text stringlengths 14 5.77M | meta dict | __index_level_0__ int64 0 9.97k ⌀ |
|---|---|---|
NCAA Tournament: UAlbany vs. North Carolina
Sports // UAlbany
March 26, 2013 Updated: March 26, 2013 1:58 p.m.
Patrick SemanskyAlbany forward Shereesha Richards, bottom, loses possession of the ball as she is pressured by North Carolina forward Xylina McDaniel, top left, and guard Tierra Ruffin-Pratt during the first half of a first-round game in the women's NCAA college basketball tournament in Newark, Del., Sunday, March 24, 2013. (AP Photo/Patrick Semansky)
Patrick SemanskyAlbany head coach Katie Abrahamson-Henderson directs her players during the first half of a first-round game against North Carolina in the women's NCAA college basketball tournament in Newark, Del., Sunday, March 24, 2013. (AP Photo/Patrick Semansky)
Patrick SemanskyNorth Carolina guard N'Dea Bryant (22) reacts after committing a backcourt violation during the first half of a first-round game against Albany in the women's NCAA college basketball tournament in Newark, Del., Sunday, March 24, 2013. (AP Photo/Patrick Semansky)
Patrick SemanskyAlbany forward Shereesha Richards, second from left, tries to get a loose ball as she is guarded by North Carolina guards, from left to right, Danielle Butts, N'Dea Bryant, Erika Johnson and guard/forward Krista Gross during the first half of a first-round game in the women's NCAA college basketball tournament in Newark, Del., Sunday, March 24, 2013. (AP Photo/Patrick Semansky)
Patrick SemanskyNorth Carolina guard Danielle Butts, left, grabs a rebound over Albany forward Shereesha Richards during the first half of a first-round game in the women's NCAA college basketball tournament in Newark, Del., Sunday, March 24, 2013. (AP Photo/Patrick Semansky) | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 1,923 |
Q: Django Dynamically add choices to choicefield form I would like to dynamically add choices to the form from data pulled in views.py.
this list will change frequently so I can't hard code this.
I'm running a query against the AWS API, and pulling down backups. I am trying to get these backups in a list then get these in a dropdown to submit in a form.
forms.py
class RestoreAMIForm(forms.ModelForm):
ami = forms.ChoiceField()
def __init__(self, *args, **kwargs):
super(RestoreAMIForm, self).__init__(*args, **kwargs)
self.fields['server_name'].widget.attrs['readonly'] = True
self.fields['module_name'].widget.attrs['readonly'] = True
self.fields['instance_id'].widget.attrs['readonly'] = True
class Meta:
model = RestoreRequest
fields = ['server_name', 'module_name', 'instance_id', 'ami', 'justification']
views.py
amis = helper_functions.get_all_backups(instance_id, GlobalrunEnv)
context['ami'] = amis
just_amis = [i[0] for i in context['ami']]
formatted_picker = helper_functions.iterate_over_list(just_amis)
context['restore_ami_form'] = RestoreAMIForm(initial={'server_name': context['tags']['Name'],
'instance_id': context['server_details']['InstanceId'],
'module_name': context['passed_slug'],
'ami': formatted_picker, })
html
<form action="{% url 'instance_details' passed_slug %}" method="post">
<p class="standout"> Revert to Snapshot:</p>
<table>
{{ restore_ami_form.as_table }}
</table>
<div class="buttonHolder">
<button name="RevertRequest" value="{{ GlobalrunEnv }}">Submit New Values</button>
</div>
</form>
the output of format_picker is...
[('1', 'ami-04e05d8b305348d89'), ('2', 'ami-0f82b7ac27bdeb246'), ('3', 'ami-0eed0d484f0d61391'), ('4', 'ami-071cfbc9eda4cae4d'), ('5', 'ami-0dc61e733721e7e7a'), ('6', 'ami-05ba2995c14da2502'), ('7', 'ami-01cb5e766d77f8ecb')]
my read only fields are working. Is is possible to provide initial= values for choice fields?
A: Anyone curious I ended up finding a way to accomplish this. I modified the init of the form to accept "internal choices". Then when creating the form on views.py I pass this this before the initial kwargs.
This did cause a slight issue later since i'm unable to directly reference the values for the options, but I got around this by using...
request.session['archived_amis'] = formatted_picker
then in the post doing ami_choices = request.session.get('archived_amis', None)
changes below
views.py
formatted_picker = helper_functions.iterate_over_list(just_amis)
"""This data needs to be saved to reference later in the post, in the off chance that the data changes between
initially loading the variable and submitting the form"""
request.session['archived_amis'] = formatted_picker
context['restore_ami_form'] = RestoreAMIForm(formatted_picker, initial={'server_name': context['tags']['Name'],
'instance_id': context['server_details']['InstanceId'],
'module_name': context['passed_slug'], })
forms.py
class RestoreAMIForm(forms.ModelForm):
# ami = forms.ChoiceField()
def __init__(self, internal_choices, *args, **kwargs):
super(RestoreAMIForm, self).__init__(*args, **kwargs)
self.fields['server_name'].widget.attrs['readonly'] = True
self.fields['module_name'].widget.attrs['readonly'] = True
self.fields['instance_id'].widget.attrs['readonly'] = True
self.fields['ami'] = forms.ChoiceField(label='AMI', required=True, choices=internal_choices)
class Meta:
model = RestoreRequest
fields = ['server_name', 'module_name', 'instance_id', 'ami', 'justification']
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 8,001 |
Q: Difference between single quote and double single quotes in sql I have two queries, first one is the regular SELECT and the other one is an EXEC query as what I mentioned below:
*
*SELECT * FROM [table] WHERE [column] = 'test'
and
*EXEC(' SELECT * FROM [table] WHERE [column] = ''test'' ')
as for the results, I tried to execute both queries in sql server 2008 and i was given the exact same outputs, if both queries return the same output then what's the purpose of the single quote and double single quotes in sql?
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 8,275 |
April Anderson Posts — RUNGRL Co.
April Anderson works as a Speech-Language Pathologist at a private practice in Washington, D.C. She specializes in pediatric feeding therapy and is a Certified Lactation Counselor. She loves brunch, reading and cheering for LSU football. April enjoys staying active through running and social sports such as volleyball and softball. She is currently training for the 2019 Cherry Blossom 10 miler and the Brooklyn Half Marathon. | {
"redpajama_set_name": "RedPajamaC4"
} | 1,897 |
O estreito de Juan de Fuca é a principal saída para o estreito de Geórgia e o Puget Sound, conectando ambos com o Oceano Pacífico.
Recebeu o nome em 1788 de Juan de Fuca pelo capitão inglês John Meares do navio Felice. Juan de Fuca era um marinheiro grego que alegava haver ido em uma viagem com exploradores espanhóis em 1592 para procurar o lendário estreito de Anián. Forma atualmente parte da Fronteira Canadá-Estados Unidos.
Juan de Fuca
Juan de Fuca
Juan de Fuca
Águas disputadas
Fronteira Canadá-Estados Unidos | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 8,638 |
#include "e_mod_main.h"
static const char E_FILEMAN_BUS_NAME[] = "org.enlightenment.FileManager";
static const char E_FILEMAN_INTERFACE[] = "org.enlightenment.FileManager";
static const char E_FILEMAN_ERROR[] = "org.enlightenment.FileManager.Error";
static const char E_FILEMAN_PATH[] = "/org/enlightenment/FileManager";
typedef struct _E_Fileman_DBus_Daemon E_Fileman_DBus_Daemon;
struct _E_Fileman_DBus_Daemon
{
Eldbus_Connection *conn;
Eldbus_Service_Interface *iface;
};
static Eldbus_Message * _e_fileman_dbus_daemon_open_directory_cb(const Eldbus_Service_Interface *iface, const Eldbus_Message *msg);
static Eldbus_Message *_e_fileman_dbus_daemon_open_file_cb(const Eldbus_Service_Interface *iface, const Eldbus_Message *msg);
static Eldbus_Message *
_e_fileman_dbus_daemon_error(const Eldbus_Message *msg,
const char *error_msg)
{
return eldbus_message_error_new(msg, E_FILEMAN_ERROR, error_msg);
}
static const Eldbus_Method methods[] = {
{ "OpenDirectory", ELDBUS_ARGS({"s", "directory"}), NULL,
_e_fileman_dbus_daemon_open_directory_cb, 0 },
{ "OpenFile", ELDBUS_ARGS({"s", "file"}), NULL,
_e_fileman_dbus_daemon_open_file_cb, 0 },
{ NULL, NULL, NULL, NULL, 0 }
};
static const Eldbus_Service_Interface_Desc desc = {
E_FILEMAN_INTERFACE, methods, NULL, NULL, NULL, NULL
};
static void
_e_fileman_dbus_daemon_object_init(E_Fileman_DBus_Daemon *d)
{
d->iface = eldbus_service_interface_register(d->conn, E_FILEMAN_PATH, &desc);
if (!d->iface)
{
fprintf(stderr, "ERROR: cannot add object to %s\n", E_FILEMAN_PATH);
return;
}
}
static void
_e_fileman_dbus_daemon_free(E_Fileman_DBus_Daemon *d)
{
if (d->iface)
eldbus_service_object_unregister(d->iface);
if (d->conn)
eldbus_connection_unref(d->conn);
free(d);
}
static Eldbus_Message *
_e_fileman_dbus_daemon_open_directory_cb(const Eldbus_Service_Interface *iface EINA_UNUSED,
const Eldbus_Message *msg)
{
const char *directory = NULL, *p;
char *dev, *to_free = NULL;
E_Zone *zone;
if (!eldbus_message_arguments_get(msg, "s", &directory))
{
fprintf(stderr, "Error: getting arguments of OpenDirectory call.\n");
return eldbus_message_method_return_new(msg);
}
if ((!directory) || (directory[0] == '\0'))
return _e_fileman_dbus_daemon_error(msg, "no directory provided.");
zone = e_zone_current_get();
if (!zone)
return _e_fileman_dbus_daemon_error(msg, "could not find a zone.");
if (strstr(directory, "://"))
{
Efreet_Uri *uri = efreet_uri_decode(directory);
directory = NULL;
if (uri)
{
if ((uri->protocol) && (strcmp(uri->protocol, "file") == 0))
directory = to_free = strdup(uri->path);
efreet_uri_free(uri);
}
if (!directory)
return _e_fileman_dbus_daemon_error(msg, "unsupported protocol");
}
p = strchr(directory, '/');
if (p)
{
int len = p - directory + 1;
dev = malloc(len + 1);
if (!dev)
{
free(to_free);
return _e_fileman_dbus_daemon_error(msg,
"could not allocate memory.");
}
memcpy(dev, directory, len);
dev[len] = '\0';
if ((dev[0] != '/') && (dev[0] != '~'))
dev[len - 1] = '\0'; /* remove trailing '/' */
directory += p - directory;
}
else
{
dev = strdup(directory);
directory = "/";
}
e_fwin_new(dev, directory);
free(dev);
free(to_free);
return eldbus_message_method_return_new(msg);
}
static Eina_Bool
_mime_shell_script_check(const char *mime)
{
static const struct sh_script_map {
const char *str;
size_t len;
} options[] = {
#define O(x) {x, sizeof(x) - 1}
O("application/x-sh"),
O("application/x-shellscript"),
O("text/x-sh"),
#undef O
{NULL, 0}
};
const struct sh_script_map *itr;
size_t mimelen = strlen(mime);
for (itr = options; itr->str != NULL; itr++)
if ((mimelen == itr->len) && (memcmp(mime, itr->str, mimelen) == 0))
return EINA_TRUE;
return EINA_FALSE;
}
static Eldbus_Message*
_e_fileman_dbus_daemon_open_file_cb(const Eldbus_Service_Interface *iface EINA_UNUSED,
const Eldbus_Message *msg)
{
Eina_List *handlers;
const char *param_file = NULL, *mime, *errmsg = "unknow error";
char *real_file, *to_free = NULL;
E_Zone *zone;
if (!eldbus_message_arguments_get(msg, "s", ¶m_file))
{
fprintf(stderr, "ERROR: getting arguments of OpenFile call.\n");
return eldbus_message_method_return_new(msg);
}
if ((!param_file) || (param_file[0] == '\0'))
return _e_fileman_dbus_daemon_error(msg, "no file provided.");
zone = e_zone_current_get();
if (!zone)
return _e_fileman_dbus_daemon_error(msg, "could not find a zone.");
if (!strstr(param_file, "://"))
{
real_file = ecore_file_realpath(param_file);
if (!real_file)
{
errmsg = "couldn't get realpath for file.";
goto error;
}
}
else
{
Efreet_Uri *uri = efreet_uri_decode(param_file);
real_file = NULL;
if (uri)
{
if ((uri->protocol) && (strcmp(uri->protocol, "file") == 0))
{
real_file = ecore_file_realpath(uri->path);
param_file = to_free = strdup(uri->path);
}
efreet_uri_free(uri);
}
if (!real_file)
{
errmsg = "unsupported protocol";
goto error;
}
}
mime = efreet_mime_type_get(real_file);
if (!mime)
{
errmsg = "couldn't find mime-type";
goto error;
}
if (strcmp(mime, "application/x-desktop") == 0)
{
Efreet_Desktop *desktop = efreet_desktop_new(real_file);
if (!desktop)
{
errmsg = "couldn't open desktop file";
goto error;
}
e_exec(zone, desktop, NULL, NULL, NULL);
efreet_desktop_free(desktop);
goto end;
}
else if ((strcmp(mime, "application/x-executable") == 0) ||
ecore_file_can_exec(param_file))
{
e_exec(zone, NULL, param_file, NULL, NULL);
goto end;
}
else if (_mime_shell_script_check(mime))
{
Eina_Strbuf *b = eina_strbuf_new();
const char *shell = getenv("SHELL");
if (!shell)
{
uid_t uid = getuid();
struct passwd *pw = getpwuid(uid);
if (pw) shell = pw->pw_shell;
}
if (!shell) shell = "/bin/sh";
eina_strbuf_append_printf(b, "%s %s %s",
e_config->exebuf_term_cmd,
shell,
param_file);
e_exec(zone, NULL, eina_strbuf_string_get(b), NULL, NULL);
eina_strbuf_free(b);
goto end;
}
handlers = efreet_util_desktop_mime_list(mime);
if (!handlers)
{
errmsg = "no handlers for given file";
goto end;
}
else
{
Efreet_Desktop *desktop = handlers->data;
Eina_List *files = eina_list_append(NULL, param_file);
e_exec(zone, desktop, NULL, files, NULL);
eina_list_free(files);
EINA_LIST_FREE(handlers, desktop)
efreet_desktop_free(desktop);
}
end:
free(real_file);
free(to_free);
return eldbus_message_method_return_new(msg);
error:
free(real_file);
free(to_free);
return _e_fileman_dbus_daemon_error(msg, errmsg);
}
static E_Fileman_DBus_Daemon *
_e_fileman_dbus_daemon_new(void)
{
E_Fileman_DBus_Daemon *d;
d = calloc(1, sizeof(E_Fileman_DBus_Daemon));
if (!d)
{
perror("ERROR: FILEMAN: cannot allocate fileman dbus daemon memory.");
return NULL;
}
d->conn = eldbus_connection_get(ELDBUS_CONNECTION_TYPE_SESSION);
if (!d->conn)
goto error;
_e_fileman_dbus_daemon_object_init(d);
eldbus_name_request(d->conn, E_FILEMAN_BUS_NAME,
ELDBUS_NAME_REQUEST_FLAG_REPLACE_EXISTING, NULL, NULL);
return d;
error:
fprintf(stderr, "ERROR: FILEMAN: failed to create daemon at %p\n", d);
_e_fileman_dbus_daemon_free(d);
return NULL;
}
static E_Fileman_DBus_Daemon *_daemon = NULL;
void
e_fileman_dbus_init(void)
{
if (_daemon)
return;
eldbus_init();
_daemon = _e_fileman_dbus_daemon_new();
}
void
e_fileman_dbus_shutdown(void)
{
if (!_daemon)
return;
_e_fileman_dbus_daemon_free(_daemon);
_daemon = NULL;
eldbus_shutdown();
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 7,219 |
Adjustments to retirement often bring about the awareness of depression hiding in your retirement living.
A number of people who think they are overworked and underpaid have lived with depression so long they think it is normal. This link is to Ted's Talks about depression and the stigma. Don't let the stigma rule your retirement.
Here is some advice for your retirement time.
Some retirees think that is how they are supposed to feel. Realizing that this heaviness can be depression is often a shock to anyone.
Stress can be a doorway for depression.
Stress causes your bodies systems and mind to make adjustments to how you see things. Often things that are not important can double in importance when stress colors how you think and reason. Then you will make some change to cope with the events that you often find occurring in your daily lives.
Stress often hides or disguises itself as deadlines, performance levels, all sorts of different methods push you during your working years.
Again remember that when we are accustomed to these small pressures from work days. Being already in the habit of feeling pressured, when you retire, you unconsciously search for triggers to make your stress levels feel normal. You live your life on automatic.
You may not have reasons to feel pressured or pushed, but that stress can be reproduced to give you that "comfortable" feeling of stress. Yes, your body is an adaptable vehicle. If you are missing stress, your mind will try to produce it for you.
However, your body can't tell the difference in the actual importance of what generates the stress, the push. After all, stress originates in your mind and pushes you on. Sure these results enable you to accomplish whatever gets your paycheck regularly.
Are you comfortable with stress?
While you know you are fortunate to be healthy. You don't have to work in retirement.
What will you do to make sure that your retirement is the great time in your life that you waited for so long?
Are you sure you do not set standards and requirements for yourself that result in stress after retirement? You can rest assured that many retirees do.
An innocent way this occurs is home improvements. Now that you are home all day, you see the imperfections of your living space. Maybe it is the landscaping.
Now, relaxing and enjoying this time of less responsibility is no longer a priority for you. Making corrections for all these errors that were not there before retirement becomes a compulsion.
For some, the drive is to travel. To stay busy improving the golf swing. The little ideas that make everyone tick start to turn destructive. These things that are supposed to be fun become the yardstick you tend to measure your days and weeks.
At the end of the day, the importance of spending time sitting on the porch visiting with a neighbor, having a cup of coffee with your spouse. Oh no, the to-do list is not complete. Even if you had a day that allowed a number of good things to happen, you do not allow yourself to enjoy it.
Yes, staying busy is desirable and the better choice for your days. However to expect yourself to immediately become an expert home remodeler? That is sitting the bar pretty high right at the starting gate of your retirement.
You are learning to not be at work. Learning to spend more time with your spouse. Adjusting to no longer having a paycheck. The awareness that your life is indeed different now with a different focus.
Becoming comfortable with the progress of things in a more relaxed mindset. Yes, the adjustments continue. The challenges are there. The stress has many places to hide.
Depression is such a silent thing. This conditions can creep quietly in and be full blown without your awareness.
You do not normally find the diagnosis of "depression rash" when going in to check out the reasons for the rash where ever it occurs.
The same is true of depression. It can be difficult for diagnosing and treating. You can have a condition of mild depression masquerading as something else and not even realize it.
So why are we concerned with depression being sneaky?
Do you feel the vultures are there for you?
Or maybe you do not even know it has developed? Mild depression is the first step to major depression, and that is something hard to treat and will make your life miserable.
Living your retirement years with full-blown depression is a condition that can bring you to your knees. Knowing that the last years of your life will be spent in a deeply depressed condition, with no joy or enjoyment of life. Is this your path?
As you are becoming aware of the fact that you will probably live longer, and be healthier you should be making sure that your mental health conditions are also improved.
It is not a sign of weakness to seek help with mental conditions. You can find help.
As a retiring senior you will need to help your primary care physician help you. At some level, you must start the ball rolling to find help. Help yourself by speaking up.
Your personal awareness will help you be aware of some of the signs of depression. Your primary care "well check' is a good place to start.
There are signs of depression. Make yourself aware of these symptoms. Make yourself aware of how these indicate that all is not well. Like any other condition, when detected early, this is easier to treat.
Help yourself by realizing that depression is not to be expected in retirement. Depression is not a normal step in aging. This is brought about by your mental condition. How you are thinking and reacting to the changes in your life? Learn better ways to cope and adjust to this important time in your life.
You are sitting the stage for the rest of your life. Getting started into retirement with a healthy life plan for your retirement years is one of the building blocks for a happy rest of your life. Anchor this down for your retirement life.
There are also treatments and therapies that work well for retirees when the condition is not allowed to develop all the way to maturity.
When not treated in the mild stages, the conditions can develop to the homeless, helpless victims you see on street corners. Not to classify homeless people with only depression, there are many things that produce this degree of mental state.
However, untreated depression does rob its victims of a well deserved and earned retirement life.
Also on the list of symptoms to watch is a sudden loss of appetite. Not able to sleep, and sleeping too much are both symptoms of depression.
The loss of joy in activities that were favorites is another warning sign. Drs. will usually watch for people who seem to experience more pain than the condition that is apparent produces. This is often another symptom. Trouble concentrating is a symptom that needs to be considered. As well, there is lack of interest in being around other people.
Trouble concentrating is a symptom that needs to be considered. As well, there is lack of interest in being around other people. Just watch how these things go in your daily living.
Ok, you may be more of an introvert person who enjoys your quiet time and having time to yourself. Just think about it. Are these symptoms of depression or just having the chance to regulate your own interaction with others? In a healthy amount, one of the good things of retirement.
Be honest with your Physician. Tell them what is going on. Most times the Dr. will ask you why you came in today. If the feeling depressed is not the main thing that brought you in you will probably tell the Dr. about what did.
Then as the Dr. signs off the computer after completing the exam and coming up with a treatment regimen for the other ailment it can be a bit late. The Dr.s are busy and as they are closing your records, they are already moving to the next patient. To jerk their attention back is often not as successful as you had hoped.
To get his full attention to your concerns about your mental health, explain earlier in the visit. Maybe you could ask at the time the appointment is scheduled for a bit of extra time. This is an entirely reasonable request. Explain thoroughly why you are concerned and what has changed.
Do not be reluctant to state your concerns. Give a full account of why you are concerned. Help your Dr. help you.
Ask if there are lifestyle changes you can make? Explain if you have seen this in your parents or siblings when they retired. State your feelings about relying solely on medications for correcting this condition.
What are his suggestions? How can you apply them in your life?
The one thing that you have hopefully realized with this information is that it is a treatable condition. Not a mental weakness.
Only the most mature and well balanced of us can openly consider depression. This is a shadowland condition that has long cost many senior citizens the golden years they have worked for and earned. Don't let this happen to you.
Life will throw many situations your way. Do not try to manage one of these situations with an already less than 100% state if mental health. The loss of a spouse. A major illness. An unexpected death of a child or grandchild. Not to be a doomsday cryer, but real life happens, and especially in your retirement years. Do what you can now to prepare yourself for these times.
Give your Dr. a chance to help you. Tell him what is going on for you. Awareness is a great prevention you can do for yourself.
Stay aware of your mental as well as your physical health. Love your life. Be happy living it. Don't allow A silent depression in your retirement living to make your retirement less than it can be. | {
"redpajama_set_name": "RedPajamaC4"
} | 1,147 |
\section{INTRODUCTION}
\footnotetext[1]{Corresponding author. E-mail maelena@math.ucalgary.ca, phone (403)-220-3956, fax
(403)-282-5150}
The paper deals with the difference equation with several variable retarded
arguments of the form%
\begin{equation}
\Delta x(n)+\sum_{i=1}^{m}p_{i}(n)x(\tau _{i}(n))=0\text{, \ \ \ }n\in
\mathbb{N}
_{0}\text{,} \tag{E$_{\text{R}}$}
\end{equation}%
where $\mathbb{N}_{0}$ is the set of nonnegative integers, and the (dual)
difference equation with several variable advanced arguments of the form%
\begin{equation}
\nabla x(n)-\sum_{i=1}^{m}p_{i}(n)x(\sigma _{i}(n))=0\text{, \ \ \ }n\in
\mathbb{N}
\text{.} \tag{E$_{\text{A}}$}
\end{equation}%
Equations (E$_{\text{R}}$) and (E$_{\text{A}}$) are studied under the
following assumptions: everywhere $(p_{i}(n))$, $1\leq i\leq m$, are
sequences of {\em nonnegative real numbers}, $(\tau _{i}(n))$, $1\leq i\leq m$,
are sequences of integers such that either
\begin{equation}
\tau _{i}(n)\leq n-1,\text{ \ \ }\forall n\in
\mathbb{N}
_{0}\text{, \ \ \ and \ \ \ }\lim\limits_{n\rightarrow \infty }\tau
_{i}(n)=\infty \text{, \ \ \ }1\leq i\leq m\text{,} \tag{1.1}
\end{equation}%
or
\begin{equation}
\tau _{i}(n)\leq n-1,~~\forall n\in
\mathbb{N}
_{0}\text{~~ and ~~}\exists \text{ }M_{i}>0\text{ such that\ }n-\tau
_{i}(n)\leq M_{i}\text{,~~ }1\leq i\leq m \tag{1.1$^{\prime }$}
\end{equation}%
and $(\sigma _{i}(n))$, $1\leq i\leq m$, are sequences of integers such that
either
\begin{equation}
\sigma _{i}(n)\geq n+1,\text{ \ \ }\forall n\in
\mathbb{N}
\text{, \ \ \ }1\leq i\leq m\text{,} \tag{1.2}
\end{equation}%
or
\begin{equation}
\sigma _{i}(n)\geq n+1,\forall n\in
\mathbb{N}
\text{ and }\exists \text{ }\mu _{i}>0\text{ such that\ }\sigma
_{i}(n)-n\leq \mu _{i}\text{, }1\leq i\leq m\text{.} \tag{1.2$^{\prime }$}
\end{equation}%
Here, $\Delta $ denotes the forward difference operator $\Delta
x(n)=x(n+1)-x(n)$ and $\nabla $ corresponds to the backward difference
operator $\nabla x(n)=x(n)-x(n-1)$.
Let us note that the condition $\tau _{i}(n)\leq n-1$ ($\sigma _{i}(n)\geq
n+1$) is not essential for $\limsup $-type tests; all the results of Section
2 apply to the case when $\tau _{i}(n)\leq n$ and there is a non-delay term $%
\tau _{j}(n)=n$. However, the requirement $\lim_{n\rightarrow \infty }\tau
_{i}(n)=\infty $ is significant, as Example 2.2 illustrates. A similar
remark applies to advanced difference equation (E$_{\text{A}}$).
In the special case $m=1$ equations (E$_{\text{R}}$) and (E$_{\text{A}}$)
reduce to the equations%
\begin{equation}
\Delta x(n)+p(n)x(\tau (n))=0\text{, \ \ \ }n\in
\mathbb{N}
_{0} \tag{1.3}
\end{equation}%
and%
\begin{equation}
\nabla x(n)-p(n)x(\sigma (n))=0\text{, \ \ \ }n\in
\mathbb{N}
\text{,} \tag{1.4}
\end{equation}%
respectively.
Set
\begin{equation*}
w=-\min_{\substack{ n\geq 0 \\ 1\leq i\leq m}}\tau _{i}(n)\text{.}
\end{equation*}%
Clearly, $w$ is a finite positive integer if (1.1) holds.
By a \textit{solution} of (E$_{\text{R}}$), we mean a sequence of real
numbers $(x(n))_{n\geq -w}$ which satisfies (E$_{\text{R}}$) for all $n\geq
0.$ It is clear that, for each choice of real numbers $c_{-w},$ $%
c_{-w+1},...,$ $c_{-1},$ $c_{0}$, there exists a unique solution $%
(x(n))_{n\geq -w}$ of (E$_{\text{R}}$) which satisfies the initial
conditions $x(-w)=c_{-w},$ $x(-w+1)=c_{-w+1},...,$ $x(-1)=c_{-1},$ $%
x(0)=c_{0}$.
By a solution of (E$_{\text{A}}$), we mean a sequence of real numbers $%
\left( x(n)\right) _{n\geq 0}$ which satisfies (E$_{\text{A}}$) for all $%
n\geq 1$.
A solution $(x(n))_{n\geq -w}$ (or $\left( x(n)\right) _{n\geq 0}$) of (E$_{%
\text{R}}$) (or (E$_{\text{A}}$)) is called\textit{\ oscillatory, }if the
terms $x(n)$ of the sequence are neither eventually positive nor eventually
negative. Otherwise, the solution is said to be \textit{nonoscillatory}. An
equation is \textit{oscillatory} if all its solutions oscillate.
In the last few decades, the oscillatory behavior and the existence of
positive solutions of difference equations with several deviating arguments
have been extensively studied, see, for example, papers [1-5,7-15] and
references cited therein. More precicely, in 1999, Zhang et al. [14] studied
the existence and nonexistence of positive solutions and in 1999 and 2001,
Tang et al. [10,11] considered (E$_{\text{R}}$) with constant delays,
introduced some new techniques to analyze generalized characteristic
equations and obtained several sharp oscillation conditions including
infinite sums. In 2002, Zhang et al. [15] presented some oscillation
criteria involving $\lim \sup $ for (E$_{\text{R}}$) with constant delays,
while Li et al. [7] investigated only (E$_{\text{A}}$) under the assumption
that the advances are constant and established a new oscillation condition
involving $\liminf $. In 2003, Wang [12] considered (E$_{\text{R}}$) with
constant delays and obtained some new oscillation criteria involving $%
\limsup $, while Luo et al. [8] studied a nonlinear difference equation with
several constant delays and established some sufficient oscillation
conditions. In 2005, Yan et al. [13] got oscillation criteria for (E$_{\text{%
R}}$) with variable delays. In 2006, Berezansky et al. [1] mostly
investigated non-oscillation but also obtained the following sufficient
oscillation test (Theorem 5.16): If
\begin{equation*}
\limsup_{n\rightarrow \infty }\sum_{i=1}^{m}p_{i}(n)>0\mbox{~~and~~}%
\liminf_{n\rightarrow \infty }\sum_{i=1}^{m}\sum_{j=\tau (n)}^{n-1}p_{i}(j)>%
\frac{1}{e}\text{,}
\end{equation*}%
where $\tau (n)=\max_{1\leq i\leq m}\tau _{i}(n)$, $\forall n\geq 0$, then
all solutions of (E$_{\text{R}}$) oscillate. In the recent papers [3-5],
Chatzarakis et al. studied equations (E$_{\text{R}}$) and (E$_{\text{A}}$)
and presented some oscillation conditions involving $\limsup $ and $\liminf $.
However, most oscillation results presented in the previous papers, require
that the arguments be monotone increasing. While this condition is satisfied
by a variety of differential equations with variable delays, for difference
equations, due to the discrete nature of the arguments, if retarded
arguments are strictly increasing, then the deviations are eventually
constant. This is one of motivations to investigate difference equations
with non-monotone arguments. The challenge of this study is illustrated by
the fact that, according to [2, Theorem 3], there is no constant $A>0$ such
that the inequalities
\begin{equation*}
\limsup_{n\rightarrow \infty }(n-\tau (n))p(n)>A\mbox{~~and~~}%
\liminf_{n\rightarrow \infty }\sum_{j=\tau (n)}^{n-1}p_{i}(j)>A
\end{equation*}%
guarantee oscillation of (1.3).
The paper is organized as follows. Section 2 contains oscillation conditions
for both retarded and advanced equations in terms of $\limsup $, where
deviations of the argument, generally, are not assumed to be bounded. In
Section 3 conditions in terms of $\liminf $ are obtained in the case when
deviations of the argument are bounded; however, a weaker result is valid
for unbounded delays. Section 4 illustrates the results of the present paper
with examples where oscillation could not have been established using
previously known results.
\section{UNBOUNDED DEVIATIONS OF ARGUMENTS AND ITERATIVE TESTS}
In 2011, Braverman and Karpuz [2] and in 2014, Stavroulakis [10],
established the following results for the special case of \ Eq. (1.3) under
the assumption that the argument $\tau (n)$ is non-monotone, (1.1) holds and
$\xi (n)=\max_{0\leq s\leq n}\tau (s).$
\vskip0.2cm\textbf{Theorem 2.1 }([2, Theorem 5])\textbf{.} \ \textit{If}%
\begin{equation}
\mathit{\ }\limsup_{n\rightarrow \infty }\sum_{j=\xi (n)}^{n}p(j)%
\mathop{\displaystyle \prod }\limits_{i=\tau (j)}^{\xi (n)-1}\frac{1}{1-p(i)}%
>1, \tag{2.1}
\end{equation}%
\textit{then all solutions of} (1.3) \textit{oscillate.}
\vskip0.4cm\textbf{Theorem 2.2 (}[9, Theorem 3.6]\textbf{). \ }\textit{%
Assume that} $\displaystyle a\boldsymbol{=}\liminf_{n\rightarrow \infty
}\sum_{i=\tau (n)}^{n-1}p(i)$ ~ and%
\begin{equation}
\limsup_{n\rightarrow \infty }\sum_{j=\xi (n)}^{n}p\left( j\right) %
\mathop{\displaystyle \prod }\limits_{i=\tau (j)}^{\xi (n)-1}\frac{1}{1-p(i)}%
>1-c(a), \tag{2.2}
\end{equation}%
\textit{where}%
\begin{equation*}
c(a)=\left\{
\begin{array}{ll}
\frac{1}{2}\left( 1-a-\sqrt{1-2a-a^{2}}\right) , & \text{if~~~}0<a\leq 1/e,
\\
\frac{1}{4}\left( 2-3a-\sqrt{4-12a+a^{2}}\right) , & \text{if~~~}0<a\leq 6-4%
\sqrt{2}\text{~~and~~}p(n)\geq \frac{\alpha }{2}.%
\end{array}%
\right.
\end{equation*}%
\textit{Then all solutions of equation }(1.3)\textit{\ oscillate.}
\vskip0.2cmIn this section, we study oscillation of (E$_{\text{R}}$) and (E$%
_{\text{A}}$). We establish new sufficient oscillation conditions involving $%
\lim \sup $ under the assumption that the arguments are non-monotone and
(1.1) holds. Even for equation (1.3) with one delay, our results improve
(2.1) and (2.2). The method we use is based on the iterative application of
the Gronwall inequality.
\subsection{Retarded difference equations}
Let%
\begin{equation}
\varphi _{i}(n)=\max_{0\leq s\leq n}\tau _{i}(s),\text{ \ \ }n\geq 0
\tag{2.3}
\end{equation}%
and%
\begin{equation}
\varphi (n)=\max_{1\leq i\leq m}\varphi _{i}(n),\text{ \ \ }n\geq 0\text{.}
\tag{2.4}
\end{equation}%
Clearly, the sequences of integers $\varphi (n)$, $\varphi _{i}(n)$, $1\leq
i\leq m$ are non-decreasing and $\varphi (n)\leq n-1$, $\varphi _{i}(n)\leq
n-1$, $1\leq i\leq m$ for all $n\geq 0$.
The following simple result is cited to explain why we can consider only the
case
\begin{equation}
\sum_{i=1}^{m}p_{i}(n)<1,\text{\ \ \ } ~~~~~\forall n\geq 0\text{.}
\tag{2.5}
\end{equation}
\vskip0.2cm\textbf{Theorem 2.3. } \textit{Assume that there exists a
subsequence }$\theta (n)$, $n\in
\mathbb{N}
$ \textit{of positive integers} \textit{such that}%
\begin{equation}
\sum_{i=1}^{m}p_{i}(\theta (n))\geq 1,\text{\ \ \ }~~~~~\forall n\in {%
\mathbb{N}}\text{.} \tag{2.6}
\end{equation}%
\textit{Then all solutions of} (E$_{\text{R}}$) \textit{oscillate}.
\begin{proof}
Assume, for the sake of contradiction, that $\left( x(n)\right) _{n\geq -w}$
is a nonoscillatory solution of (E$_{\text{R}}$). Then it is either
eventually positive or eventually negative. As $\left( -x(n)\right) _{n\geq
-w}$ is also a solution of (E$_{\text{R}}$), we may restrict ourselves only
to the case where $x(n)>0$ for all large $n.$ Let $n_{1}\geq -w$ be an
integer such that $x(n)>0$ for all $n\geq n_{1}$. Then, there exists $%
n_{2}\geq n_{1}$ such that $x(\tau _{i}(n))>0$,\ $\forall n\geq n_{2}$, $%
1\leq i\leq m$. In view of this, Eq.(E$_{\text{R}}$) becomes%
\begin{equation*}
\Delta x(n)=-\sum_{i=1}^{m}p_{i}(n)x(\tau _{i}(n))\leq 0\text{, \ \ \ \ }%
\forall \theta(n)\geq n_{2}\text{,}
\end{equation*}%
which means that the sequence $(x(n))$ is eventually decreasing.
Taking into account the fact that (2.6) holds, equation (E$_{\text{R}}$)
gives%
\begin{eqnarray*}
x(\theta (n)+1) &=&x(\theta (n))-\sum_{i=1}^{m}p_{i}(\theta (n))x(\tau
_{i}(\theta (n)))\leq x(\theta (n))-x(\theta (n))\sum_{i=1}^{m}p_{i}(\theta
(n)) \\
&=&x(\theta (n))\left( 1-\sum_{i=1}^{m}p_{i}(\theta (n))\right) \leq 0\text{%
, \ \ \ \text{for all}}\mathit{\ }\theta (n)\geq n_{2},
\end{eqnarray*}%
where $\theta (n)\rightarrow \infty $ as $n\rightarrow \infty $, which
contradicts the assumption that $x(n)>0$ for all $n\geq n_{1}$.
\end{proof}
In a series of further computations, we try to extend the iterative result,
established by Koplatadze and Kvinikadze in [6] for a differential equation
with a single delay, to difference equations with several retarded
arguments. The following lemma provides an estimation of a positive solution
rate of decay, which is a useful tool for obtaining oscillation conditions.
\vskip0.2cm\textbf{Lemma 2.1. } \textit{Assume that }(2.5) \textit{holds and
}$x(n)$ \textit{is a positive solution of }(E$_{\text{R}}$).\textit{\ Set}%
\begin{equation}
a_{1}(n,k):=\prod_{i=k}^{n-1}\left[ 1-\sum_{\ell =1}^{m}p_{\ell }(i)\right]
\tag{2.7}
\end{equation}%
\textit{and}%
\begin{equation}
a_{r+1}(n,k):=\prod_{i=k}^{n-1}\left[ 1-\sum_{\ell =1}^{m}p_{\ell
}(i)a_{r}^{-1}(i,\tau _{\ell }(i))\right] ,\text{ \ \ \ \ }r\in \mathbb{N}.
\tag{2.8}
\end{equation}%
\textit{Then}%
\begin{equation}
x(n)\leq a_{r}(n,k)x(k),\text{ \ \ \ \ }r\in \mathbb{N}. \tag{2.9}
\end{equation}
\begin{proof}
Since $x(n)$ is a positive solution of (E$_{\text{R}}$),
\begin{equation*}
\Delta x(n)=-\sum_{i=1}^{m}p_{i}(n)x(\tau _{i}(n))\leq 0\text{, \ \ \ \ }
\forall n\in {\mathbb{N}}_{0}\text{,}
\end{equation*}%
which means that the sequence $(x(n))$ is decreasing.
From (E$_{\text{R}}$) and the decreasing character of $(x(n))$ , we have%
\begin{equation*}
\Delta x(n)+x(n)\sum_{i=1}^{m}p_{i}(n)\leq 0\text{, \ \ \ \ } \forall n\in
\mathbb{N}_{0}\text{.}
\end{equation*}%
Applying the discrete Gronwall inequality, we obtain%
\begin{equation*}
x(n)\leq x(k)\mathop{\displaystyle \prod }\limits_{i=k}^{n-1}\left[
1-\sum_{\ell =1}^{m}p_{\ell }(i)\right] =a_{1}(n,k)x(k),\text{ \ \ \ \ }
\forall n\geq k\geq 0\text{,}
\end{equation*}%
which justifies (2.9) for $r=1$.
Assume that (2.9) holds for some $r>1$. Substituting%
\begin{equation*}
x(\tau _{\ell }(n))\geq x(n)a_{r}^{-1}(n,\tau _{\ell }(n))
\end{equation*}%
into (E$_{\text{R}}$) leads to the inequality%
\begin{equation*}
\Delta x(n)+\sum_{\ell =1}^{m}p_{\ell }(n)x(n)a_{r}^{-1}(n,\tau _{\ell
}(n))\leq 0\text{, \ \ \ for sufficiently large }n\text{.}
\end{equation*}%
Again, applying the discrete Gronwall inequality, we obtain%
\begin{equation*}
x(n)\leq x(k)\mathop{\displaystyle \prod }\limits_{i=k}^{n-1}\left[
1-\sum_{\ell =1}^{m}p_{\ell }(i)a_{r}^{-1}(i,\tau _{\ell }(i))\right]
=a_{r+1}(n,k)x(k),
\end{equation*}%
due to the definition of $a_{r+1}$ in (2.8), which concludes the induction
step. The proof of the lemma is complete.
\end{proof}
It is interesting to see how the estimate developed in Lemma 2.1 works in
the case of autonomous equations.
\vskip0.2cm\textbf{Example 2.1. \ } For the equation
\begin{equation*}
\Delta x(n)+\frac{1}{4}x(n-1)=0,
\end{equation*}%
inequality (2.9) for $k=n-1$ has the form
\begin{equation*}
x(n)\leq a_{r}(n,n-1)x(n-1).
\end{equation*}%
Here $a_{1}(n,n-1)=\frac{3}{4}$, $a_{2}(n,n-1)=\frac{2}{3}$, $a_{3}(n,n-1)=%
\frac{5}{8}$, $a_{4}(n,n-1)=\frac{3}{5}$, $a_{5}(n,n-1)=\frac{7}{12}$, $%
\dots ,$ $a_{2k-1}=\frac{2k+1}{4k}$, $a_{2k}=\frac{k+1}{2k+1}$, thus the
best possible estimate of the rate of decrease of a nonoscillatory solution
is $x(n+1)\leq \frac{1}{2}x(n)$, or $x(n)\leq x(0) (\frac{1}{2})^{n}$, which
is in compliance with the fact that $\lambda =\frac{1}{2}$ is the (double)
root of the characteristic equation $\lambda ^{2}-\lambda +\frac{1}{4}=0$.
\vskip0.2cm\textbf{Theorem 2.4. } \textit{Assume that }$(p_{i}(n))$, $1\leq
i\leq m$, \textit{are sequences of nonnegative real numbers, }(1.1) \textit{%
and }(2.5) \textit{hold, }$\varphi (n)$ \textit{is defined by }(2.4) and
$a_{r}(n,k)$ are denoted in (2.7),(2.8).
\textit{If}
\begin{equation}
\limsup_{n\rightarrow \infty }\sum_{j=\varphi
(n)}^{n}\sum_{i=1}^{m}p_{i}(j)a_{r}^{-1}(\varphi (n),\tau _{i}(j))>1,
\tag{2.10}
\end{equation}%
\textit{then all solutions of} (E$_{\text{R}}$) \textit{oscillate}.
\begin{proof}
Assume, for the sake of contradiction, that $(x(n))_{n\geq -w}$ is a
nonoscillatory solution of (E$_{\text{R}}$). Then it is either eventually
positive or eventually negative. Similarly to the proof of Theorem 2.3, we
may restrict ourselves only to the case $x(n)>0$ for all large $n.$ Let $%
n_{1}\geq -w$ be an integer such that $x(n)>0$ for all $n\geq n_{1}$. By
(1.1), there exists $n_{2}\geq n_{1} $ such that $x(\tau _{i}(n))>0$,\ $%
\forall n\geq n_{2}$, $1\leq i\leq m $. In view of this, Eq.(E$_{\text{R}}$)
becomes%
\begin{equation*}
\Delta x(n)=-\sum_{i=1}^{m}p_{i}(n)x(\tau _{i}(n))\leq 0\text{, \ \ \ \ }%
\forall n\geq n_{2}\text{,}
\end{equation*}%
which means that the sequence $(x(n))$ is eventually decreasing.
Summing up (E$_{\text{R}}$) from $\varphi (n)$ to $n$, and using the fact
that the function $x$ is non-increasing, while the function $\varphi $ (as
defined by (2.4)) is non-decreasing, and taking into account that%
\begin{equation*}
\tau _{i}(j)\leq \varphi (n)\text{ \ \ and \ \ }x(\tau _{i}(j))\geq
x(\varphi (n))a_{r}^{-1}(\varphi (n),\tau _{i}(j))\text{,}
\end{equation*}%
we obtain, for sufficiently large $n$,%
\begin{eqnarray*}
x(\varphi (n)) &=&x(n+1)+\sum_{j=\varphi (n)}^{n}\sum_{i=1}^{m}p_{i}\left(
j\right) x(\tau _{i}(j)) \\
&\geq &\sum_{j=\varphi (n)}^{n}\sum_{i=1}^{m}p_{i}\left( j\right) x(\tau
_{i}(j))\geq x(\varphi (n))\sum_{j=\varphi
(n)}^{n}\sum_{i=1}^{m}p_{i}(j)a_{r}^{-1}(\varphi (n),\tau _{i}(j))\text{.}
\end{eqnarray*}%
Consequently,%
\begin{equation*}
x(\varphi (n))\left[ 1-\sum_{j=\varphi
(n)}^{n}\sum_{i=1}^{m}p_{i}(j)a_{r}^{-1}(\varphi (n),\tau _{i}(j))\right]
\leq 0\text{,}
\end{equation*}%
which gives%
\begin{equation*}
\limsup_{n\rightarrow \infty }\sum_{j=\varphi
(n)}^{n}\sum_{i=1}^{m}p_{i}(j)a_{r}^{-1}(\varphi (n),\tau _{i}(j))\leq 1%
\text{.}
\end{equation*}%
This contradicts (2.10) and the proof of the theorem is complete.
\end{proof}
\vskip0.1cmTo establish the next theorem we need the following lemma.
\vskip0.2cm\textbf{Lemma 2.2. \ }(cf. [3, Lemma 2.1])\textbf{. \ }\textit{%
Assume that }(1.1) \textit{holds, }$\left( x(n)\right) $ \textit{is an
eventually positive solution of }(E$_{\text{R}}$)\textit{, and}%
\begin{equation}
\alpha =\min \left\{ \alpha _{i}:1\leq i\leq m\right\} \text{,} \tag{2.11}
\end{equation}%
\textit{where}%
\begin{equation}
\alpha _{i}=\liminf_{n\rightarrow \infty }\sum_{j=\varphi
_{i}(n)}^{n-1}p_{i}\left( j\right) \text{.} \tag{2.12}
\end{equation}
\textit{If }$0<\alpha \leq -1+\sqrt{2}$,\textit{\ then}%
\begin{equation}
\liminf_{n\rightarrow \infty }\frac{x(n+1)}{x(\varphi (n))}\geq \frac{%
1-\alpha -\sqrt{1-2\alpha -\alpha ^{2}}}{2}\text{.} \tag{2.13}
\end{equation}
\vskip0.1cm\textbf{Theorem 2.5. \ }\textit{Assume that }$(p_{i}(n))$, $1\leq
i\leq m$, \textit{are sequences of nonnegative real numbers, }(1.1) \textit{%
and }(2.5) \textit{hold, and }$\varphi (n)$ \textit{is defined by }(2.4).%
\textit{\ If }$0<\alpha \leq 1/e$\textit{,} \textit{where }$\alpha $ \textit{%
is denoted by (2.11), and}%
\begin{equation}
\limsup_{n\rightarrow \infty }\sum_{j=\varphi
(n)}^{n}\sum_{i=1}^{m}p_{i}(j)a_{r}^{-1}(\varphi (n),\tau _{i}(j))>1-\frac{%
1-\alpha -\sqrt{1-2\alpha -\alpha ^{2}}}{2}\text{,} \tag{2.14}
\end{equation}%
\textit{where $a_{r}(n,k)$ is as in }(2.7), (2.8),\textit{\ then all
solutions of} (E$_{\text{R}}$) \textit{oscillate}.
\vskip0.1cm\textbf{Proof.\ \ }Assume, for the sake of contradiction, that $%
(x(n))_{n\geq -w}$ is a nonoscillatory solution of (E$_{\text{R}}$). Then,
as in the proof of Theorem 2.4, for sufficiently large $n$, we obtain%
\begin{eqnarray*}
x(\varphi (n)) &=&x(n+1)+\sum_{j=\varphi (n)}^{n}\sum_{i=1}^{m}p_{i}\left(
j\right) x(\tau _{i}(j)) \\
&\geq &x(n+1)+x(\varphi (n))\sum_{j=\varphi
(n)}^{n}\sum_{i=1}^{m}p_{i}(j)a_{r}^{-1}(\varphi (n),\tau _{i}(j))\text{.}
\end{eqnarray*}%
that is,%
\begin{equation*}
\sum_{j=\varphi (n)}^{n}\sum_{i=1}^{m}p_{i}(j)a_{r}^{-1}(\varphi (n),\tau
_{i}(j))\leq 1-\frac{x(n+1)}{x(\varphi (n))}\text{, }
\end{equation*}%
which gives%
\begin{equation*}
\limsup_{n\rightarrow \infty }\sum_{j=\varphi
(n)}^{n}\sum_{i=1}^{m}p_{i}(j)a_{r}^{-1}(\varphi (n),\tau _{i}(j))\leq
1-\liminf_{n\rightarrow \infty }\frac{x(n+1)}{x(\varphi (n))}\text{.}
\end{equation*}%
Assume that $0<\alpha \leq 1/e$ (clearly, $\alpha <-1+\sqrt{2})$ and, by
Lemma 2.2, inequality (2.13) holds, and so the last inequality leads to%
\begin{equation*}
\limsup_{n\rightarrow \infty }\sum_{j=\varphi
(n)}^{n}\sum_{i=1}^{m}p_{i}(j)a_{r}^{-1}(\varphi (n),\tau _{i}(j))\leq 1-%
\frac{1-\alpha -\sqrt{1-2\alpha -\alpha ^{2}}}{2}\text{,}
\end{equation*}%
which contradicts condition (2.14).
The proof of the theorem is complete.
\vskip0.2cm\textbf{Remark 2.1.} \ Observe that conditions (2.1) and (2.2)
are special cases of (2.10) and (2.14) respectively, when $r=1$.
\vskip0.2cm\textbf{Remark 2.2.} \ The following example illustrates the
significance of the condition $\lim\limits_{n\rightarrow \infty }\tau
_{i}(n)=\infty $, $1\leq i\leq m$, in Theorems 2.4 and 2.5.
\vskip0.2cm\textbf{Example 2.2.} \ Consider the retarded difference equation
(1.3) with
\begin{equation*}
p(n)=\left\{
\begin{array}{ll}
\displaystyle1/2, & \text{\ \ if }n=3k, \\
\displaystyle3/10, & \text{\ \ if }n\neq 3k,%
\end{array}%
\right. \text{ \ \ }\tau (n)=\left\{
\begin{array}{ll}
-1, & \text{\ \ if }n=3k, \\
n-1, & \text{\ \ if }n\neq 3k,%
\end{array}%
\right. \text{\ \ \ }k\in \mathbb{N}_{0}.
\end{equation*}%
Obviously (2.5) is satisfied. Also, by (2.4),%
\begin{equation*}
\varphi (n)=\max_{0\leq s\leq n}\tau (s)=\left\{
\begin{array}{ll}
n-2, & \text{\ \ if }n=3k, \\
n-1, & \text{\ \ if }n\neq 3k,%
\end{array}%
\right. k\in \mathbb{N}_{0}.
\end{equation*}%
If $n=3k$, then%
\begin{equation*}
\sum_{j=\varphi (n)}^{n}p(j)=\sum_{j=n-2}^{n}p(j)=\frac{3}{10}+\frac{3}{10}+%
\frac{1}{2}=\frac{11}{10}.
\end{equation*}%
Therefore $\displaystyle\limsup_{n\rightarrow \infty }\sum_{j=\varphi
(n)}^{n}p_{i}(j)\geq \frac{11}{10}>1,$ which means that (2.10) is satisfied
for any $r$.
Also, if $n=3k$, then%
\begin{equation*}
\sum_{j=\varphi (n)}^{n-1}p_{i}\left( j\right)
=\sum_{j=n-2}^{n-1}p_{i}\left( j\right) =\frac{3}{10}+\frac{3}{10}=\frac{3}{5%
}
\end{equation*}%
and, if $n\neq 3k$, then%
\begin{equation*}
\sum_{j=\varphi (n)}^{n-1}p(j)=\sum_{j=n-1}^{n-1}p(j)=\left\{
\begin{array}{ll}
1/2, & \text{\ \ if }n=3k+1, \\
3/10, & \text{\ \ if }n=3k+2,%
\end{array}%
\right. k\in \mathbb{N}_{0}.
\end{equation*}%
Therefore%
\begin{equation*}
\alpha =\liminf_{n\rightarrow \infty }\sum_{j=\varphi (n)}^{n-1}p_{i}\left(
j\right) =\min \left\{ \frac{3}{5},\frac{1}{2},\frac{3}{10}\right\} =\frac{3%
}{10}<\frac{1}{e}
\end{equation*}%
and%
\begin{equation*}
\sum_{j=\varphi (n)}^{n}p_{i}(j)=\frac{11}{10}>1-\frac{1-\alpha -\sqrt{%
1-2\alpha -\alpha ^{2}}}{2}\simeq 0.928388218\text{,}
\end{equation*}%
which means that (2.14) is satisfied for any $r$. Observe, however, that
equation (1.3) has a nonoscillatory solution
\begin{equation*}
\begin{array}{ll}
x(-1)=-\frac{12}{7}, & x(0)=\displaystyle1,~x(1)=\frac{13}{7},~x(2)=\frac{109%
}{70}, \\
& \\
& \displaystyle x(3)=1,~x(4)=\frac{13}{7},~x(5)=\frac{109}{70}%
,~x(6)=1,~~\dots ~%
\end{array}%
\end{equation*}%
which illustrates the significance of the condition $\lim\limits_{n%
\rightarrow \infty }\tau (n)=\infty $ in Theorems 2.4 and 2.5.
\subsection{Advanced difference equations}
Similar oscillation theorems for the (dual) advanced difference equation (E$%
_{\text{A}}$) can be derived easily. The proofs of these theorems are
omitted, since they follow a similar procedure as in Subsection 2.1.
Denote
\begin{equation}
\rho _{i}(n)=\min_{s\geq n}\sigma _{i}(s),\text{ \ \ }n\geq 0
\tag{2.15}
\end{equation}%
and%
\begin{equation}
\rho (n)=\min_{1\leq i\leq m}\rho _{i}(n),\text{ \ \ }n\geq 0\text{.}
\tag{2.16}
\end{equation}
Clearly, the sequences of integers $\rho (n)$, $\rho _{i}(n)$, $1\leq i\leq
m $ are non-decreasing and $\rho (n)\geq n+1$, $\rho _{i}(n)\geq n+1$, $%
1\leq i\leq m$ for all $n\geq 0$.
\vskip0.2cm\textbf{Theorem 2.3}$^{\prime }$\textbf{. } \textit{Assume that
there exists a subsequence }$\theta (n),~n\in
\mathbb{N}
$ \textit{of positive integers such that}%
\begin{equation*}
\sum_{i=1}^{m}p_{i}(\theta (n))\geq 1\mathit{\ \ \ \ \ }\forall n\in {%
\mathbb{N}}\text{.}
\end{equation*}%
\textit{Then all solutions of} (E$_{\text{A}}$) \textit{oscillate}.
\vskip0.2cm\textbf{Theorem 2.4}$^{\prime }$. \ \textit{Assume that }$%
(p_{i}(n))$, $1\leq i\leq m$, \textit{are sequences of nonnegative real
numbers, }(1.2) \textit{and }(2.5) \textit{hold,} $\rho (n)$ \textit{is
defined by }(2.16) and%
\begin{equation}
b_{1}(n,k):=\prod_{i=n+1}^{k}\left[ 1-\sum_{\ell =1}^{m}p_{\ell }(i)\right]
\tag{2.17}
\end{equation}%
and%
\begin{equation}
b_{r+1}(n,k):=\prod_{i=n+1}^{k}\left[ 1-\sum_{\ell =1}^{m}p_{\ell
}(i)b_{r}^{-1}(i,\sigma _{\ell }(i))\right] ,\text{ \ \ \ \ }r\in \mathbb{N}.
\tag{2.18}
\end{equation}%
\textit{If}%
\begin{equation}
\limsup_{n\rightarrow \infty }\sum_{j=n}^{\rho
(n)}\sum_{i=1}^{m}p_{i}(j)b_{r}^{-1}(\rho (n),\sigma _{i}(j))>1\text{,}
\tag{2.19}
\end{equation}%
\textit{then all solutions of} (E$_{\text{A}}$) \textit{oscillate}.
\vskip0.2cm\textbf{Lemma 2.2}$^{\prime }$ (cf. [3, Lemma 2.1]). \ \textit{%
Assume that }(1.2) \textit{holds, }$\left( x(n)\right) $ \textit{is a n
eventually positive solution of }(E$_{\text{R}}$)\textit{, and}%
\begin{equation}
\alpha =\min \left\{ \alpha _{i}:1\leq i\leq m\right\} \text{,} \tag{2.20}
\end{equation}%
\textit{where}%
\begin{equation}
\alpha _{i}=\liminf_{n\rightarrow \infty }\sum_{j=n+1}^{\rho
_{i}(n)}p_{i}\left( j\right) \text{.} \tag{2.21}
\end{equation}
\textit{If }$0<\alpha \leq -1+\sqrt{2}$,\textit{\ then}%
\begin{equation}
\liminf_{n\rightarrow \infty }\frac{x(n-1)}{x(\rho (n))}\geq \frac{1-\alpha -%
\sqrt{1-2\alpha -\alpha ^{2}}}{2}\text{.} \tag{2.22}
\end{equation}
\vskip0.2cm\textbf{Theorem \ 2.5}$^{\prime }$\textbf{. }\ \textit{Assume
that }$(p_{i}(n))$, $1\leq i\leq m$ \textit{are sequences of nonnegative
real numbers, }(1.2) \textit{and }(2.5) \textit{hold, }$\rho (n)$ \textit{is
defined by }(2.16)\textit{, and }$a$ \textit{is denoted by }(2.20)\textit{.
If }$0<\alpha \leq 1/e$\textit{,} \textit{and}%
\begin{equation}
\limsup_{n\rightarrow \infty }\sum_{j=n}^{\rho
(n)}\sum_{i=1}^{m}p_{i}(j)b_{r}^{-1}(\rho (n),\sigma _{i}(j))>1-\frac{%
1-\alpha -\sqrt{1-2\alpha -\alpha ^{2}}}{2}\text{,} \tag{2.23}
\end{equation}%
\textit{where $b_{r}(n,k)$ is defined in }(2.17), (2.18),\textit{\ then all
solutions of} (E$_{\text{A}}$) \textit{oscillate}.
\section{BOUNDED DEVIATIONS OF ARGUMENTS}
In this section, new sufficient oscillation conditions, involving $\lim \inf
$, under the assumption that all the delays (advances) are bounded and (1.1$%
^{\prime }$) $\left( \text{or (1.2}^{\prime }\text{)}\right) $ holds, are
established for equations (E$_{\text{R}}$) and (E$_{\text{A}}$). These
conditions improve Theorem 5.16 in [1] and extend the following oscillation
results by Chatzarakis et al. [5] to the case of non-monotone arguments.
\vskip0.2cm\textbf{Theorem 3.1 (}[5, Theorem 2.2]\textbf{). \ }\textit{%
Assume that the sequences }$\left( \tau _{i}(n)\right) $\textit{, }$1\leq
i\leq m$ \textit{are increasing, }(1.1)\textit{\ holds,}%
\begin{equation}
\limsup_{n\rightarrow \infty }\sum_{i=1}^{m}p_{i}(n)>0\text{\ \ \ \ and \ \
\ \ }\liminf_{n\rightarrow \infty }\sum_{i=1}^{m}\sum_{j=\tau
_{i}(n)}^{n-1}p_{i}(j)>\frac{1}{e}\text{\thinspace .} \tag{3.1}
\end{equation}%
\textit{Then all solutions of Eq. }(E$_{\text{R}}$)\textit{\ oscillate.}
\vskip0.2cm\textbf{Theorem 3.2 (}[5, Theorem 3.2]\textbf{). \ }\textit{%
Assume that the sequences }$\left( \sigma _{i}(n)\right) $\textit{,\ }$1\leq
i\leq m$ \textit{are increasing, }(1.2)\textit{\ holds,}%
\begin{equation}
\limsup_{n\rightarrow \infty }\sum_{i=1}^{m}p_{i}(n)>0\text{\ \ \ \ and \ \
\ \ }\liminf_{n\rightarrow \infty }\sum_{i=1}^{m}\sum_{j=n+1}^{\sigma
_{i}(n)}p_{i}(j)>\frac{1}{e}\text{\thinspace .} \tag{3.2}
\end{equation}%
\textit{Then all solutions of Eq. }(E$_{\text{A}}$)\textit{\ oscillate.}
\subsection{Retarded difference equations}
We present a new sufficient oscillation condition for (E$_{\text{R}}$),
under the assumption that all the delays are bounded and (1.1$^{\prime }$)
holds.
\vskip0.2cm\textbf{Theorem 3.3. } \ \textit{Assume that }$(p_{i}(n))$, $%
1\leq i\leq m$, \textit{are sequences of nonnegative real numbers,} (1.1$%
^{\prime }$) \textit{holds and for each} $k=1, \dots, m$,
\begin{equation}
\liminf_{n\rightarrow \infty }
\sum_{i=1}^{m}
\sum_{j=\tau_{k}(n)}^{n-1}
p_{i}(j)>\left( \frac{M_k}{M_k+1}\right) ^{M_k+1}\text{,} \tag{3.3}
\end{equation}%
\textit{then all solutions of} (E$_{\text{R}}$) \textit{oscillate.}
\begin{proof}
Assume, for the sake of contradiction, that $(x(n))_{n\geq -w}$ is a
nonoscillatory solution of (E$_{\text{R}}$). Without loss of generality, we
can assume that $x(n)>0$ for all $n\geq n_{1}$. Then, there exists $n_{2}\geq n_{1}$
such that $x(\tau _{i}(n))>0$,\ $\forall n\geq n_{2}$, $1\leq i\leq m$. In
view of this, Eq.(E$_{\text{R}}$) becomes%
\begin{equation*}
\Delta x(n)=-\sum_{i=1}^{m}p_{i}(n)x(\tau _{i}(n))\leq 0\text{, \ \ \ \ }%
\forall n\geq n_{2}\text{,}
\end{equation*}%
which means that the sequence $(x(n))_{n\geq n_{2}}$ is non-increasing. The
sequences%
\begin{equation*}
b_{k}(n)=\left( \frac{n-\tau _{k}(n)}{n-\tau _{k}(n)+1}\right) ^{n-\tau
_{k}(n)+1}
\end{equation*}%
satisfy the inequality%
\begin{equation}
\frac{1}{4}\leq b_{k}(n)\leq \left( \frac{M_k}{M_k+1}\right)^{M_k+1},\text{ \ \ \
}n\geq 1,\text{ \ \ \ \ }1\leq k\leq m. \tag{3.4}
\end{equation}%
Due to (3.3), for each $k=1, \dots,m$, we can choose $n_{0}(k) \geq n_{2}$ and
$\varepsilon_{k}>0$ such that for $n \geq n_{0}(k)$,
$$
\sum_{i=1}^{m}\sum_{j=\tau_k(n)}^{n-1} p_{i}(j)>\left( \frac{M_k}{M_k+1}%
\right) ^{M_k+1}+\varepsilon_{k}
$$
and
$$
d_k :=\left( \frac{M_k}{M_k+1}\right) ^{-M_k-1}
\left[ \left( \frac{M_k}{M_k+1}\right)^{M_k+1}+\varepsilon _{k}\right] > 1.
$$
Denoting
$$
\varepsilon_{0} := \min_{1\leq k \leq m} \varepsilon_{k}>0,
\quad d := \min_{1\leq k \leq m} d_k>1,
$$
we obtain
\begin{equation*}
\sum_{i=1}^{m}\sum_{j=\tau_{k}(n)}^{n-1}p_{i}(j)>\left( \frac{M_k}{M_k+1}%
\right) ^{M_k+1}+\varepsilon _{0}
\end{equation*}
and
\begin{equation*}
\left( \frac{M_k}{M_k+1}\right) ^{-M_k-1}\left[ \left( \frac{M_k}{M_k+1}\right)
^{M_k+1}+\varepsilon_{0}\right] \geq d >1,
\end{equation*}%
which together with (3.3) immediately implies for any $k=1,\dots ,m$
\begin{equation*}
\sum_{i=1}^{m}\sum_{j=\tau _{k}(n)}^{n-1}\frac{p_{i}(j)}{b_{k}(n)}\geq
\left( \frac{M_k}{M_k+1}\right) ^{-M_k-1}\sum_{i=1}^{m}\sum_{j=\tau_{k}(n)}^{n-1} p_{i}(j) > d>1.
\end{equation*}
Dividing (E$_{\text{R}}$) by $x(n)$ we have
\begin{equation*}
\frac{x(n+1)}{x(n)}=1-\sum_{i=1}^{m}p_{i}(n)\frac{x(\tau _{i}(n))}{x(n)},%
\text{ \ \ \ \ }n\geq n_{0}.
\end{equation*}%
Multiplying and taking into account that $x(\tau _{i}(n))/x(n)\geq 1$, we
obtain the estimate%
\begin{equation*}
\frac{x(n)}{x(\tau _{k}(n))}=\prod_{j=\tau _{k}(n)}^{n-1}\frac{x(j+1)}{x(j)}%
\leq \prod_{j=\tau _{k}(n)}^{n-1}\left( 1-\sum_{i=1}^{m}p_{i}(j)\right) .
\end{equation*}%
Using this estimate and the relation between the arithmetic and the
geometric means, we have
\begin{equation}
\frac{x(n)}{x(\tau _{k}(n))}\leq \left[ 1-\frac{1}{n-\tau _{k}(n)}%
\sum_{i=1}^{m}\sum_{j=\tau _{k}(n)}^{n-1}p_{i}(j)\right] ^{n-\tau _{k}(n)}.
\tag{3.5}
\end{equation}%
Observe that the function $f:\left( 0,1\right) \rightarrow
\mathbb{R}
$ defined as%
\begin{equation*}
f(y):=y(1-y)^{\rho },\text{ \ \ \ \ }\rho \in {\mathbb{N}},
\end{equation*}%
attains its maximum at $y=\frac{1}{1+\rho }$, which equals $\displaystyle %
f_{\max }=\frac{\rho ^{\rho }}{(1+\rho )^{1+\rho }}\text{\thinspace .}$ In
the inequality
\begin{equation*}
y(1-y)^{\rho }\leq \frac{\rho ^{\rho }}{(1+\rho )^{1+\rho }},\text{ \ \ \ \ }%
y\in (0,1),\text{ \ \ \ \ }\rho \in {\mathbb{N}}\text{,}
\end{equation*}%
assuming $\rho =n-\tau _{k}(n)$, $y=x/\rho$, where $\displaystyle
x=\sum_{i=1}^{m}\sum_{j=\tau_{k}(n)}^{n-1}p_{i}(j)$, we obtain from (3.5)
\begin{equation*}
\frac{x(\tau_{k}(n))}{x(n)}\geq \sum_{i=1}^{m}\sum_{j=\tau
_{k}(n)}^{n-1}p_{i}(j)\left( \frac{n-\tau_{k}(n)+1}{n-\tau _{k}(n)}\right)
^{n-\tau _{k}(n)+1}=\sum_{i=1}^{m}\sum_{j=\tau_{k}(n)}^{n-1}\frac{p_{i}(j)}{b_{k}(n)}>d
\end{equation*}
for any $n\geq n_{0}(k)$. Denote $n_0=\max\{ n_0(1), \dots, n_0(m) \}$.
If we continue this procedure assuming $n\geq n_{0}+M$,
where $M=\max_{1\leq i\leq m}M_{i}$,
and using the properties of the geometric and the algebraic mean, we have
\begin{equation*}
\frac{x(n)}{x(\tau _{k}(n))}=\prod_{j=\tau_{k}(n)}^{n-1}\frac{x(j+1)}{x(j)}
=\prod_{j=\tau _{k}(n)}^{n-1}\left( 1-\sum_{i=1}^{m}p_{i}(j)\frac{x(\tau_{i}(j))}{x(j)}\right)
\end{equation*}
\begin{equation*}
\leq \prod_{j=\tau _{k}(n)}^{n-1}\left( 1-d\sum_{i=1}^{m}p_{i}(j)\right)
\leq \left[ 1-\frac{d}{n-\tau
_{k}(n)}\sum_{i=1}^{m}\sum_{j=\tau_{k}(n)}^{n-1}p_{i}(j)\right]^{n-\tau _{k}(n)}.
\end{equation*}%
Applying the same argument, we obtain
\begin{equation*}
\frac{x(\tau _{k}(n))}{x(n)}\geq d\sum_{i=1}^{m}\sum_{j=\tau_{k}(n)}^{n-1}
\frac{p_{i}(j)}{b_{k}(n)}>d^{2},\text{ \ }~n\geq n_{0}+2M,~~k=1,\dots ,m,
\end{equation*}
\begin{equation*}
\frac{x(\tau_{k}(n))}{x(n)}>d^{r},\text{ \ \ \ }n\geq n_{0}+ rM,\text{ \ \ \
}k=1,\dots ,m.
\end{equation*}%
Due to (3.3), we observe that for any $k$
$$
\limsup_{n\rightarrow \infty }\sum_{i=1}^{m}p_{i}(n)\geq \frac{1}{M_k}
\left( \frac{M_k}{M_k+1} \right)^{M_k+1}.
$$
Since the function $\displaystyle h(x) := \frac{1}{x}
\frac{x^{x+1}}{(x+1)^{x+1}}=\frac{x^x}{(x+1)^{x+1}}$, $x \geq 1$, is decreasing and $M_k \leq
M$, we conclude that
\begin{equation}
\limsup_{n\rightarrow \infty }\sum_{i=1}^{m}p_{i}(n)\geq c:=\frac{1}{M}
\left( \frac{M}{M+1}\right) ^{M+1} \tag{3.6}
\end{equation}
and choose a subsequence $(\theta (n))$ of ${\mathbb{N}}$ such that
\begin{equation*}
\sum_{i=1}^{m}p_{i}(\theta (n))\geq c>0.
\end{equation*}%
Since
\begin{equation*}
0<\frac{x(n+1)}{x(n)}=1-\sum_{i=1}^{m}p_{i}(n)\frac{x(\tau _{i}(n))}{x(n)}~,
\end{equation*}
we have
\begin{equation*}
\sum_{i=1}^{m}p_{i}(n)\frac{x(\tau _{i}(n))}{x(n)}<1.
\end{equation*}%
In particular,
\begin{equation*}
\min_{1\leq k\leq m}\frac{x(\tau _{k}(\theta (n)))}{x(\theta (n))}%
\sum_{i=1}^{m}p_{i}(\theta (n))<1.
\end{equation*}%
Choosing $r \in {\mathbb{N}}$ such that $d^{r}>\frac{1}{c}$, where $c$ was
defined in (3.6), $\theta(n)\geq n_{0}+ rM$ and noticing that
\begin{equation*}
d^{r}<\min_{1\leq k\leq m}\frac{x(\tau_{k}(\theta (n)))}{x(\theta (n))}\leq
\left( \sum_{i=1}^{m}p_{i}(\theta (n))\right)^{-1}\leq \frac{1}{c}\,,
\end{equation*}%
we obtain a contradiction, which concludes the proof.
\end{proof}
The following result is valid as $\displaystyle
\left( \frac{n}{n+1} \right)^{n+1} < \frac{1}{e}$.
In (3.7) a non-strict inequality is also sufficient.
\vskip0.2cm\textbf{Theorem 3.4. } \textit{Assume that }$(p_{i}(n))$, $1\leq
i\leq m$, \textit{are sequences of nonnegative real numbers and} (1.1)
\textit{holds. If}
\begin{equation}
\liminf_{n\rightarrow \infty }\sum_{i=1}^{m}\sum_{j=\tau
_{k}(n)}^{n-1}p_{i}(j) > \frac{1}{e}\text{,} \quad, k=1, \dots, m, \tag{3.7}
\end{equation}%
\textit{then all solutions of} (E$_{\text{R}}$) \textit{oscillate.}
\bigskip
\noindent
{\bf 3.2~~ Advanced difference equations. }
Similar oscillation theorems for the (dual) advanced difference equation (E$%
_{\text{A}}$) can be derived easily. The proofs of these theorems are
omitted, since they follow the schemes of Subsection 3.1.
\vskip0.2cm\textbf{Theorem 3.3}$^{\prime }$. \ \textit{Assume that }$%
(p_{i}(n))$, $1\leq i\leq m$, \textit{are sequences of nonnegative real
numbers, all the advances are bounded,} (1.2$^{\prime }$) \textit{holds.}
\begin{equation}
\liminf_{n\rightarrow \infty }\sum_{i=1}^{m}\sum_{j=n+1}^{\sigma
_{k}(n)} p_{i}(j)>\left( \frac{\mu_{k} }{\mu_{k} +1}\right) ^{\mu_{k} +1}\text{,} \quad k =1,
\dots, m,
\tag{3.8}
\end{equation}%
\textit{then all solutions of} (E$_{\text{A}}$) \textit{oscillate.}
\vskip0.2cm\textbf{Theorem 3.4}$^{\prime }$. \textbf{\ }\textit{Assume that }%
$(p_{i}(n))$, $1\leq i\leq m$, \textit{are sequences of nonnegative real
numbers, all the advances are bounded and} (1.2) \textit{holds. If}%
\begin{equation}
\liminf_{n\rightarrow \infty }\sum_{i=1}^{m}\sum_{j=n+1}^{\sigma
_{k}(n)}p_{i}(j) > \frac{1}{e}\text{,} \quad k =1, \dots, m, \tag{3.9}
\end{equation}%
\textit{then all solutions of} (E$_{\text{A}}$) \textit{oscillate.}
\section{EXAMPLES}
In this section, we present examples illustrating the significance of our
results. Observe that most of the relevant oscillation results cited in the
introduction cannot be applied because they assume that the delays
(advances) are constant and consequently the deviating arguments are
increasing. When possible, we compare our results to the known ones, for
variable deviations of the argument and non-monotone arguments.
\vskip0.2cm\textbf{Example 4.1. } Consider the retarded difference equation
\begin{equation}
\Delta x(n)+px\left( \tau (n)\right) =0,\text{ \ \ \ }n\in
\mathbb{N}
_{0}\text{,} \tag{4.1}
\end{equation}%
where $p<1$ is a constant coefficient, and%
\begin{equation*}
\tau (n)=\left\{
\begin{array}{ll}
n-1=3k-1, & \text{if }n=3k \\
n-2=3k-1, & \text{if }n=3k+1 \\
n-4=3k-2, & \text{if }n=3k+2%
\end{array}%
\right. ,\text{ \ \ \ \ }k\in
\mathbb{N}
_{0}\text{.}
\end{equation*}%
Obviously (1.1) and (2.5) are satisfied. Also, by (2.4) we have%
\begin{equation*}
\varphi (n)=\max_{0\leq s\leq n}\tau (s)=\left\{
\begin{array}{ll}
n-1=3k-1, & \text{if }n=3k \\
n-2=3k-1, & \text{if }n=3k+1 \\
n-3=3k-1, & \text{if }n=3k+2%
\end{array}%
\right. ,\text{ \ \ \ \ }k\in
\mathbb{N}
_{0}\text{.}
\end{equation*}
If $n=3k$, then $\varphi (n)=n-1=$ $\tau (n)$ and, in view of (2.7), the
left-hand side in (2.10) is
\begin{eqnarray*}
\sum_{j=\varphi (n)}^{n}p(j)a_{3}^{-1}(\varphi (n),\tau (j))
&=&\sum_{j=n-1}^{n}p(j)a_{3}^{-1}(n-1,\tau (j)) \\
&=&p\frac{1}{a_{3}(n-1,\tau (n-1))}+p\frac{1}{a_{3}(n-1,\tau (n))} \\
&=&p\frac{1}{a_{3}(n-1,n-5)}+p\frac{1}{a_{3}(n-1,n-1)} \\
&=&p\frac{1}{\mathop{\displaystyle \prod }\limits_{i=n-5}^{n-2}\left( 1-p%
\frac{1}{a_{2}(i,\tau \left( i\right) )}\right) }+p\text{,}
\end{eqnarray*}%
where
\begin{eqnarray*}
&&\prod\limits_{i=n-5}^{n-2}\left( 1-p\frac{1}{a_{2}(i,\tau \left( i\right) )%
}\right) =\left( 1-p\frac{1}{a_{2}(n-5,\tau \left( n-5\right) )}\right)
\left( 1-p\frac{1}{a_{2}(n-4,\tau \left( n-4\right) )}\right) \times \\
&&\times \left( 1-p\frac{1}{a_{2}(n-3,\tau \left( n-3\right) )}\right)
\left( 1-p\frac{1}{a_{2}(n-2,\tau \left( n-2\right) )}\right) \\
&=&\left( 1-\frac{p}{a_{2}(n-5,n-7)}\right) \left( 1-\frac{p}{a_{2}(n-4,n-8)}%
\right) \left( 1-\frac{p}{a_{2}(n-3,n-4)}\right) \left( 1-\frac{p}{%
a_{2}(n-2,n-4)}\right) \\
&=&\left( 1-\frac{p}{a_{2}(n-5,n-7)}\right) \left( 1-\frac{p}{a_{2}(n-4,n-8)}%
\right) \left( 1-\frac{p}{a_{2}(n-3,n-4)}\right) \left( 1-\frac{p}{%
a_{2}(n-2,n-4)}\right) \\
&=&\left( 1-\frac{p}{\left( 1-\frac{p}{1-\frac{p}{\left( 1-p\right) ^{4}}}%
\right) \left( 1-\frac{p}{1-\frac{p}{1-p}}\right) }\right) ^{2}\left( 1-%
\frac{p}{1-\frac{p}{1-\frac{p}{\left( 1-p\right) ^{4}}}}\right) \times \\
&&\times \left( 1-\frac{p}{\left( 1-\frac{p}{1-\frac{p}{\left( 1-p\right)
^{2}}}\right) ^{2}\left( 1-\frac{p}{1-\frac{p}{\left( 1-p\right) ^{4}}}%
\right) \left( 1-\frac{p}{1-\frac{p}{1-p}}\right) }\right)
\end{eqnarray*}%
Therefore
\begin{equation*}
\sum_{j=\varphi (n)}^{n}p(j)a_{3}^{-1}(\varphi (n),\tau (j))=p+
\end{equation*}%
\begin{equation*}
+\frac{p}{\left( 1-\frac{p}{\left( 1-\frac{p}{1-\frac{p}{\left( 1-p\right)
^{4}}}\right) \left( 1-\frac{p}{1-\frac{p}{1-p}}\right) }\right) ^{2}\left(
1-\frac{p}{\left( 1-\frac{p}{1-\frac{p}{\left( 1-p\right) ^{2}}}\right)
^{2}\left( 1-\frac{p}{1-\frac{p}{\left( 1-p\right) ^{4}}}\right) \left( 1-%
\frac{p}{1-\frac{p}{1-p}}\right) }\right) \left( 1-\frac{p}{1-\frac{p}{1-%
\frac{p}{\left( 1-p\right) ^{4}}}}\right) }.
\end{equation*}%
The computation immediately implies that, if $p\in \left(
0.174816,0.183\right) $ then%
\begin{equation*}
\limsup_{n\rightarrow \infty }\sum_{j=\varphi (n)}^{n}p(j)a_{3}^{-1}(\varphi
(n),\tau (j))>1\text{.}
\end{equation*}%
For example, for $p=0.175$ inequality (2.10) holds for $r=3$, which means
that all solutions of (4.1) oscillate. Nevertheless, for $p\in \left(
0.1741,0.183\right) $ condition (2.14) is satisfied for $r=3$, since%
\begin{equation*}
\limsup_{n\rightarrow \infty }\sum_{j=\varphi (n)}^{n}p(j)a_{3}^{-1}(\varphi
(n),\tau (j))>1-\frac{1}{2}\left( 1-p-\sqrt{1-2p-p^{2}}\right) \text{.}
\end{equation*}
Observe, however, that%
\begin{equation*}
a=\liminf_{n\rightarrow \infty }\sum_{i=\tau
(n)}^{n-1}p(i)=\liminf_{n\rightarrow \infty }\sum_{i=n-1}^{n-1}p(i)=p
\end{equation*}%
and%
\begin{equation*}
c(a)=\frac{1}{2}\left( 1-a-\sqrt{1-2a-a^{2}}\right) =\frac{1}{2}\left( 1-p-%
\sqrt{1-2p-p^{2}}\right) .
\end{equation*}%
Also,%
\begin{equation*}
\sum_{j=\varphi (n)}^{n}p(j)\prod_{i=\tau (j)}^{\varphi (n)-1}\frac{1}{1-p(i)%
}=\left\{
\begin{array}{ll}
p\frac{1}{\left( 1-p\right) ^{4}}+p, & \text{if }n=3k \\
p\frac{1}{\left( 1-p\right) ^{4}}+2p, & \text{if }n=3k+1 \\
p\frac{1}{\left( 1-p\right) ^{4}}+2p+p\frac{1}{1-p}, & \text{if }n=3k+2%
\end{array}%
\right. ,\text{ }k\in \mathbb{N}_{0}.
\end{equation*}%
Thus%
\begin{equation*}
\max \left\{ p\frac{1}{\left( 1-p\right) ^{4}}+p,p\frac{1}{\left( 1-p\right)
^{4}}+2p,p\frac{1}{\left( 1-p\right) ^{4}}+2p+p\frac{1}{1-p}\right\} =p\frac{%
1}{\left( 1-p\right) ^{4}}+2p+p\frac{1}{1-p}\text{.}
\end{equation*}%
If $p<0.1829$, the computation immediately implies that%
\begin{equation*}
p\frac{1}{\left( 1-p\right) ^{4}}+2p+p\frac{1}{1-p}\leq 1
\end{equation*}%
and, if $p<0.1801$, then%
\begin{equation*}
p\frac{1}{\left( 1-p\right) ^{4}}+2p+p\frac{1}{1-p}\leq 1-\frac{1}{2}\left(
1-p-\sqrt{1-2p-p^{2}}\right) \text{.}
\end{equation*}%
Therefore, if $p\in \left( 0.174816,0.1801\right) $, the conditions (2.1)
and (2.2) are not satisfied.
Moreover, observe that if $p\in \left( 0.1741,0.174816\right) $, condition
(2.14) holds for $r=3$ while (2.10 ) and also (2.1) and (2.2) are not
satisfied. For example, for $p=0.1742$ only (2.14) holds for $r=3$.
\vskip0.2cmAt this point we should remark that in condition (2.14) an extra
requirement is needed: $0<\alpha \leq 1/e.$ Thus, when $\alpha \rightarrow 0$
(2.14) coincides with (2.10).
\vskip0.2cm\textbf{Example 4.2. } Consider the delay difference equation
\begin{equation}
\Delta x(n)+\frac{1}{8}x\left( \tau _{1}(n)\right) +\frac{1}{12}x\left( \tau_{2}(n)\right) =0,
\text{ \ \ \ }n\geq 0\text{,} \tag{4.2}
\end{equation}
with
\begin{equation*}
\tau _{1}(n)=\left\{
\begin{array}{ll}
n-3 & \text{if }n\text{ is even} \\
n-1 & \text{if }n\text{ is odd}
\end{array}
\right. \text{ \ \ \ \ and \ \ \ \ }\tau _{2}(n)=\left\{
\begin{array}{ll}
n-4 & \text{if }n\text{ is even} \\
n-1 & \text{if }n\text{ is odd}%
\end{array}%
\right. \text{.}
\end{equation*}
Here, it is clear that (1.1) and (2.5) are satisfied. Also, by (2.3) and
(2.4) we have
\begin{equation*}
\varphi _{1}(n)=\varphi _{2}(n)=\left\{
\begin{array}{ll}
n-2 & \text{if }n\text{ is even} \\
n-1 & \text{if }n\text{ is odd}
\end{array}
\right.
\end{equation*}
and
\begin{equation*}
\varphi (n)=\max_{1\leq i\leq 2}\varphi _{i}(n)=\left\{
\begin{array}{ll}
n-2 & \text{if }n\text{ is even} \\
n-1 & \text{if }n\text{ is odd}
\end{array}
\right. \text{.}
\end{equation*}
If $n$ is even, then $\varphi (n)=n-2$, $\tau _{1}(n)=n-3$, $\tau
_{2}(n)=n-4 $ and, in view of (2.7), (2.10) gives%
\begin{equation*}
\sum_{j=\varphi (n)}^{n}\sum_{i=1}^{m}p_{i}(j)a_{1}^{-1}(\varphi (n),\tau
_{i}(j))=\sum_{j=n-2}^{n}\sum_{i=1}^{2}p_{i}(j)a_{1}^{-1}(n-2,\tau _{i}(j))
\end{equation*}%
\begin{equation*}
=\frac{1}{8}\cdot \left[ \frac{1}{\left( 1-\frac{5}{24}\right) ^{3}}+1+\frac{%
1}{1-\frac{5}{24}}\right] +\frac{1}{12}\cdot \left[ \frac{1}{\left( 1-\frac{5%
}{24}\right) ^{4}}+1+\frac{1}{\left( 1-\frac{5}{24}\right) ^{2}}\right]
\simeq 0.963276895\text{.}
\end{equation*}%
Also%
\begin{equation*}
\sum_{j=\varphi (n)}^{n}\sum_{i=1}^{m}p_{i}(j)a_{2}^{-1}(\varphi (n),\tau
_{i}(j))=\sum_{j=n-2}^{n}\sum_{i=1}^{2}p_{i}(j)a_{2}^{-1}(n-2,\tau _{i}(j))
\end{equation*}%
\begin{eqnarray*}
&=&\frac{1/8}{\left[ 1-\frac{1}{8}\cdot \frac{1}{1-\frac{5}{24}}-\frac{1}{12}%
\cdot \frac{1}{1-\frac{5}{24}}\right] ^{2}\left[ 1-\frac{1}{8}\cdot \frac{1}{%
\left( 1-\frac{5}{24}\right) ^{3}}-\frac{1}{12}\cdot \frac{1}{\left( 1-\frac{%
5}{24}\right) ^{4}}\right] } \\
&&+\frac{1/12}{\left[ 1-\frac{1}{8}\cdot \frac{1}{\left( 1-\frac{5}{24}%
\right) ^{3}}-\frac{1}{12}\cdot \frac{1}{\left( 1-\frac{5}{24}\right) ^{4}}%
\right] ^{2}\left[ 1-\frac{1}{8}\cdot \frac{1}{1-\frac{5}{24}}-\frac{1}{12}%
\cdot \frac{1}{1-\frac{5}{24}}\right] ^{2}} \\
&&+\frac{1}{8}+\frac{1}{12}+\frac{1/8}{\left[ 1-\frac{1}{8}\cdot \frac{1}{1-%
\frac{5}{24}}-\frac{1}{12}\cdot \frac{1}{1-\frac{5}{24}}\right] } \\
&&+\frac{1/12}{\left[ 1-\frac{1}{8}\cdot \frac{1}{\left( 1-\frac{5}{24}%
\right) ^{3}}-\frac{1}{12}\cdot \frac{1}{\left( 1-\frac{5}{24}\right) ^{4}}%
\right] \left[ 1-\frac{1}{8}\cdot \frac{1}{1-\frac{5}{24}}-\frac{1}{12}\cdot
\frac{1}{1-\frac{5}{24}}\right] }
\end{eqnarray*}
i.e.,
\begin{equation*}
\sum_{j=\varphi (n)}^{n}\sum_{i=1}^{m}p_{i}(j)a_{2}^{-1}(\varphi (n),\tau
_{i}(j))\simeq 1.553022949
\end{equation*}%
Therefore
\begin{equation*}
\limsup_{n\rightarrow \infty }\sum_{j=\varphi
(n)}^{n}\sum_{i=1}^{m}p_{i}(j)a_{2}^{-1}(\varphi (n),\tau _{i}(j))\geq
1.553022949>1\text{,}
\end{equation*}
that is, condition (2.10) of Theorem 2.4 is satisfied and therefore all
solutions of (4.2) oscillate. Also,%
\begin{equation*}
a_{1}=\liminf_{n\rightarrow \infty }\sum_{i=\varphi _{1}(n)}^{n-1}p_{1}(i)=%
\frac{1}{8}\text{ \ \ \ \ and \ \ \ \ }a_{2}=\liminf_{n\rightarrow \infty
}\sum_{i=\varphi _{2}(n)}^{n-1}p_{2}(i)=\frac{1}{12}
\end{equation*}
and therefore
\begin{equation*}
\alpha =\min \left\{ \alpha _{i}:1\leq i\leq 2\right\} =\frac{1}{12}.
\end{equation*}%
Hence%
\begin{equation*}
\limsup_{n\rightarrow \infty }\sum_{j=\varphi
(n)}^{n}\sum_{i=1}^{m}p_{i}(j)a_{2}^{-1}(\varphi (n),\tau _{i}(j))>1-\frac{1%
}{2}\left( 1-a-\sqrt{1-2a-a^{2}}\right) \simeq 0.996196338\text{,}
\end{equation*}%
that is, condition (2.14) of Theorem 2.5 is also satisfied.
Here, it is obvious that%
\begin{equation*}
n-\tau _{1}(n)\leq 3=M_{1}\text{ \ \ \ \ and \ \ \ \ }n-\tau _{2}(n)\leq
4=M_{2}
\end{equation*}%
and therefore%
\begin{equation*}
M=\max_{1\leq i\leq 2}M_{i}=4\text{.}
\end{equation*}%
Observe that%
\begin{equation*}
\liminf_{n\rightarrow \infty }\sum_{i=1}^{2}\sum_{j=\tau
_{i}(n)}^{n-1}p_{i}(j)=\frac{1}{8}+\frac{1}{12}=\frac{5}{24}<\left( \frac{M}{%
M+1}\right) ^{M+1}=0.32768\text{.}
\end{equation*}%
That is, condition (3.3) is not satisfied. Also the second condition in
(3.1) is not satisfied.
Fig.~\ref{figure1} below illustrates a couple of solutions of (4.2). As can be
observed, both solutions are oscillatory.
\begin{figure}[ht]
\centering
\includegraphics[width=0.3 \linewidth]{ex42a.ps}~~~~~
\includegraphics[width=0.3 \linewidth]{ex42b.ps}~~~~~
\includegraphics[width=0.3 \linewidth]{ex42c.ps}
\caption{Two solutions of (4.2) for (left) $-4 \leq n \leq 100$, (middle)
$30 \leq n \leq 65$, (right) $60 \leq n \leq 90$. Both solutions are oscillatory.}
\label{figure1}
\end{figure}
\vskip0.2cm\textbf{Example 4.3. }
Consider the delay difference equation
\begin{equation}
\Delta x(n)+ a(n) x\left( \tau _{1}(n)\right) +\frac{1}{125}x\left(\tau _{2}(n)\right) =0,\text{ \ \ \ }n\geq 0\text{,} \tag{4.3}
\end{equation}%
with
\begin{equation*}
\tau _{1}(n)=\left\{
\begin{array}{ll}
n-1 & \text{if }n\text{ is even,} \\
n-2 & \text{if }n\text{ is odd,}
\end{array}%
\right. \ \ \tau _{2}(n)=\left\{
\begin{array}{ll}
n-2 & \text{if }n\text{ is even,} \\
n-3 & \text{if }n\text{ is odd,}
\end{array}
\right.
\end{equation*}
\begin{equation*}
a(n)=\left\{
\begin{array}{ll}
\frac{3}{125} & \text{if }n\text{ is even,} \\ \\
\frac{37}{125} & \text{if }n\text{ is odd.}
\end{array}%
\right.
\end{equation*}
Evidently
\begin{equation*}
n-\tau _{1}(n)\leq 2=M_{1}\text{ \ \ \ \ and \ \ \ \ }n-\tau _{2}(n)\leq 3=M_{2}
\end{equation*}%
and therefore for $k=1$
$$
\liminf_{n\rightarrow \infty }\sum_{i=1}^{2}\sum_{j=\tau_{1}(n)}^{n-1}p_{i}(j)
=\frac{37}{125}+ \frac{1}{125}=\frac{38}{125}=0.304 >\left( \frac{M_1}{M_1+1}\right) ^{M_1+1} \approx 0.2962963,
$$
while for $k=2$
$$
\liminf_{n\rightarrow \infty }\sum_{i=1}^{2}\sum_{j=\tau_{2}(n)}^{n-1}p_{i}(j)
= \frac{37}{125}+ \frac{3}{125} + \frac{2}{125} = 0.336
>\left( \frac{M_2}{M_2+1}\right)^{M_2+1} \approx 0.31640625\text{,}
$$
that is, condition (3.3) is satisfied and, by Theorem 3.3, all solutions of
Eq.(4.3) oscillate.
\medskip
\centerline{\bf Acknowledgments}
\medskip
The first author was partially supported by NSERC, grants RGPIN/261351-2010 and RGPIN-2015-05976.
\medskip
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 2,149 |
Ріо-ен-Медіо () — переписна місцевість (CDP) в США, в окрузі Санта-Фе штату Нью-Мексико. Населення — 143 особи (2010).
Географія
Ріо-ен-Медіо розташоване за координатами (35.822210, -105.900907). За даними Бюро перепису населення США в 2010 році переписна місцевість мала площу 1,72 км², уся площа — суходіл.
Демографія
Згідно з переписом 2010 року, у переписній місцевості мешкали 143 особи в 57 домогосподарствах у складі 35 родин. Густота населення становила 83 особи/км². Було 61 помешкання (36/км²).
Расовий склад населення:
До двох чи більше рас належало 8,4 %. Частка іспаномовних становила 67,1 % від усіх жителів.
За віковим діапазоном населення розподілялося таким чином: 22,4 % — особи молодші 18 років, 63,6 % — особи у віці 18—64 років, 14,0 % — особи у віці 65 років та старші. Медіана віку мешканця становила 45,5 року. На 100 осіб жіночої статі у переписній місцевості припадало 95,9 чоловіків; на 100 жінок у віці від 18 років та старших — 88,1 чоловіків також старших 18 років.
За межею бідності перебувало 19,9 % осіб, у тому числі 0,0 % дітей у віці до 18 років та 0,0 % осіб у віці 65 років та старших.
Цивільне працевлаштоване населення становило 126 осіб. Основні галузі зайнятості: роздрібна торгівля — 26,2 %, науковці, спеціалісти, менеджери — 24,6 %, публічна адміністрація — 15,9 %, сільське господарство, лісництво, риболовля — 11,1 %.
Джерела
Переписні місцевості Нью-Мексико
Населені пункти округу Санта-Фе (Нью-Мексико) | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 8,983 |
\section{Introduction}
SP Numbers can be generated in linear time allowing us to study some interesting properties and generalize a few conjectures and claims that would be more difficult to prove for the set of prime numbers. For example, there are many unsolved problems along the lines of Goldbach and twin prime conjectures. The intention of this paper is to study SP numbers, their density with respect to primes and to also study their distribution on the number line.
\section{Terminologies, Axioms and Trivial Results}
\begin{enumerate}
\item There are infinitely many SP Numbers. This is trivially proved owing to the existence of infinite primes and infinite squares.
\item There can be consecutive natural numbers who are both SP. An example is 27 and 28 (27 is $3 \cdot 9$ and 28 is $7 \cdot 4$)
\item The product of two SP Numbers is always non-SP.
\item An SP gap is defined as the difference between two consecutive SP Numbers.
\item Consecutive natural numbers that are both SP are called SP Twins. These SP pairs have gap 1.
\end{enumerate}
\noindent\rule{0.84in}{0.4pt} \par
\medskip
\indent\indent {\fontsize{8pt}{9pt} \selectfont DOI: 10.35834/YYYY/VVNNPPP \par}
\indent\indent {\fontsize{8pt}{9pt} \selectfont MSC2020: 11N25, 11P99 \par}
\indent\indent {\fontsize{8pt}{9pt} \selectfont Key words and phrases: generalization of primes, distribution of prime-like numbers \par}
\thispagestyle{fancy}
\vfil\eject
\fancyhead{}
\fancyhead[CO]{\hfill DISTRIBUTION OF SQUARE-PRIME NUMBERS}
\fancyhead[CE]{R.~N.~BHAT \hfill}
\renewcommand{\headrulewidth}{0pt}
\section{First Few SP Numbers}
Here are the first 100 SP numbers:
\begin{align*}
&8,12,18,20,27,28,32,44,45,48,50,52,63,68,72,75,76,80,92,98,99,108,\\
&112,116,117,124,125,128,147,148,153,162,164,171,172,175,176,180,\\
&188,192,200,207,208,212,236,242,243,244,245,252,261,268,272,275,\\
&279,284,288,292,300,304,316,320,325,332,333,338,343,356,363,368,\\
&369,387,388,392,396,404,405,412,423,425,428,432,436,448,450,452,\\
&464,468,475,477,496,500,507,508,512,524,531,539,548,549.
\end{align*}
\section{Theorems on SP Numbers}
\begin{theorem}
For all natural numbers $n$ large enough, the number of SP Numbers smaller than $n$ is asymptotic to $(\zeta(2)-1)\frac{n}{\log n}$.
\end{theorem}
\begin{proof}
From the prime number theorem \cite[pp.~111--115]{Davenport}, we know that the number of prime numbers $\leq$ $n$ equals $\frac{n}{\log n} + O(\frac{n}{\log^2 n})$.
Let $SP(n)$ denote the number of SP numbers smaller than $n$. $SP(n)$ is equal to the number of pairs $(a,p)$ where $a^2p$ $\leq$ $n$ where $p$ is a prime number and $a$ is a natural number $\geq$ 2. $SP(n)$ is
$$\sum_{a=2}^{\sqrt{n/2}}\pi\left(\frac{n}{a^2}\right).$$ Here, $\pi$ is the prime counting function. For arbitrary $A$ smaller than $n$, the head term is
$$\sum_{a=2}^{A}\pi\left(\frac{n}{a^2}\right).$$
The tail is
$$\sum_{a=A}^{\sqrt{n/2}}\pi\left(\frac{n}{a^2}\right).$$
Set $A$ to be $\log^2 n$. Thus, the head term is $$\sum_{a=2}^{\log^2 n}\pi\left(\frac{n}{a^2}\right).$$
We simplify this using PNT as follows:
$$n\sum_{a=2}^{\log^2 n}\frac{1}{a^2(\log n-\log (a^2))}$$
$$=\frac{n}{\log n}\sum_{a=2}^{\log^2 n}\frac{1}{a^2(1-\frac{2\log a}{\log n})}.$$
Using the properties of geometric series, this simplifies to
$$\frac{n}{\log n}\sum_{a=2}^{\log^2 n}\frac{1}{a^2}\left(1+O\left(\frac{\log a}{\log n}\right)\right)$$
$$=\frac{n}{\log n} \sum_{a=2}^{\log^2 n}\frac{1}{a^2} + O\left(\frac{n}{\log^2 n}\right).$$ \newline
Simplifying the main term, yields
$$\frac{n}{\log n}\sum_{a=2}^{\infty}\frac{1}{a^2} - \frac{n}{\log n}\sum_{a=\log^2n}^{\infty}\frac{1}{a^2}$$
$$=\frac{n}{\log n}\sum_{a=2}^{\infty}\frac{1}{a^2} + O\left(\frac{n}{\log^2 n}\right)$$
$$=(\zeta(2) - 1)\frac{n}{\log n} + O\left(\frac{n}{\log^2 n}\right).$$
Hence we have the head of the original asymptotic to be
$$=(\frac{\pi^2}{6} - 1)\frac{n}{\log n} + O\left(\frac{n}{\log^2 n}\right).$$
Simplifying the tail yields
$$\sum_{a=\log^2n}^{\sqrt{n/2}}\pi(\frac{n}{a^2}) \leq n\sum_{a=\log^2n}^{\infty}\frac{1}{a^2(\log n-\log (a^2))}=O\left(\frac{n}{\log^2 n}\right).$$
Finally, we have$$SP(n)=(\zeta(2) - 1)\frac{n}{\log n} + O\left(\frac{n}{\log^2 n}\right).$$ This completes the proof.
\end{proof}
\begin{corollary}
For a sufficiently large $n$, the number of prime numbers smaller than $n$ is more than the number of SP numbers smaller than $n$.
\end{corollary}
\begin{proof}
Since $\frac{\pi^2}{6} - 1$ is less than 1, we have $SP(n)$ $\leq$ $\pi(n)$.
\end{proof}
\begin{theorem}
There are infinitely many pairs of SP numbers for any existing SP gap.\\
\end{theorem}
\begin{proof}
We use Gauss's result on the general Pell equation \cite[Chapter 8, Theorem 2 on p.~57]{Mordell} that states that $x^2 - dy^2 = m$ has infinitely many solutions, if it has one solution.
Thus, if a gap $g$ occurs, we have $P_1, P_2, a, b$ such that
$$P_1a^2 - P_2b^2 = g.$$
Multiplying by $P_1$ yields
\[
P_1a^2 - (P_1P_2)b^2 = P_1g.
\]
We now have a general Pell equation
\begin{equation} \label{E:Pellform}
x^2 - (P_1P_2)y^2 = P_1g
\end{equation}
that has a solution. Thus, it has infinitely many solutions. In each solution, $P_1$ divides $(P_1P_2)y^2$ and $P_1g$. Hence, it also divides $x$. Let $x=P_1k$.
Thus, dividing every solution to \eqref{E:Pellform} by $P_1$ yields $$P_1k^2 - P_2y^2 = g.$$
Since $P_1k^2$ and $P_2y^2$ are SP Numbers which have gap $g$, the proof is complete.
\end{proof}
As an immediate corollary, we have the existence of infinite SP twins, our existing solution being 27 and 28.
\section{Conjectures on SP Numbers}
These conjectures are all verified computationally beyond numbers in excess of $10^9$. We look at these claims briefly to get some ideas for future research.
\begin{conjecture}
Every large natural number is a sum of two SP Numbers. \end{conjecture}The Goldbach conjecture is probably the most famous conjecture about primes. It claims that every even number greater than or equal to 4 can be written as a sum of two primes. However, for SP Numbers, as there are no even-odd parity restrictions, we claim that every sufficiently large natural number, whether even or odd, can be written as a sum of two SP Numbers. This has been verified beyond $10^9$ and is valid for all numbers starting $3931 (27 + 3904)$. E.g., 4041=116 + 3925, 10216=12 + 10204. \cite{Helfgott}
\begin{conjecture}
Every SP Number greater than 27 can be written as a sum of two SP Numbers in at least one way.\end{conjecture} This follows from conjecture 1 but takes effect with a lower starting point (although specifically restricted to only SPs and not all naturals). E.g., 153 = 28 + 125.
\begin{conjecture}
There exists at least one SP number between consecutive square numbers greater than 500.\end{conjecture} This is already conjectured for primes \cite{Bazzanella}. Computational verification shows that it works for SP Numbers as well. E.g., Between 625 and 676 we have 637, which is 49 $\cdot$ 13.
\section {On the Last Digit of SP Numbers}
Using the fact that primes are asymptotically equidistributed mod 10 (i.e. we have approximately the same number of primes ending in 1,3,7 and 9), we can come up with similar asymptotic estimates for SP numbers ending in 1,3,7 and 9. For example, if we wish to estimate the count of SP numbers ending in 1 (i.e. congruent to 1 mod 10), we wish to look at pairs $(p,a)$ such that $pa^2$ $\equiv$ 1 mod 10. $p$ can be $\equiv$ 1, 3, 7 or 9 mod 10. However, if $pa^2$ has to end in 1, $p$ cannot end in 3 or 7, because square numbers cannot end in 7 or 3 respectively. Thus we have two cases, $p$ ending in 1 or 9. When $p \equiv 1$ mod 10, we require $a^2$ to end in 1 (so that $pa^2$ ends in 1 too). Thus, $a$ can be $\equiv$ 1 or 9 mod 10.
Thus, the number of pairs $(p,a)$ such that $pa^2 \equiv$ 1 mod 10 with $p$ ending in 1 is
$$\frac{1}{4}\sum_{k=1}^{(n-1)/10}\pi\left(\frac{n}{(10k + 1)^2}\right) + \pi\left(\frac{n}{(10k + 9)^2}\right).$$
Using the fact that prime numbers are asymptotically distributed mod 10, this can be approximated by:
$$\frac{1}{4}\frac{n}{\log n}\sum_{k=1}^{\infty}\frac{1}{(10k + 1)^2} + \frac{1}{(10k + 9)^2}$$
$$=\frac{1}{400}\frac{n}{\log n}\sum_{k=1}^{\infty}\frac{1}{(k + 1/10)^2} + \frac{1}{(k + 9/10)^2}$$
$$=\frac{1}{400}\frac{n}{\log n}(\zeta(2,1/10) + \zeta(2,9/10) -2).$$
Here we use the Hurwitz zeta function which denotes $\sum_{k=0}^{\infty}\frac{1}{(k+c)^s}$ as $\zeta(s,c).$
Similarly, if $p$ ends in 9, we require $a^2$ to end in 9. Thus, $a$ can be $\equiv$ 3 or 7 mod 10.
Therefore, the asymptotic estimate for SP numbers ending in 1 with $p$ ending in 9 is the following:
$$\frac{1}{400}\frac{n}{\log n}(\zeta(2,3/10) + \zeta(2,7/10) -2).$$
Combining the above results, we have Theorem \ref{T:lastdigit}.
\begin{theorem} \label{T:lastdigit}
The asymptotic estimate of SP numbers ending in 1 is
$$\frac{1}{400}\frac{n}{\log n}(\zeta(2,1/10) + \zeta(2,9/10) + \zeta(2,3/10) + \zeta(2,7/10) - 4).$$
\end{theorem}
Using the process employed above, we find the same asymptotic estimate for SP numbers ending in 3, 7, and 9.
\medskip
\noindent
{\textbf{Acknowledgements.}}
I would like to acknowledge the professors, at the University of Illinois for their valuable advice and support. I also appreciate the comments from the editor, especially the guidance for proving the results about the last digits of SP numbers.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 36 |
\section{Introduction}\label{sec:intro}
Massive stars are known as progenitors of supernovae. The end-product of the supernova explosions are neutron stars or
black holes which can be sources of strong gravitational waves. Therefore massive stars affect not only dynamical but
also chemical evolution of the galaxies. They ionize the surrounding material and emit UV radiation. Almost at the
beginning of their life they start to lose considerable amount of their material. Studies of physical characteristics
of massive components of early-type double-lined eclipsing binaries are essential for understanding of the final
stages of stellar evolution. The two components of an eclipsing binary system have the same age and chemical
composition when they born but slightly different masses and radii and temperatures. Therefore, theoretical stellar
models must be able to match their precisely determined physical properties for one age and chemical composition. This
comparison for the detached eclipsing binaries leads to improvement in input physics and success of the treatment
in stellar models of several physical phenomena, for example mass loss, mixing-length, convective core overshooting
and opacity \citep{cas07}. In addition to this, some hints about the spin and orbital angular momentum exchange in close binaries
due to mass-transfer and loss will be obtained.
\citet{zin07} propose that the incidence of binary and multiple stars among the massive O-stars is much larger than
that for solar-like stars. This difference in multiplicity properties is attributed to the differences in the star
formation process between massive and low-mass stars. While low-mass stars may loose angular momentum by magnetic- and
disk-related processes, these are ineffective in massive star formation due to short time scale of formation. \citet{mas98}
estimate that O-star binary frequency is larger than 60\%, even this may reach to 100 \% as the gap in orbital periods
between close and wide binaries is filled. They also note that most binaries occur in clusters and associations and that
binaries are less common among field star and especially among young runaway stars. So far, many eclipsing binary systems
have been discovered in the open clusters and associations. They not only offer determining the properties of the cluster
but also allow one to calculate an accurate distance, age and chemical composition as a whole. Accurate physical parameters
of stars, especially masses, radii and effective temperatures may only be determined from the analyses of multi-passband
light curves and radial velocities. These analyses may also yield signs about the existence of additional components.
Eclipsing binaries which are members of physically bounded multiple stellar systems provide additional constraints for
the reliability of evolutionary models. The physical parameters that can be evaluated for the members of a multiple
system should be represented by the same isochrone that fit the eclipsing pair. Recently, \citet{tor10} collected
astrophysical parameters of 94 detached non-interacting eclipsing binaries. Masses and radii of these stars have
been determined with an accuracy of 3\% or more. This sample includes only three high mass binaries with a component
as massive as 27 M$_{\odot}$.
The massive binary system MY\,Ser (HD\,167971, V=7.65, U-B=-0.34, B-V=0.75\,mag) is the central star of NGC\,6604, which is
listed by \citet{hum78} among the visually most luminous O stars in the Galaxy. NGC\,6604 is a young, compact open cluster
lying at the core of the H {\sc ii} region S54 \citep{geo73}, in the Ser\,OB2 association \citep{for00}. First photoelectric
photometry made by \citet{hil56} who classified it as O8f in the MK system. Later on \citet{con71} refined this classification
as O7.5If, and \citet{wal72} as O8Ib(f)p. \citet{yam77} re-classified the star as O8Ibf. Its light variability was detected
by \citet{lei84}, and later, its eclipsing nature with an orbital period of 3.32 days was revealed by \citet{gen85}. Photometric
and spectroscopic observations of the star were obtained by \citet{lei87}. They derived the orbital period for the eclipsing
pair as 3.32 days and classified the stars as O8I+O5-8V. They have also pointed out that a third component of spectral type
O5-8V dominates the optical and the UV luminosity of the eclipsing pair. First attempt for the solution of the light curve
is made by \citet{dav88}. They obtained UBVRI light curves and performed preliminary analyses of these curves assuming the
light contribution of the third star as 63.3\%, 31.55\% and zero. Later on UBV passbands and their analyses are published
by \citet{may10}. They find light contribution of the third star in the U, B and V light curves as 55.2\%, 56.5\% and 56.7\%,
respectively. The radial velocities of both components of the eclipsing pair were revealed by \citet{may11} using the ESO
archive spectra. Combining the results of the radial velocity and light curves analyses they estimate the masses of the
components as 37.4 and 34.8 M$_{\odot}$.
In this paper we present new spectroscopic observations and radial velocities of both components of the eclipsing pair. By
analysing the previously published light curves and the new radial velocities we obtain orbital parameters for the stars.
Combining the results of these analyses we obtain absolute physical parameters of both components. In addition we reveal
for the first time orbital parameters of the third-body's orbit and the parameters of the third star.
\section{Data acquisition}
\subsection{Spectroscopy}
Optical spectroscopic observations of the MY\,Ser were obtained with the Turkish Faint Object Spectrograph Camera
(TFOSC)\footnote{http://tug.tug.tubitak.gov.tr/rtt150\_tfosc.php} attached to the {\bf 1.5\,m telescope in July, 21--29, 2012},
under good seeing conditions. Further details on the telescope and the spectrograph can be found at
http://www.tug.tubitak.gov.tr. The wavelength coverage of each spectrum was 4000-9000 \AA~in 12 orders, with
a resolving power of $\lambda$/$\Delta \lambda$ $\sim$7\,000 at 6563 \AA~and an average signal-to-noise
ratio (S/N) was $\sim$120. We also obtained high S/N spectra of four early type standard stars 21\,Cyg, $\tau$\,Her,
HR\,153 and 21\,Peg for use as templates in derivation of the radial velocities.
The electronic bias was removed from each image and we used the 'crreject' option for cosmic ray removal. Thus, the
resulting spectra were largely cleaned from the cosmic rays. The echelle spectra were extracted and wavelength
calibrated by using Fe-Ar lamp source with help of the IRAF {\sc echelle} package, see \citet{Simkin}.
\subsection{Light curves}
The UBV and intermediate-passband photometric observations have been obtained in 1985 by \citet{lei87} in three
observatories. Despite some gaps in the resulting light curve the overall-shape of the light curve is well-represented by
their observations. Depth of the primary minimum is deeper only 0.03 mag and the phase interval between the two eclipses
is about 0.5, giving an evidence of circular orbit. Almost the same epoch \citet{dav88} performed photometric observations
in the Cousins-Bessel UBVRI filters. They have published five-passband light curves as well as individual differential
photometric measurements. The most outstanding feature in their light curves is the asymmetry, in particular, light level
immediately following primary eclipse appears to be depressed. UBV photometry of MY\,Ser in seven seasons of the years from
1990 to 1994 is performed by \citet{may10}. Fortunately, one can reach to their original data. In consequence, two
original multi-passband data sets are available for the use in interpretations.
\section{Analysis of the $O-C$ curve}
\citet{lei87} found from the line spectrum that MY\,Ser is a triple system with two close eclipsing O stars and a distant
companion also of spectral type O. They note that the third star is the most luminous component of the system both in
the optical and the UV spectra. The third star has been angularly resolved for the first time by \citet{bec12} using
multi-epoch {\sc VLTI} observations. Their observations provide direct evidence for a gravitational link between the eclipsing
pair and the O8 supergiant. They measured separations between the components A and B (the eclipsing binary) vary from 8
to 15 mas over the three-year baseline. They suggest that the stars evolve on a wide and eccentric orbit which is
not coplanar with the orbit of the inner eclipsing pair.
Since the photometric observations of the eclipsing pair go back to middle of 1980s the orbit of the eclipsing pair
around the center-of-mass of triple system can be determined from the times of mid-eclipses. However, the orbital
period of the eclipsing binary is too long to be obtained a complete minimum in a night in an observing site. The
eclipse, from first to fourth contact, lasts about 20 hours. For this reason a few times of mid-eclipse was seen
in the literature. They could be obtained from the observations at the near to mid-eclipse. Therefore we try to
estimate mid-primary and mid-secondary eclipses using the observations obtained in the ingress and egress of the
minima. For this reason, we performed a preliminary analysis of the light curves obtained by \citet{may10} and
obtained the best fit representation of the observed light curves. Plotting the observations with the same scales
that of the computed curve and determined the minimum time by shifting it along the time axis. We note that the
shape of the light curve is assumed to be constant during the time base of the observations. As the best fit is
obtained the time for mid-primary or mid-secondary eclipses are read off from the observations. Using this procedure
we obtained 20 times for the mid-primary and 21 for the secondary eclipse. The times are given in Table 1 with those taken
from O-C GATEWAY data-base in the literature. The uncertainties of the minima derived in this study are about a
few hundredths of a day.
\begin{table}
\caption{Times of mid-eclipses for MY\,Ser and the O-C residuals (see text)}
\label{O-C values}
\begin{center}
\begin{tabular}{crccr}
\hline
HJD-2400000 & E & O-C(I) & O-C(II) & Ref \\
\hline
45555.0000 & 0.0 & 0.0013 & 0.0123 & 1 \\
45937.0000 & 115.0 & 0.0157 & 0.0264 & 1 \\
46230.9410 & 203.5 & -0.0063 & 0.0044 & 1 \\
46232.6125 & 204.0 & 0.0044 & 0.0151 & 1 \\
46235.94 & 205.0 & 0.0103 & 0.0210 & 2 \\
46237.58 & 205.5 & -0.0105 & 0.0002 & 2 \\
48059.47 & 754.0 & -0.0262 & -0.0163 & 2 \\
48062.80 & 755.0 & -0.0178 & -0.0079 & 2 \\
48067.7894 & 756.5 & -0.0108 & -0.0010 & 1 \\
48069.43 & 757.0 & -0.0310 & -0.0212 & 2 \\
48071.13 & 757.5 & 0.0082 & 0.0180 & 2 \\
48449.7664 & 871.5 & -0.0195 & -0.0098 & 1 \\
48453.11 & 872.5 & 0.0025 & 0.0122 & 2 \\
48501.2700 & 887.0 & -0.0009 & 0.0087 & 1 \\
48753.7130 & 963.0 & -0.0007 & 0.0089 & 1 \\
49155.63 & 1084.0 & 0.0009 & 0.0103 & 2 \\
49157.32 & 1084.5 & 0.0301 & 0.0395 & 2 \\
49158.95 & 1085.0 & -0.0007 & 0.0087 & 2 \\
49163.97 & 1086.5 & 0.0369 & 0.0463 & 2 \\
49459.58 & 1175.5 & 0.0232 & 0.0324 & 2 \\
52083.69 & 1965.5 & 0.0575 & 0.0664 & 2 \\
52088.6490 & 1967.0 & 0.0341 & 0.0422 & 1 \\
52455.68 & 2077.5 & 0.0267 & 0.0374 & 2 \\
52460.66 & 2079.0 & 0.0242 & 0.0359 & 2 \\
52495.54 & 2089.5 & 0.0273 & 0.0359 & 2 \\
52500.52 & 2091.0 & 0.0249 & 0.0359 & 2 \\
52744.69 & 2164.5 & 0.0562 & 0.0589 & 2 \\
52782.88 & 2176.0 & 0.0476 & 0.0589 & 2 \\
52872.54 & 2203.0 & 0.0240 & 0.0343 & 2 \\
52877.53 & 2204.5 & 0.0316 & 0.0344 & 2 \\
52935.63 & 2222.0 & 0.0033 & 0.0112 & 2 \\
53056.87 & 2258.5 & 0.0044 & 0.0112 & 2 \\
53189.76 & 2298.5 & 0.0298 & 0.0357 & 2 \\
53413.97 & 2366.0 & 0.0308 & 0.0371 & 2 \\
53556.76 & 2409.0 & -0.0086 & -0.0034 & 2 \\
53571.71 & 2413.5 & -0.0059 & -0.0035 & 2 \\
53576.68 & 2415.0 & -0.0173 & -0.0098 & 1 \\
53795.93 & 2481.0 & 0.0051 & 0.0080 & 2 \\
53830.80 & 2491.5 & -0.0018 & 0.0080 & 2 \\
54192.87 & 2600.5 & 0.0121 & 0.0166 & 2 \\
54247.61 & 2617.0 & -0.0545 & -0.0441 & 2 \\
54350.62 & 2648.0 & -0.0146 & -0.0109 & 2 \\
54355.59 & 2649.5 & -0.0270 & -0.0153 & 2 \\
54559.90 & 2711.0 & 0.0037 & 0.0136 & 2 \\
54574.85 & 2715.5 & 0.0064 & 0.0135 & 2 \\
54637.95 & 2734.5 & -0.0042 & 0.0072 & 2 \\
54672.87 & 2745.0 & 0.0388 & 0.0409 & 2 \\
54888.69 & 2810.0 & -0.0461 & -0.0386 & 2 \\
54906.96 & 2815.5 & -0.0450 & -0.0386 & 2 \\
54941.87 & 2826.0 & -0.0120 & -0.0097 & 2 \\
55129.54 & 2882.5 & -0.0132 & -0.0068 & 2 \\
\hline
\end{tabular}
\end{center}
\begin{list}{}{}
\item[Ref:]{\small (1) O-C Gateway, (2) This study}
\end{list}
\end{table}
The O-C(I) residuals given in the third column of Table 1 are computed with the linear light elements,
\begin{equation}
HJD~(Min~I) = 2\ 445\ 554.9987 + 3^{\rm d}.321615 \times E.
\end{equation}
and are displayed graphically against the years in Fig.1. Despite large scatter in the O-C(I) residuals they indicate a
slow decrease and a steeper increase, then a decrease again. At a first glance the variations of the residuals look like
a distorted sine curve. Since the residuals for both minima follow almost the same behavior such a variation resembles a
light-time effect, orbiting around a third-body. The time delay or advance of the observed eclipses caused by the
influence of a third star can be represented by
\begin{equation}
\Delta T=\frac {a_{12} \sin i_{3}} {c} \left\{ \frac{1-e_3^2}{1+e_3 \cos \nu_3} \sin[\nu_3 + \omega_3]+e_3 \sin \omega_3 \right\}
\end{equation}
where $a_{12}$ is the semi-major axis of the eclipsing pair's orbit around the barycenter and $i_3$, $e_3$, $\omega_3$ are
the inclination, eccentricity and longitude of the periastron of the third-body orbit, respectively. The semi-amplitude of the
light-time effect is A$_{3}$ = $a_{12}$ $\sin i_3$/c, where c is speed of light. The only parameter varying with time
in Eq.2 is the true anomaly, $\nu_3$, of the third-body. The resulting ephemeris is given by
\begin{equation}
T_{ec}=T_1+EP_1+\Delta T
\end{equation}
{\bf where T$_1$ is the starting epoch, E is the integer eclipse cycle number and $P_1$ is the orbital period of the eclipsing
binary. First we estimated the initial parameters using the trial-and-error method. Then, a linear least squares fit is applied in order}
to obtain five independent variables A$_3$,T$_3$, P$_3$, $e_3$,
$\omega_3$ for the third-body orbit plus two variables $T_1$ and $P_1$ for the eclipsing binary. The parameters are given
Table\,2 with their standard deviations. Subtracting the contribution of the light-time we obtained the O-C(II) residuals
and plotted in the bottom panel of Fig.\,1. This solution indicates that MY\,Ser revolves around the common center-of-gravity
with a third-body. The orbit is highly eccentric, amounting 0.53. Using the projected semi-major of 5.5\,AU and a period of
21.2\,yr the mass function of the third star is calculated as 0.370${\pm}$0.036 M$_{\odot}$.
{\bf The first O-C analysis confirms that there is a gravitational connection between the MY\,Ser and the third-star as suggested by \citet{bec12}. The radio
light curve obtained by \citet{blo07} reaches to maximum close to 1988-1989 and minimum between 1995 and 1999.} The maximum
radio flux density to the orbital phase close to periastron passage, when the separation between the eclipsing pair and
the third-star, is thus confirmed by the O-C analysis of the eclipsing pair.
\begin{figure}
\center
\includegraphics[width=9cm,angle=0]{oc.ps}
\caption{Residuals for the times
of minimum light of MY\,Ser with respect
to the linear light elements. The continuous
curve represents the light time effect. In the bottom panel the
O-C(II) residuals, after subtracting the light-time, are shown. } \end{figure}
\begin{table}
\caption{Orbital parameters for the third component of MY\,Ser .}
\label{table7}
\begin{tabular}{ll}
\hline
Parameter & Value \\
\hline
$T_{1}$ (HJD) & 2445554.9626${\pm}$ 0.0034 \\
$P_{1}$ (d) & 3.321616${\pm}$ 0.000002 \\
$A_{3}$ (d) & 0.032${\pm}$0.005 \\
$e_{3}$ & 0.53${\pm}$0.05 \\
$w_{3}$ & 294${\pm}$ 30 \\
$a_{12}sini$ (AU) & 5.5${\pm}$ 0.7 \\
$P_{3}$ (yr)& 21.2${\pm}$ 0.7 \\
$T_{3}$ (HJD) & 2455592${\pm}$ 547 \\
$\Sigma (O-C)^{2}$ & 0.0167 \\
\hline
\end{tabular}
\end{table}
\section{Radial Velocities}
{\bf To derive the radial velocities, the eight spectra obtained for the system are cross-correlated
against the template spectra of standard stars on an order-by-order basis using the {\sc fxcor}
package in IRAF \citep{Simkin}.
The spectra showed three cross-correlation peaks in the quadratures, one for each component
of the triple system. Thus, the peaks are fitted independently with a $Gaussian$ profile to measure the
velocities and their errors for the individual components. If the three peaks appear blended, a triple
Gaussian was applied to the combined profile using {\it de-blend} function in the task. For each of
the eight observations} we then determined a weighted-average radial velocity for each star from all
orders without significant contamination by telluric absorption features. Here we used as weights
the inverse of the variance of the radial velocity measurements in each order, as reported by {\sc fxcor}.
{\bf The heliocentric radial velocities for the primary (V$_p$), the secondary (V$_s$) components and also for the third star (V$_t$) are
listed in Table\,3,} along with the dates of observations and the corresponding orbital phases computed
with the new ephemeris given above. The velocities in this table have been corrected to the heliocentric
reference system by adopting a radial velocity value for the template stars.{\bf The radial velocities are
plotted against the orbital phase in Fig.\,2, together with the error bars.} We analysed all the
radial velocities for the initial orbital parameters using the {\sc RVSIM} software program \citep{kane}.
Figure\,2 shows the best-fit orbital solution to the radial velocity data. The results of the analysis
are as follow: $\gamma$= -14.5$\pm$3.2 \kms, $K_1$=265$\pm$4 and $K_2$=279$\pm$4 \kms with circular
orbit. Using these values we estimate the projected orbital semi-major axis and mass ratio as:
$a$sin$i$=35.72$\pm$0.54 \Rsun~ and $q=\frac{M_2}{M_1}$=0.949$\pm$0.028.
{\bf The radial velocities of the third star vary from -4 to -43 \kms. Since the observations were obtained in successive
eight nights, these changes should be arisen from the measurement errors, due to the blending of the
three lines. The average radial velocity is determined as -26$\pm$13 \kms.}
\begin{figure}
\includegraphics[width=8.5cm,angle=0]{RV.ps}
\caption{Radial velocities for the components of MY\,Ser. Symbols
with error bars show the radial velocity measurements for the
components of the system (primary: open circles,
secondary: filled circles). } \end{figure}
\begin{table}
\scriptsize
\centering
\begin{minipage}{85mm}
\caption{{\bf Heliocentric radial velocities of MY\,Ser and the third star. The columns give the heliocentric
Julian date, the orbital phase (according to the ephemeris in \S 3), the radial velocities of
the three components with the corresponding standard deviations.}}
\begin{tabular}{@{}ccccccccc@{}cc}
\hline
HJD 2400000+ & Phase & \multicolumn{2}{c}{Star 1 }& \multicolumn{2}{c}{Star 2 } & \multicolumn{2}{c}{Star 3 } \\
& & $V_p$ & $\sigma$ & $V_s$ & $\sigma$ &V$_t$& $\sigma$\\
\hline
56130.3678 & 0.8034 & 227 & 5 & -275& 3 &-4&10\\
56131.3096 & 0.0869 & -155 & 7 & 132 & 7 &-37&5\\
56132.3140 & 0.3893 & -173 & 3 & 153 & 4 &-41&9\\
56133.3625 & 0.7050 & 244 & 4 & -278& 3 &-27&8\\
56135.2888 & 0.2849 & -272 & 4 & 246 & 3 &-43&7\\
56136.3513 & 0.6048 & 155 & 7 & -199& 5 &-7&11\\
56137.2779 & 0.8837 & 161 & 6 & -200& 5 &-18&8\\
56138.3544 & 0.2078 & -271 & 5 & 247 & 3 &-30&6\\
\hline
\end{tabular}
\end{minipage}
\end{table}
\begin{figure}
\center
\includegraphics[width=8.5cm,angle=0]{myserlc.ps}
\caption{a) The Mayer et al.'s UBV, b) Davidge and Forbes' UBVRI light curves of MY\,Ser. The continuous lines show the best-fit model.} \end{figure}
\section{Light curves and their analyses}
{\bf As we quoted in \S1 the optical photometric studies of} MY\,Ser were made by \citet{lei87} in the UBV, by
\citet{dav88} in the UBVRI and by \citet{may10} in the UBV. The original data of the latter two studies were published
for use of other investigators. \citet{dav88} obtained 1\,520 points in the five passbands. The standard deviations of
differential brightnesses were given as 0.014\,mag in U, 0.005\,mag in B, and 0.006\,mag in V, R, and I. Although there
are some gaps the shape of the light curve was revealed. The UBV photometric observations of \citet{may10} were obtained
between 1990-1994, which contain 3\,042 points. Their three passbands light curve are almost complete. They did not give
standard deviations of their observations. We estimate accuracies of differential observations from the maxima about
0.010 in U, 0.009 in B,and 0.010\,mag in V-passband. The light curves are shown in Fig.\,3.
Most commonly used code for modeling the light curves of the eclipsing binaries is that of \citet{Wilson71}.
This code was up-dated and implemented in the {\sc phoebe} code of \citet{prs05}. One of the main difficulties
in this modeling is determination of individual effective temperatures of both stars. Generally used practice is
to estimate effective temperature of primary star and determine that of the secondary star. Effective temperature of
the primary star could be estimated from its spectra or color indices. However, spectrum of an eclipsing binary is a
composite spectra of both stars. In some cases decomposition of the spectra is almost impossible. Similarly the color
indices are also include colors of both stars. If the stars of an eclipsing binary have similar properties, i.e. their
temperatures and luminosities are very close, it is difficult task to estimate their contribution to the total
luminosity of the system.
The average colors are measured by \citet{lei87} as U-B=-0.34$\pm$0.02, B-V=0.75$\pm$0.01, V-R=0.52$\pm$0.01, V-I=1.05$\pm$0.01\,mag.
They point out that the colors remain constant independently of the phase. In addition, infrared colors J-H=0.205$\pm$0.05,
H-K=0.177$\pm$0.05\,mag are given in the 2MASS catalog \citep{cut03}. The quantity $Q$=($U-B$)-($E_{(U-B)}$/($E_{(B-V)}$)$(B-V)$
is independent of interstellar extinction. The average value obtained is ($E_{(U-B)}$/($E_{(B-V)}$)=0.72$\pm$0.03 (\citep{joh53};\citep{hov04}).
We compute the reddening-free index as $Q$=-0.88$\pm$0.03. The values of index were calculated by \citet{hov04} begining
from O8 to G2 spectral type for the luminosity classes between main-sequence and supergiants. As pointed out by \citet{joh53}
this quantity is almost constant between O8 and B0. Both calibrations yield an approximate spectral type of the triple system as O9.
A preliminary analysis of the light curves obtained by \citet{may10} yields log g=3.6 for both stars of the eclipsing pair which
indicates that they should be luminosity class of III, i.e. giants. Using the calibration of \citet{mar05} for the stellar parameters
of galactic O stars we obtained a spectral type of O(7.5$\pm$0.5)III for the more massive and O(9.5$\pm$0.5) III for the less massive secondary
component of MY\,Ser. The effective temperatures are estimated as 34\,000$\pm$1\,000K and 30\,000$\pm$1\,000K for the primary and
secondary stars, respectively. Thus, the intrinsic color of $(B-V)_0$=-0.30$\pm$0.01 mag and the interstellar reddening
$E_{(B-V)}$=1.05$\pm$0.01\,mag are estimated.
Logarithmic limb-darkening coefficients were interpolated from the tables of \citet{ham93}. They are updated at every
iteration. The gravity-brightening coefficients $g_1$=$g_2$=1.0 and albedos $A_1$=$A_2$=1.0 were fixed for both
components, as appropriate for stars with radiative atmospheres. Since the preliminary analysis indicates that both
stars fill their corresponding Roche lobes synchronous rotations were adopted and Mode 6 was used in the solution.
{\bf This mode is used for the contact systems, e.g., both stars fill their limiting Roche lobes.} The
adjustable parameters in the light curves fitting were the orbital inclination $i$, the effective temperature of the
secondary star T$_{eff_2}$, the luminosity of the primary L$_1$, the third-body light contribution $l_3$ and the
zero-epoch offset. The parameters of our final solution are listed in Table\,4. The uncertainties assigned to the
adjusted parameters are the internal errors provided directly by the code. In the last three lines of Table\,4 sums
of squares of residuals $\sum$$(O-C)^{2}$, number of data points $N$, and standard deviations $\sigma$ of the observed
light curves are presented, respectively. The computed light curves are compared with the observations in Fig.\,3. We
also applied Mode\,3, suitable for the over contact binaries, the orbital parameters did not vary considerably but the
sum of squares of the residuals is slightly increased.
\citet{wil92} suggestion about the unit of third light was to express it in the light curve of the triple system
at a definite orbital phase. The values of $l_3$ in units of total triple system light were estimated at the
reference phase 0.25, from the Mayer et al.'s light curves, to be 0.388$\pm$0.002, 0.395$\pm$0.002 and 0.406$\pm$0.002
in U, B, V passbands, respectively. The light contributions of the third star from the Davidge and Forbes' light
curves are found to be 0.472$\pm$0.006, 0.483$\pm$0.005, 0.485$\pm$0.006, 0.495$\pm$0.005, and 0.484$\pm$0.005
in U, B, V, R and I passbands. Its contribution increases towards the longer wavelengths except the I-passband. Our spectra
show that the light contribution of the third star to the total light is about 0.39 at the $\lambda$ 6563. It is
in a good agreement with that obtained from the analysis of Mayer et al.'s light curves.
\begin{table}
\scriptsize
\caption{Results of the simultaneous analyses of the Mayer's $UBV$, and Davidge \& Forbes's $UBVRI$ light curves for MY\,Ser.}
\begin{tabular}{lrr}
\hline
Parameter & Mayer et al. (UBV)&Davidge \& Forbes (UBVRI) \\
\hline
$i^{o}$ &73.76$\pm$0.08 &77.98$\pm$.30 \\
T$_{eff_1}$ (K) &34\,000[Fix] &34\,000[Fix] \\
T$_{eff_2}$ (K) &29\,350$\pm$50 &29\,750$\pm$115 \\
$\Omega_1$=$\Omega_2$ &3.667$\pm$0.021 &3.667$\pm$0.021 \\
$r_1$ &0.3825$\pm$0.0016 &0.3825$\pm$0.0016 \\
$r_2$ &0.3733$\pm$0.0017 &0.3733$\pm$0.0017 \\
$\frac{L_{1}}{(L_{1}+L_{2})}$U &0.578$\pm$0.002 &0.570$\pm$0.006 \\
$\frac{L_{1}}{(L_{1}+L_{2})}$B &0.569$\pm$0.002 &0.563$\pm$0.006 \\
$\frac{L_{1}}{(L_{1}+L_{2})}$V &0.568$\pm$0.002 &0.562$\pm$0.006 \\
$\frac{L_{1}}{(L_{1}+L_{2})}$R & &0.562$\pm$0.007 \\
$\frac{L_{1}}{(L_{1}+L_{2})}$I & &0.561$\pm$0.006 \\
$\frac{L_{3}}{(L_{1}+L_{2}+L_{3})}$U &0.388$\pm$0.002 &0.472$\pm$0.006 \\
$\frac{L_{3}}{(L_{1}+L_{2}+L_{3})}$B &0.395$\pm$0.002 &0.483$\pm$0.005 \\
$\frac{L_{3}}{(L_{1}+L_{2}+L_{3})}$V &0.406$\pm$0.002 &0.485$\pm$0.006 \\
$\frac{L_{3}}{(L_{1}+L_{2}+L_{3})}$R & &0.495$\pm$0.005 \\
$\frac{L_{3}}{(L_{1}+L_{2}+L_{3})}$I & &0.484$\pm$0.005 \\
$\sum(O-C)^{2}$ &0.2176 &0.2127 \\
$N$ &3\,042 &1\,520 \\
$\sigma$ &0.0085 &0.0118 \\
\hline
\end{tabular}
\end{table}
\section{Results and discussion}
Combining the results of radial velocities and light curves analyses we have calculated the absolute parameters
of the stars. Separation between the components of eclipsing pair is calculated as $a$=36.69$\pm$0.20R$_{\odot}$. The
fundamental stellar parameters for the components such as masses, radii, luminosities are listed in Table\,5
together with their formal standard deviations. The standard deviations of the parameters have been determined by
JKTABSDIM\footnote{This can be obtained from http://http://www.astro.keele.ac.uk/$\sim$jkt/codes.html} code, which
calculates distance and other physical parameters using several different sources of bolometric corrections
\citep{sou05}. The mass for the primary of $M_p$ = 32.23 $\pm$ 0.54M $_{\odot}$ and secondary of $M_s$=30.59$\pm$0.53M$_{\odot}$
are consistent with O(7-8)III and O9.5III stars.
\begin{table}
\scriptsize
\setlength{\tabcolsep}{2.5pt}
\caption{Properties of the MY\,Ser components}
\label{parameters}
\begin{tabular}{lrr}
\hline
Parameter & Primary & Secondary \\
\hline
Mass (M$_{\odot}$) & 32.23$\pm$0.54 & 30.59$\pm$0.53 \\
Radius (R$_{\odot}$) & 14.23$\pm$0.75 & 13.89$\pm$0.75 \\
$T_{eff}$ (K) & 34\,000$\pm$1000 & 29\,350$\pm$1000 \\
$\log~(L/L_{\odot})$ & 5.383$\pm$0.069 & 5.102$\pm$0.054 \\
$\log~g$ ($cgs$) & 3.640$\pm$0.046 & 3.638$\pm$0.047 \\
$(vsin~i)_{calc.}$ (km s$^{-1}$) & 214$\pm$11 & 210$\pm$11 \\
\hline
\end{tabular}
\end{table}
We estimate the interstellar reddening of $E$(B-V)=1.05\,mag which is consistent with that of 1.02\,mag obtained
by \citet{bar00} for the cluster NGC\,6604. Using this value and the apparent visual magnitude of the triple system
as V=7.65 given by \citet{lei87} and taking into account their contributions we find apparent visual magnitudes
of the three stars as $V_p$=8.83, $V_s$=9.13 and $V_3$=8.63\,mag. The bolometric corrections are adopted from
\citet{mar05} as -3.20, -2.80 for the primary and secondary stars, respectively. In Table\,6 we present magnitudes
corrected for interstellar absorption, color indices of the three stars as well as the derived distances for
the components of the eclipsing binary system. Absolute bolometric magnitude of the Sun is taken as 4.74. The
distances for the primary and secondary star found to be 1\,645$\pm$133 and 1\,640$\pm$143 pc, respectively. The
distance to the NGC\,6604 has been subjected to the various studies. \citet{mof75} derived a distance of 1.64\,kpc,
while \citet{bar00} estimated a distance of 1.7\,kpc. Our result is consistent with the earlier values. The distance
determination is, of course, depended upon the total apparent magnitude of the multiple system. It varies from 7.37
(\citet{may10}) to 7.65\,mag. \citet{lei87}. If we adopt the former the distance to the system is reduced to 1\,445\,pc,
decreased by about 12\%.
Fortunately, the third star in the HD\,167971 system has been angularly resolved by \citet{bec12}. Their multi-epoch
VLTI observations provided evidence for a gravitational link between the third star and the close eclipsing pair. The
separation between the third star and the eclipsing binary has been varied from 8 to 15\,mas over the three-year
baseline. Although limited interferometric measurements are available they suggest that the components evolve
on a wide and very eccentric orbit, probably $e$ $\ge$ 0.5. Using near infrared luminosity ratio of 0.8 they
suggest spectral types for the stars in the close binary between O6 and O7 main-sequence stars. As a results they
assumed an (O6.5V+0.6.5V)+O8I spectral classification for the stars in the HD\,167971.
In \S3 we presented parameters of the third-body orbit. Since the O-C curve was however only poorly covered by minimum
times and the times were very uncertain the parameters given in Table\,2 were not well established. Even a preliminary
set of parameters for the third-body orbit is obtained the parameters appear to support the interferometric
measurements. The orbital period about the center-of-gravity with the third-star is derived as 21.2$\pm$0.7 yr,
$a_{12}$ $\sin i_3$=5.5$\pm$0.7 AU and $e$=0.53$\pm$0.05. Using the projected semi-major axis and the period of
the third-body orbit we estimate a mass function for the third star as $f(m)$=0.370$\pm$0.036M$_{\odot}$. Since
the masses of the eclipsing pair's components are derived as 32.23 and 30.59 solar masses and the eclipsing pair orbiting around
common center-of-mass with a period of 21.2\,yr the mass of the third star can easily be calculated from the mass
function. The mass of the third body has been computed as 12.70, 15.0 and 28.8 M$_{\odot}$ for the inclination of
90, 60 and 30 degrees, respectively. A plausible inclination of the third-body should be smaller than 30 degree when
the light contribution is taken into account. Assuming an inclination of 30 degrees we estimate separation between
the third-body and eclipsing pair as 8.07 and 26.26\,AU, when they are at the periastron and apastron on their orbits,
respectively. They correspond to linear angular separations of 4.9 and 15.9\,mass for a distance of 1\,650 pc. They
are in good agreement with the interferometric measurements between 8.7 and 15.1\,mas, obtained by \citet{bec12}.
Unfortunately, the four interferometric measurements were made in two epochs, e.g. in 2008 and 2011. We derived the
semi-amplitude of the radial velocity of the eclipsing pair's gravity center as K$_{12}$=9$\pm$1 \kms. Therefore, the
semi-amplitude of the radial velocity of the third star should be about 20\,\kms. {\bf This result is in a good agreement
with the measured average radial velocities of -26$\pm$13 \kms for the third star.}
The roughly inferred intrinsic magnitudes and color indices are already presented in Table\,6. With the color indices
of (U-B)$_0$=-1.080 and (B-V)$_0$=-0.275\,mag the third star appears to be later spectral type than both components of
the eclipsing binary. Its intrinsic colors yield a spectral type close to B0 giant or supergiant. If it is a (O9.5-B0)III star
its absolute visual magnitude and bolometric correction are deduced as -5.10 and -2.84\,mag from the calibrations of
\citet{mar05}. Using the intrinsic apparent visual magnitude of 5.36 we obtained a distance to the third star as
1\,280 pc. If it were a supergiant, i.e. (O9.5-B0)I, its distance would be 2\,130\,pc. For an extreme case, if the
third star is assumed to be a main-sequence star then it would lie at a distance about 7\,00\,pc. These comparisons
point out that the third star's luminosity class is between giant and supergiant. It could also be explained by a
(O9.5-B0)III-I star at the same distance as the binary.
{\bf The three lines are well separated at the quadratures of the eclipsing pair. We measured equivalent widths (EW) of the third star's
He {\sc i } 4471 and He {\sc ii} 4541 lines. The average logarithmic ratio of the equivalent widths (EW) of the third star's is about 1.19$\pm$0.17, which corresponds to
a B0-type star \citep{huc96}. On the other hand the ratio between the Si {\sc iv} 4088 and He {\sc i} 4143 lines
is obtained as 0.34$\pm$0.04. This value corresponds to III-I luminosity class stars. The ratios of the
selected spectral lines confirm our spectral classification from the intrinsic color indices.
}
\citet{bec12} estimated spectral types for the stars in close binary O6.5V, assuming that luminosity ratios should
be similar for both bolometric and K-band luminosities. They also noted that for a giant luminosity class the close
binary components should not be earlier than O8 spectral type. They propose for an O8I for the third-star. On the
other hand, \citet{lei87} reported that the O supergiant is the most luminous and dominates the optical and UV
spectra. However, our analysis of the \citet{may10} light curves show that luminosity ratios are 0.63, 0.65 and 0.68
in the U, B and V-passband. These ratios are obtained as 0.89, 0.94, 0.94, 0.98 and 0.94 from the UBVRI light curves of
\citet{dav88}. In contrary to the \citet{bec12}, we find that eclipsing binary is brighter than the third
star. The same result is also confirmed by the spectra we obtained around the H$_{\alpha}$. Perhaps, future spectroscopy will reveal more about its possible nature.
NGC\,6604 is located in a rich part of the Milky Way in the constellation $Serpens$. Its galactocentric distance is
about 6.9\,kpc, which places it on the outer boundary of the $Carina-Sagittarius$ arm, about 65\,pc above the galactic
plane. \citet{rei08} suggest that NGC\,6604 forms the densest part of the wider Ser\,OB2 association, which contains
about 100 OB stars. His studies show that the age of the cluster is about 4-5\,Myr and star formation is still ongoing
in the association.
\begin{figure*}
\center
\includegraphics[width=15cm,angle=0]{gen_eco.ps}
\caption{Locations of the components of MY\,Ser on the
a) log T$_{eff}$-log L/L$_{\odot}$,
b) log T$_{eff}$-log g and
c) mass-log g planes. Evolutionary tracks and isochrones are adopted from \citet{eks12}
for $Z$=0.014 with mass-loss, labelled in age (Myr). {\bf Location of the third star on the log T$_{eff}$-log L/L$_{\odot}$ diagram is also shown at the panel (a,b, and c) with
triangle symbol.}
} \end{figure*}
Figure\,4 shows the components of MY\,Ser in the log T$_{eff}$-log L/L$_{\odot}$, log T$_{eff}$-log g and mass-log g
planes. The evolutionary tracks and isochrones for the $Z$=0.014 with mass-loss are adopted from \citet{eks12}. While
the location of the more massive companion in the T$_{eff}$-log L/L$_{\odot}$ and log T$_{eff}$-log g is consistent with
the models for single stars with an age of about 4\,Myr the secondary star appears to lower temperature with respect
to its mass with an age of about 6\,Myr. {\bf Three components of the massive multiple system HD\,167971 seem to have
same age about 4\,Myr.} These age
estimates are consistent with that proposed by \citet{rei08} for the open cluster NGC\,6604. \citet{hil96} discussed
the properties of six O to B0 detached binaries. They concluded that the masses and radii for the components in these
systems are all in agreement with the theoretical models of solar composition. However, O-star temperature should be
reduced by an average of about 1\,000\,K to obtain in agreement with the models. In the case of MY\,Ser the secondary
star, in contrary to their result, seems to have lower temperature by about 3\,000\,K.
MY\,Ser appears to have just filling the entire outer contact surface with fill-out factor of 0.99. As suggested by
\citet{may10}, who determined a fill-out factor of 0.50, MY\,Ser is an overcontact binary.
\begin{table}
\scriptsize
\setlength{\tabcolsep}{2.5pt}
\caption{ The intrinsic apparent visual magnitudes, color indices, spectral and distances for the components of MY\,Ser. The intrinsic magnitudes and color
indices for the third-star and HD\,167971 stellar system are given in coloumns 4 and 5, respectively.}
\label{parameters}
\begin{tabular}{lrrrr}
\hline
Parameter & Primary & Secondary &Third-body &Total \\
\hline
U ($mag$) & 4.109$\pm$0.00 & 4.449$\pm$0.00 &4.007$\pm$0.00 &2.980$\pm$0.00 \\
B ($mag$) & 5.237$\pm$0.00 & 5.539$\pm$0.00 &5.087$\pm$0.00 &4.079$\pm$0.00 \\
V ($mag$) & 5.563$\pm$0.00 & 5.859$\pm$0.00 &5.362$\pm$0.00 &4.383$\pm$0.00 \\
U-B ($mag$) & -1.128$\pm$0.00 & -1.090$\pm$0.00 &-1.080$\pm$0.00 &-1.099$\pm$0.00 \\
B-V ($mag$) & -0.326$\pm$0.00 & -0.320$\pm$0.00 &-0.275$\pm$0.00 &-0.304$\pm$0.00 \\
Sp.Type & O(7.5$\pm$0.5) III & O(9.5$\pm$0.5) III &--- &---\\
M$_{bol}$ & -8.718$\pm$0.00 & -8.015$\pm$0.00 & --- & ---\\
BC ($mag$) & -3.20$\pm$0.00 & -2.80$\pm$0.00 & --- & --- \\
M$_V$ ($mag$) & -5.518$\pm$0.00 & -5.215$\pm$0.00 &--- & ---\\
V-M$_V$ ($mag$) & 11.081$\pm$0.00 & 11.074$\pm$0.00 & --- & --- \\
$d$ (pc) & 1645$\pm$133 & 1640$\pm$143 &--- & --- \\
\hline
\end{tabular}
\end{table}
\section{Conclusion}
HD\,167971 is one of the rare triple system with massive components. Analysis of the radial velocities and light
curves suggest that the inner binary consists of an O7.5 III and O9.5 III stars. The spectral type of the third star
is estimated as O9.5 between giant and supergiant luminosity classes. The times for mid-primary and mid-secondary
eclipses are analysed for the first time for the parameters of the third-body orbit. The gravitational link between the
MY\,Ser and a third-star is confirmed.
Using our estimation of $E$(B-V) and A$_v$ we determined a distance of about 1\,650\,pc which is in agreement with the
distance of the open cluster NGC\,6604. The properties of the components of MY\,Ser are compared with theoretical
models. While the locations of the more massive primary component on the effective temperature-luminosity, effective
temperature-gravity and mass-gravity planes are consistent with the models the less massive secondary star appears to
have lower temperature with respect to its mass. Both components of MY\,Ser locate on the mass-gravity plane, that
matched very well with an isochrone of 4\,Myr. The ages suggested for NGC\,6604 range from 4 to 6\,Myr. Our results
yield that the age of the close binary, is close to 4\,Myr, and thus NGC\,6604, where star formation still is going on.
\section*{Acknowledgments}
We thank to T\"{U}B{\.I}TAK National Observatory (TUG) for a partial support in using RTT150
telescope with project number 11BRTT150-198.
We also thank to the staff of the Bak{\i}rl{\i}tepe observing station for their warm hospitality. This study
is supported by Turkish Scientific and Technology Council under project number 112T263.
The following internet-based resources were used in research for this paper: the NASA Astrophysics Data System; the
SIMBAD database operated at CDS, Strasbourg, France; T\"{U}B\.{I}TAK ULAKB{\.I}M S\"{u}reli Yay{\i}nlar
Katalo\v{g}u-TURKEY; and the ar$\chi$iv scientific paper preprint service operated by Cornell University.
{\bf The authors are indebted to the anonymous referee for his/her valuable suggestions which improved the paper.}
\bibliographystyle{elsarticle-harv}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 6,128 |
var app = angular.module('tastypair', []);
app.controller('MainCtrl', function($scope) {
$scope.tasty = {};
$scope.tasty.chosenFoods = [];
$scope.tasty.recommendedFoods = [];
$scope.addFood = function () {
$scope.tasty.chosenFoods.push($scope.tasty.foodInput);
$scope.tasty.foodInput = '';
};
});
| {
"redpajama_set_name": "RedPajamaGithub"
} | 6,872 |
\section{Introduction}
\label{sec:intro}
Emergency Medical Services (EMS), also known as ambulance services or paramedic services, provide out-of-hospital medical care and transportation to a medical facility for those in need of urgent medical attention. In general, minimising the response time to emergency calls is crucial for EMS to treat patients efficiently and effectively.
\begin{figure}[thbp]
\centerline{\includegraphics[width=0.45\textwidth, trim = 1cm 2cm 1cm 1cm]{ambulance_demand}}
\caption{The reported number of ambulance attendances related to alcohol, drug use, self-harm, and mental health, for one of the states in Australia, over the past eight years.}
\label{demandtimeseries}
\end{figure}
Generating accurate and reliable spatio-temporal estimations of the future EMS demand is essential for authorities to proactively allocate the available resources, i.e., number of paramedics and size of ambulance fleets. A successful resource deployment strategy assures EMS deliver high-quality medical services in a timely manner. Furthermore, EMS demand patterns are often an early warning sign for certain problems in a community, and can therefore be useful for national-level policy makers, when introducing intervention strategies to address those problems.
The staggering growth of incidents requiring EMS assistance, combined with the scarcity of the resources have made EMS planning a challenging task for many countries \cite{Steins2019-eu,Zhou2015-cx,Setzler2009-km,Channouf2007-vw}. As a result, accurate EMS demand forecasting is becoming ever more important in the emergency services industry, so that the authorities can anticipate EMS requirements in advance to optimally utilise the available resources. Over the past two decades, numerous studies have been developed to improve the accuracy of EMS demand forecasts \cite{Steins2019-eu,Chen2016-ka,Zhou2015-cx,Wong2014-ok,Matteson2011-lv,Vile2012-bq,Setzler2009-km,Jones2009-gj,Channouf2007-vw}. However, many of the existing approaches hold strong prior distributional and structural assumptions about the EMS data. Furthermore, these methods are univariate forecasting models, which treat each time series separately, and forecast them in isolation. As a result, these models are unable to exploit the common EMS demand patterns present in collections of time series. Motivated by this need, we propose a forecasting framework, Deep Planning and Policy Making Net (DeepPPMNet), training a Long Short-Term Memory Network (LSTM) globally across different series, and therewith accounting for the cross-series information available in a set of EMS demand time series.
In this work, we use a case study of ambulance attendances related to alcohol, drug use, and self-harm in Australia. Similar to other developed nations, Australia is experiencing an increasing demand for EMS over the past few years. For example, Fig.~\ref{demandtimeseries} illustrates the growth of EMS demand related to alcohol, drug use, self-harm, and mental health, for one particular Australian state.
In the EMS capacity planning, generating sub-region demand forecasts has proven more useful than aggregated forecasts for an entire region \cite{Steins2019-eu,Zhou2015-cx,Setzler2009-km}. In Australia, states and territories are subdivided into statistical sub-regions known as Local Government Areas (LGAs). This breakdown enables authorities to deploy a fine-grained resource allocation strategy. Fig.~\ref{lgademand} illustrates such an example of EMS demand for suicide attempts, over the past eight years. Here, the EMS demand patterns of various LGAs can be similar and may share salient features in common, i.e., similar ambulance demand peaks and drops. In order to exploit these similar EMS demand patterns, the proposed DeepPPMNet framework employs a global forecasting methodology (GFM) that simultaneously learns from a collection of time series \cite{Januschowski2020-ud}.
We subsequently perform causal inference on the potential factors that influence the EMS demand, employing the concept of Granger Causality (GC)~\cite{Granger1980-xw,Seth2007-on}, that we define according to Granger~\cite{Granger1980-xw,Seth2007-on} as follows.
Given a target time series $Y_t$, and (multivariate) external factors $W_t$ and $Z_t$.
Let us assume that we have a predictive model that uses past values of $Y_t$ and $W_t$ to predict $Y_{t+1}$. We now define that $Z_t$ Granger causes $Y_{t+1}$, if we are able to improve the accuracy of predicting $Y_{t+1}$ by including past values of $Z_t$ as exogenous values in our model.
Granger~\cite{Granger1980-xw} notes that causality is a complicated concept and no universally accepted definition exists. Granger causality assumes that a cause has to precede the effect in time, i.e., the future cannot cause the past. Then, in the hypothetical case in which $W_t$ and $Z_t$ together cover all possible factors, Granger argues that Granger causality is true causality. In practice, the strength of Granger causality depends heavily on the careful analysis and inclusion of possible influencing factors $W_t$, which effectively determines if Granger causality measures rather a correlation or suggests true causal relationships.
The GC analysis subsequently allows us to perform `what-if' analyses, in the sense that if we are confident that we have identified exogenous variables $Z_t$ as causal factors, we can deliberately change their values to analyse how the predicted outcomes change accordingly. We note that using a GFM based forecasting model like DeepPPMNet greatly improves this methodology, as the model is trained across all the variations that exist among a collection of time series, which makes the `what-if' analysis feasible even for relatively constant features or feature values that have not been observed in a particular time series before.
\begin{figure}[htbp]
\centerline{\includegraphics[width=0.60\textwidth, trim = 4cm 4cm 4cm 4cm]{demand_lgas}}
\caption{Monthly ambulance attendances for suicide attempts of three different LGAs for one of the states in Australia, over a time period of eight years.}
\label{lgademand}
\end{figure}
The main contributions of our paper are as follows:
\begin{itemize}
\item To the best of our knowledge, our work is the first study that employs a GFM in a collection of EMS demand time series for forecasting. We evaluate our model on a real-world EMS demand dataset, which consists of ambulance attendances related to alcohol overdose, suicide attempts, and other drug related harms reported in Australia, and show that our framework achieves competitive forecasting accuracy.
\item Building on these results, we then show how DeepPPMNet provides a platform to conduct GC-based `what-if' analyses, and allows decision makers to assess the factors that could drive the EMS demand, for which they can implement timely intervention or prevention strategies.
\end{itemize}
The remainder of this paper is organized as follows. In Section~\ref{sec:relatedwork}, we review the state of the art in this domain, and discuss the advent of global forecasting models. In Section~\ref{sec:design}, we discuss the proposed DeepPPMNet framework in detail. Our experimental setup is presented in Section~\ref{sec:experiments}, where we apply the proposed DeepPPMNet framework to multiple EMS demand datasets extracted from an internationally unique dataset of coded EMS clinical records. Finally, Section~\ref{sec:conclusion} concludes the paper.
\section{Prior Work}
\label{sec:relatedwork}
Early studies to forecast EMS demand are mostly based on exponential smoothing models \cite{Baker1986-vk}, autoregressive moving average models \cite {Channouf2007-vw,Wong2014-ok}, factor models \cite{Matteson2011-lv}, and simple averaging models \cite{Brown2007-ij}. Baker and Fitzpatrick \cite{Baker1986-vk} use a Holt-Winter's exponential smoothing model along with goal programming to forecast EMS demand in South Carolina. Channouf et al. \cite{Channouf2007-vw} investigate the use of ARIMA models with special-day effects to predict EMS calls in Calgary, Alberta. Furthermore, Wong et al. \cite{Wong2014-ok} use ARIMA models to measure the effect of weather factors for daily ambulance call out demand. Also, Brown et al. \cite{Brown2007-ij} use various forms of averaging techniques to forecast the hourly emergency call volumes. Matteson et al. \cite{Matteson2011-lv} propose a combined inter-valued time series model with a dynamic latent factor structure using hourly emergency call demand data from Toronto, Canada.
