channel induces a significant inhibition of cell proliferation \[[@RSTB20130094C81],[@RSTB20130094C84],[@RSTB20130094C85]\]. Switching from contractile to proliferative phenotype is thus associated with changes in ion channel activity. However, one study suggests K~V~1.3 increases VSMC proliferation by voltage-dependent conformational changes of the channel that activate intracellular signalling pathways, rather than by ionic conduction \[[@RSTB20130094C54]\].
6.. K~V~11.1 {#s6}
============
The voltage-sensitive human *ether à go-go*-related gene (hERG, K~V~11.1) \[[@RSTB20130094C86]\] potassium channels have emerged as regulators of both proliferation and survival in cancer cells. K~V~11.1 (encoded by *KCNH2*) channel expression in normal adult human tissue is abundant in heart, brain, myometrium, pancreas and haematopoietic progenitors \[[@RSTB20130094C87]--[@RSTB20130094C90]\]. K~V~11.1 expression has been reported in many cancer types as well as cancer cell lines of different lineages, be it epithelial, leukemic, connective or neuronal \[[@RSTB20130094C89]--[@RSTB20130094C91]\] Various studies have demonstrated this expression to be largely confined to neoplastic cells both in solid and haematological malignancies, when compared with neighbouring normal tissues or normal bone marrow samples \[[@RSTB20130094C90]--[@RSTB20130094C94]\]. Studies over the past decade have also shown its expression to be preferential to the cancer stem cells especially in leukaemia when compared with normal haematopoietic stem cells \[[@RSTB20130094C90],[@RSTB20130094C94]\]. K~V~11.1 expression has also been linked to higher grade and worse prognosis, both in the case of solid as well as haematological malignancies \[[@RSTB20130094C89],[@RSTB20130094C91]--[@RSTB20130094C94]\]. K~V~11.1 expression is not an epiphenomenon of cancer cells and rather plays a relevant role in their proliferative capacity, for both haematological as well as solid tumours \[[@RSTB20130094C49],[@RSTB20130094C90]--[@RSTB20130094C98]\]. Studies by various groups on K~V~11.1 inhibition in cell lines derived from solid tumours or leukaemias have shown a clear reduction in proliferation \[[@RSTB20130094C49],[@RSTB20130094C90]--[@RSTB20130094C99]\]. The reduction in cell proliferation has been explained by either increase in apoptosis or an arrest at the G0/G1 phase of cell cycle \[[@RSTB20130094C49],[@RSTB20130094C90]--[@RSTB20130094C99]\]. Nevertheless, the anti-tumour effects of blockers of K~V~11.1 appear to act through a reduction in cell proliferation \[[@RSTB20130094C49],[@RSTB20130094C82],[@RSTB20130094C98],[@RSTB20130094C99]\]. Some studies have implicated the two isoforms of hERG (hERG1a and hERG1b) to play a vital role not only in cell proliferation by affecting different phases of cell cycle but also in the channel kinetics and current amplitude \[[@RSTB20130094C100]\]. Both isoforms have been shown to coexist, but hERG1b expression is more prominent in the S phase of the cell cycle and hERG1a expression in the G1 phase. Modulation of these expression patterns affects the cell proliferation \[[@RSTB20130094C95]\]. Co-assembly of hERG1a with hERG1b results in increased availability of channels on the plasma membrane and a larger current flow when compared with homomeric forms of the channel \[[@RSTB20130094C100]\]. Further insight into the hERG isoforms and its role in cancer is needed to conclusively designate hERG as a therapeutic target.
7.. K~V~10.1 {#s7}
============
K~V~10.1 (Eag1, encoded by *KCNH1*) is one of the best-studied ion channels in the context of cancer. Its oncogenic potential was first described in 1999 with the discovery that the inhibition of K~V~10.1 expression reduces proliferation of several somatic cancer cell lines \[[@RSTB20130094C101]\]. K~V~10.1 overexpression, in turn, increases cell proliferation and can confer a transformed phenotype. In the same study, our laboratory also reported that K~V~10.1 is undetectable in healthy tissues outside the brain and favours xenograft tumour progression in immunodeficient mice *in vivo*. Along these lines, K~V~10.1 has also been detected in approximately 70% of human tumour biopsies of diverse origin \[[@RSTB20130094C102]--[@RSTB20130094C113]\]. Its widespread presence in clinical samples, together with the fact that the physiological expression of K~V~10.1 is confined to the brain (with the exception of a few restricted cell populations \[[@RSTB20130094C111]\]), aroused a lot of interest in the channel owing to its potential therapeutic and diagnostic applications. It had been assumed that K~V~10.1 is present only in solid tumours but recent research has revealed its presence in leukaemias, correlating with a poor prognosis \[[@RSTB20130094C107]\]. K~V~10.1 expression also correlates with poor prognosis for patients of ovarian \[[@RSTB20130094C106]\], gastric \[[@RSTB20130094C112]\] and colon cancer \[[@RSTB20130094C114]\], and with lymph node metastasis in gastric cancer and head and neck squamous cell carcinoma, where it also correlates with the disease stage \[[@RSTB20130094C105]\]. Moreover, a number of studies have supported the observation that K~V~10.1 blockage or knockdown decreases the proliferation of many cancer cell lines and *in vivo* tumour models \[[@RSTB20130094C53],[@RSTB20130094C107],[@RSTB20130094C115],[@RSTB20130094C116]\]. An interesting exception here is glioblastoma, where the levels of K~V~10.1 are lower than that in healthy brain tissue \[[@RSTB20130094C109]\], while further silencing of channel expression increases the responsiveness to interferon gamma treatment \[[@RSTB20130094C117]\]. Although it is probably not the only relevant localization of K~V~10.1 \[[@RSTB20130094C118]\], it is also worth mentioning here that membrane localization makes K~V~10.1 an attractive target for therapy, as it is easily accessible from the extracellular side. In order to selectively induce apoptosis in cancer cells, an anti- K~V~10.1 antibody has been coupled to TNF-related apoptosis-inducing ligand**,** and this strategy has been successfully tested *in vitro* \[[@RSTB20130094C119]\].
The mechanisms of how K~V~10.1 is able to increase cell proliferation and favour tumour progression remain elusive. Ion permeation does not seem to be a necessary condition for either of the above, as non-conducting mutants retain the ability to influence proliferation and tumourigenesis \[[@RSTB20130094C52],[@RSTB20130094C53]\]. By implication, the advantage K~V~10.1 expression confers is independent of the 'classical' contributions of K^+^ channels to proliferation: regulating cell volume, maintaining the driving force for Ca^2+^ and G1/S hyperpolarization. As we already indicated earlier, this is less surprising than it may appear, because if the features associated with K^+^ permeation were enough to render a transformed phenotype, many more K^+^ channels would be oncogenic. Moreover, the loss of ionic conductances can often be compensated for by other channels, which also does not fit into the picture where removing a particular conductance drastically reduces proliferation in so many cancer cell lines, as well as tumourigenesis *in vivo*. In contrast to ion permeation, voltage-dependent conformations may be crucial for K~V~10.1 to support proliferation, as the non-conducting mutants that have a preference for the open conformation fail to influence proliferation \[[@RSTB20130094C52]\]. It is important to note that channel blockers could reduce proliferation not only by inhibiting permeation, but also by trapping the channel in a particular conformation. Hegle and co-workers also described an increase in p38-MAP kinase activity in non-cancer cells transfected with K~V~10.1, and abolishing the effect of K~V~10.1 on cell proliferation by p-38^MAPK^ inhibition. Interestingly, modulation of K~V~10.1 expression levels by p-38^MAPK^ pathway has been described in MG-63 cells from osteosarcoma \[[@RSTB20130094C102]\], so the relation between the channel and p-38^MAPK^ signalling needs further clarification. Another non-conducting function of K~V~10.1 is an increase in hypoxia resistance by boosting HIF-1 levels and VEGF secretion, eventually leading to better tumour vascularization \[[@RSTB20130094C53]\]. Nevertheless, the mechanisms described above remain insufficient to explain the benefit K~V~10.1 expression brings to the proliferation of so many different cancer cell lines.
Finally, in some models, K~V~10.1 appears to be regulated by cell cycle. Inducing the G2/M transition by progesterone in *Xenopus* oocytes heterologously expressing K~V~10.1 causes a reduction in current \[[@RSTB20130094C17]\]. This reduction is dependent on the mitosis-promoting factor (MPF, a complex of cyclin B and p34^cdc2^) and obeys a voltage-dependent block by intracellular Na^+^ \[[@RSTB20130094C16]\]. MPF induces an increase in selectivity during the M phase \[[@RSTB20130094C120]\] that results in block by Na^+^, which leads to a rectification of the current--voltage relation. The resulting net loss of K^+^ conductance at G2/M transition may be a way to achieve membrane depolarization associated with mitosis. Cell-cycle regulation of K~V~10.1 has also been studied in MCF-7 breast cancer cells. Synchronization of these cells in G0/G1 by serum starvation leads to an increase in Eag1 mRNA expression compared with unsynchronized controls, with a further increase during the progression through G1 and a decrease in the S-phase \[[@RSTB20130094C121]\]. At the functional level, this is accompanied by an increase in outward-rectifier K^+^ current that hyperpolarizes the membrane towards the S-phase \[[@RSTB20130094C121]\]. Both K~V~10.1 mRNA and K~V~10.1-mediated current in MCF-7 cells can also be increased by stimulation with insulin-like growth factor 1 (IGF-1) via the PI3 K/Akt pathway, suggesting that the progression through G1 to S triggered by IGF-1 can partially be owing to its effect on K~V~10.1 \[[@RSTB20130094C122]\]. Defective checkpoint control between G1 and S-phase can also result in K~V~10.1 upregulation. In SH-SY5Y neuroblastoma cells, K~V~10.1 expression is regulated by the p53/mir34/E2F1 pathway \[[@RSTB20130094C123]\]. Additionally, keratinocytes immortalized with human papilloma virus oncogenes E6 and E7 targeting p53 and Retinoblastoma protein (pRb) start to transcribe K~V~10.1 mRNA \[[@RSTB20130094C124]\]. One can thus expect that p53 or pRb/E2F pathway inhibition or malfunctions, which are very common in cancer, can give rise to higher K~V~10.1 expression levels. However, further research is needed to establish that K~V~10.1 expression is cell-cycle dependent and to elucidate the effect(s) of the channel on cell-cycle progression.
8.. Conclusion {#s8}
==============
Progression through the cell cycle is guarded by several checkpoint control pathways that have the ability to delay or stop further events, such as DNA synthesis or assembly of the mitotic spindle, before commitment into cell division. In accordance with the experimental data compiled in this review, there can be little doubt that K^+^ channels play an active role in cell-cycle progression. On the other hand, their expression or function can be regulated by the cell cycle. Therefore, K^+^ channels could also be viewed as effectors of the checkpoint machinery. As molecular machines that enable the passage of K^+^ ions through the membrane, they can regulate cell volume, provide driving force for Ca^2+^ entry, hyperpolarize the cell at the G1/S transition and depolarize it towards mitosis. Additionally, non-canonical, permeation-independent mechanisms may be involved, where K^+^ channels recruit or modulate signalling cascades via protein--protein interactions. It is tempting to assume that signalling cascades activated by such interactions could link the nuclear clock control with its cytoplasmic counterpart.
Unfortunately, to date we have only a rough estimate of how membrane potential changes along the cell cycle. Moreover, very little is known about the non-conducting functions of K^+^ channels. Which signalling cascades can they modify? How do they interact with other proteins? There are also more general questions that remain unanswered. How exactly does membrane potential affect the cell-cycle machinery? Further research on K^+^ channels in cell cycle and proliferation will give us better understanding of these fundamental processes and may have therapeutic implications.
[^1]: One contribution of 17 to a Theme Issue '[Ion channels, transporters and cancer](http://rstb.royalsocietypublishing.org/content/369/1638.toc)'.
|
---
abstract: |
This is a partial account of the fascinating history of Distance Geometry. We make no claim to completeness, but we do promise a dazzling display of beautiful, elementary mathematics. We prove Heron’s formula, Cauchy’s theorem on the rigidity of polyhedra, Cayley’s generalization of Heron’s formula to higher dimensions, Menger’s characterization of abstract semi-metric spaces, a result of Gödel on metric spaces on the sphere, and Schoenberg’s equivalence of distance and positive semidefinite matrices, which is at the basis of Multidimensional Scaling.\
[**Keywords**]{}: Euler’s conjecture, Cayley-Menger determinants, Multidimensional scaling, Euclidean Distance Matrix
bibliography:
- 'ibmer.bib'
---
[Six mathematical gems from the history of Distance Geometry]{}
- [*CNRS LIX, École Polytechnique, F-91128 Palaiseau, France*]{}\
Email:[liberti@lix.polytechnique.fr](liberti@lix.polytechnique.fr)
- [*IMECC, University of Campinas, 13081-970, Campinas-SP, Brazil*]{}\
Email:[clavor@ime.unicamp.br](clavor@ime.unicamp.br)
Introduction {#s:intro}
============
Distance Geometry (DG) is the study of geometry with the basic entity being distance (instead of lines, planes, circles, polyhedra, conics, surfaces and varieties). As with most everything else, it all began with the Greeks, specifically Heron, or Hero, of Alexandria sometime between 150BC and 250AD, who showed how to compute the area of a triangle given its side lengths [@heron].
After a hiatus of almost two thousand years, we reach Arthur Cayley’s: the first paper of volume I of his [*Collected Papers*]{}, dated 1841, is about the relationships between the distances of five points in space [@cayley1841]. The gist of what he showed is that a tetrahedron can only exist in a plane if it is flat (in fact, he discussed the situation in one more dimension). This yields algebraic relations on the side lengths of the tetrahedron.
Hilbert’s influence on foundations and axiomatization was very strong in the 1930s [*Mitteleuropa*]{} [@hilbert]. This pushed many people towards axiomatizing existing mathematical theories [@tarskied]. Karl Menger, a young professor of geometry at the University of Vienna and an attendee of the Vienna Circle, proposed in 1928 a new axiomatization of metric spaces using the concept of distance and the relation of congruence, and, using an extension of Cayley’s algebraic machinery (which is now known as Cayley-Menger determinant), generalized Heron’s theorem to compute the volume of arbitrary $K$-dimensional simplices using their side lengths [@menger28].
The Vienna Circle was a group of philosophers and mathematicians which convened in Vienna’s [*Reichsrat café*]{} around the nineteen-thirties to discuss philosophy, mathematics and, presumably, drink coffee. When the meetings became excessively politicized, Menger distanced himself from it, and organized instead a seminar series, which ran from 1929 to 1937 [@mengerK]. A notable name crops up in the intersection of Menger’s geometry students, the Vienna Circle participants, and the speakers at Menger’s [*Kolloquium*]{}: Kurt Gödel. Most of the papers Gödel published in the Kolloquium’s proceedings are about logic and foundations,[^1] but two, dated 1933, are about the geometry of distances on spheres and surfaces. The first [@mengerK 18 Feb. 1932, p. 198] answers a question posed at a previous seminar by Laura Klanfer, and shows that a set $X$ of four points in any metric space, congruent to four non-coplanar points in $\mathbb{R}^3$, can be realized on the surface of a three-dimensional sphere using geodesic distances. The second [@mengerK 17 May 1933, p. 252] shows that Cayley’s relationship hold locally on certain surfaces which behave locally like Euclidean spaces.
The pace quickens: in 1935, Isaac Schoenberg published some remarks on a paper [@schoenberg] by Fréchet on the [*Annals of Mathematics*]{}, and gave, among other things, an algebraic proof of equivalence between Euclidean Distance Matrices (EDM) and Gram matrices. This is almost the same proof which is nowadays given to show the validity of the classical Multidimensional Scaling (MDS) technique [@borg_10 § 12.1].
This brings us to the computer era, where the historical account ends and the contemporary treatment begins. Computers allow the efficient treatment of masses of data, some of which are incomplete and noisy. Many of these data concern, or can be reduced to, distances, and DG techniques are the subject of an application-oriented renaissance [@dgp-sirev; @dgpbook]. Motivated by the Global Positioning System (GPS), for example, the old geographical concept of [*trilateration*]{} (a system for computing the position of a point given its distances from three known points) makes its way into DG in wireless sensor networks [@eren04]. Wüthrich’s Nobel Prize for using Nuclear Magnetic Resonance (NMR) techniques in the study of proteins brings DG to the forefront of structural bioinformatics research [@wuthrich]. The massive use of robotics in mechanical production lines requires mathematical methods based on DG [@rojasthomas].
DG is also tightly connected with graph rigidity [@graverbook]. This is an abstract mathematical formulation of statics, the study of structures under the action of balanced forces [@maxwell1864b], which is at the basis of architecture [@varignon]. Rigidity of polyhedra gave rise to a conjecture of Euler’s [@euler] about closed polyhedral surfaces, which was proved correct only for some polyhedra: strictly convex [@cauchyrigid], convex and higher-dimensional [@alexandrov3], and generic[^2] [@gluck]. It was however disproved in general by means of a very special, non-generic nonconvex polyhedron [@connelly-countereg].
The rest of this paper will focus on the following results, listed here in chronological order: Heron’s theorem (Sect. \[s:heron\]), Euler’s conjecture and Cauchy’s proof for strictly convex polyhedra (Sect. \[s:eulercauchy\]), Cayley-Menger determinants (Sect. \[s:cayleymenger\]), Menger’s axiomatization of geometry by means of distances (Sect. \[s:menger\]), a result by Gödel’s concerning DG on the sphere (Sect. \[s:goedel\]), and Schoenberg’s equivalence (Sect. \[s:schoenberg\]) between EDM and Positive Semidefinite Matrices (PSD). There are many more results in DG: this is simply our own choice in terms of importance and beauty.
Heron’s formula {#s:heron}
===============
Heron’s formula, which is usually taught at school, relates the area $\mathcal{A}$ of a triangle to the length of its sides $a,b,c$ and its semiperimeter $s=\frac{a+b+c}{2}$ as follows: $$\mathcal{A} = \sqrt{s(s-a)(s-b)(s-c)}.$$ There are many ways to prove its validity. Shannon Umberger, a student of the “Foundations of Geometry I” course given at the University of Georgia in the fall of 2000, proposes, as part of his final project,[^3] three detailed proofs: an algebraic one, a geometric one, and a trigonometric one. John Conway and Peter Doyle discuss Heron’s formula proofs in a publically available email exchange[^4] from 1997 to 2001.
Our favourite proof is based on complex numbers, and was submitted[^5] to the “Art of Problem Solving” online school for gifted mathematics students by Miles Edwards[^6] when he was studying at Lassiter High School in Marietta, Georgia.
Let $\mathcal{A}$ be the area of a triangle with side lengths $a,b,c$ and semiperimeter length $s=\frac{1}{2}(a+b+c)$. Then $\mathcal{A}=\sqrt{s(s-a)(s-b)(s-c)}$. \[t:heron\]
[@milesedwards] Consider a triangle with sides $a,b,c$ (opposite to the vertices $A,B,C$ respectively) and its inscribed circle centered at $O$ with radius $r$. The perpendiculars from $O$ to the triangle sides split $a$ into $y,z$, $b$ into $x,z$ and $c$ into $x,y$ as shown in Fig. \[f:heron1\]. Let $u,v,w$ be the segments joining $O$ with $A,B,C$, respectively.
![Heron’s formula: a proof using complex numbers.[]{data-label="f:heron1"}](heron1){width="12cm"}
First, we note that $2\alpha+2\beta+2\gamma=2\pi$, which implies $\alpha+\beta+\gamma=\pi$. Next, the following complex identities are easy to verify geometrically in Fig. \[f:heron1\]: $$\begin{aligned}
r + ix &=& u e^{i\alpha} \\
r + iy &=& v e^{i\beta} \\
r + iz &=& w e^{i\gamma}.\end{aligned}$$ These imply: $$(r + ix)(r+iy)(r+iz) = (uvw) e^{i(\alpha+\beta+\gamma)} = uvw
e^{i\pi} = -uvw,$$ where the last step uses Euler’s identity $e^{i\pi}+1=0$ [@euleraninf I-VIII, § 138-140, p. 148]. Since $-uvw$ is real, the imaginary part of $(r+ix)(r+iy)(r+iz)$ must be zero. Expanding the product and rearranging terms, we get $r^2(x+y+z)=xyz$. Solving for $r$, we have the nonnegative root $$r = \sqrt{\frac{xyz}{x+y+z}}. \label{eqr}$$ We can write the semiperimeter of the triangle $ABC$ as $s=\frac{1}{2}(a+b+c)=\frac{1}{2}(y+z+x+z+x+y)=x+y+z$. Moreover, $$\begin{aligned}
s-a &=& x+y+z-y-z = x \\
s-b &=& x+y+z-x-z = y \\
s-c &=& x+y+z-x-y = z,\end{aligned}$$ so $xyz=(s-a)(s-b)(s-c)$, which implies that Eq. becomes: $$r = \sqrt{\frac{(s-a)(s-b)(s-c)}{s}}.$$ We now write the area $\mathcal{A}$ of the triangle $ABC$ by summing it over the areas of the three triangles $AOB$, $BOC$, $COA$, which yields: $$\mathcal{A} =
\frac{1}{2}(ra+rb+rc)=r\frac{a+b+c}{2}=rs=\sqrt{s(s-a)(s-b)(s-c)},$$ as claimed.
Euler’s conjecture and the rigidity of polyhedra {#s:eulercauchy}
================================================
Consider a square with unit sides, in the plane. One can shrink two opposite angles and correspondingly widen the other two to obtains a rhombus (see Fig. \[f:sqrh\]), which has the same side lengths but a different shape: no sequence of rotations, translations or reflections can turn one into the other. In other words, a square is [ *flexible*]{}. By contrast, a triangle is not flexible, or [*rigid*]{}.
![A square is flexed into a rhombus. The set of faces (the edges) are the same, and each maintains pairwise distances through the flexing, i.e. two points on the same edge have the same distance on the left as on the right figure.[]{data-label="f:sqrh"}](sqrh){width="8cm"}
Euler conjectured in 1766 [@euler1766] that all three-dimensional polyhedra are rigid. The conjecture appears at the end of the discussion about the problem [*Invenire duas superficies, quarum alteram in alteram transformare liceat, ita ut in utraque singula puncta homologa easdem inter se teneat distantias*]{}, i.e.:
> To find two surfaces for which it is possible to transform one into the other, in such a way that corresponding points on either keep the same pairwise distance. ($\dag$)
Towards the end of the paper, Euler writes [*Statim enim atque figura undique est clausa, nullam amplius mutationem patitur*]{}, which means “As soon as the shape is everywhere closed, it can no longer be transformed”. Although the wording appears ambiguous by today’s standards, scholars of Euler and rigidity agree: what Euler really meant is that 3D polyhedra are rigid [@gluck].
To better understand this statement, we borrow from [@alexandrov2] the precise definition of a [*polyhedron*]{}[^7]: a family $\mathcal{K}$ of points, open segments and open triangles is a [*triangulation*]{} if (a) no two elements of $\mathcal{K}$ have common points, and (b) all sides and vertices of the closure of any triangle of $\mathcal{K}$, and both extreme points of the closure of any segment of $\mathcal{K}$ are all in $\mathcal{K}$ themselves. Given a triangulation $\mathcal{K}$ in $\mathbb{R}^K$ (where $K\in\{1,2,3\}$), the union of all points of $\mathcal{K}$ with all points in the segments and triangles of $\mathcal{K}$ is called a [*polyhedron*]{}. Note that several triangular faces can belong to the same affine space, thereby forming polygonal faces.
Each polyhedron has an incidence structure of points on segments and segments on polygonal (not necessarily triangular) faces, which induces a partial order (p.o.) based on set inclusion. For example, the closure of the square $ABCD$ contains the closures of the segments $AB$, $BC$, $CD$, $DA$, each of which contains the corresponding adjacent points $A,B$, $B,C$, $C,D$, $D,A$. Accordingly, the p.o. is $A\subset AB,DA$; $B\subset AB,BC$; $C\subset BC,CD$; $D\subset
CD,DA$; $AB,BC,CD,DA\subset ABCD$. Since this p.o. also has a bottom element (the empty set) and a top element (the whole polyhedron), it is a [*lattice*]{}. A lattice isomorphism is a bijective mapping between two lattices which preserves the p.o. Two polyhedra $P,Q$ are [*combinatorially equivalent*]{} if their triangulations are lattice isomorphic. If, moreover, all the lattice isomorphic polygonal faces of $P,Q$ are exactly equal, the polyhedra are said to be [*facewise equal*]{}.
Under the above definition, nothing prevents a polyhedron from being nonconvex (see Fig. \[f:nonconvex\]). It is known that every closed surface, independently of the convexity of its interior, is homeomorphic (intuitively: smoothly deformable in) to some polyhedron (again [@alexandrov2 § 2.2]). This is why we can replace “surface” with “polyhedra”.
![A nonconvex polyhedron.[]{data-label="f:nonconvex"}](nonconvex){width="5cm"}
The “rigidity” implicit in Euler’s conjecture should be taken to mean that no point of the polyhedron can undergo a continuous motion under the constraint that the shape be the same at each point of the motion. As for the concept of “shape”, it is linked to that of distance, as appears clear from ($\dag$). The following is therefore a formal restatement of Euler’s conjecture: [*two combinatorially equivalent facewise equal polyhedra must be isometric under the Euclidean distance*]{}, i.e. each pair of points in one polyhedron is equidistant with the corresponding pair in the other. A natural question about the Euler conjecture stems from generalizing the example in Fig. \[f:sqrh\] to 3D (see Fig. \[f:sqrh3\]). Does this not disprove the conjecture?
![A cube can be transformed into a rhomboid, but the set of faces is not the same anymore (accordingly, corresponding point pairs may not preserve their distance, as shown).[]{data-label="f:sqrh3"}](sqrh3){width="10cm"}
The answer is no: all the polygonal faces in the cube are squares, but this does not hold in the rhomboid. The question is more complicated than it looks at first sight, which is why it took 211 years to disprove it.
Strictly convex polyhedra: Cauchy’s proof
-----------------------------------------
Although Euler’s conjecture is false in general, it is true for many important subclasses of polyhedra. Cauchy proved it true for strictly convex polyhedra.[^8] There are many accounts of Cauchy’s proof: Cauchy’s original text, still readable today [@cauchyrigid]; Alexandrov’s book [@alexandrov3], Lyusternik’s book [@lyusternik § 20], Stoker’s paper [@stoker], Connelly’s chapter [@connellych] just to name a few. Here we follow the treatment given by Pak [@pak].
We consider two combinatorially equivalent, facewise equal strictly convex polyhedra $P,Q$, and aim to show that $P$ and $Q$ are isometric.
For a polyhedron $P$ we consider its associated graph $G(P)=(V,E)$, where $V$ are the points of $P$ and $E$ its segments. Note that $G(P)$ only depends on the incidence structure of the polygonal faces, segments and points of $P$. Since $P,Q$ are combinatorially equivalent, $G(P)=G(Q)$. Consider the dihedral angles[^9] $\alpha_{uv},\beta_{uv}$ on $P,Q$ induced by the segment represented by the edge $\{u,v\}\in E$. We assign to each edge $\{u,v\}\in E$ a label $\ell_{uv}=\mbox{sgn}(\beta_{uv}-\alpha_{uv})$ (so $\ell_{uv}\in\{-1,0,1\}$), and consider, for each $v\in V$, the edge sequence $\sigma_v=(\{u,v\}\;|\;u\in N(v))$, where $N(v)$ is the set of nodes adjacent to $v$. The order of the edges in $\sigma_v$ is given by any circuit around the polygon $p(v)$ obtained by intersecting $P$ with a plane $\gamma$ which separates $v$ from the other vertices in $V$ (this is possible by strict convexity, see Fig. \[f:sephyp\]).
![The plane $\gamma$ separating $v$ from the other vertices in the strictly convex polyhedron $P$, and the intersection polygon $p(v)$ defined by $w_1,\ldots,w_4$. The line $L$ (lying in $p(v)$) separates the $+1$ and $-1$ labels applied to the points $w_1,\ldots,w_4$ of intersections between the edges of $P$ and $p(v)$.[]{data-label="f:sephyp"}](sephyp){width="9cm"}
It is easy to see that every edge $\{u,v\}\in \sigma_v$ corresponds to a vertex of $p(v)$. Therefore, a circuit over $p(v)$ defines an order over $\sigma_v$. We also assume that this order is periodic, i.e. its last element precedes the first one. Any such sequence $\sigma_v$ naturally induces a sign sequence $s_V=(\ell_{uv}\;|\;\{u,v\}\in\sigma_v)$; we let $\bar{s}_v$ be the sequence $s_v$ without the zeros, and we count the number $m_v$ of sign changes in $\bar{s}_v$, including the sign change occurring between the last and first elements.
For all $v\in V$, $m_v$ is even. \[l:even\]
Suppose $m_v$ is odd, and proceed by induction on $m_v$: if $m_v=1$, then there is only one sign change. So, the first edge $\{u,v\}$ in $\sigma_v$ to be labelled with $\ell_{uv}\not=0$ has the property that, going around the periodic sequence with only one sign change, $\{u,v\}$ is also labelled with $-\ell_{uv}$, which yields $+1=-1$, a contradiction. A trivial induction step yields the same contradiction for all odd $m_v$.
We now state a fundamental technical lemma, and provide what is essentially Cauchy’s proof, rephrased as in [@lyusternik Lemma 2 in § 20].
If $P$ is strictly convex, then for each $v\in V$ we have either $m_v=0$ or $m_v\ge 4$.\[l:signchange\]
By Lemma \[l:even\], for each $v\in V$ we have $m_v\not\in\{1,3\}$, so we aim to show that $m_v\not=2$. Suppose, to get a contradiction, that $m_v=2$, and consider the polygon $p(v)$ as in Fig. \[f:sephyp\]. By the correspondence between edges in $\sigma_v$ and vertices of $p(v)$, the labels $\ell_{uv}$ are vertex labels in $p(v)$. Since there are only two sign changes, the sequence of vertex labels can be partitioned in two contiguous sets of $+1$ and $-1$ (possibly interspersed by zeros). By convexity, there exists a line $L$ separating the $+1$ and the $-1$ vertices (see Fig. \[f:sephyp\]). Since all of the angles marked $+1$ strictly increase, the segment $\bar{L}=L\cap p(v)$ also strictly increases[^10]; but, at the same time, all of the angles marked $-1$ strictly decrease, so the segment $\bar{L}$ also strictly decreases, which means that the same segment $\bar{L}$ both strictly increases and decreases, which is a contradiction (see Fig. \[s:incrdecr\]).
![Visual representation of the contradiction in the proof of Lemma \[l:signchange\]. The angles at all vertices labelled $+1$ increase their magnitude, and those at $-1$ decrease: it follows that $\bar{L}$ both increases and decreases its length, a contradiction.[]{data-label="s:incrdecr"}](incrdecr1 "fig:"){width="4cm"} ![Visual representation of the contradiction in the proof of Lemma \[l:signchange\]. The angles at all vertices labelled $+1$ increase their magnitude, and those at $-1$ decrease: it follows that $\bar{L}$ both increases and decreases its length, a contradiction.[]{data-label="s:incrdecr"}](incrdecr2 "fig:"){width="4cm"} ![Visual representation of the contradiction in the proof of Lemma \[l:signchange\]. The angles at all vertices labelled $+1$ increase their magnitude, and those at $-1$ decrease: it follows that $\bar{L}$ both increases and decreases its length, a contradiction.[]{data-label="s:incrdecr"}](incrdecr3 "fig:"){width="4cm"}
If two closed convex polyhedra $P,Q$ are combinatorially equivalent and facewise equal, they are isometric.
We only present the proof of the base case where $$\forall v\in V\;(m_v>0) \quad \vee \quad \forall v\in
V\;(m_v=0) \label{eq:ct}$$ and $G(P)=G(Q)$ is a connected graph, and refer the reader to [@pak p. 251] for the other cases (which are mostly variations of the ideas given in the proof below).
If $m_v=0$ for all $v\in V$, it means that all of the dihedral angles in $P$ are equals to those of $Q$, which implies isometry. So we assume the alternative w.r.t. Eq. above: $\forall v\in V\;(m_v>0)$, and aim for a contradiction. Let $M=\sum_{v\in V} m_v$: by Lemma \[l:signchange\] and because $m_v>0$ for each $v$, we have $M\ge
4|V|$, a lower bound for $M$. We now construct a contradicting upper bound for $M$. For every $h\ge 3$, we let $F_h$ be the number of polygonal faces of $P$ with $h$ sides (or edges). The total number of polygonal faces in $P$ (or $Q$) is $\mathcal{F}=\sum_h F_h$, and the total number of edges is therefore $\mathcal{E}=\frac{1}{2}\sum_h h F_h$ (we divide by 2 since each edge is counted twice in the sum — one per adjacent face — given that $P,Q$ are closed). A simple term by term comparison of $\mathcal{F}$ and $\mathcal{E}$ yields $4\mathcal{E}-4\mathcal{F}=\sum_h 2(h-2)F_h$. Since each polygonal face $f$ of $P$ is itself closed, the number $c_f$ of sign changes of the quantities $\ell_{uv}$ over all edges $\{u,v\}$ adjacent to the face $f$ is even, by the same argument given in Lemma \[l:even\]. It follows that if the number $h$ of edges adjacent to the face $f$ is even, then $c_f\le h$, and $c_f\le h-1$ if $h$ is odd. This allows us to compute an upper bound on $M$: $$\begin{aligned}
M &\le& 2 F_3 + 4F_4 +4 F_5 + 6 F_6 + 6 F_7 + 8 F_8 + \dots \\
&\le& 2 F_3 + 4F_4 +6 F_5 + 8 F_6 + 10 F_7 + 12 F_8 + \dots \\
&\le& 4\mathcal{E}-4\mathcal{F} = 4|V|-8.
\end{aligned}$$ The middle step follows by simply increasing each coefficient. The last step is based on Euler’s characteristic [@eulerchar]: $|V|+\mathcal{F}-\mathcal{E}=2$. Hence we have $4|V|\le M\le
4|V|-8$, which is a contradiction.
Euler was wrong: Connelly’s counterexample
------------------------------------------
Proofs behind counterexamples can rarely be termed “beautiful” since they usually lack generality (as they are applied to one particular example). Counterexamples can nonetheless be dazzling by themselves. Connelly’s counterexample [@connelly-countereg] to the Euler’s conjecture consists in a very special non-generic nonconvex polyhedron which flexes, while keeping combinatorial equivalence and facewise equality with all polyhedra in the flex. Some years later, Klaus Steffen produced a much simpler polyhedron with the same properties[^11]. It is this polyhedron we exhibit in Fig. \[f:steffen2\].
![Steffen’s polyhedron: the flex (these two images were obtained as snapshot from the [*Mathematica*]{} [@mathematica] demonstration cited in Footnote \[fn:steffen\]). There is a rotation, in the direction showed by the arrows, around the edge which is emphasized on the right picture. The short upper right edge only appears shorter on the right because of perspective.[]{data-label="f:steffen2"}](steffen3){width="3.5cm"}
Cayley-Menger determinants and the simplex volume {#s:cayleymenger}
=================================================
The foundation of modern DG, as investigated by Menger [@menger31] and Blumenthal [@blumenthal], rests on the fact that:
> [*the four-dimensional volume of a four-dimensional simplex embedded in three dimensional space is zero,*]{} ($\ast$)
which we could also informally state as “flat simplices have zero volume”. This is related to DG because the volume of a simplex can be expressed in terms of the lengths of the simplex sides, which yields a polynomial in the length of the simplex side lengths that can be equated to zero. If these lengths are expressed in function of the vertex positions as $\|x_u-x_v\|^2$, this yields a polynomial equation in the positions $x_1,\ldots,x_5$ of the simplex vertices in terms of its side lengths. Thus, if we know the positions of $x_1,\ldots,x_4$, we can compute the unknown position of $x_5$ or prove that no such position exists, through a process called [*trilateration*]{} [@dvop].
The proof of ($\ast$) was published by Arthur Cayley in 1841 [@cayley1841], during his undergraduate studies. It is based on the following well-known lemma about determinants (stated without proof in Cayley’s paper).
If $A,B$ are square matrices having the same size, $|AB|=|A||B|$.\[clem\]
Given five points $x_1,\ldots,x_5\in\mathbb{R}^4$ all belonging to an affine 3D subspace of $\mathbb{R}^4$, let $d_{ij}=\|x_i-x_j\|_2$ for each $i,j\le 5$. Then $$\left|\begin{array}{cccccc}
0 & d_{12}^2 & d_{13}^2 & d_{14}^2 & d_{15}^2 & 1 \\
d_{21}^2 & 0 & d_{23}^2 & d_{24}^2 & d_{25}^2 & 1 \\
d_{31}^2 & d_{32}^2 & 0 & d_{34}^2 & d_{35}^2 & 1 \\
d_{41}^2 & d_{42}^2 & d_{43}^2 & 0 & d_{45}^2 & 1 \\
d_{51}^2 & d_{52}^2 & d_{53}^2 & d_{54}^2 & 0 & 1 \\
1 & 1 & 1 & 1 & 1 & 0 \end{array}\right|=0.
\label{eq:cm0}$$
We note that Cayley’s theorem is expressed for $n=5$ points in $\mathbb{R}^3$, but it also holds for $n\ge 3$ points in $\mathbb{R}^{n-2}$ [@blumenthal]. Cayley explicitly remarks that it holds for the cases $n=4$ and $n=3$ (see [@sommerville VIII, § 5] for the proof of general $n$). The determinant on the right-hand side of Eq. is called [*Cayley-Menger determinant*]{}, denoted by $\Delta$. We remark that in the proof below $x_{ik}$ is the $k$-th component of $x_i$, for each $i\le 5, k\le 4$.
We follow Cayley’s treatment. He pulls the following two matrices [$$A=\left(\begin{array}{cccccc}
\|x_1\|^2 & -2x_{11} & -2x_{12} & -2x_{13} & -2x_{14} & 1 \\
\|x_2\|^2 & -2x_{21} & -2x_{22} & -2x_{23} & -2x_{24} & 1 \\
\|x_3\|^2 & -2x_{31} & -2x_{32} & -2x_{33} & -2x_{34} & 1 \\
\|x_4\|^2 & -2x_{41} & -2x_{42} & -2x_{43} & -2x_{44} & 1 \\
\|x_5\|^2 & -2x_{51} & -2x_{52} & -2x_{53} & -2x_{54} & 1 \\
1 & 0 & 0 & 0 & 0 & 0 \end{array}\right), \quad
B=\left(\begin{array}{cccccc}
1 & 1 & 1 & 1 & 1 & 0 \\
x_{11} & x_{21} & x_{31} & x_{41} & x_{51} & 0 \\
x_{12} & x_{22} & x_{32} & x_{42} & x_{52} & 0 \\
x_{13} & x_{23} & x_{33} & x_{43} & x_{53} & 0 \\
x_{14} & x_{24} & x_{34} & x_{44} & x_{54} & 0 \\
\|x_1\|^2 & \|x_2\|^2 & \|x_3\|^2 & \|x_4\|^2 & \|x_5\|^2 & 1
\end{array}\right)$$ ]{}out of a magic hat. He performs the product $AB$, re-arranging and collecting terms, and obtains a $6\times 6$ matrix where the last row and column are $(1,1,1,1,1,0)$, and the $(i,j)$-th component is $\|x_i-x_j\|_2^2$ for every $i,j\le 5$. To see this, it suffices to carry out the computations using [*Mathematica*]{} [@mathematica]; by way of an example, the first diagonal component of $AB$ is $\|x_1\|^2-2\sum\limits_{k\le 4} x_{1k}x_{1k}+\|x_1\|^2=0$, and the component on the first row, second column of $AB$ is $\|x_1\|^2-2\sum\limits_{k\le 4}
x_{1k}x_{2k}+\|x_2\|^2=\|x_1-x_2\|^2$. In other words, $|AB|$ is the Cayley-Menger determinant in Eq. . On the other hand, if we set $x_{4k}=0$ for each $k\le 4$, effectively projecting the five four-dimensional points in three-dimensional space, it is easy to show that $|A|=|B|=0$ since the 5-th columns of both $A$ and the 5th row of $B$ are zero. Hence we have $0=|A||B|=|AB|$ by Lemma \[clem\], and $|AB|=0$ is precisely Eq. as claimed.
The missing link is the relationship of the Cayley-Menger determinant with the volume of an $n$-simplex. Since this is not part of Cayley’s paper, we only establish the relationship for $n=3$. Let $d_{12}=a$, $d_{13}=b$, $d_{23}=c$. Then: $$\left|\begin{array}{cccccc}
0 & a^2 & b^2 & 1 \\
a^2 & 0 & c^2 & 1 \\
b^2 & c^2 & 0 & 1 \\
1 & 1 & 1 & 0 \end{array}\right| = a^4 - 2 a^2 b^2 + b^4 - 2 a^2 c^2
- 2 b^2 c^2 + c^4 = -16(s(s-a)(s-b)(s-c)),$$ where $s=\frac{1}{2}(a+b+c)$ (this identity can be established by using e.g. [*Mathematica*]{} [@mathematica]). By Heron’s theorem (Thm. \[t:heron\] above) we know that the area of a triangle with side lengths $a,b,c$ is $\sqrt{s(s-a)(s-b)(s-c)}$. So, for $n=3$, the determinant on the left-hand side is proportional to the negative of the square of the triangle area. This result can be generalized to every value of $n$ [@blumenthal II, § 40, p. 98]: it turns out that the $n$-dimensional volume $V_n$ of an $n$-simplex in $\mathbb{R}^n$ with side length matrix $d=(d_{ij}\;|\;i,j\le n+1)$ is: $$V_n^2 = \frac{(-1)^{n-1}}{2^n (n!)^2} \Delta.$$
The beauty of Cayley’s proof is in its extreme compactness: it uses determinants to hide all the details of elimination theory which would be necessary otherwise. His paper also shows some of these details for the simplest case $n=3$. The starting equations, as well as the symbolic manipulation steps, depend on $n$. Although Cayley’s proof is only given for $n=5$, Cayley’s treatment goes through essentially unchanged for any number $n$ of points in dimension $n-2$.
Menger’s characterization of abstract metric spaces {#s:menger}
===================================================
At a time where mathematicians were heeding Hilbert’s call to formalization and axiomatization, Menger presented new axioms for geometry based on the notion of distance, and provided conditions for arbitrary sets to “look like” Euclidean spaces, at least distancewise [@menger28; @menger31]. Menger’s system allows a formal treatment of geometry based on distances as “internal coordinates”. The starting point is to consider the relations of geometrical figures having proportional distances between pairs of corresponding points, i.e. congruence. Menger’s definition of a congruence system is defined axiomatically, and the resulting characterization of abstract distance spaces with respect to subsets of Euclidean spaces (possibly his most important result) transforms a possibly infinite verification procedure (any subset of any number of points) into a finitistic one (any subset of $n+3$ points, where $n$ is the dimension of the Euclidean space).
It is remarkable that almost none of the results below offers an intuitive geometrical grasp, such as the proofs of Heron’s formula and Cayley’s theorem do. As formal mathematics has it, part of the beauty in Menger’s work consists in turning the “visual” geometrical proofs based on intuition into formal symbolic arguments based on sets and relations. On the other hand, Menger himself gave a geometric intuition of his results in [@mengerS p. 335], which we comment in Sect. \[s:mengerintuitive\] below.
Menger’s axioms
---------------
Let $\mathcal{S}$ be a system of sets, and for any set $S\in\mathcal{S}$ and any two (not necessarily distinct) points $p,q\in S$, denote the couple $(p,q)$ by $pq$. Menger defines a relation $\approx$ by means of the following axioms.
1. $\forall S,T\in\mathcal{S}$, $\forall p,q\in S$ and $\forall
r,s\in T$, we have either $pq\approx rs$ or $pq\not\approx rs$ but not both.\[M1\]
2. $\forall S\in\mathcal{S}$ and $\forall p,q\in S$ we have $pq\approx qp$.\[M2\]
3. $\forall S,T\in\mathcal{S}$, $\forall p\in S$ and $\forall
r,s\in T$, we have $pp\approx rs$ if and only if $r=s$.\[M3\]
4. $\forall S,T\in\mathcal{S}$, $\forall p,q\in S$ and $\forall
r,s\in T$, if $pq\approx rs$ then $rs\approx pq$.\[M4\]
5. $\forall S,T,U\in\mathcal{S}$, $\forall p,q\in S$, $\forall
r,s\in T$ and $\forall t,u\in U$, if $pq\approx rs$ and $pq\approx
tu$ then $rs\approx tu$.\[M5\]
The couple $(\mathcal{S},\approx)$ is called a [*congruence system*]{}, and the $\approx$ relation is called [*congruence*]{}.
Today, we are used to think of [*relations*]{} as defined on a single set. We remark that in Menger’s treatment, congruence is a binary relation defined on sets of ordered pairs of points, where each point in each pair belongs to the same set as the other, yet left-hand and right-hand side terms may belong to different sets. We now interpret each axiom from a more contemporary point of view.
1. Axiom \[M1\] states that Menger’s congruence relation is in fact a [*partial*]{} relation on $\mathscr{S}=(\bigcup\mathcal{S})^2$ (the Cartesian product of the union of all sets $S\in\mathcal{S}$ by itself), which is only defined for a couple $pq\in\mathscr{S}$ whenever $\exists
S\in\mathcal{S}$ such that $p,q\in S$.
2. By axiom \[M2\], the $\approx$ relation acts on sets of [ *unordered*]{} pairs of (not necessarily distinct) points; we call $\bar{\mathscr{S}}$ the set of all unordered pairs of points from all sets $S\in\mathcal{S}$.
3. By axiom \[M3\], $rs$ is congruent to a pair $pq$ where $p=q$ if and only if $r=s$.
4. Axiom \[M4\] states that $\approx$ is a symmetric relation.
5. Axiom \[M5\] states that $\approx$ is a transitive relation.
Note that $\approx$ is also reflexive (i.e. $pq\approx pq$) since $pq\approx qp\approx pq$ by two successive applications of Axiom \[M2\]. So, using today’s terminology, $\approx$ is an equivalence relation defined on a subset of $\bar{\mathscr{S}}$.
A model for the axioms
----------------------
Menger’s model for his axioms is a [*semi-metric*]{} space $S$, i.e. a set $S$ of points such that to each unordered pair $\{p,q\}$ of points in $S$ we assign a nonnegative real number $d_{pq}$ which we call [ *distance*]{} between $p$ and $q$. Under this interpretation, Axiom \[M2\] tells us that $d_{pq}=d_{qp}$ for each pair of points $p,q$, and Axiom \[M3\] tells us that $rs$ is congruent to a single point if and only if $d_{rs}=0$, which, together with nonnegativity, are the defining properties of [*semi-metrics*]{} (the remaining property, the triangular inequality, tells semi-metrics apart from [ *metrics*]{}). Thus, the set $\mathcal{S}$ of all semi-metric spaces together with the relation given by $pq\approx rs\leftrightarrow
d_{pq}=d_{rs}$ is a congruence system.
A finitistic characterization of semi-metric spaces
---------------------------------------------------
Two sets $S,T\in\mathcal{S}$ are [*congruent*]{} if there is a map (called [*congruence map*]{}) $\phi:S\to T$, such that $pq\approx
\phi(p)\phi(q)$ for all $p,q\in S$. We denote this relation by $S\approx_\phi T$, dropping the $\phi$ if it is clear from the context.
Any congruence map $\phi:S\to T$ is injective. \[l:injective\]
Suppose, to get a contradiction, that $\exists p,q\in S$ with $p\not=q$ and $\phi(p)=\phi(q)$: then $pq\approx\phi(p)\phi(q)=\phi(p)\phi(p)$ and so, by Axiom \[M3\], $p=q$ against assumption.
If $S$ is congruent to a subset of $T$, then we say that $S$ is [ *congruently embeddable*]{} in $T$.
### Congruence order
Now consider a set $S\in\mathcal{S}$ and an integer $n\ge 0$ with the following property: for any $T\in\mathcal{S}$, if all $n$-point subsets of $T$ are congruent to an $n$-point subset of $S$, then $T$ is congruently embeddable in $S$. If this property holds, then $S$ is said to have [*congruence order*]{} $n$. Formally, the property is written as follows: $$\forall T\in\mathcal{S} \; \forall T'\subseteq T \quad (\ (|T'|=n\to
\exists S'\subseteq S\;(|S'|=n\land T'\approx S')) \quad
\longrightarrow \quad \exists R\subseteq S\;(T\approx
R)\ ). \label{eq:congruenceorder}$$
If $|S|<n$ for some positive integer $n$, then $S$ can have congruence order $n$, since the definition is vacuously satisfied. So we assume in the following that $|S|\ge n$.
If $S$ has congruence order $n$ in $\mathcal{S}$, then it also has congruence order $m$ for each $m>n$. \[p:comn\]
By hypothesis, for every $T\in\mathcal{S}$, if every $n$-point subset $T'$ of $T$ is congruent to an $n$-point subset of $S$, then there is a subset $R$ of $S$ such that $T\approx_\phi R$. Now any $m$-point subset of $S$ is mapped by $\phi$ to a congruent $m$-point subset of $S$, and again $T\approx R$, so Eq. is satisfied for $S$ and $m$.
$\mathbb{R}^0$ (i.e. the Euclidean space which simply consists of the origin) has minimum congruence order $2$ in $\mathcal{S}$.
Pick any $T\in\mathcal{S}$ with $|T|>1$. None of its $2$-point subsets is congruent to any $2$-point subset of $\mathbb{R}^0$, since none exists. Moreover, $T$ itself cannot be congruently embedded in $\mathbb{R}^0$, since $|T|>1=|\mathbb{R}^0|$ and no injective congruence map can be defined, against Lemma \[l:injective\]. So the integer $2$ certainly (vacuously) satisfies Eq. for $S=\mathbb{R}^0$, which means that $\mathbb{R}^0$ has congruence order $2$. In view of Prop. \[p:comn\], it also has congruence order $m$ for each $m>2$. Hence we have to show next that the integer $1$ cannot be a congruence order for $\mathbb{R}^0$. To reach a contradiction, suppose the contrart, and let $T$ be as above. By Axiom \[M3\], every singleton subset of $T$ is congruent to a subset of $\mathbb{R}^0$, namely the subset containing the origin. Thus, by Eq. , $T$ must be congruent to a subset of $\mathbb{R}^0$; but, again, $|T|>1=|\mathbb{R}^0|$ contradicts Lemma \[l:injective\]: so $T$ cannot be congruently embedded in $\mathbb{R}^0$, which negates Eq. . Hence $1$ cannot be a congruence order for $\mathbb{R}^0$, as claimed.
### Menger’s fundamental result
The fundamental result proved by Menger in 1928 [@menger28] is that the Euclidean space $\mathbb{R}^n$ has congruence order $n+3$ but not $n+2$ for each $n>0$ in the family $\mathcal{S}$ of all semi-metric spaces. The important implication of Menger’s result is that in order to verify whether an abstract semi-metric space is congruent to a subset of a Euclidean space, we only need to verify congruence of each of its $n+3$ point subsets.
We follow Blumenthal’s treatment [@blumenthal], based on the following preliminary definitions and properties, which we shall not prove:
1. A congruent mapping of a semi-metric space onto itself is called a [*motion*]{}; \[BD1\]
2. $n+1$ points in $\mathbb{R}^{n}$ are [*independent*]{} if they are not affinely dependent (i.e. if they do not all belong to a single hyperplane in $\mathbb{R}^n$); \[BD2\]
3. two congruent $(n+1)$-point subsets of $\mathbb{R}^n$ are either both independent or both dependent; \[BP1\]
4. there is at most one point of $\mathbb{R}^n$ with given distances from an independent $(n+1)$-point subset;\[BP2\]
5. any congruence between any two subsets of $\mathbb{R}^n$ can be extended to a motion;\[BP3\]
6. any congruence between any two independent $(n+1)$-point subsets of $\mathbb{R}^n$ can be extended to a unique motion.\[BP4\]
A non-empty semi-metric space $S$ is congruently embeddable in $\mathbb{R}^n$ (but not in any $\mathbb{R}^r$ for $r<n$) if and only if: (a) $S$ contains an $(n+1)$-point subset $S'$ which is congruent with an independent $(n+1)$-point subset of $\mathbb{R}^n$; and (b) each $(n+3)$-point subset $U$ of $S$ containing $S'$ is congruent to an $(n+3)$-point subset of $\mathbb{R}^n$.
The proof of Menger’s theorem is very formal (see below) and somewhat difficult to follow. It is nonetheless a good example of a proof in an axiomatic setting, where logical reasoning is based on syntactical transformations induced by inference rules on the given axioms. An intuitive discussion is provided in Sect. \[s:mengerintuitive\].
($\Rightarrow$) Assume first that $S\approx_\phi
T\subseteq\mathbb{R}^n$, where the affine closure of $T$ has dimension $n$. Then $T$ must contain an independent subset $T'$ with $|T'|=n+1$, which we can map back to a subset $S'\subseteq S$ using $\phi^{-1}$. Since $\phi,\phi^{-1}$ are injective, $|S'|\le |T'|$, and by Axiom \[M3\] we have $|S'|\ge |T'|$, so $|S'|=n+1$, which establishes (a). Now take any $U\subseteq S$ with $|U|=n+3$ and $U\supset S'$: this can be mapped via $\phi$ to a subset $W\subseteq
T$: Lemma \[l:injective\] ensures injectivity of $\phi$ and hence $|W|=n+3$, establishing (b).\
($\Leftarrow$) Conversely, assume (a) and (b) hold. By (a), let $S'\subseteq S$ with $|S'|=n+1$ and $S'\approx_\phi T'\subseteq\mathbb{R}^n$, with $T'$ independent and $|T'|=n+1$. We claim that $\phi$ can be extended to a mapping of $S$ into $\mathbb{R}^n$. Take any $q\in S\smallsetminus S'$: by (b), $S'\cup\{q\}\approx_\psi W\subseteq \mathbb{R}^n$ with $|W|=n+2$. Note that $T'\approx_\omega W\smallsetminus\{\psi(q)\}$ by Axiom \[M5\], which implies that for any $p\in S'$, we have $\omega\phi(p)=\psi(p)$. Moreover, by Property \[BP1\] above, $W\smallsetminus\{\psi(q)\}$ is independent and has cardinality $n+1$, which by Property \[BP4\] above implies that $\omega$ can be extended to a unique motion in $\mathbb{R}^n$. So the action of $\omega$ is extended to $q$, and we can define $\phi(q)=\omega^{-1}\psi(q)$. We now show that this extension of $\phi$ is a congruence. Let $p,q\in S$: we aim to prove that $pq=\phi(p)\phi(q)$. Consider the set $U=S'\cup\{p,q\}$: since $|U|\le n+3$, by (b) there is $W\subset\mathbb{R}^n$ with $|W|=|U|$ such that $U\approx_\psi W$. As above, we note that there is a subset $W'\subseteq W$ such that $|W'|=n+1$ and $T'\approx_\omega
W'$, that $\omega\phi(r)=\psi(r)$ for each $r\in S'$, and that $\omega$ is a motion of $\mathbb{R}^n$. Hence $pq=\psi(p)\psi(q)=\omega^{-1}\phi(p)\omega^{-1}\phi(q)=\phi(p)\phi(q)$, as claimed.
An intuitive interpretation {#s:mengerintuitive}
---------------------------
Although we stated initially that part of the the beauty of the formal treatment of geometry is that it is based on symbolic manipulation rather than visual intuition, we quote from a survey paper which Menger himself wrote (in Italian, with the help of L. Geymonat) to disseminate the work carried out at his seminar [@mengerS].
> [*Affinché uno spazio metrico reale $R$ sia applicabile a un insieme parziale di $\mathbb{R}^n$ è necessario e sufficiente che per ogni $n+3$ e per ogni $n+2$ punti di esso sia $\Delta=0$ e inoltre che ogni $n+1$ punti di $R$ siano applicabili a punti di $\mathbb{R}^n$.*]{}
The translation is “a real metric space $R$ is embeddable in a subset of $\mathbb{R}^n$ if and only if $\Delta=0$ for each $(n+3)$- and $(n+2)$-point subsets or $R$, and that each $(n+1)$-point subset of $R$ is embeddable in $\mathbb{R}^n$.”