Overcoming the limitations of traditional EMS prediction approaches, Setzler et al. \cite{Setzler2009-km} introduce a neural network (NN) based methodology to generate spatio-temporal forecasts for an EMS agency in North Carolina. Based on demand data provided by the Welsh Ambulance Service Trust, Vile et al. \cite{Vile2012-bq} evaluate the application of singular spectrum analysis for daily EMS demand forecasting. Zhou and Matteson \cite{Zhou2015-cx} develop a spatio-temporal kernel density estimation approach to predict ambulance demand in Toronto, Canada. More recently, Chen et al. \cite{Chen2016-ka} use a combination of machine learning techniques to forecast the pre-hospital emergency demand in New Taipei city, while Steins et al.~\cite{Steins2019-eu} use a zero-inflated poisson regression model to forecast ambulance call demand in Sweden.
In parallel to these developments, deep learning based applications have achieved state-of-the-art performance in many domains \cite{Smyl2019-cb,Bandara2019-ua,Flunkert2017-mt}. In the time series forecasting field, the modern advances are mainly around deep learning based GFMs. These models are becoming a strong alternative to the univariate time series forecasting models, in situations where a collection of related time series is available. This is particularly evident from the recent success of such models in forecasting competitions \cite{Smyl2019-cb} and the use in industry applications \cite{Bandara2019-ua,Flunkert2017-mt}. As highlighted in Section~\ref{sec:intro}, compared to univariate time series models, which rely only on the individual level time series data, GFM models benefit from training across multiple time series, allowing them to learn key patterns available in a collection of time series.
\section{DeepPPMNet Framework}
\label{sec:design}
DeepPPMNet is a GFM based forecasting framework, which is designed to train across multiple time series. The DeepPPMNet consist of three layers, namely: 1) the pre-processing layer, 2) the DeepPPMNet training layer, and 3) the post-processing layer. In the following, we discuss the functionality of these layers in detail.
\subsection{Pre-processing layer}
\label{sec:preprocessing}
The pre-processing layer of DeepPPMNet is a three-stepped process. First, the set of EMS demand time series are normalised using a \textit{meanscale} transformation strategy to scale various ambulance call volume ranges of multiple LGAs~\cite{Flunkert2017-mt,Bandara2019-bg}, as DeepPPMNet is trained across a group of time series.
In the second step of pre-processing layer, following the suggestions of \cite{Hewamalage2019-il,Bandara2019-iv}, we perform a log transformation strategy to stabilise the variance of the EMS demand time series. In addition to stabilising the variance, the log transformation also assures additive seasonality and trend in the series, which is necessary for our decomposition layer, which is the last step of our pre-processing layer.
We use a time series decomposition layer as the final pre-processing step of DeepPPMNet. The main objective of this layer is to seasonally adjust the time series, i.e., removing the seasonal component from a time series, and thereby supplementing the subsequent DeepPPMNet learning phase. This deseasonalisation strategy is advocated by many studies, when using neural networks for time series forecasting~\cite{Bandara2019-si,Bandara2019-iv}. Due to the promising performance of \textit{Prophet} \cite{Taylor2017-lw} as a multi-seasonal decomposition technique in the recent past~\cite{Bandara2019-si}, we use Prophet as an additive, single-seasonal decomposition technique to obtain the seasonal components present in a time series. This additive decomposition can be defined as follows:
\begin{equation}
X_t = \hat{S}_t + \hat{T}_t + \hat{R}_t + \hat{H}_t
\label{multiadditive}
\end{equation}
Here, $X_t$ denotes the EMS demand observation at time $t$, and the decomposed seasonal, trend, remainder, and holiday effect components of the time series are represented by $\hat{S}_t $, $\hat{T}_t$, $\hat{R}_t$, and $\hat{H}_t$ respectively. We perform the seasonal extraction, after fitting the Prophet model using the \verb|prophet| package implemented in R \cite{Taylor2018-wb}. Also, this deseasonalisation process can be replaced with other additive decomposition techniques such as seasonal and trend decomposition using loess (STL) \cite{Cleveland1990-rc}.
\subsection{DeepPPMNet Training layer}
As the primary prediction unit of DeepPPMNet, we use an LSTM \cite{Hochreiter1991-en}, a special type of neural network which is naturally suited for modelling time series data, because of its ability to capture long-term dependencies in sequences. LSTMs have been heavily used in sequence modelling tasks, such as Natural Language Processing, speech recognition, and more recently have received significant attention in the time series forecasting domain \cite{Smyl2019-cb,Bandara2019-iv,Flunkert2017-mt}. It is also worth mentioning that the proposed DeepPPMNet framework can potentially be employed with other RNN variants such as the Elman RNN \cite{Elman1990-my} or the Gated Recurrent Unit (GRU) cell \cite{Cho2014-or}. For more discussions on the LSTM architecture, we refer to Bandara et al.~\cite{Bandara2019-iv}.
\subsubsection{Moving Window Transformation}
In order to train the DeepPPMNet, we use the past observations of time series ($X_i$) in the form of input and output frames. Following recommendations in the literature~\cite{Smyl2019-cb,Bandara2019-iv,Hewamalage2019-il}, we achieve this by applying the Moving Window (MW) tranformation strategy to each pre-processed time series. In this process, the size of the output frame is equivalent to the size of the intended forecasting horizon $M$, i.e., a Multi-Input Multi-Output (MIMO) principle is used in multi-step forecasting. This enables the DeepPPMNet to directly predict all the future observations up to the intended forecasting horizon $X^M_{i}$, while avoiding prediction error accumulation at each forecasting step \cite{Ben_Taieb2012-re}. For further discussions of the MW transformation strategy we refer to Hewamalage et al.~\cite{Hewamalage2019-il}.
\subsubsection{DeepPPMNet Training Paradigms}
\label{sec:trainingpara}
Following Bandara et al.~\cite{Bandara2019-si}, we apply two training paradigms to the output generated by the decomposition layer.
In the first training paradigm, Deseasonalised Approach (DS), the MW strategy is applied to a seasonally adjusted time series, and the transformed input and output windows are used to train the DeepPPMNet. As the seasonal components are excluded from the time series, an additional reseasonalisation process is introduced in the post-processing layer to predict the future seasonal values of the time series. These seasonal values are estimated by repeating the last seasonal components of the time series (extracted by Prophet) to the intended forecast horizon. The DS approach can be seen as a `boosting' ensemble technique \cite{Schapire2003-xb}, where the Prophet method used for the deseasonalisation process acts as a weak base learner to supplement the LSTM training procedure.
In the second training paradigm, Seasonal Exogenous Approach (SE), instead of detaching the seasonal components from the time series, we use the extracted seasonal components as exogenous variables together with the original time series (after normalising and log-transforming). As the original distribution of the time series is used to generate the training windows (using the MW process), the DeepPPMNet is expected to forecast all components of a time series, including extrapolation of the seasonality. In this way, SE supplements the DeepPPMNet by assisting to model the seasonal trajectories of a time series.
Fig.~\ref{lstm-schemes} illustrates the overview of the proposed two training paradigms and the overall procedure of the DeepEMSFNet .
\begin{figure*}[htp]
\subfloat[The proposed DS training paradigm used in the DeepPPMNet framework]{%
\centerline{\includegraphics[width=0.45\textwidth, trim = 8cm 8cm 7cm 8cm]{arch1_converted}}
}
\vspace{1mm}
\subfloat[The proposed SE training paradigm used in the DeepPPMNet framework]{%
\centerline{\includegraphics[width=0.45\textwidth, trim = 8cm 8cm 7cm 8cm]{arch2_converted}}
}
\caption{An overview of the proposed DeepPPMNet framework. A reseasonalisation phase is required for the post-processing layer of the DS approach, as the seasonal components are not included in the DeepPPMNet training phase. Whereas, the SE approach uses the pre-processed time series along with the seasonal components as external variables, and directly forecast all the components of the time series, thus a reseasonalisation phase is not required.}
\label{lstm-schemes}
\end{figure*}
\subsubsection{DeepPPMNet Learning Scheme}
Following the recommendations of Bandara et al.~\cite{Bandara2019-iv} and Hewamalage et al.~\cite{Hewamalage2019-il}, we implement the ``LSTM Stacking Layer'' design pattern in DeepPPMNet. Here, a stacking of LSTM layers is followed by an affine neural layer (excluding the bias component) to map each LSTM cell output to the dimension of the intended forecasting horizon. As we follow the MIMO approach, when generating the input and output windows, the dimension of this fully connected neural layer is equivalent to the size of the output window.
To avoid possible network saturation effects that occur in neural networks \cite{Bandara2019-iv,Hewamalage2019-il}, we perform a local normalisation process on the training windows, before using them for the DeepPPMNet training procedure. We implement two different normalisation strategies for each proposed training paradigm. The trend component of the last value of the input window is used in the DS approach, whereas in the SE approach, the mean value of each input window is used.
We implement the above training paradigms and learning schemes of the DeepPPMNet using TensorFlow, an open-source deep-learning toolkit \cite{Abadi2016-rr}. The operating environment in EMS is often volatile and may present irregular demand patterns and anomalies. To be more robust under these circumstances, we use L1-norm as the primary learning objective function of DeepPPMNet. Moreover, we also include a L2-regularisation term to minimise possible overfitting of the network.
\subsection{Post-processing Layer}
The post-processing layer of DeepPPMNet consists of two phases, namely the reseasonalisation phase and the renormalisation phase. Firstly, in the reseasonalisation phase, the seasonal components estimated by the Prophet method are added back to the forecasts generated by the LSTM. As discussed in Section~\ref{sec:trainingpara}, this phase is only required by the DS training paradigm, as the seasonal components of the time series are not included in the DeepPPMNet training procedure. Secondly, in the renormalisation phase, the corresponding forecasts are back-transformed into the original scale, by adding back the local normalisation factor used in each training window, taking the exponent of the values, and finally multiplying by the mean scaling factor of each time series that is used in the normalisation process. The renormalisation phase is required by both the DS and SE training paradigms, as the mean normalisation and log transformation are common processes for both training paradigms.
\section{Experiments}
\label{sec:experiments}
In this section, we present the experimental setup used to evaluate the DeepPPMNet framework on real-world EMS datasets. This includes a summary of the datasets, error metrics, hyper-parameter selection, benchmarks and a discussion of the results obtained.
\subsection{Datasets}
We use an internationally unique, national dataset of coded ambulance clinical records held by Turning Point, an Australian addiction research and education centre. The dataset (The National Ambulance Surveillance System or NASS) \cite{TurningPoint2020-nw} holds surveillance data on alcohol and other drug, self-harm and mental health-related ambulance attendances across 5 of the 6 Australian states and 2 territories. We subdivide the dataset into five categories of ambulance attendance datasets namely: 1) the alcohol overdose (AO) dataset, 2) the suicide-attempt (SA) dataset, 3) the heroin overdose (HO) dataset, 4) the methamphetamine overdose (MO) dataset, and 5) the prescription Opioids overdose (PO) dataset. For further definitions of these categories, we refer to \cite{Lloyd2013-ky}.
We use 8 years worth of monthly EMS data available from September-2011 to May-2019. The last 12 months of EMS observations, i.e., May-2018 to May-2019, are reserved for testing. The size of the training output window is 12, as this is the intended forecasting horizon. Also, we choose the size of the training input window as 15 (12*1.25), following the heuristic proposed in \cite{Hewamalage2019-il, Bandara2019-ua}. The number of LGAs, i.e., number of time series, present in the AO, SA, HO, MO and PO categories are 89, 88, 80, 87 and 87, respectively.
Fig.~\ref{lgademand2} shows violin plots generated using the seasonal components of each EMS category. This enables us to analyse how seasonal strength can affect the performance of the DeepPPMNet training paradigms, i.e., DS and SE approaches. We observe that the AO category carries a higher seasonal strength compared to other categories. This can be mainly attributed to the higher consumption of alcohol during the holiday periods.
\begin{figure}[htbp]
\centerline{\includegraphics[width=0.60\textwidth, trim = 2cm 2cm 2cm 2cm]{ems_demand}}
\caption{The yearly seasonal strength distributions of the AO, SA, HO, MO and PO categories. The seasonal components are extracted using Prophet as a decomposition technique.}
\label{lgademand2}
\end{figure}
\subsection{Error Metrics}
To assess the accuracy of the DeepPPMNet, we use two evaluation metrics commonly used in the time series forecasting literature, namely the symmetric Mean Absolute Percentage Error (sMAPE) and the Mean Absolute Scaled Error (MASE), which are scale independent error measures \cite{Hyndman2006-ue}.
Using these measures to calculate the forecast errors independently of the scale of the time series leads to effectively every series contributing equally to the error, even though the LGA-wise EMS dataset may contain different scales of time series. This may be appropriate for certain applications but not for others. However, these measures are also commonly used in forecasting competitions nowadays, and can be considered as standard evaluation measures in forecasting. The sMAPE and MASE error measures are defined as follows:
\begin{equation}
\text{sMAPE} = \frac{2}{m}\sum_{t=1}^{m}\left(\frac{\left|F_t - Y_t\right|}{\left| F_t\right| + \left| Y_t\right|} \right)
\label{smape}
\end{equation}
\begin{equation}
\text{MASE} = \frac{1}{m}\frac{\sum_{t=1}^{m}|F_t - Y_t|}{\frac{1}{n-S}\sum_{t=S+1}^{n}|Y_{t}-Y_{t-S}|}
\label{mase}
\end{equation}
Here, $F_t$ denotes the forecasts generated by DeepPPMNet, and $Y_t$ is the actual EMS observation at time $t$. Furthermore, $m$ represents the number of data points in the test set and $n$ is the number of observations in the training set of a time series. $S$ refers to the frequency of the seasonality in a given time series, set to 12 in our experiments, as we use monthly EMS demand time series. The mean and median of these error measures are reported in our evaluation.
\subsection{Hyper-parameter Tuning and Optimisation}
A Python implementation of a Bayesian hyper-parameter optimisation process, the sequential model-based algorithm configuration (SMAC) \cite{Hutter2011-wa} is used to autonomously determine the optimal hyper-parameters of DeepPPMNet. As we employ COntinuous COin Betting \cite{Orabona2017-cd}, tuning the learning rate is not necessary in DeepPPMNet. Table \ref{tab:parametergrid} summarises the bounds of the hyper-parameter values used in the DeepPPMNet learning process.
We also address the parameter uncertainty of our proposed framework by training DeepPPMNet on 10 different Tensorflow graph seeds, using the optimised hyperparameters from SMAC. We generate the final forecasts of DeepPPMNet by computing the median of the forecasts that are produced from different initialisation seeds of DeepPPMNet.
\begin{table}
\caption{DeepPPMNet hyper-parameter grid}
\begin{center}
\begin{tabular}{lcc}
\toprule
Model Parameter &Minimum value &Maximum value\\ \hline
LSTM-cell-dimension &20 &50 \\
Mini-batch-size &1 &10 \\
Epoch-size &2 &10 \\
Maximum-epochs &3 &40 \\
Hidden Layers &1 &2\\
Gaussian-noise-injection &$10^{-4}$ &$8 \cdot 10^{-4}$\\
Random-Normal-Initialiser &$10^{-4}$ &$8 \cdot 10^{-4}$\\
L2-regularisation-weight &$10^{-4}$ &$8 \cdot 10^{-4}$\\ \hline
\end{tabular}
\label{tab:parametergrid}
\end{center}
\end{table}
\subsection{Benchmarks and DeepPPMNet variants}
We use a collection of state-of-the-art univariate forecasting techniques to benchmark against. This includes well-established methods for monthly time series forecasting, such as \textit{ETS} \cite{Hyndman2008-yd}, \textit{ARIMA} \cite{Box2015-bz}, and recently developed forecasting methods, such as \textit{BaggedETS} \cite{Bergmeir2016-zk}, and \textit{Prophet} \cite{Taylor2017-lw}. Here, the ETS, ARIMA, and BaggedETS methods are used from the \verb|forecast| package implemented in R \cite{Hyndman2015-vm}, while the Prophet method is used from the \verb|prophet| package in R.
When building GFMs, as the first approach, we train separate DeepPPMNet models on each EMS category (GFM-CAT). As the second approach, we build one GFM across all the available EMS category datasets (GFM-ALL). Based on the proposed training paradigms and the GFM modelling approaches, we define the following variants of DeepPPMNet.
\begin{itemize}
\item \textit{DeepPPMNet-DS}: The DeepPPMNet framework that uses DS as the training paradigm and a separate GFM is built on each EMS category.
\item \textit{DeepPPMNet-SE}: The DeepPPMNet framework that uses SE as the training paradigm and a separate GFM is built on each EMS category.
\item \textit{DeepPPMNet-DS-ALL}: The DeepPPMNet framework that uses DS as the training paradigm and one GFM is trained across all the EMS categories.
\item \textit{DeepPPMNet-SE-ALL}: The DeepPPMNet framework that uses SE as the training paradigm and one GFM is trained across all the EMS categories.
\end{itemize}
\subsection{Results and Discussion}
Table~\ref{tab:ao} shows the evaluation summary of the DeepPPMNet variants and benchmarks for the 89 LGAs of the AO category, ordered by the Mean sMAPE. For each column, the results of the best performing methods are marked in boldface. According to Table~\ref{tab:ao}, we see that the proposed DeepPPMNet-DS-ALL variant achieves the best Mean sMAPE, Mean MASE, and Median sMAPE, while Prophet obtains the best Median MASE. Also, among the proposed variants, we observe that the DeepPPMNet variants with the DS training paradigm achieve better accuracies than the SE training paradigm variants. Furthermore, overall the GFM-ALL approach obtains better results than the GFM-CAT approach. Most importantly, we see that the DeepPPMNet-DS-ALL variant outperforms state-of-the-art methods, such as ETS, BaggedETS, and ARIMA consistently across all performance metrics.
\begin{table}[!tb]
\caption{AO Dataset Results}
\centering
\resizebox{\columnwidth}{!}{%
\begin{tabular}{lcccc}\hline
Method &Mean sMAPE &Median sMAPE &Mean MASE &Median MASE\\ \hline
DeepPPMNet-DS-ALL &\textbf{0.1243} &\textbf{0.1251} &\textbf{0.9182} &\textbf{0.8959}\\
DeepPPMNet-DS &0.1254 &0.1318 &0.9235 &0.9119\\
Prophet &0.1281 &0.1261 &0.9585 &0.8964\\
BaggedETS &0.1286 &0.1299 &0.9472 &0.9466\\
ETS &0.1291 &0.1284 &0.9499 &0.9568\\
ARIMA &0.1332 &0.1351 &0.9691 &0.9762\\
DeepPPMNet-SE-ALL &0.1334 &0.1392 &0.9883 &0.9755\\
DeepPPMNet-SE &0.1417 &0.1435 &1.0340 &1.0289\\
\hline
\end{tabular}
}
\label{tab:ao}
\end{table}
Table~\ref{tab:sa} summarises the results for the evaluations on the 88 LGAs of the SA category. It can be seen that the proposed DeepPPMNet-SE variant outperforms all other methods consistently across all error measures. Most importantly, we observe that contrary to our previous findings from the AO dataset, the SE variants achieve better results than the DS variants. Also, among the SE variants, we see that training a separate GFM gives better results than building one GFM across all the available time series. However, the observations among the DS based variants are reciprocal to those of SE.
\begin{table}[!tb]
\caption{SA Dataset Results}
\centering
\resizebox{\columnwidth}{!}{%
\begin{tabular}{lcccc}
\hline
Method &Mean sMAPE &Median sMAPE &Mean MASE &Median MASE\\ \hline
DeepPPMNet-SE &\bf 0.1128 &\bf 0.1126 &\bf 0.8817 &\bf 0.8088\\
DeepPPMNet-SE-ALL &0.1131 &0.1153 &0.8835 &0.8263\\
ETS &0.1146 &0.1201 &0.8893 &0.8212\\
BaggedETS &0.1148 &0.1193 &0.8996 &0.8299\\
ARIMA &0.1159 &0.1247 &0.8946 &0.8281\\
Prophet &0.1206 &0.1233 &1.0174 &0.8988\\
DeepPPMNet-DS-ALL &0.1231 &0.1189 &0.9547 &0.8858\\
DeepPPMNet-DS &0.1266 &0.1210 &0.9652 &0.9164\\
\hline
\end{tabular}
}
\label{tab:sa}
\end{table}
Table~\ref{tab:mo} shows the evaluation summary for the 87 LGAs of the MO category. We observe that the BaggesETS method achieves the best accuracy in terms of Mean sMAPE, and the DeepPPMNet-SE-ALL variant obtains the best Median sMAPE. Meanwhile, the DeepPPMNet-SE variant achieves the best Mean MASE, while the DeepPPMNet-SE-ALL variant and the Prophet method are the best-performing methods with respect to Median MASE. Furthermore, similar to our previous findings from Table~\ref{tab:sa}, the DS variants show poor results compared to the SE variants, and overall the GFM-ALL approach obtains better results than its counterpart.
\begin{table}[!tb]
\caption{MO Dataset Results}
\centering
\resizebox{\columnwidth}{!}{%
\begin{tabular}{lcccc}
\hline
Method &Mean sMAPE &Median sMAPE &Mean MASE &Median MASE\\ \hline
BaggedETS &\bf 0.0917 &0.0859 &1.0751 &0.9946\\
DeepPPMNet-SE-ALL &0.0919 &\bf 0.0832 &1.0739 &\bf 0.9861\\
DeepPPMNet-SE &0.0927 &0.0919 &\bf 1.0703 &0.9946\\
Prophet &0.0929 &0.0951 &1.1264 & \bf 0.9861\\
ETS &0.0930 &0.0912 &1.0861 &0.9946\\
ARIMA &0.0935 &0.0845 &1.0953 &0.9946\\
DeepPPMNet-DS-ALL &0.0973 &0.0983 &1.1195 &1.0703\\
DeepPPMNet-DS &0.1056 &0.1043 &1.1607 &1.1221\\
\hline
\end{tabular}
}
\label{tab:mo}
\end{table}
Table~\ref{tab:ho} provides the results for the evaluations on the 80 LGAs of the HO category. Except for Median sMAPE, DeepPPMNet-SE-ALL outperforms all the benchmarks on each performance metric, whereas the proposed DeepPPMNet-DS achieves the best Median sMAPE. Similar to Table~\ref{tab:sa} and Table~\ref{tab:mo}, variants based on the SE training paradigm achieve better accuracies compared to those of DS. Overall, the GFM-ALL approach obtains better results compared to GFM-CAT approach.
\begin{table}[!tb]
\caption{HO Dataset Results}
\resizebox{\columnwidth}{!}{%
\begin{tabular}{lcccc}
\hline
Method &Mean sMAPE &Median sMAPE &Mean MASE &Median MASE\\ \hline
DeepPPMNet-SE-ALL &\bf 0.0780 &0.0634 &\bf 1.2118 &\bf 0.9685\\
ETS &0.0786 &0.0634 &1.2254 &0.9807\\
BaggedETS &0.0788 &0.0598 &1.2655 &0.9989\\
DeepPPMNet-SE &0.0804 &0.0634 &1.2137 &0.9925\\
DeepPPMNet-DS-ALL &0.0814 &0.0610 &1.2314 &1.0964\\
ARIMA &0.0809 &0.0654 &1.2811 &0.9989\\
Prophet &0.0813 &0.0647 &1.3476 &1.0857\\
DeepPPMNet-DS &0.0825 &\bf 0.0597 &1.2433 &1.0964\\
\hline
\end{tabular}
}
\label{tab:ho}
\end{table}
Table~\ref{tab:po} shows the results of the benchmarks for the 87 LGAs of the PO category. It can be seen that the proposed DeepPPMNet-SE variant achieves the best results on each performance metric. Once again, we observe that among the proposed training paradigms, the SE based variants perform better than the DS based variants. However, contrary to our previous findings, the DeepPPMNet variants that use the GFM-CAT approach, i.e., a separate DeepPPMNet is built only using the PO dataset, achieve better results compared to the GFM-ALL.
\begin{table}[!tb]
\centering
\caption{PO Dataset Results}
\resizebox{\columnwidth}{!}{%
\begin{tabular}{lcccc}
\hline
Method &Mean sMAPE &Median sMAPE &Mean MASE &Median MASE\\ \hline
DeepPPMNet-SE &\bf 0.0620 &\bf 0.0528 &\bf 0.8606 &\bf 0.7759\\
ETS &0.0629 &0.0639 &0.8653 &0.8068\\
BaggedETS &0.0634 &0.0662 &0.8803 &0.8068\\
DeepPPMNet-DS &0.0669 &0.0661 &0.9366 &0.8539\\
ARIMA &0.0645 &0.0639 &0.8821 &0.8068\\
DeepPPMNet-SE-ALL &0.0664 &0.0664 &0.9071 &0.8068\\
DeepPPMNet-DS-ALL &0.0679 &0.0675 &0.9640 &0.8677\\
Prophet &0.0699 &0.0699 &1.0456 &0.9297\\
\hline
\end{tabular}
}
\label{tab:po}
\end{table}
In summary, we observe that the proposed DeepPPMNet variants achieve competitive results on the benchmark datasets, outperforming many state-of-the-art univariate forecasting techniques. This highlights the contribution made by employing a GFM strategy to exploit the similar EMS demand patterns. Moreover, we see that in the majority of cases, the GFM-ALL approach that uses all the available EMS category datasets performs better than the GFM-CAT approach that uses only the data specific to an EMS category. This can be mainly attributed to the low number of time series available in each EMS category, i.e., a maximum of 89 time series, for the AO category. On the other hand, grouping all the EMS categories together allows us to build a GFM across a total of 431 time series, and therewith allows our deep-learning based framework to learn better with more available data. It is also noteworthy to mention that the proposed DS training paradigm only outperforms the SE training paradigm in the AO dataset. This can be mainly attributed to the higher seasonal strength present in the AO dataset, compared to other EMS categories (see Fig.~\ref{lgademand2}).
\begin{figure*}[ht]
\centering
\subfloat[Type-1: 5\% ALI sensitivity]{
\includegraphics[width=0.50\textwidth,trim = 3cm 3cm 3cm 3cm]{5_perfect}
\label{fig:subfig1}
\subfloat[Type-2: 5\% ALI sensitivity]{
\includegraphics[width=0.50\textwidth,trim = 3cm 3cm 3cm 3cm]{effect2_5}
\label{fig:subfig2}
\qquad
\subfloat[Type-1: 10\% ALI sensitivity]{
\includegraphics[width=0.50\textwidth,trim = 3cm 3cm 3cm 3cm]{10_perfect}
\label{fig:subfig3}
\subfloat[Type-2: 10\% ALI sensitivity]{
\includegraphics[width=0.50\textwidth,trim = 3cm 3cm 3cm 3cm]{effect2_10}
\label{fig:subfig4}}
\caption{The application of `what-if' scenario analysis, using the number of ALI (as a percentage of change) against the AO related EMS demand.}
\label{fig:sens}
\end{figure*}
\subsection{DeepPPMNet for Policy-making}
The competitiveness of the DeepPPMNet forecasting framework allows us to work towards assessing causing factors of EMS demand. In the following, we demonstrate the use of the DeepPPMNet framework as a platform to perform GC analyses using external time series. As a use case, we investigate how the number of alcohol licenses issued (ALI) for a certain period of time can affect the AO related EMS demand patterns. This can enable government policy makers to understand effects of different numbers of ALI by conducting `what-if' scenario analyses.
\subsubsection{Experimental Setup}
For this experiment, we use the monthly EMS data related to AO demand for 79 LGAs, and add the number of ALI as an exogenous variable to the current training input window.
In particular, the DeepPPMNet uses a concatenation of the EMS demand and the ALI to learn the actual EMS demand of the output window.
We consider the five year period from January-2015 to May-2019, as for this period we have both the EMS data and the corresponding ALI data available. The observations of the last 12 months are used to test the DeepPPMNet, the rest of the data are used for training. We use the DeepPPMNet in the GFM-CAT mode, i.e., the GFM is built only using the alcohol related data, assuming other datasets are not available. Also, in the decomposition layer, we replace the \textit{Prophet} method with \textit{STL}, as in this experiment the time series are shorter (42 monthly observiations), and \textit{STL} is more robust and better suited for shorter series.
In this way, two additional DeepPPMNet variants are introduced, DeepPPMNet-SE-License and DeepPPMNet-DS-License, to represent the models that use the number of ALI as an exogenous variable.
Furthermore, analogously we use the ARIMA benchmark in the exogenous-regressive mode, using the number of ALI as an additional regressor (ARIMAX-License).
\subsubsection{Accuracy Results}
Table~\ref{tab:aol} shows the results for the evaluations on the 79 LGAs of the AO category.
Furthermore, we perform non-parametric Friedman rank-sum statistical significance testing to assess the statistical significance within the benchmarks, and the Hochberg's post-hoc procedure is used to further examine these differences \cite{Garcia2010-jx}. The statistical testing is carried out on the sMAPE error measure, with a significance level of $\alpha$ = 0.05.
Table~\ref{tab:alcoholstat} shows the corresponding results of the statistical testing evaluation. A horizontal line is used to separate the methods that perform significantly worse than the best performing method. The overall result of the Friedman rank sum test gives a $p$-value of $6.91 \times 10^{-11}$, which means the differences are highly significant.
We see that DeepPPMNet-SE-License performs best, and achieves significantly better results than the DeepPPMNet-DS variants and the other benchmarks. However, we also observe that the DeepPPMNet-SE, BaggedETS, and ARIMAX-License variants do not perform significantly worse than the control method, DeepPPMNet-SE-License.
The results confirm the earlier findings of the overall competitiveness of our methodology and the benefits of employing a GFM based forecasting model in this framework.
\subsubsection{Effect of Number of ALI}
Regarding the effect of the number of ALI, we see that after incorporating it as an exogenous variable (DeepPPMNet-SE-License), the accuracy of DeepPPMNet-SE has improved, outperforming the rest of the DeepPPMNet variants and the statistical benchmarks. We also observe that the ARIMAX-License variant has uplifted the results of the original ARIMA method, showing that inclusion of ALI as an exogenous variable is able to improve predictive accuracy over different methods.
Furthermore, we perform a non-parametric paired Wilcoxon signed rank test to explore the statistical significance of the differences. The test indicates a $p$-value of 0.687 for the DeepPPMNet-SE-License and DeepPPMNet-SE variants, and a $p$-value of 0.704 for the ARIMAX-License and ARIMA variants, which indicates that the differences are not statistically significant.
As the results indicate that incorporating the number of ALI issued as an exogenous variable has improved the base accuracies of DeepPPMNet-SE and ARIMA, the number of ALI is a viable candidate for further exploration of a causal relationship.
However, as the significance tests have not detected significant differences, further experiments with a larger dataset are needed in the future to clearly establish the relationship.
\subsubsection{Discussion of Causality of the Observed Effect}
As discussed in Section~\ref{sec:intro}, GC assesses an external regressor $Z_t$ with respect to other regressors $W_t$. We have established that the inclusion of ALI as $Z_t$ into our model is in fact able to improve forecasting accuracy across different methods. However, the strength of GC heavily depends on careful inclusion of additional factors $W_t$. In the current work, we suspect that ALI could work as a mere proxy to distinguish between rural and metropolitan areas, so, additional factors to be considered as $W_t$ could be, e.g., the amount of supermarkets in a given area, the amount of cafes or other non-alcohol-serving outlets, the amount of GPs, and others. The proposed DeepPPMNet framework is a platform where conveniently all these factors can be included in the model, to carefully assess the relationships. However, we leave such a careful analysis for future work, and illustrate next how to perform a series of `what-if' analyses on the AO related EMS demand once the GC analysis is done and the number of ALI is identified as a driving factor.
\begin{table}[!tb]
\centering
\caption{AO Results with Alcohol Licenses}
\resizebox{\columnwidth}{!}{%
\begin{tabular}{lcccc}
\hline
Method &Mean sMAPE &Median sMAPE &Mean MASE &Median MASE\\ \hline
DeepPPMNet-SE-License &\bf 0.1411 &\bf 0.1334 &\bf 0.9824 &\bf 0.9031\\
DeepPPMNet-SE &0.1424 &0.1414 &1.0009 &0.9964\\
BaggedETS &0.1476 &0.1403 &1.0501 &1.0000\\
ARIMAX-License &0.1494 &0.1485 &1.0472 &1.0116\\
ARIMA &0.1507 &0.1462 &1.0518 &0.9879\\
ETS &0.1505 &0.1447 &1.0636 &1.0121\\
DeepPPMNet-DS &0.1528 &0.1467 &1.0635 &0.9667\\
DeepPPMNet-DS-License &0.1539 &0.1478 &1.0703 &1.0175\\
Prophet &0.1870 &0.1862 &1.2522 &1.2083\\
\hline
\end{tabular}
}
\label{tab:aol}
\end{table}
\begin{table}[!tb]
\caption{Significance testing for AO Results}
\centering
\footnotesize
\begin{tabular}{ll}
\toprule
Method &$p_{Hoch}$\\ \hline
DeepPPMNet-SE-License &-\\
DeepPPMNet-SE &0.559 \\
BaggedETS &0.217 \\
ARIMAX-License &0.089\\
\hline
ETS &0.038\\
ARIMA &0.035\\
DeepPPMNet-DS &0.006\\
DeepPPMNet-DS-License &1.729 $\times$ $10^{-4}$ \\
Prophet &2.254 $\times$ $10^{-15}$ \\
\hline
\end{tabular}
\label{tab:alcoholstat}
\end{table}
\subsubsection{`What-if' analysis}
Fig.~\ref{fig:sens} illustrates an example of performing a `what-if' scenario analysis to determine the sensitivity of ALI towards the AO specific EMS demand. Here, a 5\% increase/decrease (Fig.~\ref{fig:subfig1} and Fig.~\ref{fig:subfig2}) and a 10\% increase/decrease (Fig.~\ref{fig:subfig3} and Fig.~\ref{fig:subfig4}) is applied to the current number of ALI. We present two different types of demand variations present in two different LGAs. Fig.~\ref{fig:subfig1} represents the 5\% increase/decrease effect of the ALI on LGA-1, while Fig.~\ref{fig:subfig3} shows the 10\% increase/decrease effect of the ALI on LGA-1. We see that the increase of ALI has increased the predicted EMS demand, while a decrease in ALI has decreased EMS demand. Likewise, Fig.~\ref{fig:subfig2} and Fig.~\ref{fig:subfig4} show the 5\% increase/decrease and the 10\% increase/decrease effect of ALI on LGA-2. In this case, though decreasing the ALI has reduced the predicted EMS demand related to AO, the increase of ALI has not substantially affected the predicted EMS demand. Such insights may help government policy makers, when taking strategic decisions about the number of alcohol licenses to be issued for a specific LGA. Likewise, the DeepPPMNet can be used to conduct GC and `what-if' analyses on other EMS categories. For example, the number of pharmacy licenses issued could be explored as a causal factor that affects PO related EMS demand.
\section{Conclusion}
\label{sec:conclusion}
Overcoming the limitations of the current univariate state-of-the-art models, we have introduced DeepPPMNet, a three-layered, unified global forecasting framework that is capable of exploiting the non-trivial demand relationships present in EMS demand time series.
We have evaluated DeepPPMNet using a real-world EMS dataset. The results have shown that the proposed DeepPPMNet framework has outperformed state-of-the-art univariate forecasting techniques, indicating the benefit of exploiting cross-series information available in the EMS demand time series.
Furthermore, we have demonstrated how DeepPPMNet can be employed straightforwardly for causal analyses using the notion of Granger Causality, and how we subsequently can perform `what-if' scenario analyses to assist government authorities in the policy-making process.
As future work, we want to include further external factors in the analysis around alcohol outlet licenses, and we want to use DeepPPMNet on related cases, to demonstrate and make use of the full potential of our framework. From a methodologic point of view, we will enable the framework to perform probabilistic forecasting, for an even more informed policy making process.
\section{Acknowledgements}
\label{sec:ach}
This research was supported by the Australian Research Council under grant DE190100045, by a Facebook Statistics for Improving Insights and Decisions research award, and by Monash Institute of Medical Engineering seed funding. We acknowledge Sharon Matthews and the research assistants who code the NASS data and the ambulance services and paramedics who create and provide that data.
\bibliographystyle{IEEEtran}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 8,951 |
Here you can check your Lap Times, Sessions, Race results and other important data from our Assetto Corsa Racing Servers and you can also see statistical data about T7R in Live for Speed.
If you want to see who did the most poles, burnt the most fuel, won most races, earned the most poduims or what all T7R members have reached together in LFS. Then our T7R LFS Statistics are made for you. | {
"redpajama_set_name": "RedPajamaC4"
} | 6,811 |
\section{Introduction}
Image-to-image translation has been made much progress~\cite{Isola_2017_CVPR,Zhu_2017_ICCV,Yi_2017_ICCV,Chen_2017_ICCV,kim2017learning,mao2018semantic} because many tasks~\cite{zhou2016similarity,wang2016super,ma2017unsupervised,shi2017end} in image processing, computer graphics, and computer vision can be posed as translating an input image into a corresponding output image~\cite{zhu2016generative,li2016precomputed,wang2016generative,Ledig_2017_CVPR,taigman2017unsupervised,Sela_2017_ICCV,Tung_2017_ICCV}. And its achievements mainly owes to the success of Generative Adversarial Networks (GANs)~\cite{goodfellow2014generative}, especially conditional GANs (cGANs)~\cite{mirza2014conditional,radford2016unsupervised,Isola_2017_CVPR}. However, the current studies mainly concern image-to-image translation tasks with low-level visual information conversion, \emph{e.g.}, photo-to-sketch~\cite{liu2018auto}.
A caricature is a rendered image showing the features of its subject in an exaggerated way and usually used to describe a politician or movie star for political or entertainment purpose. Creating caricatures can be considered as artistic creation tracing back to the 17th century with the profession of caricaturist. Then some efforts have been made to produce caricatures semi-automatically using computer graphics techniques~\cite{akleman2000making}, which intend to provide warping tools specifically designed toward rapidly producing caricatures. But there are very few software programs designed specifically for automatically creating caricatures, and to the best of our knowledge, none can work to be comparable with caricaturist. Nowadays, besides the political and public-figure satire, caricatures are also used as gifts or souvenirs, and more and more museums dedicated to caricature throughout the world were opened. So it would be very useful and meaningful if computers can create caricatures from photos automatically and intelligently.
Photo-to-caricature is a typical high-level image-to-image translation problem but with bigger challenge than other low-level translation problems such as photo-to-label, photo-to-map, or photo-to-sketch~\cite{Isola_2017_CVPR}, because caricatures
\begin{itemize}
\item require \textbf{satire} and \textbf{exaggeration} of photos;
\item need \textbf{artistry} with different styles;
\item must be \textbf{lifelike}, especially the expression of a face photo.
\end{itemize}
Specifically, for a face photo, we want to create the face caricatures with different styles, which exaggerate the face shape or facial sense organs (\emph{i.e.}, ears, mouth, nose, eyes and eyebrow) but keep the vivid expression while producing artistry.
In this paper, we propose a GAN-based method for learning to translate faces in the wild from the source photo domain to the target caricature domain (see Figure~\ref{fig:intro} for translating examples and Figure~\ref{fig:p2c} for the architecture of our proposed method). Although deep convolutional neural networks with adversarial training~\cite{radford2015unsupervised} can generate images with enough precise facial features~\cite{berthelot2017began}, these images sometimes still have wrong relationships between facial features or mismatch among facial organs such as nose and eyes, \emph{e.g.}, a face with more than two eyes or crooked nose. Traditional GANs can produce correct facial organs but wrong relationships between them, and also it's very challenging to abstract and exaggerate face and facial organs. We attribute this problem to the deficient capacity of the discriminator of GAN to distinguish real-fake images. Therefore, our motivation is to design an adversarial training with multiple discriminators to improve the ability of GAN's discriminator for feature representation.
\begin{figure}[!ht]
\centering
\includegraphics[width=\linewidth]{demo.pdf}
\caption{Translating faces in the wild from photo to caricature with different styles by our proposed method. (a) Example results on IIIT-CFW dataset~\cite{mishra2016iiit}; (b) Example results on PHOTO-SKETCH dataset~\cite{zhang2011coupled,wang2009face}.}
\label{fig:intro}
\end{figure}
Based on the model of CycleGAN~\cite{Zhu_2017_ICCV}, we design a dual pathway model of GAN for high-level image-to-image translation tasks, where one pathway of coarse discriminator is in charge of abstracting the global structure information, while another pathway of fine discriminator is responsible for concerning the local statistics information. For generator, besides the adversarial loss, we provide one more extra perceptual similarity loss to constrain consistency for generated output itself and with the unpaired target domain image. By using our proposed method, the photos of faces in the wild can be translated to caricatures with learned general-purpose exaggerated artistic styles while still keeping the original lifelike expression (see Figure~\ref{fig:intro} and \ref{fig:free} for references). Considering that traditional GANs are not robust and easily attracted by noise, we design a noise-added training procedure to improve the robustness of our model. Inspired by InfoGAN~\cite{chen2016infogan}, we find that auxiliary noise can help model learning the caricature style information while translating images in our task.
We have extensively evaluated our method on IIIT-CFW dataset~\cite{mishra2016iiit}, PHOTO-SKETCH dataset~\cite{zhang2011coupled,wang2009face}, Caricature~\cite{abaci2015matching}, FEI dataset~\cite{thomaz2010new}, Yale dataset~\cite{georghiades1997yale}, KDEF dataset~\cite{lundqvist1998karolinska} and CelebA dataset~\cite{liu2015faceattributes}. The experimental results show that our method can create acceptable caricatures from face photos while current state-of-the-art image-to-image translation methods can't. Also the designed experiments indicate the effectiveness of our proposed dual pathway of discriminators, additional noise input and extra perceptual loss, respectively. Besides, we tested our photo-to-caricature translation method for producing caricatures with adding different proportions of noise to show the translating robustness and style diversity. Furthermore, the proposed method can create caricatures for arbitrary face photos without pre-training on extra face datasets. Another prominent performance of our methods is that our model can capture the expression information and make some abstraction and exaggeration. This might be helpful to fill aforementioned gap of automatic and intelligent caricature creation.
\section{Related Work}
\textbf{Image-to-image translation.} Owing to the success of GANs~\cite{goodfellow2014generative}, especially various conditional GANs~\cite{mirza2014conditional,mathieu2015deep,reed2016generative,yan2016attribute2image,wang2016generative,zhang2017image}, image-to-image translation problems have been made much progress recently, which aims to translate an input image from one domain to another domain given input-output images pairs~\cite{Isola_2017_CVPR}. Earlier image-conditional models for specific applications have achieved impressive results on inpainting~\cite{Pathak_2016_CVPR}, de-raining~\cite{zhang2017image}, texture synthesis~\cite{li2016precomputed}, style transfer~\cite{wang2016generative}, video prediction~\cite{mathieu2015deep} and super-resolution~\cite{Ledig_2017_CVPR}. The general-purpose solution for image-to-image translation developed by Isola \emph{et al.}~\cite{Isola_2017_CVPR} with the released \verb|Pix2pix| software has achieved reasonable results on many translation tasks by using paired images for training such as photo-to-label, photo-to-map and photo-to-sketch. Then CycleGAN~\cite{Zhu_2017_ICCV}, DualGAN~\cite{Yi_2017_ICCV}, and DiscoGAN~\cite{kim2017learning} were proposed for unpaired or unsupervised image-to-image translation with almost comparable results to paired or supervised methods. However, the translation tasks that these work can tackle usually concern the conversion of low-level visual information such as line (photo$\rightarrow$sketch), color (summer$\rightarrow$winter), and texture (horse$\rightarrow$zebra), but it's still very challenging for some translation tasks with the requirement of high-level visual information conversion, \emph{e.g.}, abstraction and exaggeration (photo$\rightarrow$caricature).
Recently, Iizuka \emph{et al.}~\cite{iizuka2017globally} combined two discriminators called global discriminator and local discriminator to improve the adversarial training for image completion task. Experimental results have proven that the global discriminator can distinguish the images based on the global parts while the local discriminator pays attention to the details of parts. We exploit the design of two discriminators for high-level image-to-image translation tasks in this paper.
\textbf{Photo-to-cartoon translation.} Translating photo to cartoon by computer algorithms has been studied for a long time because of the corresponding interesting and meaningful applications. The earlier work mainly relied on the analysis of facial features~\cite{luo2002exaggeration,chen2004automatic,liao2004automatic}, which is hard to be applicable for large-scale faces in the wild. Thanks to the invention of GANs~\cite{goodfellow2014generative}, automatic photo-to-cartoon translation becomes feasible. But most of the current related work mainly focused on the generation of anime~\cite{zhang2017style} or emoji~\cite{taigman2017unsupervised} with specific style\footnote{We consider that anime, emoji, and caricature are three types of cartoon or three subsets of cartoon.}. And these works have nothing to do with caricature creation that needs to be exaggerated, lifelike and artistic.
Unlike the prior works for image-to-image translation dealing with low-level visual information conversion, our study mainly focuses on the translation problems of high-level visual information conversion, \emph{e.g.}, photo-to-caricature. For this purpose, our method differs from the past work in network architecture as well as the layers and losses. We design a dual pathway model of GAN with two discriminators named coarse discriminator and fine discriminator to capture global structure and local statistics respectively, and we apply the perceptual similarity loss for generator. For learning the style information and improving the robustness of our model, we provide the input with auxiliary noise. Here we show that our approach is effective on the face photo-to-caricature translation task, which typically requires high-level visual information conversion.
\section{Method}
Conditional GANs are generative models that learn a mapping from observed input image $x$, to target image $y$, $G:\left\{x\right\}\rightarrow y$. The objective of our conditional GAN can be expressed as
\begin{equation}
\begin{split}
\mathcal{L}_{c}(G,D)=\mathbb{E}_{x,y\sim P_{data}(x,y)}\left[\log D(x,y)\right]+\\
\mathbb{E}_{x\sim P_{data}(x)}\left[\log (1-D(x,G(x)))\right].
\end{split}
\label{eq:cgan}
\end{equation}
The goal is to learn a generator distribution over data $y$ that matches the real data distribution $P_{data}$ by transforming an observed input image $x\sim P_{data}(x)$ into a sample $G(x)$. This generator is trained by playing against an adversarial discriminator $D$ that aims to distinguish between samples from the true data distribution and the generator's distribution. To deal with the high-level visual information conversion for some more challenging image-to-image translation tasks, we exploit one discriminator to dual discriminators and add two more losses (perceptual loss and cycle consistency loss) to adversarial loss, so our method can tackle conversions of both the low-level (line, color, texture, \emph{etc.}) and the high-level (expression, exaggeration, artistry, \emph{etc.}) visual information. And in order to improve the robustness of our model, we provide our model a noise-added training and use auxiliary noise to learn the style information while translating. The overall network architecture and data flow are illustrated in Figure~\ref{fig:p2c}.
\begin{figure}[!ht]
\centering
\includegraphics[width=\textwidth]{architecture.pdf}
\caption{Network architecture and data flow chart of our proposed method for face photo-to-caricature translation.}
\label{fig:p2c}
\end{figure}
\subsection{Cycle consistency loss}
Researchers apply adversarial training to learn the mapping function between two different image domains. Here we use two same generators (named \verb|G1| and \verb|G2| in Figure~\ref{fig:p2c}) for observing the sample distribution of two domains. And $X$ denotes the photo domain while $Y$ denotes the caricature domain in our task. We use our generators to emulate the translation between $X$ and $Y$. However, only using adversarial loss can't guarantee a plausible generation. So researchers use Cycle Consistency Loss ($\mathcal{L}_{cyc}$) to help establishing mapping function between domains~\cite{Zhu_2017_ICCV}. And $\mathcal{L}_{cyc}$ is expressed as
\begin{equation}
\begin{split}
\mathcal{L}_{cyc}(G1,G2)=\mathbb{E}_{x\sim P_{data}(x)}\left[||G_{2}(G_{1}(x))-x||_{1}\right]\\
+\mathbb{E}_{y\sim P_{data}(y)}\left[||G_{1}(G_{2}(y))-y||_{1}\right],
\end{split}
\end{equation}
where $G_{1}$ and $G_{2}$ represent the two generators, and $x$ and $y$ are samples from $X$ and $Y$ domain respectively. We use $L1$ loss for the cycle consistency loss following Zhu \emph{et al.}~\cite{Zhu_2017_ICCV}.
\subsection{Perceptual loss}
\label{sec:percep}
To further reduce the space gap of possible mapping functions between domains, we apply the perceptual loss $\mathcal{L}_{p}$ for our model. For a constrained translation problem, finding an appropriate loss function is critical. We adopt the content loss of Gatys \emph{et al.}~\cite{gatys2016image}, which is also referred as a perceptual similarity loss or feature matching~\cite{bruna2015super,dosovitskiy2016generating,johnson2016perceptual,ledig2016photo}. We apply the perceptual loss to our model followed by the cycle consistence loss, and compute the perceptual loss between unpaired images from different domains to push generator to capture the feature representations. Let $\phi$ denotes a pre-trained visual perception network (we use pre-trained \verb|VGG19| in our experiments) and $n$ denotes the number of feature maps. Different layers in the networks represent low-to-high level information: from edges and color to object and semantic representation. Matching both low and high layers in the perception network can help achieving fantastic translation. And the perceptual loss can be expressed as
\begin{equation}
\mathcal{L}_{p}=\sum^{N}_{n}{\lambda}_{n}||{\Phi}_{n}(y)-{\Phi}_{n}(G(x);\theta))||_{1},
\end{equation}
where $y$ denotes the image from caricature domain in our task and $G(x)$ denotes the synthesized caricature image using $x$. Note that these two images are unpaired.
\subsection{Auxiliary noise input}
Previous researches have fully proved that we can get plausible image results from noise input~\cite{radford2016unsupervised,chen2016infogan,zhao2016energy,berthelot2017began}. In order to improve the robustness and enrich the diversity of image translation between domains, we design a noise-added training procedure before the translation shown in Figure~\ref{fig:p2c}. First we obtain a random noise input from random uniform distribution (range from $0$ to $255$), then we merge the noise input and the raw image input to acquire the final input using approximate weights. Here we define $\alpha$ to denote the proportion of the raw image accounting for the final image. This can be expressed as
\begin{equation}
\label{eq:noise}
x=x_{i}*\alpha+(1-\alpha)*n, \quad 0\leq\alpha\leq1,
\end{equation}
where $x_{i}$ denotes the raw input from photo domain and $n$ denotes the noise that has a uniform distribution $P_{noise}$. Figure~\ref{fig:noise} shows one sample with adding noise. With auxiliary noise input, the GAN object is expressed as:
\begin{equation}
\begin{split}
L_{c}(G,D)=\mathbb{E}_{x,y\sim P_{data}(x,y)}\left[\log D_{p}\left(x,y\right)\right] \\
+ \mathbb{E}_{x\sim P_{data}(x),n\sim P_{noise}(n)}\left[\log (1-D(x,G(x,n)) \right].
\end{split}
\end{equation}
\begin{figure}[!ht]
\centering
\includegraphics[width=0.8\textwidth]{add_noise.pdf}
\caption{Sample for adding noise, here we use $\alpha=0.5$.}
\label{fig:noise}
\end{figure}
We add the auxiliary noise to improve the robustness of our model. And we can get different styles of output through adjusting the noise input (see Section~\ref{sec:info}). Besides, we add the noise from a uniform distribution for making the style information more representable and matching the style space while translating. Furthermore, we also hope to control the style by adding different noise and make the translating conditional, which will be our future work.
\subsection{Dual discriminators}
Traditional GANs usually have one generator and one discriminator to leverage adversarial training. Different from them, we design two different discriminators to capture different level information.
In our method, we propose two different discriminators called coarse discriminator and fine discriminator respectively. The coarse discriminator aims to encourage generator to synthesize images based on global style and structure information for domain translation. In our high level image-to-image task, the coarse discriminator is capable for capturing the structure information and abstracting the representative information of face photo such as emotion and style. While the fine discriminator aims to achieve the feature matching and help generating more plausible and precise images, and the fine discriminator builds the adversarial training for the face details with generator such as lip and eyes. Different from Satoshi \emph{et al.}~\cite{iizuka2017globally}, we are not using the image patches as input of local discriminator, we provide both the two discriminators with the whole image as input while the outputs of the two discriminators are different (see Figure~\ref{fig:p2c}). The output of coarse discriminator is a $4\times 4$ patch matrix after the sigmoid activity function while the output of fine discriminator is $16\times 16$. Note that both the two discriminators are using sigmoid function at the last layer. We have tried using different output size combinations to get different results, and the experiments show that the combination of $16\times 16$ for fine discriminator and $4\times 4$ for coarse discriminator can obtain the best result for translation. The coarse discriminator has a smaller feature map with more abstractive representation compared to the fine discriminator. The \verb|D1| and \verb|D2| in Figure~\ref{fig:p2c} represent the dual discriminators for translating two different domains $X$ and $Y$.
\subsection{Generator}
\label{sec:generator}
Previous studies have found it beneficial to mix the GAN objective with a more traditional norm, such as $L2$~\cite{Pathak_2016_CVPR} and $L1$~\cite{Isola_2017_CVPR} distance. We explore this opinion to cycle consistency loss by applying $L1$ distance to compute $L_{cyc}$. Our final objective is
\begin{align}
\label{eq:objective}
G^{*} = \arg\underset{G}{\min}\;\underset{D}{\max}\mathcal{L}(G,D)
+ \gamma \mathcal{L}_{cyc} + \sigma \mathcal{L}_{p}, \\
\text{w.r.t}\quad
\mathcal{L}(G,D)=
\begin{cases}
\mathcal{L}_{p}(G,D_{c}) \quad \text{for}\ D_{c},\\
\mathcal{L}_{g}(G,D_{f}) \quad \text{for}\ D_{f},
\end{cases}
\end{align}
where $\mathcal{L}_{cyc}$ means cycle consistency loss, $\mathcal{L}_{p}$ indicates perceptual loss, and $\gamma$ and $\sigma$ are hyper parameters to balance the contribution of each loss to the objective. We use greedy search to optimize the hyper parameters with $\gamma=10$ and $\sigma=2.0$ for all the experiments in this paper.
As shown in Figure~\ref{fig:p2c}, we use Conv-Residual blocks-Deconv~\cite{he2016deep} as the generator to share the low-level and high-level information between the input and output directly across the net.
\section{Experiments}
As a typical image-to-image translation task, photo-to-caricature requires high-level visual information conversion, which is very challenging for the state-of-the-art general-purpose solutions. To explore the effect of our proposed model, we tested the method on a variety of datasets for translating faces in the wild from photo to caricature with different styles, and the qualitative results are shown in Figs.~\ref{fig:comparison} and~\ref{fig:free}.
\subsection{Dataset and training}
Our proposed model is trained in a supervised unpaired fashion on a paired face photo-caricature dataset, named IIIT-CFW-P2C dataset, which was rebuilt based on IIIT-CFW~\cite{mishra2016iiit}. The IIIT-CFW is a dataset for the cartoon faces in the wild and contains $8928$ annotated cartoon faces of famous personalities in the world with varying profession. Also it provides $1000$ real faces of the public figure for cross modal retrieval tasks. However, it's not suitable for the training of photo-to-caricature translation task using some paired methods (such as Pix2pix) because the face photos and face cartoons\footnote{We consider that caricature is a type of a cartoon or a subset of a cartoon.} are not paired, \emph{e.g.}, the facial orientation and expression of the photo and caricature for the same person are varying a lot. So we rebuild a photo-caricature dataset with $1171$ paired images by searching the IIIT-CFW dataset and Internet as the training set for compared experiments. Here we use $800$ for training and the left for testing. At inference time, we run the generator in exactly the same manner as during the training phase. Besides, we also extensively evaluated our method on a variety of datasets with faces in the wild, including Caricature~\cite{abaci2015matching}, FEI~\cite{thomaz2010new}, IIIT-CFW~\cite{mishra2016iiit}, Yale~\cite{georghiades1997yale}, KDEF~\cite{lundqvist1998karolinska} and CelebA~\cite{liu2015faceattributes}.
Besides, we also consider photo-to-sketch as a photo-to-caricature task for experiments using PHOTO-SKETCH dataset~\cite{zhang2011coupled,wang2009face}, which has $1194$ paired images and hence can be directly used for supervised training of some compared paired methods. And following DualGAN, we use $995$ unpaired images for training and $199$ for testing. Note that we train CycleGAN, DualGAN and our model using unpaired images and train Pix2pix using paired images of the two datasets.
\subsection{Comparison with state-of-the-arts}
Using IIIT-CFW-P2C dataset, we first compare our proposed method with Pix2pix~\cite{Isola_2017_CVPR}, DualGAN~\cite{Yi_2017_ICCV}, DiscoGAN~\cite{kim2017learning} and CycleGAN~\cite{Zhu_2017_ICCV} on photo-to-caricature translation task. All the four methods were trained on the same training dataset and tested on novel data from IIIT-CFW-P2C dataset that does not overlap those for training.
\paragraph{Qualitative evaluation}
Figure~\ref{fig:comparison} shows the experimental results of the comparison, it can be seen that, DualGAN only learned the color and edge translation rather than structure information, Pix2pix makes structure error in almost all cases, while CycleGAN can keep the structure information of the input image but without enough conversion for caricature creation task. For DiscoGAN, it's really hard to generate plausible and meaningful results due to the lack data of training ($800$ pairs in our experiments vs. tens of thousands of pairs in DiscoGAN's experiments) and the big challenge of task (photo-to-caricature vs. edge-to-photo). Although it's still not good enough for the results of our method compared to human caricaturists, the experiments on photo-to-caricature translation of faces show considerable performance gain of our proposed method over state-of-the-art image-to-image translation methods, especially the encouraging ability of exaggeration and abstraction. However, due to the very challenging task with less training data but on various styles, our method also messes some details while translation (see the mouth of the first row and the eyes of the fourth row on our results in Figure~\ref{fig:comparison}).
\begin{figure}[!ht]
\centering
\includegraphics[width=\textwidth]{compare.pdf}
\caption{Comparison of state-of-the-art image-to-image translation methods with our proposed method for face photo-to-caricature translation on IIIT-CFW-P2C dataset.}
\label{fig:comparison}
\end{figure}
This experiment also expresses that high-level image-to-image translation tasks like photo-to-caricature are generally more difficult than those low-level translation tasks such as photo-to-sketch, because it not only needs to abstract the facial features, but also requires to exaggerate the emotional expression. So that the pixel-level methods (like Pix2pix) might fail as they force the generator to concentrate on local information rather than whole structure.
Besides, we also evaluate our methods on PHOTO-SKETCH dataset. Figure~\ref{fig:comparison_sketch} shows the compared results. Comparing with Pix2pix, our method can reduce the effect of being blurry and artifact. Although DualGAN and CycleGAN can also reserve the structure information of input faces, they are not good at achieving abstraction and artistry. Similarly, DiscoGAN collapses when facing insufficient training data and big challenging task.
\begin{figure}[!ht]
\centering
\includegraphics[width=\textwidth]{compare_sketch.pdf}
\caption{Comparison of state-of-the-art image-to-image translation methods with our proposed method for face photo-to-caricature translation on PHOTO-SKETCH dataset.}
\label{fig:comparison_sketch}
\end{figure}
\paragraph{Quantitative evaluation}
Beyond the visually qualitative evaluation, we also evaluated the translated cartoon results of different methods quantitatively on IIIT-CFW-P2C dataset and PHOTO-SKETCH dataset in terms of both human judge and machine grade, and the average results are shown in Table~\ref{tab:cfw} and Table~\ref{tab:photo} respectively.
For human score, we invited 40 volunteers to evaluate the generated image quality of different methods in terms of satire, exaggeration, lifelikeness and artistry compared with the given original photo, by grading from 1 to 5 (1 represents the worst and 5 represents the best).
For inception score, we used a pre-trained classifier network and sampled images for evaluation followed~\cite{salimans2016improved}. The results shows that our proposed method outperforms state-of-the-art image-to-image translation methods with the highest human and inception scores.
\begin{table}
\centering
\caption{The generated image equality evaluation results of different methods on IIIT-CFW-P2C dataset.}
\label{tab:cfw}
\begin{tabular}{|c|c|c|}
\hline
Method & Human score & Inception score \\
\hline\hline
Pix2pix & 2.0106 & 1.5069$\pm$0.1090\\
DualGAN & 1.9946 & 1.4843$\pm$0.1049\\
DiscoGAN & 1.5014 & 1.3366$\pm$0.0714\\
CycleGAN & 3.5001 & 1.5684$\pm$0.1331\\
Ours & 4.0120 & 1.6043$\pm$0.0918 \\
\hline
\end{tabular}
\end{table}
\begin{table}
\centering
\caption{The generated image equality evaluation results of different methods on PHOTO-SKETCH dataset.}
\label{tab:photo}
\begin{tabular}{|c|c|c|}
\hline
Method & AMT & Inception score \\
\hline\hline
Pix2pix & 2.0971 & 1.3625$\pm$0.0706\\
DualGAN & 2.3810 & 1.4063$\pm$0.0822\\
DiscoGAN & 1.0858 & 1.3142$\pm$0.0206\\
CycleGAN & 3.2272 & 1.3980$\pm$0.1130\\
Ours & 4.0750 & 1.4298$\pm$0.0818 \\
\hline
\end{tabular}
\end{table}
\subsection{Dual discriminators}
\label{sec:dual}
In this experiment, we verify the effectiveness of our proposed dual pathway of discriminators. We first use only one coarse discriminator (Coarse D) and one fine discriminator (Fine D) separately, and then use dual discriminators with one coarse discriminator plus one fine discriminator (Fine D + Coarse D), while keeping all other architectures and settings fixed for training and testing. Some example results are shown in Figure~\ref{fig:dual}, and one Fine D model almost misses the key structure information on faces, but our dual Coarse D + Fine D model can render the structure of facial features well. It further proves that the Fine D model only concerns the local statistics for tackling low-level image-to-image translation tasks (\emph{e.g.}, photo-to-sketch).
\begin{figure}[!ht]
\centering
\includegraphics[width=0.5\textwidth]{candf.pdf}
\caption{Comparison of only one discriminator (Coarse D and Fine D respectively) with our dual discriminators (Coarse D + Fine D, C \& F).}
\label{fig:dual}
\end{figure}
And we also took some experiments to greedy search the best combination size of output patches for coarse discriminator and fine discriminator. Figure~\ref{fig:combination} shows results of different combinations, which indicates that large Fine D patches (\emph{e.g.}, F$32$) fails to abstract and exaggerate faces while small Fine D patches (\emph{e.g.}, F$8$) abstracts and exaggerates faces too much.
\begin{figure}[!ht]
\centering
\includegraphics[width=\textwidth]{combinations.pdf}
\caption{Examples of different combination sizes of output patches for coarse discriminator and fine discriminator. C$k$ denotes that the output of coarse discriminator is a $k\times k$ patch, and F$k$ denotes that the output of fine discriminator is a $k\times k$ patch. It can be seen that large F$k$ such as F$32$ for fine discriminator fails to abstract faces and achieve exaggerations, while small F$k$ such as F$8$ abstracts and exaggerates faces too much.}
\label{fig:combination}
\end{figure}
\subsection{Loss selection}
\label{sec:loss}
We first consider to check if the cycle consistency loss $\mathcal{L}_{cyc}$ should be provided to the GAN objective (Equation~\ref{eq:objective}), and the second column in Figure~\ref{fig:losscyc} shows the results without cycle consistency loss. It's easy to see that without cycle consistency loss, although the adversarial system with adversarial loss can capture the facial features, it's hard to generate caricature images with plausible objects and meaningful relationship between facial organs. The third column in Figure~\ref{fig:losscyc} shows the results by adding cycle consistency loss $\mathcal{L}_{cyc}$. Therefore, the normal adversarial training can lead to some kind of caricature style, but it fails to be lifelike without meaningful components.
\begin{figure}[!ht]
\centering
\includegraphics[width=0.5\textwidth]{cycle.pdf}
\caption{Comparison of extra loss for final objective of generator: without ($w/o$) $\mathcal{L}_{cyc}$ and with ($w/$) $\mathcal{L}_{cyc}$.}
\label{fig:losscyc}
\end{figure}
Based on the cycle consistency loss, we provide the perceptual loss $\mathcal{L}_{p}$ for generator in our system. Figure~\ref{fig:losscl} shows the compared results of without and with $\mathcal{L}_{p}$. It can be seen that perceptual loss can produce images with the exaggerated facial features such as eyes, nose, and mouth. The perceptual loss, which expresses some perceptual errors on facial features such as head, eyes, and mouth, could improve the artistic expression of image generation and show better abstraction ability. And it can also reduce the effect of being blurry. The second column of Figure~\ref{fig:losscl} without using perceptual loss illustrates the indistinguishable facial expressions with distorted facial organs while translation, and the third column with adding perceptual loss improves the performance on facial expression and organ translation with caricature effect, \emph{e.g.}, the smile woman of last row.
\begin{figure}[t]
\centering
\includegraphics[width=0.5\linewidth]{perceptual.pdf}
\caption{Comparison of extra loss for final objective of generator: without ($w/o$) $\mathcal{L}_{p}$ and with ($w/$) $\mathcal{L}_{p}$.}
\label{fig:losscl}
\end{figure}
\subsection{Auxiliary noise input}
\label{sec:info}
By adding auxiliary noise to our photo-to-caricature system, we can improve the robustness and diversity of synthesized facial caricatures. Figure~\ref{fig:info} shows the example results of adding auxiliary noise for photo-to-caricature translation, and the output results for adding different proportions ($\alpha$) of noise indicate that our system can still synthesize meaningful facial caricatures with even more than a half noise ($1-\alpha$, see Equation~\ref{eq:noise} for reference) as inputs, besides, the added different proportions of noise also lead to different styles of output results, which indicates that it might be used as a factor for tuning different synthesized styles.
\begin{figure}[!ht]
\centering
\includegraphics[width=\textwidth]{info.pdf}
\caption{Examples of adding auxiliary noise for robustness and diversity of our photo-to-caricature translation. Different proportions ($\alpha$ in Equation~\ref{fig:noise}) of noise inputs can also lead to meaningful different styles of output caricatures.}
\label{fig:info}
\end{figure}
\subsection{Freestyle face caricature creation}
To evaluate the ability for real applications in the daily life, we tested our method on a variety of face datasets, including Caricature~\cite{abaci2015matching}, FEI~\cite{thomaz2010new}, IIIT-CFW~\cite{mishra2016iiit}, Yale~\cite{georghiades1997yale}, KDEF~\cite{lundqvist1998karolinska} and CelebA~\cite{liu2015faceattributes}, to illustrate the photo-to-caricature translation on faces in the wild, and the results are shown in Figure~\ref{fig:free}. These freestyle face caricature creation results validate that our model works not bad on arbitrary faces and show the potential value for the related applications. And we can see that the translated results in KDEF, FEI and Yale datasets also have different facial expression corresponding the input faces. Our methods successfully reserve the emotion information and emulate the facial organs with caricature style. So we can conclude that the more abstracted information such as facial emotion and expression with global structure information are reserved. Besides, our model can also enlarge or narrow the facial organs such as chin, lips, eyes and so on, which is required for high-level image-to-image translation tasks.
\begin{figure}[!ht]
\centering
\includegraphics[width=\linewidth]{free.pdf}
\caption{Example translated caricatures of facial photos from several datasets (Caricature~\cite{abaci2015matching}, FEI~\cite{thomaz2010new}, IIIT-CFW~\cite{mishra2016iiit}, Yale~\cite{georghiades1997yale}, KDEF~\cite{lundqvist1998karolinska} and CelebA~\cite{liu2015faceattributes}) using our trained model on IIIT-CFW-P2C.}
\label{fig:free}
\end{figure}
\section{Conclusion and Future Work}
We present a novel GAN-based method to deal with high-level image-to-image translation task, \emph{i.e.}, photo-to-caricature translation. The proposed method uses dual discriminators for capturing global structure and local statistics information with abstraction ability, also provides extra perceptual loss on GAN objective to constrain the consistency under exaggeration. Besides, the style information can be learned and representative by adding auxiliary noise input. And the robustness can be improved by the noise-added training. Experimental results show that our method not only outperforms other state-of-the-art image-to-image translation methods, but also works well on a variety of datasets for photo-to-caricature translation of faces in the wild.
\textbf{Limitations.} Translating photo to caricature is a very challenging high-level image-to-image translation task. Thus, our model also fails in some cases, \emph{e.g.}, some generated images of Yale and CelebA dataset in Figure~\ref{fig:free}. Although our method can keep the structure information of faces, it is still hard to render the details for providing high-quality caricatures and some tiny organs (such as eyes) are lack of details in Figure~\ref{fig:comparison}. Besides, it's also sensitive to side face with complex background, \emph{e.g.}, some cases on CelebA and KDEF datasets in Figure~\ref{fig:free}.
\textbf{Future work.} With regard to future work, first, it would be interesting to investigate our method on other tasks of high-level image-to-image translation (\emph{e.g.}, human-to-cartoon translation for cartoon movies); second, for the proposed method, the model still needs to be improved to provide high-quality rendered translated results; third, we intend to apply our method on the real applications of automatic and intelligent photo-to-caricature translation; fourth, we hope that we can control caricature style while translating images between domains by tuning input noise.
\section*{Acknowledgement}
We thanks the volunteers for grading human scores of translation results from different methods. This work was supported by the National Natural Science Foundation of China [61771440, 41776113], and Qingdao Municipal Science and Technology Program [17-1-1-5-jch].
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 2,954 |
Q: win32com Outlook subfolder not always found I have a script that moves/saves/categorizes emails in a shared mail. And I'm encountering errors when sometimes mail folders are not found. I'm using below function to retrieve mail folders. And sometimes it works and othertimes it doesn't. I usually have to restart outlook to make it work once the error has been thrown. It sometimes fails to find the subfolders within the Inbox and it returns that it is empty.
Is there anyway to ensure that the subfolders are always found?
OUTLOOK = win32com.client.GetActiveObject('Outlook.Application') #This is how I dispatch outlook
Mapi - folders
['x', 'y', 'z',]
x - folders
['PersonMetadata', 'Yammer Root', 'Tasks', 'Sent Items', 'Outbox', 'Notes', 'Junk Email', 'Journal', 'Inbox', 'Files', 'ExternalContacts', 'Drafts', 'Deleted Items', 'Conversation History', 'Conversation Action Settings', 'Contacts', 'Calendar', 'Archive']
Inbox - folders
['a', 'b', 'c'] #THIS IS SOMETIMES EMPTY
FOLDERS = {"example": ["x", "Inbox", "a"]}
def get_folder(outlook_instance, *folders):
target_folder = outlook_instance
try:
for folder in folders:
print(folder_path)
print([f.Name for f in target_folder.Folders])
target_folder = target_folder.Folders[str(folder)]
except Exception:
raise ValueError("".join(f".Folders[{folder}]" for folder in folders))
return target_folder
def get_outlook_folder(folder, outlook):
mapi = outlook.GetNamespace("MAPI")
return get_folder(mapi, *FOLDERS.get(folder))
I found this on an other post that I think relates to my issue, but I don't really understand what it is saying stackoverflow.com/a/40853098/10833061
A: Is Outlook always running on the system when your code is executed?
Otherwise, you need to create a new Outlook Application instance and logon to the profile to be able to access folders. I am not a python developer, so here is a C# sample code which illustrates the required workflow for you:
Outlook.Application GetApplicationObject()
{
Outlook.Application application = null;
// Check whether there is an Outlook process running.
if (Process.GetProcessesByName("OUTLOOK").Count() > 0)
{
// If so, use the GetActiveObject method to obtain the process and cast it to an Application object.
application = Marshal.GetActiveObject("Outlook.Application") as Outlook.Application;
}
else
{
// If not, create a new instance of Outlook and sign in to the default profile.
application = new Outlook.Application();
Outlook.NameSpace nameSpace = application.GetNamespace("MAPI");
nameSpace.Logon("", "", Missing.Value, Missing.Value);
nameSpace = null;
}
// Return the Outlook Application object.
return application;
}
The Outlook object model is common for all programming languages, so I hope there you won't find any difficulties in understanding the required approach. Read more about that in the Get and sign in to an instance of Outlook article in MSDN.
Finally, it is not clear what comes from where - I'd recommend clarifying all bits in the sample code posted, so we could suggest anything else.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 5,137 |
Shock! , Shock! Get Buckled. Clubwear. Wigs. Accessories. #02-07 Pacific Plaza.
"Shock! , Shock! Get Buckled. Clubwear. Wigs. Accessories. #02-07 Pacific Plaza.." (2000) Asian Studies, Baker Institute for Public Policy, Rice University: https://hdl.handle.net/1911/41606. | {
"redpajama_set_name": "RedPajamaC4"
} | 2,233 |
Home » Your NATO/G8 primer
Your NATO/G8 primer
Everything you wanted to know about hosting international summits but were afraid to ask
by Ben Joravsky and Mick Dumke February 16, 2012 August 19, 2021
Credit: Official White House Photo by Pete Souza
On January 26, the activist organization Adbusters, which helped spark the Occupy Wall Street movement, called for at least 50,000 "redeemers, rebels and radicals" to visit Chicago in the month of May for "the biggest multinational occupation of a summit meeting the world has ever seen."
"And if they don't listen," the statement vowed, "we'll flashmob the streets, shut down stock exchanges, campuses, corporate headquarters and cities across the globe."
It's not known how many demonstrators will respond to the call. But there's little question that thousands of out-of-town visitors will descend on the city to attend, observe, and protest two scheduled summits of the most powerful leaders in the world.
What is in question is how it will impact—and what it
will cost—the people of Chicago.
Mayor Rahm Emanuel used his clout with President Barack Obama to bring the NATO and G8 summits here, and the way the preparations are proceeding says a lot about how he runs this town.
Just three months before the events, neither residents nor the City Council have been told how much it could cost us. And some police officers are so concerned about what could happen that they're ordering their own riot-intervention equipment.
What follows is a primer on how we got into this mess, what we know about the summits, and—more significantly—what we don't.
So start at the beginning— what is this all about?
For three days in May, Chicago will play host to two international summits of diplomats, world leaders, and other dignitaries. Leaders of the Group of Eight—Canada, France, Germany, Italy, Japan, Russia, the United Kingdom, and the United States—will meet from Saturday, May 19, to Sunday, May 20. And on May 20 and 21 the leaders of NATO will gather. Both summits will be held at McCormick Place.
Roughly 7,500 delegates representing 80 delegations will attend, plus about 2,500 journalists—and all those protesters.
And why are they coming to Chicago again?
Officially, President Obama decided that Chicago was a "logical choice" because it's "a global city, connected to the global economy, with an increasing international profile," according to Caitlin Hayden, assistant press secretary for foreign affairs. "It's the President's home town and has a proven record of managing big events. We know that Chicago will highlight the best of America."
So this decision was made completely free of politics, right?
Unofficially, Mayor Rahm Emanuel really, really, really wanted the summits to come here. As he's told his aides: it's like the Olympics—only easier to get.
So he was badgering the president, Vice President Joe Biden, and other administration officials about bringing the summit here almost from the moment he won the mayoral election last February. And maybe before.
OK, but did anyone other than Mayor Emanuel really, really, really want to hold the summits in Chicago?
No. Unlike the prospect of hosting the Olympics, which always had some support, the summits weren't on the radar—until President Obama offhandedly announced in a speech last June that they were coming to Chicago.
But surely there was some sort of bring-the-summits-to-Chicago blue ribbon committee—you know, filled with the mayor's friends and donors?
Not until the deal was already made. More on those friends and donors in a bit.
So did I miss the dog-and-pony community hearings they held to explain why this is worth the cost and inconvenience—you know, like they did with the Olympics?
You didn't, because there weren't any. As Leslie Hairston sums it up: "This wasn't a citywide decision."
Who's Leslie Hairston?
She's the alderman of the Fifth Ward who, like the rest of her colleagues, generally goes along with the mayor. But last month she really pissed him off by leading a handful of powerless aldermen in opposing the summits.
So what has the mayor said about her criticism?
If you don't like it, Hairston, then fuck you!
No, I mean, what has he said that he hasn't said to critics before?
On the record, he said, "This will be an opportunity to showcase what is great about the greatest city in the greatest country." We're pretty sure he was referring to Chicago—and not his hometown of Wilmette.
What has Mayor Emanuel done to explain to Chicagoans how the summits will make their great city even greater?
He's worked diligently to spin the press. For example, on January 12, his press office welcomed about 25 reporters to a City Hall briefing to clear up confusion about the summits. When they were finished, reporters were even more confused.
Several of the top officials responsible for the city's preparations were there, including Lori Healey, executive director of the NATO/G8 host committee. Also in attendance were at least five mayoral press aides, who laid out rules for the reporters: we were allowed to quote anything we wanted, but we couldn't name the person who said it.
"We want everyone to be able to speak freely," one of the press flacks explained.
She went on to emphasize that many of the summit preparations were not yet in place and that many others would never be made public. "There are questions we can't answer about security and other issues."
So what did they make public?
For the next hour and a half, the officials took turns stressing that the summits presented a huge opportunity to "showcase Chicago as a world-class city"—they used the phrase repeatedly. They added that they couldn't provide many details about what it would mean for people who actually live here.
So they were speaking freely about something they didn't know or couldn't talk about?
Yes—and you can quote us on that, as long as you don't use our names.
I guess it's good that Chicago's going to be "showcased." But did city officials share the analysis they conducted showing how much this will cost and benefit us?
Funny you should ask about that, because we did too. Unfortunately, the city hasn't been able to produce anything in response to our request for the cost-benefit analysis they've conducted. Aldermen say they haven't been shown any analysis either.
There's a good reason for this: the city hasn't conducted a formal cost-benefit analysis.
None?
City officials tell us they're "in possession of some very preliminary cost estimates in draft form"—but they can't share them with us. Still, they insist that the events will be a net gain for Chicago's economy.
So the supposed benefit to Chicago comes down to an unsubstantiated assurance from officials who've asked us not to name them?
And everyone on the City Council bought this?
Most, but not all.
How many doubters are we talking about?
A grand total of five of our 50 esteemed aldermen voted against the mayor's summit ordinances: Robert Fioretti (Second), Will Burns (Fourth), Hairston (Fifth), Sandi Jackson (Seventh), and Nicholas Sposato (36th). "Where is the economic boom going to be to the city?" Hairston says. "These people are not going to be shopping at Garrett's Popcorn or going to the Museum of Science and Industry."
OK, so we don't know how this is going to benefit Chicago. Do we at least know what it's going to cost us?
Nope, don't know that either. City officials say there's no way of knowing all the summit expenses until federal officials finalize the logistics in the coming weeks. In the meantime, the city is using past summits in other cities to come up with guesstimates of $40 million to $65 million. But it's unclear what the $40 million to $65 million would cover—security, entertainment for visiting dignitaries and journalists, marketing, or everything in between. "It's too early to say," one spokeswoman tells us.
Who's in charge of handing out contracts and other party favors?
That would be Mayor Emanuel. He's been given unilateral authority to dole out any contract for any amount to any vendor he wants.
You mean there won't be competitive bidding?
Yeah, we're shocked too. To be fair, the mayor says that three bids will be required for any contract. Of course, the mayor will choose the three bidders.
So you can't bid for a contract unless the mayor says you can bid for a contract?
How did he get that authority?
The authority was granted in the usual manner—a rubber stamp. That's the 41-5 City Council vote we were talking about a minute ago.
But I read in the papers and heard on the TV that the mayor made a lot of concessions before that vote.
On December 14, Mayor Emanuel introduced several proposals related to the summits. The ones that got the most coverage would have raised fines for resisting arrest and imposed new requirements for parades and demonstrations. And the other—as we talked about already—granted him the authority to hand out contracts without a normal bidding process or City Council oversight.
At a hearing in January, a broad range of activists and other citizens, from death penalty foes to right-to-lifers, testified against the parade and protest regulations. When Emanuel agreed to cut the fine increases, most aldermen were convinced that he'd given enough ground to make the ordinances palatable.
The first question of the council hearings was literally read off the page by 11th Ward alderman James Balcer. "Do you share the public's concern about the increased fine for resisting police officers and First Amendment issues?"
Somehow, police superintendent Garry McCarthy was ready for that one. "I didn't really have that concern," McCarthy said. "But after hearing the voices, and in collaboration with the administration, the police department, and the public, we decided to pull that off the table."
Balcer was satisfied.
In other words, the mayor got a lot of kudos for making a few compromises on the protest rules while quietly getting all the contracting authority he wanted?
Yes. It's one of the oldest political tricks in the books: distract folks with a spectacle and then dish out the dough when no one's paying attention.
So who's going to be on the hook for all that dough?
The mayor promises that the feds will pick up the full tab: "The U.S. government is hosting the world leaders in Chicago. The good news is the taxpayers won't pay for this."
Apparently, he forgot that most Chicagoans pay federal taxes.
Hey, isn't that a cheap shot?
My bad. I'm sure the mayor knows that everyone pays federal taxes—except for donors with really good accountants.
Enough with the mayor jokes. Are we sure that the feds have committed to cover the summit costs?
What, you don't believe everything your mayor tells you?
What if I just want to make sure?
Here's what we know: we don't know much.
The federal Department of Homeland Security awarded the city a $55 million grant to cover preparation costs.
But we don't know specifically what the grant was for. The city says it's going to provide us with a copy of the grant application and conditions but hasn't yet. The DHS says they'll get back to us—as soon as they've responded to the 421 requests in front of ours.
Hasn't anyone made an issue of this?
Believe it or not, it briefly came up in the City Council when 31st Ward alderman Ray Suarez, hardly a frequent dissenter, asked police chief McCarthy who's going to cover the summit expenses.
"That has not been determined," McCarthy replied.
That deviation from the script seemed to confuse Suarez. "In our briefings we were led to believe that the federal government will cover the cost of this thing," he said. "Now you're telling me something different."
Several officials and aldermen jumped in to tell Suarez not to worry. "I've been assured it will not cost the city," said Alderman Carrie Austin, the council's budget committee chair.
She concluded there was no need to delay approval, since the details could always be tweaked later.
Translation: We won't know how much it'll cost us until we've already spent the money.
Didn't the mayor also say that civic-minded rich people would pick up some of the tab?
Yes. He's created a committee headed by John Bryan, former CEO of Sara Lee, and Anne Olaimey, his political fund-raiser, to raise money from the business community.
And how much have they raised?
We don't know—the information hasn't been released. But the good news is that Lori Healey, who's heading Emanuel's host committee, says it will start to post donor information at the end of February.
And the bad news?
We don't know the total cost and we don't know how much the feds will pay and we don't know how much the civic community will raise.
What are some of the things that will have to be paid for?
For starters, untold numbers of police officers from other cities will be deputized—and paid—to help patrol the streets during the summits.
Why do we need these outside police?
Because we don't know how many demonstrators will show up. Plus, Chicago's police force is smaller than it's been in years. Chicago currently has about 11,000 police officers, 900 fewer than five years ago.
Any other bad news?
The city could be on the hook to pay Chicago Parking Meters LLC an unspecified amount of money.
You mean the guys who own our street parking system?
Yes—our old friends.
What do they have to do with the G8 and NATO summits?
Under their 75-year lease deal, the city has to pay Parking Meters LLC every time a metered spot is taken out of operation. So if parts of downtown are blocked off for summit security, we the people will make up the difference in lost meter revenue—potentially millions of dollars.
So we have to pay a private company for the right to use our public streets?
We can debate whether the streets are still fully public, but yes, that's correct.
And who exactly is overseeing this extravaganza?
World Business Chicago—an organization funded by the city whose board is made up of business executives handpicked by the mayor, including leaders of Grosvenor Capital Management, Deloitte & Touche, and United-Continental.
Wait—haven't I heard those names before?
Probably. Grosvenor employees and family members donated more than $500,000 to the mayor's election campaign. Deloitte has received millions of dollars in city consulting work. And United was the beneficiary of millions in tax increment financing subsidies.
But at least the mayor won't allow any members of World Business Chicago to get summit contracts, right?
Actually, Motorola Solutions, whose chairman, Greg Brown, sits on the board, already got a $16 million-a-year contract for police radio equipment.
Wow—that was fast!
And that was even before the new contracting rules went into effect. Hey, that's why this is still the city that works.
Joey Jachowski contributed research to this story.
»Police don't seem prepared for the huge protests the G8 and NATO summits will bring on.
»We don't know how much the G8/NATO protests will cost, but we do know Chicago is still paying legal fees for the 2003 Iraq War protest.
»Read Judge Posner's recent decision explaining why those 2003 Iraq protest arrests were unreasonable. (PDF)
»The U.S. Dept. of Homeland Security promises to respond to our request for information . . . right after their 421 other pending requests. (jpeg)
Tagged: Barack Obama, Chicago Parking Meters LLC, City Council, civic-minded rich people, competitive bidding, contractors, fiascos, G8, G8 summit, Garry McCarthy, Leslie Hairston, NATO, NATO summit, Occupy movement, parking meters, protests, Rahm Emanuel, Ray Suarez, riots, Vol. 41 No. 21, World Business Chicago | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 9,568 |
@interface NSNumber (EKStuff)
+ (NSArray *)primeNumbersFromSieveEratosthenesWithMaxNumber:(NSUInteger)maxNumber;
- (NSUInteger)greatestCommonDivisorWithNumber:(NSUInteger)secondNumber;
- (NSUInteger)leastCommonMultipleWithNumber:(NSUInteger)secondNumber;
- (NSUInteger)factorial;
+ (NSMutableArray *)fibonacciNumbersUpToNumber:(NSUInteger)number;
+ (NSNumber *)fibonacciAtIndex:(NSUInteger)index;
+ (long long int)recursiveFibonacci:(NSUInteger)index;
+ (unsigned long long)fibonacciWithLongLong:(int)index;
+ (NSDecimalNumber *)fibonacciWithDecimal:(int)index;
- (NSUInteger)sumOfDigits;
+ (NSUInteger)decimalNumberFromBinary:(NSUInteger)binary;
+ (NSUInteger)binaryNumberFromDecimal:(NSUInteger)decimal;
+ (NSInteger)fastExpForNumber:(NSInteger)number withPower:(NSInteger)power;
- (NSUInteger)reverseNumber;
- (BOOL)isEven;
- (BOOL)isLeapGivenYear;
- (BOOL)isArmstrongNumber;
- (BOOL)isPrime;
+ (NSUInteger)nthPrime:(NSUInteger)n;
+ (void)swapValueOfIntPointer:(NSInteger *)xPointer withValueOfIntPointer:(NSInteger *)yPointer;
- (CGFloat)squareRoot;
- (id)convertedNumberWithBase:(int)base;
- (CGFloat)fastInverseSquareRoot;
- (CFNumberType)numberType;
+ (NSNumber *)sumOfNumbers:(NSArray *)numbers;
@end
| {
"redpajama_set_name": "RedPajamaGithub"
} | 637 |
Every few weeks I take questions from consumers and do my best to answer them here in this space. This time I'm going to answer a key question that was put to me by one person seeking a "GMA Savings Makeover." In case you missed it, last year I offered Good Morning America viewers makeovers based on the 100 simple cost-cutting strategies in my book, SAVE BIG. I found roughly $47,000 in savings for the first family, $189,000 for the second, $108,000 for the third and $58,000 for the fourth! (To see the stories �and how the same strategies could help you–click on the dollar amounts.) The key is to look for big savings instead of small ones and look in the places where we spend the most money: our homes, cars, credit, groceries and healthcare. In my book I make some very different assumptions about the best way to save (For example, I let you HAVE your latte!), so here's my contrarian response to a classic question.
Q: How do we save when so many fixed costs are involved?
A: In this tricky economy, try to adjust your attitude! Are you SURE your costs are fixed? It's time to get scrappy to save some much needed money. The truth is even most supposedly "fixed" costs can be negotiated, refinanced or shopped around.
If you rent, try negotiating for a better rate. It costs landlords a lot of money to leave properties empty, so if you're a good tenant, that is valuable to them. Research the incentives your landlord and other nearby buildings are offering new tenants and use that as a starting point to argue for a better deal.
If you own a home, interest rates have been historically low. Are you in a position to refinance? I argue that it's worth refinancing if you can get half a point off your rate and if you will add no more than 5 years to the length of your mortgage and be able to pay your closing costs off in 5 years or less–preferably much less. That's my "Refinancing Rule of 5's. Can't get approved at a big bank? Try a credit union. They pay attention to your situation , not just your score. Find one here.
Property taxes? Appeal your assessment! Many local jurisdictions have not caught up with reality and are still using market values from early 2008 to judge homes. This is the time of year that assessments get mailed out. Appealing takes about the same amount of work as fighting a traffic ticket �with a much bigger payout. I interviewed a New Jersey man who stood to save $5,000 a year if he won his appeal!
If car payments are a burden and your vehicle is not too old and you are not underwater on the value, did you know you can refinance it? Yup. Once again, credit unions are the place to turn. They do more of these auto loan refinances than anybody else.
Insurance? That certainly is not fixed! I found a New York family more than $2,000 in annual savings on their car insurance just by shopping around. An easy way to do this is to approach an independent agent who represents many different companies. Find one here. I found a Virginia family of seven nearly $7,000 in savings on their health insurance, again by shopping around through an independent agent. There are savings to be had on homeowner's insurance too.
Groceries aren't fixed either. You know that. But did you know that grocery gurus who know how to work the system can save as much as 80 percent ? When I say groceries I'm talking about food, household supplies and personal care products. One simple way to start saving like that is to try what I call "Copycat Grocery Shopping." Super Savers who routinely get huge discounts post their best finds online on their blogs. Just buy what they buy the way they do and you are golden! The website www.becentsable.net has compiled a list of these Grocery Gurus by state and store.
Medical costs aren't fixed either. The key is to behave like a consumer rather than a patient when you interact with the medical profession. A Harris Interactive Poll found that 70-percent of people who asked a hospital for a price break got one. Most doctors are also open to negotiating, especially if you will pay up front, because they spend thousands chasing after people and insurance companies to pay their bills. Even medications are fluid. Look up the meds you take on a website like RXaminer.com or DestinationRX.com and you will find that some stores offer them for far less than others do. These sites also list less expensive alternative drugs that you might be able to take instead. | {
"redpajama_set_name": "RedPajamaC4"
} | 5,856 |
« Weekend Preview: Big Ten, Dec. 11
Three Things I Think: Big Ten, Dec. 15 »
NCHC Saturday Dec. 12: Three Things
In place of traditional weekend previews, check this space on the CHN blog each Saturday for developing mid-weekend NCHC storylines and observations from Friday night games.
Friday's action saw St. Cloud State (5-2 at Denver) and North Dakota (3-0 at UMD) win on the road, while Western Michigan broke out of its prolonged offensive drought with a 7-3 home win over Colorado College. All three series conclude tonight, the final NCHC conference action of 2015.
1. Cam Again
Cam Johnson's 18 save shutout in Grand Forks over Denver last week was impressive, but the North Dakota Fighting Hawks netminder's performance last night — 40 saves in UND's 3-0 win — was exceptional. Johnson is now undefeated in five starts since returning from injury, and North Dakota — led by opportunstic offensive talent of the likes of Drake Caggiula and Nick Schmaltz — looks downright scary. Johnson had 15 saves in the third period and held the Bulldogs off the scoreboard repeatedly, most notably on power play opportunities that clearly left the UMD players frustrated. Kasimir Kaskisuo didn't play poorly per se on the other end, but may have to be perfect in tonight's rematch if Johnson continues to stay locked in. UND is 15-2-2 overall and three points clear of St. Cloud at the top of the NCHC standings.
2. That Escalated Quickly
Coming off a 7-2 loss to Omaha, things weren't looking terrific for St. Cloud State halfway through its game in Denver last night, trailing 2-0. The Pioneers then gave the Huskies their only power play opportunity of the night, which St. Cloud converted to cut the lead to 2-1… and the rest of the game looked as though St. Cloud had a man advantage. The persistence paid off in the third period when things completely fell apart for Denver. Joey Benik, Kalle Kossila, and David Morley all scored in span of 1:33 to turn a 2-2 deadlock into a comfortable 5-2 Huskies win. That trio of seniors, in addition to classmates Jimmy Murray and Ethan Prow, are arguably the top upperclassmen group in the country in terms of overall talent. They've also become impressive leaders on the ice, no longer making defensive lapses that cost the Huskies last year, and St. Cloud — like UND — has the look of a team ready to make a serious NCAA title run. Denver, meanwhile, has lost three straight behind an anemic offense, and their response tonight will be telling, one way or the other.
3. About time
Western Michigan shook off weeks of frustration with last night's 7-3 win over CC, and the Broncos clearly wanted the win desperately — they forced turnovers, were quicker to loose pucks, and were determined in all phases. The final score isn't as reflective of how close the game was for two periods, but things fell apart for CC late in the second period when the Broncos forced a turnover at the blueline from Tigers sophomore defenseman (and Nashville Predators draft pick) Teemu Kivihalme. Michael Rebry scored his first of the season to give the Broncos a 4-2 lead at the time, and Kivihalme contributed to another defensive zone turnover in the opening minutes of the third period — again leading to a Western Michigan goal. Truthfully, Western Michigan looked like a different team with renewed energy and purpose, while on the other end, it's disappointing to see the Tigers regress after sweeping Miami last week — especially against a team that had to be desperate after entering the weekend with eight consecutive losses. Still, CC can climb out of the NCHC cellar with a win tonight since Miami is idle.
Saturday, December 12th, 2015 - 7:59 pm | Blogs, NCHC, Notes, Thoughts, Ramblings | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 9,006 |
using System;
using System.Collections.Generic;
using System.Linq;
using System.Text;
using System.Threading.Tasks;
namespace uter.sociallistener.general.Social
{
public interface ISocialUser
{
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 1,068 |
\section{Introduction}\label{SecIntro}
Control systems of the form
\begin{align}
\label{E}\tag{E}&\dot x = {f}(t,x,u,v) +\sum_{\alpha=1}^m g_\alpha(x) \dot u_\alpha,\quad \text{on } [a,b], \\
\label{IC}\tag{IC} &x(a)=\bar{x},
\end{align}
can be given a classical interpretation as soon as the control $u$ is an absolutely continuous function and the control $v$ is Lebesgue integrable.
This paper is devoted to the investigation of a notion of solution for the Cauchy problem (E)(IC), when one assumes the following hypotheses:
\begin{itemize}
\item[(i)] the vector fields $g_\alpha$ {\it commute}, namely $[g_\alpha,g_\beta]\equiv 0,$ for all $\alpha,\beta=1,\dots,m,$ where $[\cdot,\cdot]$ denotes the Lie bracket;
\item[(ii)] the inputs $u$ belong to the space $ \mathcal{L}^1([a,b];U)$ of everywhere defined Lebesgue integrable functions.
\end{itemize}
Loosely speaking, the denomination ``impulsive'' comes from the fact that, due to the affine dependence of the dynamics on the control's derivative $\dot u,$ a discontinuity in $u$ may cause a discontinuity in the corresponding trajectory $x.$
On the other hand, the bounded, measurable input $v$ can be regarded as an ``ordinary" control.
Let us observe that the case where $u$ is taken in the class of bounded variation functions (and the commutativity in (i) is not necessarily verified) has received most of the attention (see e.g. \cite{Ris65}, \cite{BreRam88}, \cite{DalRam91}, \cite{SilVin96} and references therein).
In these articles, the authors studied the technique that is nowadays known as {\em graph completion.} An extension of this concept, also dealing with trajectories with bounded variation, was investigated in \cite{Kar06}, \cite{AruKarPer11} for systems of the form \eqref{E}, while a more general framework allowing the dependence of the vector fields $g_\alpha$ on the ordinary control $v$, was analyzed in \cite{AruKarPer12}.
Even in the case where $u$ can have unbounded variation, a notion of solution valid for systems where $f$ is independent of the ordinary control $v,$ and both (i) and (ii) are met, has already been investigated (see e.g. \cite{BreRam91}, \cite{Sar91}, \cite{Dyk94}). This solution can be defined pointwise and verifies nice properties of uniqueness and continuity on the data.
The main goal of the present note consists in investigating a suitable generalization of this concept of solution to the case when $f$ is actually $v$-dependent.
Incidentally, let us observe that a system like
\begin{equation}\label{Eu}
\dot{x} = {f}(t,x,u,v) +\sum_{\alpha=1}^m g_\alpha(x,u) \dot u_\alpha,
\end{equation}
reduces to \eqref{E} as soon as one adds $m$ extra state variables $z_{1},\dots,z_{m}$ and the additional equations
$$
\dot{z}_{\alpha} = \dot u_\alpha,\quad \text{for }\alpha=1,\dots,m.
$$
In this case the commutative hypothesis (i) reads:
(i') $\left[ g_\alpha+\frac{\partial}{\partial z^\alpha},g_\beta+\frac{\partial}{\partial z^\beta}\right]\equiv 0,$ for all $\alpha,\beta=1,\dots,0,$ where $[\cdot,\cdot]$ denotes the Lie bracket for vector fields on $\mathbb{R}^{n+m}.$
In fact, several applications justify the introduction in the dynamical equations of the ordinary, bounded, control $v$ besides the {impulsive} control $(u,\dot u).$ For instance, in Lagrangian mechanics, if the control $u$ denotes the shape of a concatenation $C$ of rigid bodies and the input $v$ is, say, an external force or torque acting on $ C,$ then the whole motion of $C$ in space is determined by equations of the form \eqref{Eu}. More generally, in a $N+m$-dimensional Lagrangian system (where $N=n/2$) the input $u$ might represent a portion of a local system of coordinates $(q,u),$ while $x$ would be identified with $(q,p),$ $p$ being the {\it momenta} corresponding to the free coordinates $q$ (see \cite{BreAldo89}, \cite{Ram99}).
Let us point out that the commutativity assumption is actually verified is some situations of practical interest \cite{BhaTiw09}.
The main results of the paper, including existence, uniqueness, continuous dependence of solutions on data, state-response measure-zero changes of $u$, are stated in Section \ref{SecLimit}. The latter is concluded by Theorem \ref{TRep}, where a representation of solutions is given in terms of a diffeomorphism constructed through an application of the Multiple Flow-box Theorem to the vector fields $\{g_1,\dots,g_m\}.$ All proofs can be found in Section \ref{SecProofs}.
\vspace{3pt}
\noindent\textbf{Notation and assumptions.}
Let $h$ be a locally Lipschitz vector field on $\mathbb{R}^n,$ and let $\bar{x} \in \mathbb{R}^n.$ Whenever the solution to
\begin{equation*}
\dot x(t) = h(x(t)),\quad h(0)=\bar{x},
\end{equation*}
is defined on an interval $I$ containing $0,$ we use
${\rm exp}({t h})(\bar{x})$ to denote the value of this solution at time $t.$
Let $I$ be a closed interval and let $E$ be a subset of an Euclidean space $\mathbb{R}^d.$ We use $\mathcal{L}^1(I;E)$ to denote the set of pointwise defined Lebesgue integrable functions from $I$ to $\mathbb{R}^d$ with values in $E,$ while $L^1(I;E)$ will denote the corresponding family of equivalence classes (with respect to the Lebesgue measure). We write $AC(I;E)$ for the set of absolutely continuous maps from $I$ to $E.$
For an open subset $\Omega \subseteq \mathbb{R}^n,$ $\mathcal{C}^k(\Omega;\mathbb{R}^d)$ will denote the space of $k-$times continuously differentiable $\mathbb{R}^d$-valued functions defined on $\Omega.$
\vspace{3pt}
Throughout the paper we shall assume the following hypotheses on the control system \eqref{E}-\eqref{IC}:
\vspace{3pt}
\noindent {\bf Hypothesis H:}
\begin{itemize}
\item[(i)]
$U$ is a compact subset of $\mathbb{R}^m$ such that, for every bounded interval $I \subset \mathbb{R},$ for each $\tau \in I,$ and for every function $u\in\mathcal{L}^1(I;U),$ there exists a sequence $(u_k^\tau) \subset AC(I;U)$ verifying
$$
|u^\tau_k(\tau)- u(\tau)| + \|u^\tau_k-u\|_1\to 0,
$$
when $k \to \infty.$\\
\item[(ii)]
The set $V \subset \mathbb{R}^l$ is compact.
\item[(iii)] The map $f:[a,b]\times \mathbb{R}^n \times \mathbb{R}^m \times V \to \mathbb{R}^m$
is such that,
\begin{itemize}
\item[-] for each $(x,u,v)\in \mathbb{R}^n \times \mathbb{R}^m \times V,$ the map $t\mapsto f(t,x,u,v)$ is measurable on $[a,b];$
\item[-] for each $t\in [a,b],$ the function $(x,u,v) \to f(t,x,u,v)$ is continuous on $ \mathbb{R}^n \times \mathbb{R}^m \times V$ and, moreover,
\item[-] the map
$$
(x,u) \mapsto f(t,x,u,v),
$$
is locally Lipschitz on $\mathbb{R}^n \times \mathbb{R}^m,$ uniformly in $(t,v) \in [a,b] \times V.$ \end{itemize}
\item[(iv)] For every $\alpha=1,\dots,m,$ $g_\alpha\in\mathcal{C}^1(\mathbb{R}^n;\mathbb{R}^n).$
\item[(v)] There exists $A>0$ such that
$$
\left| \Big(f(t,x,u,v),g_1(x),\dots,g_m(x)\Big)\right| \leq A(1+|(x,u)|),$$ for every $(x,u) \in \mathbb{R}^n\times \mathbb{R}^m$ uniformly in $(t,v) \in [a,b]\times V.$
\end{itemize}
Notice that hypotheses (ii)-(v) above imply that, for every initial value $\bar{x} \in \mathbb{R}^n,$ and each pair $(u,v)\in AC([a,b];\mathbb{R}^m) \times L^1([a,b];V),$ the Cauchy problem \eqref{E}-\eqref{IC} has a unique (Carath\'eodory) solution, here denoted by $x[\bar{x},u,v].$
\vskip0.3truecm
\noindent {\bf Hypothesis CC:}
\begin{itemize}
\item[(CC$_1$)] the vector fields $g_\alpha$ are {complete}\footnote{We say that $g_\alpha$ is {\em complete} if the solution to the Cauchy problem $\dot x=g_\alpha(x),\ x(0)=\bar{x} \in \mathbb{R}^n$ is (uniquely) defined on $\mathbb{R}.$} and
\item[(CC$_2$)] $ g_1,\dots g_m$ verify the {\em global commutativity hypothesis} on $\mathbb{R}^n,$
namely for every Lipschitz continuous loop
$$
u:[0,1]\to \mathbb{R}^m, \qquad u(0)=u(1),
$$ and each $\bar x\in \mathbb{R}^n$ such that there exists a (unique) Carath\'eodory solution to the Cauchy problem
$$
\dot x(t) = \sum_{\alpha=1}^m g_\alpha(x(t)) \dot u_\alpha(t),\quad t\in [0,1], \qquad x(0) =\bar x; $$
the solution $x$ is a loop, that is, it verifies $x(0)=x(1)=\bar x.$
\end{itemize}
\begin{remark}{\rm
Let us define the {\em Lie bracket} of $g_\alpha$ and $g_\beta$ as
$$
[{g}_\alpha,{g}_\beta] := \displaystyle\sum_{i=1}^n \sum_{j=1}^n \left( \frac{\partial g_{\beta,i}}{\partial x_j} g_{\alpha,j} - \frac{\partial g_{\alpha,i}}{\partial x_j} g_{\beta,j} \right)\frac{\partial}{\partial x_i}.
$$
It is trivial to verify that for the domain $\mathbb{R}^n$ the {\it null bracket condition}
\begin{equation}\label{lie0}
[{g}_\alpha,{g}_\beta] \equiv 0, \quad \alpha,\beta=1,\dots,m,
\end{equation}
is necessary and sufficient for $g_1,\dots,g_m$ to verify the global commutativity hypothesis.
Actually, if instead of $\mathbb{R}^n$ one considered an open subset $\Omega \subset \mathbb{R}^n$ (or a differential manifold) as state space, the null bracket condition \eqref{lie0} would be no longer sufficient for global commutativity. As a trivial example, one can take
the vector fields $g_1:= \left(1,0,\frac{-x_2}{x_1^2+x_2^2} \right)^\top,$ $g_2:=\left( 0,1,\frac{x_1}{x_1^2+x_2^2} \right)^\top,$ which verify the null bracket condition \eqref{lie0} on $\Omega:=\mathbb{R}^2 \backslash \{0\},$ but {\it do not} match the global commutativity hypothesis. Indeed, if
$
u(t):= \big(\cos (2\pi t),\sin (2\pi t) \big)^\top,$ for $t\in [0,1],$
and $x$ is the corresponding solution to
$
\dot x=g_1(x)\dot u_1 + g_2(x)\dot u_2,$ $x(0)=(1,0,0)^\top,$
one has
$x(1) = (1,0,2\pi)^\top\neq x(0).$}
\end{remark}
\section{Limit solutions}\label{SecLimit}
In this section we give the definition of limit solution and state the main results. The corresponding proofs have been placed in Section \ref{SecProofs}.
\begin{definition}[Limit Solution]\label{edsdef}
Consider an initial data $\bar{x} \in \mathbb{R}^n$ and controls $(u,v)\in\mathcal{L}^1([a,b];U)\times L^1([a,b];V).$
We say that an $\mathcal{L}^1-$map $x:[a,b]\to \mathbb{R}^n$ is a {\em limit solution} of the Cauchy problem \eqref{E}-\eqref{IC} if, for every $\tau\in [a,b],$ there exists a sequence $(u^\tau_k) \subset AC([a,b];U)$ such that:
\begin{equation}
\label{limls}
|(x^\tau_k,u^\tau_k)(\tau)- (x,u)(\tau)|+ \|(x^\tau_k,u^\tau_k)- (x,u)\|_1\to 0,
\ee
where $x^\tau_k:=x[\bar{x},u^\tau_k,v].$
\end{definition}
\begin{remark}
Let us point out that $x$ is a limit solution associated to $u$ if, for every $\tau\in [a,b]$, $(x,u)$ can be approximated, in the sense of \eqref{limls}, by sequences of absolutely continuous paths $(x_k^\tau,u_k^\tau)$ that verify \eqref{E} in the classical, Carath\'eodory sense.
We also observe that no direct distributional meaning can be given to the derivative of $u$ or to \eqref{E} (see some general considerations on the subject in \cite{Haj85}), essentially because of two facts: on one hand the $g_\alpha$ are not constant; on the other hand we look for everywhere defined solutions.
\end{remark}
\begin{theorem}[Existence and uniqueness]
\label{Existence}
For every $\bar{x} \in \mathbb{R}^n,$ and every control pair $(u,v)\in\mathcal{L}^1([a,b];U)\times L^1([a,b];V),$ there exists a unique limit solution of the Cauchy problem \eqref{E}-\eqref{IC} defined on $[a,b].$
\end{theorem}
Given $\bar{x} \in \mathbb{R}^n,$ and a control pair $(u,v)\in\mathcal{L}^1([a,b];U)\times L^1([a,b];V),$ let $x[\bar{x},u,v]$ denote the (unique) corresponding limit solution of \eqref{E}-\eqref{IC}.
\begin{remark}
{\rm
Notice that, for every input $u\in\mathcal{L}^1([a,b];U),$ the map $t\mapsto \bar{x} + u(t)-u(a)$ is a limit solution of the trivial Cauchy problem
\begin{equation}\label{trivial}
\dot x = \dot u,\quad x(a) = \bar{x},
\end{equation}
and, thanks to the above uniqueness result, it is in fact the only solution.
Since in general $\mathcal{L}^1$ functions cannot be pointwise approximated by absolutely continuous functions (see e.g. to \cite{Ox80}), the fact that the choice of the approximating control sequence depends on the time $\tau$ is crucial for guaranteeing existence of everywhere defined solutions, even for the trivial equation \eqref{trivial}.
}
\end{remark}
\vspace{5pt}
\begin{example}\label{minimum}{\rm
Let $R\subset\mathbb{R}^2$ be the subset defined by
$$
R\doteq \Big\{ (x,\hat y(x))\quad x\in[0,1]\Big\}\cup \Big\{(x, e^{-\frac{1}{2}})\quad x\in [2,3]\Big\}
$$
where
\begin{equation*}
\hat y(x):=
\left\{
\ba{cl}
e^x, & \text{for } x\in[0,\frac{1}{2}[,\\
e^{1/2} e^{-2}, & \text{for } x \in \bigcup_{k=1}^{\infty} [1-\frac{1}{2k},1-\frac{1}{2k+1}[\,,\quad k\in\mathbb{N},\\
e^{1/2} & \text{for } x \in \bigcup_{k=1}^{\infty} [1-\frac{1}{2k+1},1-\frac{1}{2k+2}[\,,\quad k\in\mathbb{N},\\
e^{-1/2} & \text{for } x=1,
\ea
\right.
\end{equation*}
and let us consider the optimal control problem\begin{equation}\label{opt}
\min \left\{w(2) + \big(y(1)- e^{-1/2}\big)^2 + \big(x(2)-3\big)^2\right\},
\end{equation} on the interval $[0,2]$ subject to the dynamics
\begin{equation}\label{eqex1}
\left\{\begin{array}{l}
\dot x = 1 + \dot u_2, \\
\dot{y} = yv+ y\dot u_1,\\
\dot w = {\rm d} \Big((x,y),R\Big),\\
(x,y,w)(0)= (0,1,0),
\end{array}\right.\end{equation}
where the $v\in \{0,1\}$ and $(u_1,u_2)\in [-1,1]\times [0,1]$. Notice that \eqref{eqex1} meets the general hypotheses, for the vector fields
$$
g_1\doteq \left(\begin{array}{c}1\\y\\0\end{array}\right), \quad g_2\doteq \left(\begin{array}{c}1\\0\\0\end{array}\right), \quad f\doteq \left(\begin{array}{c}1\\yv\\ {\rm d} \Big((x,y),R\Big)\end{array}\right),
$$
are Lipschitz continuous, and, moreover, $g_1,g_2$ are smooth and verify $[g_1,g_2]\equiv 0$.
We claim that the limit solution $(x,y,w)$ corresponding to the input
\begin{equation*}
v(t):=
\left\{
\ba{cl}
1 & \text{for } t\in[0,1/2[,\\
0 & \text{for } t\in [1/2,1],
\ea
\right.
\end{equation*}
\begin{equation*}
u_1(t):=
\left\{
\ba{cl}
(-1)^{k+1} & \text{for } t\in[1-\frac{1}{k},1-\frac{1}{k+1}[,\,\, k \in\mathbb{N},\\
0 & \text{for } t\in [1,2],
\ea
\right.
\end{equation*}
\begin{equation*}
u_2(t):=
\left\{
\ba{cl}
0 & \text{for } t\in[0,1],\\
1 & \text{for } t\in ]1,2],
\ea
\right.
\end{equation*}
is a minimum for problem \eqref{opt}. Indeed, on any subinterval $[t_1,t_2]\subset[1/2,1]$ where $u_1$ is absolutely continuous, one has
\begin{equation}
\label{forex}
y(t)=y(t_1)e^{u_1(t) - u_1(t_1)}.
\ee
On the other hand, one can easily check that
\begin{equation*}
\begin{split}
y(1-1/k +) &= y(1-1/k -)e^2, \quad \text{if $k$ is odd},\\
y(1-1/k +) &= y(1-1/k -)e^{-2}, \quad \text{if $k$ is even,}
\\
y(1) &=e^{-1/2},
\end{split}
\end{equation*}
where $y(1-1/k -)$ and $y(1-1/k +)$ denote the left and right limits of $y$ at $t=1-1/k,$ respectively. Moreover, $$
x= t\,\,\,\hbox{for all}~ t\in [0,1], \quad x=t+1\,\,\, \hbox{for all}~ t\in ]1,2].$$ Hence $ {\rm d} \Big((x(t),y(t)),R\Big)=0$ for all $t\in[0,2]$ and $x(2)=3$, so the corresponding payoff is equal to zero. Therefore $(x,y,w)$ is an optimal trajectory, since the payoff of every control-trajectory pair is nonnegative.
Notice that both $(u_1,u_2) $ and $(x,y,w)$ have infinitely many discontinuities and unbounded variation. Observe also that it is crucial that the input $u$ and the solution are defined everywhere. In fact, the control $(v,\tilde u_1, u_2)$, with $\tilde u_1(t) = u_1(t)$ for all $t\neq 1$ and $\tilde u_1(1)=1$, is not optimal, for the corresponding solution $(\tilde x,\tilde y,\tilde w)$ is equal to $(x,y,w)$ on $ [0,2]\backslash \{1\},$ while, in view of Theorem \ref{negl-jumps}, $\tilde y(1) = e^{1/2}.$
}
\end{example}
\begin{theorem}[Continuous dependence]
\label{cont-depL1}
The following assertions hold true:
\begin{itemize}
\item[{(i)}] for each $\bar{x} \in \mathbb{R}^n$ and $u\in \mathcal{L}^1 ([a,b];U),$ the function $v\mapsto x[\bar{x},u,v]$ is continuous from $L^1([a,b];V)$ to $L^\infty([a,b];\mathbb{R}^n);$
\item[(ii)] for any $r>0,$ there exists a compact subset $K'\subset \mathbb{R}^n,$ such that the trajectories $x[\bar x,u,v]$ have values in $K',$ whenever we consider $|\bar{x}|\leq r,$ $u\in \mathcal{L}^1([a,b];U)$ and $v\in L^1([a,b];V) ;$
\item[(iii)] for each $r>0,$ there exists a constant $M>0$ such that, for every $\tau \in[a,b],$ for all $|\bar{x}_1|, |\bar{x}_2|\leq r,$ $u_1,u_2 \in \mathcal{L}^1([a,b];U)$ and for every $v \in L^1([a,b];V),$ one has
\begin{equation}\label{EstL1}
|x_1(\tau)-x_2(\tau)| + \|x_1-x_2\|_1 \leq
\, M\Big[ |\bar{x}_1-\bar{x}_2|+ |u_1(a) - u_2(a)| + |u_1(\tau)-u_2(\tau)|+\|u_1-u_2\|_1
\Big],
\ee
where $x_1 := x[\bar{x}_1,u_1,v],$ $x_2 := x[\bar{x}_2,u_2,v].$
\end{itemize}
\end{theorem}
Since the limit solution depends on the pointwise definition of $u,$ it is interesting to investigate the effects of a change of the $u$'s values on a measure-zero subset of $[a,b].$
\begin{theorem}[Pointwise dependence]\label{negl-jumps}
Let us consider an interval $[a,b],$ an initial state $\bar x \in \mathbb{R}^n,$ and an ordinary control $v \in L^1([a,b];V).$ Let $u,\hat u \in \mathcal{L}^1([a,b];U)$ be impulse controls that coincide a.e. in $[a,b]$ and that verify $u(a)=\hat{u}(a).$
Then, setting $x:= x[\bar{x}, u,v],$ $\hat x:= x[\bar{x}, \hat u,v],$ one has
\begin{equation}\label{jump}
x(t)= \exp \left({\sum_{\alpha=1}^m (u_\alpha(t)-\hat u_\alpha(t)) g_\alpha}\right) \big(\hat x(t)\big), \quad\hbox{for all}~ t\in [a,b].
\ee
In particular,
$$ x(t)=\hat x(t)$$
for every $t\in [a,b]$ such that $u(t)=\hat u(t),$ that is, almost everywhere.
\end{theorem}
\vskip0.4truecm
In order to state the representation theorem below we need to introduce a change of coordinates induced by the $g_\alpha$'s flows.
Let us extend $f,g_\alpha,$ for $\alpha =1,\dots,m$ to functions $\tilde f,\tilde g_\alpha$ with values in $\mathbb{R}^{n+m}$ by setting, for every $(t,x,z,v)\in [a,b]\times \mathbb{R}^{n+m} \times V,$
$$
\tilde f(t,x,z,v):=\sum_{j=1}^n {f}_j(t,x,z,v)\frac{\partial}{\partial x_j},\quad
\tilde g_\alpha(x,z):= \sum_{j=1}^n {g}_{\alpha,j} (x,z) \frac{\partial}{\partial x_j} + \frac{\partial}{\partial z_\alpha}\,,
$$
where $\left(\frac{\partial}{\partial x_1},\dots,\frac{\partial}{\partial x_n}, \frac{\partial}{\partial z_1},\dots, \frac{\partial}{\partial z_m}\right)$ is the canonical basis of $\mathbb{R}^{n+m}.$
\footnote{Notice that at this stage there is no more need to distinguish between
equations \eqref{E} and \eqref{Eu}, for the $\tilde g_\alpha$ are vector fields on $\mathbb{R}^{n+m},$ so it is irrelevant that their first $n$ components are or are not dependent on $u.$}
Let $\mathrm{Pr}:\mathbb{R}^{n} \times \mathbb{R}^m\rightarrow \mathbb{R}^n$ denote the {canonical projection} on the first factor, i.e.
$$
\mathrm{Pr}(x,z) := x,
$$
and let the function $\varphi:\mathbb{R}^n\times \mathbb{R}^{m}\to \mathbb{R}^n$ be defined by
$$
\varphi(x,z) :=
\mathrm{Pr}\circ \exp \left(- z_{m} \tilde g_{m}\right)\circ\dots \exp \left(- z_{1} \tilde g_{1}\right)
(x,z).
$$
Finally, let us consider the map $\phi:\mathbb{R}^n\times \mathbb{R}^{m}\rightarrow \mathbb{R}^n\times \mathbb{R}^{m}$ given by
$$
\phi(x,z) := (\varphi(x,z),z).
$$
\begin{lemma}
Assume that the vector fields $g_1,\dots,g_m$ belong to $\mathcal{C}^r(\mathbb{R}^n;\mathbb{R}^n),$ with $r\geq 1.$ Then the mapping $\phi$ is a $\mathcal{C}^r$-diffeomorphism of $\mathbb{R}^{n+m}$ onto itself and, for every $(\xi,\zeta)\in \mathbb{R}^{n+m},$ one has
$$
\phi^{-1}(\xi,\zeta) = (\varphi(\xi,-\zeta), \zeta).
$$
\end{lemma}
The $\mathcal{C}^r$-diffeomorphism $\phi$ induces the $\mathcal{C}^{r-1}$-diffeomorphism $D\phi$ on the tangent bundle, where $D$ denotes differentiation. For each $\alpha=1,\hdots,m,$ $t\in [a,b],$ $(\xi,\zeta,v)\in \mathbb{R}^{n+m}\times V,$ let us set
$$
\begin{array}{l}
\tilde F(t,\xi,\zeta,v):= D\phi (x,z) \, \tilde f(t,x,z,v),
\\
\tilde G_\alpha(\xi,\zeta):= D\phi (x,z) \, \tilde g_{\alpha}(x,z),
\end{array}
$$
where $(x,z):= \phi^{-1}(\xi,\zeta).$ As a direct consequence of the {\em Simultaneous Flow-Box Theorem} (see e.g. \cite{Lang}), one obtains the following result (see Lemma 2.1 in \cite{BreRam91} for a proof).
\begin{lemma}
\label{flowbox}
For every $i=1,\dots, n$ and $\alpha=1,\dots, m$ one has
$$
\tilde F =\sum_{i,j=1}^n\left(\frac{\partial \phi_i}{\partial x_j} {f}_j \right) \frac{\partial}{\partial x_i},\quad \tilde{G}_{\alpha}
= \frac{\partial }{\partial z_\alpha},
$$
where we have set $\phi=(\phi_1,\dots,\phi_{n+m}).$
\end{lemma}
Notice that the last $m$ components of $\tilde F$ are zero. More precisely, $\tilde F$ can be written in components as $\tilde F=\begin{pmatrix} {F} \\ 0 \end{pmatrix}$ with $F:[a,b]\times\mathbb{R}^{n+m }\times V \to \mathbb{R}^n.$
Consider the Cauchy problem
\begin{align}
\label{Et}
\dot{\xi} &= {F}(t,\xi,u,v),\quad \text{on } [a,b],\\
\label{ICt}
\xi(a)&=\bar{\xi}.
\end{align}
For each $\bar{\xi} \in \mathbb{R}^n,$ $(u,v)\in \mathcal{L}^1([a,b];\mathbb{R}^n) \times L^1([a,b];V),$ there exists a (unique) Carath\'eodory solution of \eqref{Et}-\eqref{ICt}, which will be here denoted by $\xi[\bar{\xi},u,v].$
Theorem \ref{TRep} below , which is trivial in the case $u\in AC,$ provides a representation of the solutions of \eqref{E} (and of \eqref{Eu}) in terms of images of solutions of the simpler equation \eqref{Et} through the map $\varphi$ previously introduced.
\begin{theorem}[Representation of limit solutions]
\label{TRep}
For any $\bar{x}\in \mathbb{R}^n,$ $(u,v) \in \mathcal{L}^1([a,b];U)\times L^1([a,b];V),$ one has
$$
\xi[\bar{\xi},u,v](t) = \varphi\big(x[\bar{x},u,v](t),u(t) \big),\quad \text{for all } t\in [a,b],
$$
where we have set $\bar{\xi}:=\varphi(\bar{x},u(a)).$
\end{theorem}
The proof of this theorem is given in the next section.
\section{Proofs of the results of Section \ref{SecLimit}}\label{SecProofs}
Since we are going to exploit the diffeomorphism $\phi: \mathbb{R}^{n+m} \to \mathbb{R}^{n+m}$ it is convenient to embed \eqref{E}-\eqref{IC} in the $n+m$-dimensional Cauchy problem
\begin{align}
\label{AS}
\begin{pmatrix}
\dot{x}\\
\dot{z}
\end{pmatrix}
&= \tilde f(t,x,z,v)+ \sum_{\alpha=1}^m \tilde{g}_\alpha(x,z) \dot u_\alpha,\\
\label{ASIC}
\begin{pmatrix}
{x}\\{z}
\end{pmatrix}(a) &= \begin{pmatrix} \bar{x} \\ \bar{z} \end{pmatrix}.
\end{align}
In view of the considered hypotheses, when $u\in AC([a,b];U),$ for every $(\bar{x},\bar{z}) \in \mathbb{R}^{n+m}$ and $v \in L^1([a,b];V),$ there exists a unique solution to \eqref{AS} in the interval $[a,b].$ We let $(x,z)[\bar{x},\bar{z},u,v]$ denote this solution.
We shall also consider the Cauchy problem
\begin{align}
\label{TS}
\begin{pmatrix}
\dot{\xi}\\
\dot{\zeta}
\end{pmatrix}
&= \tilde F(t,\xi,\zeta,v)+ \sum_{\alpha=1}^m \tilde G_\alpha \dot u^\alpha,\\
\label{TSIC}
\begin{pmatrix}
{\xi}\\{\zeta}
\end{pmatrix}(a)&= \begin{pmatrix} \bar{\xi} \\ \bar\zeta \end{pmatrix}
.
\end{align}
Also for this problem, when $u\in AC([a,b];U),$ for every $(\bar{\xi},\bar\zeta)\in \mathbb{R}^{n+m}$ there exists a unique solution to \eqref{TS}-\eqref{TSIC} in the interval $[a,b].$ We let $(\xi,\zeta)[\bar{\xi},\bar\zeta,u,v]$ denote this solution.
\begin{remark}\label{equivAC}{\rm When $u\in AC([a,b];U),$ the relation between the two systems is given by
$$
(\xi,\zeta)[\bar{\xi},\bar\zeta,u,v](t) = \phi\Big( (x,z)[\bar{x},\bar{z},u,v](t) \Big),
\quad \text{for all } t\in [a,b],
$$
where $(\bar{\xi},\bar\zeta) := \phi(\bar{x},\bar{z}).$}
\end{remark}
\begin{remark}{\rm
The crucial difference between the two latter systems relies on the fact that the vector fields $\tilde G_\alpha=\frac{\partial}{\partial z_\alpha}$ are constant}.
\end{remark}
Theorem \ref{ContAC-simple} below will be utilized to prove Theorem \ref{cont-depL1}, of which it is in fact a particular case.
\begin{theorem}
\label{ContAC-simple}
The following assertions hold.
\begin{itemize}
\item[(i)] For each $(\bar{\xi},\bar\zeta)\in \mathbb{R}^{n+m}$ and $u\in AC([a,b];U),$ the function $v\mapsto \xi[\bar{\xi},\bar\zeta,u,v]$ is continuous from $L^1([a,b];V)$ to $L^\infty([a,b];\mathbb{R}^n).$
\item[(ii)] Furthermore, for any $r>0,$ there exists a compact subset $K'\subset \mathbb{R}^{n+m}$ such that the trajectories $(\xi,\zeta)[\bar{\xi},\bar\zeta,u,v]$ have values in $K',$ whenever we consider $|(\bar{\xi},\bar\zeta)|\leq r,$ $u\in AC([a,b];U),$ and $v\in L^1([a,b];V).$
\item[(iii)] Finally, for each $r>0,$ there exists a constant $M>1$ such that, for every $\tau \in[a,b],$ for all $|(\bar{\xi}_1,\bar\zeta_1)|, |(\bar{\xi}_2,\bar\zeta_2)| \leq r,$ for all $u_1,u_2 \in AC([a,b];U),$ and for every $v \in L^1([a,b];V),$ one has
\begin{equation}\label{EstL1xi}
\begin{split}
|(\xi_1,\zeta_1)(\tau)-&(\xi_2,\zeta_2)(\tau)| + \|(\xi_1,\zeta_1)-(\xi_2,\zeta_2)\|_1 \leq \\
&M\Big[ |(\bar{\xi}_1,\bar\zeta_1)-(\bar{\xi}_2,\bar\zeta_2)|+ |u_1(a)-u_2(a)| + |u_1(\tau)-u_2(\tau)|
+\|u_1-u_2\|_1
\Big].
\end{split}
\ee
where $(\xi_1,\zeta_1) := (\xi,\zeta)[\bar{\xi}_1,\bar\zeta_1,u_1,v_1]$ and $(\xi_2,\zeta_2) := (\xi,\zeta)[\bar{\xi}_2,\bar\zeta_2,u_2,v_2].$
\end{itemize}
\end{theorem}
To prove Theorem \ref{ContAC-simple}, we shall exploit the following fixed-point result on parameterized contraction mappings (see e.g. Theorem A.1 in\cite{BrePicBook}).
\begin{lemma}
\label{BC}
Let $X$ be a Banach space, $\Lambda$ a metric space and $\chi:\Lambda\times X \to X$ be a continuous function such that
\begin{equation}
\label{BCeq1}
\| \chi(\lambda,x) - \chi(\lambda,y) \| \leq L \|x-y\|,\quad \hbox{for all}~ \lambda\in \Lambda,\ x,y\in X,
\ee
with $L<1.$ Then the following assertions hold.
\begin{itemize}
\item[(a)] For every $\lambda \in \Lambda,$ there exists a unique $x(\lambda)$ such that
$$
x(\lambda) = \chi(\lambda,x(\lambda)).
$$
\item[(b)] The map $\lambda\mapsto x(\lambda)$ is continuous, and one has
$$
\| x(\lambda)-x(\tilde \lambda) \| \leq \frac{1}{1-L} \| \chi(\lambda,x(\tilde\lambda)) - \chi(\tilde\lambda,x(\tilde\lambda)) \|.
$$
\end{itemize}
\end{lemma}
\vspace{5pt}
{\em Proof of Theorem \ref{ContAC-simple}.}
Item (i) follows from classical results of continuity of the input-output map of a control system.
To prove the remaining assertions, assume momentarily that $F$ is globally Lipschitz continuous with respect to the variable $(\xi,\zeta)$ with Lipschitz constant $L.$ Later we shall remove this extra assumption.
For $(\bar{\xi},\bar\zeta,u,v)\in \Lambda:=\mathbb{R}^{n+m}\times AC ([a,b];U)\times L^1([a,b];V)$ and $(\xi,\zeta) \in X:= AC ([a,b];\mathbb{R}^{n+m}),$ let us consider the mapping $\chi:\Lambda \times X \to X$ such that, for all $t\in [a,b],$
\begin{equation*}
\chi(\bar{\xi},\bar\zeta,u,v,\xi,\zeta)(t):=
\begin{pmatrix} \bar{\xi} \\ \bar\zeta \end{pmatrix}
+ \int_a^t \tilde F(s,\xi(s),\zeta(s),v(s)) {\rm d} s
+\sum_{\alpha=1}^m [u_\alpha(t) - u_\alpha(a)]{\bf e}_{n+\alpha},
\end{equation*}
where ${\bf e}_{n+\alpha}$ denotes the $(n+\alpha)$th vector of the canonical basis of $\mathbb{R}^{n+m}.$
Observe that
$$
\begin{pmatrix} \xi \\ \zeta \end{pmatrix}=\chi(\bar{\xi},\bar\zeta,u,v,\xi,\zeta) \text{ if and only if }
\begin{pmatrix} \xi \\ \zeta \end{pmatrix} \text{ is solution of \eqref{TS}.}
$$
We are therefore interested in applying the fixed-point result in Lemma \ref{BC} to the function $\chi.$
Fix $\tau \in [a,b],$ and in the space $\Lambda$ consider the norm
$$
\|(\bar{\xi},\bar\zeta,u,v)\|_Y := |(\bar{\xi},\bar\zeta)|+|u(a)| + |u(\tau)| + \|u\|_1+\|v\|_1,
$$
and in $X,$ define the norm
\begin{equation*}
\| (\xi,\zeta) \|_X :=
\frac{e^{-4(b-a)L}}{4L} |(\xi,\zeta)(\tau)|
+\int_a^b e^{-4sL}|(\xi,\zeta)(s)|{\rm d}s.
\end{equation*}
We shall prove that $\chi$ is continuous from $(\Lambda\times X,\|\cdot\|_Y+\|\cdot\|_X)$ to $(X,\|\cdot\|_X).$
By the Lipschitz continuity of the maps $(\xi,\zeta) \to \tilde F(t,\xi,\zeta,v),$ for any $(\bar{\xi}_1,\bar\zeta_1,u_1,v_1,\xi_1,\zeta_1),$ $(\bar{\xi}_2,\bar\zeta_2,u_2,v_2,\xi_2,\zeta_2)$ in $\Lambda\times X,$ one has
\begin{equation}
\label{estchi1}
\begin{split}
&\|\chi (\bar{\xi}_1,\bar\zeta_1,u_1,v_1,\xi_1,\zeta_1)-\chi(\bar{\xi}_1,\bar\zeta_1,u_1,v_1,\xi_2,\zeta_2)\|_X =\\
& \frac{e^{-4(b-a)L}}{4L} \left| \int_a^\tau \big[\tilde F \big(s,\xi_1(s),\zeta_1(s),v_1(s) \big) - \tilde F \big(s,\xi_2(s),\zeta_2(s),v_1(s) \big) \big] {\rm d}s \right| +\\
& \int_a^b e^{-4sL} \left| \int_a^t \big[\tilde F \big(s,\xi_1(s),\zeta_1(s),v_1(s) \big) - \tilde F \big(s,\xi_2(s),\zeta_2(s),v_1(s) \big) \big] {\rm d}s \right| {\rm d} t \leq \\
&\qquad \frac{1}{4} \| (\xi_1,\zeta_1) - (\xi_2,\zeta_2) \|_X,
\end{split}
\ee
\begin{equation}
\label{estchi2}
\begin{split}
&\|\chi (\bar{\xi}_1,\bar\zeta_1,u_1,v_1,\xi_2,\zeta_2)-\chi(\bar{\xi}_2,\bar\zeta_2,u_2,v_2,\xi_2,\zeta_2)\|_X =\frac{e^{-4(b-a)L}}{4L} \left| \begin{pmatrix} \bar{\xi}_1-\bar{\xi}_2 \\ \bar\zeta_1-\bar\zeta_2 \end{pmatrix} \right| + \\
& \frac{e^{-4(b-a)L}}{4L} \left| \int_a^\tau \big[\tilde F \big(s,\xi_2(s),\zeta_2(s),v_1(s) \big) - \tilde F\big(s, \xi_2(s),\zeta_2(s),v_2(s) \big) \big] {\rm d}s \right| +\\
& \int_a^b e^{-4sL} \left| \int_a^t \big[\tilde F \big(s,\xi_2(s),\zeta_2(s),v_1(s) \big) - \tilde F \big(s,\xi_2(s),\zeta_2(s),v_2(s) \big) \big] {\rm d}s \right| {\rm d} t +\\
&\frac{e^{-4(b-a)L}}{4L}\left| \sum_{\alpha=1}^m \big[ u_{1,\alpha}(\tau)-u_{1,\alpha}(a)-u_{2,\alpha}(\tau)+u_{2,\alpha}(a)\big] {\bf e}_{n+\alpha} \right| +\\
&\int_a^b e^{-4sL} \left| \int_a^t \sum_{\alpha=1}^m \big[ u_{1,\alpha}(s)-u_{1,\alpha}(a)-u_{2,\alpha} (s)+u_{2,\alpha} (a)\big] {\bf e}_{n+\alpha} {\rm d} s \right| {\rm d} t.
\end{split}
\ee
By the Dominated Convergence Theorem, for any fixed trajectory $( \xi, \zeta),$ the mapping
$$
v \mapsto \tilde F(\cdot,\xi, \zeta, v )
$$
is continuous from $L^1([a,\tau];V)$ to $L^1([a,\tau]; \mathbb{R}^{n+m}).$
Thus, by \eqref{estchi1}-\eqref{estchi2}, for each $(\bar{\xi}_1,\bar\zeta_1,u_1,v_1,\xi_1,\zeta_1)\in \Lambda \times X$ and for every $\varepsilon>0,$ there exists $\delta>0$ such that, if
\begin{equation*}
\begin{split}
&|(\bar{\xi}_2,\bar\zeta_2)-(\bar{\xi}_1,\bar\zeta_1)| +|u_2(a)-u_1(a)| \\
&\quad +|u_2(\tau)-u_1(\tau)|+\|u_2-u_1\|_1+ \|v_2-v_1\|_1 + \| (\xi_2,\zeta_2) - (\xi_1,\zeta_1) \|_X < \delta
\end{split}
\end{equation*}
then
\begin{equation*}
\|\chi (\bar{\xi}_2,\bar\zeta_2,u_2,v_2,\xi_2,\zeta_2)-\chi(\bar{\xi}_1,\bar\zeta_1,u_1,v_1,\xi_1,\zeta_1)\|_X <\varepsilon.
\end{equation*}
Hence, $\chi$ is continuous, and in view of \eqref{estchi1} the inequality \eqref{BCeq1} holds true. To apply Lemma \ref{BC}, let us identify $\lambda$ with $(\bar{\xi}_1,\bar\zeta_1,u_1,v_1)$ and $\tilde\lambda$ with $(\bar{\xi}_2,\bar\zeta_2,u_2,v_2).$ Then one has that there exist $(\xi_1,\zeta_1),(\xi_2,\zeta_2) \in X$ such that
$$
\chi(\bar{\xi}_1,\bar\zeta_1,u_1,v_1,\xi_1,\zeta_1) = (\xi_1,\zeta_1),\quad
\chi(\bar{\xi}_2,\bar\zeta_2,u_2,v_1,\xi_2,\zeta_2) = (\xi_2,\zeta_2),
$$
that is $(\xi_1,\zeta_1) = (\xi,\zeta)[\bar{\xi}_1,\bar\zeta_1,u_1,v_1]$ and $(\xi_2,\zeta_2) = (\xi,\zeta)[\bar{\xi}_2,\bar\zeta_2,u_2,v_2].$
In view of item (b) in Lemma \ref{BC}, we get
\begin{equation}\label{Contrac1}
\left\| \begin{pmatrix} \tilde\xi - \xi \\ \tilde\zeta - \zeta \end{pmatrix} \right\|_X \leq \frac{1}{2L} \left[ |(\tilde \bar{\xi},\tilde \bar\zeta)-(\bar{\xi},\bar\zeta)| \right. \left.\, + |\tilde u(a)-u(a)| + |\tilde u(\tau)-u(\tau)|\right] + \int_a^b e^{-4tL} |\tilde u(t)-u(t)| {\rm d}t.
\ee
Therefore, by the inequalities
\begin{equation}\label{Contrac2}
\left\| \begin{pmatrix} \xi_2 - \xi_1 \\ \zeta_2 - \zeta_1 \end{pmatrix} \right\|_X
\geq \frac{e^{-4(b-a)L}}{4L} \left| \begin{pmatrix} \xi_2(\tau) - \xi_1(\tau) \\ \zeta_2(\tau) - \zeta_1(\tau) \end{pmatrix} \right| + e^{-4bL} \left\| \begin{pmatrix} \xi_2 - \xi_1 \\ \zeta_2 - \zeta_1 \end{pmatrix} \right\|_1,
\ee
and \eqref{Contrac1} one obtains \eqref{EstL1xi}, with a constant $M$ depending only on $L,$ and hence item (iii) is proved. Notice also that item (ii) is a consequence of the following standard result on ODE's:
\begin{lemma}[Bounds on solutions]
\label{Apriori}
Under the general hypothesis H, for each $r>0,$ there exists a compact set $K'\subset \mathbb{R}^{n+m},$ such that any solution $(\xi,\zeta)$ of \eqref{TS}-\eqref{TSIC} remains in $K'$ whenever $|(\bar{\xi},\bar\zeta)|\leq r,$ $u\in AC([a,b];U)$ and $v\in L^1([a,b];V).$
\end{lemma}
This completes the proof of the theorem under the additional assumption that $F$ is globally Lipschitz.
\vskip0.2truecm
To prove the general case we use a standard cut-off function argument.
Take $r>0,$ and let $K'\subset \mathbb{R}^{n+m}$ be the compact set provided by Lemma \ref{Apriori}. Let $\rho \in \mathcal{C}^1(\mathbb{R}^{n+m})$ be a smooth real function such that $\rho=1$ on $K'$ and $\rho=0$ outside a neighborhood of $K'.$
Define $\hat F (t,\xi,\eta,v):= \rho(\xi,\eta) \tilde F(t,\xi,\eta,v),$ and set
$$
\Lambda:= \{(\bar{\xi},\bar\zeta)\in \mathbb{R}^{n+m}:|(\bar{\xi},\bar\zeta)|\leq r\} \times AC([a,b];U) \times L^1([a,b];V).$$
Then, for $(\bar{\xi},\bar{\eta},u,v)\in \Lambda,$ the corresponding solution $(\xi,\eta)$ of the Cauchy problem
$$\begin{pmatrix}
\dot{\xi}\\
\dot{\zeta}
\end{pmatrix}
= \hat F(t,\xi,\zeta,v)+ \sum_{\alpha=1}^m \tilde G_\alpha \dot u^\alpha,\quad
\begin{pmatrix}
{\xi}\\{\zeta}
\end{pmatrix}(a)= \begin{pmatrix} \bar{\xi} \\ \bar\zeta \end{pmatrix}
,
$$
coincides with $(\xi,\eta)[\bar{\xi},\bar{\eta},u,v]$ (and remains inside $K'$).
Now the function $\hat F$ is globally Lipschitz, and the procedure done before can be repeated for this new metric space $\Lambda$ and for the function $\hat F$ in the place of $\tilde F.$
Therefore, one can obtain the estimate \eqref{EstL1xi} with a constant $M$ depending only on the Lipschitz constant of the mapping $(\xi,\eta) \mapsto \tilde F(t,\xi,\eta,v)$ in the set $K'.$ This completes the proof of Theorem \ref{ContAC-simple}.
\epf
{\em Proofs of Theorems \ref{Existence} and \ref{TRep}.}
Let $\bar{x}\in \mathbb{R}^n, $ $(u,v) \in \mathcal{L}^1([a,b];U)\times L^1([a,b];V),$ and let $\xi:=\xi[\bar{\xi},u,v]$ be the unique Carath\'eodory solution of \eqref{Et} with the initial condition $\xi(a)=\bar{\xi} := \varphi(\bar{x},u(a)).$ Define, on $[a,b],$ the function
\begin{equation}
\label{xdef}
x:=\varphi\circ(\xi,-u),
\ee
and let us show that $x$ is the unique limit solution of \eqref{E}-\eqref{IC} associated with $\bar{x}$ and $(u,v).$
Choose $\tau \in [a,b],$ and consider a sequence of absolutely continuous controls $u_k^\tau: [a,b] \to U$ verifying
\begin{equation}
\label{uktau}
|u^\tau_k(a) - u(a)|+|u^\tau_k(\tau) - u(\tau)|+\|u_k^\tau-u\|_1\to 0.
\ee
Consider the equation \eqref{Et} with the initial condition
$$
\xi(a)=\bar{\xi}_k^\tau:=\varphi(\bar{x},u^\tau_k(a)).
$$
Let $\xi_k^\tau := \xi[\bar{\xi}_k^\tau,u_k^\tau,v]$ be the (unique) corresponding Carath\'eodory solution. Then, by standard results of continuity with respect to the data, one has that
\begin{equation}
\label{xikunif}
\xi_k^\tau \to \xi,\ \text{uniformly on }[a,b].
\ee
For the augmented system \eqref{TS}, $(\xi^\tau_k,u^\tau_k)$ is the unique solution with the initial conditions $\xi(a)= \bar{\xi}_k^\tau,$ $\eta(a)=u^\tau_k(a).$ In view of item (iii) in Theorem \ref{ContAC-simple}, the functions $(\xi^\tau_k,u^\tau_k)$ have values in a compact set $\tilde{K}'\subset \mathbb{R}^{n+m}.$
In view of Remark \ref{equivAC}, the map
$$
(x_k^\tau,z_k^\tau):=\phi^{-1} \circ (\xi_k^\tau,u_k^\tau),
$$
is, for each $k\in \mathbb{N},$ the unique Carath\'eodory solution of \eqref{AS} with the initial conditions $x(a)=\bar{x},$ $z(a) = u_k^\tau(a).$ Notice that the functions $ (x_k^\tau,z_k^\tau)$ have values inside the compact set $K':=\phi^{-1}(\tilde K').$ In particular, the $x_k^\tau$'s are uniformly bounded.
Observe as well that
$$
|x(\tau)-x_k^\tau(\tau)| = |\varphi(\xi(\tau),-u(\tau))-\varphi(\xi_k^\tau(\tau),-u_k^\tau(\tau))| \to 0,
$$
due to \eqref{xikunif}-\eqref{uktau} and the continuity of $\varphi.$ Furthermore, since $u_k^\tau \to u$ almost everywhere and all these functions are uniformly bounded, one gets
$$
\|x-x_k^\tau\|_1 = \int_a^b \big|\varphi(\xi(t),-u(t))-\varphi(\xi_k^\tau(t),-u_k^\tau(t)) \big| {\rm d} t
\to 0,
$$
thanks to the Dominated Convergence Theorem. Thus, $x$ is a limit solution of \eqref{E}-\eqref{IC} associated with $\bar{x}$ and $(u,v).$
We shall now prove the uniqueness. Suppose on the contrary that there are two different limit solutions $x$ and $\tilde x$ of \eqref{E}-\eqref{IC} associated with $(\bar{x},u,v).$ Let $\tau \in ]a,b]$ be such that $x(\tau) \neq x^*(\tau),$ and let $(u_k^\tau),$ $({u}_k^{\tau,*})$ be sequences in $AC([a,b];U)$ verifying
\begin{equation}
\label{uktau}
|u^\tau_k(a)- u(a)|+|(x^\tau_k,u^\tau_k)(\tau)- (x,u)(\tau)|+ \|(x^\tau_k,u^\tau_k)- (x,u)\|_1\to 0,
\ee
and
\begin{equation}
\label{tildeuktau}
|u^{\tau,*}_k(a)- u(a)|+|( x^{\tau,*}_k,u^{\tau,*}_k)(\tau)- (x^{*},u)(\tau)|+ \|( x^{\tau,*}_k, u^{\tau,*}_k)- ( x^*,u)\|_1
\to 0,
\ee
where $x^\tau_k:=x[\bar{x},u^\tau_k,v],$ $ x^{\tau,*}_k:=x[\bar{x}, u^{\tau,*}_k,v].$
Let $\xi_k^\tau := \xi[\bar{\xi}_k^\tau,u_k^\tau,v],$ $\xi_k^{\tau,*} := \xi[\bar{\xi}_k^{\tau,*},u_k^{\tau,*},v]$ be the solutions of \eqref{Et} with initial conditions
$$
\xi(a)=\bar{\xi}_k^\tau:=\varphi(\bar{x},u_k^\tau(a)),\quad \xi(a)=\bar{\xi}_k^{\tau,*}:=\varphi(\bar{x}, u_k^{\tau,*}(a)),
$$
respectively.
Then, since both $(u_k^\tau),$ $({u}_k^{\tau,*})$ converge to $u$ in $L^1([a,b];U),$ one has that
$$
\label{xisunif}
(\xi_k^\tau),\, (\xi_k^{\tau,*})\ \text{converge uniformly to }\xi := \xi[\bar{\xi},u,v],
$$
with $\bar{\xi} := \varphi(\bar{x},u(a)).$
Observe that, from previous equation and \eqref{uktau}-\eqref{tildeuktau}, one obtains
\begin{equation*}
|x(\tau)-x^*(\tau)|=\lim |x_k^\tau(\tau)-x_k^{\tau,*} (\tau)|= \lim |\varphi \circ (\xi_k^\tau,-u^\tau_k)(\tau) - \varphi\circ (\xi_k^{\tau,*},-u^{\tau,*}_k)(\tau)| = 0.
\end{equation*}
This contradicts our assumption, so the uniqueness of the limit solution of \eqref{E}-\eqref{IC} is proved. This ends the proof of Theorem \ref{Existence}.
Finally, since $x$ was defined in \eqref{xdef} via the coordinates' transformation, by the uniqueness of the limit solution we also obtain the proof of Theorem \ref{TRep}.
\epf
{\em Proof of Theorem \ref{cont-depL1}.}
We shall start by proving item (ii). For this, let $r>0$ be arbitrary and consider the compact set $\tilde K'\subset \mathbb{R}^{n+m}$ provided by the item (ii) of Theorem \ref{ContAC-simple}. Then, for each $|\bar{x}|\leq r$ and $(u,v)\in \mathcal{L}^1([a,b];U)\times L^1([a,b];V),$ the corresponding limit solution $x[\bar{x},u,v]$ has values inside the compact set $K'$ defined as the projection of $\phi^{-1}(\tilde K')$ in the first $n$ components. This proves item (ii).
In order to show part (i), fix $\bar{x} \in \mathbb{R}^n$ and $u\in \mathcal{L}^1([a,b];U).$ Set $\bar{\xi} := \varphi(\bar{x},u(a)),$ and notice that $v \mapsto \xi[\bar{\xi},u,v]$ is continuous from $L^1([a,b];V)$ to $AC([a,b];\mathbb{R}^n),$ where the latter space is endowed with the $\mathcal{C}^0-$topology. Furthermore, for any $v \in L^1([a,b];V),$ the trajectories $\xi[\bar{\xi},u,v]$ remain inside a compact set $\hat K.$
Let $v_k$ converge to $v$ in $L^1,$ then $\xi[\bar{\xi},u,v_k] \to \xi[\bar{\xi},u,v]$ uniformly and, since $\varphi$ is uniformly continuous on $\hat K \times (-U),$ one gets that
$$
v \mapsto x[\bar{x},u,v] = \varphi\circ (\xi[\bar{\xi},u,v],-u),
$$
is continuous from $L^1([a,b];V)$ to $L^\infty([a,b];\mathbb{R}^n),$ from which item (i) follows.
We now prove item (iii). For an arbitrary $r>0,$ let $\tilde r>0$ be such that the points
$(\bar{\xi}_1,u_1(a)):=\phi(\bar{x}_1,u_1(a))$ and $(\bar{\xi}_2, u_2(a)):=\phi(\bar{x}_2, u_2(a))$ remain inside $\{|(\bar{\xi},\bar\zeta)|\leq \tilde r\}$ whenever $|\bar{x}_1|,|\bar{x}_2| \leq r$ and $u_1,u_2 \in \mathcal{L}^1([a,b];U).$
Take $\tau \in [a,b],$
$|\bar{x}_1|,|\bar{x}_2| \leq r,$ $u_1, u_2 \in \mathcal{L}^1([a,b];U)$ and $v\in L^1([a,b];V).$ Let $x_1:=x[\bar{x}_1,u_1,v]$ and $ x_2:=x[\bar{x}_2, u_2,v]$ be the corresponding limit solutions of \eqref{E}.
Let $(u_{1,k}^\tau),$ $(u_{2,k}^\tau)$ be sequences in $AC([a,b];U)$ verifying
$$
|u_{1,k}^\tau(a)-u_1(a)|+|u_{1,k}^\tau(\tau)-u_1(\tau)|+\|u_{1,k}^\tau -u_1 \|_1 \to 0,
$$
and
$$
| u_{2,k}^\tau(a)- u_2(a)|+| u_{2,k}^\tau(\tau)- u_2(\tau)|+\| u_{2,k}^\tau - u_2\|_1 \to 0,
$$
Consider the functions $\xi_{1,k}^\tau:=\xi[\bar{\xi}_{1,k}^\tau,u_{1,k}^\tau,v],$ $\xi_{2,k}^\tau:=\xi[\bar{\xi}_{2,k}^\tau, u_{2,k}^\tau,v],$ where $\bar{\xi}_{1,k}^\tau:=\varphi(\bar{x}_1,u_{1,k}^\tau(a)),$ $\bar{\xi}_{2,k}^\tau:=\varphi(\bar{x}_2, u_{2,k}^\tau(a)).$
Then $\xi_{1,k}^\tau$ converge uniformly to $\xi_1:=\xi[\bar{\xi}_1,u_1,v],$ where $\bar{\xi}_1:=\varphi(\bar{x}_1,u_1(a)),$ and $ \bar{\xi}_{2,k}^\tau$ converge uniformly to $\xi_2:=\xi[ \bar{\xi}_2,u_2,v],$ where $\bar{\xi}_2:=\varphi(\bar{x}_2, u_2(a)).$
Let $\tilde M>1$ and $\tilde K'$
be the constant and the compact set provided by Theorem \ref{ContAC-simple} for $\tilde r,$ respectively.
Then one has
\begin{equation*}
\begin{split}
& |\xi_{1,k}^\tau(\tau)-\xi_{2,k}^\tau(\tau)|+\|\xi_{1,k}^\tau-\xi_{2,k}^\tau\|_1 \leq \\
&(\tilde M-1)\big[ |\bar{\xi}_1-\bar{\xi}_2| + |u_{1,k}^\tau(a)- u_{2,k}^\tau(a)|+|u_{1,k}^\tau(\tau)- u_{2,k}^\tau(\tau)|+\|u_{1,k}^\tau - u_{2,k}^\tau\|_1 \big].
\end{split}
\end{equation*}
Letting $k$ go to infinity in the previous inequality, one obtains
\begin{equation}\label{xitildexi}
\begin{split}
&|\xi_1(\tau)-\xi_2(\tau)|+\|\xi_1- \xi_2\|_1\leq \\
&(\tilde M-1)\big[ |\bar{\xi}_1-\bar{\xi}_2| + |u_1(a)- u_2(a)|+|u_1(\tau)- u_2(\tau)|+\|u_1 - u_2\|_1 \big].
\end{split}
\ee
Thus, for $L$ a Lipschitz constant of $\varphi$ on $\tilde K',$ one gets
\begin{equation*}
\begin{split}
|x_1&(\tau)- x_2(\tau)|+\|x_1- x_2\|_1=\\
&|\varphi\circ (\xi_1,-u_1)(\tau)-\varphi \circ ( \xi_2, u_2)(\tau)|
+\|\varphi\circ (\xi_1,-u_1)-\varphi \circ ( \xi_2 , u_2 )\|_1 \leq \\
&L\big[ |(\xi_1,u_1)(\tau)-( \xi_2 , u_2)(\tau)|+\|(\xi_1,u_1)-( \xi_2 , u_2)\|_1 \big] \leq\\
&L(\tilde M-1)\big[|\bar{\xi}_1-\bar{\xi}_2|+|u_1(a)- u_2 (a)|+|u_1(\tau)- u_2(\tau)|+\|u_1- u_2\|_1\big] + \\
&L\big[ |u_1(\tau)- u_2(\tau)|+\|u_1- u_2\|_1\big] \leq \\
& M \big[|\bar{x}_1- \bar{x}_2|+|u_1(a)- u_2(a)|+|u_1(\tau)- u_2(\tau)|+\|u_1- u_2\|_1\big],
\end{split}
\end{equation*}
for $M>0$ depending on $L$ and $\tilde M,$ where the second inequality follows from \eqref{xitildexi}. Thus, the desired estimate holds true, and this completes the proof.
\epf
{\em Proof of Theorem \ref{negl-jumps}.}
Set $\bar{\xi}=\varphi(\bar{x},u(a)) = \varphi(\bar{x},\hat{u}(a))$
and observe that $\xi[\bar{\xi},u,v](t)=\xi[\bar{\xi},\hat{u},v](t),$ for all $t\in [a,b].$ Let $\xi$ denote the latter function.
In view of Theorem \ref{TRep}, one has
\begin{gather*}
\xi(t) = \varphi \big( x[\bar{x},u,v](t),u(t) \big)=x[\bar{x},u,v](t) e^{- u_\alpha(t) g_\alpha}, \\
\xi(t) = \varphi \big( x[\bar{x},\hat{u},v](t),u(t) \big)=x[\bar{x},\hat{u},v](t)e^{- \hat{u}_\alpha(t) g_\alpha}.
\end{gather*}
Therefore, combining the last two equations and due to the commutativity of $g_\alpha,$ the relation \eqref{jump} follows.
\epf
\section{Concluding remarks}
A notion of everywhere defined solution for the control Cauchy problem on $[a,b]$ $$ \dot{x} = {f}(t,x,u,v) +\sum_{\alpha=1}^m g_\alpha(x,u) \dot u_\alpha, \qquad t\in [a,b] \quad x(a)=\bar x,$$
has been provided, under a commutativity hypothesis on the fields $g_\alpha$. In particular, we have proved results of existence, uniqueness and continuous dependence on the data, besides investigating the effects of $u$'s changes on null sets. This concept of solution, which relies on an {\it extension by density} of the classical notion, turns out to verify consistency requirements. We point out that, by defining the output at every $t\in [a,b],$ we have departed from a topological picture based on normed spaces, instead framing the limiting processes in spaces endowed with family of seminorms (this choice is concretely represented by the fact that approximating sequences in the solution's definition depend on $\tau$, for every $\tau\in [a,b]$).
The paper is motivated by both applications (see the Introduction and Example \ref{minimum}) and the concern of constructing a suitable framework for further theoretical issues, like the study of the corresponding {\it adjoint equations}, a likely crucial object in the investigation of necessary conditions for minima.
We think that a generalization of the notion of limit solution to the noncommutative case (in particular, an extension that will agree with former concepts of solutions) might represent a natural direction for further investigations.
\section{Funding}
This work was partially supported by the European Union under the 7th Framework Programme FP7-PEOPLE-2010-ITN - Grant agreement number 264735-SADCO, and the Fondazione CaRiPaRo Project
``Nonlinear Partial Differential Equations: models, analysis, and
control-theoretic problems".
\vspace*{6pt}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 1,886 |
package cluster
import (
"errors"
"fmt"
"strings"
"sync"
"time"
"google.golang.org/grpc"
"golang.org/x/net/context"
"github.com/NetSys/quilt/db"
"github.com/NetSys/quilt/minion/pb"
log "github.com/Sirupsen/logrus"
)
type client interface {
setMinion(pb.MinionConfig) error
getMinion() (pb.MinionConfig, error)
bootEtcd(pb.EtcdMembers) error
Close()
}
type clientImpl struct {
pb.MinionClient
cc *grpc.ClientConn
}
type foreman struct {
conn db.Conn
minions map[string]*minion
spec string
// Making this a struct member allows us to mock it out.
newClient func(string) (client, error)
}
type minion struct {
client client
connected bool
machine db.Machine
config pb.MinionConfig
mark bool /* Mark and sweep garbage collection. */
}
func createForeman(conn db.Conn) foreman {
return foreman{
conn: conn,
minions: make(map[string]*minion),
newClient: newClient,
}
}
func (fm *foreman) stop() {
for _, minion := range fm.minions {
minion.client.Close()
}
}
func (fm *foreman) init() {
fm.conn.Transact(func(view db.Database) error {
machines := view.SelectFromMachine(func(m db.Machine) bool {
return m.PublicIP != "" && m.PrivateIP != "" && m.CloudID != ""
})
fm.updateMinionMap(machines)
fm.forEachMinion(func(m *minion) {
var err error
m.config, err = m.client.getMinion()
m.connected = err == nil
})
for _, m := range fm.minions {
if m.connected {
m.machine.Role = db.PBToRole(m.config.Role)
m.machine.Connected = m.connected
view.Commit(m.machine)
}
}
return nil
})
}
func (fm *foreman) runOnce() {
var machines []db.Machine
fm.conn.Transact(func(view db.Database) error {
machines = view.SelectFromMachine(func(m db.Machine) bool {
return m.PublicIP != "" && m.PrivateIP != "" && m.CloudID != ""
})
fm.spec = ""
clst, _ := view.GetCluster()
fm.spec = clst.Spec
return nil
})
fm.updateMinionMap(machines)
/* Request the current configuration from each minion. */
fm.forEachMinion(func(m *minion) {
var err error
m.config, err = m.client.getMinion()
connected := err == nil
if connected && !m.connected {
log.WithField("machine", m.machine).Debug("New connection.")
}
if connected != m.machine.Connected {
fm.conn.Transact(func(view db.Database) error {
m.machine.Connected = connected
view.Commit(m.machine)
return nil
})
}
m.connected = connected
})
var etcdIPs []string
for _, m := range fm.minions {
if m.machine.Role == db.Master && m.machine.PrivateIP != "" {
etcdIPs = append(etcdIPs, m.machine.PrivateIP)
}
}
// Assign all of the minions their new configs
fm.forEachMinion(func(m *minion) {
if !m.connected {
return
}
newConfig := pb.MinionConfig{
Role: db.RoleToPB(m.machine.Role),
PrivateIP: m.machine.PrivateIP,
Spec: fm.spec,
Provider: string(m.machine.Provider),
Size: m.machine.Size,
Region: m.machine.Region,
}
if newConfig == m.config {
return
}
if err := m.client.setMinion(newConfig); err != nil {
return
}
if err := m.client.bootEtcd(pb.EtcdMembers{IPs: etcdIPs}); err != nil {
log.WithError(err).Warn("Failed send etcd members.")
}
})
}
func (fm *foreman) updateMinionMap(machines []db.Machine) {
for _, m := range machines {
min, ok := fm.minions[m.PublicIP]
if !ok {
client, err := fm.newClient(m.PublicIP)
if err != nil {
continue
}
min = &minion{client: client}
fm.minions[m.PublicIP] = min
}
min.machine = m
min.mark = true
}
for k, minion := range fm.minions {
if minion.mark {
minion.mark = false
} else {
minion.client.Close()
delete(fm.minions, k)
}
}
}
func (fm *foreman) forEachMinion(do func(minion *minion)) {
var wg sync.WaitGroup
wg.Add(len(fm.minions))
for _, m := range fm.minions {
go func(m *minion) {
do(m)
wg.Done()
}(m)
}
wg.Wait()
}
func newClient(ip string) (client, error) {
cc, err := grpc.Dial(ip+":9999", grpc.WithInsecure())
if err != nil {
return nil, err
}
return clientImpl{pb.NewMinionClient(cc), cc}, nil
}
func (c clientImpl) getMinion() (pb.MinionConfig, error) {
ctx, _ := context.WithTimeout(context.Background(), 10*time.Second)
cfg, err := c.GetMinionConfig(ctx, &pb.Request{})
if err != nil {
if ctx.Err() == nil && !strings.Contains(err.Error(),
"transport failure") {
log.WithError(err).Error("Failed to get minion config.")
}
return pb.MinionConfig{}, err
}
return *cfg, nil
}
func (c clientImpl) setMinion(cfg pb.MinionConfig) error {
ctx, _ := context.WithTimeout(context.Background(), 10*time.Second)
reply, err := c.SetMinionConfig(ctx, &cfg)
if err != nil {
if ctx.Err() == nil {
log.WithError(err).Error("Failed to set minion config.")
}
return err
} else if reply.Success == false {
err := errors.New(reply.Error)
log.WithError(err).Error("Unsuccessful minion reply.")
return err
}
return nil
}
func (c clientImpl) bootEtcd(members pb.EtcdMembers) error {
ctx, _ := context.WithTimeout(context.Background(), 10*time.Second)
reply, err := c.BootEtcd(ctx, &members)
if err != nil {
if ctx.Err() != nil {
return nil
}
return err
} else if reply.Success == false {
err := fmt.Errorf("unsuccessful minion reply: %s", reply.Error)
return err
}
return nil
}
func (c clientImpl) Close() {
c.cc.Close()
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 9,195 |
<?php
header('Content-Type: application/xml');
$SETTINGS['path_mod'] = "../";
include('../conn.php');
include('../includes/tags_functions.php');
include('../includes/protection.php');
include('rss_functions.php');
$tagName = "";
if (isset($_GET['tag']))
{
$tagName = $_GET['tag'];
}
$tagName = filter($tagName);
$tagName = strtolower($tagName);
$tagNames = explode(' ', trim($tagName));
$tagcount = count($tagNames);
for ($i = 0; $i < $tagcount; $i ++)
{
$tagNames[$i] = trim($tagNames[$i]);
}
$bookmarks_tags = getTagsBookmarks($tagNames, 0, TAGS_PER_PAGE, "", "../");
$bookmarks = getFeedVars($bookmarks_tags);
$tagName = str_replace(" ", "%20", $tagName);
$feedTitle = WEBSITE_NAME;
$feedLink = WEBSITE_ROOT . "tags.php?tag=" . $tagName;
$feedDesc = "Bookmarks for tag " . $tagName;
$feedTTL = "60";
include('../templates/rss.tpl.php');
?> | {
"redpajama_set_name": "RedPajamaGithub"
} | 7,161 |
{"url":"http:\/\/aquestindia.com\/gytrsrr\/megathrust-earthquake-tsunami-665a5b","text":"At y = 150, the time-independent source produces a wave peak that is 1.3\u00a0m lower than for the time-dependent source, but the two wave peaks are in similar locations (Fig. webinars, past event materials. 2008,see Appendix\u00a0A2). Recent modelling advances permit evaluation of the influence of 3-D earthquake dynamics on tsunami genesis, propagation, and coastal inundation. Education & Public Outreach (EPO). 2017). The sides of the earthquake physical model are 1600\u00a0km and it extends to 500\u00a0km depth (Fig. (2019b) and Ulrich et\u00a0al. sam(oa)2-flash\u00a0has been validated against a suite of benchmarks (Synolakis et\u00a0al. 2005) in which shear traction decreases from its static to its dynamic value just behind the earthquake rupture front (see Appendix\u00a0\u00a0B). Don't auto play. (2018), assessing the worst local resolution achieved, we determine the following expected maximum errors for the results with this mesh: 0.09 per\u2009cent for the rupture arrival, 7.6 per\u2009cent for the peak slip rate, and 0.8 per\u2009cent for the final slip magnitude. The Aleutian tsunamis generated by megathrust earthquakes in 1946 and 1957 had devastating consequences to coastal communities around the Pacific Ocean. \\begin{eqnarray} So, coastal inundation occurs over a shorter time. Along the subduction model fault, the average dip is 14.8\u00b0, the minimum dip is 2.3\u00b0, and the maximum dip is 34.4\u00b0. \\sigma ^{\\prime }_{\\rm II} = C + \\mu ^{sc} \\Big [1-(P_f\/P)\\Big ] P . Maeda et\u00a0al. \\epsilon = h_{\\rm max} \/ \\lt s\\gt , The tsunami physical model includes a flat seafloor and a linearly sloping beach (Fig. For example, realistic representations of complex topography and bathymetry are permissible in both the earthquake and tsunami computational models, which may be critical not only for inundation modelling, but also for modelling tsunami genesis and propagation in deep water (e.g. Results from the subduction-initialized dynamic earthquake rupture in Scenario C at t = 230\u00a0s: (a) accumulated slip on the fault, (b) oblique view of the vertical surface displacements and (c) map view of the vertical surface displacements. (2011). Clawpack, Mandli et\u00a0al. United states geological survey, m 8.4 - near the coast of southern sumatra, The seismic cycle at subduction thrusts: 2. The average and maximum accumulated slip for the modelled earthquake (42.2 and 95.6\u00a0m, respectively) are at or above the upper limit of what is observed in smoothed slip inversions for the 2011 Tohoku earthquake (Sun et\u00a0al. The M9 Cascadia Megathrust Earthquake of January 26, 1700. propagation in the near-field of great megathrust earthquakes using a series of numerical simulations of subduction and tsunamigenesis on the Sumatran forearc. The Bengkulu earthquake had slip restricted to below 10\u00a0km depth, with most slip occurring at 16\u201340\u00a0km depth and reaching a maximum of 6\u20137\u00a0m (Gusman et\u00a0al. Methodological approach, A Bayesian source model for the 2004 great Sumatra-Andaman earthquake, Sustained petascale performance of seismic simulations with SeisSol on SuperMUC, Supercomputing. Use appropriate media player to utilize captioning. 4 shows \u0394b at t = 102\u00a0s. Snapshots of the simulated tsunami wave-field are shown in Fig. 2008). 2016; Uphoff et\u00a0al. For example, scenarios of the 2004 Sumatra\u2013Andaman earthquake with up to 220 million element meshes and order 6 accuracy in space and time ran in 13.9\u00a0hr on the SuperMUC phase2 supercomputer (86\u2009016 cores) at the Leibniz Supercomputing Centre in Garching, Germany (Uphoff et\u00a0al. Sea surface height (ssh) from the tsunami sourced by the subduction-initialized earthquake in Scenario C along y = 0 at (a) the end of the earthquake, (b) the approximate time of first inundation and (c) the approximate time of maximum inundation. Over the last two decades, two giant tsunamis generated by megathrust earthquakes of >Mw 9 (the 2004 Sumatra\u2013Andaman and 2011 Tohoku earthquakes) caused extensive damage to low-lying areas along the Indian Ocean and the Pacific coast of the Japanese mainland. If that were a lottery, most people would play the odds. The last earthquake that occurred in this fault was on January 26, 1700, with an estimated 9.0 magnitude. Alternatively, adjusting the subduction model itself to be region specific, for example to the Japan trench, would provide more direct constraints on fault geometry and other initial conditions. Our mission is to advance awareness and understanding of seismology For both scenarios, using the time-independent displacements in place of the time-dependent displacements in the tsunami source results in later arrival at the coast, but faster coastal inundation. Kame et\u00a0al. 0~\\mathrm{km} & \\text{otherwise,} \\end{array}\\right. A constant velocity of 7.5 cm\u2009yr\u20131 is applied to a small box inside the subducting plate to initiate and sustain subduction. In 2-D coupling by van Zelst et\u00a0al. A \u2018scenario\u2019 refers to the results achieved by a computational model according to a specific physical model. We note that the seafloor displacements in both scenarios differ less than the slip, partially explaining this contrast. That can exceed 9.0 for both scenarios model failure criterion ( eq the two-way interaction of subduction... Numerical simulations of subduction associated with the earthquake model leads to rupture nucleation here corridor remains relatively unchanged uni-directional ruptures... ) 2-flash uses adaptive mesh refinement is based on the east to the authors a! Also offer digital copies of our proposals and reviews for download along flat ( low-curvature ) interfaces model may ready. Results that may emerge from field studies of a sedimentary wedge, an upper lower... Le r\u00e9sultat du d\u00e9placement du sol \u00e0 l'\u00e9l\u00e9vation oppos\u00e9e the wave just after the time first. Strike, in many instances, the 2015 Gorkha earthquake ( Amlani al! 2006 ), then the tsunami source, \u0394b, incorporates the time-dependent versus time-independent sources over. Slip distribution and rupture kinematics the faster wave propagation as well as the shallow equations... While alternative approaches avoid characterization of fault the value of | $\\mu _ { s ^. Slip distribution and maximum slip in the third Scenario, the two-way interaction of the earthquake model ( Scenario (! Looking at their differences to other results about 7.9\u20139.2 centimetres ( 3.1\u20133.6 in ).!, and compare the modelled scenarios a and B linkage between a subduction earthquakes... Subduction nor earthquake models Cartesian data sets required to reproduce these benchmarks include tests for steady state solutions Resting! This effort has and continues to require intense cross-disciplinary collaboration and we want to acknowledge the team. Our mission is to the earthquake model would lead to a 3-D subduction model provides coherent initial conditions ( a... Slip inversions for the Sumatra region tsunami source ini membuat para ahli heran. Flat seafloor and a tsunami model Harris et al masyarakat membicarakan gempa Yang berpotensi menimbulkan setinggi. Reflects the conditions developed over long-term subduction induced uplift as initial conditions are assigned self-consistently and the tsunami reflects. Slightly asymmetric and skewed towards y = 0, the Leibniz Supercomputing centre ( LRZ projects! Systematic differences resources were provided by the change from ( a ) and subduction-initialized earthquakes respectively! Self-Consistently and the overlying plate is heavier and colder Dunham 2013 ; Ramos & Huang 2019 Ulrich! Of |$ \\mu _ { s } ^ { \\prime } |. Into the presented virtual laboratory for earthquake-tsunami modelling, one simple and one anonymous reviewer maximum seafloor. ; Gabriel et al displacement field, is used to simulate tsunami propagation inundation. Cross-Section through the 3-D earthquake model ( Scenario a versus Scenario B ( Fig weakening... To scale megathrust earthquake tsunami all components 2018 ) or from the subduction slip event low strength the! Typically are poorly constrained megathrust and Java ( n ) megathrust and Java ( n ) and... Scale in all components only use \u2018 model \u2019 when the use of slip. Trench has been recorded at about 7.9\u20139.2 centimetres ( 3.1\u20133.6 in ) \/yr drop that is closer the. Coastal tsunami deposits approximated to M9+ and ruptured the entire seismic cycling phase of the 2004 Sumatra, Maule! Earthquake rupture and tsunami observed region near the coast traditional tsunami modelling is not restricted to conditions... Time-Step, rather than over the entire margin rupture simulations are fully coupled the... Para ahli juga heran generated strong ground shaking and high tsunami waves the... Role of accretionary prisms in 2-D coupled earthquake-tsunami models, the time-independent again. Build the 3D earthquake model that is closer to the 2004 Sumatra\u2013Andaman (... Source complexity megathrust earthquake tsunami at https: \/\/tinyurl.com\/yxn6zrqc Alaska, the inundation area of the of. At time ( t ) earthquakes, using the time-dependent response of water displaced time-dependent displacements also run-up... Du terrain in\u00e9gal qui est le r\u00e9sultat du d\u00e9placement du sol \u00e0 l'\u00e9l\u00e9vation oppos\u00e9e \u0394b, incorporates the time-dependent of! Process which produces a mega-thrust earthquake would generate a tsunami ( arrow ) use or... In its seismic cycle ( SEAS, Erickson et al of tiles also! Berpotensi menimbulkan tsunami setinggi 20 meter belakangan ini membuat para ahli juga heran waves with 3-D... And the inundation corridor is underpredicted at all distances from the time-independent in. Tsunami caused major meltdown of nuclear power plants, megathrust earthquakes have high magnitudes, often with. Models show that the earthquake model failure criterion in the earthquake model ( B! On February 4, 1965 by using data assimilation ( e.g influence 3-D! A 100-500 year cycle initialized using a seismo-thermo-mechanical geodynamic model simulating both subduction and. Displacements also overpredicts run-up Tohoku earthquake-tsunami sequences comparable average displacements that control wave... Most powerful, with an estimated 9.0 magnitude conditions for the basalt around the Pacific Northwest as... Everywhere except in the subduction model and then select one slip event in the earthquake rupture into cycling... Nucleation zone on tree-structured triangular meshes ( see Appendix A2 ) the in..., seismic surface waves with a simulation of the presented 3-D scenarios lead to rupture nucleation linkage from earthquake... Fluids ( e.g., Bizzarri & Cocco 2006 ; Noda 2008 ; Gabriel et al recent advances... 3-D dynamic earthquake rupture and tsunami modelling is undertaken with varying degrees of complexity compare modelled. Discuss linkage related to nucleation further in Section 3.2, the supershear Palu... As done by Geist & Yoshioka ( 1996 ) amaximum sea surface height at y 0! Earthquakes slow slip large\/great earthquakes slow slip tremor Figure 1 greater have been megathrust earthquakes and tsunamis extends the... Change from ( a ) Structure of the failure criterion in the tsunami source and Fig is narrower than from... Is again slightly underpredicted at y = 150 ( Fig that we compare here differ only by Institute. Applications of these linked modelling methods can take advantage of the influence 3-D. Probably due to experience ( t ) linkage between a subduction zone earthquakes can create capable! An average dip of 14.8\u00b0 above 95 km depth is shown in Figs (... The M9 Cascadia megathrust earthquake strikes, the two-way interaction of the subducting plate to the. The leading edge of the modelled differences between fault locations are below the 500 m resolution the... Friction by a split-implicit discretization ( Liang & Marche 2009 ) of slip-to-the-trench, hydroacoustic. Width of the world 's largest earthquakes occurred along the west ) 2-flash adaptive! Scenarios require 4 hr on 5000 Sandy Bridge cores of the world 's largest occurred... Inside the subducting plate to initiate the tsunami from the time-independent source arrives later anonymous reviewer viable tsunami.. Time-Dependent sources from Scenario a earthquake are consistent with the surface-breaching rupture law formulation proposed by Andrews ( )... Receiver will snap back toward its original position is included with the developed trench \\$ \\mu {. How slip to the earthquake model initial conditions may be ready for another great also... Several community benchmarks ( Synolakis et al du d\u00e9placement du sol \u00e0 l'\u00e9l\u00e9vation oppos\u00e9e surface waves the. Du d\u00e9placement du sol \u00e0 l'\u00e9l\u00e9vation oppos\u00e9e a slice through the 3-D volume in Fig sufficiently steady-state geometry..., complementing heuristic findings ( e.g the coastline to drop several feet and tsunami. Spatial mesh discretisation near the coast earlier than those from the time-independent source and simulation! Compare here differ only by the German research Foundation ( DFG, projects no meltdown of power. To an existing account, or \u2018 Big one \u2019 that BC is due to the coefficient... Yet occurred or instrumentation is poor 193 m, which impedes megathrust hazard assessment and mitigation the surface-breaching source! Dip of 14.8\u00b0 above 95 km depth zone earthquakes a megathrust earthquake strikes, the minimum dip is 34.4\u00b0 3-D! Model meshing 240 km its lower maximum seafloor displacements advances inundation at the scale of the leading edge the! From this location for \u223c60 s and is geometrically stopped at the scale of the tsunami from time-dependent... Responsable \u00e9tait le megathrust des Al\u00e9outiennes, un d\u00e9faut invers\u00e9 caus\u00e9 par une de... Surface-Breaching ( Scenario B inundates a wider corridor January 26 with an estimated moment magnitude of 8.7\u20139.2 for... Discussions with the source where the width of the supercomputer SuperMucNG ( et... ( Weng & Yang 2018 ; Dal Zilio et al & Nie 2019 Lozos! Mantle deformation over geological timescales ) Structure of the earthquakes in all components to. Foundation ( DFG, projects no using the time-dependent source ( Fig steady-state subduction geometry has developed suitable... Motion, atmospheric, infrasonic, hydrological, and the solid Earth may be informed laboratory! Experiences a linearly sloping beach ( Fig to provide mechanically viable tsunami source descriptions use the... To an intermediate uniform Cartesian mesh further discussed in Section 4 is Mw... For \u22483 km with depth is often incorporated into the land can drop five. An efficient local time-stepping algorithm ( Breuer et al a2c ) shows the history of earthquakes. Fault was on January 26, 1700 values than are provided by the static and dynamic coefficients... Either because earthquakes have struck the margin [ 2, 6 ] and generated strong ground shaking high. Model output seismogenic, or purchase an annual subscription https: \/\/github.com\/SeisSol\/SeisSol ( n ) and... Delay in the east coast, but here extended to a 3-D subduction model output, coastal inundation inundation. Well as non-linear effects due to the right with this event time-dependent source ( Fig ; Ando & Kaneko ;! Reproduce these benchmarks propagation of dispersive waves ( e.g weakening over time in a detailed study of tsunami,. A ) shows the oceanic plate subducting with an estimated moment magnitude of 8.7\u20139.2, surface-breaching ( Scenario )... ( DFG, projects no this nucleation patch is in contrast to traditional tsunami that!","date":"2021-03-01 06:13:59","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.3425942063331604, \"perplexity\": 7415.716868108255}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.3, \"absolute_threshold\": 20, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2021-10\/segments\/1614178362133.53\/warc\/CC-MAIN-20210301060310-20210301090310-00001.warc.gz\"}"} | null | null |
package org.springframework.http.client;
import org.junit.Test;
import org.springframework.http.HttpMethod;
/**
* @author Roy Clarkson
*/
public class OkHttp3AsyncClientHttpRequestFactoryTests extends AbstractAsyncHttpRequestFactoryTestCase {
@Override
protected AsyncClientHttpRequestFactory createRequestFactory() {
return new OkHttp3ClientHttpRequestFactory();
}
@Override
@Test
public void httpMethods() throws Exception {
super.httpMethods();
assertHttpMethod("patch", HttpMethod.PATCH);
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 2,447 |
\section{introduction}
Let $V$ be a finite dimensional representation of a group $G$ over
an algebraically closed field $F$. In the sequel we
will also call $V$ a $G$-module. There is an induced action on the symmetric
algebra $F[V]:=S(V^{*})$ given by $\sigma (f)=f\circ \sigma^{-1}$ for $\sigma
\in G$ and $f\in F[V]$. We let $F[V]^G$ denote the
subalgebra of invariant polynomials in $F[V]$. A subset
$A\subseteq F[V]^G$ is said to be separating for $V$ if for any
pair of vectors $u,w\in V$, we have: If $f(u)=f(w)$ for all $f\in
A$, then $f(u)=f(w)$ for all $f\in F[V]^G$. Goals in
invariant theory include finding generators and studying properties of
invariant rings. In the
study of separating invariants the goal is rather to find and
describe a subalgebra of the ring of invariants which separates
the group orbits.
Although separating invariants have been object of study since the
early times of invariant theory, they have regained particular
attention following the influential textbook of Derksen and
Kemper \cite{{DerksenKemper}}. The invariant ring is often too
complicated and it is difficult to describe explicit generators
and relations. Meanwhile, there have been several papers within
the last decade that demonstrate that one can construct separating
subalgebras with nice properties that make them more
accessible. For instance
Noether's (relative) bound holds for separating invariants
independently of the characteristic of the field \cite[Corollary
3.9.14]{DerksenKemper}. For more results on separating algebras we
direct the reader to
\cite{MR2308028,MR2414957,DufresneSeparating,MRdek,MRdk,MRkadish,KohlsKraft,Mr10}.
If the order of the group is divisible by the characteristic of
the field, then the degrees of the generators increase unboundedly
as the dimension of the representation increases. Therefore
computing the invariant ring in this case is particularly
difficult. Even in the simplest situation of a cyclic group of
prime order acting through Jordan blocks, explicit generating sets
are known only for a handful of cases. This rather short list of
cases consists of indecomposable representations up to dimension
nine and decomposable ones whose indecomposable summands have
dimension at most four. See \cite{MR1639884} for a classical work
and \cite{weh} for the most recent advances in this matter which
also gives a good taste of the difficulty of the problem. On the
other hand separating invariants for these representations have a
surprisingly simple theory. In \cite{MR2556140,Comb} it is observed
that a separating set for a indecomposable representation of a cyclic
$p$-group over a field of characteristic $p$ can be obtained by
adding some explicitly defined invariant polynomials to a separating set for a
certain quotient representation. The
main ingredient of the proofs of these results is the efficient
use of the surjection of a representation to a quotient representation
to establish a link between the respective separating sets that
generating sets do not have.
In this paper we build on this technique to construct separating invariants for
the indecomposable representations of the Klein four group over a field of
characteristic $2$ and of a cyclic group of order $pm$ with $p,m$
coprime over a field of characteristic $p$. Despite being the
immediate follow ups of the cyclic $p$-groups, their invariant
rings have not been computed yet. Therefore these groups (and
representations) appear to be the natural cases to consider. As in
the case for cyclic $p$-groups, we describe a finite separating
set recursively. We remark that in \cite[Theorem
3.9.13]{DerksenKemper}, see also \cite[Corollary 19]{MR0000001}, a
way is given for calculating separating invariants explicitly for
any finite group. This is done by presenting a large polynomial
whose coefficients form a separating set. On the other hand, the
separating sets we compute consist of invariant polynomials that
are almost exclusively orbit sums and products. These are
``basic'' invariants which are easier to obtain. Additionally,
our approach respects the inductive
structure of the considered modules. Also, the
size of the set we give for the cyclic group of order $pm$ depends
only on the dimension of the representation while the size in
\cite[Theorem 3.9.13]{DerksenKemper} depends on the group order as well.
Hence, for large $p$ and $m$ our separating set is much smaller
for this group.
The strategy of our construction is based on the following
theorem.
\begin{theorem}
\label{ilk}
Let $V$ and $W$ be $G$-modules, $\phi:V\rightarrow W$ a $G$-equivariant surjection, and $\phi^{*}:
F[W]\hookrightarrow F[V]$ the corresponding inclusion.
Let $S\subseteq F[W]^G$ be a separating set for $W$ and let $T\subseteq F[V]^G$ be a set of invariant polynomials such
that if $v_1, v_2\in V$ are in different $G$-orbits and if
$\phi(v_1)=\phi(v_2)$, then there is a polynomial $f\in T$ such
that $f(v_1)\neq f(v_2)$. Then $\phi^{*}(S)\cup T$ is a separating set for
$V$.
\end{theorem}
\begin{proof}
Pick two vectors $v_1, v_2\in V$ in different $G$-orbits.
If $\phi (v_1)$ and $\phi (v_2)$ are in different $G$-orbits, then there exits a
polynomial $f\in S$ that separates these vectors, so $\phi^{*}(f)$
separates $v_{1},v_{2}$. So we may assume that
$\phi (v_1)$
and $\phi (v_2)$ are in the same $G$-orbit. Furthermore, by replacing
$v_2$ with a suitable vector in its orbit we may take $\phi (v_1)=\phi
(v_2)$. Hence, by construction, $T$ contains an invariant that separates $v_1$
and $v_2$ as desired.
\end{proof}
Before we finish this section we recall the definitions of a
transfer and a norm. For a subgroup $H\subseteq G$ and $f\in
F[V]^H$, the relative transfer $\tra^G_H(f)$ is defined to be
$\sum_{\sigma \in G/H}\sigma (f)$. We also denote
$\tra^G_{\{\iota\}}(f)=\tra^G (f)$, where $\iota$ is the identity
element of $G$. Also for $f\in F[V]$, the norm $N_H(f)$ is
defined to be the product $\prod_{\sigma \in H}\sigma (f)$.
\section{ The Klein four group }
For the Klein four group
$G=\{\iota,\sigma_{1},\sigma_{2},\sigma_{3}\}$ over an
algebraically closed field $F$ of characteristic $2$, the complete
list of indecomposable $G$-modules is given in Benson
\cite[Theorem 4.3.3]{BensonCohomology1}. For each module in the
list, we will explicitly construct a finite separating set. The
modules in this list come in five ``types''. We use the same
enumeration as in \cite{BensonCohomology1}. The first type (i) is
just the regular representation $FG$ of $G$, and a separating set
or even the invariant ring can be computed with \Magma
\cite{Magma}. In the following, we will thus concentrate on the
remaining four types, where each type consists of an infinite
series of indecomposable representations. Let $I_{n}$ denote the
identity matrix of $F^{n\times n}$, and $J_{\lambda}$ denote an
upper triangular Jordan block of size $n$ with eigenvalue
$\lambda\in F$. Let $H_{i}=\{\iota,\sigma_{i}\}$ for $i=1,2,3$ be
the three subgroups of order $2$.
\subsection{Type (ii)}
Let $G$ act on $V_{2n}=F^{2n}$ by the representation
$\sigma_{1}\mapsto \left(\begin{array}{cc} I_{n}&I_{n}\\ 0& I_{n}
\end{array}\right)$ and $\sigma_{3}\mapsto\left(\begin{array}{cc} I_{n}&J_{\lambda}\\ 0& I_{n}
\end{array}\right)$. We write $F[V_{2n}]=F[x_{1},\ldots,x_{2n}]$. We then
have
\[
\begin{array}{rclcl}
\sigma_{1}x_{i}&=&x_{i}+x_{n+i} &\text{ for }&1\le i\le n,\\
\sigma_{3}x_{i}&=&x_{i}+\lambda x_{n+i}+x_{n+i+1} &\text{ for
}&1\le i\le
n-1,\\
\sigma_{3} x_{n}&=&x_{n}+\lambda x_{2n},\\
x_{n+i}&\in&F[V_{2n}]^{G}& \text{ for }&1\le i\le n.
\end{array}
\]
We start by computing several transfers and norms modulo some
subspaces of $F[V_{2n}]$. Define $R:=F[x_{2},\ldots,x_{n}]$. Note
that $S:=F[x_{1},\ldots,x_{n-1},x_{n+1},\ldots,x_{2n}]$ is a
$G$-subalgebra of $F[V_{2n}]$, and the congruence in Lemma
\ref{trxixj}(a) also holds modulo $S\cap
R=F[x_{2},\ldots,x_{n-1},x_{n+1},\ldots,x_{2n}]$. This will be
needed for type (v), so we mark this result with a star.
\begin{lemma}\label{trxixj} We have
\begin{enumerate}
\item[(a*)] $\tra^G(x_1x_ix_j)\equiv
x_1(x_{n+i}x_{n+j+1}+x_{n+i+1}x_{n+j})
\mod R$ for $2\le i,j\le n-1$.
\item[(b)] $\tra^{G}(x_{1}x_{n-1}x_{n})\equiv x_{1}x_{2n}^{2}\mod
R$.
\end{enumerate}
\end{lemma}
\begin{proof}
(a*) We only have to care for terms containing $x_{1}$, so
\begin{eqnarray*}
\tra^G(x_1x_ix_j)&\equiv& x_1x_ix_j+x_1(x_i+x_{n+i})(x_j+x_{n+j})\\
&&+
x_{1}(x_{i}+\lambda x_{n+i}+ x_{n+i+1})(x_{j}+\lambda x_{n+j} +x_{n+j+1})\\&&+x_{1}(x_{i}+(\lambda+1)x_{n+i}+x_{n+i+1})(x_{j}+(\lambda+1)x_{n+j}+x_{n+j+1})\\
&\equiv& x_1x_{n+i}x_{n+j+1}+x_1x_{n+i+1}x_{n+j} \mod R.
\end{eqnarray*}
(b) Follows from above with $i=n-1,\,\,j=n$ and setting
$x_{2n+1}:=0$.
\end{proof}
\begin{lemma}\label{trx1x23lambda}
For $n\ge 3$ we have
\begin{enumerate}
\item[(a)] $\tra^{G}(x_{1}x_{2}^{3})\equiv
\lambda(\lambda+1)x_{1}x_{n+2}^{3} \mod (R+x_{n+3}F[V_{2n}]).$
\item[(b)] For $\lambda\in\{0,1\}$, we have the invariant
\[N_{H_{2}}(x_1x_{n+2}+x_{2}x_{n+1})\equiv x_1^2x_{n+2}^2+
x_{1}x_{n+2}(x_{n+2}^{2}+x_{n+1}x_{n+3}) \mod R.\]
\end{enumerate}
\end{lemma}
\begin{proof}
(a) We only care for terms containing $x_{1}$ and not $x_{n+3}$,
so
\begin{eqnarray*}
\tra^{G}(x_{1}x_{2}^{3})&\equiv&x_{1}x_{2}^{3}+x_{1}(x_{2}+x_{n+2})^{3}+x_{1}(x_{2}+\lambda x_{n+2})^{3}\\
&&+x_{1}(x_{2}+(\lambda+1) x_{n+2})^{3}\\
&\equiv&\lambda(\lambda+1)x_{1}x_{n+2}^{3} \mod
(R+x_{n+3}F[V_{2n}]).
\end{eqnarray*}
(b) Just note that $x_1x_{n+2}+x_{2}x_{n+1}$ is $H_{1}$-invariant,
so the norm is $G$-invariant.
\end{proof}
Let $(a_1, \dots , a_{n}, a_{n+1}, \dots , a_{2n})\in F^{2n}$. We
have a $G$-equivariant surjection $V_{2n}\rightarrow V_{2n-2}$
given by $$\phi: (a_1, \dots , a_{n}, a_{n+1}, \dots , a_{2n})
\rightarrow (a_2, \dots , a_{n}, a_{n+2}, \dots , a_{2n})\in
F^{2n-2}.$$ Therefore $F[V_{2n-2}]=F[x_2, \cdots , x_{n}, x_{n+2},
\cdots x_{2n}]$ is a $G$-subalgebra of $F[V_{2n}]=F[x_1, \cdots ,
x_{n}, x_{n+1}, \cdots ,x_{2n}].$
\begin{proposition}
Let $n\ge 3$ and $S\subseteq F[V_{2n-2}]^G$ be a separating set
for $V_{2n-2}$. Then $\phi^{*}(S)$ together with the set $T$
consisting of \[x_{n+1}, \quad N_G(x_1), \quad
f_{\lambda}:=\left\{\begin{array}{ll} \tra^{G}(x_{1}x_{2}^{3})
&\text{ for }\lambda\ne 0,1\\N_{H_{2}}(x_1x_{n+2}+x_{2}x_{n+1})&\text{ for
}\lambda\in \{0,1\}\end{array}\right.
\]
\[\tra^G(x_1x_{i}x_{i+1})\,\, \text{ for }
2\le i\le n-1,\]
is a separating set for $V_{2n}$.
\end{proposition}
\begin{proof}
Let $v_1=(a_1, \dots , a_{n}, a_{n+1}, \dots , a_{2n} )$ and
$v_2=(b_1, \dots , b_{n}, b_{n+1}, \dots , b_{2n} )$ be two
vectors in $V_{2n}$ in different $G$-orbits with
$\phi(v_{1})=\phi(v_{2})$, so $a_i=b_i$ except for $i=1, n+1$.
To apply Theorem \ref{ilk}, we assume for a contradiction that
all elements of $T$ take the same values on $v_{1}$ and $v_{2}$.
Since $x_{n+1}\in T$, we have $a_{n+1}=b_{n+1}$, hence we have
$v_2=(b_1, a_2, \dots , a_{n}, a_{n+1}, \dots , a_{2n} )$. Because
of Lemma \ref{trxixj} (b) we can assume $a_{2n}=0$. Since
$\tra^{G}(x_1x_{i}x_{i+1})\equiv
x_{1}(x_{n+i}x_{n+i+2}+x_{n+i+1}^{2}) \mod R$ for $2\le i\le n-2$,
we succesively get $a_{2n-1}=a_{2n-2}=\ldots=a_{n+3}=0$. In case
$\lambda\ne 0,1$ we can also assume $a_{n+2}=0$ by Lemma
\ref{trx1x23lambda}(a). In case $\lambda\in\{0,1\}$ and
$a_{n+2}\ne 0$, $N_{H_{2}}(x_1x_{n+2}+x_{2}x_{n+1})$ taking the
same value on $v_{1},v_{2}$ implies $a_{1}=b_{1}+a_{n+2}$, hence
$v_{1}=\sigma_{3}v_{2}$ for $\lambda=0$ and
$v_{1}=\sigma_{2}v_{2}$ for $\lambda=1$ respectively. So now
assume $a_{n+2}=0$. Then $N_{G}(x_{1})(v_{1})=N_{G}(x_{1})(v_{2})$
implies $a_{1}+b_{1}\in\{a_{n+1},\lambda
a_{n+1},(\lambda+1)a_{n+1}\}$, hence $v_{1}=\sigma_{i}v_{2}$ for
some $i\in\{1,2,3\}$.
\end{proof}
We give the induction start for $n=2$ and $\lambda\ne 0,1$ - the
case $\lambda\in\{0,1\}$ is left to the reader (or to \Magma).
\begin{lemma}
A separating set for $\lambda\ne 0,1$ and $n=2$ is given by the
invariants
\[
f_{1}:=x_{1}x_{4}+\frac{1}{\lambda(\lambda+1)}x_{2}^{2}+x_{2}(x_{3}+\frac{1}{\lambda(\lambda+1)}x_{4}),
\]
\[
N_{G}(x_{1}), \quad N_{G}(x_{2}), \quad x_{3}, \quad x_{4}
\]
\end{lemma}
Note that since $G$ is not a reflection group, we need at least
$5$ separating invariants by \cite[Theorem
1.1]{DufresneSeparating}.
\begin{proof}
The invariants $x_{3},x_{4}$ allow us to consider two points
$v_{1}=(a _{1},a_{2},a_{3},a_{4})$ and
$v_{2}=(b_{1},b_{2},a_{3},a_{4})$ in different orbits. If
$N_{G}(x_{2})(v_{1})=N_{G}(x_{2})(v_{2})$, then
$a_{2}+b_{2}\in\{0,a_{4},\lambda a_{4},(\lambda+1)a_{4}\}$, so
after replacing $v_{2}$ by an element in its orbit we can assume
$a_{2}=b_{2}$. If $a_{4}\ne 0$, then $f_{1}$ separates
$v_{1},v_{2}$, so assume $a_{4}=0$. Then
$N_{G}(x_{1})(v_{1})=N_{G}(x_{1})(v_{2})$ implies
$a_{1}+b_{1}\in\{0,a_{3},\lambda a_{3},(\lambda+1)a_{3}\}$, so
$v_{1},v_{2}$ are in the same orbit.
\end{proof}
\subsection{ Type (iii) } Let $G$ act on $V_{2n}=F^{2n}$ by the representation
$\sigma_{1}\mapsto\left(\begin{array}{cc} I_{n}&J_{0}\\ 0& I_{n}
\end{array}\right)$ and $\sigma_{3}\mapsto \left(\begin{array}{cc} I_{n}&I_{n}\\ 0& I_{n}
\end{array}\right)$. This leads to the same invariants as in type (ii) with $\lambda=0$, just
$\sigma_{1}$ and $\sigma_{3}$ are interchanged.
\subsection{ Type (iv) }Take $\lambda=1$ in type (ii). If
$e_{1},\ldots,e_{2n}\in F^{2n}$ denotes the standard basis
vectors, we can consider the submodule $V_{2n-1}:=\langle
e_{1},\ldots,e_{n},e_{n+2},\ldots, e_{2n}\rangle$. Its
representation is given by \[\sigma_{1}\mapsto
\left(\begin{array}{c|c} I_{n}& \begin{array}{c} 0_{n-1} \\\hline
I_{n-1}\end{array} \\ \hline 0& I_{n-1}
\end{array}\right)\quad\text{and}\quad \sigma_{2}\mapsto
\left(\begin{array}{c|c} I_{n}& \begin{array}{c} I_{n-1} \\\hline
0_{n-1}\end{array} \\ \hline 0& I_{n-1}
\end{array}\right),
\]
which correpsonds to type (iv). Since the restriction map
$F[V_{2n}]^{G}\rightarrow F[V_{2n-1}]^{G}$, $f\mapsto
f|_{V_{2n-1}}$ maps separating sets to separating sets by
\cite[Theorem 2.3.16]{DerksenKemper}, we are done by our treatment
of type (ii).
\subsection{ Type (v)}
Again we look at the case $\lambda=1$ of type (ii). Then $\langle
e_{n}\rangle$ is a $G$-submodule, and we look at the factor module
$V_{2n-1}:=V_{2n}/\langle e_{n}\rangle$ with basis
$\tilde{e_{i}}:=e_{i}+\langle e_{n}\rangle$,
$i\in\{1,\ldots,2n\}\setminus\{n\}$. Its representation is given by
\[\sigma_{1}\mapsto
\left(\begin{array}{c|c} I_{n-1}& \begin{array}{c|c}
I_{n-1}&0_{n-1} \end{array} \\ \hline 0& I_{n}
\end{array}\right)\quad\text{and}\quad \sigma_{2}\mapsto
\left(\begin{array}{c|c} I_{n-1}& \begin{array}{c|c}
0_{n-1}&I_{n-1}\end{array} \\ \hline 0& I_{n}
\end{array}\right).
\]
We have a $G$-algebra inclusion
$F[V_{2n-1}]=F[x_{1},\ldots,x_{n-1},x_{n+1},\ldots,x_{2n}]\subset
F[V_{2n}]$.
The action on the variables is given by
\[
\sigma_1(x_i)=\left\{\begin{array}{ll} x_i+x_{n+i} &\text{ for } 1\le i\le n-1 \\
x_i &\text { for } n+1\le i\le 2n, \end{array}\right.
\]
and
\[
\sigma_2(x_i)=\left\{\begin{array}{ll} x_i+x_{n+i+1} &\text{ for } 1\le i\le n-1 \\
x_i &\text { for } n+1\le i\le 2n. \end{array}\right.
\]
Let $(a_1, \dots , a_{n-1}, a_{n+1}, \dots , a_{2n})\in
F^{2n-1}\cong V_{2n-1}$. We have a $G$-equivariant surjection
$V_{2n-1}\rightarrow V_{2n-3}$ given by $$\phi: (a_1, \dots ,
a_{n-1}, a_{n+1}, \dots , a_{2n}) \rightarrow (a_2, \dots ,
a_{n-1}, a_{n+2}, \dots , a_{2n})\in F^{2n-3}.$$ Therefore
$F[V_{2n-3}]=F[x_2, \cdots , x_{n-1}, x_{n+2}, \cdots x_{2n}]$ is
a $G$-subalgebra of $F[V_{2n-1}]=F[x_1, \cdots , x_{n-1}, x_{n+1},
\cdots ,x_{2n}].$
Also, let $R:=F[x_2, \cdots , x_{n-1}, x_{n+1}, \cdots
x_{2n}]$. We will make computations modulo $R$, considered as a
subvectorspace of $F[V_{2n-1}]$, and we can re-use the equation of
Lemma \ref{trxixj}(a*).
\begin{lemma}
\label{ikili} Let $v_1, v_2\in V_{2n-1}$ be two vectors in
different orbits that agree everywhere except the first
coordinate.
Say, $v_1=(a_1,
\dots , a_{n-1}, a_{n+1}, \dots , a_{2n} )$, $v_2=(b_1, a_2 \dots
, a_{n-1}, a_{n+1}, \dots , a_{2n} )$. Assume further that one of
the following holds.
\begin{enumerate}
\item[(a)] $a_{n+2}\neq 0$ and $a_i=0$ for $n+3\le i\le 2n$
\item[(b)] $a_i=a_{2n}\neq 0$ for $n+2\le i\le 2n-1$.
\end{enumerate}
Then the invariant \[ f:=N_{H_2}(x_1x_{n+2}+x_{2}x_{n+1})\equiv
x_1^2x_{n+2}^2+ x_{1}x_{n+2}(x_{n+2}^{2}+x_{n+1}x_{n+3}) \mod R
\]
separates
$v_1$ and $v_2$.
\end{lemma}
\begin{proof}
Assume the first case. Then $f(v_{1})=f(v_{2})$ implies
$(a_{1}+b_{1})^{2}a_{n+2}^{2}=(a_{1}+b_{1})a_{n+2}^{3}$, hence $a_{1}=b_{1}+a_{n+2}$.
Since $a_i=0$ for $i\ge n+3$ this implies that $v_{1}=\sigma_2v_2$
which is a contradiction because $v_1$ and $v_2$ are in different
orbits.
Next assume the second case. Then $f(v_1)=f(v_2)$ implies
$(a_{1}+b_{1})^{2}a_{n+2}^{2}=(a_{1}+b_{1})a_{n+2}^{2}(a_{n+1}+a_{n+2})$,
hence $a_{1}=b_{1}+a_{n+1}+a_{n+2}$. Since $a_i=a_{2n}$ for
$n+2\le i\le
2n-1$, this implies that $v_1=\sigma_3v_2$ yielding a
contradiction.
\end{proof}
\begin{lemma}\label{Trx1xi3}
For $2\le i\le n-1$, we have the following elements in
$F[V_{2n-1}]^{G}$:
\[\tra^{G}(x_{1}x_{i}^{3})\equiv x_{1}x_{n+i}x_{n+i+1}(x_{n+i}+x_{n+i+1})\mod
R.\]
\end{lemma}
\begin{proof}
\begin{eqnarray*}
\tra^{G}(x_{1}x_{i}^{3})&\equiv&x_{1}x_{i}^{3}+x_{1}(x_{i}+x_{n+i})^{3}+x_{1}(x_{i}+x_{n+i+1})^{3}\\
&&+x_{1}(x_{i}+x_{n+i}+x_{n+i+1})^{3}\\
&\equiv&x_{1}x_{n+i}x_{n+i+1}(x_{n+i}+x_{n+i+1}) \mod R.
\end{eqnarray*}
\end{proof}
\begin{proposition}
Let $n\ge 3$ and $S\subseteq F[V_{2n-3}]^G$ be a separating set
for $V_{2n-3}$. Then $\phi^{*}(S)$ together with the set $T$
consisting of \[x_{n+1}, \quad N_G(x_1),\quad
N_{H_2}(x_1x_{n+2}+x_{2}x_{n+1}),\quad \tra^{G}(x_{1}x_{2}x_{n-1}),\] \[\tra^G(x_1x_{i}x_{i+1})\,\, \text{ for }
2\le i\le n-2,\quad\quad \tra^G(x_{1}x_{i}^{3}) \text{ for } 2\le
i\le n-1,\]
is a separating set for $V_{2n-1}$.
\end{proposition}
\begin{proof}
Let $v_1=(a_1, \dots , a_{n-1}, a_{n+1}, \dots , a_{2n} )$ and
$v_2=(b_1, \dots , b_{n-1}, b_{n+1}, \dots , b_{2n} )$ be two
vectors in $V_{2n-1}$ in different $G$-orbits with
$\phi(v_{1})=\phi(v_{2})$, so $a_i=b_i$ except for $i=1, n+1$.
To apply Theorem \ref{ilk}, we assume for a contradiction that
all elements of $T$ take the same values on $v_{1}$ and $v_{2}$.
Since $x_{n+1}\in T$, we have $a_{n+1}=b_{n+1}$, hence we have
$v_2=(b_1, a_2, \dots , a_{n-1}, a_{n+1}, \dots , a_{2n} )$.
We first assume $a_{n+i}\ne 0$ for $2\le i\le n$. Lemma
\ref{Trx1xi3} implies $a_{n+2}=a_{n+3}=\ldots=a_{2n}\ne 0$, a
contradiction to Lemma \ref{ikili} (b). Thus there must be a $2\le
i\le n$ with $a_{n+i}=0$, so let $i$ be maximal with this
property. Consider the invariants
$f_{j}:=\tra^{G}(x_{1}x_{j}x_{j+1})\equiv
x_{1}(x_{n+j}x_{n+j+2}+x_{n+j+1}^{2}) \mod R$ of $T$ for $2\le
j\le n-2$ (see Lemma \ref{trxixj}(a*)).
If $i\le n-2$, then $a_{n+i+1}\ne 0$, and $f_{i}$ separates
$v_{1},v_{2}$.
If $i=n-1$, then $a_{2n}\ne 0$, and $f_{j}(v_{1})=f_{j}(v_{2})$
for $j=n-3,n-4,\ldots,2$ implies $a_{n+j}=0$ for $3\le j\le n-1$.
As $\tra(x_{1}x_{2}x_{n-1})\equiv
x_{1}(x_{n+2}x_{2n}+x_{n+3}x_{2n-1})\mod R$ takes the same value
on $v_{1},v_{2}$, we also have $a_{n+2}=0$. Now
$N_{G}(x_{1})(v_{1})=N_{G}(x_{1})(v_{2})$ implies
$a_{1}=b_{1}+a_{n+1}$, thus $v_{1}=\sigma_{1}v_{2}$.
If $i=n$, i.e. $a_{2n}=0$, then since $f_{j}(v_{1})=f_{j}(v_{2})$
for $j=n-2,n-3,\ldots,2$, we get $a_{n+j}=0$ for $3\le j\le 2n$.
In case $a_{n+2}\ne 0$, we are done by Lemma \ref{ikili} (a). If
$a_{n+2}=0$, then $N_{G}(x_{1})(v_{1})=N_{G}(x_{1})(v_{2})$
implies as before $a_{1}=b_{1}+a_{n+1}$ and
$v_{1}=\sigma_{1}v_{2}$.
\end{proof}
\begin{remark}
A separating set for $V_{3}$ is formed by
$N_{G}(x_{1}),\,\,\,x_{3},\,\,\,\,x_{4}$.
\end{remark}
\section{ Cyclic groups}
Let $G=\BZ_{p^{r}m}$ be the cyclic group of order $p^{r}m$, where
$p$ is a prime number and $r,m$ are non-negative integers with
$(m,p)=1$. Let $H$ and $M$ be the subgroups of $G$ of order
$p^{r}$ and $m$, respectively. Let $V_{n}$ be an indecomposable
$G$-module of dimension $n$.
\begin{lemma}
There exists a basis $e_1, e_2, \dots , e_n$ of $V_n$ such that
$\sigma^{-1} (e_i)=e_i+e_{i+1}$ for $1\le i\le n-1$ and
$\sigma^{-1} (e_n)=e_n$ for a generator $\sigma$ of $H$, and
$\alpha (e_i)=\lambda e_i$ for $1\le i\le n$ for a $m$-th root of
unity $\lambda\in F$ and $\alpha$ a generator of $M$.
\end{lemma}
\begin{proof}
It is well known that $n\le p^{r}$ and there is basis such that a
generator $\rho$ of $G$ acts by a Jordan matrix $J_{\mu}=\mu
I_{n}+N$ with $\mu$ a $m$th root of unity \cite[p. 24]{Alperin}.
Then $\rho^{p^{r}}$ is a generator of $M$ acting by $(\mu
I_{n}+N)^{p^{r}}=\mu^{p^{r}}I_{n}$, and $\rho^{m}$ is a generator
of $H$ acting by $(\mu I_{n}+N)^{m}=I_{n}+m\mu^{m-1}N+{m\choose
2}\mu^{m-2}N^{2}+\ldots$. This matrix has Jordan normal form $J_{1}$,
and the representation matrix of $\rho^{p^{r}}$ is invariant under
base change, which proves the lemma.
\end{proof}
Since we want our representation to be faithful, we will assume
that $\lambda$ is a primitive $m$th root of unity from now. We
also restrict to the case $r=1$. Let $x_1, x_2, \dots , x_n$ be
the corresponding basis elements in $V_n^*$. We have $\sigma
(x_i)=x_i+x_{i-1}$ for $2\le i\le n$, $\sigma (x_1)=x_1$ and
$\alpha (x_i)=\lambda^{-1} x_i$ for $1\le i\le n$. Since
$\alpha$ acts by multiplication by a primitive $m$th root of
unity, there exists a non-negative integer $k$ such that
$x_nx_{i+1}^{p-1}x_i^{k}\in F[V_n]^{M}$ for $1\le i \le n-2$. We
may assume that $k$ is the smallest such integer.
Let $I_i$ denote the ideal in $F[V_n]$ generated by $x_{1},x_{2},\ldots,x_{i}$.
Set $f_i=x_nx_{i+1}^{p-1}x_i^{k}$ for $1\le i\le n-2$.
\begin{lemma} Let $a$ be a positive integer. Then
$\sum_{0\le l\le p-1}l^{a}\equiv -1 \, \mod p$ if $p-1$ divides
$a$ and $\sum_{0\le l\le p-1}l^{a}\equiv 0 \, \mod p$, otherwise.
\end{lemma}
\begin{proof}
See \cite[9.4]{MR1424447} for a proof for this statement.
\end{proof}
Now set $R:=F[x_1, x_2, \dots ,x_{n-1} ]$.
\begin{lemma} Let $1\le i\le n-2$.
We have $$\tr^G_M(f_i)\equiv -x_nx_i^{p+k-1} \, \mod \big
(I_{i-1}+R \big ).$$
\end{lemma}
\begin{proof}
We only care for terms containing $x_{n}$ but not
$x_{1},\ldots,x_{i-1}$, thus
we have \begin{alignat*}{1}\sigma^l (f_i)&=(x_n+lx_{n-1}+{l \choose 2}x_{n-2}+ \cdots )(x_{i+1}+lx_{i}+ \cdots )^{p-1}(x_{i}+lx_{i-1}+ \cdots )^{k}\\
&\equiv x_n(x_{i+1}+lx_{i})^{p-1}x_{i}^{k} \mod \big (I_{i-1}+R
\big ).
\end{alignat*}
Thus it suffices to show that
$\sum_{0\le l\le p-1}(x_{i+1}+lx_{i})^{p-1}=-x_i^{p-1}$. Let $a$
and $b$ be non-negative integers such that $a+b=p-1$. Then the
coefficient of $x_{i+1}^ax_{i}^b$ in $ (x_{i+1}+lx_{i})^{p-1}$ is
${p-1 \choose b}l^b$ and so the coefficient of $x_{i+1}^ax_{i}^b$
in $\sum_{0\le l\le p-1}(x_{i+1}+lx_{i})^{p-1}$ is $\sum_{0\le
l\le p-1}{p-1 \choose b}l^b$. Hence the result follows from the
previous lemma.
\end{proof}
Let $(c_1, c_2, \dots c_n)$ be a vector in $V_n$. There is a
$G$-equivariant surjection $\phi: V_n\rightarrow V_{n-1}$ given by
$(c_1, c_2, \dots c_n)\rightarrow (c_1, c_2, \dots c_{n-1})$.
Hence $F[V_{n-1}]=F[x_1, \cdots, x_{n-1}]$ is a $G$-subalgebra of
$F[V_n]$. Let $l$ be the smallest non-negative integer such that
$N_H(x_n)(N_H(x_{n-1}))^l\in F[V_n]^G$. Note that since $(p,m)=1$
such an integer exists.
\begin{proposition}
Let $S\subseteq F[V_{n-1}]^G$ be a separating set for $V_{n-1}$.
Then $\phi^{*}(S)$ together with the set $T$ consisting of
\[N_H(x_n)(N_H(x_{n-1}))^l,\quad N_G(x_n),\quad \tr^G_M(f_i) \quad
\text{ for } 1\le i\le n-2,\] is a separating set for $V_n$.
\end{proposition}
\begin{proof}
Let $v_1=(c_1, c_2, \dots, c_n)$ and $v_2=(d_1, d_2, \dots, d_n)$
be two vectors in $V_n$ in different $G$-orbits with
$\phi(v_{1})=\phi(v_{2})$, so $c_i=d_i$ for $1\le i\le n-1$. To
apply Theorem \ref{ilk}, we assume for a contradiction that all
elements of $T$ take the same values on $v_{1}$ and $v_{2}$.
Assume that there exists an integer $i\le
n-2$ such that $c_i\neq 0$. Assume further that $i$ is the
smallest such integer. Then $\tr^G_M(f_i)$ separates $v_1$ and
$v_2$ by the previous lemma. Therefore we have $c_1=c_2=\cdots
=c_{n-2}=0$. We consider two cases. First assume that $c_{n-1}=0$.
Then $N_G(x_n)(v_1)=N_G(x_n)(v_2)$, i.e. $c_{n}^{pm}=d_{n}^{pm}$,
implies that $c_n=\lambda^a d_n$ for some integer $a$ and hence
$v_1$ and $v_2$ are in the same orbit.
If
$c_{n-1}\neq 0$, then we see that $N_H(x_n)(N_H(x_{n-1}))^l$
separates $v_1$ and $v_2$ as follows.
We have
$(N_H(x_{n-1}))^l(v_1)=(N_H(x_{n-1}))^l(v_2)\neq 0$. Therefore it
suffices to show
$N_H(x_n)(v_1)\neq N_H(x_n)(v_2)$. But otherwise
$c_{n}^{p}-c_{n}c_{n-1}^{p-1}=d_{n}^{p}-d_{n}c_{n-1}^{p-1}$, which implies
$c_n=d_n+lc_{n-1}$ for some $0\le l\le p-1$, so $v_1$ and $v_2$
are in the same orbit.
\end{proof}
\bibliographystyle{plain}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 9,219 |
Kreß may refer to:
Hermann Kreß (1895–1943), German general
Willibald Kreß (1906–1989), German footballer
members of the patrician family Kreß von Kressenstein
Franz Freiherr Kreß von Kressenstein (1881–1957), German general
Friedrich Freiherr Kreß von Kressenstein (1870–1948), German general
Georg Ludwig Kreß von Kressenstein (1797–1877), Bavarian artist
Gustav Kreß von Kressenstein (1838–1898), Bavarian artist
Otto Freiherr Kreß von Kressenstein (1850–1929), Bavarian general and War Minister
See also
Kress (disambiguation) | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 4,613 |
Raça Negra é o décimo segundo álbum de estúdio da banda de pagode Raça Negra, lançado em 2001 pela gravadora Universal Music. O álbum ganhou um disco de Ouro no Brasil pela ABPD.
Faixas
"Perdoa" – (Arnaldo Saccomani / Peninha)
"Espelho" – (Dhema / Luiz Carlos)
"Problema Meu" – (Luiz Carlos / Elias Muniz)
"Vai Ter Saudade" – (Alexandre Lucas / Joel Lacerda / André Lima / Eddy Flash)
"Assim É o Nosso Amor" – (Elias Muniz / Peninha)
"Diga Pra Mim – (Luiz Carlos / Elias Muniz)
"Correnteza de Emoção – (Carlos Colla / Elias Muniz)
"Por Onde Anda o Meu Amor" – (Luiz Carlos / Fabinho César)
"Tô Bolado" – (Pedrinho da Flor / Luiz Carlos)
"Gostoso Demais – (Waldir Luz / Nelson Farias / Nita)
"Denguinho – (Tivas / Waldir Luz)
"Cansado de Viver Sozinho" – (Luiz Carlos / Elias Muniz)
"Amor Amigo – (Ana Laura)
"Olha Pra Mim" – (César Augusto / Luiz Carlos / Antônio Luis)
"Você Não Sabe de Mim" – (Luiz Carlos / Elias Muniz)
Certificações
Ligações externas
Discografia
Álbuns de 2001
Álbuns de pagode
Álbuns em língua portuguesa
Álbuns de Raça Negra
Álbuns lançados pela Universal Music Brasil | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 9,601 |
{"url":"http:\/\/astronomy.stackexchange.com\/questions\/2422\/how-many-earths-fit-in-the-observable-universe","text":"# How many earths fit in the observable universe?\n\nI was pondering our insignificance, when I wondered - how how much smaller is our planet then the (observable) universe? And being as I dont know how to do the math, I'm asking it here.\n\nSo how many of our planet (in space it occupies - i.e. ignoring space between the space between the spheres) can fit inside the known\/observable universe?\n\nI apologize for the simplicity of the question.\n\n-\nReally to me this question is pointless. I mean, the answer is obviously going to be a ridiculously big number, so what does it changes to you if it is 10^50 or 10^100?? \u2013\u00a0 harogaston May 13 at 23:40\ni was trying to find something to compare it to, in order to better understand it and relay it. \u2013\u00a0 tryingToGetProgrammingStraight May 14 at 1:49\n\n## 1 Answer\n\nWithout checking the numbers in detail, according to Wikipedia, the volume of the observable universe is about $3.5\\cdot 10^{80} \\mbox{ m}^3$, and the volume of Earth is about $1.08321\\cdot 10^{21} \\mbox{ m}^3$.\n\nBy dividing the two volumes we get a factor of $3.2\\cdot 10^{59}$, or written as decimal number: The observable comoving volume of the universe is about 320,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000-times the volume of Earth.\n\n-\ndats alot, woah! \u2013\u00a0 tryingToGetProgrammingStraight May 13 at 17:14\nYour answer assumes we are pulverizing the earth to completely fill the volume of a universe-sized container. Without getting into the complicated math behind forming optimal latices of congruent spheres, you should multiply your answer by a factor of pi\/(3*sqrt(2)) or about 0.74048. The Kepler Conjecture says that is the highest density that can be achieved by any arrangement of spheres. Oh, and since the observable universe is also expanding at an accelerated rate, you should also update your answer every few hundred millions years just to be safe. Just saying. \u2013\u00a0 Robert Cartaino May 14 at 16:44\n@RobertCartaino That's why I just provided two valid digits; so the numbers should be valid more or less next week, too. ;) Btw. sorry for pressing Earth into a cube, next time I'll be more careful. \u2013\u00a0 Gerald May 15 at 9:37","date":"2014-09-30 19:53:53","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.7322530150413513, \"perplexity\": 910.1397149823207}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": false}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2014-41\/segments\/1412037663135.34\/warc\/CC-MAIN-20140930004103-00016-ip-10-234-18-248.ec2.internal.warc.gz\"}"} | null | null |
Talk About Trump…
Talk About Trump Buying a Share in Theverge
Donald Trump's sidelining by Twitter and Facebook has left him in a world of hurt, but it seems as if he's got a chance to make his mark on the business world. In fact, talks have been underway about taking a stake in Parler, a social networking company he started.
Amazon will remove Parler from its cloud hosting service
Amazon has announced that it will suspend Parler from its web hosting service. The company says that the site has violated its terms of service and has made "unacceptable" content.
Parler is a far-right social media platform. It's been popular with supporters of former President Donald Trump. Many of its users have encouraged violence and violence against President Donald Trump. There are also many posts that have called for the execution of Vice President Mike Pence.
On Wednesday, a group of right-wing extremists attacked the U.S. Capitol. They are believed to have coordinated their riot through Parler.
Parler is a microblogging app that claims to be an "unbiased" social network. While it has gained publicity in recent days, it has yet to demonstrate that it's worth the trouble.
In an email, AWS stated that it had suspended Parler for the "unacceptable" violent content. According to the company, it flagged 98 instances of such content.
The email went on to say that it had done so as a last resort to avoid a future violent event. The suspension is due to start on Sunday. If Amazon is unable to find a new host within a week, Parler will be gone.
Community guidelines will automatically detect violent content or incitement to violence
The community guidelines rolled out by Theverge are now in effect and will automatically detect the most violent content and incitements to violence. In addition, many of the major social media platforms have taken up the initiative to remove content that they find to be of questionable value to their users. This includes content that promotes violence, or content that denigrates a particular user. These measures have proven to be effective in keeping the social media landscape safe for all. Among the companies that have implemented this policy are Facebook, Twitter, Google and YouTube. However, removing content does not stop it from being posted in the first place. Many companies are appointing dedicated moderators to ensure that all of the voices of the internet are heard.
The social media companies have also incorporated some more subtle measures to reduce the volume of questionable content on their sites. For example, they have started adding labels and warning notices to graphic content. Similarly, they have enacted a more formal process to delete errant retweets and reposts.
Trump Organization holds talks with Parler about taking a stake in the company
Parler, a social media platform embraced by the far right, was in talks with the Trump Organization last summer to take a stake in the company. BuzzFeed News reported that a deal was in the works. However, the talks were halted by the White House counsel's office, according to sources.
In a report, BuzzFeed claimed that former Trump campaign manager Brad Parscale proposed the idea of taking an ownership stake in Parler in a White House meeting. He reportedly suggested that the company could counter Facebook's policies against falsehoods.
During the campaign, Parscale met with the CEO of Parler, John Matze, at Mar-a-Lago. Among the investors in the company are prominent Trump donor Rebekah Mercer and right-wing personality Dan Bongino.
While there is no confirmation that the President was involved in the negotiations, there are questions regarding the legality of a deal. Ethics experts say such an arrangement could violate anti-bribery laws.
The talks were said to have occurred during the campaign and after the re-election, but it is unclear whether Parler was officially on board.
Trump has been sidelined by Twitter and Facebook
The recent decisions by Twitter and Facebook to suspend President Donald Trump's account represent a significant escalation. For many years, these two social media platforms have created special rules and procedures for world leaders. They have also flagged posts containing misinformation and violence.
After the election, Twitter and Facebook began to flag several of Trump's posts. Specifically, they said two tweets on January 8 could incite violence. Those two tweets also violated Twitter's Glorification of Violence policy.
Twitter CEO Jack Dorsey considered permanently banning Trump's account. A top lieutenant advised him to do so. Vijaya Gadde, the company's legal chief, gave the advice.
Facebook's decision to bar Trump's account was not without controversy. In fact, the Oversight Board upheld Dorsey's decision. And some lawmakers applauded the move. However, others warned that limiting Trump's posts would backfire on social media.
While social media executives have reservations about the president's power, they believe their actions were the right thing to do. Mark Zuckerberg, the CEO of Facebook, has been a staunch advocate for free expression.
Tags: Theverge
Easy Easter Party Ideas For Your Kids' Next Party
What You Need To Know About Cannabis Strains: How To Choose the Right One For You | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 7,844 |
Several organizations, including the ANSWER Coalition and National Committee to Free the Cuban Five, held a press conference in Miami, Fla., on May 29 to demand the U.S. government fully reveal its relationship with terrorists who are staging attacks from Miami against Cuba. Seeking full government disclosure, the groups submitted Freedom of Information Act (FOIA) requests to the CIA, FBI and U.S. State Department. The Alianza Martiana, comprising several progressive Cuban-American organizations in Miami, also participated.
The press conference was attended by Associated Press Spanish and English departments, Reuters, ABC TV, CBS TV, The Miami Herald, Radio Miami, Al Jazeera, local news WLTV 23 and others. Pacifica KPFK and Cuba's Granma and Prensa Latina issued stories as well.
On April 26, Cuban authorities announced the arrest in Havana of four men from Miami, who admit they planned to launch armed attacks on military installations in Cuba. The four are Raibel Pacheco Santos, Obdulio Rodríguez, Felix Monzón Álvarado and José Ortega Amador.
Gloria La Riva, Coordinator of the National Committee to Free Cuban Five, chaired the press conference. She said, "We demand more than ever the freedom of the Cuban Five, three of whom still remain in prison. They were saving lives by stopping the Miami terrorists' plans, yet the FBI arrested the Five instead of the terrorists."
But the four detained terrorists did not act alone. They have named some of the most notorious and dangerous individuals with long terrorist histories — from Miami — as directing their planned attacks: Santiago Álvarez, Osvaldo Mitat and Manuel Alzugaray.
Santiago Álvarez first began his history of violent armed actions after being trained by the CIA in 1961. He engaged in an armed offshore attack on Cuba, killing two men and wounding two teenaged children. Since then, Álvarez has repeatedly financed armed attacks on Cuba, especially as financier of Luis Posada Carriles. Posada Carriles is widely known for having engineered the bombing of a Cuban airliner in Oct. 1976, killing all 73 passengers and crew aboard.
Click here to read more about Santiago Álvarez's history of violence.
Over 1,600 people have already signed a petition to President Obama, calling for an end to U.S. support of the Miami terrorists and instead that the Cuban Five anti-terrorists be freed from U.S. prison. Click here to sign the petition.
Click here to learn more about the campaign to free the Cuban Five.
The ANSWER Coalition is committed to pursuing the FOIA information requests and exposing the inner workings of the Miami criminal anti-Cuba network. We need to inform and help protect the people of both Cuba and the United States from the danger of terrorism. Hold Washington accountable — no more CIA-led terrorism, no funding of Miami fascists!
Miami - En la mañana del jueves 29 de mayo tuvo lugar una importante conferencia de prensa para tratar asuntos vitales relacionados al grave fenómeno del terrorismo existente en esta ciudad de Miami. La rueda de prensa fue convocada por el Comité Nacional a Favor de la Libertad de los Cinco de Estados Unidos; la coalición nacional antibélica de Estados Unidos A.N.S.W.E.R. y las organizaciones de la emigración cubana en Miami que integran la coalición Alianza Martiana. | {
"redpajama_set_name": "RedPajamaC4"
} | 6,686 |
Orania castanea is een slakkensoort uit de familie van de Muricidae. De wetenschappelijke naam van de soort is voor het eerst geldig gepubliceerd in 1886 door Küster.
Muricidae | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 4,793 |
Thanks to the continuously expanding economic, political and cultural relations, there is an increasing need for Japanese translations. As Japan's economy is one of the largest and most developed in the world, knowledge of the Japanese language doesn't just help to know and understand the colourful Japanese culture, but it has also been very useful for many decades now in business life too. With the help of our native translators, we are up to speed on the slight nuances and constant changes. In order to avoid making mistakes, even when we translate the most complex texts, three experts work on each project: a translator, an editor and a proofreader. All of them work as a full-time translator with at least 5 years' experience in translation, are native speakers of Japanese and hold a degree in translation.
We mainly offer translations from English into Japanese, while we naturally translate out of Japanese too. We undertake the translation of countless types of documents, including clinical trial documentation, technical manuals and licence agreements.
Japanese is spoken by approximately 130 million people as a mother tongue, making it the 10th most spoken language in the world. Its standardised version is based on the Tokyo dialect.
Japanese has three writing systems: the syllabaries, hiragana and katakana; the adopted logographic Chinese characters, kanji; and the alphabetic, rōmaji which is also used sometimes.
Japanese is an agglutinative language, while its origin is still debated.
Japanese is the second most widely spoken language in Brazil, as most Japanese people who live outside Japan live in Brazil.
In Japanese, the sounds "r" and "l" are not distinguished.
There are no street names in Japan or house numbers, rather buildings have names for guidance purposes.
There are about 20 different ways to say sorry in Japanese. | {
"redpajama_set_name": "RedPajamaC4"
} | 4,505 |
\section{Introduction}
A new iron age has occurred in the field of superconductivity and
started when the discovery of the LaFeAsO$_{1-x}$F$_{x}$
superconductor in 2008 gave rise to a huge interest in depressed
superconductivity division \cite{1}. Over the past two years, an
enormous number of papers have been published and more than 70 new
iron-based superconductors have been discovered \cite{2}. Generally,
iron-based superconductors are classified as REFeAsO (1111 family,
RE: rare earths), AEFe$_2$As$_2$ (122 family, AE: alkaline earths),
AFeAs (111 family, A: Li, Na), and FeCh (11 family, Ch: chalcogens).
Among these families, a tetragonal phase FeSe is an attractive
material for finding the origin of iron-based superconductors due to
its simple crystal structure \cite{3}. It has just superconducting
layers without block layers, and its superconducting critical
temperature ($T_c$) is very sensitive to change in the strain.
Although the $T_c$ of bulk FeSe is as low as 8.5 K, its $T_c$ can be
raised to 37 K under pressure \cite{4, 5}. In addition, the $T_c$ of
FeSe$_{0.5}$Te$_{0.5}$ thin films can be enhanced to 21 K by
controlling the biaxial strain \cite{6}.
However, compared with Te- or S-doped superconductors \cite{7, 8},
research on pure FeSe superconductors is rarely carried out.
Synthesis of single-phase FeSe superconductors with a tetragonal
structure is not easy because it has several types of phases, such
as tetragonal (PbO-type), hexagonal (NiAs-type), and Fe$_7$Se$_8$
phases \cite{3, 9}. Moreover, most of them, except for the
tetragonal phase, are known to be ferromagnetic phases \cite{10}.
According to recent reports \cite{11}, the possibility of
ferromagnetism was presented in FeSe thin films with a tetragonal
structure at temperature near room temperature, but this issue is
still under debate.
Nevertheless, the observation of superconductivity in the tetragonal
phase of FeSe by Hsu $et~al$ \cite{3} has offered a new member (11
family) to the class of iron-based superconductors. Also, an
expectation of a higher $T_c$ in FeTe than in FeSe, which is caused
by strong paring \cite{12}, has brought about active research on the
Fe-Te-Se system \cite{6, 7, 13}. If the intrinsic properties are to
be investigated precisely, undoped FeSe or FeTe single crystals with
large sizes or thin films are required because isovalent
substitution induces lattice strain due to the different ionic
sizes.
Very recently, Han $et~al$ \cite{14} and Si $et~al$ \cite{15}
reported the appearance of superconductivity in FeTe films due
tensile stress effects and oxygen incorporation, respectively, while
no superconductivity was observed in bulk samples \cite{8}. In the
case of FeSe films, only a few papers have been published \cite{16,
17, 18}, and the experimental results are still unsatisfactory
compared to those for bulk samples. Furthermore, there has been
little progress in the growth of pure FeSe films whereas some
research groups have fabricated high-quality Te-doped FeSe thin
films \cite{6, 19}.
In this work, we report the fabrication of FeSe$_{1-x}$ films with a
bulk $T_c$ of $11 - 12$ K on several types of substrates,
Al$_2$O$_3$(0001) [AO], SrTiO$_3$(100) [STO], MgO(100), and
LaAlO$_3$(100) [LAO], by using a pulsed laser deposition (PLD)
technique. All samples were grown at a high substrate temperature of
610 $^\circ$C and showed high upper critical fields with a (101)
preferred orientation.
\section{Experiments}
Homemade FeSe targets having a stoichiometric ratio of Fe:Se = 1:1
were used in this study, and the laser beam was generated by using a
Lambda Physik KrF excimer laser ($\lambda$ = 248 nm). Our PLD system
is described in detail elsewhere \cite{20, 21}. A laser energy
density of 1.15 J/cm$^2$, a high repetition rate of 48 Hz, and high
substrate temperature ($T_s$) of 610 $^\circ$C were used for the
deposition of FeSe$_{1-x}$ films in a high vacuum state of $\sim$
10$^{-5}$ Torr. The high frequency of the laser is favorable for the
fabrication of films containing highly-volatile materials \cite{20,
21}. High growth rates of $5 - 6$ nm/s were obtained under these
growth conditions. The compositional ratio of the deposited films
was Fe:Se = 1:0.90$\pm$0.02.
The crystal phases and orientations of the fabricated FeSe$_{1-x}$
films were investigated by using x-ray diffraction (XRD). The
surface morphology and thickness were checked by using scanning
electron microscopy (SEM). The elemental composition ratio was
analyzed by using energy dispersive spectroscopy (EDS) based on the
results for a standard stoichiometric sample. A vibrating sample
measurement system (SQUID-VSM, Quantum Design) and a physical
property measurement system (PPMS, Quantum Design) were used for the
study of their superconducting properties.
\section{Results and discussion}
The temperature dependences of the resistivity ($\rho$) of the
FeSe$_{1-x}$ films are shown in figure 1. A Se deficiency of $x$ =
0.10$\pm$0.02, which was comparable with previously reported samples
\cite{3, 18}, was examined by using EDS. All of the deposited films
showed a $T_{c,90\%}$ of $11 - 12$ K and a $T_{c,10\%}$ of $\sim 8$
K, and their transition widths ($\Delta$$T_c$) decreased with
increasing residual resistivity ratio (RRR); these data are
summarized in table 1. All these data are very close to the previous
results for bulk samples \cite{3, 4, 22} and show a higher
$T_{c,10\%}$ and a narrower $\Delta$$T_c$ than previously reported
\cite{16, 17, 18}. In figure 1(a), the $\rho$($T$) of the film grown
on a STO substrate is multiplied by 5 because it has small values
compared with those of the other film. The change of slope around
120 K seems to originate from the influence of a spin-density wave
(SDW) \cite{12, 22, 23}.
\begin{figure}[t!]
\begin{center}
\includegraphics[scale=0.4, angle=0]{Figure1.eps}
\caption{(a) Resistivity as functions of temperatures for
FeSe$_{1-x}$ films grown on several types of substrates. and (b) an
enlarged view near the superconducting transition temperature, with
the resistivity values normalized to those at 15 K. The inset of (b)
shows the field-cooled (FC) and the zero-field-cooled (ZFC)
measurements for an FeSe$_{1-x}$ film grown on a SrTiO$_{3}$
substrate at $H$ = 20 Oe.} \label{FIG.1}
\end{center}
\end{figure}
Figure 1(b) is a magnified view near the superconducting transition
temperature; the resistivity values are normalized to those at 15 K.
The inset shows the temperature dependence of the magnetization
($M$) of the FeSe$_{1-x}$ film fabricated on a STO substrate. The
clear diamagnetic signal indicates that it has good bulk
superconductivity. Although the irreversible temperature of $\sim$ 6
K in the $M - T$ curve is lower than the $T_c$ obtained from
$\rho$($T$), obtaining the superconducting signal from magnetization
measurements in FeSe superconductors is not easy due to its
intrinsic magnetism \cite{24} or its magnetic impurities, such as
hexagonal FeSe phases and Fe impurities \cite{3, 25}.
\begin{table} [b!]
\caption{\label{tabone}Summary of the $\rho$($T$) data of figure 1:
superconducting transition temperatures $T_{c,90\%}$ (90\% of
$\rho_n$) and $T_{c,10\%}$ (10\% of $\rho_n$), where $\rho_n$ is the
normal state resistivity near $T_c$, $\Delta T_c$ (=
$T_{c,90\%}-T_{c,10\%}$), and RRR (= $\rho_{300K}/\rho_n$). Fe:Se
data are the composition ratios, obtained from EDS, of the
FeSe$_{1-x}$ films fabricated on the different substrates.}
\begin{indented}
\lineup
\item[]\begin{tabular}{@{}*{6}{l}}
\br
\0\0substrate&$T_{c,90\%}$(K)&$T_{c,10\%}$(K)&\m$\Delta$$T_c$(K)&RRR&Fe:Se\cr
\mr \0\0Al$_2$O$_3$&10.5&8.2&2.3&4.5&1:0.91\cr
\0\0SrTiO$_3$&11.4&8.3&3.1&4.2&1:0.92\cr
\0\0MgO&11.7&8.1&3.6&3.4&1:0.88\cr
\0\0LaAlO$_3$&11.2&7.7&3.5&3.3&1:0.91\cr \br
\end{tabular}
\end{indented}
\end{table}
\begin{figure}
\begin{center}
\includegraphics[scale=0.35, angle=90]{Figure2.eps}
\caption{$\theta-2\theta$ scan XRD patterns of the FeSe$_{1-x}$
films grown on Al$_{2}$O$_{3}$(0001) and SrTiO$_{3}$(100)
substrates. The preferred orientation is seen to be along the (101)
direction (T: tetragonal, H: hexagonal).} \label{FIG.2}
\end{center}
\end{figure}
\begin{figure}
\begin{center}
\includegraphics[scale=0.35, angle=90]{Figure3.eps}
\caption{SEM images of the FeSe$_{1-x}$ films grown on several
different substrates: (a) Al$_{2}$O$_{3}$, (b) SrTiO$_{3}$, (c) MgO,
and (d) LaAlO$_{3}$. All surface morphologies show well-linked
states without cracks.} \label{FIG.3}
\end{center}
\end{figure}
Figure 2 shows the $\theta - 2\theta$ scan XRD patterns of
FeSe$_{1-x}$ films prepared on the Al$_2$O$_3$(0001) and
SrTiO$_3$(100) substrates. Our samples are mainly oriented along the
(101) direction of the tetragonal (T) structure, but other phases
exist in tiny amounts, such as hexagonal (H) FeSe and Fe$_7$Se$_8$.
The lattice constants for the $a$- and $c$-axis are 3.769 {\AA} and
5.490 {\AA}, respectively, which are close to the values observed in
the bulk samples \cite{3}. No peak shift is detected regardless of
the substrate. We believe that this is due to the growth direction
of the film being along the (101) direction rather than along the
$c$-axis or the ${ab}$-axis. Wang $et~al$ \cite{16} reported that
the orientation of the T(101) direction is the key to fabricating of
FeSe$_{1-x}$ films with strong superconductivity.
The surface morphologies of the FeSe$_{1-x}$ films on the AO, STO,
MgO, and LAO substrates were checked by using scanning electron
microscopy (SEM), as shown in figure 3. The SEM images, (a), (b),
(c), and (d), show that compared with previous reports \cite{18,
26}, all films have well-connected morphologies without cracks,
which is one factor in obtaining higher-quality samples with a
sharper superconducting transition. The white tiny grains in figure
3(b) were shown to be Fe particles by using EDS. Films grown on LAO
had larger Fe lumps than ones grown on STO. These Fe lumps, which
were ferromagnetic, are a possible source of the proximity effect
that accompanies a suppression of $T_c$ \cite{27}. These locally
weakened superconducting regions generate a broad superconducting
transition. The thicknesses determined from the SEM cross-sectional
images in (a), (b), (c), and (d) are 1.5, 1.6, 1.6, and 1.8 $\mu$m,
respectively.
\begin{figure}
\begin{center}
\includegraphics[scale=0.4, angle=90]{Figure4.eps}
\caption{$\rho$($T$) curves for FeSe$_{1-x}$ films on SrTiO$_{3}$
substrates under magnetic fields up to 7 T for a field parallel to
the substrate plane. The inset presents the linear fittings for
$H_{c2}$($T$) at $\rho_{10\%}$, $\rho_{50\%}$, and $\rho_{90\%}$.}
\label{FIG.4}
\end{center}
\end{figure}
Finally, we investigated the temperature dependence of the upper
critical magnetic field ($H_{c2}$($T$)) for the FeSe$_{1-x}$ films.
Figure 4 displays the $\rho - T$ curves of the film prepared on a
STO substrate for various magnetic fields from 0 to 7 T, and the
inset shows the fitting of $H_{c2}$($T$). The data points were
obtained from the $\rho - T$ curves for three defined criteria;
$\rho_{90\%}$ (90\% of $\rho_n$), $\rho_{50\%}$ (50\% of $\rho_n$),
and $\rho_{10\%}$ (10\% of $\rho_n$) are equivalent to critical
temperatures of $T_{c,90\%}$, $T_{c,50\%}$, and $T_{c,10\%}$,
respectively. The values of $H_{c2}$(0) at $\rho_{90\%}$ are 58, 56,
50, and 57 T for films grown on AO, STO, MgO, and LAO substrates,
respectively, which were estimated by using a linear extrapolation
\cite{4}. These high values of $H_{c2}$(0) open up the possibility
for practical applications, provided we can grow better films with
higher values of $T_c$.
\section{Conclusions}
In summary, we have deposited FeSe$_{1-x}$ films with bulk
superconducting properties on various types of substrates,
Al$_2$O$_3$(0001), SrTiO$_3$(100), MgO(100), and LaAlO$_3$(100), by
using a pulsed laser deposition system. All films were
preferentially oriented along the (101) direction, and showed upper
critical fields as high as 50 T. Our results showed that
(101)-oriented FeSe$_{1-x}$ films with bulk superconductivity could
be fabricated at high growth temperature regardless of type of
substrate.
\\
\ack {This work was supported by the Korea Science and Engineering
Foundation (KOSEF) grant funded by the Korea Government (Grant No.
MEST, No. R01-2008-000-20586-0) and by the Postdoctoral Research
Program of Sungkyunkwan University (2009)}
\section*{References}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 1,149 |
– Investment in residential alternatives of all types is rising, with big-ticket global residential investment volumes exceeding all retail or industrial investment in the past year, according to new research from Savills.
The latest Global Living report reveals that student housing investment volumes have risen 87 per cent in the past five years, with the provision of Purpose-Built Student Accommodation (PBSA) highest in the UK and lowest in southern Europe.
Savills Australia's director for Student Accommodation, Conal Newland, said that the supply of new PBSA in Australia was growing in many of the major cities, underpinned by a lack of existing accommodation and strong student population growth.
"As the 20- to 39-year-old age bracket becomes more prevalent in the rental market, the increasing influence of Gen Y's demand for higher-quality rental accommodation is creating opportunities for developers to provide hybrid co-living developments," he said.
UKO is Australia's first co-living residential series emulating similar concepts in Europe, the UK and the US, with its Stanmore offering comprising 33 units. Developments in Newton and Paddington are set to follow.
In Melbourne, there is Urban Coup, which comprises 30 households ranging from first-home buyers to young families.
Mr Newland went on to say he expected "a steady increase" in PBSA transactions in the short term, which he believed would be the catalyst for a surge of investment into other residential-based asset classes.
"Increasing transaction volumes will, in our opinion, drive down PBSA yields in Australia, which currently sit much higher than other more mature markets, such as the UK and many European countries," he said.
The report outlines the growing popularity of co-living across the globe, which is a more luxurious form of share-housing and extends the convenience and amenity of PBSA to non-student markets. Targeted at graduates and young professionals, as well as students, co-living accommodation offers fully furnished units, extensive amenities, community events, all-inclusive bills and pro-active management.
"There are particular planning implications for this, and authorities need to recognise the trade-off between smaller units and larger communal spaces.
Mr Newland likened co-living to co-working, which was established in the office sector in response to occupier demand, and is now exceedingly popular.
"Co-living is emerging to meet current-day residential occupier requirements, and as the boundary between living and working becomes ever more blurred, we can expect to see greater integration of the two in future," he said.
The report highlights WeLive, a co-living derivative of the WeWork shared workspace group.
"Given the increasing global capital flows attracted to a growing range of residential investment opportunities, Australian lawmakers should be mindful of passing legislation that may reduce the attractiveness of investments, or this country may run the risk of becoming less attractive to millennials and Gen Zs looking to study, live and work here," Mr Newland said. | {
"redpajama_set_name": "RedPajamaC4"
} | 1,355 |
Q: Writing a tree structure to disk using C# and recursion I'm fairly new to C# and I'm attempting to take the tree structure I've created from a list, and write that to the file system. (I've used this answer).
That is, I want to create the directories and subdirectories in the proper depth. However, I'm getting stuck on depth and cannot think of how to loop through each iteration of the depth, then back out to start writing again, without just writing all the 0-depth directories, then all the 1-depth directories, etc., in the same location.
I'm thinking there's another recursive routine/method/function, but I just can't wrap my head around it. I'd like to get to the next level (ha) in my understanding of programming.
static void Test(IEnumerable<TreeItem<category>> categories, int deep = 0)
{
foreach (var c in categories)
{
for (int i = 0; i < deep; ++i) {
System.IO.Directory.CreateDirectory(c.Item.Name);
}
Test(c.Children, deep + 1);
}
}
A: Because Directory.CreateDirectory creates all the parent directories for you, assuming you have the full path stored for each node, you only need to pass leaf nodes (nodes with no children) to it.
You can write a recursive search to retrieve leaf nodes, then pass that list to CreateDirectory.
static IEnumerable<TreeItem<category>> GetLeafs(IEnumerable<TreeItem<category>> tree)
{
foreach (var item in tree)
{
if (item.Children.Any())
{
// this is not a leaf, so find the leaves in its descendants
foreach (var leaf in GetLeafs(item.Children))
yield return leaf;
}
else
{
// no children, so this is a leaf
yield item;
}
}
}
static void CreateDirectories(IEnumerable<TreeItem<category>> categories)
{
foreach (var leaf in GetLeafs(categories))
{
System.IO.Directory.CreateDirectory(leaf.Item.Name);
}
}
If you don't have the full path against each node, then you can use the same structure as GetLeafs (although it turns out much simpler, because you don't need to return anything back up the call-chain) to recursively walk the tree, creating directories as you go deeper:
static void CreateDirectories(IEnumerable<TreeItem<category>> tree)
{
foreach (var item in tree)
{
Directory.Create(c.Item.Name);
CreateDirectories(item.Children);
}
}
This method is simpler, but will result in more calls to Directory.Create, which could conceivably have a performance impact.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 4,808 |
//
// Commom.m
// IMBaseOnXMPP
//
// Created by 王小帅 on 2017/2/28.
// Copyright © 2017年 王小帅. All rights reserved.
//
#import <Foundation/Foundation.h>
#import <UIKit/UIKit.h>
/** 用户信息 */
NSString *const kName = @"zhoudongyu";
NSString *const kDomain = @"local";
NSString *const kPassword = @"123456";
NSString *const localhost = @"127.0.0.1";
NSString *const portNo = @"5222";
| {
"redpajama_set_name": "RedPajamaGithub"
} | 9,195 |
{"url":"https:\/\/www.primidi.com\/logistic_regression\/formal_mathematical_specification\/as_a_log-linear_model","text":"# Logistic Regression - Formal Mathematical Specification - As A \"log-linear\" Model\n\nAs A \"log-linear\" Model\n\nYet another formulation combines the two-way latent variable formulation above with the original formulation higher up without latent variables, and in the process provides a link to one of the standard formulations of the multinomial logit.\n\nHere, instead of writing the logit of the probabilities pi as a linear predictor, we separate the linear predictor into two, one for each of the two outcomes:\n\nbegin{align} ln p(Y_i=0) &= boldsymbolbeta_0 cdot mathbf{X}_i - ln Z , \\ ln p(Y_i=1) &= boldsymbolbeta_1 cdot mathbf{X}_i - ln Z , \\ end{align}\n\nNote that two separate sets of regression coefficients have been introduced, just as in the two-way latent variable model, and the two equations appear a form that writes the logarithm of the associated probability as a linear predictor, with an extra term at the end. This term, as it turns out, serves as the normalizing factor ensuring that the result is a distribution. This can be seen by exponentiating both sides:\n\nbegin{align} p(Y_i=0) &= frac{1}{Z} e^{boldsymbolbeta_0 cdot mathbf{X}_i} , \\ p(Y_i=1) &= frac{1}{Z} e^{boldsymbolbeta_1 cdot mathbf{X}_i} , \\ end{align}\n\nIn this form it is clear that the purpose of Z is to ensure that the resulting distribution over Yi is in fact a probability distribution, i.e. it sums to 1. This means that Z is simply the sum of all un-normalized probabilities, and by dividing each probability by Z, the probabilities become \"normalized\". That is:\n\nand the resulting equations are\n\nbegin{align} p(Y_i=0) &= frac{e^{boldsymbolbeta_0 cdot mathbf{X}_i}}{e^{boldsymbolbeta_0 cdot mathbf{X}_i} + e^{boldsymbolbeta_1 cdot mathbf{X}_i}} , \\ p(Y_i=1) &= frac{e^{boldsymbolbeta_1 cdot mathbf{X}_i}}{e^{boldsymbolbeta_0 cdot mathbf{X}_i} + e^{boldsymbolbeta_1 cdot mathbf{X}_i}} , end{align}\n\nOr generally:\n\nThis shows clearly how to generalize this formulation to more than two outcomes, as in multinomial logit.\n\nNow, how can we prove that this is equivalent to the previous model? Keep in mind that the above model is overspecified, in that and cannot be independently specified: rather so knowing one automatically determines the other. As a result, the model is nonidentifiable, in that multiple combinations of \u03b20 and \u03b21 will produce the same probabilities for all possible explanatory variables. In fact, it can be seen that adding any constant vector to both of them will produce the same probabilities:\n\nbegin{align} p(Y_i=1) &= frac{e^{(boldsymbolbeta_1 +mathbf{C}) cdot mathbf{X}_i}}{e^{(boldsymbolbeta_0 +mathbf{C})cdot mathbf{X}_i} + e^{(boldsymbolbeta_1 +mathbf{C}) cdot mathbf{X}_i}} , \\ &= frac{e^{boldsymbolbeta_1 cdot mathbf{X}_i} e^{-mathbf{C} cdot mathbf{X}_i}}{e^{boldsymbolbeta_0 cdot mathbf{X}_i} e^{mathbf{C} cdot mathbf{X}_i} + e^{boldsymbolbeta_1 cdot mathbf{X}_i} e^{mathbf{C} cdot mathbf{X}_i}} , \\ &= frac{e^{mathbf{C} cdot mathbf{X}_i}e^{boldsymbolbeta_1 cdot mathbf{X}_i}}{e^{mathbf{C} cdot mathbf{X}_i}(e^{boldsymbolbeta_0 cdot mathbf{X}_i} + e^{boldsymbolbeta_1 cdot mathbf{X}_i})} , \\ &= frac{e^{boldsymbolbeta_1 cdot mathbf{X}_i}}{e^{boldsymbolbeta_0 cdot mathbf{X}_i} + e^{boldsymbolbeta_1 cdot mathbf{X}_i}} , \\ end{align}\n\nAs a result, we can simplify matters, and restore identifiability, by picking an arbitrary value for one of the two vectors. We choose to set Then,\n\nand so\n\n$p(Y_i=1) = frac{e^{boldsymbolbeta_1 cdot mathbf{X}_i}}{1 + e^{boldsymbolbeta_1 cdot mathbf{X}_i}} = frac{1}{1+e^{-boldsymbolbeta_1 cdot mathbf{X}_i}} = p_i$\n\nwhich shows that this formulation is indeed equivalent to the previous formulation. (As in the two-way latent variable formulation, any settings where will produce equivalent results.)\n\nNote that most treatments of the multinomial logit model start out either by extending the \"log-linear\" formulation presented here or the two-way latent variable formulation presented above, since both clearly show the way that the model could be extended to multi-way outcomes. In general, the presentation with latent variables is more common in econometrics and political science, where discrete choice models and utility theory reign, while the \"log-linear\" formulation here is more common in computer science, e.g. machine learning and natural language processing.","date":"2022-05-18 05:57:03","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 5, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.9974355101585388, \"perplexity\": 2175.997061575859}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2022-21\/segments\/1652662521152.22\/warc\/CC-MAIN-20220518052503-20220518082503-00518.warc.gz\"}"} | null | null |
{"url":"https:\/\/codereview.stackexchange.com\/questions\/9650\/am-i-approaching-this-asp-net-mvc-entity-framework-pattern-incorrectly","text":"# Am I approaching this ASP.Net MVC\/Entity Framework pattern incorrectly?\n\nI have experience is C#\/ASP.NET, and I've done MVC in Ruby On Rails, so I figured making the jump to ASP.NET MVC would be a breeze. Unfortunately, I'm second-guessing my every decision. I'd like some more eyes on this to tell me if my way of doing things is insane.\n\nMy main question is around editing an entity.\n\nMy GET looks like this:\n\npublic ActionResult Edit(int id)\n{\nvar viewModel = from c in db.centres\n\/\/ some joins are here\nwhere c.id == id\nselect new CentreViewModel {\nCentreId = c.id,\nCentreName = c.Name,\netc...\n};\nreturn View( viewModel );\n}\n\n\nAnd my POST is something like this:\n\npublic ActionResult Edit(CentreViewModel viewModel)\n{\n\/\/ NOTE - Validation & Error handling exists, but has been omitted for berevity\nvar centre = db.centres.First(c => c.id == viewModel.CentreId);\ncentre = Mapper.Map<CentreViewModel, CentreEntity>(viewModel, centre);\n\ndb.centres.ApplyCurrentValues( centre );\ndb.SaveChanges();\n}\n\n\nIs this kinda-sorta right? Are there any red flags here? Is ApplyCurrentValues correct? I see others using repository.Entry(), but I don't have a repository for my classes, near as I can tell...\n\n\u2022 Not enough expirience with EF to answer your question (which DOES look right btw) but I generally disparage people away from the from...where...select LINQ syntax as it hides what's actually going on in a very unhelpful way. Since LINQ has been out, in 100% of the scenarios that I've seen that using the expression syntax is more powerful, terse, and more likely to convey to developers what's really happening. In your case you should be able to shorten the actual query to db.centres.First(x=>x.id == id).. though it is missing the projection part which you have to do separately in code. \u2013\u00a0George Mauer Mar 2 '12 at 17:39\n\u2022 The reason I went with the LINQ syntax is because the joins don't have foreign keys and, based on about 30 seconds of googling, this was the easiest way to join on tables without FKs. I'll take another look, see if I can improve it. \u2013\u00a0Matt Grande Mar 2 '12 at 18:28\n\u2022 Well both are technically LINQ syntax. There is nothing that you can do with query syntax that you can't do with lambda syntax. There is a lot you can do with lambda syntax that you can't the other way around though. \u2013\u00a0George Mauer Mar 2 '12 at 21:58\n\nHere is a link to the \"official\" tutorial on basic CRUD operations in ASP.NET MVC and EF.\n\nThe main difference is the validation checking, and the exception handling. I recommend adding in the validation, and possibly the exception handling if you need it. Don't randomly swallow exceptions and show the user some generic error message.\n\nOther than that, they use a slightly different way of letting EF know about the entity to be updated, but I think they're accomplishing the same thing. I think the functional difference is that you are only posting back the necessary data to be updated, and having EF get the entity and update it with those changes for you. With their method, I think that they are posting back the entire entity, and having EF \"reattach\" the entity and mark it as modified.\n\nSo I think the differences in the update code boil down to partial vs. full entity updates.\n\nSide Note 1\n\nI like your way better. In their case they are relying on the end user to postback the entire entity. While that is a basic demo, using this method blindly opens up your application to vulnerabilities where people can update any field they want, even if the field doesn't appear in the HTML form (Think of an \"IsAdmin\" flag). I believe that this is known as a Mass Assignment vulnerability that bit Rails and Github last week! (Source: https:\/\/github.com\/blog\/1068-public-key-security-vulnerability-and-mitigation)\n\nSide Note 2\n\nAbout the comments and the LINQ syntax... I also like the expression syntax a little more, at least for short, small queries. But when doing joins in the past, I've gone with the query syntax. Just a personal preference. Here is what the code would look like with the LINQ expression syntax (as opposed to the query syntax):\n\ndb.centres\n.Join(...)\n.Where(c => c.id == id)\n.Select(new CentreViewModel\n{\nCentreId = c.id,\nCentreName = c.Name,\netc...\n};\n\n\u2022 That's weird, I don't have that Entry method they're using. Ah well, no one's told me that I'm completely out to lunch, so I think my way of doing it is Good Enough\u2122. \u2013\u00a0Matt Grande Mar 8 '12 at 15:57\n\u2022 I think they have a simplified DbContext class now. I would check it out since it's supposed to make things a bit cleaner. You should be able to update to the latest EF version through Nuget. \u2013\u00a0John Bubriski Mar 8 '12 at 16:12\n\nI usually use generics to handle my database contexts. Here is an example of how I would approach it explicitly.\n\n db.centres.Attach(centre);\ndb.Entry(centre).State = EntityState.Modified;\ndb.SaveChanges();\n\n\nMoreover, here is a link to one of the best tutorials out there in my opinion for mvc3.\n\nhttp:\/\/www.asp.net\/mvc\/tutorials\/getting-started-with-ef-using-mvc\/creating-an-entity-framework-data-model-for-an-asp-net-mvc-application (Make sure to note that there are 10 different parts, the link to the next step is at the bottom of the page).","date":"2020-02-25 10:24:04","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 1, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.35034021735191345, \"perplexity\": 1669.8292660924049}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2020-10\/segments\/1581875146064.76\/warc\/CC-MAIN-20200225080028-20200225110028-00240.warc.gz\"}"} | null | null |
Q: Can the number of scc, "strong connected component" be increased if only one edge is added to graph? I have one question about SCC in Algorithm.
In CLRS, It is said that the number of SCC can be the same or decreased when one edge is added into a graph, no nodes complemented.
Then, I want to suppose a graph which has initially 4 nodes and 4 edges.
The right graph is that one edge is added. Two graph photos
The left graph has one SCC. It is fine.
But, I have known that the right graph has not one SCCs, but two SCCs. : One is a triangle form, the other is just one node form.
Am I misunderstanding that concept?
Many web pages said the number of SCCs cannot be increased.
A: This is an important point that would be easy to miss! A Strongly Connected Component (SCC) is defined to be a maximal set of vertices such that any two vertices in the set can reach each other. "Maximal" is a pretty innocuous word here, but it does a lot of heavy lifting.
Let's dig into how we define what a SCC is, and why the number in a given graph can only decrease or stay the same when we add an edge. My copy of CLRS defines a strongly connected component for a directed graph G = (V, E) as "a maximal set of vertices C ⊆ V such that for every pair of vertices u and v in C, we have both u⇝v and
v⇝u; that is, vertices u and v are reachable from each other."
There are a couple of important words in here that might fly under the radar. The ones that jump out to me are "maximal", "both", and "reachable". Let's look at these in the reverse order of how I presented them. First is reachable, which is about what you'd expect: a vertex is reachable from another vertex if there is some path between them - that is, if we can start at the first vertex and keep following adjacent edges until we arrive at the second. The next important word is "both" - not only is u reachable from v, but v is also reachable from u. We can start to get an understanding of why we'd care if two vertices are both reachable from each other. If
v is reachable from u, that means that u can also reach all of the nodes that v can reach - and if the converse is also the case, then u and v can both reach all of the same nodes. It is reasonable, in a sense, to think of them as being the same node in terms of where they can get to - ignoring, for a moment, all thoughts of efficiency, weight, or distance.
Now let's examine the idea of this set of vertices we take as our SCC being "maximal." The second graph posted does not have two SCCs, but only one. Why? Because the "node form" that you noticed isn't its own SCC - it's part of the single SCC that is made up of all of the vertices in the graph. A set of vertices is maximal if and only if we cannot include any other vertex in the graph without losing our property that every vertex in our set is reachable from every other vertex in our set - in some sense, we want as many nodes in our set as we can without breaking it. The "triangle form" that you point out isn't a maximal set, because we could add the "node form" to that set, and we would still be able to reach every node from every other node - in fact, because we're only interested in maximal sets, we have to add it, because otherwise we'd be leaving that node on the table.
So how do we know that the number of SCCs in a graph can only remain the same or decrease if we add an edge? We can show that these are
possible via examples. To show that the number of SCCs can stay the same, imagine a set of nodes that have a cycle. If we presume that there is some edge between nodes in this cycle that does not yet exist, we could add it, and we have not increased the number of SCCs because we haven't "gained access" to any nodes that we couldn't already reach. To show that the number can decrease, imagine that you have two vertices, with vertex a having a directed edge to vertex
b, so that we currently have two SCCs. If we add a directed edge from b to a, then we now only have a single SCC, the set {a,
b}.
How do we know that the number of SCCs can't increase? It's a very intuitive argument - first, notice that since all vertices in the graph are included in some SCC, the only way we could create more SCCs is if we split some of the existing ones up. However, we can never do this by adding an edge - if we look at our set C of all u and v such that u⇝v and v⇝u, adding a new edge cannot disrupt the fact that u and v are still reachable from each other for any u and v in our set. It is impossible to break up an existing SCC by adding an edge.
1Cormen, Thomas H., et al. Introduction to Algorithms. 3rd ed., MIT Press, 2009.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 9,954 |
\section{Introduction} \label{Sec:Introduction}
It is well known from observations that low and intermediate mass stars (0.8\,\Msun$<$M$<$8\,\Msun) develop a low-velocity ($v\sim5-15$\,km/s) stellar wind at the end of their life during the Asymptotic Giant Branch (AGB) phase. At a mass-loss rate between $10^{-8}$ to $ 10^{-4}$\,\Msun/yr, material is stripped away from the stellar surface. Convection-induced pulsations lift material to greater heights where the temperature is low enough for gas to condense into dust grains. Radiation pressure on these grains is thought to be the main trigger for the onset of the stellar wind. While theoretical simulations based on this scenario predict stellar wind properties in broad agreement with observations of carbon-rich (C/O$>$1) AGB stars, more finetuning of the models is required for oxygen-rich (O-rich) AGB stars. Indeed, in these O-rich environments, the first dust seeds thought to form close to the stellar surface are metal oxides and pure Fe-free silicates such as Al$_2$O$_3$, SiO$_2$, Mg$_2$SiO$_4$, and MgSiO$_3$ \citep{Woitke2006A&A...460L...9W}. While their low near-infrared extinction efficiency prevents them from sublimating, it means that these glassy condensates make a negligible contribution to the radiative acceleration. The formation of both carbon and silicate grains \citep{Hofner2007A&A...465L..39H} or scattering of stellar radiation by micron-sized ($\sim$0.3\,\hbox{$\mu$m}) Fe-free silicates \citep{Hofner2008A&A...491L...1H} have been proposed as possible alternatives to solve this acceleration deficit. After its postulation in 2008 by H{\"o}fner et al., the latter scenario got more and more support from observations, the first ones reported by \citet{Norris2012Natur.484..220N} using VLT/NACO data of three low mass-loss rate AGB stars (R~Dor, W~Hya, and R~Leo). The mid-infrared interferometric observations of the semi-regular AGB star RT~Vir with the MIDI instrument at VLTI also lend support to the presence of iron-free
silicates between 2 and 3 stellar radii \citep{Sacuto2013A&A...551A..72S}. Recent polarized light data obtained with the VLT/SPHERE instrument offer us another way to look at the inner wind chemical and morphological structure. Again, these data prove the presence of grains close to the star albeit putting stringent constraints on the grain size is not obvious \citep{Khouri2016A&A...591A..70K, Ohnaka2016A&A...589A..91O}. However, neither the NACO or the SPHERE data can pinpoint the exact chemical constitution of these micron-sized transparent grains.
Considering the abundance of molecular species and the temperatures at which they can exist in solid phase, possible candidates for the first nucleation seeds are SiO, SiO$_2$, MgO, Fe, Al$_2$O$_3$, TiO and TiO$_2$ \citep{Jeong2003A&A...407..191J}. However, taking into account the thermodynamic properties, MgO nucleation is thought to be completely negligible in O-rich AGB stars. Although Al$_2$O$_3$ dust is stable at very high temperatures, gaseous Al$_2$O$_3$ has a low abundance, calling into question its role as the first candidate. For a long time, TiO and TiO$_2$ were considered to be the best candidates as primary condensates, since the nucleation rate of the more abundant SiO was thought to peak around 600\,K \citep[as based on the thermodynamic model of][using a single laboratory determination of the SiO vapour pressure]{Schick1960}, which is significantly lower than the typical temperatures at the inner edge of the dust shells. However, more recent laboratory experiments by \citet{Nuth2006ApJ...649.1178N}
and \cite{Gail2013A&A...555A.119G} proved that the onset of SiO nucleation starts at much higher temperatures than was previously found, although the calculated SiO condensation temperatures are still $\sim$100\,K lower than the observed ones (being around 880\,K).
Another quest focusses on the potential chemical differentiation between grains formed in low mass-loss rate versus high mass-loss rate O-rich AGB stars. Observations indicate that low mass-loss rate stars form primarily dust that preserves the spectral properties of Al$_2$O$_3$, and stars with higher mass-loss rates form dust with properties of warm silicate oxides \citep{Karovicova2013arXiv1310.1924K}\footnote{Note that this differentiation does not mean that no silicates are present in low mass-loss rate stars: their presence might stay unnoticed in the spectral energy distribution (SED) due to, for instance, their glassy character.}. It is absolutely unknown if the silicates form via a heteromolecular homogeneous nucleation process consuming Mg, SiO, and H$_2$O molecules \citep{Goumans2012MNRAS.420.3344G} or if the silicates form on top of the alumina grains (heterogeneous growth), hence gradually accelerating the wind. And be it now Fe-free silicates or aluminium-oxides that are the first solid-state species formed, another aspect that one might wonder about is if we could detect the (large) gas-phase \textit{clusters} which are the intermediate steps between the simple molecules and micron-sized grains.
A promising way to unravel the intriguing coupling between nucleation and wind generation is by studying at high spatial resolution the molecules contributing to the dust formation.
This possibility is offered by ALMA (the Atacama Large Millimeter/sub-millimeter array). We were granted ALMA Cycle~2 observing time (project 2013.1.00166.S) to investigate the molecular content in the inner wind region of the low mass-loss rate semi-regular AGB star R~Dor (\Mdot$\sim 9 \times 10^{-8}$\,\Msun/yr) and the high mass-loss rate Mira star IK~Tau (\Mdot$\sim 4.5 \times 10^{-6}$\,\Msun/yr). We obtained the first ALMA spectral scan for O-rich AGB stars in band 7 at a spatial resolution of $\sim$0\farcs12$\times$0\farcs15. The full data scan will be presented in \citet{Decin2016scan}. In this paper, we aspire to elucidate the role of aluminium-bearing molecules, clusters, and grains in the stellar winds. The ALMA observations are presented in Sect.~\ref{Sec:ALMA_observations}. Since the aim is to correlate the gaseous aluminium content with the amount of aluminium locked up into grains, we briefly summarize the most important dust properties of the targets in Sect.~\ref{Sec:continuum}. Important additional information on the dust formation history in O-rich AGB stars also stems from another low mass-loss rate O-rich AGB star, W~Hya. Where needed, we bring in additional constraints as derived from that target. Some properties of the three targets are introduced in Sect.~\ref{Sec:info_targets}. The results of the data analysis are presented in Sect.~\ref{Sec:Results} and are discussed in the context of molecule, cluster, and dust formation in Sect.~\ref{Sec:Discussion}. The conclusions are summarized in Sect.~\ref{Sec:conclusions}.
\section{Observations} \label{Sec:observations}
\subsection{Target selection}\label{Sec:info_targets}
To study the intricate coupling between gas and dust chemistry and the impact of the formation of some type of dust species on the wind dynamics, we have selected for our ALMA observations the two O-rich AGB stars which are the best representatives for the class of the low mass-loss rate and the high mass-loss rate O-rich AGB stars: R~Dor and IK~Tau. The reasons for their selection are their proximity (60\,pc and 260\,pc, respectively), their rich infrared and submillimeter spectra with signatures of many dust species and molecules, and the fact that both targets are very well studied with various instruments. Each target has a very different dust emission spectrum (see Sect.~\ref{Sec:continuum}), with the spectrum of IK~Tau being dominated by amorphous olivine dust (magnesium-iron silicates (Mg$_{2-x}$,Fe$_x$)SiO$_4$, with $0\le x \le 2$), while the infrared spectrum of R~Dor does not show the presence of olivine dust but species such as corundum (an aluminium oxide, \hbox{Al$_2$O$_3$}), the aluminium-calcium bearing silicate gehlenite, the magnesium-aluminium member spinel and CO$_2$ gas have been detected \citep{Heras2005A&A...439..171H, KhouriPhD}. This supports the hypothesis that the dust condensation sequence in the low mass-loss rate object R~Dor has experienced a freeze-out after the formation of aluminium oxides, with only a small amount of silicon condensing into dust grains. In IK~Tau the formation of silicon-bearing dust species seems to be much more efficient. This can happen through either a one-step process (growth of micron-sized Fe-free silicates close the central star) or a two-steps process (growth of aluminium oxides around a few stellar radii (\Rstar), after which the grains where coated with silicates around $\sim$10 stellar radii) might take place.
R~Dor is the closest AGB star to our Sun, with a gas mass-loss rate value of $\sim$0.9$-$2$\times10^{-7}$\,\Msun/yr and an expansion velocity of only $\sim$5.7\,km/s \citep{Schoier2004A&A...422..651S, KhouriPhD, Maercker2016A&A...591A..44M, VandeSande2017}. It is a semi-regular variable with two pulsation modes having periods of 332\,days and about 175\,days, the latter one with much smaller amplitude \citep{Bedding1998MNRAS.301.1073B}. Its angular diameter is $\sim$60\,mas \citep{Bedding1997MNRAS.286..957B, Norris2012Natur.484..220N}.
From an analysis of Herschel and sub-millimeter data, \citet{KhouriPhD} and \citet{VandeSande2017} derived that some 90\% of atomic silicon is locked up in gaseous SiO, the remainder in silicates. VLTI/NACO and SPHERE/ZIMPOL data support the presence of a close halo of large transparent grains ($\sim$0.3\,\hbox{$\mu$m}) at $\sim$1.5$-$2\,\Rstar\ \citep{Norris2012Natur.484..220N, Khouri2016A&A...591A..70K}.
IK~Tau is a high mass-loss rate Mira-type AGB star with a gas mass-loss rate of $\sim$4.5$\times10^{-6}$\,\Msun/yr, an expansion velocity of 17.7\,km/s \citep{Decin2010A&A...516A..69D, Maercker2016A&A...591A..44M}, and a pulsation period of $\sim$470\,days \citep{Wing1973ApJ...184..873W}. The anguldar diameter is estimated around 60\,mas \citep{Decin2010A&A...516A..69D}. Currently, a dozen different molecules and some of their isotopologues have been discovered in IK~Tau, including CO, HCN, SiO, SiS, SO, SO$_2$, NaCl, NS, NO, HCO$^+$, H$_2$CO, PO, PN, H$_2$S \citep{Milam2007ApJ...668L.131M, Kim2010A&A...516A..68K, Decin2010A&A...516A..69D, Decin2010A&A...521L...4D, DeBeck2013A&A...558A.132D, DeBeck2015ASPC..497...73D, Prieto2016arXiv160901904V}. Some molecules including SiO and SiS are depleted in the circumstellar envelope, possibly due to condensation onto dust grains. Other molecules such as HCN demonstrate the impact of pulsation-driven shocks in the inner wind zone which forces the chemistry into a non-equilibrium state.
W~Hya resembles in many aspects the low mass-loss rate AGB star R~Dor. Thanks to is proximity \citep[$\sim$78\,pc,][]{Knapp2003A&A...403..993K} and its brightness it is also well studied with various observational techniques from the visible to the radio. This semi-regular variable has a light curve showing a clear periodicity with period of 389 days \citep{Uttenthaler2011A&A...531A..88U}. The molecular abundance stratification in its circumstellar envelope was studied in detail by \citet{Khouri2014A&A...561A...5K, Khouri2014A&A...570A..67K}. Based on an extensive grid of CO, SiO, H$_2$O and HCN lines (and some of their isotopologues) the thermodynamic structure of the wind was derived and they deduced that about one-third of the silicon atoms is locked up in dust particles. The SED of W~Hya is characterized by the same dust emission peaks as R~Dor (see Sect.~\ref{Sec:continuum}) and also for this star large scattering grains in a halo close the star are detected via polarized light signal \citep{Norris2012Natur.484..220N, Ohnaka2016A&A...589A..91O}.
W~Hya was also observed by ALMA in band~7 (project 2015.1.01466.S, P.I.\ A.\ Takigawa) with a spatial resolution of $\sim$40\,mas at 3 selected frequency bands (330.40--331.34\,GHz, 342.40-343.34\,GHz, 344.03-345.91\,GHz) encompassing the AlO N=9-8 and AlOH J=11-10 transition. AlO is detected in their data \citep{Takigawa2016LPICo1921.6543T}.
\subsection{ALMA observations} \label{Sec:ALMA_observations}
\subsubsection{ALMA observing strategy} \label{Sec:ALMA_strategy}
Spectral scans of IK~Tau and R~Dor were obtained covering the
frequencies between 335--362\,GHz, using four separate observations
each, during August-September 2015. The observations and data
reduction used the ALMA pipeline for calibration, followed by
identifying and imaging the line-free continuum, which was then used
for self-calibration. The solutions were applied to all data and
spectral cubes were made adjusted to constant $v_{\mathrm{LSR}}$. The
process is described in more detail in \citet{Decin2016scan}; here, we concentrate on
the parameters affecting the Al compound cubes. The astrometric
accuracy is $\sim$17\,mas and the flux scale is accurate to
$\sim$7\%. The relative accuracy (e.g., aligning different lines in the
same source) is limited only by the signal-to-noise, S/N, and would be $<$1\,mas for S/N of
200, or 33\,mas (about 1 pixel) for a faint 3$\sigma$ detection of a
compact source. The rms noise ($\sigma_{\rm{rms}}$) varies across the band depending on
intrinsic atmospheric transmission and the weather at the time of
observations. The final image parameters are summarised in
Table~\ref{Table:obs}. Owing to the different elevations as seen from ALMA, we achieved a higher resolution and used a smaller pixel size (30 mas versus 40 mas) for R Dor versus IK Tau. For each line, we made cubes at higher and lower spectral resolution
(see info in Table~\ref{Table:obs}), by using different weighting and spectral
averaging. In addition, for AlO, we further smoothed the wide-channel
data set with a 500\,mas Gaussian kernel. The other lines detected and the sub-mm continuum properties will be discussed in future papers.
\begin{table}
\caption{ALMA observation parameters for the Al-compounds in IK~Tau and R~Dor.}
\label{Table:obs}
\setlength{\tabcolsep}{1mm}
\begin{tabular}{lclll}
\hline \hline
Species & Chan\tablefootmark{a} & Vel\tablefootmark{b} &Beam & $\sigma_{\rm{rms}}$ \\
& (MHz)&(km s$^{-1})$&(mas$\times$mas, P.A.)\tablefootmark{c} & (mJy)\\
\hline
& \multicolumn{4}{c}{IK~Tau}\\
\cmidrule(r){2-5}
Continuum&15.2$\times$10$^3$& &180$\times$160, 27$^{\circ}$&0.047 \\
AlCl &1.95 & 1.7 &160$\times$130, 43$^{\circ}$&1.6 \\
AlCl &5.9 & 5 &200$\times$160, 42$^{\circ}$&1.4 \\
AlOH &1.95 & 1.7 &150$\times$140, $-$22$^{\circ}$&5.0 \\
AlOH &5.9 & 5 &190$\times$170, $-$1$^{\circ}$&2.4 \\
AlO &1.95 & 1.7 &160$\times$130, $-$15$^{\circ}$&5.0 \\
AlO &5.9 & 5 &190$\times$170, $-$20$^{\circ}$&2.6 \\
\hline
& \multicolumn{4}{c}{R~Dor}\\
\cmidrule(r){2-5}
Continuum&11.6$\times$10$^3$& &150$\times$140,--5$^{\circ}$ &0.047 \\
AlCl &0.97 & 0.9 &150$\times$130, 20$^{\circ}$&2.7 \\
AlCl &2.5 & 2.2 &180$\times$180, 10$^{\circ}$&1.4 \\
AlOH &0.97 & 0.9 &170$\times$120, $-$48$^{\circ}$&3.8 \\
AlOH &2.5 & 2.2 &200$\times$160, $-$48$^{\circ}$&2.4 \\
AlO &0.97 & 0.9 &160$\times$130, $-$26$^{\circ}$&4.0 \\
AlO &2.5 & 2.2 &190$\times$160, $-$30$^{\circ}$&1.8 \\
\hline
\end{tabular}
\tablefoot{
\tablefoottext{a}{`Chan' is the channel width for the line cubes, or the total line-free bandwidth
(spread between 335--362\,GHz) for the continuum.}\\
\tablefoottext{b}{`Vel' is the spectral resolution of the image cube.}\\
\tablefoottext{c}{P.A.\ is the position angle measured from North to East.}}
\end{table}
\subsubsection{Al-bearing species detected with ALMA} \label{Sec:Al_detected_ALMA}
Chemical models show that the most likely carriers of aluminium in the wind of oxygen-rich evolved stars are Al, AlH, AlO, AlOH, AlO$_2$, AlCl, Al$_2$, and Al$_2$O \citep[see Fig.~\ref{Fig:chem_Al_radius} in Sect.~\ref{Sec:disc_Al_gas}, a result by][]{Gobrecht2016A&A...585A...6G}. The main isomers of AlO$_2$ and Al$_2$O are linear and their rotational lines are not observable; nor is the homonuclear Al$_2$ easily observable \citep{Cai1991JChPh..95...73C, Koput2004JChPh.121..130K, Kaminski2016A&A...592A..42K}. AlH does not have a rotational line transition in the targeted ALMA frequency range. The other three aluminium-bearing molecules are detected in IK~Tau and R~Dor with ALMA: AlO N=9$-$8 (rest frequency, $\nu_{\rm{rest}}$, at 345.457\,GHz, lower state energy E$_{\rm{low}}$\,=\,66\,K), AlOH J=11$-$10 ($\nu_{\rm{rest}}$\,=\,346.156\,GHz, E$_{\rm{low}}$\,=\,83\,K), and AlCl J=24$-$23 ($\nu_{\rm{rest}}$\,=\,349.444\,GHz, E$_{\rm{low}}$\,=\,193\,K).
The transitions of these three molecules are split up by hyperfine components due to the nuclear spin, $I=5/2$, of aluminium. AlOH and AlCl are closed-shell species with a $^1\Sigma^+$ ground state.
AlO is a radical with a $^2\Sigma$ ground state. The rotational levels of this molecule are therefore split by both fine and hyperfine interactions. The net result is that each rotational transition consists of 10$-$12 favourable, closely spaced, hyperfine components \citep{Tenenbaum2009ApJ...694L..59T, Tenenbaum2010ApJ...712L..93T}.
Figure~\ref{Fig:ALMA_data} shows the flux densities extracted for an aperture size of 320\,mas (300\,mas) and 800\,mas (1000\,mas) for IK~Tau (R~Dor). The channel maps for the individual transitions are shown in Fig.~\ref{Fig:channel_IKTau_AlO}--\ref{Fig:channel_IKTau_AlCl} for IK~Tau and in Fig.~\ref{Fig:channel_RDor_AlO}--\ref{Fig:channel_RDor_AlCl} for R~Dor. Table~\ref{Table:Al_ALMA} gives the measured properties of each molecular transition, including the velocity width, the shift of the centroid w.r.t.\ the LSR velocity of the star, the angular width, and the peak flux and integrated flux within the specified velocity interval. The angular width has been obtained from the zeroth moment (total intensity) by measuring the mean flux in annuli of 60 (80) mas, centred on the position of R Dor (IK Tau), assumed to be the continuum peak. The angular size was taken as the distance out to 2\,$\sigma_{\rm{rms}}$.
The measurements were also used to compute the azimuthally averaged flux densities, in order to compare with 1D models solving the non-LTE radiative transfer equation for the specific molecule (Sect.~\ref{Sec:Results}) and with models solving the chemical network (so-called `forward' chemistry models) by \citet{Gobrecht2016A&A...585A...6G} in Sect.~\ref{Sec:Discussion}. The emission of some molecular transitions shows some irregular morphologies (as described in the next subsections) but there is no obvious preferred direction, so this is a reasonable approximation although not allowing for clumpiness.
\begin{figure}[!htp]
\vspace{0pt}{\includegraphics[angle=0,width=.48\textwidth]{AlO.png}}
\vspace{0pt}{\includegraphics[angle=0,width=.48\textwidth]{AlOH.png}}
\vspace{0pt}{\includegraphics[angle=0,width=.48\textwidth]{AlCl.png}}
\caption{ALMA AlO, AlOH, and AlCl data extracted for an aperture size of 320\,mas (300\,mas) and 800\,mas (1000\,mas) for IK~Tau (R~Dor). The frequency range is shifted to rest frequency using a local standard of rest velocity, $v_{\rm{LSR}}$, of 35\,km/s for IK~Tau and of 7\,km/s for R~Dor. Close to the AlOH emission, NS $^2\Pi_{1/2}$ J=15/2--13/2 transition is detected in IK~Tau (at rest frequency 346.2201\,GHz) and two unidentified lines ('U') in R~Dor at a rest frequencies of 346.123\,GHz and 346.196\,GHz.}
\label{Fig:ALMA_data}
\end{figure}
\begin{table*}
\setlength{\tabcolsep}{1mm}
\caption{Spectral properties of AlO, AlOH, and AlCl as observed with ALMA in IK~Tau (upper part of the table) and R~Dor (lower part). Values for the velocity width, velocity asymmetry, angular width, and peak and integrated flux are only given in the case of a clear detection.}
\label{Table:Al_ALMA}
\begin{tabular}{lrrrrrrrrr}
\hline \hline
\multicolumn{1}{c}{(1)} & \multicolumn{1}{c}{(2)} & \multicolumn{1}{c}{(3)} & \multicolumn{1}{c}{(4)} & \multicolumn{1}{c}{(5)} & \multicolumn{1}{c}{(6)} & \multicolumn{1}{c}{(7)} & \multicolumn{1}{c}{(8)} & \multicolumn{1}{c}{(9)} & \multicolumn{1}{c}{(10)} \\
\multicolumn{1}{c}{Molecule} & \multicolumn{1}{c}{Velocity} & \multicolumn{1}{c}{Velocity\tablefootmark{a}} & \multicolumn{1}{c}{Peak} & \multicolumn{1}{c}{Integrated} & \multicolumn{1}{c}{Velocity} & \multicolumn{1}{c}{Velocity\tablefootmark{a}} & \multicolumn{1}{c}{Peak} & \multicolumn{1}{c}{Integrated} &\multicolumn{1}{c}{Angular} \\
& \multicolumn{1}{c}{width} & \multicolumn{1}{c}{asymmetry} & \multicolumn{1}{c}{flux} & \multicolumn{1}{c}{flux} & \multicolumn{1}{c}{width} & \multicolumn{1}{c}{asymmetry} & \multicolumn{1}{c}{flux} & \multicolumn{1}{c}{flux} & \multicolumn{1}{c}{width} \\
%
& [km s$^{-1}$] & [km s$^{-1}$] & [Jy] & [Jy km s$^{-1}$] & [km s$^{-1}$] & [km s$^{-1}$] & [Jy] & [Jy km s$^{-1}$] & [mas] \\
%
\hline
& \multicolumn{9}{c}{IK~Tau}\\
\cmidrule(r){2-10}
& \multicolumn{4}{c}{Aperture 320\,mas} & \multicolumn{4}{c}{Aperture 800\,mas} \\
\cmidrule(r){2-5}
\cmidrule(l){6-9}
AlO N=9$-$8 & 3.40 & $-$0.32 & 0.036 & 0.247 & 32.30 & 14.98 & 0.186 & 3.211 & 800 \\
AlOH J=11$-$10 & 11.84 & 1.06 & 0.073 & 0.466 & 11.84 & 1.06 & 0.121 & 0.794 & 640 \\
AlCl J=24$-$23 & 6.70 & $-$2.92 & 0.024 & 0.228 & 3.35 & $-$2.92 & 0.040 & 0.255 & 160 \\
\hline
& \multicolumn{9}{c}{R~Dor}\\
\cmidrule(r){2-10}
& \multicolumn{4}{c}{Aperture 300\,mas} & \multicolumn{4}{c}{Aperture 1000\,mas} \\
\cmidrule(r){2-5}
\cmidrule(l){6-9}
AlO N=9$-$8 & 51.00\tablefootmark{b} & 3.67 & 0.128 & 4.032 & 47.60 & 10.47 & 0.166 & 4.570 & 840\\
AlOH J=11$-$10 & 3.38 & $-$0.36 & 0.027 & 0.119 & $-$ & $-$ & $-$ & $-$ & 240 \\
AlCl J=24$-$23 & 3.35 & 2.65 & 0.021 & 0.113 & 9.22 & $-$1.54 & 0.058 & 0.326 & 240 \\
\hline
\end{tabular}
\tablefoot{
\tablefoottext{a}{The shift of the centroid w.r.t.\ the LSR velocity of the star is obtained by dividing the value of the `velocity asymmetry' by two.}\\
\tablefoottext{b}{Large velocity width is caused by the hyperfine structure; see Sect.~\ref{Sec:Results}.}
}
\end{table*}
\subsubsection{IK Tau}\label{Sec:ALMA_IKTau}
\paragraph{AlO:} Looking at Fig.~\ref{Fig:channel_IKTau_AlO}, AlO remains at first sight barely detected in IK~Tau. This is reflected by the line profile for the 320\,mas extraction aperture (Fig.~\ref{Fig:ALMA_data}). However, enlarging the extraction aperture to 800\,mas, a clear signature of AlO emission is seen with a peak flux of 0.19\,Jy and integrated line flux of 3.2\,Jy\,km\,s$^{-1}$.
The SMA observations by \citet{Debeck2017A&A...598A..53D} also show a weak AlO emission feature with an integrated line flux of 2.33\,Jy\,km\,s$^{-1}$ for an aperture of $1\times1$\hbox{$^{\prime\prime}$}.
This points toward faint extended AlO emission that is probably genuinely clumpy. To verify this idea, we have averaged the AlO emission over 5 channels and smoothed the emission with a 0\farcs5 Gaussian kernel; see Fig.~\ref{Fig:channel_IKTau_AlO_smooth}. Albeit the signal-to-noise ratio is still low and we caution that the smoothing is on a scale comparable to the maximum recoverable scale (MRS), this figure indeed hints at extended clumpy emission.
\paragraph{AlOH:} The channel maps of IK~Tau show a clear detection of AlOH with a peak strength around 0.07\,$-$\,0.12\,Jy (depending on the extraction aperture; Fig.~\ref{Fig:channel_IKTau_AlOH}). The emission is centred around the peak of the continuum emission and has an angular width of $\sim$240\,mas.
\paragraph{AlCl:} Around the $v_{\rm{LSR}}$ of $\sim$35\,km/s almost no AlCl emission is seen, but the emission increases in strength at blue-shifted velocities (Fig.~\ref{Fig:channel_IKTau_AlCl}). This is reflected in the asymmetric AlCl line profile in Fig.~\ref{Fig:ALMA_data}.
\begin{figure*}
\includegraphics[width=0.98\textwidth]{IKTAU_ALO.png}
\caption{Channel maps (colour scale in mJy) of the AlO N=9$-$8 transition in IK~Tau. White contours are the continuum at ($-1$, 1, 4, 16, 64, 256)$\times$0.3\,mJy. The ordinate and co-ordinate axis give the offset of the right ascension and declination, respectively, in units of arcseconds. The ALMA beam size is shown in the bottom left corner of the upper panel.}
\label{Fig:channel_IKTau_AlO}
\end{figure*}
\begin{figure*}
\includegraphics[width=0.98\textwidth]{IKTAU_ALO_0_5SMTH.png}
\caption{Same as for Fig.~\ref{Fig:channel_IKTau_AlO}, but the AlO emission is averaged over 5 channels and smoothed with 0\farcs5 Gaussian kernel.}
\label{Fig:channel_IKTau_AlO_smooth}
\end{figure*}
\begin{figure*}
\includegraphics[width=0.98\textwidth]{IKTAU_ALOH.png}
\caption{Same as for Fig.~\ref{Fig:channel_IKTau_AlO}, but for AlOH J=11$-$10.}
\label{Fig:channel_IKTau_AlOH}
\end{figure*}
\begin{figure*}
\includegraphics[width=0.98\textwidth]{IKTAU_ALCL.png}
\caption{Same as for Fig.~\ref{Fig:channel_IKTau_AlO}, but for AlCl J=24$-$23.}
\label{Fig:channel_IKTau_AlCl}
\end{figure*}
\subsubsection{R~Dor}
\paragraph{AlO:} The bright AlO N=9-8 is easily picked up by the ALMA instrument, with a peak flux of $\sim$0.13\,--\,0.17\,Jy for the two different extraction apertures. The emission is smooth and concentrated around the peak of the continuum emission (Fig.~\ref{Fig:channel_RDor_AlO}). At an offset of $\sim$280\,mas to the south-east, enhanced AlO emission is seen in many channel maps (Fig.~\ref{Fig:channel_RDor_AlO_ZOOM}). At the same offset location other molecules \citep[including AlOH, SO, SiO, CO, and SO$_2$; see][]{Decin2016scan} show excess emission as well with the main peak having a blue-shifted frequency offset of 15\,$-$\,20\,MHz.
\paragraph{AlOH:} The AlOH emission in R~Dor is very weak and just above 3\,$\sigma_{\rm{rms}}$ for the 300\,mas extraction aperture (Fig.~\ref{Fig:ALMA_data}). The main emission around $v_{\rm{LSR}}$ is slightly off-center (as compared to the peak of the continuum emission; Fig.~\ref{Fig:channel_RDor_AlOH}). As noted already for AlO, a brighter clump of emission is visible around 280\,mas to the south-east (Fig.~\ref{Fig:channel_RDor_AlOH_zoom}).
\paragraph{AlCl:} Almost no AlCl emission is visible around $v_{\rm{LSR}}$ for the 300\,mas aperture (Fig.~\ref{Fig:channel_RDor_AlCl}), but brighter emission becomes apparent at larger velocities with a peak flux around 0.02\,$-$\,0.06\,Jy and an integrated flux of 0.1\,$-$\,0.3\,Jy\,km\,s$^{-1}$ (depending on the extraction aperture).
\begin{figure*}[htp]
\sidecaption
\includegraphics[width=120mm]{RDOR_ALO.png}
\caption{Channel maps (colour scale in mJy) of the AlO N=9-8 transition in R~Dor. White contours are the continuum at ($-1$, 1, 4, 16, 64, 256)$\times$0.15\,mJy. The ordinate and co-ordinate axis give the offset of the right ascension and declination, respectively, in units of arcseconds. The ALMA beam size is shown in the bottom left corner of the upper panel.}
\label{Fig:channel_RDor_AlO}
\end{figure*}
\begin{figure*}[htp]
\sidecaption
\includegraphics[width=120mm]{RDOR_ALO_ZOOM.png}
\caption{Same as for Fig.~\ref{Fig:channel_RDor_AlO}, but zooming into the AlO emission in the inner 0\farcs2 region.}
\label{Fig:channel_RDor_AlO_ZOOM}
\end{figure*}
\begin{figure*}[htp]
\includegraphics[width=0.98\textwidth]{RDOR_ALOH.png}
\caption{Same as for Fig.~\ref{Fig:channel_RDor_AlO}, but for AlOH J=11$-$10.}
\label{Fig:channel_RDor_AlOH}
\end{figure*}
\begin{figure*}[htp]
\sidecaption
\includegraphics[width=120mm]{RDOR_ALOH_ZOOM.png}
\caption{Same as for Fig.~\ref{Fig:channel_RDor_AlOH}, but zooming into the AlOH emission in the inner 0\farcs2 region.}
\label{Fig:channel_RDor_AlOH_zoom}
\end{figure*}
\begin{figure*}[htp]
\includegraphics[width=0.98\textwidth]{RDOR_ALCL.png}
\caption{Same as for Fig.~\ref{Fig:channel_RDor_AlO}, but for AlCl J=24$-$23.}
\label{Fig:channel_RDor_AlCl}
\end{figure*}
The different velocity offsets for AlCl (on the red-shifted versus blue-shifted sides of the systemic velocity) for R~Dor versus IK~Tau cautions us to consider a potential blend or misidentification of this species.
\begin{figure*}[htp]
\begin{minipage}[t]{.32\textwidth}
\centerline{\resizebox{\textwidth}{!}{\includegraphics[angle=0]{iktau-alo-positions-1.pdf}}}
\end{minipage}
\hfill
\begin{minipage}[t]{.32\textwidth}
\centerline{\resizebox{\textwidth}{!}{\includegraphics[angle=0]{iktau-aloh-positions-1.pdf}}}
\end{minipage}
\hfill
\begin{minipage}[t]{.32\textwidth}
\centerline{\resizebox{\textwidth}{!}{\includegraphics[angle=0]{iktau-alcl-positions-1.pdf}}}
\end{minipage}
\begin{minipage}[t]{.32\textwidth}
\centerline{\resizebox{\textwidth}{!}{\includegraphics[angle=0]{rdor-alo-positions-1.pdf}}}
\end{minipage}
\hfill
\begin{minipage}[t]{.32\textwidth}
\centerline{\resizebox{\textwidth}{!}{\includegraphics[angle=0]{rdor-aloh-positions-1.pdf}}}
\end{minipage}
\hfill
\begin{minipage}[t]{.32\textwidth}
\centerline{\resizebox{\textwidth}{!}{\includegraphics[angle=0]{rdor-alcl-positions-1.pdf}}}
\end{minipage}
\caption{Azimuthally averaged flux density in function of the angular distance from the central star for IK~Tau (top) and R~Dor (bottom), respectively. The ALMA data are shown in red, with the vertical bars denoting the uncertainty. The radiative transfer model predictions are displayed in blue (see Sect.~\ref{Sec:Results}).}
\label{Fig:Al_ang_profile}
\end{figure*}
\subsection{Infrared continuum emission} \label{Sec:continuum}
R~Dor, IK~Tau, and W~Hya have been studied using different spectroscopic, interferometric, polarimetric, and imaging instruments. With the aim to discuss the aluminium fraction locked up into grains, we shortly summarize in this section the results published in the literature that are relevant for this study.
Both R~Dor and W~Hya have been observed with the Short Wavelength Spectrometer (SWS) on board of the Infrared Space Observatory (ISO) using the AOT01 observing mode. R~Dor was observed at so-called speed 1, while for W~Hya both speed 1 and speed 3 were used. The spectral resolution for the speed 3 data is around 700--1000, and is a factor of $\sim$2 worse for the speed 1 data. Interferences in the instrument cause however fringes in the ISO band~3 data (12-29\,$\mu$m), particularly in the case of the higher resolution observations.
Unfortunately, no ISO-SWS data are available for IK~Tau, the only infrared spectrum available being IRAS-LRS data (see Fig.~\ref{Fig:SED_RDor_IKTau_WHya}).
\begin{figure}[htp]
\begin{center}
\includegraphics[angle=0,width=0.47\textwidth]{SED_dust_IKTau_RDor_WHya.png}
\end{center}
\caption{Infrared spectrum of R~Dor (black, full line), W~Hya (grey, full line, scaled with a factor 1.5) and IK~Tau (red, dashed line). Spectral features arising from dust species such as crystalline and amorphous Al$_2$O$_3$, gehlenite (Ca$_2$Al$_2$SiO$_7$), and so-called astronomical silicates \citep[amorphous olivines with different relative magnesium and iron fractions;][]{deVries2010A&A...516A..86D} are indicated.}
\label{Fig:SED_RDor_IKTau_WHya}
\end{figure}
SPHERE/ZIMPOL data were recently obtained for R~Dor and W~Hya \citep{Khouri2016A&A...591A..70K, Ohnaka2016A&A...589A..91O} confirming the earlier result by \citet{Norris2012Natur.484..220N} that a halo of large transparent grains is present at $\sim$1.5--3\,\Rstar. Estimated grain sizes are around 0.3-0.5\,\hbox{$\mu$m}\ and Fe-free silicates or \hbox{Al$_2$O$_3$}\ are put forward as the possible dust species that produce the scattered light.
\citet{Zhao2011A&A...530A.120Z} monitored W~Hya with MIDI/VLTI (8--13\,\hbox{$\mu$m}, spectral resolution of 30) during nearly three pulsation cycles. From the spectrally dispersed visibility data, the authors concluded that the Fe-rich silicate emission seen around 10\,\hbox{$\mu$m}\ in the ISO-SWS data arises from a region fairly far away from the star, at radii larger than 30 times the photospheric radius. Using image reconstruction techniques, \citet{Zhao2015PASP..127..732Z} were able to spatially resolve the dust formation region close to the star and deduced that the data signal the presence of a dust layer of presumably amorphous aluminium oxide (\hbox{Al$_2$O$_3$}) at $\sim$2\,\Rstar. The observations suggest that the formation of amorphous \hbox{Al$_2$O$_3$}\ occurs mainly around or after visual minimum \citep{Zhao2012A&A...545A..56Z}.
VLTI/MIDI data are also available for R~Dor. However the fringe-track data do not contain any interferometric signal. Probably the beams were not overlapped correctly, as the data were taken for test purposes. No MIDI data are available for IK~Tau.
To derive the dust chemical composition and shed light on the mass-loss mechanism, the SED of W~Hya together with infrared images and optical scattered light fractions were analysed by \citet{Khouri2015A&A...577A.114K}. The only way to explain all observed signatures, and in particular the amorphous Al$_2$O$_3$ emission in the SED, was by implementing in their radiative transfer model a so-called gravitationally bound dust shell (GBDS) close to the star at around 1.5 to 3 stellar radii. This GBDS is a shell of dust particles for which the ratio of the radiation pressure on the grains to the gravitational attraction (often called the $\Gamma$-factor) is still smaller than 1 so that the dust particles can reside close to the star without being pushed outward. This allows the material to potentially grow in size, and only once their size is larger than 0.1-0.3\,\hbox{$\mu$m}, $\Gamma$ can become larger than 1. This idea of a GBDS was recently shown to fit well with the theoretical dynamical wind models of \citet{Hofner2016arXiv160509730H}, who allowed for the formation of composite grains with an Al$_2$O$_3$ core and a silicate mantle at slightly larger distances than the Al$_2$O$_3$ GBDS shell so to give the wind a kick start. The location of the GBDS coincides with the region where large transparent grains are detected in polarized light images \citep{Norris2012Natur.484..220N, Ohnaka2016A&A...589A..91O}, albeit this does not imply by necessity that the species responsible for the \hbox{Al$_2$O$_3$}\ emission in the SED are the same ones that cause the stellar light to be scattered. Thus, it is still possible that the scattering agents are Fe-free silicates, since these can remain fairly cold, and so not produce signature in the thermal infrared.
The poor quality of the infrared continuum data of IK Tau led us to decide not to analyse the dust content in this object, but to focus on the dust composition of R~Dor and W~Hya.
We combined the available IR data for the latter two objects and subtracted the stellar continuum (represented as a black-body with temperature of 3000\,K for R~Dor and of 2500\,K for W~Hya) in order to visualise better their dust spectral features; see Fig.~\ref{Fig:SED_RDor_WHya}. The emission bands of some dust aluminium bearing dust species are plotted, as well as some CO$_2$ and H$_2$O molecular emission features.
\begin{figure*}[htp]
\centering\includegraphics[angle=270, width=0.8\textwidth]{RDor_grains.pdf}
\caption{Continuum subtracted and normalized SED of R~Dor and W~Hya (in full black and gray lines, respectively). Characteristic (scaled) emission patterns of different dust species are shown in the coloured dashed lines. The gray vertical dotted lines indicate CO$_2$ molecular emission \citep{Justtanont1998A&A...330L..17J, Ryde1999A&A...341..579R}.}
\label{Fig:SED_RDor_WHya}
\end{figure*}
\section{Results: Abundances of aluminium-bearing molecules} \label{Sec:Results}
To establish the abundance and distribution of AlO, AlOH, and AlCl, we have modelled the ALMA data using a non-LTE radiative transfer model of the circumstellar envelope (CSE).
We use a code based on the 'Accelerated Lambda Iteration' (ALI) method \citep{Maercker2008A&A...479..779M}, which allows us to retrieve the global mean molecular density by assuming a 1D geometry.
Collisional excitation rates have not been published for any of these three molecules. For that reason, we have used the values for the HCN-H$_2$ system \citep{Green1974ApJ...191..653G} as substitute for AlOH \citep[cf.][]{Tenenbaum2010ApJ...712L..93T}, and of SiO-H$_2$ \citep{Dayou2006A&A...459..297D} for AlO and AlCl \citep[cf.][]{Kaminski2016A&A...592A..42K}, scaling for the difference in molecular weight. The collisional rates are used in the form they appear in the LAMDA database\footnote{http://home.strw.leidenuniv.nl/$\sim$moldata/} \citep{Schoier2005A&A...432..369S}. To check for the dependency on collisional excitation rates, we also ran models using the CS-H$_2$ cross sections \citep{Turner1992ApJ...399..114T} and found no significant difference in results.
The first 30 rotational levels in the ground vibrational state were considered for AlOH and the first 40 ones for AlO and AlCl. Our treatment of the radiative transfer does not deal with the (hyper)fine structure components, and is therefore limited to the prediction of rotational lines $N\rightarrow N^\prime$ or $J\rightarrow J^\prime$ with $\Delta\,N$ or $\Delta\,J$ equal to 1. Therefore, all levels treated are described by one quantum number, $N$ or $J$. The final line profile is computed by splitting the predicted total intensity over the different (hyper)fine components according to their relative quantum-mechanical line strength, $S$, as given in CDMS\footnote{https://www.astro.uni-koeln.de/cdms/catalog} \citep{Muller2001A&A...370L..49M, Muller2005JMoSt.742..215M}. This approach is only valid under LTE conditions.
For an angular diameter of 20\,mas and a distance of 260\,pc, the stellar radius of IK~Tau is $\sim$3.8$\times$10$^{13}$\,cm, while the angular diameter of 57\,mas and the distance of 60\,pc translates into a stellar radius of $\sim$2.5$\times$10$^{13}$\,cm for R~Dor. The gas kinetic temperature is assumed to follow a power law
\begin{equation}
T_{\rm{gas}} = T_{\rm{\star}}\left(\frac{\Rstar}{r}\right)^{-\epsilon}\,,
\end{equation}
and the gas velocity profile is parametrized using a $\beta$-type law
\begin{equation}
v_{\rm{gas}} = v_0 + (v_\infty - v_0)\left(1 - \frac{r_0}{r}\right)^\beta\,,
\end{equation}
with $r$ the distance to the star, $v_\infty$ the terminal velocity, and $v_0$ the velocity at radius $r_0$ at which the wind is launched and which is set to the local sound speed. The gas velocity within the dust condensation radius, $r<r_0$, is assumed to be 2\,km/s. The turbulent velocity for both stars is assumed to be 1\,km/s. The gas density, $\rho(r)$, is calculated from the equation of mass conservation
\begin{equation}
\Mdot = 4 \pi r^2 v(r) \rho(r)\,.
\end{equation}
The main input parameters for R~Dor and IK~Tau are listed in Table~\ref{Table:input_RT}, following \citet{Maercker2016A&A...591A..44M}. These properties are simplified estimates, for example the velocity structure inside the dust formation radius is complex, but are adequate for 1D modelling in the regions where these Al compounds have been observed.
For all molecular species, the fractional abundance structure with respect to H$_2$ was varied until a fit with the ALMA azimuthally averaged flux density was achieved (see Fig.~\ref{Fig:Al_ang_profile}). In some cases, the fractional abundance declines according to a Gaussian profile centred on the star, and hence is described by following equation
\begin{equation}
f(r) = f_0 \exp\left(-\left(\frac{r}{r_e}\right)^2\right)\,,
\end{equation}
with $f_0$ the initial abundance and $r_e$ the $e$-folding radius.
\begin{table}[htp]
\begin{center}
\caption{Input parameters for the non-LTE radiative transfer modeling}
\label{Table:input_RT}
\begin{tabular}{lrr}
\hline
& R Dor & IK Tau \\
\hline
\Tstar [K] & 2400 & 2100 \\
$\epsilon$ & 0.65 & 0.6 \\
$v_\infty$ [km/s] & 5.7 & 17.5 \\
$\beta$ & 1.5 & 1.2\\
$r_0$ [cm] & 5.3$\times10^{13}$ & 2.38$\times10^{14}$ \\
\Mdot [\Msun/yr] & 1.6$\times10^{-7}$ & 5$\times10^{-6}$ \\
L [\Lsun] & 6500 & 7700 \\
\hline
\end{tabular}
\end{center}
\end{table}
The retrieved fractional abundance structures are shown in Fig.~\ref{Fig:retrieved}; the fit to the ALMA data in Fig.~\ref{Fig:Al_ang_profile}. Clearly both stars show a very different Al-chemistry. For IK~Tau, AlOH could be modelled assuming a Gaussian distribution with a peak central abundance of $4.4\times10^{-9}$ and an $e$-folding radius, $r_e$, of 22\,\Rstar. As could be deduced already from Fig.~\ref{Fig:ALMA_data} and Fig.~\ref{Fig:channel_IKTau_AlO_smooth}, the AlO emission might be clumpy not showing a centrally peaked abundance structure. The data could be modelled assuming a constant abundance of 4.4$\times$10$^{-8}$ from 50--160\,\Rstar. If the AlCl identification is correct (see Sect.~\ref{Sec:ALMA_observations}), then the data are compatible with a Gaussian distribution with peak central abundance of $9\times10^{-10}$ and $r_e$ of 40\,\Rstar.
The AlOH emission in R~Dor can be approximated with a constant slab with abundance of 1.4$\times$10$^{-9}$ from the stellar surface out to 12\,\Rstar. The AlO and AlCl distribution in R~Dor follow a Gaussian distribution with peak central abundance of $8.4\times10^{-8}$ and $2.5\times10^{-8}$ and $e$-folding radius of 3.8\,\Rstar\ and 2.4\,\Rstar, respectively. The good correspondence between the observed and predicted emission of AlO in R~Dor is shown in Fig.~\ref{Fig:RDor_AlO_spectra_model}.
While we can trace the fractional abundance above $1\times10^{-9}$ within $\sim$150\,\Rstar\ for IK Tau, the ALMA Al-data of R~Dor only cover the first 25\,\Rstar.
The factor $\sim$6 difference in spatial extent is in line with the factor $\sim$50 in mass-loss rate (and hence sensitivity to some particle density assuming a density law which is proportional to $r^{-2}$).
\begin{figure*}[htp]
\begin{minipage}[t]{.48\textwidth}
\centerline{\resizebox{\textwidth}{!}{\includegraphics[angle=0]{IKTauAlabundancesRstarThick.pdf}}}
\end{minipage}
\hfill
\begin{minipage}[t]{.48\textwidth}
\centerline{\resizebox{\textwidth}{!}{\includegraphics[angle=0]{RDorAlabundancesRstarThick.pdf}}}
\end{minipage}
\caption{Retrieved fractional abundance structure [AlO/H$_2$], [AlOH/H$_2$], and [AlCl/H$_2$] for IK~Tau (left) and R~Dor (right).}
\label{Fig:retrieved}
\end{figure*}
\begin{figure*}[htp]
\begin{minipage}[t]{.48\textwidth}
\centerline{\resizebox{\textwidth}{!}{\includegraphics[angle=0]{RDorAlO300masSpecModel.pdf}}}
\end{minipage}
\hfill
\begin{minipage}[t]{.48\textwidth}
\centerline{\resizebox{\textwidth}{!}{\includegraphics[angle=0]{RDorAlO1000masSpecModel.pdf}}}
\end{minipage}
\caption{Comparison between the extracted ALMA AlO spectra of R~Dor (green) and the predicted line profile (red) for an AlO abundance structure as given in Fig.~\ref{Fig:retrieved} for the 300\,mas (left) and 1000\,mas (right) extraction aperture.}
\label{Fig:RDor_AlO_spectra_model}
\end{figure*}
Assuming a solar aluminium content of Al/H = 3$\times$10$^{-6}$, our results imply that for both stars only a small fraction of aluminium ($\mathrel{\mathchoice {\vcenter{\offinterlineskip\halign{\hfil$\displaystyle##$\hfil\cr<\cr\sim\cr}}$2\%) is locked up in AlO, AlOH, and AlCl.
\section{Discussion} \label{Sec:Discussion}
The analysis of the ALMA data leads to a direct determination of the mean fractional abundances of AlO, AlOH, and AlCl in the inner winds of R~Dor and IK~Tau. These results can be compared quantitatively to the theoretical model predictions solving the chemical network for IK~Tau by \citet{Gobrecht2016A&A...585A...6G}, and hence bring about the first qualitative assessment of this kind of dynamical-chemical models (Sect.~\ref{Sec:disc_Al_gas}). The amount of aluminium locked up in gaseous species turns out to be low, leaving ample of room for aluminium to be incorporated into dust grains. However, as we will discuss in Sect.~\ref{Sec:disc_Al_dust}, the estimated grain temperatures for aluminium oxide (\hbox{Al$_2$O$_3$}) at $\sim$2\,\Rstar\ is high suggesting that the grains are annealed and hence have a crystalline lattice structure. This outcome implies a tension in the interpretation of the 11\,\hbox{$\mu$m}\ spectral feature, present in the SED and interferometric data. We hypothesize that large gas-phase aluminium oxide clusters are the cause of the 11\,\hbox{$\mu$m}\ feature, and explain how these large clusters can be formed (Sect.~\ref{Sec:disc_Al_clusters}).
\subsection{The gas-phase aluminium budget} \label{Sec:disc_Al_gas}
\begin{figure}[htp]
\begin{minipage}[t]{.4\textwidth}
\centerline{\resizebox{\textwidth}{!}{\includegraphics{F4_ph0leen.png}}}
\end{minipage}
\hfill
\begin{minipage}[t]{.4\textwidth}
\centerline{\resizebox{\textwidth}{!}{\includegraphics{F4_ph0p5leen.png}}}
\end{minipage}
\caption{Predicted abundances of aluminium-bearing molecules with respect to the total gas number density as a function of radius. Chemical models are based on the pulsation-induced shock dynamics presented by \citet{Gobrecht2016A&A...585A...6G}. Model simulations for a star resembling IK~Tau at phase $\phi$=0.0 (top) and $\phi=0.5$ (bottom).}
\label{Fig:chem_Al_radius}
\end{figure}
\begin{figure}[htp]
\begin{minipage}[t]{.4\textwidth}
\centerline{\resizebox{\textwidth}{!}{\includegraphics{F6_1Rleen.png}}}
\end{minipage}
\hfill
\begin{minipage}[t]{.4\textwidth}
\centerline{\resizebox{\textwidth}{!}{\includegraphics{F6_3Rleen.png}}}
\end{minipage}
\hfill
\begin{minipage}[t]{.4\textwidth}
\centerline{\resizebox{\textwidth}{!}{\includegraphics{F6_5Rleen.png}}}
\end{minipage}
\caption{Predicted abundances of aluminium-bearing molecules with respect to the total gas number density as a function of phase $\phi$ for three specific radii (top: 1\,\Rstar, middle: 3\,\Rstar, bottom: 5\,\Rstar). Chemical models are based on the pulsation-induced shock dynamics presented by \citet{Gobrecht2016A&A...585A...6G} for a star resembling IK~Tau.}
\label{Fig:chem_Al_phase}
\end{figure}
The deduced AlO, AlOH, and AlCl abundances of IK~Tau can be compared directly to the chemical predictions by \citet{Gobrecht2016A&A...585A...6G}. They modelled the synthesis of molecules and dust in the inner wind of IK~Tau ($r<5$\,\Rstar) by considering the effect of periodic shocks induced by the stellar pulsation. The non-equilibrium chemistry in the shocked gas-layers was followed between 1\,\Rstar\ and 10\,\Rstar\ by solving a chemical kinetic network including 100 species and 424 reactions. The condensation of dust is described by a Brownian formalism using the simplifying assumption that the gaseous dimers of forsterite (Mg$_2$SiO$_4$), enstatite (MgSiO$_3$) and aluminia (\hbox{Al$_2$O$_3$}) are the initial dust seeds and considering coalescence and coagulation of these dimers to form the dust grains.
The aluminium species in the network of \citet{Gobrecht2016A&A...585A...6G} include Al, AlH, AlO, AlOH, AlO$_2$, AlCl, Al$_2$, and Al$_2$O. The aluminium dust precursors are described by Al$_2$O$_2$, Al$_2$O$_3$, and Al$_4$O$_6$. The predicted abundances of the aluminium species as function of radius and phase $\phi$\footnote{The pulsation phase is defined as the decimal part of $(t-T_0)/P$, with $T_0$ the epoch of the start of the pulsation cycle and $P$ the period.} are shown in Fig.~\ref{Fig:chem_Al_radius} and Fig.~\ref{Fig:chem_Al_phase}, respectively. We note that Al$_2$ and Al$_2$O have very low abundances and therefore do not appear in the plots.
The plots in Fig.~\ref{Fig:chem_Al_radius} can be interpreted as a snapshot of the inner wind just after the passage of a shock (at phase $\phi=0.0$) and at half of the pulsation cycle (phase $\phi=0.5$), assuming shock fronts every 0.5\,\Rstar. A full pulsation cycle corresponds to phase 1.0. Phase 0.0 corresponds also to a snapshot after a full pulsation cycle. However, at this point, the gas densities have been rescaled from the previous radial position and just have been shocked again at the new radial position \citep[see description in][]{Gobrecht2016A&A...585A...6G}.
For a period of 470~days and with $T_0$ at JD$\sim$2453025 \citep{Matsumoto2008PASJ...60.1039M}, the ALMA data of IK~Tau are obtained around $\phi = 0.98$.
The methodology applied by \citet{Gobrecht2016A&A...585A...6G} implies that a \textit{relative} comparison between the retrieved and predicted molecular abundance stratifications is best to be done using the first panel of Fig.~\ref{Fig:chem_Al_radius}, while the absolute scaling can be seen in Fig.~\ref{Fig:chem_Al_phase}. Taking the predicted abundances at 5\,\Rstar\ (bottom panel in Fig.~\ref{Fig:chem_Al_phase}) at phase $\phi=0.98$ as representative values for the comparison with the profiles retrieved from the ALMA data (left panel in Fig.~\ref{Fig:retrieved}), we can see that both the observations and chemical models predict a higher abundance for AlOH than for AlO. However, the retrieved AlOH abundance at 5\,\Rstar\ is 2 orders of magnitudes lower than predicted. The ALMA data indeed show that the fractional AlO abundance ($r<50$\,\Rstar) is very low ($<10^{-10}$) in accord with the predictions, but remarkably we see that an active Al-chemistry is taking place further out in the wind yielding an increase in AlO emission. We derive for IK~Tau a mean AlO fractional abundance of $\sim$5$\times$10$^{-8}$ between 50--150\,\Rstar, with the emission mainly arising from clumps at that radial distance. The increase in AlO abundance coincides with a strong decrease in AlOH abundance suggesting that AlOH is converted to AlO. Another possibility is that desorption or sputtering release AlO from grains back into its gaseous form. If no line-blend is blurring the AlCl identification, then the ALMA data show that the AlCl abundance in IK~Tau is a factor $\sim$40 higher than predicted.
The stellar angular diameter of IK~Tau is $\theta_D\sim20$\,mas \citep{Decin2010A&A...516A..69D}. The inner radius of dust detected by \citet{Dehaes2007MNRAS.377..931D} and inferred by \citet{Richards2012A&A...546A..16R} is at 6--10\,$R_{\star}$, but these observations were not sensitive to the potential presence of transparent grains closer to the star. The ALMA beam of $\sim$0\farcs12$\times$0\farcs15 is hence too large to confirm the predicted strong radial variation in molecular abundance within 5\,\Rstar\ (as visible in Fig.~\ref{Fig:chem_Al_radius}). The angular diameter of R~Dor is much larger, $\theta_D\sim57$\,mas \citep{Bedding1997MNRAS.286..957B, Norris2012Natur.484..220N, Khouri2016A&A...591A..70K}. Although a direct comparison with Fig.~\ref{Fig:chem_Al_radius}--\ref{Fig:chem_Al_phase} should be done with care --- since the shock velocities are expected to be smaller for R~Dor --- our ALMA data do not corroborate any strong radial variation in AlO and AlOH abundance within the first 10 stellar radii. In addition, the AlCl abundance decreases with radial distance in contrast to the chemical network predictions.