Since we know that $\Delta$, the Cayley-Menger determinant of the pairwise distances of a set $S$ of points (see Eq. ), is proportional to the volume of the simplex on $S$ embedded in $|S|-1$ dimensions, what Menger is saying is that his result on the congruence order of Euclidean spaces can be intuitively interpreted as follows.
> An abstract semi-metric space $R$ is congruently embeddable in $\mathbb{R}^n$ if and only if: (i) there are $n+1$ points in $R$ which are congruently embeddable in $\mathbb{R}^n$; (ii) the volume of the simplex on each $n+2$ points of $R$ is zero; (iii) the volume of the simplex on each $n+3$ points of $R$ is zero.
This result is exploited in the algorithm for computing point positions from distances given in [@sippl p. 2284].
Gödel on spherical distances {#s:goedel}
============================
Kurt Gödel’s name is attached to what is possibly the most revolutionary result in all of mathematics, i.e. Gödel’s incompleteness theorem, according to which any formal axiomatic system sufficient to encode the integers is either inconsistent (it proves $A$ and $\neg A$) or incomplete (there is some true statement $A$ which the system cannot prove). This shattered Hilbert’s dream of a formal system in which every true mathematical statement could be proved. Few people know that Gödel, who attended the Vienna Circle, Menger’s course in geometry, and Menger’s seminar, also contributed two results which are completely outside of the domain of logic. These results only appeared in the proceedings of Menger’s seminar [@mengerK], and concern DG on a spherical surface.
Four points on the surface of a sphere
--------------------------------------
The result we discuss here is a proof to the following theorem, conjectured at a previous seminar session by Laura Klanfer. We remark that a sphere in $\mathbb{R}^3$ is a semi-metric space whenever it is endowed with a distance corresponding to the length of a geodesic curve joining two points.
Given a semi-metric space $S$ of four points, congruently embeddable in $\mathbb{R}^3$ but not $\mathbb{R}^2$, is also congruently embeddable on the surface of a sphere in $\mathbb{R}^3$. \[t:goedel\]
Gödel’s proof looks at the circumscribed sphere around a tetrahedron in $\mathbb{R}^3$, and analyses the relationship of the geodesics, their corresponding chords, and the sphere radius. It then uses a fixed point argument to find the radius which corresponds to geodesics which are as long as the given sides.
The congruence embedding of $S$ in $\mathbb{R}^3$ defines a tetrahedron $T$ having six (straight) sides with lengths $a_1,\ldots,a_6$. Let $r$ be the radius of the sphere circumscribed around $T$ (i.e. the smallest sphere containing $T$). We shall now consider a family of tetrahedra $\tau(x)$, parametrized on a scalar $x>0$, defined as follows: $\tau(x)$ is the tetrahedron in $\mathbb{R}^3$ having side lengths $c_x(a_1),\ldots,c_x(a_6)$, where $c_x(\alpha)$ is the length of the chord subtending a geodesic having length $\alpha$ on a sphere of radius $\frac{1}{x}$. As $x$ tends towards zero, each $c_x(a_i)$ tends towards $a_i$ (for each $i\le 6$), since the radius of the sphere tends towards infinity and each geodesic length tends towards the length of the subtending chord. This means that $\tau(x)$ tends towards $T$, since $T$ is precisely the tetrahedron having side lengths $a_1,\ldots,a_6$. For each $x>0$, let $\phi(x)$ be the inverse of the radius of the sphere circumscribed about $\tau(x)$. Since $\tau(x)\to T$ as $x\to 0$, and the radius circumscribed about $T$ is $r$, it follows that $\phi(x)\to\frac{1}{r}$ as $x\to 0$. Also, since $T$ exists by hypothesis, we can define $\tau(0)=T$ and $\phi(0)=\frac{1}{r}$. Also note that it is well known by elementary spherical geometry that: $$c_x(\alpha) = \frac{2}{x}\sin\frac{\alpha x}{2}. \label{eq:cx}$$ : if $a'=\max\{a_1,\ldots,a_6\}$ then $\phi$ has a fixed point in the open interval $I=(0,\frac{\pi}{a'})$.\
. First of all notice that $\tau(0)$ exists, and $c_x(\alpha)$ is a continuous function for $x>0$ for each $\alpha$ (by Eq. ). Since $\tau(x)$ is defined by the chord lengths $c_x(a_1),\ldots,c_x(a_6)$, this also means that $\tau(x)$ varies continuously for $x$ in some open interval $J=(0,\varepsilon)$ (for some constant $\varepsilon>0$). In turn, this implies that $\bar{x}=\max \{y\in I \;|\;\tau(y)\mbox{
exists}\}$ exists by continuity. There are two cases: either $\bar{x}$ is at the upper extremum of $I$, or it is not.
- If $\bar{x}=\frac{\pi}{a'}$, then $\tau(\bar{x})$ exists, its longest edge has length $c_{\bar{x}}(a')=\frac{2a'}{\pi}$, so, by elementary spherical geometry, the radius of the sphere circumscribed around $\tau(\bar{x})$ is greater than $\frac{c_{\bar{x}}(a')}{2}$, i.e. greater than $\frac{a'}{\pi}=\frac{1}{\bar{x}}$. Thus $\phi(\bar{x})<\bar{x}$. We also have, however, that $\phi(0)=\frac{1}{r}>0$, so by the intermediate value theorem there must be some $x\in(0,\bar{x})$ with $\phi(x)=x$.
- Assume now $\bar{x}<\frac{\pi}{a'}$ and suppose $\tau(\bar{x})$ is non-planar. Then for each $y$ in an arbitrary small neighbourhood around $\bar{x}$, $\tau(y)$ must exist by continuity: in particular, there must be some $y>\bar{x}$ where $\tau(y)$ exists, which contradicts the definition of $\bar{x}$. So $\tau(\bar{x})$ is planar: this means that each geodesic is contained in the same plane, which implies that the geodesics are linear segments. It follows that the circumscribed sphere has infinite radius, or, equivalently, that $\phi(\bar{x})=0<\bar{x}$. Again, by $\phi(0)>0$ and the intermediate value theorem, there must be some $x\in(0,\bar{x})$ with $\phi(x)=x$.
This concludes the proof of the claim.\
So now let $y$ be the fixed point of $\phi$. The tetrahedron $\tau(y)$ has side lengths $c_y(a_i)$ for each $i\le 6$, and is circumscribed by a sphere $\sigma$ with radius $\frac{1}{y}$. It follows that, on the sphere $\sigma$, the geodesics corresponding to the chords given by the tetrahedron sides have lengths $a_i$ (for $i\le 6$), as claimed.
Gödel’s devilish genius {#s:genius}
-----------------------
Gödel’s proof exhibits an unusual peak of devilish genius. At first sight, it is a one-dimensional fixed-point argument which employs a couple of elementary notions in spherical geometry. Underneath the surface, the fixed-point argument eschews a misleading visual intuition.
$T$ is a given tetrahedron in $\mathbb{R}^3$ which is assumed to be non-planar and circumscribed by a sphere of finite positive radius $r$ (see Fig. \[f:goedel\], left).
![The given tetrahedron $T$ (left), and the tetrahedron $\tau(x)$ (right). Beware of this visual interpretation: it may yield misleading insights (see Sect. \[s:genius\]).[]{data-label="f:goedel"}](goedel1){width="14cm"}
The map $\tau$ sends a scalar $x$ to the tetrahedron having as side lengths the chords subtending the geodesics of length $a_i$ ($i\le 6$) on a sphere of radius $\frac{1}{x}$ (see Fig. \[f:goedel\], right). The map $\tau$ is such that $\tau(0)=T$ since for $x=0$ the radius is infinite, which means that the geodesics are equal to their chords. Moreover, the map $\phi$ sends $x$ to the inverse of the radius of the sphere circumscribing $\tau(x)$. Since every geodesic on the sphere is a portion of a great circle, it would appear from Fig. \[f:goedel\] (right) that the radius $\frac{1}{x}$ used to compute $c_x(a_i)$ ($i\le 6$) is the same as the radius $\frac{1}{\phi(x)}$ of the sphere circumscribing $\tau(x)$, which would immediately yield $\phi(x)=x$ for every $x$ — making the proof trivial. There is something inconsistent, however, in the visual interpretation of Fig. \[f:goedel\]: the given tetrahedron $T$ corresponds to the case $\tau(x)=T$, which happens when $x=0$, i.e. the radius of the sphere circumscribed around $T$ is $\infty$. But this would yield $T$ to be a planar tetrahedron, which is a contradiction with an assumption of the theorem. Moreover, if $\phi(x)$ were equal to $x$ for each $x$, this would yield $0=\phi(0)=\frac{1}{r}>0$, another contradiction.
The misleading concept is hidden in the picture in Fig. \[f:goedel\] (right). It shows a tetrahedron inscribed in a sphere, and a spherical tetrahedron [*on the same vertices*]{}. This is not true in general, i.e. the spherical tetrahedron with the given curved side lengths $a_1,\ldots,a_6$ cannot, in general, be embedded in the surface of a sphere of [*any*]{} radius. For example, the case $x=0$ yields geodesics with infinite curvatures (i.e. straight lines laying in a plane), but $\phi(x)=\frac{1}{r}>0$, and there is no flat tetrahedron with the same distances as those of $T$. The sense of Gödel’s proof is that the function $c_x$ simply transforms a set of geodesic distances into a set of linear distances, i.e. it maps scalars to scalars rather than geodesics to segments, whereas Fig. \[f:goedel\] (right) shows the special case where the geodesics are mapped to the corresponding segments, with intersections at the same points (namely the distances $a_1,\ldots,a_6$ can be embedded on the particular sphere shown in the picture). More specifically, the geodesic curves may or may not be realizable on a sphere of radius $\frac{1}{\phi(x)}$. Gödel’s proof shows exactly that there must be some $x$ for which $\phi(x)=x$, i.e. the geodesic curves become realizable.
Existential vs. constructive proofs
-----------------------------------
Like many existential proofs based on fixed-point theorems,[^12] this proof is beautiful because it asserts the truth of the theorem without any certificates other than its own logical validity. An alternative, constructive proof of Thm. \[t:goedel\] is given in [@schoenberg Thm. 3’]. The tools used in that proof, Cayley-Menger determinants and positive semidefiniteness, are discussed in Sect. \[s:schoenberg\] below.
The equivalence of EDM and PSD matrices {#s:schoenberg}
=======================================
Many fundamental innovations stem from what are essentially footnotes to apparently deeper or more important work. Isaac Schoenberg, better known as the inventor of splines [@splines], published a paper in 1935 titled [*Remarks to Maurice Fréchet’s article “Sur la définition axiomatique d’une classe d’espace distanciés vectoriellement applicable sur l’espace de Hilbert”*]{} [@schoenberg]. The impact of Schoenberg’s remarks far exceeds that of the original paper[^13]: these remarks encode what amounts to the basis of the well-known MDS techniques for visualizing high-dimensional data [@coxcox], as well as all the solution techniques for Distance Geometry Problems (DGP) based on Semidefinite Programming (SDP) [@ye; @wolkowicz].
Schoenberg’s problem
--------------------
Schoenberg poses the following problem, relevant to Menger’s treatment of distance geometry [@menger31 p. 737].
> Given an $n\times n$ symmetric matrix $D$, what are necessary and sufficient conditions such that $D$ is a EDM corresponding to $n$ points in $\mathbb{R}^r$, with $1\le r\le n$ minimum?
Menger’s solution is based on Cayley-Menger determinants; Schoenberg’s solution is much simpler and more elegant, and rests upon the following theorem. Recall that a matrix is PSD if and only if all its eigenvalues are nonnegative.
The $n\times n$ symmetric matrix $D=(d_{ij})$ is the EDM of a set of $n$ points $x=\{x_1,\ldots,x_n\}\subset\mathbb{R}^{r}$ (with $r$ minimum) if and only if the matrix $G=\frac{1}{2}(d_{1i}^2+d_{1j}^2-d_{ij}^2\;|\;2\le i,j\le n)$ is PSD of rank $r$.\[t:schoenberg\]
Instead of providing Schoenberg’s proof, we follow a more modern treatment, which also unearths the important link of this theorem with classical MDS [@coxcox § 2.2.1], an approximate method for finding sets of points $x=\{x_1,\ldots,x_n\}$ having EDM which approximates a given symmetric matrix. MDS is one of the cornerstones of the modern science of data analysis.
The proof of Schoenberg’s theorem {#s:schthm}
---------------------------------
Given a set $x=\{x_1,\ldots,x_n\}$ of points in $\mathbb{R}^r$, we can write $x$ as an $r\times n$ matrix having $x_i$ as $i$-th column. The matrix $G=x{{x}^{\top}}$ having the scalar product $x_ix_j$ as its $(i,j)$-th component is called the [*Gram matrix*]{} or [*Gramian*]{} of $x$. The proof of Thm. \[t:schoenberg\] works by exhibiting a 1-1 correspondence between squared EDMs and Gram matrices, and then by proving that a matrix is Gram if and only if it is PSD.
Without loss of generality, we can assume that the barycenter of the points in $x$ is at the origin: $$\sum_{i\le n} x_i = 0. \label{eq:mds0}$$ Now we remark that, for each $i,j\le n$, we have: $$d^2_{ij} =
\|x_i-x_j\|^2=(x_i-x_j)(x_i-x_j)=x_ix_i+x_jx_j-2x_ix_j. \label{eq:mds1}$$
### The Gram matrix in function of the EDM
We “invert” Eq. to compute the matrix $G=x{{x}^{\top}}=(x_i x_j)$ in function of the matrix $D^2=(d_{ij}^2)$. We sum Eq. over all values of $i\in\{1,\ldots,n\}$, obtaining: $$\sum_{i\le n} d^2_{ij} = \sum_{i\le n} (x_i x_i) + n (x_j x_j)
- 2\left(\sum_{i\le n} x_i\right) x_j. \label{eq:mds2}$$ By Eq. , the negative term in the right hand side of Eq. is zero. On dividing through by $n$, we have $$\frac{1}{n} \sum_{i\le n} d^2_{ij} = \frac{1}{n} \sum_{i\le n}
(x_i x_i) + x_j x_j. \label{eq:mds3}$$ Similarly for $j\in\{1,\ldots,n\}$, we obtain: $$\frac{1}{n} \sum_{j\le n} d^2_{ij} = x_i x_i + \frac{1}{n} \sum_{j\le n}
(x_j x_j). \label{eq:mds4}$$ We now sum Eq. over all $j$, getting: $$\frac{1}{n} \sum_{i\le n\atop j\le n} d^2_{ij} =
n\frac{1}{n}\sum_{i\le n} (x_i x_i) + \sum_{j\le n} (x_j
x_j) = 2\sum_{i\le n} (x_i x_i) \label{eq:mds5}$$ (the last equality in Eq. holds because the same quantity $f(k)=x_k x_k$ is being summed over the same range $\{1,\ldots,n\}$, with the symbol $k$ replaced by the symbol $i$ first and $j$ next). We then divide through by $n$ to get: $$\frac{1}{n^2} \sum_{i\le n\atop j\le n} d^2_{ij} = \frac{2}{n}
\sum_{i\le n} (x_i x_i). \label{eq:mds6}$$
We now rearrange Eq. , , as follows: $$\begin{aligned}
2 x_i x_j &=& x_i x_i + x_j x_j - d_{ij}^2
\label{eq:mds7} \\
x_i x_i &=& \frac{1}{n} \sum_{j\le n} d^2_{ij} - \frac{1}{n} \sum_{j\le n}
(x_j x_j) \label{eq:mds8} \\
x_j x_j &=& \frac{1}{n} \sum_{i\le n} d^2_{ij} - \frac{1}{n} \sum_{i\le n}
(x_i x_i), \label{eq:mds9} \end{aligned}$$ and replace the left hand side terms of Eq. - into Eq. to obtain: $$2 x_i x_j = \frac{1}{n} \sum_{k\le n} d^2_{ik} +
\frac{1}{n} \sum_{k\le n} d^2_{kj} - d_{ij}^2 - \frac{2}{n} \sum_{k\le n}
(x_k x_k),
\label{eq:mds10}$$ whence, on substituting the last term using Eq. , we have: $$2 x_i x_j = \frac{1}{n} \sum_{k\le n} (d^2_{ik} + d^2_{kj}) -
d_{ij}^2 - \frac{1}{n^2} \sum_{h\le n\atop k\le n} d^2_{hk}.
\label{eq:mds11}$$
It turns out that Eq. can be written in matrix form as: $$G = -\frac{1}{2} J D^2 J, \label{eq:mds12}$$ where $J=I_n-\frac{1}{n}\mathbf{1}{{\mathbf{1}}^{\top}}$ and $\mathbf{1}=\underbrace{(1,\ldots,1)}_n$.
### Gram matrices are PSD matrices
Any Gram matrix $G=x{{x}^{\top}}$ derived by a point sequence (also called a [*realization*]{}) $x=(x_1,\ldots,x_n)$ in $\mathbb{R}^K$ for some non-negative integer $K$ has two important properties: (i) the rank of $G$ is equal to the rank of $x$; and (ii) $G$ is PSD, i.e. ${{y}^{\top}} G y\ge 0$ for all $y\in\mathbb{R}^n$. For simplicity, we only prove these properties in the case when $x=(x_1,\ldots,x_n)$ is a $1\times n$ matrix, i.e. $x\in\mathbb{R}^n$, and $x_i$ is a scalar for all $i\le n$ (this is the case $r=1$ in Schoenberg’s problem above).
- The $i$-th column of $G$ is the vector $x$ multiplied by the scalar $x_i$, which means that every column of $G$ is a scalar multiple of a single column vector, and hence that $\mbox{\sf
rk}\,G=1$;
- For any vector $y$, ${{y}^{\top}} G
y={{y}^{\top}}(x{{x}^{\top}}) y = ({{y}^{\top}} x)({{x}^{\top}}
y)=({{x}^{\top}} y)^2\ge 0$.
Moreover, $G$ is a Gram matrix [*only if*]{} it is PSD. Let $M$ be a PSD matrix. By spectral decomposition there is a unitary matrix $Y$ such that $M=Y \Lambda {{Y}^{\top}}$, where $\Lambda$ is diagonal. By positive semidefiniteness, $\Lambda_{ii}\ge 0$ for each $i$, so $\sqrt{Y\Lambda}$ exists. Hence $M=\sqrt{Y\Lambda}{{(\sqrt{Y\Lambda})}^{\top}}$, which makes $M$ the Gram matrix of the vector $\sqrt{Y\Lambda}$. This concludes the proof of Thm. \[t:schoenberg\].
Finding the realization of a Gramian
------------------------------------
Having computed the Gram matrix $G$ from the EDM $D$ in Sect. \[s:schthm\], we obtain the corresponding realization $x$ as follows. This is essentially the same reasoning used above to show the equivalence of Gramians and PSD matrices, but we give a few more details.
Let $\Lambda=\mbox{\sf diag}(\lambda_1,\ldots,\lambda_r)$ be the $r\times r$ matrix with the eigenvalues $\lambda_1\ge\ldots\ge\lambda_r$ along the diagonal and zeroes everywhere else, and let $Y$ be the $n\times r$ matrix having the eigenvector corresponding to the eigenvalue $\lambda_j$ as its $j$-th column (for $j\le r$), chosen so that $Y$ consists of orthogonal columns. Then $G=Y\Lambda{{Y}^{\top}}$. Since $\Lambda$ is a diagonal matrix and all its diagonal entries are nonnegative (by positive semidefiniteness of $G$), we can write $\Lambda$ as $\sqrt{\Lambda}\sqrt{\Lambda}$, where $\sqrt{\Lambda}=\mbox{\sf
diag}(\sqrt{\lambda_1},\ldots,\sqrt{\lambda_r})$. Now, since $G=x{{x}^{\top}}$, $$x{{x}^{\top}} =
(Y\sqrt{\Lambda})(\sqrt{\Lambda}{{Y}^{\top}}),$$ which implies that $$x = Y\sqrt{\Lambda} \label{eq:mds13}$$ is a realization of $G$ in $\mathbb{R}^r$.
Multidimensional Scaling
------------------------
MDS can be used to find realizations of approximate distance matrices $\tilde{D}$. As above, we compute $\tilde{G}=-\frac{1}{2} J\tilde{D}^2
J$. Since $\tilde{D}$ is not a EDM, $\tilde{G}$ will probably fail to be a Gram matrix, and as such might have negative eigenvalues. But it suffices to let $Y$ be the eigenvectors corresponding to the $H$ positive eigenvalues $\lambda_1,\ldots,\lambda_H$, to recover an approximate realization $x$ of $\tilde{D}$ in $\mathbb{R}^H$.
Another interesting feature of MDS is that the dimensionality $H$ of the ambient space of $x$ is actually determined by $D$ (or $\tilde{D}$) rather than given as a problem input. In other words, MDS finds the “inherent dimensionality” of a set of (approximate) pairwise distances.
Conclusion {#s:conc}
==========
We presented what we feel are the most important and/or beautiful theorems in DG (Heron’s, Cauchy’s, Cayley’s, Menger’s, Gödel’s and Schoenberg’s). Three of them (Heron’s, Cayley’s, Menger’s) have to do with the volume of simplices given its side lengths, which appears to be [*the*]{} central concept in DG. We think Cauchy’s proof is as beautiful as a piece of classical art, whereas Gödel’s proof, though less important, is stunning. Last but not least, Schoenberg’s theorem is the fundamental link between the history of DG and its contemporary treatment.
Acknowledgments {#acknowledgments .unnumbered}
===============
The first author (LL) worked on this paper whilst working at IBM TJ Watson Research Center, and is very grateful to IBM for the freedom he was afforded. The second author (CL) is grateful to the Brazilian research agencies FAPESP and CNPq.
[^1]: The first public mention of Gödel’s completeness theorem [@goedel_compl] (which was also the subject of his Ph.D. thesis) was given at the [ *Kolloquium*]{} [@mengerK 14 May 1930, p. 135], just three months after obtaining his doctorate from the University of Vienna. As for his incompleteness theorem [@goedel_inc], F. Alt recalls [@mengerK Afterword] that Gödel’s seminar [@mengerK 22 Jan. 1931, p. 168] appears to have been the first oral presentation of its proof:
> There was the unforgettable quiet after Gödel’s presentation, ended by what must be the understatement of the century: “That is very interesting. You should publish that.” Then a question: “You use Peano’s system of axioms. Will it work for other systems?” Gödel, after a few seconds of thought: “Yes, any system broad enough to define the field of integers.” Olga Taussky (half-smiling): “The integers do not constitute a field!” Gödel, who knew this as well as anyone, and had only spoken carelessly: “Well, the…the…the domain of integrity of the integers.” And final relaxing laughter.
The incompleteness theorem was first mentioned by Gödel during a meeting in Königsberg, in Sept. 1930. Menger, who was travelling, had been notified immediately: with John Von Neumann, he was one of the first to realize the importance of Gödel’s result, and began lecturing about it immediately [@mengerK Biographical introduction].
[^2]: A polyhedron is generic if no algebraic relations on $\mathbb{Q}$ hold on the components of the vectors which represent its vertices.
[^3]: <http://jwilson.coe.uga.edu/emt668/emat6680.2000/umberger/MATH7200/HeronFormulaProject/finalproject.html>.
[^4]: <https://math.dartmouth.edu/~doyle/docs/heron/heron.txt>.
[^5]: <http://www.artofproblemsolving.com/Resources/Papers/Heron.pdf>.
[^6]: Also see <http://newsinfo.iu.edu/news/page/normal/13885.html> and <http://www.jstor.org/stable/10.4169/amer.math.monthly.121.02.149> for more recent career achievements of this gifted student.
[^7]: This definition is different from the usual definition employed in convex analysis, i.e. that a polyhedron is an intersection of half-spaces; however, a convex polyhedron in the sense given here is the same as a polytope in the sense of convex analysis.
[^8]: In fact Cauchy’s proof contained two mistakes, corrected by Steinitz [@lyusternik p. 67] and Lebesgue.
[^9]: Two half-planes in $\mathbb{R}^3$ intersecting on a line $L$ define an angle smaller than $\pi$ called the [*dihedral angle*]{} at $L$.
[^10]: This statement was also proved in Cauchy’s paper [@cauchyrigid], but this proof contained a serious flaw, later corrected by Steinitz.
[^11]: See \[fn:steffen\] <http://demonstrations.wolfram.com/SteffensFlexiblePolyhedron/>.
[^12]: Interestingly, Gödel’s famous incompleteness theorem is also a fixed-point argument (in a much more complicated set).
[^13]: A not altogether dissimilar situation arose for the Johnson-Lindenstrauss (JL) lemma [@jllemma]: the paper is concerned with extending a mapping from $n$-point subsets of a metric space to the whole metric space in such a way that the Lipschitz constant of the extension is bounded by at most a constant factor. Johnson and Lindenstrauss state on page 1 that “The main tool for proving Theorem 1 is a simply stated elementary geometric lemma”. This lemma is now known as the [*JL lemma*]{}, and postulates the existence of low-distortion projection matrices which map to Euclidean spaces of logarithmically fewer dimensions. The impact of the lemma far exceeds that of the main result.
|
Sunderland ace Steven Fletcher welcomed back to Scotland squad
GRAEME ANDERSON
STOKE City midfielder Charlie Adam says there will be no resentment towards Steven Fletcher inside the Scotland camp after the Sunderland striker’s return to the international fold.
Fletcher had a very public falling out with Scotland manager Craig Levein, which was finally settled last week, allowing the Black Cats forward the opportunity to try to salvage Scotland’s floundering World Cup qualifying campaign.
And former Blackpool and Liverpool midfielder Adam said the importance of the matches ahead far outweighs any lingering interest in the spat between manager and striker.
“You want your best players available, and he is one of our best players, so it is good to see him back involved,” said Adam.
“What happened has happened, it’s done and dusted. Let’s move on.
“It’s up to the manager if he plays, but for us it’s good to have him in the squad because he adds to the quality that we already have.
“Whoever plays up front as a pair or on their own they will be trying to do the best for their country.
“We are all in the same boat – we want to get to Brazil, and hopefully the quality that has come into the squad for these two games can enhance our chances.”
Fletcher is rooming with ex-Hibs team-mate Scott Brown, who was glad to see his old friend back in the international fold.
“It is brilliant,” said the Celtic skipper.“He’s a great guy and a good friend as well, so I am glad he is back. He’s a good lad to have around, so we will accept him no problem.
“He didn’t say a lot. He just said he is looking forward to getting the strip back on and getting back into training, and pretty much forget about everything that has happened in the last couple of years.”
Fletcher himself is just looking forward to the job of improving Scotland’s qualification hopes.
He said: “I am desperate to play for Scotland again and, like Craig Levein, I regret that it has taken this long to come about.
“I’m sure we have both done a lot of thinking during the last two years, but the country comes first.
“I think it just became apparent that we all needed to move on. The most important thing is that I can look to restart my international career.
“I am a proud Scot, and it is a privilege to play for my country.
“When I first spoke to Craig again it was fine, and was not awkward at all. We both said our piece and agreed that whatever happened, the country comes first.
“I had a good chat with the manager when he came to visit at the training ground last week and, while it was important we discussed the past and put it to bed, it was more important that we focused on the future.
“I understand why people want to know what changed, but the truth, as the manager said last week, was that the situation wasn’t helping anyone.
“I am just happy to be back as part of the squad, and looking to make a positive contribution in what are two massive World Cup qualifiers.
“I am a more experienced player than I was two years ago, and I am delighted with the way things have gone at Sunderland.
“Martin O’Neill has been very supportive, as has Phil Bardsley –who is gutted to be missing through injury.” |
Reflections on Independence and Voting
On 19 February, 2013, two weeks after Grenadians would have celebrated 39 years of independence, we will exercise our right to vote for a party to form the government for the next five years.
Every election is important; however, this election comes at a pivotal moment. The outcome of the election could determine the direction of the country at a time of regional and global uncertainty.
Beyond the necessary ritual of colourful campaign rallies and party jingles, we are about to participate in a very important process that can affect our lives in direct ways for many years to come.
Often we take the right to vote for granted but it represents a long historical struggle for freedom. To refuse to vote is to violate the sacrifices made by those who paved the way so we could enjoy basic rights today.
Let us never take the right to vote for granted. In fact, the spirit of independence should inspire us to recommit ourselves to nationhood as we vote. The act of voting should be done soberly. It should be driven by a deep sense of responsibility to both party and country.
We should use our votes to fight for jobs and bread and butter issues. However, in addition, in order to sustain genuine independence, a conscious people must think beyond where they are now and envision the future they wish to create.
The vote should also be for improved educational opportunities as we continue to break the shackles of poverty and dependency, to provide hope for the young and yet unborn.
The vote should be for improved and sustained health care, especially for the least among us. We should vote to ensure every Grenadian woman, man and child enjoy safety in the home and in the public space.
We should vote to ensure that Grenada’s natural resources, in as far as is possible are owned and controlled by Grenadians. We should vote to ensure that Grenada conducts its foreign policy strategically and with dignity as it defines and presents itself to the region and the wider world.
In essence, as we enter those polling booths, at a moment of regional and global UNCERTAINTY, the vote should be for political leadership that can CAUTIOUSLY balance job creation, long-term economic transformation and overall societal well-being, while preserving our natural resources and protecting and enhancing Grenada’s independence and sovereignty.
In February 1974, Grenadians embarked on a journey for freedom in the pursuit of our collective prosperity and common destiny. However, while we have made tremendous strides as a nation, the experience has not always been what we had anticipated: there were some missteps, shortsightedness and collective impatience.
Consequently, after almost 40 years, the promise of independence needs renewal. Therefore, this election should represent a moment of deep reflection on our collective journey and a recommitment to nationhood.
(Dr. Wendy Grenade is a Grenadian who lectures in Political Science at the University of the West Indies, Cave Hill Campus, Barbados) |
Q:
How to play a sound just one time SWIFT
I'm having a little problem with my code who consists to play a sound when I start my app. But here's the problem everytime that I go back to the first screen the sound is playing again and I want it to play just one time. When the menu screen pop ups for the fist time.
Here's my code
var bubbleSound: SystemSoundID!
bubbleSound = createBubbleSound()
AudioServicesPlaySystemSound(bubbleSound)
(...)
the function
func createBubbleSound() -> SystemSoundID {
var soundID: SystemSoundID = 0
let soundURL = CFBundleCopyResourceURL(CFBundleGetMainBundle(), "bubble", "wav", nil)
AudioServicesCreateSystemSoundID(soundURL, &soundID)
return soundID
}
A:
You can define a struct like this (source):
struct MyViewState {
static var hasPlayedSound = false
}
Then in your viewDidLoad:
if(!MyViewState.hasPlayedSound) {
var bubbleSound: SystemSoundID!
bubbleSound = createBubbleSound()
AudioServicesPlaySystemSound(bubbleSound)
MyViewState.hasPlayedSound = true
}
You can then modify MyViewState.hasPlayedSound and allow the UIViewController to play the sound again if desired.
|
David Rabeeya
David Rabeeya (born May 12, 1938) is an Israeli and American author and professor of Hebrew and Judaic Studies.
David Rabeeya is an Iraqi Jew born in Baghdad, Iraq. He and his family later moved to Israel in about 1951. In about 1970, he moved to the United States. Over the years, Dr. Rabeeya has taught countless students, in all schooling environments-high school, university, and elementary school. He has grown to accept all different cultures and religions over his lifetime, as an Arab Jew. A prolific author, Dr. Rabeeya has written many books on many subjects, but they tend to focus on the Middle East and the relationship between Jews and other groups. He currently works as a teacher for Middle and High School students.
Books
America: Criticize It But Stay
The Journey of an Arab-Jew : Through the American Maze
Baghdadi Treasures : Challenging Ideas & Humorous Sayings
Israel: Stripped Bare
Women's Struggles; Women's Dreams
Rabeeya's Reflections: Love, Sex and Wit
Sephardic Lolita: Judeo-Arabic Restoration And Reconciliation
Fruma: Caught in Her Web
A Humanistic Siddur of Spirituality And Meaning: The American Character; We Rationalize Everything
1,001 Jokes About Rabbis : And The Rest Of The World
Afifah: A Bedouin Odyssey
Fundamentalism : Roots, Causes and Implications
Zionism: Final Call
Homosexuals under Sharia Law
Visionary Memoir
Quarter in Half Time
The Journey of an Arab-Jew in European Israel
A Guide to Understanding Judaism and Islam : More Similarities Than differences
Sephardic Recipes : Delicacies from Baghdad
References
Category:1938 births
Category:Living people
Category:American people of Iraqi-Jewish descent
Category:Israeli people of Iraqi-Jewish descent
Category:Writers from Baghdad
Category:Iraqi emigrants to Israel
Category:Israeli Arab Jews
Category:Hebrew University of Jerusalem alumni
Category:Israeli emigrants to the United States
Category:American non-fiction writers
Category:American Reform rabbis
Category:20th-century rabbis
Category:21st-century rabbis
Category:Bryn Mawr College faculty |
A comparison of geostationary-orbiting satellite IR images and polar-orbiting microwave images (from the CIMSS Tropical Cyclones site) for each of the 3 tropical cyclones are shown below. Note that there is a 1-2 hour difference between the IR images and the microwave images — however, these comparisons show the utility of the microwave images for showing tropical cyclone structures that are often masked by the cold convective cloud shield.
Tropical Storm Karl: geostationary IR image + polar microwave image
Hurricane Igor: geostationary IR image + polar microwave image
Hurricane Julia: geostationary IR image + polar microwave image
An AWIPS image of EUMETSAT METOP Advanced Scatterometer (ASCAT) winds (below) indicated surface winds as high as 63 knots near the center of Hurricane Igor at 13:28 UTC; however, ASCAT winds are known to have a low speed bias (which increases as winds get to higher speeds).
EUMETSAT METOP ASCAT winds
As part of the GOES-15 Post Launch Science Test, the satellite was placed into Rapid Scan Operations (RSO) mode, providing images as frequently as every 5 minutes during the day. The evolution of the eye of Hurricane Igor is seen on GOES-15 0.63 µm visible channel images (below; also available as a QuickTime movie) — note the occasional presence of small mesovortices within the eye region. |
This series includes but may not be limited to photocopies of Travel Expense Vouchers sent to Accounting Services, receiving reports, photocopies of invoices sent to Accounting Services1 NCR copies of Internal Transactions sent to Accounting Services, and photocopies of monthly statements received from Accounting Services.
RETENTION: Retain 2 years in off ice, then 5 years in Records Center, then destroy by shredding then landfill.
ANNUAL REPORTS
This series contains photocopies of departmental annual reports
sent to Office of the Dean.
RETENTION: Retain 3 years in office, then destroy by recycling.
BUDGET FILES
This series includes but may not be limited to ledger sheets, NCR copies of Internal Transactions sent to Accounting Services, file notes, photocopies of invoices, photocopies of receipts, NCR copies of requisitions sent to Budget and Planning, photocopies of Requests for Budget Adjustment sent to Office of the Dean, and computer-generated Invoices Paid During the Period Reports received from Accounting Services.
RETENTION: Retain 3 years in office, then destroy by landfill.
CHAIRMAN WORKING FILES
This series contains campus and off-campus correspondence, photocopies of news releases, budget worksheets, photocopies of budgets sent to Office of the Dean, and file notes.
RETENTION: Correspondence: Retain 5 years in office, then destroy by shredding then landfill. Budget records: Retain 3 years in office, then destroy by recycling. All other contents:
Retain in office while active, then destroy by recycling.
COURSE AND FACULTY SCHEDULING FILES
This series includes but may not be limited to computer-generated Class Schedule Reports received from Student Records and Registration, photocopies of campus correspondence, file notes, course evaluation reports, statistical worksheets, and computer-generated Class Management Reports received from Student Records and Registration.
RETENTION: Retain in office 1 year after term distributed, then destroy by shredding then landfill.
GRADUATE COURSE MANAGEMENT FILES
This series includes but may not be limited to computer-generated
Course Offering Reports received from Student Records and
Registration, campus correspondence, and NCR copies of Request
for Dropping or Adding Sections sent to Student Records and
Registration.
RETENTION: Retain 3 years in office, then destroy by landfill.
PHILOSOPHY COURSE MANAGEMENT FILES
This series includes but may not be limited to computer-generated
Course Offering Reports received from Student Records and
Registration and NCR copies of Request for Dropping or Adding
Sections sent to Student Records and Registration.
RETENTION: Retain 3 years in office, then destroy by landfill.
RELIGIOUS STUDIES COURSE MANAGEMENT FILES
This series includes but may not be limited to computer-generated
Course Offering Reports received from Student Records and
Registration and NCR copies of Request for Dropping or Adding
Sections sent to Student Records and Registration.
RETENTION: Retain 3 years in office, then destroy by landfill.
ELECTRONIC SPREADSHEETS
This series contains spreadsheets that are recorded on electronic media such as hard disks or floppy diskettes: a) when used to produce hard copy that is maintained in organized files or b) when maintained only in electronic form.
RETENTION: a) When used to produce hard copy that is maintained in organized files: Delete when no longer needed to update or produce hard copy. b) When maintained only in electronic form:
Delete after the expiration of the retention period authorized for the hard copy by the LSU Retention and Disposal Schedule. If the electronic version replaces hard copy records with differing retention periods, and agency software does not readily permit selective deletion, delete after the longest retention period has expired.
EQUIPMENT FILES
This series includes but may not be limited to campus correspondence, security reports received from Campus Police, code number lists, photocopies of invoices, photocopies of inventory forms, equipment location sheets, NCR copies of Work Requests sent to Facility Services, Travel Blanket Orders, copies of Receiving Reports sent to Accounting Services, requisitions, NCR copies of Internal Transactions, and computer-generated ledger sheets received from Accounting Services.
RETENTION: Retain 2 years in office, then 5 years in Records Center, then destroy by landfill.
FACULTY APPLICATIONS FILES
This series contains employment applications and vitae received
from off campus.
RETENTION: Retain 1 year in office, then destroy by landfill.
PHILOSOPHY FACULTY PERSONNEL FILES
This series includes but may not be limited to teacher evaluations received from Office of the Dean, personnel appointment forms, PER-3 Personnel Status Changes continuation and separation forms, PER-ls, and PER-25s sent to Office of the Dean.
RETENTION: Retain in office 2 years after termination or until final disposition of charge or civil action, then destroy by shredding then landfill.
RELIGIOUS STUDIES FACULTY PERSONNEL FILES
This series includes but may not be limited to teacher evaluations received from Office of the Dean, personnel appointment forms, PER-3 Personnel Status Changes, continuation and separation forms, PER-ls, and PER-25s sent to Office of the Dean.
RETENTION: Retain in office 2 years after termination or until final disposition of charge or civil action, then destroy by shredding then landfill.
FACULTY PROMOTION AND TENURE FILES
This series includes but may not be limited to letters of endorsement, faculty appointment forms, faculty meeting minutes, curricula vitae, photocopies of faculty grant summary statements, and photocopies of faculty publications.
RETENTION: Retain in office 2 years after termination or until final disposition of charge or civil action, then destroy by shredding then landfill.
FACULTY PUBLICATIONS FILES
This series includes but may not be limited to curricula vitae, photocopies of faculty grant summary statements, and photocopies of faculty publications.
RETENTION: Retain in office 2 years after termination or until final disposition of charge or civil action, then destroy by shredding then landfill.
FACULTY RECRUITMENT FILES
This series includes but may not be limited to file notes, NCR copies of PER-l Requests for Permission to Fill a Vacancy sent to Human Resource Management, photocopies of job notices sent to Human Resource Management, off-campus correspondence, and letters of recommendation received from off campus.
RETENTION: Retain 3 years in office, then destroy by landfill.
GRADE BOOKS
This series contains grade books with student names and grades.
RETENTION: Retain in office 1 year after term submitted, then
destroy by shredding then landfill.
GRANTS AND CONTRACTS FILES
This series includes but may not be limited to photocopies of grant applications and photocopies of grant reports.
RETENTION: Retain in office 3 years after termination of contract, then destroy by shredding then recycling.
HONOR UNDERGRADUATE FILES
This series contains photocopies of transcripts, photocopies of test scores, and photocopies of applications sent to Admissions.
RETENTION: Retain in office 5 years after graduation term or term of last attendance, then destroy by shredding then landfill.
LSU NEAR-PRINT INFORMATIONAL MATERIAL
This series contains campus near-print informational material.
RETENTION: Retain in office while active or until superseded, then destroy by landfill.
MASTER FORMS
This series contains master forms.
RETENTION: Record (master) copy: Retain permanently in office.
Duplicates: Retain in office while active, then destroy by
landfill.
MIDTERM AND FINAL GRADE SHEETS
This series contains carbon copies of the following sent to
Student Records and Registration: Midterm Grade Sheets and Final
Grade Sheets.
RETENTION: Retain in office 1 year after term distributed, then
destroy by shredding then landfill.
MISCELLANEOUS FILES
This series contains campus and off-campus correspondence, photocopies of minutes of meetings, campus and off-campus near-print informational material, audit worksheets, NCR copies of International Transactions sent to Accounting Services, job applicant letters of reference, job applicant resumes, photocopies of PER-l Requests for Permission to Fill a Vacancy sent to Office of the Dean, student applications, visiting lectures applications, grade books, file notes, photocopies of Class Lists received from Student Record and Registration, GRE reports received from off-campus, and course schedules received from Student Records and Registration.
RETENTION: Correspondence: Retain 5 years in office, then destroy by shredding then landfill. All other contents: Retain 3 years in office, then destroy by landfill.
PHILOSOPHY UNDERGRADUATE STUDENT FILES
This series contains Degree Audit reports received from Office of the Dean, grade reports sent to Student Records and Registration, and file notes.
RETENTION: Retain in office 5 years after graduation term or term of last attendance, then destroy by shredding then landfill.
SOCIAL SECURITY NUMBER LIST
This series contains list of faculty and staff social security numbers.
RETENTION: Retain in office until superseded, then destroy by shredding then landfill.
STUDENT ASSISTANTSHIP FILES
This series includes but may not be limited to photocopies of the following sent to Student Aid and Scholarships: Form 1-9 Employee Eligibility Verifications, W-2, Loyalty Oaths, and TADs.
RETENTION: Retain in office 3 years after termination or until final disposition of charge or civil action, then destroy by shredding then landfill.
STAFF LEAVE FILES
This series includes but may not be limited to campus correspondence, computer-generated monthly PARS leave reports received from Human Resource Management, leave slips, and annual leave summaries received from Human Resource Management.
RETENTION: Retain in office 2 years after termination or until final disposition of charge or civil action, then destroy by shredding then landfill.
GRADUATE STUDENT APPLICATIONS FILES
This series contains prospective-student evaluations, off-campus correspondence, GRE reports, and transcripts received from off campus.
RETENTION: Graduate students accepted: Retain in office 5 years after graduation term or term of last attendance, then destroy by shredding then landfill. Students not accepted: Retain in office 1 year after application term, then destroy by shredding then landfill.
GRADUATE STUDENT FILES
This series includes but may not be limited to GRE reports, grade sheets, photocopies of academic probation notices received from Office of the Dean, transcripts, letters of recommendation, photocopies of final examinations, requests for examination, and campus and off-campus correspondence.
RETENTION: Retain in office 5 years after graduation term or term of last attendance, then destroy by shredding then landfill.
STUDENT ASSISTANT PAYROLL ACCOUNTING FILES
This series contains computer-generated budget sheets received from Accounting Services, Payroll Check Distribution Reports received from Accounting Services, student timesheets and photocopies of Expenditure Lists sent to Accounting Services.
RETENTION: Retain 3 years in office, then destroy by landfill.
TEXTBOOK ORDERS FILES
This series contains photocopies of book information requests sent to LSU Union Bookstore, book order worksheets, and campus correspondence.
RETENTION: Retain 3 years in office, then destroy by landfill.
RELIGIOUS STUDIES UNDERGRADUATE STUDENT FILES
This series contains Degree Audit reports received from Office of the Dean, grade reports sent to Student Records and Registration, and file notes.
RETENTION: Retain in office 5 years after graduation term or term of last attendance, then destroy by shredding then landfill.
VISA FILES
This series contains campus and off-campus correspondence and photocopies of ETA 750s sent to Human Resource Management.
RETENTION: Retain 5 years in office, then destroy by shredding then recycling.
WORD PROCESSING FILES
This series contains documents such as letters, messages, campus correspondence, reports, handbooks, and manuals recorded on electronic media such as hard disks or floppy diskettes a) when they are used to produce hard copy that is maintained in organized files or b) when they are maintained only in electronic form and they duplicate the information in and take the place of records that would otherwise be maintained in hard copy, provided that the hard copy has been authorized for destruction by the LSU Retention and Disposal Schedule.
RETENTION: a) When they are used to produce hard copy that is maintained in organized files: Delete when no longer needed to create a hard copy. b) When they are maintained only in electronic form and they duplicate the information in and take the place of records that would otherwise be maintained in hard copy, provided that the hard copy has been authorized for destruction by the LSU Retention and Disposal Schedule: Delete after the expiration of the retention period authorized for the hard copy by the LSU Retention and Disposal Schedule. |
A: Effective communication with patients, the dental team and others across dentistry, including when obtaining consent, dealing with complaints, and raising concerns when patients are at risk.B: Effective management of self and effective management of others or effective work with others in the dental team, in the interests of patients; providing constructive leadership where appropriate.C: Maintenance and development of knowledge and skill within your field of practice.
Professor of Psychology as Applied to DentistryKing's College LondonTim Newton is Professor of Psychology as Applied to Dentistry at King’s College London Dental Institute. He also holds Honorary Consultant Health Psychologist positions with Guy’s & St Thomas’ NHS Foundation Trust and King’s College Hospital NHS Foundation Trust. Tim has worked in the behavioural sciences in relation to dentistry for over 20 years, and his particular interests include the management of dental anxiety, the working life of the dental team and patients’ perceptions of treatment. He has a strong commitment to ensuring that research is of the highest scientific and ethical standards in protection of the rights of research participants and the wider public served by the research community. In pursuance of this he is Chair of the King’s College London Research Ethics Committee.
In 2016 Tim was awarded the Behavioural, Epidemiological and Health Services Research Distinguished Scientist Award by the International Association of Dental Research. |
Pages
Saturday, September 24, 2011
One Stop Craft Challenge - Easel Cards
At OSCC the newest challenge is Easel Cards....
Our sponsor for this challenge is:
Prize: 10.00 GC
My DT card for this challenge is:
The easel card I found is not the easiest card too make. But I am sure given time it will become an easier project to make. I used the cricut to make the frame using Happy Haunting Cartridge, the tree, house and ghost are from Quickutz dies. |
Cerebellar atrophy diagnosed by computed tomography and clinical data.
The diagnostic relevance of computed tomography (CT) in the classification of cerebellar atrophy or degeneration is unclear. Twenty-one patients with cerebellar atrophy at CT were studied and the findings were correlated to clinical data. Based upon such data the material was divided into two groups. In the first group (12 patients), with signs of cerebellar deficiency, 6 cases presented familial hereditary ataxia, and olivopontocerebellar atrophy of the Menzl type, 3 ataxia telangiectasia (Louis-Bar syndrome), 2 olivopontocerebellar atrophy of the sporadic type, and 1 adrenoleukodystrophy. In the second group (9 patients), without cerebellar deficit, cerebellar atrophy was found only occasionally. In all of them, there was cerebral atrophy. Clinically manifest cerebellar deficiency and cerebellar atrophy as evident at CT was mainly found in patients with familial genetic disorders. Cerebellar and/or vermian atrophy without clinical signs of cerebellar deficiency were observed only occasionally and were not specific. |
The iScore predicts total healthcare costs early after hospitalization for an acute ischemic stroke.
The ischemic Stroke risk score is a validated prognostic score which can be used by clinicians to estimate patient outcomes after the occurrence of an acute ischemic stroke. In this study, we examined the association between the ischemic Stroke risk score and patients' 30-day, one-year, and two-year healthcare costs from the perspective of a third party healthcare payer. Patients who had an acute ischemic stroke were identified from the Registry of Canadian Stroke Network. The 30-day ischemic Stroke risk score prognostic score was determined for each patient. Direct healthcare costs at each time point were determined using administrative databases in the province of Ontario. Unadjusted mean and the impact of a 10-point increase ischemic Stroke risk score and a patient's risk of death or disability on total cost were determined. There were 12,686 patients eligible for the study. Total unadjusted mean costs were greatest among patients at high risk. When adjusting for patient characteristics, a 10-point increase in the ischemic Stroke risk score was associated with 8%, 7%, and 4% increase in total costs at 30 days, one-year, and two-years. The same increase was found to impact patients at low, medium, and high risk differently. When adjusting for patient characteristics, patients in the high-risk group had the highest total costs at 30 days, while patients at medium risk had the highest costs at both one and two-years. The ischemic Stroke risk score can be useful as a predictor of healthcare utilization and costs early after hospitalization for an acute ischemic stroke. |
---
abstract: 'A description of an edge-biasing experiment conducted on the SPECTOR plasma injector is presented, along with initial results. The insertion of a disc-shaped molybdenum electrode (probe), biased at up to $+100$V, into the edge of the CT, resulted in up to 1kA radial current being drawn. Core electron temperature, as measured with a Thomson-scattering diagnostic, was found to increase by a factor of up to 2.4 in the optimal configuration tested. $\mbox{H}_{\alpha}$ intensity was observed to decrease, and CT lifetimes increased by a factor of up to 2.3. A significant reduction in electron density was observed; this is thought to be due to the effect of a transport barrier impeding CT fueling, where, as verified by MHD simulation, the fueling source is neutral gas that remains concentrated around the gas valves after CT formation.'
author:
- |
Carl Dunlea$^{1*}$, General Fusion Team$^{2}$,\
Chijin Xiao$^{1}$, and Akira Hirose$^{1}$
title: |
First results from plasma edge biasing on SPECTOR\
[ ]{}
---
$^{1}$University of Saskatchewan, Saskatoon, Canada
$^{2}$General Fusion, Vancouver, Canada
$^{*}$e-mail: cpd716@mail.usask.ca
Introduction
============
High confinement mode (H-mode) has been implemented by various means ($e.g.,$ edge biasing, neutral beams, ion or electron cyclotron heating, lower hybrid heating, and ohmic heating) on a range of magnetic confinement configurations including tokamaks, reversed field pinches, stellarators, and mirror machines. The first H-mode was produced in the ASDEX tokamak by neutral beam injection in 1982 [@wagnerASDEX]. In 1989, H-mode was first produced by electrode edge biasing on the CCT tokamak [@Taylor_CCT; @WEynants_Taylor_CCT]. In 1990, it was observed that edge impurity ion poloidal speed is modified abruptly during transitions from low to high confinement modes on the DIII-D tokamak [@Groebner]. H-mode has been produced routinely on many magnetic-fusion experiments, including practically all the large tokamaks including JET, TFTR, and JT-60. Since the initial electrode-biasing experiments on CCT, H-mode has been produced by edge biasing on many tokamaks, for example CASTOR [@CAstor; @OOst], T-10 [@OOst; @T10], STOR-M [@StorM], ISTTOK [@Figueiredo_ISTTOK], TEXTOR [@OOst; @Jachmich], and J-TEXT [@JTEXT].
Electrode biasing involves the insertion of an electrode, that is biased relative to the vessel wall near the point of insertion, into the edge of a magnetized plasma. This leads to a radially directed electric field between the probe and the wall. The resultant $\mathbf{J}_{r}\times\mathbf{B}$ force imposed on the plasma at the edge of the plasma confinement region varies with distance between the probe and the wall, because $E_{r}$, as well as the magnetic field, vary in that region. The associated torque overcomes viscous forces, spinning up the edge plasma, and results in shearing of the particle velocities between the probe and the wall. The sheared velocity profile is thought to suppress the growth of turbulent eddies that advect hot plasma particles to the wall, thereby reducing this plasma cooling mechanism. In general, H-modes induced by probe biasing share features of those initiated by various methods of heating, including a density pedestal near the wall (near the probe radius for probe biasing), diminished levels of recycling as evidenced by reduced $\mbox{H}_{\alpha}$ emission intensity, and increased particle and energy confinement times. For example, increases in energy confinement times by factors of 1.5, 1.5, 1.2, and 1.8 were reported for CCT, STOR-M, TEXTOR and T-10 respectively. Core electron density increased by a factor of four on CCT, while line-averaged electron density increased by factors of 2, 2, 1.5, and 1.8 on STOR-M, TEXTOR, T-10, and CASTOR respectively. Of these five examples, a biasing-induced temperature increase was noted only for the T-10 experiment, with an increase in core ion temperature by a factor of 1.4 reported, while reduced $\mbox{H}_{\alpha}$ emission intensity was recorded in each case.
Positive as well as negative electrode biasing works well on some machines; in other instances only one biasing polarity has the desired effect. Most biasing experiments have used passive electrodes, while some have implemented electron-emitting electrodes. Emissive electrodes have, in addition to a circuit to bias the electrode relative to the vacuum vessel, a separate heating circuit, and are heated until they emit electrons. Materials traditionally used for emissive electrodes include lanthanum hexaboride (LaB6) and tungsten (W). Generally speaking, emissive electrodes add complexity to an experiment, but may be beneficial when the edge plasma electron density is so low that dangerously high voltages (which could initiate a current arc that could damage the electrode and vessel) would be required in order to draw an edge current sufficiently high enough for the $\mathbf{J}_{r}\times\mathbf{B}$ force to overcome inertial effects (viscosity, friction) and drive edge rotation. In the CCT tokamak [@Taylor_CCT], LaB6 cathodes heated by carbon rods drew edge current up to $40\mbox{A}$ when the voltage measured between the electrode (probe) and vessel wall was $V_{probe}\sim-250\mbox{V}$. On CCT, for negative bias, it was found that both electron-emissive electrodes and passive graphite electrodes produced similar results, as long as the electrode was large enough to draw sufficient current ($\sim20\mbox{A}$), and small enough not to form a limiter [@Taylor_CCT]. For negative biasing on ISTTOK, it was not possible to draw more than $2\mbox{ to }3\mbox{A}$ with a passive electrode, $cf.\sim20\mbox{A}$ with an emissive electrode, while the current drawn with positive biasing was the same for emissive and non-emissive electrode ($I_{probe}\sim28\mbox{A}$ at $V_{probe}\sim+130\mbox{V}$).
Pre-biasing conditions of radial electric field and an extensive range of plasma parameters play roles in determining the beneficial polarity and the level of bias-induced plasma confinement improvement [@Tendler]. A reduction of radial transport at the edge would be beneficial for confinement not only because of reduced outward thermal transport, but also due to reduced inward transport of cold wall-recycled particles to the core. This latter effect is especially relevant on small machines (such as SPECTOR, $R\sim11\mbox{cm},\,a\sim8\mbox{cm}$, no limiter or divertor) for which the surface area to volume ratio of the magnetically confined plasma is large, particularly in configurations without a limiter or divertor, where the recycling process is more important.
An overview of the experiment setup with a description of the biasing electrode assembly is presented in section \[sec:Experiment-setup\]. Circuit analysis, leading to an estimate for the resistance of the plasma between the electrode and flux conserver, which was useful for optimising the circuit, is the focus of section \[sec:Circuit-analysis\]. Main results are presented in section \[sec:Main-results\]. A discussion of principal findings, conclusions and possible further improvements to the experiment is presented in section \[sec:Discussion-and-conclusions\]. Results from simulations of neutral and plasma fluid interaction during CT formation in SPECTOR geometry is presented in appendix \[sec:Neutral\_SPECTOR\]. The simulations indicate that neutral gas, which remains concentrated around the locations of the machine gas valves after CT formation, can diffuse up the gun as a source of CT fueling. It may be partly due to the effect of a transport barrier impeding the CT fueling process that the improvements in confinement times and electron temperatures observed with the initial edge biasing tests on SPECTOR appear especially significant.