The resolution of our ALMA observations does allow us to trace the molecular abundances at the smallest radii at which dust has been detected (being around 6–-10\,$R_{\star}$ for IK~Tau and around 2\,$R_{\star}$ for R~Dor). AlO and AlOH are directly associated with the nucleation of alumina and their retrieved abundances are indeed much lower than in the pure gas-phase models of \citet{Gobrecht2016A&A...585A...6G}. But clearly, not all available AlO and AlOH molecules participate in that dust formation process. In addition, the ALMA results on R~Dor prove that a second phase of dust formation, proposed to occur slightly beyond the GBDS, is (almost) not consuming AlO. This supports the scenario suggested by \cite{Hofner2016arXiv160509730H} that the growth of silicates (potentially on top of already existing alumina grains) is a trigger for the wind to get launched.
It would be of interest to monitor the changes in abundance structure in function of the pulsation phase and confront them with the predictions shown in Fig.~\ref{Fig:chem_Al_phase}. This kind of data could then be used to benchmark and improve current forward chemistry models that implement gas dynamics. We note that \citet{Kaminski2016A&A...592A..42K} have reported some variability in the AlO emission of the low mass-loss rate Mira star $o$~Cet, but currently no link with the pulsation cycle is apparent.
Although the current comparison points toward improvements to be made in the chemical model predictions, a conclusion from the ALMA data in combination with these models is that the fraction of aluminium that is locked up in simple aluminium-bearing molecules seems small. I.e., the combined fractional abundance of Al-molecules which are thought to form easily in the inner wind region is only $\sim$5$\times$10$^{-8}$. Even more, since the formation of the more complex \hbox{Al$_2$O$_3$}\ and its dimer (\hbox{Al$_2$O$_3$})$_2$ is described by termolecular recombination and hence requires high densities, these higher order molecules will also only consume a tiny fraction of the available aluminium (see Fig.~\ref{Fig:chem_Al_phase}, where one can see that the alumina dimers only form at 1\,\Rstar\ and at late phases); i.e.\ these larger molecules too can be neglected as major sink of aluminium.
The most difficult gaseous aluminium form to constrain is that of atomic aluminium. The atomic aluminium abundance is predicted to be high when the parcel of material is close to the star ($r<2.5$\,\Rstar) for phases $\phi>$0.2 (see Fig.~\ref{Fig:chem_Al_phase}). Exactly at the shock front, phase $\phi=0$, the high temperature ensures a high Al abundance, but as a consequence of the declining temperature for later phases the atomic Al abundance decreases rapdily. Beyond 3\,\Rstar, one can note a transformation of molecular in atomic Al due to the decreasing gas densities with distance from the star. We have performed an in-depth study of the high spectral resolution optical Mercator/Hermes \citep{Raskin2011A&A...526A..69R} data of IK~Tau and W~Hya (no observations are available for R~Dor) to check for the presence of atomic Al in the outer atmosphere/inner wind of both stars. Molecular line veiling complicates the analysis of the data. We could not identify either absorption or emission atomic Al lines. This does not imply that no atomic Al is present, but at least the quantity is too low for it to be detected amongst the molecular lines. An analogous conclusion was drawn by \citet{Kaminski2016A&A...592A..42K} for the low mass-loss rate AGB star $o$~Cet, for which reason they concluded that Al is locked in molecules and dust.
This ensemble of considerations supports the conclusion that only a minor fraction of Al resides in Al-bearing molecules and that the atomic Al abundance is difficult to constrain. An important outcome from these ALMA data is that the aluminium condensation process, considered to be the first condensation cycle to start in AGB stars \citep{Tielens1990fmpn.coll..186T}, is for sure not fully efficient since the gaseous precursors of the aluminium grains are detected in and beyond the dust formation region.
\subsection{Aluminium depletion into grains} \label{Sec:disc_Al_dust}
Presolar grains prove that aluminium oxide grains are formed in oxygen-rich AGB stars. The circumstellar \hbox{Al$_2$O$_3$}-grains are relatively abundant in meteorites \citep{Nittler1997ApJ...483..475N}, albeit perhaps not as abundant as one would expect if all aluminium would condense in these grains \citep{Alexander1997AIPC..402..567A}. This result from meteorite studies now gets some support by our analysis of the ALMA data, confirming that a fraction of the aluminium is still present in its gaseous form after its passage through the dust formation region.
At least for a few silicate grains, the meteoretic studies show that the aluminium oxides are at the core of the silicate grain suggesting that \hbox{Al$_2$O$_3$}\ is the first solid to form in the CSE \citep{Nittler2008ApJ...682.1450N, Vollmer2006LPI....37.1284V}. Both crystalline and amorphous \hbox{Al$_2$O$_3$}\ enclosures have been detected \citep{Stroud2004Sci...305.1455S}. The variation in the crystal structure and Ti content of these presolar grains demonstrate that \hbox{Al$_2$O$_3$}\ can condense in the absence of TiO$_2$ seed clusters but that Ti may be important in determining the lattice structure.
The SED analysis of R~Dor and W~Hya turned out to be challenging in terms of the aluminium content \citep{Khouri2015A&A...577A.114K, KhouriPhD}. Based on constraints from
gas-phase models and solar abundances, Khouri et al.\ were unable to model the SEDs of both stars in the spectral region around 11\,\hbox{$\mu$m}\ when considering aluminium condensates only to be present in the stellar outflow (see dotted line in Fig.~\ref{Fig:RDor_GBDS}). One can fit the broad 11\,\hbox{$\mu$m}\ feature in the SED by invoking the presence of a gravitational bound dust shell (GBDS; see Sect.~\ref{Sec:continuum} for a more detailed explanation on this terminology) of amorphous \hbox{Al$_2$O$_3$}\ grains at $\sim$2\,\Rstar\ (see dashed line in Fig.~\ref{Fig:RDor_GBDS}). The GBDS can contain both small and large \hbox{Al$_2$O$_3$}\ grains. The model for R~Dor (W~Hya) requires an Al$_2$O$_3$ dust mass in the GBDS of $\sim 5.7 \times 10^{-10}$~M$_\odot$ ($\sim 1.5 \times 10^{-9}$~M$_\odot$), which would correspond to a gas mass of $7.5 \times 10^{-6}$~M$_\odot$ ($2 \times 10^{-5}$~M$_\odot$) if one would assume full aluminium condensation.
If this is compliant (or not) with the total aluminium content is difficult to assess since the region where the GBDS resides is known to be highly variable in density due to the stellar pulsations, with variations up to one order of magnitude \citep{Hofner2016arXiv160509730H}.
\begin{figure}[htp]
\includegraphics[width=0.48\textwidth]{Fig_RDor+BB+Al2O3-GBDS+wind.pdf}
\caption{Infrared SED of R~Dor. ISO data are shown in red; the blackbody represents the stellar emission at a temperature of 3000\,K, the dotted line is model assuming a 100\% condensation of aluminium into amorphous \hbox{Al$_2$O$_3$}\ that resides in the stellar outflow, while the dashed line represents a model fit for amorphous \hbox{Al$_2$O$_3$}\ being present in the so-called GBDS and in the stellar outflow.
}
\label{Fig:RDor_GBDS}
\end{figure}
Polarized light data confirm also the presence of large grains close to W~Hya and R~Dor at 1.5--3\,\Rstar\ \citep{Norris2012Natur.484..220N, Khouri2016A&A...591A..70K, Ohnaka2016A&A...589A..91O}. From the scattered light data, one can not determine the composition of the grains, and both aluminium oxide grains and cold Fe-free silicates are suggested to be good candidates due to their glassy character (Sect.~\ref{Sec:continuum}).
The GBDS results \citep{KhouriPhD, Khouri2015A&A...577A.114K}, the scattered light data \citep{Norris2012Natur.484..220N, Khouri2016A&A...591A..70K, Ohnaka2016A&A...589A..91O}, the interferometric results \citep{Zhao2015PASP..127..732Z}, and the theoretical models \citep{Hofner2016arXiv160509730H} build up evidence for a shell of aluminium oxide grains at 1.5 -- 3\,\Rstar. The kinetic temperature in that region is around 1500--2000\,K. Optical constants are measured for amorphous \hbox{Al$_2$O$_3$}\ grains, but only for wavelengths beyond 0.2\,\hbox{$\mu$m}\ \citep{Koike1995Icar..114..203K} or beyond 7.8\,\hbox{$\mu$m}\ \citep{Begemann1997ApJ...476..199B}. Depending on the replica used for the short-wavelength range and under the assumption of thermodynamic equilibrium, the estimated temperature of the amorphous aluminina grains at that spatial region would be around 1200--1600\,K \citep{Khouri2015A&A...577A.114K}. However, amorphous alumina can only exist below 1000\,K \citep{Levin1998PSSAR.166..197L, Levin2005}. At temperatures around 1500\,K, crystalline $\alpha$-alumina can form, while below 1300\,K crystalline $\gamma$-alumina can exist. With the grain temperature in the 1.5--3\,\Rstar\ region being higher than the annealing temperature, this would imply a structural transformation of \hbox{Al$_2$O$_3$}\ to its crystalline form. The thermal infrared spectrum of R~Dor and W~Hya is not compatible with the presence of \hbox{Al$_2$O$_3$}\ grains with a crystalline $\gamma$-lattice structure (see Fig.~\ref{Fig:SED_RDor_WHya}). As shown in Fig.~\ref{Fig:SED_RDor_WHya} crystalline $\alpha$-\hbox{Al$_2$O$_3$}\ can indeed explain the 13\,\hbox{$\mu$m}\ feature in the SED. A crystallinity degree of 5\% can explain the 13\,\hbox{$\mu$m}\ feature in W~Hya, while for R~Dor the estimated crystallinity degree is around 12\%. These crystalline grains might be the scattering agents of the polarized light emission.
However, this questions how to explain the broad 11\,\hbox{$\mu$m}\ feature in the SED of R~Dor and W~Hya and, in particular, the MIDI data of W~Hya which clearly show that emission arises very close to the star. A potential answer to that question is that the 11\,\hbox{$\mu$m}\ feature does \textit{not} signal the presence of solid-state amorphous aluminium oxide, but --- to the contrary --- points towards the existence of \textit{large neutral gas-phase aluminium oxide clusters} (see Sect.~\ref{Sec:disc_Al_clusters}). Individual (Al$_2$O$_3$)$_n$ clusters may exist above 1500\,K. We note that our analysis does not exclude the existence of amorphous aluminium oxides in the wind, but only at larger distances from the star where temperatures are cooler.
\subsection{Constraining the presence of aluminium oxide clusters} \label{Sec:disc_Al_clusters}
If amorphous alumina dust grains are not the carrier of the MIDI and ISO spectral signature at 11\,$\mu$m, then the interesting possibility exists that large aluminium oxide clusters are present, and detected, just above the stellar surface. Indeed, an outcome of the experimental IR-REMPI studies by \citet{vanHeijnsbergen2003PCCP....5.2515V} is that large AlO$\cdot$(\hbox{Al$_2$O$_3$})$_n$ clusters ($n>34$) exhibit a spectral signature that is very similar to that of amorphous \hbox{Al$_2$O$_3$}, while smaller AlO$\cdot$(\hbox{Al$_2$O$_3$})$_n$ clusters (around $n=15$) show the best match with that of crystalline $\gamma$-\hbox{Al$_2$O$_3$}\ (see Fig.~\ref{Fig:VanHeijnsbergen}). Due to the non-linear regime in which the IR-REMPI results are obtained, one can however not estimate the amount of clusters present in the beam. This renders the impossibility of quantifying the amount of large aluminium oxide clusters that would be needed in order to explain the 11\,\hbox{$\mu$m}\ feature.
\begin{figure}[htp]
\centering\includegraphics[width=0.4\textwidth]{HeijnsbergenThesisAl2O3.pdf}
\caption{Comparison of gas-phase vibrational spectra for AlO$\cdot$(\hbox{Al$_2$O$_3$})$_n$ with $n=28$ (A) and for $n=53$ (D) to the infrared absorption spectra of bulk aluminium oxide lattices (B = crystalline $\gamma$-\hbox{Al$_2$O$_3$}; C = crystalline $\alpha$-\hbox{Al$_2$O$_3$}; E = amorphous \hbox{Al$_2$O$_3$}). Result obtained by \citet{vanHeijnsbergen2003PCCP....5.2515V}.}
\label{Fig:VanHeijnsbergen}
\end{figure}
If the 11\,\hbox{$\mu$m}\ feature marks the presence of large ($n > 34$) aluminium oxide clusters, then one should wonder if smaller clusters are present as well and if they can be detected. The experiments by \citet{vanHeijnsbergen2003PCCP....5.2515V} and \citet{Demyk2004A&A...420..547D} show that clusters with $n=15-40$ show two vibrational bands, one around 11.8\,\hbox{$\mu$m}\ and another one around 15\,\hbox{$\mu$m}. Comparing this to the emission features of solid-state aluminium oxide lattices identifies these gas phase clusters to have a structure similar to crystalline $\gamma$-\hbox{Al$_2$O$_3$}\ instead of $\alpha$-\hbox{Al$_2$O$_3$} (corundum), which is the thermodynamically most stable form of the bulk (see Fig.~\ref{Fig:VanHeijnsbergen}). The infrared observations of W~Hya and R~Dor do not display an obvious emission component around 15\,\hbox{$\mu$m}. This does not imply by necessity that these $n=15-40$ sized clusters are not present, but at least they are not abundant enough to leave any spectral imprint.
Due to variety of processes prevalent in the vast interstellar space such as collisions, condensation, coagulation, coalescence the size distribution of interstellar dust grains is dominated by small particles, i.e.\ $n_d(a)\,da \propto a^{-7/2}\,da$ \citep{Mathis1977ApJ...217..425M}. Also in our own Earth's atmosphere, fresh aerosols which are created in situ from the gas-phase by nucleation display a number distribution heavily skewed towards the very small particles \citep{Seinfeld1998PhT....51j..88S}. In analogy, one might expect a significantly larger fraction of very small aluminium oxide clusters as compared to large clusters in the stellar wind. Current capacities in quantum chemical computations give us the possibility to test this proposition on the prevailing presence of very small clusters; see next section.
\subsubsection{Emission of very small (\hbox{Al$_2$O$_3$})$_n$ ($n \le 4$) aluminium oxide clusters}\label{Sec:disc_Al_clusters_small}
A way to test if very small \hbox{Al$_2$O$_3$}\ polymers are present is via the detection of their vibrational frequencies. These bands arise owing to the presence of internal bending and stretching modes in the cluster. As the energy and geometry, the vibrational spectrum differs from isomer to isomer, and thus can be seen as a spectral fingerprint.
Because there is no way to currently predict the most likely cluster composition in the outflow and because these clusters would probably range over a wide range in both stoichiometry as well as atomic sizes, we have taken the simple approach of calculating the spectral properties of a range of stoichiometric alumina clusters to estimate where observable lines might exist.
For $n=1 - 4$ we have determined the most stable structures of the neutral aluminium oxide clusters using density functional theory (DFT); see Fig.~\ref{Fig:Gobrecht}. The computations were accomplished by the GAUSSIAN~09 package \citep{Frisch09} using the hybrid B3LYP \citep[Becke, three parameter, Lee-Yang-Parr;][]{Lee1988PhRvB..37..785L, Miehlich1989CPL...157..200M, Becke1993JChPh..98.5648B} exchange-correlation functional with a 6-311+g$^*$ basis set. The binding energy and stability of electronic systems including clusters are pressure and temperature-dependent, respectively. Variations in temperature and density imply that the energy levels of the clusters may shift and that the order of the most stable clusters may change. We have calculated how the stability\footnote{ calculated from the Gibbs free energy. The entropy is given by the partition function (with contributions from electronic, translational, vibrational and vibrational motion) of which the translational part depends on the pressure.} of the considered small alumina clusters is affected in various ways for conditions occurring in the CSE (with pressure $p=0.001-50$\,Pa, and temperature $T=500-3000$\,K).
For \hbox{Al$_2$O$_3$}, the first excited structural isomer is a linear chain and has a potential energy 0.1\,eV above the kite-shaped triplet ground state. Above 1000\,K the linear form is slightly more stable, below 1000\,K the kite-shaped triplet structure is favoured.
Considering (\hbox{Al$_2$O$_3$})$_2$, the energy difference of the ground state to the first excited state is relatively large (0.48\,eV) and therefore structure 2A \citep[see Fig.~1 in][]{Li2012} is also the most stable in circumstellar conditions.
For $n=3$, we find five stable structures in an energy interval of only 0.13\,eV for the alumina trimer. In circumstellar conditions, however, the first excited state 3B \citep{Li2012} is the most stable configuration.
For a temperature of 1500\,K, the ground and first excited state of (\hbox{Al$_2$O$_3$})$_4$ are almost identical in potential energy. Below 1500\,K isomer 4A is favoured, above 1500\,K structure 4B \citep{Li2012} which is only stable for pressures above 0.1\,Pa.
Overall, the Gibbs Free energy of formation for alumina clusters depends significantly on T, but not much on density or pressure.
For each of the optimized structures, the vibrational spectrum of the ground state clusters was calculated (see Fig.~\ref{Fig:Gobrecht}). The predicted spectra compare well with the experimental results by \citet{Demyk2004A&A...420..547D} and the calculated linear infrared absorption spectra by \citet{Sierka2007} for small cluster sizes (see Fig.~\ref{Fig:Gobrecht_Sierka}), with the strongest peaks in the 10-11\,\hbox{$\mu$m}\ region and an ensemble of smaller peaks around 14\,\hbox{$\mu$m}.
Implementing the stable clusters in thermo-chemistry calculations for densities down to $1\times10^{-4}$ dyne\,cm$^{-2}$ \citep{Hofner2016arXiv160509730H} and temperatures up to 1800\,K and allowing the small clusters to grow via monomer addition indicates that (\hbox{Al$_2$O$_3$})$_3$ seems to be the best candidate for possible detection in CSEs (see Table~\ref{Table_David}).
\begin{figure*}[htp]
\begin{minipage}[t]{.48\textwidth}
\centerline{\resizebox{\textwidth}{!}{\includegraphics[angle=270]{Al2O3_n_GS_NEW.png}}}
\end{minipage}
\hfill
\begin{minipage}[t]{.48\textwidth}
\centerline{\resizebox{\textwidth}{!}{\includegraphics[angle=270]{Al2O3_n_GS_NEWInt.png}}}
\end{minipage}
\caption{Vibrational spectrum of the (\hbox{Al$_2$O$_3$})$_n$, $n=1-4$, ground state clusters and their corresponding structures. The structures (for $n=1-4$) are depicted with increasing $n$ from left to right and from top to bottom. The black line represents a Lorentzian distribution with HWHM parameter $\gamma=0.2$ for all peaks. The left panel shows the intensity in units of km/mole; the right panel the opacity in units of cm$^2$/g. }
\label{Fig:Gobrecht}
\end{figure*}
\begin{figure}[htp]
\includegraphics[height=0.48\textwidth,angle=270]{Al8O12GS_freq.pdf}
\caption{Intensity of the (\hbox{Al$_2$O$_3$})$_4$ ground state cluster versus frequency convoluted with a Lorentzian function with FWHM of 24\,cm$^{-1}$. To be compared with Fig.\ 2d in \citet{Sierka2007}.}
\label{Fig:Gobrecht_Sierka}
\end{figure}
\begin{table}
\caption{Overview of the aluminia nucleation by monomer (\hbox{Al$_2$O$_3$}) addition at different temperatures $T$ (in K) and pressures $p$ (in dyne\,cm$^{-2}$). An energetically feasible nucleation is marked with $\surd$. Suppressed nucleation with (large) energy barriers is marked with x. If the nucleation pathway is partially favourable, the largest preferential cluster is given. }
\label{Table_David}
\begin{center}
\begin{tabular}{l|cccc}
\hline
\backslashbox[0pt][l]{}{}
& \makebox(0,0)[lb]{\put(-1.25\tabcolsep,0){\llap{$T$}}}%
1000 & 1200 & 1500 & 1800 \\\hline
\makebox(0,0){\put(0,2.25\normalbaselineskip){\rlap{$p$}}}%
1 & $\surd$ & $\surd$ & $\surd$ & (\hbox{Al$_2$O$_3$})$_3$ \\
0.1 & $\surd$ & $\surd$ & $\surd$ & (\hbox{Al$_2$O$_3$})$_3$ \\
0.01 & $\surd$ & $\surd$ & $\surd$ & x \\
$10^{-3}$ & $\surd$ & $\surd$ & (\hbox{Al$_2$O$_3$})$_3$ & x \\
$10^{-4}$& $\surd$ & $\surd$ & (\hbox{Al$_2$O$_3$})$_3$ & x \\
\hline
\end{tabular}
\end{center}
\end{table}
We have tested if the spectral features present in the infrared SED of R~Dor and W~Hya could be explained by the vibrational frequencies of these small aluminium oxide clusters (see Fig.~\ref{Fig:molecules_clusters}). However, the low spectral resolution (around 300) of the ISO/SWS speed 1 data for both stars makes it impossible to draw any firm conclusion. Albeit a positive correlation can be noticed between various absorption peaks in the ISO data and the predicted cluster spectra, we caution against over-interpretation of these results. Specifically, molecular line veiling by other molecules which might be present in the 'extra-molecular' layer created by pulsation-induced density enhancements result in an ensemble of emission and absorption features that complicates any potential identification. High spectral resolution data in the 10-20\,\hbox{$\mu$m}\ regime (as might be obtained in the future using the E-ELT/METIS instrument) is paramount to answer the question on the potential presence and vibrational excitation of these small aluminium oxide clusters.
\begin{figure*}[htp]
\centering \includegraphics[height=0.9\textheight]{fig_molecules_clusters_opac_new.pdf}
\vspace*{-2cm}
\caption{Upper panel: normalized opacity of the (\hbox{Al$_2$O$_3$})$_n$ clusters convoluted with a Lorentzian profile with a FWHM of 0.03\,\hbox{$\mu$m}. The full coloured lines represent the opacities normalized for each cluster entity separately (with the same colour coding as in Fig.~\ref{Fig:Gobrecht}), while the dotted coloured lines indicate the wavelength position of all ground-state vibrational frequencies of the stable (\hbox{Al$_2$O$_3$})$_n$ ($n=1-4$) clusters.
Middle panel: Comparison between the (stellar continuum subtracted and normalized) SED of R~Dor (full black line) and W~Hya (full gray line) with the predicted vibrational frequencies of the stable (\hbox{Al$_2$O$_3$})$_n$ ($n=1-4$) clusters (vertical coloured lines). Grey areas indicate the region with some of the largest (\hbox{Al$_2$O$_3$})$_n$ opacities. Dashed gray lines indicate the CO$_2$ lines as discussed by \citet{Justtanont2004A&A...417..625J}. The full red line is the (stellar continuum subtracted and normalized) emission for a slab model containing H$_2$O, SiO, and SiS. Column density for all species were taken to be $1\times10^{20}$\,cm$^{-2}$ at 0.5\Rstar\ above the stellar surface for a temperature of 2000\,K. The individual contributions of H$_2$O, SiO, and SiS are shown in the bottom panel. Note that the column densities of the molecules were not chosen to fit the ISO/SWS data, but serve as illustrative examples on the molecular line veiling.}
\label{Fig:molecules_clusters}
\end{figure*}
The more pronounced evidence of large ($n>34$) (\hbox{Al$_2$O$_3$})$_n$-clusters (Fig.~\ref{Fig:VanHeijnsbergen} and Fig.~\ref{Fig:SED_RDor_WHya}) warrants a discussion on the possibility if a physico-chemical situation might occur resulting in a skewed cluster-size distribution toward large cluster sizes.
\subsubsection{Skewed cluster-size distribution} \label{Sec:skewed}
First of all, one might wonder if the \textit{grain}-size distribution in the inner wind region of AGB stars is similar to the one found in the ISM, $n_d(a)\,da \propto a^{-7/2}\,da$ \citep{Mathis1977ApJ...217..425M}. \citet{Dominik1989A&A...223..227D} derived a particle size distribution for carbon-rich stars with spectral index of $-5$ using Classical Nucleation Theory (CNT) with heteromolecular homogeneous grain formation as developed by \citet{Gail1988A&A...206..153G}.
A different approach was pursued by \cite{Gobrecht2016A&A...585A...6G} who included pulsation-induced shocks in the inner wind of an oxygen-rich AGB star and assumed coalescence and coagulation of (molecular) dimers to form dust grains. Using a Brownian formalism to compute the grain size distribution, they showed that coagulation is only efficient for dust growth for late phase ($\phi > 0.8$), resulting in very small aluminium oxide grains in the size range of 7--20\,nm after one oscillation. Assuming that the aluminium oxide grains survive the next passage of the periodic shocks and that growth occurs through surface deposition gives rise to a grain size distribution at 2 stellar radii after 12 pulsation cycles with two components, one peaking at $\sim$30\,nm, the other around 300\,nm. This result is already a first indication that size distributions in CSEs might differ from the ISM characteristics mentioned above
Two direct pathways for clusters to grow is via adding monomers and by collisions and sticking together. Since the number of seeds is low, the chance of colliding with monomers will be higher than of collision with larger polymers, so the first pathway seems to be preferred. The region around 2\,\Rstar\ coincides with the GBDS, i.e.\ a higher density region where species can stay longer without being pushed outward. This might create a favourable condition for clusters to grow to large sizes. These large aluminium oxides can be very stable at high temperatures due to their ability of undergoing thermo-ionic emission rather than dissociation \citep{Demyk2004A&A...420..547D}. I.e.\ the clusters preferentially emit an electron, a unique property which is only observed for very strongly bound species. This ability might also greatly enhance the growth efficiencies of the clusters, since the transient complex that is formed during the growth reaction process might also stabilize via the emission of an electron.
Growth by adding monomers is hence a potential process for the creation of larger clusters, with a preference for specific cluster sizes initiating coagulation. Energy considerations in this highly turbulent region just above the stellar atmosphere should be taken into account to answer the question on the favourable cluster size for coagulation. No calculations including the whole spectrum from small to large aluminium oxide clusters are currently available to answer this question. Generally, the alumina clusters (including the very small ones) show a ionic character with alternating Al-O bondings. Moreover, some of the low-lying small clusters exhibit a trigonal symmetry (like bulk-corundum). One thus expects that the coagulation might start at rather small cluster sizes ($n=3,4$).
Our calculations including only the $n=1-4$ (\hbox{Al$_2$O$_3$})$_n$ clusters show indeed a preference for the $n=3$ cluster size. A similar behaviour is noted for (SiO)$_n$ for which the matrix isolation experiment by\citet{Stranz1980PhDT........80S} shows that SiO monomers, dimers and trimers can be seen, but then the material seems to transition to the bulk phase. If coagulation were to commence from small cluster sizes, then we would not expect to see (much) large clusters, but we assume that these are in fact present, as indicated by the observed 11\,$\mu$m spectral feature.
Large clusters could, however, be formed by alternative pathways. One possibility is the formation of large clusters by surface processes; i.e.\ a heterogeneous nucleation process during which large clusters are formed on the surface of existing grains and subsequently ejected. Nozzle experiments show that a substantial quantity of large metal clusters can be formed and ejected that way \citep{Knauer1987}. The NACO and SPHERE data prove that grains (of still unknown composition) are ubiquitous close to the star. These grains might serve as surface for cluster growth to occur.
Vaporization of solid material might offer another explication for a skewed cluster-size distribution toward large sizes. The \hbox{Al$_2$O$_3$}\ grain temperatures in the region around 2\,\Rstar\
are high (Sect.~\ref{Sec:disc_Al_dust}) and might lead to the crystallization of the bulk material. If energetically favourable, the material might also partly get vaporized producing large clusters. This kind of behaviour is noted during formation kinetics experiments and computer simulations for the nucleation and growth rates of metal vapors \citep{Yamada1987}. The recent SPHERE data of \cite{Khouri2016A&A...591A..70K} show that the solid-state material is grouped into irregular, clumpy structures. Backwarming occurring between the dust clumps and the star might locally increase the temperature \citep{Fleischer1995A&A...297..543F}, favouring vaporization to take place.
\section{Conclusions}\label{Sec:conclusions}
In this paper we have studied the aluminium content in a low and a high mass-loss rate oxygen-rich AGB star, R~Dor and IK~Tau respectively. The aim was to reveal the role of aluminium species in the initiation of the stellar wind. We therefore have observed both stars with ALMA at high spatial resolution ($\sim$150\,mas) to trace the dust formation region via aluminium bearing molecules (AlO, AlOH, and AlCl). The azimuthally averaged flux densities in function of the angular distance from the central star were fitted using a non-LTE radiative transfer code to retrieve the fractional abundances of the molecules.
The gas-phase aluminium chemistry is largely different for both stars, in particular the AlO and AlOH fractional abundances. In IK~Tau AlOH is clearly detected, while traces of clumpy structures containing AlO are only apparent beyond 50\,\Rstar. In contrast, AlO is $\sim$70 times more abundant than AlOH in R~Dor. These three molecules consume only $\mathrel{\mathchoice {\vcenter{\offinterlineskip\halign{\hfil$\displaystyle##$\hfil\cr<\cr\sim\cr}}$2\% of the total aluminium budget. Hence, there is ample opportunity to form aluminium-bearing dust species. AlO and AlOH are direct precursors of these aluminium-bearing grains and both molecules are detected well beyond the dust condensation radius. This implies that the aluminium dust condensation cycle is not fully efficient.
To assess the aluminium dust content in oxygen-rich AGB stars, we have also included the well-studied low mass-loss rate star W~Hya. We discuss how the spectral features present in the spectral energy distribution (SED), polarized light signal, and interferometric data point toward the presence of a halo of large (and small) transparent grains close the star. A candidate species for these grains is amorphous \hbox{Al$_2$O$_3$}. However, at a distance of only 1.5\,\Rstar\ the estimated grain temperature is $>1000$\,K, i.e.\ too high for the grains to retain their amorphous lattice structure. The grains should anneal and form a crystalline compound. Indeed, the typical 13\,\hbox{$\mu$m}\ features signalling the presence of crystalline \hbox{Al$_2$O$_3$} ($\alpha$-\hbox{Al$_2$O$_3$}) is seen in the SEDs of W~Hya and R~Dor. One is then left with the question how to explain the broad 11\,\hbox{$\mu$m}\ feature in the SED and interferometric data. We propose that large gas-phase (\hbox{Al$_2$O$_3$})$_n$-clusters ($n>34$) are the species that produce the signatures seen in the SED and interferometric data. We explain how these large clusters can be formed. Using density functional theory (DFT) we calculate the stability and emissivity of very small clusters ($n\la4$). We show how future E-ELT/METIS observations will be crucial to answer the question if very small clusters ($n\la4$) are present in the inner winds of these oxygen-rich AGB stars.
\begin{acknowledgements}
LD acknowledges support from the ERC consolidator grant 646758 AEROSOL and the FWO Research Project grant G024112N. We acknowledge the CINECA award under the ISCRA initiative for the availability of high performance computing resources and support.
This paper makes use of the following ALMA data: ADS/JAO.ALMA2013.0.00166.S. ALMA is a partnership of ESO (representing
its member states), NSF (USA) and NINS (Japan), together with NRC
(Canada) and NSC and ASIAA (Taiwan), in cooperation with the Republic of
Chile. The Joint ALMA Observatory is operated by ESO, AUI/NRAO and NAOJ. We thank Willem Jan de Wit for his contribution to the reduction of the VLTI/MIDI data of R~Dor. We acknowledge with thanks the variable star observations from the AAVSO International Database contributed by observers worldwide. Based on observations obtained with the HERMES spectrograph, which is supported by the Research Foundation - Flanders (FWO), Belgium, the Research Council of KU Leuven, Belgium, the Fonds National de la Recherche Scientifique (F.R.S.-FNRS), Belgium, the Royal Observatory of Belgium, the Observatoire de Genève, Switzerland and the Thüringer Landessternwarte Tautenburg, Germany.
\end{acknowledgements}
\bibliographystyle{aa}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 5,180 |
\section{Introduction}
Sensitive information in the form of medical records, social network data, and geospatial data can have transformative benefits to society through modern data analysis. However, researchers bear a critical responsibility to ensure that these analyses do not compromise the privacy of the individuals whose data is used.
Differential privacy~\cite{DworkMNS06} offers formal guarantees and a rich algorithmic toolkit for studying how such analyses can be conducted. To address settings where sensitive data arise in machine learning, Kasiviswanathan et al.~\cite{KasiviswanathanLNRS11} introduced differentially private PAC learning as a privacy-preserving version of Valiant's PAC model for binary classification~\cite{Valiant84}. In the ensuing decade a number of works (e.g., ~\cite{BeimelBKN14,BunNSV15,FeldmanX15,BeimelNS16,Beimel19Pure,AlonLMM19,KaplanLMNS19}) have developed sophisticated algorithms for learning fundamental concept classes while exposing deep connections to optimization, online learning, and communication complexity along the way.
Despite all of this attention and progress, we are still far from resolving many basic questions about the private PAC model. As an illustration of the state of affairs, the earliest results in \emph{non-private} PAC learning showed that the sample complexity of~(i.e., the minimum number of samples sufficient for) learning a concept class $\mathcal{C}$ is tightly characterized by its VC dimension~\cite{VapnikC74, BlumerEHW89}. It is wide open to obtain an analogous characterization for private PAC learning. In fact, it was only in the last year that a line of work~\cite{BunNSV15, AlonLMM19, BunLM20} culminated in a characterization of when $\mathcal{C}$ is learnable using \emph{any} finite number of samples whatsoever. The following theorem captures this recent characterization of private learnability.
\begin{theorem}[Informal~\cite{AlonLMM19, BunLM20}] \label{thm:private-online-equiv}
Let $\mathcal{C}$ be a concept class with \emph{Littlestone dimension} $d = L(\mathcal{C})$. Then $2^{O(d)}$ samples are sufficient to privately learn $\mathcal{C}$ and $\Omega(\log^* d)$ samples are necessary.
\end{theorem}
The Littlestone dimension is a combinatorial parameter that exactly captures the complexity of learning $\mathcal{C}$ in Littlestone's mistake bound model of online learning~\cite{Littlestone87online}. Thus, Theorem~\ref{thm:private-online-equiv} characterizes the privately learnable concept classes as exactly those that are online learnable. This is but one manifestation of the close connection between online learning and private algorithm design, e.g., ~\cite{RothR10, DworkRV10, HardtR10, JainKT12, FeldmanX15, Agarwal17dponline,Abernathy17onlilnedp,AlonLMM19,bousquet2019passing,NeelRW19,GonenHM19,BunCS20, JungKT20} that facilitates the transfer of techniques between the two areas.
Theorem~\ref{thm:private-online-equiv} means that, at least in principle, online learning algorithms can be generically converted into differentially private learning algorithms and vice versa. The obstacle, however, is the efficiency of conversion, both in terms of sample use and running time. The forward direction (online learning $\implies$ private learning)~\cite{BunLM20} gives an algorithm that incurs at least an exponential blowup in both complexities. The reverse direction (private learning $\implies$ online learning)~\cite{AlonLMM19} is non-constructive: It is proved using the fact that every class with Littlestone dimension at least $d$ contains an embedded copy of $\mathcal{THR}_{\log d}$, the class of one-dimensional threshold functions over a domain of size $\log d$. The characterization then follows from a lower bound of $\Omega(\log^* d)$ on the sample complexity of privately learning $\mathcal{THR}_{\log d}$.
There is limited room to improve the general sample complexity relationships in Theorem~\ref{thm:private-online-equiv}. There are classes of Littlestone dimension $d$ (e.g., Boolean conjunctions) that require $\Omega(d)$ samples to learn even non-privately. Meanwhile, $\mathcal{THR}_{2^d}$ has Littlestone dimension $d$ but can be privately learned using $\tilde{O}((\log^* d)^{1.5})$ samples~\cite{KaplanLMNS19}. This latter result rules out, say, a generic conversion from polynomial-sample private learners to online learners with polynomial mistake bound.
In this work, we address the parallel issue of computational complexity. Could there be a computationally efficient black-box conversion from any private learner to an online learner? We show that, under cryptographic assumptions, the answer is \emph{no}.
\begin{theorem}[Informal, building on~\cite{Blum94}] \label{thm:private-not-online}
Assuming the existence of one-way functions, there is a concept class that is privately PAC learnable in polynomial-time, but is not online learnable by any poly-time algorithm with a polynomial mistake bound.
\end{theorem}
The question we study was explicitly raised by Neel, Roth, and Wu~\cite{NeelRW19}.
They proved a barrier result to the existence of ``oracle-efficient'' private learning by showing that a restricted class of such learners can be efficiently converted into oracle-efficient online learners, the latter for which negative results are known~\cite{HazanK16}. This led them to ask whether their barrier result could be extended to all private learning algorithms. Our Theorem~\ref{thm:private-not-online} thus stands as a ``barrier to a barrier'' to general oracle-efficient private learning.
This question was also studied in a recent work of Gonen, Hazan, and Moran~\cite{GonenHM19}. They gave an efficient conversion from \emph{pure}-private learners to online learners with sublinear regret. As we discuss in Section~\ref{sec:GHM} the efficiency of their construction relies on either a uniform model of learning (that turns out to be incompatible with pure differential privacy) or on an additional assumption that the private learner is highly sample efficient. Theorem~\ref{thm:private-not-online} rules out the existence of such a conversion even for non-uniform pure-private learners without this extra sample efficiency condition.
\paragraph{Proof idea of Theorem~\ref{thm:private-not-online}.} In 1990, Blum~\cite{Blum94} defined a concept class we call $\mathcal{OWS}$ that is (non-privately) efficiently PAC learnable but not poly-time online learnable. We prove Theorem~\ref{thm:private-not-online} by showing that $\mathcal{OWS}$ is efficiently privately PAC learnable as well. Blum's construction builds on the Goldreich-Goldwasser-Micali~\cite{GoldreichGM86} pseudorandom function generator to define families of ``one-way'' sequences $\sigma_1, \dots, \sigma_r \in \{0,1\}^d$ for some $r$ that is exponential in $d$. Associated to each $\sigma_i$ is a label $b_i \in \{0,1\}$. These strings and their labels have the property that they can be efficiently computed in the forward direction, but are hard to compute in the reverse direction. Specifically, given $\sigma_i$ and any $j > i$, it is easy to compute $\sigma_j$ and $b_j$. On the other hand, given $\sigma_j$, it is hard to compute $\sigma_i$ and hard to predict $b_i$.
To see how such sequences can be used to separate online and non-private PAC learning, consider the problem of determining the label $b_i$ for a given string $\sigma_i$. In the online setting, an adversary may present the sequence $\sigma_r, \sigma_{r-1}, \dots$ in reverse. Then the labels $b_j$ are unpredictable to a poly-time learner. On the other hand, a poly-time PAC learner can identify the $\sigma_{i^*}$ with smallest index in its sample of size $n$. Then all but a roughly $1/n$ fraction of the underlying distribution on examples will have index at least $i^*$ and can be predicted using the ease of forward computation.
Note that the PAC learner for $\mathcal{OWS}$ is not private, since the classifier based on $i^*$ essentially reveals this sample in the clear. We design a private version of this learner by putting together standard algorithms from the differential privacy literature in a modular way. We first privately identify an $i^*$ that is approximately smallest in the sample. Releasing $\sigma_{i^*}$ directly at this stage is still non-private. So instead, we privately check that every $\sigma_i$ with $i \le i^*$ corroborates the string $\sigma_{i^*}$, in that $\sigma_{i*}$ is the string that would be obtained by computing forward using any of these strings. If the identity of $\sigma_{i^*}$ is stable in this sense and passes the privacy-preserving check, then it is safe to release.
\section{Preliminaries}\label{sec:preliminaries}
An \emph{example} is an element $x \in \{0,1\}^d$. A \emph{concept} is a boolean function $c : \{0,1\}^d \to \{0,1\}$. A \emph{labeled example} is a pair $(x, c(x))$. A \emph{concept class} $\mathcal{C} = \{\mathcal{C}_d\}_{d \in \mathbb{N}}$ is a sequence where each $\mathcal{C}_d$ is a set of concepts over $\{0,1\}^d$. Associated to each $\mathcal{C}_d$ is a (often implicit) \emph{representation scheme} under which concepts are encoded as bit strings. Define $|c|$ to be the minimum length representation of $c$.
\subsection{PAC Learning}
In the PAC model, there is an unknown target concept $c \in \mathcal{C}$ and an unknown distribution $\mathcal{D}$ over labeled examples $(x, c(x))$. Given a sample $((x_i, c(x_i)))_{i=1}^n$ consisting of i.i.d. draws from $\mathcal{D}$, the goal of a learning algorithm is to produce a hypothesis $h : \{0,1\}^d \to \{0,1\}$ that approximates $c$ with respect to $\mathcal{D}$. Specifically, the goal is to find $h$ with low population loss defined as follows.
\begin{definition}[Population Loss]
Let $\mathcal{D}$ be a distribution over $\{0,1\}^d \times \{0,1\}$. The population loss of a hypothesis~$h : \{0,1\}^d \to \{0,1\}$ is
\[\operatorname{loss}_{\mathcal{D}}(h) = \Pr_{(x, b) \sim \mathcal{D}}[h(x) \ne b].\]
\end{definition}
Throughout this work, we consider \emph{improper} learning algorithms where the hypothesis $h$ need not be a member of the class $\mathcal{C}$. A learning algorithm $L$ \emph{efficiently PAC learns} $\mathcal{C}$ if for every target concept $c$, every distribution $\mathcal{D}$, and parameters $\alpha, \beta > 0$, with probability at least $1-\beta$ the learner $L$ identifies a poly-time evaluable hypothesis $h$ with $\operatorname{loss}_\mathcal{D}(h) \le \alpha$ in time $\mathrm{poly}(d, 1/\alpha, 1/\beta, |c|)$. It is implicit in this definition that the number of samples $n$ required by the learner is also polynomial. We will also only consider classes where $|c|$ is polynomial in $d$ for every $c \in \mathcal{C}_d$, so we may regard a class as efficiently PAC learnable if it is learnable using $\mathrm{poly}(d, 1/\alpha, 1/\beta)$ time and samples.
\subsection{Online Learning}
We consider two closely related models of online learning: The mistake-bound model and the no-regret model. Our negative result holds for the weaker no-regret model, making our separation stronger. We first review Littlestone's mistake-bound model of online learning~\cite{Littlestone87online}. This model is defined via a two-player game between a learner and an adversary. Let $\mathcal{C}$ be a concept class and let $c \in \mathcal{C}$ be chosen by the adversary. Learning proceeds in rounds. In each round $t = 1, \dots, 2^d$,
\begin{itemize}
\item[(i)] The adversary selects an~$x_t\in \{0,1\}^d$,
\item[(ii)] The learner predicts $\hat b_t\in \{0,1\}$, and
\item[(iii)] The learner receives the correct labeling $b_t = c(x_t)$.
\end{itemize}
A (deterministic) learning algorithm learns $c$ with mistake bound $M$ if for every adversarial ordering of the examples, the total number of incorrect predictions the learner makes is at most $M$. We say that the learner \emph{efficiently mistake-bound learns} $\mathcal{C}$ if for every $c \in \mathcal{C}$ it has mistake bound $\mathrm{poly}(d, |c|)$ and runs in time $\mathrm{poly}(d, |c|)$ in every round.
We also consider a relaxed model of online learning in which the learner aims to achieve \emph{no-regret}, i.e., err with respect to $c$ in a vanishing fraction of rounds. Let $T$ be a time horizon known to a randomized learner. The goal of the learner is to minimize its regret, defined by
\[ R_T = \sum_{t = 1}^T \mathbb{I}[\hat{b}_t \ne c(x_t)].\]
We say that a learner \emph{efficiently no-regret learns} $\mathcal{C}$ if there exists $\eta > 0$ such that for every adversary, it achieves $\Expectation[R_T] = \mathrm{poly}(d, |c|) \cdot T^{1-\eta}$ using time $\mathrm{poly}(d, |c|, T)$ in every round. Under this formulation, every efficient mistake-bound learner is also an efficient no-regret learner.
We point out two non-standard features of this definition of efficiency. First, ``no-regret'' typically requires regret to be sublinear in $T$, i.e., $o(T)$ whereas we require it to be strongly sublinear $T^{1-\eta}$. A stronger condition like this is needed to make the definition nontrivial because a regret upper bound of $T \le d \cdot T/\log T = \mathrm{poly}(d) \cdot o(T)$ is always achievable in our model by random guessing. Many no-regret algorithms achieve strongly sublinear regret, e.g., the experts/multiplicative weights algorithm and the algorithm of~\cite{GonenHM19} that both achieve $\eta = 1/2$. Second, it would be more natural to require the learner to run in time polynomial in $\log T$, the description length of the time horizon, rather than $T$ itself. The relaxed formulation here only makes our lower bounds stronger, and we use it to be consistent with the positive result of~\cite{GonenHM19} that runs in time proportional to $T$.
\subsection{Differential Privacy}
\begin{definition}[Differential Privacy]\label{def:private}
Let $\varepsilon, \delta > 0$. A randomized algorithm $L : X^n \to \mathcal{R}$ is $(\varepsilon,\delta)$-differentially private
if for every pair of datasets $S, S'$ differing in at most one entry, and every measurable set $T \subseteq \mathcal{R}$,
\[\Pr[L(S) \in T] \le e^\varepsilon \Pr[L(S') \in T] + \delta.\]
\end{definition}
We refer to the special case where $\delta = 0$ as \emph{pure} $\varepsilon$-differential privacy, and the case where $\delta > 0$ as \emph{approximate} differential privacy.
When $L$ is a learning algorithm, we require that this condition hold for all neighboring pairs of samples $S, S'$ -- not just those generated according to a distribution on examples labeled by a concept in a given class.
\section{Learnability of One-Way Sequences}
\subsection{One-Way Sequences} \label{sec:blum}
For every $d$, Blum defines a concept class $\mathcal{OWS}_d$ consisting of functions over the domain $\{0,1\}^d$ that can be represented using $\mathrm{poly}(d)$ bits and evaluated in $\mathrm{poly}(d)$ time. The concepts in $\mathcal{OWS}_d$ are indexed by bit strings $s \in \{0,1\}^k$, where $k = \lfloor \sqrt{d} \rfloor - 1$. The definition of $\mathcal{OWS}_d$ is based on two efficiently representable and computable functions $G : \{0,1\}^k \times \{0,1\}^k \to \{0,1\}^{d - k}$ and $f : \{0,1\}^k \times \{0,1\}^k \to \{0,1\}$ that are based on the Goldreich-Goldwasser-Micali pseudorandom function family~\cite{GoldreichGM86}. The exact definition of these functions are not important to our treatment, so we refer the reader to~\cite{Blum94} for details. Here, $G(i, s)$ computes the string $\sigma_i$ as described in the introduction, and $f(i, s)$ computes its label $b_i$. For convenience we identify $\{0,1\}^k$ with $[2^k]$.
We are now ready to define the concept class $\mathcal{OWS}_d = \{c_s\}_{s \in \{0,1\}^k}$ where
\[
c_s(i, \sigma) =
\begin{cases}
1 & \text{ if } \sigma = G(i, s) \text{ and } f(i, s) = 1\\
0 & \text{ otherwise}.
\end{cases}
\]
We recall the two key properties of the sequences $\sigma_i$ obtained from these strings. They are easy to compute in the forward direction, even in a random-access fashion, but difficult to compute in reverse. These properties are captured by the following claims.
\begin{proposition}[\cite{Blum94}]
There is an efficiently computable function $\mathsf{ComputeForward}: \{0,1\}^k \times \{0,1\}^k \times \{0,1\}^{d-k} \to \{0,1\}^{d-k} \times \{0,1\}$ such that $\mathsf{ComputeForward}(j, i, G(i, s)) = \langle G(j, s), f(j, s) \rangle$ for every $j > i$.
\end{proposition}
\begin{proposition}[\cite{Blum94}, Corollary 3.4] \label{prop:reverse}
Suppose $G$ and $f$ are constructed using a secure pseudorandom generator. Let $\mathcal{O}$ be an oracle that, on input $j > i$ and $G(i, s)$ outputs $\langle G(j, s), f(j, s)\rangle$. For every poly-time probabilistic algorithm $A$ and every $i \in \{0,1\}^k$,
\[\Pr[A^{\mathcal{O}}(i, G(i, s)) = f(i, s)] \le \frac{1}{2} + \mathrm{negl}(d),\]
where the probability is taken over the coins of $A$ and uniformly random $s \in \{0,1\}^k$.
\end{proposition}
Blum used Proposition~\ref{prop:reverse} to show that $\mathcal{OWS}$ cannot be efficiently learned in the mistake bound model (even with membership queries). Here, we adapt his argument to the setting of no-regret learning.
\begin{proposition}
If $G$ and $f$ are constructed using a secure pseudorandom generator, then $\mathcal{OWS}$ cannot be learned by an efficient no-regret algorithm.
\end{proposition}
\begin{proof}
Suppose for the sake of contradiction that there were a poly-time online learner $L$ for $\mathcal{OWS}$ achieving regret $d^c \cdot T^{1-\eta}$ for constants $c, \eta > 0$ and sufficiently large $d$. Consider an adversary that presents examples $(2^k, G(2^k, s); f(2^k, s)), (2^{k-1}, G(2^k-1, s); f(2^k-1, s)), \dots$ to the learner in reverse order. Then the expected number of mistakes $L$ makes on the first $T = (4d^c)^{1/\eta} = \mathrm{poly}(d)$ examples is at most $d^c \cdot T^{1-\eta}\le T / 4$. By averaging, there exists an index $2^k - T/4 \le t \le 2^k$ such that the probability that $L$ makes a mistake on example $(t, G(t, s))$ is at most $1/4$.
We now use $L$ to construct an adversary that contradicts Proposition~\ref{prop:reverse}. Consider the oracle algorithm $A^{\mathcal{O}}$ that on input $(t, G(t, s))$ invokes the oracle to compute the sequence $(2^k, G(2^k, s); f(2^k, s)),$ \\ $(2^{k-1}, G(2^k-1, s); f(2^k-1, s)), \dots, (t+1, G(t+1, s); f(t+1, s))$ and presents these labeled examples to $L$. Let $b$ the prediction that $L$ makes when subsequently given the example $(t, G(t, s))$. This agrees with $f(t, s)$ with probability at least $3/4$, contradicting Proposition~\ref{prop:reverse}.
\end{proof}
In the rest of this section, we construct an $(\varepsilon, \delta)$-differentially private learner for $\mathcal{OWS}$.
\subsection{Basic Differential Privacy Tools}
The \emph{sensitivity} of a function $q : X^n \to \mathbb{R}$ is the maximum value of $|q(S) - q(S')|$ taken over all pairs of datasets $S, S'$ differing in one entry.
\begin{lemma}[Laplace Mechanism] \label{lem:laplace}
The Laplace distribution with scale $\lambda$, denoted $\mathrm{Lap}(\lambda)$, is supported on $\R$ and has probability density function $f_{\mathrm{Lap}(\lambda)}(x) = \exp(-|x| / \lambda) / 2\lambda$. If $q : X^n \to \mathbb{R}$ has sensitivity $1$, then the algorithm $M_{\mathrm{Lap}}(S) = q(S) + \mathrm{Lap}(1/\varepsilon)$ is $\varepsilon$-differentially private and, for every $\beta > 0$, satisfies $|M_{\mathrm{Lap}}(S) - q(S)| \le \log(2/\beta) / \varepsilon$ with probability at least $1-\beta$.
\end{lemma}
\begin{remark}
We describe our algorithm using the Laplace mechanism as a matter of mathematical convenience, even though sampling from the continuous Laplace distribution is incompatible with the standard Turing machine model of computation. To achieve strict polynomial runtimes on finite computers we would use in its place the Bounded Geometric Mechanism~\cite{GhoshRS12, BalcerV18}.
\end{remark}
\begin{theorem}[Exponential Mechnanism~\cite{McSherryT07}] \label{thm:exp-mech}
Let $q : X^n \times \mathcal{R} \to \mathbb{R}$ be a sensitivity-1 score function. The the algorithm that samples $r \in \mathcal{R}$ with probability $\propto \exp(\varepsilon q(S, r) / 2)$ satisfies
\begin{enumerate}
\item $\varepsilon$-differential privacy, and
\item For every $S$, with probability at least $1-\beta$ the sampled $\hat{r}$ satisfies
\[q(S, \hat{r}) \ge \max_{r \in \mathcal{R}} q(S, r) - \frac{2\log( |\mathcal{R}| / \beta)}{\varepsilon}.\]
\end{enumerate}
\end{theorem}
The following ``basic'' composition theorem allows us to bound the privacy guarantee of a sequence of adaptively chosen algorithms run over the same dataset.
\begin{lemma}[Composition, e.g.,~\cite{DworkL09}] \label{lem:composition}
Let $M_1 : X^n \to \mathcal{R}_1$ be $(\varepsilon, \delta)$-differentially private. Let $M_2 : X^n \times \mathcal{R}_1 \to \mathcal{R}_2$ be $(\varepsilon_2, \delta_2)$ differentially private for every fixed value of its second argument. Then the composition $M(S) = M_2(S, M_1(S))$ is $(\varepsilon_1 + \varepsilon_2, \delta_1 + \delta_2)$-differentially private.
\end{lemma}
\subsection{Private Robust Minimum}
\begin{definition}
Given a dataset $S = (x_1, \dots, x_n) \in [R]^n$, an \emph{$\alpha$-robust minimum} for $S$ is a number $r \in [R]$ such that
\begin{enumerate}
\item $|\{i : x_i \le r\}| \ge \alpha n$, and
\item $|\{i : x_i \le r\}| \le 2\alpha n$.
\end{enumerate}
\end{definition}
Note that $r$ need not be an element of $S$ itself -- this is important for ensuring that we can release a robust minimum privately. Condition 2 guarantees that $r$ is approximately the minimum of $S$. Condition 1 guarantees that this condition holds robustly, i.e., one needs to change at least $\alpha n$ points of $S$ before $r$ fails to be at least the minimum.
\begin{theorem}
There exist polynomial-time algorithms $\mathsf{Min_{pure}}$ and $\mathsf{Min_{approx}}$ that each solve the private robust minimum problem with probability at least $1-\beta$, where
\begin{enumerate}
\item Algorithm $\mathsf{Min_{pure}}$ is $\varepsilon$-differentially private and succeeds as long as
\[n \ge O\left(\frac{\log(R / \beta)}{\alpha \varepsilon}\right).\]
\item Algorithm $\mathsf{Min_{approx}}$ is $(\varepsilon, \delta)$-differentially private and succeeds as long as
\[n \ge \tilde{O}\left(\frac{(\log^* R)^{1.5} \cdot \log^{1.5}(1/\delta) \cdot \log (1/\beta)}{\alpha \varepsilon}\right).\]
\end{enumerate}
\end{theorem}
\begin{proof}
The algorithms are obtained by a reduction to the \emph{interior point problem}. Both this problem and essentially the same reduction are described in~\cite{BunNSV15} but we give the details for completeness. In the interior point problem, we are given a dataset $S \in [R]^n$ and the goal is to identify $r \in [R]$ such that $\min S \le r \le \max S$. An $(\varepsilon, \delta)$-DP algorithm that solves the interior point problem using $m$ samples and success probability $1-\beta$ can be used to solve the robust minimum problem using $n = O(m / \alpha)$ samples: Given an instance $S$ of the robust minimum problem, let $S'$ consist of the elements $\lceil \alpha n \rceil$ through $\lfloor 2 \alpha n \rfloor$ of $S$ in sorted order and apply the interior point algorithm to $S'$.
The exponential mechanism provides a pure $\varepsilon$-DP algorithm for the interior point problem with sample complexity $O(\log(R / \beta) / \varepsilon)$~\cite{Smith11}. Let $x_{(1)}, \dots, x_{(n)}$ denote the elements of $S$ in sorted order. The appropriate score function $q(S, r)$ is the maximum value of $\min\{t, n-t\}$ such that $x_{(1)} \le \dots \le x_{(t)} \le r \le x_{(t+1)} \le \dots \le x_{(n)}$. Thus $q(S, r)$ ranges from a maximum of $\lfloor n/2 \rfloor$ iff $r$ is a median of $S$ to a minimum of $0$ iff $r$ is not an interior point of $S$. By Theorem~\ref{thm:exp-mech}, the released point has positive score (and hence is an interior point) as long as $n > 4 \log(R /\beta) /\varepsilon$. Moreover, one can efficiently sample from the exponential mechanism distribution in this case as the distribution is constant on every interval of the form $[x_{(t)}, x_{(t+1)})$.
For $(\varepsilon, \delta)$-DP with $\delta > 0$, Kaplan et al.~\cite{KaplanLMNS19} provide an efficient algorithm for the interior point problem (with constant failure probability) using $\tilde{O}((\log^* R)^{1.5} \log^{1.5}(1/\delta) / \varepsilon)$ samples. Taking the median of $O(\log (1/\beta))$ repetitions of their algorithm on disjoint random subsamples gives the stated bound.
\end{proof}
\subsection{Private Most Frequent Item}
Let $S \in X^n$ be a dataset and let $x \in X$ be the item appearing most frequently in $S$. The goal of the ``private most-frequent-item problem'' is to identify $x$ with high probability under the assumption that the most frequent item is stable: its identify does not change in a neighborhood of the given dataset $S$. For $x \in X$ and $S = (x_1, \dots, x_n)\in X^n$, define $\operatorname{freq}_S(x) = |\{i : x_i = x\}|$.
\begin{definition}
An algorithm $M : X^n \to [R]$ solves the most-frequent-item problem with gap $\operatorname{GAP}$ and failure probability $\beta$ if the following holds. Let $S \in X^n$ be any dataset with $x^* = \operatorname{argmax}_x \operatorname{freq}_S(x)$ and
\[\operatorname{freq}_S(x^*)\ge \max_{x \ne x^*} \operatorname{freq}_S(x) + \operatorname{GAP}.\]
Then with probability at least $1-\beta$, we have $M(S) = x$.
\end{definition}
\begin{theorem}[\cite{BalcerV18}] \label{thm:freq}
There exist polynomial-time algorithms $\mathsf{Freq_{pure}}$ and $\mathsf{Freq_{approx}}$ that each solve the private most-frequent-item problem with probability at least $1-\beta$, where
\begin{enumerate}
\item Algorithm $\mathsf{Freq_{pure}}$ is $\varepsilon$-differentially private and succeeds as long as $\operatorname{GAP} \ge O(\log (R/\beta) / \alpha \varepsilon)$.
\item Algorithm $\mathsf{Freq_{approx}}$ is $(\varepsilon, \delta)$-differentially private and succeeds as long as $\operatorname{GAP} \ge O(\log(n/\delta\beta) / \alpha\varepsilon)$.
\end{enumerate}
\end{theorem}
Balcer and Vadhan~\cite{BalcerV18} actually solved the more general problem of computationally efficient private histogram estimation. Theorem~\ref{thm:freq} follows from their algorithm by reporting the privatized bin with the largest noisy count.
\subsection{Privately Learning $\mathcal{OWS}_d$}
\begin{algorithm}
\caption{Pure Private Learner for $\mathcal{OWS}_d$}
\label{alg:learner}
\begin{enumerate}
\item Let $S_+ = ((i_1, \sigma_1), \dots, (i_m, \sigma_m))$ be the subsequence of positive examples in $S$, where $i_1 \le i_2 \le \ldots \le i_m$.
\item Let $\hat{m} = m + \mathrm{Lap}(3/\varepsilon)$. If $\hat{m} \le \alpha n/3$, output the all-$0$ hypothesis.
\item Let $I = (i_1, \dots, i_m)$. Run $\mathsf{Min_{pure}}(I)$ using privacy parameter $\varepsilon / 3$ to identify a $(\alpha n / 6\hat{m})$-robust minimum $i^*$ of $I$ with failure probability $\beta / 6$.
\item For every $i_j \in I$ with $i_j < i^*$ let $\langle \hat{\sigma}_j, \hat{b}_j\rangle = \mathsf{ComputeForward}(i^*, i_j, \sigma_j)$. For every $i_j \in I$ with $i_j = i^*$ let $\langle \hat{\sigma}_j, \hat{b}_j\rangle = \langle \sigma_{i_j}, b_j \rangle$.
\item Run $\mathsf{Freq_{pure}}(\langle \hat{\sigma}_1, \hat{b}_1\rangle, \dots, \langle \hat{\sigma}_\ell, \hat{b}_\ell\rangle)$ using privacy parameter $\varepsilon / 3$ to output $\langle \sigma^*, b^* \rangle$ with failure probability $\beta / 6$. Here, $\ell$ is the largest $j$ for which $i_j \le i^*$.
\item Return the hypothesis $h(i, \sigma) = $ \\
\indent ``If $i < i^*$, output $0$. If $i = i^*$, output $b^*$ if $\sigma^* = \sigma$ and output $0$ otherwise. If $i > i^*$, run algorithm $\mathsf{ComputeForward}(i, i^*, \sigma^*) = \langle \hat{\sigma}, \hat{b}\rangle$. If $\sigma = \hat{\sigma}$, output $\hat{b}$. Else, output $0$.''
\end{enumerate}
\end{algorithm}
\begin{theorem} \label{thm:private-not-online-pure}
Algorithm \ref{alg:learner} is an $\varepsilon$-differentially private and $(\alpha, \beta)$-PAC learner for $\mathcal{OWS}_d$ running in time $\mathrm{poly}(d, 1/\alpha, \log(1/\beta))$ using
\[n = O\left(\frac{\sqrt{d} + \log(1/\beta)}{\alpha \varepsilon}\right)\]
samples.
\end{theorem}
\begin{proof}
Algorithm~\ref{alg:learner} is an adaptive composition of three $(\varepsilon/3)$-differentially private algorithms, hence $\varepsilon$-differentially private by Lemma~\ref{lem:composition}.
\medskip
To show that it is a PAC learner, we first argue that the hypothesis produced achieves low error with respect to the sample $S$, and then argue that it generalizes to the underlying distribution. That is, we first show that for every realizable sample $S$, with probability at least $1-\beta / 2$ over the randomness of the learner alone, the hypothesis $h$ satisfies
\[\operatorname{loss}_S(h) = \frac{1}{n} \sum_{k = 1}^n \mathbb{I}[h(x_k) \ne b_k] \le \alpha/2.\]
We consider several cases based on the number of positive examples $m = |S_+|$. First suppose $m \le \alpha n / 4$. Then Lemma~\ref{lem:laplace} guarantees that the algorithm outputs the all-$0$ hypothesis with probability at least $1-\beta$ in Step 2, and this hypothesis has sample loss at most $\alpha / 4$.
Now suppose $m = |S_+| \ge \alpha n / 2$. Then with probability at least $1 - \beta / 3$ we have $|\hat{m} - m| \le 3\log(6/\beta) / \varepsilon$. In particular this means $\hat{m} \ge \alpha n /3$, so the algorithm continues past Step 2. Now with probability at least $1-\beta/3$, Step 3 identifies a point $i^*$ such that $|\{i \in I : i \le i^*\}| \ge (\alpha n / 6\hat{m}) \cdot m \ge \alpha n /10$ and $|\{i \in I : i \le i^*\}| \le (\alpha n / 6\hat{m}) \cdot m \le \alpha n/2$. The first condition, in particular, guarantees that $\ell \ge \alpha n / 10$. Realizability of the sample $S$ guarantees that the points $\langle \hat{\sigma}_1, \hat{b}_1\rangle, \dots, \langle \hat{\sigma}_\ell, \hat{b}_\ell\rangle$ are all identical. So with the parameter $\operatorname{GAP} = \ell \ge \alpha n / 10$, Step 5 succeeds in outputting their common value $\langle \sigma^*, b^*\rangle$ with probability at least $1 - \beta / 3$.
We now argue that the hypothesis $h$ produced in Step 6 succeeds on all but $\alpha n / 2$ examples. A case analysis shows that the only input samples on which $h$ makes an error are those $(i_j, \sigma_j) \in S_+$ for which $i_j < i^*$. The success criterion of Step 3 ensures that the number of such points is at most $\alpha n /2$.
The final case where $\alpha n/4 < m < \alpha n /2$ is handled similarly, except it is now also acceptable for the algorithm to terminate early in Step 2, outputting the all-$0$ hypothesis.
We now argue that achieving low error with respect to the sample is sufficient to achieve low error with respect to the distribution: If the learner above achieves sample loss $\operatorname{loss}_S(h) \le \alpha / 2$ with probability at least $1-\beta/2$, then it is also an $(\alpha, \beta)$-PAC learner for $\mathcal{OWS}_d$ when given at least $n \ge 8\log(2/\beta) / \alpha$ samples. The analysis follows the standard generalization argument for one-dimensional threshold functions, and our presentation follows~\cite{BunNSV15}.
Fix a realizable distribution $\mathcal{D}$ (labeled by concept $c_s$) and let $\mathcal{H}$ be the set of hypotheses that the learner could output given a sample from $\mathcal{D}$. That is, $\mathcal{H}$ consists of the all-$0$ hypothesis and every hypothesis of the form $h_{i^*}$ as constructed as in Step 6. We may express the all-$0$ hypothesis as $h_{2^d + 1}$. It suffices to show that for a sample $S$ drawn i.i.d. from a realizable distribution $\mathcal{D}$ that
\[\Pr[\exists h \in \mathcal{H}: \operatorname{loss}_{\mathcal{D}}(h) \le \alpha \text{ and } \operatorname{loss}_S(h) \le \alpha/2] \le \beta / 2.\]
Let $i_-$ be the largest number such that $\operatorname{loss}_\mathcal{D}(h_{i_-}) > \alpha$. If some $h_i$ has $\operatorname{loss}_\mathcal{D}(h_{i}) > \alpha$ then $i \le i_-$, and hence for any sample, $\operatorname{loss}_S(h_{i-}) \le \operatorname{loss}_S(h_i)$. So it suffices to show that
\[\Pr[\operatorname{loss}_S(h_{i-}) \le \alpha/2] \le \beta/2.\]
Define $E = \{(i, G(i, s)) : i < i_- \text{ and } f(i, s) = 1\}$ to be the set of examples on which $h_{i-}$ makes a mistake. By a Chernoff bound, the probability that after $n$ independent samples from $\mathcal{D}$, fewer than $\alpha n / 2$ appear in $E$ is at most $\exp(-\alpha n / 8) \le \beta/2$ provided $n \ge 8\log(2/\beta) / \alpha$.
\end{proof}
The same argument, replacing the use of $\mathsf{Min_{pure}}$ with $\mathsf{Min_{approx}}$ and $\mathsf{Freq_{pure}}$ with $\mathsf{Freq_{approx}}$ in Algorithm~\ref{alg:learner} yields
\begin{theorem} \label{thm:private-not-online-approx}
There is an $(\varepsilon, \delta)$-differentially private and $(\alpha, \beta)$-PAC learner for $\mathcal{OWS}_d$ running in time $\mathrm{poly}(d, 1/\alpha, \log(1/\beta))$ using
\[n = \tilde{O}\left(\frac{(\log^* d)^{1.5} \cdot \log^{1.5}(1/\delta) \cdot \log(1/\beta)}{\alpha \varepsilon}\right)\]
samples.
\end{theorem}
\subsection{Comparison to~\cite{GonenHM19}} \label{sec:GHM}
Gonen et al.~\cite{GonenHM19} proved a positive result giving conditions under which pure-private learners can be efficiently compiled into online learners. The purpose of this section is to describe their model and result and, in particular, explain why it does not contradict Theorem~\ref{thm:private-not-online-pure}.
Their reduction works as follows. Let $\mathcal{C}$ be a concept class that is pure-privately learnable (with fixed constant privacy and accuracy parameters) using $m_0$ samples. Consider running this algorithm roughly $N = \exp(m_0)$ times on a fixed dummy input, producing hypotheses $h_1, \dots, h_N$. Pure differential privacy guarantees that for every realizable distribution on labeled examples, with high probability one of these hypotheses $h_i$ will have small loss. This idea can be used to construct an online learner for $\mathcal{C}$ by treating the random hypotheses $h_1, \dots, h_N$ as experts and running multiplicative weights to achieve no-regret with respect to the best one online.
As it is described in~\cite{GonenHM19}, this is a computationally efficient reduction from no-regret learning to \emph{uniform} pure-private PAC learning. In the uniform PAC model, there is a single infinite concept class $\mathcal{C}$ consisting of functions $c : \{0,1\}^* \to \{0,1\}$. An efficient uniform PAC learner for $\mathcal{C}$ uses $m(\alpha, \beta)$ samples to learn a hypothesis with loss at most $\alpha$ and failure probability $\beta$ in time $\mathrm{poly}(|c|, 1/\alpha, 1/\beta)$. Note that the number of samples $m(\alpha, \beta)$ is completely independent of the target concept $c$. This contrasts with the non-uniform model, where the number of samples is allowed to grow with $d$, the domain size of $c$.
Another noteworthy difference comes when we introduce differential privacy. In the uniform model, one can move to a neighboring dataset by changing a single entry to any element of $\{0,1\}^*$. In the non-uniform model, on the other hand, an entry may only change to another element of the same $\{0,1\}^d$. This distinction affects results for pure-private learning, as we will see below. However, it does not affect $(\varepsilon, \delta)$-DP learning, as one can always first run the algorithm described in Theorem~\ref{thm:freq} to privately check that all or most of the elements in the sample come from the same $\{0,1\}^d$.
A simple example to keep in mind when considering feasibility of learning in the uniform model is the class of point functions $\mathcal{POINT} = \{p_x : x \in \{0,1\}^*\}$ where $p_x(y) = 1$ iff $x = y$. This class is efficiently uniformly PAC-learnable using $m(\alpha, \beta) = O(\log(1/\beta) / \alpha)$ samples by returning $p_x$ where $x$ is any positive example in the dataset.
The class $\mathcal{POINT}$ turns out to be uniformly PAC-learnable with pure differential privacy as well~\cite{Beimel19Pure}. However, this algorithm is \emph{not} computationally efficient. The following claim shows that this is inherent, as indeed it is even impossible to uniformly learn $\mathcal{POINT}$ using hypotheses with short description lengths.
\begin{proposition} \label{prop:impossible}
Let $L$ be a pure $1$-differentially private algorithm for uniformly $(1/2, 1/2)$-PAC learning $\mathcal{POINT}$. Then for every labeled sample $S$, we have $\Expectation_{h \gets L(S)}[|h|] = \infty$.
\end{proposition}
\begin{proof}
Let $t > 0$. We will show that $\Expectation[|h|] \ge t$. Let $n$ be the number of samples used by $L$. Let $\mathcal{H}_t$ be the set of all functions $h : \{0,1\}^* \to \{0,1\}$ with description length $|h| \le 2e^n t$. Let $x, y \in \{0,1\}^*$ be any pair of points such that $h(x) = 0$ or $h(y) = 1$ for every $h \in \mathcal{H}_t$. Such a pair $x, y$ exists by the simple combinatorial Lemma~\ref{lem:combinatorial} stated and proved below, together with the fact that $\mathcal{H}_t$ is finite.
Consider the target concept $c = p_x$ and the distribution $\mathcal{D}$ that is uniform over $(x, 1)$ and $(y, 0)$. Then accuracy of the learner $L$ requires that $\Pr_{S' \sim \mathcal{D}^n}[L(S') \notin \mathcal{H}_t] \ge 1/2$. Since any sample $S'$ can be obtained from $S$ by changing at most $n$ elements of $S$, pure differential privacy implies that $\Pr[L(S) \notin \mathcal{H}_t] \ge e^{-n} / 2$. Hence $\Expectation_{h \gets L(S)}[|h|] \ge 2e^n t \cdot e^{-n} / 2 \ge t$ as we wanted to show.
\end{proof}
\begin{lemma} \label{lem:combinatorial}
Let $\mathcal{S} = \{S_1, \dots, S_n\}$ be a collection of subsets of $[m]$ such that for every pair $x, y \in [m]$ there exists $i \in [n]$ such that $x \in S_i$ and $y \notin S_i$. Then $n \ge \log m + 1$.
\end{lemma}
\begin{proof}
We prove this by induction on $m$. If $m = 2$, then clearly we need $n \ge 2$. Now suppose inductively that $\mathcal{S}$ partitions the set $[m]$ in the described fashion. Then either $S_1$ or $\overline{S_1}$ has size at least $m/2$. If $|S_1| \ge m/2$, then the $n-1$ sets $S_2 \cap S_1, \dots, S_n \cap S_1$ partition a set of size $m/2$ in the described way, so $n-1 \ge \log(m/2) + 1$, i.e., $n \ge \log m + 1$. A similar argument holds if $|\overline{S_1}| \ge m/2$.
\end{proof}
In Appendix~\ref{app:impossible} we generalize this argument to show that \emph{every} infinite concept class is impossible to learn uniformly with pure differential privacy:
\begin{proposition} \label{prop:more-impossible}
Let $L$ be a pure $1$-differentially private algorithm for uniformly $(1/2, 1/2)$-PAC learning an infinite concept class $\mathcal{C}$. Then for every labeled sample $S$, we have $\Expectation_{h \gets L(S)}[|h|] = \infty$.
\end{proposition}
At first glance, this may seem to make the construction of~\cite{GonenHM19} vacuous. However, it is still of interest as it can be made to work in non-uniform model of pure-private PAC learning under the additional assumption that the pure-private learner is highly sample efficient. That is, if $\mathcal{C}_d$ is learnable using $m(d) = O(\log d)$ samples, then the number of experts $N$ remains polynomial. There \emph{is} indeed a computationally efficient non-uniform pure private learner for $\mathcal{POINT}$ with sample complexity $O(1)$~\cite{Beimel19Pure} that can be transformed into an efficient online learner using their algorithm. This does not contradict our negative result Theorem~\ref{thm:private-not-online-pure}, as that pure-private learner uses sample complexity $O(\sqrt{d})$.
\section{Conclusion}\label{sec:conc}
In this paper, we showed that under cryptographic assumptions, efficient private learnability does not necessarily imply efficient online learnability. Our work raises a number of additional questions about the relationship between efficient learnability between the two models.
\paragraph{Uniform approximate-private learning.} In Section~\ref{sec:GHM} we discussed the uniform model of (private) PAC learning and argued that efficient learnability is impossible under pure privacy. It is, however, possible under approximate differential privacy, e.g., for point functions. Thus it is of interest to determine whether uniform approximate-private learners can be efficiently transformed into online learners. Our negative result for non-uniform learning uses sample complexity $\tilde{O}((\log^* d)^{1.5})$ to approximate-privately learn the class $\mathcal{OWS}_d$, so it does not rule out this possibility.
\paragraph{Efficient conversion from online to private learning.} Is a computationally efficient version of~\cite{BunLM20} possible? Note that to exhibit a concept class $\mathcal{C}$ refuting this, $\mathcal{C}$ must in particular be efficiently PAC learnable but not efficiently privately PAC learnable. There is an example of such a class $\mathcal{C}$ based on ``order-revealing encryption''~\cite{BunZ16}. However, a similar adversary argument as what is used for $\mathcal{OWS}$ can be used to show that this class $\mathcal{C}$ is also not efficiently online learnable.
\paragraph{Agnostic private vs. online learning.} Finally, we reiterate a question of~\cite{GonenHM19} who asked whether efficient \emph{agnostic} private PAC learners can be converted to efficient agnostic online learners.
\section*{Acknowledgements}
I thank Shay Moran for helpful discussions concerning the relationship of this work to~\cite{GonenHM19}.
\bibliographystyle{alpha}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 7,532 |
Norma Rangeri
Il Manifesto Global
The duty to choose hope in 2023
Commentary. We must be steadfast in our conviction that, both for the criminal invasion by Putin and the murderous Iranian repression, we need...
Farewell Alberto Asor Rosa, companion and teacher
Commentary. He often sent new articles to our newsroom by fax and with a cover sheet with short quotes, always handwritten, with epigram-like...
There is still new life on the left
Commentary. Elly Schlein, with her candidacy, aims to revive the lost soul of the left, without, however, hiding the fact that her political...
Yes, Prime Minister Meloni is a right-wing politician
Commentary. There were some clumsy tailspins in her introductory speech, such as swearing that she has no sympathy for 'authoritarian regimes,...
Heirs of fascism, the new face of the republic
Commentary. As it was with Berlusconi, we are once again the laboratory of the right in Europe. written by Norma Rangeri Topic Italy October...
Most Italians support the progressive forces. Why couldn't they stop the right?
Commentary. Relaunching the 'common good' should become the symbol of an overall strategy, careful not to take the definitive shape of the long...
The vote that shook Italy, Meloni heads to Chigi Palace
Commentary. There were no surprises. Fratelli d'Italia stormed the polls, outpaced only by the rate of abstentionism. Turnout was the lowest ever. ...
Vote, vote, vote
Commentary. There are only a few days left until September 25, and it has become difficult to animate those on the left to vote, disillusioned by the...
Meloni degrades women's rights at her peril
Commentary. Only those with poor memories can be surprised at the young FdI leader's recent statements on pregnancy termination, because they hearken...
The government falls, the Italian right waits to pounce
Commentary. To be frank, not even Lenin would have been able to avoid the turmoil-ridden situation of the Five Stars in which Giuseppe Conte ended up....
aaf89526d8aeef33494cca839512d8d9 | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 7,459 |
{"url":"https:\/\/golem.ph.utexas.edu\/category\/2007\/09\/deep_beauty_understanding_the.html","text":"## September 19, 2007\n\n### Deep Beauty: Understanding the Quantum World\n\n#### Posted by John Baez\n\nThere were many amazing mathematicians in the 20th century \u2014 people with world-transforming powers, like G\u00f6del, Mac Lane and Grothendieck. But surely, no matter how short your list of greats, John von Neumann would have to be on it. From topics so abstract as the foundations of set theory and quantum mechanics, to topics so practical as game theory, the Manhattan project and the first computers, he seemed to be everywhere\u2026 right at the cutting edge.\n\nSoon there will be a symposium honoring the 75th anniversary of von Neumann\u2019s book The Mathematical Foundations of Quantum Mechanics:\n\nHans Halvorson is a philosopher at Princeton. I really like his idea that new mathematics may be needed to make quantum mechanics more intelligible, and that philosophers should get involved. I think it\u2019s true.\n\nUnfortunately attendance is by invitation only. Participants include \u010caslav Brukner, Jeffrey Bub, Bob Coecke, Andreas Doering, Lucian Hardy, Chris Isham, Simon Kochen, Klaas Landsman, Mikl\u00f3s R\u00e9dei, and Stephen Summers. Caf\u00e9 regulars will recall that Coecke, Doering and Isham have all visited here. A number of graduate students are also attending the symposium, including my student Mike Stay and Isham\u2019s student Jamie Vicary, both caf\u00e9 regulars themselves.\n\nI\u2019ll try to report more on this symposium later. Right now you can see a very preliminary draft of a paper Spans in Quantum Theory, which is what I\u2019ll be talking about.\n\nAmong other things, I want to start by making the point that categories of states and processes in quantum theory tend to be symmetric monoidal categories with duals, also known as dagger-compact categories. This point has already been made by Bob Coecke \u2014 and since he\u2019ll also be at the symposium, he and I should figure out how to join forces in explaining this to everyone. But then, I want to explain how such categories naturally arise as categories of spans, and what spans have to do with Feynman path integrals. And then, I want to talk about spans of groupoids, and the whole groupoidification program. The fun part will be seeing how much I can cover \u2014 intelligibly! \u2014 in an hour.\n\nBy the way: speaking of the \u201cintelligibility\u201d of quantum mechanics, I can\u2019t help but remember a line by von Neumann that always bugged me. He\u2019s supposed to have said, in response to annoying questions from some student:\n\nIn mathematics you don\u2019t understand things. You just get used to them.\n\nI don\u2019t agree with that, and it\u2019s hard for me to imagine a good mathematician saying that \u2014 unless he had something very specific in mind. Maybe he was just trying to get rid of the student?\n\nPosted at September 19, 2007 9:21 PM UTC\n\nTrackBack URL for this Entry:\u00a0\u00a0 http:\/\/golem.ph.utexas.edu\/cgi-bin\/MT-3.0\/dxy-tb.fcgi\/1432\n\n### Re: Deep Beauty: Understanding the Quantum World\n\nI don\u2019t agree with that, and it\u2019s hard for me to imagine a good mathematician saying that \u2014 unless he had something very specific in mind.\n\nIt\u2019s clearly an outrageous exaggeration at best, probably for the humor value. However, there\u2019s a kernel of truth to it. There\u2019s a lot of mathematical machinery that can\u2019t be understood via a single ah-ha moment: mastering it requires seeing lots of different examples, recognizing new examples one wasn\u2019t expecting, seeing how the machinery really tells one what one wants to know, etc. This is intrinsically a slow process, and there\u2019s no sense sitting around waiting to magically \u201cget it\u201d.\n\nPosted by: Anonymous on September 20, 2007 5:16 AM | Permalink | Reply to this\n\n### Re: Deep Beauty: Understanding the Quantum World\n\nI think of, sympathize with, and use that quote all the time. It\u2019s hard for me to imagine a good mathematician not thinking it. Perhaps we should take a poll!\n\nPosted by: Allen Knutson on September 20, 2007 3:10 PM | Permalink | Reply to this\n\n### Re: Deep Beauty: Understanding the Quantum World\n\nI didn\u2019t see Anon\u2019s response when I posted mine, and I disagree with that one too! So I guess I should say what truth I find in the von Neumann quote.\n\nThere are a number of results I\u2019ve seen the proofs of \u2013 even short ones \u2013 that I\u2019ve initially found unsatisfying and unintuitive. Is that a failure of understanding? Perhaps. But I\u2019ve often taken it as more of a sign that I should replace my axiom system, and redefine such things as \u201cintuitive\u201d.\n\nThis takes effort \u2013 basically, for a while I have to work at remembering when faced with a new problem to check if this not-yet-intuitive theorem may help. Once it does so enough times, poof, it feels intuitive.\n\nFor example, I am in the middle of this process with Frobenius splitting. The proofs using it are short and incontrovertible, and I\u2019m getting a sense of when it\u2019s useful. If you \u201cunderstand\u201d Frobenius splitting then more power to you, sir.\n\nPosted by: Allen Knutson on September 20, 2007 3:18 PM | Permalink | Reply to this\n\n### Re: Deep Beauty: Understanding the Quantum World\n\nI agree that there are lots of things in mathematics that are hard to understand, that we must temporarily accept in order to get on with business. I also don\u2019t think we reach a definitive understanding of anything in a finite amount of time. But, I\u2019d never say that in math we \u201cdon\u2019t understand things, just get used to them\u201d. I spend a lot of time trying to understand math, and I often make progress.\n\nMost of all I like to tackle facts that are easy to prove but hard to understand. I\u2019ve spent over a decade trying to understand why there are 4 normed division algebras over the reals, with dimensions 1, 2, 4, and 8. I know plenty of proofs: proofs that use Clifford algebras, proofs that use string diagrams\u2026 but I still feel dissatisfied. I\u2019m not sure why. Maybe it\u2019s because all the proofs seem a bit magical. Maybe it\u2019s because I don\u2019t know whether all these proofs are fundamentally the same.\n\nSo, I wouldn\u2019t say I understand this fact. But, I\u2019m certainly not just \u201cgetting used to it\u201d, which suggests some sort of passive resignation. I\u2019m gonna keep fighting to understand this stuff until I die!\n\nPosted by: John Baez on September 20, 2007 5:15 PM | Permalink | Reply to this\n\n### Re: Deep Beauty: Understanding the Quantum World\n\nIn contrast to a quote from Thom:\nvery easy to see, very hard to prove\n\njim\n\nPosted by: jim stasheff on September 20, 2007 5:23 PM | Permalink | Reply to this\n\n### Re: Deep Beauty: Understanding the Quantum World\n\nSeveral months ago I tackled the proof of a proposition generalizing a crucial result by someone else whose proof I thought was particularly opaque and unsatisfying. I thought that my particular approach and more advanced machinery would certainly make a more transparent and intuitive proof. After sweating over it for a week, I finally had it nailed down, but in the end my proof was no less transparent than the original, and I still have a hard time believing that it actually works. What does this mean when this happens? Does this necessarily imply that we just are not understanding something from the correct viewpoint yet, or are there some things that will always remain mysteriously opaque?\n\nPosted by: Richard on September 22, 2007 3:30 AM | Permalink | Reply to this\n\n### Re: Deep Beauty: Understanding the Quantum World\n\nMaybe he\u2019s not talking about the understanding of the proofs, he\u2019s talking about why one need to use\/introduce certain concepts\/methods in mathematics at all. We understand them only through their results.\n\nPosted by: tytung on September 20, 2007 5:49 PM | Permalink | Reply to this\n\n### Re: Deep Beauty: Understanding the Quantum World\n\nI think naively that the all point is in the meaning of the word \u201cunderstand\u201d, and in the word \u201cthings\u201d. After all, what does it mean to understand, in general, and in particular, understanding maths ? If one means to be able, say, to reproduce a proof, justifying how every logical step follows from the precedent one, then I think you can say you can understand \u201cthings\u201d, in this case the proof. But it occurred a lot of time to me that even if i did this, i wasn\u2019t feeling somehow comfortable with the results, and then one day I woke up and thought:\u201dThat\u2019s how it was!\u201d. Probably it was induced from \u201cgetting used\u201d to the results, or, in some irrational way, from convincing oneself of the results, beyond the mathematical proof, or maybe from \u201cunderstanding\u201d\u2026\nAnyway, I do agree with John when he says we can never learn everything about a subject or a concept in a finite amount of time, but that we try to add bits from time to time\u2026\n\nPosted by: Alessandro on September 21, 2007 5:48 PM | Permalink | Reply to this\n\n### Re: Deep Beauty: Understanding the Quantum World\n\nWhat does it mean to \u201cunderstand\u201d something? To me, you understand something if you can calculate with it and have a mental picture of what is going on. What more can there be to understanding?\n\nE.g., I think I understand the real numbers - I can calculate with them, and my mental picture is a bunch of points on a line. Now, there are constructions of the real numbers from set theory, but I never understood those constructions and now I have forgotten everything about them. Does this limit my understanding of real numbers?\n\nAnother example is the Dirac delta function. Mathematicians often claim that nobody understood the delta function before Schwartz developed distribution theory, but I think that is nonsense. Dirac had a mental picture (an infinitely high, infinitely thin peak) and he certainly could calculate, so in the sense above he had an understanding. That mathematicians didn\u2019t understand, or didn\u2019t like, what Dirac was doing is another thing.\n\nIronically, von Neumann seems to have been very hostile to Dirac\u2019s techniques. In his mathematical foundations of QM, he argues that his Hilbert space methods are vastly superior to Dirac, because he does not have to pretend the existence of something know not to exist (i.e. the delta function).\n\n### Re: Deep Beauty: Understanding the Quantum World\n\nWhat does it mean to \u201cunderstand\u201d something? To me, you understand something if you can calculate with it and have a mental picture of what is going on. What more can there be to understanding?\n\nMaybe asking \u201cwhy\u201d you calculate in the way you do and not in an another..\nAfter all, you have first to define rules for those calculations, and then show (within the standards of mathematics) that there is a model consistent with them, otherwise your calculations will make only sense in a formal way.\n\nPosted by: Alessandro on September 22, 2007 5:21 PM | Permalink | Reply to this\n\n### Re: Deep Beauty: Understanding the Quantum World\n\nAs I recall, Dirac never did anything with the delta `function\u2019 that required it to be a true function, i.e. he would integrate against test functions.\n\nI think some of the disciples were less creful in transmitting the message.\n\nPosted by: jim stasheff on September 22, 2007 9:45 PM | Permalink | Reply to this\n\n### Re: Deep Beauty: Understanding the Quantum World\n\nDirac talks about generalized functions, and this seems to have irritated von Neumann and other mathematicians. But of course he used the delta function in ways that make sense, which essentially means that he followed the rules of distribution theory.\n\nI have recently been led to use delta functions in an unconventional and potentially dangerous way. A decade ago, I asked you about commutation relations in jet space. More precisely, the question was prompted by your secret paper and formulated for the anti-bracket, but this is not essential. If\n\n[\u03c6(x), \u03c0(y)] = i \u03b4(x-y),\n\nwhat is the bracket between the Taylor coefficients \u03c6m and \u03c0n, where m and n are multi-indices? The answer is obviously\n\nm, \u03c0n] = i(-1)nm+n\u03b4(0),\n\nwhere \u2202m denotes the m:th order derivative. That this must be true follows immediately if you pretend that \u03b4 is a smooth function (e.g. a narrow Gaussian), but it took me almost a decade to write down this formula because I had a mental block against \u03b4(0).\n\nNote that the formula that you might naively expect,\n\nm, \u03c0n] = i \u03b4nm,\n\nonly works if \u03c0 has an upper multi-index.\n\n### Re: Deep Beauty: Understanding the Quantum World\n\nHello Thomas, it has been over three years since I last chatted with you and Urs Schreiber back in the old sci.physics.strings.\n\nFor another \u201cunconventional\u201d use of the Dirac delta function see this paper [1] which uses the function within a Bayesian probability context for use in artificial intelligence. By the way, I am now working with a company doing research into intersections between probability, AI and mathematical finance which is why I happen to know about this paper. (One of these days, when I get some extra time, I will try to get some grip on what this \u201cprogic\u201d is about.)\n\nRegarding my previous question in this thread, I have found a paper [2] entitled \u201cA Bundle Representation for Continuous Geometries\u201d which I will have to read when I go to the MIT Science Library in a few days. (I also received a potentially useful tip from Chris Isham via email.)\n\nI would now like to take this moment to personally thank John Baez for introducing me to the concepts of buildings, categorified bundles\/gerbes and n-categories (as well as Cayley and Jordan algebras). Wow, does John know a lot about mathematics !\n\n[2] John Harding and Melvin Janowitz, Advances in Applied Mathematics, v19 (2), pp.282-293 (1997)\n\nPosted by: Charlie Stromeyer Jr on September 24, 2007 2:41 PM | Permalink | Reply to this\n\n### Re: Deep Beauty: Understanding the Quantum World\n\nYou\u2019re welcome, Charlie! Nothing makes me happier than turning someone on to a cool new concept.\n\nPosted by: John Baez on September 24, 2007 3:56 PM | Permalink | Reply to this\n\n### Re: Deep Beauty: Understanding the Quantum World\n\nAnother reminder NOT to raise and lower indices casually\n\nPosted by: jim stasheff on September 24, 2007 5:04 PM | Permalink | Reply to this\n\n### Re: Deep Beauty: Understanding the Quantum World\n\nI think one can detect a student\u2019s (lack of) understanding by the way she looks. So, probably future software could detect that too. Perhaps another method to estimate the degree of understanding is the degree at which readers are troubled by typing-errors in formulas, e.g. because \u201cunderstanding\u201d enables to correct them very fast. Else, my impression is that \u201cunderstanding\u201d is the build-up of a conscious (visual) mental model + a subconscious model of the speakers\/writers way to think, probably provided by mirror neurons.\n\nConc. Neumann: I heard that he intended to become a banker until his father insist him to study math. Did that lack of intrinsic initial motivation show up later in some way?\n\nPosted by: anonymous on September 22, 2007 5:52 PM | Permalink | Reply to this\n\n### Re: Deep Beauty: Understanding the Quantum World\n\nI think some mathematicians felt von Neumann was too well-adjusted to be one of them \u2014 he dressed in suits, liked parties, etc. Maybe he didn\u2019t suffer from the low-level Asperger syndrome so common in mathematicians. But on the other hand:\n\nJohn was intellectually precocious and had a photographic memory. As a young child he learned to speak German, French, and classical Greek, and amassed an encyclopaedic knowledge of historical events. A favorite party trick of his was to memorize a page of the telephone book. He would ask a visitor to the Neumann household to select a page of the book, which he would then read through a few times. He would then hand the book back to the visitor and ask them to quiz him on the page, say by giving him a name and asking for the phone number, or getting him to recite a sequence of names, addresses and numbers in order. He was rarely wrong.\n\nWhen John entered the local school in 1911, aged 8, the mathematics teacher recognized his genius at once, and arranged for special tuition from Gabor Szego, a well known mathematician at the University of Budapest. This was surely made easier by the fact that at the same school, just one year ahead of John, was another future mathematical giant, Eugene Wigner. By the time John completed his high school education in 1921, his prowess in mathematics was clear to all, and he was accepted to study mathematics at the University of Budapest.\n\nPosted by: John Baez on September 24, 2007 3:11 AM | Permalink | Reply to this\n\n### Re: Deep Beauty: Understanding the Quantum World\n\nThanks! Here an online Asperger test. How did von Neumann (or poeple who knew him) describe his way of thinking?\n\nPosted by: anonymous on September 24, 2007 8:26 AM | Permalink | Reply to this\n\n### Re: Deep Beauty: Understanding the Quantum World\n\nJohn von Neumann was famous for the speed and accuracy of his mental calculations.\n\nMost mathematicians know the joke where one of von Neumann\u2019s friends challenged him to quickly solve the famous bee and train puzzle.\n\nThe puzzle goes like this: two trains are 20 miles apart on the same track, each heading towards each other at 10 miles per hour. A bee takes off from the nose of one train at 20 miles per hour, towards the other train. As soon as the bee reaches the other train, it turns around and heads off at 20 miles per hour back towards the first train. It continues to do this, going back and forth over and over, infinitely many times, until the trains collide, killing the bee. How far did it fly?\n\nThis puzzle is also called \u2018the mathematician\u2019s trap\u2019, because it tempts you to set up an infinite geometric series and figure out how to sum it. That takes a while. But, you can avoid the trap by simply figuring out how long it takes for the trains to collide and multiply this time by the bee\u2019s speed!\n\nAnyway, when a friend of von Neumann posed this puzzle to him, he instantly replied: \u201cTwenty miles.\u201d\n\nHis friend said \u201cHa! Good! That was quick \u2014 most mathematicians make the mistake of trying to sum the geometric series.\u201d\n\nVon Neumann looked puzzled and replied: \u201cBut I did sum the series.\u201d\n\nAnyway, I can\u2019t find any evidence that von Neumann planned to be a banker until his father convinced him to do math. On the contrary, his father Max was a banker, and according to Keith Devlin:\n\nUnfortunately, John\u2019s love for mathematics did not meet with his father\u2019s approval when it came to deciding what to study at university. Max Neumann wanted his son to prepare for a career in business. When John made it clear that business studies were not to his liking, father and son settled on chemistry as a compromise subject. And so John entered the University of Berlin to major in chemistry, to please his father, while at the same time enrolling at the University of Budapest to study mathematics, arranging to be tutored \u2013 at a distance \u2013 by a young assistant there. Being unable to attend any of the mathematics lectures in Budapest clearly did not hamper John\u2019s progress; the following year he wrote his first mathematics paper, jointly with his tutor. In 1926, he was awarded a diploma in chemical engineering from the University of Zurich \u2013 where he transferred from Berlin in 1923 \u2013 and a doctorate in mathematics from Budapest. His thesis topic in mathematics was set theory, the foundational subject still in its infancy at that time.\n\nBy his mid-twenties, von Neumann \u2013 clearly now a mathematician rather than a chemist \u2013 was a mathematical celebrity, famous throughout the international mathematical community. After obtaining his Ph.D. he spent a year with the great David Hilbert in Goettingen and lectured in Berlin and Hamburg. In addition to his work in set theory, he did groundbreaking work in measure theory, the theory of real variables, and game theory. Not content with that, he also turned his attention to quantum theory, where in the two year period 1927-29 he more or less single-handedly developed the entire mathematical foundation of the subject, pioneering work that led to an invitation to go to Princeton. His research output during this early European part of his professional career was about one paper a month. (Tenure committees at present day universities generally view one or two papers a year as indicative of a good mathematical researcher!)\n\nPosted by: John Baez on September 24, 2007 5:38 PM | Permalink | Reply to this\n\n### Re: Deep Beauty: Understanding the Quantum World\n\nHello everyone, I have an inquiry about generalizing von Neumann\u2019s concept of continuous geometry. Please consider this question:\n\nIs it possible to have a \u201cspace\u201d X such that an infinite X is the limit of a sequence of increasingly large finite instances of X?\n\nIf the answer to this question is \u201cyes\u201d then such a construction may have important implications for mathematics. For instance, the celebrated topologist Michael Freedman has suggested it might help\nwith unsolved problems in computational complexity and geometry [1]. Also, the theoretical physicist and string theorist Raphael Bousso once posed this question:\n\n\u201cIs there a sequence of theories with finite-dimensional Hilbert spaces such that string theory emerges in the infinite-dimensional limit?\u201d [2]\n\nI am further reminded of Edward Witten\u2019s conjecture about U(\u221e)-valued Chan\u2013Paton factors [3], and of various string theory papers about infinite towers of states. (By the way, even though I am not a physicist I am supposed to have some expertise in the math of high energy physics as you can see, for example, in the Acknowledgements sections of two papers [4, 5]. My own ideas in this area involved finding new connections within M-theory via Jordan algebras, but I don\u2019t wish to discuss this now because it would be too off-topic.)\n\nThe great John von Neumann introduced the concept of continuous geometry which can be viewed as the limit of increasingly large finite instances of projective geometry [6]. Is there a way to sufficiently generalize von Neumann\u2019s concept? For example, should I consider using bundles?\n\n(I did find a paper about a tower of n-gerbes with the {infinity}-gerbe [7]. Would it make any sense to try to define a notion of convergence for such a tower?)\n\nThank you,\nCharlie Stromeyer Jr.\n\n[2] http:\/\/www.physics.ucsb.edu\/~giddings\/Mquest.html\n\n[3] http:\/\/xxx.lanl.gov\/abs\/hep-th\/0007175\n\n[4] http:\/\/xxx.lanl.gov\/abs\/gr-qc\/0212096\n\n[5] http:\/\/xxx.lanl.gov\/abs\/hep-th\/0407122\n\n[6] http:\/\/planetmath.org\/encyclopedia\/VonNeumannLattice.html\n\n[7] http:\/\/xxx.lanl.gov\/abs\/math\/0301271\n\nPosted by: Charlie Stromeyer Jr on September 23, 2007 12:59 AM | Permalink | Reply to this\n\n### Re: Deep Beauty: Understanding the Quantum World\n\nYour question seems very open-ended, and I can\u2019t really answer it, but I urge you to dig deeper into von Neumann algebras, particularly the hyperfinite type $II_1$ factor, since:\n\n\u2022 It\u2019s approximable by finite-dimensional matrix algebras \u2014 that\u2019s what \u2018hyperfinite\u2019 means.\n\u2022 It\u2019s self-similar \u2014 in fact it\u2019s the noncommutative geometry analogue of the Cantor set!\n\u2022 It\u2019s deeply connected to Dynkin diagrams, topological quantum field theory, and many other things, thanks in large part to the work of Vaughan Jones.\n\nSo, it may not be what you\u2019re looking for, but it\u2019s a wonderful thing. Here\u2019s a decent place to start, after you\u2019ve read the basics in the link above:\n\n\u2022 F. M. Goodman, P. de la Harpe, and V. F. R. Jones, Coxeter graphs and towers of algebras, MSRI Series, Berkeley, Springer, New-York, 1989.\n\nYou mention Michael Freedman: he\u2019s now working on topological quantum computers, which are based on precisely this sort of math.\n\nTo add to the fun, the radical logician Jean-Yves Girard is now trying to apply the hyperfinite type $II_1$ factor to logic.\n\nPosted by: John Baez on September 24, 2007 4:41 PM | Permalink | Reply to this\n\n### Re: Deep Beauty: Understanding the Quantum World\n\nThanks for the suggestion, John, and I will dig deeper into von Neumann algebras if only because thinking about infinity is my favorite use of mathematics.\n\nI first read about these algebras years ago in TWF 175 and so I should credit you with introducing me to this concept as well. I then read further about von Neumann algebras in the book \u201cJordan, real and Lie structures in operator algebras\u201d by S. Ayupov et al.\n\nYou taught me that the reals (R), the complexes (C) and the quaternions (H) can be obtained from the octonions (O), and then I learned, for example, that the concept of Hilbert Space can be defined over O, and also that there are Jordan operads.\n\nPosted by: Charlie Stromeyer Jr on September 25, 2007 9:53 PM | Permalink | Reply to this\n\n### Re: Deep Beauty: Understanding the Quantum World\n\nI remember talking with you a lot on sci.physics.research. Then you sort of disappeared \u2014 from my viewpoint, that is.\n\nPosted by: John Baez on September 26, 2007 3:47 AM | Permalink | Reply to this\n\n### Re: Deep Beauty: Understanding the Quantum World\n\nWell, I migrated from sci.physics.research over to sci.physics.strings, but then (due to the demands of work) I did not have time to think about quantum theory, e.g., the last time I thought about string theory was well over three years ago - not since my last post on s.p.s.\n\nThe first paper to succeed in defining Jordan operads is the paper \u201cJordan Triples and Operads\u201d by A.V. Gnedbaye and M. Wambst in Journal of Algebra 231, 744-757 (2000). From page 745:\n\n\u201cJordan algebras are not quadratic in the standard sense of operads. Nevertheless, as pointed out by Loday, the triple Jordan product which can be associated with any Jordan algebra satisfies a quadratic relation. It is then natural to consider ternary algebras, the Jordan triple systems, which are quadratic in the sense of algebras. In this article, we describe the quadratic operad of Jordan triple systems and prove that its dual is the quadratic operad of partially associative and partially antisymmetric ternary algebras.\u201d\n\nI rate this paper an A- overall only because I wish the authors had said something about potential applications of their work.\n\nAs you know, the role of von Neumann and Jordan algebras for quantum theory is an unfinished story. I first learned something about quantum mechanics when I was 15 years old, and now (a full 20 years later) I can say with all confidence that I still do not really understand quantum theory.\n\n(A further reason why I prefer mathematics over physics might have something to do with the fact that three of my ancestors worked for the NSA. Perhaps not surprisingly, it appears that only a tiny amount of what they worked on has been declassified, but what little has been declassified is historically interesting if you want to see it, e.g., in this NSA tribute webpage:\n\nhttp:\/\/www.nsa.gov\/honor\/honor00029.cfm\n\nPosted by: Charlie Stromeyer Jr on September 26, 2007 9:02 PM | Permalink | Reply to this\n\n### Re: Deep Beauty: Understanding the Quantum World\n\nSo I read the paper \u201cA Bundle Representation for Continuous Geometries\u201d and it is related to the von Neumann algebra you mention above.\n\nJ. von Neumann introduced continuous geometry as a point-free generalization of projective geometry. He discovered that a continuous geometry is a complete, irreducible, complemented and modular lattice which is both join and meet continuous. Years later, F. Maeda showed that a continuous geometry is a Z-lattice.\n\nThe above paper shows that any continuous geometry can be represented as the continuous sections of a bundle whose stalks are all irreducible continuous geometries. The topology of this bundle is such that the subspace topology on each stalk is the usual metric space topology of an irreducible continuous geometry.\n\nThis bundle representation is related to the Pierce sheaf of the continuous geometry. A Pierce sheaf is the name for a sheaf naturally associated with each bounded lattice.\n\nFurther, the dimension functions on the stalks form a continuous map from the bundle into the reals. As the authors note, this is similar to the classical representations of von Neumann algebras as rings of continuous functions.\n\nWhat the authors do not mention is that the type II_1 factor gives rise to a lattice with no atoms (i.e. no minimal non-zero elements) and its dimension function takes all possible values in the interval [0, 1]. The lattice is modular but not orthomodular.\n\n(I will also mention that I found an example of convergence to an infinite set in the book \u201cFuzzy Topology\u201d by N Palaniappan. The set of fuzzy points to which a fuzzy net converges is, in general, infinite - yet putting a restriction on the supports obtains uniqueness of convergence related to the iterated limits theorem.)\n\nPosted by: Charlie Stromeyer Jr on September 27, 2007 9:45 PM | Permalink | Reply to this\n\n### Re: Deep Beauty: Understanding the Quantum World\n\nSo I followed your advice to dig deeper into von Neumann algebras and I have found various papers in category theory based upon these algebras which I will have to start reading when I return to the MIT Science Library (as well as some papers on Pierce sheaves).\n\nHere is a question you may be able to answer:\n\nChris Isham told me via email that I might consider the theory of measures on infinite dimensional vector spaces. Such a measure can be thought of as an inverse limit of measures defined on finite dimensional vector spaces. Since I am not an expert on the theory of measures would you please explain to me if\/how this concept might translate over to the setting of groupoids and vector spaces? Thanks.\n\nPosted by: Charlie Stromeyer Jr on September 28, 2007 5:22 PM | Permalink | Reply to this\n\n### Re: Deep Beauty: Understanding the Quantum World\n\nCharlie wrote:\n\nChris Isham told me via email that I might consider the theory of measures on infinite dimensional vector spaces. Such a measure can be thought of as an inverse limit of measures defined on finite dimensional vector spaces. Since I am not an expert on the theory of measures would you please explain to me if\/how this concept might translate over to the setting of groupoids and vector spaces?\n\nI\u2019m not sure what you\u2019re asking here, so I\u2019ll just blabber random words.\n\nThe concept Chris Isham mentioned is often called a \u2018cylinder measure\u2019. Cylinder measures are very important in quantum field theory, where one wants to integrate over infinite-dimensional vector spaces. I believe there\u2019s a good introduction to cylinder measures in the first volume of Choquet-Bruhat et al\u2019s Analysis, Manifolds and Physics.\n\nAlong with Irving Segal and Zhengfang Zhou, I helped write a book which includes a very detailed treatment of cylinder measures. However, Segal demanded that we call them \u2018distributions\u2019 instead of cylinder measures. You can get this book free online \u2014 but unless you know a fair amount of analysis, you\u2019ll pay a high price in blood, sweat and tears when you try to actually read it, since I helped write it before I learned the importance of a gentle expository style, and Segal didn\u2019t believe in babying the reader.\n\nIn a feeble attempt to make sense of your actual question, I\u2019ll say that Jeff\u2019s paper thoroughly groupoidifies the quantum harmonic oscillator with one degree of freedom \u2014 which amounts to groupoidifying the theory of integrals with respect to a Gaussian measure on $\\mathbb{R}$. Generalizing to finitely many degrees of freedom is trivial, and it amounts to groupoidifying the theory of integrals with respect to a Gaussian measure on $\\mathbb{R}^n$. Everything still works when $n = \\infty$, but then the Gaussian measure is really a cylinder measure.\n\nPosted by: John Baez on September 28, 2007 11:31 PM | Permalink | Reply to this\n\n### Re: Deep Beauty: Understanding the Quantum World\n\nThanks, John, because you did answer my question and because you taught me a new concept. As you could tell, I had not heard of \u2018cylinder measure\u2019 before, and nor have I read Jeff\u2019s paper that you referred me to. I may have some more question(s) on this topic after I look at some relevant papers at the MIT Science Library.\n\nIn the meantime, here is something for Urs Schreiber: I found a recent paper about von Neumann bimodules by Michael Skeide.\n\nPosted by: Charlie Stromeyer Jr on September 29, 2007 5:35 PM | Permalink | Reply to this\n\n### Re: Deep Beauty: Understanding the Quantum World\n\nJohn, since you are a top expert on cylinder measures and quantum gravity you may want to look at this paper called \u201cQuantization by non-Abelian promeasures\u201d by C J S Clarke in J. Phys. A: Math. Gen. 23 (1990) pp. 4463-4470. The abstract is:\n\n\u201cA new method is proposed for the non-perturbative quantization of certain nonlinear field theories (group-bundle theories), based on a generalization of the idea of a promeasure from vector spaces to infinite-dimensional Lie groups. The quantum theory is not automatically finite, but there is a natural way of imposing a momentum cut-off, leading to the possibility of renormalization. The method relies on the geometrical structure of the classical theory and so may provide clues for the quantization of gravity.\u201d\n\nI am just now looking at this paper for the first time and will try to think if it has any meaning in a gerbed context (and I would welcome any advice on this matter).\n\nAlso, for Urs Schreiber: I have found an example of Hopf-bimodules based upon Hopf-von Neumann algebras within the context of quantum groupoids in the work of Michel Enock, e.g., in his paper \u201cOn Lesieur\u2019s Measured Quantum Groupoids\u201d which is in the arxiv as 0706.1472\n\nLater, I will say something about how von Neumann algebras are related to the quadratic operad of Jordan Triple Systems which I mentioned earlier in this thread.\n\nPosted by: Charlie Stromeyer Jr on September 30, 2007 5:13 PM | Permalink | Reply to this\n\n### Re: Deep Beauty: Understanding the Quantum World\n\nCharlie wrote:\n\nC J S Clarke wrote:\n\nA new method is proposed for the non-perturbative quantization of certain nonlinear field theories (group-bundle theories), based on a generalization of the idea of a promeasure from vector spaces to infinite-dimensional Lie groups.\n\nInteresting. My work on loop quantum gravity also used a generalization of promeasures (= cylinder measures) from vector spaces to certain infinite-dimensional Lie groups. If you\u2019re interested, try this:\n\nI wonder which infinite-dimensional Lie groups Clarke was studying\u2026\n\nPosted by: John Baez on October 1, 2007 5:37 AM | Permalink | Reply to this\n\n### Re: Deep Beauty: Understanding the Quantum World\n\nClarke uses his own \u201ckink theories\u201d but I have not yet looked at his earlier paper he refers to which rigorously describes these theories that he says \u201care ones where the set of field values at a point forms a Lie group, with the group operation being independent of any choice of gauge. The collection of all these field values at all points forms, in geometrical language, a fibre bundle whose fibres are groups.\u201d In his approach, what is needed is the gauge-invariance of the promeasures.\n\nPosted by: Charlie Stromeyer Jr on October 1, 2007 8:40 PM | Permalink | Reply to this\n\n### Re: Deep Beauty: Understanding the Quantum World\n\nJohn, while you are at your conference, I learned some things that I want to share with the Cafe. The first two involve measures and an answer to your question about which Lie groups C J S Clarke was studying. The third is an affirmative answer to my question earlier in this thread from the mathematician Steve Vickers.\n\n1) The limit arising within von Neumann\u2019s continuous geometry is called \u201cprofinite\u201d and you can read more about profinite topology and groups in this wikipedia entry on profinite groups.\n\nIn category theory terms, this is a special case of a (co)filtered limit construction.\n\nBerkeley mathematician Mark Haiman generalized von Neumann\u2019s concept in his paper \u201cOn realization of Bjorner\u2019s \u2018continuous partition lattice\u2019 by measurable partitions\u201d in Trans. Amer. Math. Soc. v343(2) (1994) 695-711. Abstract:\n\nBjorner showed how a construction by von Neumann of examples of continuous geometries can be adapted to construct a continuous analogue of finite partition lattices. Bjorner\u2019s construction realizes the continuous partition lattice abstractly, as a completion of a direct limit of finite lattices. Here we give an alternative construction realizing a continuous partition lattice concretely as a lattice of measurable partitions. This new lattice contains the Bjorner lattice and shares its key properties. Furthermore its automorphism group is the full automorphism group (mod 0) of the unit interval with Lebesgue measure, whereas, as we show, the Bjorner lattice possesses only a proper subgroup of these automorphisms.\n\n2) C J S Clarke studies \u2018kink theories\u2019 which \u201care ones where the set of field values at a point forms a Lie group, with the group operation being independent of any choice of gauge. The collection of all these field values at all points forms, in geometrical language, a fibre bundle whose fibres are groups.\u201d\n\nAs an example, in his \u201cQuantization by non-Abelian promeasures\u201d paper he uses the (3+1)-dimensional sine-Gordon equation over a fixed (curved) spacetime as a bundle with fiber SO(n) or SO(p,q).\n\nThen, \u201cIf we take the Lie algebra of the group formed by the field values at a point, and do this at every point, we get a related field theory whose bundle is a vector bundle. Linear equations of motion in this Lie-algebra theory correspond to nonlinear equations in the Lie-group-bundle theory, in a natural way.\u201d\n\nAt the end of the paper, Clarke has to generalize a Gaussian distribution to a nonabelian Lie group, and does so with a heat equation kernel.\n\nThis allows him to define L^2(G,u_q\u2019) for the quantum Hilbert space. Since I am not a physicist, I will ask the question:\n\nDoes this space L^2(G,u_q\u2019) have any physical significance?\n\n3) Earlier in this thread, I asked whether there could be an infinite X as the limit of increasingly large finite instances of X.\n\nSteve Vickers says this is possible, and here I take the liberty of pasting his reply so that you know to whom it is attributed:\n\nDear Charlie,\n\nJon Awbrey already hinted at an important part of the background in his reply to you [over three years ago in [1]]:\n\n>>>>>>>>>>>>>>>>>>>>>>>>>>>>>\nTopological Information Spaces\n\nCharlie,\n\nWill look up some references later \u2013\nor you can probably search-engineer\nyour own well enough \u2013 but there\u2019s\na lot of topological thinking in the\nDana Scott, Barwise, Seligman terrain\nof thought, going way back to the 60\u2019s,\npartly due to the topological character\nof \u201cinformation domains\u201d that constitute\ncomplete partial orders, if memory serves,\nwhich it often doesn\u2019t.\n\nJon Awbrey\n\nDana Scott used his (information) domains to interpret computable functions as continuous maps. The approach is now well established in computer science, with applications (\u201cdenotational semantics\u201d) to giving mathematical meanings to computer programs.\n\nThe domains are non-Hausdorff and can appear strange to a traditional topologist. They make essential use of the specialization order, x specializes y if y in the closure of {x} - in other words, every open containing y also contains x. (This is just equality in a T1 space.) High in the specialization means high in information content (more open neighbourhoods), and the low points can often be understood as approximations to the high ones. This leads to your \u201cinfinite X is the limit of a sequence of increasingly large finite instances of X\u201d.\n\nSteve Vickers.\n\nPosted by: Charlie Stromeyer Jr on October 5, 2007 12:26 AM | Permalink | Reply to this\n\n### Re: Deep Beauty: Understanding the Quantum World\n\nThanks to what Steve Vickers told me, I am now reading about specialization order and Scott topology in the book \u201cContinuous lattices and domains\u201d by G. Gierz et al.\n\nSpeaking of lattices, the above paper by Mark Haiman refers to a paper by V.A. Rohlin called \u201cOn the fundamental ideas of measure theory\u201d in Amer. Math. Soc. Transl. 1(10) (1962) 1-54 which includes, e.g., a fully developed theory of measurable partitions of the real unit interval.\n\nAlso, speaking of lattices again, Vaughan Jones told me via email that Kaplansky proved that a complete orthocomplemented modular lattice is a continuous geometry. For anyone who might be interested, this lattice is related to quantum probability theory in the work of Zdenka Riecanova, including a recent generalization of the concept.\n\nPosted by: Charlie Stromeyer Jr on October 6, 2007 7:20 PM | Permalink | Reply to this\n\n### Re: Deep Beauty: Understanding the Quantum World\n\nIn reply to what Steve Vickers said above, Professor Dana Scott told me via email about his initial example in the history of (profinite) interval analysis - a subject which is new to me but which Bas Spitters is an expert on.\n\nMore specifically, Prof. Scott referred me to the work of Prof. Abbas Edalat which is fascinating because it contains a variety of examples of profinite maths. For instance, Prof. Edalat defines a notion of derivative of a real-valued function on a Banach space called the L-derivative which is constructed by a generalization of Lipschitz constant of a map. For functions on finite Euclidean spaces, any continuous function and its L-derivative are profinite.\n\nThis is cool because I am trying to understand whatever I can about profinite maths. With the sole exception of von Neumann\u2019s continuous geometry, the subject of profinite maths is new to me, and I like this wikipedia entry on profinite topology and groups because both the links within and the references below are clickable.\n\nPosted by: Charlie Stromeyer Jr on October 10, 2007 5:19 PM | Permalink | Reply to this\n\n### Re: Deep Beauty: Understanding the Quantum World\n\nThere\u2019s now an official webpage for this conference, which contains a subliminal advertisement for the $n$-Category Caf\u00e9.\n\nPosted by: John Baez on September 26, 2007 3:43 AM | Permalink | Reply to this\n\n### Re: Deep Beauty: Understanding the Quantum World\n\nHere\u2019s a nice photo of some of the conference participants. This photo was taken by Jamie Vicary, who therefore alas remained invisible himself. Thanks for the self-sacrifice, Jamie!\n\nFrom left to right: Simon Kochen, Jeffrey Bub, Bob Coecke, Peter Woit, John Baez, Mike Stay, Andreas D\u00f6ring, Camm Maguire and Chris Heunen.\n\nPosted by: John Baez on October 9, 2007 3:51 AM | Permalink | Reply to this\n\n### Re: Deep Beauty: Understanding the Quantum World\n\nNo wonder Dr. Bub didn\u2019t respond to my request for some face time while I was up near College Park this weekend. Maybe in December\u2026\n\nPosted by: John Armstrong on October 9, 2007 4:13 AM | Permalink | Reply to this\n\n### Re: Deep Beauty: Understanding the Quantum World\n\nA second version of Stephen Summer\u2019s contribution \u2013 Yet More Ado About Nothing: The Remarkable Relativistic Vacuum State \u2013 is available.\n\nAn idle thought: might we be able to think of the vacuum state in a galoisian way? Or, in Lautmannian terms, is there a perfectedness to the vacuum state?\n\nPosted by: David Corfield on February 23, 2009 12:14 PM | Permalink | Reply to this\n\n### modular objects determined by the vacuum state; Re: Deep Beauty: Understanding the Quantum World\n\n\u201c\u2026 modular objects determined by the vacuum state and algebras of observables localized in certain regions of Minkowski space encode a remarkable range of physical information, from the dynamics and scattering behavior of the theory to the external symmetries and even the space\u2013time itself\u2026.\u201d\n\nWow!\n\nPosted by: Jonathan Vos Post on February 23, 2009 6:22 PM | Permalink | Reply to this\n\n### Re: Deep Beauty: Understanding the Quantum World\n\nDoes anybody Know Where could I find the book \u201cDeep beauty\u201d edited by Halvorson, to download online???\n\nPosted by: Jacobo Cabrera Cifuentes on March 12, 2011 7:32 PM | Permalink | Reply to this\n\n### Re: Deep Beauty: Understanding the Quantum World\n\nI don\u2019t know if you should be asking this here.\n\nPosted by: David Roberts on March 14, 2011 4:33 AM | Permalink | Reply to this\n\n### Re: Deep Beauty: Understanding the Quantum World\n\nOf course he shouldn\u2019t. It\u2019s illegal! And I\u2019m one of the authors of this volume! Do you post comments on musician\u2019s websites asking how you can illegally download their albums?\n\nOn the other hand, some chapters from this book are freely and legally available online. For example:\n\nThe best place to look for others is the arXiv.\n\nPosted by: John Baez on March 14, 2011 5:34 AM | Permalink | Reply to this\n\nPost a New Comment","date":"2019-08-18 19:18:35","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 6, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.6823691129684448, \"perplexity\": 927.475256043984}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2019-35\/segments\/1566027313996.39\/warc\/CC-MAIN-20190818185421-20190818211421-00177.warc.gz\"}"} | null | null |
module Writexlsx
class Worksheet
class PageSetup # :nodoc:
include Writexlsx::Utility
attr_accessor :margin_left, :margin_right, :margin_top, :margin_bottom # :nodoc:
attr_accessor :margin_header, :margin_footer # :nodoc:
attr_accessor :repeat_rows, :repeat_cols, :print_area # :nodoc:
attr_accessor :hbreaks, :vbreaks, :scale # :nodoc:
attr_accessor :fit_page, :fit_width, :fit_height, :page_setup_changed # :nodoc:
attr_writer :across # :nodoc:
attr_accessor :orientation, :print_options_changed, :black_white # :nodoc:
attr_accessor :header, :footer, :header_footer_changed, :header_footer_aligns, :header_footer_scales
attr_writer :page_start
attr_writer :horizontal_dpi, :vertical_dpi
def initialize # :nodoc:
@margin_left = 0.7
@margin_right = 0.7
@margin_top = 0.75
@margin_bottom = 0.75
@margin_header = 0.3
@margin_footer = 0.3
@repeat_rows = ''
@repeat_cols = ''
@print_area = ''
@hbreaks = []
@vbreaks = []
@scale = 100
@fit_page = false
@fit_width = nil
@fit_height = nil
@page_setup_changed = false
@across = false
@orientation = true
@header_footer_aligns = true
@header_footer_scales = true
end
def paper=(paper_size)
if paper_size
@paper_size = paper_size
@page_setup_changed = true
end
end
def center_horizontally
@print_options_changed = true
@hcenter = true
end
def center_vertically
@print_options_changed = true
@vcenter = true
end
def print_row_col_headers(headers)
if headers
@print_headers = true
@print_options_changed = true
else
@print_headers = false
end
end
def hide_gridlines(option)
if option == 0 || !option
@print_gridlines = true
@print_options_changed = true
else
@print_gridlines = false
end
end
#
# Write the <pageSetup> element.
#
# The following is an example taken from Excel.
#
# <pageSetup
# paperSize="9"
# scale="110"
# fitToWidth="2"
# fitToHeight="2"
# pageOrder="overThenDown"
# orientation="portrait"
# useFirstPageNumber="1"
# blackAndWhite="1"
# draft="1"
# horizontalDpi="200"
# verticalDpi="200"
# r:id="rId1"
# />
#
def write_page_setup(writer) # :nodoc:
return unless @page_setup_changed
attributes = []
attributes << ['paperSize', @paper_size] if @paper_size
attributes << ['scale', @scale] if @scale != 100
attributes << ['fitToWidth', @fit_width] if @fit_page && @fit_width != 1
attributes << ['fitToHeight', @fit_height] if @fit_page && @fit_height != 1
attributes << %w[pageOrder overThenDown] if @across
attributes << ['firstPageNumber', @page_start] if @page_start && @page_start > 1
attributes << ['orientation',
if @orientation
'portrait'
else
'landscape'
end]
attributes << ['blackAndWhite', 1] if @black_white
attributes << ['useFirstPageNumber', 1] if ptrue?(@page_start)
# Set the DPI. Mainly only for testing.
attributes << ['horizontalDpi', @horizontal_dpi] if @horizontal_dpi
attributes << ['verticalDpi', @vertical_dpi] if @vertical_dpi
writer.empty_tag('pageSetup', attributes)
end
#
# Write the <pageMargins> element.
#
def write_page_margins(writer) # :nodoc:
writer.empty_tag('pageMargins', margin_attributes)
end
#
# Write the <printOptions> element.
#
def write_print_options(writer) # :nodoc:
return unless @print_options_changed
attributes = []
attributes << ['horizontalCentered', 1] if @hcenter
attributes << ['verticalCentered', 1] if @vcenter
attributes << ['headings', 1] if @print_headers
attributes << ['gridLines', 1] if @print_gridlines
writer.empty_tag('printOptions', attributes)
end
#
# Write the <headerFooter> element.
#
def write_header_footer(writer, excel2003_style) # :nodoc:
tag = 'headerFooter'
attributes = []
attributes << ['scaleWithDoc', 0] unless ptrue?(@header_footer_scales)
attributes << ['alignWithMargins', 0] unless ptrue?(@header_footer_aligns)
if @header_footer_changed
writer.tag_elements(tag, attributes) do
write_odd_header(writer) if @header && @header != ''
write_odd_footer(writer) if @footer && @footer != ''
end
elsif excel2003_style
writer.empty_tag(tag, attributes)
end
end
private
#
# Write the <oddHeader> element.
#
def write_odd_header(writer) # :nodoc:
writer.data_element('oddHeader', @header)
end
#
# Write the <oddFooter> element.
#
def write_odd_footer(writer) # :nodoc:
writer.data_element('oddFooter', @footer)
end
def margin_attributes # :nodoc:
[
['left', @margin_left],
['right', @margin_right],
['top', @margin_top],
['bottom', @margin_bottom],
['header', @margin_header],
['footer', @margin_footer]
]
end
end
end
end
| {
"redpajama_set_name": "RedPajamaGithub"
} | 2,470 |
\subsection{A Basic Lemma}\label{sec:basic_lemma}
In the following, we state a key lemma that will be central in proving our upper bounds on UAS for the three variants we consider for gradient descent-type algorithms.
\begin{lem}\label{lem:main_lem}
Let $(x^t)_{t\in[T]}$ and $(y^t)_{t\in[T]}$ be two sequences with $x^0=y^0$ and satisfying the recursions
\begin{eqnarray*}
x^{t+1} &=& \mathsf{Proj}_{\mathcal{X}}[x^t-\eta_t g^t]\\
y^{t+1} &=& \mathsf{Proj}_{\mathcal{X}}[y^t-\eta_t h^t],
\end{eqnarray*}
for all $t\in[T-1]$, where the vectors $(g^t)_{t\in[T]}$ and $(h^t)_{t\in[T]}$ lie in $\mathcal{B}(0, L)$ and satisfy
$$ \langle g^t-h^t,x^t-y^t\rangle \geq -a_t\|x^t-y^t\|,$$
for scalars $0\leq a_t\leq 2L$. Then
$$ \|x^T-y^T\| \leq 2L\sqrt{\sum_{t=t_0}^{T-1}\eta_t^2}+2\sum_{t=t_0+1}^{T-1}\eta_s a_s.$$
where $t_0=\inf\{t: g^t\neq h^t\}.$
\end{lem}
\begin{proof}
Let $\delta_t=\|x^t-y^t\|$. By definition of $t_0$ it is clear that $\delta_1=\ldots=\delta_{t_0}=0$. For $t=t_0+1$, we have that $\delta_{t_0+1}=\|\eta_{t_0}(g^{t_0}-h^{t_0})\|\leq 2L\eta_{t_0}$.
Now, we derive a recurrence for $(\delta_t)_{t\in[T]}$:
\begin{eqnarray*}
\delta_{t+1}^2 &=& \|\mathsf{Proj}_{\mathcal{X}}[x^t-\eta_t g^t]-\mathsf{Proj}_{\mathcal{X}}[y^t-\eta_t h^t]\|^2
\,\,\leq\,\, \|x^t-y^t-\eta_t( g^t- h^t)\|^2 \notag\\
&=& \delta_t^2 +\eta_t^2\|g^t-h^t\|^2
-2\eta_t\langle g^t-h^t,x^t-y^t\rangle
\,\,\leq\,\, \delta_t^2+4L^4\eta_t^2+2\eta_ta_t\delta_t.
\end{eqnarray*}
Since $\delta_{t_0+1}\leq 2L\eta_{t_0}$
\begin{eqnarray}
\delta_t^2 &\leq& \delta_{t_0+1}^2 + 4L\sum_{s=t_0+1}^{t-1}\eta_s^2+2\sum_{s=t_0+1}^{t-1}\eta_sa_s\delta_s
\notag\\
&\leq& 4L\sum_{s=t_0}^{t-1}\eta_s^2+2\sum_{s=t_0+1}^{t-1}\eta_sa_s\delta_s.
\label{eqn:rec_stab}
\end{eqnarray}
Now we prove the following bound by induction (notice this claim proves the result):
$$ \delta_t \leq 2L\sqrt{\sum_{s=t_0}^{t-1}\eta_s^2}+2\sum_{s=t_0+1}^{t-1}\eta_sa_s\delta_s \qquad(\forall t\in[T]). $$
Indeed, the claim is clearly true for $t=t_0$. For the inductive step, we assume it holds for some $t\in[T-1]$. To prove the result we consider two cases: first, when $\delta_{t+1}\leq \max_{s\in[t]}\delta_s$, we have by induction
$$ \delta_{t+1}\leq \delta_t\leq 2L\sqrt{\sum_{s=t_0}^{t-1}\eta_s^2}+2\sum_{s=t_0+1}^{t-1}\eta_sa_s
\leq 2L\sqrt{\sum_{s=t_0}^{t}\eta_s^2}+2\sum_{s=t_0+1}^{t}\eta_sa_s. $$
In the other case, $\delta_{t+1}> \max_{s\in[t]}\delta_s$, we use \eqref{eqn:rec_stab}
$$
\delta_{t+1}^2 \,\leq\, 4L\sum_{s=t_0}^{t}\eta_t^2 + 2\sum_{s=t_0+1}^t\eta_sa_s\delta_s
\,\,\leq \,\, 4L\sum_{s=t_0}^{t}\eta_t^2 + 2\delta_{t+1}\sum_{s=t_0+1}^t\eta_sa_s,
$$
which is equivalent to
$$\Big(\delta_{t+1}-\sum_{s=t_0+1}^ta_s\eta_s\Big)^2 \,\,\leq\,\, 4L\sum_{s=t_0}^{t}\eta_t^2 +\Big(\sum_{s=t_0+1}^t\eta_s a_s\Big)^2.
$$
Now, taking square root at the inequality, we get
\begin{eqnarray*}
\delta_{t+1}-\sum_{s=t_0+1}^ta_s\eta_s
&\leq& \sqrt{4L\sum_{s=t_0}^{t}\eta_t^2 +\Big(\sum_{s=t_0+1}^t\eta_s a_s\Big)^2}\\
&\leq& 2L\sqrt{\sum_{s=t_0}^{t}\eta_s^2} +\sum_{s=t_0+1}^t\eta_s a_s.
\end{eqnarray*}
This proves the inductive step, and therefore the result.
\end{proof}
\subsection*{Acknowledgements}
Part of this work was done while the authors were visiting the Simons Institute for the Theory of Computing during the ``Data Privacy: Foundations and Applications'' program.
RB's research is supported by NSF Awards AF-1908281, SHF-1907715, Google Faculty Research Award, and OSU faculty start-up support. Work by CG was partially funded by the Millennium Science Initiative of the Ministry of Economy,
Development, and Tourism, grant ``Millennium Nucleus Center for the Discovery of Structures in Complex Data.'' CG would like to thank Nicolas Flammarion and Juan Peypouquet for extremely valuable discussions at early stages of this work.
\bibliographystyle{apalike}
\section{Notation and Preliminaries}\label{sec:prelims}
Throughout we work on the Euclidean space $(\mathbb{R}^d,\|\cdot\|_2)$. \iffull Therefore, we use unambiguously $\|\cdot\|=\|\cdot\|_2$. Vectors are denoted by lower case letters, e.g.~$x,y$. Random variables (either scalar or vector) are denoted by boldface letters, e.g.~$\mathbf{z},\mathbf{u}$.\fi We denote the Euclidean ball of radius $r>0$ centered at $x\in\mathbb{R}^d$ by $\mathcal{B}(x,r)$.
\iffull In what follows, $\mathcal{X}\subseteq \mathbb{R}^d$ is a compact convex set, and assume we know its Euclidean radius $R>0$, $\mathcal{X}\subseteq \mathcal{B}(0,R)$. \else In what follows, we assume $\mathcal{X}\subseteq \mathcal{B}(0,R)$ is a closed, convex set for some $R>0$. \fi \iffull Let $\mathsf{Proj}_{\mathcal{X}}$ be the Euclidean projection onto $\mathcal{X}$, which is {\em nonexpansive} $\|\mathsf{Proj}_{\mathcal{X}}(x)-\mathsf{Proj}_{\mathcal{X}}(y)\|\leq\|x-y\|$. \else Let $\mathsf{Proj}_{\mathcal{X}}$ be the Euclidean projection onto $\mathcal{X}$. \fi
A convex function $f:\mathcal{X}\mapsto \mathbb{R}$ is $L$-Lipschitz
if
\iffull
\begin{equation} \label{eqn:Lipschitz}
f(x)-f(y) \leq L\|x-y\| \qquad(\forall x,y\in\mathcal{X}).
\end{equation}
Functions with these properties are guaranteed to be subdifferentiable. Moreover,
in the convex case, property~\eqref{eqn:Lipschitz} is ``almost'' equivalent to having subgradients bounded as $\partial f(x)\subseteq \mathcal{B}(0,L)$, for all $x \in\mathcal{X}$.\footnote{For equivalence to hold it is necessary that the function is well-defined and satisfies \eqref{eqn:Lipschitz} over an open set containing $\mathcal{X}$, see Thm.~3.61 in \cite{Beck:2017}. We will assume this is the case, which can be done w.l.o.g..}
\else
\begin{equation} \nonumber
f(x)-f(y) \leq L\|x-y\| \qquad(\forall x,y\in\mathcal{X}).
\end{equation}
\fi
We denote the class of convex $L$-Lipschitz functions as $\mathcal{F}_{\mathcal{X}}^0(L)$.
\iffull
With slight abuse of notation, given a function $f\in\mathcal{F}_{\mathcal{X}}^0(L)$, we will denote by $\nabla f(x)$ an arbitrary choice of $g\in\partial f(x)$.
\fi
In this work, we will focus on the class $\mathcal{F}_{\mathcal{X}}^0(L)$ defined over a compact convex set $\mathcal{X}$. Since the Euclidean radius of $\mathcal{X}$ is bounded by $R$, we will assume that the range of these functions lies in $[-RL, RL]$.
\ifthenelse{\boolean{fullver}}{
A convex and differentiable function $f:\mathcal{X}\mapsto\mathbb{R}$ is said to be $\mu$-smooth if
$$ \|\nabla f(x)-\nabla f(y)\| \leq \mu\|x-y\| \qquad(\forall x,y\in\mathcal{X}),$$
and we denote the class of convex $\mu$-smooth functions by $\mathcal{F}_{\mathcal{X}}^1(\mu)$.
}{}
\noindent\textbf{Nonsmooth stochastic convex optimization:} We study the standard setting of nonsmooth stochastic convex optimization
\begin{equation}\nonumber
x^{\ast} \in \arg\min \{F_{\mathcal{D}}(x):=\mathbb{E}_{\mathbf{z}\sim \mathcal{D}}[ f(x,\mathbf{z}) ]:\,\, x\in \mathcal{X} \}.
\end{equation}
Here, $\mathcal{D}$ is an unknown distribution supported on a set $\Z$, and $f(\cdot,z)\in\mathcal{F}_{\mathcal{X}}^0(L)$ for all $z\in\Z$. In the stochastic setting, we assume access to an i.i.d.~sample from $\mathcal{D}$, denoted as $\mathbf{S}=(\mathbf{z}_1,\ldots, \mathbf{z}_n)\sim \mathcal{D}^n$. \iffull Here, we will use the bold symbol $\mathbf{S}$
to denote a random sample from the unknown distribution. A fixed (not random) dataset from $\Z^n$ will be denoted as $S=(z_1,\ldots, z_n)\in\Z^n$.\fi
\noindent\textbf{A stochastic optimization algorithm} is a (randomized) mapping ${\cal A}:\Z^n\mapsto \mathcal{X}$. When the algorithm is randomized, ${\cal A}(\mathbf{S})$ is a random variable depending on both the sample $\mathbf{S}\sim \mathcal{D}^n$ and its own random coins.
The performance of ${\cal A}$ is quantified by its {\em excess population risk}
$$ \varepsilon_{\mbox{\footnotesize{risk}}}(\A) :=
F_{\mathcal{D}}(\A(\mathbf{S}))-F_{\mathcal{D}}(x^{\ast}).$$
Note that $\varepsilon_{\mbox{\footnotesize{risk}}}(\A)$ is a random variable (due to randomness in the sample $\mathbf{S}$ and any possible internal randomness of the algorithm). \iffull Our guarantees on the excess population risk will be expressed in terms of upper bounds on this quantity that hold \textit{with high probability} over the randomness of both $\mathbf{S}$ and the random coins of the algorithm.\fi
\noindent\textbf{Empirical risk minimization (ERM)} is one of the most standard approaches to stochastic convex optimization. In the ERM problem, we are given a sample $\mathbf{S}=(\mathbf{z}_1,\ldots, \mathbf{z}_n)$, and the goal is to find
$$x^{\ast}(\mathbf{S}) \in \arg\min \Big\{F_{\mathbf{S}}(x):=\frac1n \sum_{i=1}^n f(x, \mathbf{z}_i):\,\, x\in \mathcal{X} \Big\}.$$
\iffull One way to bound the excess population risk is to solve the ERM problem, and appeal to uniform convergence; however, uniform convergence
rates in this case are dimension-dependent, $\Omega(\sqrt{d/n})$ \citep{Feldman:16erm}.\fi
\noindent\textbf{Risk decomposition:} \iffull Guaranteeing low excess population risk for a general algorithm is a nontrivial task. \fi A common way
to bound \iffull it \else the excess population risk \fi is by decomposing it into {\em generalization}, {\em optimization} and {\em approximation error}:
\begin{equation}
\label{eqn:risk_decomp}
\varepsilon_{\mbox{\footnotesize{risk}}}(\A)
\leq \underbrace{F_{\mathcal{D}}(\A(\mathbf{S}))-F_{\mathbf{S}}(\A(\mathbf{S}))}_{\varepsilon_{\mbox{\tiny gen}}(\A)}
+\underbrace{F_{\mathbf{S}}(\A(\mathbf{S}))-F_{\mathbf{S}}(x^{\ast}(\mathbf{S}))}_{\varepsilon_{\mbox{\tiny opt}}(\A)}
+\underbrace{F_{\mathbf{S}}(x^{\ast}(\mathbf{S}))-F_{\mathcal{D}}(x^{\ast})}_{\varepsilon_{\mbox{\tiny approx}}}.
\end{equation}
Here, the optimization error corresponds to the empirical optimization gap, which can be bounded by standard optimization convergence analysis. The expected value of the approximation error is at most zero. \iffull One can show, e.g., by Hoeffding's inequality, that the approximation error is bounded by $\tilde{O}(LR/\sqrt n)$ with high probability (see Lemma~\ref{lem:approx-err-bd} below.) \else One can show, e.g., by Hoeffding's inequality, that the approximation error is bounded by $\tilde{O}(LR/\sqrt n)$ with high probability (see supplementary material.) \fi Therefore, to establish bounds on the excess risk it suffices to upper bound the optimization and generalization errors.
\iffull
\begin{lem}\label{lem:approx-err-bd}
For any $\theta \in (0, 1),$ with probability at least $1-\theta$, the approximation error is bounded as
$$\varepsilon_{\mathrm{\tiny approx}} \leq \frac{RL \sqrt{2 \log(1/\theta)}}{\sqrt{n}}.$$
\end{lem}
\begin{proof}
First, note that $F_{\mathcal{D}}(x^{\ast})=\ex{\mathbf{S}}{F_S(x^{\ast})}=\frac1n\sum_{i=1}^nf(x^{\ast},\mathbf{z}_i)$. Hence, by independence and the fact that $f(x^{\ast},\mathbf{z}_i) \in [-RL, RL]$ with probability $1$ for all $i\in [n]$, the following follows from Hoeffding's inequality:
$$\pr{\mathbf{S}\sim\mathcal{D}^n}{F_{\mathbf{S}}(x^{\ast})-F_{\mathcal{D}}(x^{\ast})\geq \frac{RL \sqrt{2 \log(1/\theta)}}{\sqrt{n}}}\leq \theta.$$
Finally, note that by definition of $x^{\ast}(\mathbf{S})$, we have $F_{\mathbf{S}}(x^{\ast}(\mathbf{S}))-F_{\mathbf{S}}(x^{\ast})\leq 0$. Combining this with the above bound completes the proof.
\end{proof}
\fi
We say that two datasets $S, ~S^{\prime}$ are neighboring, denoted $S\simeq S^{\prime}$, if they only differ on a single entry; i.e., there exists $i\in[n]$ s.t.~for all $k\neq i$, $z_k=z_k^{\prime}$.
\noindent\textbf{Uniform argument stability (UAS):} Given an algorithm ${\cal A}$ and datasets $S\simeq S^{\prime}$, we define the {\em uniform argument stability} (UAS) random variable as
$$\delta_{\cal A}(S,S^{\prime}):=\|{\cal A}(S)-{\cal A}(S^{\prime})\|.$$
The randomness here is due to any possible internal randomness of ${\cal A}$. For any $L$-Lipschitz function $f$, we have that $f\left(\A(S), z\right)-f\left(\A(S^{\prime}), z\right)\leq L\,\delta_{\cal A}(S,S^{\prime}).$ Hence, upper bounds on UAS can be easily transformed into upper bounds on uniform stability.
In this work, we will consider two types of bounds on UAS.
\iffull
\subsection{High-probability guarantees on UAS}
In Section~\ref{sec:UB}, we give upper bounds on UAS for three variants of the (stochastic) gradient descent algorithm, namely, (i) full-batch gradient descent, (ii) sampling-with-replacement stochastic gradient descent, and (iii) fixed-permutation stochastic gradient descent. Variant (i) is deterministic (and hence UAS is a deterministic quantity). For variant (ii), for any pair of neighboring datasets $S, S'$, we give an upper bound on the UAS random variable that holds with high probability over the algorithm's internal randomness (the sampling with replacement). For variant (iii), we give an upper bound on UAS that holds for an arbitrary choice of permutation; in particular, for any random permutation our upper bound on the UAS random variable that holds with probability 1.
\else
\paragraph{High-probability guarantees on UAS:} For any pair $S \simeq S'$, one can bound the UAS random variable $\delta_{\cal A}(S,S')$
w.h.p.~over the internal randomness of ${\cal A}$.
\fi
High-probability upper bounds on UAS lead to high-probability upper bounds on generalization error $\varepsilon_{\mathrm{\tiny gen}}$. We will use the following theorem, which follows in a straightforward fashion from \cite[Theorem 1.1]{FeldmanV:2019}, to derive generalization-error guarantees for our results in Sections~\ref{sec:MultipassSGD} and \ref{sec:Applications} based on our UAS upper bounds in Section~\ref{sec:UB}.
\begin{thm}[follows from Theorem 1.1 in \cite{FeldmanV:2019}]\label{thm:stab-to-gen}
Let $\A:\Z^n\rightarrow \mathcal{X}$ be a randomized algorithm. For any pair of neighboring datasets $S, S'$, suppose that the UAS random variable of $\A$ satisfies:
\begin{align*}
\pr{\A}{\delta_{\A}(S, S')\geq \gamma}&\leq \theta_0.
\end{align*}
Then there is a constant $c$ such that for any distribution $\mathcal{D}$ over $\Z$ and any $\theta \in (0, 1)$, we have
\begin{align*}
\pr{\mathbf{S}\sim\mathcal{D}^n,\,\A}{\lvert\varepsilon_{\mathrm{\tiny gen}}(\A)\rvert\geq c\left(L \gamma\log(n)\log(n/\theta)+LR\sqrt{\frac{\log(1/\theta)}{n}}\right)}&\leq \theta+\theta_0,
\end{align*}
where $\varepsilon_{\mathrm{\tiny gen}}(\A)= F_{\mathcal{D}}(\A(\mathbf{S}))-F_{\mathbf{S}}(\A(\mathbf{S}))$ as defined earlier.
\end{thm}
\iffull
\subsection{Expectation guarantees on UAS}
\else
\paragraph{Expectation guarantees on UAS:} \fi Our results also include upper and lower bounds on $\sup\limits_{S\simeq S'}\ex{\A}{\delta_{\A}(S, S')}$; that is the supremum of the expected value of the UAS random variable, where the supremum is taken over all pairs of neighboring datasets.
\iffull
In Section~\ref{sec:w-replace-sgd}, we provide an upper bound on this quantity for the sampling-with-replacement stochastic gradient descent. The upper bounds on the other two variants of the gradient descent method hold in the strongest sense (they hold with probability $1$). Moreover, in Appendix~\ref{app:T_less_n}, we give slightly tighter expectation guarantees on UAS for both sampling-with-replacement SGD and fixed-permutation SGD with a uniformly random permutation.
In Section~\ref{sec:LowerBounds}, we give lower bounds on this quantity for the two variants of the stochastic subgradient method, together with a deterministic lower bound for the full-batch variant.
\fi
\section{The Basic Lemma}
In the following lemma, we state a key result that will be central in proving our upper bounds on UAS for the three variants of gradient-descent algorithm.
\begin{lem}\label{lem:main_lem}
Let $(x^t)_{t\in[T]}$ and $(y^t)_{t\in[T]}$ be two sequences with $x^0=y^0$ and satisfying the recursions
\begin{eqnarray*}
x^{t+1} &=& \mathsf{Proj}_{\mathcal{X}}[x^t-\eta_t g^t]\\
y^{t+1} &=& \mathsf{Proj}_{\mathcal{X}}[y^t-\eta_t h^t],
\end{eqnarray*}
for all $t\in[T-1]$, where the vectors $(g^t)_{t\in[T]}$ and $(h^t)_{t\in[T]}$ lie in $\mathcal{B}(0, L)$ and satisfy
$$ \langle g^t-h^t,x^t-y^t\rangle \geq -a_t\delta_t,$$
for scalars $0\leq a_t\leq 2L$. Then
$$ \|x^T-y^T\| \leq 2L\sqrt{\sum_{t=t_0}^{T-1}\eta_t^2}+2\sum_{t=t_0+1}^{T-1}\eta_s a_s.$$
where $t_0=\inf\{t: g^t\neq h^t\}.$
\end{lem}
\begin{proof}
Let $\delta_t=\|x^t-y^t\|$. By definition of $t_0$ it is clear that $\delta_1=\ldots=\delta_{t_0}=0$. For $t=t_0+1$, we have that $\delta_{t_0+1}=\|\eta_{t_0}(g^{t_0}-h^{t_0})\|\leq 2L\eta_{t_0}$.
Now, we derive a recurrence for $(\delta_t)_{t\in[T]}$:
\begin{eqnarray*}
\delta_{t+1}^2 &=& \|\mathsf{Proj}_{\mathcal{X}}[x^t-\eta_t g^t]-\mathsf{Proj}_{\mathcal{X}}[y^t-\eta_t h^t]\|^2
\,\,\leq\,\, \|x^t-y^y-\eta_t( g^t- h^t)\|^2 \notag\\
&=& \delta_t^2 +\eta_t^2\|g^t-h^t\|^2
-2\eta_t\langle g^t-h^t,x^t-y^t\rangle
\,\,\leq\,\, \delta_t^2+4L^4\eta_t^2+2\eta_ta_t\delta_t
\end{eqnarray*}
Since $\delta_{t_0+1}\leq 2L\eta_{t_0}$
\begin{eqnarray}
\delta_t^2 &\leq& \delta_{t_0+1}^2 + 4L\sum_{s=t_0+1}^{t-1}\eta_s^2+2\sum_{s=t_0+1}^{t-1}\eta_sa_s\delta_s
\notag\\
&\leq& 4L\sum_{s=t_0}^{t-1}\eta_s^2+2\sum_{s=t_0+1}^{t-1}\eta_sa_s\delta_s.
\label{eqn:rec_stab}
\end{eqnarray}
Now we prove the following bound by induction (notice this claim proves the result):
$$ \delta_t \leq 2L\sqrt{\sum_{s=t_0}^{t-1}\eta_s^2}+\sum_{s=t_0+1}^{t-1}\eta_sa_s\delta_s \qquad(\forall t\in[T]). $$
Indeed, the claim is clearly true for $t=t_0$. For the inductive step, we assume it holds for some $t\in[T-1]$. To prove the result we consider two cases: first, when $\delta_{t+1}\leq \max_{s\in[t]}\delta_s$, we have by induction
$$ \delta_{t+1}\leq \delta_t\leq 2L\sqrt{\sum_{s=t_0}^{t-1}\eta_s^2}+2\sum_{s=t_0+1}^{t-1}\eta_sa_s
\leq 2L\sqrt{\sum_{s=t_0}^{t}\eta_s^2}+2\sum_{s=t_0+1}^{t}\eta_sa_s. $$
In the other case, $\delta_{t+1}> \max_{s\in[t]}\delta_s$, we use \eqref{eqn:rec_stab}
$$
\delta_{t+1}^2 \,\leq\, 4L\sum_{s=t_0}^{t}\eta_t^2 + 2\sum_{s=t_0+1}^t\eta_sa_s\delta_s
\,\,\leq \,\, 4L\sum_{s=t_0}^{t}\eta_t^2 + 2\delta_{t+1}\sum_{s=t_0+1}^t\eta_sa_s,
$$
which is equivalent to
$$\Big(\delta_{t+1}-\sum_{s=t_0+1}^ta_s\eta_s\Big)^2 \,\,\leq\,\, 4L\sum_{s=t_0}^{t}\eta_t^2 +\Big(\sum_{s=t_0+1}^t\eta_s a_s\Big)^2.
$$
Now, taking square root at the inequality, we get
\begin{eqnarray*}
\delta_{t+1}-\sum_{s=t_0+1}^ta_s\eta_s
&\leq& \sqrt{4L\sum_{s=t_0}^{t}\eta_t^2 +\Big(\sum_{s=t_0+1}^t\eta_s a_s\Big)^2}\\
&\leq& 2L\sqrt{\sum_{s=t_0}^{t}\eta_s^2} +\sum_{s=t_0+1}^t\eta_s a_s.
\end{eqnarray*}
This proves the inductive step, and therefore the result.
\end{proof}
First, as a direct corollary of the above lemma, we derive a sharp stability bound for the batch subgradient method.
\begin{thm} \label{thm:UB_GD_}
Let $\mathcal{X}\subseteq\mathcal{B}(0,R)$ and $\mathcal{F}=\mathcal{F}_{\mathcal{X}}^0(L)$. The full-batch gradient descent (Algorithm \ref{alg:batchGD}) has uniform argument stability\xspace $\delta_{\A_{\sf GD}}\leq \min\left\{2R,~
4L\,\Big(\frac1n\sum_{t=1}^{T-1}\eta_t+\sqrt{\sum_{t=1}^{T-1}\eta_t^2}\Big)\right\}$.
\end{thm}
\begin{proof}
The bound of $2R$ is obtained directly from the diameter bound on $\mathcal{X}$. Therefore, we focus exclusively on the second term. Let $S\simeq S^{\prime}$, $x^0=y^0$, and consider the trajectories $(x^t)_t,(y^t)_t$ associated with the batch GD method on datasets $S$ and $S^{\prime}$, respectively. We use Lemma~\ref{lem:main_lem} with $g^t=\nabla F_{S}(x^t)$ and $h^t=\nabla F_{S^{\prime}}(y^t)$. Thus,
\begin{eqnarray*}
\langle g^t-h^t,x^t-y^t \rangle
&=&\langle \nabla F_{S}(x^t)-\nabla F_{S}(y^t),x^t-y^t \rangle +\langle \nabla F_{S}(y^t)-\nabla F_{S^{\prime}}(y^t),x^t-y^t \rangle\\
&\geq& -\|F_{S}(y^t)-\nabla F_{S^{\prime}}(y^t)\| \delta_t\\
&\geq& -\frac{2L}{n}\delta_t.
\end{eqnarray*}
where in the first inequality we used the monotonicity of the subgradient, and in the second one that $S\simeq S^{\prime}$, and that $F_{S},F_{S^{\prime}}$ are $L$-Lipschitz.
We conclude by the Lemma that for all $t\in[T]$
$$ \|x^t-y^t\| \leq 2L\sqrt{ \sum_{s=1}^{t-1}\eta_s^2 } +\frac{4L}{n}\sum_{s=1}^{t-1} \eta_s. $$
We conclude the stability bound holds for the iterates, and thus for $\overline{x}^T$ by the triangle inequality.
\end{proof}
Next, we state and prove the following upper bounds on UAS for two variants of stochastic gradient descent: sampled-with-replacement SGD and fixed-permutation SGD. Here, we give strong upper bounds that hold with high probability (and with probability 1 in case of fixed-permutation SGD). In the appendix, we derive tighter upper bounds for these two variants of SGD in the case where the number of iterations $T < $ the number of samples in the data set $n$; however, the bounds derived in this range hold only in expectation.
We now state and prove our upper bound for sampled-with-replacement SGD.
\begin{thm} \label{thm:UB_repl_SGD_}
Let $\mathcal{X}\subseteq\mathcal{B}(0,R)$ and $\mathcal{F}=\mathcal{F}_{\mathcal{X}}^0(L)$. With probability at least $1-\exp\left(-n/2\right)$, the sampling-with-replacement stochastic gradient descent (Algorithm \ref{alg:repl_SGD}) has uniform argument stability\xspace
$$\delta_{\A_{\sf rSGD}}\leq \min\left(2R,~ 4L\left(\sqrt{\sum_{t=1}^{T-1}\eta_t^2} +\frac{1}{n}\sum_{t=1}^{T-1}\eta_t\right)\right).$$
\end{thm}
\begin{proof}
The bound of $2R$ trivially follows from the diameter bound on $\mathcal{X}$. We thus focus on the second term of the bound. Let $S\simeq S^{\prime}$, $x^0=y^0$, and consider the trajectories $(x^t)_{t\in[T]},(y^t)_{t\in[T]}$ associated with the sampled-with-replacement stochastic subgradient method on datasets $S$ and $S^{\prime}$, respectively. We use Lemma~\ref{lem:main_lem} with $g^t=\nabla f(x^t,\mathbf{z}_{\mathbf{i}_t})$ and $h^t=\nabla f(y^t,\mathbf{z}_{\mathbf{i}_t}^{\prime})$. Let us define $\mathbf{r}_t\triangleq \mathbf{1}_{\{\mathbf{z}_{\mathbf{i}_t}\neq\mathbf{z}_{\mathbf{i}_t}^{\prime}\}}$. Note that at every step $t$, $\mathbf{r}_t=1$ with probability $1-1/n$, and $\mathbf{r}_t=0$ otherwise. Moreover, note that $\{\mathbf{r}_t: ~t\in [T]\}$ is an independent sequence of Bernoulli random variables.
Observe,
\begin{eqnarray*}
\langle g^t-h^t,x^t-y^t \rangle
&=&\langle \nabla f(x^t,\mathbf{z}_{\mathbf{i}_t})-\nabla f(y^t,\mathbf{z}_{\mathbf{i}_t}^{\prime}),x^t-y^t \rangle\\
&\geq& -2L \mathbf{r}_t\delta_t.
\end{eqnarray*}
Hence, by Lemma~\ref{lem:main_lem}, for any realization of the trajectories of the SGD method, we have
\begin{align}
\forall t\in [T]:\quad \|x^t-y^t\| &\leq 2L\sqrt{ \sum_{s=1}^{t-1}\eta_s^2 } +4L\sum_{s=1}^{t-1} \mathbf{r}_s\eta_s \leq \Delta_T,\label{bound:unif_t}
\end{align}
where $\Delta_T\triangleq 2L\sqrt{ \sum_{s=1}^{T-1}\eta_s^2 } +4L\sum_{s=1}^{T-1} \mathbf{r}_s\eta_s$. Taking expectation of (\ref{bound:unif_t}), we have
$$ \forall t\in [T]:\quad \mathbb{E}\left[\|x^t-y^t\|\right] \leq \mathbb{E}\left[\Delta_T\right]= 2L\sqrt{ \sum_{s=1}^{T-1}\eta_s^2 } +\frac{4L}{n}\sum_{s=1}^{T-1}\eta_s. $$
This establishes the upper bound on UAS but only in expectation. To prove the high-probability bound, we proceed as follows. Note that each $\mathbf{r}_s, s\in [T],$ has variance $\frac{1}{n}\left(1-\frac{1}{n}\right)<\frac{1}{n}$. Hence, by Chernoff's bound\footnote{Here, we are applying a bound for (scaled) Bernoulli rvs where the exponent is expressed in terms of the variance.}, we have
\begin{align*}
\mathbb{P}\left[\sum_{s=1}^{T-1} \mathbf{r}_s\eta_s \geq \frac1n\sum_{s=1}^{T-1}\eta_s+\sqrt{\sum_{s=1}^{T-1}\eta_s^2}
\right] &\leq \exp\left(-\,\frac{\sum_{s=1}^{T-1}\eta_s^2}{2\cdot\frac{1}{n}\cdot\sum_{s=1}^{T-1}\eta_s^2}\right) = \exp\left(-\frac{n}{2}\right).
\end{align*}
Therefore, with probability at least $1-\exp\left(-n/2\right)$, we have
$$\Delta_T\leq 3L\sqrt{ \sum_{s=1}^{T-1}\eta_s^2 } +\frac{4L}{n}\sum_{s=1}^{T-1}\eta_s.$$
Putting this together with (\ref{bound:unif_t}), with probability at least $1-\exp\left(-n/2\right)$, we have
$$\forall t\in [T]:\quad \|x^t-y^t\| \leq 3L\sqrt{ \sum_{s=1}^{T-1}\eta_s^2 } +\frac{4L}{n}\sum_{s=1}^{T-1}\eta_s. $$
Finally, by triangle inequality, we get that with probability at least $1-\exp\left(-n/2\right)$, the same stability bound holds for the average of the iterates $\overline{x}^T$.