Experiment setup\[sec:Experiment-setup\]
========================================
A schematic of the SPECTOR [@spectPoster] plasma injector is depicted in figure \[fig:Spector\](a), where the red dots along the flux conserving wall of the CT containment region represent the locations of magnetic probes. SPECTOR is a magnetized Marshall gun that produces compact tori (CTs). It has, in addition to the formation circuit that drives up to $0.8\mbox{MA}$ formation current over around $80\upmu$s, a separate circuit to produce an approximately constant shaft current of up to $0.5\mbox{MA}$, which flows up the outer walls of the machine and down the central shaft, increasing CT toroidal field and making the CT more robust against MHD instability. Shaft current duration is extended to around 3ms with a crowbar inductor/diode circuit, which is indicated schematically in figure \[fig:Spector\](a). Toroidal field at the CT core is typically around $0.5\mbox{T}$. The high CT aspect ratio, and the $q$ profile, define the CTs as spherical tokamaks. Coaxial helicity injection produces plasma currents in the range $300-800\mbox{kA}$. A selection of Thomson-scattering (TS) system-produced electron temperature and electron density measurements [@TS_GF] (both taken at $300$$\upmu\mbox{s}$ after CT formation), electron density measurements obtained with a far-infrared (FIR) interferometer [@polarimetryGF], spectral data, and magnetic probe data, will be presented in the following. Principal diagnostics are indicated in figure \[fig:Spector\](b).
Figure \[fig:bias probe assembly\] indicates a top-view of the electrode (probe) assembly with extendable vacuum bellows. The biasing electrode can be retracted behind the gate valve and isolated from the machine vacuum. The approximately disc-shaped electrode is machined from molybdenum and has a 30mm diameter. Molybdenum was chosen for its high work function against sputtering, high melting point, and its resilience against the corrosive action of lithium, which is used as a gettering agent on SPECTOR. A pyrolytic boron nitride (PBN) tube is used as a plasma-compatible insulator around the M3 stainless steel rod that connects the electrode to a tapered aluminum rod, which is in turn connected to the $0.5"$ diameter copper rod that forms part of the $8\mbox{kV}$ $2.75"$ CF vacuum feedthrough. The electrode can be inserted up to 45mm into the vacuum vessel; insertion depth was 11mm for the results presented here.
Circuit analysis\[sec:Circuit-analysis\]
========================================
Figure \[fig:Biasing-probe-circuit\] indicates the most optimal of the biasing probe circuit configurations tested. The biasing circuit was kept open-circuited until well after CT formation, in order to protect biasing circuit components. A thyratron switch (indicated in the figure) is robust against large amplitude negative voltage spikes that can appear on the probe during CT formation when initially open stuffing-field lines, that are resistively pinned to the injector inner and outer electrodes, intersect the probe (see left subfigure). These thyratron switches are designed to operate at several kilovolts, and usually require several kiloamps of current to remain closed, but careful setting of switch temperature enabled operation at moderate voltages and currents. The biasing capacitor voltage setting $V_{bc0}$, parallel and series resistors $R_{1}$ and $R_{2}$, and $R_{p}$, the plasma resistance between the electrode ($i.e.,$ probe) and flux conserver, determine $V_{probe}$, the voltage measured between the probe and flux conserver, and $I_{probe},$ the radial current drawn through the plasma edge. The radial current leads, in the classical edge biasing scenario, to $\mathbf{J}_{r}\times\mathbf{B}$ driven edge velocity shearing and consequential decorrelation of turbulence cells and confinement improvement. For the circuit with the $3\mbox{mF}$ capacitor depicted in figure \[fig:Biasing-probe-circuit\], optimal circuit resistances were found to be $R_{1}\sim0.2\Omega$, and $R_{2}\sim0.5\Omega$. Negative electrode biasing was briefly tested; the results presented in this paper were obtained with positive biasing. The effective resistance $R_{e}$, comprised of $R_{p}$ and $R_{1}$ in parallel (see figure \[fig:Biasing-probe-circuit\], right subfigure), is given by $$R_{e}(t)=\frac{R_{p}(t)\,R_{1}}{R_{p}(t)+R_{1}}\label{eq:1-1}$$ The voltage applied by the capacitor on the probe is $$V_{applied}(t)=V_{bc}(t)\left(\frac{R_{e}(t)}{R_{e}(t)+R_{2}}\right)\label{eq:2}$$ where $V_{bc}(t)$ is the voltage across the biasing capacitor. Equations \[eq:1-1\] and \[eq:2\] can be combined to provide an expression for $R_{p}$: $$R_{p}(t)=\frac{R_{1}R_{2}V_{applied}(t)}{R_{1}(V_{bc}(t)-V_{applied}(t))-V_{applied}(t)\,R_{2}}\label{eq:3}$$
Figure \[26400\](a) shows the voltage measured between the probe and the vacuum vessel, and the current drawn through the plasma edge, as measured with the Rogowski coil indicated in figure \[fig:Biasing-probe-circuit\], for shot 26400. At the biasing capacitor voltage found to be most optimal for CT lifetime and electron temperature (as obtained with the TS system), the voltage measured between the probe and vacuum vessel was typically $V_{probe}\sim+50\mbox{V}$ to $+80\mbox{V}$, and the maximum radial current drawn to the probe from the wall was $I_{probe}\sim700\mbox{A}$ to $\sim1\mbox{kA}$ shortly after firing the biasing capacitor(s). For shot 26400, the electrode was inserted 11mm into the edge plasma, and biased at $t_{bias}=230\upmu\mbox{s}$ after firing the formation capacitor banks, as indicated in figure \[26400\](a). Note that current is already flowing through the plasma edge, and through resistor $R_{1},$ before $t_{bias}$, as a result of the plasma-imposed potential on the electrode, which typically led to a measurement of $V_{probe}\sim-100\mbox{V}$ when magnetized plasma first enters the CT confinement area at around $20\upmu$s. $V_{probe}$ and $I_{probe}$ decrease over time at a rate that depends on plasma and circuit parameters. Figure \[26400\](b) indicates, for shot 26400, the poloidal field measured at the magnetic probes indicated as red dots in figure \[fig:Spector\](a). It is interesting that the fluctuations in $B_{\theta}$, which are thought to be associated with internal reconnection events, are also manifested on the biasing voltage and current measurements, $e.g.,$ at $\sim845\upmu$s in figures \[26400\](a) and (b). This observation is enabled by the presence of the small parallel $R_{1}$. As edge plasma impedance varies, as determined by internal MHD events, the system can divert varying proportions of capacitor driven current through $R_{1}$. In future studies, it may be possible to influence the behaviour of the internal modes that cause the $B_{\theta}$ fluctuations, by driving an edge current that is resonant with the fluctuations.
When the plasma is considered as a time-dependent voltage source, which biases the probe to floating potential $V_{float}(t)$, a more complete circuit diagram is as depicted in figure \[fig:Circuit\_plasma\](a). The inclusion of $R_{1}$, a small external resistance in parallel with $R_{p}$ (the plasma resistance between the probe and wall), allows current driven by the floating potential to flow in the circuit in the case where the thyratron switch is open (see figure \[fig:Circuit\_plasma\](b)). When the switch is closed, a proportion of the biasing capacitor-driven current may divert to flow through $R_{1}$, see figure \[fig:Circuit\_plasma\](c). This proportion increases as $R_{p}$ increases with reducing electron temperature as the CT decays, thereby allowing $I_{probe}$ to decrease at a rate roughly in proportion to the rate of decrease of the main CT plasma currents. The presence of an appropriately sized $R_{1}$ also prevents development of a sustained arc, which could damage the wall and probe, through the ambient plasma that remains between the probe and wall after the CT has extinguished. In previous edge biasing studies on tokamaks, the standard is to maintain approximately constant $V_{applied}$ and $I_{probe}$ for an extended time which is a segment of the duration over which the approximately constant externally driven toroidal plasma current flows. On SPECTOR plasmas, the plasma currents are not driven and are allowed to decay naturally after formation, so a circuit configuration that establishes constant $V_{applied}$ and $I_{probe}$ would not be compatible.
The differential voltage measured between the probe and flux conserver is $$V_{probe}(t)=V_{applied}(t)+V_{float}(t)\label{eq:2.1}$$ If the bias capacitor is not fired, and $R_{1}$ is removed from the circuit, then in the open circuit condition $V_{probe}(t)=V_{float}(t)$. Note that $V_{float}$ is not measured directly on each shot, however, looking at the $V_{probe}$ measurements taken during several open circuit, probe-in shots, the floating potential can be approximated as an RC rise of the form $$V_{float}(t)=V_{f0}\,e^{-\frac{t}{\tau_{RCf}}}\label{eq:2.11}$$ with $V_{f0}\sim-80\mbox{V}$, and, (depending on CT lifetime) $\tau_{RCf}\sim1\mbox{ms}$. $V_{float}(t)$ rises from $\sim-80\mbox{V}$ at the time when plasma enters the CT confinement region, to $0\mbox{V}$ when the CT has decayed away. With this, an approximation for $V_{applied}$ can be made using equation \[eq:2.1\]. $V_{bc}(t)$, the voltage across the biasing capacitor, was not measured directly in the experiment, but can be estimated as $$V_{bc}(t)=V_{bc0}\,e^{-\frac{t}{\tau_{RCb}}}\label{eq:2.12}$$ where, for shot 26400, $V_{bc0}=700\mbox{V}$ and $\tau_{RCb}=0.5\Omega*3\mbox{mF}=1.5\mbox{ms}$ (resistance $R_{2}=0.5\Omega\gg R_{e})$. With these approximations for $V_{applied}(t)$ and $V_{bc}(t)$, an estimate of the plasma resistance along a path that has a principal component along the helical magnetic field between the probe (with insertion depth $11\mbox{mm}$) and flux conserver is evaluated, between $t_{bias}=250\upmu\mbox{s}$ until the time when the CT has decayed, using equation \[eq:3\]:
The approximations (from equations \[eq:2.1\], \[eq:2.11\], and \[eq:2.12\]) for $V_{applied}(t)$, $V_{float}(t)$, and $V_{bc}(t)$, and measured $V_{probe}(t)$, for shot 26400, are shown in figure \[26400VRI\_bias\](a). Figure \[26400VRI\_bias\](b) shows the estimation, from equation \[eq:3\], for $R_{p}(t)$. $R_{p}(t)\sim0.15\Omega$ to $0.2\Omega$, and rises as $T_{e}$ decreases ($\eta_{plasma}$ increases) over CT decay, then drops as the edge current path length $L$ (recall $R(t)=\eta(t)L(t)/A(t)$) decreases. Path length decreases because $B_{\theta}$ decreases faster than $B_{\phi}$ (CT toroidal field is maintained at a relatively constant level by the crow-barred external shaft current) as the CT decays, $i.e.,$ $q$ increases - there are fewer poloidal transits for each toroidal transit along the path which defines $R_{p}$. The sharp dip in $R_{p}$ at $t\sim845\upmu\mbox{s}$ coincides with the fluctuations in $I_{probe}(t)$ and $B_{\theta}$ seen in figures \[26400\](a) and (b). Note that the current through the path enclosed by the Rogowski coil depicted in figures \[fig:Biasing-probe-circuit\] and \[fig:Circuit\_plasma\] can be calculated using basic circuit theory as: $$I_{rog\,(calc.)}=\frac{1}{R_{p}(t)}\left[\frac{R_{2}\left(V_{bc}(t)\,R_{1}+V_{float}(t)\,R_{1}+V_{bc}(t)\,R_{p}(t)\right)}{R_{1}R_{2}+R_{2}R_{p}(t)+R_{p}(t)\,R_{1}}-V_{bc}(t)-V_{float}(t)\right]\label{eq:4}$$ Figure \[26400VRI\_bias\](c) compares measured $I_{probe}(t)$ (black trace) with calculated parameters, to verify the calculation of $R_{p}(t)$. Referring to figure \[fig:Circuit\_plasma\](c), it is seen that $V_{applied}(t)/R_{p}(t)$ should, as is confirmed in figure \[26400VRI\_bias\](c) (dark blue trace), give the measured $I_{probe}(t)$ current when the switch is closed after $t=t_{bias}$. Referring to figure \[fig:Circuit\_plasma\](b), $V_{probe}(t)/R_{1}\sim I_{probe}(t)$ when the switch is open before $t=t_{bias}$ (red trace in \[26400VRI\_bias\](c)). A good match to measured $I_{probe}(t)$ is found by using calculated $R_{p}(t)$ and the estimated profile of $V_{float}(t)$ in equation \[eq:4\] (after $t=t_{bias}$, cyan trace). A good estimate of $R_{p}(t)$ is useful for optimizing external circuit resistances.
Main results\[sec:Main-results\]
=================================
Figure \[fig:life\_3mF\_se\](a) indicates how CT lifetimes varied with $V_{bc0}$ (coloured circles) for shots taken with the biasing probe inserted $11\mbox{mm}$ into the plasma edge in the configuration using the $3\mbox{mF}$ biasing capacitor circuit, with $R_{1}=0.1\Omega$ and $R_{2}=0.5\Omega$, compared with shots taken with the probe removed (black squares). Figure \[fig:life\_3mF\_se\](b) indicates the average of CT lifetimes for the probe-out configuration (black squares), and the averages for the probe-in configuration (red circles) for the setpoints $V_{bc0}=0\mbox{V},\,400$V, and $700$V. It is indicated that CT lifetime increased from around 450 to 600eV even when the biasing capacitor was not fired - in that case, the presence of the resistor ($R_{1}$) in parallel with the biasing capacitor enables current, driven by the potential applied by the plasma, to flow from the electrode to the wall. At $V_{bc0}=400$V, CT lifetime increased by a factor of around 2.3, from $\sim460\upmu$s to $\sim1070\upmu$s. Note that TS data is not available for the configuration with the 3mF capacitor in the biasing circuit.
Figure \[fig:life\_100uF\_se\] shows equivalent information for shots taken with a $100\upmu$F, 5kV capacitor in the biasing circuit, with $R_{1}=0.4\Omega$ and $R_{2}=3\Omega$ (TS data is available for this configuration). CT lifetimes were approximately doubled in this configuration, with an optimal biasing capacitor setpoint of $V_{bc0}\sim2$kV.
Figure \[fig:TS\_2profile\] shows shot data indicating the temperature and density profiles obtained with the TS system at $300\upmu$s after firing the formation capacitor banks, for the configuration with the $100\upmu$F, 5kV biasing capacitor. Note that the TS sampling points are indicated in figure \[fig: Spect\_n\_Te\](b). With $V_{bc0}\sim2$kV, the measurements indicate that temperature is more than doubled at the inner sampling points, increasing by a factor of around 2.4 at the sampling point at $r=140$mm, (black squares $cf.$ red circles) and the proportional increase in temperature falls off towards the CT edge. Note that current drawn through the CT edge leads to a temperature increase even when no voltage is externally applied to the electrode (black squares $cf.$ blue circles). Referring to figure \[fig:TS\_2profile\](b), electron density is markedly reduced when the electrode is inserted and the reduction is enhanced when the electrode is externally biased. The proportional decrease in density is greater towards the CT edge, consistent with the theory that edge fueling impediment due to an edge transport barrier is largely responsible for the density reduction (see appendix \[sec:Neutral\_SPECTOR\]) . The diagnostic indicates an electron density reduction by factors of approximately 1.5 and 2.3 at $r=130$mm and $r=170$mm respectively.
Figure \[fig:Halphavert\](a) indicates how $\mbox{H}_{\alpha}$ intensity, along a vertical chord located at $r=88$mm, is reduced when the electrode is inserted and biased. $\mbox{H}_{\alpha}$ intensity reduction is a sign of reduced recombination at the vessel walls, and is associated with improved confinement. The purple traces are from shots with the electrode removed from the vacuum vessel. As shown in figure \[fig:Halphavert\](b), time-resolved FIR interferometer data from the chord at 140mm (FIR chord locations are indicated in figure \[fig: Spect\_n\_Te\](a)) confirms the reduction in electron density when the biased probe is inserted into the plasma edge. Again, the purple traces are from shots taken with the electrode retracted. This density reduction is thought to be due to the effect of the transport barrier impeding the level of CT fueling associated with neutral gas diffusing up the gun, as discussed in appendix \[sec:Neutral\_SPECTOR\]. Note that the fueling effect is not entirely eliminated by the biasing effect - density starts to increase at around $500$ to 600$\upmu$s ($cf.$ figure \[fig:Spector\_nN\](b)). Note that H$_{\alpha}$ intensity increases dramatically at around the same time (figure \[fig:Halphavert\](a)). The CTs associated with the purple traces (probe-out configuration) in figure \[fig:Halphavert\](a) and (b) do not last for long enough to enable observation of the density and H$_{\alpha}$ intensity increases at that time.
Discussion and conclusions\[sec:Discussion-and-conclusions\]
============================================================
Significant increases in CT lifetime and electron temperature, and reductions in electron density and $\mbox{H}_{\alpha}$ intensity, were observed when the electrode was inserted into the plasma edge, even when the biasing capacitor was not fired. In that case, the presence of the resistor ($R_{1}$) in parallel with the biasing capacitor enables current, driven by the potential applied by the plasma, to flow from the electrode to the wall. Note that in cases where the biasing capacitor was not fired, the enhanced performance was eliminated when $R_{1}$ was removed from the circuit. In terms of enhanced CT lifetime, which was observed to increase by a factor of up to 2.3, the optimal biasing circuit tested was with the 3mF capacitor in place, but TS data was not available in that configuration. CT lifetimes and electron temperatures were observed to increase by factors of around 2 and 2.4 (temperature near the CT core) respectively in the configuration with the 100$\upmu$F capacitor charged to 2.1kV, while density decreased by a factor of around 2.3 near the CT edge. This density reduction is thought to be due to the effect of the transport barrier impeding level of CT fueling associated with neutral gas diffusing up the gun. The consequent reduction of cool particle influx to the CT is thought to partially responsible for the particularly significant increases in observed temperature, as compared with prior edge biasing experiments. Up to $\sim1200$A was drawn $11\mbox{\mbox{mm}}$ through the edge plasma, while improving CT lifetime and temperature.
Note that the biasing experiment was conducted without a fresh lithium coating on the inside of the SPECTOR flux conserver. With a fresh coating, CT lifetimes are typically around 2ms. The biasing experiment may be run again with a fresh coating. The experiment was conducted over a short period (less than two weeks). As the majority of the probe-out shots were taken at the beginning of each day, there is likely some data skew due to cleaning effects. The improvement shown with biasing may be extended with further circuit optimization. Negative biasing was tested briefly - a slight increase of electron temperature and a peaking of the electron temperature profile was observed, but there was no evidence of lifetime increase. It may be that the ion-sputtering of the probe associated with negative biasing lead to performance degradation associated with plasma impurities that offset the improvement associated with the establishment of a transport barrier. Perhaps more cleaning shots are required to see a significant improvement with negative biasing - the efficacy of negative biasing hasn’t been confirmed. An IV curve was produced with the electrode biased to a range of positive and negative voltages on a shot to shot basis. Langmuir analysis indicated $T_{e}\sim130$$\mbox{\mbox{eV}}$ and $n_{e}\sim10^{19}[\mbox{m}^{-3}]$ at the probe location at $300$$\upmu\mbox{s}$, and $T_{e}\sim85$$\mbox{\mbox{eV}}$ and $n_{e}\sim5\times10^{18}[\mbox{m}^{-3}]$ at $600$$\upmu\mbox{s}$. Compared with TS data, the electron temperature estimates in particular appear too high. The fact that probe biasing affects electron temperature and electron density makes the Langmuir analysis results dubious at best, but it may be possible to correct for this effect.
It would be worth repeating the experiment with a fresh lithium coating on the inner flux conserver. Circuit parameters, probe insertion depth, and machine operation settings should be optimized further. The effects of biasing on edge conditions should be characterised using Langmuir and Mach probes, and ion Doppler diagnostics. Negative biasing may be tested more rigorously. It would be interesting to look at the effects of driving edge current resonant to the MHD behaviour that manifests itself in the form of fluctuations on measurements including CT poloidal field.
The biasing experiment was especially noteworthy because it has generally been found that insertion of foreign objects, such as thin alumina tubes containing magnetic probes, into SPECTOR CTs, leads to performance degradation associated with plasma impurities. After the extensive problems encountered relating to plasma/material interaction and impurities during the magnetic compression experiment [@thesis; @exppaper], special care was taken to choose a plasma-compatible material for the biasing electrode assembly. The pyrolytic boron nitride tube and molybdenum electrode combination seems to have been a good choice - at least the benefit due to drawing a current through the CT edge outweighed any performance degradation that may have been associated with impurities introduced to the system.
Acknowledgments
===============
Funding was provided in part by General Fusion Inc., Mitacs, University of Saskatchewan, and NSERC. Particular thanks to Alex Mossman, Kelly Epp, Akbar Rohollahi, Russ Ivanov, and Adrian Wong for machine operation support, to Russ Ivanov, Ivan Khalzov, and Meritt Reynolds for useful discussions, and to Wade Zawalski, Curtis Gutjahr, Blake Rablah, James Wilkie, Alan Read, Mark Bunce, Pat Carle and Bill Young for hardware and diagnostics support.
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Simulation of interaction between neutral and plasma fluids in SPECTOR geometry\[sec:Neutral\_SPECTOR\]
========================================================================================================
It is usual to observe a significant rise in electron density at around $500\upmu$s on the SPECTOR machine, and it is thought that this may be a result of neutral gas, that remains around the gas valve locations after CT formation, diffusing up the gun. Ionization of the neutral particles would lead to CT fueling and an increase in observed electron density. An energy, particle, and toroidal flux conserving finite element axisymmetric MHD code was developed to study CT formation into a levitation field, and magnetic compression [@thesis; @exppaper; @SIMpaper]. The Braginskii MHD equations with anisotropic heat conduction were implemented. As described in [@thesis; @Neut_paper], a plasma-neutral interaction model including ionization, recombination, charge-exchange reactions, and a neutral particle source, was implemented to the MHD code and used to study the effect of neutral gas on simulated CT formation in SPECTOR geometry.
Figures \[fig:SPEC\_psi\_n\_nN\](a), (b) and (c) show $\psi$ contours and profiles of $n_{e}$ and $n_{n}$ at $20\upmu$s, as plasma enters the CT containment region. Profiles of the same quantities are shown in figures \[fig:SPEC\_psi\_n\_nN\](d), (e) and (f) at $500\upmu$s, around the time when the rise in measured electron density is usually observed. It can be seen how neutral fluid density is highest at the bottom of the gun barrel - any neutral gas advected or diffusing upwards is ionized. A region of particularly high electron density is apparent just above, and outboard of, the entrance to the containment region - this is due to the fueling effect arising from neutral gas diffusion.\
The region of particularly high electron density is more defined in figure \[fig: Spect\_n\_Te\](a), in which cross-sections of the horizontal chords representing the lines of sight of the FIR (far-infrared) interferometer [@polarimetryGF] are also depicted. The electron temperature profile at 500$\upmu$s is shown in figure \[fig: Spect\_n\_Te\](b). Referring to figure \[fig:SPEC\_psi\_n\_nN\](f), it can be seen how neutral fluid density is low in regions of high $T_{e}$ as a result of ionization.
Figure \[fig:Spector\_nN\](a) shows measured line-averaged electron density along the chord at $r=140$mm from a selection of several shots in SPECTOR. It can be seen how density starts to rise at around 500 to $600\upmu$s. Figure \[fig:Spector\_nN\](b) shows the simulated diagnostic for line-averaged electron density along the chords indicated in figure \[fig: Spect\_n\_Te\](a). The density rise is qualitatively reproduced when a neutral fluid is included in the simulation. Similar simulations without the inclusion of neutral fluid do not indicate this density rise (dashed lines in figure \[fig:Spector\_nN\](b)). Note that the simulations presented in figure \[fig:Spector\_nN\](b) were run with artificially high plasma density in order to allow for an increased timestep and moderately short simulation run-times. Hence, the electron temperatures indicated in figure \[fig: Spect\_n\_Te\](b) are underestimations of the actual temperatures due to the overestimation of density in the simulation. The main goal of these simulations was to demonstrate that the inclusion of neutral fluid interaction can qualitatively model the observed electron density increase.
|
China confirms visit of North Korean leader - report
By IBT Staff Reporter08/30/10 AT 10:36 AM
China confirmed to foreign diplomats in Beijing on Monday that North Korean leader Kim Jong-il had visited the country, Japan's Kyodo news agency said, as Chinese state media praised ties with the isolated country.
A source told Reuters at the weekend that Kim and his youngest son -- his presumed heir -- were on a trip to China but there has been no official confirmation from either government.
Kyodo, citing diplomatic sources, said the Chinese Foreign Ministry explained to some Beijing-based diplomats the visit of North Korean leader Kim Jong-il to the country.
The report said that Kim had met Chinese President Hu Jintao, adding that Kim was expected to go back to North Korea later in the day.
In the past, China and North Korea have only publicly confirmed Kim's visits after he has returned home.
A Chinese-language newspaper nonetheless lauded relations between the two countries on Monday.
Maintaining and stabilising the current relationship between China and North Korea is of maximum benefit to China, the popular tabloid the Global Times said in an editorial.
China is the only major supporter for North Korea, which is largely isolated from the international community over its nuclear weapons programme and which has come under further condemnation after South Korea accused it of sinking one of its warships earlier this year.
China's official Xinhua news agency also praised ties between the two, especially the bonds forged between their people during the 1950-53 Korean War.
Those who sacrificed their lives for the China-DPRK (North Korea) friendship should be remembered generation after generation, particularly at a time of changing and complicated regional situations, it said in an English-language commentary.
Kim, 68 and who rarely travels abroad, is reportedly in China for the second time this year. This time he is thought to have brought along his youngest son Kim Jong-un, widely seen as the next head of the family dynasty that has led North Korea since its founding more than 60 years ago.
NO DEFINITE SIGHTINGS
On Monday, police lined the streets in Tumen, a city on China's border with North Korea, in a sign that Kim may visit there or pass through on his way home.
The clues of Kim's travels have been pursued by legions of reporters, especially from the North's neighbours, South Korea and Japan, who study the comings and goings of trains and motorcades, some of them possibly meant to bamboozle the press.
But there have been no definite sightings of Kim.
Kim may be lining up China behind succession plans involving his son, foreign analysts have said. The Workers' Party (WPK), which rubber-stamps big decisions in the North, is due to hold a rare meeting in September that could set in motion succession steps.
The Chinese newspaper blamed outside forces for pressuring North Korea as a way to create trouble for China, the sole major economic and diplomatic supporter of its much weaker neighbour.
The sinking of the South Korean navy ship, in which 46 sailors died, deepened tensions between Pyongyang and Seoul and strained Chinese ties with South Korea. |
Docfaustino
Previously: The family has broken off in to two factions: The Amiti faction, led by Daut and Pirro Amiti, who oppose the idea of a female ruling the family. And Besmir's faction, led by Kaltrina Krasniq with the support of Dardan Petrela and Bledar Morina, who support Preston's decision of having his wife rule the family. Kaltrina lured Pirro to her bedroom only to beat him severely, and ordered the death of one of his men. In response, the Amiti faction will try (and fail) to get Besmir to join them. No matter who comes out on top, one thing is clear, Kaltrina is running the family in to a ground with her high profile behavior, severe punishment for anyone who opposes her, and unwillingness to talk anything out.
Kaltrina's Theme
Returning Characters.
Besmir Krasniq
Kaltrina Krasniq
Dardan Petrela
Kalem Vulaj
New Characters
The Messenger
Dardan Petrela Sits in the front booth of the Belly Deli. He is having a good time, though the men that surround him, Besmir has not seen before. Next to him is his beloved brother, Kalem Vulaj. Kalem was never too bright. But he had recently won the crew's respect by knocking out two men who tried to encroach on their small bit of turf. The rest of the men are likely possible suitors here to see Kaltrina, Besmir thinks.
Dardan: Kalem, little brother. I just want to eat you up.
Unknown Man: Get a room, Dardan.
Dardan: What's wrong with you? He's got brain damage.
[Besmir walks in]
Dardan: There he is, the man who missed this week's tribute.
Besmir: Funny guy. I'm broke.
Unknown Man: Besmir. So good to see you. Can we talk?
Besmir: I suppose. In the other room.
Unknown Man: Look, kid. I know why you don't kick up to Dardan. You don't respect him. Dardan himself, is not respectable.
Besmir: Yes, yes, it's fun to hate on Dardan. Can I go now?
Unknown Man: Do you remember one 'Joey Becker'?
Besmir: What is this, some kind of joke? Guards!
Unknown Man: It's all in the past, my friend. We'd be willing to forget your past offenses if you could switch your allegiances to a more professional crew. The Amiti crew.
Besmir: You're even more treacherous than you look. Asking me to turn on my own mother? You some kind of maniac? Kaltrina is violent, yeah. But that's the way Preston wants it.
Amiti Messenger: The highest earning captains since '02 are Pirro and Daut Amiti. That's two world class earners leading our side. And who leads your side? Kaltrina is brutal. Dardan Petrela is a bum, and his brother's a f*cking retard. If some negro came in here and said, "I'm the boss now," Dardan would say: "Gee what can I do for you, boss?" He's muscle. You're better than that.
Besmir: You better show my mother the same respect I always showed the Amiti brothers. For now, you stay out of my way!
Dardan: What was that, Besmir?
Besmir: Your average bum. How you been?
Dardan: Better since your mother came around. She asked to personally accompany me on a deal for some hot merchandise. Our current boss wants to see me in action!
(Lyrics: Here she comes, you better watch your step)Walking down the street, Besmir and Dardan working as bodyguards. Kaltrina silently smiles at the passerbys who are amazed by her charm. The trio are followed with a long tracking shot, a la Goodfellas.
(Lyrics: It's not hard to realize, just look in to her false colored eyes)Extreme closeup on Kaltrina's eyes. At first, her gaze appears genuine, warm. Once all the pedestrian's have passed, her face turns to something more sinister. The eyes of crazed killer.
(Cause everybody knows, she's a Femme Fatale)The trio walk in to a nearby store, and Kaltrina begins to harass the owner. After attempting to seduce him, he hands over a wad of protection money. They exit the store.
(Hear the way she talks) 1:11The music suddenly stops. A sharp ringing noise begins, and a slow-motion splatter of blood across Besmir's face. The halting tires of a Vapid Redwood cause a sharp screech that only adds to the confusion. You can just barely make out the man behind the wheel. It's the man who insulted Dardan at the bar. He orders a gunman to shoot.
Amiti Messenger: Now's the time!
Kaltrina: [painful scream]
Besmir: MOTHER!
Dardan: These sons of bitches! Stay with me Besmir, Hold them off with a glock while I get some hardware from the car.
The Amiti gunmen speed away in the Redwood. A few men across the street disguised as ordinary civilians stand up and pull weapons. The hit was planned well.
Aim for the head to dispatch enemies quickly.
Dardan: I'm sorry for what happened to her, but goddammit, I'm getting out of here alive!
Besmir: You animals! You hit a lady!
Dardan: Don't ask any questions, this is the best I got! It's a flamethrower, don't waste the gas.
Hitman: You have no where left to go! That stupid bitch deserved to get popped, and so do you.
Besmir: [steadying flamethrower] f*ck YOU!
Dardan: Take cover behind the cars, that's the best we've got in an open street.
Press [LB] to take cover
Besmir: We've got more on the way, watch out for those trucks!
Dardan: Burn them out of the cars, I can't hit 'em from within there.
Kill the reinforcements .
Dardan: Besmir! I've got her in the Willard, it's time to go!
Drive Kaltrina to Bohan Medical and Dental Center .
Dardan: Drive!
Once at the destination, we cut to a brightly lit private room with a single hospital bed. Dardan paces around the room nervously, and Driton appears to be saying a prayer. Her husband is nowhere to be seen, but Besmir kneels beside her, nearly tearing
Tyla
Mean Streets and Goodfellas references in the same mission?? Beautiful!
I loved the tracking shot echoed by Kaltrina's theme. This is as much to me a movie script as it is a story, every scene playing through my mind as if it were live action. Your concept never fails to draw me in, Ped. Excellent work!
Docfaustino
If you check the mission table, I've listed the rest of the missions in the first chapter. Every chapter will be followed by a voice-over to clear up any confusion, figured its an easy way to break everything down for people who don't want to read every mission. Two fairly major plot twists are coming up, and Besmir will make a snap decision to try and desperately erase the burden he feels from almost single handedly keeping a crime family he didn't ask to inherit from becoming a sinking ship. Everything up through the 13th mission is sort of a standalone story, so if you read through all of that, you should still be satisfied without spending days looking through a whole concept! Glad everyone enjoys this, I'm always looking for suggestions.
Docfaustino
Blood And a Four Leaf Clover is what gave me the idea to start this thread, so it means a lot coming from you as well as MOB. Do you mind if I send one of you a PM on IV continuity? Jedi, props for repping that Albanian flag.
Akavari
Blood And a Four Leaf Clover is what gave me the idea to start this thread, so it means a lot coming from you as well as MOB. Do you mind if I send one of you a PM on IV continuity? Jedi, props for repping that Albanian flag.
Continuity regarding what? Like the mission timeline? If so, I think MOB has better knowledge of it since he made the Possible Trinity and all.
Blood And a Four Leaf Clover is what gave me the idea to start this thread, so it means a lot coming from you as well as MOB. Do you mind if I send one of you a PM on IV continuity? Jedi, props for repping that Albanian flag.
Continuity regarding what? Like the mission timeline? If so, I think MOB has better knowledge of it since he made the Possible Trinity and all.
Docfaustino
The dream begins with a flashback to Besmir's earlier years. His family was Muslim once. Besmir is haunted by the mission 'Femme Fatale', where he was forced to burn many men alive, and in his dreams, begins to come to terms with that, as well as his partial dedication to his one-time religion.
Kaltrina: We cannot take Besmir to a mosque! Every priest in this city is getting charged with rape, they must have wiretaps in the church!
Preston: I will not deprive of our religion because of a Christian scandal.
Kaltrina: No more of this praying to Mecca, Preston, we will never fit in to this country!
Preston: Is that all you care about? Fitting in? By all means, spend some more time at your white people country club. That should help.
Kaltrina: I love you Preston. But we do not bring Islam to the city of terrorist paranoia. Don't let it rub off on the children.
Cut to a plane landing in Mecca, the holiest city in Islam. Besmir exits the plane in traditional Muslim garb, before covering his head with a hijab. The crowd appears to be making the yearly pilgrimage to Mecca, to honor the prophet Mohammed. Beside Besmir is his friend and corrupt FIB contact, Marvin Walsh.
Besmir: What are you here for?
Marvin: My new broad's a Paki. She wanted me to experience this pilgrimage thing. Plus, I figured I could make up for my avarice.
Besmir: I cut down 7 men who attacked my mother. I used a flamethrower. They burned alive. I can't do this any more, Marvin. I was raised too long on religion. It really gets to me.
As Besmir attempts to come to terms with his sins, a voiceover appears over a black screen in Besmir's voice.
Voiceover: Muslim hell is a personal hell. In my hell, you burn for every one of your victims. All nine of them. And more on the way.
Claude4Catalina
I'm sure there's a reason that I instantly feel jealous when I read something you write...probably due to the fact hat your writing can't get much better, but in terms of the main page, I could help with the weapons modifications section if you wish and possibly try my hand at creating an icon for the AN-94.
Docfaustino
A lot of Albanians are Muslim because of the Ottomans. I figured it was a good way for Besmir to express his regret for burning people alive. Imagine you reluctantly kill your first victim, and then a week later, you face 7 men trying to kill your mother with only a flamethrower at your disposal. Thanks for the compliments.
I had to use the closest thing to an AN-94. Would be awesome if someone could design it. Which reminds me I need to credit EdwardKoeleJuck...
Docfaustino
That's fine as well. From now on Marvin Walsh will be appearing a bit more. Since I know this stuff gets complicated, basically he's an FIB guy who's laying low by helping out the organized crime force at the Hove Beach station. Remember, the hierarchy chart of Kaltrina's organization that Petrovic leaked to the feds are basically the diamonds of this story. And Marvin Walsh will be furiously trying to discredit this chart since the last thing he wants is for Besmir to get indicted. Marvin sort of knows that if Besmir goes to jail, he'll blame Marvin for doing a poor job as a corrupt cop, and he'll be killed for it. In the next mission, you'll finally see why Preston was protesting in the opening cutscene. More exciting than it sounds, I think.
In other news, a DYOM guy offered to make a DYOM pack based on Balkan Valor. Let's hope it goes well!
Akavari
That's fine as well. From now on Marvin Walsh will be appearing a bit more. Since I know this stuff gets complicated, basically he's an FIB guy who's laying low by helping out the organized crime force at the Hove Beach station. Remember, the hierarchy chart of Kaltrina's organization that Petrovic leaked to the feds are basically the diamonds of this story. And Marvin Walsh will be furiously trying to discredit this chart since the last thing he wants is for Besmir to get indicted. Marvin sort of knows that if Besmir goes to jail, he'll blame Marvin for doing a poor job as a corrupt cop, and he'll be killed for it. In the next mission, you'll finally see why Preston was protesting in the opening cutscene. More exciting than it sounds, I think.
In other news, a DYOM guy offered to make a DYOM pack based on Balkan Valor. Let's hope it goes well!
Was it Satournfan? Because I got a PM from that guy regarding DYOM as well.
Mati
That's fine as well. From now on Marvin Walsh will be appearing a bit more. Since I know this stuff gets complicated, basically he's an FIB guy who's laying low by helping out the organized crime force at the Hove Beach station. Remember, the hierarchy chart of Kaltrina's organization that Petrovic leaked to the feds are basically the diamonds of this story. And Marvin Walsh will be furiously trying to discredit this chart since the last thing he wants is for Besmir to get indicted. Marvin sort of knows that if Besmir goes to jail, he'll blame Marvin for doing a poor job as a corrupt cop, and he'll be killed for it. In the next mission, you'll finally see why Preston was protesting in the opening cutscene. More exciting than it sounds, I think.
In other news, a DYOM guy offered to make a DYOM pack based on Balkan Valor. Let's hope it goes well!
Was it Satournfan? Because I got a PM from that guy regarding DYOM as well.
Docfaustino
Right, it was SatournFan. Sure thing, Mati. I don't have San Andreas on PC but I read the outlines occasionally. Editing it in to the front page, here's the current version. Pirro and Daut Amiti are the men who just shot Kaltrina. Daut is a longterm friend of Preston, and Preston being the 'fair' boss, he's used to being treated like a bigshot. Which is why he wasn't happy when a woman was named interim boss over him. Gez Istogu will play a pivotal role in the story, he is also a friend you can call up for activities and will next appear in 'Petrela Wars'. Other men on this chart play minor roles, but you'll still recognize them.
lindsayjames10
loving it so far, especily the dreams/flashabacks of his life, this game shows how evil the albanians are..albanians are known for brutal methods like cutting peoples organs out why they still alive then sowing them back but repeat the progress to they beg for a quick death.
i wonder if in your story, the albanians buy a club or go in buissness with gay tony buying the maschette club and remaking the club then have as front for illegal activites.
will yill dusku be in any more missions and will ray bulgarin make a return.
Docfaustino
Maisonette 9 won't be available as for the most part, this game takes place during the middle of Niko's story, not the end. However, a major part of it is about purchasing various interiors where you can buy cigarettes, shots, champagne minigame, play pool, etc. So that should make up for the lack of Maisonette. Read Mafia Management if you want to know more about that. Yili Dushku will play a big role, but not in the beginning of the story. In the end we will see a lot of him. Ray Bulgarin? I don't have him planned to make a return. If you have any ideas, though, by all means.
LuisBellic
The new missions are fantastic. May I borrow your dream idea? There's some background about my character, and him almost being a victim of Organ Trafficing and watching people get slaughtered is one of the dream ideas I have. |
The US suspension of and withdrawal from the INF-Treaty is an irresponsible move that opens the path for a new-nuclear arms race and highlights the importance of real multilateral, binding solutions like the UN Treaty on the Prohibition of Nuclear Weapons. With Russia and the US putting the entire world at risk, it is urgent for all responsible states to stand up and join the Nuclear Ban Treaty.
Beatrice Fihn, Executive Director of the International Campaign on Nuclear Weapons: “ Trump has fired the starting pistol on Cold War II. Only this one could be bigger, more dangerous, and the world may not be so lucky this time around. European leaders and all NATO allies, must make it clear that withdrawing from the INF treaty is a threat to European security. European governments must be working toward removing all nuclear weapons from European soil by joining the Treaty on the Prohibition of Nuclear Weapons.”
The importance of the INF Treaty
The INF Treaty was the first agreement between Russia and the US that eliminated entire categories of nuclear weapons. For over 30 years, both sides agreed to the elimination of all nuclear and conventional ground-launched ballistic and cruise missiles with ranges of between 500 and 5,500 kilometre. As a result, the US destroyed 846 of its missiles and 32 launch sites, and the USSR destroyed 1,846 missiles and 117 sites [1].
Suspension and next steps
As per today’s announcement, the US is suspending its compliance with the treaty and handing in its notice of withdrawal from the Treaty, on the basis that the Russian Novator 9M729 missile is within the prohibited missile range (more than 500 km). This means the INF-Treaty will terminate on 2 August unless US revokes this notice. Russia would officially still be bound by the treaty until August 2nd, but has warned that if the United States developed INF-range missiles, Russia would do so as well.
If Russia and the US are honest about their commitment to nuclear disarmament, both parties should do everything they can to save the Treaty in the coming 6 month period. But over the past few weeks, months, and even years, t both Russia and the US have signaled an apparent interest in a new nuclear arms race. Last week, Trump began building new nuclear missiles and Putin has said he will do the same. Both countries are spending millions in modernising their arsenals.
Europe at risk
While the INF Treaty only bound two countries, its demise endangers the entire world. The only ones applauding the decision to tear up this Treaty are the nuclear weapons manufacturers, eagerly anticipating the kickoff of Cold War II.
While announcing the withdrawal, US Secretary of State Pompeo stated that Russia’s violations put millions of Europeans and Americans at greater risk, and speaks of the unity among the US’ allies in support for US withdrawal. Yet the EU have actually called for both the US and Russia to do everything they can to save the INF-Treaty. European governments know a new nuclear arms race poses an unacceptable security risk for Europe, which is once again put at the literal center of potential conflict.
In 1987, the INF eliminated the category of nuclear weapons that put European countries most at risk, due to the distances the missiles could travel, rescuing Europe from one of the most dangerous escalations of the Cold War. The collapse of the INF-Treaty opens the doors for the US to station these intermediate range missiles in Europe again, alongside the other nuclear weapons it already stores in Germany, Italy, Belgium, Netherlands and Turkey. General Valery Gerasimov, the chief of staff of the Russian military, indicated that US missile sites on allied territory could become “the targets of subsequent military exchanges.” In short, pulling out of the INF will move the world in reverse and endanger all Europeans.
The Nuclear Ban Treaty as an alternative to the new nuclear arms race
While nuclear armed states appear hell-bent on increasing and modernising their arsenals, there is a growing global resistance to nuclear weapons. 70 nations have signed the UN Treaty on the Prohibition of Nuclear Weapons, and 21 have ratified it. All over the world, support for the Treaty is growing and inspiring action at the national and local level: more than 1200 active members of parliament across all continents have already pledged their support for the Treaty, and just today Hiroshima and Nagasaki (Japan) and Trondheim (Norway) have called on their governments to join it.
With 50 States Party, the treaty will ban all nuclear weapons under international law, like chemical weapons and landmines before them. For the sake of sanity and humanity, responsible leaders must counteract the destruction of the INF Treaty with a clear signal that nuclear weapons are not acceptable. They must join the Treaty on the Prohibition of Nuclear Weapons.
–
Media inquiries: press@icanw.org
[1] Arms Control Association, ‘The Intermediate-Range Nuclear Forces (INF) Treaty at a Glance’, Updated December 2018, at: http://bit.ly/2T6lziN
[2] T. Grove, ‘Putin Threatens Arms Race as U.S. Prepares to Exit Nuclear Treaty’, Wall Street Journal, 5 December 2018, at: https://on.wsj.com/2FLWdUk; and ‘Russian Compliance with the Intermediate Range Nuclear Forces (INF) Treaty: Background and Issues for Congress’, Updated 18 January 2019, Congressional Research Service, Washington DC, p. 3, at: http://bit.ly/2RMFA1n. |
Mechanisms of disease: the hygiene hypothesis revisited.
In industrialized countries the incidence of diseases caused by immune dysregulation has risen. Epidemiologic studies initially suggested this was connected to a reduction in the incidence of infectious diseases; however, an association with defects in immunoregulation is now being recognized. Effector T(H)1 and T(H)2 cells are controlled by specialized subsets of regulatory T cells. Some pathogens can induce regulatory cells to evade immune elimination, but regulatory pathways are homeostatic and mainly triggered by harmless microorganisms. Helminths, saprophytic mycobacteria, bifidobacteria and lactobacilli, which induce immunoregulatory mechanisms in the host, ameliorate aberrant immune responses in the setting of allergy and inflammatory bowel disease. These organisms cause little, if any, harm, and have been part of human microecology for millennia; however, they are now less frequent or even absent in the human environment of westernized societies. Deficient exposure to these 'old friends' might explain the increase in immunodysregulatory disorders. The use of probiotics, prebiotics, helminths or microbe-derived immunoregulatory vaccines might, therefore, become a valuable approach to disease prevention. |
On the other hand, sometimes you see other peoples or any person you know suffering from a bad situation or may be treated badly and wants to take the stand and raise for himself or herself, but he or she can't in that case if you think you can take stand for her/him then go for it, don't just step back.You know what, just remain silent all the time is not enough. Ok, sometimes you need to be silent when it needed in any particular situation. But when it needed to take the stand and raise voice then you have to go for it. sometimes, if you see other people need help then you have to raise voice and take the stand for himself/herself.That's the way you should live the life. that's the way you can solve the problem. because if you want to get out of something unfair or bad circumstances, you have to do by taking the stand, and this is one of the effective wayAll and all just take stand every time you think you have to then just go for it maybe for you, maybe for another person that you don't know. don't be hesitate, so when situations call for you just take the stand and raise voice don't step back.Thank youHave a nice dayFollow us |
264 S.W.3d 222 (2008)
Ruscel Lovel BATTISE, Appellant,
v.
The STATE of Texas, Appellee.
No. 01-06-00935-CR.
Court of Appeals of Texas, Houston (1st Dist.).
January 31, 2008.
Rehearing Overruled March 14, 2008.
Discretionary Review Refused October 1, 2008.
*224 James F. Keegan, Houston, TX, for appellant.
Scott Ray Peal, Assistant District Attorney, Anahuac, TX, for State.
Panel consists of Justices NUCHIA, JENNINGS, and KEYES.
OPINION
EVELYN V. KEYES, Justice.
A jury convicted appellant, Ruscel Lovel Battise, of unauthorized use of a motor vehicle,[1] and the trial court assessed his punishment at imprisonment for 12 months. In four issues, appellant argues that the evidence is legally and factually insufficient to prove that he (1) intentionally or knowingly operated a motor vehicle without the effective consent of its owner and (2) operated the same motor vehicle in Chambers County.
We affirm.
Background
At approximately 1:00 p.m. on February 3, 2007, in Chambers County, Texas, Hubert Thomas ("Thomas") loaned his 1998 Lincoln Town Car to appellant, instructing him to wash the car and return it before 5:00 p.m. that same day. Thomas had known appellant for several years because appellant had been dating Thomas's sister, Jennifer Thomas. Appellant and Jennifer Thomas had borrowed vehicles from Thomas on several occasions before this incident. Upon receiving the car on February 3, 2007, appellant drove from Winnie in Chambers County to Beaumont and failed to return the car to Thomas by 5:00 p.m. Thomas waited at work for several hours for appellant to return with his car, but he was eventually forced to call his wife for a ride. Around 10:30 that evening, Thomas called the Chambers County Sheriff's Department to report the car stolen. Deputy S. Eldridge responded to the call and indicated in his report that Thomas *225 had loaned appellant the car with instructions to wash it and return it. Deputy Eldridge was not aware of there being any specific time in which the car was to be returned. Deputy Eldridge testified at trial that Thomas reported that he only gave appellant permission to take the car to a car wash in Winnie. Thomas then signed an affidavit of non-consent that stated that he did not give anyone consent to use his car,[2] and Deputy Eldridge reported the car stolen.
Appellant and Jennifer Thomas, however, testified that appellant was unaware that the car was to be returned by 5:00 p.m. and that Thomas never specified a time for its return. Regarding the purpose for which Thomas loaned the car to appellant, appellant testified that Thomas was "very vague," and only told him to make sure to bring the car back clean. Although she was not actually present when Thomas loaned the car to appellant, Jennifer Thomas testified that Thomas loaned appellant the car so that appellant could get the car washed and look for a job. Appellant testified that he drove the car to the Dairy Queen in Winnie where Jennifer Thomas was working, and then he drove it to Beaumont. Once in Beaumont, appellant looked into getting a birth certificate so that he could get an identification card and drove to a few other places looking for a job. Then he drove the car to a car wash in Beaumont and attempted to find someone to detail the car.
Once Jennifer Thomas realized that her brother was angry that the car had not been returned she contacted appellant while he was in Beaumont. Appellant left the car unlocked with the keys under the floor mat at a car wash in Beaumont so that Jennifer Thomas could pick it up. When Jennifer Thomas went to retrieve the car, it was not there. The next day, a Beaumont police officer recovered the stolen car from three men who had been found riding around in it. Appellant was not one of the men discovered with the stolen vehicle. The car had been damaged on the right front side, the headliner was torn out, and the back seat was ripped out.
Thomas had to pay a fine to get the car out of impoundment and a $500 insurance deductible. Shortly after the incident, appellant's attorney prepared an affidavit of non-prosecution stating that there was a "mutual misunderstanding" regarding the terms of Thomas's consent, and appellant presented it to Thomas. Thomas reviewed the affidavit and signed it.[3] Appellant and Thomas agreed that Thomas would not *226 prosecute appellant in exchange for appellant's paying the $500 insurance deductible. Thomas testified at trial that he had not received any payment from appellant. Appellant and Jennifer Thomas, however, both testified that appellant paid Thomas $200. Appellant testified that he paid the $200 to Thomas prior to the signing of the affidavit, and Jennifer Thomas testified that appellant paid Thomas the money on the day the affidavit was signed.
Appellant was tried before a jury in July 2006. The State subpoenaed Thomas to testify as its key witness. Deputy Eldridge also testified for the State. Appellant testified on his own behalf, and Jennifer Thomas also testified on appellant's behalf. The jury found appellant guilty, and the trial judge assessed punishment at imprisonment for 12 months with credit for time served. This appeal followed.
Standard of Review
When an appellant challenges both the legal and factual sufficiency of the evidence, we must first determine whether the evidence was legally sufficient to support the verdict. Harmond v. State, 960 S.W.2d 404, 406 (Tex.App.-Houston [1st Dist.] 1998, no pet.). We review the legal sufficiency of the evidence by viewing the evidence in the light most favorable to the verdict to determine whether any rational trier of fact could have found the essential elements of the crime beyond a reasonable doubt. King v. State, 29 S.W.3d 556, 562 (Tex.Crim.App.2000). Although our analysis considers all the evidence presented at trial, we may not re-weigh the evidence and substitute our judgment for that of the fact finder. Id.
Factual sufficiency analysis is broken down into two prongs. First, we must ask whether the evidence introduced to support the verdict, although legally sufficient, is so weak that the jury's verdict seems clearly wrong and manifestly unjust. Watson v. State, 204 S.W.3d 404, 414-15 (Tex.Crim.App.2006) (quoting Johnson v. State, 23 S.W.3d 1, 11 (Tex. Crim.App.2000)). Second, we must ask whether, considering the conflicting evidence, the jury's verdict, although legally sufficient, is nevertheless against the great weight and preponderance of the evidence. Id. at 415. In conducting this review, we view all of the evidence in a neutral light. Id. at 414. We are also mindful that a jury has already passed on the facts, and that we cannot order a new trial simply because we disagree with the verdict. Id. What weight to give contradictory testimonial evidence is within the sole province of the jury because it turns on an evaluation of credibility and demeanor. Cain v. State, 958 S.W.2d 404, 408-09 (Tex.Crim. App.1997). Therefore, we must defer appropriately to the fact finder and avoid substituting our judgment for its judgment, and we may find evidence factually insufficient only when necessary to prevent manifest injustice. Id. at 407; see also Johnson, 23 S.W.3d at 12.
Analysis
Effective Consent
In his first and second issues, appellant argues that the evidence was legally and factually insufficient to convict him of intentionally or knowingly operating a motor vehicle without the effective consent of its owner because appellant believed Thomas gave him consent to use the vehicle without any limitations.
A person is guilty of unauthorized operation of a motor vehicle if he "intentionally or knowingly operates another's boat, airplane, or motor-propelled vehicle without the effective consent of the owner." TEX. PEN.CODE ANN. § 31.07(a) (Vernon 2003). Effective consent includes *227 "consent by a person legally authorized to act for the owner." Id. § 31.01(3) (Vernon Supp.2007). Thus, operating a vehicle is unlawful only if the accused is aware that the operation of the vehicle is without the owner's consent. McQueen v. State, 781 S.W.2d 600, 603 (Tex.Crim.App.1989); Edwards v. State, 178 S.W.3d 139, 144 (Tex. App.-Houston [1st Dist.] 2005, no pet.). Testimony that the car owner did not give consent to operate his vehicle can be sufficient to support a finding that an appellant knew he did not have consent to operate the vehicle. McQueen, 781 S.W.2d at 604-05; Edwards, 178 S.W.3d at 145. When an appellant asserts a mistake-of-fact defense concerning the circumstances surrounding the operation of the vehicle, the finder of fact is free to reject the evidence. McQueen, 781 S.W.2d at 604-05; see also Sharp v. State, 707 S.W.2d 611, 614 (Tex. Crim.App.1986) (holding that trier of fact is exclusive judge of facts, credibility of witnesses, and weight to be given to testimony).
The evidence was legally sufficient to show beyond a reasonable doubt that appellant intentionally or knowingly operated Thomas's car without effective consent. Thomas testified at trial that he told appellant to return the car by 5:00 p.m. and that appellant's use of the car after that time was without consent. This testimony was sufficient to establish that appellant knew he no longer had consent to operate the vehicle after 5:00 p.m. See McQueen, 781 S.W.2d at 604-05; Edwards, 178 S.W.3d at 145. The jury was free to believe Thomas's testimony in spite of contrary testimony from appellant and Jennifer Thomas. See McQueen, 781 S.W.2d at 604-05; Sharp, 707 S.W.2d at 614. Therefore, viewing the evidence in the light most favorable to the verdict, we hold that the evidence was legally sufficient for a rational trier of fact to find that appellant intentionally or knowingly operated the vehicle without Thomas's effective consent. See King, 29 S.W.3d at 562.
The evidence is also factually sufficient. Appellant testified that Thomas did not limit his consent to a certain time frame. Although Jennifer Thomas was not present when Thomas loaned appellant the car, she testified that Thomas had loaned appellant the car for the limited purpose of looking for a job and getting the car washed, but that Thomas had not mentioned a time for the car's return. Appellant also presented the contradiction between Thomas's affidavit of non-consent, in which Thomas said that he did not give anyone permission to use his car, and the affidavit of non-prosecution, in which Thomas said that there was a misunderstanding between himself and appellant over when appellant was to return the car.
The State introduced the testimony of Thomas and of Deputy Eldridge that Thomas's permission to appellant was limited in nature. The State also introduced evidence that appellant exceeded that consent by keeping the car much longer than he was supposed to and eventually abandoning it at a car wash in Beaumont instead of returning it to Thomas in Winnie.
Because the evidence relied on by both appellant and the State is almost exclusively testimonial and directly contradictory, the outcome depends on whose testimony seems most credible. Therefore, we must defer to the jury in this case. See Cain, 958 S.W.2d at 407. The jury was free to reject appellant's evidence that there was a mistake or misunderstanding regarding when he was to return the car. See McQueen, 781 S.W.2d at 604-05. Viewing all the evidence in a neutral light, we cannot say that the jury's verdict was against the great weight and preponderance of the evidence. See Watson, 204 S.W.3d at 415. Likewise, the conflicting *228 evidence presented by appellant at trial does not show the jury's verdict to be contrary to the great weight of the evidence. See id. at 415.
We overrule appellant's first and second issues.
Venue
In his third and fourth issues, appellant argues that the evidence was legally and factually insufficient to establish that Chambers County was the proper venue. Venue for the prosecution of an unauthorized use of a motor vehicle case is proper either "where the unauthorized use occurred or in the county in which the vehicle was reported stolen." TEX.CODE CRIM. PROC. ANN. art. 13.23 (Vernon 2003). An objection to venue must be raised in the trial court, or venue will be presumed proven at trial unless the record affirmatively shows the contrary. TEX.R.APP. P. 44.2(c)(1).
Here, appellant did not object to venue at trial, nor does the record affirmatively show that venue was not proper in Chambers County. In fact, the record establishes that the State introduced evidence that the vehicle was reported stolen in Chambers County, where the trial took place. Therefore, we must presume that the State met its burden proving venue was proper in Chambers County. See id.
We overrule appellant's third and fourth issues.
Conclusion
We affirm the judgment of the trial court.
Justice JENNINGS, dissenting.
TERRY JENNINGS, Justice, dissenting.
I agree with the majority that the evidence is legally sufficient to support the conviction of appellant, Ruscel Lovel Battise, of the offense of unauthorized use of a vehicle.[1] However, the majority misapplies the standard of review for determining the factual sufficiency of the evidence. It then erroneously concludes that the evidence in this case is factually sufficient to support the jury's implied finding that appellant operated the complainant's car "without the complainant's effective consent." Accordingly, I respectfully dissent.
In his second point of error, appellant argues that the evidence is factually insufficient to support his conviction because the evidence "did not prove that [he] knew that his operation of [the complainant's car] was without the effective consent of [the complainant]."
In a factual sufficiency review, an appellate court must view all the evidence in a neutral light, both for and against the finding, and set aside the verdict if the proof of guilt is so obviously weak as to undermine confidence in the jury's determination, i.e., that the verdict seems "clearly wrong and manifestly unjust," or the proof of guilt, although legally sufficient, is nevertheless against the great weight and preponderance of the evidence. Watson v. State, 204 S.W.3d 404, 414-15 (Tex.Crim.App.2006).
It is true, as emphasized by the majority, that we should always be "mindful" that a jury is in the best position to pass on the facts and that we should not order a new trial "simply because [we] disagree" with the verdict. See id. at 414. However, being mindful of these principles does not end our analysis. If so, appellate courts could never reverse a trial court judgment on the ground that evidence is *229 factually insufficient to support it. As explained by the Texas Court of Criminal Appeals,
It is in the very nature of a factual-sufficiency review that it authorizes an appellate court, albeit to a very limited degree, to act in the capacity of a so-called "thirteenth juror." Indeed, it is this characteristic of a factual-sufficiency review that justifies the conclusion that a reversal on the basis of factually insufficient evidence has no jeopardy consequences.
Id. at 416-17. Thus, when an appellate court can "say, with some objective basis in the record, that the great weight and preponderance of the (albeit legally sufficient) evidence contradicts the jury's verdict[,]. . . it is justified in exercising its appellate fact jurisdiction to order a new trial." Id. at 417.
A person commits the offense of unauthorized use of a vehicle if he "intentionally or knowingly" operates a vehicle "without the effective consent of the owner." TEX. PENAL CODE ANN. § 31.07(a) (Vernon 2003). "[W]hat separates lawful operation of another's motor vehicle from unauthorized use is the actor's knowledge of a `crucial circumstance surrounding the conduct' that such operation is done without the effective consent of the owner." McQueen v. State, 781 S.W.2d 600, 604 (Tex.Crim.App.1989). Thus, the State had the burden to prove to the jury that appellant was actually aware that his operation of the complainant's car was without the complainant's effective consent. See id.
Here, a neutral review of the record reveals that the complainant, Hubert Thomas, testified that on February 3, 2007, he loaned his car to appellant, who had been dating Thomas's sister, Jennifer, for several years. Appellant was to wash the car and return it to Thomas by 5:00 p.m. When appellant failed to return the car, Thomas reported the car stolen at about 10:30 p.m. Chambers County Sheriff's Deputy S. Eldridge testified that he met with Thomas, who told Eldridge that he had loaned the car to appellant with instructions to wash it and return it.
However, Thomas did not tell Deputy Eldridge that he instructed appellant to return the car by a certain time. Thomas actually told Eldridge that he only gave appellant permission to take the car to a car wash in Winnie, Texas. More importantly, Thomas testified in an affidavit, on a form provided to him by Eldridge, that,
I, [Thomas], the undersigned affiant, do solemnly s[w]ear that I did not give anyone any permission, consent[,] or authority to take from my possession at. . . 547 N. McDaniel, Winnie, TX 77665 in Chambers County, Texas, the following described property[:] . . . 1998 Lincoln Town[ ]Car . . . [,] [value listed at $11,000,] which offense occurred on or about the 3rd day of February, 2005.
(Emphasis added). The preprinted form was apparently used by the Chambers County Sheriff's Office to make sure that its law enforcement authority was not used by individuals to resolve what are essentially civil, and not criminal, disputes. Although Thomas had just told Eldridge that he in fact loaned his car to appellant, Eldridge, inexplicably, accepted Thomas's sworn affidavit to the contrary and then reported the car "stolen."
Under any objective standard, based upon the complainant's perjured[2] testimony in his affidavit to Deputy Eldridge, a criminal case against appellant should never *230 have been filed in the first place. Moreover, after appellant's arrest, the complainant, in an affidavit of non-prosecution, further testified,
When I signed the form stating [that] I did not give [appellant] consent to use my 1998 Lincoln Town[][C]ar, I[,] in good faith[,] believed that [appellant] was deliberately disobeying my instructions as to when to return my car. After talking with family members and remembering how I had loaned other cars to [appellant] for long trips over long periods of time, and after learning from my sister that, after I signed the non-consent form, [appellant] did call her and tell my sister where I could find my car, I[,] in good faith[,] have come to believe that [appellant] and I had a mutual misunderstanding as to where [appellant] was permitted to go with my car and when [appellant] was to return my car. I[,] in good faith[,] believe that this matter should be settled in a civil court of law rather than in a criminal court of law.
(Emphasis added). The complainant's testimony that he and appellant had a "mutual misunderstanding" as to where appellant was permitted to go with the complainant's car and when appellant was to return the car went uncontradicted at trial.
The majority, emphasizing that we "must defer to the jury in this case," holds that the jury's verdict was not against the great weight and preponderance of the evidence. In doing so, the majority allows the principle that we must normally defer to the jury's fact-finding role to trump any objective assessment of the evidence before the jury. Under the majority's reasoning, as noted above, an appellate court could never reverse a judgment on the ground that the evidence is factually insufficient to support it.
The bottom line is that the State had the burden to prove beyond a reasonable doubt that appellant was actually aware that his operation of the complainant's car was without the complainant's effective consent. See McQueen, 781 S.W.2d at 604. Although the State, through the complainant, presented some evidence to support such an implied finding, an objective review of the record reveals two vital facts that logically preclude such a finding: (1) the complainant, himself, testified that he and appellant had a "mutual misunderstanding" as to where appellant was permitted to go with the complainant's car and when appellant was to return the car, and (2) the complainant committed perjury in his initial affidavit to Deputy Eldridge when he testified that he "did not give anyone any permission, consent[,] or authority" to take his car when he, in fact, had done exactly that.
I would hold that the great weight and preponderance of the evidence contradicts the jury's implied finding that appellant was aware that his operation of the complainant's car was without the complainant's effective consent. See Watson, 204 S.W.3d at 417 (explaining when appellate court has "objective basis in the record[,] that the great weight and preponderance of the (albeit legally sufficient) evidence contradicts the jury's verdict[,] . . . it is justified in exercising its appellate fact jurisdiction to order a new trial"). The jury's finding that appellant was so aware is clearly wrong and manifestly unjust. Accordingly, I would reverse the judgment of the trial court and remand the case for a new trial. The majority's holding to the contrary and affirmance of the trial court's judgment are in serious error.
NOTES
[1] See TEX. PEN.CODE ANN. § 31.07(a) (Vernon 2003).
[2] This affidavit was a pre-printed form from Chambers County Sheriff's Department, and the entire text reads, "I, Thomas, Hubert Otis Jr., the undersigned affiant, do solemnly s[w]ear that I did not give anyone any permission, consent[,] or authority to take from my possession at . . . 547 N. McDaniel, Winnie, TX 77665 in Chambers County, Texas, the following described property[:] . . . 1998 Lincoln Town[ ]Car . . .[,] [value listed at $11,000,] which offense occurred on or about the 3rd day of February, 2005." Thomas also designated that the property had not yet been recovered.
[3] The pertinent part of this affidavit stated, "When I signed the form stating [that] I did not give [appellant] consent to use my 1998 Lincoln Town[ ][C]ar, I[,] in good faith[,] believed that [appellant] was deliberately disobeying my instructions as to when to return my car. After talking with family members and remembering how I had loaned other cars to [appellant] for long trips over long periods of time, and after learning from my sister that, after I signed the non-consent form, [appellant] did call her and tell my sister where I could find my car, I[,] in good faith[,] have come to believe that [appellant] and I had a mutual misunderstanding as to where [appellant] was permitted to go with my car and when [appellant] was to return my car. I[,] in good faith[,] believe that this matter should be settled in a civil court of law rather than in a criminal court of law."
[1] See Tex. Penal Code Ann. § 31.07(a) (Vernon 2003).
[2] A person commits the Class A misdemeanor offense of perjury if, "with intent to deceive and with knowledge of the statement's meaning[,]. . . he makes a false statement under oath." TEX. PENAL CODE ANN. § 37.02(a) (Vernon 2003).
|
Tumour detection in the liver: role of multidetector-row CT.
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Identification of a crustacean β-1,3-glucanase related protein as a pattern recognition protein in antibacterial response.
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Trial of Vasopressin and Epinephrine to Epinephrine Only for In-Hospital Pediatric Cardiopulmonary Resuscitation (Vasopressin)
The safety and scientific validity of this study is the responsibility of the study sponsor and investigators. Listing a study does not mean it has been evaluated by the U.S. Federal Government.
Read our disclaimer for details.
Combination vasopressin and epinephrine will decrease the time to ROSC [ Time Frame: Immediate ]
Vasopressin and epinephrine will improve the proportion of CPA survivors with favorable neurologic outcome (short-term Pediatric Overall Performance Category) [POPC] score discharge of 1-3 or unchanged from hospital admission at the time of hospital . [ Time Frame: Period of hospitalization ]
Vasopressin and epinephrine will improve the proportion of CPA survivors with favorable neurologic outcome (short-term Pediatric Cerebral Performance Category) [PCPC] score of 1-3 or unchanged from hospital admission at time of hospital discharge. [ Time Frame: Period of Hospitalization ]
Combination vasopressin and epinephrine will decrease the time to ROSC [ Time Frame: Immediate ]
Vasopressin and epinephrine will improve the proportion of CPA survivors with favorable neurologic outcome (short-term Pediatric Overall Performance Category) [POPC] scordischargee of 1-3 or unchanged from hospital admission at the time of hospital . [ Time Frame: Period of hospitalization ]
Vasopressin and epinephrine will improve the proportion of CPA survivors with favorable neurologic outcome (short-term Pediatric Cerebral Performance Category) [PCPC] score of 1-3 or unchanged from hospital admission at time of hospital discharge. [ Time Frame: Period of Hospitalization ]
Trial of Vasopressin and Epinephrine to Epinephrine Only for In-Hospital Pediatric Cardiopulmonary Resuscitation
Official Title ICMJE
A Prospective, Randomized, Controlled Trial of Combination Vasopressin and Epinephrine to Epinephrine Only for In-Intensive Care Unit Pediatric Cardiopulmonary Resuscitation
Brief Summary
Cardiac arrest has a very poor prognosis, especially with prolonged efforts at resuscitation, and unfortunately, survivors are often severely neurologically impaired. CPA in children is often the result of a prolonged illness rather than a sudden, primary cardiac event as is frequent in adults. This necessitates that resuscitation research must be conducted separately for pediatric and adult patients. Authorities currently endorse the use of epinephrine for restoring spontaneous circulation based on its ability to maintain diastolic blood pressure and subsequent blood flow to the heart during resuscitation. However, human studies have shown no clear survival benefit of epinephrine and have elucidated concerning adverse effects. Recently, both the European Resuscitation Council and the American Heart Association have recognized the use of vasopressin as a promising vasoconstrictor and an alternative or adjunct to epinephrine in the resuscitation of adults. Vasopressin causes profound vasoconstriction without the adverse effects of epinephrine and is associated with improved blood flow to the heart and brain. This increased cerebral blood flow has been associated with better neurologic outcome in animal studies. In light of compelling animal and human studies of combined vasopressin and epinephrine, pediatric trials are indicated for vasopressin usage in pediatric CPR. This study will evaluate the addition of the administration of vasopressin to standard advanced CPR therapy (epinephrine alone) for pediatric patients that experience in-intensive care unit CPA to assess for improved time to return of spontaneous circulation (ROSC), survival to 24 hours, survival to hospital discharge, and neurologic outcome. When a patient experiences a CPA, standard Pediatric Advanced Life Saving (PALS) protocols as endorsed by the American Heart Association will be initiated. This will include receiving epinephrine as the first vasopressor medication. Patients will then be randomized to receive vasopressin (treatment group) or epinephrine (control group) as the second vasopressor medication, if needed. If more then two doses of vasopressor medication is required in either group, epinephrine will be administered according to the PALS algorithm until the end of the event. All CPA events meeting inclusion criteria will be entered into the National Registry of Cardiopulmonary Resuscitation (NRCPR) Database, which tracts all CPA events at Children's Medical Center Dallas. Prior to commencement of the RCT, a pilot trial of 10 patients will be completed to assess preliminary safety, feasibility, and effectiveness of combination epinephrine-vasopressin for pediatric in-intensive care unit CPA refractory to initial epinephrine dosing. All pilot patients will receive vasopressin as the second vasopressor medication.
Detailed Description
CONCISE SUMMARY OF PROJECT:
The study design will be a prospective, randomized, controlled clinical trial to be conducted in the PICU of CMC Dallas (UT Southwestern Medical Center) following a pilot trial enrolling 10 patients. This study will be undertaken after consultation with and acceptance by the resuscitation committee and PICU at CMC. Pediatric patients that experience in-hospital CPA who remain in cardiac arrest despite CPR and an initial, standard dose of epinephrine (0.01 mg/kg), will be randomly assigned to receive either standard dose epinephrine (0.01 mg/kg) or vasopressin (0.8 units/kg) rescue as the second vasopressor medication.
SUMMARY OF STUDY PROCEDURES:
When a patient experiences a CPA, standard Pediatric Advanced Life Saving (PALS) protocols will be followed. This will include: establishment of an airway, support of breathing including supplemental oxygen, evaluation of cardiac rhythm, chest compressions, electrical defibrillation if appropriate, and administration of epinephrine as the first vasopressor medication. The quality of CPR will be monitored and reported by the documenting nurse for the event, to include rate of ventilation, rate and depth of chest compressions, and no-flow time (time without chest compressions). In addition, monitoring of end-tidal CO2, diastolic blood pressure via arterial line if present, and human observation and coaching will be employed to track CPR quality. The patient will then be randomized to receive either epinephrine (control group) or vasopressin (treatment group) as the second vasopressor medication if needed. If further doses of medication are required in either group, epinephrine will be administered according to the PALS algorithm until the end of the CPA event as defined below. Thus, the only difference between the groups will be the replacement of epinephrine with vasopressin as the second vasopressor medication in the algorithm. A total of 120 patients will be enrolled in the randomized, controlled trial portion of the study.
After completion of PICU staff training and prior to the randomized, controlled trial, a pilot trial of vasopressin resuscitation involving 10 patients will be conducted to test the feasibility and safety of study methodology. Pilot participants who meet inclusion criteria will be enrolled serially from the PICU at CMC. Study protocol for the treatment group of the randomized, control trial (vasopressin + epinephrine) will be followed. Collected information will be reviewed by members of a Data Safety and Monitoring Board (DSMB) before proceeding to the next phase of the trial. These patients will not be included in the final analysis.
The requirement of two doses of resuscitation medications will provide adequate time for randomization and use of vasopressin. It will also exclude those children with rapidly reversible conditions who would not have time to benefit from vasopressin versus epinephrine intervention. Stratified randomization technique will be used to control for the effect of vasopressor infusions the patient is receiving at the time of cardiac arrest. Stratification will be based on 4 groups i.e., epinephrine, vasopressin, both, or neither. In order to avoid extreme imbalance in the size of the treatment arms, a permuted block design will be used and the size of each block will be set at 6. SPSS pseudo-random number generator will be used to design the randomization charts. Randomization will be accomplished by the PICU pharmacist via sealed envelopes designating study arm assignment that will be available on every code cart in the PICU. Thus, the study medication will be blinded to all but the pharmacist.
On admission to the PICU, families of all patients will be informed and educated of this ongoing study with exception from informed consent (EFIC) via posters in the waiting rooms and a brochure regarding the study clearly explaining how to "opt out" of inclusion. The number of patients who "opt out" of inclusion will be documented and available to the IRB at their request. Representatives of the study will be available by phone 24 hours a day and in person in the waiting room daily to discuss the study and answer questions. Parents will be informed of inclusion within 24 hours in person or by phone or letter if unavailable. This notification will be documented and consent will be elicited for follow up data collection. CPA events will be limited to those occurring in the PICU only. Providers that respond to CPA events will be in-serviced regarding the study protocol prior to implementation via didactic sessions. Input from pharmacists and providers in the PICU will be sought to assure the easiest implementation possible. Vasopressin is currently available for administration on all resuscitation (code) carts at CMC. Inclusion and exclusion criteria will be posted on all code carts to assist providers. Current protocol at CMC to enter all CPA events into the National Registry of Cardiopulmonary Resuscitation (NRCPR) will be followed. This data is a complete account of the events of the CPA and will be sufficient to meet all of the study's stated goals and objectives. Only data pertinent to the outcomes of this study will be reviewed from the database. This data will also be reviewed to assure standardization of execution of the study protocol.
Time to Completion Given that 89 CPA events met inclusion criteria from January 2005 to June 2006 in the PICU at CMC, approximately 30 months will be required to enroll 130 total patients into this study (10 patients in pilot trial, 120 patients in main study). Subjects will be enrolled in the study until discharge or in-hospital death.
Definition CPA End of Event
. ROSC that is sustained for > 20mins with no further need for chest compressions, including with a pacemaker or extracorporeal membrane oxygenation OR
. Resuscitation event is terminated and patient is declared dead (unresponsive to advanced life support, medical futility, advance directive, restriction by family member)
SPECIAL PRECAUTIONS:
A full resuscitation team will be present whenever vasopressin is administered including a physician, nurse, respiratory technician, clinical technician, and pharmacist. A sign will be clearly posted on the bed of any patient whose proxy has "opted" the patient out of the study or who meets any other exclusion criteria. The DSMB will evaluate the study for clear benefit or harm, lack of efficacy, or unacceptable toxicity of vasopressin.
SOURCES OF RESEARCH MATERIALS/ COLLECTION OF FOLLOW UP DATA:
Data from CPA events will be collected from the NRCPR database which is completed at the conclusion of every CPA event currently at CMC. Data is collected in six major categories of variables: (1) Facility data, (2) Patient demographic data, (3) Pre-event data, (4) Event data, (5) Outcome data, and (6) Quality improvement data. All patient identifiers will be destroyed at the earliest possible opportunity. Data will be de-identified for analysis. This data will include laboratory and treatment data from the hospitalization in the PICU at CMC before, during, and after resuscitation. Specifically, this data will include: age, gender, vital signs, treatments, laboratory results, neurologic exam (PCPC and POPC scores) and physical exam. The data collection is based on in-hospital Utstein-style guidelines.
One dose of vasopressin (0.8 units/kg) intravenously rescue as the second vasopressor medication.
Other Name: Pitressin
Drug: Epinephrine
One standard dose epinephrine (0.01 mg/kg) intravenously rescue as the second vasopressor medication.
Other Name: Adrenaline
Study Arms
Experimental: 1
Pediatric patients that experience in-hospital CPA who remain in cardiac arrest despite CPR and an initial, standard dose of epinephrine (0.01 mg/kg), will be randomly assigned to receive vasopressin (0.8 units/kg) rescue as the second vasopressor medication.
Intervention: Drug: Vasopressin
Active Comparator: 2
Pediatric patients that experience in-hospital CPA who remain in cardiac arrest despite CPR and an initial, standard dose of epinephrine (0.01 mg/kg), will be randomly assigned to receive standard dose epinephrine (0.01 mg/kg)rescue as the second vasopressor medication.
Intervention: Drug: Epinephrine
Publications *
Not Provided
* Includes publications given by the data provider as well as publications
identified by ClinicalTrials.gov Identifier (NCT Number) in Medline.
Recruitment Information
Recruitment Status ICMJE
Unknown status
Estimated Enrollment ICMJE (submitted: March 4, 2008)
130
Original Estimated Enrollment ICMJE
Same as current
Estimated Study Completion Date
December 2011
Estimated Primary Completion Date
April 2011 (Final data collection date for primary outcome measure)
Eligibility Criteria ICMJE
Inclusion Criteria:
All children, ages 0 to 18 years, admitted to the PICU who experience CPA requiring either chest compressions and/or defibrillation. This will include males, females and Spanish speaking individuals.
Patients must require at least 2 doses of vasopressor medication during the CPA event (all patients would receive epinephrine as first dose, followed by either epinephrine or vasopressin as second dose depending on randomization, all subsequent doses required would be epinephrine) given via any route (intravenous, intraosseous, or endotracheal).
Exclusion Criteria:
Do Not Attempt Resuscitate (DNAR) patients
Chemical code only (i.e., no CPR/defibrillation)
Events not requiring chest compressions and/or defibrillation
Events with a pulse requiring synchronized or unsynchronized cardioversion |
Q:
Can't delete a folder win 10
I have a folder on my desktop called con.
15/02/2016 11:29 <DIR> con
I cannot open it
I have tried the following
run cmd as administrator
kill explorer.exe
rmdir con
C:\Users\sean\Desktop>rmdir con /s
con, Are you sure (Y/N)? Y
Access is denied.
Set the security settings
The only option remaining that I have seen is to use a gparted on a linux distribution. However, there must be a way to do it in windows?
Does anyone know why this is happening and how to delete an unopenable and undeletable folder in Winodws 10?
A:
Source: http://answers.microsoft.com/en-us/windows/forum/windows_8-files/a-folder-that-refused-to-be-deleted-invalid-file/a8506e19-d623-4af0-ab19-0fd17a672a3a
Solution quote modified to suit your folder location
Solution
You can remove this using the command line, but will want to reference the location differently than you normally would (using UNC).
If the folder is C:\Users\sean\Desktop\con, then this command entered on the command line will remove it:
rd \\.\\c:\Users\sean\Desktop\con /S /Q
rd is the command line tool to remove the directory.
\\. refers to the current computer.
\c:\Users\sean\Desktop\con is the path of the offending folder entry.
/S is a switch that tells rd to remove all subdirectories and files (like the old DELTREE command).
/Q is a switch that tells rd to this removal silently (you won't be prompted for removing the contents).
Change the \c:\Users\sean\Desktop\con path to whatever the location is for your CON folder.
This deletes the entire folder and its contents. It may be possible to rename or copy the folder to another name. I didn't have much luck with the rename, but I also didn't spend too much time on it.
Note that although you can use Windows Explorer to browse to the location \., Explorer doesn't let you delete the folder. You need to use the command line to remove the CON folder.
|
Lentivirus-mediated gene transfer to the respiratory epithelium: a promising approach to gene therapy of cystic fibrosis.
Gene therapy of cystic fibrosis (CF) lung disease needs highly efficient delivery and long-lasting complementation of the CFTR (cystic fibrosis transmembrane conductance regulator) gene into the respiratory epithelium. The development of lentiviral vectors has been a recent advance in the field of gene transfer and therapy. These integrating vectors appear to be promising vehicles for gene delivery into respiratory epithelial cells by virtue of their ability to infect nondividing cells and mediate long-term persistence of transgene expression. Studies in human airway tissues and animal models have highlighted the possibility of achieving gene expression by lentiviral vectors, which outlasted the normal lifespan of the respiratory epithelium, indicating targeting of a 'stem cell' compartment. Modification of the paracellular permeability and pseudotyping with heterologous envelopes are the strategies currently used to overcome the paucity of specific viral receptors on the apical surface of airway epithelial cells and to reach the basolateral surface receptors. Preclinical studies on CF mice, demonstrating complementation of the CF defect, offer hope that lentivirus gene therapy can be translated into an effective treatment of CF lung disease. Besides a direct targeting of the stem/progenitor niche(s) in the CF airways, an alternative approach may envision homing of hematopoietic stem cells engineered to express the CFTR gene by lentiviral vectors. In the context of lentivirus-mediated CFTR gene transfer to the CF airways, biosafety aspects should be of primary concern. |
Obama aims to avoid a 'cycle of escalation' in cyberattacks by countries
U.S. President Barack Obama said his country has had problems with cyber intrusions from Russia and other countries in the past, but aims to establish some norms of behavior rather than let the issue escalate as happened in arms races in the past.
Obama’s statement on the sidelines of the G20 summit in China, after he met with Russian President Vladimir Putin, did not refer specifically to a recent hack of the Democratic National Committee of the Democratic Party that the U.S. Federal Bureau of Investigation is probing.
Politically embarrassing emails from the breach were leaked ahead of the convention of the party, with many security experts holding that the hack had the backing of Russian intelligence services. Whistleblowing website WikiLeaks released the emails but did not disclose their source. The U.S. government hasn’t blamed Russia for the incident.
“I am not going to comment on specific investigations that are still live and active, but I'll tell you that we had problems with cyber intrusions from Russia in the past, from other countries in the past," Obama told reporters, according to news outlets.
The president described moving into a new era where a number of countries have significant cyber capacities, and “frankly we have got more capacity than anybody both offensively and defensively,” he said in an apparent warning to the Russians.
He said the goal of the U.S. was not to duplicate in cyberspace a “cycle of escalation” akin to other arms races in the past, but to start “instituting some norms" so that everybody is acting responsibly.
Obama said there are going to be enough problems in cyberspace with "non-state actors" using the Internet for theft and other illegal purposes, which creates the need for protecting critical infrastructure and securing financial systems.
"What we cannot do is have a situation in which, certainly, this becomes the wild, wild West, where countries that have significant cyber capacity start engaging in unhealthy competition or conflict through these means," Obama said. He said he had discussed the topic of cybersecurity norms with Putin and earlier with some other countries, and is already seeing some willingness from a lot of countries to adopt the rules, though it will have to be seen whether they are following them. |
Gertrude Lippincott Award
The Gertrude Lippincott Award is an annual award for the best English-language article in the field of dance studies. Previously it was awarded by the Society of Dance History Scholars; since 2017 it has been awarded by the Dance Studies Association.
The $500 award was named after modern dance teacher and mentor Gertrude Lippincott and honors exemplary dance scholarship. Ms. Lippincott was herself honored in 1973 with the National Dance Association's Heritage Award for her contributions to dance education. She was one of the founders of the Congress on Research in Dance and of the Modern Dance Center of Minneapolis. She was also an editor for the periodicals Dance Observer and Dance Magazine.
Award Winners
2018 - VK Preston, "Baroque Relations: Performing Silver and Gold in Daniel Rabel’s Ballet of the Americas," in The Oxford Handbook of Dance and Reenactment, edited by Mark Franko (New York: Oxford, 2017), pp. 285-310.
2017 - Kareem Khubchandani, “Snakes on the Dance Floor: Bollywood, Gesture, and Gender,” The Velvet Light Trap 77 (2016), pp. 69-85.
2016 - Brandon Shaw, “Phantom Limbs and the Weight of Grief in Sasha Waltz’s noBody” Theatre Journal 67 (2015), pp. 21-42.
2016 (Honorable Mention) - Andrea Harris, “Sur la Pointe on the Prairie: Giuseppina Morlacchi and the Urban Problem in the Frontier Melodrama,” in The Journal of American Drama and Theatre 27:1 (Winter 2015).
2015 - Sherril Dodds, “The Choreographic Interface: Dancing Facial Expression in Hip-Hop and Neo-Burlesque Striptease” in Dance Research Journal 46:Special Issue 02 (2014), pp. 39–56.
2014 - Alexandra Kolb, “The Migration and Globalization of Schuhplattler Dance: A Sociological Analysis” in Cultural Sociology 7:1, pp. 39–55
2013 - Anurima Banerji, “Dance and the Distributed Body: Odissi, Ritual Practice, and Mahari Performance” in About Performance 11: 7–39
2013 - J. Lorenzo Perillo, “‘If I was not in prison, I would not be famous’: Discipline, Choreography, and Mimicry in the Philippines” in Theatre Journal 63: 607–621.
2011 - Selby Wynn Schwartz, “Martha@Martha: A Séance with Richard Move” in Women & Performance: A Journal of Feminist Theory 20.1 (2010): 61–87.
2011 (Honorable Mention) - Öykü Potuoglu-Cook, “The Uneasy Vernacular: Choreographing Multiculturalism and Dancing Difference Away in Globalised Turkey” in Anthropological Notebooks 16.3 (2010): 93–105.
2010 - Kate Elswit, "'Berlin ... Your Dance Partner is Death" in TDR: The Drama Review, 53:1 2009, pp. 73–92.
2009 - Cindy Garcia, "Don't leave me Celia: Salsera homosociality and pan-Latina corporealities" in Women and Performance: A Journal of Feminist Theory, 18:3, pp. 199–213.,
2009 - honorable mention to Victoria Phillips Geduld, "Performing Communism in the American Dance: Culture, Politics, and the New Dance Group" in American Communist History 7:1, 2008, pp. 39–65.
2009 - honorable mention to Melissa Blanco Borelli, "Yahora que vas a hacer, mulata? Hip choreographies in the Mexican cabaretera film 'Mulata'" in Women and Performance: A Journal of Feminist Theory, 18:3, pp. 215–233.
2008 - Priya Srinivasan, "The Bodies Beneath the Smoke or What's Behind the Cigarette Poster: Unearthing Kinesthetic Connections in American Dance History" in Discourse in Dance, Ramsey Burt and Susan Leigh Foster, editors, Volume 4 Issue 1 2007, pp. 7–48.
2008 - Rebekah Kowal, "Dance Travels: 'Walking With Pearl'" Performance Research, 12(2), pp. 85–94, 2007.
2007 - Anthea Kraut, "Recovering Hurston, Reconsidering the Choreographer" which appeared in Women and Performance: A Journal of Feminist Theory, 16/1 (March 2006).
2007 - honorable mention to April K. Henderson, "Dancing Between Islands: Hip Hop and the Samoan Diaspora" which appeared in The Vinyl Ain't Final: Hip Hop and the Globalization of Black Popular Culture, Dipannita Basuand and Sidney J. Lemelle, eds., London: Pluto Press, 2006.
2006 - Kimerer LeMothe, " 'A God Dances through Me': Isadora Duncan on Friedrich Nietzsche's Revaluation of Values," Journal of Religion 85 (2), 2005, pp. 241–266.
2005 - No prize awarded.
2004 - Danielle Goldman, "Ghostcatching: An Intersection of Technology, Labor, and Race," Dance Research Journal v. 35/2 & 36/2 (Winter 2003 & Summer 2004 combined issue) pp. 68–87. 2004.
2003 - No prize awarded.
2002 - Theresa Jill Buckland, "Th'Owd Pagan Dance": Ritual, Enchantment, and an Enduring Intellectual Paradigm", in Journal for the Anthropological Study of Human Movement, vols 11, no. 4 and 12, no. 1, Fall 2001/Spring 2002.
2001 - Petra Kuppers, "Deconstructing Images: Performing Disability," Contemporary Theatre Review 11.3&4 (2001).
2000 - Anne Flynn and Lisa Doolittle, "Dancing in the Canadian Wasteland: A Post-Colonial Reading of Regionalism in the 1960s and 1970s," in Dancing Bodies, Living Histories: New Writing about Dance and Culture, edited by Anne Flynn and Lisa Doolittle (Banff Centre Press, 2000).
1999 - Susan C. Cook, "Watching Our Step: Embodying Research, Telling Stories," in Audible Traces: Gender, Identity, and Music, edited by Elaine Barkin and Lydia Hamessley (Zurich: Carciofoli Verlagshaus, 1999).
1998 - Ananya Chatterjea, "Chandralekha: Negotiating the Female Body and Movement in Cultural/Political Signification," Dance Research Journal 30.2 (Spring 1998).
1997 - Jody Bruner, "Redeeming Giselle: Making a Case for the Ballet We Love to Hate," in Rethinking the Sylph: New Perspectives on the Romantic Ballet, edited by Lynn Garafola (Middletown, Conn.: Wesleyan University Press, 1997).
1996 - Linda J. Tomko, "Fête Accompli: Gender, 'Folk Dance,' and Progressive-Era Political Ideals in New York City," in Corporealities: Dancing Knowledge, Culture, and Power, edited by Susan Leigh Foster (London and New York: Routledge, 1996).
Resources
Gertrude Lippincott Award
SDHS Award Winners
Society of Dance History Scholars
Category:Dance awards
Category:Humanities awards |
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Hi,
until now I haven't checked Exchange.hasOut() in Processor.process(Exchange)
and always used only Exchange.getIn(). However, I started using
BindingComponent which creates BindingEndpoint and this BindingEndpoint
calls delegate Processor with Exchange that has both in and out Messages
set.
My question is if this behaviour of BindingEndpoint is correct or if
BindingEndpoint shall set only in Message and the out Message shall be null.
The code snippet is from the BindingEndpoint and the bold statement copies
the in Message into out Message. It is not clear to me why the statement is
there if createNextExchange(bindingExchange) before already takes care of
the message copying and sets the out Message to null.
Thank you.
public void pipelineBindingProcessor(Processor bindingProcessor,
Exchange exchange, Processor delegateProcessor) throws Exception {
// use same exchange - seems Pipeline does these days
Exchange bindingExchange = exchange;
bindingProcessor.process(bindingExchange);
Exchange delegateExchange = createNextExchange(bindingExchange);
* ExchangeHelper.copyResults(bindingExchange, delegateExchange);*
delegateProcessor.process(delegateExchange);
}
protected Exchange createNextExchange(Exchange previousExchange) {
Exchange answer = previousExchange;
// now lets set the input of the next exchange to the output of the
// previous message if it is not null
if (answer.hasOut()) {
answer.setIn(answer.getOut());
answer.setOut(null);
}
return answer;
}
--
View this message in context: http://camel.465427.n5.nabble.com/Must-Processor-check-if-Exchange-has-out-message-already-tp5739977.html
Sent from the Camel - Users mailing list archive at Nabble.com. |
import DS from 'ember-data';
export default DS.RESTSerializer.extend({});
|
nd -6*l**2 + 4*l**2 + 3*l**2 + (3*l - 4*l + 2*l)*(-10*l + 7*l + 17*l) + (5*l + 11*l + 3*l)*(0 - 2*l + 0) + 2*l**2 + l**2 - 5*l**2.
-25*l**2
Expand (-66*b + 66*b + 17*b**4)*(-2 + 66*b + 2 - 20*b).
782*b**5
Expand -16967*w**2 + 34181*w**2 - 16476*w**2 + (0*w + 4*w - 2*w)*(2*w + 4*w - 4*w) - w**2 + 0 + 0.
741*w**2
Expand ((s**3 + s**3 - s**3)*(-1 + 5 - 2) + 3*s**3 - 2*s**3 - 23*s**3)*(s + 28*s + 15*s).
-880*s**4
Expand (-9*o - 4 + 4)*(171 + 177 - 450 + 146).
-396*o
Expand 106*g + 315*g**5 - 106*g + (5*g**4 - 5*g**4 + 3*g**4)*(0*g + 0*g - 2*g).
309*g**5
Expand 31 - 31 - 5*d + (-3*d - 2*d - 5*d)*(-1 - 8 - 6).
145*d
Expand -1 + 1 + 46147*x - 29533*x + (-2 + 2 - 1)*(-4*x - 2*x + 7*x) - 2*x - 2*x + 3*x.
16612*x
Expand (-296*r + 148*r + 152*r)*(29 + 18 + 70 - 24).
372*r
Expand (-5*g + 2*g + 0*g)*(-112 + 5358*g**3 + 278 - 166).
-16074*g**4
Expand (4*q**4 + 0*q**4 - 3*q**4)*(979 + 1552 - 862) - q**4 - 5*q**4 + 7*q**4.
1670*q**4
Expand (-8*b + 2*b + 0*b)*(2 - 5 + 2 - 4 + 3 + 2 + (2 - 4 + 4)*(22 + 8 + 22)).
-624*b
Expand -8*i**3 + 4*i**3 - 5*i**3 + (-3 + 2 + 0)*(-3*i**3 + 0*i**3 + i**3) + 2*i + 2*i**3 + 6*i**3 + 7*i**3.
8*i**3 + 2*i
Expand -1 + u + 1 + (-1 + 3 + 0)*(-5*u + 4*u + 3*u) + 2*u + 3*u - 3*u + 2 + u - 2 - 1115*u - 1317*u + 3770*u.
1346*u
Expand (107*a - 89*a + 37*a)*(-4 + 4 + 2)*(6*a - 15*a - 3*a).
-1320*a**2
Expand (-2 + 0 + 3)*(4*w + 4*w - 10*w)*(-5*w**3 + 5*w**3 + 7*w**4)*(-7 - 3 + 0).
140*w**5
Expand (-8*z - 17*z + 0*z)*(202*z**4 + 108*z**4 - 155*z**4).
-3875*z**5
Expand -724*r**2 - 596*r**2 + 1459*r**2 + (16*r**2 - 269*r + 269*r)*(1 + 2 - 1).
171*r**2
Expand (-139*b + 7*b - 330*b)*(0 + 5 - 4).
-462*b
Expand (2 - 5 - 4)*(-6*g - 4*g + 3*g + 2*g + 0*g + 2*g + (0*g - 2*g + 3*g)*(-4 + 2 + 3)).
14*g
Expand (-3*o + 0*o + o + 28*o + 4905 - 4905 + (0 - 1 + 3)*(2*o + 2 - 2))*(4*o + 5*o - 23*o**2 + 24*o**2).
30*o**3 + 270*o**2
Expand (6 - 2 - 3)*(-2*b + 4*b - 3*b)*(9*b + 15*b + 42*b).
-66*b**2
Expand (-2*o**2 + 10*o - 522*o**2 + 1 + 624*o**2)*(-2*o + 1 - 1).
-200*o**3 - 20*o**2 - 2*o
Expand (23*m + 15*m - 21*m)*(36 - 19 - 143)*(-m + 4*m - 4*m)*(0*m + 4*m - m).
6426*m**3
Expand (-997 + 2629 + 2066)*(9*o - o + 0*o).
29584*o
Expand (-28*m**2 - 10*m**2 - 24*m**2)*(-1 - 2 + 4 - m**2).
62*m**4 - 62*m**2
Expand 38*j - 23*j + 3*j + 9*j - 30*j + 14*j + (1 + 3 - 2)*(-3*j - j - j).
j
Expand (-1103 + 1119 - r + 18*r)*(10*r**2 - 7*r**2 + 8*r**2)*(-4*r - 2*r + 5*r).
-187*r**4 - 176*r**3
Expand (24*b - 6*b - 14*b)*(60 - 40 + 41)*(-4 + 4 + 4).
976*b
Expand 55*p**3 - 11*p**3 - 20*p**3 + 982*p**3 + 911*p**3 - 2145*p**3 + (3*p - 3*p + 2*p)*(3*p**2 - 5*p**2 + 4*p**2).
-224*p**3
Expand (299 + 448 - 576)*(-3 + 0 + 5)*(-x + 2 - 2).
-342*x
Expand (1 + 5*b**2 - 1)*(7 + 5 + 9)*(-2 + 2 - 3*b)*(-5 + 5 + 3).
-945*b**3
Expand (2*k - 2*k + k**2)*(5*k**3 + 0*k**3 - 3*k**3) + (10*k + 2*k**3 - 10*k)*(-3*k**2 - 5*k**2 + 31*k**2).
48*k**5
Expand (4*q**2 - 12*q**2 - 6*q**2)*(10*q - 10*q + 3*q**2) + (2*q**3 - 3*q**3 - q**3)*(3*q - 2*q + q).
-46*q**4
Expand (44*r - 69*r - 54*r)*(0 + 0 - r**3) - 7*r**2 + 2722*r**3 - 2721*r**3 + 3*r**4 - 4*r**4.
78*r**4 + r**3 - 7*r**2
Expand (3*n - n - n)*(3 + 2 - 7) - n + 0*n - n + n - 7*n - 14*n + 2 + 3*n - 2 + 5*n - n - 2*n + (-n + 2*n - 3*n)*(3 - 6 + 1).
-15*n
Expand (1349*b**2 + 18994 + 18998 - 37987)*(0 + 0 - b**3).
-1349*b**5 - 5*b**3
Expand h**2 - 1 + 1 + (h + 3 - 3*h + 0*h)*(-10 - 2*h + 3 + 0) + 5*h**2 - 2*h**2 - 2*h**2.
6*h**2 + 8*h - 21
Expand (-18 - 24*n + 18)*(4 - 6 + 1)*(50 + 33 - 6)*(1 - 1 - 2).
-3696*n
Expand (40*i - 32*i**2 + 165*i - 26 - 204*i)*(4 - 2 + 1).
-96*i**2 + 3*i - 78
Expand (6*b + 2*b - 17*b)*(0 + 1 + 0) - 16*b - 16*b + 24*b.
-17*b
Expand -f + f + f - 5*f + 0*f + 4*f + (2*f - 3 + 3)*(-2 + 1 + 3) - 103815*f + 23577*f + 78006*f - 41104*f - 153745*f - 116622*f.
-313699*f
Expand (51*l - 51*l - 189*l + (-l + 0*l - l)*(3 - 6 + 2) + 3817 - 3817 - 38*l)*(-l - l + 3*l).
-225*l**2
Expand 65192 - c**4 - c**2 - 71*c**3 - 130383 + 65193 + (-c - 1 + 1)*(-2 + c**3 + 2).
-2*c**4 - 71*c**3 - c**2 + 2
Expand (3*d - 20 + 20 + (4 - 1 - 4)*(-d + 6*d - 3*d))*(-2*d + 2*d - d**3) + (-8 + 8 + 2*d)*(1 - 1 - d)*(2*d**2 + 0 + 0).
-5*d**4
Expand (4 - 9*o**2 + o + 1 + 8*o**2)*(13538 - 13538 + 888*o).
-888*o**3 + 888*o**2 + 4440*o
Expand ((0 - n + 0)*(-2*n - 3*n + 4*n) + n**2 + n**2 + 2*n**2)*(-17*n - 1 - 30*n + 2).
-235*n**3 + 5*n**2
Expand (-4*n**2 - 3*n**2 + 5*n**2)*(2 + 0 - 1) - 6 + 793*n - 791*n + 5 - 183*n**2.
-185*n**2 + 2*n - 1
Expand ((-8 - 7 + 1)*(13*q + 5*q - 2*q) - 1 + 3*q - 4 + 4)*(-5*q**2 - q**2 + 3*q**2).
663*q**3 + 3*q**2
Expand w**3 - 4*w**3 + 0*w**3 + (-w + 0*w + 2*w)*((0 + 3 - 5)*(8 + 6 - 5) + 4 - 6 + 4)*(6*w**2 + 0*w**2 + 39*w**2).
-723*w**3
Expand (9*o + 0 + 0)*(44*o + 2*o - 12*o)*(-2*o**2 + 5*o**2 - o**2 + (-o + 1 - 1)*(o - 2*o + 2*o) + 2*o**2 - 6*o**2 + 2*o**2).
-306*o**4
Expand ((-2*q + 2*q + 3*q)*(1 - 1 + 2) - 1 - 2*q + 1)*(8 - 12 - 42).
-184*q
Expand -r**2 - 3*r**2 + 6*r**2 + (-1 + 1 + r)*(0*r - r + 3*r) + 2 - 2*r**2 - 2 - 9*r + 15744*r**2 + 5487*r**2 + 9*r + 517*r**2.
21750*r**2
Expand ((-4 + 1 + 2)*(t - 4*t + 2*t) + 3*t - 3*t + 2*t - 5*t + 0*t + 0*t)*(-6 + 0 + 3)*(-2 + 4 - 4)*(2*t + 0*t + 3*t).
-60*t**2
Expand (-3*r + 6*r - r)*(r**4 - 3*r**4 + r**4) + 26*r**5 + 329*r**5 + 19*r**5.
372*r**5
Expand 2*u - 2*u - 3*u**3 - 4*u**2 + 4*u**2 - u**3 + (-2 - u + 2)*(-40*u**2 - 139*u**2 - 49*u**2) + (1 - u - 1)*(u - u**2 - u).
225*u**3
Expand (4*h**2 + h**2 - 3*h**2)*(18817*h + 3727 + 3615 - 7344).
37634*h**3 - 4*h**2
Expand d - 2*d + 4*d + (202 - 71 - 49)*(1 - 1 - 2*d) - 7*d + 5*d + 7*d.
-156*d
Expand 2*i - i**3 - 3*i + 3*i**3 + (-i + 0 + 0)*(2*i**2 - i**2 + i**2) + i**3 + i - i + 15270*i**3 - 62604*i**3 - 7574*i**3.
-54907*i**3 - i
Expand (2*y - y - 3*y)*(-4 + 2 + 0)*(-6 + 3 + 4)*(594 - 1088 + 892).
1592*y
Expand (-5 - 3 + 5)*(-18 + 18 + 38*w)*(0*w**2 + 0*w**2 - w**2)*(2 - 2 + 2*w).
228*w**4
Expand (-11*q - q - 5*q)*((-4*q + 2*q + 0*q)*(-2*q + 4*q + 0*q) - 15*q**2 + 5*q**2 + 33*q**2).
-323*q**3
Expand (0 - 3 + 0)*(-7*p**2 + 3*p**2 - 3*p**2) + 97*p**2 - 6174 + 6174.
118*p**2
Expand (8300 + 637 - 2973 + 3995)*(-3*m - 4*m + 6*m).
-9959*m
Expand (5*l**2 - 3*l**2 + 5*l**2)*(-127 - 45 - 107).
-1953*l**2
Expand (-4*q**4 + 11*q**4 + 6*q**4)*(3*q - 2*q + 0*q + (3 - 5 + 4)*(-8*q + 5*q + 6*q)).
91*q**5
Expand 0 + d**4 + 0 + 0 - d**4 + 0 + (-3*d + 2*d + 2*d)*(-d**3 - 2*d**3 + 2*d**3) - 4*d**3 - 2*d**4 + 4*d**3 + (98*d - 58*d - 52*d)*(5*d - 5*d - 1 - 4*d**3).
45*d**4 + 12*d
Expand (-2*j**4 + 2*j**4 + 6*j**4)*(-8*j - 17*j - 37*j) - j**5 + 2*j**5 + j**5 + 0*j**5 - j**5 + 2*j**5 + (-3*j**2 + j**2 + 0*j**2)*(j**3 + 0*j**3 + j**3).
-373*j**5
Expand -135*s**4 + 135*s**4 + 172*s**5 + 0 - s**5 + 0 + (-2*s**5 + 5*s**5 - 5*s**5)*(-1 + 2 + 1) - 5*s**5 + 4*s**5 - s**5.
165*s**5
Expand (6*n**2 + 2*n**2 - 4*n**2)*(1610 - 170 + 2988 + 2256).
26736*n**2
Expand -57*m**4 + 14*m**5 + 57*m**4 - 3*m**5 + 6*m**5 - m**5 + (-m + 2*m + 2*m)*(2*m**4 + 0 + 0) - 7*m**5 - 4*m**5 + 0*m**5 + 2*m**4 - 2*m**4 + 2*m**5.
13*m**5
Expand (a**4 + 3*a**4 - 6*a**4)*(-5*a + 0*a + 4*a) + (-130*a**3 + 130*a**3 + 24*a**4)*(-49 - a + 49)*(1 + 3 - 3).
-22*a**5
Expand (2*o - 31 + 526 - 1889)*(0*o**2 - o**2 + 3*o**2).
4*o**3 - 2788*o**2
Expand (146*k**2 + 177*k**2 - 370*k**2)*(-3 + 3 + 2*k**2) + 5*k - k - 26*k**4 + 19*k**4.
-101*k**4 + 4*k
Expand (-2*m**2 + 0*m**2 + m**2)*(-43 + 54 - 215) + (-4*m - 5*m + 4*m)*(0*m + 0*m + 4*m).
184*m**2
Expand (i**3 + 4*i**2 - 4*i**2)*(-15*i - 2010*i**2 - 9*i + 2031*i**2).
21*i**5 - 24*i**4
Expand (0 - 1 - 4)*(45*g + 23*g - 40*g) + (2 + 2 - 2)*(-g - 2*g + g) + g - g - 2*g - 3*g - 4*g + 6*g - 2*g + 3*g - 2*g.
-148*g
Expand (0*x + x + x)*(36*x - 42*x + 84*x) + 2*x**2 - x**2 - 2*x**2.
155*x**2
Expand (-8 + 6 + 29)*(-4*k**3 + 9*k**3 - k**3) + (6*k**2 + 0*k**2 - 4*k**2)*(-3 - k + 3).
106*k**3
Expand -13*y**2 - 2*y**2 + 7*y**2 + (-y - y + 4*y)*(-533 + 9*y - 12*y + 602).
-14*y**2 + 138*y
Expand (3 - 5 + 4)*(-18*y + 27*y + 12*y + (-1 + 1 + 2*y)*(2 + 4 - 4)) - 5 + 6*y + 5 + (-1 + 0 + 2)*(-4*y + y + 2*y) + y - 5 + 5.
56*y
Expand (-4*k - 13*k - 2 - 17*k)*(6*k**4 - 12*k**4 + 5*k**4).
34*k**5 + 2*k**4
Expand (-3991 + 2663 - 1225)*(-h + 2*h - 6*h**3 + 5*h**3).
2553*h**3 - 2553*h
Expand 4 - 4 + 7*y + (1 - y - 1)*(-1 + 3 - 1) + (-y - 2*y + 4*y)*(-30 - 12 + 1)*(1 + 0 + 0).
-35*y
Ex |
1. Field of the Invention
The present invention relates to an apparatus that assists in selecting a shaft for a golf club.
2. Description of the Related Art
In recent years, a tendency among golfers to want golf clubs more suitable for them is growing. Hence, a method of measuring the head speed and struck ball data upon a test strike, and selecting a golf club in accordance with the measurement results (for example, Japanese Patent Laid-Open No. 2003-102892), etc. have been proposed. A tendency to want parts of a golf club, which are individually, exclusively suitable for each golfer, is also growing, and many golfers want especially shafts suitable for them. Hence, a golf club with an easily exchangeable shaft (for example, Japanese Patent Laid-Open No. 2009-178296), etc. have also been proposed.
A wide variety of shafts have been distributed to the market, so it is becoming important for golf shops to carefully select and recommend shafts suitable for individual golfers. In the conventional recommended shaft selection, it is often the case that shafts are classified mainly in accordance with their flexes (stiffnesses), and shafts with flexes corresponding to individual golfers are selected and recommended in consideration of, for example, their head speeds. However, even shafts with nearly the same flex may give greatly different swing feels and produce greatly different test strike results, so a new method of selecting a recommended shaft is required. |
Q:
Download a file through Spring MVC controller using streams
I am using spring MVC with REST service for one of my project. I am having a service to attach and download user files.
I am using below service API for upload and save file into server directory
http://myrestserver/attachmentService/attach/userKey
And below service API for download files from server directory
http://myrestserver/attachmentService/download/userKey/fileKey
The issue is that when a file is downloaded, the downloaded URL shows the REST service API URL. To avoid this, I thought of write a controller for attach and download file.
I wrote a spring controller which handle file attachment process. Even I wrote a controller(say download.do) for download a file, but when a file downloaded, the file name shows as the same name of the controller(downloaded file name shows "download.do" always) instead of original file name.
Below code is from my download.do controller
WebResource resource = null;
resource = client.resource("http://myrestserver/attachmentService/download/userKey/fileKey");
clientResponse = resource.accept(MediaType.APPLICATION_OCTET_STREAM).get(
ClientResponse.class);
InputStream inputStream = clientResponse.getEntityInputStream();
if(inputStream != null){
byteArrayOutputStream = new ByteArrayOutputStream();
try {
IOUtil.copyStream(inputStream, byteArrayOutputStream);
} catch (IOException e) {
log.error("Exception in download:"+ e);
}
}
And, in my service API, the code is
file = new File(directory, attachmentFileName);
fileOutputStream = new FileOutputStream(file);
fileOutputStream.write(attachmentContent);
fileOutputStream.close();
response = Response.ok((Object) file).type(MediaType.APPLICATION_OCTET_STREAM);
response.header("Content-Disposition", "attachment; filename=" + "\"" + attachmentFileName
+ "\"");
return response.build();
By analyzing the issue, I understood that, am not setting file header in downloaded file through download.do controller.
If I am using outstream in download.do controller, I will not be able to set the file header.
Can any one help me to resolve this issue. My primary aim is to hide my rest service URL from downloaded file by stream through a MVC controller.
I found a post (Downloading a file from spring controllers )in stack overflow almost like my question, but the file type is previously known. Please note that, in my application user can attach any type of file.
A:
You have to set the Content-Disposition prior to writing the file to the output stream. Once you start writing to the output stream, you cannot set headers any longer.
|
---
abstract: |
When ambiguous beliefs are represented by multiple priors, a decision maker (DM)’s updating rule may also include a step of refining the initial belief. Maximum Likelihood (ML) updating indicates one example of such a refinement, whereas Full Bayesian (FB) updating does not allow for any refinement. In fact, ML and FB are the two extremes of refining beliefs with likelihood in updating. To capture behaviors between these two extremes, the present paper proposes and axiomatizes a new updating rule, Relative Maximum Likelihood (RML), to represent conditional preferences that are not as extreme as either case. RML updates an intermediate set of priors, which can be expressed as a linear contraction of the initial set with respect to the set of maximum likelihood priors. The linear contraction parameter captures the DM’s relative inclination towards ML with respect to FB, it is also the threshold of a relative likelihood ratio test. Moreover, RML includes both FB and ML as the two extreme special cases.
Additionally, the present paper provides an axiomatization of ML for preferences admit Maxmin Expected Utility (MEU) representation as a preliminary result for the exposition of RML.\
*JEL: D80, D81*
*Keywords: ambiguity, MEU, updating, full Bayesian, maximum likelihood, likelihood ratio test, contingent reasoning, ambiguous signals, robustness*
author:
- 'Xiaoyu Cheng[^1]'
bibliography:
- 'references.bib'
title: 'Relative Maximum Likelihood Updating of Ambiguous Beliefs[^2]'
---
Introduction
============
For decisions under uncertainty, when sufficient information to pin down a unique probability for uncertainty is lacking, the decision maker (DM)’s revealed preference sometimes is not consistent with any probabilistic belief but is consistent with a belief of multiple priors (Ellsberg 1961[@ellsberg1961risk], Machina and Schmeidler 1992[@machina1992more]). When this DM learns additional information, the updating of her multiple priors belief may actually involve two steps: first, she could use this information to make an inference about the plausibility of each prior and refine her initial belief by discarding the priors that are deemed to be implausible; then second, updates every prior in the refined belief using Bayes’ rule conditional on the information she receives.