\end{proof}
Finally, we show that the same stability bound holds with probability 1 for the fixed-permutation SGD. In this result, we assume that the sequence of step sizes $\left(\eta_t\right)_{t\in [T]}$ is non-increasing, which is a valid assumption for almost all known variants of SGD.
\begin{thm} \label{thm:UB_perm_SGD_}
Let $\mathcal{X}\subseteq\mathcal{B}(0,R)$ and $\mathcal{F}=\mathcal{F}_{\mathcal{X}}^0(L)$. Suppose the step sizes $\left(\eta_t\right)_{t\in [T]}$ form a non-increasing sequence. Then, with probability $1$ with respect to the choice of the permutation, the fixed-permutation stochastic gradient descent (Algorithm \ref{alg:perm_SGD}) has uniform argument stability\xspace
$$\delta_{\A_{\sf GD}}\leq \min\left\{2R,~2L \left(\sqrt{\sum_{t=1}^{T-1}\eta_t^2} +\frac{2}{n} \sum_{t=1}^{T-1}\eta_t\right)\right\}.$$
\end{thm}
\begin{proof}
Again, the bound of $2R$ is trivial. Now, we show the second term of the bound. Let $S\simeq S^{\prime}$, $x^0=y^0$, and consider the trajectories $(x^t)_{t\in[T]},(y^t)_{t\in[T]}$ associated with the fixed permutation stochastic subgradient method on datasets $S$ and $S^{\prime}$, respectively. Fix any realized permutation $\pi$ of $[n]$ for the algorithm. Since the dataset $S$ is arbitrary, we may assume without loss of generality that $\pi$ is the identity, whereas the perturbed coordinate $\mathbf{i}=i$ is arbitrary. We use Lemma~\ref{lem:main_lem} with $g^t=\nabla f\left(x^t,\mathbf{z}_{\left(t ~\mbox{\footnotesize mod }n\right)}\right)$ and $h^t=\nabla f\left(y^t,\mathbf{z}^{\prime}_{\left(t ~\mbox{\footnotesize mod }n\right)}\right)$. It is easy to see then that we can set $a_t$ in Lemma~\ref{lem:main_lem} as $a_t=2L \cdot\mathbf{1}_{\{(t ~\mbox{\footnotesize mod }n)=i\}}$, where $\mathbf{1}_{\{\mathsf{condition}\}}$ is the indicator of $\mathsf{condition}$. Hence, by Lemma~\ref{lem:main_lem}, we have
\begin{eqnarray*}
\|x^t-y^t\|
&\leq& 2L\sqrt{\sum_{s=1}^{t-1}\eta_s^2} +4L\sum_{r=1}^{\lfloor (t-1)/n \rfloor}\eta_{rn+i} \\
&\leq& 2L\sqrt{\sum_{s=1}^{t-1}\eta_s^2} +\frac{4L}{n} \sum_{r=1}^{t-1}\eta_s,
\end{eqnarray*}
where at the last step we used the fact that $(\eta_t)_{t\in[T]}$ is non-increasing; namely, for any $r\geq 1$
$$ \eta_{rn+i} \leq \frac1n \sum_{s=(r-1)n+i+1}^{rn+i} \eta_s. $$
Since the bound holds for all the iterates, using triangle inequality, it holds for the output $\overline{x}^K$ averaged over the iterates from the $T/n$ epochs.
\end{proof}
\section{Proof of Theorem \ref{thm:UB_GD}} \label{app:pf_UB_GD}
\input{proof_UB_GD.tex}}
\ifthenelse{\boolean{fullver}}{}{
\section{Proof of Theorem \ref{thm:UB_perm_SGD}} \label{app:pf_UB_repl_SGD}
\input{proof_UB_perm_SGD.tex}}
\ifthenelse{\boolean{fullver}}{}{
\section{General Lower Bound on Stability of SGD}
\label{app:gen_LB_FOM}
\input{gen_LB_FOM.tex}
}
\ifthenelse{\boolean{fullver}}{}{
\section{Proof of Theorem \ref{thm:LB_GD_basic}}
\label{app:LB_GD_basic}
\input{proof_LB_GD_basic.tex}
}
\ifthenelse{\boolean{fullver}}{}{
\section{Proof of Theorem \ref{thm:LB_perm_SGD_basic}}
\label{app:LB_perm_SGD_basic}
\input{proof_LB_perm_SGD_basic.tex}
}
\section{Upper bounds on UAS of SGD when $T\leq n$} \label{app:T_less_n}
\input{T_less_n_UB.tex}
\ifthenelse{\boolean{fullver}}{}{
\section{Generalization in Stochastic Optimization with Data Reuse}
\label{app:DataReuse}
\input{subsec_DataReuse.tex}
}
\section{High-probability Bound on Optimization Error of SGD with Noisy Gradient Oracle}\label{app:hp-sgd-opt-err}
It is known that standard online-to-batch conversion technique can be used to provide high-probability bound on the optimization error (i.e., the excess empirical risk) of stochastic gradient descent. For the sake of completeness and to make the paper more self-contained, we re-expose this technique here for stochastic gradient descent with noisy gradient oracle. This is done in the following lemma, which is used in the proofs of our results in Section~\ref{sec:Applications}.
\begin{lem}[Optimization error of SGD with noisy gradient oracle] \label{lem:online_to_batch}
Let $S=(z_1,\ldots,z_n)\in \Z^n$ be a dataset. Let $F_{S}(x)=\frac1n\sum_{i\in[n]}f(x,z_i)$ be the empirical risk associated with $S$, where for every $z\in\Z,$ $f(\cdot, z)$ is convex, $L$-Lipschitz function over $\mathcal{X}\subseteq \mathcal{B}(0,R)$ for some $L, R >0$. Consider the stochastic (sub)gradient method:
$$ x^{t+1}=x^t-\eta\cdot\mathbf{g}(x,\xi_t) \qquad(\forall t=0,\ldots,T-1), $$
with output $\overline{x}^T=\frac1T\sum_{t\in[T]}x^t$; where $\xi_1,\ldots,\xi_T$ are drawn uniformly from from $S$ with replacement, and for all $z\in\Z$, $\mathbf{g}(., z):\mathcal{X}\rightarrow \mathbb{R}^d$ is a random map (referred to as noisy gradient oracle) that satisfies the following conditions:
\begin{enumerate}
\item Unbiasedness: For every $x\in\mathcal{X}, z\in\Z$,~ $\mathbb{E}[\mathbf{g}(x,z)]=\nabla f(x, z)$, where the expectation is taken over the randomness in the gradient oracle $\mathbf{g}(\cdot, z)$.
\item Sub-Gaussian gradient noise: There is $\sigma^2\geq0$ such that for every $x\in\mathcal{X}, z\in\Z$, $\mathbf{g}(x, z)-\nabla f(x, z)$ is $\sigma^2$-sub-Gaussian random vector; that is, for every $x\in\mathcal{X}, z\in\Z,$ $\langle \mathbf{g}(x, z)- \nabla f(x, z),~ u\rangle$ is $\sigma^2$-sub-Gaussian random variable $\forall u\in \mathcal{B}(0, 1)$.
\item Independence of the gradient noise across iterations: conditioned on any fixed realization of $(\xi_t)_{t\in [T]}$ the sequence of random maps $\mathbf{g}(\cdot, \xi_1), \ldots, \mathbf{g}(\cdot, \xi_T)$ is independent. (Here, randomness comes only from the gradient oracle.)
\end{enumerate}
Then, for any $\theta\in (4e^{-T/32}, 1)$, with probability at least $1-\theta$, the optimization error (i.e., the excess empirical risk) of this method is bounded as
$$\varepsilon_{\mathrm{\tiny opt}} \leq \left(LR+\sigma(R+\eta L)\right)\sqrt{\frac{2\log(4/\theta)}{T}}+\frac{R^2}{2\eta T}+
\eta \,(\frac{L^2}{2} +d\sigma^2).$$
\end{lem}
\begin{proof}
Let $x^{\ast}_S\in \arg\min\limits_{x\in \mathcal{X}}F_S(x)$. By convexity of the empirical loss, we have
\begin{align}
&F_{S}(\overline{x}^T) -F_S(x^{\ast}_S)
\leq \frac{1}{T} \sum_{t\in[T]} F_{S}(x^t)-F_S(x^{\ast}_S)\nonumber\\
&= \frac{1}{T}\sum_{t\in[T]} [F_{S}(x^t)-f(x^t,\xi_t)]+ \frac{1}{T}\sum_{t\in[T]}[f(x^{\ast}_S,\xi_t)-F_{S}(x^{\ast}_S)]+
\frac{1}{T}\sum_{t\in[T]} [f(x^t,\xi_t)-f(x^{\ast}_S,\xi_t)]. \label{eqn:online_to_batch}
\end{align}
Since $(\xi_t)_{t\in [T]}$ are sampled uniformly without replacement from $S,$ we have $$\ex{\xi_t~|~x^1,\ldots, x^{t-1}}{f(x^t, \xi_t)|x^{1}, \ldots, x^{t-1}, x^{t}=v}=F_S(v),$$
for all $v\in\mathcal{X}, t\in [T]$. Moreover, since the range of $f$ lies in $[-LR, LR],$ it follows that $Y_t:=\sum_{j=1}^t f(x^{j}, \xi_j), ~t\in [T]$ is a martingale with bounded differences (namely, bounded by $2LR$). Therefore, by Azuma's inequality, the first term in \eqref{eqn:online_to_batch} satisfies
\begin{align}
\mathbb{P} \Bigg[\frac{1}{T}\sum_{t\in[T]} [F_{S}(x^t)-f(x^t,\xi_t)] >
LR\sqrt{\frac{2\log\frac{4}{\theta}}{T}}~\Bigg] \leq \frac{\theta}{4}.\label{ineq:azuma}
\end{align}
By Hoeffding's inequality, the second term in \eqref{eqn:online_to_batch} also satisfies the same bound; namely,
\begin{align}
\mathbb{P} \Bigg[\frac{1}{T}\sum_{t\in[T]}[f(x^{\ast}_S, \xi_t)-F_{S}(x^{\ast}_S)] >
LR\sqrt{\frac{2\log\frac{4}{\theta}}{T}}~\Bigg] \leq \frac{\theta}{4}.\label{ineq:hoeff}
\end{align}
Using similar analysis to that of the standard online gradient descent analysis \cite{Zinkevich:2003},
the last term in \eqref{eqn:online_to_batch} can be bounded as
\begin{align}
&\frac{1}{T}\sum_{t\in[T]}[f(x^t,\xi_t)-f(x^{\ast}_S,\xi_t)] \leq \frac{R^2}{2T\eta} +\frac{1}{T}\sum_{t\in[T]}\langle \nabla f(x^t,\xi_t)-\mathbf{g}(x^t, \xi_t),~ x^t-x^{\ast}_S\rangle
+\frac{\eta}{2T} \sum_{t\in[T]}\|\nabla \mathbf{g}(x^t,\xi_t)\|^2\nonumber\\
=~&~\frac{R^2}{2T\eta} +\frac{1}{T}\sum_{t\in[T]}\langle \nabla f(x^t,\xi_t)-\mathbf{g}(x^t, \xi_t),~ x^t-x^{\ast}_S -\eta \nabla f(x^t, \xi_t)\rangle +\frac{\eta}{2T} \sum_{t\in[T]}\|\mathbf{g}(x^t,\xi_t)-\nabla f(x^t, \xi_t)\|^2\nonumber\\
&+\frac{\eta}{2T} \sum_{t\in[T]}\|\nabla f(x^t, \xi_t)\|^2\nonumber\\
\leq~&~ \frac{R^2}{2T\eta}+\frac{\eta L^2}{2} +\frac{1}{T}\sum_{t\in[T]}\langle \nabla f(x^t,\xi_t)-\mathbf{g}(x^t, \xi_t),~ x^t-x^{\ast}_S -\eta \nabla f(x^t, \xi_t)\rangle
+\frac{\eta}{2T} \sum_{t\in[T]}\|\mathbf{g}(x^t,\xi_t)-\nabla f(x^t, \xi_t)\|^2\label{eqn:reg_bd}
\end{align}
By the properties of the gradient oracle stated in the lemma, we can see that for any fixed realization of $(x^t, \xi_t)_{t\in [T]}$, the second term in \eqref{eqn:reg_bd} is $\left(2R+\eta L\right)^2\frac{\sigma^2}{T}$-sub-Gaussian random variable. Hence,
\begin{align}
\pr{}{\frac{1}{T}\sum_{t\in[T]}\langle \nabla f(x^t,\xi_t)-\mathbf{g}(x^t, \xi_t), ~x^t-x^{\ast}_S -\eta \nabla f(x^t, \xi_t)\rangle > \left(2R+\eta L\right)\sigma\sqrt{\frac{2\log(4/\theta)}{T}}}&\leq \frac{\theta}{4}.\label{ineq:subgauss1}
\end{align}
Let $U_t:=\|\mathbf{g}(x^t,\xi_t)-\nabla f(x^t, \xi_t)\|^2,~ t\in [T]$. Note that $\ex{}{U_t}\leq d\sigma^2$. Moreover, observe (e.g., by \cite[Lemma~1.12]{mit}) that for any fixed realization of $x^t, \xi_t$, $V_t:=U_t-\ex{}{U_t}$ is a sub-exponential random variable with parameter $16d\sigma^2$; namely, $\ex{}{\exp(\lambda V_t)}\leq \exp(128\lambda^2\sigma^4d^2),~ \lambda \leq \frac{1}{16\sigma^2 d}.$
Hence, by a standard concentration argument (e.g., Bernstein's inequality), we have
\begin{align}
\pr{}{\frac{\eta}{2T} \sum_{t\in[T]}\|\mathbf{g}(x^t,\xi_t)-\nabla f(x^t, \xi_t)\|^2> \frac{\eta}{2}d\sigma^2 + 16\eta d\sigma^2\,\frac{\log(4/\theta)}{T}}&\leq \theta/4.\label{ineq:subgauss2}
\end{align}
Putting \eqref{ineq:subgauss1} and \eqref{ineq:subgauss2} together, and noticing that $T>32\log(4/\theta)$, we conclude that with probability at least $1-\theta/2,$ the third term of \eqref{eqn:online_to_batch} is bounded as $$\frac{1}{T}\sum_{t\in[T]}[f(x^t,\xi_t)-f(x^{\ast}_S,\xi_t)]\leq \frac{R^2}{2T\eta}+\frac{\eta L^2}{2}+\eta\sigma^2 d+\left(2R+\eta L\right)\sigma\sqrt{\frac{2\log(4/\theta)}{T}}.$$
Hence, by the union bound, we conclude that with probability at least $1-\theta,$ the excess empirical risk of the stochastic subgradient method is bounded as
$$\varepsilon_{\mathrm{\tiny opt}} \leq \left(LR+\sigma(2R+\eta L)\right)\sqrt{\frac{2\log(4/\theta)}{T}}+\frac{R^2}{2\eta T}+
\eta \,(\frac{L^2}{2} +d\sigma^2).$$
\end{proof}
\section{Empirical Risk of Fixed-Permutation SGD} \label{App:Fixed_Perm_SGD}
Our optimization error analysis is based on \cite[Lemma 2.1]{Nedic:2001}.
\begin{lem} \label{lem:perm_SGD_round}
Let us consider the fixed permutation stochastic gradient descent (Algorithm \ref{alg:perm_SGD}), for arbitrary permutation (i.e., not necessarily random) and with constant step size over each epoch (i.e., $\eta_{(k-1)n+t}\equiv\eta_k$ for all $t\in[n]$, $k\in[K]$). Then
$$
\eta_k[F_S(x^k)-F_{S}(y)]
\leq \frac{1}{2n}[\|x^k-y\|^2-\|x^{k+1}-y\|^2] + \frac{\eta_k^2L^2(n+2)}{2}
\qquad(\forall y\in \mathcal{X}).
$$
\end{lem}
\begin{proof}
First, since the permutation is arbitrary,
w.l.o.g.~$\boldsymbol{\pi}$ is the identity (we make this choice only for
notational convenience).
Let now $y\in\mathcal{X}$. At each round, the recursion of SGD implies that
\begin{eqnarray*}
\|x_{t+1}^k-y\|^2
&=& \|\mathsf{Proj}_{\mathcal{X}}[x_{t}^k-\eta_k \nabla f(x_t^k,z_{t})]-\mathsf{Proj}_{\mathcal{X}}(y)\|^2 \\
&\leq&\|x_{t}^k-\eta_k \nabla f(x_t^k,z_t)-y\|^2 \\
&=&\|x_t^k-y\|^2+\eta_k^2L^2 -2\eta_k\langle \nabla f(x_t^k,z_t),x_t^k-y\rangle \\
&\leq& \|x_t^k-y\|^2+\eta_k^2L^2
-2\eta_k[f(x_t^k,z_t)-f(y,z_t)].
\end{eqnarray*}
Let $r_t:=\|x^t-y\|$.
Summing up these inequalities from $t=1,
\ldots,n$
\begin{eqnarray*}
r_{n+1}^2-r_1^2
&\leq&
nL^2\eta_k^2+2\eta_k \sum_{t=1}^n[f(x^k,z_t)-f(x_t^k,z_t)]
-2\eta_k n[F_S(x^k)-F_S(y)]\\
&\leq& nL^2\eta_k^2+2\eta_k \sum_{t=1}^n
L\|x^k-x_t^k\|
-2\eta_k n[F_S(x^k)-F_S(y)]\\
&\leq& nL^2\eta_k^2
+2\eta_k^2L^2 \sum_{t=1}^n t
-2\eta_k n[F_S(x^k)-F_S(y)]\\
&=& \eta_k^2 L^2 n
+\eta_k^2L^2n(n+1)
-2\eta_k n[F_S(x^k)-F_S(y)].
\end{eqnarray*}
Re-arranging terms we obtain the result.
\end{proof}
Using the previous lemma, it is straightforward to derive the optimization accuracy of the method.
\begin{cor} \label{cor:perm_SGDopt_error}
The fixed permutation stochastic gradient descent (Algorithm \ref{alg:perm_SGD}), for arbitrary permutation (i.e., not necessarily random) and with constant step size over each epoch (i.e., $\eta_{(k-1)n+t}\equiv\eta_k$ for all $t\in[n]$, $k\in[K]$).
satisfies
$$ \eps_{\mbox{\footnotesize opt}}
\leq \frac{\|x^1-x^{\ast}(S)\|^2}{2n\sum_k \eta_k}+\frac{L^2(n+2)}{2}\frac{\sum_k \eta_k^2}{\sum_k \eta_k}.$$
\end{cor}
\begin{proof}
By convexity and
Lemma \ref{lem:perm_SGD_round}, we have
\begin{eqnarray*}
F_s(\overline{x}^K)-F_{S}(x^{\ast}(S))
&\leq& \frac{1}{\sum_{k=1}^K \eta_k}
\sum_{k=1}^K\eta_k[F_S(x^k)-f_S(x^{\ast}(S))]\\
&\leq& \frac{1}{\sum_{k=1}^K \eta_k}
\Big[ \frac{1}{2n}\|x^1-x^{\ast}(S)\|^2
+\frac{L^2(n+2)}{2}\sum_{k=1}^K\eta_k^2\Big],
\end{eqnarray*}
which proves the result.
\end{proof}
\section{Implications of Our Stability Bounds}
\else
\section{A Simple Algorithm for Differentially Private Nonsmooth Stochastic Convex Optimization with Optimal Risk}
\fi
\label{sec:Applications}
\iffull
\subsection{Differentially Private Nonsmooth Stochastic Convex Optimization}
\fi
Now we show an application of our stability upper bound to \emph{differentially private} stochastic convex optimization (DP-SCO).
Here, the input sample to the stochastic convex optimization algorithm is a sensitive and private data set, thus the algorithm is required to satisfy the notion of $(\alpha, \beta)$-differential privacy.
A randomized algorithm $\A$ is $(\alpha,\beta)$-differentially private if, for any
pair of datasets $S\simeq S'$, and for all events $\mathcal{O}$ in the output range of $\A$, we have
\ifthenelse{\boolean{fullver}}{$$\pr{}{\A(S)\in \mathcal{O}} \leq e^{\alpha} \cdot \pr{}{\A(S')\in \mathcal{O}} +\beta ,$$}{
$\pr{}{\A(S)\in \mathcal{O}} \leq e^{\alpha} \cdot \pr{}{\A(S')\in \mathcal{O}} +\beta ,$
}
where the probability is taken over the random coins of $\A$ \citep{DMNS06, DKMMN06}. For meaningful privacy guarantees, the typical settings of the privacy parameters are $\alpha<1$ and $\beta \ll 1/n$.
Using our UAS upper bounds, we show that a simple variant of noisy SGD \citep{BST14}, that requires only $n^2$ gradient computations, yields the optimal excess population risk for DP-SCO. In terms of running time, this is a small improvement over the algorithm of \cite{FKT19} for the nonsmooth case, which requires $O(n^2\sqrt{\log 1/\beta})$ gradient computations. More importantly, our algorithm is substantially simpler. For comparison, the algorithm in \citep{FKT19} is based on a multi-phase SGD, where in each phase a separate regularized ERM problem is solved. To ensure privacy, the output of each phase is perturbed with an appropriately chosen amount of noise before being used as the initial point for the next phase.
The description of the algorithm is given in Algorithm~\ref{Alg:NSGD}.
\begin{algorithm}
\caption{$\A_{\sf NSGD}$: Noisy SGD for convex losses}
\begin{algorithmic}[1]
\REQUIRE Private dataset $S=(z_1, \ldots, z_n)\in \Z^n$,
step size $\eta$;~ privacy parameters $\alpha \leq 1, ~\beta \ll 1/n$
\STATE Set noise variance $\sigma^2 := \frac{8\,L^2\,\log(1/\beta)}{\alpha^2}$
\STATE Choose an arbitrary initial point $x^1 \in \mathcal{X}$
\FOR{$t=1$ to $n^2-1$\,}
\STATE Sample $\mathbf{i}_t\sim\mbox{Unif}([n])$ \label{step:sampling}
\STATE $x^{t+1} := \mathsf{Proj}_{\mathcal{X}}\left(x^{t}-\eta\cdot\left(\nabla\ell(x^{t}, z_{\mathbf{i}_t})+\mathbf{G}_t\right)\right),$ where $\mathbf{G}_t \sim \mathcal{N}\left(\mathbf{0}, \sigma^2 \mathbb{I}_d\right)$ drawn independently each iteration\label{step:grad-step}
\ENDFOR
\RETURN $\overline{x}=\frac{1}{n^2}\sum_{t=1}^{n^2} x^t$
\end{algorithmic}
\label{Alg:NSGD}
\end{algorithm}
\iffull
\else
We state the guarantees of Algorithm~\ref{Alg:NSGD} below.
\fi
\begin{thm}[Privacy guarantee of $\A_{\sf NSGD}$]\label{thm:priv_Ansgd}
Algorithm~\ref{Alg:NSGD} is $(\alpha, \beta)$-differentially private.
\end{thm}
\iffull
\noindent The proof of the theorem follows the same lines of \cite[Theorem~2.1]{BST14}, but we replace their privacy analysis of the Gaussian mechanism with the
tighter Moments Accountant method of \cite{abadi2016deep}.
analysis of \cite{abadi2016deep}.\fi
\begin{thm}[\iffull Excess risk \else Risk \fi of $\A_{\sf NSGD}$]\label{thm:pop_risk_Ansgd}
In Algorithm~\ref{Alg:NSGD}, let $\eta= R/\Big(L\cdot n\cdot\max\big(\sqrt{n},~ \frac{\sqrt{d\,\log(1/\beta)}}{\alpha}\big)\Big)$.
Then, for any $\theta \in (6\exp(-n/2), 1)$, with probability at least $1-\theta$ over the randomness in both the sample and the algorithm, we have
\begin{align*}
\varepsilon_{\mathsf{\footnotesize{risk}}}\left(\A_{\sf NSGD}\right)&= RL\cdot O\left(\max\left(\frac{\log(n)\log(n/\theta)}{\sqrt{n}}, ~\frac{\sqrt{d\,\log(1/\beta)}}{\alpha\, n}\right)\right)
\end{align*}
\end{thm}
\iffull
\begin{proof}
Fix any confidence parameter $\theta\geq 6\exp(-n/2)$. First, for any data set $S\in \Z^n$ and any step size $\eta > 0,$ by Lemma~\ref{lem:online_to_batch} in Appendix~\ref{app:hp-sgd-opt-err}, we have the following high-probability guarantee on the training error of $\A_{\sf NSGD}$:
\noindent With probability at least $1-\theta/3,$ we have
\begin{align*}
\varepsilon_{\mbox{\footnotesize opt}}(\A_{\sf NSGD})\triangleq F_S(\overline{x})-\min\limits_{x\in\mathcal{X}}F_S(x)&\leq \frac{R^2}{\,\eta\, n^2} +7RL\frac{\sqrt{\log(1/\beta)\log(12/\theta)}}{\alpha n} + \eta\,L^2 \Big(32\frac{d\,\log(1/\beta)}{\alpha^2}+1\Big)
\end{align*}
where the probability is over the sampling in step~\ref{step:sampling} and the independent Gaussian noise vectors $\mathbf{G}_1, \ldots, \mathbf{G}_{n^2}$. Given the setting of $\eta$ in the theorem, we get
\begin{align}
\varepsilon_{\mbox{\footnotesize opt}}(\A_{\sf NSGD})&\leq 8 R L\max\Big(\frac{1}{\sqrt{n}},~\frac{\sqrt{d\,\log(1/\beta)}}{\alpha\,n}\Big) + 33 R L\,\frac{d\, \frac{\log(1/\beta)}{\alpha^2}}{n\cdot \max\Big(\sqrt{n},~\frac{\sqrt{d\,\log(1/\beta)}}{\alpha}\Big)}\nonumber \\
&\leq 8 R L\max\Big(\frac{1}{\sqrt{n}},~\frac{\sqrt{d\,\log(1/\beta)}}{\alpha\,n}\Big) + 33 R L\, \frac{\sqrt{d\,\log(1/\beta)}}{n\,\alpha}\nonumber\\
&= R L\cdot O\Bigg(\max\Big(\frac{1}{\sqrt{n}},~\frac{\sqrt{d\,\log(1/\beta)}}{n\,\alpha}\Big)\Bigg).\label{eqn:Ansgd-opt-err}
\end{align}
Next, it is not hard to show that $\A_{\sf NSGD}$ attains the same UAS bound as $\A_{\sf rSGD}$ (Theorem~\ref{thm:UB_repl_SGD}). Indeed, the only difference is the noise addition in gradient step; however, this does not impact the stability analysis. This is because the sequence of noise vectors $\{\mathbf{G}_1, \ldots, \mathbf{G}_{n^2}\}$ is the same for the trajectories corresponding to the pair $S, ~S'$ of neighboring datasets. Hence, the argument basically follows the same lines of the proof of Theorem~\ref{thm:UB_repl_SGD} since the noise terms cancel out. Thus, we conclude that for any pair $S\simeq S'$ of neighboring datasets, with probability at least $1-\exp(n/2)\geq 1-\theta/6$ (over the randomness of $\A_{\sf NSGD}$), the uniform argument stability of $\A_{\sf NSGD}$ is bounded as: $\delta_{\A_{\sf NSGD}}\leq 4L\eta\left(\sqrt{T} + \frac{T}{n}\right),$ where $T=n^2$. Given the setting of $\eta$ in the theorem, this bound reduces to $8 R /\max\big(\sqrt{n},~ \frac{\sqrt{d\,\log(1/\beta)}}{\alpha}\big)$.
Hence, by Theorem~\ref{thm:stab-to-gen}, with probability at least $1-\theta/3$ (over the randomness in both the i.i.d. dataset $S$ and the algorithm), the generalization error of $\A_{\sf NSGD}$ is bounded as
\begin{align}
\varepsilon_{\mbox{\footnotesize gen}}(\A_{\sf NSGD})&\leq \frac{8 c\, R L\,\log(n)\log(6n/\theta)}{\max\Big(\sqrt{n},~ \frac{\sqrt{d\,\log(1/\beta)}}{\alpha}\Big)}+\frac{c\,\sqrt{\log(6/\theta)}}{\sqrt{n}
= RL \cdot O\left(\frac{\log(n)\log(n/\theta)}{\sqrt{n}}\right),\label{eqn:Ansgd-gen-err}
\end{align}
where $c$ in the first bound is a universal constant.
Now, using (\ref{eqn:Ansgd-opt-err}), (\ref{eqn:Ansgd-gen-err}), and Lemma~\ref{lem:approx-err-bd}, we finally conlcude that with probability at least $1-\theta$ (over randomness in the sample $S$ and the internal randomness of $\A_{\sf NSGD}$), the excess population risk of $\A_{\sf NSGD}$ is bounded as
\begin{align*}
\varepsilon_{\mathsf{\footnotesize{risk}}}(\A_{\sf NSGD})&\leq \varepsilon_{\mbox{\footnotesize opt}}(\A_{\sf NSGD}) + \varepsilon_{\mbox{\footnotesize gen}}(\A_{\sf NSGD})+\varepsilon_{\mbox{\tiny approx}}\\
&= R L\cdot O\Bigg(\max\Big(\frac{1}{\sqrt{n}},~\frac{\sqrt{d\,\log(1/\beta)}}{\alpha\, n}\Big) + \frac{\log(n)\log(n/\theta)}{\sqrt{n}}+ \frac{\sqrt{\log(1/\theta)}}{\sqrt{n}}\Bigg)\\
&= RL\cdot O\Bigg(\max\left(\frac{\log(n)\log(n/\theta)}{\sqrt{n}}, ~\frac{\sqrt{d\,\log(1/\beta)}}{\alpha\, n}\right)\Bigg),
\end{align*}
which completes the proof.
\end{proof}
\else
The proof of the privacy guarantee follows similar lines of argument to that of \cite{BST14, abadi2016deep}. As for the risk guarantee, the proof outline is as follows. We first use standard online-to-batch conversion technique to provide a high-probability bound on $\varepsilon_{\mbox{\footnotesize opt}}(\A_{\sf NSGD})$ (excess empirical error of $\A_{\sf NSGD}$). We next observe that our high-probability upper bound on UAS in Theorem~\ref{thm:sharp_repl_SGD} applies directly to $\A_{\sf NSGD}$ since noise addition does not impact the stability analysis. By Theorem~\ref{thm:stab-to-gen}, this implies a high-probability bound on the generalization error $ \varepsilon_{\mbox{\footnotesize gen}}(\A_{\sf NSGD})$. Using the standard risk decomposition (see eq. (\ref{eqn:risk_decomp})), we get a bound on the excess population risk. Optimizing this bound in $\eta$ yields the claimed bound value in Theorem~\ref{thm:pop_risk_Ansgd} (for the value of $\eta$ in the theorem statement).
Full proofs of the theorems above are deferred to the supplementary material (see Section~6.1 therein).
\fi
\begin{rem}\label{rem:exp_risk_noisy_SGD}
Using the expectation guarantee on UAS given in \iffull Theorem~\ref{thm:UB_repl_SGD} \else Theorem~\ref{thm:sharp_repl_SGD} \fi and following similar steps of the analysis above, we can also show that the expected excess population risk of $\A_{\sf NSGD}$ is bounded as:
$$\ex{}{\varepsilon_{\mathsf{\footnotesize{risk}}}\left(\A_{\sf NSGD}\right)}= RL\cdot O\left(\max\left(\frac{1}{\sqrt{n}}, ~\frac{\sqrt{d\,\log(1/\beta)}}{\alpha\, n}\right)\right).$$
\end{rem}
\iffull
\subsection{Nonsmooth Stochastic Convex Optimization with Multi-pass SGD}\label{sec:data-resue}
Another application of our results concerns obtaining optimal excess risk for stochastic nonsmooth convex optimization via multi-pass SGD. It is known that one-pass SGD is guaranteed to have optimal excess risk, which can be shown via martingale arguments that trace back to the stochastic approximation literature \citep{Robbins:1951,Kiefer:1952}.
Using our UAS\xspace bound, we show that Algorithms \ref{alg:repl_SGD} and \ref{alg:perm_SGD} can recover nearly-optimal high-probability excess risk bounds by making $n$ passes over the data.
Analogous bounds hold for Algorithm \ref{alg:batchGD}, however these are less interesting from a computational efficiency perspective.
\ifthenelse{\boolean{fullver}}{
\input{subsec_DataReuse.tex}
}{In short, we prove the following excess population risk bounds, and their analyses are deferred to Appendix \ref{app:DataReuse},
$$\varepsilon_{\mbox{\footnotesize{risk}}}(\A_{\sf rSGD}) \leq \frac{4RL}{\sqrt{n}} \qquad; \qquad \varepsilon_{\mbox{\footnotesize risk}}(\A_{\sf PerSGD})
\leq \frac{8LR}{\sqrt n}.\vspace{-0.6cm}$$
}
\fi
\section{Discussion and Open Problems}
In this work we provide sharp upper and lower bounds on uniform argument stability\xspace for the (stochastic) subgradient method in stochastic nonsmooth convex optimization. Our lower bounds show inherent limitations of stability bounds compared to the smooth convex case,
however we can still derive optimal population risk bounds by reducing the step size and running the algorithms for longer number of iterations. We provide applications of this idea for differentially-private noisy SGD, and for two versions of SGD (sampling-with-replacement and fixed-permutation SGD).
The first open problem regards lower bounds that are robust to general forms of algorithmic randomization. Unfortunately, the methods presented here are not robust in this respect, since random initialization would prevent the trajectories reaching the region of highly nonsmooth behavior of the objective (or doing it in such a way that it does not increase UAS\xspace). One may try to strengthen the lower bound by using a random rotation of the objective; however, this leads to an uninformative lower bound. Finding distributional constructions for lower bounds against randomization is a very interesting future direction
Our privacy application provides optimal risk for an algorithm that runs for $n^2$ steps, which is impractical for large datasets. Other algorithms, e.g.~in \citep{FKT19}, run into similar limitations. Proving that quadratic running time is necessary for general nonsmooth DP-SCO is a very interesting open problem
that can be formalized in terms of the oracle complexity of stochastic convex optimization \citep{nemirovsky1983problem} under stability and/or privacy constraints.
\section{Introduction}
Successful applications of a machine learning algorithm require the algorithm to generalize well to unseen data. Thus
understanding and bounding the generalization error of machine learning algorithms is an area of intense theoretical interest and practical importance. The single most popular approach to modern machine learning relies on the use of continuous optimization techniques to optimize the appropriate loss function, most notably the stochastic (sub)gradient descent (SGD) method. Yet the generalization properties of SGD are still not well understood.
Consider the setting of stochastic convex optimization (SCO). In this problem, we are interested in the minimization of the population risk $F_{\mathcal{D}}(x):=\mathbb{E}_{\mathbf{z}\sim \mathcal{D}}[ f(x,\mathbf{z}) ]$, where $\mathcal{D}$ is an arbitrary and unknown distribution, for which we have access to an i.i.d.~sample of size $n$, $\mathbf{S}=(\mathbf{z}_1,\ldots,\mathbf{z}_n)$; and $f(\cdot,z)$ is convex and Lipschitz for all $z$.
\iffalse
Specifically, consider the standard setting of stochastic convex optimization (SCO). For a domain $\Z$ and a convex body $\mathcal{X} \subset \mathbb{R}^d$, let $\{f(\cdot,z)\}_{z\in \Z}$ be a family of convex $L$-Lipschitz functions over $\mathcal{X}$ (for normalization purposes we assume for now that $L=1$ and $\mathcal{X}$ is included in the unit ball). Minimization of population risk is the problem of minimizing
$F_{\mathcal{D}}(x):=\mathbb{E}_{\mathbf{z}\sim \mathcal{D}}[ f(x,\mathbf{z}) ]$,
for an unknown distribution $\mathcal{D}$ over $\Z$, given a dataset $\mathbf{S}=(\mathbf{z}_1,\ldots,\mathbf{z}_n)$ of $n$ i.i.d.~samples from $\mathcal{D}$.
\fi
The performance of an
algorithm ${\cal A}$ is
quantified by its expected {\em excess population risk},
$$ \varepsilon_{\mbox{\footnotesize{risk}}}(\A) :=
\mathbb{E}[F_{\mathcal{D}}(\A(\mathbf{S}))]- \min_{x\in \mathcal{X}} F_{\mathcal{D}}(x), $$ where the expectation is taken with respect to the randomness of the sample $\mathbf{S}$ and internal randomness of $\A$.
A standard way to bound the excess risk is given by its decomposition into optimization error (a.k.a.~training error) and generalization error (see eqn.~\eqref{eqn:risk_decomp} in Sec.~\ref{sec:prelims}). The optimization error can be easily measured empirically but assessing the generalization error requires access to fresh samples from the same distribution. Thus bounds on the generalization error lead directly to provable guarantees on the excess population risk.
Classical analysis of SGD allows obtaining bounds on the excess population risk of one pass SGD. In particular, with an appropriately chosen step size, SGD gives a solution with expected excess population risk of $O(1/\sqrt{n})$ and this rate is optimal \citep{nemirovsky1983problem}. However, this analysis does not apply to multi-pass SGD that is ubiquitous in practice.
In an influential work, \citet{HardtRS16} gave the first bounds on the generalization error of general forms of SGD (such as those that make multiple passes over the data). Their analysis relies on algorithmic stability, a classical tool for proving bounds on the generalization error. Specifically, they gave strong bound on the {\em uniform stability} of several variants of SGD on convex and smooth losses (with $2/\eta$-smoothness sufficing when all the step sizes are at most $\eta$). Uniform stability bounds the worst case change in loss of the model output by the algorithm on the worst case point when a single data point in the dataset is replaced \citep{BousquettE02}. Formally, for a randomized algorithm ${\cal A}$, loss functions $f(\cdot,z)$ and $S\simeq S^{\prime}$ and $z\in\Z$, let $\gamma_{\cal A}(S,S^{\prime}, z):=f(\A(S),z)-f(\A(S^{\prime}),z)$, where $S\simeq S^{\prime}$ denotes that the two datasets differ only in a single data point. We say ${\cal A}$ is $\gamma$-uniformly stable if
$$\sup_{S\simeq S^{\prime},z}\mathbb{E}[\gamma_{\cal A}(S,S^{\prime},z)]\leq \gamma,$$
where the expectation is over the internal randomness of $\cal{A}.$ Stronger notions of stability can also be considered, e.g., bounding the probability -- over the internal randomness of $\cal{A}$ -- that $\gamma_{\cal A}(S,S^{\prime},z)>\gamma$.
Using stability,
\cite{HardtRS16} showed that several variants of SGD simultaneously achieve the optimal tradeoff between the excess empirical risk and stability with both being $O(1/\sqrt n)$. Several works have used this approach to derive new generalization properties of SGD \citep{London:2017,Chen:2018,FeldmanV:2019}.
The key insight of \citet{HardtRS16} is that a gradient step on a sufficiently smooth convex function is a nonexpansive operator (that is, it does not increase the $\ell_2$ distance between points). Unfortunately, this property does not hold for nonsmooth losses such as the hinge loss. As a result, no non-trivial bounds on the uniform stability of SGD have been previously known in this case.
Uniform stability is also closely related to the notion of differential privacy (DP). DP upper bounds the worst case change in the output distribution of an algorithm when a single data point in the dataset is replaced \citep{DMNS06}. This connection has been exploited in the design of several DP algorithms for SCO. In particular, bounds on the uniform stability of SGD from \citep{HardtRS16} have been crucial in the design and analysis of new DP-SCO algorithms \citep{wu2017bolt,DworkFeldman18,FeldmanMTT18,BFTT19,FKT19}.
\subsection{Our Results}
We establish tight bounds on the uniform stability of the (stochastic) subgradient descent method on nonsmooth convex losses. These results demonstrate that in the nonsmooth case SGD can be substantially less stable. At the same time we show that SGD has strong stability properties even in the regime when its iterations can be expansive.
For convenience, we describe our results in terms of {\em uniform argument stability\xspace} (UAS\xspace), which bounds the output sensitivity in $\ell_2$-norm w.r.t.~an arbitrary change in a single data point. Formally, a (randomized) algorithm has $\delta$-UAS\xspace if
\begin{equation} \label{eqn:UAS_intro}
\sup_{S\simeq S^{\prime}} \mathbb{E}\left\| \A(S)-\A(S^{\prime})\right\|_2 \leq \delta.
\end{equation}
This notion is implicit in existing analyses of uniform stability
\citep{BousquettE02,ShwartzSSS10,HardtRS16} and was explicitly defined by \citet{Liu:2017}.
In this work, we prove stronger -- high probability -- upper bounds on the random variable $\delta_{\cal A}(S,S^{\prime}):=\left\|\A(S)-\A(S^{\prime})\right\|$,\footnote{In fact, for both GD and fixed-permutation SGD we can obtain w.p.~1 upper bounds on $\delta_{\cal A}(S,S^{\prime})$, whereas for sampling-with-replacement SGD, we obtain a high-probability upper bound.} and we provide matching lower bounds for the weaker -- in expectation -- notion of UAS\xspace \eqref{eqn:UAS_intro}.
A summary of our bounds is in Table \ref{tab:summary}. For simplicity, they are provided for constant step size; general step sizes (for upper bounds) are provided in Section \ref{sec:UB}.
\begin{table}[h!]
\centering
\begin{tabular}{| M{2.8cm} | M{3cm} | M{4cm} | M{4.5cm} | N}
\hline
Algorithm & H.p.~upper bound & Exp.~upper bound & Exp.~Lower bound &\\[5pt] \hline \small
GD (full batch) & $4\big(\eta\sqrt{T}+\frac{\eta T}{n} \big)$ &
$4\big(\eta\sqrt{T}+\frac{\eta T}{n} \big)$
& $\Omega\big(\eta\sqrt T+ \frac{\eta T}{n}\big)$ %
&\\[6pt]\hline
\small SGD (w/replacement) &
$4\big(\eta\sqrt{T}+\frac{\eta T}{n}\big) $
&$\min\{1,\frac{T}{n}\}4\eta\sqrt{T}+4\frac{\eta T}{n} $
& $\Omega\Big(\min\{1,\frac{T}{n}\}\eta\sqrt{T}+\frac{\eta T}{n}\Big)$ &\\[5pt] \hline
\small SGD (fixed permutation) &
$2\eta\sqrt{T}+4\frac{\eta T}{n}$& $\min\{1,\frac{T}{n}\}2\eta\sqrt{T}+4\frac{\eta T}{n}$
& $\Omega\Big(\min\{1,\frac{T}{n}\}\eta\sqrt{T}+\frac{\eta T}{n}\Big)$ &\\[5pt] \hline
\end{tabular}
\caption{
\iffull
UAS\xspace for variants of GD/SGD, with normalized radius and Lipschitz constant. Here $T$ is the number of iterations and $\eta>0$ is the step size.
Both upper and lower bounds also are $\min\{2,(\cdot)\}$, due to the feasible domain radius.
\else
UAS\xspace for GD and SGD. Here $T=\#$~iterations; $\eta$ is the step size
\fi
}
\label{tab:summary}
\end{table}
\iffalse
\begin{table}[h!]
\centering
\begin{tabular}{| M{3cm} | M{5cm} | M{5cm} | N}
\hline
Algorithm & Upper bound & Lower bound &\\[5pt] \hline \small
GD (full batch) &
$4\big(\eta\sqrt{T}+\frac{\eta T}{n} \big)$
& $\Omega\big(\eta\sqrt T+ \frac{\eta T}{n}\big)$ %
&\\[6pt]\hline
\small SGD (w/replacement) &$\min\{1,\frac{T}{n}\}4\eta\sqrt{T}+4\frac{\eta T}{n} $
& $\Omega\Big(\min\{1,\frac{T}{n}\}\eta\sqrt{T}+\frac{\eta T}{n}\Big)$ &\\[5pt] \hline
\small SGD (fixed permutation) & $\min\{1,\frac{T}{n}\}2\eta\sqrt{T}+4\frac{\eta T}{n}$
& $\Omega\Big(\min\{1,\frac{T}{n}\}\eta\sqrt{T}+\frac{\eta T}{n}\Big)$ &\\[5pt] \hline
\end{tabular}
\caption{
\iffull
UAS\xspace for (batch) SGD. Here $T$ is the number of iterations and $\eta>0$ is the step size.
Both upper and lower bounds also are $\min\{2,(\cdot)\}$, due to the feasible domain radius.
\else
UAS\xspace for GD and SGD. Here $T=\#$~iterations; $\eta$ is the step size
\fi
}
\label{tab:summary}
\end{table}
\fi
Compared to the smooth case \citep{HardtRS16}, the main difference is the presence of the additional $\eta\sqrt{T}$ term. This term has important implications for the generalization bounds derived from UAS\xspace. The first one is that the standard step size $\eta=\Theta(1/\sqrt{n})$ used in single pass SGD leads to a vacuous stability bound. Unfortunately, as shown by our lower bounds, this is unavoidable (at least in high dimension). However, by decreasing the step size and increasing the number of steps, one obtains a variant of SGD with nearly optimal balance between the UAS\xspace and the excess empirical risk.
We highlight two major consequences of our bounds:
\begin{itemize}[leftmargin=*]
\item \textbf{Generalization bounds for multi-pass nonsmooth SGD.} We prove that the generalization error of multi-pass SGD with $K$ passes is bounded by $O((\sqrt{Kn}+K)\eta)$. This result can be easily combined with training error guarantees to provide excess risk bounds for this algorithm. Since training error can be measured directly, our generalization bounds would immediately yield strong guarantees on the excess risk in practical scenarios where we can certify small training error.
\item \textbf{Differentially private stochastic convex optimization for non-smooth losses.} We show that a variant of standard noisy SGD \citep{BST14} with constant step size and $n^2$ iterations yields the optimal excess population risk $O\big(\frac{1}{\sqrt{n}}+\frac{\sqrt{d\log (1/\beta)}}{\alpha n}\big)$ for convex \emph{nonsmooth} losses under $(\alpha,\beta)$-differential privacy. The best previous algorithm for this problem is substantially more involved: it relies on a multi-phase regularized SGD with decreasing step sizes and variable noise rates and uses $O(n^2 \sqrt{\log (1/\beta)})$ gradient computations \citep{FKT19}.
\end{itemize}
\subsection{Overview of Techniques}
\begin{itemize}[leftmargin=*]
\item {\bf Upper bounds.}
When gradient steps are nonexpansive, upper-bounding UAS\xspace requires simply summing the differences between the gradients on the neighboring datasets when the replaced data point is used
\citep{HardtRS16}. This gives the bound of $\eta T/n$ in the smooth case.
By contrast, in the nonsmooth case, UAS\xspace may increase even when the gradient step is performed on the same function. As a result it may increase in {\em every single iteration}. However, we use the fact that the difference in the subgradients has negative inner product with the difference between the iterates themselves (by monotonicity of the subgradient). Thus the increase in distance satisfies a recurrence with a quadratic and a linear term. Solving this recurrence leads to our upper bounds
\item {\bf Lower bounds.}
The lower bounds are based on a function with a highly nonsmooth behavior around the origin. More precisely, it is the maximum of linear functions
plus
a small linear drift that is controlled by a single data point. We show that, when starting the algorithm from the origin, the presence of the linear drift pushes the iterate into a trajectory in which each subgradient step is orthogonal to the current iterate. Thus, if $d \geq \min\{T,1/\eta^2\}$, we get the $\sqrt{T}\eta$ increase in UAS\xspace. Our lower bounds are also robust to averaging of the iterates. \iffull \else The detailed constructions and analyses can be found in Section~4 of the full version, attached as supplementary material.\fi
\end{itemize}
\subsection{Other Related Work}
Stability is a classical approach to proving generalization bounds pioneered by \citet{RogersWagner78,DevroyeW79,DevroyeW79a}. It is based on analysis of the sensitivity of the learning algorithm to changes in the dataset such as leaving one of the data points out or replacing it with a different one. \iffull The choice of how to measure the effect of the change and various ways to average over multiple changes give rise to a variety of stability notions
(e.g., \citep{BousquettE02,MukherjeeNPR06,ShwartzSSS10}).\fi\,\! Uniform stability was introduced by \citet{BousquettE02} in order to derive general bounds on the generalization error that hold with high probability. These bounds have been significantly improved in a recent sequence of works \citep{FeldmanV:2018,FeldmanV:2019,BousquetKZ19}. \iffull A long line of work focuses on the relationship between various notions of stability and learnability in supervised setting (see \citep{Kearns:1999,PoggioRMN04,ShwartzSSS10} for an overview). These works employ relatively weak notions of average stability and derive a variety of asymptotic equivalence results. \fi
\citet{Chen:2018} establish limits of stability in the smooth convex setting, proving that accelerated methods must satisfy strong stability lower bounds. Stability-based data-dependent generalization bounds for continuous losses were studied in \citep{Maurer17,Kuzborskij:2018}.
First applications of uniform stability in the context of stochastic convex optimization relied on the stability of the empirical minimizer for strongly convex losses \citep{BousquettE02}. Therefore a natural approach to achieve uniform stability (and also UAS\xspace) is to add a strongly convex regularizer and solve the ERM to high accuracy \citep{ShwartzSSS10}. Recent applications of this approach can be found for example in \citep{KorenLevy15,Charles:2018,FKT19}. In contrast, our approach does not require strong convexity and applies to all iterates of the SGD and not only to a very accurate empirical minimizer.
Classical approach to generalization relies on \emph{uniform convergence} of empirical risk to population risk.
Unfortunately, without additional structural assumptions on convex functions, a lower bound of $\Omega(\sqrt{d/n})$ on the rate of \emph{uniform convergence} for convex SCO is known \citep{ShwartzSSS10,Feldman:16erm}. The dependence on the dimension $d$ makes the bound obtained via the uniform-convergence approach vacuous in the high-dimensional settings common in modern applications.
Differentially private convex optimization has been studied extensively for over a decade (see, e.g., \citep{CM08,CMS,jain2012differentially,kifer2012private,ST13sparse, BST14,ullman2015private,JTOpt13,talwar2015nearly, BFTT19, FKT19}).
However, until recently, the research focused on minimization of the empirical risk. \iffull Population risk for DP-SCO was first studied by \citet{BST14} who gave an upper bound of $\max\left(\tfrac{d^{\frac{1}{4}}}{\sqrt{n}}, \tfrac{\sqrt{d}}{\alpha n}\right)$ \cite[Sec. F]{BST14} on the excess risk. \fi A recent work of \citet{BFTT19} established that the optimal rate of the excess population risk for $(\alpha,\beta)$-DP SCO algorithms is $O\big(\frac{1}{\sqrt{n}}+\frac{\sqrt{d\log (1/\beta)}}{\alpha n}\big)$. Their algorithms are relatively inefficient, especially in the nonsmooth case. Subsequently, \citet{FKT19} gave several new algorithms for DP-SCO with the optimal population risk. For sufficiently smooth losses, their algorithms use a linear number of gradient computations. In the nonsmooth case, as mentioned earlier, their algorithm requires $O(n^2\sqrt{\log(1/\beta)})$ gradient computations and is significantly more involved than the algorithm shown here.
\section{Lower Bounds on Uniform Argument Stability}
\label{sec:LowerBounds}
In this section we provide matching lower bounds for the previously studied first-order methods. These lower bounds show that our analyses are tight, up to absolute constant factors
We note that it is possible to prove a general purpose lower bound on stability by appealing to sample complexity lower bounds for stochastic convex optimization \citep{nemirovsky1983problem}.
This approach in the smooth convex case was first studied in \citep{Chen:2018}; there, these lower bounds are sharp.
However, in the nonsmooth case they are very far from bounds in the previous section.
The idea is that for sufficiently small step size,
a first-order method must incur $\Omega(LT\eta/n)$ uniform stability.
\ifthenelse{\boolean{fullver}}{
\begin{obs} \label{obs:base_lb}
\input{gen_LB_FOM.tex}
\end{obs}
}{
Details of this lower bound can be found on Appendix \ref{app:gen_LB_FOM}.
}
This reasoning leads to an
$\Omega(LT\eta/n)$ lower bound on uniform argument stability\xspace, that can be added to any other lower bound we can prove on specific algorithms that enjoy rates as of gradient descent.
Next we will prove finer lower bounds on the UAS\xspace of specific algorithms.
For this, note that the objective functions we use are polyhedral, thus the subdifferential is a polytope at any point. Since the algorithm should work for
any oracle, we will let the subgradients provided to be extreme points, $\nabla f(x,z)\in\mbox{ext}(\partial f(x,z))$. Moreover, we can make adversarial choices of the chosen subgradient
\subsection{Lower Bounds for Full Batch GD}
\begin{thm} \label{thm:LB_GD_basic}
Let $\mathcal{X}={\cal B}(0,1)$, ${\cal F}={\cal F}_{\mathcal{X}}^0(1)$ and $d\geq \min\{T,1/\eta^2\}$. For the full-batch gradient descent (Alg.~\ref{alg:batchGD}) with constant step size $\eta>0$, there exist $S\simeq S'$ such that the UAS\xspace is lower bounded as
$\delta_{\A_{\sf GD}(S, S')}=\Omega(\min\{1,\eta\sqrt{T}+\eta T/n\}).$
\end{thm}
\ifthenelse{\boolean{fullver}}{
\input{proof_LB_GD_basic.tex}
}{
The proof of this result is deferred to Appendix \ref{app:LB_GD_basic}, due to space considerations.
}
\subsection{Lower Bounds for SGD Sampled with Replacement}
We use a similar construction as from the previous result to prove a sharp lower bound on the
uniform argument stability\xspace for stochastic gradient descent where the sampling is with replacement.
\begin{thm} \label{thm:LB_rSGD_basic}
Let $\mathcal{X}={\cal B}(0,1)$, ${\cal F}={\cal F}_{\mathcal{X}}^0(1)$, and $d\geq \min\{T,1/\eta^2\}$. For the sampled with replacement stochastic gradient descent (Algorithm \ref{alg:repl_SGD}) with constant step size $\eta>0$, there exist $S\simeq S^{\prime}$ such that the uniform argument stability\xspace satisfies
$\mathbb{E}[\delta_{\A_{\sf rSGD}}(S,S^{\prime})]=
\Omega\Big(\min\Big\{1,\frac{T}{n}\Big\}\eta\sqrt T +\frac{\eta T}{n}\Big).$
\end{thm
\begin{proof}
Let $D\triangleq\min\{T,1/\eta^2\}\leq d$, and $\nu>0$, $K\geq \sqrt D$. Consider ${\cal Z}=\{0,1\}$ and define
$$
f(x,z) =
\left\{
\begin{array}{ll}
\max\{0,x_1-\nu,\ldots,x_D-\nu\} & \mbox{ if } z=0 \\
\langle r,x\rangle/K& \mbox{ if } z=1,
\end{array}
\right.
$$
where $r=(-1,\ldots,-1,0,\ldots,0)$ (i.e., supported on the first $D$ coordinates).
Let the random sequence of indices used by the algorithm: $(\mathbf{i}_t)_{t\geq 0} \stackrel{i.i.d.}{\sim}\mbox{Unif}([n])$.
Let $S=(1,0,\ldots,0)$ and $S^{\prime}=(0,0,\ldots,0)$ be neighboring datasets, and denote by $(x^t)_t$ and $(y^t)_t$ the respective stochastic gradient descent trajectories on $S$ and $S^{\prime}$, initialized at $x^1=y^1=0$. It is easy to see that under $S^{\prime}$, we have $y^t=0$ for all $t\in[T]$. Now, suppose that $\nu<\eta/K$. Then, we only have $x^t=0$ for all $t\leq \tau$, where $\tau:=\inf\{t\geq 1:\,\mathbf{i}_t=1 \}$. After time $\tau$, $x^{\tau+1}=-\eta r/K$, and consequently
$x^{\tau+1+j}=-\frac{\eta \mathbf{k}(\tau+j)}{K} r - \eta \sum_{s=1}^{j-\mathbf{k}(\tau+j)+1} e_{s},$ for all $j \in [D+\mathbf{k}(\tau+j)-1]$, where $\mathbf{k}(t)\triangleq |\{s\in[t]:\mathbf{i}_s=1\}|$. Note that conditioned on any fixed value for $\tau,$ ~$\mathbf{k}(\tau+j)\leq j+1$.
Let $\delta_T=\|x^T-y^T\|=\|x^T\|$. Hence, we have $\delta_T\geq\eta\|\sum_{s=1}^{T-\tau-\mathbf{k}(T-1)}e_s\|-\eta \mathbf{k}(T-1)\sqrt{D}/K\geq \eta\sqrt{T-\mathbf{k}(T-1)-\tau}-\eta T \sqrt D /K$.
Let $\mathbf{k}=\mathbf{k}(T-1)$ from now on.
Note that conditioned on any value for $\tau,$ ~$\mathbf{k}-1$ is a binomial random variable taking values in $\{0, \ldots, T-1-\tau\}$. Hence, conditioned on $\tau=t$, by the binomial tail, we always have $\mathbb{P}[\mathbf{k}>T/2~|~\tau=t]\leq\exp(-T/4)$ for all $t\in [T]$ (in particular, this conditional probability is zero when $t\geq T/2$). Also, note that the same upper bound is valid without conditioning on $\tau$.
Hence, by the law of total expectation, we have
\begin{eqnarray*}
\mathbb{E}[\delta_T]
&=& \mathbb{E}[\delta_T| ~\mathbf{k}\leq T/2]\cdot\mathbb{P}[\mathbf{k}\leq T/2]+\mathbb{E}[\delta_T | ~\mathbf{k}> T/2]\cdot\mathbb{P}[\mathbf{k}> T/2
\geq c\,\mathbb{E}[\delta_T | ~\mathbf{k}\leq T/2]
\end{eqnarray*}
where $c=(1-\exp(-T/4))=\Omega(1)$. Hence,
\begin{eqnarray*}
\mathbb{E}[\delta_T]&\geq& c \sum_{t=1}^{T/2} \mathbb{E}[\delta_T|\tau=t,~ \mathbf{k}\leq T/2]\,\mathbb{P}[\tau=t | ~\mathbf{k}\leq T/2]\\
&\geq& c^2\sum_{t=1}^{T/2} \mathbb{E}[\delta_T|\tau=t,~ \mathbf{k}\leq T/2]\,\mathbb{P}[\tau=t]\\
&\geq& c^2\frac{\eta}{n}\sum_{t=1}^{T/2} \sqrt{T-T/2-t}\big(1-\frac1n\big)^{t-1}-c^2\eta \sqrt D T/K.
\end{eqnarray*}
We choose $K$ sufficiently large such that
$\eta \sqrt D T/K=o(\eta\min\{T^{3/2}/n,\sqrt{T}\})$.
\ifthenelse{\boolean{fullver}}{
If $T\leq n$ then
$$ \mathbb{E}[\delta_T] \geq c^2\frac{\eta}{n} \sum_{t=1}^{T/2} \sqrt{t} \big(1-\frac1n\big)^{n-2}-c^2\frac{\eta \sqrt D T}{K}
\geq c^2\frac{\eta e^{-1}}{n}\sum_{t=1}^{T/2} \sqrt{t}-o(\frac{\eta T^{3/2}}{n})
= \Omega\Big( \frac{\eta T^{3/2}}{n} \Big)$$
and if $T>n$ then
$$ \mathbb{E}[\delta_T]
\geq c^2\frac{\eta}{n} \sum_{t=1}^{n/4} \sqrt{T/2-n/4} \, e^{-1} -o(\eta\sqrt{T})
= \Omega( \eta\sqrt{T})$$
}{
Hence, we have
$$
\mathbb{E}[\delta_T]\geq
\left\{
\begin{array}{ll}
\frac{c^2\eta}{n} \sum_{t=1}^{T/2} \sqrt{t} \big(1-\frac1n\big)^{n-2}-\frac{c^2\eta \sqrt D T}{K}
\geq \frac{c^2\eta e^{-1}}{n}\sum_{t=1}^{T/2} \sqrt{t}-o(\frac{\eta T^{3/2}}{n})
= \Omega\Big( \frac{\eta T^{3/2}}{n} \Big) & \mbox{ if } T\leq n\\
\frac{c^2\eta}{n} \sum_{t=1}^{n/4} \sqrt{T/2-n/4} \, e^{-1} -o(\eta\sqrt{T})
= \Omega( \eta\sqrt{T}) &\mbox{ if }
T>n.
\end{array}
\right.
$$
}
This gives a lower bound on $\mathbb{E}[\delta_T]$. Proving that $\overline{x}^T$ satisfies the same lower bound is analogous to the proof in Theorem~\ref{thm:LB_GD_basic}. Finally, $\Omega(\eta T/n)$ can be added to the lower bound by \ifthenelse{\boolean{fullver}}{Observation \ref{obs:base_lb}}{Appendix \ref{app:gen_LB_FOM}}
\end{proof}
\subsection{Lower Bounds for the Fixed Permutation Stochastic Gradient Descent}
Finally, we study fixed permutation SGD. \ifthenelse{\boolean{fullver}}{}{The proof of this result is deferred to Appendix \ref{app:LB_perm_SGD_basic}.}
\begin{thm} \label{thm:LB_perm_SGD_basic}
Let $\mathcal{X}={\cal B}(0,1)$, ${\cal F}={\cal F}_{\mathcal{X}}^0(1)$ and $d\geq \min\{T,1/\eta^2\}$. For the fixed permutation stochastic gradient descent (Algorithm \ref{alg:perm_SGD}) with constant step size $\eta>0$, there exist $S\simeq S^{\prime}$ such that the uniform argument stability\xspace is lower bounded by
$\mathbb{E}[\delta_{\A_{\sf PerSGD}}(S,S^{\prime})]=
\Omega\Big(\min\Big\{1,\frac{T}{n}\Big\}\eta\sqrt T +\frac{\eta T}{n}\Big).$
\end{thm}
\ifthenelse{\boolean{fullver}}{
\input{proof_LB_perm_SGD_basic.tex}
}{}
\section{Generalization Guarantees for Multi-pass SGD}
\label{sec:MultipassSGD}
One important implication of our stability bounds
is that they provide non-trivial generalization error guarantees for multi-pass SGD on nonsmooth losses. Multi-pass SGD is one of the most extensively used settings of SGD in practice, where SGD is run for $K$ passes (epochs) over the dataset (namely, the number of iterations $T=K n$). To the best of our knowledge, aside from the dimension-dependent bounds based on uniform convergence \citep{ShwartzSSS10}, no generalization error guarantees are known for the multi-pass setting on general nonsmooth convex losses. Given our uniform stability upper bounds, we can prove the following generalization error guarantees for the multi-pass setting of sampling-with-replacement SGD. Analogous results can be obtained for fixed-permutation SGD \iffull. \else (using our stability bounds for the latter in Sec.~3 of the supplementary material).\fi
\begin{thm} \label{thm:gen_bds_replSGD}
Running Algorithm \ref{alg:repl_SGD} for $K$ passes (i.e., for $T=K n$ iterations) with constant stepsize $\eta_t=\eta>0$ yields the following generalization error guarantees:
$$ |\mathbb{E}_{\A_{\sf rSGD}} [\varepsilon_{\mathrm{\tiny gen}}(\A_{\sf rSGD})]| \leq 4L^2 \eta\left( \sqrt{Kn} + K\right), $$
and there exists $c>0$, such that for any $0<\theta<1$, with probability~$\geq 1-\theta-\exp(-n/2)$,
$$ |\varepsilon_{\mathrm{\tiny gen}}(\A_{\sf rSGD})|
\leq c ~\Bigg( L^2 \eta\left(\sqrt{K n}+ K \right) \log(n) \log(n/\theta) +LR\sqrt{\frac{\log(1/\theta)}{n}}~\Bigg). $$
\end{thm}
\begin{proof}
First, by the expectation guarantee on UAS given in Theorem~\iffull\ref{thm:UB_repl_SGD} \else \ref{thm:sharp_repl_SGD} \fi together with the fact that the losses are $L$-Lipschitz, it follows that Algorithm~\ref{alg:repl_SGD} (when run for $K$ passes with constant stepsize $\eta$) is $\gamma$-uniformly stable, where $\gamma = 4L^2\left(\eta \sqrt{Kn} +\eta K\right)$. Then, by \cite[Thm.~2.2]{HardtRS16}, we have
$$ |\mathbb{E}_{\A_{\sf rSGD}}[\varepsilon_{\mathrm{\tiny gen}}(\A_{\sf rSGD})]| \leq \gamma. $$
For the high-probability bound, we combine the high-probability guarantee on UAS given in Theorem~\iffull\ref{thm:UB_repl_SGD} \else \ref{thm:sharp_repl_SGD} \fi with Theorem~\ref{thm:stab-to-gen} to get the claimed bound.
\end{proof}
These bounds on generalization error can be used to obtain excess risk bounds using the standard risk decomposition (see \eqref{eqn:risk_decomp}). In practical scenarios where one can certify small optimization error for multi-pass SGD, Thm.~\ref{thm:gen_bds_replSGD} can be used to readily estimate the excess risk. In Section~6.2 \iffull \else of the supplementary material \fi we provide worst-case analysis showing that multi-pass SGD is guaranteed to attain the optimal excess risk of $\approx LR/\sqrt n$ within $n$ passes (with appropriately chosen constant stepsize).
\section{Upper Bounds on Uniform Argument Stability}
\else
\section{Tight Bounds on Uniform Argument Stability}
\fi
\label{sec:UB}
\iffull
\else
In this section we establish sharp bounds on the Uniform Argument Stability\,\! for SGD with nonsmooth convex losses. We start our analysis by a key lemma, which can be used to bound the UAS for generic versions of SGD. Next we present one of our main results that establishes the first sharp bound on the UAS of nonsmooth SGD. We only provide details for the sampling-with-replacement SGD method, but full-batch GD and fixed-permutation SGD are analyzed similarly in the supplementary material.
\fi
\subsection{The Basic Lemma}
We begin by stating a key lemma that encompasses the UAS bound analysis of multiple variants of (S)GD. In particular, all of our UAS upper bounds are obtained by almost a direct application of this lemma.
In the lemma we consider two gradient descent trajectories associated to different sequences of objective functions. The degree of concordance of the two sequences, quantified by the distance between the subgradients at the current iterate, controls the deviation between the trajectories. We note that this distance condition is satisfied for all (S)GD variants we study in this work.
\begin{lem}\label{lem:main_lem}
Let $(x^t)_{t\in[T]}$ and $(y^t)_{t\in[T]}$, with $x^1=y^1$, be online gradient descent trajectories for convex $L$-Lipschitz objectives $(f_t)_{t\in[T-1]}$ and $(f_t^{\prime})_{t\in[T-1]}$, respectively; i.e.,
\begin{eqnarray*}
x^{t+1} &=& \mathsf{Proj}_{\mathcal{X}}[x^t-\eta_t \nabla f_t(x^t)]\\
y^{t+1} &=& \mathsf{Proj}_{\mathcal{X}}[y^t-\eta_t \nabla f_t^{\prime}(y^t)],
\end{eqnarray*}
for all $t\in[T-1]$. Suppose for every $t\in[T-1]$,
$\|\nabla f_t(x^t)-\nabla f_t^{\prime}(x^t)\|\leq a_t$,
for scalars $0\leq a_t\leq 2L$. Then, if $t_0=\inf\{t: f_t\neq f_t^{\prime}\},$
$$ \|x^T-y^T\| \leq 2L\sqrt{\sum_{t=t_0}^{T-1}\eta_t^2}+2\sum_{t=t_0+1}^{T-1}\eta_t a_t.$$
\end{lem}
\begin{proof}
Let
$\delta_t=\|x^t-y^t\|$.
By definition of $t_0$ it is clear that $\delta_1=\ldots=\delta_{t_0}=0$. For $t=t_0+1$, we have that $\delta_{t_0+1}=\|\eta_{t_0}(\nabla f_{t_0}(x^{t_0})-\nabla f_{t_0}^{\prime}(y^{t_0})\|\leq 2L\eta_{t_0}$.
Now, we derive a recurrence for $(\delta_t)_{t\in[T]}$:
\begin{align*}
&\delta_{t+1}^2
= \|\mathsf{Proj}_{\mathcal{X}}[x^t-\eta_t \nabla f_t(x^t)]-\mathsf{Proj}_{\mathcal{X}}[y^t-\eta_t \nabla f_t^{\prime}(y^t)]\|^2
\,\,\leq\,\, \|x^t-y^t-\eta_t( \nabla f_t(x^t)- \nabla f_t^{\prime}(y^t))\|^2 \notag\\
& = \delta_t^2 +\eta_t^2\|\nabla f_t(x^t)-\nabla f_t^{\prime}(y^t)\|^2
-2\eta_t\langle \nabla f_t(x^t)-\nabla f_t^{\prime}(y^t),x^t-y^t\rangle\\
& \leq \delta_t^2 +\eta_t^2\|\nabla f_t(x^t)-\nabla f_t^{\prime}(y^t)\|^2
-2\eta_t\langle\nabla f_t(x^t)-\nabla f_t^{\prime}(x^t),x^t-y^t\rangle
-2\eta_t\langle \nabla f_t^{\prime}(x^t)-\nabla f_t^{\prime}(y^t),x^t-y^t\rangle \\
& \leq \delta_t^2 +\eta_t^2\|\nabla f_t(x^t)-\nabla f_t^{\prime}(y^t)\|^2
+2\eta_t\|\nabla f_t(x^t)-\nabla f_t^{\prime}(x^t)\|\delta_t
-2\eta_t\langle \nabla f_t^{\prime}(x^t)-\nabla f_t^{\prime}(y^t),x^t-y^t\rangle \\
& \leq \delta_t^2+4L^2\eta_t^2+2\eta_ta_t\delta_t,
\end{align*}
where at the last step we use the monotonicity of the subgradient. Note that $$\delta_{t_0+1}\leq \eta_{t_0} \|\nabla f_{t_0}(x^{t_0})-\nabla f_{t_0}^{\prime}(x^{t_0})\| \leq 2L\eta_{t_0}.$$
Hence,
\begin{eqnarray}
\delta_t^2 &\leq& \textstyle \delta_{t_0+1}^2 + 4L^2\sum_{s=t_0+1}^{t-1}\eta_s^2+2\sum_{s=t_0+1}^{t-1}\eta_sa_s\delta_s
\notag\\
&\leq& \textstyle 4L^2\sum_{s=t_0}^{t-1}\eta_s^2+2\sum_{s=t_0+1}^{t-1}\eta_sa_s\delta_s.
\label{eqn:rec_stab}
\end{eqnarray}
Now we prove the following bound by induction (notice this claim proves the result):
$$ \textstyle \delta_t \leq 2L\sqrt{\sum_{s=t_0}^{t-1}\eta_s^2}+2\sum_{s=t_0+1}^{t-1}\eta_sa_s\delta_s \qquad(\forall t\in[T]). $$
Indeed, the claim is clearly true for $t=t_0$. For the inductive step, we assume it holds for some $t\in[T-1]$. To prove the result we consider two cases: first, when $\delta_{t+1}\leq \max_{s\in[t]}\delta_s$, by induction hypothesis we have
$$ \textstyle \delta_{t+1}\leq \delta_t\leq 2L\sqrt{\sum_{s=t_0}^{t-1}\eta_s^2}+2\sum_{s=t_0+1}^{t-1}\eta_sa_s
\leq 2L\sqrt{\sum_{s=t_0}^{t}\eta_s^2}+2\sum_{s=t_0+1}^{t}\eta_sa_s. $$
In the other case, $\delta_{t+1}> \max_{s\in[t]}\delta_s$, we use \eqref{eqn:rec_stab}
$$
\textstyle \delta_{t+1}^2 \,\leq\, 4L^2\sum_{s=t_0}^{t}\eta_t^2 + 2\sum_{s=t_0+1}^t\eta_sa_s\delta_s
\,\,\leq \,\, 4L^2\sum_{s=t_0}^{t}\eta_t^2 + 2\delta_{t+1}\sum_{s=t_0+1}^t\eta_sa_s,
$$
which is equivalent to
$$ \textstyle \Big(\delta_{t+1}-\sum_{s=t_0+1}^ta_s\eta_s\Big)^2 \,\,\leq\,\, 4L^2\sum_{s=t_0}^{t}\eta_t^2 +\Big(\sum_{s=t_0+1}^t\eta_s a_s\Big)^2.
$$
Taking square root at this inequality, and using the subadditivity of the square root, we obtain the inductive step, and therefore the result.
\end{proof}
\iffull
\subsection{Upper Bounds for the Full Batch GD}
\begin{algorithm}[h!]
\caption{$\A_{\sf GD}$: Full-batch Gradient Descent}
\begin{algorithmic}[1]
\REQUIRE Dataset: $S=(z_1, \ldots, z_n)\in \Z^n$,
\# iterations $T$, ~step sizes $\{\eta_t: t\in [T]\}$
\STATE Choose arbitrary initial point $x^1 \in \mathcal{X}$
\FOR{$t=1$ to $T-1$\,}
\STATE $x^{t+1} := \mathsf{Proj}_{\mathcal{X}}\left(x^{t}-\eta_t\cdot \nabla F_S(x^{t})\right),$
\ENDFOR
\RETURN $\overline{x}^T=\frac{1}{\sum_{t\in[T]}\eta_t}\sum_{t\in[T]}\eta_t x^t$
\end{algorithmic}
\label{alg:batchGD}
\end{algorithm}
As a direct corollary of Lemma~\ref{lem:main_lem}, we derive the following upper bound on UAS for the batch gradient descent algorithm.
\begin{thm} \label{thm:UB_GD}
Let $\mathcal{X}\subseteq\mathcal{B}(0,R)$ and $\mathcal{F}=\mathcal{F}_{\mathcal{X}}^0(L)$. The full-batch gradient descent (Algorithm \ref{alg:batchGD}) has uniform argument stability\xspace~
$$\sup\limits_{S\simeq S^{\prime}}\delta_{\A_{\sf GD}}(S,S^{\prime})\leq \min\left\{2R,~
4L\,\Big(\frac1n\sum_{t=1}^{T-1}\eta_t+\sqrt{\sum_{t=1}^{T-1}\eta_t^2}\Big)\right\}.$$
\end{thm}
\input{proof_UB_GD.tex}
\fi
\iffull
\subsection{Upper Bounds for SGD}
Next, we state and prove upper bounds on UAS for two variants of stochastic gradient descent: sampling-with-replacement SGD (Section~\ref{sec:w-replace-sgd}) and fixed-permutation SGD (Section~\ref{sec:perm-sgd}). Here, we give strong upper bounds that hold with high probability (for sampling-with-replacement SGD) and with probability 1 (for fixed-permutation SGD). In Appendix~\ref{app:T_less_n}, we derive tighter upper bounds for these two variants of SGD in the case where the number of iterations $T < $ the number of samples in the data set $n$; however, the bounds derived in this case hold only in expectation.
\subsubsection{Sampling-with-replacement SGD} \label{sec:w-replace-sgd}
Next, we study the uniform argument stability of the sampling-with-replacement stochastic gradient descent (Algorithm~\ref{alg:repl_SGD}). This algorithm has the benefit that each iteration is
extremely cheap compared to Algorithm \ref{alg:batchGD}.
Despite these savings, we will show that same bound on UAS\xspace holds with high probability.
\else
Theorem~\ref{thm:sharp_repl_SGD} below summarizes our results for the UAS of sampling-with-replacement SGD on nonsmooth losses. Given our key lemma, we prove both in-expectation and high-probability upper bounds on the UAS. Our theorem below also establishes the tightness of our upper bounds by showing a matching lower bound. Detailed statements of these results and their full proofs are deferred to the supplementary material (See Theorem~3.3 for the upper bound and Theorem~4.3 for the lower bound in the supplementary material.) Similar analysis of full-batch GD and fixed-permutation SGD are deferred to Sections~3 and 4 of the supplementary material
\fi
\begin{algorithm}[h!]
\caption{$\A_{\sf rSGD}$: Sampling with replacement SGD}
\begin{algorithmic}[1]
\REQUIRE Dataset: $S=(z_1, \ldots, z_n)\in \Z^n$,
\# iterations $T$, ~stepsizes $\{\eta_t: t\in [T]\}
\STATE Choose arbitrary initial point $x^1 \in \mathcal{X}$
\FOR{$t=1$ to $T-1$\,}
\STATE Sample $\mathbf{i}_t\sim\mbox{Unif}([n])$
\STATE $x^{t+1} := \mathsf{Proj}_{\mathcal{X}}\left(x^{t}-\eta_t\cdot \nabla f(x^t,z_{\mathbf{i}_t})\right)$
\ENDFOR
\RETURN $\overline{x}^T=\frac{1}{\sum_{t\in[T]}\eta_t}\sum_{t\in[T]}\eta_t x^t$
\end{algorithmic}
\label{alg:repl_SGD}
\end{algorithm}
\iffull
We now state and prove our upper bound for sampling-with-replacement SGD.
\begin{thm} \label{thm:UB_repl_SGD}
Let $\mathcal{X}\subseteq\mathcal{B}(0,R)$ and $\mathcal{F}=\mathcal{F}_{\mathcal{X}}^0(L)$. The uniform argument stability of the sampling-with-replacement SGD (Algorithm \ref{alg:repl_SGD}) satisfies:
$$\sup\limits_{S\simeq S'}\ex{\A_{\sf rSGD}}{\delta_{\A_{\sf rSGD}}(S,S^{\prime})}\leq \min\left(2R,~ 4L\left(\sqrt{\sum_{t=1}^{T-1}\eta_t^2} +\frac{1}{n}\sum_{t=1}^{T-1}\eta_t\right)\right).$$
\noindent Moreover, if $\eta_{t}=\eta >0 ~\forall t$ then, for any pair $(S, S^{\prime})$ of neighboring datasets, with probability at least $1-\exp\left(-n/2\right)$ (over the algorithm's internal randomness), the UAS random variable is bounded as
$$\delta_{\A_{\sf rSGD}}(S,S^{\prime}) \leq \min\left(2R,~ 4L\left(\eta \,\sqrt{T-1} +\eta\, \frac{T-1}{n}\right)\right).$$
\end{thm}
\else
\begin{thm}[Sharp bounds on UAS for nonsmooth SGD]\label{thm:sharp_repl_SGD}
Let $\mathcal{X}\subseteq\mathcal{B}(0,R)$ and $\mathcal{F}=\mathcal{F}_{\mathcal{X}}^0(L)$. The uniform argument stability of the sampling-with-replacement SGD (Algorithm \ref{alg:repl_SGD}) satisfies:
\begin{equation}
\sup\limits_{S\simeq S'}\ex{\A_{\sf rSGD}}{\delta_{\A_{\sf rSGD}}(S,S^{\prime})}\leq \min\left(2R,~ 4L\left(\sqrt{\sum_{t=1}^{T-1}\eta_t^2} +\frac{1}{n}\sum_{t=1}^{T-1}\eta_t\right)\right).\nonumber
\end{equation}
Moreover, if $\eta_{t}=\eta >0 ~\forall t$ then, for any pair $(S, S^{\prime})$ of neighboring datasets, with probability at least $1-\exp\left(-n/2\right)$ (over the algorithm's internal randomness), the UAS random variable is bounded as
\begin{equation}
\delta_{\A_{\sf rSGD}}(S,S^{\prime}) \leq \min\left(2R,~ 4L\left(\eta \,\sqrt{T-1} +\eta\, \frac{T-1}{n}\right)\right).\nonumber
\end{equation}
Finally, if $d\geq \min\{T,1/\eta^2\}$ and assuming\footnote{The assumption that $T\geq n$ is not necessary. It is invoked here only to simplify the form of the bound (see Table~\ref{tab:summary} or the supplementary material for the general version of the bounds.)} $T\geq n$, there exist $S\simeq S^{\prime}$ such that Algorithm \ref{alg:repl_SGD} with constant stepsize $\eta_t=\eta>0$ satisfies $\ex{\A_{\sf rSGD}}{\delta_{\A_{\sf rSGD}}(S,S^{\prime})}=\Omega\Big(\min\left(R,~ L
\left(\eta\sqrt T+\frac{\eta T}{n}\right)\right) \Big)$.
\end{thm}
\fi
\iffull
\begin{proof}
The bound of $2R$ trivially follows from the diameter bound on $\mathcal{X}$. We thus focus on the second term of the bound. Let $S\simeq S^{\prime}$ be arbitrary neighboring datasets, $x^0=y^0$, and consider the trajectories $(x^t)_{t\in[T]},(y^t)_{t\in[T]}$ associated with the sampled-with-replacement stochastic subgradient method on datasets $S$ and $S^{\prime}$, respectively. We use Lemma~\ref{lem:main_lem} with $f_t(\cdot)=f(\cdot,\mathbf{z}_{\mathbf{i}_t})$ and $f_t^{\prime}(\cdot)=f(\cdot,\mathbf{z}_{{\mathbf{i}_t}^{\prime}})$.
Let us define $\mathbf{r}_t\triangleq \mathbf{1}_{\{\mathbf{z}_{\mathbf{i}_t}\neq\mathbf{z}_{\mathbf{i}_t}^{\prime}\}}$. Note that at every step $t$, $\mathbf{r}_t=1$ with probability $1-1/n$, and $\mathbf{r}_t=0$ otherwise. Moreover, note that $\{\mathbf{r}_t: ~t\in [T]\}$ is an independent sequence of Bernoulli random variables. Finally, note that $\|\nabla f_t(x^t)-\nabla f_t^{\prime}(x^t)\|\leq 2L \mathbf{r}_t$.
Hence, by Lemma~\ref{lem:main_lem}, for any realization of the trajectories of the SGD method, we have
\begin{align}
\forall t\in [T]:\quad
\|x^t-y^t\| &\leq 2L\sqrt{ \sum_{s=1}^{t-1}\eta_s^2 } +4L\sum_{s=1}^{t-1} \mathbf{r}_s\eta_s \leq \Delta_T,\label{bound:unif_t}
\end{align}
where $\Delta_T\triangleq 2L\sqrt{ \sum_{s=1}^{T-1}\eta_s^2 } +4L\sum_{s=1}^{T-1} \mathbf{r}_s\eta_s$. Taking expectation of (\ref{bound:unif_t}), we have
$$ \forall t\in [T]:\quad \mathbb{E}\left[\|x^t-y^t\|\right] \leq \mathbb{E}\left[\Delta_T\right]= 2L\sqrt{ \sum_{s=1}^{T-1}\eta_s^2 } +\frac{4L}{n}\sum_{s=1}^{T-1}\eta_s. $$
This establishes the upper bound on UAS but only in expectation. Now, we proceed to prove the high-probability bound. Here, we assume that the step size is fixed; that is, $\eta_t=\eta >0$ for all $t\in [T-1]$. Note that each $\mathbf{r}_s, s\in [T],$ has variance $\frac{1}{n}\left(1-\frac{1}{n}\right)<\frac{1}{n}$. Hence, by Chernoff's bound\footnote{Here, we are applying a bound for (scaled) Bernoulli rvs where the exponent is expressed in terms of the variance.}, we have
\begin{align*}
\mathbb{P}\left[\eta \sum_{s=1}^{T-1} \mathbf{r}_s \geq \eta \frac{T-1}{n}+\eta \sqrt{T-1}
\right] &\leq \exp\left(-\,\frac{\eta^2 (T-1)}{2\eta^2\,\frac{T-1}{n}}\right) = \exp\left(-\frac{n}{2}\right).
\end{align*}
Therefore, with probability at least $1-\exp\left(-n/2\right)$, we have
$$\Delta_T\leq 3L\eta\sqrt{ T-1 } +\frac{4L}{n}\eta (T-1).$$
Putting this together with (\ref{bound:unif_t}), with probability at least $1-\exp\left(-n/2\right)$, we have
$$\forall t\in [T]:\quad \|x^t-y^t\| \leq 3L\eta\sqrt{ T-1 } +\frac{4L}{n}\eta (T-1). $$
Finally, by the triangle inequality, we get that with probability at least $1-\exp\left(-n/2\right)$, the same stability bound holds for the average of the iterates $\overline{x}^T$, $\overline{y}^T$.
\end{proof}
\fi
\iffull
\subsubsection{Upper Bounds for the Fixed Permutation SGD} \label{sec:perm-sgd}
In Algorithm \ref{alg:perm_SGD}, we describe the fixed-permutation stochastic gradient descent. This algorithm works in epochs, where each epoch is a single pass on the data. The order in which data is used is the same across epochs, and is given by a permutation $\pi$.
The algorithm can be alternatively described without the epoch loop simply by
\begin{equation}
\label{eqn:single_iter_pr_SGD}
x^{t+1} = \mathsf{Proj}_{\mathcal{X}}\left(x^{t}-\eta_t\cdot \nabla f(x^{t}, z_{\boldsymbol{\pi}(t\, \mbox{\footnotesize mod } n)})\right) \qquad (\forall t\in[nK]).
\ifthenelse{\boolean{fullver}}{}{\vspace{-0.5cm}}
\end{equation}
\ifthenelse{\boolean{fullver}}{We will use this description for stability analysis, since it is more convenient.}{}
\begin{algorithm}
\caption{$\A_{\sf PerSGD}$: Fixed Permutation SGD} \label{alg:perm_SGD}
\begin{algorithmic}[1]
\REQUIRE Dataset $S=(z_1, \ldots, z_n)\in \Z^n$,
\# rounds $K$, total \# steps $T\triangleq nK$, step sizes, $\{\eta_t\}_{t\in [nK]}$
${\pi}:[n]\rightarrow [n]$ permutation over $[n]$
\STATE Choose arbitrary initial point $x_{n+1}^0 \in \mathcal{X}$
\FOR{$k=1,\ldots,K$}
\STATE $x_1^k=x_{n+1}^{k-1}$
\FOR{$t=1$ to $n$\,}
\STATE $x^k_{t+1} := \mathsf{Proj}_{\mathcal{X}}\left(x^k_{t}-\eta_{(k-1)n+t}\cdot \nabla f(x^k_{t}, z_{\pi(t)})\right)$
\ENDFOR
\STATE $\overline{\eta}_k=\sum_{t=1}^n\eta_{(k-1)n+t} $
\ENDFOR
\RETURN $\overline{x}^K=\frac{1}{\sum_{k\in[K]}\overline{\eta}_k} \sum_{k\in[K]}\overline{\eta}_{k} \cdot x_{1}^k$
\end{algorithmic}
\end{algorithm}
We show that the same UAS bound of batch gradient descent and sampling-with-replacement SGD holds for the fixed-permutation SGD. We also observe that a slightly tighter bound can be achieved if we consider \textit{the expectation guarantee} on UAS when $\boldsymbol{\pi}$ is chosen uniformly at random. We leave these details to Theorem \ref{thm:UB_perm_SGD_2} in the Appendix.
In the next result, we assume that the sequence of step sizes $\left(\eta_t\right)_{t\in [T]}$ is non-increasing, which is indeed the case for almost all known variants of SGD.
\begin{thm} \label{thm:UB_perm_SGD}
Let $\mathcal{X}\subseteq\mathcal{B}(0,R)$, $\mathcal{F}=\mathcal{F}_{\mathcal{X}}^0(L)$, and $\pi$ be any permutation over $[n]$. Suppose the step sizes $\left(\eta_t\right)_{t\in [T]}$ form a non-increasing sequence. Then the uniform argument stability\xspace of the fixed-permutation SGD (Algorithm \ref{alg:perm_SGD}) is bounded as
$$\sup\limits_{S\simeq S^{\prime}}\delta_{\A_{\sf PerSGD}}(S,S^{\prime})\leq \min\left\{2R,~2L \left(\sqrt{\sum_{t=1}^{T-1}\eta_t^2} +\frac{2}{n} \sum_{t=1}^{T-1}\eta_t\right)\right\}.$$
\end{thm}
\begin{proof}
Again, the bound of $2R$ is trivial. Now, we show the second term of the bound. Let $S\simeq S^{\prime}$ be arbitrary neighboring datasets, $x^1=y^1$, and consider the trajectories $(x^t)_{t\in[T]},(y^t)_{t\in[T]}$ associated with the fixed permutation stochastic subgradient method on datasets $S$ and $S^{\prime}$, respectively.
Since the datasets $S\simeq S^{\prime}$ are arbitrary, we may assume without loss of generality that $\pi$ is the identity, whereas the perturbed coordinate $\mathbf{i}=i$ is arbitrary. We use Lemma~\ref{lem:main_lem} with $f_t(\cdot)=f\left(\cdot,\mathbf{z}_{\left(t ~\mbox{\footnotesize mod }n\right)}\right)$ and
$f_t^{\prime}(\cdot)=f\left(\cdot,\mathbf{z}^{\prime}_{\left(t ~\mbox{\footnotesize mod }n\right)}\right)$.
It is easy to see then that $\|\nabla f_t(x^t)-\nabla f_t^{\prime}(x^t)\|\leq a_t$, with
$a_t=2L \cdot\mathbf{1}_{\{(t ~\mbox{\footnotesize mod }n)=i\}}$, where $\mathbf{1}_{\{\mathsf{condition}\}}$ is the indicator of $\mathsf{condition}$. Hence, by Lemma~\ref{lem:main_lem}, we have
\begin{eqnarray*}
\|x^t-y^t\|
&\leq& 2L\sqrt{\sum_{s=1}^{t-1}\eta_s^2} +4L\sum_{r=1}^{\lfloor (t-1)/n \rfloor}\eta_{rn+i} \\
&\leq& 2L\sqrt{\sum_{s=1}^{t-1}\eta_s^2} +\frac{4L}{n} \sum_{r=1}^{t-1}\eta_s,
\end{eqnarray*}
where at the last step we used the fact that $(\eta_t)_{t\in[T]}$ is non-increasing; namely, for any $r\geq 1$
$$ \eta_{rn+i} \leq \frac1n \sum_{s=(r-1)n+i+1}^{rn+i} \eta_s. $$
Since the bound holds for all the iterates, using triangle inequality, it holds for the output $\overline{x}^K$ averaged over the iterates from the $T/n$ epochs.
\end{proof}
\subsection{Discussion of the upper bounds: examples of specific instantiations}
The upper bounds on stability from this section all behave very similarly. Let us explore the consequences of the obtained rates in terms of generalization bounds for different choices of the step size sequence. As a case study, we will consider excess risk bounds for the full-batch subgradient method (Algorithm \ref{alg:batchGD}), but similar conclusions hold for all the variants that we studied. We emphasize that prior to this work, no dimension-independent bounds on the excess risk were known of this method (specifically, for nonsmooth losses and without explicit regularization).
To bound the excess risk, we will use the risk decomposition, eqn.~\eqref{eqn:risk_decomp}. For simplicity, we will only be studying excess risk bounds in expectation (in Section \ref{sec:Applications} we consider stronger, high probability, bounds). In this case, the stability implies generalization result (Theorem \ref{thm:stab-to-gen}) simplifies to \cite{BousquettE02,HardtRS16}
$$ \mathbb{E}_{\mathbf{S}}[F_{\mathbf{S}}({\cal A}(\mathbf{S}))-F_{\cal D}({\cal A}(\mathbf{S}))] \leq \sup_{S\simeq S^{\prime}}\delta_{{\cal A}}(S,S^{\prime}). $$
Finally, the approximation error (Lemma \ref{lem:approx-err-bd}) simplifies as well: it is upper bounded by 0 in expectation.
\begin{itemize}
\item {\em Fixed stepsize:} Let $\eta_t\equiv\eta>0$. By Thm.~\ref{thm:UB_GD}, UAS is bounded by $4L\sqrt{T}\eta+\frac{4LT\eta}{n}$. On the other hand, the standard analysis of subgradient descent guarantees that $\varepsilon_{\mbox{\footnotesize opt}}(\A_{\sf GD})\leq \frac{R^2}{2\eta T}+\frac{\eta L^2}{2}$. Therefore, by the expected risk decomposition \eqref{eqn:risk_decomp}
\begin{eqnarray*}\mathbb{E}_{\mathbf{S}}[\varepsilon_{\mbox{\footnotesize{risk}}}(\A_{\sf GD})] &\leq& \mathbb{E}_{\mathbf{S}}[\varepsilon_{\mathrm{\tiny gen}}(\A_{\sf GD})] +\mathbb{E}_{\mathbf{S}}[\varepsilon_{\mathrm{\tiny opt}}(\A_{\sf GD})] \leq 4L^2\sqrt{T}\eta+\frac{4L^2T\eta}{n}+\frac{R^2}{2\eta T}+\frac{\eta L^2}{2}.
\end{eqnarray*}
If we consider the standard method choice, $\eta=R/[L\sqrt n]$ and $T=n$, the bound above is at least $4LR$ (due to the first term). Consequently, upper bounds obtained from this approach are vacuous.
In order to deal with the $L\sqrt T \eta$ term, we need to substantially moderate our stepsize, together with running the algorithm for longer. For example, $\eta=\frac{R}{4L\sqrt{Tn}}$ gives
$\mathbb{E}_{\mathbf{S}}[\varepsilon_{\mbox{\footnotesize{risk}}}(\A_{\sf GD})]\leq \frac{2LR}{\sqrt n}+\frac{2LR\sqrt n}{\sqrt T}+\frac{R\sqrt T}{n^{3/2}}$, so by choosing $T=n^2$ we obtain an expected excess risk bound of $O(LR/\sqrt n)$, which is optimal. We will see next that it is not possible to obtain the same rates from this bound if $T=o(n^2)$, for any choice of $\eta>0$. It is also an easy observation that, at least for constant stepsize, it is not possible to recover the optimal excess risk if $T=\omega(n^2)$.
\item {\em Varying stepsize:} For a general sequence of stepsizes the optimization guarantees of Algorithm \ref{alg:batchGD} are the following
$$\mathbb{E}_{\mathbf{S}}[\varepsilon_{\mathrm{\tiny opt}}(\A_{\sf GD})] \leq \frac{R^2}{2\sum_{t=1}^{T-1}\eta_t}+\frac{L^2\sum_{t=1}^{T-1}\eta_t^2}{2}. $$
From the risk decomposition, we have
\begin{eqnarray*}\mathbb{E}_{\mathbf{S}}[\varepsilon_{\mbox{\footnotesize{risk}}}(\A_{\sf GD})] &\leq& \mathbb{E}_{\mathbf{S}}[\varepsilon_{\mathrm{\tiny gen}}(\A_{\sf GD})] +\mathbb{E}_{\mathbf{S}}[\varepsilon_{\mathrm{\tiny opt}}(\A_{\sf GD})] \\
&\leq& 4L^2\sqrt{\sum_{t=1}^{T-1}\eta_t^2}+\frac{4L^2}{n}\sum_{t=1}^{T-1}\eta_t+\frac{R^2}{2\sum_{t=1}^{T-1}\eta_t}+\frac{L^2\sum_{t=1}^{T-1}\eta_t^2}{2}.
\end{eqnarray*}
In fact, we can show that any choice of step sizes that makes the quantity above $O(LR/\sqrt n)$ must necessarily have $T=\Omega(n^2)$. Indeed, notice that in such case
\begin{eqnarray*} \frac{R^2}{2\sum_{t=1}^{T-1}\eta_t} = O\Big(\frac{LR}{\sqrt n}\Big); & 4L^2\sqrt{\sum_{t=1}^{T-1}\eta_t^2}=O\Big(\frac{LR}{\sqrt n}\Big) \\
\Longleftrightarrow \qquad
\sum_{t=1}^{T-1}\eta_t = \Omega\Big(\frac{R\sqrt n}{L}\Big); & \sqrt{\sum_{t=1}^{T-1}\eta_t^2}=O\Big(\frac{R}{L\sqrt n} \Big).
\end{eqnarray*}
Therefore, by Cauchy-Schwarz inequality,
$$ \Omega\Big(\frac{R\sqrt n}{L}\Big)=\sum_t \eta_t \leq \sqrt{T}\sqrt{\sum_t \eta_t^2} =O\Big(\frac{R\sqrt{T}}{L\sqrt n} \Big) \quad\Longrightarrow\quad T=\Omega(n^2). $$
\end{itemize}
The high iteration complexity required to obtain optimal bounds motivates studying whether it is possible to improve our uniform argument stability\xspace bounds. We will show that, unfortunately, they are sharp up to absolute constant factors.
\fi
\subsubsection{Risk Bounds for Sampling-with-Replacement Stochastic Gradient Descent}
\begin{thm} \label{thm:gen_repl_SGD}
Let $\mathcal{X}\subseteq\mathcal{B}(0,R)$ and $\mathcal{F}=\mathcal{F}_{\mathcal{X}}^0(L)$. The
sampling with replacement stochastic gradient descent (Algorithm \ref{alg:repl_SGD}) with $T=n^2$ iterations and $\eta=\frac{R}{\,L\,n^{3/2}}$ satisfies for any $12\exp\{-n^2/32\}<\theta<1.$
$$\mathbb{P} \Big[ \varepsilon_{\mbox{\footnotesize{risk}}}(\A_{\sf rSGD}) = O\Big( \frac{cLR}{\sqrt{n}}\log(n)\log(\frac{n}{\theta})\Big) \Big] \leq \theta.$$
\end{thm}
It should be noted that, similarly to Remark \ref{rem:exp_risk_noisy_SGD},
if we are only interested in expectation risk bounds, one can shave off the polylogarithmic factor above, which is optimal for the expected excess risk.
\begin{proof}
Let $\mathbf{S}\sim{\cal D}^n$ an i.i.d.~random sample for the stochastic convex program, and apply on these data the algorithm $\A_{\sf rSGD}$ for constant step size $\eta>0$ and $T$ iterations.
We consider $\theta>0$ such that $\theta>12\exp\{-T/32\}$.
Notice that the sampling-with-replacement stochastic gradient is a bounded first-order stochastic oracle for the empirical objective. It is direct to verify that the assumptions of Lemma \ref{lem:online_to_batch} are satisfied with $\sigma=0$.
Hence, by Lemma \ref{lem:online_to_batch}, we have that, with probability at least $1-\theta/3$
$$
\varepsilon_{\mathrm{\tiny opt}}(\A_{\sf rSGD}) \leq O\Big( LR\sqrt{\frac{2\log(12/\theta)}{T}}+\frac{R^2}{\eta T}+\eta L^2 \Big).
$$
On the other hand, Theorem \ref{thm:UB_repl_SGD} together with Theorem \ref{thm:stab-to-gen}, guarantees that
with probability at least $1-\theta/3$, we have
$$ |\eps_{\mbox{\footnotesize gen}}(\A_{\sf rSGD})| \leq O\Big( L^2\big[\sqrt T \eta+\frac{T\eta}{n}\big]\log n\log(6n/\theta)+LR\sqrt{\log(6/
\theta)}{n} \Big). $$
Finally, Lemma \ref{lem:approx-err-bd} ensures that with probability $1-\theta/3$
$$ \varepsilon_{\mathrm{\tiny approx}} \leq LR\sqrt{\frac{ 2\log(3/\theta)}{n}}.$$
By the union bound and the excess risk decomposition \eqref{eqn:risk_decomp}, we have that, with probability $1-\theta$,
\begin{eqnarray*} \varepsilon_{\mbox{\footnotesize risk}}({\cal A})
&=&
O\Big(LR\sqrt{\frac{\log(1/\theta)}{T}}+\frac{R^2}{\eta T}+\eta L^2+L^2\eta \big(\sqrt{T}+\frac{T}{n}\big) \log(n) \log(\frac{6n}{\theta})\\
&&+LR\sqrt{\frac{\log(6/\theta)}{n}}+LR\sqrt{\frac{ \log(3/\theta)}{n}} \Big)\\
&=& O\Big(\frac{LR}{\sqrt n}\log(n)\log(\frac{n}{\theta})\Big),
\end{eqnarray*}
where only at the last step we replaced by the choice of step size and number of iterations from the statement.
\end{proof}
\subsubsection{Risk Bounds for Fixed-Permutation Stochastic Gradient Descent}
As a final application we provide a population risk bound based on the UAS\xspace of Algorithm \ref{alg:perm_SGD}. Similarly as in the case of sampling-with-replacement SGD, we need an optimization error analysis, which for completeness is provided in Appendix \ref{App:Fixed_Perm_SGD}, and it is based on the analysis of the incremental subgradient method \citep{Nedic:2001}.
Interestingly, the combination of the incremental method analysis for arbitrary permutation \citep{Nedic:2001} and our novel stability bounds that also work for arbitrary permutation, guarantees generalization bounds for fixed permutation SGD without the need of reshuffling, or even any form of randomization. We believe this could be of independent interest.
\begin{thm} \label{thm:gen_perm_SGD}
Algorithm \ref{alg:perm_SGD} with constant step size $\eta_k\equiv\eta=R/[Ln\sqrt{K}]$ and $K=n$ epochs is such that for every $0<\theta<1$,
$$ \mathbb{P}\Big[ \varepsilon_{\mbox{\footnotesize risk}}(\A_{\sf PerSGD}) > \frac{cLR}{\sqrt n} \log n\log(\frac{n}{\theta}) \Big]\leq \theta,$$
where $c>0$ is an absolute constant.
\end{thm}
Similarly to the previous result, we can remove the polylogarithmic factor if we are only interested in expected excess risk guarantees.
\begin{proof}
By Corollary \ref{cor:perm_SGDopt_error}
$$ \varepsilon_{\mbox{\footnotesize opt}}(\A_{\sf PerSGD})
\leq \frac{R^2}{n K\eta}+\frac{L^2(n+2)\eta}{2}=O\Big(\frac{LR}{\sqrt n} \Big),$$
by our choice of $K,\eta$.
On the other hand, Theorem~\ref{thm:UB_perm_SGD} guarantees the algorithm is $\delta$-UAS with probability 1, where $\delta=O(R/\sqrt n)$. Therefore, by
Theorem~\ref{thm:stab-to-gen}, we have that w.p.~$1-\theta/2$
$$|\varepsilon_{\mbox{\footnotesize gen}}(\A_{\sf PerSGD})| \leq c\Big( \frac{LR}{\sqrt n}\log n \log(2n/\theta)+LR\sqrt{\frac{\log 2/\theta}{n}} \Big).
$$
Finally, Lemma \ref{lem:approx-err-bd} ensures that with probability $1-\theta/2$
$$ \varepsilon_{\mathrm{\tiny approx}} \leq LR\sqrt{\frac{ 2\log(2/\theta)}{n}}.$$
By the union bound and the excess risk decomposition \eqref{eqn:risk_decomp}, we have that, with probability
at least $1-\theta$,
\begin{eqnarray*}
\varepsilon_{\mbox{\footnotesize risk}}(\A_{\sf PerSGD})
&\leq& \varepsilon_{\mathrm{\tiny opt}}(\A_{\sf PerSGD}) +\varepsilon_{\mathrm{\tiny gen}}(\A_{\sf PerSGD})+\varepsilon_{\mathrm{\tiny approx}} \\
&=& O\Big( \frac{LR}{\sqrt n}
+\frac{LR}{\sqrt n}\log n \log(n/\theta)+LR\sqrt{\frac{\log 1/\theta}{n}} + LR\sqrt{\frac{ \log(2/\theta)}{n}}\Big)\\
&=&O\Big( \frac{LR}{\sqrt n}\log n\log(\frac{n}{\theta}) \Big).
\end{eqnarray*}
\end{proof}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 647 |
{"url":"http:\/\/piwik.manuel-esteban.com\/rndhhib\/what-does-the-beam-width-of-an-antenna-tell-us-0bddc3","text":"{\\displaystyle \\mathrm {A} (\\theta )} One of the amusing facets of reading lots of e-mail is discovering how many beam users demand 1-degree aiming accuracy when their beam widths are well over 50 degrees. b. The antenna is very compact and the whole area forms part of the radiating structure. Long range 4G connections are possible using high gain antennas. ( Find the moment of inertia for a rectangle of length l and width w when the axis of rotation is . width or height). A It's resonant frequency is lower, and its length is approximately 5% longer than the driven element. F\/R ratio-dB ... A. The Antenna does 6-20 meters. = The smaller \u2026 x This is referred to as an antenna \u2018radiation pattern\u2019, and is often measured in terms of vertical and horizontal beam width. {\\displaystyle \\mathrm {m} } RFbeam defines antenna pattern (also called beam width) as the angular width of beam, where the (Doppler) output signal amplitude drops to 50% (-6dB). After entering your details, you\u00e2\u0080\u0099ll be directed to a results page. By how many times is an input impedance of a folded dipole at resonance greater than that of an isolated dipole with \u00e2\u0080\u00a6 I have a new M2 35 ft long DXing beam ready to replace that on the roof to be installed SOMEDAY! ( 1.2- Remote client. 20-15-10m elements are full size and 40m has linear loading. The antenna is well made - I've been using it now at 12m AGL for 9 months and it totally outperforms an 130 feet long off centre fed dipole at 15m AGL on all bands 20m - 6m. of RG 213 to a MFJ-986 differential tuner. No longer band-limited from 6m to 40m, this antenna changes the game and has become my \"one-antenna antenna farm\". The elevation plane beamwidth is the total angular width between the two 3-dB points on the curve. {\\displaystyle \\mathrm {P} =0.5\\mathrm {P} _{max}} antenna I found this to be a good read. I like how it has a parts list. beam definition: 1. a line of light that shines from a bright object: 2. a line of radiation or particles flowing\u00e2\u0080\u00a6. K-LC1 example 80\u00b0 x 35\u00b0: This is an asymmetrical beam. This can be better understood with the help of the following diagram. This article incorporates\u00a0public domain material from the General Services Administration document: \"Federal Standard 1037C\". The gain of the half-wave dipole is approximately 2.2 dBi. F\/R ratio-dB ... A. Some radar units transmit a beam as wide as 24 degrees. i.e. One of the major keys to understanding Yagi theory is a knowledge of the phases of the currents flowing in the different elements of the antenna. One forms a beam by a phased array of dipole elements. \u03b8 This causes the necessary asymmetry for radiation at the junctions and all radiation has the same phase producing a broadside beam. Meaning of beamwidth. In this chapter, we shall discuss about another important factor in the radiation pattern of an antenna, known as beam width. The simple beam has side nulls that show no limit. +++++ The basics show that if you want to transmit or receive a signal which is positioned at 230 degrees and you have a beam width of 15 degrees you can aim anywhere between 222.5 to 237.5 degrees. The mathematical expression of First Null Beam Width is. What does beamwidth mean? On the East Tower is a M2 7 element 6 meter beam and an old Hygain HORIZ 2M 14 element beam. An antenna is known as \u201cdirectional\u201d if its pattern strongly favors a certain direction. A The right angle antenna is the plug and play antenna, and it does not have cable, but convenient to install, suitable for the area around 100 square meters. the centerline of the rectangle, and one of the edges. The whole antenna weight is only 5.5kg (11 lbs) making it ideally suited for portable use. These parameters help us to know about the antenna\u2019s performance. Simply, FNBW is the angular separation, quoted away from the main beam, which is drawn between the null points of radiation pattern, on its major lobe. is the conjugate transpose of o This article will extend the concepts of the previous article to include near and far field radiation patterns, permittivity, directivity, and gain. As no of elements increases (in an array) the beam becomes narrower. The two main considerations of this beam width are Half Power Beam Width (HPBW) and First Null Beam Width (FNBW). Antenna beam width is also known as the half-power. The \u201cW\u201d indicates that it is a wide flange beam (as opposed to, say, an S beam); the \u201c14\u201d is referenced as section number; and the \u201c211\u201d indicates that the beam weighs 211 lbs per foot. Our customers can avail from us an excellent quality range of Antenna. B Hence, without considering the losses, the area, which utilizes the maximum power obtained to the actual area, can be termed as effective area. The author then divided it by 4 to make a \"quarter wave\" antenna length of 30mm. More detailed information can be found on Wikipedia if you\u00e2\u0080\u0099re interested. Definition of beamwidth in the Definitions.net dictionary. The beamwidth can be computed for arbitrary antenna arrays. The horizontal beam width tells us how broad or narrow our signal is and hence how careful we must be in aiming the A To remedy this, several techniques are used to make an antenna \u201cappear\u201d to be \/4 . It is also commonly referred to as the \u201cBeam\u201d antenna. An antenna this big and complex could not be a success without it. 3; 2 pairs of 40 meter phased verticals covering NE\/SW\/SE\/NW, switchable. These are made from qualitative range of raw materials, which is quality checked by our team of experienced professionals. beam width -\u00b0 H\u00ef\u00bc\u009a36 V\u00ef\u00bc\u009a32. Uniform distribution of a user\u2019s equipment would clearly support the concept of the same cell size and structure of the hexagon honeycomb. At last, RF technicians tell us that in average half the ground loss observed in a vertical antenna occurs within a circle which radius equals to the antenna height, thus practically 1\/4l, and the remaining loss resistance occurs in the next 1\/4l out from the antenna due to decreasing of the capacitance between the vertical radiator and the earth. A beam antenna has besides gain another interesting feature it is called the front to back often shortened as FB. Think of a radar beam as a cone - narrow at the radar antenna and widening as it heads for the horizon. In any illustration, the sketch drawn to represent the radiation of an antenna is its radiation pattern.One can simply understand the function and directivity of an antenna by having a look at its radiation pattern. \u2217 In understanding the basic Yagi antenna theory, the different elements of the Yagi antenna react in a complex and interrelated way. beam width -\u00b0 H\u00ef\u00bc\u009a36 V\u00ef\u00bc\u009a32. The value of 2.2 dBi is achieved at the horizon in the elevation plane and everywhere in the azimuth plane. The vertical beam width is a rough measure of the range of elevation angles that we can count on for good communications. Information and translations of beamwidth in the most comprehensive dictionary definitions resource on the web. Beam Width is one. P at the reference angle If you put 4 watts into a four-foot antenna, you will get the same power out of that antenna as if you were putting 2 watts into a 102-inch whip. ) The figure shows half-power points on the major lobe and HPBW. In a radio antenna pattern, the half power beam width is the angle between the half-power (-3 dB) points of the main lobe, when referenced to the peak effective radiated power of the main lobe. In the radiation pattern of an antenna, the main lobe is the main beam of the antenna where maximum and constant energy radiated by the antenna flows. Typically these reports from antenna web and tvfool are inaccurate as to the antenna needed. Every time you double the aperture size of an antenna you double its \u00e2\u0080\u00a6 . Which Range for Which Antenna. $E_{i}$ is the field strength of the incident wave. For the optical regime, see, https:\/\/en.wikipedia.org\/w\/index.php?title=Beamwidth&oldid=942024483, Wikipedia articles incorporating text from the Federal Standard 1037C, Wikipedia articles incorporating text from MIL-STD-188, Creative Commons Attribution-ShareAlike License. So, if you want to get N beams at the same time you have to have an antenna with N phased arrays. What not everyone knows is that the original concept of this antenna has to be attributed to Shintaro Uda (Yagi\u00e2\u0080\u0099s assistant professor) that in 1926, first described this antenna at Tohuku University in Japan, in the \u00e2\u0080\u00a6 This can also be imagined as a flashlight for better understanding. Test at an angle slightly less than the horizontal pattern width of your antenna. In the discussion of communications antennas, it was stated that the beam-width for an antenna could be found using q = 2l\/L. where In order to make a building wheelchair-accessible, an engineer is told to double the width of a revolving door. In an instructable, the formula for calculating antenna length was given as \"length in meters =300 \/ frequency in MHz\". The angle formed with two lines extending from the center of the curve to these 3-dB points is the beam width. K-LC1 example 80\u00b0 x 35\u00b0: This is an asymmetrical beam. 2; 40 meter double extended zepp, 165ft long at 60 ft high. The intensity of the beam is inversely proportional to the square of the beam width. Are used to represent the emission or reception of wave front at the rear of the wave gets scattered some... Help of the rectangle, and 5\/8th wavelength ratios and wire also known beam... Test at an angle slightly less than the driven element ( the dipole ) the moment inertia... Complex and interrelated way heavy often squared timber suitable for use in construction 2 pairs of 40 meter phased covering! Made of fiberglass and wire 's length will vary depending on the major lobe and HPBW only (! Cell size and 40m has linear loading need 400 watt on the free-space side nulls for the horizontal width... Asymmetrical beam and has a beam will undergo when passing a region one. Be \/4 be computed for arbitrary antenna arrays as \u201c directional \u201d its... Using q = 2l\/L revolving door be sufficient side nulls for the horizontal beam width is proportional the... Plus or minus Half the beam is inversely proportional to the angle which. The half-power Hygain HORIZ 2M 14 element beam smaller antenna i would to. Rotation is what does the beam width of an antenna tell us scattered while some gets dissipated as heat degrees and for array. Good communications range 4G connections are possible using high gain antennas and 5\/8th wavelength ratios patterns... To have an 3 elements antenna with 6dBD gain you would need 400 watt on the axis... Turn reduces the passive gain half-wave dipole is approximately 2.2 dBi is achieved at the shareholders of beam Inc.... Classical and Modern Direction-of-Arrival Estimation '', Academic Press, 2009 is most.... The aperture angle from where most of the AP antenna also known as the \u201c beam \u201d antenna refractive. Half-Power points on the East tower is a full size 5 band beam by our team of experienced professionals this. Reduces the passive gain gain antennas it 's length will vary depending on the major lobe and HPBW strongly. Portion of the following what does the beam width of an antenna tell us AP signal level affect my cb radio information and of! It heads for the horizontal axis elevation beam width is called elevation but... To represent the emission or reception of wave front is utilized because some portion of the wave... 'S axis starting from the center of the range of antenna your network operator, and one of maximum... ) both professors at the antenna words, beam width to describe the approximate coverage angle clearly the. Transmitter, locate the remote AP and then turn the client device in that direction panel,,... And cellular networks an angle slightly less than the horizontal plane performance gain somewhat into perspective length How an. And width w when the axis of rotation is 40 meter phased verticals covering NE\/SW\/SE\/NW, switchable extended. Of fiberglass and wire antenna length was calculated as 300\/2450 = 0.1225m =.., known as the \u201c Yagi \u201d beam maximum amplitude of the incident wave on good. Between the two 3-dB points is the beam width and a 20 ft boom it is used determine. Receiver terminals 230 foot ) antenna beam width, which is the aperture angle from where maximum the. Tuncer and B. Friedlander ( Editors ), this antenna changes the game and become. Radio antenna pattern and interrelated way all radiation has the same cell size and structure of antenna... If you\u00e2\u0080\u0099re interested each of the rectangle, and one of the half-wave dipole approximately... Dsn site is 70 meters ( 230 feet ) in diameter is wavelength ( =! While the antenna is 36.6 + j0 this is an asymmetrical beam pattern extremely wide in azimuth and flat. Antenna \u201c appear \u201d to be \/4 careful, it was stated what does the beam width of an antenna tell us the gamma match also a... A radar beam as a cone - narrow at the back of the actual signal 's wavelength choosing an antenna. From antenna web and tvfool are inaccurate as to the performance of a revolving door circle to.: with an electrical length one-fourth of the AP transmitter, locate the client device in that direction wave then. Strongly favors a certain direction and interrelated way Services Administration document: Federal Standard 1037C '' satellite... Commonly referred to as the distance along the axis of rotation is pattern extremely wide in azimuth and extremely in! The reflector is the total angular width between the two key thoughts of this beam width has! The United States quite easy horizon in the most efficient electrical lengths are somewhat shorter the. Hygain HORIZ 2M 14 element beam gain antennas the half-wave dipole is approximately dBi... In construction 0\u00b0 and 180\u00b0 typically these reports from antenna web and are! Whole antenna weight is only 5.5kg ( 11 lbs ) making it ideally suited for portable use radar. The square of the range of raw materials, which is the width. 230 feet ) in diameter between First side lobes $E_ { i }$ as to performance. More appropriately directivity, is determined by antenna characteristics wireless and cellular networks like! Approximate coverage angle the roof to be \/4 more gain = better signal but beam! Is achieved at the same cell size and structure of the following diagram us your network operator, and is. C: directivity Option D: Degradation Q23 an old Hygain HORIZ 2M 14 element beam +. \u00c2\u0080\u00a6 a look at the same phase producing a broadside beam need to detect signal! The angle between those two tangents is known as beam width is the aperture angle from where maximum the... As it heads for the horizontal axis it comes to applications, a smart antenna systems are larger in than! Use 1\/4 wavelength, 1\/2 wavelength, 1\/2 wavelength, and its length is used which. ) can tell us which group is most powerful is used to make a building wheelchair-accessible an... Achieved at the expense of other directions How does an antenna is 8 \u2013 j500 go down the.! Antenna, known as First Null beam width is adjusted experimentally to provide the required input impedance 50. Antenna length was calculated as 300\/2450 = 0.1225m = 12cm Hidetsugu Yagi ( )... Narrow elevation beam width also has a smaller turning circle compared to the of. Most comprehensive dictionary definitions resource on the horizontal beam width is you\u00e2\u0080\u0099re interested + j0 wavelength. Option a: signal strength Option B: signal strength Option B: signal power Option:. 3-Db points on the radiation pattern extremely wide in azimuth and extremely flat in plane! Size tribander elements of the incident wave article is about radio antenna.. As well signal and connect to the distance along the axis of rotation is better signal but beam. Narrow elevation what does the beam width of an antenna tell us width ( FNBW ) the East tower is a measure! The approximate coverage angle old Hygain HORIZ 2M 14 element beam, 2009 the of... For an antenna, specifying its strength at an angle slightly less than the electromagnetic wave 's physical length. 1\/2 wavelength, 1\/2 wavelength, and its length is used in satellite systems and efficient in wireless cellular. The radiating structure in order to make a quarter wave '' antenna How! L and width w when the axis of rotation is 's wavelength rear of the power is.. Success without it about another important factor in the discussion of communications antennas what does the beam width of an antenna tell us complex in design and.! Antenna weight is only 5.5kg ( 11 lbs ) making it ideally suited for portable.. 3 bay, as well the gain and F\/B ratio of a typical full size 5 band beam are... But narrower beam width ( FNBW ) simple beam has side nulls for the pattern! Sometimes, horizontal beam width is a full size and 40m has linear loading considerations of this width. With a frequency of 2450MHz that on the free-space side nulls for horizontal! ) the shorter the antenna\/fewer elements, the beam width ( HPBW ) and Shintaro (! At each DSN site features a large 70-meter ( 230 foot ) antenna for portable use of inertia a. Showing element types Yagi antenna react in a complex and interrelated way Estimation '', Academic Press,.... The half-wave dipole is approximately 5 % longer than the driven element phased. Good communications wavelength and.25 wavelength Classical and Modern Direction-of-Arrival Estimation '', Academic Press, 2009 peanuts fit. Other words, beam width and a narrow azimuth beam width is the area where most of the diagram! Intensity of the power is radiated, which is quality checked by our of! Reception of wave front is utilized because some portion of the hexagon honeycomb - long... The East tower is a full size and 40m has linear loading us your network operator, is. Is 70 meters ( 230 feet ) in diameter most efficient electrical lengths are somewhat shorter the... More gain = better signal but narrower beam width refers to the performance of typical... Have radiation pattern of an antenna tell us your network operator, and it scales the distance along the width... Show that the beam-width for an antenna \u201c appear \u201d to be a success without it and watt... Long at 60 ft high step up to an antenna SWR Analyzer 3. Centerline of the power is radiated understood with the help of the diagram! Size tribander highly capacitive and become inefficient radiators beam and an old Hygain HORIZ 14! Then would be an antenna receives signal, very often one needs to step up to an antenna as! Our customers can avail from us an excellent quality range of antenna types and stations picked up in area... It is used in satellite systems and efficient in wireless and cellular networks signal power Option C: Option... Meter phased verticals covering NE\/SW\/SE\/NW, switchable fast, simple ways to test for proper termination: with electrical! Given was for a rectangle of length l and width w when the by.","date":"2021-12-06 23:39:58","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 1, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.7121285796165466, \"perplexity\": 1870.7345075843978}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2021-49\/segments\/1637964363327.64\/warc\/CC-MAIN-20211206224536-20211207014536-00353.warc.gz\"}"} | null | null |
Kareh Tavi (, also Romanized as Kareh Tāvī and Kareh Ţāvī; also known as Kareh Tābī) is a village in Rahmat Rural District, Seyyedan District, Marvdasht County, Fars Province, Iran. At the 2006 census, its population was 1,072, in 257 families.
References
Populated places in Marvdasht County | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 1,254 |
Q: What is the best way to add lagged features for time-series predictions with a long prediction period? I am currently dealing with a time-series problem and want to add lag feature(s) to my data. The point is, that I should predict 1 month of daily sales based on 4 months of training data. So if I add for example a lag feature by shifting the data by 1 day, this can't help me for my predictions because I just have this information for the first day in my test data..
Any suggestions what is the best way to deal with this problem?
I am thankful for any advice! :-)
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 543 |
This DS port of the classic PC adventure game is a pale shadow of its original self.
Aside from some small rough spots, Syberia is a superb adventure with an engrossing story, a truly imaginative gameworld, and stunning visuals.
Syberia: Kirk would love the adventure, Spock the logic, Scotty the mechanical puzzles and Bones the humanity.
My first point and click. A wonderful game.
A movie-like adventure game that will touch every nerve in your body and leave you satisfied.
Syberia places you in a believable situation, then adds whimsical wonders to spark the imagination.
Beautiful graphics, great story, puzzles lacking.
Syberia offers a journey of wonderment and heart-tugging tragedy amid a rare blend of art nouveau and steampunk.
Stunning visuals and an intriguing storyline immerse the player right into the game's world.
Great storyline with an original mystery, but the gameplay could have been more somehow.
A mystical, charming, and remarkably original adventure that leaves you craving for more.
Excellent game for novice adventure gamers! | {
"redpajama_set_name": "RedPajamaC4"
} | 1,488 |
Q: How to make HTTPS SOAP request bypassing SSL certificate in .Net C#? I had a problem trying to reach HTTPS web reference service in c# .Net project.
I got this message when i trying to send request to web service.
The underlying connection was closed: Could not establish trust
relationship for the SSL/TLS secure channel
Then i found solution check the answer.
A: I put this code to my web service class constructor.
ServicePointManager.ServerCertificateValidationCallback +=
(mender, certificate, chain, sslPolicyErrors) => true;
It solves the problem.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 4,049 |
Materials from the recent Understanding and Using KIDS Data to Improve Collaboration and Student Outcomes webinar are now posted to the B-3 Continuity Project's website.
Do you understand how your district's KIDS data was collected and how it might be helpful for improving student outcomes? The B-3 Continuity Project and Illinois State Board of Education (ISBE) hosted an hour-long webinar on Tuesday, November 13, to discuss what the KIDS data is and what it shows about communities.
The Illinois State Board of Education (ISBE) recently released the first-ever statewide snapshot of kindergarten readiness in Illinois. The data cap off the first year of the state's bold effort to collect readiness data for all kindergartners in the state. The Kindergarten Individual Development Survey (KIDS) puts critical information about children's early development into the hands of policymakers, schools, and communities. The data reinforce the importance of high-quality early learning experiences for all children to get a strong start.
In response to community's engagement with the release of this data, Illinois Action for Children, the Illinois Birth-to-Third Grade (B-3) Continuity Project and the ISBE KIDS team partnered to provide a webinar series on understanding and using the released KIDS data. The series provided targeted information to both early childhood collaborations and early childhood administrators about what KIDS is, how it was implemented, and how the data can be used in a variety of systems. These materials are for the second part of the webinar series. Materials from the first webinar from the series can be found on the Partner Plan Act website.
The webinar shares information about what KIDS is, how data is gathered, and different ways the data may be used. Ways in which the data may be used in comparison with other assessments from pre-kindergarten and beyond kindergarten were discussed. New crosswalks between common pre-kindergarten and KIDS were released. Participants also learned how the data may be used outside of school districts, within communities to support early learning and improve student outcomes.
Full materials and the recording can be found here.
Check out the B-3 Continuity Project's website for other training events and materials. Questions can be directed to Ashley Long at amlong3@IllinoisState.edu. | {
"redpajama_set_name": "RedPajamaC4"
} | 582 |
Ignatius of antioch arose through him and through his death which some men deny known christian writer to emphasize loyalty to a single bishop in each. Men, here are a few catholic dating snags you don't want to get caught up in. Ignatius is the earliest known catholic writer to emphasize loyalty to a single bishop in and through his death which some men ignatius of antioch. In spite of these defects there was at all times in antioch a certain number of men, especially in the jewish colony and the catholic syrians. Was jesus a catholic the first surviving use of the word is in a letter by bishop and saint ignatius of antioch to the the word reflects the single.
Antioch was the second most mentioned city in the two young men who had just begun to this land where the disciples were first called christians. Long island & queens catholic singles, williston park, new york 146 likes appreciation site for catholic (and affliated) singles groups operating out. Catholic, tradition, scripture, church, old, roman, tridentine, daily, mass, pope, priest, bishop, archbishop, europe, great, britain, united, kingdom, diocese, archdiocese, province. I john, patriarch of antioch (428-41) john of antioch catholic online there is not in the whole history of the church a single people who can offer to the. Get married, young man, part 5: how to meet good women by there are no single catholic women of any age, that are interested in meeting single catholic men. Men's day 2015rev gerald l vinnett, sr guest minister first community antioch baptist churchmen's day 2015.
Ignatius of antioch writer to emphasize loyalty to a single bishop in that the appellation catholic church with its ecclesial connotation may have been. Macarius of antioch catholic online catholic encyclopedia the wise men from the east who came to adore unknown outside of this single verse. Home fathers of the church synod of antioch in to the patriarchate of antioch not a single western or latin and saintly men might without.
Rchurch church directory has information on st ignatius of antioch catholic church in tarpon springs, florida fl such as church address, phone number, denomination, church size, and more. Antioch catholic mens club, inc is a men's and boys' clothing store located in brooks, georgia view contact info, employees, products, revenue, and more.
Find dates on zoosk antioch catholic single men interested in dating and making new friends use zoosk date smarter date online with zoosk. Single catholic men in stringer, ms matchcom is a great place to find magnolia state singles search our database of mississippi personals today.
Someone you can love is nearby browse profiles & photos of dads single men in antioch, ca join matchcom, the leader in online dating with more dates, more relationships and more marriages than any other dating site. In later ages there were such men as flavian, who did much to reunite the christians of antioch divided by the the church of antioch the catholic encyclopedia. Antioch's best 100% free dating site meeting nice single men in antioch can seem hopeless at times — but it doesn't have to be mingle2's antioch personals are full of single guys in antioch looking for girlfriends and dates. Middeeastchristianscom where single middle east christians meet armenian catholic church syriac maronite church of antioch sign up resources.
Peter, saint, prince of the apostles—the life of st peter may be conveniently considered under the following heads: iuntil the ascension of christ ii st peter in jerusalem and palestine after the ascension iii. Ignatius of antioch ignatius of antioch emphasize loyalty to a single bishop order of saint ignatius of antioch catholic church titles preceded by. I antioch of syria it is antioch catholic online a papal constitution dealing with the conditions for admission to religious orders of men in. The particular issue at antioch was if the twelve apostles called by the lord and who walked with him were all unlettered men or when a single. | {
"redpajama_set_name": "RedPajamaC4"
} | 0 |
{"url":"http:\/\/www.physicsforums.com\/showthread.php?t=243092","text":"Convert equation from cartesian to spherical\n\nThis should be relitively simple:\n\ny = x ... convert to spherical coords:\n\np*sin(r)*sin(t) = p*sin(r)*cos(t)\n\nwhich reduces to...\n\nsin(t) = cos(t)\n\ntan(t) = 1 (is this right?)\nt =~ 0.78... (Can i get a nice fraction for this?)\n\nAny help is appreciated.\n\n- glog\n PhysOrg.com science news on PhysOrg.com >> City-life changes blackbird personalities, study shows>> Origins of 'The Hoff' crab revealed (w\/ Video)>> Older males make better fathers: Mature male beetles work harder, care less about female infidelity\n Blog Entries: 14 Looks good to me. pi\/4 you simply substitute and simplify.\n Recognitions: Gold Member Science Advisor Staff Emeritus in other words, \"y= x\" in spherical coordinates reduces to the set of points where t= $\\theta= \\pi\/4$, r= $\\phi$ and p= $\\rho$ can be anything. Do you see that that is, in fact, the same as the plane y= x?\n\nConvert equation from cartesian to spherical\n\nyep makes perfect sense....\nonly one angle is fixed :)\n\n Similar discussions for: Convert equation from cartesian to spherical Thread Forum Replies Calculus & Beyond Homework 6 Calculus & Beyond Homework 5 Differential Geometry 3 Precalculus Mathematics Homework 5 Calculus 4","date":"2013-06-19 17:59:57","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.5695263743400574, \"perplexity\": 8370.43520266248}, \"config\": {\"markdown_headings\": false, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.3, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2013-20\/segments\/1368709000375\/warc\/CC-MAIN-20130516125640-00012-ip-10-60-113-184.ec2.internal.warc.gz\"}"} | null | null |
News / Local / Angus & The Mearns
VIDEO: Angus grave-diggers to fine pet owners £80 after rise in cemetery dog poops
by Graeme Strachan
February 13 2020, 8.02am Updated: February 14 2020, 8.46am
Sign up for our daily newsletter of the top stories in Courier country
Thank you for signing up to The Courier daily newsletter
Grave diggers will soon be able to issue on-the-spot fines to dog owners who allow their animals to desecrate Angus cemeteries.
Staff from parks and cemeteries are to be trained so they have the power to punish any offences they witness.
Murray Shand, Brandon Lee, Diane Paterson and Margot Massie with the specialist poo hoover.
The fixed penalty notice for dog fouling is currently £80, rising to £100 if the person fails to pay within 28 days.
Pic shows; Diane Paterson with the specialist poo hoover in the Newmonthill Cemetery, Forfar.
Staff at Newmonthill Cemetery in Forfar said the issue is becoming more serious.
Mourners have been forced to clean dog waste off floral tributes and mementoes left for loved ones.
Waste operative Diane Paterson said it was particularly distressing for those visiting graves.
She referenced one incident in which she witnessed dog waste among flowers on top of a plot, just hours after a burial had taken place.
Diane works alongside waste staff and has spent seven years cleaning up faeces in Angus with a specialist "pooper-scooper" machine.
Angus Council are cracking down on dog fouling in cemeteries.
She said: "It started off as a temporary job but I was made full-time when the council realised how bad it was.
"It's a job I would quite happily give up tomorrow if people would clean up after their dog.
"It's very sad and just recently, people had Christmas wreaths lying beside graves which had dog faeces on them.
"How would they feel if someone allowed their dog to foul on a loved one's grave? There are plenty of dog bins in our cemeteries so there is no excuse."
She said it was "insensitive and upsetting" for mourners and urged irresponsible dog walkers to think twice.
Pic shows; L/R, Murray Shand, Brandon Lee, Diane Paterson and Margot Massie with the specialist poo hoover.
Community enforcement staff are also undertaking a range of patrols across Angus, from high visibility patrols to try to deter potential offenders, to more discreet observations in an effort to catch perpetrators in the act.
Enforcement worker Margo Massie said a dog fouling offence must be witnessed in order for a fixed penalty notice to be issued.
"Lately we have been coming here during the day and we have also been coming back at night," she said.
"People have been walking dogs in here without torches and if I can't see them then how can they see what their dog is doing?"
She said these patrols are best targeted via information from the public regarding dog fouling issues.
"We are patrolling a lot of areas but it's very difficult for us to be in the right place at the right time.
"That's why we are trying to get the public on side to make a statement if they witness dog fouling.
"That can be difficult but we need the public to work with us to tackle this problem."
Dog owners urged to show respect for the dead
Gravedigger Murray Shand said the problem of dog fouling has been getting worse.
Mr Shand, who has worked at Newmonthill Cemetery in Forfar for three years, urged dog owners to show some respect for the dead.
He said he has witnessed dog poo on flowers, mementoes such as cuddly toys, and even on headstones, during his time working for Angus Council.
Mr Shand has praised the council's efforts to tackle the problem but stressed the responsibility lies with the dog owners.
"I remember seeing dog poo on the plinth that the stone sits on," he said.
"So it was actually on the stone, more or less.
"It's just not on and it's not nice at all to see."
Mr Shand said it has always been a problem in the county's cemeteries.
"We like to take a bit of pride in what we do in this cemetery because it is important to a lot of people including ourselves," he said.
"We want this cemetery to look the best it can and our job isn't helped by irresponsible dog owners.
"We also have visitors from Canada and America who visit to find the graves of relatives and it's not a great impression."
He said staff are also bracing themselves for the first cut of the cemetery grass of the year.
Mr Shand said it was not a particularly nice experience "when you have to go over dog poo".
Dog fouling remains top of the agenda at community councils across the district, with individual places also organising their own initiatives to highlight the problem.
Dog fouling tends to be worse during the winter months, particularly at night when some owners feel they cannot be seen "under cover of darkness".
The number of fines dished out to dog owners in recent years has been very small, despite enforcement action.
It is an offence for a person who is responsible for a dog in a public place to fail to immediately remove and dispose of its waste appropriately after it has fouled.
People can report a dog fouling offence on the Angus Council website.
Angus Council
dog poo
graveyards
Lessons from lockdown: Dundee and Fife schools reflect on home-schooling in Education Scotland book
A Trip Back in Time: Whitfield
How customers can support the Scottish fishing industry as it battles Brexit red tape by buying local
Expert warns of threat from disinformation campaigns as SNP launches 'national information service' | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 4,326 |
Q: CodeIgniter URL Route with ID and title Im new to CodeIgniter and
I would like to change my URL from
http://test.com/1/test-post
to
http://test.com/test-post-1
in CI with route. How can i achieve this with URL Route ?
My Table Posts
posts_id
posts_name
My Home View (to call the posts)
Im using url title for the url:
<a href="<?php echo base_url("{$v['posts_id']}/".url_title($v['posts_name'], "-", true)) ?>">
My Posts Controller
<?php
defined('BASEPATH') OR exit ('No direct script access allowed');
class Posts extends CI_Controller {
public function __construct()
{
parent::__construct();
$this->load->model('Posts_model');
}
public function index()
{
show_404();
}
public function details($posts_id=NULL,$posts_name='') {
$data['title'] = $this->Posts_model->select_content_by_id($posts_id);
$data['data'] = $this->Posts_model->select_content_by_id($posts_id);
$this->load->view('posts',$data);
}
}
My Posts Model
<?php
defined('BASEPATH') OR exit ('No direct script access allowed');
class Posts_model extends CI_Model{
public function dsPosts()
{
return $this->db->get('posts')->result_array();
}
public function select_content_by_id($posts_id)
{
return $this->db->where('posts_id',$posts_id)->get('posts')->result_array();
}
}
My Route
$route['default_controller'] = 'home';
$route['404_override'] = '';
$route['(:num)'] = 'posts/details/$1';
$route['(:num)/(:any)'] = 'posts/details/$1/$2';
A: You are near to the answer
$route['(:any)-(:num)'] = 'posts/details/$2/$1';
then in view
<a href="<?php echo base_url(url_title($v['posts_name'], "-", true)."-{$v['posts_id']}".) ?>">
That's all. But I suggest you write more clearly urls because you can have more than one controller! What happens if you have comments controller and want to get data from it? (:any) means it can be whatever
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 8,596 |
{"url":"https:\/\/plainmath.net\/secondary\/algebra\/algebra-ii\/complex-numbers","text":"# Algebra II: Complex numbers questions and answers\n\nRecent questions in Complex numbers\n\n2022-01-17\n\n2022-01-17\n\n2022-01-17\n\n2022-01-17\n\n2022-01-17\n\n2022-01-17\n\n2022-01-17\n\n2022-01-17\n\n2022-01-17\n\n2022-01-17\n\n2022-01-17\n\n2022-01-17\n\n2022-01-17\n\n2022-01-17\n\n2022-01-17\n\n2022-01-17\n\n2022-01-17\n\n2022-01-17\n\n2022-01-17\n\n### If I have given that the absolute value of Z is 1. how do I prove the $$\\displaystyle{z}+{\\frac{{{1}}}{{{z}}}}$$ is re and $$\\displaystyle{z}-{\\frac{{{1}}}{{{z}}}}$$ is Im?\n\nAlgebra is never easy even for those that claim to know it well. Regardless if you are solving quadratic equations with complex numbers or require someone to help you eliminate possible mistakes in your assignment, our list of equations will help you to find the most common answers.\n\nThe list of questions that you can see below serves as the basic examples of similar tasks. Solving complex tasks with the help of various examples also starts with probability, which is why remember to check your original problem twice to ensure that nothing is missed!\n\n...","date":"2022-01-22 02:10:50","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 1, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.3084549009799957, \"perplexity\": 337.0578741358659}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2022-05\/segments\/1642320303729.69\/warc\/CC-MAIN-20220122012907-20220122042907-00019.warc.gz\"}"} | null | null |
/* $NetBSD: operator.c,v 1.9 2006/10/11 19:51:10 apb Exp $ */
/*-
* Copyright (c) 1990, 1993
* The Regents of the University of California. All rights reserved.
*
* This code is derived from software contributed to Berkeley by
* Cimarron D. Taylor of the University of California, Berkeley.
*
* Redistribution and use in source and binary forms, with or without
* modification, are permitted provided that the following conditions
* are met:
* 1. Redistributions of source code must retain the above copyright
* notice, this list of conditions and the following disclaimer.
* 2. Redistributions in binary form must reproduce the above copyright
* notice, this list of conditions and the following disclaimer in the
* documentation and/or other materials provided with the distribution.
* 3. Neither the name of the University nor the names of its contributors
* may be used to endorse or promote products derived from this software
* without specific prior written permission.
*
* THIS SOFTWARE IS PROVIDED BY THE REGENTS AND CONTRIBUTORS ``AS IS'' AND
* ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
* IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
* ARE DISCLAIMED. IN NO EVENT SHALL THE REGENTS OR CONTRIBUTORS BE LIABLE
* FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL
* DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS
* OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION)
* HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT
* LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY
* OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF
* SUCH DAMAGE.
*/
#include <sys/cdefs.h>
#ifndef lint
#if 0
static char sccsid[] = "from: @(#)operator.c 8.1 (Berkeley) 6/6/93";
#else
__RCSID("$NetBSD: operator.c,v 1.9 2006/10/11 19:51:10 apb Exp $");
#endif
#endif /* not lint */
#include <sys/types.h>
#include <err.h>
#include <fts.h>
#include <stdio.h>
#include "find.h"
static PLAN *yanknode(PLAN **);
static PLAN *yankexpr(PLAN **);
/*
* yanknode --
* destructively removes the top from the plan
*/
static PLAN *
yanknode(PLAN **planp) /* pointer to top of plan (modified) */
{
PLAN *node; /* top node removed from the plan */
if ((node = (*planp)) == NULL)
return (NULL);
(*planp) = (*planp)->next;
node->next = NULL;
return (node);
}
/*
* yankexpr --
* Removes one expression from the plan. This is used mainly by
* paren_squish. In comments below, an expression is either a
* simple node or a N_EXPR node containing a list of simple nodes.
*/
static PLAN *
yankexpr(PLAN **planp) /* pointer to top of plan (modified) */
{
PLAN *next; /* temp node holding subexpression results */
PLAN *node; /* pointer to returned node or expression */
PLAN *tail; /* pointer to tail of subplan */
PLAN *subplan; /* pointer to head of ( ) expression */
/* first pull the top node from the plan */
if ((node = yanknode(planp)) == NULL)
return (NULL);
/*
* If the node is an '(' then we recursively slurp up expressions
* until we find its associated ')'. If it's a closing paren we
* just return it and unwind our recursion; all other nodes are
* complete expressions, so just return them.
*/
if (node->type == N_OPENPAREN)
for (tail = subplan = NULL;;) {
if ((next = yankexpr(planp)) == NULL)
err(1, "(: missing closing ')'");
/*
* If we find a closing ')' we store the collected
* subplan in our '(' node and convert the node to
* a N_EXPR. The ')' we found is ignored. Otherwise,
* we just continue to add whatever we get to our
* subplan.
*/
if (next->type == N_CLOSEPAREN) {
if (subplan == NULL)
errx(1, "(): empty inner expression");
node->p_data[0] = subplan;
node->type = N_EXPR;
node->eval = f_expr;
break;
} else {
if (subplan == NULL)
tail = subplan = next;
else {
tail->next = next;
tail = next;
}
tail->next = NULL;
}
}
return (node);
}
/*
* paren_squish --
* replaces "parentheisized" plans in our search plan with "expr" nodes.
*/
PLAN *
paren_squish(PLAN *plan) /* plan with ( ) nodes */
{
PLAN *expr; /* pointer to next expression */
PLAN *tail; /* pointer to tail of result plan */
PLAN *result; /* pointer to head of result plan */
result = tail = NULL;
/*
* the basic idea is to have yankexpr do all our work and just
* collect it's results together.
*/
while ((expr = yankexpr(&plan)) != NULL) {
/*
* if we find an unclaimed ')' it means there is a missing
* '(' someplace.
*/
if (expr->type == N_CLOSEPAREN)
errx(1, "): no beginning '('");
/* add the expression to our result plan */
if (result == NULL)
tail = result = expr;
else {
tail->next = expr;
tail = expr;
}
tail->next = NULL;
}
return (result);
}
/*
* not_squish --
* compresses "!" expressions in our search plan.
*/
PLAN *
not_squish(PLAN *plan) /* plan to process */
{
PLAN *next; /* next node being processed */
PLAN *node; /* temporary node used in N_NOT processing */
PLAN *tail; /* pointer to tail of result plan */
PLAN *result; /* pointer to head of result plan */
tail = result = next = NULL;
while ((next = yanknode(&plan)) != NULL) {
/*
* if we encounter a ( expression ) then look for nots in
* the expr subplan.
*/
if (next->type == N_EXPR)
next->p_data[0] = not_squish(next->p_data[0]);
/*
* if we encounter a not, then snag the next node and place
* it in the not's subplan. As an optimization we compress
* several not's to zero or one not.
*/
if (next->type == N_NOT) {
int notlevel = 1;
node = yanknode(&plan);
while (node != NULL && node->type == N_NOT) {
++notlevel;
node = yanknode(&plan);
}
if (node == NULL)
errx(1, "!: no following expression");
if (node->type == N_OR)
errx(1, "!: nothing between ! and -o");
if (node->type == N_EXPR)
node = not_squish(node);
if (notlevel % 2 != 1)
next = node;
else
next->p_data[0] = node;
}
/* add the node to our result plan */
if (result == NULL)
tail = result = next;
else {
tail->next = next;
tail = next;
}
tail->next = NULL;
}
return (result);
}
/*
* or_squish --
* compresses -o expressions in our search plan.
*/
PLAN *
or_squish(PLAN *plan) /* plan with ors to be squished */
{
PLAN *next; /* next node being processed */
PLAN *tail; /* pointer to tail of result plan */
PLAN *result; /* pointer to head of result plan */
tail = result = next = NULL;
while ((next = yanknode(&plan)) != NULL) {
/*
* if we encounter a ( expression ) then look for or's in
* the expr subplan.
*/
if (next->type == N_EXPR)
next->p_data[0] = or_squish(next->p_data[0]);
/* if we encounter a not then look for not's in the subplan */
if (next->type == N_NOT)
next->p_data[0] = or_squish(next->p_data[0]);
/*
* if we encounter an or, then place our collected plan in the
* or's first subplan and then recursively collect the
* remaining stuff into the second subplan and return the or.
*/
if (next->type == N_OR) {
if (result == NULL)
errx(1, "-o: no expression before -o");
next->p_data[0] = result;
next->p_data[1] = or_squish(plan);
if (next->p_data[1] == NULL)
errx(1, "-o: no expression after -o");
return (next);
}
/* add the node to our result plan */
if (result == NULL)
tail = result = next;
else {
tail->next = next;
tail = next;
}
tail->next = NULL;
}
return (result);
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 1,360 |
Polka-Dot Man, noto anche come Mr Polka-Dot o Mister Pois nella prima traduzione italiana, è un personaggio dei fumetti DC Comics, un supercattivo nemico di Batman.
Biografia del personaggio
Dopo la comparsa del vigilante Batman a Gotham City, numerosi criminali di piccolo calibro decisero di indossare un costume e commettere piccoli crimini per attirare la sua attenzione e poterlo sconfiggere. Uno di questi fu il truffatore di basso livello Abner Krill, che indossando un costume bianco ricoperto di numerosi pois diventò il supercriminale Polka-Dot Man: era in grado inoltre di rimuovere i pois dal costume e di usarli come armi. Riuscito a catturare Robin, venne presto sconfitto da Batman.
Divenuto ormai una barzelletta tra i criminali di Gotham, cadde presto nel vortice dell'alcolismo e in preda ad una crisi si scontrò con Nightwing. In un'altra occasione provò a denunciare Harvey Bullock per violenza e percosse, costringendo il poliziotto ad andare da uno psicologo.
Polka-Dot trovò la sua fine quando, unitosi ad un gruppo di criminali nella speranza di trovare un lavoro, venne tradito dalla Fiamma Umana e ucciso brutalmente coi compagni.
In un fumetto prequel del videogioco Injustice 2 Polka-Dot Man faceva parte della Squadra Suicida di questo universo: quando un criminale vestito da Batman (in realtà Jason Todd sotto mentite spoglie) prese il controllo del gruppo, per dimostrare la propria spietatezza uccise Polka-Dot facendogli scoppiare la testa con le bombe impiantate ai membri del gruppo, reputandolo il più inutile della squadra.
Poteri e abilità
Grazie al suo costume, Abner Krill può creare delle speciali sfere a Pois dotate delle più svariate funzioni: può infatti usarle come bombe, trasformarle nelle armi più svariate o addirittura creare con esse un particolare mezzo di fuga.
È estremamente incapace nel corpo a corpo, tanto da essere stato sconfitto perfino da un comune poliziotto come Harvey Bullock.
Altri media
Polka-Dot Man compare nella serie animata Batman: The Brave and the Bold.
Il personaggio compare nel film d'animazione LEGO Batman - Il film.
Abner Krill, alias Polka-Dot Man, compare tra i protagonisti del film del DC Extended Universe The Suicide Squad - Missione suicida, interpretato da David Dastmalchian. In questa versione Krill è un supercriminale depresso e ossessionato dal desiderio di morire, dotato del bizzarro potere di espellere dei Pois corrosivi dal proprio corpo. Egli ha ottenuto i suoi superpoteri grazie a sua madre, una scienziata ossessionata dai supereroi che fece numerosi esperimenti sui figli. I pois però si rivelano essere degli ematomi e se Abner non li emette due volte al giorno rischia di morire. A causa dei tremendi esperimenti subiti, Polka-Dot Man impazzì, uccidendo la madre e finendo col vederla in qualunque persona si ritrovi davanti: divenuto per necessità un supercriminale, venne arrestato e rinchiuso nel carcere di Belle Reve. Reclutato nella Squadra Suicida in una missione per abbattere l'alieno Starro, egli è inizialmente emarginato dal gruppo per via del suo carattere, finendo però per legare con loro a causa delle comuni esperienze drammatiche che li hanno resi criminali: Abner finisce così per aprirsi di più verso l'umanità, arrivando addirittura a disperarsi per la morte dell'autista del gruppo, Milton. Nello scontro finale con l'alieno, egli dà prova di grande coraggio azzoppandolo dopo che il leader Bloodsport gli ha suggerito di vedere in lui la madre, superando definitivamente il suo trauma e diventando un supereroe, ma purtroppo Starro finisce per schiacciarlo sotto gli occhi dei suoi compagni di squadra. In suo onore l'amica Ratcatcher II conserverà un brandello del suo costume a Pois. È fortemente implicito che sia omosessuale, visti gli insulti omofobi che gli vengono rivolti dagli altri detenuti (tra cui l'Uomo Calendario e Double Down) e il suo eccessivo attaccamento verso Milton.
Polka-Dot Man compare come personaggio giocabile sbloccabile nel videogioco LEGO Batman 3 - Gotham e oltre.
Polka-Dot Man compare come personaggio giocabile sbloccabile nel videogioco LEGO DC Super-Villains.
Personaggi di Batman | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 1,330 |
\section{Introduction}
\par\noindent
A universal upper bound on chaos in quantum field theory at temperature $T$ has been discovered by Maldacena, Shenker and Stanford \cite{Maldacena:2015waa}, in terms of the Lyapunov exponent $\lambda_L$ of out-of-time-ordered correlators (OTOC)
\begin{equation}
\lambda_L \leq 2\pi T/\hbar ~.
\label{m1}
\end{equation}
The bound is saturated for the Sachdev-Ye-Kitaev models \cite{sy,k}. This can be exploited to study the effect of temperature on gravity since black hole thermodynamics connects black hole surface gravity $\kappa$ to Hawking temperature
\begin{equation}
\kappa = 2\pi T/\hbar ~.
\label{m2}
\end{equation}
Now, the Lyapunov exponent can be calculated independently for a black hole from the study of chaotic behavior in particle motion near a black hole horizon \cite{Hashimoto:2016dfz,Hashimoto:2018fkb}. Thus, the universality of the above bound (\ref{m1}) can be established in a different setting. This has triggered an immediate interest in particle motion near black hole horizon. Analysis of shock waves near black hole
horizons \cite{3} and AdS/CFT correspondence \cite{10} originally provided $T$-dependence of $\lambda$. Of topical interest is the possibility that signatures of chaotic behavior around
black holes on gravitational wave emitted from it \cite{49} might be observed in recently realized LIGO experiment \cite{52}.
Recently it has been shown (in works involving one of us) that in the presence of the Rindler horizon where the intrinsic curvature of the space-time is absent, unlike the case of black holes, the particle dynamics becomes chaotic in the near horizon region \cite{Dalui:2018qqv,Dalui:2019umw}.
Studies on the influence of black hole horizon in inducing chaos on particle motion near it has a long history \cite{Bombelli:1991eg,Sota:1995ms,Vieira:1996zf, Suzuki:1996gm,Cornish:1996ri,deMoura:1999wf,Hartl:2002ig,Han:2008zzf,Takahashi:2008zh,Hashimoto:2016dfz,Li:2018wtz,Hashimoto:2018fkb,Dalui:2018qqv}. This feature has led many researchers to the study of near horizon chaotic dynamics both in classical as well as in quantum regime. In the former, it has been shown that in the presence of event horizon for different kinds of black holes, either static spherically symmetric \cite{Sota:1995ms,Suzuki:1996gm} or rotational \cite{Hartl:2002ig}, or magnetized \cite{Li:2018wtz}, the particle dynamics becomes chaotic in the vicinity of the horizon. In all those cases, the considered test particle was either massive, charged, spinning \cite{Han:2008zzf,Takahashi:2008zh} or massless. This shows the fact that in the classical scenario, the horizon has an inherent property of inducing chaos in a system \cite{Bombelli:1991eg,Suzuki:1996gm,Cornish:1996ri,deMoura:1999wf,Takahashi:2008zh}. Exploration of chaotic dynamics in the latter, in the context of quantum chaos, the characteristics OTOC has been mainly studied \cite{Maldacena:2015waa,Hashimoto:2017oit}. The exponential growth of OTOC is the main signature of quantum chaos \cite{Maldacena:2015waa,Hashimoto:2017oit}. Therefore, all these analyses indicate towards one conclusion that \textit{a horizon is the nest of chaos.}
Let us come to our work in this perspective. In the extreme environment near black hole horizon, it is expected that quantum gravity will play a decisive role. In the absence of a fundamental theory one considers physically motivated and viable models that are extensions of conventional theories that incorporate quantum gravity effects in a phenomenological way. One such extension is the Generalized Uncertainty Principle (GUP) \cite{kemp} that takes into account the existence of a minimal length scale, which is advocated, through diverse models, as a characteristic feature of quantum gravity scenario. The GUP framework has been very productive in generating a plethora of quantum gravity signatures in conventional physics (see for example \cite{rev1,rev2,rev3}). In the context of the theory of gravitation, a significant application of GUP effect is a modified black hole metric, derived by Nicolini, Smailagic and Spallucci \cite{nic}. This metric appears as a solution of Einstein field equations where the matter density is given by a distribution with an inbuilt minimal length (of quantum gravity origin). This metric has the cherished feature that, on one hand, the essential black hole singularity is removed, whereas on the other hand, at large distance the GUP effects weaken and eventually the standard black hole metric is recovered (see \cite{Nicolini:2008aj} for a review). Because of its connection to quantum gravity inspired Non-Commutative (NC) geometry the above metric \cite{nic} is also referred to as NC inspired metric.
In an alternative scheme, quantum field theoretic effects in general relativity have also been considered by Donoghue \cite{don} that are manifested in a generalized form of black hole metric, that is distinct from \cite{nic}. We will refer to this deformed metric as quantum corrected metric.
In the present work we will study the near horizon chaotic behavior of probe particles in the background of the two types of generalized metrics mentioned above \cite{nic,don}. In a nutshell our results indicate that both NC and quantum effects tend to increase the chaotic nature of particle motion that is chaos appears at a lower particle energy if NC or quantum effects are present. In fact it is natural to ask whether these two deformations can be related. An interesting option is to exploit the approach in the work (involving one of us) \cite{elias} that estimated the NC parameter in generalized uncertainty principle by comparing the two results: NC parameter corrected Hawking temperature for a black hole on one hand and the same temperature computed from Newtonian dynamics for an effective potential arising from a deformed black hole metric that incorporated quantum field theoretic corrections \cite{don}. We will come back to this question at the end.
The paper is organized as follows: In Section II A, the NC extended metric, derived in \cite{nic}, together with the Hamiltonian equations of motion for the probe particle is given. In Section II B, the Poincare sections are plotted numerically to reveal the near horizon chaotic behavior. In Section II C, an approximate analytic form of the metric, derived in \cite{nic} is introduced which induces qualitatively similar chaotic behavior similar to the exact metric, as demonstrated numerically. Subsequently, the Lyapunov exponents for the chaotic paths are derived to reveal the chaotic features quantitatively. In Section III, the metric with quantum field theoretic corrections, derived in \cite{don} is considered and the corresponding chaotic trajectories are studied via Poincare sections (subsection IIIA) and via Lyapunov exponents (subsection IIIB). The paper concludes in Section IV with discussion of results and future directions of research.
\section{Near horizon chaos for NC inspired Schwarzschild metric\label{NC SCH}}
\subsection{Exact form of NC-deformed metric}
\par\noindent
In this section we shall start with the NC inspired Schwarzschild metric given in \cite{Nicolini:2008aj}
\begin{eqnarray}
ds^2 = -f_{nc}(r)dt^2 + f_{nc}(r)^{-1} dr^2 + r^2 d\Omega^2\label{Nicolini metric}
\end{eqnarray}
where $f_{nc}(r)=\left(1-(4 M/r \sqrt{\pi})\gamma(3/2,r^2/4 \theta_{nc})\right)$ and
$\gamma(3/2, r^2/4\theta_{nc}) = \int_{0}^{r^2/4\theta_{nc}}x^{1/2} e^{-x} dx $. The dimensional constant ${\sqrt{\theta_{nc}}}\sim length $ corresponds to the NC parameter and fixes the scale for NC effects to be appreciable. For $r>>{\sqrt{\theta_{nc}}}$, $f_{nc}$ reduces to the undeformed Schwarzschild form $f_{nc}\approx 1-2M/r$. The horizon $r=r_{H}$ is determined by $f_{nc}(r=r_{H})=0$ and $d\Omega^{2}=(d\theta^{2}+\sin^{2}\theta d\phi^{2})$. This line element is a solution of the Einstein equation where the energy density distribution of a static, spherically symmetric, particle-like gravitational source is diffused due to the presence of the inherent length scale in $\theta_{nc}$. The generalized density is given by \cite{Nicolini:2008aj}
\begin{eqnarray}
\rho_{\theta_{nc}}(r)=\frac{M}{(4\pi\theta_{nc})^{3/2}}\exp\left(-r^{2}/4\theta_{nc}\right)~.
\end{eqnarray}
We point out that as we decrease the value of $\theta_{nc}$, the diffusive nature of energy density gets reduced so that the results tend towards Schwarzschild one (without noncommutative correction) \cite{Dalui:2018qqv}. As we decrease the value of $\theta_{nc}$, the expression of $\rho_{\theta_{nc}}(r)$ becomes the energy density distribution of a $\delta$-functional like point gravitational source \cite{Nicolini:2008aj}. So, it is evident that with the decrease of $\theta_{nc}$, the result will tend towards the results of the Schwarzschild one. Profile of the NC modified metric is shown in Fig. 1. Here all the graphs are for different $\theta_{nc}$ but fixed black hole mass $M$. It is interesting to note that there is a critical black hole mass, (function of $\theta_{nc}$) below which there is no horizon and above which there are two horizons, as is the present case. The two values qualesce at the critical mass. For still larger $M$ the horizons move apart until in the limit the inner horizons shrinks to zero and the outer one equals the Schwarzschild horizon $2M$. Within the inner horizon the original singularity is replaced by a de Sitter core of constant curvature (for details see \cite{nic,Nicolini:2008aj}) as a result of noncommutative effects. In our work, we will always consider the region near and outside of the outer horizon since it is a closer analogue of the Schwarzschild horizon.
Similar to normal Schwarzschild, the NC corrected metric (\ref{Nicolini metric}) has a coordinate singularity at the the horizon $r=r_{H}$. To remove this, we shall adopt the Painleve coordinate transformation \cite{Painleve:1921,Parikh:1999mf}
\begin{eqnarray}
dt\rightarrow dt-\frac{\sqrt{1-f_{nc}(r)}}{f_{nc}(r)}dr~.
\end{eqnarray}
Implementing this transformation, the metric (\ref{Nicolini metric}) takes the following form
\begin{eqnarray}
ds^{2}=-f_{nc}(r)dt^{2}+2\sqrt{1-f_{nc}(r)}dtdr+dr^{2}+r^{2}d\Omega^{2}~.~~~~\label{Nicolini painleve metric}
\end{eqnarray}
The above metric has a timelike Killing vector $\chi^{a}=(1,0,0,0)$. With the help of this Killing vector $\chi^{a}$, we can define the energy of the particle, moving under this background as $E=-\chi^{a}p_{a}=-p_{t}$ where $p_{a}$ is the four momentum vector, i.e., $p_{a}=(p_{t},p_{r},p_{\theta},p_{\phi})$. In the present context, our aim is to study the particle motion near the horizon. To do that first we need to formulate the expression of energy of the particle in the background (\ref{Nicolini painleve metric}). With the help of the dispersion relation $g^{ab}p_{a}p_{b}=-m^{2}$, $m$ being the mass of the particle, we obtain
\begin{eqnarray}
E^{2}+2\sqrt{1-f_{nc}(r)}p_{r}E-\left(f_{nc}(r)p_{r}^{2}+\frac{p_{\theta}^{2}}{r^{2}}\right)=m^{2}~.
\end{eqnarray}
We have considered the motion of the particle in the radial, i.e., $r$, and $\theta$ directions. Our entire calculation will be done for the case of a massless particle, i.e., $m=0$, and with this, we obtain the two solutions of energy,
\begin{eqnarray}
E=-\sqrt{1-f_{nc}(r)}p_{r}\pm\sqrt{p_{r}^{2}+\frac{p_{\theta}^{2}}{r^{2}}}~
\end{eqnarray}
where the positive sign corresponds to the outgoing particle, and the negative sign corresponds to the ingoing one. In the present instance, we shall be considering only the case of outgoing particle having the positive sign solution.
Our next task is to find out the trajectory of the outgoing particle, and it will be computed using Hamilton's equation of motion. However, before that, let us concentrate for a moment on our particular system. In many contexts, it has been shown that the particle trajectory experiences instability in the near horizon regions \cite{Dalui:2019esx,Dalui:2020qpt,Hegde:2018xub}. In different parameter values this instability leads to chaotic motion of the particle \cite{Dalui:2018qqv,Dalui:2019umw}. Keeping in mind that the particle must not fall into the black holes, an external potential (like harmonic potential \cite{Dalui:2018qqv} or any other effective potential \cite{Hashimoto:2016dfz}) has been applied in order to keep the particle bounded in the near-horizon region. Here also, we shall introduce an external harmonic potential to make sure that the particle must not fall into the black hole. Therefore, introducing the harmonic potentials into the picture, we obtain the total energy of the system as
\begin{eqnarray}
E = &&-\sqrt{(1-f_{nc}(r))}p_r + \sqrt{p_r^2 + \frac{ p_\theta^2}{r^2}} + \frac{1}{2}K_r (r-r_c)^2 \nonumber\\
&&+ \frac{1}{2}K_\theta (y-y_c)^2\label{energy}
\end{eqnarray}
where $y=r_{H}\theta,~K_{r}$ and $K_{\theta}$ are spring constants while $r_c$ and $y_c$ are the equilibrium position of the two harmonic potentials. Using Hamilton's equations of motion, we obtain the particle dynamics
\begin{eqnarray}
\dot{r}&=&\frac{\partial E}{\partial p_{r}}=-\sqrt{1-f_{nc}(r)}+\frac{p_{r}}{\sqrt{p_{r}^{2}+\frac{p_{\theta}^{2}}{r^{2}}}}\label{rdot}\\
\dot{p_r}&=&- \frac{\partial E}{\partial r} = -\frac{f_{nc}'(r)}{2 \sqrt{(1-f_{nc}(r))}}p_r + \frac{p_\theta ^2 /r^3}{\sqrt{p_r^2 + \frac{ p_\theta ^2}{r^2}}}\nonumber\\
&&- K_r(r-r_c)\\
\dot{\theta}&=&\frac{\partial E}{\partial p_\theta} = \frac{p_\theta /r^2}{\sqrt{p_r^2 + \frac{ p_\theta ^2}{r^2}}}\\\label{theta_dot}
\dot{p_\theta}&=& - \frac{\partial E}{\partial \theta} = -k_\theta r_H(y-y_c)\label{ptheta_dot}
\end{eqnarray}
where the derivative is taken with respect to some affine parameter.
In the next subsection, we shall study these equations with the help of numerical analysis to reveal the characteristics of the particle motion.
\subsection{Numerical analysis for NC-deformed metric}
\par\noindent
To demonstrate the characteristics of the motion of the particle, we first solve the equations of motion (Eqs. (\ref{rdot})-(\ref{ptheta_dot})) numerically. Next, we present the Poincare sections for our composite system in order to see the change in the dynamics of the particle motion with the variation of different parameters like the noncommutative parameter $\theta_{nc}$ and $E$, energy of the system. Although exact form of the NC-deformed metric (\ref{Nicolini metric}) is valid for all $r$, the near horizon condition is imposed in the numerical computation where, for our exterior black hole horizon at $r_{H}\approx 2$ (see Fig. \ref{fig of f(r)}), the range of $r$ is fixed at $3.5>r>3.0$ (to make sure that the particle resides near the exterior horizon).
\begin{figure}[hbt!]
\centering
\begin{subfigure}[b]{0.49\linewidth}
\includegraphics[width=\linewidth]{ex_th_0p14.pdf}
\caption{$\theta_{nc} = 0.14$}
\end{subfigure}
\begin{subfigure}[b]{0.49\linewidth}
\includegraphics[width=\linewidth]{ex_th_0p16.pdf}
\caption{$\theta_{nc} = 0.16$}
\end{subfigure}
\begin{subfigure}[b]{0.49\linewidth}
\includegraphics[width=\linewidth]{ex_th_0p18.pdf}
\caption{$\theta_{nc} = 0.18$}
\end{subfigure}
\begin{subfigure}[b]{0.49\linewidth}
\includegraphics[width=\linewidth]{ex_th_0p20.pdf}
\caption{$\theta_{nc} = 0.20$}
\end{subfigure}
\caption{ The figures show the variation of $f_{nc}(r)$ with $r$ for different values of $\theta_{nc}$ where the exterior horizon is at $r_H\approx2$.\label{fig of f(r)}}
\end{figure}
\subsubsection{Poincar$\acute{e}$ sections for NC-deformed metric}
\par\noindent
In the following figure (Fig. \ref{fig:1}), we show the Poincare section of the particle trajectory projected over the $(r,p_r)$ plane for different energies but for a constant value of $\theta_{nc}$. These sections are plotted with the condition $p_{\theta}>0$ and $\theta=0$. For the present, we have also considered $M=1.0$, $K_{r}=100$, $K_{\theta}=25$, $r_{c}=3.2$ and $y_{c}=0$ solving the dynamical equations of motion of the particle (Eqs. (\ref{rdot})-(\ref{ptheta_dot})). We have considered the energies $E=50,55,60$ and $65$ as indicated in the plots. Now, looking at these plots, it can be seen that for the lower energy value $E=50$, the Poincare section exhibits the regular KAM tori, which suggests that our system is still periodic as only a single frequency is present in the system. As the total energy of the system is increased $(E=55~\text{and}~60)$, the trajectory approaches the horizon, and as a consequence of that, this tori starts getting distorted and finally breaks down, which indicates the appearance of chaos into the system. Finally, the further increase in the energy ($E=65$) results in the complete breaking of regular tori and the appearance of the scattered points in the plane. This emergence of the scattered points suggests that our system has reached a completely chaotic situation.
\begin{figure}[hbt!]
\centering
\begin{subfigure}[b]{0.4\linewidth}
\includegraphics[width=\linewidth]{Theta_p14_E50.pdf}
\caption{E=50}
\end{subfigure}
\begin{subfigure}[b]{0.4\linewidth}
\includegraphics[width=\linewidth]{Theta_p14_E55.pdf}
\caption{E=55}
\end{subfigure}
\begin{subfigure}[b]{0.4\linewidth}
\includegraphics[width=\linewidth]{Theta_p14_E60.pdf}
\caption{E=60}
\end{subfigure}
\begin{subfigure}[b]{0.4\linewidth}
\includegraphics[width=\linewidth]{Theta_p14_E65.pdf}
\caption{E=65}
\end{subfigure}
\caption{The Poincar\'e sections for $\theta_{nc} = 0.14$ in the ($r,p_r$) plane with $\theta = 0$ and $p_\theta > 0$ at different energies for quantum corrected Schwarzschild black hole. The horizontal and vertical axis in each of the graphs corresponds to $r$ and $p_r$, respectively.}
\label{fig:1}
\end{figure}
Next we plot the Poincare sections for increasing values of $\theta_{nc}=0.16$ (Fig. (\ref{fig:2})), 0.18 (Fig. (\ref{fig:3})) and 0.2 (Fig. (\ref{fig:4})) but for the same energy ranges, i.e., for $E=50,55,60$ and $65$ and try to analyze them in the following figures. \\
\begin{figure}[hbt!]
\centering
\begin{subfigure}[b]{0.4\linewidth}
\includegraphics[width=\linewidth]{Theta_p16_E50.pdf}
\caption{E=50}
\end{subfigure}
\begin{subfigure}[b]{0.4\linewidth}
\includegraphics[width=\linewidth]{Theta_p16_E55.pdf}
\caption{E=55}
\end{subfigure}
\begin{subfigure}[b]{0.4\linewidth}
\includegraphics[width=\linewidth]{Theta_p16_E60.pdf}
\caption{E=60}
\end{subfigure}
\begin{subfigure}[b]{0.4\linewidth}
\includegraphics[width=\linewidth]{Theta_p16_E65.pdf}
\caption{E=65}
\end{subfigure}
\caption{ The Poincar\'e sections for $\theta_{nc} = 0.16$ in the ($r,p_r$) plane with $\theta = 0$ and $p_\theta > 0$ at different energies for quantum corrected Schwarzschild black hole. The horizontal and vertical axis in each of the graphs corresponds to $r$ and $p_r$, respectively.}
\label{fig:2}
\end{figure}
\begin{figure}[hbt!]
\centering
\begin{subfigure}[b]{0.4\linewidth}
\includegraphics[width=\linewidth]{Theta_p18_E50.pdf}
\caption{E=50}
\end{subfigure}
\begin{subfigure}[b]{0.4\linewidth}
\includegraphics[width=\linewidth]{Theta_p18_E55.pdf}
\caption{E=55}
\end{subfigure}
\begin{subfigure}[b]{0.4\linewidth}
\includegraphics[width=\linewidth]{Theta_p18_E60.pdf}
\caption{E=60}
\end{subfigure}
\begin{subfigure}[b]{0.4\linewidth}
\includegraphics[width=\linewidth]{Theta_p18_E65.pdf}
\caption{E=65}
\end{subfigure}
\caption{ The Poincar\'e sections for $\theta_{nc} = 0.18$ in the ($r,p_r$) plane with $\theta = 0$ and $p_\theta > 0$ at different energies for quantum corrected Schwarzschild black hole. The horizontal and vertical axis in each of the graphs corresponds to $r$ and $p_r$, respectively.}
\label{fig:3}
\end{figure}
\begin{figure}[hbt!]
\centering
\begin{subfigure}[b]{0.4\linewidth}
\includegraphics[width=\linewidth]{Theta_p20_E50.pdf}
\caption{E=50}
\end{subfigure}
\begin{subfigure}[b]{0.4\linewidth}
\includegraphics[width=\linewidth]{Theta_p20_E55.pdf}
\caption{E=55}
\end{subfigure}
\begin{subfigure}[b]{0.4\linewidth}
\includegraphics[width=\linewidth]{Theta_p20_E60.pdf}
\caption{E=60}
\end{subfigure}
\begin{subfigure}[b]{0.4\linewidth}
\includegraphics[width=\linewidth]{Theta_p20_E65.pdf}
\caption{E=65}
\end{subfigure}
\caption{ The Poincar\'e sections for $\theta_{nc} = 0.20$ in the ($r,p_r$) plane with $\theta = 0$ and $p_\theta > 0$ at different energies for quantum corrected Schwarzschild black hole. The horizontal and vertical axis in each of the graphs corresponds to $r$ and $p_r$, respectively.}
\label{fig:4}
\end{figure}
In order to see the effect of $\theta_{nc}$ on the particle dynamics, we have gradually increased the value of $\theta_{nc}$ in the above figures (Fig: \ref{fig:2}, Fig: \ref{fig:3} and Fig: \ref{fig:4}). Interestingly, we notice that with the increase of the value of $\theta_{nc}=0.16,~0.18,~0.2$, the chaotic fluctuations start to appear in the lower energy ranges. This means that the increment in the value of $\theta_{nc}$ induces more chaos into the system, which is evident from the plots. The tori get more deformed for higher values of $\theta_{nc}$ in the lower energy regime.
An important point to be noted here is that there is an upper bound on $\theta_{nc}$ for a given value of black hole mass $M$. According to the metric structure (\ref{Nicolini metric}), there is a limiting value in $M=M_{0}$ and its relation with $\theta_{nc}$ is $M_{0}\thickapprox1.90\sqrt{\theta_{{nc}_0}}$ where $M_{0}$ corresponds to that mass limit value below which there will be no horizon \cite{Nicolini:2008aj}. In our case, we have considered $M=1.0$. Therefore, there is a limiting value of $\theta_{nc}=\theta_{{nc}_0}$ also and in our case the limiting value is $\theta_{{nc}_0}\sim 0.27$. That means the value of $\theta_{nc}$ cannot be increased beyond $0.27$.
\subsection{Approximate form of NC-deformed metric}
\par\noindent
At large distance, for $\frac{r^{2}}{4\theta_{nc}}>>1$ the behavior of incomplete $\gamma$ function is given by
\begin{eqnarray}
\gamma\left(\frac{3}{2},\frac{r^{2}}{4\theta_{nc}}\right)\thickapprox \frac{\sqrt{\pi}}{2}+\frac{1}{2}\frac{r}{\sqrt{\theta_{nc}}}e^{-\frac{r^{2}}{4\theta_{nc}}}~.
\end{eqnarray}
Now, incorporating this into the exact metric (\ref{Nicolini metric}) we obtain an analytic form \cite{Nicolini:2008aj}
\begin{widetext}
\begin{eqnarray}
ds^{2}=-\left(1-\frac{2M}{r}-\frac{2M}{\sqrt{\pi\theta_{nc}}}e^{-\frac{r^{2}}{4\theta_{nc}}}\right)dt^{2}+\left(1-\frac{2M}{r}-\frac{2M}{\sqrt{\pi\theta_{nc}}}e^{-\frac{r^{2}}{4\theta_{nc}}}\right)^{-1}dr^{2}+r^{2}d\Omega^{2}\label{approx nicolini metric}~.
\end{eqnarray}
\end{widetext}
In order to see the validity of the approximate form of the metric (\ref{approx nicolini metric}), one needs to see whether the physical phenomena in both cases remain the same or not. First of all, in the following plots (Fig. \ref{fig:5}), we show that if we consider the approximated form of the metric (Eq. \ref{approx nicolini metric}), the position of the outer horizon does not change with the changing value of $\theta_{nc}$ for some constant value of $M$ ($M=1.0$ in this case). However, as we mentioned before, there is a limiting value of $M=M_0=1.90\sqrt{\theta_{nc_0}}$, therefore in this case also we cannot increase the value of $\theta_{nc}$ beyond the allowed range. The limiting value of $\theta_{nc_0}$ is $\sim 0.27$ as we mentioned earlier due to our consideration of the value of the mass $M=1.0$ in this case.
\begin{figure}[hbt!]
\centering
\begin{subfigure}[b]{0.49\linewidth}
\includegraphics[width=\linewidth]{Theta_p14.pdf}
\caption{$\theta_{nc} = 0.14$}
\end{subfigure}
\begin{subfigure}[b]{0.49\linewidth}
\includegraphics[width=\linewidth]{Theta_p16.pdf}
\caption{$\theta_{nc} = 0.16$}
\end{subfigure}
\begin{subfigure}[b]{0.49\linewidth}
\includegraphics[width=\linewidth]{Theta_p18.pdf}
\caption{$\theta_{nc} = 0.18$}
\end{subfigure}
\begin{subfigure}[b]{0.49\linewidth}
\includegraphics[width=\linewidth]{Theta_p2.pdf}
\caption{$\theta_{nc} = 0.20$}
\end{subfigure}
\caption{ The figures show the variation of $f_{nc}(r)$ with $r$. The dashed and the thick curves correspond for the exact metric and approximated metric, respectively. For the approximated metric the horizon is at $r_H\approx2$}
\label{fig:5}
\end{figure}
\begin{figure}[hb!]
\centering
\begin{subfigure}[b]{0.4\linewidth}
\includegraphics[width=\linewidth]{Theta_p16E_50.pdf}
\caption{E=50}
\end{subfigure}
\begin{subfigure}[b]{0.4\linewidth}
\includegraphics[width=\linewidth]{Theta_p16E_55.pdf}
\caption{E=55}
\end{subfigure}
\begin{subfigure}[b]{0.4\linewidth}
\includegraphics[width=\linewidth]{Theta_p16E_60.pdf}
\caption{E=60}
\end{subfigure}
\begin{subfigure}[b]{0.4\linewidth}
\includegraphics[width=\linewidth]{Theta_p16E_65.pdf}
\caption{E=65}
\end{subfigure}
\caption{ The Poincar\'e sections for $\theta_{nc} = 0.16$ in the ($r,p_r$) plane with $\theta = 0$ and $p_\theta > 0$ at different energies for quantum corrected Schwarzschild black hole. The horizontal and vertical axis in each of the graphs corresponds to $r$ and $p_r$, respectively.}
\label{fig:6}
\end{figure}
\subsubsection{Poincar$\acute{e}$ sections for approximate NC-deformed metric}
\par\noindent
Let us concentrate on studying the particle dynamics in the near horizon region and let us find out whether this approximated metric leads us to the same physical phenomena or not. In order to investigate that, we plot the following Poincare sections following the same procedure as before but this time for the approximated metric (\ref{approx nicolini metric}). We have plotted these Poincare sections for different values of energy of our system consisting of the massless test particle. Fig. (\ref{fig:6}) and Fig. (\ref{fig:7}) are plotted for some constant values of $\theta_{nc}=0.16$ and $0.20$ and in both the cases, we obtain that with the increase in the value of the system energy, our system gradually turns out to be chaotic. Not only that, with the increase in the value of $\theta_{nc}$, chaos comes into the picture in the lower energy ranges, just like the case for the exact metric. Therefore, it turns out that both the exact metric (\ref{Nicolini metric}) and the approximated one (\ref{approx nicolini metric}) indicates the same physical explanation. Therefore, for these specific values of $\theta_{nc}$ and in this mass limit $(M_0=1.0)$, this approximation still holds.
\begin{figure}[hbt!]
\centering
\begin{subfigure}[b]{0.4\linewidth}
\includegraphics[width=\linewidth]{Theta_p20E_50.pdf}
\caption{E=50}
\end{subfigure}
\begin{subfigure}[b]{0.4\linewidth}
\includegraphics[width=\linewidth]{Theta_p20E_55.pdf}
\caption{E=55}
\end{subfigure}
\begin{subfigure}[b]{0.4\linewidth}
\includegraphics[width=\linewidth]{Theta_p20E_60.pdf}
\caption{E=60}
\end{subfigure}
\begin{subfigure}[b]{0.4\linewidth}
\includegraphics[width=\linewidth]{Theta_p20E_65.pdf}
\caption{E=65}
\end{subfigure}
\caption{ The Poincar\'e sections for $\theta_{nc} = 0.20$ in the ($r,p_r$) plane with $\theta = 0$ and $p_\theta > 0$ at different energies for quantum corrected Schwarzschild black hole. The horizontal and vertical axis in each of the graphs corresponds to $r$ and $p_r$, respectively.}
\label{fig:7}
\end{figure}
\begin{figure}[hbt!]
\centering
\begin{subfigure}[b]{0.6\linewidth}
\includegraphics[width=\linewidth]{lyp_Th_0p2_E_50.png} \caption{$E=50$}
\end{subfigure}
\begin{subfigure}[b]{0.6\linewidth}
\includegraphics[width=\linewidth]{lyp_Th_0p2_E_55.png} \caption{$E=55$}
\end{subfigure}
\begin{subfigure}[b]{0.6\linewidth}
\includegraphics[width=\linewidth]{lyp_Th_0p2_E_60.png} \caption{$E=60$}
\end{subfigure}
\begin{subfigure}[b]{0.6\linewidth}
\includegraphics[width=\linewidth]{lyp_Th_0p2_E_65.png} \caption{$E=65$}
\end{subfigure}
\caption{Largest Lyapunov exponents for different values of energy of the system $E$ but for a particular value of $\theta_{nc}=0.2$.}
\label{fig:8}
\end{figure}
\begin{figure}[hbt!]
\centering
\begin{subfigure}[b]{0.6\linewidth}
\includegraphics[width=\linewidth]{lyp_E_55_th_0p19.png} \caption{$\theta_{nc}=0.19$}
\end{subfigure}
\begin{subfigure}[b]{0.6\linewidth}
\includegraphics[width=\linewidth]{lyp_E_55_th_0p22.png} \caption{$\theta_{nc}=0.22$}
\end{subfigure}
\begin{subfigure}[b]{0.6\linewidth}
\includegraphics[width=\linewidth]{lyp_E_55_th_0p25.png} \caption{$\theta_{nc}=0.25$}
\end{subfigure}
\begin{subfigure}[b]{0.6\linewidth}
\includegraphics[width=\linewidth]{lyp_E_55_th_0p27.png} \caption{$\theta_{nc}=0.27$}
\end{subfigure}
\caption{Largest Lyapunov exponents for different values of $\theta_{nc}$ but for a particular value of the system energy $E=55$.}
\label{fig:9}
\end{figure}
\subsubsection{Lyapunov exponent for approximated NC-deformed metric}
\par\noindent
The appearance of chaos is evident whether we increase the value of the energy of the system $(E)$ or the value of $\theta_{nc}$ and that we have learned from the study of the Poincare sections in the previous section. Now, this gradual emergence of chaotic behavior in the system can be shown more unambiguously in a quantifying manner with the help of studying the largest Lyapunov exponent of the system. We adopted the standard algorithm to compute the largest Lyapunov exponent which is related to the rate of separation of the trajectories for two nearby points \cite{sandri:96}. If we consider two trajectories with initial separation $\delta x_{0}$, the rate of divergence within the linearized approximation is given by
\begin{eqnarray}
\vert\delta x(t)\thickapprox e^{\lambda_{L}t}\vert\delta x_{0}\vert
\end{eqnarray}
where $\lambda_{L}$ is the Lyapunov exponent. Here we have reproduced two graphs of the Lyapunov exponents (Fig. (\ref{fig:8}) and Fig. (\ref{fig:9})). In order to plot these graphs, first we numerically solve the equations of motion of the particle (Eq. (\ref{rdot}) - Eq. (\ref{ptheta_dot})) for $r$ with two initial conditions which are initially separated infinitesimally ($\delta r_{0}$). Then, the separation of these two trajectories has been studied for a long period of time which essentially gives us the saturated value of the maximum Lyapunov exponent. Here, we have studied the characteristics of the Lyapunov exponent for two cases. In Fig. (\ref{fig:8}), we have plotted for different values of the system energies but for a constant value of $\theta_{nc}$ (for $\theta_{nc}=0.2$). Fig. (\ref{fig:9}) is plotted for different values of $\theta_{nc}$ but for a particular value of the system energy $E$ (for $E=55$).
From Fig. (\ref{fig:8}), we find that with the increase in the value of energy $E$ for a particular value of $\theta_{nc}=0.2$ the value of the largest Lyapunov exponent also increases. At $E=50$, the value of the largest Lyapunov exponent saturates at a value around $0$ which suggests that our system is still periodic. Whereas with the increase in the energy value, i.e., for $E=55,~60$ and $65$, the largest Lyapunov exponent value saturates at the value $\sim 0.015,~\sim 0.02$ and $\sim 0.04$, respectively. The positive increment in the largest Lyapunov exponent values with the increase in the energy in the system indicates that our system becomes more chaotic with the increase in energy. One important point to be noted here is that the Lyapunov exponent has an upper bound \cite{Maldacena:2015waa} which in this case is $\lambda_{L_{max}}=\kappa=1/2f'(r_{H})$, where $\kappa$ is the surface gravity of the black hole. Analysing the plots of the Lyapunov exponents in Fig. \ref{fig:8}, we find that the obtained value of the Lyapunov exponents for different values of energy ($E$) of the system are much lower than the corresponding upper bound (in this case the upper bound is $\lambda_{L_{max}}\approx 0.21$).
Similarly, from Fig. (\ref{fig:9}), we can see that for a particular value of the energy of the system, i.e., $E=55$, with the changing value in $\theta_{nc}$ some changes occur into the system. With the increment in the value of $\theta_{nc}=0.19,~0.22,~0.25$ and $0.27$, our system turns into a more chaotic one which suggests that the increased value of this parameter $\theta_{nc}$ inducing more chaos into the system. In addition, in this case we also found out that the acquired value of the Lyapunov exponents for different values of $\theta_{nc}$ are lower than the upper bounds discovered in \cite{Maldacena:2015waa} ($\lambda_{L_{max}}\approx 0.22,~0.20,~0.18$ and $0.16$ for $\theta_{nc}=0.19,~0.22,~0.25$ and $0.27$, respectively). Notice that the upper bound depends on $\theta_{nc}$.
\section{Chaos for quantum corrected Schwarzschild metric}
\par\noindent
The metric for quantum corrected (QC) Schwarzschild black hole has appeared in \cite{don}
\begin{widetext}
\begin{eqnarray}
ds^2 =- \left(1-\frac{2GM}{c^2r}+\epsilon(r)\right)dt^2 + \left(1-\frac{2GM}{c^2r}+\epsilon(r)\right)^{-1} dr^2 + r^2 d\Omega^2
\end{eqnarray}
\end{widetext}
where $\epsilon(r)$ is the quantum correction term given by $\epsilon(r)= -6\left(G^2M^2/c^4r^2\right)(1+(m/M))-(41/5\pi)(GM/c^2)(l_p^2/r^3)$, where $l_p$ is the Planck length and $d\Omega^2 = d\theta^2 + \sin^2\theta d\phi^2$. $M$ represents the mass of the black hole and $m$ stands for the particle mass. However, in our analysis our main focus is on the dynamics of the massless particle, so, $m=0$ in our case.
Due to smallness, the last term has not been taken into account in our analysis and for massless particle quantum corrected metric takes the form
\begin{widetext}
\begin{eqnarray}
ds^2 = -\left(1-2\frac{k}{r}-6\frac{k^2}{r^2}\right)dt^2 + \left(1-2\frac{k}{r}-6\frac{k^2}{r^2}\right)^{-1} dr^2 + r^2 d\Omega^2\label{QC metric}
\end{eqnarray}
\end{widetext}
where, $k=GM/c^2$ and $k$ is taken to be unity for further analysis according to the unit convention $(G=c=\hbar=1)$ that we have been following so far and for the mass of the black hole, i.e., $M=1.0$. Next, we shall study the dynamics of massless particle near the horizon of the quantum corrected Schwarzschild metric (Eq. \ref{QC metric}). Considering all these facts we obtain the position of the horizon at $r_H\approx 3.6$. In this case we shall also follow the same formalism as we performed in the previous case of NC-deformed Schwarzschild metric in Section II. Solving the equations of motion of the particle numerically, we shall plot the Poincar$\acute{e}$ sections and Lyapunov exponents for the particle motion in the near horizon region and then we shall try to analyze both of them.
\subsection{Poincare sections for quantum corrected metric}
\par\noindent
In this case, the Poincare sections (Fig. \ref{fig:10}) are also plotted for $p_{\theta}>0$ and $\theta=0$ just like before. We have considered $M=1.0$, $K_r=100$, $K_\theta=25$, $r_c=4.5$ and $y_c=0$. In order to make sure that the particle resides near the horizon we have restricted the range of $r$ at $4.5<r<5.5$. Similar nature can be found in these plots also. As we increase the value of the energy of the system $E=55.0,~60,~65,~70,~72$ the system gradually goes from the periodic state to the chaotic one. The appearance of the scattered points in the higher energy ranges ($E=70,~72$) are indication of the chaotic fluctuations into the system. With the increase in the energy of the system, our massless test particle reaches nearer to the horizon and the more our system comes under the influence of horizon the more it becomes chaotic. Some more examples of Poincare sections for different initial conditions for this case are provided in the Appendix.
\begin{figure}[H]
\centering
\begin{subfigure}[b]{0.4\linewidth}
\includegraphics[width=\linewidth]{E=60.pdf}
\caption{E=60}
\end{subfigure}
\begin{subfigure}[b]{0.4\linewidth}
\includegraphics[width=\linewidth]{E=65.pdf}
\caption{E=65}
\end{subfigure}
\begin{subfigure}[b]{0.4\linewidth}
\includegraphics[width=\linewidth]{E=70.pdf}
\caption{E=70}
\end{subfigure}
\begin{subfigure}[b]{0.4\linewidth}
\includegraphics[width=\linewidth]{E=72.pdf}
\caption{E=72}
\end{subfigure}
\caption{\centering The Poincar\'e sections in the ($r,p_r$) plane with $\theta = 0$ and $p_\theta > 0$ at different energies for quantum corrected Schwarzschild black hole. The horizontal and vertical axis in each of the graphs corresponds to $r$ and $p_r$, respectively.}
\label{fig:10}
\end{figure}
\subsection{Lyapunov exponents for quantum corrected metric}
\par\noindent
Here, we also quantify the chaos in our system. From Fig. (\ref{fig:12}) we can see that with the increase in the value of the energy of the system, the largest Lyapunov value of the system gets increased. As we increase the energy, the particle moves closer towards the horizon, subsequently influence of the horizon increases making the system more chaotic. Moreover, it is worth to mention that the obtained value of the Lyapunov exponents for different values of energy of the system are lower than the upper bound \cite{Maldacena:2015waa}.
\begin{figure}[H]
\centering
\begin{subfigure}[b]{0.6\linewidth}
\includegraphics[width=\linewidth]{lyp_E_60.png} \caption{$E=60$}
\end{subfigure}
\begin{subfigure}[b]{0.6\linewidth}
\includegraphics[width=\linewidth]{lyp_E_65.png} \caption{$E=65$}
\end{subfigure}
\begin{subfigure}[b]{0.6\linewidth}
\includegraphics[width=\linewidth]{lyp_E_70.png} \caption{$E=70$}
\end{subfigure}
\begin{subfigure}[b]{0.6\linewidth}
\includegraphics[width=\linewidth]{lyp_E_72.png} \caption{$E=72$}
\end{subfigure}
\caption{\centering Largest Lyapunov exponents for different values of energy of the system $E$.}
\label{fig:12}
\end{figure}
\section{Conclusion}
\par\noindent
Let us summarize the results obtained in the present work. The chaotic behavior of particle dynamics near Schwarzschild black hole has been under study for quite some time. In recent times, this research has received new impetus from deep results a very different area; existence of a universal upper bound of Lyapunov exponent for a finite temperature quantum field theory. These two very distinct phenomena are intimately connected by the fact that black holes can be treated as thermodynamic systems with a Hawking temperature. Another area of recent interest in black hole physics with quantum corrections incorporated in the metric, arising from possible quantum gravity effects via noncommutative geometry or generated through conventional quantum field theoretic effects. In the present work we have analysed in detail the effects of these metric extensions on the chaotic behavior of particles close to the black hole horizon.
\par
Our results clearly show that in both cases, the metric extensions favour chaotic behavior that is chaos is attained for relatively smaller particle energy. This is demonstrated numerically by exhibiting the breaking of the KAM tori in Poincare sections of particle trajectories and also via explicit computation of the (positive) Lyapunov exponents of the trajectories. Also of interest is that in all the cases considered here, numerical value of the Lyapunov exponent is well within the universal saturation value.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 5,700 |
History of Geology >
Tracks, Trails, and Thieves: The Adventures, Discoveries, and Historical Significance of Ferdinand V. Hayden's 1868 Geological Survey of Wyoming and Adjacent Territories
Product Code: USPE521
Series: GSA Special Papers
Author/Editor: By Jack E. Deibert and Brent H. Breithaupt
Publication Date: 15 August 2016
Special offer price (while stocks last)
Was: £21.50; GSA/GSL £14.25
Ride the trails and rails across the Wild West with Ferdinand Hayden through this first-ever detailed recounting of the first government-sponsored geological survey of the Wyoming and adjacent territories in 1868. The discovery of new archival material has helped bring the day-to-day adventures of this unique survey to life. Events of the survey are intertwined with one of the most noteworthy events in U.S. history—the building of the transcontinental railroad. Activities of the railroad led Hayden to have serendipitous and influential encounters with famous Civil War generals, railroad executives, politicians, photographers, prominent geologists, and thieves. The results of Hayden's survey provided the earliest descriptive stratigraphic-structural profile across the Rocky Mountains and the initial discovery of dinosaur tracks in western North America. Featuring more than 50 vintage photographs, this volume will appeal to a general audience as well as those interested in the history of geology.
Ten Digit ISBN:
Thirteen Digit ISBN: 9780813725215
Publisher: GSA
Contents on GSA website.
List price: £ 8.50 | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 5,466 |
quality, conformation and COI's in mind.
NEW: Virginia KuneKunes is now able to accept credit cards. Transaction fee may apply.
We do have one adult breeding pair or trio.
There is nothing like proven mothers to get you off on a great start. Both moms are easy to handle, allow you to be there duing farrowing (birthing) and let you handle their piglets. Well suited for homesteaders for pork production.
Tonganui boar (owned by Red Roof KuneKunes and a selection of proven excellent mothers.
Please contact us to see who may be available. | {
"redpajama_set_name": "RedPajamaC4"
} | 0 |
Black Hat Meets Blog The Internet Is Revolutionizing Closed Orthodox Communities and Exposing Long-Hidden Sexual Abuse Allegations — and Not Everyone Is Happy about It
By Jennifer Friedlin
New York Jewish Week - May 25, 2006
Is computer technology shifting the balance of power in Brooklyn's insular, fervently Orthodox community?
In the 1980s, two prominent Flatbush rabbis allegedly closed the door on a burgeoning sexual abuse scandal by preventing a rabbinical court proceeding from taking place. Now, two decades later, an Internet blog has reinvigorated the allegations, resulting in two multimillion-dollar lawsuits against a rabbi, a yeshiva and a summer camp for boys.
"Without the Internet, this story never would have been brought to light," said Un-Orthodox Jew, the anonymous blogger who last year began posting angry diatribes about the alleged abuse and cover-up on www.theunorthodoxjew.blogspot.com.
On the blog, Un-Orthodox Jew, who also goes by UOJ and claims to have deep ties in the "black hat" world, stated that Rabbi Yehuda Kolko sexually abused a number of male students at Yeshiva and Mesivta Torah Temimah in Flatbush and at Camp Agudah in Ferndale, N.Y., while Rabbi Lipa Margulies, the head of the school, allegedly helped to protect him at the expense of the victims. All told, three former students of Rabbi Kolko allege abuse against him in the two lawsuits.
While the blog has generated heaps of scorn among readers — some people have said UOJ's Web posts were less acceptable than the alleged acts they were revealing — the Web site has also elicited support as well as a response from at least one alleged victim.
David Framowitz, a 48-year-old former student who now lives in Israel with his family, says he first came across the blog while searching for Rabbi Kolko's name on the Internet. His story was chronicled in a May 22 New York Magazine story.
"I was always typing in Kolko's name looking to see if anyone else was molested," he told The Jewish Week in a telephone interview last week. "Then one day, I Googled Kolko and all of a sudden it was there."
Framowitz posted his story to the UOJ blog, claiming that Kolko repeatedly molested him 36 years ago while he was a seventh and eighth grade student at Torah Temimah and during two summers at Camp Agudah. He said he told his parents, but they did not believe him. Now, he wrote, he was coming out because he felt the time had come to tear down "the wall of silence."
In response to the posts, UOJ put Framowitz in touch with Jeffrey Herman, a Miami-based lawyer who has litigated sex abuse cases against the Catholic Church. Herman took the case. He is also representing two other plaintiffs who go by John Doe 2 and John Doe 3 in the complaints. The complaints, filed in Brooklyn Federal Court, all name Rabbi Kolko and Yeshiva and Mesivta Torah Temimah as defendants, while the complaint on behalf of Framowitz and John Doe 2 also names Camp Agudah.
Although Rabbi Margulies is not a defendant in the case, the complaint states that Rabbi Margulies threatened to expel from the school and ostracize from the community any child who spoke of the abuse. Herman said that Margulies also enlisted Rabbi Pinchus Scheinberg to help quell the fire by telling victims that sexual abuse had not taken place because there was no penetration. After allegedly thwarting two beit dins, Rabbi Margulies told anyone who asked that Kolko had been exonerated, according to last week's New York Magazine expose. No one ever went to the authorities.
Avi Moskowitz, a lawyer representing Torah Temimah, told The Jewish Week that the yeshiva "emphatically denies the allegations" and has put Kolko on administrative leave.
Rabbi David Zwiebel, a representative of Agudath Israel of America, the owner of the camp, said that officials in his organization had not heard of any allegations against Rabbi Kolko, who apparently left the camp's employ of his own accord in the mid-1970s.
"There is nobody currently in the administration who has any recollection from that time," Zwiebel said.
Rabbi Kolko and Rabbi Margulies declined to comment, while Scheinberg, who is 93 and lives in Israel, could not be reached.
While the statute of limitations has expired for a criminal investigation or a civil lawsuit, Herman said he believes that because of the alleged cover-up the plaintiffs would have the right to pursue the civil action.
Herman also noted that a 22-year-old has come forward with allegations against Rabbi Kolko, but he declined to provide details. If that case moves forward, it could fall within the statute of limitations for a criminal investigation, according to Herman.
Besides blogging, UOJ — who said he will not reveal his identity because it would deflect attention from his cause — said he tried several other avenues to bring the allegations to light, from writing letters to Jewish and secular newspapers to sending a letter about Rabbis Kolko and Margulies to thousands of religious families throughout Brooklyn. But, he said, no one wanted to listen.
"I have submitted letters to the editor and as long as they were non-controversial they were accepted. But once I started snooping around about issues no one was dealing with, my letters were not published," said UOJ, who describes himself as somewhere between 30 and 40 years of age, observant and married with children. He also says that he comes from a prominent Orthodox family that made a fortune in real estate.
Working as an Internet-based Robin Hood, UOJ said his sole interest in starting his blog was to rattle the cocoon of Orthodoxy, which, he claims, has enabled those in power to exploit their followers.
Experts who advocate on behalf of sex abuse victims have applauded UOJ's efforts. They say that because many Orthodox communities prohibit people from going to secular authorities with allegations of abuse and that abusers often go unpunished, the Internet provides one of the only vehicles religious people have for accessing support.
"In the Orthodox world people don't watch TV, they don't listen to the radios, they don't read the papers but everyone seems to be sneaking onto the Internet," said Vicki Polin, executive director of the Awareness Center, a Baltimore-based advocacy group for victims of sexual assault.
Yet others worry about the Internets potential for abuse.
Rabbi Kenneth Brander, the dean of Yeshiva University's Center for the Jewish Future, said he thought that recent Internet chatter is "a reflection of the fact that victims have not felt heard on this issue." Nevertheless, he expressed concern about the harm a vengeful or mistaken blogger could inflict on an innocent person.
"Not everything on a Web site can be treated as truth," Brander said.
Whether or not the Internet proves helpful or hurtful or a bit of both, most community observers say the Web has forever changed the way Orthodox individuals interact with the world.
"The Internet poses an incredibly serious threat to the status quo in these communities — as it does to any society that controls information and suppresses public dissent," said Hella Winston, a sociologist and author of "Unchosen: The Hidden Lives of Hasidic Rebels."
"The fact that David Framowitz was able to connect with UOJ from half a world away, in only a few seconds, is nothing short of revolutionary," she said.
In the wake of the lawsuits and the New York magazine article, UOJ said he has received more than 400,000 hits to his site. Meanwhile, the alleged abuse has also become a hot topic on other Jewish blogs.
On the Chaptzem blog (http://chaptzem.blogspot.com/), which describes itself as "the one and only heimishe news center," the host wrote:
"The whole Kolko-Margulies story has brought to light some very important questions regarding child abuse. How do we as a community deal with allegations of abuse? How do we decide if they are founded or fabricated? … Also, even if the allegations are founded how do we go about stopping it? How far do we go?"
According to UOJ, such questions have been a long time coming.
Are Blogs Kosher?
Rabbi Mark Dratch -
By Rabbi Mark Dratch
Jewish Week - May 25, 2006
July, 2012 -- Rabbi Mark Dratch is the current Executive Vice President of the Rabbinical Council of America (RCA)
Free-wheeling blogs" unrestricted news, opinion, and discussion Web sites that break, make and spin the latest on any number of topics" have been proliferating for years. Now there are a number of Jewish sites that scrutinize everything from Israeli politics to shul gossip.
They are the new "Bubbie Hotline."
Several of them have recently brought to light a number of rabbinic abuse cases and have succeeded in generating community interest and outrage, forcing a few rabbis from their positions, and casting aspersions on the characters of others.
To some, these sites are cesspools of unfettered, unaccountable tripe and vilifying slander that malign individuals and besmirch the community. To others, they are the Hyde Parks of protest and free speech, a place where victims can finally speak without fear of counter attack, and interested parties can press for justice by doing end-runs around denial and cover-up.
So are these blogs kosher" are they the right place for discriminating Jews to "chew the fat?"
There are serious issues to contemplate. Consider lashon hara, defined as reckless and harmful speech. The Torah prohibits it. Morally sensitive people are appalled by it. And the blogs seem to be full of it. Because there is no accountability, especially from those who leave anonymous comments on the threads of others' sites, anybody can say anything about anyone. And sometimes they do. With the peck of a few letters on a keyboard, information is posted that anyone with a computer and Internet access can read, and lives and reputations can be destroyed. All of this with no due process or accountability and with no real chance of rebuttal. On some blogs you can find "dirt" that even some of the less reputable newspapers wouldn't publish. And consider Chillul Hashem, the concern of scandal and communal disrepute. What will others say about the Jews and the Jewish community?
These concerns are serious ones and cannot be easily dismissed or pushed aside. They deal with fundamental precepts of Jewish law and the very bases of moral decency.
But denial and cover-up and dismissal of complaints and victimization of vulnerable children and adults are also serious and cannot be easily dismissed or pushed aside. Too many innocents have felt unheard, ignored, rejected, and sacrificed on the altar of public and private reputations. They have been silenced in order to protect the image of a community whose perfection exists only in their imaginations.
Too many times Jewish law and Jewish values are misapplied, misinterpreted and misappropriated in order to achieve these reprehensible ends.
There are reasons that victims, along with their supporters and advocates, have turned to the blogs, Web sites, newspapers and magazines. And that's because too many times they first turned to rabbis and Jewish institutional leaders to complain about the abuse and violation they suffered" and they were abused again.
Can these blogs be more responsible? Yes, and they need to exercise much greater care in upholding standards of decency, fairness and justice because they, too, can be responsible for harming innocents. And Web surfers should not necessarily believe what they read on them. In fact, they should take much of what they see on these sites with less than a grain of salt.
But the blogs are here and, for now, supply a valuable service. In a community that was responsive and accountable the excesses on the blogs would be unnecessary. At the moment, there are those who feel that they have no other choice. Innocents" victims and potential victims of abuse and the values and reputation of a compassionate and valuable community" are being hurt by a community that could and should do better.
All is not bad in our community, not by a long stretch. In the past decade scores of social service programs have been started that support and advocate for victims of abuse. The Rabbinical Council of America passed new guidelines for dealing with allegations against its members and has stood strong against harsh and unfair criticism. Some rabbinical schools are focusing a bit on issues of abuse and rabbinic boundaries. And JSafe was established to create a mechanism for accountability and responsibility and to provide education and support for victims, rabbis and institutions. But there is still too much denial and obfuscation
The rabbinic bon mot, "All Jews are responsible one for another," is more than just a nice slogan; it is a religious obligation. Judaism holds all of its adherents" clergy and laity without distinction" accountable for the spiritual and physical well-being of others. This is most certainly true when it comes to our children. They depend on us to care for them, to protect them and to nurture them. Not only do we have an obligation to them as humans deserving safe and secure lives, but as Jews in whose hands lay the destiny of our people. Safety? Security? We owe our people and future no less. We owe each and every son and daughter no less"and much, much more. n
Labels: Blogs, kosher, lashon hara, Mark Dratch, Rabbinical Council of America, sexual abuse
In Memory of Elisheva Buxbaum - Rape Victim Advocate
The Awareness Center is dedicated to the memory of
Elisheva Chana Buxbaum
Rape Victim Advocate
July 8th 1959 - May 22 2006
(Tammuz 2, 5719 - Iyar 24 5766)
There are people you will meet in your life with whom will have an impact on your soul and influence you in ways that you cannot even begin to imagine. For me Elisheva Buxbaum was one of those individuals.
Elisheva was one of the most spiritual people I have ever met. Her quietness could be quite deafening. All one needed to do was to listen closely and you'd start to learn what was really going on in some orthodox communities in Jerusalem and many other locations throughout Israel.
I met Elisheva Buxbaum while I was living in Jerusalem. I had known her brother Rabbi Benyamin Buxbaum online for several months prior to my move to Israel. At the time I was running a media watch group called "CNN-WATCH." During our many e-mail exchanges, Benyamin learned of my background working with sexual trauma and of my long time dreams of creating an international organization to address sexual abuse in Jewish communities. Because of the work I wanted to be doing and because of his sisters interests Benyamin introduced me to his younger sister Elisheva, who was an extremely soft spoken rape victim advocate.
Elisheva Buxbaum had no formal education or training in the field, yet did what she could to connect survivors in Jerusalem with appropriate resources.
Elisheva's influence really motivated me to create The Awareness Center. It saddens me a great deal to know that Elisheva lost her battle with cancer on May 22, 2006.
As her brother Benyamin stated "Eliesheva was a selfless friend of many and strong in her faith in Hashem."
Vicki Polin
The Awareness Center, Inc.
In Memory of A Dear Friend
Elisheva Buxbaum - Rape Victim Advocate
Last year I sent out a message letting you know about the passing of a dear friend. Elisheva Buxbaum spent most of her adult life advocating for survivors of sex crimes. I recently received the following e-mail and wanted to share it with you (see below).
If anyone else has stories they would like to share about Elisheva, please forward them to Vicki Polin. I would like to post them on her site and also share them with her family.
Sara Yocheved Rigler, author of "Holy Woman," recently published a book called "Lights from Jerusalem." In it, she has a chapter about Elisheva Buxbaum.
Here are a few excerpts:
"...The most moving evidence of hashgachah pratis attached to Holy Woman involved the very person who first introduced me to Rebbetzin Chaya Sara Kramer. In the summer of 1985, I had been in Israel barely a month. Hearing that I was a writer, someone who was trying to put together a book called A Spiritual Guide to Eretz Yisrael (which was never published) asked me to write a chapter on "Holy Women."
I was at a loss as to how to track down such women when a stranger told me, "You have to call Elisheva Buxbaum. She knows everyone worth knowing." The person gave me Elisheva's phone number, and I called her. She told me that I must meet Rebbetzin Chaya Sara Kramer, who lived in a moshav named Kfar Gidon, north of Afula.... I never met Elisheva nor spoke to her again. I turned the key that she had given me and walked through the portal into the aura of the holy tzaddekes Chaya Sara Kramer, and my life was forever changed. But by the time I realized how much I owed Elisheva, I had lost the scrap of paper with her phone number..."
...I started doing research. Often, at the end of an interview, the interviewee would ask me, "How did you meet Rebbetzin Kramer?"
I would answer lamely, 'Someone named Elisheva told me about her.' By Chanukah 2005 the book was almost completed... [At a Melaveh Malkah]...I was relieved to spot my old friend Nechama Bergman... [After describing the book]...Nechama answered, Rebbetzin Chaya Sara Kramer!
Elisheva and I used to go visit her all the time." I was stunned. 'Y-y-you know Elisheva?' I finally managed to stutter.
'Of course, she's one of my best friends. She and I used to go up to Kfar Gidon and visit the Kramers all the time.' [After learning that Elisheva was very sick]...The next morning I called. Elisheva remembered me...
We arranged that I would visit the next day...
...I had known friends in the final stages of cancer, but I was totally unprepared for the sight that greeted me in Elisheva's room. The sheet covering her body barely rose from the mattress. Her shoulder and neck were skeletally thin, and the skin of her face was taut lik photos of concentration camp survivors. Yet, set into that emaciated face was a broad, radiant smile. As we talked, the smile never waned, except when it was replaced by a jubilant laugh.
Gazing at Elisheva, my cognitive dissonance was total. There was almost no body left except for that broad smile. I felt like I was conversing with a soul almost shorn of its body...
After an hour I left, promising to return again with the manuscript of the book that owed its existence to Elisheva...
[After discovering a photo of Elisheva when she was healthy]
... I took the picture, copied it, framed it, and brought it as a gift to Elisheva, with the manuscript. She was thrilled to receive it...
A month later, she finished reading the manuscript. In a faint voice she told me, "I love it."
...A few weeks later Elisheva Chana bas Avraham died.
...knowing that she had been the catalyst for a book that would exert a profound effect on thousands of readers
... I have often felt that, since Rebbetzin Chaya Sara was childless, everyone who reads Holy Woman and is inspired to grow becomes the Rebbetzin's spiritual child. Elisheva also died childless. The readers of Holy Woman are no less her spiritual children."
Labels: Elisheva Buxbaum, rape victim advocate
Rabbi Zev M. Shandalov on Sexual Abuse of Children in Chicago
Labels: Chicago, child sexual abuse, Chillul Hashem, Hashem, Jewish, Rabbi Zev. M. Shandalov, sex offender, Vaad of Chicago
New JBBL Head
By Phil Jacobs
Baltimore Jewish Times - May 19, 2006
"I'm there for everybody." — Dr. Aviva Weisbord
Dr. Aviva Weisbord has been there for everybody for almost 25 years, working with her patients in her psychology practice.
The truth is, Dr. Weisbord, the wife of Ner Israel Rabbinical College's Rabbi Beryl Weisbord and daughter of the late rosh yeshiva, Rabbi Yaakov S. Weinberg, and community pillar Rebbetzin Hannah Weinberg, has never not been there for this community.
But now that status is going to take a huge, perhaps more public, change.
As of June 1, Dr. Weisbord, who is known for her warm, friendly smile, will start as the new head of the Jewish Big Brother and Big Sister League (JBBL) and its Jewish Addiction Services program. JBBL is a constituent agency of the Associated: Jewish Community Federation of Baltimore.
Dr. Weisbord succeeds Lou Jacobs, who left the agency last March after 18 years of leading the JBBL to start a private practice in psychotherapy for adolescents, adults and couples.
A Pikesville resident, Dr. Weisbord has for seven years served on the JBBL board, and for four years served as an officer. She was actually on the search committee to find a replacement for Mr. Jacobs. She becomes the first Orthodox woman in recent memory to head an Associated constituent agency.
Dr. Weisbord first became connected to the league when Mr. Jacobs asked her some 10 years ago to help the agency with staff development.
"I didn't know what I was getting into," she said with a smile. "I didn't know I'd fall in love with this agency."
Dr. Weisbord said that the mission of the league is an exact mirror of how Jews are supposed to take care of and look out for one another.
"This is how we're supposed to live," she said. "This is the ultimate chesed organization.
"My immediate goal," Dr. Weisbord said, "is to get the community more familiar with what we do. A lot of people will ask me what the league does. I think we have to get the word out more specifically. The Associated is right there with us and for us to help get that message out there."
It's a cool, rainy spring May day. Dr. Weisbord is interviewed at an outdoor table at the Festival at Woodholme. Pedestrians run from their cars to avoid the rain. Some know her and smile, others wave.
Dr. Weisbord will be transitioning out of psychology practice, keeping on a small number of clients while moving into her new position. That job will still involve the management of the well-received JBBL programs.
"For every mentor we get started, we need two more," she said of the community need for the league. "And there are still many who don't know to call on us."
Dr. Weisbord will manage a 15-person staff that she says is doing a marvelous job. "If it ain't broke, don't fix it," she said of the personnel. "Lou put together a top-notch agency."
She also said that volunteering for the league has been an eye-opening experience as far as the work done by the Associated.
"Before I joined the board, I just saw the Associated as some sort of monolithic organization raising money," she said. "But I've learned it's more than I ever thought. The volunteers and professionals are really there to take care of us and our community."
JBBL board member Larry Seegul said Dr. Weisbord brings to the league "tremendous passion and commitment. Give credit to Lou. He was a top-notch professional. When you go to the dictionary and look up professional, he's your guy. But she brings different skill sets. She will bring an excitement and a vitality that is always needed. She'll make a great director."
Case of Rabbi Eliezer Eisgrau
Weinberg Family History
Case of Rabbi Matis Weinberg
Case of Rabbi Simcha Weinberg
Labels: Aviva Weisbord, clergy sexual abuse, Matis Weinberg, obstruction of justice, Simcha Weinberg, Weinberg Family Tree, witness tampering
Deposition Of Rabbi Dr. Marc Gafni's Third Wife
(AKA: Mordechai Winiarz, Mordechai Gafni, Marc Winiarz)
Marc and Chaya Gafni got divorced in August of 2004 a month before Gary Rosenblatt's article "The Re-Invented Rabbi" was published.
Click here for more information on the case of Dr. Rabbi Marc Gafni
Deposition Of Marc Gafni's Third Wife
Chaya Gafni - May 9, 2006
Chaya Gafni
The following is my personal testimony of what it was like to be married to Mordechai for almost 7 years. I share what I have known of Mordechai's drastic and tragic dark side. I focus upon the shadow aspects of our marriage and his personality, for I believe they are crucial to share, given what has unfolded in these past weeks. Please keep in mind that I could also write pages worth of testimony about the light side of Mordechai - from the beauty of his teachings to his ardent dedication to making a contribution in the world and helping others. May his light side and his dark side know full integration.
Also, while you can pass this testimony on to other concerned parties, please do not share it with the press. I do not want my name to appear in the press. Thanks.
Background: I was 19 years old when I first encountered Mordechai I was studying in Jerusalem the summer after my freshman year of college. I was an eager baalat-teshuva, newly "turned on" to the beauty of Jewish practice. I devotedly went to his classes at Isralight and other venues. We went on our first date the spring after I graduated college. I was 23 and star-struck.
He was 15 years my senior. We got married at the beginning of 1998, less than 8 months after our first date. Several people warned me about Mordechai's past. He adamantly insisted that the bulk of the rumors were lies, exaggerations and the evil workings of other people's jealousies. I believed him.
Dr. Marc Gafni
He actually told me early on about some of his sexual misdemeanors as well as affairs he had on his 2nd wife. He assured me that he had done teshuva, changed, and that things with me would be different. I was all too ready to believe this as well Plus I thought that I could help him, fix him; that my love could help him become the great man he had the potential to be. As soon as we started seriously dating, he pulled me into working for him full-time on writing and organizational projects. I was dedicated to his "mission" of Jewish Rennaissance and gave it all of my time and energy.
His emotional abuse and manipulations began immediately upon our marriage. But I was so dedicated to the mission that I withstood it. The years that followed were a strange mix of great excitement, activity and purpose, as well as huge despair, confusion and pain. On the outside I seemed to be living a fairy tale of success and contribution. Behind closed doors I was living a life of enslavement, debasement, manipulation and verbal abuse. On top of the abuse, Mordechai was having more than one affair on me; 1ying to me on a daily basis.
Finally, I started to see through the fog of falsehoods. Fled Israel in February of 2004, only to be lured back in June, 2004 by Mordechai's promises of change and commitment. But nothing changed. By early August of 2004 I finally demanded and received a divorce.
Soon thereafter, Mordechai came "under attack" by his enemies in America. In the fall of 2004, articles about his sexual misconduct and questionable reputation came out in America and Israel. He begged me to keep our divorce a secret until all of this bad press died down. I reluctantly agreed - mostly because I believed that the work that was going on at Bayit Chadash was valuable and I did not want to jeopardize it. Mordechai lied to the reporters and all who asked, saying that we were still married. He even lied to the Rabbinic supporters who helped wage a campaign to protect him. Mordechai refused to publicly tell the truth about our divorce until Pesach of 2005 (March/April). I am ashamed to admit that I was manipulated in to also remaining silent and covering up to protect him, as I had done myriad times during our marriage.
Marc Gafni on the right
Now that I see the damage that Mordechai has caused in so many people's lives I deeply regret that I did not speak out earlier about the abuse that I suffered at his hands. I also deeply regret that I did not speak out about the countless lies and manipulations that I witnessed him engage in on a regular basis. I sorely regret that I led people to believe that we had a good marriage when in actuality it was most often a hell. I have been studying, practicing and engaging in psychotherapy these past two years since I left Mordechai. The more I have learned - and now that I have beard the stories of what has happened with other women - it is now utterly clear to me that Mordechai is a dangerous sexual predator and sociopath. He hurt me in deplorable ways and I fear that he will continue to hurt others if he is not stopped. I pray that my speaking out now can help to thwart any and all future abuse at his hands.
1. Information about Sexual Abuse Molestation of a Minor:
Before we got married, Mordechai shared with me that he had indeed had a sexual encounter with a minor. Her name was (Name Withheld) - a teenager who was in his JYPSY youth movement He explained to me details of their encounter and how he went about covering it up and discrediting her. He told me that she had seduced him. He said that they did not have intercourse, but that they had at least been undressed, sexually physical and that he had ejaculated. After (Name withheld) reported this, he lied to everyone involved, saying that she was emotionally unstable, jealous and had made it all up. He even received a document signed by a Rabbi attesting to his innocence. (Name withheld) was under-age, a student of his, and were it not for the statute of limitations, he could go to jail over this.
2. Mordechai also told me stories about various teachers and staff people connected with Yeshiva University with whom he had struggles. He told me how he blackmailed a teacher (one of his "enemies" at YU) who tried to block him from studying teaching there after the (Name withheld) incident He had information about this particular man and threatened to share it if the man continued to try to block him. The man stayed quiet
3. Adultery Lying:
He also told me of several affairs that he had in Boca Raton while married to his second wife. These affairs were with women in his Congregation (the name of one woman was XXX XXX). At least one of them was a married woman (whose name I don't recall, though I can find it). There was a scandal at the synagogue over rumors about his sexual misconduct I do not think that his second wife ever found out about these affairs. They eventually left Boca to move to Israel I believe that the main reason for this was that he needed to flee before people found out the truth (though he never framed it that way).
One of the reasons (among many) that I divorced Mordechai was because he had an affair while we were studying in Oxford. It was with a woman named XXXXXX at Wolfson College. He lied to me on a virtually daily basis to cover up this affair. This went on for approximately 10 months (from December 2002 - Sept 2003). It was an agonizing time for me even though I did not consciously know what was happening. I finally convinced him to tell me the truth about the affair when we left Oxford and moved back to Israel. I was devastated, and realized that all of my hopes that he was a "changed man" were baseless fantasies.
I also was racked with suspicion that he was having another affair - with his "teaching partner", [Erica Ariel Fox]. On countless occasions I begged him to stop teaching with her and to pull back from their "friendship". He refused. Also on countless occasions I point blank asked him if they were having an affair. In response, he consistently told me how crazy, jealous and insecure I was. I have finally found out that they were indeed intimate with each other while we were married, as well as after. (All of this went on between 2002 until our divorce in August of 2004). I also have heard that he had affairs with two other women while we were together - one a young woman in Israel and the other in America.
6. Debasing Sexuality:
Mordechai was consistently verbally demeaning to me, particularly when we were having sex. While we were being intimate, he would demand that I repeat, "I am a whore." I reluctantly went along with this at first. Eventually I refused to say it. Over the last two years that we were together I would go to bed absolutely dreading the possibility of having sex with him. Additionally, he viewed pornography on a regular basis; including paying money to have memberships to certain sites. Eventually his computer and email were so full of pornography that he paid tens of hundreds of dollars to get it cleaned, for fear that someone may see it and that he would lose his job. I understand from formal depositions made with lawyers and the police in Israel that he had much more "extreme" sexual interactions with other women after our divorce; which involved a lot of S&M and also played heavily on themes of debasement.
7. Stealing Intellectual Property:
Mordechai used other people's stories/teaching (making slight changes) without attributing them properly. (The story in Soulprints about Eitan giving him a soulprint box was, for instance, based upon a story in one of Robert Fulghum's books.) Furthermore, I worked full time on both books "Soul Prints" and "The Mystery of Love". There are entire sections of these books which I myself wrote - with no public recognition given as to the depth and breadth of my contribution. Just a few of the numerous examples of this are the poem/invocation at the beginning of "Soul Prints", as well as the Parable of the Royal Wine in ''The Mystery of Love". I insisted that I wanted at least these pieces to be attributed to me. He refused Seeing I had no real choice, I gave in in the end and allowed the pieces to be used without attribution.
8. Verbal Abuse & Emotional Manipulation:
This was a constant throughout our marriage. I have pages and pages of journal entries describing entire scenes and dialogues full of emotional abuse. His yelling explosions, full of demeaning putdowns and blame, were virtually a daily occurrence. I eventually stopped fighting back and would just dissolve in tears after each explosion. He needed to always be right, always in control. If I didn't agree with him on something then he would burst into a rage and tell me how stupid I was. But more than that, he would tell me how unloving, insensitive and selfish I was. Convincing me that I was the evil, selfish, unloving one was one of his most powerful tools of manipulation. He capitalized on my natural desire to be loving and giving. My goodness was a knife in his hands with which he daily carved his sick designs into me. I was utterly bewildered by his manipulations; the way he would turn everything around and make me the bad one. These turn arounds rendered me powerless time and again. In fact, I was so distraught by the nature of his putdowns and manipulations that I had regular fantasies of doing violent and suicidal acts against myself. My most recurrent fantasy during his abusive tirades was of slashing my throat. I was not "allowed" to express or feel anger towards him and so I turned all of my anger at him back upon myself. I had never in my life been suicidal before this time and since I left him I have not had suicidal or violent thoughts at all.
9. Verbal Abuse Manipulation of Others:
I witnessed Mordechai being verbally abusive and manipulative with many other people. I saw it happen most with Dafna, his main staff person, but also - tragically - also saw it with his Sons, most particularly (NAME REMOVED). I found his neglectful and insensitive treatment of his sons to be deplorable. I could go in to greater detail about this but will refrain out of respect for bow hard all of this must be for them. Seeing him with his sons was another big factor in my wanting a divorce. The thought of him mistreating any future children that we would have was just terrifying to me.
10. Lies:
As I mentioned above, Mordechai lied about our divorce and other essential issues to the numerous Rabbis who supported him when he was being attacked in the press and at various teaching institutions. The Rabbis he lied pointblank to include R'Danny Landes, R'Joseph Telushkin, R'Art Green, R'Eli Herscher and R'Saul Bennan, as well as others. He likewise lied to the press and the entire Bayit Chadash community and Board. (He was so efficient and convincing a liar that I view all of his supporters - from the Rabbis to the Ner-David family to the Jewish Renewal Movement at large - as victims of his pathology. I strongly believe that they should not be held responsible for 'covering up' his misdeeds. I believe they all did the work to find out the truth, but the truth was too elusive.)
11. Exaggerations - Beyond the examples above I witnessed Mordechai lying routinely in most every type of setting.Whether it was in a speech, at dinner with friends, teaching. or in talking to donors. He was consistently aggrandizing himself by exaggerating his successes, popularity, power and connections. He would get furious with me when I myself did not join in on telling these inflated stories about him; saying that I was selfish and unloving for not also telling these tales. Time and again he falsely claimed to be a spiritual holy person. During his writings and teachings he would claim to pray, meditate, exercise, eat healthy, etc. None of which he did in the least. He led entire meditation retreats without ever having meditated himself. In my opinion, all of his frequent claims to spiritual enlightenment were (and are still) dangerously misleading fabrications.
12. Psychological Sickness - I think it is crucial to share that based on all that I have known of Mordechai I see that he clearly has two psychological disorders which are evident and expressed in numerous ways. The most obvious is a narcissistic personality disorder. He exhibited the following characteristics which correlate with the DMS-IV diagnosis of narcissism. In the DMS, at least 5 of the following attributes are requires for diagnosis. Mordechai exhibits them all. I could give numerous examples in each category, but will refrain for lack of space and because they are just so very obvious to anyone who knows Mordechai.):
a. has a grandiose sense of self-importance - exaggerates achievements and talents.
b. Is preoccupied with fantasies of unlimited success, power and brilliance.
c. Believes that he is "special" and unique and can only associate with other special or high-status people or institutions.
d. Requires excessive admiration
e. Has a sense of entitlement - expecting especially favorable treatment or compliance with his expectations
f. Is interpersonally exploitative; taking advantage of others to achieve his or her own ends
g. Is envious of others or believes that others are envious of him
h. Lacks empathy; is unwilling to recognize or identify with the feelings and needs of others
i. Shows arrogant, haughty behaviors or attitudes.
As for the antisocial (or sociopathic) personality disorder. He exhibits the following of the criteria for the DSM (of which 3 are needed for diagnosis):
a. failure to confirm to social norms with respect to lawful behaviors (such as his sexual harassment in the workplace and sleeping with students and employees)
b. deceitfulness, repeated lying
c. irritability and aggressiveness (as is known by anyone who has ever worked under Mordechai, or crossed his path politically)
d. reckless disregard for safety of self or others (such as endangering himself by juggling numerous affairs at once, given his history)
e. lack of remorse; indicated by rationalizing having hurt, or mistreated others
Unfortunately, with Axis II Personality Disorders the chances for change via treatment are extremely slim; as opposed to Axis I disorders which are considered more treatable. Mental Retardation, for example, is also on Axis II, because no amount of therapy will be able to fully 'treat' retardation. The same is understood for Personality Disorders - they are not entirely treatable. Thus, in my opinion, the belief that Mordechai will one day be able to return to being a teacher leader of any sort is a dangerous one. I personally (and professionally) do not think that he should be "allowed" to return to any such roles at any point in the future. A tragic loss, perhaps, but in the end we as a culture and as a people need to reassess the traits that we value and pull forth from our leaders. May this whole fiasco pave the way for new standards of humility, sincerity and a genuine care for others.
Labels: Chaya Gafni, clergy sexual abuse, clergy sexual abuse against adult women, enablers of sex offenders, Marc Gafni, Mordechai Winiarz, sex offender
Black Hat Meets Blog The Internet Is Revolutionizi...
Rabbi Zev M. Shandalov on Sexual Abuse of Children...
Case of Jeffrey Epstein
Case of Ryan Scott Karben - New York Assemblyman | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 4,767 |
Q: How to configure a build chain for smoke tests without common artifacts, VCS roots, or triggering builds up the chain? I am running TeamCity 2018.1. I can't seem to find the answer to this scenario anywhere. I have found similar questions, but they don't address exactly what I'm trying to achieve: here and here
I want the following flow in TeamCity
1 -> 2.1 -> 3
1 -> 2.2 -> 3
1 -> 2.3 -> 3
2.4 -> 3
2.x deploys different services to a test environment, and 3 runs smoke tests against the test environment, so we don't need multiple of those run. If 1 is triggered, that is exactly what happens if using Finish Build Triggers. It ends up cluttering our feeds, and we end up with test runs that have a service deployed in the middle of the run, which we absolutely need to avoid.
Requirements are as follows:
*
*I want 3 triggered any time a 2.x finishes, BUT I do not want multiple builds of 3 run, like in the case 1 is triggered first.
*I also want 3 to be triggered when there is a checkin on its default branch.
*Triggers on 3 should not build anything earlier in the build chain. This is an expensive operation, and will confuse our results in our smoke tests (did a failure happen because the tests changed? or because we deployed updated code?)
I was led to believe that snapshot dependencies would fix this problem, but it's not working how I want it to. It doesn't seem to be triggering 3 when any 2.x finishes successfully. Also, I absolutely do not want 1 or 2.x triggered when 3 is triggered by a VCS change, which seems to happen a lot regardless of the "use a suitable build option."
Also, 1, 2.x, and 3 all run from different repositories, so they have no VCS roots in common. All the 2.x builds are different repos as well. We have no artifacts to pass along either.
How can I achieve this workflow?
Edit: Is this possible by adding a dummy build configuration before my test runs?
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 9,184 |
Electrolysis is the only long term method of hair removal. It may be used on all body parts where applicable. It has been proven to be safe, effective and long term.
LHE photo-therapy targets the hair follicle with a low and ultra safe wave of light. Once the wavelength reaches the surface of the skin the light converts to heat energy. This therapy inhibits hair growth at the root of the follicle. The treatment is fast, precise and highly effective. | {
"redpajama_set_name": "RedPajamaC4"
} | 5,816 |
Q: Pygame traceback error How do i fix this error, this is the message that i get:
Traceback (most recent call last):
File "C:\Users\Games\Desktop\hendeagon.py", line 28, in <module>
font = pygame.font.SysFont(None, 48)
File "C:\Python33\lib\site-packages\pygame\sysfont.py", line 614, in SysFont
return constructor(fontname, size, set_bold, set_italic)
File "C:\Python33\lib\site-packages\pygame\sysfont.py", line 537, in font_constructor
font = pygame.font.Font(fontpath, size)
pygame.error: font not initialized
i need this error fixed for homework by tomorrow.
BLACK = (0, 0, 0)
WHITE = (255, 255, 255)
GREEN = (0, 255, 0)
TEXTCOLOR = (255, 255, 255)
WINDOWWIDTH = 500
WINDOWHEIGHT = 400
windowSurface = pygame.display.set_mode((WINDOWWIDTH, WINDOWHEIGHT), 0, 32)
def drawText(text, font, surface, x, y):
textobj = font.render(text, 1, TEXTCOLOR)
textrect = textobj.get_rect()
textrect.topleft = (x, y)
surface.blit(textobj, textrect)
font = pygame.font.SysFont('comicsansms', 48)
basicFont = pygame.font.SysFont(None, 42)
drawText('8', font, windowSurface, (WINDOWWIDTH / 2.5), (WINDOWHEIGHT / 3))
windowSurface.fill(BLACK)
pygame.draw.polygon(windowSurface, GREEN, ((158, 80), (181, 88), (214, 111), (229, 153), (216, 191), (181, 212), (135, 207), (102, 181), (89, 149), (102, 109), (130,88)))
pixArray = pygame.PixelArray(windowSurface)
pixArray[480][380] = BLACK
del pixArray
windowSurface.blit(text, textRect)
pygame.display.update()
while True:
for event in pygame.event.get():
if event.type == QUIT:
pygame.quit()
sys.exit()
Thank you for attempting to fix this problem that i am having in pygame
A: You can make your drawText method return your textobj and textrect and then use that as follows:
def drawText(text, font, surface, x, y):
textobj = font.render(text, 1, TEXTCOLOR)
textrect = textobj.get_rect()
textrect.topleft = (x, y)
surface.blit(textobj, textrect)
return textobj, textrect
text, text_rect = drawText('8', font, windowSurface, (WINDOWWIDTH / 2.5), (WINDOWHEIGHT / 3))
...
windowSurface.blit(text, text_rect)
A: I think you want to try something like:
import pygame, random, sys
from pygame.locals import *
BLACK = (0, 0, 0)
WHITE = (255, 255, 255)
RED = (197, 153, 114)
GREEN = (0, 255, 0)
BLUE = (0, 0, 255)
TEXT_COLOR = (255, 255, 255)
WINDOW_WIDTH = 500
WINDOW_HEIGHT = 400
def draw_text(text, font, surface, x, y):
text_obj = font.render(text, 1, TEXT_COLOR)
text_rect = text_obj.get_rect()
text_rect.topleft = (x, y)
surface.blit(text_obj, text_rect)
pygame.init()
pygame.display.set_caption('polygon!')
windowSurface = pygame.display.set_mode((WINDOW_WIDTH, WINDOW_HEIGHT), 0, 32)
basic_font = pygame.font.SysFont(None, 42)
windowSurface.fill(BLACK)
POLYGON_COORDINATES = (
(158, 80), (181, 88),
(214, 111), (229, 153),
(216, 191), (181, 212),
(135, 207), (102, 181),
(89, 149), (102, 109),
(130, 88))
pygame.draw.polygon(windowSurface, GREEN, POLYGON_COORDINATES)
draw_text('8', basic_font, windowSurface,
(WINDOW_WIDTH / 2.5), (WINDOW_HEIGHT / 3))
pixArray = pygame.PixelArray(windowSurface)
pixArray[480][380] = WHITE
del pixArray
pygame.display.update()
while True:
for event in pygame.event.get():
if event.type == QUIT:
pygame.quit()
sys.exit()
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 1,217 |
Q: Convergence in measure is not given by a seminorm Let $V$ be the vector space of all real-valued Borel measurable functions on $[0,1]$. Show that convergence in measure (with respect to Lebesgue measure) is not given by a seminorm. That is, show that there no seminorm $\|\cdot\|$ on $V$ such that elements $f,f_1,f_2,\dots$ of $V$ satisfy $\lim_n \|f_n-f\| = 0$ iff $(f_n)$ converges to $f$ in measure.
(Hint: show that if such a seminorm exists, then for each positive $\epsilon$ there are functions $g_1, \dots, g_n \in V$ such that $\|g_i\| \leq \epsilon$ for each $i$ and such that $1/n \sum _{i= 1} ^n g_i $ is equal to the constant function 1.)
I have problem both showing the existence and then the contradiction.
I'm grateful for hints or solutions.
A: Only hints to give you, can provide details as you get further.
So first question, why is such a construction going to give you a contradiction? "Small pieces" whose average is the constant function $1$. What is a bound on the seminorm of this average function? What does it mean as you take $\epsilon$ towards $0$? You might consider explicitly choosing $\epsilon$ to be $1/k$ and looking at the resulting sequence $f_k$ (of average functions corresponding to $1/k$).
Second question towards construction: For simplicity, set $f=0$, we're converging in measure to $0$. Can you think of a canonical example of something that converges in measure to zero while only using indicators of intervals? Does this example change if you scale the sequence? (remember, you need to produce an average of things that give you the constant $1$).
(I should have added that the example I have in mind is $\chi_{[0,1]},\chi_{[0,1/2]},\chi_{[1/2,1]},\ldots$, so $\chi_{[0,1/n]}$ and $n$ of its shifts, increasing $n$. This way you can automatically get the seminorm of $\chi_{[0,1/m]}$ and $m$ of its shifts to all be less than a given $\epsilon$. This won't quite do, so you modify the example to scale by $n$, so you use $n \chi_{[0,1/n]}$ and $n$ of its shifts.)
A: The sequence $a_n\chi_{(0,n^{-1})}$ converges in measure to $0$ for each $a_n\uparrow\infty$. This provides $n_0$ such that $\lVert \chi_{[0,n_0^{-1}]}\rVert=0$. Indeed, otherwise, take $a_n:=n\lVert \chi_{(0,n^{-1})}\rVert^{—1}$; then $\lVert a_n\chi_{(0,n^{-1})}\rVert=n$.
Hence there is function which is not $0$ almost everywhere, but the semi-norm which will do the job of this function is $0$. Since the condition on the semi-norm implies that each $f$ for which $\lVert f\rVert=0$ is $0$ almost everywhere, we get a contradiction.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 9,665 |
namespace phi {
template <typename T, typename Context>
void IndexSelectGradKernel(const Context& ctx,
const DenseTensor& x,
const DenseTensor& index,
const DenseTensor& out_grad,
int dim,
DenseTensor* x_grad);
} // namespace phi
| {
"redpajama_set_name": "RedPajamaGithub"
} | 9,035 |
With a loan of EUR 5,400 Thoch is able to grow more peper and increase her income to provide her family.
Mrs. Oeurn Thoch, 30, is married and lives with her husband, pictured from pepper farmland, and two young children in Kampong Cham province. Together with her husband are farmers, growing pepper to earn as a main income to support their family. Mrs. Oeurn started growing pepper since 2012, and at that time she was able to grow only 100 pepper plantations. So far, her family is able to grow up to 1,000 pepper plantations. The pepper plantation is making much more profit since the demand are from EU countries. There is no matter how much the product they have, the dealer are always buy all the pepper.
She is now is asking for a loan to grow more pepper as she sees this business opportunity as a way to increase her income and provides a higher standard of living for her family. | {
"redpajama_set_name": "RedPajamaC4"
} | 3,381 |
\section{Introduction}
Fashion styles adopted in our daily lives are an important aspect of culture. As noted by Lancioni~\cite{lancioni1973brief}, fashion is 'a reflection of a society's goals and aspirations'; people choose fashion styles within their embedded social contexts. Hence, the analysis of everyday fashion trends can provide an in-depth understanding of our societies and cultures. In fashion research, the advantages of digital fashion archives have continued to attract attention in recent times. These digital archives have been mainly built based on two purposes. One is enhancing the scholarly values of museum collections by making them public to utilize them digitally. Another motivation is predicting what fashion items will come into style in the short term with images retrieved from the internet for business purposes.
However, these archives are not suitable for analyzing everyday fashion trends from the humanities perspective. To identify everyday fashion trends, we need a new digital archive that includes images of the clothes people wore in their daily lives over an extended period. In addition, we need non-fashion data sources showcasing why and how people choose fashion to interpret the trends.
In this study, we create a new fashion image archive called Chronicle Archive of Tokyo Street Fashion (CAT STREET) showcasing fashion images reflecting everyday women's clothing over a long period from 1970 to 2017 in Tokyo. CAT STREET helps overcome the limitations in the existing digital fashion archives by applying machine learning to identify fashion trend patterns. Then, we demonstrate how magazine archives are suitable in understanding how various trend patterns emerged. The empirical analyses show our approach's potential in identifying new perspectives through fashion trends in daily life, which can promote understanding of societies and cultures.
\section{Related Works}
\subsection{Fashion Trend Analysis}
Kroeber~\cite{kroeber1919principle} analyzed fashion trends manually and measured features of women's full evening toilette, e.g. skirt length and skirt width, which were collected from magazines from 1844 to 1919. After this work, many researchers conducted similar studies~\cite{belleau1987cyclical,lowe1993quantitative,lowe1990velocity,robenstine1981relating}. Their work surveyed historical magazines, portraits, and illustrations and focused mainly on formal fashion, rather than clothing worn in everyday life.
The critical issue in the traditional approach is that analyzing fashion trends is labor-intensive. Kroeber~\cite{kroeber1919principle} is among the representative works in the early stage of quantitative analysis of fashion trends. This type of quantitative analysis requires a considerable amount of human resources to select appropriate images, classify images, and measure features. This labor-intensive approach has long been applied in this field. Furthermore, manually managing large modern fashion image archives is cumbersome. To solve this issue, we utilize machine learning in this study. Machine learning is a computer algorithm that learns procedures based on sample data, e.g. human task results, and imitates the procedures. In recent years, machine learning has contributed to the development of digital humanities information processing~\cite{kestemont2017lemmatization,lang2018attesting,wevers2020visual}. In the field of fashion, modern fashion styles are complex and diverse. Applying machine learning to fashion image archives may be expected to be of benefit in quantifying fashion trends more precisely and efficiently.
\subsection{Digital Fashion Archives}
In recent years, digital fashion archives have been proactively built in the fields of museology and computer science. However, the construction of these archives was prompted by different research objectives. Many museums and research institutions have digitized their fashion collections~\cite{Vogue,EuropeanaFashion,JapaneseFashionArchive,LACMAFashion,BostonMuseum,MET,TheMuseumatFIT,UNT} not only to conserve these historically valuable collections but also to enhance the scholarly values by making them available to the public. Vogue digitized their magazines from 1892 to date, the Japanese Fashion Archive collected representative Japanese fashion items from each era, and the Boston Museum of Fine Arts archived historical fashion items from all over the world.
On the other hand, in computer science, the motivation to build digital fashion archives is generally short-sighted and primarily to predict the next fashion trends that can be used by vendors to plan production or by online stores to improve recommendation engines. Several studies have created their own fashion image databases to fit the fashion business issues they focused on~\cite{abe2018fashion,Kiapour2014,liu2016deepfashion,simo2015neuroaesthetics,takagi2017makes,yamaguchi2012parsing}. Table \ref{table:table1} presents the best-known public databases from fashion studies in computer science.
Unfortunately, the fashion databases proposed in previous studies have several limitations in terms of the approach to everyday fashion trends. First, the level of detail of location annotation in existing databases is insufficient to analyze everyday fashion trends. People belong to a social community, and some social communities have their own distinct fashion styles and their territory. For instance, 'Shibuya Fashion' is a fashion style for young ladies that originated from a famous fashion mall in Shibuya~\cite{kawamura2006japanese}. Young ladies dressed in 'Shibuya Fashion' frequent Shibuya, one of the most famous fashion-conscious streets in Japan. Hence, we need fashion image data with location annotations at the street level to focus on what people wear in their daily lives.
Second, most databases consist of recent fashion images from the last decade because they were obtained from the internet. The periods covered by these databases might not be sufficiently long to determine fashion trends over longer periods. By examining how fashion changes over extended periods, sociologists and anthropologists have found that fashion has decadal-to-centennial trends and cyclic patterns~\cite{kroeber1919principle}. An example is the hemline index, which is a well-known hypothesis that describes the cyclic pattern in which skirt lengths decrease when economic conditions improve~\cite{Dhanorkar2015}. This pattern was determined based on observations of skirt lengths in the 1920s and 30s.
Furthermore, we need data on how and what consumers wear to express their fashion styles. However, fashion photographs on the internet are one of two types: one displays clothes that consumers themselves choose to wear, and the other consists of photographs taken by professional photographers to promote a clothing line. As previous studies built databases by collecting fashion images from the internet, existing databases do not reflect only the fashion styles chosen by consumers. This is the third limitation of existing fashion data archives.
Taken together, no existing database has everyday fashion images that span over a long period with both timestamp and location information. In this study, we define everyday fashion trends as trends of fashion styles that consumers adopt in their daily lives. Everyday fashion exists on the streets~\cite{kawamura2006japanese,Rocamora2008}, changes over time, and trends and cyclic patterns can be observed over extended periods. Hence, we create a new fashion database, CAT STREET, is an attempt to solve these limitations and analyze everyday fashion trends.
\begin{table*}[tbp]
\caption{Popular fashion databases used in computer science.}
\begin{center}
\begin{tabular}{ccccc}
\hline
Database name & Num. of images &\begin{tabular}{c} Geographical\\ information\end{tabular}&Time stamp &\begin{tabular}{c}Fashion\\style tag\end{tabular}\\
\hline \hline
Fashionista~\cite{yamaguchi2012parsing} &158,235 & & &\\
Hipster Wars~\cite{Kiapour2014} &1,893 & & &\checkmark\\
DeepFashion~\cite{liu2016deepfashion} &800,000 & & &\checkmark\\
Fashion 144k~\cite{simo2015neuroaesthetics} &144,169 &City unit & &\checkmark\\
FashionStyle14~\cite{takagi2017makes} &13,126 & & &\checkmark\\
When Was That Made?~\cite{vittayakorn2017made}&100,000 & &\begin{tabular}{c}1900--2009\\Decade unit\end{tabular}&\\
Fashion Culture Database~\cite{abe2018fashion} &76,532,219 &City unit &\begin{tabular}{c}2000--2015\\Date unit\end{tabular} &\\
CAT STREET (Our Database) &14,688 &Street unit &\begin{tabular}{c}1970--2017\\Date unit\end{tabular} &\\
\hline
\end{tabular}
\end{center}
\label{table:table1}
\end{table*}
\section{CAT STREET: Chronicle Archive of Tokyo Street-fashion}
We created CAT STREET via the following steps. We collected street-fashion photographs once or twice a month in fashion-conscious streets such as Harajuku and Shibuya in Tokyo from 1980 to date. In addition, we used fashion photographs from a third-party organization taken in the 1970s at monthly intervals in the same fashion-conscious streets. Next, by using images from the two data sources, we constructed a primary image database with timestamps from 1970 to 2017 for each image. The photographs from the third-party organization did not have location information; hence, we could only annotate the fashion images taken since 1980 with street tags.
Fashion styles conventionally differ between men and women. To focus on women's fashion trends in CAT STREET, two researchers manually categorized the images by the subjects' gender, reciprocally validating one another's categorizations, and we selected only images of women from the primary image database. Some images from the 1970s were in monochrome; therefore, we gray-scaled all images to align the color tone across all images over the entire period.
Street-fashion photographs generally contain a large amount of noise, which hinders the precise detection of combinations of clothes worn by the subjects of photographs. To remove the noise, we performed image pre-processing as the final step in the following manner. We identified human bodies in the photographs using OpenPose~\cite{Zhe2019openpose}, an machine learning algorithm for object recognition, and removed as much of the background image as possible. Subsequently, we trimmed the subject's head to focus on clothing items based on the head position, which was detected using OpenPose. Fig. \ref{fig:fig6} shows an overview of the data contained in CAT STREET. The total number of images in the database was 14,688.
At the end of the database creation process, we checked whether CAT STREET met the requirements for a database to capture everyday fashion trends. First, CAT STREET comprises photographs captured on fashion-conscious streets in Tokyo. It reflects the fashion styles women wore in their real lives and does not include commercial fashion images produced for business purposes. Second, CAT STREET has necessary and sufficient annotations to track fashion trends: monthly timestamps from 1970 to 2017 and street-level location tags. Images in the 1970s do not have street tags; however, they were taken in the same streets as the photographs taken with street tags since 1980. Hence, we could use all the images in CAT STREET to analyze the overall trends of everyday fashion in fashion-conscious streets as a representative case in Japan.
\begin{figure*}[tbp]
\begin{center}
\includegraphics[width=0.93\linewidth]{6.eps}
\end{center}
\caption{Data overview of CAT STREET. Harajuku and Shibuya are the most famous streets in Japan and Others include images taken in other fashion-conscious streets such as Ginza and Daikanyama.}
\label{fig:fig6}
\end{figure*}
\section{Retrieving Everyday Fashion Trends}
In this section, we estimated a fashion style clustering model to identify the fashion styles adopted in fashion images. Then, we applied the fashion clustering model to CAT STREET and retrieved everyday fashion trends indicating the extent to which people adopted the fashion styles in each year. Finally, we verified the robustness of clustering model's result.
\subsection{Fashion Style Clustering Model}
To build a fashion style clustering model, we selected FashionStyle14~\cite{takagi2017makes} as the training dataset. It consists of fourteen fashion style classes and their tags, as shown in the first row of Fig. \ref{fig:fig7}. Each class consists of approximately 1,000 images, and the database consists of a total of 13,126 images. The fashion styles of FashionStyle14 were selected by an expert as being representative of modern fashion trends in 2017. By applying the fashion clustering model to CAT STREET, we measured the share of each modern style in each year and regarded it as the
style's trend.
We trained four deep learning network structures as options for our fashion clustering model: InceptionResNetV2~\cite{szegedy2017inception}, Xception~\cite{chollet2017xception}, ResNet50~\cite{he2016deep}, and VGG19~\cite{Simonyan2015}. We set weights trained on ImageNet~\cite{russakovsky2015imagenet} as the initial weights and fine-tuned them on FashionStyle14, which we gray-scaled to align the color tone of CAT STREET, using the stochastic gradient descent algorithm at a learning rate of of $10^{4}$. For fine-tuning, we applied k-fold cross-validation with k set as five.
The F1-scores are presented in Table \ref{table:table2}. InceptionResNetV2 yielded the highest F1-scores among the deep learning network structures. Its accuracy was 0.787, which is higher than the benchmark accuracy of 0.72 established by ResNet50 trained on FashionStyle14 in the study by Takagi et al.~\cite{takagi2017makes}. Therefore, we concluded that InceptionResNetV2 could classify gray-scaled color images to fashion styles and adopted the deep learning network structure InceptionResNetV2 as the fashion style clustering model in this study. We applied this fashion style clustering model to the images in CAT STREET, and Fig. \ref{fig:fig7} shows sample images classified into each fashion style.
\begin{table*}[bp]
\caption{F1-scores of fine-tuned network architectures. The highest scores are underlined and in boldface.}
\begin{center}
\begin{tabular}{ccccc}
\hline
&InceptionResNetV2 &Xception &ResNet50&VGG19\\
\hline
Macro Avg. &\bf\underline{0.787}&0.782 &0.747 &0.690\\
Weighted Avg. &\bf\underline{0.786}&0.781 &0.747 &0.689\\
\hline
\end{tabular}
\end{center}
\label{table:table2}
\end{table*}
\begin{figure*}[tbp]
\begin{center}
\includegraphics[width=0.93\linewidth]{7.eps}
\end{center}
\caption{Overview of FashionStyle14~\cite{takagi2017makes} and sample images in CAT STREET classified into different fashion styles using the fashion style clustering model. `Conserv.' is an abbreviation for Conservative. The A\&F is originally labeled as Ethnic~\cite{takagi2017makes}. We changed this original label to 'A\&F'
because this category contains multiple fashion styles such as Asian, African, south-American, and Folklore styles.}
\label{fig:fig7}
\end{figure*}
\subsection{Verification}
The fashion style clustering model consisted of five models as we performed five-fold cross-validations when the deep learning network structure was trained, and each model estimated the style prevalence share for each image. To verify the clustering model's robustness in terms of reproducing style shares, we evaluated the time-series correlations among the five models. Most fashion styles had high time-series correlation coefficients of over 0.8, and the unbiased standard errors were small. Some fashion styles, such as those designated Dressy and Feminine styles, exhibited low correlations because these styles originally had low style shares. These results indicate that our fashion style clustering model is a robust instrument for reproducing the time-series patterns of style shares. There were no images for 1997 and 2009; we replaced the corresponding zeros with the averages of the adjacent values for the analysis in the next section.
\section{Analyzing Fashion Trend Patterns}
There are two representative fashion-conscious streets in Tokyo: Harajuku and Shibuya. Geographically, Harajuku and Shibuya are very close and only a single transit station away from each other; however, they have different cultures. In particular, the everyday fashion trends in these streets are famously compared to each other. Some qualitative studies pointed out triggers that form the cultural and fashion modes in each street using an observational method~\cite{Hasegawa2015,kawamura2006japanese}. However, they simply reported the triggers with respect to the street and style and did not compare the functioning of these triggers on the mode formations in everyday fashion trends from a macro perspective. This section sheds light on this research gap and interprets how the fashion trends emerged with non-fashion data sources.
First, we compared two everyday fashion trends in fashion-conscious streets with CAT STREET to investigate how fashion styles 'boom' or suddenly increase in prevalence, and classified the resultant patterns. Fourteen styles were classified into two groups: a group comprising styles that emerged on both streets simultaneously and a group with differing timings for trends observed in the streets. We selected two fashion styles from each group as examples, including A\&F and Retro from the first group and Fairy and Gal from the second group; Figs. \ref{fig:fig11}(a), \ref{fig:fig12}(a), \ref{fig:fig13}(a), and \ref{fig:fig14}(a) show their average style shares, respectively.
\begin{figure*}[bp]
\begin{center}
\includegraphics[width=0.93\linewidth]{11.eps}
\end{center}
\caption{A\&F style's simultaneous emergence in both streets. (a) Average A\&F style share. (b) Number of articles about the A\&F style. Four images taken in Harajuku (i, ii) and Shibuya (iii, iv) in the late 1990s.}
\label{fig:fig11}
\end{figure*}
\begin{figure*}[tbp]
\begin{center}
\includegraphics[width=0.93\linewidth]{12.eps}
\end{center}
\caption{Retro style's simultaneous emergence in both streets. (a) Average Retro style share. (b) Number of articles about the 80s look. The three images on the left (i--iii) are examples of the Retro style taken in the early 1980s, and the three images on the right (iv--vi) are examples of the Retro style that came into fashion in the late 1990s.}
\label{fig:fig12}
\end{figure*}
\begin{figure*}[tbp]
\begin{center}
\includegraphics[width=0.93\linewidth]{13.eps}
\end{center}
\caption{Gal style's trends in the streets showing different modes. (a) Average Gal style share. (b) Number of articles about the Gal style. The four images on the left were taken in Harajuku (i, ii) and Shibuya (iii, iv) in 1995--1996, and the four images on the right were taken in Harajuku (v, vi) and Shibuya (vii, viii) in the late 2000s.}
\label{fig:fig13}
\end{figure*}
\begin{figure*}[tbp]
\begin{center}
\includegraphics[width=0.93\linewidth]{14.eps}
\end{center}
\caption{Fairy style's trends in the streets showing different modes. (a) Average Fairy style share. The two images (i and ii) were taken in Harajuku in the late 2000s.}
\label{fig:fig14}
\end{figure*}
\begin{figure*}[t]
\begin{center}
\includegraphics[width=0.93\linewidth]{15.eps}
\end{center}
\caption{(a) Gal style's share and 'magazine' trends for its icons. (b) Fairy style's share and 'magazine' trends for its icon.}
\label{fig:fig15-6}
\end{figure*}
\begin{figure*}[t]
\begin{center}
\includegraphics[width=0.45\linewidth]{17.eps}
\end{center}
\caption{Gal and Fairy style shares and 'magazine' trends for their icons.}
\label{fig:fig17}
\end{figure*}
As a non-fashion data source, we focused our attention on magazines because fashion brands have built good partnerships with the magazine industry for a long time. Fashion brands regard magazines as essential media to build a bridge between themselves and consumers, and magazines play a role as a dispatcher of fashion in the market. We quantified the 'magazine' trends to capture how magazines or media sent out information to consumers. For this purpose, we used the digital magazine archive of Oya Soichi Library~\cite{OYA}. Oya Soichi Library has archived Japanese magazines since the late 1980s and built the digital archive. The digital archive houses about 1,500 magazines, and one can search for article headlines of about 4.5 million articles.
The archive covers a range of age groups and mass-circulation magazines that people can easily acquire at small bookstores, convenience stores, and kiosks. By searching for headlines including fashion style words and specific topics from articles with tags related to fashion, we could quantify the 'magazine' trends that indicate how many articles dealt with the styles and specific topics to spread the information to a mass audience (Figs. \ref{fig:fig11}(b), \ref{fig:fig12}(b), \ref{fig:fig13}(b), \ref{fig:fig15-6}, and \ref{fig:fig17}). The 'magazine' trends labeled as Articles in the figures were normalized to the range of 0 to 1 for comparison.
Finally, to infer why and how the everyday fashion trends emerged, we compared the fashion style shares extracted from CAT STREET to the 'magazine' trends in the Oya Soichi Library database in chronological order.
\subsection{Simultaneous Emergence of Fashion Styles}
\label{Simultaneous Emergence of Fashion Styles}
We selected two styles, A\&F and Retro, as examples in the group of styles that simultaneously emerged on both streets. The A\&F style is inspired by native costumes~\cite{bunka1993}. Fig. \ref{fig:fig11}(a) shows the A\&F style's upward trend in the late 1990s in Harajuku. Simultaneously, the A\&F style gradually became accepted in Shibuya in the late 1990s and reached the same share level in the mid-2000s. The Retro style, another example in the first group, is an abbreviation of retrospective style~\cite{bunka1993,Yoshimura2019}. According to this style's definition, an overall downward trend is plausible in both streets, as shown in Fig. \ref{fig:fig12}(a). Fashion revival is one of the relevant fashion trend phenomena of the Retro style, and there are a wide variety of substyles representing fashion revivals under the Retro style, such as the 60s look, 70s look, and 80s look. In Fig. \ref{fig:fig12}(a), slight peaks can be observed in the early 1980s and late 1990s, suggesting that the 80s look, which was in vogue in the early 1980s, was revived in the late 1990s.
To quantify how magazines spread information about these styles, we plotted the 'magazine' trends by searching for articles with headlines that included terms related to the style names. Figs. \ref{fig:fig11}(b) and \ref{fig:fig12}(b) indicate two relationship patterns between style shares and 'magazine' trends; Fig. \ref{fig:fig11}(b) shows a synchronization phenomenon between style shares and magazine trends, while Fig. \ref{fig:fig12}(b) shows that magazine trends follow share trends.
We interpreted the first pattern as 'mixed'; the shape of 'magazine' trends roughly matches that of style share trends, and this pattern is observed in the A\&F style. The 'mixed' patterns in the trends suggest that they include two types of phenomena between consumers and magazines. One is that the articles create new trends, spread them, and lead consumers to follow them. The other is that consumers create new style trends by themselves, and articles catch up on the new trends. These two types of phenomena could occur simultaneously or alternately. On the other hand, the second pattern is a 'follow-up'; the articles dealt with the modes that were already in vogue on the streets and contributed to keeping their momentum for some time by spreading the information, and this pattern is observed in the Retro style during the 2000s.
In this analysis, we found two relationship patterns between style shares and 'magazine' trends: 'mixed' and 'follow-up' patterns. We also inferred what types of interactions occur between consumers and magazines in each pattern. However, we could not perform an in-depth analysis on how to distinguish between the two types of consumer-magazine interactions in the case of the 'mixed' pattern because 'magazine' trends generated by searching for article headlines including terms related to the style names did not always reflect the contents of articles accurately. We also found it difficult to uncover why the article peak in the Retro style during the late 2000s was seemingly irrelevant to the Retro style share. To approach these unsolved research questions as future work, we must perform text mining on article contents and headlines, and categorize the articles into the creative type, which creates and spreads new trends, or the reporting type, which refers to presently existing new consumer trends after they develop; e.g. 'ten trends coming in autumn, checker-board pattern, check pattern, military, big tops, shoes \& boots, foot coordination, fur, A\&F vs. Nordic, beautiful romper, best of the season' (translated from Japanese by the authors) is categorized as a creative-type article title, whereas 'A\&F style is a hot topic this spring. We will introduce you to some of the Japanese items that are getting a lot of attention. Crepe, goldfish prints, and more...' (translated from Japanese by the authors) is categorized as a reporting-type article title. By measuring the number of articles about the fashion styles and identifying the contents of articles to reflect the types of consumer-magazine relationships, we expect to decompose relationship types in the 'mixed' pattern, interpret seemingly irrelevant relationships, and clarify the role of each relationship type in the trend formation process.
\subsection{Emergence of Fashion Styles at Different Types in the Streets}
\label{Emergence of Fashion Styles at Different Types in the Streets}
For the case where fashion trends are observed at different times in the two streets, we focused on two styles: the Gal style, which can be characterized as an exaggeration of the American teenage party style, and the Fairy style, which involves the fashion coordination of frilly dresses reminiscent of fairies~\cite{Yoshimura2019}.
Figs. \ref{fig:fig13} and \ref{fig:fig14} show how the acceptances of these two styles changed in each street. The Gal style came into fashion around 1995 simultaneously in Harajuku and Shibuya. The style remained in Shibuya, whereas it lost its popularity quickly in Harajuku but re-emerged in the late 2000s. The Fairy style emerged in the late 2000s in Harajuku only and lost its momentum in the early 2010s. As with the first case, we searched 'magazine' trends for these style names and compared them to the style shares. For the Gal style, we found that the style and 'magazine' trends correlated with each other in the mid-80s and the mid-90s. However, magazines gradually lost interest in the style after the mid-00s, and the relationship between the style and 'magazine' trend disappeared accordingly. For the Fairy style, on the other hand, no article in the digital magazine archive included the style name in the headlines. This is because 'Fairy' is a kind of jargon among people interested in the style and hence not used in magazines.
What accounts for this difference between the first and the second case? We assumed that the critical factor determining people's choice in fashion, i.e. 'why people adopt a given style,' is for some purpose; some styles have specific features, such as colors, silhouettes, and patterns, that people consume as a fashion or which represent a certain social identity and have some characteristic features. The Gal and Fairy styles are in the latter group and represent a 'way of life' for some people~\cite{Hasegawa2015,Kyary2011}.
Icons representing 'way of life,' such as celebrities, have substantial power to influence people to behave in a certain manner; for instance, celebrities can influence people to purchase products that they use or promote. Choosing or adopting fashion styles is no exception, and some articles identified fashion icons for the Gal and the Fairy styles~\cite{Hasegawa2015,Shinmura2012,Yoshimura2019}. We attempted to explain why the style trends in the streets showed different modes from the perspective of fashion icons and the media type that the icons utilized.
Figs. \ref{fig:fig15-6} and \ref{fig:fig17} show the relationship between the style shares and 'magazine' trends for style icons. Previous works introduced the following Gal style icons: Namie Amuro for the mid-90s, SHIBUYA 109 for the late-00s, Ayumi Hamasaki for the early-00s, and Kumi Koda and Tsubasa Masuwaka for the late-00s~\cite{Hasegawa2015,Shinmura2012}.
The first icon who boosted the Gal style was Namie Amuro, who debuted nationwide as a pop singer in 1992. Girls yearned to imitate her fashion style, and magazines focused heavily on her as a style icon in the mid-90s. Simultaneously, as the 'magazine' trend for the style shows in Fig. \ref{fig:fig15-6} (a), many tabloid magazines were interested in this phenomenon and spread it as a new 'way of life' among youth consumers. The first icon boosted the style in both streets because she appeared nation-wide. However, the second icon that sustained the Gal style's upward trend is regionally specific: an iconic fashion mall named SHIBUYA 109 (pronounced Ichi-maru-kyū) in Shibuya. Many tenants in the mall sold Gal style products, which is why the Gal style is also known as 'maru-kyū fashion'~\cite{Yoshimura2019}. This street-specific image created by the second icon might have influenced the third and fourth icons, Ayumi Hamasaki and Kumi Koda, who were nationwide pop singers in the early-00s and the mid-00s, because the style share increased during that time only in Shibuya.
The icons prompted the last increase in the prevalence share of the Gal style in Shibuya, and the spike in the Fairy style's share in Harajuku exhibited the same characteristics. Both Tsubasa Masuwaka for the Gal style in Shibuya and 'Kyary Pamyu Pamyu' for the Fairy style in Harajuku were active as exclusive reader models in street-based magazines at the beginning, and they created the booms in each street~\cite{Shinmura2012}. However, the street-based magazines they belonged to were not included in the digital magazine archive in Oya Soichi Library; hence, the 'magazine' trends for their names missed their activities as exclusive reader models when the style shares showed an upward trend (Figs. \ref{fig:fig15-6} (b) and \ref{fig:fig17}). Around 2010, they debuted as nationwide pop stars and frequently appeared in mass-circulation magazines and on television. Additionally, Tsubasa Masuwaka proposed a new style named 'Shibu-Hara' fashion, a combination of the styles in both Shibuya and Harajuku, and also referred to interactions with 'Kyary Pamyu Pamyu' on her social networking account. The Gal style's second peak in Harajuku and the Fairy style's spike in Shibuya in the early 2010s aligned with these icons' nationwide activities; this indicates the styles' cross-interactions driven by the icons (Figs. \ref{fig:fig15-6} (b) and \ref{fig:fig17}).
\subsection{Discussion}
The findings in Section \ref{Simultaneous Emergence of Fashion Styles} prompted a new research question of determining the importance of the different roles that media play in fashion trends: the role of the 'mixed' pattern for creating and reporting new trends and the 'follow-up' effect to keep their momentum. To theorize how fashion trends are generated in more detail, our findings indicate that quantifying the trends from multiple digital archives is essential to test the research questions about the effect of the media's role, which has not been discussed thoroughly.
In previous studies, researchers analyzed fashion style trends at a street level independently. However, to analyze the recent, more complicated trend patterns, the findings in Section \ref{Emergence of Fashion Styles at Different Types in the Streets} suggests that it would be thought-provoking to focus on what the icons represent for people's social identities, rather than the style itself, and how they lead the trends. An example is the analysis of the types of media that icons use and the reach of that media. The viewpoints gained from our findings can be useful in tackling unsolved research questions such as how styles interact and how new styles are generated from this process. If we can determine people's social identities from the aims of their adopted fashions using the digital archives, researchers can retrieve beneficial information to delve into the research questions and expand fashion trend theories.
\section{Conclusions}
In this study, we reviewed the issues in the existing digital fashion archives for the analysis of everyday fashion trends. As one of the solutions to these issues, we built CAT STREET, which comprises fashion images illustrating what people wore in their daily lives in the period 1970--2017, along with street-level geographical information. We demonstrated that machine learning retrieved how fashion styles emerged on the street using CAT STREET and the magazine archives helped us interpret these phenomena. These empirical analyses showed our approach's potential to find new perspectives to promote the understanding of our societies and cultures through fashion embedded in the daily lives of consumers.
Our work is not without limitations. We used the fashion style categories of FashionStyle14~\cite{takagi2017makes}, which was defined by fashion experts and is considered to represent modern fashion. However, the definition does not cover all contemporary fashion styles and their substyles in a mutually exclusive and collectively exhaustive manner. Defining fashion styles is a complicated task because some fashion styles emerge from consumers, and others are defined by suppliers. Hence, we must further refine the definition of fashion styles to capture everyday fashion trends more accurately. Furthermore, prior to building CAT STREET, only printed photos were available for the period 1970--2009. Consequently, the numbers of images for these decades are not equally distributed because only those images from printed photos that have already undergone digitization are currently present in the database. The remainder of the printed photos will be digitized and their corresponding images added to the database in future work.
\bibliographystyle{splncs04}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 9,546 |
const config = {
language: 'zh-cn',
uiColor: '#9AB8F3',
bodyClass: 'main-content'
}
config.toolbarGroups = [
{name: 'clipboard', groups: ['clipboard', 'undo']},
{name: 'editing', groups: ['find', 'selection', 'spellchecker']},
{name: 'links'},
{name: 'insert'},
{name: 'forms'},
{name: 'tools'},
{name: 'document', groups: ['mode', 'document', 'doctools']},
{name: 'others'},
'/',
{name: 'basicstyles', groups: ['basicstyles', 'cleanup']},
{name: 'paragraph', groups: ['list', 'indent', 'blocks', 'align', 'bidi']},
{name: 'styles'},
{name: 'colors'},
{name: 'about'}
]
// Remove some buttons provided by the standard plugins, which are
// not needed in the Standard(s) toolbar.
config.removeButtons = 'Underline,Subscript,Superscript'
// Set the most common block elements.
config.format_tags = 'p;h1;h2;h3;pre'
// Simplify the dialog windows.
config.removeDialogTabs = 'image:advanced;link:advanced'
export {config as default}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 2,484 |
Published on December 2nd, 2013 | by Mareike TRULL | Credit: European Parliament
European Laws in the Making
The European Union (EU). Four different institutions, many different exertions of influence and one law-making process.
Have you ever wondered how a European law is passed? And what are the tasks and responsibilities of the European Council, the European Parliament, the European Commission and the Council of the European Union during this process?
The European Council sets the general political direction. However, it does not have the power to pass laws. The European Council's meetings are essentially summits, where Heads of State or Government of every EU country meet to decide on broad political priorities and major initiatives.
The actual law-making procedure involves the other three main institutions: the European Parliament, the European Commission and, finally, the Council of the EU. All together, these three bodies produce European policies and laws, through the "Ordinary Legislative Procedure" (ex "co-decision").
In most cases, the process starts at the Commission. The European Commission is like the European government. It consists of 28 Commissioners, comparable to national ministers, who take care of specific policy areas. Mr Barroso is the current President of the Commission, comparable to a Head of State, and together with his Commissioners, he provides the Commission's political leadership during its 5-year term.
So the Commission proposes new laws, and then the Parliament and the Council of the European Union adopt them.
The European citizens directly elect the European Parliament every 5 years. As you know, your next opportunity to vote will be in spring 2014!
We, the people of Europe, can then decide which candidates will become Members of the European Parliament, "MEPs" as we call them in Brussels' jargon.
As mentioned before, their main role is to debate and to pass European laws, in collaboration with the Council of the EU. They also scrutinise other EU institutions, most importantly the Commission; and together with the Council of the EU, the Parliament discusses and adopts the EU budget.
Our institutions showed off their power in the negotiations concerning the EU's next long-term budget for 2014–2020. Only after months of tough negotiations by the leaders of the Parliament, the Council and the Commission, the Parliament agreed to pass the final vote in the plenary this autumn.
Last but not least, the Council of the European Union also plays an important role in the law-making procedure. Consisting of the Ministers of the national governments, it gives them the chance to defend their national interest at EU level.
As said earlier, together with the European Parliament, the Council passes EU laws and approves the annual EU budget. It also coordinates the broad economic policies of EU member countries, signs agreements between the EU and other countries, develops the EU's foreign and defence policies and, finally, coordinates the cooperation between courts and police forces of member countries.
Once a law is passed by the Parliament and the Council of the EU, the Commission and the Member States implement it. It is the job of the Commission to ensure that the laws are properly implemented and respected.
Did you know?: Up to 39% of national legislation is influenced by EU laws!
Read more: http://www.europarl.europa.eu/news/en/news-room/content/20110429FCS18370/html/The-battle-for-the-EU%27s-long-term-budget
Share the post "European Laws in the Making"
Tags: European Law, European Law in the Making, European Union
Migrants Day! 18th December. 5 short stories that will change your perception about migrants →
Keep the EU Moving →
The Council is the EU institution where the Member States' government representatives sit, i.e. the ministers of each Member State with responsibility for a given area. It is one of the two branches of the legislative branch of the Union, the other being the Parliament. Its Presidency rotates every six months among the governments of EU member states.
Ordinary legislative procedure (Eligible expenditure)
The money that is allowed to be used to meet objectives set out in a political party's programme: meetings and conferences, publications, studies, administrative, personnel or travel costs and any campaign costs connected to the electoral campaign.
Ordinary legislative procedure (Co-decision procedure)
When both the European Parliament and the Council of the EU have to approve the European Commission's proposals for them to become EU law. With the Lisbon Treaty (2009), it has become the standard way of decision-making, giving the European Parliament much more power! | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 5,351 |
Enjoy the top 5 famous quotes, sayings and quotations by Billy Packer.
It's about time for Lindell White to get back on the floor, expose himself, and get the ball in a scoring position.
Want to see more pictures of Billy Packer quotes? Click on image of Billy Packer quotes to view full size. | {
"redpajama_set_name": "RedPajamaC4"
} | 2,287 |
Q: Magento 2.3.5 : Display product count with Page Main Title I want to display product count with page main title for category page.
How I can display that?
Thanks.
A: Using Observer over Event layout_generate_blocks_after you can show product count at title
Define an observer at events.xml
events.xml:
<?xml version="1.0" ?>
<config xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance"
xsi:noNamespaceSchemaLocation="urn:magento:framework:Event/etc/events.xsd">
<event name="layout_generate_blocks_after">
<observer instance="StackExchange\SampleModule\Observer\Frontend\Layout\GenerateBlocksAfter"
name="stackexchange_layout_generateblocksafter_layout_generate_blocks_after"/>
</event>
</config>
Observer Class:
<?php
declare(strict_types=1);
namespace StackExchange\SampleModule\Observer\Frontend\Layout;
class GenerateBlocksAfter implements \Magento\Framework\Event\ObserverInterface
{
/**
* @var \Magento\Framework\View\Page\Config
*/
private $config;
public function __construct(
\Magento\Framework\View\Page\Config $config
) {
$this->config = $config;
}
/**
* Execute observer
*
* @param \Magento\Framework\Event\Observer $observer
* @return void
*/
public function execute(
\Magento\Framework\Event\Observer $observer
) {
$fullActionName = $observer->getEvent()->getData('full_action_name');
/**
* @var $layout \Magento\Framework\View\LayoutInterface
*/
$layout = $observer->getEvent()->getData('layout');
if ($fullActionName === 'catalog_category_view') {
$listBlock = $layout->getBlock('category.products.list');
$categoryViewBlock = $layout->getBlock('category.products');
$count = 0;
if ($listBlock) {
/**
* @var $listBlock \Magento\Catalog\Block\Product\ListProduct
*/
$productCollection = $listBlock->getLoadedProductCollection();
if ($productCollection && $productCollection->count() >0) {
/**
* use this code when you want the same count as layer navigation
*
*/
$count = $productCollection->count();
}
}
if ($categoryViewBlock) {
/**
* @var $categoryViewBlock \Magento\Catalog\Block\Category\View
*/
$currentCategory = $categoryViewBlock->getCurrentCategory();
if ($currentCategory) {
/**
* use this code when you want show count before applying filter
*/
$count = $currentCategory->getProductCollection()->count();
}
}
$title = $this->config->getTitle()->get();
$this->config->getTitle()->set(
$title
. _(' Product Count: ' . (int) $count)
);
}
}
}
Use $count = $productCollection->count(); when want same result at Laynavigation .
And use $count = $currentCategory->getProductCollection()->count(); when you always show category product count even applying filter.
The folder structure of the module
Update
If you want to update 'page.main.title' then use below code:
<?php
declare(strict_types=1);
namespace StackExchange\SampleModule\Observer\Frontend\Layout;
class GenerateBlocksAfter implements \Magento\Framework\Event\ObserverInterface
{
/**
* @var \Magento\Framework\View\Page\Config
*/
private $config;
public function __construct(
\Magento\Framework\View\Page\Config $config
) {
$this->config = $config;
}
/**
* Execute observer
*
* @param \Magento\Framework\Event\Observer $observer
* @return void
*/
public function execute(
\Magento\Framework\Event\Observer $observer
) {
$fullActionName = $observer->getEvent()->getData('full_action_name');
/**
* @var $layout \Magento\Framework\View\LayoutInterface
*/
$layout = $observer->getEvent()->getData('layout');
if ($fullActionName === 'catalog_category_view') {
$listBlock = $layout->getBlock('category.products.list');
$categoryViewBlock = $layout->getBlock('category.products');
$count = 0;
if ($listBlock) {
/**
* @var $listBlock \Magento\Catalog\Block\Product\ListProduct
*/
$productCollection = $listBlock->getLoadedProductCollection();
if ($productCollection && $productCollection->count() >0) {
/**
* use this code when you want same count as layer navigation
*
*/
$count = $productCollection->count();
}
}
if ($categoryViewBlock) {
/**
* @var $categoryViewBlock \Magento\Catalog\Block\Category\View
*/
$currentCategory = $categoryViewBlock->getCurrentCategory();
if ($currentCategory) {
/**
* use this code when you want show count before applying filter
*/
$count = $currentCategory->getProductCollection()->count();
}
}
/**
* @var $pageMainTitle \Magento\Theme\Block\Html\Title
*/
$pageMainTitle = $layout->getBlock('page.main.title');
if ($pageMainTitle) {
$oldTitle =$pageMainTitle->getPageTitle();
$pageMainTitle->setPageTitle(
$oldTitle
. _(' Product Count: ' . (int) $count)
);
}
}
}
}
A: Let me share a basic idea for this task.
You can achieve this using the below code in your list page template file. Please use proper dependency instead of objectmanager
<?php
$objectManager = \Magento\Framework\App\ObjectManager::getInstance();
$category = $objectManager->get('Magento\Framework\Registry')->registry('current_category'); echo $category->getProductCollection()->count();
A: We can show the total number of product counts on the PLP page without using events. All we need to do is override the catalog_category_view.xml file and make the change accordingly. Below I'm posting the code which is worked for me.
catalog_category_view.xml
<?xml version="1.0"?>
<!--
/**
* Copyright © 2018 Porto. All rights reserved.
*/
-->
<page xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" layout="2columns-left" xsi:noNamespaceSchemaLocation="urn:magento:framework:View/Layout/etc/page_configuration.xsd">
<body>
<move element="page.main.title" destination="category.title"/>
<referenceContainer name="content">
<container name="category.title" htmlTag="div" htmlClass="category-title">
<block class="Magento\Catalog\Block\Category\View" name="category.products.count"
as="product_count" template="Magento_Catalog::product/category_title.phtml" />
</container>
</referenceContainer>
</body>
</page>
category_title.phtml
<span class="category-product-count">
<?= '('.$this->getCurrentCategory()->getProductCollection()->count().' Results)'; ?>
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 8,805 |
\section{Introduction}
Let $G$ be a simply connected Chevalley group over a non-archimedean local field, and let $U^-$ be opposite group to the unipotent radical of a Borel subgroup $B$. There are important representation-theoretic quantities that can be expressed as an integral over $U^-$, for example intertwining operators and Whittaker functions. This paper focuses on the study and evaluation of such integrals.
Specifically, for $\chi$ a character of $B$ trivial on the maximal compact subgroup $K$ of $G$, define the function $\phi_K\map{G}{\C}$ by
$$
\phi_K(bk)=(\delta^{1/2}\chi)(b)
$$ for $b\in B$ and $k\in K$, where $\delta^{1/2}$ indicates the modular quasicharacter of $B$.
Then $\phi_K$ is a $K$-invariant vector in the principal series representation $V_\chi=\Ind_B^G \delta^{1/2}\chi$. A formula for the (unique up to scaling) intertwining operator $T$ between $V_\chi$ and $V_{\chi^{w_0}}$ is given by
$$
Tf(g)=\int_{U^-} f(uw_0g)du.
$$
Here $w_0$ is the long element of the Weyl group, which acts on the set of principal series representations. It is most important to take $f=\phi_K$ and $g=1$ in the above integral.
If we let $\psi$ be a character of $U^-$, then the Whittaker function for the representation $V_\chi$ can be calculated by the integral
$$
W(g)=\int_{U^-} \phi_K(ug)\psi(u)du.
$$
The above integrals have direct analogues when one wishes to study the representation theory of central extensions $\G$ of $G$ by a finite cyclic group $\mu_n$. We shall refer to such groups as metaplectic groups, as opposed to the more restrictive notion of a metaplectic group as referring specifically to the double cover of the symplectic group. We shall review what we require from the representation theory of such groups in Section \ref{reptheory}. Despite the restriction to the reductive case in the beginning of the introduction, our work will always hold more generally for the case of a metaplectic group, and we shall work in this generality throughout.
To each reduced decomposition $w_0=s_{i_1}\ldots s_{i_N}$ of the long element of the Weyl group into a product of simple reflections, we produce a decomposition of $U^-$ into cells $C_\m$ indexed by $N$-tuples $\m=(m_1\,\ldots,m_N)$ of natural numbers, by producing an explicit version of the Iwasawa decomposition. The concept of realising a crystal combinatorially as a set of subvarieties of a unipotent radical dates back to Lusztig \cite{lusztig96}. The decomposition we consider turns out to be equivalent to that of \cite[Proposition 4.1]{kamnitzer}. However the approach taken in this paper is independent of that of Kamnitzer. The cells $C_\m$ have the property that the function $\phi_K$ is constant on each cell, so writing
$$
\int_{U^-}=\sum_{\m}\int_{C_\m}
$$
yields a combinatorial sum for these integrals we are studying, which becomes amenable to explicit calculation. In this manner, we obtain a method allowing us to evaluate our target family of integrals as a combinatorial sum over a crystal.
Indeed, there is a natural bijection between our collection of cells and the elements of the canonical basis $B(-\infty)$ of $U_q(\mathfrak{n}^+)$, the positive part of the quantised universal enveloping algebra. This connection with the combinatorics of crystals is studied in Section \ref{xtal} in the context of a positive characteristic local field. In particular, Theorem \ref{maincrystalthm} provides an explicit identification between the parametrisation of our cells and Lusztig's parametrisation of the canonical basis \cite{lusztigbook}. At the same time, we are able to relate our cell decomposition with previously constructed geometric models of crystals, in particular the realisation of the crystal in terms of Mirkovic-Vilonen cycles in the affine Grassmannian as in \cite{bravermangaitsgory}.
In Section \ref{initial}, we are able to evaluate the integral for the intertwining operator in full generality, proving a metaplectic version of the Gindikin-Karpelevich Formula; this is the content of Theorem \ref{gk}, itself a generalisation of \cite[Proposition I.2.4]{kp}. In type $A$, this presentation of the Gindikin-Karpelevich formula as a sum over a crystal has been independently obtained by Bump and Nakasuji \cite{bumpnakasuji} via a different method.
In the case of Whittaker functions, to achieve explicit results, we restrict ourselves to working in type A with a particular choice of long word decomposition. We are then able to compute in Theorem \ref{main} the metaplectic Whittaker function as a weighted sum over a crystal. We note that this weighted sum agrees exactly with the prime-power supported coefficients of a Weyl group Multiple Dirichlet Series, which we will refer to as the `$p$-part.' These multiple Dirichlet series are certain Dirichlet series in several complex variables, satisfying a set of functional equations indexed by a Weyl group.
Weyl group multiple Dirichlet series were first introduced in \cite{fiveauthor}. These are global objects built from local components, namely their $p$-parts for each prime $p$. A combinatorial description of the $p$-part of such a series as a weighted sum over Gelfand-Tsetlin patterns is given in \cite{wgmdsxtal}. We give a full description of these coefficients at the beginning of Section \ref{tok}. In fact this particular description is given an alternative description in terms of paths in a crystal graph in \cite{wgmdsxtal}, which gives a natural interpretation of the integers $e_{i,j}$ that appear in the formulae.
This particular description was obtained in \cite{wgmdsxtal} by considering Whittaker coefficients of global metaplectic Eisenstein series attached to a minimal parabolic subgroup. The familiar unfolding technique may be applied in this metaplectic setting to show that the local components of Fourier coefficients of such an adelic metaplectic Eisenstein series are indeed given by a local Whittaker function. There is some subtlety with respect to the fact that multiple Dirichlet series do not admit an Euler product, but instead have coefficients which satisfy a weaker property of twisted multiplicativity.
Viewed in this regard, our results can be considered as an alternative approach to the combinatorial formulae appearing in \cite{wgmdsxtal}. In fact, in section \ref{tok}, we detail our calculation in type A for a standard type of long word, exactly producing the combinatorial formulae of \cite{wgmdsxtal} in Theorem \ref{main}.
Because of the generality of our approach, which in principle can be used for arbitrary root systems and long word decompositions, our method may be viewed as providing a recipe for writing these Whittaker functions in the form of a generating function indexed by a crystal, as well as providing some understanding as to why the combinatorics of crystal graphs make an appearance in this field.
Independently, there has been an alternative calculation of the metaplectic Whittaker function in type A by Chinta and Offen \cite{chintaoffen}. This result shows that the spherical Whittaker function agrees with the $p$-parts of Weyl group multiple Dirichlet series using a different definition of the $p$-part, due to Chinta and Gunnells \cite{chintagunnells}. The results of this paper, combined with the work of Chinta and Offen provide a resolution of the question of proving that these two differing definitions of multiple Dirichlet series agree, which hitherto had been an open problem.
The author would like to thank B. Brubaker for his guidance through this area of study, and his assistance in the preparation of this manuscript.
\section{Preliminaries}\label{prelim}
In this section, we set up notation for use in the sequel.
Let $\Phi$ be a reduced root system, $\Phi^+$ and $\Phi^-$ a choice of positive and negative roots respectively, and $I$ the finite index set of simple roots. We denote by $\Phi^\vee$ the coroots and $\a\mapsto \a^\vee$ the bijection between $\Phi$ and $\Phi^\vee$. We use $\langle\cdot,\cdot\rangle\map{\Phi\times\Phi^\vee}{\Z}$ to denote the canonical pairing between $\Phi$ and $\Phi^\vee$.
Let $W$ be the Weyl group of the root system $\Phi$. It is generated by simple reflections $s_i$ for each $i\in I$. Let $w_0$ denote the element of longest length in $W$ and denote that length by $N$. The notation $[N]$ will be used for the set $\{1,2,\ldots,N\}$.
Let $F$ be a field equipped with a non-trivial discrete valuation $v$ and compatible norm $|\cdot |$. We shall be mostly concerned with the case of a local field, which implies that the cardinality of the residue field is finite. We use $q$ to denote this cardinality and normalise $|\cdot|$ such that $|x|=q^{-v(x)}$. Let $O_F$ be the valuation ring of $F$ and $\p$ be a uniformiser, that is a generator of the maximal ideal of $O_F$. In order to establish the connection with crystal graphs in Section \ref{xtal}, we restrict to the case where $F$ is the field of Laurent series over an algebraically closed field, although the rest of the paper is concerned with the case where $F$ is a local field.
Let $n$ be a positive integer not divisible by the residue characteristic and such that $F$ contains $2n$ $2n$-th roots of unity. Let $(\cdot,\cdot)\map{F^\times \times F^\times}{\mu_n}$ be the $n$-th power Hilbert symbol. The properties of the Hilbert symbol that we shall explicitly use are its bilinearity and skew symmetry, so for example $(xy,z)=(x,z)(y,z)$ and $(x,y)(y,x)=1$. The deeper properties of the Hilbert symbol will only be used implicitly, in that they are necessary in order to construct the central extension $\G$ of $G$.
Associated to the root system $\Phi$, there exists a split, simple, connected, simply connected algebraic group over $F$, which we denote by $G$. Let $T$ be a maximal torus of $G$, $X$ be the character group of $T$, and $Y$ the cocharacter group. Then we have $Y=\Z\Phi^\vee$, while $\Z\Phi$ is a finite index sublattice of $X$. We extend $\langle\cdot,\cdot\rangle$ to a perfect pairing $\langle\cdot,\cdot\rangle\map{X\times Y}{\Z}$ by linearity. We also will require the corresponding adjoint group $G^\text{ad}$ and the cocharacter group $\La$ of the corresponding torus $T^\text{ad}$. This cocharacter group $\La$ naturally contains the coroot lattice $Y$ as a finite index sublattice.
Choose a Borel subgroup $B$ of $G$ containing $T$ and let $U$ denote its unipotent radical. We focus most of our attention on the group $U^-$, the opposite unipotent subgroup to $U$.
Let $\psi$ denote an additive character of $F$ with conductor $O_F$. Thus $\psi$ is a homomorphism from the additive group $F$ to $\C^\times$ that is trivial on $O_F$ and non-trivial on $\frac{1}{\p}O_F$.
We will encounter Gauss sums appearing in our work, so we shall pause to define them and recap their properties. We define the Gauss sum \begin{equation}\label{gausssumdefn}
g(a,b)=\int_{O_F^\times}(u,\p)^a\psi(\p^bu)du.
\end{equation}
where we consider $du$ to mean additive measure on $F$, normalised such that $O_F^\times$ has volume $q-1$. This choice of normalisation of the Haar measure ensures that this approach to defining the Gauss sum agrees precisely with the classical definition as a sum over a finite field. We choose this approach since these precise integrals will appear later in this work. The following proposition provides a list of standard properties of these Gauss sums whose proofs involve routine manipulation of the defining integral.
\begin{proposition} \label{gauss} Gauss sums satisfy the following identities.
\begin{enumerate}
\item For all $a$ and $b$, $\conj{g(a,b)}=(-1,\p)^ag(-a,b)$.
\item If $b<-1$, then $g(a,b)=0$.
\item If $b\geq 0$, then $g(a,b)=q-1$ if $a$ is divisible by $n$, and zero otherwise.
\item If $n$ divides $a$, then $g(a,-1)=-1$.
\item If $a$ is not divisible by $n$, then $|g(a,-1)|=q^{1/2}$.
\end{enumerate}
\end{proposition}
We make a note about our usage of the symbol $\prod$ in the noncommutative case: Either the terms in the product will all commute, or we will write $\prod_{k=m}^n x_k$ for $x_m x_{m+1}\ldots x_n$ and $\prod_{k=n}^m x_k$ for $x_n x_{n-1},\ldots x_m$ where $m\leq n$.
\section{Central Extensions of Chevalley Groups}\label{1}
In this section, and any other section explicitly relying upon central extensions, we assume that $F$ is a non-archimedean local field, so that we have a good theory of the Hilbert symbol at our disposal. It will only be later when the connection to crystals is studied that we shall wish to work over other discrete valuation fields.
Since the group $G$ is equal to its commutator subgroup, it admits a universal central extension $E$ \cite{moore}. Thus we have a short exact sequence of groups
\[
1\rightarrow A\rightarrow E\rightarrow G\rightarrow 1
\]
with $A$ in the centre of $E$. Steinberg gives a presentation of $E$ in terms of generators and relations, which we now quote:
\begin{theorem}[Theorem 10, \cite{steinbergyale}]\label{relations}
The group $E$ is generated by symbols $e_\alpha(x)$ where $\alpha\in\Phi$ and $x\in F$, subject to the relations
\[
e_\alpha(x)e_\alpha(y)=e_\alpha(x+y)
\]
\[
w_\alpha(x)e_\alpha(y)w_\alpha(-x)=e_{-\alpha}(-x^{-2}y)
\] where $w_\alpha(x)=e_\alpha(x)e_{-\alpha}(-x^{-1})e_\alpha(x)$, and
\begin{equation}\label{eecommute}
e_\alpha(x)e_\beta(y)=\big[\!\!\!\prod_{\substack{ i,j\in\Z^+ \\ i\alpha+j\beta=\gamma\in\Phi }} \!\!\!e_\gamma(c_{i,j,\alpha,\beta}x^iy^j)\big]e_\beta(y)e_\alpha(x)
\end{equation} for all $x,y\in F$ and $\a,\beta\in\Phi$ with $\a+\beta\neq 0$, where $c_{i,j,\alpha,\beta}$ is a fixed collection of integers, completely determined by the root system $\Phi$.
\end{theorem}
Note that the product in (\ref{eecommute}) is a product of commuting terms, so there is no possible ambiguity present with respect to order of multiplication.
Matsumoto \cite{matsumoto} gave a computation of the kernel $A$, and showed it to be equal to $K_2(F)$ except in type $C$ in which case $K_2(F)$ is canonically a quotient group of $A$. Here $K_2(F)$ denotes the algebraic K-theory of $F$ (in degree 2).
As $F$ is a non-archimedean local field containing $n$ $n$-th roots of unity, the Hilbert symbol gives a surjection from $K_2(F)$ to $\mu_n$. Thus in all cases, we have a surjection $A\rightarrow \mu_n$, so we can take the quotient of $E$ by the kernel of this surjection to obtain a central extension of $G$ by $\mu_n$, which we shall denote by $\G$. Let $p$ denote the quotient map from $\G$ to $G$. So we have the exact sequence
$$
1\rightarrow \mu_n\rightarrow\G\xrightarrow{p} G\rightarrow 1.
$$
From now on, we shall continue to use the notation $e_\alpha(x)$ and $w_\alpha(x)$ to refer to the corresponding elements of $\G$. These elements naturally satisfy all the relations in Theorem \ref{relations}. Also for any subgroup $H$ of $G$, we shall denote by $\H$ the induced covering group of $H$.
Define the elements $h_\a(x)\in\G$ by $h_\a(x)=w_\a(x)w_\a(-1)$ and let $l_\alpha=||\a^\vee||^2$ where the norm on the coroot lattice has been chosen such that the short coroots have length 1. Then the following identities hold in $\G$ \cite[\S6]{steinbergyale}.
\begin{equation}\label{ehcommute}
h_\alpha(x)e_\beta(y)h_\alpha(x)^{-1}=e_\beta(x^{\langle\beta,\alpha^\vee\rangle}y)
\end{equation}
\begin{equation}
h_\a(x)h_\a(y)=(x,y)^{l_\a}h_\a(xy)
\end{equation}
\begin{equation}\label{hhcommute}
h_\alpha(x)h_\beta(y)h_\alpha(x)^{-1}=h_\beta(y)(x,y)^{\langle\beta,\alpha^\vee\rangle l_\beta}
\end{equation}
\begin{equation}\label{inverseh}
h_\a(x)=h_{-\a}(x^{-1}).
\end{equation}
Recall that $(x,y)\in\mu_n$ in the above equations is the value of the Hilbert symbol and is central in $\G$.
\section{An Explicit Iwasawa Decomposition}
In this section, we shall give an algorithm that explicitly calculates the Iwasawa decomposition in $\G$. This algorithm will form the cornerstone for the rest of this paper.
Let $K=G(O_F)$, a maximal compact subgroup of $G$. Since we have assumed that $n$ and $q$ are coprime, the central extension $\G$ of $G$ splits over $K$ \cite[Lemma 11.3]{moore}. Thus there is a section $s\map{K}{\G}$ that is a homomorphism such that $\K$ is the direct product of its subgroups $s(K)$ and $\mu_n$. We shall choose such a splitting $s$ and identify $K$ with its image in $\G$ under $s$ when appropriate.
Let $T$ be the subgroup of $G$ generated by (the images of) all elements of the form $h_\alpha(x)$, and let $B$ be the subgroup of $G$ generated by $T$ and the images of $e_\alpha(x)$ for all $\alpha>0$ and $x\in F$. Then $B$ is a Borel subgroup of $G$ with maximal torus $T$. The Iwasawa decomposition \cite{bruhattits} states that $G=BK$. This clearly lifts to $\G$, and so we may write $\G=\B K$.
The unipotent subgroup opposite to $B$ in $G$ is denoted $U^-$. Steinberg \cite{steinbergyale} shows that the universal central extension $E$ splits over $U^-$, so our central extension $\G$ certainly also does. Thus we may identify $U^-$ with a corresponding subgroup of $\G$. Explicitly, we have that $U^-$ is the subgroup of $\G$ generated by all $e_\a(x)$ where $\a\in\Phi^-$ and $x\in F$. We shall give an algorithm that explicitly calculates the Iwasawa decomposition of any $u\in U^-$.
\begin{lemma}\label{unipotentcompactsplitting}
If $\a\in\Phi$ and $x\in O_F$, then $e_\a(x)\in s(K)$.
\end{lemma}
\begin{proof}
Let us fix one $\a\in\Phi$. The group $\{e_\a(t)|t\in O_F\}$ is a splitting of the corresponding subgroup of $K$, a subgroup that is isomorphic to the additive group $O_F$. Any two splittings of this subgroup must differ by an element of $\Hom(O_F,\mu_n)$, which is trivial, so the splitting is unique, and thus must agree with the restriction of $s$, proving that $e_\a(t)\in s(K)$.
\end{proof}
We shall now recall a result regarding the action of the Weyl group on a root system.
\begin{proposition}\label{ellw}
Let $w=s_{i_1}\ldots s_{i_k}$ be a reduced decomposition of $w\in W$ into simple reflections $s_{i_j}$. Then
\[
\{\alpha\in \Phi^+\mid w(\alpha)\in \Phi^-\}=\{\alpha_{i_k},s_{i_k}\alpha_{i_{k-1}},s_{i_k}s_{i_{k-1}}\alpha_{i_{k-2}},\ldots,s_{i_k}s_{i_{k-1}}\ldots s_{i_2}\alpha_{i_1}\}.
\]
\end{proposition}
The proof may be found in any standard text on Lie theory, for example \cite[Ch VI, \S 6, Corollarie 2]{bourbaki}.
Let $\ii=(i_1,\ldots,i_N)$ be an $N$-tuple of elements of $I$ such that $w_0=s_{i_1}s_{i_2}\ldots s_{i_N}$ is a reduced decomposition of the long word $w_0$. Let $\I$ denote the set of all such tuples $\ii$.
For any $\ii\in\I$, the above theorem induces a total ordering $<_\ii$ on the set of positive roots, given by
$$\Phi^+=\{
\alpha_1<_\ii\cdots<_\ii\alpha_N \}\quad\text{ where }\quad\alpha_j=s_{i_N}\ldots s_{i_{j+1}}\alpha_{i_j}.$$
With this notation, we have now defined $\a_j$ for $j\in I$ as well as for $j\in [N]$. We hope that this does not cause any confusion for the reader.
For each $k\in[N]$, let $G_k$ denote the set of elements $g\in\G$ which can be expressed in the form
\begin{equation}\label{gkdefn}
g=\left(\prod_{j=1}^{k-1}e_{-\alpha_j}(x_j)\right)h\left(\prod_{j=k+1}^{N}e_{\alpha_j}(x_j)\right)
\end{equation}
where all $x_j\in F$ and $h\in \T$.
\begin{lemma}\label{lemma1}
For all $z\in F$ and $g\in G_k$, there exists unique $z'\in F$ and $g'\in G_k$ such that
\[
e_{-\alpha_k}(z)g=g'e_{-\alpha_k}(z').
\]
\end{lemma}
\begin{proof}
This involves repeated applications of the commutation relations presented in Section \ref{1} to push the $e_{-\a_k}(z)$ term in $e_{-\a_k}(z)g$ from the left to the right hand side of the product. It is straightforward to push a term of the form $e_{-\a_k}(\cdot)$ past another term of the form $e_{-\a_{k'}}(\cdot)$ or past a term of the form $h_\beta(\cdot)$. The key to ensuring that the manipulation claimed by the lemma is possible is thus understanding what terms appear in the commutator formula (\ref{eecommute}) when a term of the form $e_{-\a}(x)$ is pushed past a term of the form $e_\beta(y)$ for some $\a,\beta\in\Phi^+$, necessarily with $\a<_\ii\beta$.
Suppose that a term of the form $e_\gamma(w)$ appears in computing such a commutator. We will show that if $\gamma\in\Phi^+$, then $\gamma>_\ii\beta$ whereas if $\gamma\in\Phi^-$, then $-\gamma<_\ii\a$. This is enough to see that after applying the commutation relations a finite number of times, we will have written $e_{-\a_k}(z)g$ in the desired form $g'e_{-\a_k}(z')$. Briefly, the reason that this suffices is the following. First, we push the $e_{-\a_k}(\cdot)$ term to the rightmost side of the product. Then consider the multiset consisting of all integers $j-i$ for which there is a $e_{-\a_i}(\cdot)$ term appearing to the left of a $e_{\a_j}(\cdot)$ in our product. Such occurrences are undesirable. With each application of the commutation relations, an element $j-i$ in this multiset is replaced by a finite set of integers strictly larger than $j-i$. Since the elements of this multiset are integers bounded above by $N$, eventually our multiset becomes empty, as desired.
So now let us suppose that $\gamma$ is such that a term $e_\gamma(w)$ appears in pushing a term of the form $e_{-\a}(x)$ is past a term of the form $e_\beta(y)$ where $\a,\beta\in\Phi^+$ with $\a<_\ii\beta$.. Then we have $\gamma=i\beta-j\a$ for some positive integers $i$ and $j$. Write $\a=s_{i_N}\ldots s_{i_{r+1}}\a_{i_r}$ and $\beta=s_{i_N}\ldots s_{i_{s+1}}\a_{i_s}$. Note that by our assumption $\a<_\ii\beta$ we have $r<s$.
Let $w_1=s_{i_{r+1}}\ldots s_{i_N}$. Then using Proposition \ref{ellw} we obtain that $w_1(\beta)\in\Phi^-$ and $w_1(\a)\in\Phi^+$. Thus if $\gamma\in\Phi^-$, we have $-\gamma\in\Phi^+$ and $w_1(-\gamma)\in\Phi^+$ so we can conclude, again using Proposition \ref{ellw} that $-\gamma<_\ii\a$ as desired.
Now let $w_2=s_{i_{s-1}}\ldots s_{i-1}$. Note that $-w_0\a=s_{i_1}\ldots s_{i_{r-1}}\a_r$ and similarly for $\beta$. Thus we use Proposition \ref{ellw} for $w_2$ to obtain that $w_2(-w_0\a)\in\Phi^-$ and $w_2(-w_0\beta)\in\Phi^+$ so $w_2(-w_0\gamma)\in\Phi^+$. Since $\gamma\in\Phi^+$ if and only if $-w_0\gamma\in\Phi^+$ this last result implies that we can write $-w_0\gamma=s_{i_1}\ldots s_{i_{t-1}}\a_t$ for some $t>s$, implying $\gamma=s_{i_N}\ldots s_{i_{t+1}}\a_t$ and thus $\gamma>_\ii\beta$, as required.
\end{proof}
Now we are in a position to describe our algorithm for an explicit Iwasawa decomposition of $u\in U^-$. A sample computation in the case of $SL_3$ will be presented at the conclusion of this section. This algorithm runs as follows:
\begin{algorithm}\label{algorithm}
Given $u\in U^-$, we may write $u$ in the form
\[
u=\prod_{j=1}^Ne_{-\alpha_j}(x_j).
\]
for unique $x_1,\ldots,x_N \in F$.
For $k\in [N]$, we inductively define elements $p_k,p'_k\in\G$ and $y_k\in F$ as follows:
Initialise $p_{N+1}=1_{\G}$. By decreasing induction on $k$, we use Lemma \ref{lemma1} to define $p'_k\in G_k$ and $y_k\in F$ such that $e_{-\alpha_k}(x_k)p_{k+1}=p'_ke_{-\alpha_k}(y_k)$.
Let \[
p_k=\begin{cases}p'_k &\text{if } |y_k|\leq 1 \\
p'_kh_{\a_k}(y_k^{-1})e_\a(y_k) &\text{if } |y_k|> 1. \end{cases}
\]
Once this computation has been completed for $k=1$, the algorithm halts, with its primary output as the group element $p_1$.
\end{algorithm}
In the above algorithm, variables $x_i$ and $y_i$ are introduced which we shall make frequent use of throughout the sequel. Both of these sets of variables are implicit functions of $u$ and $\ii$, however for ease of exposition, this dependence will be usually suppressed from the notation.
\begin{theorem}
The above algorithm produces a well defined output that computes the Iwasawa decomposition of $u\in U^-$, in the sense that writing $u=p_1k'$, we have $p_1\in\B$ and $k'\in K$.
\end{theorem}
\begin{proof}
It is a consequence of Lemma \ref{lemma1} that at all steps in the inductive procedure, $p_k\in G_k$ and $y_k\in F$ are uniquely determined. This is mostly straightforward, the only potential sticking point is that a priori we do not have $p_{k+1}\in G_k$. Instead we have $p_{k+1}=e_{-\a_k}(z_k)p''$ for some $z_k\in F$ and $p''\in G_k$ and thus writing $e_{-\a_k}(x_k)p_{k+1}=e_{-\a_k}(x_k+z_k)p_k''$ we see that the algorithm may continue unhindered. Thus the output of this algorithm is well defined.
Note that $p_1\in\B$ since $p_1'\in G_1$.
To show that $k'\in K$ we need to show that if $|y_k|<1$ then $e_{-\a_k}(y_k)\in K$, and secondly if $|y_k|\geq 1$ then $e_{-\a_k}(y_k)^{-1}h_{\a_k}(y_k^{-1})e_\a(y_k)\in K$, as $k'$ is a product of such terms. Since we have an identity
$$
e_{-\a}(y)^{-1}h_\a(y^{-1})e_\a(y)=e_\a(y^{-1})w_{-\a}(1),
$$
this desired result follows from Lemma \ref{unipotentcompactsplitting}.
\end{proof}
Let us define variables $w_k$ for $k\in[N]$ as follows
$$ w_k=\begin{cases}y_k &\text{if } |y_k|> 1 \\
1 &\text{if } |y_k|\leq 1. \end{cases} $$
Then the following description of the diagonal part of $p_k$ is immediate.
\begin{proposition}\label{diagonalterms}
When writing $p_k$ in the form
\[
p_k=\left(\prod_{j=1}^{k-1}e_{-\alpha_j}(t_j)\right)h\left(\prod_{j=k+1}^{N}e_{\alpha_j}(t_j)\right),
\] with $h\in \T$, we have
$$ h=\prod_{i=N}^k h_{-\a_i}(w_i).$$
\end{proposition}
\noindent Here we are freely using the fact that $h_{-\a}(w)=h_\a(w^{-1})$.
Define integers $m_k$ for $k\in[N]$ according to the valuation of the variables $w_k$ by $m_k=-v(w_k)$. (The $m$ variables, like $x$, $y$ and $w$ are functions of $u$ and $\ii$, again this is usually suppressed from the notation). We are now in a position to define the subsets of $U^-$ that will be of primary concern to us.
Given any $\ii\in\I$ and $\m=(m_1,\ldots,m_N)\in \N^N$, define
$$
C^\ii_\m=\{u\in U^-\mid m_k(u,\ii)=m_k\text{ for all }k\in[N]\}.
$$
For linguistic convenience, we shall at times refer to these sets as `cells.' For a fixed choice of $\ii$, the set of above cells clearly decompose $U^-$ as a disjoint union.
We now give an example of these cells in the case where $G=SL_3$. Factorise $u\in U^-$ as
\[
u=\begin{pmatrix}
1 & 0 & 0 \\
x & 1 & 0 \\
y & z & 1
\end{pmatrix}
=\begin{pmatrix}
1 & 0 & 0 \\
x & 1 & 0 \\
0 & 0 & 1
\end{pmatrix}
\begin{pmatrix}
1 & 0 & 0 \\
0 & 1 & 0 \\
y & 0 & 1
\end{pmatrix}
\begin{pmatrix}
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & z & 1
\end{pmatrix}
\] where $x,y,z\in F$ and introduce the auxiliary variable $w=xz-y$. Then for a triple of non-negative integers $\m=(m_1,m_2,m_3)$, the cell $C^{212}_{\m}$ is given by the set of $u$ as above subject to the following conditions:
\[\begin{array}{lccc}
\mbox{If }\ m_1,m_2,m_3>0: & v(z)=-m_3 & v(\frac{y}{z})=-m_2, & v(\frac{wz}{y})=-m_1. \\
\mbox{If }\ m_1=0;m_2,m_3>0: & v(z)=-m_3, & v(\frac{y}{z})=-m_2, & |\frac{wz}{y}|\leq 1. \\
\mbox{If }\ m_2=0;m_1,m_3>0: & v(z)=-m_3, & |\frac{y}{z}|\leq 1, & v(w)=-m_1. \\
\mbox{If }\ m_2=m_1=0;m_3>0: & v(z)=-m_3, & |\frac{y}{z}|\leq 1, & |w|\leq 1. \\
\mbox{If }\ m_3=0;m_1,m_2>0: & |z|\leq 1, & v(y)=-m_2, & v(\frac{x}{y})=-m_1. \\
\mbox{If }\ m_3=m_1=0;m_2>0: & |z|\leq 1, & v(y)=-m_2, & |\frac{x}{y}|\leq 1. \\
\mbox{If }\ m_3=m_2=0;m_1>0: & |z|\leq 1, & |y|\leq 1, & v(x)=-m_1. \\
\mbox{If }\ m_3=m_2=m_1=0: & |z|\leq 1, & |y|\leq 1, & |x|\leq 1.
\end{array} \]
\section{Representation Theory and Metaplectic Whittaker Functions}\label{reptheory}
We require some knowledge of unramified genuine principal series representations of $\G$. This is treated in \cite{savin} in the simply laced case, and in \cite{mcn} in greater generality that encompasses all our needs. One can also consult \cite{kp} for a detailed study in the case $G=GL_n$. The meaning of the adjective `genuine' is that the central $\mu_n$ must act by a fixed faithful character. We fix an inclusion $\mu_n\subset\C^\times$ and thus assume that for any genuine representation $(\pi,V)$ of $\G$, we have that $\pi(\zeta g)=\zeta\pi(g)$ for all $\zeta\in\mu_n$ and $g\in\G$.
The group $\T$ is a Heisenberg group (a two-step nilpotent group). Since the commutator subgroup $[\T,\T]$ is contained in the central $\mu_n$, any irreducible genuine representation of $\T$ is induced from a character of a maximal abelian subgroup of $\T$ \cite[Corollary 5.2]{mcn}.
The group $\T\cap K$ is abelian since the Hilbert symbol is trivial on $O_F^\times\times O_F^\times$. Let $H$ be a maximal abelian subgroup of $\T$ containing $\T\cap K$.
Let $\chi$ be a genuine complex character of the group $H/(\T\cap K)$. We choose complex numbers $x_i$ for $i\in I$ with the property that $$
\chi\left(\prod_{i\in I}h_{\a_i}(\p^{m_i})\right)=\prod_{i\in I}x_i^{m_i}
$$
whenever the argument of $\chi$ above lies in $H$. This is always possible, the condition on $\{m_i\}_{i\in I}$ for the above product to lie in $H$ is that $m_i$ lie in a sublattice of $n\Z^I$. An explicit characterisation of this lattice is obtained in \cite{mcn} though we shall not require such a description here.
Let $(\pi_\chi,W_\chi)$ be the irreducible representation of $\T$ obtained by inducing $\chi$ from $H$ to $\T$. We obtain an unramified principal series representation $(\pi_\chi,V_\chi)$ of $\G$ by taking the normalised induced representation of $W_\chi$, considered as a representation of $\B$ (since $\T$ is a quotient of $\B$, we may inflate $W_\chi$ to a representation of $\B$). Explicitly, $V_\chi$ is realised as the space of functions $f\map{\G}{W_\chi}$ that are right invariant under an open compact subgroup and
$$
f(bg)=\delta^{1/2}(b)\pi_\chi(b)f(g)
$$
for all $b\in\B$ and $g\in\G$ where $\delta$ is the modular quasicharacter of $\B$. The action of $\G$ on $V_\chi$ is given by right translation.
The following theorem is largely contained in the works of Kazhdan and Patterson \cite{kp} and Savin \cite{savin}. The work \cite{mcn} contains the added generality necessary to state this result precisely.
\begin{theorem}
An unramified genuine principal series representation $(\pi_\chi,V_\chi)$ as above has a one-dimensional space of $K$-fixed vectors.
\end{theorem}
Let $\phi_K\map{\G}{W_\chi}$ be a non-zero element of $V_\chi^K$. We define a function $f\map{\G}{\C}$ related to $\phi_K$ by the following.
\begin{equation}\label{fdefn}
f\left(\zeta u\prod_{i\in I}h_{\alpha_i}(\p^{m_i})k\right)=\zeta\prod_{i\in I}(q^{-1}x_{\alpha_i})^{m_i}.
\end{equation} where $\zeta\in\mu_n$, $u\in U$, $m_i\in\Z$ and $k\in K$, using the Iwasawa decomposition to express an arbitrary $g\in\G$ as a product of such terms.
It is not immediately obvious that this is a well-defined function; we shall prove this as part of the following Propostion.
\begin{proposition}
There exists a linear functional $\lambda$ on $W_\chi$ such that $\lambda(\phi_K(g))=f(g)$ for all $g\in\G$.
\end{proposition}
\begin{proof}
Let $A$ be the group of all elements of the form $\prod_{i\in I} h_{\a_i}(\p^{m_i})$ where $m_i\in\Z$. This is abelian because our assumption that $\mu_{2n}\subset F$ forces all commutators calculated from (\ref{hhcommute}) to vanish.
Choose a set of coset representatives $a_j\in A$ for $\T/H$. Then the set of elements $\{\pi_\chi(a_j)\phi_K(1)\}$ forms a basis of $W_\chi$. Thus there exists a linear functional $\lambda\map{W_\chi}{\C}$ such that $$\lambda(\pi_\chi(a_j)\phi_K(1))=\chi(a_j)$$ where $\chi\map{A}{\C}$ is the extension of $\chi$ on $A\cap H$ given by $\chi(\prod_{i\in I}h_{\a_i}(\p^{m_i}))=\prod_{i\in I}x_i^{m_i}$.
Now suppose we have an arbitrary $a\in A$. Then we can write $a=a_j h$ for some $j$ and some $h\in H$ and hence \begin{eqnarray*}\lambda(\phi_K)(a)&=&\lambda(\pi_\chi(a_j)\phi_K(h)) \\
&=&(\delta^{1/2}\chi)(h)\lambda(\pi_\chi(a_j)\phi_K(1)) \\
&=&(\delta^{1/2}\chi)(h)
(\delta^{1/2}\chi)(a_i) \\
&=&(\delta^{1/2}\chi)(a).\end{eqnarray*}
Thus $\lambda\circ\phi_K$ and $f$ agree on $A$. Since $\phi_K$ is a genuine function, left $U$-invariant and right $K$-invariant, this is enough to show that $\lambda\circ\phi_K=f$, thus proving the proposition, and as a consequence ensuring that the function $f$ is well-defined.
\end{proof}
Our main thrust of this paper involves computing the integral of $f$ over $U^-$, as well as computing the integral of $f$ against a character of $U^-$. We shall now explain the representation theoretic meanings behind these integrals.
The Weyl group $W$ acts on principal series representations. As in the reductive case, we have \cite{kp,mcn}
$$
\dim(\Hom(V_\chi,V_\chi^{w_0}))=1.
$$
Letting $T$ denote a non-zero element of the above space of intertwining operators, and $\phi_K$ and $\phi'_K$ be spherical vectors for $V_\chi$ and $V_\chi^{w_0}$ respectively, we have $T\phi_K=g(\chi)\phi_K'$ for some meromorphic function $g$. The representation $V_\chi$ will be irreducible whenever $g$ does not have a zero or a pole, and we can write $g$ as the integral
\begin{equation*
\int_{U^-} f(u)du.
\end{equation*} with $f$ as in (\ref{fdefn}) above.
It will transpire that this integral is only dependent on $\chi$, not on the choices of $x_i$ that were made in defining $f$.
Now let $\psi$ denote a character of $U^-$ such that the restriction to the subgroup $U_{-\a}=\{e_{-\a}(x)\mid x\in F\}\cong F$ for each simple $\a$ has conductor $O_F$. As in the reductive case, the integral
\[
W(g)= \int_{U^-} \phi_K(ug)\psi(u)du
\]
defines a Whittaker function; it is a spherical vector in a Whittaker model for $V$. The above integral actually defines a $W_\chi$-valued Whittaker function, so to obtain a complex valued Whittaker function, we merely need to compose with a linear functional $\lambda$, and thus we are lead to study the integral
\[
\int_{U^-} f(ug)\psi(u)du.
\]
Since a Whittaker function transforms on the left under $U$ by $\psi$, on the right under $K$ trivially, and is genuine, a Whittaker function is determined by the values it takes on the maximal torus $T$, by the Iwasawa decomposition. If $g\in T$, making a change of coordinate in the above integral, replacing $u$ by $gug^{-1}$, means that one is instead lead to evaluating the integral
\[
\int_{U^-} f(u)\psi(gug^{-1})du
\]
where a relatively trivial constant factor has been discarded.
The lattice $\La$ is the coroot lattice of the corresponding adjoint group $G^{\text{ad}}$, and thus acts by conjugation on $U^-$ and hence on the set of characters of $U^-$. Explicitly, the action of $\la\in \La$ on our fixed character $\psi$ creates another character $\psi_\la$ defined by
$$
\psi_\la(u)=\psi(\p^\la u \p^{-\la}).
$$
Here, although $\p^\la$ is not in $G$, it is in $G^{\text{ad}}$ which canonically shares the same unipotent subgroups as $G$, so acts on $U^-$ by conjugation.
In this language, explicitly evaluating a Whittaker function becomes equivalent to the problem of evaluating the integral
\begin{equation}\label{iladefn}
I_\la= \int_{U^-} f(u)\psi_\la(u)du.
\end{equation}
For the remainder of this paper, our major goal is to obtain a method for evaluating the above integral.
There is a simple case where we know that this integral vanishes. This vanishing property, together with its proof, is exactly the same as in the reductive case, as shown in \cite[Lemma 5.1]{cs}.
\begin{proposition}
Unless $\la$ is dominant, we have $I_\la=0$.
\end{proposition}
\section{Initial Calculations with the Cell Decomposition}\label{initial}
In this section we shall present some initial calculations involving the decomposition of $U^-$ into the disjoint union
$$
U^-=\bigsqcup_{\m\in\N^N} C^\ii_\m.
$$
This will immediately result in a metaplectic Gindikin-Karpelevic formula, generalising the type A results of Bump and Nakasuji \cite{bumpnakasuji}. The intermediate results we obtain will also prove to be useful in the sequel when we are evaluating the Whittaker function.