The type of inference in the first step is widely used in non-Bayesian statistics. For example, maximum likelihood estimation makes an inference about the parameter values based on the likelihood of generating the observed data. Its counterpart in the ambiguity literature is **Maximum Likelihood (ML)** updating, in which the DM updates only the priors attain maximum likelihood of the observed event in her initial belief; i.e., only those priors are deemed to be plausible. In contrast, another well-known updating rule, **Full Bayesian (FB)**[^3], updates all the priors in an initial belief; in other words, it allows no such inference.
Hence, ML and FB, as two popular updating rules for multiple priors belief, are actually the two polar extremes in terms of making an inference with respect to the likelihood of observed information in updating. The conditional preferences that result from some intermediate inferences, for example, one in which the priors that attain almost maximum likelihood are also deemed to be plausible, cannot be captured by either of these two extremes. In fact, this type of intermediate behaviors are largely missing from the updating literature with axiomatization. In other words, the preference behaviors that corresponding to such an intermediate inference have not been extensively studied.
The present paper proposes an updating rule, **Relative Maximum Likelihood (RML)**, which is able to capture a full range of behaviors between FB and ML. Furthermore, its behavioral foundation is characterized when the decision criterion under multiple priors follows the Maxmin Expected Utility (MEU), in which the DM evaluates her decision according to the worst expected utility among all the priors (Gilboa and Schmeidler 1989[@gilboa1989maxmin]).
It is known in the literature that, FB is characterized by a behavioral axiom under MEU preferences (Pires 2002[@pires2002rule]), however ML is axiomatized only when preferences admit both MEU and Choquet Expected Utility (CEU) representations (Gilboa and Schmeidler 1993[@gilboa1993updating]), which is a strict special case of MEU preferences. As a preliminary result for the exposition of RML, Theorem \[thm1\] provides an axiomatization of ML under general MEU preferences[^4].
Relative Maximum Likelihood
---------------------------
RML captures an intermediate updating behavior between FB and ML in the sense that it updates a refined set of prior that is both the subset of the initial priors and also a superset of the maximum likelihood priors.
Formally, let a convex and closed set $C$ denote the initial set of priors, and for each conditional event $E$, let $C^{*}(E)$ denote the set of priors in $C$ that attain maximum likelihood of event $E$.[^5] For some parameter $\alpha \in [0,1]$, RML selects the following set of priors $C_{\alpha}(E)$ for updating when event $E$ occurs: $$C_{\alpha}(E) = (1-\alpha)C + \alpha C^{*}(E) = \{(1-\alpha)p + \alpha q: p \in C \text{ and } q \in C^{*}(E) \}.$$ Namely, $C_{\alpha}(E)$ is a linear mixture of the two sets $C$ and $C^{*}(E)$, and the linear mixture is defined as an element-wise mixture.
Since $C^{*}(E) \subseteq C$, thus for all $\alpha \in [0,1]$ one has $C^{*}(E) \subseteq C_{\alpha}(E) \subseteq C$. Geometrically, the set $C_{\alpha}(E)$ is a linear contraction of the set $C$ with $C^{*}(E)$ being the center[^6]. Furthermore, it reduces to FB or ML when $\alpha$ equals to 0 or 1 respectively. Indeed, the parameter $\alpha$ captures a *relative* inclination towards ML with respect to FB, as $\alpha = 0$ and $\alpha = 1$ capture the two extreme inclinations.
To better illustrate how RML works and its relation with FB and ML, consider a version of the Ellsberg’s three-colored urn problem: an urn contains 30 red (R) balls, 60 black (B) and yellow (Y) balls, nature randomly draws a ball from the urn. Let $\{R, B, Y\}$ denote the color of the drawn ball, suppose the DM forms a multiple priors belief that coincides with the information of the urn: $$C = \{p \in \Delta(\{R,B,Y\}): p(R) = 1/3 \}$$ Then let $E = \{R,B\}$ be the observed event, i.e. the DM later learns that the drawn ball is not yellow. Under FB, the DM updates every prior in $C$ conditional on $E$ and results in the following set of posteriors: $$\Pi^{FB} = \{p \in \Delta(\{R,B\}): 1/3 \leq p(R) \leq 1 \}$$ Under ML, since the maximum likelihood of event $E$ is attained by the prior $p\in C$ such that $p(B) = 2/3$, then $C^{*}(E) = \{p \in \Delta(\{R,B, Y\}): p(R) = 1/3, p(B) = 2/3\}$ and its posterior is also: $$\Pi^{ML} = \{p \in \Delta(\{R,B\}): p(R) = 1/3\}$$ Under RML for some $\alpha \in [0,1]$, the updated set by definition is given by $$C_{\alpha}(E) = \{p \in \Delta(\{R,B, Y\}): p(R) = 1/3, p(B) \geq 2\alpha/3\}$$ and the set of posteriors is therefore: $$\Pi^{RML} = \{p \in \Delta(\{R,B\}): 1/3 \leq p(R) \leq 1/(1+2\alpha) \}$$ Consider a bet $f_{B}$ that pays 1 on state $B$ and nothing on the other states. With MEU preference, notice that the conditional evaluation of the bet under RML is given by $\frac{2\alpha}{1+2\alpha}$, which is increasing with respect to $\alpha$. Given the observed event $\{R,B\}$, the DM may infer that there cannot be too many yellow balls in the urn. A DM who is more willing to make such an inference would be more willing to conclude that there actually should be even fewer yellow balls, therefore the evaluation of the bet $f_{B}$ should also be higher. Hence, the value of parameter $\alpha$ calibrates exactly such an attitude. In section 3, it can be further shown that, $\alpha$ is also the threshold of a relative likelihood ratio test.
Furthermore, ranging over $\alpha \in [0,1]$, the conditional evaluation of the bet $f_{B}$ under RML traces out the full range of all possible values between $0$ and $2/3$, the conditional evaluations under FB and ML respectively. In this sense, RML is able to capture a full range of intermediate behaviors between FB and ML.
Representation Theorems
-----------------------
The main contribution of the present paper is identifying the behavioral axioms that characterize preferences represented by RML. Especially, two different representation theorems that feature in two different strength of identification of the parameter $\alpha$ are provided (Theorem \[thm2\], Theorem \[thm4\]).
The stronger one, Theorem \[thm4\], is the ultimate representation theorem that provides a characterization of RML with constant $\alpha$. Whereas, by dropping one of the axioms in Theorem \[thm4\], Theorem \[thm2\] characterizes the **Weak RML**, in which the parameter $\alpha[E]$ depends on $E$ such that it could be different across events.
Both representation theorems are important in their own right. A constant $\alpha$ would be more convenient for applications of RML and also provide sharper predictions for updating behaviors. On the other hand, an $\alpha[E]$ that depends on the conditional event $E$ would allow for more flexibility in terms of interpreting updating behaviors. For example, the experimental results shown by Table 4.2 in Liang (2019)[@liang2019] correspond to a preference under MEU with RML such that $\alpha[E] < 1/2$ when $E = $“good news” and $\alpha[E] > 1/2$ when $E = $“bad news”[^7].
The axiomatization of RML leverages on weakening of the axioms that characterize FB and ML, while the latter axioms are themselves weakening of a property so-called **Contingent Reasoning (CR)**.
In dynamic choice, CR relates a DM’s ex-ante preference to her conditional preferences in the same fashion as Savage’s P2 axiom (Sure-thing principle). Suppose a DM, conditional on an event $E$, is indifferent between an act[^8] $f$ and some constant act $x$: $f\sim_{E}x$. Consider then she compares between two acts ex-ante: $f_{E}g$ and $x_{E}g$, where $f_{E}g$ denotes an act that pays according to the act $f$ when event $E$ occurs, and pays according to $g$ otherwise.
CR requires that, her ex-ante preference could be recovered by evaluating her choices contingent on whether event $E$ occurs. In the above case, contingent on $E$ occurring, the DM is conditional indifferent between $f$ and $x$; while contingent on $E$ not occurring, she receives the same act $g$ in both acts. From this type of reasoning, the DM should further conclude that she is indifferent between $f_{E}g$ and $x_{E}g$ no matter what the act $g$ is[^9]. As a matter of fact, an expected utility maximizer with Bayesian updating always satisfies this CR condition.
Under MEU preferences, both FB and ML are well-known to sometimes violate CR (e.g. in the Ellsberg’s three-colored example). However, when the act $g$ pays on the event $E^{c}$ is restricted to be some specific act, it can be shown that FB and ML will satisfy CR in different special cases, and more importantly, those special cases are also sufficient to characterize FB and ML respectively.
Pires (2002)[@pires2002rule] characterizes FB by an axiom that imposes CR only when the act $g$ is restricted to be the **conditional certainty equivalence**, i.e. $f \sim_{E}x$ implies that $f_{E}x \sim x$. The present paper shows that ML would satisfy CR when the act $g$ is restricted to be **large consequences**, i.e. for all sufficiently large consequences $x^{*}$, $f \sim_{E}x$ implies that $f_{E}x^{*} \sim x_{E}x^{*}$. Furthermore, Theorem \[thm1\] establishes the necessary and sufficient relation between ML under MEU and this special CR condition.
The two axioms that characterize FB and ML are called **Contingent Reasoning for Conditional Certainty Equivalence (CR-CCE)** and **Contingent Reasoning for Large Consequences (CR-LC)** respectively, summarizing their contents.
As RML captures an intermediate behavior between FB and ML, the two axioms CR-CCE and CR-LC, will sometimes be violated under RML. By definition, RML deviates from FB and ML in two different directions (in the sense of subset and superset), which further implies that the two special CR conditions will also be violated in two different directions. Indeed, the **Undershooting and Overshooting (U-O)** axiom, which says that if $f\sim_{E}x$, then $f_{E}x \precsim x$ and $f_{E}x^{*} \succsim x_{E}x^{*}$ for all sufficiently large $x^{*}$, reflects the two directions of deviation from the CR conditions. Proposition \[prop\] suggest that the U-O axiom is necessary and sufficient for any updating rule that is intermediate between FB and ML just like the way in RML.
All intuitions about a DM not willing to be as extreme as either FB and ML are captured by the U-O axiom. However, Proposition \[prop\] further implies that this intuition per se is not sufficient to pin down any specific set that represents the DM’s conditional preferences. To achieve more concrete characterization of conditional preferences, more restrictions are needed to impose on the behaviors.
The **Weak CR-CCE** axiom features another direction of weakening the CR-CCE axiom, orthogonal to the direction of weakening in the U-O axiom. For any two acts $f$ and $g$, CR-CCE implies that if $f\sim_{E}g\sim_{E}x$, then $f_{E}x \sim g_{E}x$. Instead of imposing this CR condition on all acts that are conditionally indifferent, Weak CR-CCE imposes it only when these two acts are also indifference when sufficiently large consequences are paid on $E^{c}$, i.e. $f_{E}x^{*} \sim g_{E}x^{*}$ for all sufficiently large $x^{*}$.[^10]
Theorem \[thm2\] shows that the U-O and Weak CR-CCE axiom are necessary and sufficient for the preferences to be represented by the Weak RML with $\alpha[E]$ that may depend on the observed event $E$. To further characterize a constant $\alpha$ across different events, an additional axiom, **Event Consistency (EC)** will be shown to be sufficient. The EC axiom restricts the DM’s conditional preferences across different events by a similar requirement as in the Weak CR-CCE. Eventually, imposing U-O, Weak CR-CCE and EC axiom is necessary and sufficient for the preferences to be represented by RML with a constant $\alpha$ (Theorem \[thm4\]).
(Naive) Consequentialist Approach
---------------------------------
A fundamental issue for updating ambiguous beliefs is the fact that, for ambiguity sensitive choices, an updating rule cannot preserve both dynamic consistency and consequentialism at the same time (Hanany and Klibanoff 2007[@hanany2007updating], Siniscalchi 2009[@siniscalchi2009two]). By the definition in Hanany and Klibanoff (2007)[@hanany2007updating], dynamic consistency means that the ex-ante most preferred act should remain to be preferred after updating to any other acts that are both feasible and agree with it on the event not occurs; consequentialism means that the conditional preferences should not depend on event not occurs as well as the context of the decision problem (e.g. feasible acts etc.). Therefore, any updating rule for ambiguous beliefs needs to relax one of the two properties.
As RML is characterized by axioms that relaxing the contingent reasoning property which is related to dynamic consistency, and it also incorporates FB and ML as special cases, both of which are well-known to violate dynamic consistency, its violation of dynamic consistency thus can be expected. That being said, RML takes the consequentialist approach which keeps the consequentialism property but relaxes dynamic consistency.
For updating rules take the consequentialist approach, a common potential problem is that when dynamic consistency is violated, the DM would find that her ex-ante preferred course of action will not be carried out since she will prefer another action after updating. Concerning this problem, the naive approach considers the DM to be not sophisticated enough to anticipate that, so she would just follow her updating rule no matter what. In contrast, the sophisticated approach considers the DM to be able to anticipate that and uses consistent planning to commit herself to the ex-ante preferred action that is feasible under her updating.
In order to make the exposition of RML as clear as possible, the present paper takes the naive approach to highlight the updating rule per se. Using the framework developed in Siniscalchi (2011)[@siniscalchi2011dynamic], one will be able to apply RML in a sophisticated approach and consider its predictions.
Robust Applications of Ambiguity
--------------------------------
One important feature of RML is the fact that it unifies FB and ML to the same family of updating rules. The additional parameter $\alpha$ provides an abundant amount of freedom to model DMs’ different reactions with respect to information in the context of ambiguity. Especially, this freedom creates an additional lens to look at robustness questions related to applications of ambiguity.
Many recent applications of ambiguity to information economic models assume that players’ preferences are represented by MEU with FB updating (Bose and Renou 2014[@bose2014mechanism], Beauchene, Li and Li 2019[@beauchene2019ambiguous], Kellner and Le Quement 2018[@kellner2018endogenous]). Although all of these papers present results that the introduction of ambiguity is strictly beneficial for some player so that she can exploit the ambiguity aversion of the other players, their rather strong behavioral assumptions create an caveat that the strict benefits might be restricted only to this specific functional form instead of the general idea of ambiguity.
As FB is a special case of RML, relaxing the behavioral assumptions to MEU with RML updating for any $\alpha \in [0,1]$ provides one way to consider the robustness of the strict benefits by allowing the other players to have arbitrary attitude towards making an inference with respect to the likelihood of information.
In section 5, one example of such a type of robustness analysis is presented. I take the illustrative example in Beauchene, Li and Li (2019)[@beauchene2019ambiguous], in which they show that the sender can gain strictly more payoff than Bayesian persuasion by using ambiguous persuasion. Once one relaxes the behavioral assumption on the receiver from FB ($\alpha = 0$) to RML for any $\alpha \in [0,1]$, first of all, the device that approaching the sender’s optimal payoff[^11] will no longer induce the same action from a receiver with any $\alpha > 0$. Furthermore, the sender’s payoff when receiver’s $\alpha$ is non-zero will be strictly worse than Bayesian persuasion. Seemingly, the strict benefits do crucially depend on the FB assumption here.
Nonetheless, section 5 further shows that there still exists an ambiguous persuasion scheme in which the sender will be able to induce the same action from the receiver for any $\alpha \in [0,1]$, and more importantly, the sender’s payoff will be strictly higher than using Bayesian persuasion. It thus suggests that, the strict gain from using ambiguous signals in this example is robust to allowing for arbitrary attitudes toward inference with respect to the likelihood of information.\
The remainder of the paper is organized as follows: Section 2 sets up the environment and provides the characterization of ML under MEU preferences; Section 3 gives the formal definition of RML and Weak RML; Section 4 provides preference foundation for RML and Weak RML; Section 5 looks at a special case of the current setting and illustrates the predictions and interpretations that RML is able to offer; Section 6 shows an example of robustness analysis using RML in the context of information design; Section 7 talks about related literature; Section 8 concludes.
Preliminaries
=============
Set up
------
I adopt the Anscombe-Aumann framework. Let $\Omega$ be the set of *states of the world* with at least three states[^12], endowed with a sigma-algebra $\Sigma$ of *events* with generic element $E$. Let $X$ be the set of all simple (i.e. finite-support) lotteries over a set of consequences $Z$ and let $x$ denote a generic element of $X$. Let $\mathcal{F}$ denote the set of *bounded acts*, meaning that each $f \in \mathcal{F}$ is a bounded $\Sigma$-measurable function from $\Omega$ to $X$. With conventional abuse of notation, denote a constant act which maps all states $\omega \in \Omega$ to $x$ simply by $x$.
The primitive is a family of preferences $\{\succsim_{E}\}_{E\in \Sigma}$ over all acts $f\in \mathcal{F}$. Let $\succsim_{\Omega} \equiv \succsim$ denote the ex-ante preference and for all the other $E\in \Sigma$, let $\succsim_{E}$ denote the conditional preference when event $E$ occurs. ($\succsim_{\emptyset}$ is irrelevant.)
First of all, assume that the ex-ante preference $\succsim$ admits a Maxmin Expected Utility representation and is represented by a set $C \subseteq \Delta(\Omega)$ and an affine utility function $u$ such that for all $f,g \in \mathcal{F}$: $$f \succsim g \Leftrightarrow \min\limits_{p \in C} \int_{\Omega} u(f)dp \geq \min\limits_{p \in C} \int_{\Omega} u(g) dp$$ where $u(f)$ denote a function $Y : \Omega \rightarrow {\mathbb{R}}$ such that $Y(\omega) = u(f(\omega))$ for all $\omega \in \Omega$ and call $u(f)$ the *utility profile* of act $f$. In addition, assume that $\succsim$ has finitely many plausible priors[^13].
The finitely many plausible priors assumption is extremely useful since it helps simplify the statements of the axioms to a great extent while conveying the same intuitions. All characterizations can be achieved without this assumption by different axioms as shown in the appendix. On the other hand, this assumption is also arguably reasonable as it still covers a tremendous number of popular cases in applications of multiple priors, for example the $\epsilon$-contamination.
Assume that $X$ is unbounded under the ex-ante preference in the sense of the following: there exists $x \succ y$ in $X$ such that, for all $\alpha \in (0,1)$, there exists $z \in X$ that satisfies either $y \succ \alpha z + (1-\alpha)x$ or $\alpha z + (1-\alpha)y \succ x$. By Lemma 29 in Maccheroni et al. (2006)[@maccheroni2006ambiguity], it is equivalent to say that the set $u(X)$ is unbounded.
For each $E\in \Sigma$, for any $f,g \in \mathcal{F}$, let $f_{E}g$ denote an act that maps all $\omega \in E$ to $f(\omega)$ and maps all $\omega \in E^{c}$ to $g(\omega)$. An event $E$ is $\succsim$-null if for all $f,g,h \in \mathcal{F}$, one has $f_{E}h \sim g_{E}h$. Under MEU, an event $E$ is $\succsim$-null if and only if $p(E) = 0$ for all $p \in C$. Thus, $E$ is $\succsim$-nonnull if there exists $p\in C$ such that $p(E) > 0$.
Furthermore, define an event $E$ to be strict $\succsim$-nonnull if for all $x,x' \in X$ such that $x \succ x'$, one has $x_{E}x' \succ x'$. Under MEU, an event $E$ is strict $\succsim$-nonnull if and only if $p(E) > 0$ for all $p \in C$. Therefore, an event $E$ is $\succsim$-nonnull but is also strict $\succsim$-null if there exists $p, p'\in C$ such that $p(E) =0$ and $p'(E) >0$.
For each $\succsim$-nonnull $E\in \Sigma$, assume that the conditional preference $\succsim_{E}$ also admits a Maxmin Expected Utility representation and is represented by a set $C_{E} \subseteq \Delta(\Omega)$ and the same utility function $u$ such that for all $f,g \in \mathcal{F}$: $$f \succsim_{E} g \Leftrightarrow \min\limits_{p \in C_{E}} \int_{\Omega} u(f)dp\geq \min\limits_{p \in C_{E}}\int_{\Omega} u(g)dp$$ Meanwhile, the conditional preference $\succsim_{E}$ for $\succsim$-null $E$ is unrestricted.
Finally, assume that the conditional preferences satisfy *consequentialism*: first, for all $p \in C_{E}$, $p(E) = 1$, i.e. the complement of the conditional event is irrelevant for conditional preference; and second, ex-ante preference and conditional event $E$ completely determine the conditional preference $\succsim_{E}$, which rules out the possibility that the updating rule may depend on the context of the decision problem.\
All properties assumed for the primitive have axiomatize foundations:
- Maxmin Expected Utility representation: axioms from Gilboa and Schmeidler (1989)[@gilboa1989maxmin].
- Finitely many plausible priors: no local hedging axiom from Siniscalchi (2006)[@siniscalchi2006behavioral].
- $u$ is independent of $E$: state independence axiom from Pires (2002)[@pires2002rule] or unchanged tastes axiom from Hanany and Klibanoff (2007)[@hanany2007updating].
- Consequentialism: null complement axiom and independence from feasible sets axiom from Hanany and Klibanoff (2007)[@hanany2007updating].
Setting up in this way enables the establishing of a clean if and only if connection between the key axioms and the updating rules. Especially, the most important behavioral foundations that characterize different updating rules will be highlighted in the representation theorems.
For example, FB updating is defined in Pires (2002)[@pires2002rule] by the following:
The primitive $\{ \succsim_{E}\}_{E\in \Sigma}$ is represented by FB updating if the following holds for all $f\in\mathcal{F}$: for all $\succsim$-nonnull $E \in \Sigma$, $$\min\limits_{p \in C_{E}} \int_{\Omega} u(f)dp \leq \min\limits_{p \in C: p(E) \neq 0} \int_{E} u(f)\frac{dp}{p(E)}$$ and especially for all strict $\succsim$-nonnull event $E\in \Sigma$, $$\min\limits_{p \in C_{E}} \int_{\Omega} u(f)dp = \min\limits_{p \in C} \int_{E} u(f)\frac{dp}{p(E)}$$
Her axiomatization result can be simplified, in this framework, to the statement that FB is necessary and sufficient to the following axiom (which is exactly her A9 axiom):\
**Axiom CR-CCE** (Contingent Reasoning for Conditional Certainty Equivalence). For all $\succsim$-nonnull event $E\in \Sigma$, for all $f \in \mathcal{F}$ and $x\in X$, if $f \sim_{E}x$ then $f_{E}x \sim x$.
$\{ \succsim_{E}\}_{E\in \Sigma}$ is represented by FB updating if and only if the CR-CCE axiom holds.
This statement of the theorem highlights that the behavioral foundation of FB is the relaxation of the following Contingent Reasoning (CR) property:\
**Contingent Reasoning** For all $\succsim$-nonnull $E\in \Sigma$, for all $f\in \mathcal{F}$ and $x\in X$, if $f\sim_{E}x$ then $f_{E}g \sim x_{E}g$ for all $g\in \mathcal{F}$.\
CR-CCE relaxes CR in a way that only requires ex-ante indifference when the act $g$ pays on $E^{c}$ is restricted to be the conditional certainty equivalence $x$.
Maximum Likelihood Updating
---------------------------
The primitive $\{\succsim_{E}\}_{E\in \Sigma}$ is represented by ML updating if the following holds for all $\succsim$-nonnull $E \in \Sigma$ and for all $f\in \mathcal{F}$: $$\min\limits_{p \in C_{E}} \int_{\Omega} u(f)dp = \min\limits_{p \in C^{*}(E)} \int_{E} u(f)\frac{dp}{p(E)}$$ where $C^{*}(E) = \arg\max\limits_{p \in C} p(E)$.
In other words, the conditional preference $\succsim_{E}$ for $\succsim$-nonnull $E$ is represented by the set of posteriors that updated from the priors attain maximum likelihood of event $E$.
When the ex-ante preference $\succsim$ admits both MEU and CEU representations and the set $X$ is bounded such that there exists a maximal element $x^{*}$: $x^{*} \succsim x$ for all $x\in X$, Gilboa and Schmeidler (1993)[@gilboa1993updating] show that, ML is characterized by **Contingent Reasoning for Best Consequence**:\
**Axiom CR-BC** (Contingent Reasoning for Best Consequence). For all $\succsim$-nonnull event $E\in \Sigma$, for all $f \in \mathcal{F}$ and $x\in X$, if $f \sim_{E}x$ then $f_{E}x^{*} \sim x_{E}x^{*}$.\
Notice that CR-BC also relaxes CR in a similar manner as CR-CCE. CR-BC claims that, the DM’s ex-ante evaluation of the act $f_{E}x^{*}$ is equivalently given by her contingent reasoning with respect to the event $E$. Gilboa and Schmeidler (1993)[@gilboa1993updating] offer a “pessimistic” interpretation of this behavior: the DM’s conditional preference of an act $f$ comes from the consideration that the best consequence would have been received had the complement event happened. In other words, her conditional evaluation of the act $f$ reflects a disappointment that event $E$ actually occurs. Moreover, such a disappointment reflected in her conditional preference is triggered by the fact that she assigns maximum probability to the event $E$ under her ex-ante preference.
Although their intuition about the behavioral foundation of ML updating is qualitatively correct, yet the intuition captured by CR-BC would become imprecise when one enlarges the domain of preferences to the more general MEU case.
The following example illustrates exactly a case where the ex-ante preference admits MEU but not CEU representation, and the CR-BC axiom is violated under ML updating.
\[exp3\] Consider a state space $\Omega = \{\omega_{1}, \omega_{2}, \omega_{3}\}$, denote any $p \in \Delta(\Omega)$ by a vector with three coordinates $p = (p_{1}, p_{2}, p_{3})$ such that $p_{1} + p_{2} + p_{3} = 1$. Consider a set $C \subseteq \Delta(\Omega)$ that is a convex hull of the following three points: $(1/2, 0, 1/2), (0, 1/2, 1/2)$ and $(1/3, 1/3, 1/3)$.
Let $X = [0,1]$ with utility function $u(x) = x$. An act $f$ pays $1$ at $\omega_{1}$, pays $0$ at $\omega_{2}$ and is undetermined on state $\omega_{3}$. Let the conditional event $E$ be the two states $\{\omega_{1}, \omega_{2}\}$.
First of all, it is straightforward to verify that a MEU preference with respect to this $C$ does not admit a CEU representation[^14].
If the conditional preference is given by ML updating, as $C^{*}(E)$ contains only the extreme point $(1/3,1/3,1/3)$ then $$f \sim_{E} 1/2$$ The CR-BC axiom implies that $f_{E}x^{*}$ should be indifferent to $1/2 _{E} x^{*}$ where $x^{*} = 1$. However, it is actually the case that $$f_{E}1 \prec 1/2_{E}1$$ i.e. the CR-BC axiom is false.
One might notice that the breakdown of CR-BC axiom results from the fact that $f_{E}1$ and $1/2_{E}1$ are evaluated at different extreme points, especially, the act $f_{E}1$ is not evaluated at the extreme point $(1/3, 1/3, 1/3)$. In other words, her ex-ante evaluation of the act $f_{E}1$ **does not** trigger the disappointment reflected in her conditional evaluation of $f$ that event $E$ occurs. Moreover, the graphical illustration given in figure \[fig:2\] suggests that it is exactly because the consequence $x^{*} = 1$ is not good enough to trigger such a disappointment for this act $f$.
![Graphical Illustration of Example \[exp3\][]{data-label="fig:2"}](2){width="1.0\linewidth"}
In figure \[fig:2\], the blue arrow indicates how the act $f_{\omega_{1}, \omega_{2}}x$ would be changing (in angle) if one increases the consequence $x$. When $x = 1$, the act $f_{\omega_{1}, \omega_{2}}1$ is evaluated at the extreme point $(0, 1/2, 1/2)$ according to MEU and when $x = 2$, the act $f_{\omega_{1}, \omega_{2}}2$ is indifferent between evaluating at the two extreme points $(1/3, 1/3, 1/3)$ and $(0, 1/2, 1/2)$. If one keeps increasing $x$, apparently for all $x \geq 2$, the act $f_{\omega_{1}, \omega_{2}}x$ will be evaluated at the extreme point $(1/3, 1/3, 1/3)$ (When $x$ goes to infinity, the act eventually will be parallel to the bottom line of the triangle).
Therefore, for this act $f$ and conditional event $E = \{\omega_{1}, \omega_{2}\}$, the consequence $x$ paid on the event $E^{c}$ needs to be at least better than $x = 2$ in order to trigger the disappointment reflected in her conditional preference. Consequently, the assumption that $X$ is bounded by $x^{*} = 1$ would prevent that from happening. In fact, one can also verify that $f_{E}2 \sim 1/2_{E}2$ in this example.
At a higher level, Gilboa and Schmeidler (1993)[@gilboa1993updating]’s intuition about the behavioral foundation of ML updating is qualitatively correct, i.e. the conditional evaluation of an act $f$ under ML reflects a disappointment that event $E$ occurs while some sufficiently good consequence would have been paid had event $E$ not occur. However under general MEU, the best consequence $x^{*}$ sometimes is not good enough to trigger such a disappointment in the ex-ante preference. Example \[exp3\] shows exactly a case where the threshold of sufficiently good consequence $(x=2)$ is strictly greater than the best consequence $(x^{*} = 1)$.
Hence, in the current setting where $X$ is assumed to be unbounded, the behavioral intuition for ML updating under MEU is more precisely given by the following axiom:\
**Axiom CR-LC** (Contingent Reasoning for Large Consequences). For all $\succsim$-nonnull event $E \in \Sigma$, for all $f \in \mathcal{F}$ and $x\in X$, there exists $\bar{x}_{E, f} \in X $ such that if $f \sim_{E}x$ then $f_{E}x^{*} \sim x_{E}x^{*}$ for all $x^{*} \succsim \bar{x}_{E, f}$.\
The CR-LC axiom explicitly states that the threshold for large consequences $\bar{x}_{E,f}$ would depend on the conditional event $E$ and act $f$. More importantly, the existence of such a threshold for every $E$ and $f$ is guaranteed by assumptions on the primitives: $X$ is unbounded and $C$ has finitely many extreme points.
The reason why $X$ being unbounded is important for the existence of $\bar{x}_{E,f}$ has been explained above. For the other assumption, when $C$ contains infinitely many extreme points, for example, is a circle in the three-dimensional simplex, for the act $f$ in example $\ref{exp3}$, $f_{E}x$ is evaluated at the extreme point that maximizes the likelihood of event $E$ only if $x$ goes to infinity. In that case, a threshold $\bar{x}_{E,f}$ will not exist. In contrast, when $C$ contains finitely many extreme points, such a threshold always exists[^15].
The following theorem shows that CR-LC axiom is necessary and sufficient for ML updating under MEU.
\[thm1\] $\{\succsim_{E}\}_{E\in \Sigma}$ is represented by ML updating if and only if the CR-LC axiom holds.
**Remark.** If one drops the finitely many extreme points assumption, as the above discussion suggested, the threshold $\bar{x}_{E,f}$ would not always exist. When it does not exist, one can find a sequence of thresholds such that the difference between the evaluations of $f_{E}x^{*}$ and $x_{E}x^{*}$ for all $x^{*}$ greater than the threshold vanishes as the threshold increases. Formal statement of the axiom and characterization of ML in this case can be found in appendix \[apx1\].
Relative Maximum Likelihood
===========================
The present paper proposes a new updating rule for multiple priors beliefs called Relative Maximum Likelihood (RML) in order to capture the intermediate behaviors between FB and ML. It features a parameter $\alpha \in [0,1]$ that is able to calibrate an extent of *relative* inclination towards ML with respect to FB, and it is also the threshold of a *relative likelihood ratio test*, related to the (absolute) likelihood ratio test used in statistics.
Formally, let $C$ denote the initial set of priors and for some event $E \in \Sigma$ let $C^{*}(E)$ denote the subset of $C$ assigning maximum likelihood of event $E$. For some $\alpha \in [0,1]$, RML updates the following set of priors: $$C_{\alpha}(E) = (1-\alpha)C + \alpha C^{*}(E) = \{(1-\alpha)p + \alpha q: p \in C \text{ and } q \in C^{*}(E) \}$$ Geometrically, as illustrated in Figure \[fig:1\], the set $C_{\alpha}(E)$ is a linear contraction of the set $C$ with $C^{*}(E)$ being the center. In Figure \[fig:1\], the triangle represents the simplex of probability distributions over the three states $\omega_{1}, \omega_{2}$ and $\omega_{3}$. The larger hexagon represents the set of priors $C$ and when $E = \{\omega_{1}, \omega_{2}\}$ the bottom line of it represents $C^{*}(E)$. Then the blue shaded area represents $C_{\alpha}(E)$ for some $\alpha \in [0,1]$. More specifically, $C_{\alpha}(E)$ is the closed convex hull of points that are linear mixtures of extreme points in $C$ and extreme points in $C^{*}(E)$.
{width="1.1\linewidth"} \[fig:1\]
{width="1.1\linewidth"} \[fig:3\]
Observe that when $\alpha = 0$, $C_{\alpha}(E)$ coincides with $C$ and RML reduces to FB, which means that the corresponding conditional preference does not express any inclination towards ML. On the contrary, when $\alpha = 1$, $C_{\alpha}(E)$ becomes $C^{*}(E)$ and RML reduces to ML. In this case, the corresponding conditional preference shows an extreme inclination towards ML.
All the other $\alpha \in (0,1)$ capture an intermediate behavior between FB and ML where the conditional preference has *some* inclination towards ML. Moreover, the set $C_{\alpha}(E)$ shrinks as $\alpha$ increases reflects the fact that larger $\alpha$ corresponds to a DM being more willing to make an inference with respect to likelihood and thus discards more priors from her initial belief.
This fact further suggests a connection between the parameter $\alpha$ and some threshold for refining priors that depends on likelihood. (Absolute) Likelihood ratio test is commonly used in statistics to determine whether a statistical model with fewer parameters is “good enough” compared with a model with maximum number of parameters. The former model is good enough if the likelihood ratio of generating the observed data compared with the latter model exceeds some threshold. In other words, a model is plausible only if it generates the observed data with some likelihood that its ratio with respect to the maximum likelihood exceeds the threshold.
Accordingly, in updating multiple priors, a DM may find a prior to be plausible only if it generates the observed event with some likelihood that its ratio with respect to the maximum likelihood exceed a threshold, for example $$\frac{p(E)}{\max\limits_{p \in C}p(E)} \geq \lambda$$ In this sense, the level of this threshold ($\lambda$) also reflects the extent of willingness to make an inference using likelihood. In fact, the updating rule proposed in Epstein and Schneider (2007)[@epstein2007learning] explicitly uses such a likelihood ratio test to determine whether a prior is plausible or not.
For RML, its functional form suggests a related but different test that is used for selecting priors. Notice that, for all $p \in C_{\alpha}(E)$, $$\frac{p(E)}{\max\limits_{p \in C}p(E)} \geq \alpha + (1-\alpha) \frac{\min\limits_{p \in C}p(E)}{\max\limits_{p \in C}p(E)}$$ or equivalently $$\frac{p(E)-\min\limits_{p \in C}p(E) }{\max\limits_{p \in C}p(E) - \min\limits_{p \in C}p(E)} \geq \alpha$$ from the second formula it is obvious that $\min\limits_{p\in C}p(E)$ becomes an additional benchmark for conducting likelihood ratio test. That being said, a prior is plausible only if it generates the information with some likelihood that its relative difference with respect to the minimum likelihood compared with the difference between maximum likelihood and minimum likelihood exceeds the threshold. Therefore, it is called a **relative likelihood ratio test**. Apparently, when the minimum likelihood of event $E$ is zero, the two likelihood ratio tests coincide.
However, RML does not require only the relative likelihood ratio test for prior selection. Passing the relative likelihood ratio test is only necessary for a prior to be updated under RML, the linear contraction suggests that the shape of the set of initial priors $C$ also plays a crucial role in determining whether a prior will be updated. In fact, the set $C_{\alpha}(E)$ sometimes is a strict subset of the set of priors passing the relative likelihood ratio test.
Let $\hat{C}_{\alpha}(E)$ denote the latter set of priors: $$\hat{C}_{\alpha}(E) = \left\{p\in C: \frac{p(E)-\min\limits_{p \in C}p(E) }{\max\limits_{p \in C}p(E) - \min\limits_{p \in C}p(E)} \geq \alpha\right\}$$ Figure \[fig:3\] shows a case where $C_{\alpha}(E) \subsetneqq \hat{C}_{\alpha}(E)$. For the same scenario as in figure \[fig:1\], notice that the area below the red dashed line and in the set $C$ is the set $\hat{C}_{\alpha}(E)$ with the same $\alpha$ as in $C_{\alpha}(E)$. Hence the area below the red dashed line yet is not in $C_{\alpha}(E)$ represents the priors that pass the relative likelihood ratio test with some $\alpha$, but are not updated under RML with this $\alpha$.\
Finally, as all the discussions above look at a single conditional event $E$, and the DM’s updating across different events is actually not restricted. It is different from FB or ML, where as the DM’s attitudes towards ML is fixed to be a constant: $\alpha = 0$ or $1$ respectively, her updating across different events also needs to be consistent with this constant such that will always be FB or ML.
For RML, two different definitions will be provided (Weak RML, RML) differs exactly in the sense that whether the updating rule, or more specifically the parameter $\alpha$, depends on the conditional event $E$.
The primitive $\{\succsim_{E}\}_{E\in \Sigma}$ is represented by Weak RML if for all $\succsim$-nonnull $E\in \Sigma$, either there exists $\alpha[E] \in (0,1]$ such that for all $f\in \mathcal{F}$: $$\min\limits_{p \in C_{E}} \int_{\Omega} u(f)dp = \min\limits_{p \in C_{\alpha[E]}(E)} \int_{E} u(f)\frac{dp}{p(E)}$$ or $\succsim_{E}$ is represented by FB ($\alpha[E] = 0$).
It should be noted that this definition separates the case $\alpha[E] \in (0,1]$ and FB ($\alpha[E] = 0$) only because the definition of FB treats events that are both $\succsim$-nonnull and strict $\succsim$-null differently. Whereas, for any $\succsim$-nonnull event $E$, when $\alpha[E] \in (0,1]$ every $p \in C_{\alpha[E]}(E)$ satisfies $p(E) > 0$.
The definition of Weak RML allows for the possibility that $\alpha[E]$ may be different across events. For example, it is able to capture an updating behavior in which the DM updates with FB ($\alpha[E] = 0$) for some event $E$ and updates with ML ($\alpha[E'] = 1$) for another event $E'$. Such inconsistency across events may come from different likelihood of events, different context (good news, bad news), or even just different labels. Although one can argue whether this inconsistency is desirable or not, by offering such flexibility, Weak RML is able to accommodate a broader class of updating behaviors. Especially, some experimental evidence also reflects such inconsistency (Liang 2019[@liang2019]).
On the other hand, Weak RML may not be strong enough in applications when sharper predictions of the updating behaviors are needed. Therefore accordingly, RML will be defined to restrict that $\alpha$ needs to be a constant across events.
The primitive $\{\succsim_{E}\}_{E\in \Sigma}$ is represented by RML if either there exists $\alpha\in (0,1]$ such that for all $\succsim$-nonnull $E\in \Sigma$ and for all $f\in \mathcal{F}$: $$\min\limits_{p \in C_{E}} \int_{\Omega} u(f)dp = \min\limits_{p \in C_{\alpha}(E)} \int_{E} u(f)\frac{dp}{p(E)}$$ or $\{\succsim_{E}\}_{E\in \Sigma}$ is represented by FB ($\alpha[E] = 0$).
Preference Foundation
=====================
For a DM who finds both FB and ML to be too extreme in terms of updating beliefs, she will sometimes deviate from the behaviors that characterize FB or ML. Recall that FB and ML correspond to two different relaxations of CR: if $f \sim_{E}x$, FB implies CR-CCE, i.e. $f_{E}x \sim x$; whereas ML implies CR-LC, i.e. $f_{E}x^{*} \sim x_{E}x^{*}$ for all sufficiently large $x^{*}$.
In order to be not as extreme as either FB or ML, the DM would necessarily update some set $C(E)$ such that $C^{*}(E) \subseteq C(E) \subseteq C$. It further implies her conditional evaluation of any act $f$ would be weakly higher than under FB and weakly lower than under ML. Let $x, x^{FB}$ and $x^{ML}$ denote the conditional certainty equivalence of an act $f$ under different updating rules such that $x^{ML} \succsim x \succsim x^{FB}$.
First notice that CR-CCE implies $f_{E}x^{FB} \sim x^{FB}$. As one replaces $x^{FB}$ by $x$ on both sides of this indifference, since $x \succsim x^{FB}$ and the better consequence is received only on event $E^{c}$ for the left hand side, yet it will be received on all states for the right hand side. Thus it would imply that the increase in terms of utility will be higher for the right hand side, i.e. $f_{E}x \precsim x$.
Intuitively, this DM’s ex-ante comparison between the acts $f_{E}x$ and $x$ can no longer be recovered exactly from contingent reasoning. Instead, for the reason that she discards some priors for conditional evaluation of the act $f$, her ex-ante evaluation of $f_{E}x$ would be **undershooting**, meaning that she is now only confident in the fact that $f_{E}x$ should be weakly worse than $x$.
On the other hand, as CR-LC implies $f_{E}x^{*} \sim x^{ML}_{E}x^{*}$ for all sufficiently large $x^{*}$, replacing $x^{ML}$ by the worse consequence $x$ would imply that $f_{E}x^{*} \succsim x_{E}x^{*}$ for all sufficiently large $x^{*}$. Similarly, the DM’s ex-ante comparison between $f_{E}x^{*}$ and $x_{E}x^{*}$ cannot be recovered exactly by contingent reasoning as well. Since now her conditional evaluation of the act $f$ depends on more priors than under ML, her ex-ante evaluation of $f_{E}x^{*}$ would be **overshooting**. It means that instead of knowing she should be indifferent, now she is only confident in that $f_{E}x^{*}$ should be weakly better than $x_{E}x^{*}$.
The following axiom summarizes exactly the above intuitions:\
**Axiom U-O** (Undershooting and Overshooting) For all $\succsim$-nonnull event $E \in \Sigma$, for all $f \in \mathcal{F}$ and $x\in X$, there exists $\bar{x}_{E,f} \in X $ such that if $f \sim_{E}x$, then $f_{E}x \precsim x$ and $f_{E}x^{*} \succsim x_{E}x^{*}$ for all $x^{*} \succsim \bar{x}_{E,f}$.\
The previous discussions suggest that the U-O axiom is necessary for a DM who updates some intermediate set between FB and ML. Moreover, this intuition is also sufficient such that the U-O axiom is able to characterize such an updating rule.
\[prop\] For each $\succsim$-nonnull $E$, the conditional preference $\succsim_{E}$ is represented by the set of posteriors of the set $C(E)$ such that $C^{*}(E) \subseteq C(E) \subseteq C$ if and only if the U-O axiom holds.
The proof of this proposition relies on separating hyperplane arguments, similar to the proof of Proposition 12 in Hanany and Klibanoff (2007)[@hanany2007updating] thus omitted in the present paper.\
The U-O axiom captures exactly the intuitions about a DM who is not willing to be as extreme as either FB or ML. However, Proposition \[prop\] claims that this intuition is not strong enough to further pin down some specific set to represent the DM’s conditional preference. In order to get more concrete characterization of conditional preferences, consider a different direction of weakening the CR-CCE axiom:
For any two acts $f,g \in \mathcal{F}$, CR-CCE implies that if $f \sim_{E}g \sim_{E}x$, then $f_{E}x \sim g_{E}x$, i.e. contingent reasoning is also true for comparison between the two acts $f_{E}x$ and $g_{E}x$. However, when the conditional preference is not as extreme as FB, the U-O axiom implies that the DM now can only conclude $f_{E}x \precsim x$ and $g_{E}x \precsim x$, but the comparison between $f_{E}x$ and $g_{E}x$ is undetermined.
To impose some restrictions on the DM’s ex-ante preference between $f_{E}x$ and $g_{E}x$, suppose that a DM would conclude $f_{E}x \sim g_{E}x$ only when the two acts $f$ and $g$ are sufficiently “similar”. In particular, suppose these two acts are “pessimistically similar” such that when sufficiently large consequences is paid on event $E^{c}$, it would trigger the same extent of disappointment reflected in the conditional evaluations of the two acts. To be more specific, it means that $f_{E}x^{*} \sim g_{E}x^{*}$ for all sufficiently large $x^{*}$.
Therefore the next axiom, **Weak CR-CCE**, is a relaxation of CR-CCE such that only when both $f \sim_{E} g \sim_{E} x$ and $f_{E}x^{*} \sim g_{E}x^{*}$ are true then it would imply that $f_{E}x \sim g_{E}x$. However, notice that there is a special case for RML where the second quantifier is redundant: when $\alpha = 1$, i.e. RML reduces to ML, $f\sim_{E} g$ would necessarily imply that $f_{E}x^{*} \sim g_{E}x^{*}$ by the CR-LC axiom. In this case, the additional restriction power from the second quantifier is no longer meaningful, and thus the implication should not always be true as well. Therefore, in the formal statement of Weak CR-CCE, I separate this special case from the general statement:\
**Axiom Weak CR-CCE.** For all $\succsim$-nonnull event $E\in \Sigma$, at least one of the following scenarios hold:
(i) For all $f, g \in \mathcal{F}$, there exists $\bar{x}_{E,f,g} \in X$ such that if $f \sim_{E} g$, then $f_{E}x^{*} \sim g_{E}x^{*}$ for all $x^{*} \succsim \bar{x}_{E,f,g}$.
(ii) For all $f, g \in \mathcal{F}$ and $x\in X$, there exists $\bar{x}_{E,f,g} \in X$ such that if $f \sim_{E} g \sim_{E}x $ and $f_{E}x^{*} \sim g_{E}x^{*} $ for all $x^{*} \succsim \bar{x}_{E,f,g}$, then $f_{E}x \sim g_{E}x$.
The following theorem shows that the additional behavioral restriction imposed by the Weak CR-CCE axiom together with the U-O axiom are sufficient to characterize conditional preferences that are represented by Weak RML.
\[thm2\] $\{\succsim_{E}\}_{E\in \Sigma}$ is represented by Weak RML if and only if the U-O and Weak CR-CCE axioms hold. Furthermore, for each $\succsim$-nonnull $E \in \Sigma$, $\alpha[E]$ is unique if there exists $p, p' \in C$ s.t. $p(E) \neq p'(E)$.
For sufficiency of the U-O and Weak CR-CCE axiom, fix any $\succsim$-nonnull $E\in \Sigma$, the proof proceeds by the following three steps:
For all $f\in \mathcal{F}$, there exists $\alpha[E,f]$ such that for all sufficiently large consequences $x^{*} \in X$ one has $$\label{eq}
(1-\alpha[E,f])\text{CE}(f_{E}x) + \alpha[E,f] \text{CE}(f_{E}x^{*}) \sim (1-\alpha[E,f])x + \alpha[E,f] \text{CE}(x_{E}x^{*})$$ where CE$(f)$ denotes the certainty equivalence of the act $f$. The quantifier “sufficiently large” means that $x^{*}$ needs to be greater than the following three thresholds: (i)$x$ (ii) $\bar{x}_{E,f}$ and (iii) $\hat{x}_{E,f}$, where $\hat{x}_{E,f}$ denotes the consequence such that for all $x^{*} \succsim \hat{x}_{E,f}$ one has $$\min\limits_{p\in C} \int_{\Omega} u(f_{E}x^{*}) dp = \min\limits_{p\in C^{*}(E)} \int_{\Omega} u(f_{E}x^{*}) dp$$
Once $x^{*}$ is sufficiently large, the existence of an $\alpha[E,f]$ such that does not depend on $x^{*}$ is guaranteed. Furthermore, this $\alpha[E,f]$ is unique if either $f_{E}x \prec x$ or $f_{E}x^{*} \succ x_{E}x^{*}$ holds.
Equation (\[eq\]) implies that the DM’s conditional evaluation of any $f \in \mathcal{F}$ can be represented by $$\min\limits_{p\in C_{\alpha[E, f]}(E)} \int_{E} u(f) \frac{dp}{p(E)}$$
By plugging into the expression of each certainty equivalence in equation (\[eq\]), one is able to show that under another MEU preference $\succsim'$, which is represented by the set $C_{\alpha[E,f]}(E)$, the following indifference holds: $f_{E}x \sim' x$. Then it further implies that the conditional evaluation of the act $f$, equals to $u(x)$, also can be represented by a FB updating of preference $\succsim'$. Notice that when $\alpha \in (0,1]$, $C_{\alpha}(E)$ is always strict $\succsim'$-nonnull.
The Weak CR-CCE axiom implies that $\alpha[E,f]$ needs to be the same across all $f \in \mathcal{F}$ and it is unique if $C \neq C^{*}(E)$, i.e. there exists $p, p' \in C$ such that $p(E) \neq p'(E)$.
First, if there does not exist any $f\in \mathcal{F}$ such that $\alpha[E,f]$ is unique, then by step 1, it is the case both $f_{E}x \sim x$ and $f_{E}x^{*} \sim x_{E}x^{*}$ hold. In other words, the conditional preference are given by both FB and ML, which further implies $C = C^{*}(E)$. Thus, if $C \neq C^{*}(E)$, then there exists at least one $f\in \mathcal{F}$ such that $\alpha[E,f]$ is unique.
Fix any $f\in \mathcal{F}$ such that $\alpha[E,f]$ is unique. For any $l \in {\mathbb{R}}_{++}$, let $f_{l}$ denote an act such that $u(f_{l}(\omega)) = l \cdot u(f(\omega))$ for all $\omega \in E$. For any $\lambda \in [0,1]$ and $y\in X$, let $f_{\lambda}y$ denote the mixture $\lambda f + (1-\lambda)y$ of an act $f$ and the consequence $y$. Equation (\[eq\]) implies that for all $l\in {\mathbb{R}}_{++}$, $\lambda \in [0,1]$ and $y \in X$, $\alpha[E, f_{l\lambda}y] = \alpha[E,f]$. Namely, these two operations on $f$ preserves $\alpha[E,f]$.
If the first scenario of the Weak CR-CCE axiom holds, since it is equivalent to CR-LC axiom, it would imply that $\alpha[E,f] = 1$ for all $f \in \mathcal{F}$.
If the first scenario is false, then the second scenario must be true. For any $g\in \mathcal{F}$ that cannot be obtained from $f$ by the two operations that preserves $\alpha[E,f]$, one can always construct an $g_{l\lambda}y$ such that the following two conditions hold: $$f \sim_{E} g_{l\lambda}y \text{ and } f_{E}x^{*} \sim [g_{l\lambda}y]_{E}x^{*}$$ By the implication of the second scenario of Weak CR-CCE, it is necessarily the case that: first, $\alpha[E,f] \neq 1$ and second, $\alpha[E, g_{l\lambda}y] = \alpha[E,g] = \alpha[E,f]$. This argument applies to arbitrary $g\in \mathcal{F}$, thus $\alpha[E,f]$ needs to be a constant in this case as well.
To conclude, combining both scenarios ($\alpha[E,f] = 1$ and $\alpha[E,f]\neq 1$) implies that $\alpha[E,f]$ is a constant across all $f\in \mathcal{F}$ for each $\succsim$-nonnull $E\in \Sigma$.
Theorem \[thm2\] shows that the U-O and Weak CR-CCE axioms characterize the Weak RML updating rule where the parameter $\alpha[E]$ may be different across events. It is obviously because of the fact that both U-O and Weak CR-CCE only restrict behaviors within each conditional event $E$, yet does not explicitly restrict behaviors across events. To impose such an additional restriction, again consider another different direction of weakening the CR-CCE axiom:
For any two $\succsim$-nonnull events $E_{1}$ and $E_{2}$, if there are two acts $f,g$ such that $f \sim_{E_{1}}x$ and $g\sim_{E_{2}}x$, i.e. the conditional evaluation of the two acts under the two events $E_{1}$ and $E_{2}$ respectively are the same, then CR-CCE further implies that $f_{E_{1}}x \sim g_{E_{2}}x$. Thus CR-CCE actually implicitly imposes such a restriction on behaviors across different events, and therefore characterizes FB where the conditional preference is consistent across events.
For conditional preferences that are not as extreme as FB, only $f_{E_{1}}x \precsim x$ and $g_{E_{2}}x \precsim x$ are known. Hence to impose some restrictions on behaviors across events in the similar manner as in Weak CR-CCE, one can also restrict that $f_{E_{1}}x \sim g_{E_{2}}x $ only when the two acts $f$ and $g$ are in addition “pessimistically similar”. However notice that, since the conditional events are different here, the sufficiently large consequences need to satisfy some additional restriction to be able to calibrate the exact same extent of disappointment that event $E_{1}$ or $E_{2}$ occurs. To be more specific, in order to claim that $f_{E_{1}}x_{1}^{*} \sim g_{E_{2}}x_{2}^{*}$, one needs to find $x_{1}^{*}$ and $x_{2}^{*}$ that are not only sufficiently large, but also satisfies $x_{E_{1}}x_{1}^{*} \sim x_{E_{2}}x_{2}^{*}$.
Furthermore, the same special case ($\alpha = 1$) needs to be handled separately. Thus the axiom that further restricts behaviors across different events is given by the following:\
**Axiom EC** (Event Consistency). For all $\succsim$-nonnull events $E_{1}$ and $E_{2}$, at least one of the following scenarios hold:
(i) For all $f, g\in \mathcal{F}$ and for all $x\in X$, there exists $\bar{x}_{E_{1}, E_{2}, f,g} \in X$ such that if $f\sim_{E_{1}}x$ and $g \sim_{E_{2}} x$, then $ f_{E_{1}}x_{1}^{*} \sim g_{E_{2}}x_{2}^{*}$ for all $x_{1}^{*}, x_{2}^{*} \succsim \bar{x}_{E_{1}, E_{2}, f,g} $ whenever $x_{E_{1}}x_{1}^{*} \sim x_{E_{2}}x_{2}^{*}$.
(ii) For all $f, g\in \mathcal{F}$ and for all $x\in X$, there exists $\bar{x}_{E_{1}, E_{2}, f,g} \in X$ such that if $f\sim_{E_{1}}x$, $g \sim_{E_{2}} x$ and$ f_{E_{1}}x_{1}^{*} \sim g_{E_{2}}x_{2}^{*}$ for all $x_{1}^{*}, x_{2}^{*} \succsim \bar{x}_{E_{1}, E_{2}, f,g} $ whenever $x_{E_{1}}x_{1}^{*} \sim x_{E_{2}}x_{2}^{*}$, then $f_{E_{1}}x \sim g_{E_{2}}x$.
The following theorem shows that the additional restriction imposed by the EC axiom exactly characterizes a constant $\alpha$ for Weak RML.
\[thm4\] $\{\succsim_{E}\}_{E\in \Sigma}$ is represented by RML if and only if the U-O, AC and EC axioms hold. Furthermore, $\alpha$ is unique if there exists $\succsim$-nonnull $E \in \Sigma$ such that $p(E) \neq p'(E)$ for some $p,p' \in C$.
Given Theorem \[thm2\], the only remaining proof is to show EC axiom is necessary and sufficient to characterize a constant $\alpha$ across all events. While a detailed proof can be found in the appendix, from the similarity between Weak CR-CCE and EC, the argument here is similar to the step 3 in the proof of Theorem \[thm2\].
Updating Ambiguous Signals
==========================
One special case in the current setting is when the state space $\Omega$ can be divided into payoff-relevant states $\Theta$ and payoff-irrelevant signals $S$. Namely, the grand state space $\Omega$ is given by the Cartesian product $\Theta \times S$ and a generic element is denoted by $\theta \times s$. The conditional event in which signal $s$ realizes is given by $E = \Theta \times s$. The set of available acts denoted by $\mathcal{F}_{\Theta}$ contains acts that depend only on $\Theta$: for all $f\in \mathcal{F}_{\Theta}$, $f(\theta\times s) = f(\theta \times s')$ for all $\theta \in \Theta$ and $s, s' \in S$.