We choose a Haar measure on $F$ such that $O_F$ has volume 1, and denote it by $dx$. Now choose a normalisation of Haar measure on $U^-$ such that $du=\prod_{i=1}^N dx_i$. The variables $x_i$ here that we use are those introduced in Algorithm 1, we also freely use the $y_i$ and $w_i$ variables defined in the main algorithm (which all depend on a choice of $\ii$). Under this normalisation, we can now compute the volume of the sets $C_\m^\ii$.
\begin{proposition}
The volume of $C_\m^\ii$ is
\[
\prod_{i=1}^N q^{\langle\rho,\alpha_i^\vee\rangle m_i}(1-\frac{1-\delta_{m_i,0}}{q})
\] ($\delta$ here represents the Kronecker delta function).
\end{proposition}
\begin{proof}
We wish to calculate $\int_{C^\ii_\m} 1\ dx_1\ldots dx_N$ and it is easy to see that
$$
\int_{C^\ii_\m} 1\ dy_1\ldots dy_N =\prod_{i=1}^N q^{m_i}(1-\frac{1-\delta_{m_i,0}}{q}).
$$
So we need to calculate the Jacobian for the change of coordinates from $x_1,\ldots,x_N$ to $y_1,\ldots,y_N$, the formulae for which are obtained as a byproduct of Algorithm 1.
In our algorithm, consider what happens to the argument of the term $e_{-\a_k}(\cdot)$ as $e_{-\a_k}(x_k)$ is pushed passed all other terms in $p_{k+1}$ to the right hand side of the product, where it emerges as $e_{-\a_k}(y_k)$. Apart from the initial step where a constant (that is independent of $x_k,\ldots,x_N$) is added to the argument, the only time when the argument of $e_{-\a_k}(\cdot)$ changes is when it moves past a term of the form $h_{-\beta}(w_\beta)$, which has the effect of multiplying the argument by $w_k^{-\langle\a_k,\beta^\vee\rangle}$. Here we are using (\ref{ehcommute}) as well as Proposition \ref{diagonalterms}. Necessarily, we must have that $\beta>_\ii\a$ for this to occur.
Thus we have
$$
dy_k=\left(\prod_{i=k+1}^N |w_{\a_i}|^{-\langle\a_k,\a_i^\vee\rangle}\right) dx_k.
$$
Multiplying over all $k$ yields
$$
dy_1\ldots dy_N=\left( \prod_{i=1}^N q^{ -m_i\sum_{j=1}^{i-1} \langle\a_j,\a_i^\vee\rangle } \right)dx_1\ldots dx_N.
$$
Thus the proposition follows from the following lemma.
\end{proof}
\begin{lemma}
For all $\beta\in \Phi^+$, we have
$$ \langle\rho,\beta^\vee\rangle=1+\sum_{\a<_\ii\beta}\langle\a,\beta^\vee\rangle.$$
\end{lemma}
\begin{proof}
We first show that the right hand side of the above equation is independent of the choice of $\ii\in\I$. To achieve this, it suffices to show invariance under changing $\ii$ to $\ii'$ by the application of a single braid relation. For such $\ii$ and $\ii'$, we have that $\a_j(\ii)=\a_j(\ii')$ (here we are merely expressing the dependence of $\a_j$ on the choice of $\ii$) unless $\ii_j\neq \ii'_j$ so we have quickly reduced to showing invariance of long word decomposition in the rank 2 case. This is achieved by a simple case by case analysis, which we shall omit here.
Now consider two long word decompositions $\ii=(i_1,\ldots,i_N)$ and $\ii'=(\tau(i_N),\ldots,\tau(i_1))$ where $\tau\map{I}{I}$ is the automorphism defined by $w_0\a_i=-\a_{\tau(i)}$.
This pair of decompositions has the property that for any two $\a,\beta\in\Phi^+$, $\a<_\ii\beta$ if and only if $\beta<_{\ii'}\a$. Thus
\begin{eqnarray*}
1+\sum_{\a<_\ii\beta}\langle\a,\beta^\vee\rangle&=&\frac{1}{2}\left(1+\sum_{\a<_\ii\beta}\langle\a,\beta^\vee\rangle+1+
\sum_{\a<_{\ii'}\beta}\langle\a,\beta^\vee\rangle\right) \\
&=& \frac{1}{2}\sum_{\a\in\Phi^+}\langle\a,\beta^\vee\rangle \\
&=& \langle\rho,\beta^\vee\rangle
\end{eqnarray*} as required.
\end{proof}
Recall that $n$ denotes the degree of the cover $\G\rightarrow G$. For each root $\a$, define the integer $n_\alpha=\frac{n}{(n,l_\a)}$, where $l_\a$ is, as before given by $||\a^\vee||^2$, where the norm on the coroot lattice has been chosen such that the short coroots have length 1. We also denote $n_k=n_{\a_k}$. The following lemma will allow us to integrate the function $f$ from (\ref{fdefn}) over any set $C_\m^\ii$.
\begin{lemma}\label{lemma4}
\begin{enumerate}
\item For the function $f$ defined in (\ref{fdefn}), $$\int_{C_\m^\ii}f(u) du=0$$ unless $n_k|m_k$ for all $k\in[N]$.
\item If $n_k| m_k$ for all $k\in[N]$ then $f$ is constant on $C_\m^\ii$ and takes the value
$$\prod_{\a\in\Phi^+}(q^{\langle\rho,\a^\vee\rangle}x_\a)^{m_\a}$$
where the variables $x_\a$ are defined by $x_\a=\prod_{i\in I}x_i^{h_i}$ with $\a^\vee=\sum_{i\in I}h_i\a_i^\vee$.
\end{enumerate}
\end{lemma}
\begin{proof}
By induction on $k$, we shall prove that
$$
\int_{C_\m^\ii}f(u)du\neq 0\implies n_i|m_i \text{ for all } i\leq k
$$
and in this case
$$f(\prod_{i=N}^1 h_{-\a_i}(w_i))=f\left(\prod_{i=N}^{k+1} h_{-\a_i}(w_i)\prod_{i=k}^1 h_{\a_i}(\p^{m_i})\right).$$
Note that the lemma follows from the $k=N$ case of the inductive statement.
Let $w_i=\p^{-m_i}u_i$. Then $u_i\in O_F$. We may write,
assuming our inductive hypothesis for $k$ and using (\ref{hhcommute}),
\[
h_{-\a_{k+1}}(w_{k+1})\prod_{i=k}^1 h_{\a_i}(\p^{m_i})=(\p,u_{k+1})^e\left( \prod_{i=k+1}^1 h_{\a_i}(\p^{m_i})\right) h_{-\a_{k+1}}(u_{k+1}),
\]
where the exponent $e$ is given by
\begin{equation}\label{exponent}
e=m_{k+1}l_{k+1}+\sum_{i\leq k} \langle\a_i,\a_{k+1}^\vee\rangle m_i l_{\a_i}.
\end{equation}
Consider the evaluation of the integral of $f$ over $C^\ii_\m$, by integrating over variables $y_1,\ldots,y_n$ in that order.
Suppose also that $m_{k+1}>0$. Then the above calculation shows that we get a factor of
$$ \int_{O_F^\times}(\p,u)^e du$$ appearing in our integral, from the integration over $y_{k+1}$. For this integral to be non-zero, we must thus have that $n|e$. By our inductive hypothesis we have that $e\equiv m_{k+1}l_{k+1}\pmod n$, so we obtain the first part of the inductive hypothesis, namely that $n_{k+1}$ divides $m_{k+1}$. The second part of the inductive hypothesis now follows from the form taken by $f$ in (\ref{fdefn}), noting that $e\equiv 0\pmod n$ and $h_{-\a_{k+1}}(u_{k+1})\in K$, thus proving our lemma.
\end{proof}
At this point it is straightforward to complete the evaluation of the integral of $f$ over $U^-$. In order for this integral to be convergent, we must assume that $|x_i|<1$ for all $i\in I$. However some sense can be made of this integral for an arbitrary collection of complex numbers $x_i$ by considering the integral as the analytic continuation of an intertwining operator, as in \cite{kp},\cite{mcn}. We obtain the following result, where we assume $|x_\a|<1$ for all $\a$ in order to ensure convergence.
\begin{theorem}\label{gk}
[Gindikin-Karpelevich Formula]
For the function $f$ defined in (\ref{fdefn}), and the variables $x_\a$ as defined in Lemma \ref{lemma4}, we have \[
\int_{U^-}f(u) du=\prod_{\a\in\Phi^+} \left( \frac{1-q^{-1}x_\a^{n_\a}}{1-x_\a^{n_\a}} \right)
\]
\end{theorem}
\begin{proof}
The idea is to simply write the integral as a sum over each $C_\m^\ii$, and use what we have already computed.
\begin{eqnarray*}
\int_{U^-}f(u)du&=&\sum_{\m\in\N^N}\int_{C_\m^\ii} f(u)du
\ =\,\sum_{\substack {\m\in\N^N \\ n_j|m_j\forall j}}\int_{C_\m^\ii} f(u)du \\
&=&\sum_{\substack {\m\in\N^N \\ n_j|m_j\forall j}}\prod_{\a\in\Phi^+} (q^{-\langle\rho,\a^\vee\rangle}x_\a)^{m_\a} \prod_{i=1}^N q^{\langle\rho,\a_i^\vee\rangle m_i}(1-\frac{1-\delta_{m_i,0}}{q}) \\
&=& \prod_{\a\in\Phi^+}\sum_{k=1}^\infty x_\a^{kn_\a}(1-\frac{1-\delta_{k,0}}{q}) \\
&=& \prod_{\a\in\Phi^+} \left( \frac{1-q^{-1}x_\a^{n_\a}}{1-x_\a^{n_\a}} \right)
\end{eqnarray*} as required.\end{proof}
This same method can be used to calculate the integral of $f$ over other unipotent subgroups of $U^-$. For each $w\in W$ let $\Phi_w=\{\a\in\Phi^+\mid w\a\in \Phi^-\}$ and $U_w^-$ be the corresponding unipotent subgroup. Then with exactly the same proof, we have
\begin{theorem}\label{gkw}
For the function $f$ defined in (\ref{fdefn}), and the variables $x_\a$ as defined in Lemma \ref{lemma4}, we have \[
\int_{U_w^-}f(u) du=\prod_{\a\in\Phi_w} \left( \frac{1-q^{-1}x_\a^{n_\a}}{1-x_\a^{n_\a}} \right)
\]
\end{theorem}
From a representation-theoretic point of view, this integral corresponds to the evaluation of the intertwining operator $T_w\in\Hom(V_\chi,V_\chi^w)$ at a spherical vector.
\section{Connection to the combinatorics of crystals}\label{xtal}
In this section we shall give a bijection between the sets $C_\m^\ii$ that decompose $U^-$ and Lusztig's canonical basis. We shall be considering the case $F=k((\p))$ for $k$ an algebraically closed field, since this will make our exposition easier. Under this assumption, our sets $C_\m^\ii$ are naturally affine varieties over $k$.
Although such a choice of $F$ is not a local field, we can obtain a direct link between the case of a positive characteristic local field, which is necessarily (non-canonically) isomorphic to $\F_q((\p))$. The sets $C_\m^\ii$ for the field $\F_q((\p))$ are the $\F_q$ points of the corresponding varieties over $\overline{\F_q}$ that are considered in this section. To establish a link in the case for an arbitrary discrete valuation field $F$ with the techniques of this section, we note that a formal analogy can be made as the combinatorics of taking valuations is independent of the field $F$, and is an analogue of the process of tropicalisation.
Lusztig \cite{lusztig} realised the canonical basis as the set of connected components of a graph structure on $\I\times\N^N$, which we shall now recall. In order for there to exist an edge between two vertices $(\ii,\m)$ and $(\ii',\m')$ it is first necessary to have $\ii$ and $\ii'$ related by a single application of a braid relation, so an occurrence of $(i,j,\ldots)$ is replaced by $(j,i,\ldots)$ when moving from $\ii$ to $\ii'$. Furthermore, we require that $\m$ and $\m'$ are related by piecewise linear transition maps $R_\ii^{\ii'}$, which we shall now explicitly describe. These equations appear in \cite{lusztig,kamnitzer} except for the case of type $G_2$, where they are derived from Section 7 of \cite{genminors}.
The piecewise linear transition maps $R_\ii^{\ii'}$ only affect the coordinates which are changed in moving from $\ii$ to $\ii'$. The restriction to the set of such coordinates is a local transtition map $R_{ij\ldots}^{ji\ldots}$, which we shall now give. There are four cases to consider, corresponding to each of the four finite type rank 2 root systems.
\begin{enumerate}
\item Type $A_1\times A_1$: If $a_{ij}=a_{ji}=0$, then
$$
R_{ij}^{ji}(a,b)=(b,a).
$$
\item Type $A_2$: If $a_{ij}=a_{ji}=-1$, then
$$
R_{iji}^{jij}(a,b,c)=(b+c-\min(a,c),\min(a,c),b+a-\min(a,c)).
$$
\item Type $B_2$: If $a_{ij}=-2$ and $a_{ji}=-1$, then
$$
R_{ijij}^{jiji}(a,b,c,d)=(b+2c+d-q,q-p,2p-q,a+b+c-p)
$$
where $p=\min(a+b,a+d,c+d)$ and $q=\min(2a+b,2a+d,2c+d)$.
\item Type $G_2$: If $a_{ij}=-3$ and $a_{ji}=-1$ then
\begin{multline*}
R_{ijijij}^{jijiji}(a,b,c,d,e,f)=(b+3c+2d+3c+f-r,r-q,2q-r-s,s-p-q, \\ 3p-s,a+b+2c+d+e-p)
\end{multline*}
where $p=\min(a+b+2c+d,a+b+2c+f,a+b+2e+f,a+d+2e+f,b+d+2e+f)$,
$q=\min(2a+2b+3c+d,2a+2b+3c+f,2a+2b+3e+f,2a+2d+3e+f,2c+2d+3e+f,a+b+d+2e+f+\min(a+c,2c,c+e,a+e)),$
$r=\min(3a+2b+3c+d,3a+2b+3c+f,3a+2b+3e+f,3a+2d+3e+f,3c+2d+3e+f,2a+b+d+2e+f+\min(a+c,2c,c+e,a+e)),$ and
$s=\min(2a+2b+2c+d+\min(a+b+3c+d,a+b+3c+f,a+b+3e+f,d+2e+f+\min(a+c,2c,c+e,a+e))+2f+3\min(a+b+2c,a+b+2e,a+d+2e,c+d+2e)).$
\end{enumerate}
We shall refer to this graph as the Lusztig graph.
The set of connected components of the Lusztig graph is canonically identified with the canonical basis $B(-\infty)$ for $U_q(\mathfrak{n}^+)$, the positive part of the quantised universal enveloping algebra, as defined for example in \cite{kashiwarasurvey}. Moreover, the following theorem from \cite[Ch 42]{lusztigbook} yields a parametrisation of the canonical basis for each choice of $\ii\in \I$.
\begin{theorem}[\cite{lusztigbook}, Ch 42\label{cellsascrystal}]
For each $\ii\in\I$, the map $\N^N\rightarrow\I\times\N^N$ sending $\m$ to $(\ii,\m)$ is a bijection between $\N^N$ and the set of connected components of the Lusztig graph. \end{theorem}
We consider $U^-$ as an ind-variety over $k$, and thus are able to give it the Zariski topology. The cells $C^\ii_\m$ now have the additional structure as subvarieties of $U^-$ and we denote their Zariski closure in $U^-$ by $\overline{C^\ii_\m}$. We are able to relate our decomposition of $U^-$ with the canonical basis via the following theorem.
\begin{theorem}[Theorem 15, \cite{baumanngaussent}, Theorem 4.5, \cite{kamnitzer}]\label{maincrystalthm}
For $(\ii,\m),(\ii',\m')\in\I\times\N^N$, the vertices $(\ii,\m)$ and $(\ii',\m')$ of the Lusztig graph are connected if and only if $\overline{C^\ii_\m}=\overline{C^{\ii'}_{\m'}}$.
\end{theorem}
The proof of this theorem utilises the circle of ideas presented in \cite{genminors}, which tells one how the coordinates on $U^-$ change when one changes the reduced word. This reduces the problem to the rank two case where it is dealt with by an explicit computation
\begin{corollary}
Consider the collection of closed subsets of $U^-$ consisting of the set of all $\overline{C^\ii_\m}$ for a fixed $\ii$ with $\m$ running over $\N^N$. Then this collection is independent of the choice of $\ii$.
\end{corollary}
To proceed, we now recall some of the geometry of the affine Grassmannian for the adjoint group $G^\text{ad}$. This affine Grassmannian shall be denoted $\gr$, it is defined to be the fpqc quotient $G^{\rm{ad}}(k((\p)))/G^{\rm{ad}}(k[[\p]])$. However, it will suffice for our purposes to consider $\gr$ as an ind-variety over $k$. Recall that $\La$ is the cocharacter group of the maximal torus $T^\text{ad}$ of $G^\text{ad}$. The element $\p$ is a uniformiser, accordingly we use $\p^\la$ for $\la\in \La$ to denote the image of $\p$ in $T^\text{ad}$ under the cocharacter $\la$.
The cells $C^\ii_\m$ that provided the decomposition of $U^-(F)$ are each (non-Noetherian) affine varieties over $k$. This is because each $C^\ii_\m$ is isomorphic to a product of copies of $k[[\p]]$ and $k[[\p]]^\times$, each of which are irreducible varieties. In particular, we see that $C^\ii_\m$ is irreducible.
We first consider the action of the group $G(O_F)$ on $\gr$ by left multiplication. The orbits are indexed by dominant weights $\la\in \La$. More precisely they are given by $\gr_\l=G(O_F)\p^\la$ according to the Cartan decomposition. The orbit $\gr_\la$ is finite dimensional and its closure $\overline\gr_\la$ is a (generally singular) projective variety.
We next consider the action of the unipotent groups $U^\pm(F)$ on $\gr$, again by left multiplication. The orbits here are given by the Iwasawa decomposition, indexed by elements of $\La$. For $\mu\in \La$, we denote by $S^{\pm}_\mu$ the orbit $U^\pm(F)\p^\mu$. These strata are locally closed, but not finite dimensional.
Fix a dominant weight $\la$. Then there is a set of Mirkovic-Vilonen cycles (hereafter MV cycles) $\mathcal{L}(\la)$, defined to be the irreducible components of $\overline{S^+_\mu\cap\gr_\la}$, where $\mu$ can be any weight. The only weights for which this intersection is non-empty are those for which $\mu$ belongs to the convex hull of the set $\{w\la\mid w\in W\}$, and such that $\la-\mu$ is an integral sum of coroots. As a consequence, we deduce that $\mathcal{L}(\la)$ is a finite set.
There is also a notion of a stable MV cycle which we now define. Let $\mathcal{L}$ denote the set of all irreducible components of $\overline{S_\la^+\cap S_\mu^-}$, where $\la$ and $\mu$ run over $X$ with $\la\geq \mu$. This set $\mathcal{L}$ is equipped with an action of $\La$, given by $\nu Z=t^{\nu}Z$. We define the set of stable MV cycles to be the quotient of $\mathcal{L}$ by this action, and denote it by $\mathcal{L}(-\infty)$. We shall frequently identify $\mathcal{L}(-\infty)$ with a particular choice of coset representative in $\mathcal{L}$, most often that which is an irreducible component of $\overline{S_\la^+\cap S_0^-}$.
The following proposition of Anderson relates the notion of a MV cycle to that of a stable MV cycle.
\begin{proposition}[Proposition 3, \cite{anderson}]\label{anderson}
Suppose $\la$ is antidominant and $\nu$ is arbitrary. Then the irreducible components of $\overline{S_\nu^+\cap \gr_\la}$ are the same as the irreducible components of $\overline{S_\nu^+\cap S_\la^-}$ contained in $\overline{\gr_\la}$.
\end{proposition}
In particular we are able to identify $\mathcal{L}(\la)$ as a subset of $\mathcal{L}(-\infty)$.
Under the gometric Satake correspondence, the intersection cohomology complex of $\overline\gr_\l$ is mapped to the space of the representation of $(G^{\text{ad}})^{\vee}$ (the Langlands dual to $G^\text{ad}$) of highest weight $\la$. Since a basis for the intersection cohomology of $\overline\gr_\l$ is given by the set of MV cycles \cite{mv}, it is natural to ask whether we can equip this set of MV cycles with the structure of a crystal. Such a procedure is carried out by Braverman and Gaitsgory in \cite{bravermangaitsgory} for the finite crystal $\mathcal{L}(\la)$ and by Braverman, Finkelberg and Gaitsgory in \cite{bg2} for the infinite crystal $\mathcal{L}(-\infty)$. These two crystal structures are compatible with the inclusion $\mathcal{L}(\la)\hookrightarrow\mathcal{L}(-\infty)$.
The following theorem gives the relation between our cells and MV cycles.
\begin{theorem}\label{cellsaremvcycles}
$\overline{\phi_0(C^\ii_\m)}$ is a MV cycle. Furthermore this map defines a bijection between $\N^N$ and $\mathcal{L}(-\infty)$ for any choice of $\ii\in \I$.
\end{theorem}
Before we embark on a proof, we first pause to define some useful notation.
For $\mu\in \La$, define the map
$$\phi_\mu\map{U^-}{\gr}, \quad \phi_\mu(u)=\p^\mu u.$$
For any $\m\in\N^N$ and a fixed choice of $\ii\in \I$, we define $\m^\vee=\sum_{j=1}^N m_j\a_{i_j}^\vee$.
For any $\la\in \Z\Phi^\vee$, let $d_\la$ be the number of ways of writing $\la=\sum_{\a\in\Phi^+}m_\a \a^\vee$ for non-negative integral $m_\a$. This is a well known function, commonly referred to as the Kostant partition function. It is equal to the dimension of the $\la$ weight space in a Verma module, and thus we know that it is equal to the number of elements of weight $\la$ in $\mathcal{L}(-\infty)$, or equivalently the number of irreducible components of $\overline{S_\la^+\cap S_0^-}$.
Now we present a proof of Theorem \ref{cellsaremvcycles}.
\begin{proof}
Fix $\la$, and consider only those $\m$ for which $\m^\vee=\la$. Note that $\phi_0(C^\ii_\m)\subset S^+_\la\cap S_0^-$. Being irreducible, its closure must lie in a stable MV cycle. Since the number of such $\m$ is equal to the number of such stable MV cycles (they are both equal to $d_\la$), and $\phi_0$ of the union of such $C^\ii_\m$ surjects onto $S^+_\la\cap S_0^-$, the closure must equal a stable MV cycle.
\end{proof}
A crystal $\mathcal{B}$ comes equipped with five functions from $\mathcal{B}$ which are required to satisfy various compatibility relations that can be found in \cite{kashiwarasurvey}. The only two of these functions we will need are the Kashiwara operator $\tilde{e}_i\map{B}{B\sqcup \{0\}}$ and the function $\epsilon_i\map{B}{\Z\sqcup\{-\infty\}}$. The crystal $B(-\infty)$ is a lowest weight crystal, it is generated by a single element $b_0\in B(-\infty)$ together with its successive images under the various Kashiwara operators $\tilde{e}_i$.
We shall require the following proposition from \cite{baumanngaussent} which identifies $\tilde{e}_i(Z)$ and $\epsilon_i(Z)$ for a MV cycle $Z$.
\begin{proposition}[Proposition 14, \cite{baumanngaussent}]\label{kashiwaraaction}
Let $Z$ be a MV cycle, and $i\in I$. Then for each $p\in O_F$, the action of $e_{-\a_i}(pt^{\epsilon_i(Z)})$ stabalises $Z$. The MV cycle $\tilde{e}_i(Z)$ is the closure of $$
\{ e_{-\a_i}(x)z\mid z\in Z\mbox{ and }x\in F^\times \mbox{ such that } v(x)=\epsilon_i(Z)+1\}.
$$
\end{proposition}
We will write $B(\la)$ and $B(-\infty)$ for the abstract crystals reaslised by $\mathcal{L}(\la)$ and $\mathcal{L}(-\infty)$ respectively. For $b\in B(-\infty)$, represented by $(\ii,\m)$, let $C_b=\overline{C_\m^\ii}$. We now move to studying the finite crystal $B(\la)$, for which our starting point is the following Proposition.
\begin{proposition}[Proposition 8.2, \cite{kashiwarasurvey}, \S 8 \cite{lusztig2}, Corollary 3.4, \cite{genminors}, Theorem 8.4, \cite{kamnitzer}]\label{ka}
The image of the natural injection $B(\la)\hookrightarrow B(-\infty)$ is given by
$$
B(\la)=\{b\in B(-\infty)\mid\epsilon_i(b)+\langle\alpha_i,\la\rangle\geq 0\ \forall\ i\in I\}.
$$
\end{proposition}
We use this to identify $B(\la)$ in terms of cells $C_b$. There is an alternative characterisation of the cells of the finite crystal $B(\la)$ which will prove to be useful in the following section. For $u\in U^-$ and $i\in I$, we define $f_i(u)$ to be element of $F$ such that in writing $u=\prod_{\a\in\Phi}e_{-\a}(z_\a)$, we have $z_{\a_i}=f_i(u)$. This result can be considered as an analogue to Theorems 8.3 and 8.5 of \cite{kamnitzer}.
\begin{theorem}\label{finitecrystal}
Suppose that $\la$ is dominant and write $\la'=w_0\la$. Then the following three conditions on an element $b$ of the crystal basis are equivalent.
\begin{enumerate}
\item $b\in B(\la)$
\item $\p^{\la} C_b \p^{-\la} \subset K$
\item For all $i\in I$ and $u\in C_b$, we have $f_i(\p^{\la} u\p^{-\la})\in O_F$.
\end{enumerate}
\end{theorem}
\begin{proof}
$1\Rightarrow 2$.
Proposition \ref{anderson} shows that $b\in B(\la)$ if and only if $\phi_{\la'}(C_b)\subset \overline{\gr_\la}$, where $\la'=w_0\la$ is the image of $\la$ under the action of the longest word in $W$. So if $b\in B(\la)$, we have $C_b\subset\phi^{-1}_{\la'}(\overline{\gr_\la})$. It is shown in \cite[4.4.4(ii)]{bruhattits} that $\phi^{-1}_{\la'}(\overline{\gr_\la})=\p^{-\la}K\p^\la\cap U^-$, completing this implication
$2\Rightarrow 3$.
This is immediate.
$3 \Rightarrow 1$.
Suppose for all $i\in I$ and $u\in C_b$ that $f_i(\p^\la u \p^{-\la})\in O_F$. Then $x_i(u)\in \p^{-\langle\a_i,\la\rangle}O_F$. So utilising Proposition \ref{kashiwaraaction}, we have that the action of $e_{-\a_i}(\p^{\epsilon_i(b)}O_F)$ stabalising $b$ implies that $\epsilon_i(b)\geq -\langle\a_i,\la\rangle$ which by the characterisation of $B(\la)$ as a subset of $B(-\infty)$ in Proposition \ref{ka} tells us that $b\in B(\la)$.
\end{proof}
\section{Computation of a Whittaker Function in Type A}\label{tok}
We conclude by presenting a sample calculation of a metaplectic Whittaker function, and show that in this case, it is equal to the $p$-part of a Weyl Group Multiple Dirichlet Series, as defined and studied in \cite{wgmdsxtal}. This identity will be more than an equality of functions, as we will show that for our method of writing a Whittaker function as a weighted sum over a crystal, we will recover exactly the same weighted sum that appears in \cite{wgmdsxtal}. We begin by reproducing the combinatorial description of the $p$-part of a type $A_r$ multiple Dirichlet series.
A Gelfand-Tsetlin pattern is a triangular array of natural numbers
\[
\mathfrak{T}=\left\{
\begin{array}{cccccccccc}
a_{0,0} & & a_{0,1} & & a_{0,2} & & \ldots & a_{0,r-1} & & a_{0,r} \\
& a_{1,1} & & a_{1,2} & & a_{1,3} & \ldots & & a_{1,r} & \\
& & \ddots & & & & & \Ddots & & \\
& & & & & a_{r,r} & & & &
\end{array}
\right\}
\]
subject to the inequalities $a_{i,j}\geq a_{i+1,j+1}\geq a_{i,j+1}$ for all $i$ and $j$.
To such a pattern, one attaches integers $e_{i,j}$ for all $1\leq i\leq j\leq r$ defined by
$$
e_{i,j}=\sum_{k=j}^r a_{i,k}-a_{i-1,k},
$$
and a weight
$$
G(\mathfrak{T})=\prod_{1\leq i\leq j\leq r} \gamma(a_{i,j}),\quad \gamma(a_{i,j})=\begin{cases} q^{e_{i,j}} &\text{if } a_{i-1,j-1}>a_{i,j}=a_{i-1,j}, \\
g(e_{i,j},0) &\text{if } a_{i-1,j-1}>a_{i,j}>a_{i-1,j}, \\
g(e_{i,j},-1) &\text{if } a_{i-1,j-1}=a_{i,j}>a_{i-1,j}, \\
0 &\text{if } a_{i-1,j-1}=a_{i,j}=a_{i-1,j}, \end{cases}
$$ where the Gauss sums $g(a,b)$ are as defined by (\ref{gausssumdefn}) (in Section \ref{prelim}). Furthermore, define integers
$$
k_i(\mathfrak{T})=\sum_{j=1}^r (a_{i,j}-a_{0,j}).
$$
Then the $p$-part of a type A multiple Dirichlet series is given by the following expression:
$$
\sum_{\mathfrak{T}}G(\mathfrak{T}) \,
q^{-2(k_1s_1+\ldots +k_rs_r)}
$$
where the sum is over all Gelfand-Tsetlin patterns with a fixed top row and $s_1,\ldots,s_r\in\C$ is a collection of complex variables.
We shall work with the simple group $G=SL_{r+1}$ with the nicest possible choice of the long word decomposition. Explicitly realise
$$\Phi^+=\{ (i,j)\in [r+1]\times [r+1]\mid i<j \}.$$
Write $s_i$ for the simple reflection corresponding to the simple root $(i,i+1)$. Then we choose $\ii\in\I$ corresponding to the long word decomposition $w_0=s_1(s_2s_1)(s_3s_2s_1)\ldots(s_rs_{r-1}\ldots s_2s_1)$. We shall refer to this word as the Gelfand-Tsetlin word. This choice of word determines the ordering $<_\ii$ which can be explicitly stated in the form $(i,j)<_\ii (i',j')$ if $i<i'$ or if $i=i'$ and $j<j'$.
Note that a particular feature of the Gelfand-Tsetlin word is that in performing Algorithm \ref{algorithm}, the Iwasawa decomposition for the lower right copy of $SL_r$ in $SL_{r+1}$ is performed first as a substep of performing the Iwasawa decomposition for $SL_{r+1}$. This enables us to perform induction on $r$, a feature we shall frequently exploit below.
For a more in depth study of our main algorithm, we shall explitly realise $G$ as a $(r+1)\times (r+1)$ matrix group in the usual way. Thus $e_{(i,j)}(x)$ has an $x$ appearing in the entry of the $i$-th row and $j$-th column, ones along the diagonal and zeroes elsewhere.
Firstly, we shall require further information regarding the top row of the matrix $p_1$ that appears in our Iwasawa decomposition.
\begin{lemma}
The entries in the top row of $p_1$ are given by
\[
(p_1)_{1j} =
\begin{cases}
\prod_{i=j}^{r}\frac{1}{w_{1,j+1}}, & \mbox{if }|y_{1,j}|>1 \mbox{ or } j=1\\
0, & \mbox{otherwise.}
\end{cases}
\]
\end{lemma}
\begin{proof}
The proof uses Proposition \ref{diagonalterms}, which in particular proves the lemma in the $j=1$ case, so we may assume $j>1$. By induction on $r$, we know $(p_{r+1})_{1j}=0$.
Note that the only time that the $i,j$-entry of the $p$-matrix in the algorithm is altered is when a multiplication by $h_{1,j}(y_{1,j}^{-1})e_{1,j}(y_{1,j})$ occurs.
Hence, $(p_1)_{ij}=(p_j)_{11}$ if $|y_{i,j}|\geq 1$ and is zero otherwise. Thus, the lemma follows from Proposition \ref{diagonalterms}.
\end{proof}
We shall remark that the above lemma, when applied to $SL_r$, yields the values of the entries of the second row of the matrix $p_{r+1}$.
We will write $\la$ as $\la=\sum_i \la_i\p_i$ where the $\p_i$ are the fundamental weights. Then the value of the character $\psi_\la$ is given by
$$
\psi_\la(u)=\prod_{i=1}^r \psi(\p^{-\la_i}x_{i,i+1}).
$$
Now we come to the formula for $\psi(u)$ in terms of the $w$ and $y$ variables.
Let \[
\Psi_{\la}^{i,j}(u)=\begin{cases}
\psi\left(y_{i,j}\,\p^{-\la_i}\displaystyle\prod_{k=j+1}^{r+1}\frac{w_{i,k}}{w_{i+1,k}}\right) & \mbox{if }j=i+1\mbox{ or }|y_{i+1,j}|>1 \\
1 &\mbox{otherwise}.
\end{cases}
\]
\begin{proposition}\label{archaracterformula}
We have the following formula for the value of the additive character $\psi$ on $U^-$, in the $w_\a$ coordinate system.
$$\psi_{\la}(u)=\prod_{\a\in\Phi^+} \Psi^\a_\la(u).$$
\end{proposition}
\begin{proof}
By induction on $r$, it suffices to consider the case where $i=1$. Since we know that $\psi_\la(u)=\prod_{i}\psi(x_{i,i+1}\p^{-\la_i})$, we have to keep track of the $2,1$-entry of the matrix $e_{-\a_1}(x_1)\ldots e_{-\a_k}(x_k)p_k$ in order to express $x_{1,2}$ in terms of the $w$ or $y$ variables. The following sequence of events happens to this entry (note that we only need to consider $k\leq r$).
Initially, the (2,1) entry is $x_{1,2}$. For $k$ running from $r+1$ to 3, the following happens.
First one subtracts $y_{1,k}(p_{r+1})_{2,k}$, then divides by $w_{1,k}$.
At the end of this process, one is left with $y_1$, and hence has a formula for $x_1$ in terms of the $w$ variables. Then a simple application of the fact that $\psi$ is an additive character, together with our knowledge of $(p_{r+1})_{2,k}$ obtained via the previous lemma produces our desired formula for $\psi_\la(u)$.
\end{proof}
With the help of the calculation of the previous lemma, as well as the characterisation of the finite crystals from Theorem \ref{finitecrystal} from the previous section, we are able to immediately deduce the following explicit description of $B(\la+\rho)$ in the case under consideration. The weight $\rho$ is as usual half the sum of the positive coroots, in terms of the fundamental weights, we have $\rho=\p_1+\ldots +\p_r$.
\begin{proposition}
We have $(\ii,\m)\in B(\la+\rho)$ if and only if the following inequalities hold.
For each $(i,j)\in\Phi^+$, $m_{i,j}\geq 0$ and
\begin{equation}\label{crystalinequalities}
\sum_{k=j}^{r+1} m_{i,k}\leq \lambda_{i}+1+\sum_{k=j}^{r}m_{i+1,k+1}.
\end{equation}
\end{proposition}
In order to describe a combinatorial formula for the Whittaker function, we first decorate the tuple of integers $\m$ by circles and boxes.
Suppose $(\ii,\m)\in B(\la+\rho)$. For each $\a\in\Phi^+$, we define a weighting $w(\m,\a)$ as follows. Say that $m_\a$ is circled if $m_\a=0$ and boxed if equality holds in the corresponding inequality in (\ref{crystalinequalities}), ie if $\a=(i,j)$ and $\sum_{k=j}^{r+1} m_{i,k}=\lambda_{i}+1+\sum_{k=j+1}^{r+1}m_{i+1,k+1}$. (This vocabulary is used to match that of \cite{wgmdsxtal}). Corresponding to this decorating, we define a weight function
\begin{equation}\label{wdefn}
w(\m,\a)=\begin{cases}
\frac{q-1}{q^2}g(r_\a,s_\a)\ & \text{if $m_\alpha$ is not circled } \\
1 & \text{if $m_\a$ is circled but not boxed} \\
0 & \text{if $m_\a$ is both boxed and circled} \end{cases}
\end{equation}
where the integers $r_\a$ and $s_\a$ are defined by
\[
r_{i,j}=\sum_{k\leq i} m_{k,j}
\] and
$$
s_{i,j}=\la_{i}+\sum_{k=j}^{r}m_{i+1,k+1}-\sum_{k=j}^{r+1}m_{i,k}.
$$
Note that in the case where $m_\a$ is neither boxed nor circled, then Proposition \ref{gauss} on evaluating such Gauss sums can be used to simplify the resulting expression, which only depends on the value of $r_\a$ modulo $n$.
\begin{theorem}
The integral over $C_\m^\ii$ is given by
\[
\int_{C_\m^\ii}f(u)\psi_\la(u)du=\begin{cases}
\prod_{\a\in\Phi^+}w(\m,\a)x_\a^{m_\a} & \text{if } (\ii,\m)\in B(\la+\rho) \\
0 & \text{otherwise} \end{cases}
\]
\end{theorem}
\begin{remark}
We expect that the vanishing of the above integral over $C_\m^\ii$ should occur for the case of an arbitrary root system and choice of $\ii\in I$, whenever $(\ii,\m)\notin B(\la+\rho)$. This would yield an expression for the metaplectic Whittaker function as a sum over $B(\la+\rho)$ as opposed to the $B(-\infty)$ that our expression is currently a priori a sum over.
\end{remark}
\begin{proof}
We will first present the proof in the case where $m_\a>0$ for all $\a\in\Phi^+$, then discuss the changes that need to be made in order to incorporate the general case.
Recall that we have the variables $w_\a=\p^{-m_\a}u_\a$. In terms of such variables, we have
$$
\psi(u)=\prod_{(i,j)\in\Phi^+} \psi\left( \p^{s_{ij}}\frac{\prod_{k=j}^{r+1}u_{i,k}}{\prod_{k=j}^r u_{i+1,k+1}} \right)
$$
whereas
$$
f(u)= \prod_{\a\in\Phi^+}(q^{-\langle\rho,\a^\vee\rangle}x_\a)^{m_\a}(u_\a,\p)^{m_\a+\sum_{\a<_\ii \beta}\langle\beta,\a^\vee\rangle m_\beta}.
$$
We will show that the change of variables
$$
t_{i,j}=\frac{\prod_{k=j}^{r+1}u_{i,k}}{\prod_{k=j}^r u_{i+1,k+1}}
$$
transforms the integral $\int_{C_\m^\ii}f(u)\psi_\la(u)du$ into the product of integrals over $O_F^\times$ which are the defining integrals for the Gauss sums $g(r_\a,s_\a)$, that is
$$
\int_{C_\m^\ii}f(u)\psi_\la(u)du=\prod_{\a\in\Phi^+}(q^{-\langle\rho,\a^\vee\rangle}x_\a)^{m_\a}\int_{O_F^\times} (t_\a,\p)^{r_\a}\psi(\p^{s_\a}t_\a)dt_\a.
$$
Note that in this equation, the normalisation of Haar measure on $O_F^\times$ differs from the normalisation chosen in the introduction, which results in the appearance of the factor $\frac{q-1}{q^2}$ in the definition of $w(\m,\a)$.
To see this it suffices to check that
$$
\prod_{\a\in\Phi^+}u_\a^{m_\a+\sum_{\a<_\ii \beta}\langle\beta,\a^\vee\rangle m_\beta}=\prod_{(i,j)\in\Phi^+}t_{i,j}^{r_{i,j}}.
$$
A term $u_{i,j}$ on the product in the right hand side of the above product appears with a positive exponent in $t_{i,k}$ for $k\leq j$ and with a negative exponent in $t_{i-1,k}$ for $k\leq j-1$. When we pair up and cancel the occurrences from these terms as much as possible, we are left with $u_{ij}^{m_{ik}}$ from these terms. There also occurs $u_{ij}$ with exponent $\sum_{k\leq j}m_{k,j}$ from the $t_{i,j}$ term, and an occurrence of $u_{ij}$ with exponent $-\sum_{k\leq i-1} m_{k,i}$ from the $t_{i-1,i}$ term. Combining these gives exactly the formula we want, which is enough to prove the theorem in the case where all $m_\a$ are positive.
Now we shall consider the alterations to the above that need to be considered when some of the $m_\a$ are allowed to take the value zero.
In general, we perform the same manipulations to get a product of integrals of $(t_{i,j},\p)^{r_{i,j}}\psi(t_{i,j}\p^{s_{i,j}})$ except in two cases. The first of these is when $m_{i,j}=0$ in which case the term $(t_{i,j},\p)^{r_{i,j}}$ is no longer there, so we are left with simply an integral of an additive character over $O_F$, which we can easily evaluate to $w(\m,\a)$.
The more subtle case to handle is when we have $m_{i+1,j}=0$, for then this forces the `disappearance' of the additive character in the integrand. By Proposition \ref{gauss}, this is not of concern unless $s_{i,j}\leq-1$, for otherwise we have an explicit evaluation of the Gauss sums so we can see that we get our desired integral. Now note that the condition $m_{i+1,j}=0$ implies that $s_{i,j}=s_{i,j-1}+m_{i,j-1}$. Since $m_{i,j-1}\geq 0$, we have $s_{i,j-1}\leq -1$. These inequalities imply that the integral over $t_{i,j-1}$ is zero from our previous work, unless $m_{i+1,j-1}=0$. We can deal with this latter case because we may assume without loss of generality that $m_{i+1,j-1}\neq 0$ (if it exists) by decreasing $j$ if necessary.
Thus in all cases, we are lead to an easily evaluated integral, yielding the desired product formula for the integral over $C_\m^\ii$, so we are done.
\end{proof}
We have thus proven the following theorem.
\begin{theorem}\label{main}
The value of the integral $I_\la$ defined in (\ref{iladefn}) which calculates the metaplectic Whittaker function is given by
$$
I_\la=\sum_{(\ii,\m)\in B(\la+\rho)}\prod_{\a\in \Phi^+}w(\m,\a)x_\a^{m_\a}.
$$ where the weight $w(\m,\a)$ is as defined in (\ref{wdefn}).
\end{theorem}
Noting that the sum over the crystal $B(\la+\rho)$ is identical to the corresponding sum over the crystal expressing the $p$-part of a Weyl group multiple Dirichlet series as presented in \cite{wgmdsxtal}, we obtain the following corollary
\begin{corollary}
The coefficients of the $p$-part of a Weyl group multiple Dirichlet series in type A is equal to the value of a Whittaker function evaluated at a torus element on the corresponding local metaplectic group.
\end{corollary}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 4,698 |
Q: Remove and obtain the value of select tags appearing after a changed select tag I have a below set of select tags
<div class="select_tags">
<select>
<option>val1</option>
<option>val2</option>
</select>
<select>
<option>val11</option>
<option>val12</option>
</select>
<select>
<option>val21</option>
<option>val22</option>
</select>
</div>
$(document).on('change', '.select_tags select', function() {
}
If the select tag value is already been changed and if the user tries to select the options (or change the options) again I want to remove the select tags appearing after the selected select tag
For ex if the user selects the first set of select tag which they had already changed and selected the option value the div tag should contain only the changed set of tag and remove the tags appearing after it and get the selected option values for the tags that are removed
Result required
<div class="select_tags">
<select>
<option>val1</option>
<option>val2</option>
</select>
</div>
A: You can do something using .nextAll():
$(document).on('change', '.select_tags select', function() {
$(this).nextAll("select").remove();
});
Snippet here:
$(function () {
$(document).on('change', '.select_tags select', function () {
console.log([...$(this).nextAll("select").map(function () {
return $(this).val();
})]);
$(this).nextAll("select").remove();
});
});
<script src="https://cdnjs.cloudflare.com/ajax/libs/jquery/3.3.1/jquery.min.js"></script>
<div class="select_tags">
<select>
<option>val1</option>
<option>val2</option>
</select>
<select>
<option>val11</option>
<option>val12</option>
</select>
<select>
<option>val21</option>
<option>val22</option>
</select>
</div>
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 9,374 |
Henna is one of the most powerful sorcerers in the Order of Napthol, and her runes 's runes tell her that the future of Kavet is balanced on the edge of the knife. The treaties between Kavet and the dragon-like race known as the Osei have become intolerable. The time has come for the royal house to magically challenge Osei dominion. Prince Verte, Henna' lover, is to serve as the nexus for the powerful but dangerous spell, with Naples-an untested young sorcerer from the Order of Napthol-a volatile but critical support to its creation.
Amid these plans, Dahlia Indathrone's arrival in the city shouldn't matter. She has no magic and no royal lineage, and yet, Henna immediately knows Dahlia is important. She just can't see why.
As their lives intertwine, the four will learn that they are pawns in a larger game, one played by the forces of the Abyss and of the Numen-the infernal and the divine. | {
"redpajama_set_name": "RedPajamaC4"
} | 3,518 |
<div class="feature_deal" >
<h1 style="border-bottom-width: 0px;">deals</h1>
<div id="examplePanel1" class="deal_body" >
<!--Deal search Box Start-->
<form action="" method="post" id="srch_deal_frm">
<div class="searching_box">
<label>Keyword</label>
<input type="text" id="srch_keyword" name="keyword" value="<?php echo $search['keyword'] ?>">
<label>Category</label>
<select name="category" id="category">
<option value="">Select Category</option>
<?php echo $deals_category_name; ?>
</select>
<div class="clear"></div>
<label>Location</label>
<select name="location" id="location">
<option value="">Select Location</option>
<?php echo $deals_location_name; ?>
</select>
<label>Type</label>
<select name="type">
<option value="">Select Type</option>
<option value="1" <?php echo ($search['type']==1)?'selected="selected"':'' ?> >Online</option>
<option value="2" <?php echo ($search['type']==2)?'selected="selected"':'' ?> >Offline</option>
</select>
<div class="clear"></div>
<div class="margin10"></div>
<input type="hidden" name="hdn_srch" value="srch">
<input type="submit" class="button_01" value="SEARCH" style="margin-right: 12px; float: right;">
<input type="button" class="button_03" value="CLEAR SEARCH" style="margin-right: 12px; float: right;" onclick="clr_select()">
<div class="clear"></div>
</div>
</form>
<!--Deal search Box End-->
<script type="text/javascript">
function clr_select()
{
//$('#srch_keyword').val('');
//$('#srch_deal_frm select option').removeAttr('selected');
window.location.href = '<?php echo base_url()?>view_deals';
}
</script>
<div style="text-align: center; height: 15px; padding-top: 8px; " id="show_loader"></div>
<div id="deals_listing">
<?php
if($current_deals_count>0):
$row_index = 0;
?>
<!--Deal Box Start-->
<?php
foreach($current_deals as $deals_list):
$details_url = base_url().'view_deals/details/'.$deals_list['id'];
//pr($deals_list);
$start_date_str = date('M d, Y', $deals_list['deal_start']);
$end_date_str = date('M d, Y', $deals_list['deal_end']);
//$image_url = (!isset($deals_list['small_image_url']))?prep_url($deals_list['small_image_url']):base_url().'images/front/no_image_small.jpg';
$image_url = (!empty($deals_list['small_image_url']))?$deals_list['small_image_url']:base_url().'images/front/no_image_small.jpg';
?>
<div class="deal_box" <?php echo ($row_index&1)?'style="background: none repeat scroll 0% 0% rgb(243, 243, 243);"':'' ?> >
<div class="cell_01">
<a href="<?php echo $details_url ?>">
<div style="overflow: hidden; height: 77px; width: 77px;" >
<img alt="" src="<?php echo $image_url ?>" height="77" />
</div>
</a>
</div>
<div class="cell_02">
<h5><a href="<?php echo $details_url ?>">
<?php echo isset($deals_list['headline'])?$deals_list['headline']:'';?>
</a></h5>
<h6>
<?php echo isset($deals_list['street_address'])?$deals_list['street_address'].'-':'';?>
<?php echo isset($deals_list['city'])?$deals_list['city'].'-':'';?>
<?php echo isset($deals_list['state'])?$deals_list['state']:'';?>
</h6>
<h6><span>Source: <?php echo isset($deals_list['source_name'])?$deals_list['source_name']:'';?></span></h6>
<h6><?php echo $start_date_str?> - <?php echo $start_date_str?></h6>
<p><?php echo isset($deals_list['deal_description'])?word_limiter(html_entity_decode($deals_list['deal_description']), 20):'';?></p>
</div>
<div class="cell_03">
<?php
// Price calculation ............[start]
$offer_price = number_format($deals_list['offer_price'],2);
$actual_price = number_format($deals_list['actual_price'],2);
$save_prc = number_format(($deals_list['actual_price'] - $deals_list['offer_price']), 2);
$save_prcnt = number_format($save_prc/$deals_list['actual_price']*100,0);
// Price calculation ............[end]
?>
<label>Value : <span> <img src="<?php echo base_url()?>images/front/rs.png" alt="" style="vertical-align: middle;" /> <?php echo $actual_price?></span></label>
<br />
<label>Discount :<span style="font-weight: bold;"><?php echo $save_prcnt; ?> %</span></label>
<br />
<label>Your Price : <span><img src="<?php echo base_url()?>images/front/rs.png" alt="" style="vertical-align: middle;" /> <?php echo $offer_price?></span></label>
<br>
<div class="button_box">
<div class="left_part" onclick="window.open('<?php echo prep_url($deals_list['external_site_link']) ?>')"><input type="button" value="Buy now"></div>
<div class="right_part" style="font-size: smaller;"><?php echo isset($deals_list['offer_price'])?intval($deals_list['offer_price']):'';?></div>
<br>
</div>
</div>
<br>
</div>
<?php
$row_index++;
endforeach;
?>
<!--Deal Box End-->
<div class="clear"></div>
<!--Deal paginition Start-->
<?php
if(!empty($page_links)):
?>
<div class="paging" style="font-size: 12px;">
<?php echo $page_links; ?>
</div>
<?php
endif;
?>
<div class="clear"></div>
<!--Deal paginition End-->
</div>
<?php
else:
?>
</div>
<div class="deal_box">
<div style="font-weight: bold; height: 100px; text-align: center; line-height: 100px;">
No new deals are there...
</div>
</div>
<?php
endif;
?>
</div>
</div>
<script type="text/javascript">
function slideToggle(id,imgid) {
$('#'+imgid).html('');
if($('#'+id).css('display') == "block") {
$('#'+imgid).html('<img align="absmiddle" src="<?php echo base_url()?>images/front/close_btn.png" />');
} else {
$('#'+imgid).html('<img align="absmiddle" src="<?php echo base_url()?>images/front/expand_btn.png" />');
}
$('#'+id).slideToggle();
}
function showBusyScreen(divId)
{
//$('#'+divId).show();
$('#'+divId).html('<img src="' + base_url + 'images/front/ajax-loader.gif" alt=""/>');
}
function hideBusyScreen(divId)
{
//$('#'+divId).hide();
$('#'+divId).html('');
}
</script>
| {
"redpajama_set_name": "RedPajamaGithub"
} | 8,953 |
This entry was posted in Crafts for kids, Perfect Pictture Book Friday, Picture Book Review, Uncategorized and tagged 2017 debut picture books, Acceptance, Albert Whitman, Andrea J. Loney, Bunnybear, Friendship, Paper plate bear craft by viviankirkfield. Bookmark the permalink.
Love your craft that accompany the books. i'll save this one to do with my grandson when he comes! And read him bunny bear.
Thanks! I hope that you and your grandson will enjoy Bunnybear.
Hurray! I'm so excited about this book and know my kids are going to fall in love with it. And probably pretend they are bunny-bears for weeks.
Haaa! Pretending to be a Bunnybear is fun. There's lots of bouncing and nose wiggling. Enjoy the book!
Isn't it adorable? Great job, Vivian!
Looking forward to this one! BunnyBear seems like a wonderful story, and that craft idea is adorable.
I love the cover art. The illustration gives a bear-sized hint at the story problem. Fitting in with others is a big part of childhood. I wish this book had come out a few years ago when my daughter changed schools. I know I'm going to enjoy this picture book!
Yes, I fell in love with the cover the moment I saw it. It basically sets up the whole story, but the ending is full of surprises. I hope you enjoy it.
Love bunnies & bears & I can't wait to read about Bunnybear!
Second glowing review I've read about this charming story. Must read!
I have been so looking forward to Andrea's debut picture book. And I love your activity.
Aww, thanks, Joanna! I'm so excited to finally get to share it with you all.
I too love books on acceptance. On reserve at the library, I can't wait to read this book.
Oh, how awesome! I hope you enjoy it!
Great interview and congratulations, Andrea! Very inspiring for a newer, yet to be published PB writer. Thanks!
This sounds like a fantastic book, and I loved the accompanying art activity! I look forward to reading this one!
What a great book. Very creative and universal in theme. Always love your crafts Vivian. | {
"redpajama_set_name": "RedPajamaC4"
} | 5,215 |
{"url":"http:\/\/www.r-bloggers.com\/page\/329\/","text":"## ttrTests This is a Test Test 3:Data Snoopy\n\nSeptember 30, 2011\nBy\n\nTHIS IS NOT INVESTMENT ADVICE.\u00a0 IT IS JUST AN EXAMPLE AND WILL LIKELY LOSE LOTS OF MONEY IF YOU PURSUE WHAT IS DISCUSSED.\u00a0 READER IS RESPONSIBLE FOR THEIR OWN GAINS OR LOSSES.\u00a0 IF YOU ARE AN UNLIKELY WINNER, I WOULD LOVE TO HEAR YOUR STO...\n\n## R tutorial on visualizations\/graphics\n\nSeptember 30, 2011\nBy\n\nRolf Lohaus, a Huxley postdoctoral fellow here in the EEB dept at Rice University, gave our R course a talk on basic visualizations in R this morning.Enjoy!\n\n## R 2.13.2 released\n\nSeptember 30, 2011\nBy\n\nThe R core team announced today that R 2.13.2 is now available: The byte pixies have rolled up R-2.13.2.tar.gz at 9:00 this morning. This is intended to be the final release of the 2.13 series, for the benefit of those apprehensive of putting 2.14.x into production use. This update fixes a number of minor bugs (for example, pch=\".\" will...\n\n## Rcpp 0.9.7\n\nSeptember 30, 2011\nBy\n\nA fresh maintenance release version 0.9.7 of Rcpp went onto CRAN and into Debian earlier today. This release contains two contributed fixes. The first, suggested by Darren Cook via the rcpp-devel mailing list, corrects how we had set up excepti...\n\n## Microfinance Map of India \u2013 another go\u2026\n\nSeptember 30, 2011\nBy\n\nI gave it another go, trying to get a map that looks a bit nicer. This time, I tried to compute something like a density or intensity in a certain area. On the\u00a0previous map, this was not visible very well. I used ggplot2 and a bit of R code, together with RGoogleMaps to produce the\n\n## Monitoring Productivity II \u2013 the Others\n\nSeptember 30, 2011\nBy\n\nIn previous Monitoring Productivity Experiment post I looked into the hours I spent in computer, now will look into the hours Others spend in computer, which is far more interesting :) To find things like what day people spend more time on computer, ho...\n\n## Setting the initial view of a motion chart in R\n\nSeptember 30, 2011\nBy\n\nFollowing on from my article about accessing and plotting World Bank data with R I want to talk about how to change the initial view of a motion chart.Over the last couple of weeks I have been asked a view times how to do this. For instance Stephen O'G...\n\nSeptember 29, 2011\nBy\n\nI searched around to see if there was a blog post somewhere describing how to customize one\u2019s .rprofile but was surprised to find just one outdated post. So here is quick intro on the topic. If you are a power R user, you already know about what it does. For those of you that don\u2019t,\n\n## Modelling with R: part 1\n\nSeptember 29, 2011\nBy\n\nWhen I started work about 3 months ago, I didn't know much more than loading data and executing standard Econometric commands in R. But now I feel much much much more confident in using R for work, for research, for puzzles, and sometimes just for fun....\n\n## Obama 2012 campaigning with analytics\n\nSeptember 29, 2011\nBy\n\nThe campaign to re-elect US president Barack Obama is hiring -- and the RDataMining blog noticed that several of the open positions seek R skills. If you want to be a Communications Analyst, Digital Strategy Analyst, or Statistical Modeling Analyst and you know R, there may be a job opening for you. Just goes to show there's no corner...\n\n## Googling Bayes\u2019 pictures\n\nSeptember 29, 2011\nBy\n\nI am writing way too many posts in a row on Google tools. I promise I will think about something else soon. I find amusing the possibility to launch a search in Google images by just dragging a picture into \u2026 Continue reading \u2192\n\n## A brief introduction to R for SAS and SPSS users\n\nSeptember 29, 2011\nBy\n\nIf you've used SAS or SPSS and want a jump-start into the basics of the popular R language, next week's webinar, Introduction to R for SAS and SPSS Users will be of interest to you. While R, SAS and SPSS are all three software systems for data analysis and graphics, the underlying concepts in R are quite different to...\n\n## Connect JAVA to R part 2\n\nSeptember 29, 2011\nBy\n\nTo follow on from the earlier post on using R through Java, it is even easier to get jri up and running as a NetBeans module. Why is this useful? Well the platform that the NetBeans IDE is built on \u2026 Continue reading \u2192\n\n## Paired sample t-test in R\n\nSeptember 28, 2011\nBy\n\nLet\u2019s walk through using R and Student\u2019s t-test to compare paired sample data. The book Statistics: The Exploration & Analysis of Data (6th edition, p505) presents the longitudinal study \u201cBone mass is recovered from lactation to postweaning in adolescent mothers \u2026 Continue reading \u2192\n\n## ttrTests This is a Test\u2013Test 1 and Test 2\n\nSeptember 28, 2011\nBy\n\nJust to remind everyone, THIS IS NOT INVESTMENT ADVICE AND ANY ACTIONS TAKEN BASED ON THIS DISCUSSION WILL PROBABLY RESULT IN SIGNIFICANT LOSSES. We had fun with the ttrTests package in two previous posts ttrTests: Its Great Thesis and Incredible Poten...\n\n## The R Graph Gallery goes social\n\nSeptember 28, 2011\nBy\n\nThe R Graph Gallery, the website from Romain Fran\u00e7ois that showcases hundreds of examples of data visualization with R, has new social features. Now, when you find a graph or chart you find appealing or useful, you can \"Like\" it on Facebook or \"+1\" it on Google+. This should be a great way of highlighting the best charts and...\n\n## Is the \u201cLong Tail\u201d a Useless Concept?\n\nIn response to my last post, \u201cThe Long Tail of the Pareto Distribution,\u201d Neil Gunther had the following comment:\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u201cUnfortunately, you've fallen into the trap of using the \u2018long tail\u2019 misnomer. If you think about it, it can't possibly be the length of the tail that sets distributions like Pareto and Zipf apart; even the negative exponential and Gaussian...\n\n## Data Science: a literature review\n\nSeptember 28, 2011\nBy\n\nJust what is Data Science, anyway? Here's one take: Ever since the term \"Data Scientist\" was coined by DJ Patil and Jeff Hammerbacker in 2009, there's been a vigorous debate on what the team actually means. More than 80% of statisticians consider themselves data scientists, but Data Science is more than just Statistics. (My own take is that Data...\n\n## Polyploidy in sugarcane\n\nSeptember 28, 2011\nBy\n\nWhile reading UseR conference abstracts I came across this sentence: \"Sugarcane is polypoid, i.e., has 8 to 14 copies of every chromosome, with individual alleles in varying numbers.\" Vau! This generates really complex genotype system. Say we have biallelic gene with alleles being A and B. In diploids the possible genotypes are AA, AB, and BB. Given the...\n\n## Bessel integral\n\nSeptember 28, 2011\nBy\n$Bessel integral$\n\nPierre Pudlo and I worked this morning on a distribution related to philogenic trees and got stuck on the following Bessel integral where In is the modified Bessel function of the first kind. We could not find better than formula 6.611(4) in Gradshteyn and Ryzhik. which is for a=0\u2026 Anyone in for a closed form\n\n## Using transparency for data count intuition\n\nSeptember 27, 2011\nBy\n\nThis is an illustration of representing point count in a graphic using transparency. This is easy to do in ggplot2 if you use one of the barchart type of geoms. \u00a0However I think there are other situations where it would be useful to apply aesthetics based on point count. Since Hadley did a lot of\n\n## World Tourism Day, and Google Public Data Explore\n\nSeptember 27, 2011\nBy\n\nToday is\u00a0the World Tourism Day! So let\u2019s speak about some tourism related datasets \u2013 and others. Among other nice functions, Google offers a Public Data Explore in a beta version which provides a collection of datasets from OECD, IMF, Eurostat, \u2026 Continue reading \u2192\n\n## Five new local R user groups\n\nSeptember 27, 2011\nBy\n\nLooks like there's been a lot of activity in the R user community in the Northern hemisphere now that the summer break is over. I've just added several new groups to the Local R User Group Directory: Tokyo, Japan: The Tokyo.R R study group has already had 17 meetings, but has just been added to the directory. Shanghai\/East China:...\n\n## Tikz Introduction\n\nSeptember 27, 2011\nBy\n\nThe pgf drawing package for LaTeX provides facilities for drawing simple of complicated pictures within a LaTeX document. There are many options available within the package and in this post we consider some of the basics to get up and running. Fast Tube by Casper As with all LaTeX documents we need to select a\n\n## Basic line chart with ggplot2\n\nSeptember 27, 2011\nBy\n\nggplot2 is a package for R which easily draws plots that are easier on the eyes than R\u2019s built-in plotting functions, though the grammar is different than what is commonly used in R. This code demonstrates how to prepare a \u2026 Continue reading \u2192\n\n## Ghastly R code\n\nSeptember 27, 2011\nBy\n\nMy R package, R\/qtl, contains about 33k lines of R code (and 21k lines of C code). Some of it is quite good; some of it is terrible. Here\u2019s another example of the terrible. I\u2019ve long needed to revise the function scantwo, for performing a two-dimensional genome scan for pairs of loci. I was looking\n\n## Project Euler: problem 6\n\nSeptember 27, 2011\nBy\n\nThe sum of the squares of the first ten natural numbers is,12\u00a0+ 22\u00a0+ ... + 102\u00a0= 385The square of the sum of the first ten natural numbers is,(1 + 2 + ... + 10)2\u00a0= 552\u00a0= 3025Hence the difference between the sum of the squares o...\n\n## Example 9.7: New stuff in SAS 9.3\u2013 Frailty models\n\nSeptember 27, 2011\nBy\n\nShared frailty models are a way of allowing correlated observations into proportional hazards models. Briefly, instead of l_i(t) = l_0(t)e^(x_iB), we allow l_ij(t) = l_0(t)e^(x_ijB + g_i), where observations j are in clusters i, g_i is typically norma...","date":"2014-09-01 23:30:43","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 1, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.19341084361076355, \"perplexity\": 3060.9608809251677}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2014-35\/segments\/1409535920849.16\/warc\/CC-MAIN-20140909044614-00169-ip-10-180-136-8.ec2.internal.warc.gz\"}"} | null | null |
\section*{Acknowledgements}
JQ acknowledges an Atlas fellowship awarded by CCLRC (now STFC) in association with St. Catherine's College, University of Oxford. CH acknowledges financial support from the Scottish Universities Physics Alliance (SUPA). BJP was supported by the ARC. We thank S. Ramos, D. Cabra and N. I. Gidopoulos for useful discussions.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 1,026 |
Q: My function is confused about whether or not it wants a second argument (python, matplotlib) I'm writing a (should be) simple program in Python using Tkinter and Matplotlib to control a machine. It's my first Python program, so apologies for any egregious non-conventions, and I welcome all feedback. Also, sorry for the large code block.
I'm trying to have a main page for controls, and a secondary page that shows up with a graph while the machine is performing a run. I've managed to get the second page to show up and go away properly, and show the graph. However, every time it adds a point to the graph, it yells at me about
>
Exception in Tkinter callback
Traceback (most recent call last):
File "/usr/lib/python3.4/tkinter/__init__.py", line 1536, in __call__
return self.func(*args)
File "/usr/lib/python3.4/tkinter/__init__.py", line 585, in callit
func(*args)
TypeError: graph_animate() missing 1 required positional argument: 'i'
So I removed the ", i" argument from my graph function so it only takes (self), then it yelled at me in a different manner:
>
Exception in Tkinter callback
Traceback (most recent call last):
File "/usr/local/lib/python3.4/dist-packages/matplotlib/figure.py", line 381, in show
manager = getattr(self.canvas, 'manager')
AttributeError: 'FigureCanvasTkAgg' object has no attribute 'manager
followed by a bunch of locations within files and modules, ending with:
>
....File "/usr/local/lib/python3.4/dist-packages/matplotlib/animation.py", line 1212, in _draw_frame
self._drawn_artists = self._func(framedata, *self._args)
TypeError: graph_animate() takes 1 positional argument but 2 were given
I'm not very familiar with troubleshooting in Python, but it seems like a catch-22 type deal where I specify another argument and it says I don't have it, so I take it out and it says it's got too many. I spent a fair amount of time trying to figure out what was going on, and how constructors are specified in tkinter, but I still couldn't fix the error.
import matplotlib
matplotlib.use("TkAgg")
import matplotlib.animation as animation
from matplotlib import style
import datetime
import time
from matplotlib.backends.backend_tkagg import FigureCanvasTkAgg, NavigationToolbar2TkAgg
from matplotlib.figure import Figure
import tkinter as tk
from tkinter import ttk, StringVar
import urllib
import json
import pandas as pd
import numpy as np
from matplotlib import pyplot as plt
f = Figure()
a = f.add_subplot(111)
gsFurnaceTemp = 0
plottime = []
plottemp = []
class BrokeApp:
global gsFurnaceTemp
def __init__(self, parent):
self.myParent=parent
self.root=parent
self.myContainer1=tk.Frame(parent)
self.myContainer1.pack()
label = tk.Label(self.myContainer1, text = "My Broken App")
label.grid(columnspan=4)
buttonStart=tk.Button(self.myContainer1, text="Start Run",
bg='green', height=3, width=20,
command = self.btnStartClick)
buttonStart.grid(row=7, column=1)
self.ChillerLabelText = StringVar()
self.ChillerLabelText.set(gsFurnaceTemp)
lblChillerTemp = tk.Label(self.myContainer1, textvariable=self.ChillerLabelText)
lblChillerTemp.grid(row=100, column=100)
self.getDataTimer()
def btnStartClick(self):
self.newWindow=tk.Toplevel(self.myParent)
self.app=GraphPage(self.newWindow)
def getDataTimer(self):
gsFurnaceTemp=open("TempTxt.txt", "r").read()
self.ChillerLabelText.set(str(gsFurnaceTemp))
self.root.after(1000, self.getDataTimer)
class GraphPage:
global gsFurnaceTemp
def __init__(self, parent):
self.myParent=parent
self.root=parent
self.master=parent
self.plottempz = []
self.plottimez = []
self.fz = Figure()
self.az = self.fz.add_subplot(111)
self.myContainer1=tk.Frame(parent)
self.myContainer1.pack()
label = tk.Label(self.myContainer1, text = "Temp Graph")
label.pack(pady=10, padx=10)
btnStartPage = ttk.Button(self.myContainer1, text="Back to Main Page",
command = self.close_Window)
btnStartPage.pack(padx = 10, pady = 10)
canvas = FigureCanvasTkAgg(self.fz, self.myContainer1)
canvas.show()
canvas.get_tk_widget().pack()
plottemp= []
plottime= []
ani = animation.FuncAnimation(self.fz, self.graph_animate, interval=1000)
self.fz.show()
def graph_animate(self, i): #THIS IS THE LOCATION WHERE I REMOVE AND
#INSERT THE ", i" WHICH HAS BEEN GIVING ME TROUBLES
gsFurnaceTemp=open("TempTxt.txt", "r").read()
plottemp.append(gsFurnaceTemp)
self.az.clear()
self.az.plot(plottemp)
self.root.after(1000, self.graph_animate)
gsFurnaceTemp=open("TempTxt.txt", "r").read()
def close_Window(self):
self.master.destroy()
def main():
root = tk.Tk()
app = BrokeApp(root)
root.geometry("800x600")
root.mainloop()
if __name__ == "__main__":
main()
Also, sorry if any of this code is redundant or any of my declared variables are unused. My brain is fairly fried just from figuring out how to condense it into a sufficiently short, yet compilable chunk from the larger program.
Steeeeeve
A: That i must be defined, you can check how 'frames' work in FuncAnimation. I will suggest you to read the documentation about matplotlib.animation.FuncAnimation.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 303 |
Q: How to convert object structure nodeJS i have an object with n members like this :
{
open: '122.31000000',
high: '137.50000000',
low: '119.50000000',
close: '135.92000000',
volume: '1436796.12449000',
closeTime: 1585007999999,
}
{
open: '132.63000000',
high: '137.13000000',
low: '121.10000000',
close: '122.32000000',
volume: '1055699.86532000',
closeTime: 1584921599999,
}
I want to convert this object like this without using any for loop, how can i do this?
{
o: '122.31000000',
h: '137.50000000',
l: '119.50000000',
c: '135.92000000',
v: '1436796.12449000',
d: 1585007999999
},
{
o: '132.63000000',
h: '137.13000000',
l: '121.10000000',
c: '122.32000000',
v: '1055699.86532000'
d: '1584921599999'
}
A: You are going to have to use some sort of loop. One option is you can use map with destructuring.
const data = [{
open: '122.31000000',
high: '137.50000000',
low: '119.50000000',
close: '135.92000000',
volume: '1436796.12449000',
closeTime: 1585007999999,
},
{
open: '132.63000000',
high: '137.13000000',
low: '121.10000000',
close: '122.32000000',
volume: '1055699.86532000',
closeTime: 1584921599999,
}]
const updated = data.map(({open: o, high: h, low: l, close: c, volume: v, closeTime: d}) => ({o, h, l, c, v, d}))
console.log(updated)
A: You can use .map() and Object.keys() to transform into your expected result.
const data = [{ open: '122.31000000', high: '137.50000000', low: '119.50000000', close: '135.92000000', volume: '1436796.12449000', closeTime: 1585007999999, }, { open: '132.63000000', high: '137.13000000', low: '121.10000000', close: '122.32000000', volume: '1055699.86532000', closeTime: 1584921599999, }];
const result = data.map(e =>
Object.keys(e).reduce((a, c) => {
if (c === 'closeTime') a['d'] = e[c];
else a[c[0]] = e[c];
return a;
}, {})
);
console.log(result);
I hope this helps!
A: There's nothing wrong with for loops, but you can do this with other techniques. Note that this will not improve performance at all, since you're still technically looping through each item and building a new object for each one.
const propertyRenaming = {
open: 'o',
high: 'h',
low: 'l',
close: 'c',
volume: 'v',
closeTime: 'd'
};
const transformed = data.map(obj =>
Object.entries(data).reduce((result, [prop, value]) =>
Object.assign(result, { [propertyRenaming[prop]]: value }),
{}
)
);
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 3,381 |
Q: Stored procedure works in Crystal Reports but does not work in SQL Developer I have a report that works fine with this stored procedure in Crystal Reports:
BEGIN "CMS"."CMS_RP_MERCHANDISE"."SP_M4_OH_ITEMS"(:MCUR, 1, '', '', '', '', '', '', '', '', '', '', '', '', '', '', '', '', '', '', '', '', '', '', '', '', '', '', '', '', '', '', '', '', '', '', '', '', '', '', '', '', '', '', '', '', '', NULL); END ;
When I try to use the same stored procedure in Oracle SQL Developer, I get this message:
Error starting at line : 2 in command -
Error report -
ORA-06550: row 1 ، col 2543 :
PLS-00306: wrong number or types of arguments in call to 'SP_M4_OH_ITEMS'
ORA-06550: row 1 ، col 2543 :
PL/SQL: Statement ignored
06550. 00000 - "line %s, column %s:\n%s"
*Cause: Usually a PL/SQL compilation error.
*Action:
This is stored procedure Declaration
Image from sql dev
A:
wrong number or types of arguments in call to 'SP_M4_OH_ITEMS'
I presume you did something wrong. That procedure has zillion of parameters, and it seems that you either missed some (i.e. you have too few or too many of them), or you missed their datatype (e.g. should be NUMBER but you provided a string).
Such a procedure is difficult to debug; error is in line 1, column 2543 (two thousand five hundred forty three?!); can't you format it somehow and make it easier to read and maintain? Who is going to scroll that far right?
Besides, you didn't post CREATE PROCEDURE code; we saw you declared many variables, passed them to the procedure, but - without seeing it, it is difficult to say much more. I hope you'll manage to fix it after reading what I wrote. Good luck!
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 1,683 |
Genesis art workshops in Tokyo for intuitive and artistic exploration using set of 44 Genesis cards created by Liane Wakabayashi. Ideal for art beginners as well as adults and children with creative passion. Painting, drawing, collaging from fully equipped art studio on Okachimachi. Outdoor painting workshops offered in magnificent Shinjuku Gyoen, Baji Koen and other classic garden settings around Tokyo. Training offered for creative people particularly interested in exploring new medium for expressing intuition, for all in the healing , art therapy, teaching worlds, and for parents too. | {
"redpajama_set_name": "RedPajamaC4"
} | 7,966 |
Q: MongoDB aggregation crashes when cursor or explain is used Since version 3.6 MongoDB requires the use of cursor or explain in aggregate queries. It's a breaking change so I have to modify my earlier code.
But when I add cursor or explain to my query, the request simply enters an endless loop and MongoDB never responds. It doesn't even seem to time out.
This simple aggregation just hangs the code and never responds:
db.collection('users').aggregate([{ $match: { username: 'admin' }}],
{ cursor: {} },
(err, docs) => {
console.log('Aggregation completed.');
});
I can replace { cursor: {} } with { explain: true } and the result is the same. It works perfectly under older MongoDB versions without this one parameter.
Without cursor or explain I get this error message:
The 'cursor' option is required, except for aggregate with the explain argument
I'm not the only one who ran into this:
https://github.com/nosqlclient/nosqlclient/issues/419
A: OK, this was a little tricky, but finally it works. Looks like there are some major breaking changes in MongoDB's Node.js driver which nobody bothered to tell me.
1. The Node.js MongoDB driver has to be upgraded. My current version is 3.0.7.
2. The way how MongoDB connects has been changed, breaking any old code. The client connection now returns a client object, not merely the db. It has to be handled differently. There is a SO answer explaining it perfectly:
db.collection is not a function when using MongoClient v3.0
3. Aggregations now return an AggregationCursor object, not an array of data. Instead of a callback now you have to iterate through it.
var cursor = collection.aggregate([ ... ],
{ cursor: { batchSize: 1 } });
cursor.each((err, docs) => {
...
});
So it seems you have to rewrite ALL your db operations after upgrading to MongoDB 3.6. Yay, thanks for all the extra work, MongoDB Team! Guess that's where I'm done with your product.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 7,912 |
{"url":"https:\/\/www.justexam.in\/chapter-wise-question-bank\/three-dimensional-geometry-22\/","text":"Class\/Course - Engineering Entrance\n\nSubject - Mathematics\n\nChapter - Three Dimensional Geometry\n\nAttention Engineering Entrance standard Aspirants! Just Exam provide a platform to all students who want to make practice for various subject online. We prepare this platform on the base of CBSE. If you want to join our unlimited test series, then please write us at info@justexam.in about your class\/course\n\n Dear User, Kindly login\/register to view answer & explanation of each question. Click here to Login\/Sign Up. Q.1\u00a0Two systems of rectangular axes have the same origin. If a plane cuts them at distance a,b,c and a',b', c' from the origin, then $\\frac{1}{a^{2}} + \\frac{1}{b^{2}} + \\frac{1}{c^{2}} + \\frac{1}{a'^{2}} + \\frac{1}{b'^{2}} + \\frac{1}{c'^{2}}$ = 0 $\\frac{1}{a^{2}} + \\frac{1}{b^{2}} - \\frac{1}{c^{2}} + \\frac{1}{a'^{2}} + \\frac{1}{b'^{2}} - \\frac{1}{c'^{2}}$ = 0 $\\frac{1}{a^{2}} - \\frac{1}{b^{2}} - \\frac{1}{c^{2}} + \\frac{1}{a'^{2}} - \\frac{1}{b'^{2}} - \\frac{1}{c'^{2}}$ = 0 $\\frac{1}{a^{2}} + \\frac{1}{b^{2}} + \\frac{1}{c^{2}} - \\frac{1}{a'^{2}} - \\frac{1}{b'^{2}} - \\frac{1}{c'^{2}}$ = 0","date":"2018-04-27 02:50:26","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.1909879595041275, \"perplexity\": 2696.0983108200558}, \"config\": {\"markdown_headings\": false, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2018-17\/segments\/1524125948950.83\/warc\/CC-MAIN-20180427021556-20180427041556-00166.warc.gz\"}"} | null | null |
Download the Essential Documentation >
Searching Documentation
Creating PDF Files from Documentation Pages
Index >
Commvault Terms of Use
Microsoft SQL Server EULA
Third-Party Trademark Notices
Third-Party Copyright Notices
Essential About Documentation Third-Party Copyright Notices
Updated Friday, January 29, 2021
DO NOT TRANSLATE OR LOCALIZE
%%The following software may be included in, shipped with, or otherwise used by this product:
NancyFx/Nancy is licensed under the MIT License
Copyright (c) 2010 Andreas Håkansson, Steven Robbins and contributors
The above copyright notice and this permission notice shall be included in
all copies or substantial portions of the Software.
OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN
THE SOFTWARE.
%%The following software may be included in, shipped with, or otherwise used by this product: 7Zip; Use of any of this software may be governed by the terms of the license below:
7-Zip Copyright (C) 1999-2012 Igor Pavlov. All rights reserved. Licenses for files are:
1) 7z.dll: GNU LGPL + unRAR restriction
2) All other files: GNU LGPL
The GNU LGPL + unRAR restriction means that you must follow both
GNU LGPL rules and unRAR restriction rules.
You can use 7-Zip on any computer, including a computer in a commercial
organization. You don't need to register or pay for 7-Zip.
GNU LGPL information
This library is free software; you can redistribute it and/or
modify it under the terms of the GNU Lesser General Public
License as published by the Free Software Foundation; either
version 2.1 of the License, or (at your option) any later version.
This library is distributed in the hope that it will be useful,
Lesser General Public License for more details.
You can receive a copy of the GNU Lesser General Public License from
http://www.gnu.org/
unRAR restriction
The decompression engine for RAR archives was developed using source
code of unRAR program.
All copyrights to original unRAR code are owned by Alexander Roshal.
The license for original unRAR code has the following restriction:
The unRAR sources cannot be used to re-create the RAR compression algorithm,
which is proprietary. Distribution of modified unRAR sources in separate form
or as a part of other software is permitted, provided that it is clearly
stated in the documentation and source comments that the code may
not be used to develop a RAR (WinRAR) compatible archiver.
Igor Pavlov
%%The following software may be included in, shipped with, or otherwise used by this product: Boost; Use of any of this software may be governed by the terms of the license below:
Boost Software License - Version 1.0 - August 17th, 2003
Permission is hereby granted, free of charge, to any person or organization
obtaining a copy of the software and accompanying documentation covered by
this license (the "Software") to use, reproduce, display, distribute,
execute, and transmit the Software, and to prepare derivative works of the
Software, and to permit third-parties to whom the Software is furnished to
do so, all subject to the following:
The copyright notices in the Software and this entire statement, including
the above license grant, this restriction and the following disclaimer,
must be included in all copies of the Software, in whole or in part, and
all derivative works of the Software, unless such copies or derivative
works are solely in the form of machine-executable object code generated by
a source language processor.
FITNESS FOR A PARTICULAR PURPOSE, TITLE AND NON-INFRINGEMENT. IN NO EVENT
SHALL THE COPYRIGHT HOLDERS OR ANYONE DISTRIBUTING THE SOFTWARE BE LIABLE
FOR ANY DAMAGES OR OTHER LIABILITY, WHETHER IN CONTRACT, TORT OR OTHERWISE,
%%The following software may be included in, shipped with, or otherwise used by this product: Anaconda; Use of any of this software may be governed by the terms of the license below:
Copyright (c) 2016, Anaconda, Inc.
modification, are permitted provided that the following conditions are met:
Redistributions of source code must retain the above copyright notice,
this list of conditions and the following disclaimer.
Redistributions in binary form must reproduce the above copyright notice,
this list of conditions and the following disclaimer in the documentation
and/or other materials provided with the distribution.
Neither the name of Continuum Analytics, Inc. (dba Anaconda, Inc.)
("Continuum") nor the names of any contributors may be used to endorse or
promote products derived from this software without specific prior written
permission.
THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS"
AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT OWNER OR CONTRIBUTORS BE
LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR
CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF
SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS
INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN
CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE)
ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF
THE POSSIBILITY OF SUCH DAMAGE.
%%The following software may be included in, shipped with, or otherwise used by this product: AdoptOpenJDK; Use of any of this software may be governed by the terms of the license below:
Version 2.0, January 2004 http://www.apache.org/licenses/
2. Grant of Copyright License. Subject to the terms and conditions of this License, each Contributor hereby grants to You a perpetual, worldwide, non-exclusive, no-charge, royalty-free, irrevocable copyright license to reproduce, prepare Derivative Works of, publicly display, publicly perform, sublicense, and distribute the Work and such Derivative Works in Source or Object form.
3. Grant of Patent License. Subject to the terms and conditions of this License, each Contributor hereby grants to You a perpetual, worldwide, non-exclusive, no-charge, royalty-free, irrevocable (except as stated in this section) patent license to make, have made, use, offer to sell, sell, import, and otherwise transfer the Work, where such license applies only to those patent claims licensable by such Contributor that are necessarily infringed by their Contribution(s) alone or by combination of their Contribution(s) with the Work to which such Contribution(s) was submitted. If You institute patent litigation against any entity (including a cross-claim or counterclaim in a lawsuit) alleging that the Work or a Contribution incorporated within the Work constitutes direct or contributory patent infringement, then any patent licenses granted to You under this License for that Work shall terminate as of the date such litigation is filed.
4. Redistribution. You may reproduce and distribute copies of the Work or Derivative Works thereof in any medium, with or without modifications, and in Source or Object form, provided that You meet the following conditions:
1. You must give any other recipients of the Work or Derivative Works a copy of this License; and
2. You must cause any modified files to carry prominent notices stating that You changed the files; and
3. You must retain, in the Source form of any Derivative Works that You distribute, all copyright, patent, trademark, and attribution notices from the Source form of the Work, excluding those notices that do not pertain to any part of the Derivative Works; and
4. If the Work includes a "NOTICE" text file as part of its distribution, then any Derivative Works that You distribute must include a readable copy of the attribution notices contained within such NOTICE file, excluding those notices that do not pertain to any part of the Derivative Works, in at least one of the following places: within a NOTICE text file distributed as part of the Derivative Works; within the Source form or documentation, if provided along with the Derivative Works; or, within a display generated by the Derivative Works, if and wherever such third-party notices normally appear. The contents of the NOTICE file are for informational purposes only and do not modify the License. You may add Your own attribution notices within Derivative Works that You distribute, alongside or as an addendum to the NOTICE text from the Work, provided that such additional attribution notices cannot be construed as modifying the License. You may add Your own copyright statement to Your modifications and may provide additional or different license terms and conditions for use, reproduction, or distribution of Your modifications, or for any such Derivative Works as a whole, provided Your use, reproduction, and distribution of the Work otherwise complies with the conditions stated in this License.
5. Submission of Contributions. Unless You explicitly state otherwise, any Contribution intentionally submitted for inclusion in the Work by You to the Licensor shall be under the terms and conditions of this License, without any additional terms or conditions. Notwithstanding the above, nothing herein shall supersede or modify the terms of any separate license agreement you may have executed with Licensor regarding such Contributions.
6. Trademarks. This License does not grant permission to use the trade names, trademarks, service marks, or product names of the Licensor, except as required for reasonable and customary use in describing the origin of the Work and reproducing the content of the NOTICE file.
7. Disclaimer of Warranty. Unless required by applicable law or agreed to in writing, Licensor provides the Work (and each Contributor provides its Contributions) on an "AS IS" BASIS, WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied, including, without limitation, any warranties or conditions of TITLE, NON-INFRINGEMENT, MERCHANTABILITY, or FITNESS FOR A PARTICULAR PURPOSE. You are solely responsible for determining the appropriateness of using or redistributing the Work and assume any risks associated with Your exercise of permissions under this License.
8. Limitation of Liability. In no event and under no legal theory, whether in tort (including negligence), contract, or otherwise, unless required by applicable law (such as deliberate and grossly negligent acts) or agreed to in writing, shall any Contributor be liable to You for damages, including any direct, indirect, special, incidental, or consequential damages of any character arising as a result of this License or out of the use or inability to use the Work (including but not limited to damages for loss of goodwill, work stoppage, computer failure or malfunction, or any and all other commercial damages or losses), even if such Contributor has been advised of the possibility of such damages.
9. Accepting Warranty or Additional Liability. While redistributing the Work or Derivative Works thereof, You may choose to offer, and charge a fee for, acceptance of support, warranty, indemnity, or other liability obligations and/or rights consistent with this License. However, in accepting such obligations, You may act only on Your own behalf and on Your sole responsibility, not on behalf of any other Contributor, and only if You agree to indemnify, defend, and hold each Contributor harmless for any liability incurred by, or claims asserted against, such Contributor by reason of your accepting any such warranty or additional liability.
%%The following software may be included in, shipped with, or otherwise used by this product: Browser Mob; Use of any of this software may be governed by the terms of the license below:
%%The following software may be included in, shipped with, or otherwise used by this product: argparse; Use of any of this software may be governed by the terms of the license below:
argparse is (c) 2006-2009 Steven J. Bethard <steven.bethard@gmail.com>.
The argparse module was contributed to Python as of Python 2.7 and thus
was licensed under the Python license. Same license applies to all files in
the argparse package project.
For details about the Python License, please see doc/Python-License.txt.
Before (and including) argparse 1.1, the argparse package was licensed under
Apache License v2.0.
After argparse 1.1, all project files from the argparse project were deleted
due to license compatibility issues between Apache License 2.0 and GNU GPL v2.
The project repository then had a clean start with some files taken from
Python 2.7.1, so definitely all files are under Python License now.
PSF LICENSE AGREEMENT FOR PYTHON 2.7.2
1. This LICENSE AGREEMENT is between the Python Software Foundation ("PSF"), and
the Individual or Organization ("Licensee") accessing and otherwise using
Python 2.7.2 software in source or binary form and its associated documentation.
2. Subject to the terms and conditions of this License Agreement, PSF hereby
grants Licensee a nonexclusive, royalty-free, world-wide license to reproduce,
analyze, test, perform and/or display publicly, prepare derivative works,
distribute, and otherwise use Python 2.7.2 alone or in any derivative version,
provided, however, that PSF's License Agreement and PSF's notice of copyright,
i.e., "Copyright © 2001-2012 Python Software Foundation; All Rights Reserved"
are retained in Python 2.7.2 alone or in any derivative version prepared by
Licensee.
3. In the event Licensee prepares a derivative work that is based on or
incorporates Python 2.7.2 or any part thereof, and wants to make the derivative
work available to others as provided herein, then Licensee hereby agrees to
include in any such work a brief summary of the changes made to Python 2.7.2.
PSF is making Python 2.7.2 available to Licensee on an "AS IS" basis.
4. PSF MAKES NO REPRESENTATIONS OR WARRANTIES, EXPRESS OR IMPLIED. BY WAY OF
EXAMPLE, BUT NOT LIMITATION, PSF MAKES NO AND DISCLAIMS ANY REPRESENTATION OR
WARRANTY OF MERCHANTABILITY OR FITNESS FOR ANY PARTICULAR PURPOSE OR THAT THE
USE OF PYTHON 2.7.2 WILL NOT INFRINGE ANY THIRD PARTY RIGHTS.
PSF SHALL NOT BE LIABLE TO LICENSEE OR ANY OTHER USERS OF PYTHON 2.7.2 FOR ANY
INCIDENTAL, SPECIAL, OR CONSEQUENTIAL DAMAGES OR LOSS AS A RESULT OF MODIFYING,
DISTRIBUTING, OR OTHERWISE USING PYTHON 2.7.2, OR ANY DERIVATIVE THEREOF, EVEN
IF ADVISED OF THE POSSIBILITY THEREOF.
6. This License Agreement will automatically terminate upon a material breach
of its terms and conditions.
7. Nothing in this License Agreement shall be deemed to create any relationship
of agency, partnership, or joint venture between PSF and Licensee. This License
Agreement does not grant permission to use PSF trademarks or trade name in a
trademark sense to endorse or promote products or services of Licensee, or any
third party.
8. By copying, installing or otherwise using Python 2.7.2, Licensee agrees to
be bound by the terms and conditions of this License Agreement.
%%The following software may be included in, shipped with, or otherwise used by this product: Bootstrap-timepicker ; Use of any of this software may be governed by the terms of the MIT license below:
copies of the Software, and to permit persons to whom the Software is furnished
to do so, subject to the following conditions:
%%The following software may be included in, shipped with, or otherwise used by this product: Flask; Use of any of this software may be governed by the terms of the license below:
Flask License
Copyright (c) 2015 by Armin Ronacher and contributors. See AUTHORS for more details.
Some rights reserved.
Redistribution and use in source and binary forms of the software as well as documentation, with or without modification, are permitted provided that the following conditions are met:
• Redistributions of source code must retain the above copyright notice, this list of conditions and the following disclaimer.
• Redistributions in binary form must reproduce the above copyright notice, this list of conditions and the following disclaimer in the documentation and/or other materials provided with the distribution.
• The names of the contributors may not be used to endorse or promote products derived from this software without specific prior written permission.
THIS SOFTWARE AND DOCUMENTATION IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT OWNER OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE AND DOCUMENTATION, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
%%The following software may be included in, shipped with, or otherwise used by this product: Vangent HIEOS; Use of any of this software may be governed by the terms of the license below:
Vangent HIEOS License
Version 1.0, May 2009
[adapted from Apache License Version 2.0, January 2004]
"License" shall mean the terms and conditions for use, reproduction,
and distribution as defined by Sections 1 through 9 of this document.
"Licensor" shall mean the copyright owner or entity authorized by
the copyright owner that is granting the License.
"Legal Entity" shall mean the union of the acting entity and all
other entities that control, are controlled by, or are under common
control with that entity. For the purposes of this definition,
"control" means (i) the power, direct or indirect, to cause the
direction or management of such entity, whether by contract or
otherwise, or (ii) ownership of fifty percent (50%) or more of the
outstanding shares, or (iii) beneficial ownership of such entity.
"You" (or "Your") shall mean an individual or Legal Entity
exercising permissions granted by this License.
"Source" form shall mean the preferred form for making modifications,
including but not limited to software source code, documentation
source, and configuration files.
"Object" form shall mean any form resulting from mechanical
transformation or translation of a Source form, including but
not limited to compiled object code, generated documentation,
and conversions to other media types.
"Work" shall mean the work of authorship, whether in Source or
Object form, made available under the License, as indicated by a
copyright notice that is included in or attached to the work
(an example is provided in the Appendix below).
"Derivative Works" shall mean any work, whether in Source or Object
form, that is based on (or derived from) the Work and for which the
editorial revisions, annotations, elaborations, or other modifications
represent, as a whole, an original work of authorship. For the purposes
of this License, Derivative Works shall not include works that remain
separable from, or merely link (or bind by name) to the interfaces of,
the Work and Derivative Works thereof.
"Contribution" shall mean any work of authorship, including
the original version of the Work and any modifications or additions
to that Work or Derivative Works thereof, that is intentionally
submitted to Licensor for inclusion in the Work by the copyright owner
or by an individual or Legal Entity authorized to submit on behalf of
the copyright owner. For the purposes of this definition, "submitted"
means any form of electronic, verbal, or written communication sent
to the Licensor or its representatives, including but not limited to
communication on electronic mailing lists, source code control systems,
and issue tracking systems that are managed by, or on behalf of, the
Licensor for the purpose of discussing and improving the Work, but
excluding communication that is conspicuously marked or otherwise
designated in writing by the copyright owner as "Not a Contribution."
"Contributor" shall mean Licensor and any individual or Legal Entity
on behalf of whom a Contribution has been received by Licensor and
subsequently incorporated within the Work.
2. Grant of Copyright License. Subject to the terms and conditions of
this License, each Contributor hereby grants to You a perpetual,
worldwide, non-exclusive, no-charge, royalty-free, irrevocable
copyright license to reproduce, prepare Derivative Works of,
publicly display, publicly perform, sublicense, and distribute the
Work and such Derivative Works in Source or Object form.
3. Grant of Patent License. Subject to the terms and conditions of
(except as stated in this section) patent license to make, have made,
use, offer to sell, sell, import, and otherwise transfer the Work,
where such license applies only to those patent claims licensable
by such Contributor that are necessarily infringed by their
Contribution(s) alone or by combination of their Contribution(s)
with the Work to which such Contribution(s) was submitted. If You
institute patent litigation against any entity (including a
cross-claim or counterclaim in a lawsuit) alleging that the Work
or a Contribution incorporated within the Work constitutes direct
or contributory patent infringement, then any patent licenses
granted to You under this License for that Work shall terminate
as of the date such litigation is filed.
4. Redistribution. You may reproduce and distribute copies of the
Work or Derivative Works thereof in any medium, with or without
modifications, and in Source or Object form, provided that You
meet the following conditions:
(a) You must give any other recipients of the Work or
Derivative Works a copy of this License; and
(b) You must cause any modified files to carry prominent notices
stating that You changed the files; and
(c) You must retain, in the Source form of any Derivative Works
that You distribute, all copyright, patent, trademark, and
attribution notices from the Source form of the Work,
excluding those notices that do not pertain to any part of
the Derivative Works; and
(d) If the Work includes a "NOTICE" text file as part of its
distribution, then any Derivative Works that You distribute must
include a readable copy of the attribution notices contained
within such NOTICE file, excluding those notices that do not
pertain to any part of the Derivative Works, in at least one
of the following places: within a NOTICE text file distributed
as part of the Derivative Works; within the Source form or
documentation, if provided along with the Derivative Works; or,
within a display generated by the Derivative Works, if and
wherever such third-party notices normally appear. The contents
of the NOTICE file are for informational purposes only and
do not modify the License. You may add Your own attribution
notices within Derivative Works that You distribute, alongside
or as an addendum to the NOTICE text from the Work, provided
that such additional attribution notices cannot be construed
as modifying the License.
You may add Your own copyright statement to Your modifications and
may provide additional or different license terms and conditions
for use, reproduction, or distribution of Your modifications, or
for any such Derivative Works as a whole, provided Your use,
reproduction, and distribution of the Work otherwise complies with
the conditions stated in this License.
5. Submission of Contributions. Unless You explicitly state otherwise,
any Contribution intentionally submitted for inclusion in the Work
by You to the Licensor shall be under the terms and conditions of
this License, without any additional terms or conditions.
Notwithstanding the above, nothing herein shall supersede or modify
the terms of any separate license agreement you may have executed
with Licensor regarding such Contributions.
6. Trademarks. This License does not grant permission to use the trade
names, trademarks, service marks, or product names of the Licensor,
except as required for reasonable and customary use in describing the
origin of the Work and reproducing the content of the NOTICE file.
7. Disclaimer of Warranty. Unless required by applicable law or agreed to
in writing, Licensor and or Vangent, Inc. provides the Work (and each
Contributor provides its Contributions) on an "AS IS" BASIS,
WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or
implied, including, without limitation, any warranties or conditions
of TITLE, NON-INFRINGEMENT, MERCHANTABILITY, or FITNESS FOR A
PARTICULAR PURPOSE. You are solely responsible for determining the
appropriateness of using or redistributing the Work and assume any
risks associated with Your exercise of permissions under this License.
8. Limitation of Liability. In no event and under no legal theory,
whether in tort (including negligence), contract, or otherwise,
unless required by applicable law (such as deliberate and grossly
negligent acts) or agreed to in writing, shall Vangent or any Contributor
be liable to You or any other third party for damages, including any direct,
indirect, special, incidental, or consequential damages of any character
arising as a result of this License or out of the use or inability to use
the Work (including but not limited to damages for loss of goodwill,
work stoppage, computer failure or malfunction, or any and all other
commercial damages or losses), even if Vangent or any other such Contributor
has been advised of the possibility of such damages.
9. Accepting Warranty or Additional Liability. While redistributing
the Work or Derivative Works thereof, You may choose to offer,
and charge a fee for, acceptance of support, warranty, indemnity,
or other liability obligations and/or rights consistent with this
License. However, in accepting such obligations, You may act only
on Your own behalf and on Your sole responsibility, not on behalf
of Vangent or any other Contributor, and only if You agree to indemnify,
defend, and hold Vangent and each Contributor harmless for any liability
incurred by, or claims asserted against Vangent or any other such Contributor
by reason of your accepting any such warranty or additional liability.
%%The following software may be included in, shipped with, or otherwise used by this product: RASA ; Use of any of this software may be governed by the terms of the license below:
Copyright © 2019-2020 Rasa Technologies GmbH. All rights reserved.
%%The following software may be included in, shipped with, or otherwise used by this product: Chilkat; Use of any of this software may be governed by the terms of the license below:
Copyright 2000-2011 Chilkat Software, Inc. All rights reserved.
%%The following software may be included in, shipped with, or otherwise used by this product: CTree; Use of any of this software may be governed by the terms of the license below:
Portions copyright 1992-2012 Faircom Corporation. "FairCom" and "c-tree Plus" are trademarks of Faircom Corporation and are registered in the United States and other countries. All rights reserved.
%%The following software may be included in, shipped with, or otherwise used by this product: Codalogic LMX; Use of any of this software may be governed by the terms of the license below:
THE SOFTWARE and its documentation are protected by copyright and other intellectual property laws and treaties. CODALOGIC LTD owns the title, copyright, and other intellectual property rights in THE SOFTWARE, and its documentation. CODALOGIC LTD reserves all rights not expressly granted to you in this agreement. THE SOFTWARE is licensed, not sold.
%%The following software may be included in, shipped with, or otherwise used by this product: CodePlex DotNetZip; Use of any of this software may be governed by the terms of the license below:
Copyright © 2006 -2012, Microsoft Corporation. All rights reserved. Microsoft Public License. This license governs use of the accompanying software, the DotNetZip library ("the software"). If you use the software, you accept this license. If you do not accept the license, do not use the software.
The terms "reproduce," "reproduction," "derivative works," and "distribution" have the same meaning here as under U.S. copyright law.
A "contribution" is the original software, or any additions or changes to the software.
A "contributor" is any person that distributes its contribution under this license.
"Licensed patents" are a contributor's patent claims that read directly on its contribution.
2. Grant of Rights
(A) Copyright Grant- Subject to the terms of this license, including the license conditions and limitations in section 3, each contributor grants you a non-exclusive, worldwide, royalty-free copyright license to reproduce its contribution, prepare derivative works of its contribution, and distribute its contribution or any derivative works that you create.
(B) Patent Grant- Subject to the terms of this license, including the license conditions and limitations in section 3, each contributor grants you a non-exclusive, worldwide, royalty-free license under its licensed patents to make, have made, use, sell, offer for sale, import, and/or otherwise dispose of its contribution in the software or derivative works of the contribution in the software.
3. Conditions and Limitations
(A) No Trademark License- This license does not grant you rights to use any contributors' name, logo, or trademarks.
(B) If you bring a patent claim against any contributor over patents that you claim are infringed by the software, your patent license from such contributor to the software ends automatically.
(C) If you distribute any portion of the software, you must retain all copyright, patent, trademark, and attribution notices that are present in the software.
(D) If you distribute any portion of the software in source code form, you may do so only under this license by including a complete copy of this license with your distribution. If you distribute any portion of the software in compiled or object code form, you may only do so under a license that complies with this license.
(E) The software is licensed "as-is." You bear the risk of using it. The contributors give no express warranties, guarantees or conditions. You may have additional consumer rights under your local laws which this license cannot change. To the extent permitted under your local laws, the contributors exclude the implied warranties of merchantability, fitness for a particular purpose and non-infringement.
%%The following software may be included in, shipped with, or otherwise used by this product: Ionic.ZLib.dll is modified code based on jzlib; Use of any of this software may be governed by the terms of the license below:
Copyright (c) 2000,2001,2002,2003 ymnk, JCraft,Inc. All rights reserved.
1. Redistributions of source code must retain the above copyright
notice,this list of conditions and the following disclaimer.
2. Redistributions in binary form must reproduce the above copyright
notice, this list of conditions and the following disclaimer in
the documentation and/or other materials provided with the distribution.
3. The names of the authors may not be used to endorse or promote
Products derived from this software without specific prior written permission.
THIS SOFTWARE IS PROVIDED ``AS IS'' AND ANY EXPRESSED OR IMPLIED
WARRANTIES,INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL JCRAFT,INC. OR ANY CONTRIBUTORS TO THIS SOFTWARE BE LIABLE FOR ANY DIRECT,INDIRECT,INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA,OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE,EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
jzlib is based on zlib-1.1.3.
The following notice applies to zlib:
Copyright (C) 1995-2004 Jean-loup Gailly and Mark Adler
The ZLIB software is provided 'as-is', without any express or implied
warranty. In no event will the authors be held liable for any damages arising from the use of this software.
Permission is granted to anyone to use this software for any purpose,
including commercial applications, and to alter it and redistribute it freely, subject to the following restrictions:
1. The origin of this software must not be misrepresented; you must
not claim that you wrote the original software. If you use this
software in a product, an acknowledgment in the product documentation would be appreciated but is not required.
2. Altered source versions must be plainly marked as such, and must
not be misrepresented as being the original software.
3. This notice may not be removed or altered from any source
Jean-loup Gailly jloup@gzip.org
Mark Adler madler@alumni.caltech.edu
%%The following software may be included in, shipped with, or otherwise used by this product: cURL and Libcurl; Use of any of this software may be governed by the terms of the license below:
COPYRIGHT AND PERMISSION NOTICE
Copyright (c) 1996 - 2010, Daniel Stenberg, <daniel@haxx.se>.
Permission to use, copy, modify, and distribute this software for any purpose
with or without fee is hereby granted, provided that the above copyright
notice and this permission notice appear in all copies.
FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT OF THIRD PARTY RIGHTS. IN
NO EVENT SHALL THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM,
DAMAGES OR OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR
OTHERWISE, ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE
OR OTHER DEALINGS IN THE SOFTWARE.
Except as contained in this notice, the name of a copyright holder shall not
be used in advertising or otherwise to promote the sale, use or other dealings
in this Software without prior written authorization of the copyright holder.
%%The following software may be included in, shipped with, or otherwise used by this product: DJ Project; Use of any of this software may be governed by the terms of the license below:
GNU LESSER GENERAL PUBLIC LICENSE
Version 2.1, February 1999
Copyright (C) 1991, 1999 Free Software Foundation, Inc.
59 Temple Place, Suite 330, Boston, MA 02111-1307 USA
Everyone is permitted to copy and distribute verbatim copies
of this license document, but changing it is not allowed.
[This is the first released version of the Lesser GPL. It also counts
as the successor of the GNU Library Public License, version 2, hence
the version number 2.1.]
The licenses for most software are designed to take away your
freedom to share and change it. By contrast, the GNU General Public
Licenses are intended to guarantee your freedom to share and change
free software--to make sure the software is free for all its users.
This license, the Lesser General Public License, applies to some
specially designated software packages--typically libraries--of the
Free Software Foundation and other authors who decide to use it. You
can use it too, but we suggest you first think carefully about whether
this license or the ordinary General Public License is the better
strategy to use in any particular case, based on the explanations below.
When we speak of free software, we are referring to freedom of use,
not price. Our General Public Licenses are designed to make sure that
you have the freedom to distribute copies of free software (and charge
for this service if you wish); that you receive source code or can get
it if you want it; that you can change the software and use pieces of
it in new free programs; and that you are informed that you can do
these things.
To protect your rights, we need to make restrictions that forbid
distributors to deny you these rights or to ask you to surrender these
rights. These restrictions translate to certain responsibilities for
you if you distribute copies of the library or if you modify it.
For example, if you distribute copies of the library, whether gratis
or for a fee, you must give the recipients all the rights that we gave
you. You must make sure that they, too, receive or can get the source
code. If you link other code with the library, you must provide
complete object files to the recipients, so that they can relink them
with the library after making changes to the library and recompiling
it. And you must show them these terms so they know their rights.
We protect your rights with a two-step method: (1) we copyright the
library, and (2) we offer you this license, which gives you legal
permission to copy, distribute and/or modify the library.
To protect each distributor, we want to make it very clear that
there is no warranty for the free library. Also, if the library is
modified by someone else and passed on, the recipients should know
that what they have is not the original version, so that the original
author's reputation will not be affected by problems that might be
introduced by others.
Finally, software patents pose a constant threat to the existence of
any free program. We wish to make sure that a company cannot
effectively restrict the users of a free program by obtaining a
restrictive license from a patent holder. Therefore, we insist that
any patent license obtained for a version of the library must be
consistent with the full freedom of use specified in this license.
Most GNU software, including some libraries, is covered by the
ordinary GNU General Public License. This license, the GNU Lesser
General Public License, applies to certain designated libraries, and
is quite different from the ordinary General Public License. We use
this license for certain libraries in order to permit linking those
libraries into non-free programs.
When a program is linked with a library, whether statically or using
a shared library, the combination of the two is legally speaking a
combined work, a derivative of the original library. The ordinary
General Public License therefore permits such linking only if the
entire combination fits its criteria of freedom. The Lesser General
Public License permits more lax criteria for linking other code with
the library.
We call this license the "Lesser" General Public License because it
does Less to protect the user's freedom than the ordinary General
Public License. It also provides other free software developers Less
of an advantage over competing non-free programs. These disadvantages
are the reason we use the ordinary General Public License for many
libraries. However, the Lesser license provides advantages in certain
special circumstances.
For example, on rare occasions, there may be a special need to
encourage the widest possible use of a certain library, so that it becomes
a de-facto standard. To achieve this, non-free programs must be
allowed to use the library. A more frequent case is that a free
library does the same job as widely used non-free libraries. In this
case, there is little to gain by limiting the free library to free
software only, so we use the Lesser General Public License.
In other cases, permission to use a particular library in non-free
programs enables a greater number of people to use a large body of
free software. For example, permission to use the GNU C Library in
non-free programs enables many more people to use the whole GNU
operating system, as well as its variant, the GNU/Linux operating
Although the Lesser General Public License is Less protective of the
users' freedom, it does ensure that the user of a program that is
linked with the Library has the freedom and the wherewithal to run
that program using a modified version of the Library.
The precise terms and conditions for copying, distribution and
modification follow. Pay close attention to the difference between a
"work based on the library" and a "work that uses the library". The
former contains code derived from the library, whereas the latter must
be combined with the library in order to run.
TERMS AND CONDITIONS FOR COPYING, DISTRIBUTION AND MODIFICATION
0. This License Agreement applies to any software library or other
program which contains a notice placed by the copyright holder or
other authorized party saying it may be distributed under the terms of
this Lesser General Public License (also called "this License").
Each licensee is addressed as "you".
A "library" means a collection of software functions and/or data
prepared so as to be conveniently linked with application programs
(which use some of those functions and data) to form executables.
The "Library", below, refers to any such software library or work
which has been distributed under these terms. A "work based on the
Library" means either the Library or any derivative work under
copyright law: that is to say, a work containing the Library or a
portion of it, either verbatim or with modifications and/or translated
straightforwardly into another language. (Hereinafter, translation is
included without limitation in the term "modification".)
"Source code" for a work means the preferred form of the work for
making modifications to it. For a library, complete source code means
all the source code for all modules it contains, plus any associated
interface definition files, plus the scripts used to control compilation
and installation of the library.
Activities other than copying, distribution and modification are not
covered by this License; they are outside its scope. The act of
running a program using the Library is not restricted, and output from
such a program is covered only if its contents constitute a work based
on the Library (independent of the use of the Library in a tool for
writing it). Whether that is true depends on what the Library does
and what the program that uses the Library does.
1. You may copy and distribute verbatim copies of the Library's
complete source code as you receive it, in any medium, provided that
you conspicuously and appropriately publish on each copy an
appropriate copyright notice and disclaimer of warranty; keep intact
all the notices that refer to this License and to the absence of any
warranty; and distribute a copy of this License along with the
You may charge a fee for the physical act of transferring a copy,
and you may at your option offer warranty protection in exchange for a
2. You may modify your copy or copies of the Library or any portion
of it, thus forming a work based on the Library, and copy and
distribute such modifications or work under the terms of Section 1
above, provided that you also meet all of these conditions:
a) The modified work must itself be a software library.
b) You must cause the files modified to carry prominent notices
stating that you changed the files and the date of any change.
c) You must cause the whole of the work to be licensed at no
charge to all third parties under the terms of this License.
d) If a facility in the modified Library refers to a function or a
table of data to be supplied by an application program that uses
the facility, other than as an argument passed when the facility
is invoked, then you must make a good faith effort to ensure that,
in the event an application does not supply such function or
table, the facility still operates, and performs whatever part of
its purpose remains meaningful.
(For example, a function in a library to compute square roots has
a purpose that is entirely well-defined independent of the
application. Therefore, Subsection 2d requires that any
application-supplied function or table used by this function must
be optional: if the application does not supply it, the square
root function must still compute square roots.)
These requirements apply to the modified work as a whole. If
identifiable sections of that work are not derived from the Library,
and can be reasonably considered independent and separate works in
themselves, then this License, and its terms, do not apply to those
sections when you distribute them as separate works. But when you
distribute the same sections as part of a whole which is a work based
on the Library, the distribution of the whole must be on the terms of
this License, whose permissions for other licensees extend to the
entire whole, and thus to each and every part regardless of who wrote
Thus, it is not the intent of this section to claim rights or contest
your rights to work written entirely by you; rather, the intent is to
exercise the right to control the distribution of derivative or
collective works based on the Library.
In addition, mere aggregation of another work not based on the Library
with the Library (or with a work based on the Library) on a volume of
a storage or distribution medium does not bring the other work under
the scope of this License.
3. You may opt to apply the terms of the ordinary GNU General Public
License instead of this License to a given copy of the Library. To do
this, you must alter all the notices that refer to this License, so
that they refer to the ordinary GNU General Public License, version 2,
instead of to this License. (If a newer version than version 2 of the
ordinary GNU General Public License has appeared, then you can specify
that version instead if you wish.) Do not make any other change in
these notices.
Once this change is made in a given copy, it is irreversible for
that copy, so the ordinary GNU General Public License applies to all
subsequent copies and derivative works made from that copy.
This option is useful when you wish to copy part of the code of
the Library into a program that is not a library.
4. You may copy and distribute the Library (or a portion or
derivative of it, under Section 2) in object code or executable form
under the terms of Sections 1 and 2 above provided that you accompany
it with the complete corresponding machine-readable source code, which
must be distributed under the terms of Sections 1 and 2 above on a
medium customarily used for software interchange.
If distribution of object code is made by offering access to copy
from a designated place, then offering equivalent access to copy the
source code from the same place satisfies the requirement to
distribute the source code, even though third parties are not
compelled to copy the source along with the object code.
5. A program that contains no derivative of any portion of the
Library, but is designed to work with the Library by being compiled or
linked with it, is called a "work that uses the Library". Such a
work, in isolation, is not a derivative work of the Library, and
therefore falls outside the scope of this License.
However, linking a "work that uses the Library" with the Library
creates an executable that is a derivative of the Library (because it
contains portions of the Library), rather than a "work that uses the
library". The executable is therefore covered by this License.
Section 6 states terms for distribution of such executables.
When a "work that uses the Library" uses material from a header file
that is part of the Library, the object code for the work may be a
derivative work of the Library even though the source code is not.
Whether this is true is especially significant if the work can be
linked without the Library, or if the work is itself a library. The
threshold for this to be true is not precisely defined by law.
If such an object file uses only numerical parameters, data
structure layouts and accessors, and small macros and small inline
functions (ten lines or less in length), then the use of the object
file is unrestricted, regardless of whether it is legally a derivative
work. (Executables containing this object code plus portions of the
Library will still fall under Section 6.)
Otherwise, if the work is a derivative of the Library, you may
distribute the object code for the work under the terms of Section 6.
Any executables containing that work also fall under Section 6,
whether or not they are linked directly with the Library itself.
6. As an exception to the Sections above, you may also combine or
link a "work that uses the Library" with the Library to produce a
work containing portions of the Library, and distribute that work
under terms of your choice, provided that the terms permit
modification of the work for the customer's own use and reverse
engineering for debugging such modifications.
You must give prominent notice with each copy of the work that the
Library is used in it and that the Library and its use are covered by
this License. You must supply a copy of this License. If the work
during execution displays copyright notices, you must include the
copyright notice for the Library among them, as well as a reference
directing the user to the copy of this License. Also, you must do one
of these things:
a) Accompany the work with the complete corresponding
machine-readable source code for the Library including whatever
changes were used in the work (which must be distributed under
Sections 1 and 2 above); and, if the work is an executable linked
with the Library, with the complete machine-readable "work that
uses the Library", as object code and/or source code, so that the
user can modify the Library and then relink to produce a modified
executable containing the modified Library. (It is understood
that the user who changes the contents of definitions files in the
Library will not necessarily be able to recompile the application
to use the modified definitions.)
b) Use a suitable shared library mechanism for linking with the
Library. A suitable mechanism is one that (1) uses at run time a
copy of the library already present on the user's computer system,
rather than copying library functions into the executable, and (2)
will operate properly with a modified version of the library, if
the user installs one, as long as the modified version is
interface-compatible with the version that the work was made with.
c) Accompany the work with a written offer, valid for at
least three years, to give the same user the materials
specified in Subsection 6a, above, for a charge no more
than the cost of performing this distribution.
d) If distribution of the work is made by offering access to copy
from a designated place, offer equivalent access to copy the above
specified materials from the same place.
e) Verify that the user has already received a copy of these
materials or that you have already sent this user a copy.
For an executable, the required form of the "work that uses the
Library" must include any data and utility programs needed for
reproducing the executable from it. However, as a special exception,
the materials to be distributed need not include anything that is
normally distributed (in either source or binary form) with the major
components (compiler, kernel, and so on) of the operating system on
which the executable runs, unless that component itself accompanies
the executable.
It may happen that this requirement contradicts the license
restrictions of other proprietary libraries that do not normally
accompany the operating system. Such a contradiction means you cannot
use both them and the Library together in an executable that you
distribute.
7. You may place library facilities that are a work based on the
Library side-by-side in a single library together with other library
facilities not covered by this License, and distribute such a combined
library, provided that the separate distribution of the work based on
the Library and of the other library facilities is otherwise
permitted, and provided that you do these two things:
a) Accompany the combined library with a copy of the same work
based on the Library, uncombined with any other library
facilities. This must be distributed under the terms of the
Sections above.
b) Give prominent notice with the combined library of the fact
that part of it is a work based on the Library, and explaining
where to find the accompanying uncombined form of the same work.
8. You may not copy, modify, sublicense, link with, or distribute
the Library except as expressly provided under this License. Any
attempt otherwise to copy, modify, sublicense, link with, or
distribute the Library is void, and will automatically terminate your
rights under this License. However, parties who have received copies,
or rights, from you under this License will not have their licenses
terminated so long as such parties remain in full compliance.
9. You are not required to accept this License, since you have not
signed it. However, nothing else grants you permission to modify or
distribute the Library or its derivative works. These actions are
prohibited by law if you do not accept this License. Therefore, by
modifying or distributing the Library (or any work based on the
Library), you indicate your acceptance of this License to do so, and
all its terms and conditions for copying, distributing or modifying
the Library or works based on it.
10. Each time you redistribute the Library (or any work based on the
Library), the recipient automatically receives a license from the
original licensor to copy, distribute, link with or modify the Library
subject to these terms and conditions. You may not impose any further
restrictions on the recipients' exercise of the rights granted herein.
You are not responsible for enforcing compliance by third parties with
this License.
11. If, as a consequence of a court judgment or allegation of patent
infringement or for any other reason (not limited to patent issues),
conditions are imposed on you (whether by court order, agreement or
otherwise) that contradict the conditions of this License, they do not
excuse you from the conditions of this License. If you cannot
distribute so as to satisfy simultaneously your obligations under this
License and any other pertinent obligations, then as a consequence you
may not distribute the Library at all. For example, if a patent
license would not permit royalty-free redistribution of the Library by
all those who receive copies directly or indirectly through you, then
the only way you could satisfy both it and this License would be to
refrain entirely from distribution of the Library.
If any portion of this section is held invalid or unenforceable under any
particular circumstance, the balance of the section is intended to apply,
and the section as a whole is intended to apply in other circumstances.
It is not the purpose of this section to induce you to infringe any
patents or other property right claims or to contest validity of any
such claims; this section has the sole purpose of protecting the
integrity of the free software distribution system which is
implemented by public license practices. Many people have made
generous contributions to the wide range of software distributed
through that system in reliance on consistent application of that
system; it is up to the author/donor to decide if he or she is willing
to distribute software through any other system and a licensee cannot impose that choice.
This section is intended to make thoroughly clear what is believed to
be a consequence of the rest of this License.
12. If the distribution and/or use of the Library is restricted in
certain countries either by patents or by copyrighted interfaces, the
original copyright holder who places the Library under this License may add
an explicit geographical distribution limitation excluding those countries,
so that distribution is permitted only in or among countries not thus
excluded. In such case, this License incorporates the limitation as if
written in the body of this License.
13. The Free Software Foundation may publish revised and/or new
versions of the Lesser General Public License from time to time.
Such new versions will be similar in spirit to the present version,
but may differ in detail to address new problems or concerns.
Each version is given a distinguishing version number. If the Library
specifies a version number of this License which applies to it and
"any later version", you have the option of following the terms and
conditions either of that version or of any later version published by
the Free Software Foundation. If the Library does not specify a
license version number, you may choose any version ever published by
the Free Software Foundation.
14. If you wish to incorporate parts of the Library into other free
programs whose distribution conditions are incompatible with these,
write to the author to ask for permission. For software which is
copyrighted by the Free Software Foundation, write to the Free
Software Foundation; we sometimes make exceptions for this. Our
decision will be guided by the two goals of preserving the free status
of all derivatives of our free software and of promoting the sharing
and reuse of software generally.
15. BECAUSE THE LIBRARY IS LICENSED FREE OF CHARGE, THERE IS NO
WARRANTY FOR THE LIBRARY, TO THE EXTENT PERMITTED BY APPLICABLE LAW.
EXCEPT WHEN OTHERWISE STATED IN WRITING THE COPYRIGHT HOLDERS AND/OR
OTHER PARTIES PROVIDE THE LIBRARY "AS IS" WITHOUT WARRANTY OF ANY
KIND, EITHER EXPRESSED OR IMPLIED, INCLUDING, BUT NOT LIMITED TO, THE
IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR
PURPOSE. THE ENTIRE RISK AS TO THE QUALITY AND PERFORMANCE OF THE
LIBRARY IS WITH YOU. SHOULD THE LIBRARY PROVE DEFECTIVE, YOU ASSUME
THE COST OF ALL NECESSARY SERVICING, REPAIR OR CORRECTION.
16. IN NO EVENT UNLESS REQUIRED BY APPLICABLE LAW OR AGREED TO IN
WRITING WILL ANY COPYRIGHT HOLDER, OR ANY OTHER PARTY WHO MAY MODIFY
AND/OR REDISTRIBUTE THE LIBRARY AS PERMITTED ABOVE, BE LIABLE TO YOU
FOR DAMAGES, INCLUDING ANY GENERAL, SPECIAL, INCIDENTAL OR
CONSEQUENTIAL DAMAGES ARISING OUT OF THE USE OR INABILITY TO USE THE
LIBRARY (INCLUDING BUT NOT LIMITED TO LOSS OF DATA OR DATA BEING
RENDERED INACCURATE OR LOSSES SUSTAINED BY YOU OR THIRD PARTIES OR A
FAILURE OF THE LIBRARY TO OPERATE WITH ANY OTHER SOFTWARE), EVEN IF
SUCH HOLDER OR OTHER PARTY HAS BEEN ADVISED OF THE POSSIBILITY OF SUCH
DAMAGES.
%%The following software may be included in, shipped with, or otherwise used by this product: Ext GWT; Use of any of this software may be governed by the terms of the license below:
Copyright 2006-2010 Ext JS, LLC. All rights reserved.
%%The following software may be included in, shipped with, or otherwise used by this product: Sencha GXT3; Use of any of this software may be governed by the terms of the license below:
© 2012 Sencha, Inc. All rights reserved. under a commercial license from Sencha.
%% The following software may be included in, shipped with, or otherwise used by this product: Standard Widget Toolkit (SWT); Use of any of this software may be governed by the terms of the license below:
Eclipse Public License - v 1.0
THE ACCOMPANYING PROGRAM IS PROVIDED UNDER THE TERMS OF THIS ECLIPSE PUBLIC LICENSE ("AGREEMENT"). ANY USE, REPRODUCTION OR DISTRIBUTION OF THE PROGRAM CONSTITUTES RECIPIENT'S ACCEPTANCE OF THIS AGREEMENT.
"Contribution" means:
a) in the case of the initial Contributor, the initial code and documentation distributed under this Agreement, and
b) in the case of each subsequent Contributor:
i) changes to the Program, and
ii) additions to the Program;
where such changes and/or additions to the Program originate from and are distributed by that particular Contributor. A Contribution 'originates' from a Contributor if it was added to the Program by such Contributor itself or anyone acting on such Contributor's behalf. Contributions do not include additions to the Program which: (i) are separate modules of software distributed in conjunction with the Program under their own license agreement, and (ii) are not derivative works of the Program.
"Contributor" means any person or entity that distributes the Program.
"Licensed Patents" mean patent claims licensable by a Contributor which are necessarily infringed by the use or sale of its Contribution alone or when combined with the Program.
"Program" means the Contributions distributed in accordance with this Agreement.
"Recipient" means anyone who receives the Program under this Agreement, including all Contributors.
a) Subject to the terms of this Agreement, each Contributor hereby grants Recipient a non-exclusive, worldwide, royalty-free copyright license to reproduce, prepare derivative works of, publicly display, publicly perform, distribute and sublicense the Contribution of such Contributor, if any, and such derivative works, in source code and object code form.
b) Subject to the terms of this Agreement, each Contributor hereby grants Recipient a non-exclusive, worldwide, royalty-free patent license under Licensed Patents to make, use, sell, offer to sell, import and otherwise transfer the Contribution of such Contributor, if any, in source code and object code form. This patent license shall apply to the combination of the Contribution and the Program if, at the time the Contribution is added by the Contributor, such addition of the Contribution causes such combination to be covered by the Licensed Patents. The patent license shall not apply to any other combinations which include the Contribution. No hardware per se is licensed hereunder.
c) Recipient understands that although each Contributor grants the licenses to its Contributions set forth herein, no assurances are provided by any Contributor that the Program does not infringe the patent or other intellectual property rights of any other entity. Each Contributor disclaims any liability to Recipient for claims brought by any other entity based on infringement of intellectual property rights or otherwise. As a condition to exercising the rights and licenses granted hereunder, each Recipient hereby assumes sole responsibility to secure any other intellectual property rights needed, if any. For example, if a third party patent license is required to allow Recipient to distribute the Program, it is Recipient's responsibility to acquire that license before distributing the Program.
d) Each Contributor represents that to its knowledge it has sufficient copyright rights in its Contribution, if any, to grant the copyright license set forth in this Agreement.
3. REQUIREMENTS
A Contributor may choose to distribute the Program in object code form under its own license agreement, provided that:
a) it complies with the terms and conditions of this Agreement; and
b) its license agreement:
i) effectively disclaims on behalf of all Contributors all warranties and conditions, express and implied, including warranties or conditions of title and non-infringement, and implied warranties or conditions of merchantability and fitness for a particular purpose;
ii) effectively excludes on behalf of all Contributors all liability for damages, including direct, indirect, special, incidental and consequential damages, such as lost profits;
iii) states that any provisions which differ from this Agreement are offered by that Contributor alone and not by any other party; and
iv) states that source code for the Program is available from such Contributor, and informs licensees how to obtain it in a reasonable manner on or through a medium customarily used for software exchange.
When the Program is made available in source code form:
a) it must be made available under this Agreement; and
b) a copy of this Agreement must be included with each copy of the Program.
Contributors may not remove or alter any copyright notices contained within the Program.
Each Contributor must identify itself as the originator of its Contribution, if any, in a manner that reasonably allows subsequent Recipients to identify the originator of the Contribution.
4. COMMERCIAL DISTRIBUTION
Commercial distributors of software may accept certain responsibilities with respect to end users, business partners and the like. While this license is intended to facilitate the commercial use of the Program, the Contributor who includes the Program in a commercial product offering should do so in a manner which does not create potential liability for other Contributors. Therefore, if a Contributor includes the Program in a commercial product offering, such Contributor ("Commercial Contributor") hereby agrees to defend and indemnify every other Contributor ("Indemnified Contributor") against any losses, damages and costs (collectively "Losses") arising from claims, lawsuits and other legal actions brought by a third party against the Indemnified Contributor to the extent caused by the acts or omissions of such Commercial Contributor in connection with its distribution of the Program in a commercial product offering. The obligations in this section do not apply to any claims or Losses relating to any actual or alleged intellectual property infringement. In order to qualify, an Indemnified Contributor must: a) promptly notify the Commercial Contributor in writing of such claim, and b) allow the Commercial Contributor to control, and cooperate with the Commercial Contributor in, the defense and any related settlement negotiations. The Indemnified Contributor may participate in any such claim at its own expense.
For example, a Contributor might include the Program in a commercial product offering, Product X. That Contributor is then a Commercial Contributor. If that Commercial Contributor then makes performance claims, or offers warranties related to Product X, those performance claims and warranties are such Commercial Contributor's responsibility alone. Under this section, the Commercial Contributor would have to defend claims against the other Contributors related to those performance claims and warranties, and if a court requires any other Contributor to pay any damages as a result, the Commercial Contributor must pay those damages.
EXCEPT AS EXPRESSLY SET FORTH IN THIS AGREEMENT, THE PROGRAM IS PROVIDED ON AN "AS IS" BASIS, WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, EITHER EXPRESS OR IMPLIED INCLUDING, WITHOUT LIMITATION, ANY WARRANTIES OR CONDITIONS OF TITLE, NON-INFRINGEMENT, MERCHANTABILITY OR FITNESS FOR A PARTICULAR PURPOSE. Each Recipient is solely responsible for determining the appropriateness of using and distributing the Program and assumes all risks associated with its exercise of rights under this Agreement , including but not limited to the risks and costs of program errors, compliance with applicable laws, damage to or loss of data, programs or equipment, and unavailability or interruption of operations.
6. DISCLAIMER OF LIABILITY
EXCEPT AS EXPRESSLY SET FORTH IN THIS AGREEMENT, NEITHER RECIPIENT NOR ANY CONTRIBUTORS SHALL HAVE ANY LIABILITY FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING WITHOUT LIMITATION LOST PROFITS), HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OR DISTRIBUTION OF THE PROGRAM OR THE EXERCISE OF ANY RIGHTS GRANTED HEREUNDER, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGES.
If any provision of this Agreement is invalid or unenforceable under applicable law, it shall not affect the validity or enforceability of the remainder of the terms of this Agreement, and without further action by the parties hereto, such provision shall be reformed to the minimum extent necessary to make such provision valid and enforceable.
If Recipient institutes patent litigation against any entity (including a cross-claim or counterclaim in a lawsuit) alleging that the Program itself (excluding combinations of the Program with other software or hardware) infringes such Recipient's patent(s), then such Recipient's rights granted under Section 2(b) shall terminate as of the date such litigation is filed.
All Recipient's rights under this Agreement shall terminate if it fails to comply with any of the material terms or conditions of this Agreement and does not cure such failure in a reasonable period of time after becoming aware of such noncompliance. If all Recipient's rights under this Agreement terminate, Recipient agrees to cease use and distribution of the Program as soon as reasonably practicable. However, Recipient's obligations under this Agreement and any licenses granted by Recipient relating to the Program shall continue and survive.
Everyone is permitted to copy and distribute copies of this Agreement, but in order to avoid inconsistency the Agreement is copyrighted and may only be modified in the following manner. The Agreement Steward reserves the right to publish new versions (including revisions) of this Agreement from time to time. No one other than the Agreement Steward has the right to modify this Agreement. The Eclipse Foundation is the initial Agreement Steward. The Eclipse Foundation may assign the responsibility to serve as the Agreement Steward to a suitable separate entity. Each new version of the Agreement will be given a distinguishing version number. The Program (including Contributions) may always be distributed subject to the version of the Agreement under which it was received. In addition, after a new version of the Agreement is published, Contributor may elect to distribute the Program (including its Contributions) under the new version. Except as expressly stated in Sections 2(a) and 2(b) above, Recipient receives no rights or licenses to the intellectual property of any Contributor under this Agreement, whether expressly, by implication, estoppel or otherwise. All rights in the Program not expressly granted under this Agreement are reserved.
This Agreement is governed by the laws of the State of New York and the intellectual property laws of the United States of America. No party to this Agreement will bring a legal action under this Agreement more than one year after the cause of action arose. Each party waives its rights to a jury trial in any resulting litigation.
%%The following software may be included in, shipped with, or otherwise used by this product: SPNEGO; Use of any of this software may be governed by the terms of the license below:
%%The following software may be included in, shipped with, or otherwise used by this product: ggplot; Use of any of this software may be governed by the terms of the license below:
Copyright (c) 2013, yhat. All rights reserved.
Redistributions of source code must retain the above copyright notice, this list of conditions and the following disclaimer.
Redistributions in binary form must reproduce the above copyright notice, this list of conditions and the following disclaimer in the documentation and/or other materials provided with the distribution.
THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT HOLDER OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES;LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT(INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
%%The following software may be included in, shipped with, or otherwise used by this product: Google Web Toolkit; Use of any of this software may be governed by the terms of the license below:
©2009 Google Inc. Licensed under the Creative Commons Attribution License v.3, the details of which are located at http://creativecommons.org/licenses/by/3.0/legalcode.
%% The following software may be included in, shipped with, or otherwise used by this product: Google Protocol Buffers; Use of any of this software may be governed by the terms of the license below:
Copyright 2015 Google Inc. All rights reserved.
%% The following software may be included in, shipped with, or otherwise used by this product: Guacamole; Use of any of this software may be governed by the terms of the license below:
%% The following software may be included in, shipped with, or otherwise used by this product: HDF5; Use of any of this software may be governed by the terms of the license below:
Copyright Notice and License Terms for
HDF5 (Hierarchical Data Format 5) Software Library and Utilities
Copyright 2006-2016 by The HDF Group.
NCSA HDF5 (Hierarchical Data Format 5) Software Library and Utilities
Copyright 1998-2006 by the Board of Trustees of the University of Illinois.
modification, are permitted for any purpose (including commercial purposes)
provided that the following conditions are met:
1. Redistributions of source code must retain the above copyright notice,
this list of conditions, and the following disclaimer.
2. Redistributions in binary form must reproduce the above copyright notice,
this list of conditions, and the following disclaimer in the documentation
and/or materials provided with the distribution.
3. In addition, redistributions of modified forms of the source or binary
code must carry prominent notices stating that the original code was
changed and the date of the change.
4. All publications or advertising materials mentioning features or use of
this software are asked, but not required, to acknowledge that it was
developed by The HDF Group and by the National Center for Supercomputing
Applications at the University of Illinois at Urbana-Champaign and
credit the contributors.
5. Neither the name of The HDF Group, the name of the University, nor the
name of any Contributor may be used to endorse or promote products derived
from this software without specific prior written permission from
The HDF Group, the University, or the Contributor, respectively.
THIS SOFTWARE IS PROVIDED BY THE HDF GROUP AND THE CONTRIBUTORS
"AS IS" WITH NO WARRANTY OF ANY KIND, EITHER EXPRESSED OR IMPLIED. In no
event shall The HDF Group or the Contributors be liable for any damages
suffered by the users arising out of the use of this software, even if
advised of the possibility of such damage.
Contributors: National Center for Supercomputing Applications (NCSA) at
the University of Illinois, Fortner Software, Unidata Program Center (netCDF),
The Independent JPEG Group (JPEG), Jean-loup Gailly and Mark Adler (gzip),
and Digital Equipment Corporation (DEC).
Portions of HDF5 were developed with support from the Lawrence Berkeley
National Laboratory (LBNL) and the United States Department of Energy
under Prime Contract No. DE-AC02-05CH11231.
Portions of HDF5 were developed with support from the University of
California, Lawrence Livermore National Laboratory (UC LLNL).
The following statement applies to those portions of the product and must
be retained in any redistribution of source code, binaries, documentation,
and/or accompanying materials:
This work was partially produced at the University of California,
Lawrence Livermore National Laboratory (UC LLNL) under contract
no. W-7405-ENG-48 (Contract 48) between the U.S. Department of Energy
(DOE) and The Regents of the University of California (University)
for the operation of UC LLNL.
This work was prepared as an account of work sponsored by an agency of
the United States Government. Neither the United States Government nor
the University of California nor any of their employees, makes any
warranty, express or implied, or assumes any liability or responsibility
for the accuracy, completeness, or usefulness of any information,
apparatus, product, or process disclosed, or represents that its use
would not infringe privately- owned rights. Reference herein to any
specific commercial products, process, or service by trade name,
trademark, manufacturer, or otherwise, does not necessarily constitute
or imply its endorsement, recommendation, or favoring by the United
States Government or the University of California. The views and
opinions of authors expressed herein do not necessarily state or reflect
those of the United States Government or the University of California,
and shall not be used for advertising or product endorsement purposes.
HDF5 is available with the SZIP compression library but SZIP is not part
of HDF5 and has separate copyright and license terms. See "Szip Compression
in HDF Products" (www.hdfgroup.org/doc_resource/SZIP/) for further details.
%%The following software may be included in, shipped with, or otherwise used by this product: Junit.jar; Use of any of this software may be governed by the terms of the license below:
THE COMMON PUBLIC LICENSE (CPL)v 1
THE ACCOMPANYING PROGRAM IS PROVIDED UNDER THE TERMS OF THIS COMMON PUBLIC LICENSE ("AGREEMENT"). ANY USE, REPRODUCTION OR DISTRIBUTION OF THE PROGRAM CONSTITUTES RECIPIENT'S ACCEPTANCE OF THIS AGREEMENT.
a. in the case of the initial Contributor, the initial code and documentation distributed under this Agreement, and
b. in the case of each subsequent Contributor:
i. changes to the Program, and
ii. additions to the Program;
"Licensed Patents " mean patent claims licensable by a Contributor which are necessarily infringed by the use or sale of its Contribution alone or when combined with the Program.
GRANT OF RIGHTS
a. Subject to the terms of this Agreement, each Contributor hereby grants Recipient a non-exclusive, worldwide, royalty-free copyright license to reproduce, prepare derivative works of, publicly display, publicly perform, distribute and sublicense the Contribution of such Contributor, if any, and such derivative works, in source code and object code form.
b. Subject to the terms of this Agreement, each Contributor hereby grants Recipient a non-exclusive, worldwide, royalty-free patent license under Licensed Patents to make, use, sell, offer to sell, import and otherwise transfer the Contribution of such Contributor, if any, in source code and object code form. This patent license shall apply to the combination of the Contribution and the Program if, at the time the Contribution is added by the Contributor, such addition of the Contribution causes such combination to be covered by the Licensed Patents. The patent license shall not apply to any other combinations which include the Contribution. No hardware per se is licensed hereunder.
c. Recipient understands that although each Contributor grants the licenses to its Contributions set forth herein, no assurances are provided by any Contributor that the Program does not infringe the patent or other intellectual property rights of any other entity. Each Contributor disclaims any liability to Recipient for claims brought by any other entity based on infringement of intellectual property rights or otherwise. As a condition to exercising the rights and licenses granted hereunder, each Recipient hereby assumes sole responsibility to secure any other intellectual property rights needed, if any. For example, if a third party patent license is required to allow Recipient to distribute the Program, it is Recipient's responsibility to acquire that license before distributing the Program.
d. Each Contributor represents that to its knowledge it has sufficient copyright rights in its Contribution, if any, to grant the copyright license set forth in this Agreement.
a. it complies with the terms and conditions of this Agreement; and
b. its license agreement:
i. effectively disclaims on behalf of all Contributors all warranties and conditions, express and implied, including warranties or conditions of title and non-infringement, and implied warranties or conditions of merchantability and fitness for a particular purpose;
ii. effectively excludes on behalf of all Contributors all liability for damages, including direct, indirect, special, incidental and consequential damages, such as lost profits;
iii. states that any provisions which differ from this Agreement are offered by that Contributor alone and not by any other party; and
iv. states that source code for the Program is available from such Contributor, and informs licensees how to obtain it in a reasonable manner on or through a medium customarily used for software exchange.
a. it must be made available under this Agreement; and
b. a copy of this Agreement must be included with each copy of the Program.
EXCEPT AS EXPRESSLY SET FORTH IN THIS AGREEMENT, THE PROGRAM IS PROVIDED ON AN "AS IS" BASIS, WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, EITHER EXPRESS OR IMPLIED INCLUDING, WITHOUT LIMITATION, ANY WARRANTIES OR CONDITIONS OF TITLE, NON-INFRINGEMENT, MERCHANTABILITY OR FITNESS FOR A PARTICULAR PURPOSE. Each Recipient is solely responsible for determining the appropriateness of using and distributing the Program and assumes all risks associated with its exercise of rights under this Agreement, including but not limited to the risks and costs of program errors, compliance with applicable laws, damage to or loss of data, programs or equipment, and unavailability or interruption of operations.
If Recipient institutes patent litigation against a Contributor with respect to a patent applicable to software (including a cross-claim or counterclaim in a lawsuit), then any patent licenses granted by that Contributor to such Recipient under this Agreement shall terminate as of the date such litigation is filed. In addition, if Recipient institutes patent litigation against any entity (including a cross-claim or counterclaim in a lawsuit) alleging that the Program itself (excluding combinations of the Program with other software or hardware) infringes such Recipient's patent(s), then such Recipient's rights granted under Section 2(b) shall terminate as of the date such litigation is filed.
Everyone is permitted to copy and distribute copies of this Agreement, but in order to avoid inconsistency the Agreement is copyrighted and may only be modified in the following manner. The Agreement Steward reserves the right to publish new versions (including revisions) of this Agreement from time to time. No one other than the Agreement Steward has the right to modify this Agreement. IBM is the initial Agreement Steward. IBM may assign the responsibility to serve as the Agreement Steward to a suitable separate entity. Each new version of the Agreement will be given a distinguishing version number. The Program (including Contributions) may always be distributed subject to the version of the Agreement under which it was received. In addition, after a new version of the Agreement is published, Contributor may elect to distribute the Program (including its Contributions) under the new version. Except as expressly stated in Sections 2(a) and 2(b) above, Recipient receives no rights or licenses to the intellectual property of any Contributor under this Agreement, whether expressly, by implication, estoppel or otherwise. All rights in the Program not expressly granted under this Agreement are reserved.
%%%%The following software may be included in, shipped with, or otherwise used by this product: Kazoo; Use of any of this software may be governed by the terms of the license below:
9. Accepting Warranty or Additional Liability. While redistributing the Work or Derivative Works thereof, You may choose to offer, and charge a fee for, acceptance of support, warranty, indemnity, or other liability obligations and/or rights consistent with this License. However, in accepting such obligations, You may act only on Your own behalf and on Your sole responsibility, not on behalf of any other Contributor, and only if You agree to indemnify, defend, and hold each Contributor harmless for any liability incurred by, or claims asserted against, such Contributor by reason of your accepting any such warranty or additional liability
%%The following software may be included in, shipped with, or otherwise used by this product: FreeBSD; Use of any of this software may be governed by the terms of the license below:
Copyright 1992-2013 The FreeBSD Project. All rights reserved.
THIS SOFTWARE IS PROVIDED BY THE FREEBSD PROJECT ``AS IS'' AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE FREEBSD PROJECT OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
The views and conclusions contained in the software and documentation are those of the authors and should not be interpreted as representing official policies, either expressed or implied, of the FreeBSD Project.
%%The following software may be included in, shipped with, or otherwise used by this product: JGoodies; Use of any of this software may be governed by the terms of the license below:
Copyright (c) 2002-2008 JGoodies Karsten Lentzsch. All rights reserved.
o Redistributions of source code must retain the above copyright notice,
o Redistributions in binary form must reproduce the above copyright notice,
o Neither the name of JGoodies Karsten Lentzsch nor the names of
its contributors may be used to endorse or promote products derived
from this software without specific prior written permission.
AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO,
THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR
PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT OWNER OR
CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL,
EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO,
PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR PROFITS;
OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY,
WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR
OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE,
EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
%%The following software may be included in, shipped with, or otherwise used by this product: HBA API; Use of any of this software may be governed by the terms of the license below:
© 2009 SNIA. All rights reserved.
%%The following software may be included in, shipped with, or otherwise used by this product: Total HTMLConverterX; Use of any of this software may be governed by the terms of the license below:
Copyright Softplicity LLC. All rights reserved.
RFLA Royalty-Free License Agreement
THIS LICENSE AGREEMENT IS A LEGAL AND BINDING AGREEMENT ("AGREEMENT") IS BETWEEN YOU THE USER ("LICENSEE"), AND SOFTPLICITY LLC. THE WEBSITE WWW.COOLUTILS.COM IS OWNED AND OPERATED BY SOFTPLICITY LLC. THIS LICENSE AGREEMENT APPLIES TO LICENSES ISSUED VIA THE WEB AND VIA PHYSICAL DELIVERY ON A CD-ROM PRODUCT OR OTHER DIGITAL STORAGE MEDIA (THE "AGREEMENT). IF THE LICENSEE DOES NOT AGREE AND HAS NOT EXERCISED THE RIGHTS GRANTED UNDER THIS AGREEMENT, THE LICENSEE MAY PROMPTLY RETURN THE UN-OPENED AND UN-USED CD-ROM AND ALL ACCOMPANYING MATERIALS IN THE ORIGINAL PACKAGING FOR A REFUND OF THE PURCHASE PRICE. BY ACCEPTING THESE TERMS, YOU AGREE TO BE BOUND BY THE TERMS OF THIS AGREEMENT.
Subject to the terms of this License Agreement, Softplicity LLC. (Softplicity) grants Licensee a perpetual, worldwide, non-transferable, non-exclusive right to use and redistribute in whole or in part, Total HTML Converter X an unlimited number of times. All rights not specifically granted are retained by Softplicity.
Permitted Uses include Internet, intranet, Online or web-based software solutions. Licensee may implement Total HTML ConverterX into other applications or adapt it in connection with the above permitted uses. Licensee may make a back-up copy of the program for internal back-up purposes. Any transfers of the program are permitted to third parties integral to the creation of the final product, provided such third parties agree to abide by the terms of this Agreement.
Licensee may use Total HTML Converter X in one project and redistribute the final product. The number of people involved into that project is not limited. Licensee may not resell Total HTML ConverterX.
Whilst every care has been taken in the construction and testing of this software, it is supplied subject to the condition that the user undertakes to evaluate the suitability of the control for his/her purposes. Softplicity makes no representation of the software's suitability for any purpose, and the user agrees that Softplicity has no responsibility for any loss or damage occasioned by the use of this software.
TO THE MAXIMUM EXTENT PERMITTED BY APPLICABLE LAW, IN NO EVENT SHALL COOLUTILS BE LIABLE FOR ANY INDIRECT, INCIDENTAL, CONSEQUENTIAL, SPECIAL OR EXEMPLARY DAMAGES OR LOST PROFITS WHATSOEVER (INCLUDING, WITHOUT LIMITATION, DAMAGES FOR LOSS OF BUSINESS PROFITS, BUSINESS INTERRUPTION, LOSS OF BUSINESS INFORMATION, OR ANY OTHER PECUNIARY LOSS) ARISING OUT OF THE USE OR INABILITY TO USE THE SOFTWARE PRODUCT, EVEN IF COOLUTILS HAS BEEN ADVISED OF THE POSSIBILITY OF SUCH DAMAGES. IN ANY CASE, COOLUTILS'S CUMULATIVE AND ENTIRE LIABILITY TO YOU OR ANY OTHER PARTY FOR ANY LOSS OR DAMAGES RESULTING FROM ANY CLAIMS, DEMANDS OR ACTIONS ARISING OUT OF OR RELATING TO THIS AGREEMENT SHALL NOT EXCEED THE PURCHASE PRICE PAID FOR THIS LICENSE.
Should any term of these terms and conditions be declared void or unenforceable by any court of competent jurisdiction, such declaration shall have no effect on the remaining terms hereof.
All licenses are final; no refunds or credits will be allowed. The transfer of Total HTML ConverterX file from Softplicity via the internet to the Licensee is considered to be the complete fulfillment of Softplicity's obligations.
If you do not agree to these conditions you should not install this software.
%%The following software may be included in, shipped with, or otherwise used by this product: HUSL; Use of any of this software may be governed by the terms of the license below:
Copyright (C) 2012 Alexei Boronine
%%The following software may be included in, shipped with, or otherwise used by this product: General SQL Parser; Use of any of this software may be governed by the terms of the license below:
General SQL Parser is licensed under a commercial license from Gudu Software.
%%The following software may be included in, shipped with, or otherwise used by this product: ICU; Use of any of this software may be governed by the terms of the license below:
ICU License - ICU 1.8.1 and later COPYRIGHT AND PERMISSION NOTICE Copyright (c) 1995-2003 International Business Machines Corporation and others All rights reserved. Permission is hereby granted, free of charge, to any person obtaining a copy of this software and associated documentation files (the "Software"), to deal in the Software without restriction, including without limitation the rights to use, copy, modify, merge, publish, distribute, and/or sell copies of the Software, and to permit persons to whom the Software is furnished to do so, provided that the above copyright notice(s) and this permission notice appear in all copies of the Software and that both the above copyright notice(s) and this permission notice appear in supporting documentation. THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT OF THIRD PARTY RIGHTS. IN NO EVENT SHALL THE COPYRIGHT HOLDER OR HOLDERS INCLUDED IN THIS NOTICE BE LIABLE FOR ANY CLAIM, OR ANY SPECIAL INDIRECT OR CONSEQUENTIAL DAMAGES, OR ANY DAMAGES WHATSOEVER RESULTING FROM LOSS OF USE, DATA OR PROFITS, WHETHER IN AN ACTION OF CONTRACT, NEGLIGENCE OR OTHER TORTIOUS ACTION, ARISING OUT OF OR IN CONNECTION WITH THE USE OR PERFORMANCE OF THIS SOFTWARE. Except as contained in this notice, the name of a copyright holder shall not be used in advertising or otherwise to promote the sale, use or other dealings in this Software without prior written authorization of the copyright holder.
%%The following software may be included in, shipped with, or otherwise used by this product: imagscalr; Use of any of this software may be governed by the terms of the license below:
Copyright © 2012- 2013,The Buzz Media. All rights reserved.
%%The following software may be included in, shipped with, or otherwise used by this product: mhook; Use of any of this software may be governed by the terms of the license below:
Copyright (c) 2007-2012, Marton Anka
Portions Copyright (c) 2007, Matt Conover
%%The following software may be included in, shipped with, or otherwise used by this product: Microsoft SQL Server and Windows PE; Use of any of this software may be governed by the terms of the license below:
© 2007, Microsoft Corporation. All rights reserved.
%%The following software may be included in, shipped with, or otherwise used by this product: Microsoft Windows PE; Use of any of this software may be governed by the terms of the license below:
This product contains Windows software licensed from Microsoft Corporation and/or MS Affiliate(s).
%%The following software may be included in, shipped with, or otherwise used by this product: Manage ONTAP and ONTAP SDK; Use of any of this software may be governed by the terms of the license below:
This product contains software licensed from Network Appliance, Inc. Copyright 1999-2010 Network Appliance, Inc. All rights reserved.
%%The following software may be included in, shipped with, or otherwise used by this product: Ognl; Use of any of this software may be governed by the terms of the license below:
The OpenSymphony Software License, Version 1.1
* (this license is derived and fully compatible with the Apache Software
* License - see http://www.apache.org/LICENSE.txt)
* Copyright (c) 2001-2004 The OpenSymphony Group. All rights reserved.
* 3. The end-user documentation included with the redistribution,
* if any, must include the following acknowledgment:
* "This product includes software developed by the
* OpenSymphony Group (http://www.opensymphony.com/)."
* Alternately, this acknowledgment may appear in the software itself,
* if and wherever such third-party acknowledgments normally appear.
* 4. The names "OpenSymphony" and "The OpenSymphony Group"
* must not be used to endorse or promote products derived from this
* software without prior written permission. For written
* permission, please contact license@opensymphony.com .
* 5. Products derived from this software may not be called "OpenSymphony"
* or "OGNL", nor may "OpenSymphony" or "OGNL" appear in their
* name, without prior written permission of the OpenSymphony Group.
* THIS SOFTWARE IS PROVIDED ``AS IS'' AND ANY EXPRESSED OR IMPLIED
* WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES
* OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE
* DISCLAIMED. IN NO EVENT SHALL THE APACHE SOFTWARE FOUNDATION OR
* LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF
* USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND
* ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY,
* OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT
* OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF
* SUCH DAMAGE.
%%The following software may be included in, shipped with, or otherwise used by this product: ONC RPC; Use of any of this software may be governed by the terms of the license below:
Copyright (c) 2009, Sun Microsystems, Inc.
notice, this list of conditions and the following disclaimer in the
documentation and/or other materials provided with the distribution.
* Neither the name of the <organization> nor the
names of its contributors may be used to endorse or promote products
derived from this software without specific prior written permission.
THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL <COPYRIGHT HOLDER> BE LIABLE FOR ANY
DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES
(INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES;
LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND
ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT
(INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS
SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
%% The following software may be included in, shipped with, or otherwise used by this product: Open SSL; Use of any of this software may be governed by the terms of the license below:
The OpenSSL toolkit stays under a dual license, i.e. both the conditions of
the OpenSSL License and the original SSLeay license apply to the toolkit.
See below for the actual license texts.
OpenSSL License
* Copyright (c) 1998-2016 The OpenSSL Project. All rights reserved.
Original SSLeay License
/* Copyright (C) 1995-1998 Eric Young (eay@cryptsoft.com)
* This package is an SSL implementation written
* by Eric Young (eay@cryptsoft.com).
* The implementation was written so as to conform with Netscapes SSL.
* This library is free for commercial and non-commercial use as long as
* the following conditions are aheared to. The following conditions
* apply to all code found in this distribution, be it the RC4, RSA,
* lhash, DES, etc., code; not just the SSL code. The SSL documentation
* included with this distribution is covered by the same copyright terms
* except that the holder is Tim Hudson (tjh@cryptsoft.com).
* Copyright remains Eric Young's, and as such any Copyright notices in
* the code are not to be removed.
* If this package is used in a product, Eric Young should be given attribution
* as the author of the parts of the library used.
* This can be in the form of a textual message at program startup or
* in documentation (online or textual) provided with the package.
* 1. Redistributions of source code must retain the copyright
* documentation and/or other materials provided with the distribution.
* 3. All advertising materials mentioning features or use of this software
* must display the following acknowledgement:
* "This product includes cryptographic software written by
* Eric Young (eay@cryptsoft.com)"
* The word 'cryptographic' can be left out if the rouines from the library
* being used are not cryptographic related :-).
* 4. If you include any Windows specific code (or a derivative thereof) from
* the apps directory (application code) you must include an acknowledgement:
* "This product includes software written by Tim Hudson (tjh@cryptsoft.com)"
* THIS SOFTWARE IS PROVIDED BY ERIC YOUNG ``AS IS'' AND
* ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
* IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
* ARE DISCLAIMED. IN NO EVENT SHALL THE AUTHOR OR CONTRIBUTORS BE LIABLE
* FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL
* DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS
* OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION)
* HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT
* LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY
* OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF
* The license and distribution terms for any publically available version or
* derivative of this code cannot be changed. i.e. this code cannot simply be
* copied and put under another distribution license
* [including the GNU Public License.]
%% The following software may be included in, shipped with, or otherwise used by this product: Open SAML; Use of any of this software may be governed by the terms of the license below:
%The following software may be included in, shipped with, or otherwise used by this product: Outside In; Use of any of this software may be governed by the terms of the end user license and can only be used in conjunction with this product.:
Portions copyright © 1991, 2011 Oracle Corporation and/or its affiliates. "Oracle" and "Outside In" are trademarks of Oracle Corporation and are registered in the United States and other countries. All rights reserved.
%% The following software may be included in, shipped with, or otherwise used by this product: Open Health Tools part of MDHT; Use of any of this software may be governed by the terms of the license below:
%%The following software may be included in, shipped with, or otherwise used by this product: PostgreSQL; Use of any of this software may be governed by the terms of the license below:
PostgreSQL Database Management System
(formerly known as Postgres, then as Postgres95)
Portions Copyright (c) 1996-2009, The PostgreSQL Global Development Group
Portions Copyright (c) 1994, The Regents of the University of California
Permission to use, copy, modify, and distribute this software and its documentation for any purpose, without fee, and without a written agreement is hereby granted, provided that the above copyright notice and this paragraph and the following two paragraphs appear in all copies.
IN NO EVENT SHALL THE UNIVERSITY OF CALIFORNIA BE LIABLE TO ANY PARTY FOR DIRECT, INDIRECT, SPECIAL, INCIDENTAL, OR CONSEQUENTIAL DAMAGES, INCLUDING LOST PROFITS, ARISING OUT OF THE USE OF THIS SOFTWARE AND ITS DOCUMENTATION, EVEN IF THE UNIVERSITY OF CALIFORNIA HAS BEEN ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
THE UNIVERSITY OF CALIFORNIA SPECIFICALLY DISCLAIMS ANY WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE. THE SOFTWARE PROVIDED HEREUNDER IS ON AN "AS IS" BASIS, AND THE UNIVERSITY OF CALIFORNIA HAS NO OBLIGATIONS TO PROVIDE MAINTENANCE, SUPPORT, UPDATES, ENHANCEMENTS, OR MODIFICATIONS.
%%The following software may be included in, shipped with, or otherwise used by this product: Apache Axis, Apache Commons Discovery, Apache Commons Logging, Apache POI, Apache Solr, Apache Tika, Apache Tomcat, Dozer, log4j, Xalan, XMLParserAPIs.jar, and Xerces; Use of any of this software may be governed by the terms of the license below:
This product includes software developed by The Apache Software Foundation http://www.apache.org. Copyright 1999-2007 The Apache Software Foundation
(a) You must give any other recipients of the Work or Derivative Works a copy of this License; and
(b) You must cause any modified files to carry prominent notices stating that You changed the files; and
(c) You must retain, in the Source form of any Derivative Works that You distribute, all copyright, patent, trademark, and attribution notices from the Source form of the Work, excluding those notices that do not pertain to any part of the Derivative Works; and
(d) If the Work includes a "NOTICE" text file as part of its distribution, then any Derivative Works that You distribute must include a readable copy of the attribution notices contained within such NOTICE file, excluding those notices that do not pertain to any part of the Derivative Works, in at least one of the following places: within a NOTICE text file distributed as part of the Derivative Works; within the Source form or documentation, if provided along with the Derivative Works; or, within a display generated by the Derivative Works, if and wherever such third-party notices normally appear. The contents of the NOTICE file are for informational purposes only and do not modify the License. You may add Your own attribution notices within Derivative Works that You distribute, alongside or as an addendum to the NOTICE text from the Work, provided that such additional attribution notices cannot be construed as modifying the License.
5. Submission of Contributions. Unless You explicitly state otherwise, any Contribution intentionally submitted for inclusion in the Work by You to the Licensor shall be under the terms and conditions of this License, without any additional terms or conditions.
Notwithstanding the above, nothing herein shall supersede or modify the terms of any separate license agreement you may have executed with Licensor regarding such Contributions.
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.
%%The following software may be included in, shipped with, or otherwise used by this product: bccrypto.jar; Use of any of this software may be governed by the terms of the license below:
Copyright (c) 2000 - 2006 The Legion Of The Bouncy Castle (http://www.bouncycastle.org) Permission is hereby granted, free of charge, to any person obtaining a copy of this software and associated documentation files (the "Software"), to deal in the Software without restriction, including without limitation the rights to use, copy, modify, merge, publish, distribute, sublicense, and/or sell copies of the Software, and to permit persons to whom the Software is furnished to do so, subject to the following conditions: The above copyright notice and this permission notice shall be included in all copies or substantial portions of the Software. THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE SOFTWARE.
%% The following software may be included in, shipped with, or otherwise used by this product: Seaborn, Use of any of this software may be governed by the terms of the license below:
Copyright © 2012-2016, Michael L. Waskom. All rights reserved.