Any environment involves states and signals can be modeled using this special framework. In general, a DM’s ambiguous belief may express ambiguity both on the marginal belief over states[^16] $\Theta$ and signals $S$. This section focuses on a further special case where the DM has an unambiguous marginal belief over the states. Formally, $p( A \times S) = p'(A \times S)$ for all $A\times S \in \Sigma$ and $p, p' \in C$.
It is the case where the DM has probabilistic belief about the states whereas the signals are ambiguous in the sense that the exact correlation between states and signals is unknown. In reality, one can imagine a situation in which a decision problem may have been encountered repeatedly such that the distribution over the states could be learned from historical data. Meanwhile for the current problem, an additional new data set (signal) is available to the DM. It is potentially informative for the true state from her knowledge about the data; however, the exact correlation between the data and the states is unknown.
This special case has been adopted in various applications: Bose and Renou (2014)[@bose2014mechanism], Beauchene, Li and Li (2019)[@beauchene2019ambiguous], Kellner and Le Quement (2018)[@kellner2018endogenous]. In addition, several recent experimental papers also focus on understanding subjects’ response to ambiguous signals in this case: Liang (2019)[@liang2019], Shishkin Ortoleva (2019)[@shishkin2019ambiguous], Kellner, Quement and Riener (2019)[@kellner2019reacting]. In the following, I will use a stylized example to illustrate the theoretical predictions RML is able to offer here.
\[exp1\] There are two payoff-relevant states: $\Theta = \{ \theta_{1},\theta_2\}$ with unambiguous marginal prior $p(\theta_{1}) = \beta \in [0,1]$. The DM is evaluating a bet $f$ that pays one at $\theta_{1}$ and nothing at $\theta_{2}$. There are two signals: $S = \{s_{1}, s_{2}\}$ and the signaling structure is ambiguous, means that there are two possible correlations between signals and states: $$\begin{aligned}
p(s_{i}| \theta_{i}) = \lambda_{1} \\
p(s_{i}| \theta_{i}) = \lambda_{2} \end{aligned}$$ and which correlation generates the signal is unknown. Without loss of generality, assume $\lambda_{1} \geq \lambda_{2}$ and $\frac{\lambda_{1} + \lambda_{2}}{2} \geq \frac{1}{2}$. Namely, the signal $s_{1}$ is “on average” an informative signal for the state $\theta_{1}$. Moreover the first signaling device governed by $\lambda_{1}$ is more accurate than $\lambda_{2}$.
Suppose a DM forms a multiple priors belief coincides exactly to the information given in this example. Her set of priors $C$ can be easily parameterized by a parameter $\mu \in [0,1]$ which denotes the probability that the first signaling device governed by $\lambda_{1}$ is the one generates the signals. Namely, $C = \{p_{\mu} \in \Delta(\Theta \times S) : \mu \in [0,1]\}$ where $p_{\mu}$ is defined as in the following: $$\begin{aligned}
&p_{\mu} (\theta_{1} \times \{s_{1}, s_{2}\}) = \beta \\
&p_{\mu} (\theta_{2} \times \{s_{1}, s_{2}\}) = 1-\beta\\
&p_{\mu}(\theta_{1}\times s_{1}) = \beta[\mu\cdot \lambda_{1} + (1-\mu) \cdot \lambda_{2}]\\
&p_{\mu}(\theta_{2}\times s_{1}) = (1-\beta)[\mu\cdot (1-\lambda_{1}) +(1-\mu)\cdot (1-\lambda_{2}) ]\\
&p_{\mu} (\{\theta_{1}, \theta_{2} \}\times s_{1}) = p_{\mu}(\theta_{1}\times s_{1}) + p_{\mu}(\theta_{2}\times s_{1})\end{aligned}$$
To fully characterize the DM’s behavior in this example, there are in total six different cases: $\beta > 1/2, \beta = 1/2$ and $\beta < 1/2$ combined with signal realization $s_{1}$ or $s_{2}$. Only one of the cases will be derived here in detail and a summary of behaviors in all these cases will be provided afterwards in Table \[table1\].
Consider the case $\beta > 1/2$ and signal $s_{1}$ is realized. Notice that when $\beta > 1/2$, the likelihood of signal $s_{1}$, given by $p_{\mu} (\{\theta_{1}, \theta_{2} \}\times s_{1})$ is maximized when $\mu = 1$. Moreover, thanks to this simple parametrization of the set $C$, for any $\alpha \in [0,1]$ the set $C_{\alpha}(\{\theta_{1}, \theta_{2} \}\times s_{1}) $ is simply given by $$C_{\alpha}(\{\theta_{1}, \theta_{2} \}\times s_{1}) = \{p_{\mu} \in C: \mu \in [\alpha, 1] \}$$ Under RML, the DM updates only the priors $p_{\mu}$ for $\mu \in [\alpha, 1]$ and for each $p_{\mu}$ the posterior about states after observing $s_{1}$ is given by $$\pi_{\mu}(\theta_{1} | s_{1}) = \frac{p_{\mu}(\theta_{1}\times s_{1})}{p_{\mu} (\{\theta_{1}, \theta_{2} \}\times s_{1})}$$
Therefore the DM’s set of posteriors will be the following interval[^17]: $\left[ \pi_{\alpha}(\theta_{1} | s_{1}), \pi_{1}(\theta_{1} | s_{1})\right]$. Apparently, under FB, the DM’s set of posteriors will be $\left[ \pi_{0}(\theta_{1} | s_{1}), \pi_{1}(\theta_{1} | s_{1})\right]$; and under ML, the DM will end up with a singleton posterior: $\pi_{1}(\theta_{1} | s_{1})$.
Furthermore under MEU, the DM’s conditional evaluation of the bet $f$ will be given by the lowest posterior of state $s_{1}$. Let $x_{f}^{FB}, x_{f}^{RML}$ and $x_{f}^{ML}$ denote the conditional certainty equivalence of the bet $f$ under FB, RML and ML updating, then one has $x_{f}^{FB} \leq x_{f}^{RML} \leq x_{f}^{ML}$ for all $\alpha \in [0,1]$. Especially, $x_{f}^{RML}$ is increasing with respect to $\alpha$ and coincides with $x_{f}^{FB}$ and $x_{f}^{ML}$ when $\alpha = 0$ and $\alpha = 1$ respectively.
Table \[table1\] summarizes the predictions of behaviors under RML in all six cases. The first two rows are the only cases where the DM’s evaluation of $f$ depends on the value of $\alpha$. In other words, those are the cases where different inferences with respect to likelihood might affect one’s conditional evaluation of $f$ after updating, and thus RML is able to provide richer predictions than FB and ML.
The 3rd and 4th row are the cases where the MEU evaluation of $f$ are given by the posteriors updated from the maximum likelihood priors, thus even the updated beliefs are different under FB and ML, MEU makes their evaluations the same. For this reason, although the updated belief under RML is also different from FB and ML for any $\alpha \in (0,1)$, the MEU evaluation under RML is still the same.
Finally, the 5th and 6th rows represent the same type of special case where every prior agrees with the likelihood of the signal. It is the case where FB coincides with ML, and therefore RML does not have an extra bite. All the priors will be updated under RML for all $\alpha \in [0,1]$, thus the updated belief will always be the same.
--------- --------- ----------- ---------------------------------- -----------------------
$\beta$ Signal ML prior Evaluation Comparison with
of $f$ probabilistic signal
$\alpha <1/2$: lower
$>1/2$ $s_{1}$ $\mu = 1$ $\pi_{\alpha}(\theta_{1}|s_{1})$ $\alpha = 1/2$: equal
$\alpha >1/2$: higher
$\alpha <1/2$: lower
$>1/2 $ $s_{2}$ $\mu = 0$ $\pi_{\alpha}(\theta_{1}|s_{2})$ $\alpha = 1/2$: equal
$\alpha >1/2$: higher
$<1/2 $ $s_{1}$ $\mu = 0$ $\pi_{0}(\theta_{1}|s_{1})$ All $\alpha$: lower
$<1/2 $ $s_{2}$ $\mu = 1$ $\pi_{1}(\theta_{1}|s_{2})$ All $\alpha$: lower
$=1/2 $ $s_{1}$ All $\mu$ $\pi_{0}(\theta_{1}|s_{1})$ All $\alpha$: lower
$=1/2 $ $s_{2}$ All $\mu$ $\pi_{1}(\theta_{1}|s_{2})$ All $\alpha$: lower
--------- --------- ----------- ---------------------------------- -----------------------
: Summary of Example \[exp1\] under RML[]{data-label="table1"}
The last column of Table \[table1\] presents a comparison that is highlighted in experiments by Liang (2019)[@liang2019]. It compares the DM’s s conditional evaluations of $f$ under this ambiguous signaling structure with a probabilistic signaling structure where the correlation equals to $\frac{\lambda_{1} + \lambda_{2}}{2}$. The latter is designed to reflect an average accuracy of the two signaling devices in the ambiguous signaling structure. For probabilistic signals, assume that the DM follows Bayesian updating.
In the case where $s_{1}$ is realized, notice that the DM’s posterior under the probabilistic signaling structure is exactly given by $\pi_{1/2}(\theta_{1} | s_{1})$. Therefore, for $\beta \leq 1/2$, the DM’s conditional evaluation under ambiguous signal is always **lower** than her conditional evaluation under probabilistic signal. On the other hand, when $\beta > 1/2$, this comparison would depend on the value of $\alpha$. More specifically, $\alpha < 1/2$ implies that the conditional evaluation under ambiguous signal is lower, $\alpha > 1/2$ implies that under ambiguous signal is higher and $\alpha = 1/2$ implies they are the same. Therefore, according to RML, this type of comparison sometimes actually reflect the DM’s attitudes towards making an inference with respect to likelihood captured by $\alpha$.
For the experimental results shown by the Table 4.2 in Liang (2019)[@liang2019], each row there can be categorized into some case in the Table \[table1\] here, as $s_{1}$ is “good news” and $s_{2}$ is “bad news” in his terminology. A rather interesting pattern emerged from the choice data is that, when $\beta > 1/2$, the subjects’ comparison is lower after receives good news ($s_{1}$) and higher after receives bad news ($s_{2}$). Notice that from Table \[table1\], the two directions of the comparisons are aligned with $\alpha < 1/2$ and $\alpha > 1/2$ respectively. It implies that the subjects may have different attitudes towards making inferences when the signal realization is different, especially when they are associates with meanings such as “good news” and “bad news”. This pattern can be captured by the **Weak RML**, which is characterized exactly to reflect such a flexibility in behaviors.
Moreover, the choice data shown in the other rows corresponding to cases where $\alpha$ is irrelevant for this comparison, is also aligned with behaviors under RML such that the DM finds her evaluation of $f$ is lower under ambiguous signal[^18]. Therefore, the Weak RML with $\alpha[\{\theta_{1}, \theta_{2} \}\times s_{1}] <1/2$ and $\alpha[\{\theta_{1}, \theta_{2} \}\times s_{2}] >1/2$, not only offers an interpretation for the different directions of comparison across signals, but in fact also provides an approach to accommodate almost all the behavioral patterns in this experiment[^19].\
In summary, for this special case where DM has probabilistic belief over states and ambiguous belief over signals, RML is able to provide richer predictions when inferences with respect to likelihoods affects behaviors. Moreover, such richness on the other hand would also be useful for considering the robustness of behaviors under ambiguous signals when the DM can potentially make an inference with respect to likelihoods.
Robust Ambiguous Persuasion: An Example
=======================================
This section elaborates the last comment from the previous section in the context of information design, or more specifically, persuasion. The scope here is restricted to an illustrative example, more general results are left for future work.
Consider a persuasion environment where the sender can commit to some signaling device to induce actions from the receiver and suppose the sender and receiver have a common probabilistic belief about the states.
Bayesian persuasion describes the case where the available signaling devices for the sender are only probabilistic. If the sender also has access to ambiguous devices, which specifies a set of probabilistic devices and the probability of using any one of them is unknown[^20], Beauchene, Li and Li (2019)[@beauchene2019ambiguous] show that when the receiver’s preference is represented by MEU with FB, the sender is able to gain strictly more payoff by sending ambiguous signals compared with the optimal probabilistic device.
Notice that once the sender commits to an ambiguous device, from the receiver’s point of view, it is exactly the updating ambiguous signals situation extensively discussed in the previous section. Then apparently the FB assumption excludes the possibility that the receiver may use the likelihood of signals to further infer about the plausibility of each probabilistic device. This fact thus suggests a caveat that the strict gain from using ambiguous signals may only come from the FB assumption instead of the general idea of ambiguity. In other words, it might seem to be that the ambiguous signals could help the sender exploit ambiguity aversion of the receiver only when inferences about the initial belief are not allowed.
RML provides exactly the tool to address such an issue. In the following, I take the illustrative example from Beauchene, Li and Li (2019) [@beauchene2019ambiguous] and relax their FB assumption to RML with any $\alpha \in [0,1]$. Then for this particular example, first of all, the ambiguous device that approaches the sender’s optimal payoff under FB assumption is not robust. In particular, for RML with any $\alpha > 0$, their ambiguous device cannot induce the same action from the receiver as under FB ($\alpha = 0$), and the sender’s payoff is strictly worse than using the optimal probabilistic device.
However on the other hand, for this example there still exists a *robust* ambiguous device such that induce the same action from the receiver and generates the same payoff for the sender for all $\alpha \in [0,1]$ under RML. Moreover, the sender gains strictly more payoff from this robust ambiguous device compared with the optimal probabilistic device. In other words, the strict benefits from using ambiguous signals in this example is robust to any possible inferences with respect to the likelihood of signals in updating.
\[exp4\] There are two states $\{ \omega_{l}, \omega_{h}\}$ with a uniform marginal prior, and the receiver has three feasible actions: $\{a_{l}, a_{m}, a_{h}\}$. The payoff of sender and receiver for each state and action is given as follows:
$\omega_{l}$ $\omega_{h}$
--------- -------------- --------------
$a_{l}$ (-1,3) (-1,-1)
$a_{m}$ (0,2) (0,2)
$a_{h}$ (1,-1) (1,3)
where in each cell, the first number is sender’s payoff and the second is receiver’s.
The payoff structure here is standard for persuasion, where the sender always prefers the receiver to take higher action yet the receiver prefers to choose the action that matches the state.
First consider the FB assumption, the following replicates the discussions from Beauchene, Li and Li (2019)[@beauchene2019ambiguous]. For more in depth explanations, readers are advised to refer to the original paper.
The ambiguous device they offer under FB assumption can be constructed by the following two steps: first, identify the *base probabilistic devices* that generate the desired set of posteriors, and then construct the probabilistic devices from these base devices in a way that is able to hedge against the sender’s own ambiguity.
In this example, let $\{m_{l}, m_{h}\}$ be the set of signals and then the base probabilistic devices $\pi_{1}$ and $\pi_{2}$ are:
$\pi_{1}(m|\omega)$ $\omega_{l}$ $\omega_{h}$
--------------------- -------------- --------------
$m_{l}$ $2/3$ 0
$m_{h}$ $1/3$ $1$
$\pi_{2}(m|\omega)$ $\omega_{l}$ $\omega_{h}$
--------------------- -------------- --------------
$m_{l}$ $3/4$ $1/4$
$m_{h}$ $1/4$ $3/4$
If the sender’s ambiguous device consists of these two base devices, then the receiver would form the following set of posteriors: $$\begin{split}
&p(\omega_{h}|m_{l}) = \{0, 1/4\} \\
&p(\omega_{h}|m_{h}) = \{3/4, 3/4\}
\end{split}$$ where the first and second posterior in each set is updated from the first and second base device respectively. Given these posteriors, the receiver with MEU preference would take action $a_{m}$ when signal $m_{l}$ is realized and takes action $a_{h}$ when signal $m_{h}$ is realized. The posterior $p(\omega_{h}|m_{l}) = 1/4$ is crucial since any posterior that assigns less probability on $\omega_{h}$ would induce the receiver to take action $a_{l}$ and makes the sender worse off.
Given the receiver’s action for each signal, the sender’s evaluation of the ambiguous device may also be affected by the existence of ambiguity. If the sender uses an ambiguous device which contains exactly these two base devices, then under MEU her ex-ante payoff is given by $$\frac{1}{2} u_{s}(a_{m}) + \frac{1}{2} u_{s}(a_{h})$$ where $u_{s}$ stands for the sender’s payoff function and it is exactly the payoff from using the optimal probabilistic device. Thus the sender needs to hedge against this ambiguity to get a higher payoff.
Consider the following construction: first increase the number of signals to four such that now the signals are $\{m_{l}, m_{h}, m_{l}', m_{h}'\}$. Then consider a probabilistic device generated by a mixture of the two base devices: $\pi_{1}' = \lambda \pi_{1} \oplus (1-\lambda)\pi_{2}$, which represents a device sending signals $\{m_{l}, m_{h}\}$ with probability $\lambda$ according to the base device $\pi_{1}$ and sending signals $\{m_{l}', m_{h}'\}$ with probability $(1-\lambda)$ according to $\pi_{2}$. One can verify that given this device, the receiver’s posteriors coincide with $\pi_{1}$ when $m \in \{m_{l}, m_{h}\}$ and coincides with $\pi_{2}$ when $m \in \{m_{l}', m_{h}'\}$.
Moreover, consider the following two probabilistic devices constructed in the same manner $\pi_{1}' = \lambda \pi_{1} \oplus (1-\lambda)\pi_{2}$ and $\pi_{2}' = (1-\lambda)\pi_{2}\oplus \lambda \pi_{1} $:
$\pi_{1}'(m|\omega)$ $\omega_{l}$ $\omega_{h}$
---------------------- ------------------------ ------------------------
$m_{l}$ $\lambda \cdot 2/3$ 0
$m_{h}$ $\lambda \cdot 1/3$ $\lambda$
$m_{l}'$ $(1-\lambda)\cdot 3/4$ $(1-\lambda)\cdot 1/4$
$m_{h}'$ $(1-\lambda)\cdot 1/4$ $(1-\lambda)\cdot 3/4$
$\pi_{2}'(m|\omega)$ $\omega_{l}$ $\omega_{h}$
---------------------- ------------------------ ------------------------
$m_{l}$ $(1-\lambda)\cdot 3/4$ $(1-\lambda)\cdot 1/4$
$m_{h}$ $(1-\lambda)\cdot 1/4$ $(1-\lambda)\cdot 3/4$
$m_{l}'$ $\lambda \cdot 2/3$ 0
$m_{h}'$ $\lambda \cdot 1/3$ $\lambda$
When $m \in \{m_{l}, m_{h}\}$, the posterior of $\pi_{1}'$ coincides with $\pi_{1}$ and the posterior of $\pi_{2}'$ coincides with $\pi_{2}$, so that the set of posteriors generated by the ambiguous device $\Pi' = \{\pi_{1}', \pi_{2}'\}$ remains the same as the base devices. Then signals $m_{l}$ and $m_{l}'$ would induce the receiver to take action $a_{m}$ and signals $m_{h}$ and $m_{h}'$ would induce action $a_{h}$.
Furthermore notice that, the difference between these two probabilistic devices is **only** the label of the signals. Thus, given the receiver’s action is the same across signal $m_{l}$ and $m_{l}'$ as well as across signals $m_{h}$ and $m_{h}'$. The two probabilistic devices induce each action with the same frequency, hence the sender’s ex-ante payoff will be the same across these two devices. Therefore using an ambiguous device that contains these two probabilistic devices will not generate any ambiguity for sender’s ex-ante payoffs.
Especially, the sender’s ex-ante payoff from this ambiguous device is given by $$\label{ap}
\left[\frac{1}{2}(1-\lambda) + \frac{1}{3}\lambda\right]u_{s}(a_{m}) + \left[\frac{1}{2}(1-\lambda) + \frac{2}{3}\lambda \right]u_{s}(a_{h})$$ which is strictly higher than using the optimal probabilistic device when $\lambda > 0$ and it is also increasing in $\lambda$.
Therefore, the optimal ambiguous persuasion can be approached by letting $\lambda \rightarrow 1$, notice that $\lambda$ cannot be exactly one.\
Above is the Beauchene, Li and Li (2019)[@beauchene2019ambiguous]’s construction of the ambiguous device under FB assumption. The fact that their optimal device relies on letting $\lambda$ approaches one but cannot be exactly one already suggests its fragility when FB assumption is slightly deviated.
More formally, such a concern is exactly originated from the fact that the likelihood of generating the signals by each device also depends on $\lambda$. If letting $\lambda \rightarrow 1$, the likelihood of generating the same signal by each device might be severely different.
Let $l_{i}(m)$ denote the likelihood of generating signal $m$ under device $\pi_{i}'$:
$l_{1}(m)$ $p(\omega_{h}|m)$
---------- ------------------------ -------------------
$m_{l}$ $\lambda \cdot 1/3$ 0
$m_{h}$ $\lambda \cdot 2/3$ 3/4
$m_{l}'$ $(1-\lambda)\cdot 1/2$ 1/4
$m_{h}'$ $(1-\lambda)\cdot 1/2$ 3/4
$l_{2}(m)$ $p(\omega_{h}|m)$
---------- ------------------------ -------------------
$m_{l}$ $(1-\lambda)\cdot 1/2$ 1/4
$m_{h}$ $(1-\lambda)\cdot 1/2$ 3/4
$m_{l}'$ $\lambda \cdot 1/3$ 0
$m_{h}'$ $\lambda \cdot 2/3$ 3/4
When $\lambda \rightarrow 1$, notice that the likelihood of generating signal $m_{l}$ by device $\pi_{2}'$ goes to $0$, whereas the likelihood of $m_{l}$ by device $\pi_{1}'$ goes to $1/3$. Thus intuitively, knowing $\lambda \rightarrow 1$, whenever signal $m_{l}$ is observed, the receiver should be almost sure that it is generated by device $\pi_{1}'$. However, notice that the crucial posterior $1/4$ is in fact generated by the other device $\pi_{2}'$.
Indeed, for RML with any $\alpha > 0$, since the device $\pi_{2}'$ has the minimum likelihood of generating signal $m_{l}$, the crucial posterior $1/4$ is always excluded from the set of posteriors. Then the receiver would find $a_{l}$ is strictly better than $a_{m}$, thus the desired action $a_{m}$ can no longer induced. Furthermore, one can also verify that the sender’s payoff is equivalent to some non-optimal probabilistic device, thus she instead becomes strict worse off using ambiguous signals compared with using the optimal probabilistic device when the receiver slightly deviates from FB in the direction of ML.
Therefore the first observation is that, ambiguous devices that are strictly profitable than the optimal probabilistic device under FB may sharply depend on this assumption. Any slight deviation from FB in the direction of ML could actually result in a complete reversal of the comparison.
Nonetheless notice that, when $\lambda = 3/5$, the likelihood of generating signal $m_{l}$ by device $\pi_{1}'$ and $\pi_{2}'$ becomes the same and it is also true for the signal $m_{l}'$. Consequently, the receiver cannot use likelihood to make any inference about the devices. It is the case where FB and ML coincides and the receiver will always update with respect to both devices under RML no matter what the $\alpha$ is. As the crucial posterior $1/4$ will always be updated, the sender is able to induce the action $a_{m}$ from the receiver when signal $m_{l}$ or $m_{l}'$ realizes.
Furthermore, even though the likelihood of generating $m_{h}$ and $m_{h}'$ is not the same across the two devices, as the corresponding posteriors are the same, these signals can always induce the same action $a_{h}$ from the receiver as well. That being said, the ambiguous device $\pi'$ with $\lambda = 3/5$ is able to always induce action $a_{m}$ and $a_{h}$ with corresponding signals regardless of the receiver’s attitudes towards making inferences, i.e. for any RML receiver with any $\alpha \in [0,1]$. In addition, the sender’s payoff will also be given by equation (\[ap\]) after plugging into $\lambda = 3/5$, which is strictly higher than using the optimal probabilistic device.
Therefore, this ambiguous device is *robustly better* than probabilistic devices in the sense that it is able to generate strictly more payoff for the sender regardless of the receiver’s attitudes between FB and ML. More importantly, it further suggests that the strict gain from using ambiguous signals is also robust to the concern that the receiver may use likelihoods to make inferences.\
This section presents an example of considering the robustness question for applications of ambiguity, especially it illustrates how RML can be applied as a tool to look at robustness with respect to updating ambiguity. Although the scope here is restricted only to this example, more results along this line hopefully may help resolve some potential doubts and concerns about applications of ambiguity. Undoubtedly, RML considers only one dimension of robustness, the other dimensions captured by other updating models such as dynamic consistency (Hanany and Klibanoff 2007[@hanany2007updating]) are also equally important.
Related Literature
==================
The present paper adds to the literature on dynamic choice under ambiguity by characterizing the RML updating rule. As mentioned in the introduction, for ambiguity sensitive choice, one cannot preserve both dynamic consistency and consequentialism at the same time (Hanany and Klibanoff 2007[@hanany2007updating], Siniscalchi 2009[@siniscalchi2009two]). RML takes the consequentialist approach such that relaxes dynamic consistency. Pires (2002)[@pires2002rule] and Gilboa and Schmeidler (1993)[@gilboa1993updating] are the two most closely related papers that also take the consequentialist approach. Along another route, Hanany and Klibanoff (2007, 2009)[@hanany2007updating][@hanany2009updating] axiomatize updating rules that preserve dynamic consistency yet do not require consequentialism.
The idea of refining initial belief as new information arrives is essential in many different non-Bayesian updating rules, and it is also not an exclusive feature for beliefs with multiple priors. When the initial belief is a singleton, for example, Ortoleva (2012)[@ortoleva2012modeling] characterizes a hypothesis testing updating rule such that if the likelihood of information received is too low under the initial belief, then the DM will revise that initial belief and find a different prior for updating. Since the initial belief is probabilistic, the hypothesis testing updating rule emphasizes on dealing with unexpected events such as those with a probability of 0.
Zhao (2017)[@zhao2017] also considers the probabilistic belief and focuses on unexpected information, especially when the information takes the form “event $A$ is more likely than event $B$” such that it contradicts to the DM’s initial belief. He characterizes the Pseudo-Bayesian updating rule in which the DM updates another prior that is closest to the initial belief in terms of Kullback-Leibler divergence and is subject to the constraint specified in the unexpected information.
When the information is ambiguous yet the DM is ambiguity neutral and forms a single prior belief, Suleymanov (2018)[@suleymanov2018a] characterizes the Robust Maximum Likelihood updating rule in which the DM revises her initial belief according to the maximum likelihood of the observed event. Namely, the DM is an expected utility maximizer both ex-ante and conditionally, yet the posterior is not updated from the prior. Thus the main difference between the Robust Maximum Likelihood and ML as well as RML (Relative Maximum Likelihood) is that the latter updating rules require that the posteriors have to be updated from the subset of those priors that represent the DM’s ex-ante preference. In other words, RML necessarily reduces to Bayesian updating when the DM’s ex-ante preference is represented by expected utility, which is not true for Robust Maximum Likelihood updating.
In cases where the initial belief is a set of priors, one way of refining is to rule out priors from the initial set. Epstein and Schneider (2007)[@epstein2007learning] proposes an updating rule without characterization, such that the refining is done according to the likelihood ratio test. RML belongs to this category of ruling out priors, and more importantly, relative likelihood ratio test is the necessary criterion for selecting priors under RML. The dynamic consistent updating rule characterized in Hanany and Klibanoff (2007)[@hanany2007updating] also features ruling out priors, and the criteria there is to maintain the optimality of the ex-ante preferred act.
Yet another way of refining is to consider a different set of priors, where it is possible that some priors are not included in the initial belief. Ortoleva (2014)[@Ortoleva2014] characterizes the hypothesis testing updating rule for multiple priors: if the likelihood of the observed event is too low under some prior in the initial belief, then the DM will revise her initial belief and change to another set of priors for updating.
When the initial belief further involves confidence ranking proposed by Hill (2013)[@hill2013confidence], based on a similar motivation, Hill (2019)[@hill2019] characterizes an updating rule under this framework which also features in refining initial belief by the information received, and it include FB and ML as special cases as well.
Except for the theoretical development on this idea of revising and refining initial belief, De Filippis et al. (2019)[@de2018non] also identifies such behavior in a social learning experiment. Their finding suggests that the non-Bayesian behavior observed in the experiment is consistent with a generalized ML updating rule where subjects revise their initial belief according to the information they receive. As mentioned in section 5, there are several recent experimental papers, Liang (2019)[@liang2019], Shishkin Ortoleva (2019)[@shishkin2019ambiguous] and Kellner, Quement and Riener (2019)[@kellner2019reacting] testing updating behaviors under ambiguity in a special environment.
Gul and Pesendorfer (2019)[@gul2017evaluating] observe a common unintuitive feature of both FB and ML that is “all news is bad news”. Namely, a DM sometimes find that the ex-ante preferred alternative is dominated by another alternative no matter what the realization of the signal is. Imposing the “not all news can be bad news” axiom, they characterize an updating by proxy rule for preferences admitting CEU representation with capacities that are totally monotone, which is a strict subset of the preferences considered in the present paper.\
Concluding Remarks
==================
Recall the updating procedure for multiple priors beliefs proposed at the beginning of introduction. Bayesian updating of single prior beliefs is a special case where the first step is absent. In the case where the true probability law that governs the uncertainty is known, such absence is reasonable since one cannot further refine the belief but can only update it conditional on the information received.
However, in scenarios where the underlying probability law is unknown, the DM needs to form a conjecture about the uncertainty for decision making. Whether the conjecture is a singleton or a set of probabilities, it all seems too stringent to require the DM to always stick with her initial conjecture despite new information she might receive. Thus Bayesian updating of the conjecture belief actually reflects a confidence about her initial belief for updating. Accordingly, updating rules that do not reflect such a confidence and allow for refining initial belief should also be reasonable.
As mentioned in the related literature, several different updating rules have been proposed to capture the situation in which initial conjecture is a singleton and it may be revised after seeing new information. For when the initial beliefs are multiple priors, the present paper proposes RML updating rule, in which the initial beliefs are revised based on likelihood of the information observed. More importantly, the present paper pinpoints the behaviors that are equivalent to such an updating rule, which provides a preference foundation for using likelihood as a criterion for updating.
One application of RML highlighted in the present paper is using it for robustness analysis in applications involve ambiguity. As only one example is illustrated here, more general results are left for future research.
Proofs of the results
=====================
Throughout all the proofs, let $C$ denote the set of priors representing the ex-ante preference; $C^{*}(E)$ denote the subset of $C$ assigning maximum likelihood of event $E$: $C^{*}(E) \equiv \{p \in C: p(E) \geq p'(E) \forall p' \in C\}$ and $p^{*}(E)$ denote the maximum likelihood of event $E$: $p^{*}(E) \equiv \max\limits_{p \in C}p(E)$.
Proof of Theorem \[thm1\]
-------------------------
**Necessity.** The necessity of CR-LC axiom for ML updating is proved via the following three lemmas:
\[lem1\] Suppose the conditional preferences are represented by ML. For all $\succsim$-nonnull $E\in \Sigma$, for all $f\in \mathcal{F}$ and for all $x, x^{*} \in X$, if $f \sim_{E} x $, $x^{*} \succsim x$ and $$\min\limits_{p \in C} \int_{\Omega} u(f_{E}x^{*})dp = \min\limits_{p \in C^{*}(E)} \int_{\Omega} u(f_{E}x^{*})dp$$ then $f_{E}x^{*} \sim x_{E}x^{*}$.
Suppose the conditional preferences are represented by ML. For any $\succsim$-nonnull $E$, $f \sim_{E}x$ implies that $\min\limits_{p \in C^{*}(E)} \int_{E} u(f)\frac{dp}{p(E)}= u(x)$, then one can further derive $$\begin{aligned}
\min\limits_{p \in C} \int_{\Omega} u(f_{E}x^{*}) dp &= \min\limits_{p \in C^{*}(E)} \int_{\Omega} u(f_{E}x^{*}) dp \\
&= \min\limits_{p\in C^{*}(E)} \left[ \int_{E} u(f_{E}x^{*}) \frac{dp}{p(E)} \cdot p(E) + (1-p(E))u(x^{*}) \right] \\
& = p^{*}(E)\cdot \min\limits_{p\in C^{*}(E)} \int_{E} u(f)\frac{dp}{p^{*}(E)} + (1-p^{*}(E))u(x^{*}) \\
&= p^{*}(E) u(x) + (1-p^{*}(E))u(x^{*}) \\
&= \min\limits_{p \in C} \int_{\Omega}u(x_{E}x^{*})dp\end{aligned}$$ where the third equality follows from $p(E) = p^{*}(E)$ for all $p \in C^{*}(E)$, the last equality follows from the fact that $u(x^{*}) \geq u(x)$ as $x^{*} \succsim x$.
\[lem2\] For all $\succsim$-nonnull $E\in \Sigma$, for all $f\in \mathcal{F}$, if there exists $\bar{x}\in X$ such that $$\min\limits_{p \in C} \int_{\Omega}u(f_{E}\bar{x})dp = \min\limits_{p \in C^{*}(E)} \int_{\Omega} u(f_{E}\bar{x})dp$$ then for all $x^{*} \succsim \bar{x}$ one has $$\min\limits_{p \in C} \int_{\Omega}u(f_{E}x^{*})dp = \min\limits_{p \in C^{*}(E)} \int_{\Omega}u(f_{E}x^{*})dp$$
For any $\succsim$-nonnull $E \in \Sigma$ and any $f\in \mathcal{F}$. Suppose there exists $x\in X$ such that $\min\limits_{p \in C} \int_{\Omega}u(f_{E}x)dp = \min\limits_{p \in C^{*}(E)} \int_{\Omega} u(f_{E}x)dp$. Towards a contradiction suppose there also exists $x'$ such that $x' \succsim x$ as well as $$\min\limits_{p\in C}\int_{\Omega}u(f_{E}x')dp < \min\limits_{p\in C^{*}(E)} \int_{\Omega}u(f_{E}x')dp$$ Then this strict inequality further implies: $$\begin{split}
&\min\limits_{p\in C} \left[ \int_{E} u(f)dp + u(x')(1-p(E)) \right]< \min\limits_{p\in C^{*}(E)} \left[\int_{E} u(f)dp + u(x') (1-p(E)) \right] \\
&\min\limits_{p\in C} \left[ \int_{E} u(f)dp + u(x')(1-p(E)) \right]< \min\limits_{p\in C^{*}(E)}\left[ \int_{E} u(f)dp+ u(x)(1-p^{*}(E)) + (u(x')-u(x))(1-p^{*}(E)) \right]\\
&\min\limits_{p\in C} \left[ \int_{E} u(f)dp + u(x')(1-p(E)) \right] - (u(x')-u(x))(1-p^{*}(E)) < \min\limits_{p\in C^{*}(E)} \int_{\Omega} u(f_{E}x) dp\\
&\min\limits_{p\in C} \left[\int_{E} u(f)dp + u(x)(1-p(E)) + (u(x')-u(x))(p^{*}(E)- p(E)) \right] < \min\limits_{p\in C^{*}(E)} \int_{\Omega} u(f_{E}x) dp\\
&\min\limits_{p\in C} \left[ \int_{\Omega} u(f_{E}x) dp + (u(x')-u(x))(p^{*}(E)- p(E)) \right]< \min\limits_{p\in C^{*}(E)} \int_{\Omega} u(f_{E}x) dp
\end{split}$$ Notice that, for the LHS of the last inequality, minimum of $\int_{\Omega} u(f_{E}x) dp $ can be obtained at some $p \in C^{*}(E)$ which also minimizes the second term $(u(x')-u(x))(p^{*}(E)- p(E))$, since for all $p\in C$, $p^{*}(E) - p(E) \geq 0$. Thus the minimum of LHS is obtained at some p with $p(E) = p^{*}(E)$ and it implies $$\min\limits_{p\in C} \int_{\Omega} u(f_{E}x) dp = \min\limits_{p\in C^{*}(E)} \int_{\Omega} u(f_{E}x) dp < \min\limits_{p\in C^{*}(E)}\int_{\Omega} u(f_{E}x) dp$$ which is a contradiction.
\[lem4\] If $\succsim$ admits MEU representation with finitely many plausible priors, then for all $\succsim$-nonnull $E\in \Sigma$, for all $f\in \mathcal{F}$, an $\bar{x}_{E,f} \in X$ such that both $$\min\limits_{p \in C} \int_{\Omega}u(f_{E}\bar{x}_{E,f})dp = \min\limits_{p \in C^{*}(E)} \int_{\Omega} u(f_{E}\bar{x}_{E,f})dp$$ and $\bar{x}_{E,f} \succsim_{E} f$ always exists.
First show that when $C$ contains only finitely many extreme points, an $\bar{x}_{E,f}$ such that $f_{E}\bar{x}_{E,f}$ is evaluated at some extreme point in $C^{*}(E)$ always exists.
Let $q$ be any extreme point in $C^{*}(E)$ and let $p$ be any extreme point in $C$. The act $f_{E}\bar{x}_{E,f}$ is evaluated at $q$ if for all $p \in C$, $$\int_{\Omega} u(f_{E}\bar{x}_{E,f}) dq \leq \int_{\Omega} u(f_{E}\bar{x}_{E,f}) dp$$ It can be further derived as $$\label{eq1}
\int_{E} u(f) dq + (1-q(E)) u(\bar{x}_{E,f}) \leq \int_{E} u(f) dp + (1-p(E)) u(\bar{x}_{E,f})$$ Notice that the first term of both LHS and RHS does not depend on $\bar{x}_{E,f}$, furthermore, $(1-q(E)) \leq (1-p(E))$ as $q \in C^{*}(E)$. When $p$ is also in $C^{*}(E)$, the value of $\bar{x}_{E,f}$ does not matter and one can pin down the $q \in C^{*}(E)$ such that minimizes the evaluation of $f_{E}\bar{x}_{E,f}$ among all extreme points in $C^{*}(E)$.
Fix that $q$, and then for extreme points not in $C^{*}(E)$, the following inequality becomes strict: $(1-q(E)) < (1-p(E))$. Then for all $f, p, E$, since $X$ is unbounded under the ex-ante preference, there always exists an $\bar{x}_{E,f,p}$ such that the inequality (\[eq1\]) holds. As inequality (\[eq1\]) also holds for any $x \succsim \bar{x}_{E,f,p}$, it suffice to let $\bar{x}_{E,f}$ be $ \max \{ \max\limits_{p} \bar{x}_{E,f,p}, x\}$ for $x \sim_{E} f$ where the “max” is according to the ex-ante preference. The existence of such an $\bar{x}_{E,f}$ is guaranteed by $X$ being unbounded under the ex-ante preference and there are only finitely many extreme points in $C$
In summary, for all $\succsim$-nonnull $E\in \Sigma$, for all $f \in \mathcal{F}$ and $x \in X$, Lemma \[lem2\] and \[lem4\] together show the existence of a threshold $\bar{x}_{E,f}$ such that for all $x^{*} \succsim \bar{x}_{E,f}$, the evaluation of the act $f_{E}x^{*}$ is given by some prior in $C^{*}(E)$, meanwhile $x^{*} \succsim x$. Then Lemma \[lem1\] shows that, when $\succsim_{E}$ is given by ML updating, for all $x^{*} \succsim \bar{x}_{E,f}$ one has $f_{E}x^{*} \sim x_{E}x^{*}$.\
**Sufficiency.** For sufficiency of CR-LC axiom, fix any $\succsim$-nonnull $E\in \Sigma$, consider the contra positive statement: not ML updating implies not CR-LC.
Let $C_{E}$ be the closed and convex set of posteriors represents the conditional preference $\succsim_{E}$. Not ML updating implies that $C_{E} \neq \{\frac{p}{p(E)}: p \in C^{*}(E)\}$. In other words, either there exists $\tilde{p} \in C^{*}(E)$ such that $\frac{\tilde{p}}{\tilde{p}(E)} \notin C_{E}$, or there exists $q \in C_{E}$ such that $q \notin \{\frac{p}{p(E)}: p \in C^{*}(E)\}$ or both.
Not CR-LC means that there exists $f\in \mathcal{F}$ and $x\in X$ such that $f\sim_{E}x$ and for all $\bar{x} \in X$, there exists $x^{*} \in X$ such that $x^{*} \succsim \bar{x}$ and it is not the case $f_{E}x^{*} \sim x_{E}x^{*}$.
For the two different cases of not ML, since both $C_{E}$ and $\{\frac{p}{p(E)}: p \in C^{*}(E)\}$ are convex and closed set, the same type of separating hyperplane argument can be applied to both cases. Thus the proof here only shows the implication of the first case, while the same argument applies to the other case.
Formally, in the first case, there exists $\tilde{p} \in C^{*}(E)$ such that $\frac{\tilde{p}}{\tilde{p}(E)} \notin C_{E}$, strong separating hyperplane theorem implies that there exists an act $f\in \mathcal{F}$ such that $$\int_{E} u(f) \frac{d\tilde{p}}{\tilde{p}(E)} < \min\limits_{p\in C_{E}} \int_{E} u(f)dp$$ Then first as $\tilde{p} \in C^{*}(E)$, one has $\min\limits_{p\in C^{*}(E)} \int_{E} u(f) \frac{dp}{p(E)} \leq \int_{E} u(f) \frac{d\tilde{p}}{\tilde{p}(E)}$. Second, for any $x\in X$, $f \sim_{E} x$ implies that $\min\limits_{p\in C_{E}} \int_{E} u(f)dp = u(x)$. These inequalities and equality together imply that $$\min\limits_{p\in C^{*}(E)} \int_{E} u(f) \frac{dp}{p(E)} < u(x)$$
On the other hand by Lemma \[lem4\], there always exists an $\bar{x}_{E,f}$ such that $\bar{x}_{E,f} \succsim x$ and for all $x^{*} \succsim \bar{x}_{E,f}$, $$\min\limits_{p \in C} \int_{\Omega} u(f_{E}x^{*}) dp = \min\limits_{p \in C^{*}(E)} \int_{\Omega} u(f_{E}x^{*})dp$$ Then for any $x^{*} \succsim \bar{x}_{E,f}$, the following is true: $$\begin{split}
\min\limits_{p \in C} \int_{\Omega} u(f_{E}x^{*}) dp & = \min\limits_{p \in C^{*}(E)} \left[ \int_{E} u(f) \frac{dp}{p(E)} \cdot p(E) + x^{*}(1-p(E)) \right]\\
& =p^{*}(E) \min\limits_{p \in C^{*}(E)} \int_{E} u(f) \frac{dp}{p(E)} + x^{*}(1-p^{*}(E))\\
& < p^{*}(E) \cdot u(x) + u(x^{*})(1-p^{*}(E)) \\
&= \min\limits_{p \in C} \int_{\Omega} u(x_{E}x^{*}) dp
\end{split}$$ It means that, if $f\sim_{E}x$ then $f_{E}x^{*} \prec x_{E}x^{*}$ for all $x^{*} \succsim \bar{x}_{E,f}$. That is, for this $f\in \mathcal{F}$, for any $\bar{x} \in X$, there always exists $x^{*} \succsim \bar{x}$ such that it is not the case $f_{E}x^{*} \sim x_{E}x^{*}$, i.e. the CR-LC axiom is not true.
The argument for the second case is analogously the same and combining both cases shows that not ML updating implies not CR-LC.
Proof of Theorem \[thm2\]
-------------------------
The necessity of the U-O axiom is given before introducing the axiom. For the Weak CR-CCE axiom, $\alpha[E] = 1$ implies the first scenario, $\alpha[E] \neq 1$ implies the second, and both scenarios are true when $\alpha[E]$ is not unique. This is immediate by noticing that equation (\[equ1\]) holds with $\alpha[E]$ is necessary under Weak RML, which is further given by reversing the arguments in step 2 for sufficiency.\
For sufficiency, fix any $\succsim$-nonnull $E\in \Sigma$, the proof proceeds by the following steps:
: Show that for all $f\in \mathcal{F}$, there exists $\alpha[E,f] \in [0,1]$ such that for all sufficiently large consequence $x^{*} \in X$ one has $$\label{equ1}
(1-\alpha[E,f])\text{CE}(f_{E}x) + \alpha[E,f] \text{CE}(f_{E}x^{*}) \sim (1-\alpha[E,f])x + \alpha[E,f] \text{CE}(x_{E}x^{*})$$ where CE$(f)$ denote the certainty equivalence of the act $f$. Furthermore, $\alpha[E,f]$ is unique if either $f_{E}x \prec x$ or $f_{E}x^{*} \succ x_{E}x^{*}$ hold.\
When the U-O axiom is true, $f \sim_{E} x$ implies that $f_{E}x \precsim x$ and there exists $\bar{x}_{E,f}\in X$ such that $f_{E}x^{*} \succsim x_{E}x^{*}$ for all $x^{*} \succsim \bar{x}_{E,f}$.
First consider all $x^{*} \in X$ such that $x^{*} \succsim \bar{x}_{E,f}$ and $x^{*} \succsim x$, then the following inequalities hold: $f_{E}x^{*} \succsim x_{E}x^{*} \succsim x \succsim f_{E}x$, which further implies that, **for each $x^{*}$**, there always exists an $\alpha[E,f] \in [0,1]$ such that the following equation holds: $$(1-\alpha[E,f]) u(\text{CE}(f_{E}x)) + \alpha[E,f] u(\text{CE}(f_{E}x^{*})) = (1-\alpha[E,f]) u(x) + \alpha[E,f] u(\text{CE}(x_{E}x^{*}))$$ Notice that, $\alpha[E,f]$ here may depend on the value of $x^{*}$ because the act $f_{E}x^{*}$ could be evaluated at different extreme points for different $x^{*}$. However, in the case where $f_{E}x^{*}$ is always evaluated at the extreme points in $C^{*}(E)$, as both $u(\text{CE}(f_{E}x^{*}))$ and $u(\text{CE}(x_{E}x^{*}))$ have the common term $u(x^{*})(1-\max\limits_{p\in C} p(E))$ which cancels out, $\alpha[E,f]$ does not depend on the value of $x^{*}$ any more.
By Lemma \[lem2\] and \[lem4\], for all $f\in \mathcal{F}$, there exists another threshold $\hat{x}_{E,f}$ such that for all $x^{*} \succsim \hat{x}_{E,f}$, one has $$\min\limits_{p \in C} \int_{\Omega}u(f_{E}x^{*})dp = \min\limits_{p \in C^{*}(E)} \int_{\Omega} u(f_{E}x^{*})dp$$
Therefore, there exists $\alpha[E,f]$ such that the following is true for all $x^{*} \succsim \max\{\bar{x}_{E,f}, x, \hat{x}_{E,f}\}$:[^21] $$\label{equ}
(1-\alpha[E,f]) u(\text{CE}(f_{E}x)) + \alpha[E,f] u(\text{CE}(f_{E}x^{*})) = (1-\alpha[E,f]) u(x) + \alpha[E,f] u(\text{CE}(x_{E}x^{*}))$$ Furthermore, this $\alpha[E,f]$ is unique if either $f_{E}x \prec x$ or $f_{E}x^{*} \succ x_{E}x^{*}$ hold.\
Equation (\[equ\]) implies that the DM’s conditional evaluation of any $f\in \mathcal{F}$ can be represented by $$\min\limits_{p\in C_{\alpha[E, f]}(E)} \int_{E} u(f) \frac{dp}{p(E)} = u(x)$$ for some $\alpha[E,f] \in (0,1]$ or is represented by FB.
For all sufficiently large $x^{*}$, the LHS of equation (\[equ\]) can further be derived as: $$\begin{aligned}
&\quad (1-\alpha[E, f]) u(CE(f_{E}x)) + \alpha[E, f] u(\text{CE}(f_{E}x^{*}))\\
& =(1-\alpha[E, f]) \min\limits_{p\in C} \int_{\Omega} u(f_{E}x) dp + \alpha[E, f] \min\limits_{p\in C} \int_{\Omega} u(f_{E}x^{*})dp \\
& = (1-\alpha[E, f]) \min\limits_{p\in C} \int_{\Omega} u(f_{E}x) dp + \alpha[E, f] \min\limits_{q\in C^{*}(E)} \int_{\Omega} u(f_{E}x^{*})dq \\
& = (1-\alpha[E, f]) \min\limits_{p\in C} \int_{\Omega} u(f_{E}x) dp+ \alpha[E, f]\left[ \min\limits_{q\in C^{*}(E)} \int_{E} u(f) \frac{dq}{p^{*}(E)}\cdot p^{*}(E) +\right. \\
&\left.\quad (1-p^{*}(E)) u(x^{*}) \right] \\
& = (1-\alpha[E, f]) \min\limits_{p\in C} \int_{\Omega} u(f_{E}x) dp+ \alpha[E, f]\left[ \min\limits_{q\in C^{*}(E)} \int_{E} u(f) \frac{dq}{p^{*}(E)}\cdot p^{*}(E) +\right. \\
&\left.\quad (1-p^{*}(E)) u(x) \right] + \alpha[E, f] [u(x^{*}) - u(x)][1-p^{*}(E)]\end{aligned}$$ where the second equality follows from $f_{E}x^{*}$ is evaluated at some $p\in C^{*}(E)$.
On the other hand, the RHS of equation (\[equ\]) can also be derived as $$\begin{aligned}
&\quad (1-\alpha[E, f]) u(x) + \alpha[E, f] u( \text{CE}(x_{E}x^{*}))\\
& = (1-\alpha[E, f])u(x) + \alpha[E, f]\min\limits_{p \in C} \int_{\Omega} u(x_{E}x^{*})dp\\
& = (1-\alpha[E, f])u(x) + \alpha[E, f][u(x)p^{*}(E) + u(x^{*})(1-p^{*}(E))] \\
& = u(x) + \alpha[E, f] [u(x^{*}) - u(x)][1-p^{*}(E)]\end{aligned}$$ Observe that now equalizing the LHS and RHS of equation (\[equ\]) implies $$\begin{aligned}
u(x) & = (1-\alpha[E, f]) \min\limits_{p\in C} \int_{\Omega} u(f_{E}x) dp+ \alpha[E, f]\left[ \min\limits_{q\in C^{*}(E)} \int_{E} u(f) \frac{dq}{p^{*}(E)}\cdot p^{*}(E) +(1-p^{*}(E)) u(x) \right] \\
& = (1-\alpha[E, f]) \min\limits_{p\in C} \int_{\Omega} u(f_{E}x) dp + \alpha[E, f] \min\limits_{q\in C^{*}(E)} u(f_{E}x)dq \\
& = \min\limits_{p\in C} \min\limits_{q\in C^{*}(E)} \int_{\Omega} u(f_{E}x) d((1-\alpha[E, f]) p + (\alpha[E, f])q)\\
& = \min\limits_{p \in C_{\alpha[E, f]}(E)} \int_{\Omega} u(f_{E}x)dp\end{aligned}$$ where $C_{\alpha[E,f]}(E) = (1-\alpha[E,f])C + \alpha[E,f]C^{*}(E)$.
As $\alpha[E,f] \in [0,1]$, if $\alpha[E,f] = 0$, then by definition the CR-CCE axiom holds, thus the conditional evaluation of $f$ will be represented by FB. If $\alpha \in (0,1]$, notice that from the last equation one can further derive: $$\begin{aligned}
0 & = \min\limits_{p \in C_{\alpha[E, f]}(E)} \int_{\Omega} u(f_{E}x)dp - u(x) \\
& = \min\limits_{p \in C_{\alpha[E, f]}(E)} \int_{\Omega} [u(f_{E}x) - u(x)]dp \\
& = \min\limits_{p \in C_{\alpha[E, f]}(E)} \int_{E} [u(f) - u(x)]dp \\
& = \min\limits_{p \in C_{\alpha[E, f]}(E)} \left[ \int_{E} u(f)dp - u(x)p(E) \right]\\\end{aligned}$$
When $E$ is $\succsim$-nonnull, $p^{*} (E) > 0$ thus $p(E) > 0$ for all $p \in C_{\alpha[E,f]}(E)$, then the last equality further implies $$0 = \min\limits_{p \in C_{\alpha[E, f]}(E)} \left[ \int_{E} u(f)\frac{dp}{p(E)} - u(x) \right]$$ i.e. $$\min\limits_{p\in C_{\alpha[E, f]}(E)} \int_{E} u(f) \frac{dp}{p(E)} = u(x)$$ which represents the conditional evaluation of $f$ under $\succsim_{E}$ since $f \sim_{E}x$.\
In summary, the conditional evaluation of $f$ is either given by FB ($\alpha[E,f] = 0$) or is given by $$\min\limits_{p\in C_{\alpha[E, f]}(E)} \int_{E} u(f) \frac{dp}{p(E)} = u(x)$$ for some $\alpha \in (0,1]$.\
The Weak CR-CCE axiom implies $\alpha[E, f]$ is constant across all $f\in \mathcal{F}$, and it is unique if $C \neq C^{*}(E)$.\
**Equivalent Class.** First, for any two acts $f, f'$, denote them by $f \equiv_{E} f'$ if $f(\omega) = f'(\omega)$ for all $\omega \in E$. Then an equivalence class of acts can be accordingly defined: $$[f] = \{ f' \in \mathcal{F}: f' \equiv_{E} f \}$$ By step 1, $\alpha [E, f] = \alpha[E, f']$ whenever $f' \in [f]$. Thus in the following, I abuse notation to use $f$ denote the whole class of acts $[f]$.\
**When $\alpha[E]$ is not unique.** If there does not exists any $f\in \mathcal{F}$ such that $\alpha[E, f]$ is unique. Then by step 1, for all $f\in \mathcal{F}$ and $x\in X$ such that $f \sim_{E}x$, it implies that both $f_{E}x \sim x$ and $f_{E}x^{*} \sim x_{E}x^{*}$ for all sufficiently large $x^{*}$ hold.
Then it is the case in which both the CR-CCE axiom and the CR-LC axiom hold at the same time. Namely, the conditional preference $\succsim_{E}$ can be represented by both FB and ML. When $E$ is strict $\succsim$-nonnull, it implies that $C = C^{*}(E)$; when $E$ is $\succsim$-nonnull but not strict, it is impossible since $C^{*}(E) \subsetneqq C$ implies that the conditional preference under ML is represented by a strict subset of the set under FB. In other words, $C \neq C^{*}(E)$ implies that there exists at least an $f\in \mathcal{F}$ such that $\alpha[E,f]$ is unique.\
When unique $\alpha[E,f]$ exists, fix some $f\in \mathcal{F}$ such that $\alpha[E, f]$ is unique.
**Two operations on acts that preserve $\alpha[E,f]$.** For any $l\in {\mathbb{R}}_{++}$, let $f_{l}$ denote an act such that $u(f_{l}(\omega)) = l \cdot u(f(\omega))$ for all $\omega \in \Omega$. i.e. the utility profile of $f_{l}$ is a positive linear transformation of the utility profile of $f$.
Then $f\sim_{E}x$ implies that $f_{l} \sim_{E} x_{l}$ for $u(x_{l}) = l \cdot u(x)$. For all $x^{*}$ that are sufficiently large for both $f$ and $f_{l}$, equation (\[equ\]) for $f_{l}$ implies that $$\begin{aligned}
&~~~~~~~~(1-\alpha[E, f_{l}]) u(\text{CE}(f_{lE}x_{l})) + \alpha[E, f_{l}] u(\text{CE}(f_{lE}x^{*})) = (1-\alpha[E, f_{l}]) u(x_{l}) + \alpha[E, f_{l}] u(\text{CE}(x_{lE}x^{*})) \\
&\Rightarrow (1-\alpha[E, f_{l}]) l \cdot u(\text{CE}(f_{E}x)) + \alpha[E, f_{l}] \min\limits_{p\in C^{*}(E)} \int_{E} l \cdot u(f) dp =(1-\alpha[E, f_{l}]) l \cdot u(x) \\
& + \alpha[E, f_{l}] l \cdot u(x)p^{*}(E)\\
&\Rightarrow(1-\alpha[E, f_{l}]) u(\text{CE}(f_{E}x)) + \alpha[E, f_{l}] \min\limits_{p\in C^{*}(E)} \int_{E}u(f) dp = (1-\alpha[E, f_{l}]) u(x) + \alpha[E, f_{l}] u(x)p^{*}(E)\\
&\Rightarrow(1-\alpha[E, f_{l}]) u(\text{CE}(f_{E}x)) + \alpha[E, f_{l}] u(\text{CE}(f_{E}x^{*})) = (1-\alpha[E, f_{l}]) u(x) + \alpha[E, f_{l}] u(\text{CE}(x_{E}x^{*}))\end{aligned}$$ on the other hand, $f\sim_{E} x$ also implies that $$(1-\alpha[E, f]) u(\text{CE}(f_{E}x)) + \alpha[E, f] u(\text{CE}(f_{E}x^{*})) = (1-\alpha[E, f]) u(x) + \alpha[E, f] u(\text{CE}(x_{E}x^{*}))$$ Therefore, $\alpha[E,f_{l}] = \alpha[E,f]$ for all $l \in {\mathbb{R}}_{++}$ since $\alpha[E,f]$ is unique.
Next, for any $y \in X$, for any $\lambda \in [0,1]$ consider the act $f_{\lambda}y = \lambda f + (1-\lambda)y $, which is an Anscombe-Aumann mixture of acts. By certainty independence, $f\sim_{E}x$ implies that $f_{\lambda} y \sim_{E} \lambda x + (1-\lambda)y$. Let $x_{\lambda}y\in X$ denote the consequence on the RHS. Then for all $x^{*}$ that are sufficiently large for $f$ and $f_{\lambda}y$, equation (\[equ\]) for $f_{\lambda}y$ implies that $$\begin{aligned}
&(1-\alpha[E, f_{\lambda}y]) u(\text{CE}([f_{\lambda}y]_{E}x_{\lambda}y) + \alpha[E, f_{\lambda}y] u(\text{CE}([f_{\lambda}y]_{E}x^{*})) \\
&= (1-\alpha[E, f_{\lambda}y]) u(x_{\lambda}y) + \alpha[E, f_{\lambda}y] u(\text{CE}([x_{\lambda}y]_{E}x^{*}))\end{aligned}$$ Notice that $$\begin{aligned}
u(\text{CE}([f_{\lambda}y]_{E}x_{\lambda}y) = u(\text{CE}([f_{E}x]_{\lambda}y)) = \lambda u(\text{CE}(f_{E}x)) + (1-\lambda)u(y)\end{aligned}$$ where the first equality follows from Anscombe-Aumann mixture, the second equality also follows from certainty independence.
Then one can further derive $$\begin{aligned}
&(1-\alpha[E, f_{\lambda}y]) [\lambda u(\text{CE}(f_{E}x)) + (1-\lambda)u(y)] + \alpha[E, f_{\lambda}y]
\min\limits_{p\in C^{*}(E)} \int_{E} u(f_{\lambda}y)dp \\
&= (1-\alpha[E, f_{\lambda}y]) u(x_{\lambda}y) + \alpha[E, f_{\lambda}y] u(x_{\lambda}y) p^{*}(E)\end{aligned}$$ which further implies that $$(1-\alpha[E, f_{\lambda}y]) u(\text{CE}(f_{E}x) + \alpha[E, f_{\lambda}y] u(\text{CE}(f_{E}x^{*})) = (1-\alpha[E, f_{\lambda}y]) u(x) + \alpha[E, f_{\lambda}y] u(\text{CE}(x_{E}x^{*}))$$ Then since $f\sim_{E}x$ and $\alpha[E,f]$ is unique, one can also conclude that $\alpha[E,f_{\lambda}y] = \alpha[E,f]$ for all $\lambda\in [0,1]$ and $y\in X$.
In summary, for all $l \in {\mathbb{R}}_{++}$, $y \in X$ and $\lambda \in [0,1]$, $\alpha[E, f_{l\lambda} y] = \alpha[E,f]$.\
**The Weak CR-CCE axiom implies $\alpha[E,f]$ is constant across acts.** First consider the case where the first scenario of Weak CR-CCE is true. Since the fist scenario is equivalent to the CR-LC axiom. Thus it would imply that $\alpha[E,f] = 1 $ for all $f\in \mathcal{F}$. Then $\alpha[E,f]$ is indeed a constant in this case.\
Next, suppose the first scenario of Weak CR-CCE is false, then it further implies that the second scenario must be true.
For any $g\in \mathcal{F}$ such that cannot be obtained from $f$ by the two operations preserve $\alpha[E,f]$. Without loss of generality, I normalize its utility profile such that $u(g(\omega)) \geq 0$ for all $\omega \in E$. Consider the act $g_{l\lambda}y$ for any $l \in {\mathbb{R}}_{++}$, $\lambda \in (0,1]$ and $y \in X$. By previous result, $\alpha[E, g_{l\lambda}y] = \alpha[E,g]$, thus it suffice to show that $\alpha[E,f] = \alpha[E,g_{l\lambda}y]$ for some $l \in {\mathbb{R}}_{++}$, $\lambda \in (0,1]$ [^22]and $y \in X$.
As the conditional preference $\succsim_{E}$ admits MEU representation, let it be represented by the set $C_{E}$. Let $p_{f}$ and $p_{g}$ denote the extreme points in $C_{E}$ that evaluate the act $f$ and $g$ respectively. By certainty independence, the act $g_{l\lambda}y$ is also evaluated at the same extreme point as $g$.
Then $f \sim_{E} g_{l\lambda}y$ when $$u(g_{l\lambda}y) \cdot p_{g} = \lambda u(g_{l})\cdot p_{g} + (1-\lambda) u(y) = u(f) \cdot p_{f}$$
Next, for all $x^{*}$ that are sufficiently large for $f$ and $g_{l\lambda}y$, consider the condition $g_{l\lambda E}x^{*} \sim f_{E}x^{*}$, which is equivalent to $$\begin{aligned}
&~~~~~~~~\min\limits_{p\in C} \int_{\Omega} u([g_{l\lambda}y]_{E}x^{*}) = \min\limits_{p\in C} \int_{\Omega} u(f_{E}x^{*})\\
&\Rightarrow \min\limits_{p\in C^{*}(E)} \int_{E} u(g_{l\lambda}y) dp + (1-p^{*}(E))u(x^{*}) = \min\limits_{p\in C^{*}(E)} \int_{E} u(f) dp + (1-p^{*}(E))u(x^{*})\\
&\Rightarrow \min\limits_{p\in C^{*}(E)} \int_{E} u(g_{l\lambda}y) dp = \min\limits_{p\in C^{*}(E)} \int_{E} u(f) dp \\
& \Rightarrow \min\limits_{p\in C^{*}(E)} \int_{E} u(g_{l\lambda}y) \frac{dp}{p(E)} = \min\limits_{p\in C^{*}(E)} \int_{E} u(f) \frac{dp}{p(E)} \end{aligned}$$ where the last equality follows from $p(E) = p^{*}(E)$ for all $p \in C^{*}(E)$.
For the set of posteriors of $C^{*}(E)$, let $q_{f}$ and $q_{g}$ denote the two extreme points that evaluate the act $f$ and $g$ respectively. Again, the act $g_{l\lambda}y$ is also evaluated at the same extreme point as $g$. Thus, the above condition can also be written as
$$u(g_{l\lambda}y) \cdot q_{g} = \lambda u(g_{l})\cdot q_{g}+ (1-\lambda) u(y) = u(f) \cdot q_{f}$$
Consider the following construction:
In a two dimensional space, draw the two points $(1, u(f) \cdot q_{f})$ and $(1,u(f) \cdot p_{f})$ as well as the two points $(2, u(g_{l}) \cdot q_{g})$ and $(2,u(g_{l}) \cdot p_{g})$. Notice that, by the U-O axiom, the set $C_{E}$ is a superset of the set of posteriors of $C^{*}(E)$, thus it is necessarily the case that $u(f) \cdot q_{f} \geq u(f) \cdot p_{f}$ and $u(g_{l}) \cdot q_{g} \geq u(g_{l}) \cdot p_{g}$. Now for different possibilities of the two inequalities:
- $u(f) \cdot q_{f} = u(f) \cdot p_{f}$ and $u(g_{l}) \cdot q_{g} \geq u(g_{l}) \cdot p_{g}$.
In this case, consider when $g=x$ in the second scenario of the Weak CR-CCE axiom. The first equality implies that $u(f) \cdot q_{f} = u(f) \cdot p_{f} = u(x)$, i.e. $f_{E}x \sim x_{E}x^{*}$ as well. Then the second scenario of the Weak CR-CCE axiom would imply that $f_{E}x \sim x$. By step 1, both $f_{E}x \sim x_{E}x^{*}$ and $f_{E}x \sim x$ holds imply that $\alpha[E,f]$ is not unique, a contradiction to the current assumption.
- $u(f) \cdot q_{f} > u(f) \cdot p_{f}$ and $u(g_{k}) \cdot q_{g} = u(g_{k}) \cdot p_{g}$.
This case is symmetry to the first case, and it necessarily implies that $\alpha[E,g]$ is not unique. Then it suffice to let $\alpha[E, g] = \alpha[E, f]$ for this case.
- $u(f) \cdot q_{f} > u(f) \cdot p_{f}$ and $u(g_{k}) \cdot q_{g} > u(g_{k}) \cdot p_{g}$.
In this case, $\alpha[E,f] \neq 1$ because $p_{f} \neq q_{f}$. In the following one can further show that the second scenario of the Weak CR-CCE axiom implies that $\alpha[E,f] = \alpha[E,g]$.
Consider in the Case 3, find $l \in {\mathbb{R}}_{++}$ such that the following two strict inequalities hold: $$\begin{aligned}
&u(g_{l})\cdot p_{g} > u(f) \cdot p_{f} \\
&u(g_{l})\cdot q_{g} - u(g_{l})\cdot p_{g} > u(f) \cdot q_{f} - u(f) \cdot p_{f} \end{aligned}$$
Since both LHS of the inequalities are increasing with respect to $l$, thus one can always find such $l$ (recall our normalization $u(g(\omega)) \geq 0$ for all $\omega \in E$). Once $l$ is identified, draw two straight lines, one connects $(1, u(f) \cdot q_{f}), (2, u(g_{k}) \cdot q_{g})$ and the other one connects $(1,u(f) \cdot p_{f}), (2,u(g_{k}) \cdot p_{g})$. By construction, these two lines must intersect at some point on the southwest of the point $(1,u(f) \cdot p_{f})$, and denote it by $(z, u(y))$. Then the desired $\lambda$ is given by $\frac{1-z}{2-z}$ and this $g_{l\lambda}y$ satisfies the two conditions: $$f \sim_{E} g_{l\lambda}y \text{ and } f_{E}x^{*} \sim [g_{l\lambda}y]_{E}x^{*}$$
In the following, I abuse notation to use $g$ denote the act $g_{l\lambda}y$ such that two conditions above hold.
By step 1, $f\sim_{E} x$ implies that $$\label{equ11}
(1-\alpha[E,f]) u(\text{CE}(f_{E}x)) + \alpha[E,f] u(\text{CE}(f_{E}x^{*})) = (1-\alpha[E,f]) u(x) + \alpha[E,f] u(\text{CE}(x_{E}x^{*}))$$ and $g \sim_{E} x$ also implies that $$\label{equ10}
(1-\alpha[E,g]) u(\text{CE}(g_{E}x)) + \alpha[E,g] u(\text{CE}(g_{E}x^{*})) = (1-\alpha[E,g]) u(x) + \alpha[E,g] u(\text{CE}(x_{E}x^{*}))$$
Consider the LHS of equation (\[equ10\]) and denote it by $L$: $$\begin{aligned}
L & = (1-\alpha[E, g]) u(\text{CE}(g_{E}x)) + \alpha[E, g] u(\text{CE}(g_{E}x^{*})) \\
& = (1-\alpha[E, f]) u(\text{CE}(g_{E}x)) + \alpha[E, f] u(\text{CE}(g_{E}x^{*})) + [\alpha[E, g] - \alpha[E, f]](u(\text{CE}(g_{E}x^{*})) - u(\text{CE}(g_{E}x)))\\
& = L' +[\alpha[E, g] - \alpha[E, f]] M_{1} \end{aligned}$$ Meanwhile the RHS of equation (\[equ10\]) denoted by $R$ can be further derived to $$\begin{aligned}
R & = (1-\alpha[E, g]) u(x) + \alpha[E, g] u(\text{CE}(x_{E}x^{*})) \\
& = (1-\alpha[E, f]) u(x) + \alpha[E, f] u(\text{CE}(x_{E}x^{*})) + [\alpha[E, g] - \alpha[E, f]](u(\text{CE}(x_{E}x^{*})) - u(x))\\
& = R' + [\alpha[E, g] - \alpha[E, f]] M_{2} \end{aligned}$$ Notice that by equation (\[equ11\]), $R'$ also equals to $(1-\alpha[E, f])\text{CE}(f_{E}x) + \alpha[E, f] \text{CE}(f_{E}x^{*})$. Then as $\alpha[E, f] \neq 1$, the second scenario of Weak CR-CCE implies that $L' = R'$ as $f_{E}x \sim g_{E}x$ and $f_{E}x^{*} \sim g_{E}x^{*}$ hold.
Then as $L = R$, one has $$\begin{aligned}
&L - R = L' +[\alpha[E, g] - \alpha[E, f]] M_{1} - R' - [\alpha[E, g] - \alpha[E, f]] M_{2} = 0 \\
&\Rightarrow L' - R' = [\alpha[E, f] - \alpha[E, g]] [M_{1} - M_{2}]\end{aligned}$$ Further notice that $$\begin{aligned}
M_{1} - M_{2}& = [u(\text{CE}(g_{E}x^{*})) - u(\text{CE}(g_{E}x))] - [ u(\text{CE}(x_{E}x^{*})) - u(x)] \\
& = [\min\limits_{p\in C} \int_{\Omega}u(g_{E}x^{*}) - \min\limits_{p\in C} \int_{\Omega}u(g_{E}x)] - [u(x)p^{*}(E) + u(x^{*})(1-p^{*}(E)) - u(x)] \\
& = [\min\limits_{p\in C^{*}(E)} \int_{\Omega}u(g_{E}x^{*}) - \min\limits_{p\in C} \int_{\Omega}u(g_{E}x)] - [u(x^{*}) - u(x)](1-p^{*}(E)) \\
& = [\min\limits_{p\in C^{*}(E)} \int_{\Omega}u(g_{E}x^{*}) - \min\limits_{p\in C^{*}(E)} \int_{\Omega}u(g_{E}x)] - [u(x^{*}) - u(x)](1-p^{*}(E)) \\
& + [\min\limits_{p\in C^{*}(E)} \int_{\Omega}u(g_{E}x) - \min\limits_{p\in C} \int_{\Omega}u(g_{E}x) ] \\
& = \min\limits_{p\in C^{*}(E)} \int_{\Omega}u(g_{E}x) - \min\limits_{p\in C} \int_{\Omega}u(g_{E}x)\end{aligned}$$ where the last equality follows from $$\begin{aligned}
\min\limits_{p\in C^{*}(E)} \int_{\Omega}u(g) - \min\limits_{p\in C^{*}(E)} \int_{\Omega}u(g) &= [\min\limits_{p\in C^{*}(E)} \int_{E}u(g_{E}x^{*}) - \min\limits_{p\in C^{*}(E)} \int_{E}u(g_{E}x)]p^{*}(E) \\
&\quad+ [u(x^{*}) - u(x)](1-p^{*}(E)) \end{aligned}$$
\[lem5\] For any $f\in \mathcal{F}$ such that $\alpha[E, f]$ is unique, if $f\sim_{E}x$ then $$\min\limits_{p\in C} \int_{\Omega} u(f_{E}x) dp < \min\limits_{p\in C^{*}(E)} \int_{\Omega} u(f_{E}x) dp$$
Since if $\alpha[E, f]$ is unique, it implies that for $f\sim_{E}x$, one of the inequalities $f_{E}x^{*} \succsim x_{E}x^{*}$ and $x \succsim f_{E}x$ is strict. Notice that the first inequality implies $$\min\limits_{p\in C^{*}(E)} \int_{\Omega} u(f)dp \geq u(x)p^{*}(E)$$ and the second implies that $$u(x) \geq \min\limits_{p\in C} \int_{\Omega} u(f_{E}x)dp$$
Add the term $u(x)(1-p^{*}(E))$ to both sides of the first inequality and combine both inequalities and recall that one of them has to be strict yield: $$\min\limits_{p\in C^{*}(E)} \int_{\Omega} u(f)dp + u(x)(1-p^{*}(E)) > \min\limits_{p\in C} \int_{\Omega} u(f_{E}x)dp$$ which is equivalent to $\min\limits_{p\in C^{*}(E)} \int_{\Omega} u(f_{E}x) dp > \min\limits_{p\in C} \int_{\Omega} u(f_{E}x) dp $.
Denote it by $$\Delta f_{E}x \equiv \min\limits_{p\in C^{*}(E)} \int_{\Omega} u(f_{E}x)dp - \min\limits_{p\in C} \int_{\Omega} u(f_{E}x)dp$$
If $\alpha[E,g]$ is not unique, then it suffice to let $\alpha[E,g] = \alpha[E,f]$. If it is also unique, given Lemma \[lem5\], $g\sim_{E}x$ implies that $$\Delta g_{E}x > 0$$
Therefore the difference $$L' - R' = [\alpha[E, f] - \alpha[E, g]] [M_{1} - M_{2}] = [\alpha[E, f] - \alpha[E, g]]\cdot \Delta g_{E}x$$ is 0 if and only if $\alpha[E, f] = \alpha[E, g]$.
To conclude, when the first scenario of the Weak CR-CCE axiom is false, it implies that the second scenario must be true. Fix $f\in \mathcal{F}$ such that $\alpha[E,f]$ is unique, for any $g\in \mathcal{F}$, when $\alpha[E,g]$ is not unique, it suffice to let $\alpha[E,g] = \alpha[E,f]$. If $\alpha[E,g]$ is unique, the second scenario implies that it cannot be the case that $\alpha[E,f] = 1$. Then when $\alpha[E,f] \neq 1$, by construction of $g_{l\lambda}y$, the second scenario of the Weak CR-CCE axiom further suggests that it is necessarily the case $\alpha[E, f] = \alpha[E,g]$.
Finally, combining the two cases ($\alpha[E, f] = 1$ and $\alpha[E, f] \neq 1$) yields that $\alpha[E,f]$ needs to be a constant across all $f\in \mathcal{F}$ for each $\succsim$-nonnull $E\in \Sigma$.
Proof of Theorem \[thm4\]
-------------------------
Given Theorem \[thm2\], the only remaining proof here is to show $\alpha[E]$ is a constant across all events if and only if the EC axiom holds.
The necessity of the EC axiom is also immediate when one plug a constant $\alpha[E]$ into equation (\[equ1\]), $\alpha[E] = 1$ implies the first scenario, and $\alpha[E] \neq 1$ implies the second. Both scenarios are true when $\alpha[E]$ is not unique.\
In the following I show that, EC implies $\alpha[E]$ needs to be a constant across all $\succsim$-nonnull events.
First of all, consider the case there does not exists any $\succsim$-nonnull $E\in \Sigma$ such that $C \neq C^{*}(E)$. Then it implies that all $p \in C$ agree with the probability of all $\succsim$-nonnull event $E$. Namely, all $p\in C$ agree on the probability of all $\succsim$-nonnull events.
Thus, if there exists one $\succsim$-nonnull event $E$ such that $C \neq C^{*}(E)$, i.e. $\alpha[E]$ is unique, then it suffice to let this $\alpha[E]$ to be the constant $\alpha$ across all events.
Next, when there exists at least two $\succsim$-nonnull events, $E_{1}$ and $E_{2}$, such that both $\alpha[E_{1}]$ and $\alpha[E_{2}]$ are unique. Find any $f,g\in \mathcal{F}$ and $x\in X$ such that $f \sim_{E_{1}} x$ and $g \sim_{E_{2}} x$. For all $x^{*}\in X$ that are sufficiently large for act $f$ under event $E_{1}$ and for act $g$ under event $E_{2}$. (The same definition as in the proof of Theorem \[thm2\]), identify two consequences $x_{1}^{*}$ and $x_{2}^{*}$ such that $x_{E_{1}}x_{1}^{*} \sim x_{E_{2}}x_{2}^{*}$.
By Theorem \[thm2\], $f \sim_{E_{1}} x$ implies that $$\label{eq2}
(1-\alpha[E_{1}]) u(\text{CE}(f_{E_{1}}x)) + \alpha[E_{1}] u(\text{CE}(f_{E_{1}}x_{1}^{*})) = (1-\alpha[E_{1}]) u(x) + \alpha[E_{1}] u(\text{CE}(x_{E_{1}}x_{1}^{*}))$$
Meanwhile $g \sim_{E_{2}} x$ also implies $$\label{eq3}
(1-\alpha[E_{2}]) u(\text{CE}(g_{E_{2}}x)) + \alpha[E_{2}] u(\text{CE}(g_{E_{2}}x_{2}^{*})) = (1-\alpha[E_{2}]) u(x) + \alpha[E_{2}] u(\text{CE}(x_{E_{2}}x_{2}^{*}))$$
If the first scenario of the EC axiom is true, equation (\[eq2\]) and (\[eq3\]) always imply $f_{E_{1}}x_{1}^{*} \sim g_{E_{2}}x_{2}^{*}$ for all such $f,g$ only when $\alpha[E_{1}] = \alpha[E_{2}] = 1$. The same applies to all $\succsim$-nonnull $E$ with unique $\alpha[E]$, thus in this case $\alpha[E] = 1$ for all $\succsim$-nonnull $E$.
If the first scenario is false, the EC axiom further implies that the second scenario must be true. Consider a similar construction as in the proof of Theorem \[thm2\]. Notice that since $\alpha[E]$ is a constant across all $f\in \mathcal{F}$ for each $\succsim$-nonnull $E$, thus all operations on acts preserve $\alpha[E]$.
Fix any $f\in \mathcal{F}$ such that $f\sim_{E_{1}}x$. For any $g \in \mathcal{F}$, let $g_{l}$ denote the act such that $u(g_{l}(\omega)) = l \cdot u(g(\omega))$ where normalize $u$ such that $u(g(\omega))$ is non-negative for all $\omega \in E_{2}$. Then for any consequence $y\in X$, for any $\lambda \in [0,1]$, let the act $g_{l\lambda}y$ denote the mixture $\lambda g_{l} + (1-\lambda)y $ of the act $g_{l}$ and consequence $y$. It is important to notice that the act $g_{l\lambda}y$ is always evaluated at the same extreme point as $g$.
Then let $p_{f}$ denote the extreme point in $C_{E_{1}}$ that evaluates the act $f$, and let $p_{g}$ denote the extreme point in $C_{E_{2}}$ that evaluates the act $g$, where $C_{E_{1}}$ and $C_{E_{2}}$ are the two sets representing the conditional preference $\succsim_{E_{1}}$ and $\succsim_{E_{2}}$ respectively.
Then $g_{l\lambda}y \sim_{E_{2}} x$ when $$u(g_{l\lambda}y)\cdot p_{g} = \lambda u(g_{l})\cdot p_{g} + (1-\lambda) u(y) = u(x)$$
Further, since $f \sim_{E_{1}} x$, it is also suffice to let $$\lambda u(g_{l})\cdot p_{g} + (1-\lambda) u(y) = u(f)\cdot p_{f}$$
Next, for all $x^{*}$ that are sufficiently large for $f$ under $E_{1}$ and $g_{l\lambda}y$ under $E_{2}$, consider the condition $[g_{l\lambda}y]_{E}x^{*}_{2} \sim f_{E_{1}}x^{*}_{1}$, which is equivalent to $$\begin{aligned}
&~~~~~~~~\min\limits_{p\in C} \int_{\Omega} u([g_{l\lambda}y]_{E}x_{2}^{*}) dp = \min\limits_{p\in C} \int_{\Omega} u(f_{E_{1}}x^{*}_{1}) dp \\
&\Rightarrow \min\limits_{p\in C^{*}(E_{2})} \int_{E_{2}} u(g_{l\lambda}y) dp + u(x_{2}^{*}) (1- p^{*}(E_{2})) = \min\limits_{p\in C^{*}(E_{1})} \int_{E_{1}} u(f) dp + u(x^{*}_{1}) (1- p^{*}(E_{1}))\end{aligned}$$ where $p^{*}(E_{1})$ denotes $\max\limits_{p\in C} p(E_{1})$ and $p^{*}(E_{2})$ is defined accordingly.
Let $q_{f}$ and $q_{g}$ be the two extreme points in $C^{*}(E_{1})$ and $C^{*}(E_{2})$ that evaluates $f$ and $g_{l\lambda}y$ respectively, then the last equality can be written as: $$u(g_{l\lambda}y) \cdot q_{g} + u(x_{2}^{*}) (1- p^{*}(E_{2})) = u(f)\cdot q_{f} + u(x^{*}_{1}) (1- p^{*}(E_{1}))$$
Furthermore as the condition $x_{E_{1}}x_{1}^{*} \sim x_{E_{2}}x_{2}^{*}$ implies that $$u(x) p^{*}(E_{1}) + u(x_{1}^{*})(1-p^{*}(E_{1})) = u(x) p^{*}(E_{2}) + u(x_{2}^{*}) (1-p^{*}(E_{2}))$$ i.e. $$u(x_{1}^{*})(1-p^{*}(E_{1})) - u(x_{2}^{*}) (1-p^{*}(E_{2})) = u(x)[p^{*}(E_{2}) - p^{*}(E_{1})]$$ As $E_{1}$ and $E_{2}$ are chosen arbitrarily, without loss of generality to let $p^{*}(E_{2}) - p^{*}(E_{1}) \geq 0$ and also normalize $u$ such that $u(x) \geq 0$.
Then $[g_{l\lambda}y]_{E}x^{*}_{2} \sim f_{E_{1}}x^{*}_{1}$ is equivalent to $$\lambda u(g_{l})\cdot q_{g} + (1-\lambda)u(y) = u(f)\cdot q_{f} + u(x)[p^{*}(E_{2}) - p^{*}(E_{1})]$$
Again, since both $\alpha[E_{1}]$ and $\alpha[E_{2}]$ are unique, the set represents the conditional preference is a superset of the corresponding set of posteriors of maximum likelihood priors. Thus the following two inequality are still necessarily true: $u(f) \cdot q_{f} \geq u(f) \cdot p_{f}$ and $u(g_{k}) \cdot q_{g} \geq u(g_{k}) \cdot p_{g}$. For different possibilities of the two inequalities:
- Either $u(f) \cdot q_{f} = u(f) \cdot p_{f}$ or $u(g_{k}) \cdot q_{g} = u(g_{k}) \cdot p_{g}$.
Under this case, without loss of generality assume the first equality is true. Consider $g=x$ in the second scenario of the EC axiom. The equality implies that when $f \sim_{E_{1}}x$, one has $f_{E_{1}}x_{2}^{*} \sim x_{E_{2}}x_{2}^{*}$ as well. Then the second scenario of EC would imply that $f_{E_{1}}x \sim x$. By step 1 in the proof of Theorem \[thm2\], it further implies that $\alpha[E_{1}]$ is not unique, a contradiction to the current assumption.
- $u(f) \cdot q_{f} > u(f) \cdot p_{f}$ and $u(g_{k}) \cdot q_{g} > u(g_{k}) \cdot p_{g}$.
In this case, both $\alpha[E_{1}]$ and $\alpha[E_{2}]$ cannot be 1.
Consider in the Case 2, given our normalization, it is also the case that $u(f) \cdot q_{f}+ u(x)[p^{*}(E_{2}) - p^{*}(E_{1})] > u(f) \cdot p_{f}$ and $u(g_{l}) \cdot q_{g} > u(g_{l}) \cdot p_{g}$. Now for a slight different construction from the one in proof of Theorem \[thm2\]:
In a two dimensional space, draw the two points $(1, u(f) \cdot q_{f} + u(x)[p^{*}(E_{2}) - p^{*}(E_{1})])$ and $(1,u(f) \cdot p_{f})$ as well as the two points $(2, u(g_{l}) \cdot q_{g})$ and $(2,u(g_{l}) \cdot p_{g})$.
First find $l \in {\mathbb{R}}_{++}$ such that the following two strict inequalities hold: $$\begin{aligned}
&u(g_{l})\cdot p_{g} > u(f) \cdot p_{f} \\
&u(g_{l})\cdot q_{g} - u(g_{l})\cdot p_{g} > u(f) \cdot q_{f} + u(x)[p^{*}(E_{2}) - p^{*}(E_{1})]- u(f) \cdot p_{f} \end{aligned}$$
Since both LHS of the inequalities are increasing with respect to $l$, thus one can always find such $l$. Once $l$ is identified, draw two straight lines, one cross $(1, u(f) \cdot q_{f}), (2, u(g_{l}) \cdot q_{g})$ and the other one cross $(1,u(f) \cdot p_{f} + u(x)[p^{*}(E_{2}) - p^{*}(E_{1})]), (2,u(g_{l}) \cdot p_{g})$. These two lines must intersect at some point on the southwest of the point $(1,u(f) \cdot p_{f})$, and denote it by $(z, u(y))$. Then the desired $\lambda$ is given by $\frac{1-z}{2-z}$, and such $g_{l\lambda}y$ satisfies the two conditions in the second scenario of EC.
From this point on, apply exactly the same argument in step 3 of the proof of Theorem \[thm2\] would imply that $\alpha[E_{1}] = \alpha[E_{2}]$, and furthermore $\alpha[E]$ needs to be a constant across all $\succsim$-nonnull $E\in \Sigma$.
To conclude, combining the two cases ($\alpha[E_{1}] = 1$ and $\alpha[E_{1}] \neq 1$) imply that $\alpha$ is a constant across all $\succsim$-nonnull event $E$. Furthermore, $\alpha$ is unique if there exists $\succsim$-nonnull $E\in \Sigma$ such that not all $p\in C$ agree on $p(E)$.
ML under general MEU {#apx1}
====================
The characterization results in the main text rely on the finitely many plausible priors assumption. As mentioned, such an assumption helps simplify the axioms and conveys the same intuition. This section in the appendix provides characterization results without this assumption.
For the ex-ante preference $\succsim$, this section assumes only that it admits MEU representation with some closed and convex set $C$, which may have infinitely many plausible priors. In addition, endow the space $\Delta(\Omega)$ with weak topology. Except for these, every other assumption about the primitive $\{\succsim_{E}\}_{E\in \Sigma}$ is the same as in Section 2.
The definition is the same as before, $\{\succsim_{E}\}_{E\in \Sigma}$ is represented by ML updating if for all $\succsim$-nonnull $E\in \Sigma$ and for all $f\in \mathcal{F}$: $$\min\limits_{p\in C_{E}} \int_{\Omega}u(f) dp = \min\limits_{p\in C^{*}(E)} \int_{E} u(f) \frac{dp}{p(E)}$$
For the general MEU case where $C$ may contain infinitely many plausible priors, consider the following axiom:\
**Axiom Approximate CR-LC**. For all $\succsim$-nonnull $E\in \Sigma$, for all $f\in \mathcal{F}$ and for all $x, z, w \in X$ such that $z \succ w$, there exists $\bar{x}_{E,f,z,w}$ such that if $f \sim_{E}x$ then $$\frac{1}{2} f_{E}x^{*} + \frac{1}{2} w \prec \frac{1}{2} x_{E}x^{*} + \frac{1}{2} z$$ and $$\frac{1}{2} f_{E}x^{*} + \frac{1}{2}z \succ \frac{1}{2} x_{E}x^{*} + \frac{1}{2}w$$ for all $x^{*} \succsim \bar{x}_{E,f,z,w}$.\
Notice that when $\bar{x}_{E,f}$ exists, CR-LC implies Approximate CR-LC. Whereas, in the case where $\bar{x}_{E,f}$ does not exist such that CR-LC is silent, Approximate CR-LC imposes additional restriction on behaviors. It restricts that the difference between $f_{E}x^{*}$ and $x_{E}x^{*}$ should be arbitrarily small when $x^{*}$ is sufficiently large.
Hence, Approximate CR-LC conveys essentially the same intuition as CR-LC. The following representation theorem claims that Approximate CR-LC is equivalent to ML in the current setting.
\[thm5\] $\{\succsim_{E}\}_{E\in \Sigma}$ is represented by ML updating if and only if the Approximate CR-LC\* axiom holds.
First consider the following lemma:
\[alem\] For any $\succsim$-nonnull $E\in \Sigma$ and $f\in \mathcal{F}$, for any $\epsilon > 0$ there exists $\bar{x}_{E,f,\epsilon} \in X$ such that $$\min\limits_{p\in C^{*}(E)} \int_{\Omega} u(f_{E}x^{*})dp - \min\limits_{p\in C}\int_{\Omega} u(f_{E}x^{*}) dp < \epsilon$$ for all $x^{*} \succsim \bar{x}_{E,f,\epsilon}$.
For any $\succsim$-nonnull $E\in \Sigma$ and $f\in \mathcal{F}$, either there exists $\bar{x}_{E,f}$ such that $$\min\limits_{p\in C} \int_{\Omega} u(f_{E}x^{*}) dp = \min\limits_{p\in C^{*}(E)} \int_{\Omega} u(f_{E}x^{*}) dp$$ for all $x^{*} \succsim \bar{x}_{E,f}$ or not. If it is the first case, then this lemma is trivially true.
Consider the case there does not exist $\bar{x}_{E,f}$ for some $E$ and $f$. For each $x \in X$, let $p_{x}$ be the probability measure in $C$ that evaluates the act $f_{E}x$, i.e. $p_{x} \equiv \arg\min\limits_{p\in C} \int_{\Omega} u(f_{E}x)dp $. Let $q$ denote the probability measure in $C^{*}(E)$ that evaluates the act $f_{E}x$ and notice that it does not depend on the value of $x$.
Then one has $$\begin{aligned}
&\quad \min\limits_{p\in C^{*}(E)} \int_{\Omega} u(f_{E}x)dp - \min\limits_{p\in C}\int_{\Omega} u(f_{E}x) dp \\
& = \int_{E} u(f)dq + u(x)(1-p^{*}(E)) - \int_{E} u(f)dp_{x} - u(x)(1-p_{x}(E))
\end{aligned}$$
Take derivative with respect to $u(x)$ and apply envelope theorem yields $$\frac{d}{du(x)} \left[ \min\limits_{p\in C^{*}(E)} \int_{\Omega} u(f_{E}x)dp - \min\limits_{p\in C}\int_{\Omega} u(f_{E}x) dp \right] = p_{x}(E) - p^{*}(E)$$
The current assumption $p_{x} \notin C^{*}(E)$ implies that $p_{x}(E) - p^{*}(E) < 0$, i.e. the difference is decreasing with respect to $u(x)$. Furthermore, since the difference is bounded below by zero, monotone convergence theorem implies that $$\min\limits_{p\in C^{*}(E)} \int_{\Omega} u(f_{E}x)dp - \min\limits_{p\in C}\int_{\Omega} u(f_{E}x) dp \rightarrow 0$$ for $u(x) \rightarrow \infty$, i.e. the lemma holds.
For necessity of the Approximate CR-LC axiom, recall Lemma \[lem1\] implies that if $f\sim_{E}x$ then under ML updating $$\min\limits_{p\in C^{*}(E)} \int_{\Omega} u(f_{E}x^{*})dp = \min\limits_{p\in C} \int_{\Omega} u(x_{E}x^{*})dp$$ Thus for any $\epsilon > 0$ there exists $\bar{x}_{E,f,\epsilon}$ such that $$\min\limits_{p\in C} \int_{\Omega} u(x_{E}x^{*})dp - \min\limits_{p\in C}\int_{\Omega} u(f_{E}x^{*}) dp < \epsilon$$ for all $x^{*} \succsim \bar{x}_{E,f,\epsilon}$. Then for each $z \succ w$, it suffice to let $\epsilon = u(z) - u(w)$ and Approximate CR-LC axiom will hold.\
For sufficiency, fix any $\succsim$-nonnull $E$ and consider the contra positive statement: not ML updating implies not Approximate CR-LC axiom. Not ML means that $C_{E} \neq \{ \frac{p}{p(E)}: p \in C^{*}(E) \}$. In other words, either thre exists $\tilde{p} \in C^{*}(E)$ such that $\frac{\tilde{p}}{\tilde{p}(E)} \notin C_{E}$ or there exists $q \in C_{E}$ such that $q \notin \{\frac{p}{p(E)}: p \in C^{*}(E)\}$ or both.
Consider the first case, by the same strong separating hyperplane argument in the proof of Theorem \[thm1\], there exists $f\in \mathcal{F}$ and $x\in X$ such that $f\sim_{E}x$ and $$\min\limits_{p\in C^{*}(E)} \int_{E}u(f)\frac{dp}{p(E)} < u(x)$$ Then for all $x^{*} \succsim x$, $$\begin{aligned}
\min\limits_{p\in C} \int_{\Omega} u(x_{E}x^{*})dp - \min\limits_{p\in C^{*}(E)} \int_{\Omega} u(f_{E}x^{*})dp = u(x)p^{*}(E) - \min\limits_{p\in C^{*}(E)} \int_{E}u(f)dp> 0
\end{aligned}$$ where $p^{*}(E) = \max\limits_{p\in C} p(E)$. That is, there exists $\delta_{E,f} > 0 $ such that $$\begin{aligned}
\min\limits_{p\in C} \int_{\Omega} u(x_{E}x^{*})dp - \min\limits_{p\in C^{*}(E)} \int_{\Omega} u(f_{E}x^{*})dp > \delta_{E,f} > 0
\end{aligned}$$ Now for all $x^{*} \succsim x$, $$\begin{aligned}
&\quad \min\limits_{p\in C} \int_{\Omega} u(x_{E}x^{*})dp - \min\limits_{p\in C}\int_{\Omega} u(f_{E}x^{*})dp \\
&= \min\limits_{p\in C} \int_{\Omega} u(x_{E}x^{*})dp - \min\limits_{p\in C^{*}(E)} \int_{\Omega} u(f_{E}x^{*})dp + \min\limits_{p\in C^{*}(E)} \int_{\Omega} u(f_{E}x^{*})dp - \min\limits_{p\in C}\int_{\Omega} u(f_{E}x^{*})dp \\
& > \delta_{E,f} > 0
\end{aligned}$$ the last inequality comes from the fact that $\min\limits_{p\in C^{*}(E)} \int_{\Omega} u(f_{E}x^{*})dp - \min\limits_{p\in C}\int_{\Omega} u(f_{E}x^{*})dp \geq 0$. Then it suffice to find $z \succ w$ such that $u(z) - u(w) < \delta_{E,f}$ and it will imply that for all $x^{*} \succsim x$: $$\frac{1}{2} x_{E}x^{*} + \frac{1}{2} w \succ \frac{1}{2}f_{E}x^{*} + \frac{1}{2}z$$ i.e. the Approximate CR-LC axiom fails.
Now consider the second case, there exists $q \in C_{E}$ such that $q \notin \{\frac{p}{p(E)}: p \in C^{*}(E)\}$. By strong separating hyperplane, there exists $f \sim_{E}x$ such that $$u(x) = \min\limits_{p\in C_{E}} \int_{\Omega} u(f) dp \leq \int_{\Omega} u(f) dq < \min\limits_{p\in C^{*}(E)} \int_{E}u(f)\frac{dp}{p(E)}$$ Then it further implies for all $x^{*} \succsim x$, $$\min\limits_{p\in C^{*}(E)} \int_{\Omega} u(f_{E}x^{*})dp - \min\limits_{p\in C} \int_{\Omega} u(x_{E}x^{*})dp = \min\limits_{p\in C^{*}(E)} \int_{E}u(f)dp - u(x)p^{*}(E) > 0$$ i.e. there exists $\delta_{E,f}$ such that $$\min\limits_{p\in C^{*}(E)} \int_{\Omega} u(f_{E}x^{*})dp - \min\limits_{p\in C} \int_{\Omega} u(x_{E}x^{*})dp > \delta_{E,f} > 0$$ On the other hand, by Lemma \[alem\], for any $\epsilon >0$ there exists $\bar{x}_{E,f,\epsilon}$ such that $$\min\limits_{p\in C^{*}(E)} \int_{\Omega} u(f_{E}x^{*})dp - \min\limits_{p\in C}\int_{\Omega} u(f_{E}x^{*}) dp < \epsilon$$ for all $x^{*} \succsim \bar{x}_{E,f,\epsilon}$. Now for all $x^{*} \succsim \max\{x, \bar{x}_{E,f,\epsilon}\}$, $$\begin{aligned}
&\quad \min\limits_{p\in C}\int_{\Omega} u(f_{E}x^{*})dp - \min\limits_{p\in C} \int_{\Omega} u(x_{E}x^{*})dp \\
&= \min\limits_{p\in C}\int_{\Omega} u(f_{E}x^{*})dp - \min\limits_{p\in C^{*}(E)} \int_{\Omega} u(f_{E}x^{*})dp + \min\limits_{p\in C^{*}(E)} \int_{\Omega} u(f_{E}x^{*})dp - \min\limits_{p\in C} \int_{\Omega} u(x_{E}x^{*})dp\\
& > -\epsilon + \delta_{E,f}
\end{aligned}$$ Now find any $z \succ w$ such that $u(z) - u(w) = \eta < \delta_{E,f}$ and let $\epsilon = \delta_{E,f} - \eta > 0$. Then for all $x^{*} \succsim \max\{x, \bar{x}_{E,f,\epsilon}\}$ the previous result implies that $$\min\limits_{p\in C}\int_{\Omega} u(f_{E}x^{*})dp - \min\limits_{p\in C} \int_{\Omega} u(x_{E}x^{*})dp > -\epsilon + \delta_{E,f} = \eta > 0$$ Then $$\begin{aligned}
\frac{1}{2} \min\limits_{p\in C}\int_{\Omega} u(f_{E}x^{*})dp + \frac{1}{2}u(w) - \min\limits_{p\in C} \int_{\Omega} u(x_{E}x^{*})dp - \frac{1}{2}u(z) > \frac{1}{2}[\eta - \eta] = 0
\end{aligned}$$ i.e. $\frac{1}{2} f_{E}x^{*} + \frac{1}{2} w \succ \frac{1}{2}x_{E}x^{*} + \frac{1}{2}z$ for all $x^{*} \succsim \bar{x}_{E,f,\epsilon}$. Thus the Approximate CR-LC axiom fails in this case as well. Combine both cases shows that, not ML implies not Approximate CR-LC.
**Remark.** The characterization results for Weak RML and RML can be extended similarly to this general case. The statement of the axioms and also the additional steps in the proof will be analogously the same as in ML, thus omitted in the present paper.
[^1]: Department of Managerial Economics and Decision Sciences, Kellogg School of Management, Northwestern University, Evanston, IL, USA. E-mail: xiaoyu.cheng@kellogg.northwestern.edu
[^2]: I am grateful to Peter Klibanoff and Marciano Siniscalchi for invaluable guidance and discussions throughout the completion of this paper. I thank Eddie Dekel, Eran Hanany, Jingyi Xue, and especially Lorenzo Stanca for insightful discussions. I thank all participants at Northwestern Strategy Bag Lunch and Theory Lunch for comments. All remaining errors are my own.
[^3]: Some may also refer to it as prior-by-prior updating
[^4]: With additional technical assumption, the characterization of ML without this assumption is given by Theorem \[thm5\].
[^5]: These notations will be used throughout the present paper.
[^6]: A graphical illustration can be found in section 3.
[^7]: Details will be provided in section 4.
[^8]: An act $f$ is a function maps from the state space to the consequence space.
[^9]: Some may refer to this condition also as **Dynamic Consistency**. The present paper follows the terminology in Hanany and Klibanoff (2007)[@hanany2007updating], in which dynamic consistency is defined to require the other direction of implication: ex-ante indifference implies conditional indifference.
[^10]: In the formal statement of the axiom, a special case needs to be stated separately, which is the case $f\sim_{E}g$ always implies $f_{E}x^{*} \sim g_{E}x^{*}$, i.e. when RML reduces to ML ($\alpha = 1$).
[^11]: The optimal payoff cannot be achieved but can only be approached arbitrarily close.
[^12]: When there are only two states, the conditional preferences are trivial.
[^13]: Plausible priors are defined in Siniscalchi (2005), under MEU, it equivalently means that the set $C$ is a polytope in $\Delta(\Omega)$, i.e. it is a convex and closed set with finitely many extreme points.
[^14]: This $C$ is not a core of any convex capacity.
[^15]: See Lemma \[lem4\].
[^16]: I abuse terminology to use states to denote payoff-relevant states thereafter.
[^17]: Notice that $\pi_{\mu}(\theta_{1} | s_{1})$ is increasing in $\mu$.
[^18]: Only row 8 of Table 4.2 in Liang (2019) is different from the prediction of RML. The last two rows do not specify the signals thus are ignored.
[^19]: The same conclusion can be drawn also from a within-subject comparison shown in Table B.4 of Liang (2019).
[^20]: The sender can choose a set of probabilistic devices and delegate the choice from this set to a third party or to the draw from an Ellsberg urn to make the signals ambiguous.
[^21]: Throughout this proof, whenever I say “for all $x^{*}$ that are sufficiently large for some acts”, it means that for each one of those acts, say $f$, $x^{*} \succsim \max\{\bar{x}_{E,f}, x, \hat{x}_{E,f}\}$ holds.
[^22]: The case $\lambda = 0$ is excluded since when $\lambda = 0$, $g_{l\lambda}y$ coincides with $y$, and $\alpha[E,y]$ is not unique.
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Monday, December 5, 2011
Living in the USA: My Culture Shock Top 10
I read with great interest and was amused by the directness of an essay by Sophia Angelique about the culture shock of moving to and living in the USA, after being in the UK, and I recognized much of what she said - although I must admit that I was a little troubled that she was still undergoing culture shock a full eight years after arriving there.
Some of the videos she posted up on there are excellent too. The one of Stephen Fry (Brit) talking to Clive James (Ausie who lives in the UK) stood out as particularly outstanding and again, much of what they say is very recognizable to me. (I’ve included the vid at the end of the top 10).
I thought it would fun to summarize some of the top 10 things that can be surprising and difficult for a typical Brit living in the USA, based on a variety of sources including my own experiences and those observations mentioned above.
I’m writing from a purely British perspective, of course, so I don’t know how much you will relate to it, if at all, if you’re non-British, or if you have never experienced living in the USA. It also occurs to me that younger Brits might find it easier than older gits like me to cope with the culture shock, as I am probably stuck in my ways to some degree. But anyway, in no particular order, here goes with my top 10.
As Stephen Fry points out, coming from a monarchy, you kind of expect the US to have more republican values of “all men are equal”, but in fact American values are quite different from that. Individual freedom is seen as more valuable than equality and justice in the US which gives it a different value system to the UK (and indeed the majority of republics). Fry argues in the video that it’s rooted in the way that the American constitution prioritizes things.
As I mentioned in a previous blog, it’s difficult to underestimate and understate the importance of religion in American life, whether it’s in everyday life, or in general society and politics. Outside of respectable religion, I would also say that superstition and hokey beliefs are also much more common too.
It’s much more commercial and materialistic here compared to Blighty. Practices that would be considered mercenary back in the UK are much more commonplace. Money seems to trump all in the USA, maybe even religion. It certainly runs the political system.
As a Brit, you get told that we’re a class ridden society - and we certainly are compared to places like Germany, The Netherlands, Scandanavia, but in many ways the British class system is nothing compared to the US where there is a stronger sense of hierarchy, and social and economic status really is the be all and end all for most people.
American politics. Things are shifted so far to the right here compared to the UK, I still have trouble working out who’s who. Often a ranting politician that I think is a rightwinger, turns out to be Democrat. The loony, foaming at the mouth, religious nut, I discover later is a respected Republican Senator.
The lack of infrastructure can be frustrating here, if you’re not used to it. Outside of the big cities, there is very little public transport. Whereas you will get shops and pubs in the suburbs of towns and in rural areas in the UK, there is very little of that in the USA. Not only is it a convenience thing, but local shops, post offices, pubs are where a British community would meet and builds ties. (It was interesting to read that this was one thing that Brits found difficult about moving to Australia in a BBC article that I read).
As Fry points out, America is an enormous place with lots of semi-autonomous states that often have a strong sense of self-identity. People often have more in common with and identify more with their state than the country as a whole. (In the UK, of course, we have Scotland, Wales, England and Ulster, but it is different.)
Americans really believe in things. British people tend to be skeptical about pretty much everything. We make good scientists but poor dreamers.
The right to bear arms and the gun thing is difficult to understand as an outsider. I am not particularly comfortable with all the violence that you get in America movie and drama either, although sometimes I think it just acts as a lazy plot device, it can seem to come uncomfortably close to romanticizing violence. I am no pacifist, by the way, I just don’t think real violence is in any way romantic.
As Fry says, the American ideas of “liberty” and “freedom” are very difficult for a Brit to understand. For us, “liberty” and “freedom” are essentially concepts, which makes them essentially wooly. That doesn’t mean that we see them as bad ideas, we just don’t understand them as being solid things.
Ironically, after a year of living in the USA, I quite often feel that I understand it less now than I did when I arrived.
14 comments:
I read the affiliate post on Hubpages and I wish I hadn't in some ways. That woman is extremely bitter and locked in to one particular way of thinking. There's not a little irony in the way she expresses that she is uncomfortable with the idea of America as 'the best country in the world' when she is clearly indicating in her entire people about the superiority of life in Britain. Weird...anyway...
Yes, there are problems in America. But again, I see another reference to America not being a place of refined tastes. We have produced great musicians, artists, writers and actors, so obviously it's not some hellish backwater. Not to mention, we are gracious and generally embrace foreigners who openly and admittedly come to this country to do what they cannont do in their own: Make a ton of money.
What I don't get about the individual freedom thing in America is that is really just a concept, and not a concrete thing. People will pay lip service to personal freedom and then dress the same, buy the same, drive the same as everybody else and if you don't tow the line you're 'weird' or 'subversive'. So much for personal freedom.
And I bemoan the idea of justice being down on the list, but I have to admit after much soul-searching that you may be correct about that. It just pains me, because I come from a family that prizes justice (social and legal) and intellectualism/critical thinking. So, it's hard to accept that perhaps we aren't living up to what I once believed was an ideal or value that I did view once as American.
I like the Stephen Fry video, much better than the post. Although both give me context for you being somewhat more reasonable and in the middle ground than I previously viewed you as being. As an American, I appreciate a different perspective, as painful as it can be to be viewed through another culture's eyes as money-grubbing, uncouth, thoughtless, unprincipled, theocratic, dogmatic, unreasonable, illogical and mercenary, not to mention mendacious.
My article is really a subjective article about British culture shock rather than what is objectively good or bad about the USA. I am certainly no expert on the US, just fascinated by the Brit in the USA experience.
I would tend to agree that Sophia Angelique appears to be more critical about the US, but obviously I can't speak on her behalf.
I agree that the USA has produced great music and literature, it is also the world's pioneer in things such as computers and technology and American beer is far better than its international reputation, but that sort of stuff doesn't generally cause culture shock to a Brit, so it isn't included.
Some of the arguments and discussions in the US, don't exist in the UK because Americans are coming at issues from a different place (both physically and psychologically!).
Not a surprising list. These things surprised (and shocked) me while living in the US, and I'm an American. In regard to justice, the US still maintains a certain lawless sensibility. People who might be squashed get squashed if it's convenient for the system, not necessarily because they've done anything to warrant punishment. I'm not saying it always happens this way in the "justice" system, but sometimes it does. I've seen it.
America is very much about who you know, too. This determines everything--where you come from, who your family is, etc.
All the negatives you've listed are things I've always struggled with as an American. If you don't agree with those values, it's very alienating, and you will be an outsider. Oscar Wilde had it right when he said that "America is the only country that went from barbarism to decadence without civilization in between."
That said, I liked the note about the dreamers. We are good at that--don't know where it's getting us these days though. (I just finished watching all 6 parts of that Frontline God in America thing and it addresses this quality a number of times.)
Thanks for your comment taramoyle. By scientists and dreamers, I tend to think of the Dyson vacuum cleaner and the Steve Jobs ipod. The UK has great practical inventions, but the US is great at coming out with visionary leaps.
I'm surprised by #4. I don't think I've really ever experienced a class divide in America. I still have this romantic notion that anyone from any background can achieve anything in America. Maybe that's not the reality anymore.
As for that original article being negative, it was written by a Brit. We are the best complainers. We'd much rather point out the negatives than extol the positives. The American notion of positivity is perhaps my #1 'shock'.
The class thing is complex, but I would agree with Fry overall. I think I will denote an entire post to social class at some point, as it's a fascinating area and certainly different to the UK.
I do feel that the poorer and more excluded sections of society can seem much more passive here in the US, and I get the impression that they don't see themselves as being as good as other Americans who are wealthier, whereas as the Brits at the bottom of the pack have a more: 'how dare you treat me like this, I'm as good as the next man' mentality. (The French are the archetypal republicans though, of course).
Yes, British people see complaining as virtually an art form, whereas it is considered almost taboo in the US. My wife still finds it difficult that I moan about things without expecting any resolution.
Having recently (today) discovered your blog, I believe you offer an important and underrepresented perspective on 21st century American culture from a British perspective. There's a bit of a spark of interest in Britain these days, thanks as I mentioned in a prior comment probably to "Downton Abbey's" tremendous success. There are many themes covered by that series, but even disregarding that fictional period drama completely, the non-fictional subject of "class" (i.e., socioeconomic status) endlessly intrigues so many of us -- across the centuries, and across "the pond." Please, please do write about it... It is especially interesting to read about the contemporary American class system as experienced through you (i.e., a British expat) because those of us who are totally engulfed in the American system truly cannot see what exactly we're engulfed in :) It would be very interesting to read your observations, and see if they mesh with our gut feelings. Don't worry about "offending" your American readers, we want to see our "world" from fresh outside eyes ... not simply because of self-absorbed narcissism, but also because we suspect our system does have certain built-in flaws which we could possibly correct, or at least try, if only we were a little more certain we knew what they are! :)
Thanks for your comment. I am working myself up to dealing with this topic area. I am a big fan of Downton Abbey, which I think is great drama. Writing 4oo or 500 words on a topic can sometimes feel a little unsatisfying as I know that I am just scratching the surface.
(Come to think of it, the US does have "complaining as virtually an art form", but it was developed by the minority, socially excluded black population, not the majority whites, and they call it "The Blues")
@Iota. The politics here is difficult. It's not just that the US is far to the right of the UK (which it is) but that they have some very different ideas about what is and isn't important - it brings home how different US society is. I know that my parents friend who moved here 25 years ago spent about 15 years struggling with the politics. |
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In bas |
Background
==========
Obesity is a global healthcare issue that is associated with significant morbidity and mortality \[[@b1-amjcaserep-21-e924432]\]. Various therapeutic measures have been developed for the management of obesity. The use of bariatric surgical procedures is increasing owing to their efficacy in weight reduction and improved management of obesity-associated co-morbidities \[[@b2-amjcaserep-21-e924432]\]. Sleeve gastrectomy is the most frequently performed bariatric procedure worldwide \[[@b3-amjcaserep-21-e924432]\].
Although comprehensive preoperative assessment is essential for patients undergoing sleeve gastrectomy, performing upper endoscopy routinely prior to the operation is a matter of debate \[[@b4-amjcaserep-21-e924432]\]. Herein, we describe a case of gastric schwannoma, an uncommon type of gastric neoplasm, discovered incidentally in a patient scheduled for a sleeve gastrectomy procedure.
Case Report
===========
A 27-year-old woman was referred to a bariatric clinic after several attempts at lifestyle modification for weight loss yielded suboptimal outcomes. In addition to class III obesity (height 152 cm, weight 101 kg, and BMI 43.7 kg/m^2^), her medical records revealed hypertension, diabetes mellitus, and no surgical history. Her hypertension was well-controlled by valsartan and hydrochlorothiazide, and her diabetes mellitus was being treated with metformin, sitagliptin, and insulin therapy. Her physical examination results were unremarkable, and routine laboratory investigations findings were normal. She agreed to undergo laparoscopic sleeve gastrectomy when presented with surgical management options. However, a gastric mass was incidentally discovered during the standard diagnostic exploration. Hence, the surgery was aborted, and the patient was referred to our institution for further evaluation and management.
Here, upper gastrointestinal endoscopy under conscious sedation was performed and revealed an elevated submucosal mass in the gastric antrum with normal overlying mucosa ([Figure 1A](#f1-amjcaserep-21-e924432){ref-type="fig"}). The mass was further characterized by performing an endoscopic ultrasound examination, which revealed a hypoechoic homogenous oval-shaped mass lesion with well-demarcated walls that appeared to arise from the muscularis propria and lacked adjacent lymphadenopathy ([Figure 1B](#f1-amjcaserep-21-e924432){ref-type="fig"}). Cytological examination of the biopsy specimens, obtained by fine-needle aspiration using a 19-gauge needle, revealed a spindle cell neoplasm, giving the impression of a gastric gastrointestinal stromal tumor (GIST). However, the tissue sample was not sufficient for further assessment and complete excision was recommended for definitive diagnosis.
Subsequently, contrast-enhanced computed tomography (CT) of the abdomen revealed an exophytic, well-circumscribed, homogenously hypodense mass bulging into the gastric lumen on the lesser curvature of the pyloric antrum anteriorly ([Figure 2](#f2-amjcaserep-21-e924432){ref-type="fig"}). It measured approximately 3.4×5.3×4.0 cm in its maximum dimensions and had no definite invasion. The case was discussed in the oncology multidisciplinary meeting and surgical management in the form of distal subtotal gastrectomy was planned.
The surgery was performed laparoscopically under general anesthesia with the patient in a supine and leg-split position. Five ports were inserted for carrying out the procedure. After establishing pneumoperitoneum and introducing trocars, standard diagnostic exploration was performed. The gastric mass was identified on the lesser curvature of the gastric antrum, 3 cm away from the pylorus, and measured 5×4 cm ([Figure 3](#f3-amjcaserep-21-e924432){ref-type="fig"}). The stomach was mobilized by dividing the gastrocolic ligament. The distal stomach was resected at the level of the first part of the duodenum distally with an endoscopic linear stapler (Ethicon, NJ, USA). The resected part was retrieved using an Endobag (Ethicon, NJ, USA) through extending the supraumbilical port incision, and the specimen was sent for histopathological analysis. Given the small size of the gastric remnant, the gastrointestinal continuity was restored using Billroth II gastrojejunostomy, which is a technically feasible and relatively simple procedure. The lymph nodes along the greater curvature (station 4d) were removed during preparation for the anastomosis. Methylene blue was flushed into the stomach through the nasogastric tube and no leak was seen. The total operative time was approximately 120 minutes and the estimated blood loss was less than 50 ml. The patient tolerated the procedure well.
The pathological examination of the resected specimen showed a submucosal encapsulated tumor composed of spindle cells arranged in interlacing fascicles, interspersed with collagen fibers microscopically observed ([Figure 4](#f4-amjcaserep-21-e924432){ref-type="fig"}). The tumor was infiltrating the muscularis propria. The neoplastic cells had ill-defined eosinophilic cytoplasm and slender or wavy, elongated, bland-appearing nuclei. The surgical margins and all the 15 lymph nodes obtained with the specimen were free of neoplasia. Immunohistochemically, the neoplastic cells were diffusely reactive for S-100 protein (nuclear and cytoplasmic), and vimentin, and focally reactive for GFAP and CD34, but lacked immunoreactivity for c-Kit, DOG-1, smooth-muscle actin, h-Caldesmon, and β-catenin. The histopathologic features and the immunohistochemical staining pattern were consistent with the diagnosis of gastric schwannoma.
On the first postoperative day, the patient had a Gastrografin meal study, which demonstrated no evidence of a leak. The patient started on a liquid diet, which was well-tolerated. The patient was discharged on the sixth postoperative day. During the follow-up visits over 6 months, the patient was asymptomatic and was able to tolerate a regular diet.
Discussion
==========
Schwannoma is a neurogenic tumor originating from Schwann cells, that can occur anywhere along the course of peripheral nerves. However, it is rarely seen in the gastrointestinal tract \[[@b5-amjcaserep-21-e924432]\]. The most common site for schwannoma in the gastrointestinal tract is the stomach, where it arises from the sheaths of Auerbach's or Meissner's plexuses, although it accounts for only 0.2% of all gastric neoplasms \[[@b6-amjcaserep-21-e924432]\]. It was first described by Daimaru et al. in 1988 in a series of 24 cases \[[@b7-amjcaserep-21-e924432]\]. Schwannoma predominantly affects adults above the age of 40 years and it has a female predilection \[[@b8-amjcaserep-21-e924432]\].
Gastric schwannoma frequently arises from the gastric body, followed by the gastric antrum and fundus, and rarely in the gastric cardia \[[@b9-amjcaserep-21-e924432]\]. It has a slow growth pattern and is often discovered incidentally, as in our case. If symptomatic, non-specific abdominal discomfort is the most commonly reported symptom \[[@b9-amjcaserep-21-e924432]\]. Gastrointestinal bleeding is another frequently reported symptom that is related to ulceration of the gastric mucosa \[[@b10-amjcaserep-21-e924432]\]. Rarely, a palpable mass may be observed \[[@b6-amjcaserep-21-e924432]\].
Schwannoma is generally a benign neoplasm with an excellent prognosis. Simple complete surgical excision is curative \[[@b8-amjcaserep-21-e924432]\]. However, malignant schwannoma has been reported in a few cases. Recurrence is only observed in cases of malignant schwannoma \[[@b6-amjcaserep-21-e924432]\].
Gastric schwannoma may be misdiagnosed as a GIST, as in our case. This is due to the lack of distinct features unique to these neoplasms on preoperative investigations. In a CT scan, gastric schwannoma typically manifests as an ovoid, well-circumscribed, exophytic mass with homogenous progressive enhancement \[[@b5-amjcaserep-21-e924432]\]. Compared to the GIST, gastric schwannoma usually has oval morphology, extraluminal growth, homogenous enhancement, lack of necrosis, and presence of perigastric lymph nodes \[[@b9-amjcaserep-21-e924432]\]. Post-gadolinium sequences show slow but relatively uniform enhancement. Fluorodeoxyglucose positron emission tomography is of limited value in the preoperative evaluation, as various neoplasms can give similar findings, but it may be used to detect recurrence or metastasis of malignant schwannoma \[[@b11-amjcaserep-21-e924432]\]. Immunohistochemistry allows the differentiation of gastric schwannoma from other spindle cell neoplasms, as neoplastic cells that are positive for S-100 protein and negative for actin, c-Kit, and CD34 are consistent with the diagnosis of schwannoma \[[@b5-amjcaserep-21-e924432]\].
In the present case, schwannoma was diagnosed incidentally during the operation, which led to cancelation of the procedure. Although the role of upper endoscopy prior to bariatric surgeries is controversial, many surgeons advocate its routine use among these patients as it can help in the selection of bariatric procedures appropriate to the patient \[[@b4-amjcaserep-21-e924432]\]. For example, the use of restrictive bariatric procedures might worsen gastroesophageal reflux disease, if present. Furthermore, some parts of the gastrointestinal tract might be inaccessible after certain bariatric procedures, and preoperative endoscopy would be useful to rule out any existing pathology. On the contrary, other surgeons consider routine upper endoscopy unnecessary due to cost, sedation, and risk associated with endoscopy; hence, they do not recommend performing upper endoscopy routinely in all patients, including asymptomatic ones. However, there is a poor correlation between endoscopic findings and the presence of clinical symptoms \[[@b12-amjcaserep-21-e924432]\]. In our institution, we perform an endoscopic examination in all patients prior to bariatric procedures, and our case is an example of the value of the evaluation, even in asymptomatic patients.
The increase in the number of bariatric surgeries has led to a rise in the discovery of incidental findings. The GIST is among the most frequently encountered lesions \[[@b13-amjcaserep-21-e924432],[@b14-amjcaserep-21-e924432]\]. Chiappetta et al. \[[@b13-amjcaserep-21-e924432]\] reported 8 patients with incidental gastric GIST discovered incidentally during bariatric surgeries.
Interestingly, all these patients underwent preoperative upper endoscopy, and the tumor was not seen. Thus, the bariatric surgeon should perform a careful inspection during laparoscopy. Crouthamel et al. \[[@b14-amjcaserep-21-e924432]\] reported 17 cases of incidental gastric mesenchymal tumors identified during a laparoscopic sleeve gastrectomy, including 2 cases of gastric schwannoma. The surgical plan was not altered because the tumor was within the area of resection in all cases. For tumors located at the cardia or lesser curvature, cancelation of sleeve gastrectomy procedure is required to allow appropriate tumor resection. In our case, the planned sleeve gastrectomy was aborted so that we could discuss treatment options with the patient.
Conclusions
===========
Gastric schwannoma is an uncommon neurogenic neoplasm. The present case should remind physicians of the importance of comprehensive preoperative evaluation in patients planned for bariatric procedures, including the performance of upper endoscopy. In addition, bariatric surgeons should have standard operating procedure protocols for dealing with gastric lesions discovered intraoperatively.
**Conflict of interest**
None.
{#f1-amjcaserep-21-e924432}
{#f2-amjcaserep-21-e924432}
{#f3-amjcaserep-21-e924432}
{#f4-amjcaserep-21-e924432}
[^1]: Authors' Contribution:
[^2]: Study Design
[^3]: Data Collection
[^4]: Statistical Analysis
[^5]: Data Interpretation
[^6]: Manuscript Preparation
[^7]: Literature Search
[^8]: Funds Collection
[^9]: **Conflict of interest:** None declared
|
Molecular assays for quantitative and qualitative detection of influenza virus and oseltamivir resistance mutations.
Sensitive and reproducible molecular assays are essential for influenza virus diagnostics. This manuscript describes the design, validation, and evaluation of a set of real-time RT-PCR assays for quantification and subtyping of human influenza viruses from patient respiratory material. Four assays are included for detection of oseltamivir resistance mutations H275Y in prepandemic and pandemic influenza A/H1N1 and E119V and R292K in influenza A/H3N2 neuraminidase. The lower limits of detection of the quantification assay were determined to be 1.7 log(10) virus particles per milliliter (vp/mL) for influenza A and 2.2 log(10) vp/mL for influenza B virus. The lower limits of quantification were 2.1 and 2.3 log(10) vp/mL, respectively. The RT-PCR efficiencies and lower limits of detection of the quantification assays were only marginally affected when tested on the most dissimilar target sequences found in the GenBank database. Finally, the resistance RT-PCR assays detected at least 5% mutant viruses present in mixtures containing both wild-type and mutant viruses with approximated limits of detection of 2.4 log(10) vp/mL. Overall, this set of RT-PCR assays is a powerful tool for enhanced influenza virus surveillance. |
Transformers: Dark of the Moon, the third film in the blockbuster Transformers franchise, is returning to 246 IMAX domestic locations for an extended two-week run from Friday, Aug. 26 through Thursday, Sept. 8. During those two weeks, the 3-D film will play simultaneously with other films in the IMAX network. Since its launch on June 29, Transformers: Dark of the Moon has grossed $1.095 billion globally, with $59.6 million generated from IMAX theatres globally. ”Transformers: Dark of the Moon: An IMAX 3D Experience” has been digitally re-mastered into the image and sound quality of The IMAX Experience® with proprietary IMAX DMR® (Digital Re-mastering) technology for presentation in IMAX 3D. The crystal-clear images, coupled with IMAX’s customized theatre geometry and powerful digital audio, create a unique immersive environment that will make audiences feel as if they are in the movie. |
[Cultural adaptation to Spanish and validation of the Gastrointestinal Short Form Questionnaire].
To describe the process followed for the cultural and psychometric adaptation (validation) to Spanish of the Gastrointestinal Short Form Questionnaire (GSFQ), used to measure the interference of symptoms of gastroesophageal reflux disease GERD and to report the psychometric properties of this instrument. The adaptation process was supervised by a five-member expert panel. After forward and backward translations in duplicate, a Spanish version was obtained, which was administered to two samples; a five-patient pilot sample to check comprehension and face validity, and a 4,000-patient sample to check structural validity (factor analysis and reliability), construct validity, and discriminative validity. The questionnaire showed a unique dimension that matched that of the original questionnaire. Reliability was high (alpha=0.83), and the correlation between even-odd items was good (r=0.69). The overall score correlated with generic health-related quality of life measures evaluated by the EQ-5D tariff (r=0.499) and VAS (r=-0.481). The scale discriminated between GERD severity levels (p<0.008) as measured by the Savary-Miller scale, except for the most severe level with respect to the levels immediately below. The questionnaire was able to detect differences between diverse concomitant diseases and antecedents. Sensitivity with respect to the GERD clinician criterion was 60.5% and specificity was 68.3%. Normative comparison scaling values are reported. The results show acceptable psychometric properties. A new instrument to assess the interference of GERD symptoms is thus available to health professionals. This instrument takes the patient's perspective into account. |
Kees Akerboom Jr.
Kees Akerboom Jr. (born 20 December 1983) is a retired Dutch basketball player. Kees is the son of Kees Akerboom Sr., a former successful professional basketball player as well. During the majority of his career, Akerboom played with Den Bosch, which he won three DBL championships with. He also played for the Netherlands national basketball team during his career.
Professional career
Akerboom started his professional career with the same team as his father did, EiffelTowers Den Bosch in the Dutch Eredivisie.
On 29 May 2004, Akerboom signed with MPC Capitals from Groningen.
After two seasons with the Capitals, Akerboom returned to EiffelTowers. He won his first Dutch championship in 2007. He won his second in 2012, after beating ZZ Leiden 4–1 in the Finals. In 2015, Akerboom won his third championship after beating Donar 4–1 in the Finals.
On 31 May 2018, Akerboom announced his retirement at age 34. His jersey number 12 was retired by Den Bosch.
International career
Akerboom played 99 games for the Netherlands national basketball team. On 5 July 2016, Akerboom's retirement from the Dutch national team was announced.
Honours
Club
Den Bosch
Dutch Basketball League: 2006–07, 2011–12, 2014–15
NBB Cup: 2007–08, 2008–09, 2012–13
Dutch Supercup: 2013
Individual
DBL All-Star (11): 2004, 2006, 2008, 2009, 2010, 2011, 2012, 2014, 2015, 2016, 2017
DBL All-First Team: 2010–11
DBL Rookie of the Year: 2002–03
DBL MVP Under 23: 2003–04
DBL Three points percentage leader: 2007–08, 2008–09, 2010–11, 2011–12
DBL All Star Game: 3 Point Contest Winner: 2008, 2009
DBL All-Star Game MVP: 2010
References
Category:1983 births
Category:Living people
Category:Donar (basketball club) players
Category:Dutch Basketball League players
Category:Dutch men's basketball players
Category:People from Sint-Michielsgestel
Category:Small forwards
Category:Heroes Den Bosch players |
Last week I had the chance to speak with Jeff Seibert, Senior Director of Product at Twitter and Co-Founder of Crashlytics (acquired by Twitter for $259m) and one of the main individuals leading the new guard at Twitter.
In what turned out to be a reflective interview we discussed a range of topics including what it was like raising VC money in 2009, why being overly transparent with your team can be a hindrance and why Jeff decided selling Crashlytics to Twitter was the best option, allowing them to be deployed on over 1 billion devices.
Seibert also discussed the need for startups to negotiate for continued investment in their product by their acquirer in acquisition talks; at what stage runway startups should look to go and raise again; what life is like at Twitter following the acquisition of Crashlytics — and the next 5 years for him and the Twitter product? |
let Magix = require('magix');
let Base = require('__test__/example');
let $ = require('$');
module.exports = Base.extend({
tmpl: '@1.html',
render() {
this.updater.digest({
list: [{
text: '左对齐（默认）',
value: 'left'
}, {
text: '居中对齐',
value: 'center'
}, {
text: '右对齐',
value: 'right'
}],
current: 'left',
bizCodes: [{
text: 'adStrategy（策略中心）',
value: 'adStrategy'
}, {
text: 'unionMedia（联盟媒体测）',
value: 'unionMedia'
}, {
text: 'unionMerchant（联盟商家测）',
value: 'unionMerchant'
}],
types: [{
text: 'alimama（默认）',
value: 'alimama'
}, {
text: 'taobao',
value: 'taobao'
}, {
text: 'etao',
value: 'etao'
}, {
text: 'tanx',
value: 'tanx'
}, {
text: 'iconfont',
value: 'iconfont'
}],
currentType: 'alimama',
currentBizcode: ''
});
},
'change<change>'(e) {
this.updater.digest({
current: e.params.value
})
},
'changeType<change>'(e) {
this.updater.digest({
currentType: e.params.value,
currentBizcode: ''
})
},
'changeBizCode<change>'(e) {
this.updater.digest({
currentType: '',
currentBizcode: e.params.value
})
}
}); |
/*
* Licensed to the Apache Software Foundation (ASF) under one
* or more contributor license agreements. See the NOTICE file
* distributed with this work for additional information
* regarding copyright ownership. The ASF licenses this file
* to you under the Apache License, Version 2.0 (the
* License); you may not use this file except in compliance
* with the License. You may obtain a copy of the License at
*
* http://www.apache.org/licenses/LICENSE-2.0
*
* Unless required by applicable law or agreed to in writing,
* software distributed under the License is distributed on an
* AS IS BASIS, WITHOUT WARRANTIES OR CONDITIONS OF ANY
* KIND, either express or implied. See the License for the
* specific language governing permissions and limitations
* under the License.
*/
/*
* Copyright (c) 2020, OPEN AI LAB
* Author: haoluo@openailab.com
*/
#ifndef __WINO_CONV_KERNEL_ARM_H_
#define __WINO_CONV_KERNEL_ARM_H_
#include "tengine_ir.h"
#include "convolution_param.h"
#include "../conv_hcl_kernel.h"
int wino_conv_hcl_prerun(struct ir_tensor* input_tensor, struct ir_tensor* filter_tensor,
struct ir_tensor* output_tensor, struct conv_priv_info* info, struct conv_param* param)
__attribute__((weak));
int wino_conv_hcl_postrun(struct conv_priv_info* info) __attribute__((weak));
int wino_conv_hcl_run(struct ir_tensor* input_tensor, struct ir_tensor* filter_tensor, struct ir_tensor* bias_tensor,
struct ir_tensor* output_tensor, struct conv_priv_info* conv_info, struct conv_param* param,
int num_thread, int affinity) __attribute__((weak));
#endif
|
Brushstroke Cross with A Ransom for Many
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This brushstroke image of the cross is complemented with the words A Ransom for Many. This was taken from Mark 10:45 which says that "The Son of Man did not come to be served, but to serve, and to give His life a ransom for many." In other words, Jesus became the sinless sacrifice to pay the price for the sins of mankind -- past, present and future. |
Q:
UIPageControl lagging on retina devices
I've implemented an iphone app with a page controller following Apple's code example. There are only 3 views on the page controller. The app works fine on normal screens, but when I test it on retina devices, there is a visible lag when scrolling either horizontally or vertically in any of the views. Any ideas of what the problem might be or how can I "debug" this?
The content of 2 of the views is a table view and on the third view I'm using CorePlot. The network is not used while scrolling.
A:
I was adding a shadow around the 3 views and that's what was making the scrolling slow.
|
Short Story Saturday: Clipping Bud
For the next few Saturdays, we’ll be posting some tales by local writer, Charles Wilson. He takes a nugget of local color and adds some imagination (maybe a lot of imagination–you be the judge.) We haven’t posted fiction for a long time (since the well-received SoHumBorn stories–go here to check them out.) Let us know what you think about adding a touch of invention to the world of fact. If you have a story you would like to submit, send it to mskymkemp@gmail.com.
There they were, three young women standing by the road with backpacks in a pile behind them. One of them, a short blonde in jeans and a tight tee shirt was holding up a big piece of cardboard cut to resemble scissors while another had a sign saying “got work?”
Yeah, I had work!
I had been on a town run to get lunch for the 2 local gals I had trimming weed back at the cabin. One wouldn’t eat anything with gluten (whatever that is) and the other was a vegan so none of the grub I had at home was satisfactory. They were trustworthy but their diets were a giant pain in the ass! So I asked them to make me a list of what they wanted for their midday meal, took the list and some cash and headed for town to keep them happy.
That was essential since I had a greenhouse full of pot that urgently needed harvesting. Far more than I could deal with myself so I did what any grower in my shoes did, I put out word that I was hiring at 20 an hour and got a couple local pros. These girls were fast and thorough, they had been doing this job for years and knew what the market wanted. They earned every dollar of the 20 per that they demanded!
Since they were locals I only had to provide their lunches and drinks, at day’s end they drove back to their homes and dealt with their dietary peculiarities themselves. Thank Jah for that! But as good as they were I had far more than the three of us could process and these chicks looked interesting. And pretty good too… At best they were 6 more willing hands and just maybe a little sex. At worst I could always play Donald Trump and say “Yer fired!” then drive em back to town. It certainly didn’t hurt that they were young and pretty good looking either.
So I’m a sexist pig. Got a problem with that?
I pulled the shiny black king cab up next to them and stopped, toggled the power window on the passenger side of the truck down and said “Got any experience?”
“Oh yeah, do we ever!” It was the blonde that said this as she smiled at her friends. They smiled back. Her eyes were pale blue, the color of sunlight viewed through ice.
“20 per hour and a place to lay your bedrolls. I provide 3 meals a day if your dietary needs aren’t too weird. I can probably keep you busy for a week or more.”
“Sounds good, yer on! Right girls?” Again it was the blonde that spoke, she seemed to be the leader of the group. The others agreed enthusiastically.
“I’ll have to ask you for your cellphones.”
“Whoa dude, what the fuck?? Are we supposed to trust you that much?” This time it was the tallest gal, a willowy brunette, that spoke and she was clearly pissed off. “Do we really want to work for this guy or just wait for someone else.”
“I need the money, let’s do it!” It was the third woman that spoke up this time as she looked into the faces of her companions.
I said “Look, where I live the damn things don’t work anyway but I don’t want pictures taken so it’s gimme the damn phones or I’m outta here-by myself. Next question-any weird dietary restrictions or can you gals eat normal food?”
“Whatever dude! And what do you mean by normal food”
“Ham sandwiches for example. Or pizza, chicken, steak, pasta, vegies. You know, normal food?” They laughed, the tension lifted, and all agreed that they did indeed eat normal food. Then the blonde dug into her pack and removed a smart phone and handed it to me through the window with a grin. The other two did the same. All 3 phones went in the glove box.
“My name’s Buddy. Toss your packs in the back and hop in” I hit the switch that unlocked the back doors on the crew cab.
“Climb in you mean!” One of them said. This sucker is big enough you almost need a ladder to get into it.”
They smelled nice, pretty clean for three girls supposedly hitchhiking so I figured they had split a room at one of the motels in town.
A couple minutes later we were heading up the Alderpoint road. The big diesel ate up the miles about as fast as it drank up petroleum and within an hour we were back at the ranch.
The two groups of women eyed each other warily at first. The locals were not happy at what they saw as competition but they had also seen the inside of my greenhouses. They knew there was more work, urgent work, than we three could hope to accomplish. It was a warm autumn day so the gals had been trimming outdoors at a picnic table. I pulled a couple plastic chairs over to the ends of the table, went in to prepare lunch for the newcomers and myself and invited the locals to come in and prepare their own delicacies from the stuff I had just gotten in town.
I carved some thin slices off a ham I had cooked a couple days before. I had been craving ham so I had purchased and baked a small one but… Some wit once defined eternity as two people and a ham. Well I was only one so having 3 other mouths was actually kind of a welcome addition. In a few minutes I had constructed 4 ham sandwiches and a pot of strong coffee. By the time I was done the locals were back at the picnic tables and all the women were starting to talk and laugh. I set down the plate with sandwiches, another with condiments, then went back in for the coffee and cups.
45 minutes later the food was gone as was most of the coffee. The conversation was getting animated and scissors were snipping their way through big baskets of buds. I was constantly bringing fresh-cut stuff from the greenhouses or taking the clipped buds and laying them on screens in my drying sheds. These were just a couple of portable mini-barns loaded with screen trays that spanned the width of the structures. These were spaced 6 inches apart and took up both ends of the sheds. The narrow gap in the middle contained two dehumidifiers, an array of fans and a small electric heater controlled by a thermostat that kept the interior at 90. An hour after loading the screens the places felt like aromatic jungles and the dehumidifiers needed draining every couple hours but after three days the product was dry enough to bag for sale.
At the end of the work day I would go out to the cash stash and get enough to pay everyone for their days work. The locals would take their money and go home for the evening and the trim sisters would stay at the ranch. I would make dinner for all of us, we’d smoke a few joints then they would retire to the deck and their bedrolls. I hadn’t tried to hook up with any of them yet but figured it would be easy after a few days. I wasn’t bad looking, I had a bitchin new truck and it was clear I had the peacock’s tail of the human species-money. When I made my move it should be easy, like taking candy from a baby.
With just the three of us working we were alternating sheds, one was being filled while the other was drying out product but with 6 of us I wasn’t sure they would be able to keep up. I was right! First I started pulling the stuff out in 2 days and putting it in open cardboard boxes in my loft to finish drying then I went and rented a big U-haul van for a couple weeks and bought some more fans, dehumidifiers and another heater. Plus a lot of screen and 2x4s. While the women clipped I franticly kludged the back of the box van into a drying shed. Soon it was full too.
It was getting time to convert a bunch of the dried stuff into cash. The cabin loft was filling up with weighed pounds so I called my broker and a price range was agreed upon. He would be up in a couple days and in the meantime the harvest was slowing but not over. I took my stuff out of the rented box truck, swept the interior and returned it. It was still pretty aromatic in the back but the guy that took the keys and put the charge on my card said nothing about it. After all, this was SoHum and I did return it in spotless condition…
I figured when my man came up and we did the deal I would take the scissor sisters out to dinner at the cove, feed em a couple bottles of champagne and then make my move. I had my eye on the tall one. Tall she was but then so am I. She had eyes dark as chocolate and a nice athletic body. I’d been flirting with her a little and she was giving me these knowing up from under smiles in return so things seemed mighty hopeful. She was starting to get me incredibly horny. I was figuring I could charm her into my bed and maybe even get her to stick around. I could use a pretty woman around the place and she was just my type. I was playing it cool while there was work to be done though. I didn’t want any complications with my staff while the crop needed clipping.
I gave my man a buzz, we discussed prices and it seemed like we could come to a deal pretty easily. He of course wanted to be able to inspect the product before committing to a price. I was confident he’d like it, I grow good shit!. He showed up two days later. I loved the rig he used to run weed back to the city, a tow truck with “City Long Distance Towing and Delivery” on the doors! The weed went in the diamond-plate aluminum toolboxes in the back that were then padlocked. Who’s gonna suspect a tow truck?
He sampled my products, we haggled a little, agreed on a price then started weighing, bagging, and heat-sealing the stuff up. Thirty pounds later we called it a day. He paid me a tidy chunk and set off, tow hook swinging jauntily. There was more weed drying in the sheds, more in bags, but most of the stuff had been picked. All told there might be another shed full to pick then I was done for another year.
After I paid the women I tossed in five hundred bucks each as a tip. And maybe as an aid in bedding that sloe-eyed gal I hoped. Then I invited the sisters out to dinner to celebrate my good fortune. And loosen them up a little with champagne…
I seemed to be the one that drank most of the champagne however. The sisters each had a glass of the first bottle but on the second they all said they had drunk enough so… I was pretty fucked up by the time dinner was over with and it was time for the two hour drive home. The women all rebelled at me driving so I surrendered the keys, the tall gal got behind the wheel and I got in beside her. I told her how to adjust the power seat and when it was where she wanted I told her how to start the diesel. The big engine rattled into life, she put it in drive and started out smoothly. I promptly fell asleep.
To awaken with something prodding me in the back of the head. Something hard. It was a short ugly pistol in the hand of the short blonde. It didn’t tremble a bit, it was as steady as her voice. We were back at my place and I didn’t think this was just a bad dream.
“OK pal, we’re going to walk to your cash stash. It’s payday! And don’t get cute, I assure you I know how to use this thing and you wouldn’t have a snowball’s chance. Two tours in Iraq Buddy Boy, as an army communications officer. Before they sent me there they taught me how to shoot. Now, shall we both get out of this rig and take a walk?”
She produced a flashlight from her jacket pocket, turned it on. Then said “Tell me where we’re going Buddy Boy and I’ll light the way”. I pointed at a path through the grass and we set off. A couple minutes later we were standing by the top of a plastic garbage can sticking about 6 inches above the dirt.
“Open it and get out the cash. SLOWLY! If I think you’re pulling a weapon you’re dead so be nice and smooth.”
I got on my knees, pulled the lid off and slowly put my arm into the container. I grabbed the first bag of cash, said “coming up” and slowly brought the bag to the surface and set it on the ground.
“Get the other one too Buddie boy. Then put it next to the first bag, stand and walk 3 steps over there then sit down! Do what I say and you’re safe!”
I stood, stepped, and sat. She bent, picked up both bags by their tops, stood, then told me to stand up again and head back to the cabin. The muzzle of her automatic followed me on the walk like a faithful dog.
When we got back to the cabin the other gals had all their stuff packed up and in the back of the truck. They were ready to go. The tall one came over , took the cash and put it in one of the packs, Then the short one asked “got his phone? The tall one grinned and flourished it like a trophy.
“We got it. This was as simple as taking candy from a baby.”
She turned to me and said “River and Jane will be here in the morning and we’ll leave your truck will be in town.”
Could have made it a double twist. “You won’t hurt a hair on my head. Remember I took a few polaroids of you all when you arrived, those aren’t here. We’re prepared for your kind in this neck of the woods. Look at my phone you just took, I’ve got the sheriff on speed dial. Get the fuck out of here, army brat. You know damn well you’d never get away with it [edit] Get lost and don’t come back, everybody’s going to know who you are.”
This was probably written in one sitting and is less controversial this way. Maybe if you were writing an action story, or a movie. What your stating sounds like new age movie lines. Which this is not. Fact of the matter, commenters would be saying rude things if those things you said were in the story. I think this isn’t controversial this way.
Obviously this was written by someone who is not in the business…the guy called his ‘broker’ to sell his weed? Really? In 40+ years I have never heard a person refer to their buyer as a broker. Other than that, it was truly horrible.
A little sex? That is just asking for trouble. Never mix business with pleasure. At least to not create favoritism and unrest on a job site. That has always seemed the height of unethical treatment of employees under ones care.
Good fiction is really hard to conjure. Non fiction is much easier as the fact that it is based in reality can make stories compelling just based on there truth.
Creating compelling characters and worlds made from scratch require the highest of talent and ability to channel hidden meanings and undiscovered truths of life similar to good songwriting and musical composition.
IMO.
Bravo… (condescending golf clap)… Phenomenal, fantasy juvenile fiction writing ready for the NY Times best seller list under “fiction Y/A” ! Entertaining, yet so intellectual, and adding an elusive depth of credence to our community that is not often found and oh so refreshing. Did you study literature at Yale? Or was it Oxford Mr. Wilson (aka… JK Rowling, I know it’s you!)?
Every one of these short stories has enough of the truth in it to stand as quite entertaining and illustrative of some of the downsides of our little ‘Behind the Redwood Curtain ‘ culture. And why some of y’all just don’t get it is ‘fiction’ is not the author’s fault. Few grains of truth and the focus to turn it into a story……..Just not good enough for the braggarts who spew threats and machismo all year, but in fact, do NOTHING to ‘police ourselves’ – when it comes to the ‘turkeybaggers’ (ha! Pot carpetbaggers! Perfect!); abusive macho men who destroy lives and land. Nope, nothing, nada – just try to get through in the shadows (happy, in fact, to have bigger, distracting targets than themselves!).
Many disappointing responses ……If you can write, how about doing better? |
Archive for the ‘School Reform’ category
So how are you spending your 4th of July? I am here enjoying some time off. Looking forward to a barbecue and fireworks this evening at my daughter’s middle school. I am also responding to Calling All Bloggers! by Dr. Scott Mcleod from Dangerously Irrelevant. I have been an educator for the last 12 years. I had the opportunity to step out of the classroom for four years. I just came back this past school year. I was able to work closely with the leadership of my school and directly with the principal. As I reflect on my experience, I consider myself fortunate. I worked with a principal that was able to listen to new and innovative ideas and help in making them happen. This process is not something that happened overnight. It was a professional relationship built on trust. We (leadership of the school and teachers) were there for the same reasons. Our mission (not just the cliche) was to provide the best education possible for our students. Sometimes adult conflicts came up around the implementation of programs or the decisions involved in moving a school forward. Once the dust cleared we were able to ground ourselves on the same basic premise about taking it back to student learning. In regards to technology in schools. I tend to agree that it isn’t about the tools. It is more about the effective use of whatever tools are available to us. We can spin our wheels complaining about what we don’t have or wish we had. At one point I was able to “convince” our school to provide the hardware. We bought over 100 computers to be distributed to 30 classrooms. We bought other “cool” toys also. I quickly learned that the desired effect or use of technology wasn’t going to happen by osmosis. It wasn’t until the tech committee and the leadership team agreed on an implementation plan. This implementation plan was to be long term and focused on professional development not only for teachers but also for the administrators. We spent a week on retreat planning for the initial steps of this program. This was only the beginning. It was not all success. The challenges still persisted as we went back to our school. We spent a lot of energy and time resolving basic issues regarding the check out of hardware. We started to “embed” the use of tech tools into our leadership meetings. We supported the use of web 2.0 tools through workshops for teachers and parents. What effect did this have on our students? That kind of data isn’t easy to quantify. I have now moved on to a different school. Our principal was also transferred. I feel that the effect has been that I took these skills to my classroom. I also know that my principal took this vision on the use of technology to her new school. It really does go back to our kids. Whoever they may be now or in the future. |
Works great with our Wave shield or class D amplifier board. Just be careful not to overdrive it if you're using the class D amp breakouts as it can put 1.7W into the speaker, blowing it out! Start with lower volume and increase carefully. |
---
params: [m]
solution: np.sqrt((m ** 2).sum())
tests:
- [ [50] ]
- [ [10, 20, 30] ]
- [ [[1, 2, 3], [4, 5, 6], [7, 8, 9]] ]
---
Good job! Now let's implement something known as the Frobenius norm:
math`\| m \|_F = \left( \sum_{i,j=1}^n {m_{ij}}^2 \right)^{1/2}`
**Hint:** You can implement this a number of ways, using operators (`**`), functions (`np.sqrt`), and methods (`.sum`).
|
Evaluation of bioactive saponins and triterpenoidal aglycons for their binding properties on human endothelin ETA and angiotensin AT1 receptors.
Different types of triterpenes including saponins and aglycons were evaluated for their ability to inhibit [3H] BQ-123 and [3H] angiotensin II binding to the human endothelin 1 ETA and angiotensin II AT1 receptors, respectively. Selectivity for only one of the two receptors was exhibited by asiatic acid and its saponins (ETA) and oleanolic acid (AT1). To a lesser extent betulinic acid, beta-amyrin and friedelin also showed selectivity for the ETA receptor. To address the question whether the effect of saponins on cell membranes might interfere with the normal binding of specific radioligands to their receptors, the activity of saponins with different haemolytic properties were compared. Highly haemolytic saponins such as alpha-hederin and beta-escine showed partial (60%) inhibition of radioligand-binding to the ETA receptor and complete inhibition (100%) to the AT1 receptor. Moreover, the haemolytically inactive kryptoescine, at the same concentration, caused complete inhibiton of radioligand-binding to both receptors, indicating that inhibition of receptor binding was not related to the membrane-interacting properties of saponins. |
Panthers back from the dead to stun Sea Eagles
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Penrith's pre-finals freefall has been arrested in the most utterly Panthers fashion possible, with four tries in seven minutes completing a record 28-24 comeback victory over Manly on their own turf.
With 13 minutes to play Penrith were staring down the barrel of a fifth loss in six games, and a likely eighth-placed finish by the end of the weekend, having sat atop the ladder just two months at the start of the Origin period.
The Sea Eagles led 24-6 thanks to a second-half Brian Kelly blitzkrieg before the Panthers pyrotechnics started - with no side in rugby league history overcoming an 18-point deficit with so little time remaining on the clock.
Josh Mansour helped himself first after 67 minutes, flying over in the corner from a Waqa Blake offload, before Manly's flimsy right edge defence couldn't stop Isaah Yeo from reaching the line with defenders hanging on.
Still trailing by eight but with the mother of all wet sails, Viliame Kikau busted through down the left edge once more, offloading for Nathan Cleary and then Blake to race away.
When Kikau once more found himself in the thick of it, forcing the ball free from Tom Trbojevic as he returned a kick, Cleary found himself in under the posts and Penrith in for the most remarkable of leads late in the piece.
For well over an hour of the contest, Penrith's 51 missed tackles and 8 errors were deserving of a side battling to avoid the wooden spoon, Manly's own commitment to the cause a stark improvement on last week's thrashing from the Roosters.
Match: Sea Eagles v Panthers
With yet another loss at Lottoland – their sixth straight on home turf now their second-worst in history – the Sea Eagles sit just two points ahead of last-placed Parramatta and somehow have to pick themselves up from the most heartbreaking of defeats.
When Kelly fired into gear after the break, Manly looked home and hosed.
A hard fought 8-6 halftime lead soon blew out as the Ballina product sent Daly Cherry-Evans in under the posts before crossing himself soon after from a Trbojevic short ball.
When he dotted down from another Trbojevic grubber that left new Penrith fullback Tyrone Peachey blushing, Manly led 24-6 and the Panthers obituaries were being finished in the press box.
But with Manly defending in the same manner and this Panthers side more than accustomed to a fast finish, the Sea Eagles faithful that booed their side from Lottoland last week were left devastated once more.
News & notes:Assistant referee Gavin Badger was forced from the field by a calf injury with 10 minutes remaining, handing NSW InTrust Super Premiership whistleblower Todd Smith his first grade debut... Manly's loss at Lottoland was their sixth on the trot – their second worst streak behind the eight straight lost across 2003-04 ... Despite their contrasting seasons, the Sea Eagles have scored 61 tries, just one behind Penrith's 62 this year... Next week Manly travel to the other end of town to face Cronulla while Penrith host the Raiders at the foot of the mountains... Crowd: 6134. |
Genome-wide methylation changes in the brains of suicide completers.
Gene expression changes have been reported in the brains of suicide completers. More recently, differences in promoter DNA methylation between suicide completers and comparison subjects in specific genes have been associated with these changes in gene expression patterns, implicating DNA methylation alterations as a plausible component of the pathophysiology of suicide. The authors used a genome-wide approach to investigate the extent of DNA methylation alterations in the brains of suicide completers. Promoter DNA methylation was profiled using methylated DNA immunoprecipitation (MeDIP) followed by microarray hybridization in hippocampal tissue from 62 men (46 suicide completers and 16 comparison subjects). The correlation between promoter methylation and expression was investigated by comparing the MeDIP data with gene expression profiles generated through mRNA microarray. Methylation differences between groups were validated on neuronal and nonneuronal DNA fractions isolated by fluorescence-assisted cell sorting. The authors identified 366 promoters that were differentially methylated in suicide completers relative to comparison subjects (273 hypermethylated and 93 hypomethylated). Overall, promoter methylation differences were inversely correlated with gene expression differences. Functional annotation analyses revealed an enrichment of differential methylation in the promoters of genes involved, among other functions, in cognitive processes. Validation was performed on the top genes from this category, and these differences were found to occur mainly in the neuronal cell fraction. These results suggest broad reprogramming of promoter DNA methylation patterns in the hippocampus of suicide completers. This may help explain gene expression alterations associated with suicide and possibly behavioral changes increasing suicide risk. |
1973 Queen's Club Championships – Men's Doubles
Jim McManus and Jim Osborne were the defending champions, but did not participate this year.
Tom Okker and Marty Riessen won the men's doubles title at the 1973 Queen's Club Championships tennis tournament, defeating Ray Keldie and Raymond Moore 6–4, 7–5 in the final.
Draw
Finals
Top Half
Bottom Half
References
Draw
Category:1973 Queen's Club Championships |
Microsoft to acquire Nokia for $7.17 billion
Nokia and Microsoft jointly announced that the Redmond company will acquire the Finnish handset maker for EUR 5.44 billion ($7.17 billion USD). As part of the transaction, Nokia will sell all of its Devices & Services business as well as license its patents to Microsoft for EUR 5.44 billion in cash. Microsoft is paying 3.70 billion for the handset division and 1.65 billion for the patent licenses.
The deal is expected to close in Q1 2014 and is subject to both regulatory approval and approval by Nokia’s and Microsoft’s boards. Not surprisingly, Nokia CEO Stephen Elop will rejoin Microsoft. Joining him in the exodus to Redmond will be senior Nokia executives Jo Harlow, Juha Putkiranta, Timo Toikkanen, and Chris Weber.
With its devices and services division gone, Nokia will focus on its three remaining core businesses — NSN network infrastructure and services; HERE mapping and location services; and Advanced Technologies, its technology development and licensing arm. Nokia will continue to operate from its Finland headquarters. |
Readers Set Me Straight: The Love Parade Tragedy
Since I wrote about the stampede at Germany’s Love Parade on Saturday, a clearer picture of the event has emerged. Eyewitnesses, including some readers of this blog, have stated that the deaths were not due to a panicked stampede, but rather to the simple force of human bodies pressing forward into a dead-end space. Writes Keith Martin:
It wasn’t fear. It was necessity. I was in there. It was poor planning and far too many people. We were all stuck in a tunnel… NO WAY OUT. There was a mile long line of people behind us and when the venue filled, they simply closed the gates. We had nowhere to go and people kept pushing. Once exhaustion/dehydration set in people could no longer stand or remain conscious so they would collapse and people would fall on them and a body pile would assemble, with those at the body never getting back up. It wasnt fear… People had no choice but to crush each other.
Reader Mats writes:
I also was there, and have to agree with Keith. There was no panic and no stampede, there was just a slow grind as the enclosed area filled up with more and more people, and the ones in front were told to move back again against the people coming in, and people falling trying to climb out… I was in the crowd well before the big crush happened – I was into the festival area at 15:00 – but even then the crowd was intense and I saw with my own eyes a lifeless body being carried out on a stretcher from the tunnel. Ironically, the first thing I did when getting into the entrance area was what you recommend, taking note of exits and escape routes with the intention of getting out ASAP – only to find there was not a single one. There was really no way out, not from the entrance area, the festival area or from the crowd. Even if the entrance had worked, in my mind there is no question there would be an equal incident on the actual parade grounds – even there every single exit was locked down and not opened before the disaster was a fact.
I was careful to point out in my original post that the psychology of panic is only half the story when it comes to crowd stampedes; once the mass shoving is underway, the question of automatic versus deliberate action becomes irrelevant. In the case of the Duisburg tragedy, it seems that what happened wasn’t really the result of a stampede at all, in the strict sense, but rather a kind of slow-motion build up of pressure onto a crowd with no avenue for escape. At any rate, an investigation into the incident is currently underway, so hopefully in due time fuller answers will emerge.
The Love Parade tragedy is definitely something that could have been avoided only if the organizers were keen on their crowd control plans/ strategies. This event has been going on for several years and so far the event has been running without a hitch.
What went wrong this year? Is it the change of venue? Or was the crowd rowdier than the previous years? At any rate, whatever change implemented, this should not have happened. Memorials are now being offered to those who died. Germany and other countries are in mourning because of lives lost. What’s sad is that these people went to the Love Parade to show their love for music and for life. It is such an unfortunate event!
Thinking About Fear & the Brain
If I find myself in a severe crisis, will I be able to keep it together? How can I control anxiety and panic? Is it possible to lead a life less bounded by fear? These are the sorts of questions that I'll be exploring in this blog, an offshoot of my book, Extreme Fear: The Science of Your Mind in Danger, published on December 8, 2009 by Palgrave Macmillan. |
The Diagnostic Value of Troponin T Level in the Determination of Cardiac Damage in Perinatal Asphyxia Newborns.
Perinatal asphyxia is a clinical condition which results from oxygen deprivation of the fetus or newborn and the breakdown of perfusion in various organs. The aim of this study was to evaluate and compare troponin T levels over time as a marker of cardiac injury in cases of perinatal asphyxia and healthy newborns. The study included a total of 30 newborns diagnosed with perinatal asphyxia with a gestational age of 32-41 weeks, based on the last menstruation date, and 30 healthy newborns with a gestational age of 34-40 weeks, as the control group. Levels of troponin T and creatinin kinase MB were recorded for all participants. No difference was determined between the groups in terms of gestational age, manner of birth, electrocardiographic findings, and PaO2 and PaCO2 values. The umbilical artery pH levels and bicarbonate levels in the study group were found to be statistically lower than those in the control group (p < 0.001). The troponin T and creatinin kinase MB levels in the patients in the study group were higher than those within the control group, at all times. The periods when specificity and sensitivity were highest together for troponin T were the 12th and 24th h. Specificity for troponin T reached the highest value at the 24th h and sensitivity reached the highest value in the cord blood. A positive correlation was found between the troponin T and creatinin kinase MB values at the 6th and 12th h. However, no correlation could be found in the blood between the serum troponin T and creatinin kinase MB levels at the 3rd and 24th h. The troponin T level is a useful test for showing cardiac damage in hypoxic patients in the neonatal period. The sensitivity and specificity of cardiac specific troponin T levels in detecting cardiac damage are much higher according to telecardiography and electrocardiography, while the implementation of the method is simple. |
Q:
Create groups chat react-native using firebase
I'm beginner in react-native and I'm actually developing a real-time chat application in React-Native.I wanted to know if it is possible to create different channels (like discord) using Firebase ?
Thanks !
A:
Of course, you can. The one keep in mind is structuring your firebase database structure.
Structuring your Firebase Data correctly for a Complex App
Structure Your Database
For chatting app, you can reference from chat-sdk-ios
|
Egeler/Shelhamer in Montana
Shelhamer genealogy published....Eastern Pa. lines
This 244 page family history with index has been produced on acid free paper by Masthof publications of Morgantown, Pa. for David Silcox of Shillington, Pa. Originally from Pottsville, Pa., Silcox has been researching the Shellhammer, Shelhamer family for over a 48 year period. His mother was a Shelhamer, the daughter of Charles Albert Shelhamer. His earliest ancestor was Hans Jurg Schellhammer who arrived in Philadelphia in 1753 and eventually settled in what is now West Penn Township. This book contains many photos and copies of early land warrants and surveys of land obtained by the Shellhammer family of West Penn Township and information relating to many of the families that married into the Shellhamer family.
The cost of this publication is $21.99 plus $3.50 for shipping via media mail. Add $1.00 extra for mailing for each additional copy. The book will be available for shipping sometime around October 20, 2011 and can be ordered from David R. Silcox, 404 E. Broad Street, Shillington, Pa. 19607. Call or e-mail with any questions to 610-777-2167 or silx404@aol.com |
Advertima is an artificial intelligence software solution service from Switzerland for real world advertising to upgrade digital signage using Advertima Engine to detect people's age, gender, mood, motion behaviour and even fashion style. Advertima enables content to be catered to specific groups of people such as advertisements, articles, or entertainment for more relevant content. The service can also be used in train stations to change the displays for peak hours and to optimize advertising space through the digital signage market. |
Chronic cocaine treatment induces dysregulation in the circadian pattern of rats' feeding behavior.
The effects of protracted cocaine administration (15 mg/kg i.p., twice a day for 9 days) on the circadian pattern of feeding behavior was studied in individually housed male Sprague-Dawley rats, maintained under a 12:12 light:dark cycle. Water and food were available ad libitum and food intake was measured twice a day before, during and after withdrawal of cocaine (or saline) treatment. Neither total 24-h food intake, nor body mass at the end of the experiment, was significantly different between cocaine-treated and control animals. However, cocaine administration affected the temporal distribution of food consumption. During the dark (activity) phase, rats receiving cocaine injections consumed significantly less food than control animals, and this effect persisted for up to 3 days of cocaine withdrawal. During the light (rest) phase, cocaine administration promoted food consumption and a significantly higher food intake was also observed during the first five cocaine withdrawal days. Continuous monitoring of locomotor activity did not reveal significant changes in the circadian pattern of activity between the two experimental groups during different treatment periods, except for an acute increase in locomotion within an hour after daytime cocaine injection. The results of this study demonstrate that sub-chronic cocaine administration alters the circadian pattern of rats' feeding behavior. |
Algorithm for removing scalp signals from functional near-infrared spectroscopy signals in real time using multidistance optodes.
A real-time algorithm for removing scalp-blood signals from functional near-infrared spectroscopy signals is proposed. Scalp and deep signals have different dependencies on the source-detector distance. These signals were separated using this characteristic. The algorithm was validated through an experiment using a dynamic phantom in which shallow and deep absorptions were independently changed. The algorithm for measurement of oxygenated and deoxygenated hemoglobins using two wavelengths was explicitly obtained. This algorithm is potentially useful for real-time systems, e.g., brain-computer interfaces and neuro-feedback systems. |
Q:
DataTables' Bootstrap Not showing in Laravel
I want to use Datatable's Bootstrap for advance search in laravel.But it's not showing in my required page.
Here is my app.blade.php code.
<!DOCTYPE html>
<html lang="en">
<head>
<meta charset="utf-8">
<meta http-equiv="X-UA-Compatible" content="IE=edge">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Todo App</title>
<link href="{{asset('/css/app.css')}}" rel="stylesheet">
<!-- Fonts -->
<link href='//fonts.googleapis.com/css?family=Roboto:400,300' rel='stylesheet' type='text/css'>
<link rel="stylesheet" href="http://maxcdn.bootstrapcdn.com/bootstrap/3.3.6/css/bootstrap.min.css">
<link rel="stylesheet" href="https://cdn.datatables.net/1.10.11/css/dataTables.bootstrap.min.css">
<link rel="stylesheet" href="//code.jquery.com/ui/1.11.2/themes/smoothness/jquery-ui.css">
<!-- HTML5 shim and Respond.js for IE8 support of HTML5 elements and media queries -->
<!-- WARNING: Respond.js doesn't work if you view the page via file:// -->
<!--[if lt IE 9]>
<script src="https://oss.maxcdn.com/html5shiv/3.7.2/html5shiv.min.js"></script>
<script src="https://oss.maxcdn.com/respond/1.4.2/respond.min.js"></script>
<![endif]-->
</head>
<body>
<nav class="navbar navbar-default">
<div class="container-fluid">
<div class="navbar-header">
<button type="button" class="navbar-toggle collapsed" data-toggle="collapse" data-target="#bs-example-navbar-collapse-1">
<span class="sr-only">Toggle Navigation</span>
<span class="icon-bar"></span>
<span class="icon-bar"></span>
<span class="icon-bar"></span>
</button>
<a class="navbar-brand" href="#">Laravel</a>
</div>
<div class="collapse navbar-collapse" id="bs-example-navbar-collapse-1">
<ul class="nav navbar-nav">
<li><a href="/">Todo Application</a></li>
</ul>
<ul class="nav navbar-nav navbar-right">
@if (Auth::guest())
<li><a href="/auth/login">Login</a></li>
<li><a href="/auth/register">Register</a></li>
@else
<li class="dropdown">
<a href="#" class="dropdown-toggle" data-toggle="dropdown" role="button" aria-expanded="false">{{ Auth::user()->name }} <span class="caret"></span></a>
<ul class="dropdown-menu" role="menu">
<li><a href="/auth/logout">Logout</a></li>
</ul>
</li>
@endif
</ul>
</div>
</div>
</nav>
@if(count($errors)>0)
<div class="alert alert-danger">
<strong>Whoops!</strong> Enter Valid Input</br>
<ul>
@foreach($errors->all() as $error)
<li>{{$error}}</li>
@endforeach
</ul>
</div>
@endif
@if(Session::has('success'))
<div class="alert alert-success">
{{Session::get('success')}}
</div>
@endif
@yield('content')
<!-- Scripts -->
<script src="//cdnjs.cloudflare.com/ajax/libs/jquery/2.1.3/jquery.min.js"></script>
<script src="//cdnjs.cloudflare.com/ajax/libs/twitter-bootstrap/3.3.1/js/bootstrap.min.js"></script>
<script src="code.jquery.com/jquery-1.12.0.min.js"></script>
<script src="https://cdn.datatables.net/1.10.11/js/jquery.dataTables.min.js"></script>
<script src="https://cdn.datatables.net/1.10.11/js/dataTables.bootstrap.min.js</script>
</body>
</html>
And here is my home.blade.php where I want to use Datatables for searching:
@extends('app')
@section('content')
<div class="container">
<div class="row">
<div class="col-md-12 ">
<div class="col-lg-12">
<div class="col-lg-6">
<form action="" method="POST" class="form-inline">
<input type="hidden" name="_token" value="{{ csrf_token() }}">
<div class="form-group">
<input type="text" name="name" class="form-control" placeholder="Enter a task name"></br>
</br>
<input type="submit" class="btn btn-primary" value="Save">
</div>
</form>
</div>
</div>
<h3> Todo Application </h3>
<table class="table table-striped table-bordered" id="example">
<tr>
<td>Serial No</td>
<td>Task Name</td>
<td>Status</td>
<td>Action</td>
</tr>
<?php $i=1; ?>
@foreach($data as $row)
<tr>
<td>{{$i}}</td>
<td>{{$row->name }}</td>
<td>{{$row->status}}</td>
<td>
<a href="{{route('getEditRoute',$row->id)}}" class="btn btn-warning">Edit</a>
<form action="{{route('postDeleteRoute',$row->id)}}" method="POST" style="display:inline;"
onsubmit="if(confirm('Are you sure?')) {return true} else {return false};">
<input type="hidden" name="_token" value="{{ csrf_token() }}">
<input type="submit" class="btn btn-danger" value="Delete">
</form>
</td>
</tr>
<?php $i++; ?>
@endforeach
</table>
<?php echo $data->render(); ?>
</div>
</div>
</div>
<script>
$(document).ready(function() {
$('#example').DataTable();
} );
</script>
<script src="//code.jquery.com/jquery-1.10.2.js"></script>
<script src="//code.jquery.com/ui/1.11.2/jquery-ui.js"></script>
<script>
$(function() {
$( "#datepicker" ).datepicker();
});
</script>
@endsection
A:
Place all our script in master blade which in your case app.blade.php.
Then use @yield('script')
After that add partial view and add your script for initializing datatable
@section('script')
<script>
$(document).ready(function() {
$('#example').DataTable();
} );
</script>
@endsection
and finally don't forget to add thead and tbody in your given table.
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High-power, cascaded random Raman fiber laser with near complete conversion over wide wavelength and power tuning.
Cascaded Raman fiber lasers based on random distributed feedback (RDFB) are proven to be wavelength agile, enabling high powers outside rare-earth doped emission windows. In these systems, by simply adjusting the input pump power and wavelength, high-power lasers can be achieved at any wavelength within the transmission window of optical fibers. However, there are two primary limitations associated with these systems, which in turn limits further power scaling and applicability. Firstly, the degree of wavelength conversion or spectral purity (percentage of output power in the desired wavelength band) that can be achieved is limited. This is attributed to intensity noise transfer of input pump source to Raman Stokes orders, which causes incomplete power transfer reducing the spectral purity. Secondly, the output power range over which the high degree of wavelength conversion is maintained is limited. This is due to unwanted Raman conversion to the next Stokes order with increasing power. Here, we demonstrate a high-power, cascaded Raman fiber laser with near complete wavelength conversion over a wide wavelength and power range. We achieve this by culmination of two recent developments in this field. We utilize our recently proposed filtered feedback mechanism to terminate Raman conversion at arbitrary wavelengths, and we use the recently demonstrated technique (by J Dong and associates) of low-intensity noise pump sources (Fiber ASE sources) to achieve high-purity Raman conversion. Pump-limited output powers >34W and wavelength conversions >97% (highest till date) were achieved over a broad - 1.1μm to 1.5μm tuning range. In addition, high spectral purity (>90%) was maintained over a broad output power range (>15%), indicating the robustness of this laser against input power variations. |