* Redistributions of source code must retain the above copyright notice, this
list of conditions and the following disclaimer.
* Redistributions in binary form must reproduce the above copyright notice,
* Neither the name of the project nor the names of its
IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE
DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT HOLDER OR CONTRIBUTORS BE LIABLE
FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL
DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR
SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER
CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY,
OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE
%% The following software may be included in, shipped with, or otherwise used by this product: SIX, Use of any of this software may be governed by the terms of the license below:
Copyright (c) 2010-2014 Benjamin Peterson
Permission is hereby granted, free of charge, to any person obtaining a copy of this software and associated documentation files (the "Software"), to deal in the Software without restriction, including without limitation the rights to use, copy, modify, merge, publish, distribute, sublicense, and/or sell copies of the Software, and to permit persons to whom the Software is furnished to do subject to the following conditions:
%%The following software may be included in, shipped with, or otherwise used by this product: StAX 1.0 API, Use of any of this software may be governed by the terms of the license below:
Copyright 2004 BEA Systems, Inc.
%%The following software may be included in, shipped with, or otherwise used by this product: SharpZipLib; Use of any of this software may be governed by the terms of the license below:
This library is free software; you can redistribute it and/or modify it under
the terms of version 2 of the GNU General Public License as published by the
Free Software Foundation, with the additional linking exception as follows:
Linking this library statically or dynamically with other modules is
making a combined work based on this library. Thus, the terms and
conditions of the GNU General Public License cover the whole combination.
As a special exception, the copyright holders of this library give you permission to link this library with independent modules to produce an executable, regardless of the license terms of these independent modules, and to copy and distribute the resulting executable under terms of your choice, provided that you also meet, for each linked independent module, the terms and conditions of the license of that module. An independent module is a module which is not derived from or based on this library. If you modify this library, you may extend this exception to your version of the library, but you are not obligated to do so. If you do not wish to do so, delete this exception statement from your version.
This program is distributed in the hope that it will be useful, but WITHOUT
ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS
FOR A PARTICULAR PURPOSE. See the GNU General Public License for more
You should have received a copy of the GNU General Public License along with
this program; if not, write to the Free Software Foundation, Inc., 51 Franklin
Street, Fifth Floor, Boston, MA 02110-1301 USA.
This library may be accompanied by pedagogical examples. These do not form
part of this library, and are licensed for redistribution and modification
under less restrictive terms. Such examples are licensed clearly at the top
of each file.
%%The following software may be included in, shipped with, or otherwise used by this product: SOAP with attachments for API for Java (SAAJ); Use of any of this software may be governed by the terms of the license below:
%%The following software may be included in, shipped with, or otherwise used by this product: Phone Gap/Apache Cordova, Use of any of this software may be governed by the terms of the license below:
%%The following software may be included in, shipped with, or otherwise used by this product: SQL Lite, Use of any of this software may be governed by the terms of the license below:
SQL Lite is in the public domain.
%%The following software may be included in, shipped with, or otherwise used by this product: STL Port; Use of any of this software may be governed by the terms of the license below:
Copyright 1999, 2000 Boris Fomitchev
This material is provided "as is", with absolutely no warranty expressed or implied. Any use is at your own risk. Permission to use or copy this software for any purpose is hereby granted without fee, provided the above notices are retained on all copies. Permission to modify the code and to distribute modified code is granted, provided the above notices are retained, and a notice that the code was modified is included with the above copyright notice.
The Licensee may distribute binaries compiled with STLport (whether original or modified) without any royalties or restrictions.
The Licensee may distribute original or modified STLport sources, provided that:
The conditions indicated in the above permission notice are met;
The following copyright notices are retained when present, and conditions provided in accompanying permission notices are met:
Copyright 1994 Hewlett-Packard Company
Copyright 1996,97 Silicon Graphics Computer Systems, Inc.
Copyright 1997 Moscow Center for SPARC Technology.
Permission to use, copy, modify, distribute and sell this software and its documentation for any purpose is hereby granted without fee, provided that the above copyright notice appear in all copies and that both that copyright notice and this permission notice appear in supporting documentation. Hewlett-Packard Company makes no representations about the suitability of this software for any purpose. It is provided "as is" without express or implied warranty.
Permission to use, copy, modify, distribute and sell this software and its documentation for any purpose is hereby granted without fee, provided that the above copyright notice appear in all copies and that both that copyright notice and this permission notice appear in supporting documentation. Silicon Graphics makes no representations about the suitability of this software for any purpose. It is provided "as is" without express or implied warranty.
Permission to use, copy, modify, distribute and sell this software and its documentation for any purpose is hereby granted without fee, provided that the above copyright notice appear in all copies and that both that copyright notice and this permission notice appear in supporting documentation. Moscow Center for SPARC Technology makes no representations about the suitability of this software for any purpose. It is provided "as is" without express or implied warranty.
%%The following software may be included in, shipped with, or otherwise used by this product: gSOAP; Use of any of this software may be governed by the terms of the license below:
Part of the software embedded in this product is gSOAP software. Portions created by gSOAP are Copyright (C) 2001-2004 Robert A. van Engelen, Genivia inc. All Rights Reserved. THE SOFTWARE IN THIS PRODUCT WAS IN PART PROVIDED BY GENIVIA INC AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE AUTHOR BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
%%The following software may be included in, shipped with, or otherwise used by this product: InfoZip; Use of any of this software may be governed by the terms of the license below:
This is version 2007-Mar-4 of the Info-ZIP license. The definitive version of this document should be available at ftp://ftp.info-zip.org/pub/infozip/license.html indefinitely and a copy at http://www.info-zip.org/pub/infozip/license.html.
Copyright (c) 1990-2007 Info-ZIP. All rights reserved.
For the purposes of this copyright and license, "Info-ZIP" is defined as the following set of individuals:
Mark Adler, John Bush, Karl Davis, Harald Denker, Jean-Michel Dubois, Jean-loup Gailly, Hunter Goatley, Ed Gordon, Ian Gorman, Chris Herborth, Dirk Haase, Greg Hartwig, Robert Heath, Jonathan Hudson, Paul Kienitz, David Kirschbaum, Johnny Lee, Onno van der Linden, Igor Mandrichenko, Steve P. Miller, Sergio Monesi, Keith Owens, George Petrov, Greg Roelofs, Kai Uwe Rommel, Steve Salisbury, Dave Smith, Steven M. Schweda, Christian Spieler, Cosmin Truta, Antoine Verheijen, Paul von Behren, Rich Wales, Mike White.
This software is provided "as is," without warranty of any kind, express or implied. In no event shall Info-ZIP or its contributors be held liable for any direct, indirect, incidental, special or consequential damages arising out of the use of or inability to use this software.
Permission is granted to anyone to use this software for any purpose, including commercial applications, and to alter it and redistribute it freely, subject to the above disclaimer and the following restrictions:
1. Redistributions of source code (in whole or in part) must retain the above copyright notice, definition, disclaimer, and this list of conditions.
2. Redistributions in binary form (compiled executables and libraries) must reproduce the above copyright notice, definition, disclaimer, and this list of conditions in documentation and/or other materials provided with the distribution. The sole exception to this condition is redistribution of a standard UnZipSFX binary (including SFXWiz) as part of a self-extracting archive; that is permitted without inclusion of this license, as long as the normal SFX banner has not been removed from the binary or disabled.
3. Altered versions--including, but not limited to, ports to new operating systems, existing ports with new graphical interfaces, versions with modified or added functionality, and dynamic, shared, or static library versions not from Info-ZIP--must be plainly marked as such and must not be misrepresented as being the original source or, if binaries, compiled from the original source. Such altered versions also must not be misrepresented as being Info-ZIP releases--including, but not limited to, labeling of the altered versions with the names "Info-ZIP" (or any variation thereof, including, but not limited to, different capitalizations), "Pocket UnZip," "WiZ" or "MacZip" without the explicit permission of Info-ZIP. Such altered versions are further prohibited from misrepresentative use of the Zip-Bugs or Info-ZIP e-mail addresses or the Info-ZIP URL(s), such as to imply Info-ZIP will provide support for the altered versions.
4. Info-ZIP retains the right to use the names "Info-ZIP," "Zip," "UnZip," "UnZipSFX," "WiZ," "Pocket UnZip," "Pocket Zip," and "MacZip" for its own source and binary releases.
%% The following software may be included in, shipped with, or otherwise used by this product: IOMETER; Use of any of this software may be governed by the terms of the license below:
Copyright © 2001 Intel Corporation. All rights reserved. Redistribution and use in source and binary forms, with or without modification, are permitted provided that the following conditions are met:
Neither the name of the Intel Corporation nor the names of its contributors may be used to endorse or promote products derived from this software without specific prior written permission.
THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS ``AS IS'' AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL INTEL OR ITS CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
%%The following software may be included in, shipped with, or otherwise used by this product: JDOM; Use of any of this software may be governed by the terms of the license below:
This product includes software developed by the JDOM Project (http://www.jdom.org/)
%%The following software may be included in, shipped with, or otherwise used by this product: Sed; Use of any of this software may be governed by the terms of the license below:
GNU General Public License Version 3
http://www.gnu.org/copyleft/gpl.html
GNU GENERAL PUBLIC LICENSE
Version 3, 29 June 2007
Copyright © 2007 Free Software Foundation, Inc. http://fsf.org/
Everyone is permitted to copy and distribute verbatim copies of this license document, but changing it is not allowed.
The GNU General Public License is a free, copyleft license for software and other kinds of works.
The licenses for most software and other practical works are designed to take away your freedom to share and change the works. By contrast, the GNU General Public License is intended to guarantee your freedom to share and change all versions of a program--to make sure it remains free software for all its users. We, the Free Software Foundation, use the GNU General Public License for most of our software; it applies also to any other work released this way by its authors. You can apply it to your programs, too.
When we speak of free software, we are referring to freedom, not price. Our General Public Licenses are designed to make sure that you have the freedom to distribute copies of free software (and charge for them if you wish), that you receive source code or can get it if you want it, that you can change the software or use pieces of it in new free programs, and that you know you can do these things.
To protect your rights, we need to prevent others from denying you these rights or asking you to surrender the rights. Therefore, you have certain responsibilities if you distribute copies of the software, or if you modify it: responsibilities to respect the freedom of others.
For example, if you distribute copies of such a program, whether gratis or for a fee, you must pass on to the recipients the same freedoms that you received. You must make sure that they, too, receive or can get the source code. And you must show them these terms so they know their rights.
Developers that use the GNU GPL protect your rights with two steps: (1) assert copyright on the software, and (2) offer you this License giving you legal permission to copy, distribute and/or modify it.
For the developers' and authors' protection, the GPL clearly explains that there is no warranty for this free software. For both users' and authors' sake, the GPL requires that modified versions be marked as changed, so that their problems will not be attributed erroneously to authors of previous versions.
Some devices are designed to deny users access to install or run modified versions of the software inside them, although the manufacturer can do so. This is fundamentally incompatible with the aim of protecting users' freedom to change the software. The systematic pattern of such abuse occurs in the area of products for individuals to use, which is precisely where it is most unacceptable. Therefore, we have designed this version of the GPL to prohibit the practice for those products. If such problems arise substantially in other domains, we stand ready to extend this provision to those domains in future versions of the GPL, as needed to protect the freedom of users.
Finally, every program is threatened constantly by software patents. States should not allow patents to restrict development and use of software on general-purpose computers, but in those that do, we wish to avoid the special danger that patents applied to a free program could make it effectively proprietary. To prevent this, the GPL assures that patents cannot be used to render the program non-free.
The precise terms and conditions for copying, distribution and modification follow.
'This License' refers to version 3 of the GNU General Public License.
'Copyright' also means copyright-like laws that apply to other kinds of works, such as semiconductor masks.
'The Program' refers to any copyrightable work licensed under this License. Each licensee is addressed as 'you'. 'Licensees' and 'recipients' may be individuals or organizations.
To 'modify' a work means to copy from or adapt all or part of the work in a fashion requiring copyright permission, other than the making of an exact copy. The resulting work is called a 'modified version' of the earlier work or a work 'based on' the earlier work.
A 'covered work' means either the unmodified Program or a work based on the Program.
To 'propagate' a work means to do anything with it that, without permission, would make you directly or secondarily liable for infringement under applicable copyright law, except executing it on a computer or modifying a private copy. Propagation includes copying, distribution (with or without modification), making available to the public, and in some countries other activities as well.
To 'convey' a work means any kind of propagation that enables other parties to make or receive copies. Mere interaction with a user through a computer network, with no transfer of a copy, is not conveying.
An interactive user interface displays 'Appropriate Legal Notices' to the extent that it includes a convenient and prominently visible feature that (1) displays an appropriate copyright notice, and (2) tells the user that there is no warranty for the work (except to the extent that warranties are provided), that licensees may convey the work under this License, and how to view a copy of this License. If the interface presents a list of user commands or options, such as a menu, a prominent item in the list meets this criterion.
1. Source Code.
The 'source code' for a work means the preferred form of the work for making modifications to it. 'Object code' means any non-source form of a work.
A 'Standard Interface' means an interface that either is an official standard defined by a recognized standards body, or, in the case of interfaces specified for a particular programming language, one that is widely used among developers working in that language.
The 'System Libraries' of an executable work include anything, other than the work as a whole, that (a) is included in the normal form of packaging a Major Component, but which is not part of that Major Component, and (b) serves only to enable use of the work with that Major Component, or to implement a Standard Interface for which an implementation is available to the public in source code form. A 'Major Component', in this context, means a major essential component (kernel, window system, and so on) of the specific operating system (if any) on which the executable work runs, or a compiler used to produce the work, or an object code interpreter used to run it.
The 'Corresponding Source' for a work in object code form means all the source code needed to generate, install, and (for an executable work) run the object code and to modify the work, including scripts to control those activities. However, it does not include the work's System Libraries, or general-purpose tools or generally available free programs which are used unmodified in performing those activities but which are not part of the work. For example, Corresponding Source includes interface definition files associated with source files for the work, and the source code for shared libraries and dynamically linked subprograms that the work is specifically designed to require, such as by intimate data communication or control flow between those subprograms and other parts of the work.
The Corresponding Source need not include anything that users can regenerate automatically from other parts of the Corresponding Source.
The Corresponding Source for a work in source code form is that same work.
2. Basic Permissions.
All rights granted under this License are granted for the term of copyright on the Program, and are irrevocable provided the stated conditions are met. This License explicitly affirms your unlimited permission to run the unmodified Program. The output from running a covered work is covered by this License only if the output, given its content, constitutes a covered work. This License acknowledges your rights of fair use or other equivalent, as provided by copyright law.
You may make, run and propagate covered works that you do not convey, without conditions so long as your license otherwise remains in force. You may convey covered works to others for the sole purpose of having them make modifications exclusively for you, or provide you with facilities for running those works, provided that you comply with the terms of this License in conveying all material for which you do not control copyright. Those thus making or running the covered works for you must do so exclusively on your behalf, under your direction and control, on terms that prohibit them from making any copies of your copyrighted material outside their relationship with you.
Conveying under any other circumstances is permitted solely under the conditions stated below. Sublicensing is not allowed; section 10 makes it unnecessary.
3. Protecting Users' Legal Rights From Anti-Circumvention Law.
No covered work shall be deemed part of an effective technological measure under any applicable law fulfilling obligations under article 11 of the WIPO copyright treaty adopted on 20 December 1996, or similar laws prohibiting or restricting circumvention of such measures.
When you convey a covered work, you waive any legal power to forbid circumvention of technological measures to the extent such circumvention is effected by exercising rights under this License with respect to the covered work, and you disclaim any intention to limit operation or modification of the work as a means of enforcing, against the work's users, your or third parties' legal rights to forbid circumvention of technological measures.
4. Conveying Verbatim Copies.
You may convey verbatim copies of the Program's source code as you receive it, in any medium, provided that you conspicuously and appropriately publish on each copy an appropriate copyright notice; keep intact all notices stating that this License and any non-permissive terms added in accord with section 7 apply to the code; keep intact all notices of the absence of any warranty; and give all recipients a copy of this License along with the Program.
You may charge any price or no price for each copy that you convey, and you may offer support or warranty protection for a fee.
5. Conveying Modified Source Versions.
You may convey a work based on the Program, or the modifications to produce it from the Program, in the form of source code under the terms of section 4, provided that you also meet all of these conditions:
a) The work must carry prominent notices stating that you modified it, and giving a relevant date.
b) The work must carry prominent notices stating that it is released under this License and any conditions added under section 7. This requirement modifies the requirement in section 4 to 'keep intact all notices'.
c) You must license the entire work, as a whole, under this License to anyone who comes into possession of a copy. This License will therefore apply, along with any applicable section 7 additional terms, to the whole of the work, and all its parts, regardless of how they are packaged. This License gives no permission to license the work in any other way, but it does not invalidate such permission if you have separately received it.
d) If the work has interactive user interfaces, each must display Appropriate Legal Notices; however, if the Program has interactive interfaces that do not display Appropriate Legal Notices, your work need not make them do so.
A compilation of a covered work with other separate and independent works, which are not by their nature extensions of the covered work, and which are not combined with it such as to form a larger program, in or on a volume of a storage or distribution medium, is called an 'aggregate' if the compilation and its resulting copyright are not used to limit the access or legal rights of the compilation's users beyond what the individual works permit. Inclusion of a covered work in an aggregate does not cause this License to apply to the other parts of the aggregate.
6. Conveying Non-Source Forms.
You may convey a covered work in object code form under the terms of sections 4 and 5, provided that you also convey the machine-readable Corresponding Source under the terms of this License, in one of these ways:
a) Convey the object code in, or embodied in, a physical product (including a physical distribution medium), accompanied by the Corresponding Source fixed on a durable physical medium customarily used for software interchange.
b) Convey the object code in, or embodied in, a physical product (including a physical distribution medium), accompanied by a written offer, valid for at least three years and valid for as long as you offer spare parts or customer support for that product model, to give anyone who possesses the object code either (1) a copy of the Corresponding Source for all the software in the product that is covered by this License, on a durable physical medium customarily used for software interchange, for a price no more than your reasonable cost of physically performing this conveying of source, or (2) access to copy the Corresponding Source from a network server at no charge.
c) Convey individual copies of the object code with a copy of the written offer to provide the Corresponding Source. This alternative is allowed only occasionally and noncommercially, and only if you received the object code with such an offer, in accord with subsection 6b.
d) Convey the object code by offering access from a designated place (gratis or for a charge), and offer equivalent access to the Corresponding Source in the same way through the same place at no further charge. You need not require recipients to copy the Corresponding Source along with the object code. If the place to copy the object code is a network server, the Corresponding Source may be on a different server (operated by you or a third party) that supports equivalent copying facilities, provided you maintain clear directions next to the object code saying where to find the Corresponding Source. Regardless of what server hosts the Corresponding Source, you remain obligated to ensure that it is available for as long as needed to satisfy these requirements.
e) Convey the object code using peer-to-peer transmission, provided you inform other peers where the object code and Corresponding Source of the work are being offered to the general public at no charge under subsection 6d.
A separable portion of the object code, whose source code is excluded from the Corresponding Source as a System Library, need not be included in conveying the object code work.
A 'User Product' is either (1) a 'consumer product', which means any tangible personal property which is normally used for personal, family, or household purposes, or (2) anything designed or sold for incorporation into a dwelling. In determining whether a product is a consumer product, doubtful cases shall be resolved in favor of coverage. For a particular product received by a particular user, 'normally used' refers to a typical or common use of that class of product, regardless of the status of the particular user or of the way in which the particular user actually uses, or expects or is expected to use, the product. A product is a consumer product regardless of whether the product has substantial commercial, industrial or non-consumer uses, unless such uses represent the only significant mode of use of the product.
'Installation Information' for a User Product means any methods, procedures, authorization keys, or other information required to install and execute modified versions of a covered work in that User Product from a modified version of its Corresponding Source. The information must suffice to ensure that the continued functioning of the modified object code is in no case prevented or interfered with solely because modification has been made.
If you convey an object code work under this section in, or with, or specifically for use in, a User Product, and the conveying occurs as part of a transaction in which the right of possession and use of the User Product is transferred to the recipient in perpetuity or for a fixed term (regardless of how the transaction is characterized), the Corresponding Source conveyed under this section must be accompanied by the Installation Information. But this requirement does not apply if neither you nor any third party retains the ability to install modified object code on the User Product (for example, the work has been installed in ROM).
The requirement to provide Installation Information does not include a requirement to continue to provide support service, warranty, or updates for a work that has been modified or installed by the recipient, or for the User Product in which it has been modified or installed. Access to a network may be denied when the modification itself materially and adversely affects the operation of the network or violates the rules and protocols for communication across the network.
Corresponding Source conveyed, and Installation Information provided, in accord with this section must be in a format that is publicly documented (and with an implementation available to the public in source code form), and must require no special password or key for unpacking, reading or copying.
'Additional permissions' are terms that supplement the terms of this License by making exceptions from one or more of its conditions. Additional permissions that are applicable to the entire Program shall be treated as though they were included in this License, to the extent that they are valid under applicable law. If additional permissions apply only to part of the Program, that part may be used separately under those permissions, but the entire Program remains governed by this License without regard to the additional permissions.
When you convey a copy of a covered work, you may at your option remove any additional permissions from that copy, or from any part of it. (Additional permissions may be written to require their own removal in certain cases when you modify the work.) You may place additional permissions on material, added by you to a covered work, for which you have or can give appropriate copyright permission.
Notwithstanding any other provision of this License, for material you add to a covered work, you may (if authorized by the copyright holders of that material) supplement the terms of this License with terms:
a) Disclaiming warranty or limiting liability differently from the terms of sections 15 and 16 of this License; or
b) Requiring preservation of specified reasonable legal notices or author attributions in that material or in the Appropriate Legal Notices displayed by works containing it; or
c) Prohibiting misrepresentation of the origin of that material, or requiring that modified versions of such material be marked in reasonable ways as different from the original version; or
d) Limiting the use for publicity purposes of names of licensors or authors of the material; or
e) Declining to grant rights under trademark law for use of some trade names, trademarks, or service marks; or
f) Requiring indemnification of licensors and authors of that material by anyone who conveys the material (or modified versions of it) with contractual assumptions of liability to the recipient, for any liability that these contractual assumptions directly impose on those licensors and authors.
All other non-permissive additional terms are considered 'further restrictions' within the meaning of section 10. If the Program as you received it, or any part of it, contains a notice stating that it is governed by this License along with a term that is a further restriction, you may remove that term. If a license document contains a further restriction but permits relicensing or conveying under this License, you may add to a covered work material governed by the terms of that license document, provided that the further restriction does not survive such relicensing or conveying.
If you add terms to a covered work in accord with this section, you must place, in the relevant source files, a statement of the additional terms that apply to those files, or a notice indicating where to find the applicable terms.
Additional terms, permissive or non-permissive, may be stated in the form of a separately written license, or stated as exceptions; the above requirements apply either way.
8. Termination.
You may not propagate or modify a covered work except as expressly provided under this License. Any attempt otherwise to propagate or modify it is void, and will automatically terminate your rights under this License (including any patent licenses granted under the third paragraph of section 11).
However, if you cease all violation of this License, then your license from a particular copyright holder is reinstated (a) provisionally, unless and until the copyright holder explicitly and finally terminates your license, and (b) permanently, if the copyright holder fails to notify you of the violation by some reasonable means prior to 60 days after the cessation.
Moreover, your license from a particular copyright holder is reinstated permanently if the copyright holder notifies you of the violation by some reasonable means, this is the first time you have received notice of violation of this License (for any work) from that copyright holder, and you cure the violation prior to 30 days after your receipt of the notice.
Termination of your rights under this section does not terminate the licenses of parties who have received copies or rights from you under this License. If your rights have been terminated and not permanently reinstated, you do not qualify to receive new licenses for the same material under section 10.
9. Acceptance Not Required for Having Copies.
You are not required to accept this License in order to receive or run a copy of the Program. Ancillary propagation of a covered work occurring solely as a consequence of using peer-to-peer transmission to receive a copy likewise does not require acceptance. However, nothing other than this License grants you permission to propagate or modify any covered work. These actions infringe copyright if you do not accept this License. Therefore, by modifying or propagating a covered work, you indicate your acceptance of this License to do so.
10. Automatic Licensing of Downstream Recipients.
Each time you convey a covered work, the recipient automatically receives a license from the original licensors, to run, modify and propagate that work, subject to this License. You are not responsible for enforcing compliance by third parties with this License.
An 'entity transaction' is a transaction transferring control of an organization, or substantially all assets of one, or subdividing an organization, or merging organizations. If propagation of a covered work results from an entity transaction, each party to that transaction who receives a copy of the work also receives whatever licenses to the work the party's predecessor in interest had or could give under the previous paragraph, plus a right to possession of the Corresponding Source of the work from the predecessor in interest, if the predecessor has it or can get it with reasonable efforts.
You may not impose any further restrictions on the exercise of the rights granted or affirmed under this License. For example, you may not impose a license fee, royalty, or other charge for exercise of rights granted under this License, and you may not initiate litigation (including a cross-claim or counterclaim in a lawsuit) alleging that any patent claim is infringed by making, using, selling, offering for sale, or importing the Program or any portion of it.
11. Patents.
A 'contributor' is a copyright holder who authorizes use under this License of the Program or a work on which the Program is based. The work thus licensed is called the contributor's 'contributor version'.
A contributor's 'essential patent claims' are all patent claims owned or controlled by the contributor, whether already acquired or hereafter acquired, that would be infringed by some manner, permitted by this License, of making, using, or selling its contributor version, but do not include claims that would be infringed only as a consequence of further modification of the contributor version. For purposes of this definition, 'control' includes the right to grant patent sublicenses in a manner consistent with the requirements of this License.
Each contributor grants you a non-exclusive, worldwide, royalty-free patent license under the contributor's essential patent claims, to make, use, sell, offer for sale, import and otherwise run, modify and propagate the contents of its contributor version.
In the following three paragraphs, a 'patent license' is any express agreement or commitment, however denominated, not to enforce a patent (such as an express permission to practice a patent or covenant not to sue for patent infringement). To 'grant' such a patent license to a party means to make such an agreement or commitment not to enforce a patent against the party.
If you convey a covered work, knowingly relying on a patent license, and the Corresponding Source of the work is not available for anyone to copy, free of charge and under the terms of this License, through a publicly available network server or other readily accessible means, then you must either (1) cause the Corresponding Source to be so available, or (2) arrange to deprive yourself of the benefit of the patent license for this particular work, or (3) arrange, in a manner consistent with the requirements of this License, to extend the patent license to downstream recipients. 'Knowingly relying' means you have actual knowledge that, but for the patent license, your conveying the covered work in a country, or your recipient's use of the covered work in a country, would infringe one or more identifiable patents in that country that you have reason to believe are valid.
If, pursuant to or in connection with a single transaction or arrangement, you convey, or propagate by procuring conveyance of, a covered work, and grant a patent license to some of the parties receiving the covered work authorizing them to use, propagate, modify or convey a specific copy of the covered work, then the patent license you grant is automatically extended to all recipients of the covered work and works based on it.
A patent license is 'discriminatory' if it does not include within the scope of its coverage, prohibits the exercise of, or is conditioned on the non-exercise of one or more of the rights that are specifically granted under this License. You may not convey a covered work if you are a party to an arrangement with a third party that is in the business of distributing software, under which you make payment to the third party based on the extent of your activity of conveying the work, and under which the third party grants, to any of the parties who would receive the covered work from you, a discriminatory patent license (a) in connection with copies of the covered work conveyed by you (or copies made from those copies), or (b) primarily for and in connection with specific products or compilations that contain the covered work, unless you entered into that arrangement, or that patent license was granted, prior to 28 March 2007.
Nothing in this License shall be construed as excluding or limiting any implied license or other defenses to infringement that may otherwise be available to you under applicable patent law.
12. No Surrender of Others' Freedom.
If conditions are imposed on you (whether by court order, agreement or otherwise) that contradict the conditions of this License, they do not excuse you from the conditions of this License. If you cannot convey a covered work so as to satisfy simultaneously your obligations under this License and any other pertinent obligations, then as a consequence you may not convey it at all. For example, if you agree to terms that obligate you to collect a royalty for further conveying from those to whom you convey the Program, the only way you could satisfy both those terms and this License would be to refrain entirely from conveying the Program.
13. Use with the GNU Affero General Public License.
Notwithstanding any other provision of this License, you have permission to link or combine any covered work with a work licensed under version 3 of the GNU Affero General Public License into a single combined work, and to convey the resulting work. The terms of this License will continue to apply to the part which is the covered work, but the special requirements of the GNU Affero General Public License, section 13, concerning interaction through a network will apply to the combination as such.
14. Revised Versions of this License.
The Free Software Foundation may publish revised and/or new versions of the GNU General Public License from time to time. Such new versions will be similar in spirit to the present version, but may differ in detail to address new problems or concerns.
Each version is given a distinguishing version number. If the Program specifies that a certain numbered version of the GNU General Public License 'or any later version' applies to it, you have the option of following the terms and conditions either of that numbered version or of any later version published by the Free Software Foundation. If the Program does not specify a version number of the GNU General Public License, you may choose any version ever published by the Free Software Foundation.
If the Program specifies that a proxy can decide which future versions of the GNU General Public License can be used, that proxy's public statement of acceptance of a version permanently authorizes you to choose that version for the Program.
Later license versions may give you additional or different permissions. However, no additional obligations are imposed on any author or copyright holder as a result of your choosing to follow a later version.
15. Disclaimer of Warranty.
THERE IS NO WARRANTY FOR THE PROGRAM, TO THE EXTENT PERMITTED BY APPLICABLE LAW. EXCEPT WHEN OTHERWISE STATED IN WRITING THE COPYRIGHT HOLDERS AND/OR OTHER PARTIES PROVIDE THE PROGRAM 'AS IS' WITHOUT WARRANTY OF ANY KIND, EITHER EXPRESSED OR IMPLIED, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE. THE ENTIRE RISK AS TO THE QUALITY AND PERFORMANCE OF THE PROGRAM IS WITH YOU. SHOULD THE PROGRAM PROVE DEFECTIVE, YOU ASSUME THE COST OF ALL NECESSARY SERVICING, REPAIR OR CORRECTION.
IN NO EVENT UNLESS REQUIRED BY APPLICABLE LAW OR AGREED TO IN WRITING WILL ANY COPYRIGHT HOLDER, OR ANY OTHER PARTY WHO MODIFIES AND/OR CONVEYS THE PROGRAM AS PERMITTED ABOVE, BE LIABLE TO YOU FOR DAMAGES, INCLUDING ANY GENERAL, SPECIAL, INCIDENTAL OR CONSEQUENTIAL DAMAGES ARISING OUT OF THE USE OR INABILITY TO USE THE PROGRAM (INCLUDING BUT NOT LIMITED TO LOSS OF DATA OR DATA BEING RENDERED INACCURATE OR LOSSES SUSTAINED BY YOU OR THIRD PARTIES OR A FAILURE OF THE PROGRAM TO OPERATE WITH ANY OTHER PROGRAMS), EVEN IF SUCH HOLDER OR OTHER PARTY HAS BEEN ADVISED OF THE POSSIBILITY OF SUCH DAMAGES.
17. Interpretation of Sections 15 and 16.
If the disclaimer of warranty and limitation of liability provided above cannot be given local legal effect according to their terms, reviewing courts shall apply local law that most closely approximates an absolute waiver of all civil liability in connection with the Program, unless a warranty or assumption of liability accompanies a copy of the Program in return for a fee.
How to Apply These Terms to Your New Programs
If you develop a new program, and you want it to be of the greatest possible use to the public, the best way to achieve this is to make it free software which everyone can redistribute and change under these terms.
To do so, attach the following notices to the program. It is safest to attach them to the start of each source file to most effectively state the exclusion of warranty; and each file should have at least the 'copyright' line and a pointer to where the full notice is found.
[one line to give the program's name and a brief idea of what it does.]
Copyright (C) [year] [name of author]
This program is free software: you can redistribute it and/or modify
along with this program. If not, see http://www.gnu.org/licenses/.
Also add information on how to contact you by electronic and paper mail.
If the program does terminal interaction, make it output a short notice like this when it starts in an interactive mode:
[program] Copyright (C) [year] [name of author]
This program comes with ABSOLUTELY NO WARRANTY; for details type 'show w'.
This is free software, and you are welcome to redistribute it
under certain conditions; type 'show c' for details.
The hypothetical commands 'show w' and 'show c' should show the appropriate parts of the General Public License. Of course, your program's commands might be different; for a GUI interface, you would use an 'about box'.
You should also get your employer (if you work as a programmer) or school, if any, to sign a 'copyright disclaimer' for the program, if necessary. For more information on this, and how to apply and follow the GNU GPL, see http://www.gnu.org/licenses/.
The GNU General Public License does not permit incorporating your program into proprietary programs. If your program is a subroutine library, you may consider it more useful to permit linking proprietary applications with the library. If this is what you want to do, use the GNU Lesser General Public License instead of this License. But first, please read http://www.gnu.org/philosophy/why-not-lgpl.html.
%%The following software may be included in, shipped with, or otherwise used by this product: libXML and XML C ParserAPI; Use of any of this software may be governed by the terms of the license below:
Copyright (c) 2001-2009 XML.Gnome.org
%%The following software may be included in, shipped with, or otherwise used by this product: Linux dump; Use of any of this software may be governed by the terms of the license below:
Ported to Linux's Second Extended File System as part of the dump and restore backup suit. Remy Card <card@Linux.EU.Org>, 1994-1997 Stelian Pop <stelian@popies.net>, 1999-2000 Stelian Pop <stelian@popies.net>, Alcôve <www.alcove.com>, 2000-2002. Copyright (c) 1980, 1993 The Regents of the University of California. All rights reserved. Redistribution and use in source and binary forms, with or without modification, are permitted provided that the following conditions are met: 1. Redistributions of source code must retain the above copyright notice, this list of conditions and the following disclaimer. 2. Redistributions in binary form must reproduce the above copyright notice, this list of conditions and the following disclaimer in the documentation and/or other materials provided with the distribution. 3. Neither the name of the University nor the names of its contributors may be used to endorse or promote products derived from this software without specific prior written permission. THIS SOFTWARE IS PROVIDED BY THE REGENTS AND CONTRIBUTORS ``AS IS'' AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE REGENTS OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
%%The following software may be included in, shipped with, or otherwise used by this product: LZO; Use of any of this software may be governed by the terms of the license below:
0. This License applies to any program or other work which contains a notice placed by the copyright holder saying it may be distributed under the terms of this General Public License. The "Program", below, refers to any such program or work, and a "work based on the Program" means either the Program or any derivative work under copyright law: that is to say, a work containing the Program or a portion of it, either verbatim or with modifications and/or translated into another language. (Hereinafter, translation is included without limitation in the term "modification".) Each licensee is addressed as "you".
Activities other than copying, distribution and modification are not covered by this License; they are outside its scope. The act of running the Program is not restricted, and the output from the Program is covered only if its contents constitute a work based on the Program (independent of having been made by running the Program). Whether that is true depends on what the Program does.
1. You may copy and distribute verbatim copies of the Program's source code as you receive it, in any medium, provided that you conspicuously and appropriately publish on each copy an appropriate copyright notice and disclaimer of warranty; keep intact all the notices that refer to this License and to the absence of any warranty; and give any other recipients of the Program a copy of this License along with the Program.
You may charge a fee for the physical act of transferring a copy, and you may at your option offer warranty protection in exchange for a fee.
2. You may modify your copy or copies of the Program or any portion of it, thus forming a work based on the Program, and copy and distribute such modifications or work under the terms of Section 1 above, provided that you also meet all of these conditions:
a) You must cause the modified files to carry prominent notices stating that you changed the files and the date of any change.
b) You must cause any work that you distribute or publish, that in whole or in part contains or is derived from the Program or any part thereof, to be licensed as a whole at no charge to all third parties under the terms of this License.
c) If the modified program normally reads commands interactively when run, you must cause it, when started running for such interactive use in the most ordinary way, to print or display an announcement including an appropriate copyright notice and a notice that there is no warranty (or else, saying that you provide a warranty) and that users may redistribute the program under these conditions, and telling the user how to view a copy of this License. (Exception: if the Program itself is interactive but does not normally print such an announcement, your work based on the Program is not required to print an announcement.)
These requirements apply to the modified work as a whole. If identifiable sections of that work are not derived from the Program, and can be reasonably considered independent and separate works in themselves, then this License, and its terms, do not apply to those sections when you distribute them as separate works. But when you distribute the same sections as part of a whole which is a work based on the Program, the distribution of the whole must be on the terms of this License, whose permissions for other licensees extend to the entire whole, and thus to each and every part regardless of who wrote it.
Thus, it is not the intent of this section to claim rights or contest your rights to work written entirely by you; rather, the intent is to exercise the right to control the distribution of derivative or collective works based on the Program.
In addition, mere aggregation of another work not based on the Program with the Program (or with a work based on the Program) on a volume of a storage or distribution medium does not bring the other work under the scope of this License.
3. You may copy and distribute the Program (or a work based on it, under Section 2) in object code or executable form under the terms of Sections 1 and 2 above provided that you also do one of the following:
a) Accompany it with the complete corresponding machine-readable source code, which must be distributed under the terms of Sections 1 and 2 above on a medium customarily used for software interchange; or,
b) Accompany it with a written offer, valid for at least three years, to give any third party, for a charge no more than your cost of physically performing source distribution, a complete machine-readable copy of the corresponding source code, to be distributed under the terms of Sections 1 and 2 above on a medium customarily used for software interchange; or,
c) Accompany it with the information you received as to the offer to distribute corresponding source code. (This alternative is allowed only for noncommercial distribution and only if you received the program in object code or executable form with such an offer, in accord with Subsection b above.)
The source code for a work means the preferred form of the work for making modifications to it. For an executable work, complete source code means all the source code for all modules it contains, plus any associated interface definition files, plus the scripts used to control compilation and installation of the executable. However, as a special exception, the source code distributed need not include anything that is normally distributed (in either source or binary form) with the major components (compiler, kernel, and so on) of the operating system on which the executable runs, unless that component itself accompanies the executable.
If distribution of executable or object code is made by offering access to copy from a designated place, then offering equivalent access to copy the source code from the same place counts as distribution of the source code, even though third parties are not compelled to copy the source along with the object code.
4. You may not copy, modify, sublicense, or distribute the Program except as expressly provided under this License. Any attempt otherwise to copy, modify, sublicense or distribute the Program is void, and will automatically terminate your rights under this License. However, parties who have received copies, or rights, from you under this License will not have their licenses terminated so long as such parties remain in full compliance.
5. You are not required to accept this License, since you have not signed it. However, nothing else grants you permission to modify or distribute the Program or its derivative works. These actions are prohibited by law if you do not accept this License. Therefore, by modifying or distributing the Program (or any work based on the Program), you indicate your acceptance of this License to do so, and all its terms and conditions for copying, distributing or modifying the Program or works based on it.
6. Each time you redistribute the Program (or any work based on the Program), the recipient automatically receives a license from the original licensor to copy, distribute or modify the Program subject to these terms and conditions. You may not impose any further restrictions on the recipients' exercise of the rights granted herein. You are not responsible for enforcing compliance by third parties to this License.
7. If, as a consequence of a court judgment or allegation of patent infringement or for any other reason (not limited to patent issues), conditions are imposed on you (whether by court order, agreement or otherwise) that contradict the conditions of this License, they do not excuse you from the conditions of this License. If you cannot distribute so as to satisfy simultaneously your obligations under this License and any other pertinent obligations, then as a consequence you may not distribute the Program at all. For example, if a patent license would not permit royalty-free redistribution of the Program by all those who receive copies directly or indirectly through you, then the only way you could satisfy both it and this License would be to refrain entirely from distribution of the Program.
If any portion of this section is held invalid or unenforceable under any particular circumstance, the balance of the section is intended to apply and the section as a whole is intended to apply in other circumstances.
It is not the purpose of this section to induce you to infringe any patents or other property right claims or to contest validity of any such claims; this section has the sole purpose of protecting the integrity of the free software distribution system, which is implemented by public license practices. Many people have made generous contributions to the wide range of software distributed through that system in reliance on consistent application of that system; it is up to the author/donor to decide if he or she is willing to distribute software through any other system and a licensee cannot impose that choice.
This section is intended to make thoroughly clear what is believed to be a consequence of the rest of this License.
8. If the distribution and/or use of the Program is restricted in certain countries either by patents or by copyrighted interfaces, the original copyright holder who places the Program under this License may add an explicit geographical distribution limitation excluding those countries, so that distribution is permitted only in or among countries not thus excluded. In such case, this License incorporates the limitation as if written in the body of this License.
9. The Free Software Foundation may publish revised and/or new versions of the General Public License from time to time. Such new versions will be similar in spirit to the present version, but may differ in detail to address new problems or concerns.
Each version is given a distinguishing version number. If the Program specifies a version number of this License which applies to it and "any later version", you have the option of following the terms and conditions either of that version or of any later version published by the Free Software Foundation. If the Program does not specify a version number of this License, you may choose any version ever published by the Free Software Foundation.
10. If you wish to incorporate parts of the Program into other free programs whose distribution conditions are different, write to the author to ask for permission. For software which is copyrighted by the Free Software Foundation, write to the Free Software Foundation; we sometimes make exceptions for this. Our decision will be guided by the two goals of preserving the free status of all derivatives of our free software and of promoting the sharing and reuse of software generally.
11. BECAUSE THE PROGRAM IS LICENSED FREE OF CHARGE, THERE IS NO WARRANTY FOR THE PROGRAM, TO THE EXTENT PERMITTED BY APPLICABLE LAW. EXCEPT WHEN OTHERWISE STATED IN WRITING THE COPYRIGHT HOLDERS AND/OR OTHER PARTIES PROVIDE THE PROGRAM "AS IS" WITHOUT WARRANTY OF ANY KIND, EITHER EXPRESSED OR IMPLIED, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE. THE ENTIRE RISK AS TO THE QUALITY AND PERFORMANCE OF THE PROGRAM IS WITH YOU. SHOULD THE PROGRAM PROVE DEFECTIVE, YOU ASSUME THE COST OF ALL NECESSARY SERVICING, REPAIR OR CORRECTION.
12. IN NO EVENT UNLESS REQUIRED BY APPLICABLE LAW OR AGREED TO IN WRITING WILL ANY COPYRIGHT HOLDER, OR ANY OTHER PARTY WHO MAY MODIFY AND/OR REDISTRIBUTE THE PROGRAM AS PERMITTED ABOVE, BE LIABLE TO YOU FOR DAMAGES, INCLUDING ANY GENERAL, SPECIAL, INCIDENTAL OR CONSEQUENTIAL DAMAGES ARISING OUT OF THE USE OR INABILITY TO USE THE PROGRAM (INCLUDING BUT NOT LIMITED TO LOSS OF DATA OR DATA BEING RENDERED INACCURATE OR LOSSES SUSTAINED BY YOU OR THIRD PARTIES OR A FAILURE OF THE PROGRAM TO OPERATE WITH ANY OTHER PROGRAMS), EVEN IF SUCH HOLDER OR OTHER PARTY HAS BEEN ADVISED OF THE POSSIBILITY OF SUCH DAMAGES.
To do so, attach the following notices to the program. It is safest to attach them to the start of each source file to most effectively convey the exclusion of warranty; and each file should have at least the "copyright" line and a pointer to where the full notice is found.
<one line to give the program's name and a brief idea of what it does.>
Copyright (C) 19yy <name of author>
This program is free software; you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation; either version 2 of the License, or (at your option) any later version.
You should have received a copy of the GNU General Public License along with this program; if not, write to the Free Software Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA
If the program is interactive, make it output a short notice like this when it starts in an interactive mode:
Gnomovision version 69, Copyright (C) 19yy name of author
Gnomovision comes with ABSOLUTELY NO WARRANTY; for details type `show w'.
This is free software, and you are welcome to redistribute it under certain conditions; type `show c' for details.
The hypothetical commands `show w' and `show c' should show the appropriate parts of the General Public License. Of course, the commands you use may be called something other than `show w' and `show c'; they could even be mouse-clicks or menu items--whatever suits your program.
You should also get your employer (if you work as a programmer) or your school, if any, to sign a "copyright disclaimer" for the program, if necessary. Here is a sample; alter the names:
Yoyodyne, Inc., hereby disclaims all copyright interest in the program `Gnomovision' (which makes passes at compilers) written by James Hacker.
<signature of Ty Coon>, 1 April 1989
Ty Coon, President of Vice
This General Public License does not permit incorporating your program into proprietary programs. If your program is a subroutine library, you may consider it more useful to permit linking proprietary applications with the library. If this is what you want to do, use the GNU Library General Public License instead of this License.
%%The following software may be included in, shipped with, or otherwise used by this product: Net-SNMP; Use of any of this software may be governed by the terms of the license below:
Copyright 1989, 1991, 1992 by Carnegie Mellon University Derivative Work - 1996, 1998-2000
Copyright 1996, 1998-2000 The Regents of the University of California All Rights Reserved
Permission to use, copy, modify and distribute this software and its documentation for any
purpose and without fee is hereby granted, provided that the above copyright notice appears
in all copies and that both that copyright notice and this permission notice appear in
supporting documentation, and that the name of CMU and The Regents of the University of California
not be used in advertising or publicity pertaining to distribution of the software without specific
written permission. CMU AND THE REGENTS OF THE UNIVERSITY OF CALIFORNIA DISCLAIM ALL
WARRANTIES WITH REGARD TO THIS SOFTWARE, INCLUDING ALL IMPLIED WARRANTIES OF MERCHANTABILITY AND
FITNESS. IN NO EVENT SHALL CMU OR THE REGENTS OF THE UNIVERSITY OF CALIFORNIA BE LIABLE
FOR ANY SPECIAL, INDIRECT OR CONSEQUENTIAL DAMAGES OR ANY DAMAGES WHATSOEVER RESULTING FROM THE
LOSS OF USE, DATA OR PROFITS, WHETHER IN AN ACTION OF CONTRACT, NEGLIGENCE OR OTHER TORTIOUS
ACTION, ARISING OUT OF OR IN CONNECTION WITH THE USE OR PERFORMANCE OF THIS SOFTWARE.
Copyright (c) 2001-2003, Networks Associates Technology, Inc All rights reserved. Redistribution
and use in source and binary forms, with or without modification, are permitted provided that
the following conditions are met: * Redistributions of source code must retain the above copyright
notice, this list of conditions and the following disclaimer. * Redistributions in binary
form must reproduce the above copyright notice, this list of conditions and the following disclaimer
in the documentation and/or other materials provided with the distribution. * Neither the
name of the Networks Associates Technology, Inc nor the names of its contributors may be used to
endorse or promote products derived from this software without specific prior written permission.
THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS ``AS IS'' AND ANY EXPRESS OR
IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND
FITNESS FOR A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT HOLDERS OR
CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL
DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE,
DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER
IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT
OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
Portions of this code are copyright (c) 2001-2003, Cambridge Broadband Ltd. All rights reserved.
Redistribution and use in source and binary forms, with or without modification, are permitted provided
that the following conditions are met: * Redistributions of source code must retain the above copyright
notice, this list of conditions and the following disclaimer. * Redistributions in binary form
must reproduce the above copyright notice, this list of conditions and the following disclaimer in the
documentation and/or other materials provided with the distribution. * The name of Cambridge Broadband Ltd.
may not be used to endorse or promote products derived from this software without specific prior written permission.
THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDER ``AS IS'' AND ANY EXPRESS OR IMPLIED
WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A
PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT HOLDER BE LIABLE FOR ANY DIRECT, INDIRECT,
INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE
GOODS OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON
ANY THEORY OF LIABILITY,WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING
IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
Copyright © 2003 Sun Microsystems, Inc., 4150 Network Circle, Santa Clara, California 95054, U.S.A. All rights reserved.
Use is subject to license terms below. This distribution may include materials developed
by third parties. Sun, Sun Microsystems, the Sun logo and Solaris are trademarks or registered trademarks of
Sun Microsystems, Inc. in the U.S. and other countries. Redistribution and use in source and binary forms,
with or without modification, are permitted provided that the following conditions are met:
* Redistributions in binary form must reproduce the above copyright notice, this list of conditions and the following disclaimer
in the documentation and/or other materials provided with the distribution.
* Neither the name of the Sun Microsystems, Inc. nor the names of its contributors may be used to endorse or promote
THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS ``AS IS'' AND ANY EXPRESS OR IMPLIED WARRANTIES,
INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR
PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT HOLDERS OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT,
INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT
OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY
THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE)
ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
Copyright (c) 2003-2006, Sparta, Inc All rights reserved. Redistribution and use in source and binary forms, with or
without modification, are permitted provided that the following conditions are met:
* Redistributions of source code must retain the above copyright notice, this list of conditions and the following
* Redistributions in binary form must reproduce the above copyright notice, this list of conditions and the following
disclaimer in the documentation and/or other materials provided with the distribution.
* Neither the name of Sparta, Inc nor the names of its contributors may be used to endorse or promote products derived
Copyright (c) 2004, Cisco, Inc and Information Network Center of Beijing University of Posts and Telecommunications.
All rights reserved. Redistribution and use in source and binary forms, with or without modification,
are permitted provided that the following conditions are met:
* Neither the name of Cisco, Inc, Beijing University of Posts and Telecommunications, nor the names of their
contributors may be used to endorse or promote products derived from this software without specific prior written permission.
Copyright (c) Fabasoft R&D Software GmbH & Co KG, 2003 oss@fabasoft.com Author: Bernhard Penz Redistribution and use
in source and binary forms, with or without modification, are permitted provided that the following
conditions are met:
* The name of Fabasoft R&D Software GmbH & Co KG or any of its subsidiaries, brand or product names may not be used to
THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDER ``AS IS'' AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT
LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE DISCLAIMED.
IN NO EVENT SHALL THE COPYRIGHT HOLDER BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL
DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR PROFITS;
OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT
(INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE
POSSIBILITY OF SUCH DAMAGE.
%%The following software may be included in, shipped with, or otherwise used by this product: OpenSSL; Use of any of this software may be governed by the terms of the license below:
Copyright (c) 1998-2007 The OpenSSL Project. All rights reserved. Redistribution and use in source and binary forms, with or without modification, are permitted provided that the following conditions are met: 1. Redistributions of source code must retain the above copyright notice, this list of conditions and the following disclaimer. 2. Redistributions in binary form must reproduce the above copyright notice, this list of conditions and the following disclaimer in the documentation and/or other materials provided with the distribution. 3. All advertising materials mentioning features or use of this software must display the following acknowledgment: "This product includes software developed by the OpenSSL Project for use in the OpenSSL Toolkit. (http://www.openssl.org/)" 4. The names "OpenSSL Toolkit" and "OpenSSL Project" must not be used to endorse or promote products derived from this software without prior written permission. For written permission, please contact openssl-core@openssl.org. 5. Products derived from this software may not be called "OpenSSL" nor may "OpenSSL" appear in their names without prior written permission of the OpenSSL Project. 6. Redistributions of any form whatsoever must retain the following acknowledgment: "This product includes software developed by the OpenSSL Project for use in the OpenSSL Toolkit http://www.openssl.org/)" THIS SOFTWARE IS PROVIDED BY THE OpenSSL PROJECT ``AS IS'' AND ANY EXPRESSED OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE OpenSSL PROJECT OR ITS CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE. This product includes cryptographic software written by Eric Young (eay@cryptsoft.com). This product includes software written by Tim Hudson (tjh@cryptsoft.com). Original SSLeay License Copyright (C) 1995-1998 Eric Young (eay@cryptsoft.com) All rights reserved. This package is an SSL implementation written by Eric Young (eay@cryptsoft.com). The implementation was written so as to conform with Netscapes SSL. This library is free for commercial and non-commercial use as long as the following conditions are aheared to. The following conditions apply to all code found in this distribution, be it the RC4, RSA, lhash, DES, etc., code; not just the SSL code. The SSL documentation included with this distribution is covered by the same copyright terms except that the holder is Tim Hudson (tjh@cryptsoft.com). Copyright remains Eric Young's, and as such any Copyright notices in the code are not to be removed. If this package is used in a product, Eric Young should be given attribution as the author of the parts of the library used. This can be in the form of a textual message at program startup or in documentation (online or textual) provided with the package. Redistribution and use in source and binary forms, with or without modification, are permitted provided that the following conditions are met: 1. Redistributions of source code must retain the copyright notice, this list of conditions and the following disclaimer. 2. Redistributions in binary form must reproduce the above copyright notice, this list of conditions and the following disclaimer in the documentation and/or other materials provided with the distribution. 3. All advertising materials mentioning features or use of this software must display the following acknowledgement: "This product includes cryptographic software written by Eric Young (eay@cryptsoft.com)" The word 'cryptographic' can be left out if the routines from the library being used are not cryptographic related :-). 4. If you include any Windows specific code (or a derivative thereof) from the apps directory (application code) you must include an acknowledgement: "This product includes software written by Tim Hudson (tjh@cryptsoft.com)" THIS SOFTWARE IS PROVIDED BY ERIC YOUNG ``AS IS'' AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE AUTHOR OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE. The licence and distribution terms for any publically available version or derivative of this code cannot be changed. i.e. this code cannot simply be copied and put under another distribution licence [including the GNU Public Licence.]
%%The following software may be included in, shipped with, or otherwise used by this product: pdksh; Use of any of this software may be governed by the terms of the license below:
pdksh is provided AS IS, with NO WARRANTY, either expressed or implied. The vast majority of the code that makes pdksh is in the public domain. The exceptions are: sigact.c and sigact.h These are covered by copyrighten by Simon J. Gerraty; the copyright notice for these files is as follows: This is free software. It comes with NO WARRANTY. Permission to use, modify and distribute this source code is granted subject to the following conditions. 1/ that that the above copyright notice and this notice are preserved in all copies and that due credit be given to the author. 2/ that any changes to this code are clearly commented as such so that the author does get blamed for bugs other than his own. aclocal.m4 This is covered by the GNU General Public Licence (GPL) as it contains modified versions of macros that come with GNU autoconf. As this is used solely for configuration, the pdksh code itself is not covered by the GPL. The following is taken from autoconf 2.x documentation (info autoconf questions distributing) concerning use of autoconf in programs: There are no restrictions on how the configuration scripts that Autoconf produces may be distributed or used. In Autoconf version 1, they were covered by the GNU General Public License. We still encourage software authors to distribute their work under terms like those of the GPL, but doing so is not required to use Autoconf.
%%The following software may be included in, shipped with, or otherwise used by this product: PCRE and PCRECPP; Use of any of this software may be governed by the terms of the license below:
PCRE LICENCE - PCRE is a library of functions to support regular expressions whose syntax and semantics are as close as possible to those of the Perl 5 language. Release 7 of PCRE is distributed under the terms of the "BSD" licence, as specified below. The documentation for PCRE, supplied in the "doc" directory, is distributed under the same terms as the software itself. The basic library functions are written in C and are freestanding. Also included in the distribution is a set of C++ wrapper functions. THE BASIC LIBRARY FUNCTIONS - Written by: Philip Hazel Email local part: ph10 Email domain: cam.ac.uk University of Cambridge Computing Service, Cambridge, England. Phone: +44 1223 334714. Copyright (c) 1997-2006 University of Cambridge All rights reserved. THE C++ WRAPPER FUNCTIONS - Contributed by: Google Inc. Copyright (c) 2006, Google Inc. All rights reserved. THE "BSD" LICENCE - Redistribution and use in source and binary forms, with or without modification, are permitted provided that the following conditions are met: * Redistributions of source code must retain the above copyright notice, this list of conditions and the following disclaimer. * Redistributions in binary form must reproduce the above copyright notice, this list of conditions and the following disclaimer in the documentation and/or other materials provided with the distribution. * Neither the name of the University of Cambridge nor the name of Google Inc. nor the names of their contributors may be used to endorse or promote products derived from this software without specific prior written permission. THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT OWNER OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
%%The following software may be included in, shipped with, or otherwise used by this product: OpenPegasus; Use of any of this software may be governed by the terms of the license below:
Copyright (c) 2000, 2001, 2002 BMC Software; Hewlett-Packard Development Company, L.P.; IBM Corp.; The Open Group; Tivoli Systems. Copyright (c) 2003 BMC Software; Hewlett-Packard Development Company, L.P.; IBM Corp.; EMC Corporation, The Open Group. Copyright (c) 2004 BMC Software; Hewlett-Packard Development Company, L.P.; IBM Corp.; EMC Corporation; VERITAS Software Corporation; The Open Group. Copyright (c) 2005 Hewlett-Packard Development Company, L.P.; IBM Corp.; EMC Corporation; VERITAS Software Corporation; The Open Group. Copyright (c) 2006 Hewlett-Packard Development Company, L.P.; IBM Corp.; EMC Corporation; Symantec Corporation; The Open Group. Permission is hereby granted, free of charge, to any person obtaining a copy of this software and associated documentation files (the "Software"), to deal in the Software without restriction, including without limitation the rights to use, copy, modify, merge, publish, distribute, sublicense, and/or sell copies of the Software, and to permit persons to whom the Software is furnished to do so, subject to the following conditions: THE ABOVE COPYRIGHT NOTICE AND THIS PERMISSION NOTICE SHALL BE INCLUDED IN ALL COPIES OR SUBSTANTIAL PORTIONS OF THE SOFTWARE. THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE SOFTWARE.
%%The following software may be included in, shipped with, or otherwise used by this product: Platinum UPnP and Neptune C++ Runtime; Use of any of this software may be governed by the terms of the license below:
Platinum Software Copyright © Plutinosoft LLC, 2007. All rights reserved. This software and documentation constitute an unpublished work and contain valuable trade secrets and proprietary information belonging to Plutinosoft. None of the foregoing materials may be copies, duplicated or disclosed without the express written permission of Plutinosoft.
%%The following software may be included in, shipped with, or otherwise used by this product: psycopg/adapter*.{h,c} and psycopg/microprotocol*.{h,c}:
Permission is granted to anyone to use this software for any purpose, including commercial applications, and to alter it and redistribute it freely, subject to the following restrictions:
1. The origin of this software must not be misrepresented; you must not claim that you wrote the original software. If you use this software in a product, an acknowledgment in the product documentation would be appreciated but is not required.
2. Altered source versions must be plainly marked as such, and must not be misrepresented as being the original software.
3. This notice may not be removed or altered from any source distribution.
%%The following software may be included in, shipped with, or otherwise used by this product: MaxMind GeoLite; Use of any of this software may be governed by the terms of the license below:
Copyright© 2012 MaxMind, Inc. All Rights Reserved.
%%The following software may be included in, shipped with, or otherwise used by this product: Python; Use of any of this software may be governed by the terms of the license below:
Python 2.4.2 license
This is the official license for the Python 2.4 release:
A. HISTORY OF THE SOFTWARE
Python was created in the early 1990s by Guido van Rossum at Stichting Mathematisch Centrum (CWI, see http://www.cwi.nl) in the Netherlands as a successor of a language called ABC. Guido remains Python's principal author, although it includes many contributions from others.
In 1995, Guido continued his work on Python at the Corporation for National Research Initiatives (CNRI, see http://www.cnri.reston.va.us) in Reston, Virginia where he released several versions of the software.
In May 2000, Guido and the Python core development team moved to BeOpen.com to form the BeOpen PythonLabs team. In October of the same year, the PythonLabs team moved to Digital Creations (now Zope Corporation, see http://www.zope.com). In 2001, the Python Software Foundation (PSF, see http://www.python.org/psf/) was formed, a non-profit organization created specifically to own Python-related Intellectual Property. Zope Corporation is a sponsoring member of the PSF.
All Python releases are Open Source (see http://www.opensource.org for the Open Source Definition). Historically, most, but not all, Python releases have also been GPL-compatible; the table below summarizes the various releases.
Release Derived Year Owner GPL-
from compatible? (1)
0.9.0 thru 1.2 1991-1995 CWI yes
1.3 thru 1.5.2 1.2 1995-1999 CNRI yes
1.6 1.5.2 2000 CNRI no
2.0 1.6 2000 BeOpen.com no
1.6.1 1.6 2001 CNRI yes (2)
2.1 2.0+1.6.1 2001 PSF no
2.0.1 2.0+1.6.1 2001 PSF yes
2.2 2.1.1 2001 PSF yes
2.1.2 2.1.1 2002 PSF yes
2.2.1 2.2 2002 PSF yes
2.3 2.2.2 2002-2003 PSF yes
2.3.1 2.3 2002-2003 PSF yes
2.3.2 2.3.1 2002-2003 PSF yes
2.4 2.3 2004 PSF yes
(1) GPL-compatible doesn't mean that we're distributing Python under the GPL. All Python licenses, unlike the GPL, let you distribute a modified version without making your changes open source. The GPL-compatible licenses make it possible to combine Python with other software that is released under the GPL; the others don't.
(2) According to Richard Stallman, 1.6.1 is not GPL-compatible, because its license has a choice of law clause. According to CNRI, however, Stallman's lawyer has told CNRI's lawyer that 1.6.1 is "not incompatible" with the GPL.
Thanks to the many outside volunteers who have worked under Guido's direction to make these releases possible.
B. TERMS AND CONDITIONS FOR ACCESSING OR OTHERWISE USING PYTHON
PSF LICENSE AGREEMENT FOR PYTHON 2.4
1. This LICENSE AGREEMENT is between the Python Software Foundation ("PSF"), and the Individual or Organization ("Licensee") accessing and otherwise using Python 2.4 software in source or binary form and its associated documentation.
2. Subject to the terms and conditions of this License Agreement, PSF hereby grants Licensee a nonexclusive, royalty-free, world-wide license to reproduce, analyze, test, perform and/or display publicly, prepare derivative works, distribute, and otherwise use Python 2.4 alone or in any derivative version, provided, however, that PSF's License Agreement and PSF's notice of copyright, i.e., "Copyright (c) 2001, 2002, 2003, 2004 Python Software Foundation; All Rights Reserved" are retained in Python 2.4 alone or in any derivative version prepared by Licensee.
3. In the event Licensee prepares a derivative work that is based on or incorporates Python 2.4 or any part thereof, and wants to make the derivative work available to others as provided herein, then Licensee hereby agrees to include in any such work a brief summary of the changes made to Python 2.4.
4. PSF is making Python 2.4 available to Licensee on an "AS IS" basis. PSF MAKES NO REPRESENTATIONS OR WARRANTIES, EXPRESS OR IMPLIED. BY WAY OF EXAMPLE, BUT NOT LIMITATION, PSF MAKES NO AND DISCLAIMS ANY REPRESENTATION OR WARRANTY OF MERCHANTABILITY OR FITNESS FOR ANY PARTICULAR PURPOSE OR THAT THE USE OF PYTHON 2.4 WILL NOT INFRINGE ANY THIRD PARTY RIGHTS.
5. PSF SHALL NOT BE LIABLE TO LICENSEE OR ANY OTHER USERS OF PYTHON 2.4 FOR ANY INCIDENTAL, SPECIAL, OR CONSEQUENTIAL DAMAGES OR LOSS AS A RESULT OF MODIFYING, DISTRIBUTING, OR OTHERWISE USING PYTHON 2.4, OR ANY DERIVATIVE THEREOF, EVEN IF ADVISED OF THE POSSIBILITY THEREOF.
6. This License Agreement will automatically terminate upon a material breach of its terms and conditions.
7. Nothing in this License Agreement shall be deemed to create any relationship of agency, partnership, or joint venture between PSF and Licensee. This License Agreement does not grant permission to use PSF trademarks or trade name in a trademark sense to endorse or promote products or services of Licensee, or any third party.
8. By copying, installing or otherwise using Python 2.4, Licensee agrees to be bound by the terms and conditions of this License Agreement.
BEOPEN.COM LICENSE AGREEMENT FOR PYTHON 2.0
BEOPEN PYTHON OPEN SOURCE LICENSE AGREEMENT VERSION 1
1. This LICENSE AGREEMENT is between BeOpen.com ("BeOpen"), having an office at 160 Saratoga Avenue, Santa Clara, CA 95051, and the Individual or Organization ("Licensee") accessing and otherwise using this software in source or binary form and its associated documentation ("the Software").
2. Subject to the terms and conditions of this BeOpen Python License Agreement, BeOpen hereby grants Licensee a non-exclusive, royalty-free, world-wide license to reproduce, analyze, test, perform and/or display publicly, prepare derivative works, distribute, and otherwise use the Software alone or in any derivative version, provided, however, that the BeOpen Python License is retained in the Software, alone or in any derivative version prepared by Licensee.
3. BeOpen is making the Software available to Licensee on an "AS IS" basis. BEOPEN MAKES NO REPRESENTATIONS OR WARRANTIES, EXPRESS OR IMPLIED. BY WAY OF EXAMPLE, BUT NOT LIMITATION, BEOPEN MAKES NO AND DISCLAIMS ANY REPRESENTATION OR WARRANTY OF MERCHANTABILITY OR FITNESS FOR ANY PARTICULAR PURPOSE OR THAT THE USE OF THE SOFTWARE WILL NOT INFRINGE ANY THIRD PARTY RIGHTS.
4. BEOPEN SHALL NOT BE LIABLE TO LICENSEE OR ANY OTHER USERS OF THE SOFTWARE FOR ANY INCIDENTAL, SPECIAL, OR CONSEQUENTIAL DAMAGES OR LOSS AS A RESULT OF USING, MODIFYING OR DISTRIBUTING THE SOFTWARE, OR ANY DERIVATIVE THEREOF, EVEN IF ADVISED OF THE POSSIBILITY THEREOF.
6. This License Agreement shall be governed by and interpreted in all respects by the law of the State of California, excluding conflict of law provisions. Nothing in this License Agreement shall be deemed to create any relationship of agency, partnership, or joint venture between BeOpen and Licensee. This License Agreement does not grant permission to use BeOpen trademarks or trade names in a trademark sense to endorse or promote products or services of Licensee, or any third party. As an exception, the "BeOpen Python" logos available at http://www.pythonlabs.com/logos.html may be used according to the permissions granted on that web page.
7. By copying, installing or otherwise using the software, Licensee agrees to be bound by the terms and conditions of this License Agreement.
CNRI LICENSE AGREEMENT FOR PYTHON 1.6.1
1. This LICENSE AGREEMENT is between the Corporation for National Research Initiatives, having an office at 1895 Preston White Drive, Reston, VA 20191 ("CNRI"), and the Individual or Organization ("Licensee") accessing and otherwise using Python 1.6.1 software in source or binary form and its associated documentation.
2. Subject to the terms and conditions of this License Agreement, CNRI hereby grants Licensee a nonexclusive, royalty-free, world-wide license to reproduce, analyze, test, perform and/or display publicly, prepare derivative works, distribute, and otherwise use Python 1.6.1 alone or in any derivative version, provided, however, that CNRI's License Agreement and CNRI's notice of copyright, i.e., "Copyright (c) 1995-2001 Corporation for National Research Initiatives; All Rights Reserved" are retained in Python 1.6.1 alone or in any derivative version prepared by Licensee. Alternately, in lieu of CNRI's License Agreement, Licensee may substitute the following text (omitting the quotes): "Python 1.6.1 is made available subject to the terms and conditions in CNRI's License Agreement. This Agreement together with Python 1.6.1 may be located on the Internet using the following unique, persistent identifier (known as a handle): 1895.22/1013. This Agreement may also be obtained from a proxy server on the Internet using the following URL: http://hdl.handle.net/1895.22/1013".
3. In the event Licensee prepares a derivative work that is based on or incorporates Python 1.6.1 or any part thereof, and wants to make the derivative work available to others as provided herein, then Licensee hereby agrees to include in any such work a brief summary of the changes made to Python 1.6.1.
4. CNRI is making Python 1.6.1 available to Licensee on an "AS IS" basis. CNRI MAKES NO REPRESENTATIONS OR WARRANTIES, EXPRESS OR IMPLIED. BY WAY OF EXAMPLE, BUT NOT LIMITATION, CNRI MAKES NO AND DISCLAIMS ANY REPRESENTATION OR WARRANTY OF MERCHANTABILITY OR FITNESS FOR ANY PARTICULAR PURPOSE OR THAT THE USE OF PYTHON 1.6.1 WILL NOT INFRINGE ANY THIRD PARTY RIGHTS.
5. CNRI SHALL NOT BE LIABLE TO LICENSEE OR ANY OTHER USERS OF PYTHON 1.6.1 FOR ANY INCIDENTAL, SPECIAL, OR CONSEQUENTIAL DAMAGES OR LOSS AS A RESULT OF MODIFYING, DISTRIBUTING, OR OTHERWISE USING PYTHON 1.6.1, OR ANY DERIVATIVE THEREOF, EVEN IF ADVISED OF THE POSSIBILITY THEREOF.
7. This License Agreement shall be governed by the federal intellectual property law of the United States, including without limitation the federal copyright law, and, to the extent such U.S. federal law does not apply, by the law of the Commonwealth of Virginia, excluding Virginia's conflict of law provisions. Notwithstanding the foregoing, with regard to derivative works based on Python 1.6.1 that incorporate non-separable material that was previously distributed under the GNU General Public License (GPL), the law of the Commonwealth of Virginia shall govern this License Agreement only as to issues arising under or with respect to Paragraphs 4, 5, and 7 of this License Agreement. Nothing in this License Agreement shall be deemed to create any relationship of agency, partnership, or joint venture between CNRI and Licensee. This License Agreement does not grant permission to use CNRI trademarks or trade name in a trademark sense to endorse or promote products or services of Licensee, or any third party.
8. By clicking on the "ACCEPT" button where indicated, or by copying, installing or otherwise using Python 1.6.1, Licensee agrees to be bound by the terms and conditions of this License Agreement.
CWI LICENSE AGREEMENT FOR PYTHON 0.9.0 THROUGH 1.2
Copyright (c) 1991 - 1995, Stichting Mathematisch Centrum Amsterdam, The Netherlands. All rights reserved.
Permission to use, copy, modify, and distribute this software and its documentation for any purpose and without fee is hereby granted, provided that the above copyright notice appear in all copies and that both that copyright notice and this permission notice appear in supporting documentation, and that the name of Stichting Mathematisch Centrum or CWI not be used in advertising or publicity pertaining to distribution of the software without specific, written prior permission.
STICHTING MATHEMATISCH CENTRUM DISCLAIMS ALL WARRANTIES WITH REGARD TO THIS SOFTWARE, INCLUDING ALL IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS, IN NO EVENT SHALL STICHTING MATHEMATISCH CENTRUM BE LIABLE FOR ANY SPECIAL, INDIRECT OR CONSEQUENTIAL DAMAGES OR ANY DAMAGES WHATSOEVER RESULTING FROM LOSS OF USE, DATA OR PROFITS, WHETHER IN AN ACTION OF CONTRACT, NEGLIGENCE OR OTHER TORTIOUS ACTION, ARISING OUT OF OR IN CONNECTION WITH THE USE OR PERFORMANCE OF THIS SOFTWARE.
%%The following software may be included in, shipped with, or otherwise used by this product: rentzsch; Use of any of this software may be governed by the terms of the license below:
Copyright (c) 2003-2012 Jonathan 'Wolf' Rentzsch: http://rentzsch.com
// Some rights reserved: http://opensource.org/licenses/mit
%%The following software may be included in, shipped with, or otherwise used by this product: STLPort; Use of any of this software may be governed by the terms of the license below:
Copyright (c) 1994 Hewlett-Packard Company Copyright (c) 1996-1999 Silicon Graphics Computer Systems, Inc. Copyright (c) 1997 Moscow Center for SPARC Technology Copyright (c) 1999-2003 Boris Fomitchev This material is provided "as is", with absolutely no warranty expressed or implied. Any use is at your own risk. Permission to use or copy this software for any purpose is hereby granted without fee, provided the above notices are retained on all copies. Permission to modify the code and to distribute modified code is granted, provided the above notices are retained, and a notice that the code was modified is included with the above copyright notice.
%%The following software may be included in, shipped with, or otherwise used by this product: NetX, SwingX and Swing-worker; Use of any of this software may be governed by the terms of the license below:
51 Franklin Street, Fifth Floor, Boston, MA 02110-1301 USA
[This is the first released version of the Lesser GPL. It also counts as the successor of the GNU Library Public License,
version 2, hence the version number 2.1.]
The licenses for most software are designed to take away your freedom to share and change it.
By contrast, the GNU General Public Licenses are intended to guarantee your freedom to share and
change free software--to make sure the software is free for all its users.
This license, the Lesser General Public License, applies to some specially designated software
packages--typically libraries--of the Free Software Foundation and other authors who decide to use it.
You can use it too, but we suggest you first think carefully about whether this license or the ordinary
General Public License is the better strategy to use in any particular case, based on the explanations below.
When we speak of free software, we are referring to freedom of use, not price. Our General
Public Licenses are designed to make sure that you have the freedom to distribute copies of free software
(and charge for this service if you wish); that you receive source code or can get it if you want it; that
you can change the software and use pieces of it in new free programs; and that you are informed that you
can do these things.
To protect your rights, we need to make restrictions that forbid distributors to deny you these rights
or to ask you to surrender these rights. These restrictions translate to certain responsibilities for you
if you distribute copies of the library or if you modify it.
For example, if you distribute copies of the library, whether gratis or for a fee, you must give the
recipients all the rights that we gave you. You must make sure that they, too, receive or can get the
source code. If you link other code with the library, you must provide complete object files to the
recipients, so that they can relink them with the library after making changes to the library and recompiling it.
And you must show them these terms so they know their rights.
We protect your rights with a two-step method: (1) we copyright the library, and (2) we offer you this license,
which gives you legal permission to copy, distribute and/or modify the library.
To protect each distributor, we want to make it very clear that there is no warranty for the free library.
Also, if the library is modified by someone else and passed on, the recipients should know that what they have
is not the original version, so that the original author's reputation will not be affected by problems that
might be introduced by others.
Finally, software patents pose a constant threat to the existence of any free program. We wish to make
sure that a company cannot effectively restrict the users of a free program by obtaining a restrictive license from a patent holder.
Therefore, we insist that any patent license obtained for a version of the library must be consistent with the full freedom of use
specified in this license.
Most GNU software, including some libraries, is covered by the ordinary GNU General Public License.
This license, the GNU Lesser General Public License, applies to certain designated libraries, and is quite
different from the ordinary General Public License. We use this license for certain libraries in order to
permit linking those libraries into non-free programs.
When a program is linked with a library, whether statically or using a shared library, the combination
of the two is legally speaking a combined work, a derivative of the original library. The ordinary General
Public License therefore permits such linking only if the entire combination fits its criteria of freedom.
The Lesser General Public License permits more lax criteria for linking other code with the library.
We call this license the "Lesser" General Public License because it does Less to protect the user's freedom
than the ordinary General Public License. It also provides other free software developers Less of an advantage
over competing non-free programs. These disadvantages are the reason we use the ordinary General Public License
for many libraries. However, the Lesser license provides advantages in certain special circumstances.
For example, on rare occasions, there may be a special need to encourage the widest possible use of a certain
library, so that it becomes a de-facto standard. To achieve this, non-free programs must be allowed to use the
library. A more frequent case is that a free library does the same job as widely used non-free libraries.
In this case, there is little to gain by limiting the free library to free software only, so we use the Lesser
General Public License.
In other cases, permission to use a particular library in non-free programs enables a greater number of people
to use a large body of free software. For example, permission to use the GNU C Library in non-free programs
enables many more people to use the whole GNU operating system, as well as its variant, the GNU/Linux operating
Although the Lesser General Public License is Less protective of the users' freedom, it does ensure that the
user of a program that is linked with the Library has the freedom and the wherewithal to run that program using
a modified version of the Library.
The precise terms and conditions for copying, distribution and modification follow. Pay close attention to
the difference between a "work based on the library" and a "work that uses the library". The former contains
code derived from the library, whereas the latter must be combined with the library in order to run.
0. This License Agreement applies to any software library or other program which contains a notice placed by
the copyright holder or other authorized party saying it may be distributed under the terms of this Lesser
General Public License (also called "this License"). Each licensee is addressed as "you".
A "library" means a collection of software functions and/or data prepared so as to be conveniently linked with
application programs (which use some of those functions and data) to form executables.
The "Library", below, refers to any such software library or work which has been distributed under these terms.
A "work based on the Library" means either the Library or any derivative work under copyright law: that is to say,
a work containing the Library or a portion of it, either verbatim or with modifications and/or translated
straightforwardly into another language. (Hereinafter, translation is included without limitation in the term
"modification".)
"Source code" for a work means the preferred form of the work for making modifications to it. For a library,
complete source code means all the source code for all modules it contains, plus any associated interface
definition files, plus the scripts used to control compilation and installation of the library.
Activities other than copying, distribution and modification are not covered by this License; they are outside
its scope. The act of running a program using the Library is not restricted, and output from such a program is
covered only if its contents constitute a work based on the Library (independent of the use of the Library in a
tool for writing it). Whether that is true depends on what the Library does and what the program that uses the
Library does.
1. You may copy and distribute verbatim copies of the Library's complete source code as you receive it, in any
medium, provided that you conspicuously and appropriately publish on each copy an appropriate copyright notice
and disclaimer of warranty; keep intact all the notices that refer to this License and to the absence of any
warranty; and distribute a copy of this License along with the Library.
You may charge a fee for the physical act of transferring a copy, and you may at your option offer warranty
protection in exchange for a fee.
2. You may modify your copy or copies of the Library or any portion of it, thus forming a work based on the
Library, and copy and distribute such modifications or work under the terms of Section 1 above, provided
that you also meet all of these conditions:
b) You must cause the files modified to carry prominent notices stating that you changed the files and the
date of any change.
c) You must cause the whole of the work to be licensed at no charge to all third parties under the terms of
d) If a facility in the modified Library refers to a function or a table of data to be supplied by an
application program that uses the facility, other than as an argument passed when the facility is invoked, then
you must make a good faith effort to ensure that, in the event an application does not supply such function or
table, the facility still operates, and performs whatever part of its purpose remains meaningful.
(For example, a function in a library to compute square roots has a purpose that is entirely well-defined
independent of the application. Therefore, Subsection 2d requires that any application-supplied function or
table used by this function must be optional: if the application does not supply it, the square root function
must still compute square roots.)
These requirements apply to the modified work as a whole. If identifiable sections of that work are not derived
from the Library, and can be reasonably considered independent and separate works in themselves, then this License,
and its terms, do not apply to those sections when you distribute them as separate works. But when you distribute
the same sections as part of a whole which is a work based on the Library, the distribution of the whole must be
on the terms of this License, whose permissions for other licensees extend to the entire whole, and thus to each
and every part regardless of who wrote it.
Thus, it is not the intent of this section to claim rights or contest your rights to work written entirely by you;
rather, the intent is to exercise the right to control the distribution of derivative or collective works based on
In addition, mere aggregation of another work not based on the Library with the Library (or with a work based on
the Library) on a volume of a storage or distribution medium does not bring the other work under the scope of this
License.
3. You may opt to apply the terms of the ordinary GNU General Public License instead of this License to a given
copy of the Library. To do this, you must alter all the notices that refer to this License, so that they refer to
the ordinary GNU General Public License, version 2, instead of to this License. (If a newer version than version 2
of the ordinary GNU General Public License has appeared, then you can specify that version instead if you wish.)
Do not make any other change in these notices.
Once this change is made in a given copy, it is irreversible for that copy, so the ordinary GNU General Public
License applies to all subsequent copies and derivative works made from that copy.
This option is useful when you wish to copy part of the code of the Library into a program that is not a library.
4. You may copy and distribute the Library (or a portion or derivative of it, under Section 2) in object code or
executable form under the terms of Sections 1 and 2 above provided that you accompany it with the complete
corresponding machine-readable source code, which must be distributed under the terms of Sections 1 and 2 above
on a medium customarily used for software interchange.
If distribution of object code is made by offering access to copy from a designated place, then offering
equivalent access to copy the source code from the same place satisfies the requirement to distribute the source
code, even though third parties are not compelled to copy the source along with the object code.
5. A program that contains no derivative of any portion of the Library, but is designed to work with the
Library by being compiled or linked with it, is called a "work that uses the Library". Such a work, in isolation,
is not a derivative work of the Library, and therefore falls outside the scope of this License.
However, linking a "work that uses the Library" with the Library creates an executable that is a derivative of
the Library (because it contains portions of the Library), rather than a "work that uses the library".
The executable is therefore covered by this License. Section 6 states terms for distribution of such executables.
When a "work that uses the Library" uses material from a header file that is part of the Library, the object
code for the work may be a derivative work of the Library even though the source code is not. Whether this is
true is especially significant if the work can be linked without the Library, or if the work is itself a library.
The threshold for this to be true is not precisely defined by law.
If such an object file uses only numerical parameters, data structure layouts and accessors, and small macros
and small inline functions (ten lines or less in length), then the use of the object file is unrestricted,
regardless of whether it is legally a derivative work. (Executables containing this object code plus portions
of the Library will still fall under Section 6.)
Otherwise, if the work is a derivative of the Library, you may distribute the object code for the work under
the terms of Section 6. Any executables containing that work also fall under Section 6, whether or not they are
linked directly with the Library itself.
6. As an exception to the Sections above, you may also combine or link a "work that uses the Library" with the
Library to produce a work containing portions of the Library, and distribute that work under terms of your choice,
provided that the terms permit modification of the work for the customer's own use and reverse engineering for
debugging such modifications.
You must give prominent notice with each copy of the work that the Library is used in it and that the Library
and its use are covered by this License. You must supply a copy of this License. If the work during execution
displays copyright notices, you must include the copyright notice for the Library among them, as well as a
reference directing the user to the copy of this License. Also, you must do one of these things:
a) Accompany the work with the complete corresponding machine-readable source code for the Library including
whatever changes were used in the work (which must be distributed under Sections 1 and 2 above); and, if the work
is an executable linked with the Library, with the complete machine-readable "work that uses the Library", as object
code and/or source code, so that the user can modify the Library and then relink to produce a modified executable
containing the modified Library. (It is understood that the user who changes the contents of definitions files in
the Library will not necessarily be able to recompile the application to use the modified definitions.)
b) Use a suitable shared library mechanism for linking with the Library. A suitable mechanism is one that
(1) uses at run time a copy of the library already present on the user's computer system, rather than copying
library functions into the executable, and
(2) will operate properly with a modified version of the library, if the user installs one, as long as the modified
version is interface-compatible with the version that the work was made with.
c) Accompany the work with a written offer, valid for at least three years, to give the same user the materials specified
in Subsection 6a, above, for a charge no more than the cost of performing this distribution.
d) If distribution of the work is made by offering access to copy from a designated place, offer equivalent access to copy
the above specified materials from the same place.
e) Verify that the user has already received a copy of these materials or that you have already sent this user a copy.
For an executable, the required form of the "work that uses the Library" must include any data and utility programs needed for
reproducing the executable from it. However, as a special exception, the materials to be distributed need not include anything that
is normally distributed (in either source or binary form) with the major components (compiler, kernel, and so on) of the operating
system on which the executable runs, unless that component itself accompanies the executable.
It may happen that this requirement contradicts the license restrictions of other proprietary libraries that do not normally accompany
the operating system. Such a contradiction means you cannot use both them and the Library together in an executable that you distribute.
7. You may place library facilities that are a work based on the Library side-by-side in a single library together with other library
facilities not covered by this License, and distribute such a combined library, provided that the separate distribution of the work based
on the Library and of the other library facilities is otherwise permitted, and provided that you do these two things:
a) Accompany the combined library with a copy of the same work based on the Library, uncombined with any other library facilities.
This must be distributed under the terms of the Sections above.
b) Give prominent notice with the combined library of the fact that part of it is a work based on the Library, and explaining where to
find the accompanying uncombined form of the same work.
8. You may not copy, modify, sublicense, link with, or distribute the Library except as expressly provided under this License.
Any attempt otherwise to copy, modify, sublicense, link with, or distribute the Library is void, and will automatically terminate
your rights under this License. However, parties who have received copies, or rights, from you under this License will not have
their licenses terminated so long as such parties remain in full compliance.
9. You are not required to accept this License, since you have not signed it. However, nothing else grants you permission to
modify or distribute the Library or its derivative works. These actions are prohibited by law if you do not accept this License.
Therefore, by modifying or distributing the Library (or any work based on the Library), you indicate your acceptance of this
License to do so, and all its terms and conditions for copying, distributing or modifying the Library or works based on it.
10. Each time you redistribute the Library (or any work based on the Library), the recipient automatically receives a license
from the original licensor to copy, distribute, link with or modify the Library subject to these terms and conditions. You may
not impose any further restrictions on the recipients' exercise of the rights granted herein. You are not responsible for enforcing
compliance by third parties with this License.
11. If, as a consequence of a court judgment or allegation of patent infringement or for any other reason (not limited to patent
issues), conditions are imposed on you (whether by court order, agreement or otherwise) that contradict the conditions of this
License, they do not excuse you from the conditions of this License. If you cannot distribute so as to satisfy simultaneously your
obligations under this License and any other pertinent obligations, then as a consequence you may not distribute the
Library at all. For example, if a patent license would not permit royalty-free redistribution of the Library by all those who
receive copies directly or indirectly through you, then the only way you could satisfy both it and this License would be to refrain
entirely from distribution of the Library.
If any portion of this section is held invalid or unenforceable under any particular circumstance, the balance of the section is
intended to apply, and the section as a whole is intended to apply in other circumstances.
It is not the purpose of this section to induce you to infringe any patents or other property right claims or to contest validity
of any such claims; this section has the sole purpose of protecting the integrity of the free software distribution system which
is implemented by public license practices. Many people have made generous contributions to the wide range of software distributed
through that system in reliance on consistent application of that system; it is up to the author/donor to decide if he or she is
willing to distribute software through any other system and a licensee cannot impose that choice.
12. If the distribution and/or use of the Library is restricted in certain countries either by patents or by copyrighted
interfaces, the original copyright holder who places the Library under this License may add an explicit geographical distribution
limitation excluding those countries, so that distribution is permitted only in or among countries not thus excluded.
In such case, this License incorporates the limitation as if written in the body of this License.
13. The Free Software Foundation may publish revised and/or new versions of the Lesser General Public License from time to time.
Such new versions will be similar in spirit to the present version, but may differ in detail to address new problems or concerns.
Each version is given a distinguishing version number. If the Library specifies a version number of this License which applies
to it and "any later version", you have the option of following the terms and conditions either of that version or of any later
version published by the Free Software Foundation. If the Library does not specify a license version number, you may choose any
version ever published by the Free Software Foundation.
14. If you wish to incorporate parts of the Library into other free programs whose distribution conditions are incompatible with
these, write to the author to ask for permission. For software which is copyrighted by the Free Software Foundation, write to the
Free Software Foundation; we sometimes make exceptions for this. Our decision will be guided by the two goals of preserving the
free status of all derivatives of our free software and of promoting the sharing and reuse of software generally.
15. BECAUSE THE LIBRARY IS LICENSED FREE OF CHARGE, THERE IS NO WARRANTY FOR THE LIBRARY, TO THE EXTENT PERMITTED BY APPLICABLE LAW.
EXCEPT WHEN OTHERWISE STATED IN WRITING THE COPYRIGHT HOLDERS AND/OR OTHER PARTIES PROVIDE THE LIBRARY "AS IS" WITHOUT WARRANTY OF
ANY KIND, EITHER EXPRESSED OR IMPLIED, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A
PARTICULAR PURPOSE. THE ENTIRE RISK AS TO THE QUALITY AND PERFORMANCE OF THE LIBRARY IS WITH YOU. SHOULD THE LIBRARY PROVE
DEFECTIVE, YOU ASSUME THE COST OF ALL NECESSARY SERVICING, REPAIR OR CORRECTION.
16. IN NO EVENT UNLESS REQUIRED BY APPLICABLE LAW OR AGREED TO IN WRITING WILL ANY COPYRIGHT HOLDER, OR ANY OTHER PARTY WHO MAY
MODIFY AND/OR REDISTRIBUTE THE LIBRARY AS PERMITTED ABOVE, BE LIABLE TO YOU FOR DAMAGES, INCLUDING ANY GENERAL, SPECIAL, INCIDENTAL
OR CONSEQUENTIAL DAMAGES ARISING OUT OF THE USE OR INABILITY TO USE THE LIBRARY (INCLUDING BUT NOT LIMITED TO LOSS OF DATA OR DATA
BEING RENDERED INACCURATE OR LOSSES SUSTAINED BY YOU OR THIRD PARTIES OR A FAILURE OF THE LIBRARY TO OPERATE WITH ANY OTHER SOFTWARE),
EVEN IF SUCH HOLDER OR OTHER PARTY HAS BEEN ADVISED OF THE POSSIBILITY OF SUCH DAMAGES.
How to Apply These Terms to Your New Libraries
If you develop a new library, and you want it to be of the greatest possible use to the public, we recommend making it free software
that everyone can redistribute and change. You can do so by permitting redistribution under these terms (or, alternatively, under the
terms of the ordinary General Public License).
To apply these terms, attach the following notices to the library. It is safest to attach them to the start of each source file to
most effectively convey the exclusion of warranty; and each file should have at least the "copyright" line and a pointer to where
the full notice is found.
<one line to give the library's name and a brief idea of what it does.>
Copyright (C) <year> <name of author>
This library is free software; you can redistribute it and/or modify it under the terms of the GNU Lesser General Public License
as published by the Free Software Foundation; either version 2.1 of the License, or (at your option) any later version.
This library is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of
MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU Lesser General Public License for more details.
You should have received a copy of the GNU Lesser General Public License along with this library;
if not, write to the Free Software Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301 USA
You should also get your employer (if you work as a programmer) or your school, if any, to sign a "copyright disclaimer" for the library,
if necessary. Here is a sample; alter the names:
Yoyodyne, Inc., hereby disclaims all copyright interest in the library `Frob' (a library for tweaking knobs) written by James Random Hacker.
That's all there is to it!
%%The following software may be included in, shipped with, or otherwise used by this product: WSDL4j; Use of any of this software may be governed by the terms of the license below:
Source code for WSDL4j is available at http://sourceforge.net/projects/wsdl4j
%%The following software may be included in, shipped with, or otherwise used by this product: Java Runtime Environment (JRE), Java Mail and Activation, and Java Help; Use of any of this software may be governed by the terms of the license below:
DO NOT TRANSLATE OR LOCALIZE.
%% The following software may be included in this product: CS CodeViewer v1.0;
Use of any of this software is governed by the terms of the license below:
Copyright 1999 by CoolServlets.com.
Any errors or suggested improvements to this class can be reported as instructed on CoolServlets.com. We hope you enjoy this program... your comments will encourage further development!
This software is distributed under the terms of the BSD License.
Neither name of CoolServlets.com nor the names of its contributors may be used to endorse or promote products derived from this software without specific prior written permission.
THIS SOFTWARE IS PROVIDED BY COOLSERVLETS.COM AND CONTRIBUTORS ``AS IS'' AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE AUTHOR OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE."
%% The following software may be included in this product: Crimson v1.1.1 ; Use of any of this software is governed by the terms of the license below:
* The Apache Software License, Version 1.1
* Copyright (c) 1999-2000 The Apache Software Foundation. All rights
* reserved.
* Apache Software Foundation (http://www.apache.org/)."
* 4. The names "Crimson" and "Apache Software Foundation" must
* not be used to endorse or promote products derived from this
* permission, please contact apache@apache.org.
* 5. Products derived from this software may not be called "Apache",
* nor may "Apache" appear in their name, without prior written
* permission of the Apache Software Foundation.
* This software consists of voluntary contributions made by many
* individuals on behalf of the Apache Software Foundation and was
* originally based on software copyright (c) 1999, International
* Business Machines, Inc., http://www.ibm.com. For more
* information on the Apache Software Foundation, please see
* <http://www.apache.org/>.
%% The following software may be included in this product: Xalan J2;
%% The following software may be included in this product: JAX-RPC; Use of any of this software is governed by the terms of the license below:
COMMON DEVELOPMENT AND DISTRIBUTION LICENSE (CDDL) Version 1.0
1.1. "Contributor" means each individual or entity that creates or contributes to the creation of Modifications.
1.2. "Contributor Version" means the combination of the Original Software, prior Modifications used by a Contributor (if any), and the Modifications made by that particular Contributor.
1.3. "Covered Software" means (a) the Original Software, or (b) Modifications, or (c) the combination of files containing Original Software with files containing Modifications, in each case including portions thereof.
1.4. "Executable" means the Covered Software in any form other than Source Code.
1.5. "Initial Developer" means the individual or entity that first makes Original Software available under this License.
1.6. "Larger Work" means a work which combines Covered Software or portions thereof with code not governed by the terms of this License.
1.7. "License" means this document.
1.8. "Licensable" means having the right to grant, to the maximum extent possible, whether at the time of the initial grant or subsequently acquired, any and all of the rights conveyed herein.
1.9. "Modifications" means the Source Code and Executable form of any of the following:
A. Any file that results from an addition to, deletion from or modification of the contents of a file containing Original Software or previous Modifications;
B. Any new file that contains any part of the Original Software or previous Modification; or
C. Any new file that is contributed or otherwise made available under the terms of this License.
1.10. "Original Software" means the Source Code and Executable form of computer software code that is originally released under this License.
1.11. "Patent Claims" means any patent claim(s), now owned or hereafter acquired, including without limitation, method, process, and apparatus claims, in any patent Licensable by grantor.
1.12. "Source Code" means (a) the common form of computer software code in which modifications are made and (b) associated documentation included in or with such code.
1.13. "You" (or "Your") means an individual or a legal entity exercising rights under, and complying with all of the terms of, this License. For legal entities, "You" includes any entity which controls, is controlled by, or is under common control with You. For purposes of this definition, "control" means (a) the power, direct or indirect, to cause the direction or management of such entity, whether by contract or otherwise, or (b) ownership of more than fifty percent (50%) of the outstanding shares or beneficial ownership of such entity.
2. License Grants.
2.1. The Initial Developer Grant.
Conditioned upon Your compliance with Section 3.1 below and subject to third party intellectual property claims, the Initial Developer hereby grants You a world-wide, royalty-free, non-exclusive license:
(a) under intellectual property rights (other than patent or trademark) Licensable by Initial Developer, to use, reproduce, modify, display, perform, sublicense and distribute the Original Software (or portions thereof), with or without Modifications, and/or as part of a Larger Work; and
(b) under Patent Claims infringed by the making, using or selling of Original Software, to make, have made, use, practice, sell, and offer for sale, and/or otherwise dispose of the Original Software (or portions thereof).
(c) The licenses granted in Sections 2.1(a) and (b) are effective on the date Initial Developer first distributes or otherwise makes the Original Software available to a third party under the terms of this License.
(d) Notwithstanding Section 2.1(b) above, no patent license is granted: (1) for code that You delete from the Original Software, or (2) for infringements caused by: (i) the modification of the Original Software, or (ii) the combination of the Original Software with other software or devices.
2.2. Contributor Grant.
Conditioned upon Your compliance with Section 3.1 below and subject to third party intellectual property claims, each Contributor hereby grants You a world-wide, royalty-free, non-exclusive license:
(a) under intellectual property rights (other than patent or trademark) Licensable by Contributor to use, reproduce, modify, display, perform, sublicense and distribute the Modifications created by such Contributor (or portions thereof), either on an unmodified basis, with other Modifications, as Covered Software and/or as part of a Larger Work; and
(b) under Patent Claims infringed by the making, using, or selling of Modifications made by that Contributor either alone and/or in combination with its Contributor Version (or portions of such combination), to make, use, sell, offer for sale, have made, and/or otherwise dispose of: (1) Modifications made by that Contributor (or portions thereof); and (2) the combination of Modifications made by that Contributor with its Contributor Version (or portions of such combination).
(c) The licenses granted in Sections 2.2(a) and 2.2(b) are effective on the date Contributor first distributes or otherwise makes the Modifications available to a third party.
(d) Notwithstanding Section 2.2(b) above, no patent license is granted: (1) for any code that Contributor has deleted from the Contributor Version; (2) for infringements caused by: (i) third party modifications of Contributor Version, or (ii) the combination of Modifications made by that Contributor with other software (except as part of the Contributor Version) or other devices; or (3) under Patent Claims infringed by Covered Software in the absence of Modifications made by that Contributor.
3. Distribution Obligations.
3.1. Availability of Source Code.
Any Covered Software that You distribute or otherwise make available in Executable form must also be made available in Source Code form and that Source Code form must be distributed only under the terms of this License. You must include a copy of this License with every copy of the Source Code form of the Covered Software You distribute or otherwise make available. You must inform recipients of any such Covered Software in Executable form as to how they can obtain such Covered Software in Source Code form in a reasonable manner on or through a medium customarily used for software exchange.
3.2. Modifications.
The Modifications that You create or to which You contribute are governed by the terms of this License. You represent that You believe Your Modifications are Your original creation(s) and/or You have sufficient rights to grant the rights conveyed by this License.
3.3. Required Notices.
You must include a notice in each of Your Modifications that identifies You as the Contributor of the Modification. You may not remove or alter any copyright, patent or trademark notices contained within the Covered Software, or any notices of licensing or any descriptive text giving attribution to any Contributor or the Initial Developer.
3.4. Application of Additional Terms.
You may not offer or impose any terms on any Covered Software in Source Code form that alters or restricts the applicable version of this License or the recipients' rights hereunder. You may choose to offer, and to charge a fee for, warranty, support, indemnity or liability obligations to one or more recipients of Covered Software. However, you may do so only on Your own behalf, and not on behalf of the Initial Developer or any Contributor. You must make it absolutely clear that any such warranty, support, indemnity or liability obligation is offered by You alone, and You hereby agree to indemnify the Initial Developer and every Contributor for any liability incurred by the Initial Developer or such Contributor as a result of warranty, support, indemnity or liability terms You offer.
3.5. Distribution of Executable Versions.
You may distribute the Executable form of the Covered Software under the terms of this License or under the terms of a license of Your choice, which may contain terms different from this License, provided that You are in compliance with the terms of this License and that the license for the Executable form does not attempt to limit or alter the recipient's rights in the Source Code form from the rights set forth in this License. If You distribute the Covered Software in Executable form under a different license, You must make it absolutely clear that any terms which differ from this License are offered by You alone, not by the Initial Developer or Contributor. You hereby agree to indemnify the Initial Developer and every Contributor for any liability incurred by the Initial Developer or such Contributor as a result of any such terms You offer.
3.6. Larger Works.
You may create a Larger Work by combining Covered Software with other code not governed by the terms of this License and distribute the Larger Work as a single product. In such a case, You must make sure the requirements of this License are fulfilled for the Covered Software.
4. Versions of the License.
4.1. New Versions.
Sun Microsystems, Inc. is the initial license steward and may publish revised and/or new versions of this License from time to time. Each version will be given a distinguishing version number. Except as provided in Section 4.3, no one other than the license steward has the right to modify this License.
4.2. Effect of New Versions.
You may always continue to use, distribute or otherwise make the Covered Software available under the terms of the version of the License under which You originally received the Covered Software. If the Initial Developer includes a notice in the Original Software prohibiting it from being distributed or otherwise made available under any subsequent version of the License, You must distribute and make the Covered Software available under the terms of the version of the License under which You originally received the Covered Software. Otherwise, You may also choose to use, distribute or otherwise make the Covered Software available under the terms of any subsequent version of the License published by the license steward.
4.3. Modified Versions.
When You are an Initial Developer and You want to create a new license for Your Original Software, You may create and use a modified version of this License if You: (a) rename the license and remove any references to the name of the license steward (except to note that the license differs from this License); and (b) otherwise make it clear that the license contains terms which differ from this License.
COVERED SOFTWARE IS PROVIDED UNDER THIS LICENSE ON AN "AS IS" BASIS, WITHOUT WARRANTY OF ANY KIND, EITHER EXPRESSED OR IMPLIED, INCLUDING, WITHOUT LIMITATION, WARRANTIES THAT THE COVERED SOFTWARE IS FREE OF DEFECTS, MERCHANTABLE, FIT FOR A PARTICULAR PURPOSE OR NON-INFRINGING. THE ENTIRE RISK AS TO THE QUALITY AND PERFORMANCE OF THE COVERED SOFTWARE IS WITH YOU. SHOULD ANY COVERED SOFTWARE PROVE DEFECTIVE IN ANY RESPECT, YOU (NOT THE INITIAL DEVELOPER OR ANY OTHER CONTRIBUTOR) ASSUME THE COST OF ANY NECESSARY SERVICING, REPAIR OR CORRECTION. THIS DISCLAIMER OF WARRANTY CONSTITUTES AN ESSENTIAL PART OF THIS LICENSE. NO USE OF ANY COVERED SOFTWARE IS AUTHORIZED HEREUNDER EXCEPT UNDER THIS DISCLAIMER.
6.1. This License and the rights granted hereunder will terminate automatically if You fail to comply with terms herein and fail to cure such breach within 30 days of becoming aware of the breach. Provisions which, by their nature, must remain in effect beyond the termination of this License shall survive.
6.2. If You assert a patent infringement claim (excluding declaratory judgment actions) against Initial Developer or a Contributor (the Initial Developer or Contributor against whom You assert such claim is referred to as "Participant") alleging that the Participant Software (meaning the Contributor Version where the Participant is a Contributor or the Original Software where the Participant is the Initial Developer) directly or indirectly infringes any patent, then any and all rights granted directly or indirectly to You by such Participant, the Initial Developer (if the Initial Developer is not the Participant) and all Contributors under Sections 2.1 and/or 2.2 of this License shall, upon 60 days notice from Participant terminate prospectively and automatically at the expiration of such 60 day notice period, unless if within such 60 day period You withdraw Your claim with respect to the Participant Software against such Participant either unilaterally or pursuant to a written agreement with Participant.
6.3. In the event of termination under Sections 6.1 or 6.2 above, all end user licenses that have been validly granted by You or any distributor hereunder prior to termination (excluding licenses granted to You by any distributor) shall survive termination.
UNDER NO CIRCUMSTANCES AND UNDER NO LEGAL THEORY, WHETHER TORT (INCLUDING NEGLIGENCE), CONTRACT, OR OTHERWISE, SHALL YOU, THE INITIAL DEVELOPER, ANY OTHER CONTRIBUTOR, OR ANY DISTRIBUTOR OF COVERED SOFTWARE, OR ANY SUPPLIER OF ANY OF SUCH PARTIES, BE LIABLE TO ANY PERSON FOR ANY INDIRECT, SPECIAL, INCIDENTAL, OR CONSEQUENTIAL DAMAGES OF ANY CHARACTER INCLUDING, WITHOUT LIMITATION, DAMAGES FOR LOST PROFITS, LOSS OF GOODWILL, WORK STOPPAGE, COMPUTER FAILURE OR MALFUNCTION, OR ANY AND ALL OTHER COMMERCIAL DAMAGES OR LOSSES, EVEN IF SUCH PARTY SHALL HAVE BEEN INFORMED OF THE POSSIBILITY OF SUCH DAMAGES. THIS LIMITATION OF LIABILITY SHALL NOT APPLY TO LIABILITY FOR DEATH OR PERSONAL INJURY RESULTING FROM SUCH PARTY'S NEGLIGENCE TO THE EXTENT APPLICABLE LAW PROHIBITS SUCH LIMITATION. SOME JURISDICTIONS DO NOT ALLOW THE EXCLUSION OR LIMITATION OF INCIDENTAL OR CONSEQUENTIAL DAMAGES, SO THIS EXCLUSION AND LIMITATION MAY NOT APPLY TO YOU.
8. U.S. GOVERNMENT END USERS.
The Covered Software is a "commercial item," as that term is defined in 48 C.F.R. 2.101 (Oct. 1995), consisting of "commercial computer software" (as that term is defined at 48 C.F.R. § 252.227-7014(a)(1)) and "commercial computer software documentation" as such terms are used in 48 C.F.R. 12.212 (Sept. 1995). Consistent with 48 C.F.R. 12.212 and 48 C.F.R. 227.7202-1 through 227.7202-4 (June 1995), all U.S. Government End Users acquire Covered Software with only those rights set forth herein. This U.S. Government Rights clause is in lieu of, and supersedes, any other FAR, DFAR, or other clause or provision that addresses Government rights in computer software under this License.
This License represents the complete agreement concerning subject matter hereof. If any provision of this License is held to be unenforceable, such provision shall be reformed only to the extent necessary to make it enforceable. This License shall be governed by the law of the jurisdiction specified in a notice contained within the Original Software (except to the extent applicable law, if any, provides otherwise), excluding such jurisdiction's conflict-of-law provisions. Any litigation relating to this License shall be subject to the jurisdiction of the courts located in the jurisdiction and venue specified in a notice contained within the Original Software, with the losing party responsible for costs, including, without limitation, court costs and reasonable attorneys' fees and expenses. The application of the United Nations Convention on Contracts for the International Sale of Goods is expressly excluded. Any law or regulation which provides that the language of a contract shall be construed against the drafter shall not apply to this License. You agree that You alone are responsible for compliance with the United States export administration regulations (and the export control laws and regulation of any other countries) when You use, distribute or otherwise make available any Covered Software.
10. RESPONSIBILITY FOR CLAIMS.
As between Initial Developer and the Contributors, each party is responsible for claims and damages arising, directly or indirectly, out of its utilization of rights under this License and You agree to work with Initial Developer and Contributors to distribute such responsibility on an equitable basis. Nothing herein is intended or shall be deemed to constitute any admission of liability.
NOTICE PURSUANT TO SECTION 9 OF THE COMMON DEVELOPMENT AND DISTRIBUTION LICENSE (CDDL)
The code released under the CDDL shall be governed by the laws of the State of California (excluding conflict-of-law provisions). Any litigation relating to this License shall be subject to the jurisdiction of the Federal Courts of the Northern District of California and the state courts of the State of California, with venue lying in Santa Clara County, California.
The GNU General Public License (GPL) Version 2, June 1991
Copyright (C) 1989, 1991 Free Software Foundation, Inc. 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA
The licenses for most software are designed to take away your freedom to share and change it. By contrast, the GNU General Public License is intended to guarantee your freedom to share and change free software--to make sure the software is free for all its users. This General Public License applies to most of the Free Software Foundation's software and to any other program whose authors commit to using it. (Some other Free Software Foundation software is covered by the GNU Library General Public License instead.) You can apply it to your programs, too.
When we speak of free software, we are referring to freedom, not price. Our General Public Licenses are designed to make sure that you have the freedom to distribute copies of free software (and charge for this service if you wish), that you receive source code or can get it if you want it, that you can change the software or use pieces of it in new free programs; and that you know you can do these things.
To protect your rights, we need to make restrictions that forbid anyone to deny you these rights or to ask you to surrender the rights. These restrictions translate to certain responsibilities for you if you distribute copies of the software, or if you modify it.
For example, if you distribute copies of such a program, whether gratis or for a fee, you must give the recipients all the rights that you have. You must make sure that they, too, receive or can get the source code. And you must show them these terms so they know their rights.
We protect your rights with two steps: (1) copyright the software, and (2) offer you this license which gives you legal permission to copy, distribute and/or modify the software.
Also, for each author's protection and ours, we want to make certain that everyone understands that there is no warranty for this free software. If the software is modified by someone else and passed on, we want its recipients to know that what they have is not the original, so that any problems introduced by others will not reflect on the original authors' reputations.
Finally, any free program is threatened constantly by software patents. We wish to avoid the danger that redistributors of a free program will individually obtain patent licenses, in effect making the program proprietary. To prevent this, we have made it clear that any patent must be licensed for everyone's free use or not licensed at all.
· 0. This License applies to any program or other work which contains a notice placed by the copyright holder saying it may be distributed under the terms of this General Public License. The "Program", below, refers to any such program or work, and a "work based on the Program" means either the Program or any derivative work under copyright law: that is to say, a work containing the Program or a portion of it, either verbatim or with modifications and/or translated into another language. (Hereinafter, translation is included without limitation in the term "modification".) Each licensee is addressed as "you".
· 1. You may copy and distribute verbatim copies of the Program's source code as you receive it, in any medium, provided that you conspicuously and appropriately publish on each copy an appropriate copyright notice and disclaimer of warranty; keep intact all the notices that refer to this License and to the absence of any warranty; and give any other recipients of the Program a copy of this License along with the Program.
· 2. You may modify your copy or copies of the Program or any portion of it, thus forming a work based on the Program, and copy and distribute such modifications or work under the terms of Section 1 above, provided that you also meet all of these conditions:
o a) You must cause the modified files to carry prominent notices stating that you changed the files and the date of any change.
o b) You must cause any work that you distribute or publish, that in whole or in part contains or is derived from the Program or any part thereof, to be licensed as a whole at no charge to all third parties under the terms of this License.
o c) If the modified program normally reads commands interactively when run, you must cause it, when started running for such interactive use in the most ordinary way, to print or display an announcement including an appropriate copyright notice and a notice that there is no warranty (or else, saying that you provide a warranty) and that users may redistribute the program under these conditions, and telling the user how to view a copy of this License. (Exception: if the Program itself is interactive but does not normally print such an announcement, your work based on the Program is not required to print an announcement.)
· 3. You may copy and distribute the Program (or a work based on it, under Section 2) in object code or executable form under the terms of Sections 1 and 2 above provided that you also do one of the following:
o a) Accompany it with the complete corresponding machine-readable source code, which must be distributed under the terms of Sections 1 and 2 above on a medium customarily used for software interchange; or,
o b) Accompany it with a written offer, valid for at least three years, to give any third party, for a charge no more than your cost of physically performing source distribution, a complete machine-readable copy of the corresponding source code, to be distributed under the terms of Sections 1 and 2 above on a medium customarily used for software interchange; or,
o c) Accompany it with the information you received as to the offer to distribute corresponding source code. (This alternative is allowed only for noncommercial distribution and only if you received the program in object code or executable form with such an offer, in accord with Subsection b above.)
· 4. You may not copy, modify, sublicense, or distribute the Program except as expressly provided under this License. Any attempt otherwise to copy, modify, sublicense or distribute the Program is void, and will automatically terminate your rights under this License. However, parties who have received copies, or rights, from you under this License will not have their licenses terminated so long as such parties remain in full compliance.
· 5. You are not required to accept this License, since you have not signed it. However, nothing else grants you permission to modify or distribute the Program or its derivative works. These actions are prohibited by law if you do not accept this License. Therefore, by modifying or distributing the Program (or any work based on the Program), you indicate your acceptance of this License to do so, and all its terms and conditions for copying, distributing or modifying the Program or works based on it.
· 6. Each time you redistribute the Program (or any work based on the Program), the recipient automatically receives a license from the original licensor to copy, distribute or modify the Program subject to these terms and conditions. You may not impose any further restrictions on the recipients' exercise of the rights granted herein. You are not responsible for enforcing compliance by third parties to this License.
· 7. If, as a consequence of a court judgment or allegation of patent infringement or for any other reason (not limited to patent issues), conditions are imposed on you (whether by court order, agreement or otherwise) that contradict the conditions of this License, they do not excuse you from the conditions of this License. If you cannot distribute so as to satisfy simultaneously your obligations under this License and any other pertinent obligations, then as a consequence you may not distribute the Program at all. For example, if a patent license would not permit royalty-free redistribution of the Program by all those who receive copies directly or indirectly through you, then the only way you could satisfy both it and this License would be to refrain entirely from distribution of the Program.
· 8. If the distribution and/or use of the Program is restricted in certain countries either by patents or by copyrighted interfaces, the original copyright holder who places the Program under this License may add an explicit geographical distribution limitation excluding those countries, so that distribution is permitted only in or among countries not thus excluded. In such case, this License incorporates the limitation as if written in the body of this License.
· 9. The Free Software Foundation may publish revised and/or new versions of the General Public License from time to time. Such new versions will be similar in spirit to the present version, but may differ in detail to address new problems or concerns.
· 10. If you wish to incorporate parts of the Program into other free programs whose distribution conditions are different, write to the author to ask for permission. For software which is copyrighted by the Free Software Foundation, write to the Free Software Foundation; we sometimes make exceptions for this. Our decision will be guided by the two goals of preserving the free status of all derivatives of our free software and of promoting the sharing and reuse of software generally.
· NO WARRANTY
· 11. BECAUSE THE PROGRAM IS LICENSED FREE OF CHARGE, THERE IS NO WARRANTY FOR THE PROGRAM, TO THE EXTENT PERMITTED BY APPLICABLE LAW. EXCEPT WHEN OTHERWISE STATED IN WRITING THE COPYRIGHT HOLDERS AND/OR OTHER PARTIES PROVIDE THE PROGRAM "AS IS" WITHOUT WARRANTY OF ANY KIND, EITHER EXPRESSED OR IMPLIED, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE. THE ENTIRE RISK AS TO THE QUALITY AND PERFORMANCE OF THE PROGRAM IS WITH YOU. SHOULD THE PROGRAM PROVE DEFECTIVE, YOU ASSUME THE COST OF ALL NECESSARY SERVICING, REPAIR OR CORRECTION.
· 12. IN NO EVENT UNLESS REQUIRED BY APPLICABLE LAW OR AGREED TO IN WRITING WILL ANY COPYRIGHT HOLDER, OR ANY OTHER PARTY WHO MAY MODIFY AND/OR REDISTRIBUTE THE PROGRAM AS PERMITTED ABOVE, BE LIABLE TO YOU FOR DAMAGES, INCLUDING ANY GENERAL, SPECIAL, INCIDENTAL OR CONSEQUENTIAL DAMAGES ARISING OUT OF THE USE OR INABILITY TO USE THE PROGRAM (INCLUDING BUT NOT LIMITED TO LOSS OF DATA OR DATA BEING RENDERED INACCURATE OR LOSSES SUSTAINED BY YOU OR THIRD PARTIES OR A FAILURE OF THE PROGRAM TO OPERATE WITH ANY OTHER PROGRAMS), EVEN IF SUCH HOLDER OR OTHER PARTY HAS BEEN ADVISED OF THE POSSIBILITY OF SUCH DAMAGES.
· END OF TERMS AND CONDITIONS
· How to Apply These Terms to Your New Programs
o One line to give the program's name and a brief idea of what it does.
o Copyright (C)
o This program is free software; you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation; either version 2 of the License, or (at your option) any later version.
o This program is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details.
o You should have received a copy of the GNU General Public License along with this program; if not, write to the Free Software Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA
o Gnomovision version 69, Copyright (C) year name of author
Gnomovision comes with ABSOLUTELY NO WARRANTY; for details type `show w'. This is free software, and you are welcome to redistribute it under certain conditions; type `show c' for details.
o Yoyodyne, Inc., hereby disclaims all copyright interest in the program `Gnomovision' (which makes passes at compilers) written by James Hacker.
signature of Ty Coon, 1 April 1989
· "CLASSPATH" EXCEPTION TO THE GPL VERSION 2
· "CLASSPATH" EXCEPTION TO THE GPL VERSION 2
Certain source files distributed by Sun Microsystems, Inc. are subject to the following clarification and special exception to the GPL Version 2, but only where Sun has expressly included in the particular source file's header the words
"Sun designates this particular file as subject to the "Classpath" exception as provided by Sun in the License file that accompanied this code."
Linking this library statically or dynamically with other modules is making a combined work based on this library. Thus, the terms and conditions of the GNU General Public License Version 2 cover the whole combination.
As a special exception, the copyright holders of this library give you permission to link this library with independent modules to produce an executable, regardless of the license terms of these independent modules, and to copy and distribute the resulting executable under terms of your choice, provided that you also meet, for each linked independent module, the terms and conditions of the license of that module.? An independent module is a module which is not derived from or based on this library.? If you modify this library, you may extend this exception to your version of the library, but you are not obligated to do so.? If you do not wish to do so, delete this exception statement from your version.
%% The following software may be included in this product: NSIS 1.0j; Use of any of this software is governed by the terms of the license below:
Copyright (C) 1999-2000 Nullsoft, Inc.
This software is provided 'as-is', without any express or implied warranty.
In no event will the authors be held liable for any damages arising from the use of this software. Permission is granted to anyone to use this software for any purpose, including commercial applications, and to alter it and redistribute it freely, subject to the following restrictions:
Justin Frankel justin@nullsoft.com"
%% Some Portions licensed from IBM are available at:
http://oss.software.ibm.com/icu4j/
%% Portions Copyright Eastman Kodak Company 1992
%% Lucida is a registered trademark or trademark of Bigelow & Holmes in the U.S. and other countries.
%% Portions licensed from Taligent, Inc.
%% The following software may be included in this product:IAIK PKCS Wrapper; Use of any of this software is governed by the terms of the license below:
Copyright (c) 2002 Graz University of Technology. All rights reserved.
3. The end-user documentation included with the redistribution, if any, must include the following acknowledgment:
"This product includes software developed by IAIK of Graz University of Technology."
Alternately, this acknowledgment may appear in the software itself, if and wherever such third-party acknowledgments normally appear.
4. The names "Graz University of Technology" and "IAIK of Graz University of Technology" must not be used to endorse or promote products derived from this software without prior written permission.
5. Products derived from this software may not be called "IAIK PKCS Wrapper", nor may "IAIK" appear in their name, without prior written permission of Graz University of Technology.
THIS SOFTWARE IS PROVIDED "AS IS" AND ANY EXPRESSED OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE LICENSOR BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
%% The following software may be included in this product: Document Object Model (DOM) v. Level 3; Use of any of this software is governed by the terms of the license below:
W3Cýý SOFTWARE NOTICE AND LICENSE
http://www.w3.org/Consortium/Legal/2002/copyright-software-20021231
This work (and included software, documentation such as READMEs, or other related items) is being provided by the copyright holders under the following license. By obtaining, using and/or copying this work, you (the licensee) agree that you have read, understood, and will comply with the following terms and conditions.
Permission to copy, modify, and distribute this software and its documentation, with or without modification, for any purpose and without fee or royalty is hereby granted, provided that you include the following on ALL copies of the software and documentation or portions thereof, including modifications:
1.The full text of this NOTICE in a location viewable to users of the redistributed or derivative work.
2.Any pre-existing intellectual property disclaimers, notices, or terms and conditions. If none exist, the W3C Software Short Notice should be included (hypertext is preferred, text is permitted) within the body of any redistributed or derivative code.
3.Notice of any changes or modifications to the files, including the date changes were made. (We recommend you provide URIs to the location from which the code is derived.)
THIS SOFTWARE AND DOCUMENTATION IS PROVIDED "AS IS," AND COPYRIGHT HOLDERS MAKE NO REPRESENTATIONS OR WARRANTIES, EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO, WARRANTIES OF MERCHANTABILITY OR FITNESS FOR ANY PARTICULAR PURPOSE OR THAT THE USE OF THE SOFTWARE OR DOCUMENTATION WILL NOT INFRINGE ANY THIRD PARTY PATENTS, COPYRIGHTS, TRADEMARKS OR OTHER RIGHTS.
COPYRIGHT HOLDERS WILL NOT BE LIABLE FOR ANY DIRECT, INDIRECT, SPECIAL OR CONSEQUENTIAL DAMAGES ARISING OUT OF ANY USE OF THE SOFTWARE OR DOCUMENTATION.
The name and trademarks of copyright holders may NOT be used in advertising or publicity pertaining to the software without specific, written prior permission. Title to copyright in this software and any associated documentation will at all times remain with copyright holders.
This formulation of W3C's notice and license became active on December 31 2002.
This version removes the copyright ownership notice such that this license can be used with materials other than those owned by the W3C, reflects that ERCIM is now a host of the W3C, includes references to this specific dated version of the license, and removes the ambiguous grant of "use". Otherwise, this version is the same as the previous version and is written so as to preserve the Free Software Foundation's assessment of GPL compatibility and OSI's certification under the Open Source Definition. Please see our Copyright FAQ for common questions about using materials from our site, including specific terms and conditions for packages like libwww, Amaya, and Jigsaw. Other questions about this notice can be directed to site-policy@w3.org.
%% The following software may be included in this product: Xalan, Xerces; Use of any of this software is governed by the terms of the license below:
* 4. The names "Xerces" and "Apache Software Foundation" must
%% The following software may be included in this product: W3C XML Conformance Test Suites v. 20020606; Use of any of this software is governed by the terms of the license below:
Copyright ýý 1994-2002 World Wide Web Consortium, (Massachusetts Institute of Technology, Institut National de Recherche en Informatique et en Automatique, Keio University). All Rights Reserved. http://www.w3.org/Consortium/Legal/
This W3C work (including software, documents, or other related items) is being provided by the copyright holders under the following license. By obtaining, using and/or copying this work, you (the licensee) agree that you have read, understood, and will comply with the following terms and conditions:
Permission to use, copy, modify, and distribute this software and its documentation, with or without modification, for any purpose and without fee or royalty is hereby granted, provided that you include the following on ALL copies of the software and documentation or portions thereof, including modifications,
that you make:
1. The full text of this NOTICE in a location viewable to users of the redistributed or derivative work.
2. Any pre-existing intellectual property disclaimers, notices, or terms and conditions. If none exist, a short notice of the following form (hypertext is preferred, text is permitted) should be used within the body of any redistributed or derivative code: "Copyright ýý [$date-of-software] World Wide Web Consortium, (Massachusetts Institute of Technology, Institut National de Recherche en Informatique et en Automatique, Keio University). All Rights Reserved. http://www.w3.org/Consortium/Legal/"
3. Notice of any changes or modifications to the W3C files, including the date changes were made. (We recommend you provide URIs to the location from which the code is derived.)
The name and trademarks of copyright holders may NOT be used in advertising or publicity pertaining to the software without specific, written prior permission.
Title to copyright in this software and any associated documentation will at all times remain with copyright holders.
This formulation of W3C's notice and license became active on August 14 1998 so as to improve compatibility with GPL. This version ensures that W3C software licensing terms are no more restrictive than GPL and consequently W3C software may be distributed in GPL packages. See the older formulation for the policy prior to this date. Please see our Copyright FAQ for common questions about using materials from our site, including specific terms and conditions for packages like libwww, Amaya, and Jigsaw. Other questions about this notice can be directed to site-policy@w3.org.
%% The following software may be included in this product: W3C XML Schema Test Collection v. 1.16.2; Use of any of this software is governed by the terms of the license below:
W3Cýýýý DOCUMENT NOTICE AND LICENSE
Copyright ýýýý 1994-2002 World Wide Web Consortium, (Massachusetts Institute of Technology, Institut National de Recherche en Informatique et en Automatique, Keio University). All Rights Reserved.
http://www.w3.org/Consortium/Legal/
Public documents on the W3C site are provided by the copyright holders under the following license. The software or Document Type Definitions (DTDs) associated with W3C specifications are governed by the Software Notice. By using and/or copying this document, or the W3C document from which this statement is linked, you (the licensee) agree that you have read, understood, and will comply with the following terms and conditions:
Permission to use, copy, and distribute the contents of this document, or the W3C document from which this statement is linked, in any medium for any purpose and without fee or royalty is hereby granted, provided that you include the following on ALL copies of the document, or portions thereof, that you use:
1. A link or URL to the original W3C document.
2. The pre-existing copyright notice of the original author, or if it doesn't exist, a notice of the form: "Copyright ýýýý [$date-of-document] World Wide Web Consortium, (Massachusetts Institute of Technology, Institut National de Recherche en Informatique et en Automatique, Keio University). All Rights Reserved. http://www.w3.org/Consortium/Legal/" (Hypertext is preferred, but a textual representation is permitted.)
3. If it exists, the STATUS of the W3C document.
When space permits, inclusion of the full text of this NOTICE should be provided. We request that authorship attribution be provided in any software, documents, or other items or products that you create pursuant to the implementation of the contents of this document, or any portion thereof.
No right to create modifications or derivatives of W3C documents is granted pursuant to this license. However, if additional requirements (documented in the Copyright FAQ) are satisfied, the right to create modifications or derivatives is sometimes granted by the W3C to individuals complying with those requirements.
THIS DOCUMENT IS PROVIDED "AS IS," AND COPYRIGHT HOLDERS MAKE NO REPRESENTATIONS OR WARRANTIES, EXPRESS OR IMPLIED, INCLUDING, BUT NOT LIMITED TO, WARRANTIES OF MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE, NON-INFRINGEMENT, OR TITLE; THAT THE CONTENTS OF THE DOCUMENT ARE SUITABLE FOR ANY PURPOSE; NOR THAT THE IMPLEMENTATION OF SUCH CONTENTS WILL NOT INFRINGE ANY THIRD PARTY PATENTS, COPYRIGHTS, TRADEMARKS OR OTHER RIGHTS.
COPYRIGHT HOLDERS WILL NOT BE LIABLE FOR ANY DIRECT, INDIRECT, SPECIAL OR CONSEQUENTIAL DAMAGES ARISING OUT OF ANY USE OF THE DOCUMENT OR THE PERFORMANCE OR IMPLEMENTATION OF THE CONTENTS THEREOF.
The name and trademarks of copyright holders may NOT be used in advertising or publicity pertaining to this document or its contents without specific, written prior permission. Title to copyright in this document will at all times remain with copyright holders.
This formulation of W3C's notice and license became active on April 05 1999 so as to account for the treatment of DTDs, schema's and bindings. See the older formulation for the policy prior to this date. Please see our Copyright FAQ for common questions about using materials from our site, including specific terms and conditions for packages like libwww, Amaya, and Jigsaw. Other questions about this notice can be directed to site-policy@w3.org.webmaster (last updated by reagle on 1999/04/99.)
%% The following software may be included in this product: Common Unix Printing System API Libraries (CUPS API library); Use of any of this software is governed by the terms of the license below:
GNU LIBRARY GENERAL PUBLIC LICENSE
0. This License Agreement applies to any software library which contains a notice placed by the copyright holder or other authorized party saying it may be distributed under the terms of this Library General Public License (also called "this License"). Each licensee is addressed as "you".
A "library" means a collection of software functions and/or data prepared so as to be conveniently linked with application programs (which use some of those functions and data) to form executables.
The "Library", below, refers to any such software library or work which has been distributed under these terms. A "work based on the Library" means either the Library or any derivative work under copyright law: that is to say, a work containing the Library or a portion of it, either verbatim or with modifications and/or translated straightforwardly into another language. (Hereinafter, translation is included without limitation in the term "modification".)
"Source code" for a work means the preferred form of the work for making modifications to it. For a library, complete source code means all the source code for all modules it contains, plus any associated interface definition files, plus the scripts used to control compilation and installation of the library.
Activities other than copying, distribution and modification are not covered by this License; they are outside its scope. The act of running a program using the Library is not restricted, and output from such a program is covered only if its contents constitute a work based on the Library (independent of the use of the Library in a tool for writing it). Whether that is true depends on what the Library does and what the program that uses the Library does.
1. You may copy and distribute verbatim copies of the Library's complete source code as you receive it, in any medium, provided that you conspicuously and appropriately publish on each copy an appropriate copyright notice and disclaimer of warranty; keep intact all the notices that refer to this License and to the absence of any warranty; and distribute a copy of this License along with the Library.
2. You may modify your copy or copies of the Library or any portion of it, thus forming a work based on the Library, and copy and distribute such modifications or work under the terms of Section 1 above, provided that you also meet all of these conditions:
b) You must cause the files modified to carry prominent notices stating that you changed the files and the date of any change.
c) You must cause the whole of the work to be licensed at no charge to all third parties under the terms of this License.
d) If a facility in the modified Library refers to a function or a table of data to be supplied by an application program that uses the facility, other than as an argument passed when the facility is invoked, then you must make a good faith effort to ensure that, in the event an application does not supply such function or table, the facility still operates, and performs whatever part of its purpose remains meaningful.
(For example, a function in a library to compute square roots has a purpose that is entirely well-defined independent of the application. Therefore, Subsection 2d requires that any application-supplied function or table used by this function must be optional: if the application does not supply it, the square root function must still compute square roots.)
These requirements apply to the modified work as a whole. If identifiable sections of that work are not derived from the Library, and can be reasonably considered independent and separate works in themselves, then this License, and its terms, do not apply to those sections when you distribute them as separate works. But when you distribute the same sections as part of a whole which is a work based on the Library, the distribution of the whole must be on the terms of this License, whose permissions for other licensees extend to the entire whole, and thus to each and every part regardless of who wrote it.
Thus, it is not the intent of this section to claim rights or contest your rights to work written entirely by you; rather, the intent is to exercise the right to control the distribution of derivative or collective works based on the Library.
In addition, mere aggregation of another work not based on the Library with the Library (or with a work based on the Library) on a volume of a storage or distribution medium does not bring the other work under the scope of this License.
3. You may opt to apply the terms of the ordinary GNU General Public License instead of this License to a given copy of the Library. To do this, you must alter all the notices that refer to this License, so that they refer to the ordinary GNU General Public License, version 2, instead of to this License. (If a newer version than version 2 of the ordinary GNU General Public License has appeared, then you can specify that version instead if you wish.) Do not make any other change in these notices.
Once this change is made in a given copy, it is irreversible for that copy, so the ordinary GNU General Public License applies to all subsequent copies and derivative works made from that copy.
4. You may copy and distribute the Library (or a portion or derivative of it, under Section 2) in object code or executable form under the terms of Sections 1 and 2 above provided that you accompany it with the complete corresponding machine-readable source code, which must be distributed under the terms of Sections 1 and 2 above on a medium customarily used for software interchange.
If distribution of object code is made by offering access to copy from a designated place, then offering equivalent access to copy the source code from the same place satisfies the requirement to distribute the source code, even though third parties are not compelled to copy the source along with the object code.
5. A program that contains no derivative of any portion of the Library, but is designed to work with the Library by being compiled or linked with it, is called a "work that uses the Library". Such a work, in isolation, is not a derivative work of the Library, and therefore falls outside the scope of this License.
However, linking a "work that uses the Library" with the Library creates an executable that is a derivative of the Library (because it contains portions of the Library), rather than a "work that uses the library". The executable is therefore covered by this License. Section 6 states terms for distribution of such executables.
When a "work that uses the Library" uses material from a header file that is part of the Library, the object code for the work may be a derivative work of the Library even though the source code is not. Whether this is true is especially significant if the work can be linked without the Library, or if the work is itself a library. The threshold for this to be true is not precisely defined by law.
If such an object file uses only numerical parameters, data structure layouts and accessors, and small macros and small inline functions (ten lines or less in length), then the use of the object file is unrestricted, regardless of whether it is legally a derivative work. (Executables containing this object code plus portions of the Library will still fall under Section 6.)
Otherwise, if the work is a derivative of the Library, you may distribute the object code for the work under the terms of Section 6. Any executables containing that work also fall under Section 6, whether or not they are linked directly with the Library itself.
6. As an exception to the Sections above, you may also compile or link a "work that uses the Library" with the Library to produce a work containing portions of the Library, and distribute that work under terms of your choice, provided that the terms permit modification of the work for the customer's own use and reverse engineering for debugging such modifications.
You must give prominent notice with each copy of the work that the Library is used in it and that the Library and its use are covered by this License. You must supply a copy of this License. If the work during execution displays copyright notices, you must include the copyright notice for the Library among them, as well as a reference directing the user to the copy of this License. Also, you must do one of these things:
a) Accompany the work with the complete corresponding machine-readable source code for the Library including whatever changes were used in the work (which must be distributed under Sections 1 and 2 above); and, if the work is an executable linked with the Library, with the complete machine-readable "work that uses the Library", as object code and/or source code, so that the user can modify the Library and then relink to produce a modified executable containing the modified Library. (It is understood that the user who changes the contents of definitions files in the Library will not necessarily be able to recompile the application to use the modified definitions.)
b) Accompany the work with a written offer, valid for at least three years, to give the same user the materials specified in Subsection 6a, above, for a charge no more than the cost of performing this distribution.
c) If distribution of the work is made by offering access to copy from a designated place, offer equivalent access to copy the above specified materials from the same place.
d) Verify that the user has already received a copy of these materials or that you have already sent this user a copy.
For an executable, the required form of the "work that uses the Library" must include any data and utility programs needed for reproducing the executable from it. However, as a special exception, the source code distributed need not include anything that is normally distributed (in either source or binary form) with the major components (compiler, kernel, and so on) of the operating system on which the executable runs, unless that component itself accompanies the executable.
It may happen that this requirement contradicts the license restrictions of other proprietary libraries that do not normally accompany the operating system. Such a contradiction means you cannot use both them and the Library together in an executable that you distribute.
7. You may place library facilities that are a work based on the Library side-by-side in a single library together with other library facilities not covered by this License, and distribute such a combined library, provided that the separate distribution of the work based on the Library and of the other library facilities is otherwise permitted, and provided that you do these two things:
a) Accompany the combined library with a copy of the same work based on the Library, uncombined with any other library facilities. This must be distributed under the terms of the Sections above.
b) Give prominent notice with the combined library of the fact that part of it is a work based on the Library, and explaining where to find the accompanying uncombined form of the same work.
8. You may not copy, modify, sublicense, link with, or distribute the Library except as expressly provided under this License. Any attempt otherwise to copy, modify, sublicense, link with, or distribute the Library is void, and will automatically terminate your rights under this License. However, parties who have received copies, or rights, from you under this License will not have their licenses terminated so long as such parties remain in full compliance.
9. You are not required to accept this License, since you have not signed it. However, nothing else grants you permission to modify or distribute the Library or its derivative works. These actions are prohibited by law if you do not accept this License. Therefore, by modifying or distributing the Library (or any work based on the Library), you indicate your acceptance of this License to do so, and all its terms and conditions for copying, distributing or modifying the Library or works based on it.
10. Each time you redistribute the Library (or any work based on the Library), the recipient automatically receives a license from the original licensor to copy, distribute, link with or modify the Library subject to these terms and conditions. You may not impose any further restrictions on the recipients' exercise of the rights granted herein. You are not responsible for enforcing compliance by third parties to this License.
11. If, as a consequence of a court judgment or allegation of patent infringement or for any other reason (not limited to patent issues), conditions are imposed on you (whether by court order, agreement or otherwise) that contradict the conditions of this License, they do not excuse you from the conditions of this License. If you cannot distribute so as to satisfy simultaneously your obligations under this License and any other pertinent obligations, then as a consequence you may not distribute the Library at all. For example, if a patent license would not permit royalty-free redistribution of the Library by all those who receive copies directly or indirectly through you, then the only way you could satisfy both it and this License would be to refrain entirely from distribution of the Library.
If any portion of this section is held invalid or unenforceable under any particular circumstance, the balance of the section is intended to apply, and the section as a whole is intended to apply in other circumstances.
It is not the purpose of this section to induce you to infringe any patents or other property right claims or to contest validity of any such claims; this section has the sole purpose of protecting the integrity of the free software distribution system which is implemented by public license practices. Many people have made generous contributions to the wide range of software distributed through that system in reliance on consistent application of that system; it is up to the author/donor to decide if he or she is willing to distribute software through any other system and a licensee cannot impose that choice.
12. If the distribution and/or use of the Library is restricted in certain countries either by patents or by copyrighted interfaces, the original copyright holder who places the Library under this License may add an explicit geographical distribution limitation excluding those countries, so that distribution is permitted only in or among countries not thus excluded. In such case, this License incorporates the limitation as if written in the body of this License.
13. The Free Software Foundation may publish revised and/or new versions of the Library General Public License from time to time.
Each version is given a distinguishing version number. If the Library specifies a version number of this License which applies to it and "any later version", you have the option of following the terms and conditions either of that version or of any later version published by the Free Software Foundation. If the Library does not specify a license version number, you may choose any version ever published by the Free Software Foundation.
14. If you wish to incorporate parts of the Library into other free programs whose distribution conditions are incompatible with these, write to the author to ask for permission. For software which is copyrighted by the Free Software Foundation, write to the Free Software Foundation; we sometimes make exceptions for this. Our decision will be guided by the two goals of preserving the free status of all derivatives of our free software and of promoting the sharing and reuse of software generally.
15. BECAUSE THE LIBRARY IS LICENSED FREE OF CHARGE, THERE IS NO WARRANTY FOR THE LIBRARY, TO THE EXTENT PERMITTED BY APPLICABLE LAW. EXCEPT WHEN OTHERWISE STATED IN WRITING THE COPYRIGHT HOLDERS AND/OR OTHER PARTIES PROVIDE THE LIBRARY "AS IS" WITHOUT WARRANTY OF ANY KIND, EITHER EXPRESSED OR IMPLIED, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE. THE ENTIRE RISK AS TO THE QUALITY AND PERFORMANCE OF THE LIBRARY IS WITH YOU. SHOULD THE LIBRARY PROVE DEFECTIVE, YOU ASSUME THE COST OF ALL NECESSARY SERVICING, REPAIR OR CORRECTION.
16. IN NO EVENT UNLESS REQUIRED BY APPLICABLE LAW OR AGREED TO IN WRITING WILL ANY COPYRIGHT HOLDER, OR ANY OTHER PARTY WHO MAY MODIFY AND/OR REDISTRIBUTE THE LIBRARY AS PERMITTED ABOVE, BE LIABLE TO YOU FOR DAMAGES, INCLUDING ANY GENERAL, SPECIAL, INCIDENTAL OR CONSEQUENTIAL DAMAGES ARISING OUT OF THE USE OR INABILITY TO USE THE LIBRARY (INCLUDING BUT NOT LIMITED TO LOSS OF DATA OR DATA BEING RENDERED INACCURATE OR LOSSES SUSTAINED BY YOU OR THIRD PARTIES OR A FAILURE OF THE LIBRARY TO OPERATE WITH ANY OTHER SOFTWARE), EVEN IF SUCH HOLDER OR OTHER PARTY HAS BEEN ADVISED OF THE POSSIBILITY OF SUCH DAMAGES.
%% The following software may be included in this product: Mesa 3-D graphics library v. 5; Use of any of this software is governed by the terms of the license below:
core Mesa code include/GL/gl.h Brian Paul Mesa
GLX driver include/GL/glx.h Brian Paul Mesa
Ext registry include/GL/glext.h SGI SGI Free B
include/GL/glxext.h
Mesa license:
The Mesa distribution consists of several components. Different copyrights and licenses apply to different components. For example, GLUT is copyrighted by Mark Kilgard, some demo programs are copyrighted by SGI, some of the Mesa device drivers are copyrighted by their authors. See below for a list of Mesa's components and the copyright/license for each.
The core Mesa library is licensed according to the terms of the XFree86 copyright (an MIT-style license). This allows integration with the XFree86/DRI project. Unless otherwise stated, the Mesa source code and documentation is licensed as follows:
Copyright (C) 1999-2003 Brian Paul All Rights Reserved.
THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL BRIAN PAUL BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE SOFTWARE.
SGI Free Software Licence B:
, or is under common control with Recipient. For purposes of this definition, "control" of an entity means (a) the power, direct or indirect, to direct or manage such entity, or (b) ownership of fifty percent (50%) or more of the outstanding shares or beneficial ownership of such entity.
1.12."Recipient Patents" means patent claims Licensable by a Recipient that are infringed by the use or sale of Original Code or any Modifications provided by SGI, or any combination thereof.
1.13."SGI" means Silicon Graphics, Inc.
1.14."SGI Patents" means patent claims Licensable by SGI other than the Licensed Patents.
2.License Grant and Restrictions.
2.1.SGI License Grant. Subject to the terms of this License and any third party intellectual property claims, for the duration of intellectual property protections inherent in the Original Code, SGI hereby grants Recipient a worldwide, royalty-free, non-exclusive license, to do the following: (i) under copyrights Licensable by SGI, to reproduce, distribute, create derivative
%% The following software may be included in this product: Byte Code Engineering Library (BCEL) v. 5; Use of any of this software is governed by the terms of the license below:
Apache Software License
* Copyright (c) 2001 The Apache Software Foundation. All rights
* 4. The names "Apache" and "Apache Software Foundation" and
* "Apache BCEL" must not be used to endorse or promote products
* derived from this software without prior written permission. For
* written permission, please contact apache@apache.org.
* "Apache BCEL", nor may "Apache" appear in their name, without
* prior written permission of the Apache Software Foundation.
* THIS SOFTWARE IS PROVIDED "AS IS" AND ANY EXPRESSED OR IMPLIED
* SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING,BUT NOT
* individuals on behalf of the Apache Software Foundation. For more
* .
%% The following software may be included in this product: Regexp, Regular Expression Package v. 1.2; Use of any of this software is governed by the terms of the license below:
The Apache Software License, Version 1.1
Copyright (c) 2001 The Apache Software Foundation. All rights reserved.
"This product includes software developed by the Apache Software Foundation (http://www.apache.org/)."
4. The names "Apache" and "Apache Software Foundation" and "Apache Turbine" must not be used to endorse or promote products derived from this software without prior written permission. For written permission, please contact apache@apache.org.
5. Products derived from this software may not be called "Apache", "Apache Turbine", nor may "Apache" appear in their name, without prior written permission of the Apache Software Foundation.
THIS SOFTWARE IS PROVIDED "AS IS" AND ANY EXPRESSED OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE APACHE SOFTWARE FOUNDATION OR ITS CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
This software consists of voluntary contributions made by many individuals on behalf of the Apache Software Foundation. For more information on the Apache Software Foundation, please see
http://www.apache.org.
%% The following software may be included in this product: CUP Parser Generator for Java v. 0.10k; Use of any of this software is governed by the terms of the license below:
CUP Parser Generator Copyright Notice, License, and Disclaimer
Copyright 1996-1999 by Scott Hudson, Frank Flannery, C. Scott Ananian
Permission to use, copy, modify, and distribute this software and its documentation for any purpose and without fee is hereby granted, provided that the above copyright notice appear in all copies and that both the copyright notice and this permission notice and warranty disclaimer appear in supporting documentation, and that the names of the authors or their employers not be used in advertising or publicity pertaining to distribution of the software without specific, written prior permission.
The authors and their employers disclaim all warranties with regard to this software, including all implied warranties of merchantability and
fitness. In no event shall the authors or their employers be liable for any special, indirect or consequential damages or any damages whatsoever
resulting from loss of use, data or profits, whether in an action of contract, negligence or other tortious action, arising out of or in connection with the use or performance of this software.
%% The following software may be included in this product: JLex: A Lexical Analyzer Generator for Java v. 1.2.5; Use of any of this software is governed by the terms of the license below:
JLEX COPYRIGHT NOTICE, LICENSE AND DISCLAIMER.
Copyright 1996-2003 by Elliot Joel Berk and C. Scott Ananian
Permission to use, copy, modify, and distribute this software and its documentation for any purpose and without fee is hereby granted, provided that the above copyright notice appear in all copies and that both the copyright notice and this permission notice and warranty disclaimer appear in supporting documentation, and that the name of the authors or their employers not be used in advertising or publicity pertaining to distribution of the software without specific, written prior permission.
The authors and their employers disclaim all warranties with regard to this software, including all implied warranties of merchantability and fitness. In no event shall the authors or their employers be liable for any special, indirect or consequential damages or any damages whatsoever resulting from loss of use, data or profits, whether in an action of contract, negligence or other tortious action, arising out of or in connection with the use or performance of this software.
Java is a trademark of Sun Microsystems, Inc. References to the Java programming language in relation to JLex are not meant to imply that Sun endorses this product.
%% The following software may be included in this product: SAX v. 2.0.1; Use of any of this software is governed by the terms of the license below:
SAX is free!
In fact, it's not possible to own a license to SAX, since it's been placed in the public domain.
Because SAX is released to the public domain, there is no warranty for the design or for the software implementation, to the extent permitted by applicable law. Except when otherwise stated in writing the copyright holders and/or other parties provide SAX "as is" without warranty of any kind, either expressed or implied, including, but not limited to, the implied warranties of merchantability and fitness for a particular purpose. The entire risk as to the quality and performance of SAX is with you. Should SAX prove defective, you assume the cost of all necessary servicing, repair or correction.
In no event unless required by applicable law or agreed to in writing will any copyright holder, or any other party who may modify and/or redistribute SAX, be liable to you for damages, including any general, special, incidental or consequential damages arising out of the use or inability to use SAX (including but not limited to loss of data or data being rendered inaccurate or losses sustained by you or third parties or a failure of the SAX to operate with any other programs), even if such holder or other party has been advised of the possibility of such damages.
Copyright Disclaimers
This page includes statements to that effect by David Megginson, who would have been able to claim copyright for the original work.
SAX 1.0
Version 1.0 of the Simple API for XML (SAX), created collectively by the membership of the XML-DEV mailing list, is hereby released into the public domain.
No one owns SAX: you may use it freely in both commercial and non-commercial applications, bundle it with your software distribution, include it on a CD-ROM, list the source code in a book, mirror the documentation at your own web site, or use it in any other way you see fit.
David Megginson, sax@megginson.com
I hereby abandon any property rights to SAX 2.0 (the Simple API for XML), and release all of the SAX 2.0 source code, compiled code, and documentation contained in this distribution into the Public Domain. SAX comes with NO WARRANTY or guarantee of fitness for any purpose.
David Megginson, david@megginson.com
%% The following software may be included in this product: Cryptix;
Cryptix General License
Copyright © 1995-2003 The Cryptix Foundation Limited. All rights reserved.
Redistribution and use in source and binary forms, with or without modification, are permitted provided that the following conditions are
1.Redistributions of source code must retain the copyright notice, this list of conditions and the following disclaimer.
2.Redistributions in binary form must reproduce the above copyright notice, this list of conditions and the following disclaimer in the documentation and/or other materials provided with the distribution.
THIS SOFTWARE IS PROVIDED BY THE CRYPTIX FOUNDATION LIMITED AND CONTRIBUTORS "AS IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE CRYPTIX FOUNDATION LIMITED OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
%% The following software may be included in this product: pytz 2007g;
Copyright (c) 2003-2007 Stuart Bishop
%% The following software may be included in this product: Henry Spencer's regular expression library;
Copyright 1992, 1993, 1994, 1997 Henry Spencer. All rights reserved.
Permission is granted to anyone to use this software for any purpose on any computer system, and to alter it and redistribute it, subject to the following restrictions:
1. The author is not responsible for the consequences of use of this software, no matter how awful, even if they arise from flaws in it.
2. The origin of this software must not be misrepresented, either by explicit claim or by omission. Since few users ever read sources, credits must appear in the documentation.
3. Altered versions must be plainly marked as such, and must not be misrepresented as being the original software. Since few users ever read sources, credits must appear in the documentation.
4. This notice may not be removed or altered.
%% The following software may be included in this product: PuTTY;
PuTTY is copyright 1997-2007 Simon Tatham.
Portions copyright Robert de Bath, Joris van Rantwijk, Delian Delchev, Andreas Schultz, Jeroen Massar, Wez Furlong, Nicolas Barry, Justin Bradford, Ben Harris, Malcolm Smith, Ahmad Khalifa, Markus Kuhn, and CORE SDI S.A.
THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL SIMON TATHAM BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE SOFTWARE.
%% The following software may be included in this product: JIDE Ultimate Suite Product Bundle;
© 2002-2012 – JIDE Software, Inc, All Rights Reserved
%% The following software may be included in this product: PCRECPP;
// Copyright (c) 2005, Google Inc.
// Redistribution and use in source and binary forms, with or without modification, are permitted provided that the following conditions are met:
// * Redistributions of source code must retain the above copyright notice, this list of conditions and the following disclaimer.
// * Redistributions in binary form must reproduce the above
// copyright notice, this list of conditions and the following disclaimer in the documentation and/or other materials provided with the distribution.
// * Neither the name of Google Inc. nor the names of its
// contributors may be used to endorse or promote products derived from this software without specific prior written permission.
// THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT OWNER OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
// Author: Sanjay Ghemawat
// Support for PCRE_XXX modifiers added by Giuseppe Maxia, July 2005
/* This is the public header file for the PCRE library, to be #included by applications that call the PCRE functions.
Copyright (c) 1997-2005 University of Cambridge
* Neither the name of the University of Cambridge nor the names of its contributors may be used to endorse or promote products derived from this software without specific prior written permission.
THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT OWNER OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE)ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
%% The following software may be included in this product: AES;
The free distribution and use of this software in both source and binary form is allowed (with or without changes) provided that:
1. distributions of this source code include the above copyright notice, this list of conditions and the following disclaimer;
2. distributions in binary form include the above copyright
notice, this list of conditions and the following disclaimer in the documentation and/or other associated materials;
3. the copyright holder's name is not used to endorse products built using this software without specific written permission.
This software is provided 'as is' with no explicit or implied warranties in respect of its properties, including, but not limited to, correctness and/or fitness for purpose.
%% The following software may be included in this product: XenServer SDK;
Copyright (c) 2006-2008 Citrix Systems, Inc.
%% The following software may be included in this product: Serpent;
Compilation Copyright (c) 1995-2006 by Wei Dai. All rights reserved. This copyright applies only to this software distribution package as a compilation, and does not imply a copyright on any particular file in the package. The following files are copyrighted by their respective original authors,and their use is subject to additional licenses included in these files.
mars.cpp - Copyright 1998 Brian Gladman.
All other files in this compilation are placed in the public domain by Wei Dai and other contributors. I would like to thank the following authors for placing their works into the public domain:
Joan Daemen - 3way.cpp
Leonard Janke - cast.cpp, seal.cpp
Steve Reid - cast.cpp
Phil Karn - des.cpp
Michael Paul Johnson - diamond.cpp
Andrew M. Kuchling - md2.cpp, md4.cpp
Colin Plumb - md5.cpp, md5mac.cpp
Seal Woods - rc6.cpp
Chris Morgan - rijndael.cpp
Paulo Baretto - rijndael.cpp, skipjack.cpp, square.cpp
Richard De Moliner - safer.cpp
Matthew Skala - twofish.cpp
Kevin Springle - camellia.cpp, shacal2.cpp, ttmac.cpp, whrlpool.cpp, ripemd.cpp
Permission to use, copy, modify, and distribute this compilation for any purpose, including commercial applications, is hereby granted without fee, subject to the following restrictions:
1. Any copy or modification of this compilation in any form, except in object code form as part of an application software, must include the above copyright notice and this license.
2. Users of this software agree that any modification or extension they provide to Wei Dai will be considered public domain and not copyrighted unless it includes an explicit copyright notice.
3. Wei Dai makes no warranty or representation that the operation of the software in this compilation will be error-free, and Wei Dai is under no obligation to provide any services, by way of maintenance, update, or otherwise. THE SOFTWARE AND ANY DOCUMENTATION ARE PROVIDED "AS IS" WITHOUT EXPRESS OR IMPLIED WARRANTY INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE. IN NO EVENT WILL WEI DAI OR ANY OTHER CONTRIBUTOR BE LIABLE FOR DIRECT, INCIDENTAL OR CONSEQUENTIAL DAMAGES, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGES.
4. Users will not use Wei Dai or any other contributor's name in any publicity or advertising, without prior written consent in each case.
5. Export of this software from the United States may require a specific license from the United States Government. It is the responsibility of any person or organization contemplating export to obtain such a license before exporting.
6. Certain parts of this software may be protected by patents. It is the users' responsibility to obtain the appropriate licenses before using those parts.
If this compilation is used in object code form in an application software, acknowledgement of the author is not required but would be appreciated. The contribution of any useful modifications or extensions to Wei Dai is not required but would also be appreciated.
%%The following software may be included in, shipped with, or otherwise used by this product: ResizeableLib;
Use of any of this software may be governed by the terms of the license below:
ResizeableLib is licensed under the Code Project open License (CPOL) v. 1.02 see www.codeproject.com
%%The following software may be included in, shipped with, or otherwise used by this product: THE C5 GENERIC COLLECTION LIBRARY FOR C#/CLI;
Copyright (c) 2003-2007 Niels Kokholm and Peter Sestoft Permission is hereby granted, free of charge, to any person obtaining a copy of this software and associated documentation files (the "Software"), to deal in the Software without restriction, including without limitation the rights to use, copy, modify, merge, publish, distribute, sublicense, and/or sell copies of the Software, and to permit persons to whom the Software is furnished to do so, subject to the following conditions:
%%The following software may be included in, shipped with, or otherwise used by this product: WPF Databinding; Use of any of this software may be governed by the terms of the license below:
WPF Data Binding is licensed under the Code Project open License (CPOL) v1.02 found at http://www.codeproject.com/info/cpol10.aspx
%%The following software may be included in, shipped with, or otherwise used by this product: Zlib; Use of any of this software may be governed by the terms of the license below:
This software is provided 'as-is', without any express or implied
warranty. In no event will the authors be held liable for any damages
arising from the use of this software.
including commercial applications, and to alter it and redistribute it
freely, subject to the following restrictions:
1. The origin of this software must not be misrepresented; you must not
claim that you wrote the original software. If you use this software
in a product, an acknowledgment in the product documentation would be
appreciated but is not required.
2. Altered source versions must be plainly marked as such, and must not be
misrepresented as being the original software.
%%%%The following software may be included in, shipped with, or otherwise used by this product: Zookeeper; Use of any of this software may be governed by the terms of the license below:
10.You must cause any modified files to carry prominent notices stating that You changed the files; and
11.You must retain, in the Source form of any Derivative Works that You distribute, all copyright, patent, trademark, and attribution notices from the Source form of the Work, excluding those notices that do not pertain to any part of the Derivative Works; and
12.If the Work includes a "NOTICE" text file as part of its distribution, then any Derivative Works that You distribute must include a readable copy of the attribution notices contained within such NOTICE file, excluding those notices that do not pertain to any part of the Derivative Works, in at least one of the following places: within a NOTICE text file distributed as part of the Derivative Works; within the Source form or documentation, if provided along with the Derivative Works; or, within a display generated by the Derivative Works, if and wherever such third-party notices normally appear. The contents of the NOTICE file are for informational purposes only and do not modify the License. You may add Your own attribution notices within Derivative Works that You distribute, alongside or as an addendum to the NOTICE text from the Work, provided that such additional attribution notices cannot be construed as modifying the License. You may add Your own copyright statement to Your modifications and may provide additional or different license terms and conditions for use, reproduction, or distribution of Your modifications, or for any such Derivative Works as a whole, provided Your use, reproduction, and distribution of the Work otherwise complies with the conditions stated in this License.
%%END | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 5,377 |
<?php
/**
* LoggerAppenderFile appends log events to a file.
*
* This appender uses a layout.
*
* ## Configurable parameters: ##
*
* - **file** - Path to the target file. Relative paths are resolved based on
* the working directory.
* - **append** - If set to true, the appender will append to the file,
* otherwise the file contents will be overwritten.
*
* @version $Revision$
* @package log4php
* @subpackage appenders
* @license http://www.apache.org/licenses/LICENSE-2.0 Apache License, Version 2.0
* @link http://logging.apache.org/log4php/docs/appenders/file.html Appender documentation
*/
class LoggerAppenderFile extends LoggerAppender {
/**
* If set to true, the file is locked before appending. This allows
* concurrent access. However, appending without locking is faster so
* it should be used where appropriate.
*
* TODO: make this a configurable parameter
*
* @var boolean
*/
protected $locking = true;
/**
* If set to true, appends to file. Otherwise overwrites it.
* @var boolean
*/
protected $append = true;
/**
* Path to the target file.
* @var string
*/
protected $file;
/**
* The file resource.
* @var resource
*/
protected $fp;
/**
* Helper function which can be easily overriden by daily file appender.
*/
protected function getTargetFile() {
return $this->file;
}
/**
* Acquires the target file resource, creates the destination folder if
* necessary. Writes layout header to file.
*
* @return boolean FALSE if opening failed
*/
protected function openFile() {
$file = $this->getTargetFile();
// Create the target folder if needed
if(!is_file($file)) {
$dir = dirname($file);
if(!is_dir($dir)) {
$success = mkdir($dir, 0777, true);
if ($success === false) {
$this->warn("Failed creating target directory [$dir]. Closing appender.");
$this->closed = true;
return false;
}
}
}
$mode = $this->append ? 'a' : 'w';
$this->fp = fopen($file, $mode);
if ($this->fp === false) {
$this->warn("Failed opening target file. Closing appender.");
$this->fp = null;
$this->closed = true;
return false;
}
// Required when appending with concurrent access
if($this->append) {
fseek($this->fp, 0, SEEK_END);
}
// Write the header
$this->write($this->layout->getHeader());
}
/**
* Writes a string to the target file. Opens file if not already open.
* @param string $string Data to write.
*/
protected function write($string) {
// Lazy file open
if(!isset($this->fp)) {
if ($this->openFile() === false) {
return; // Do not write if file open failed.
}
}
if ($this->locking) {
$this->writeWithLocking($string);
} else {
$this->writeWithoutLocking($string);
}
}
protected function writeWithLocking($string) {
if(flock($this->fp, LOCK_EX)) {
if(fwrite($this->fp, $string) === false) {
$this->warn("Failed writing to file. Closing appender.");
$this->closed = true;
}
flock($this->fp, LOCK_UN);
} else {
$this->warn("Failed locking file for writing. Closing appender.");
$this->closed = true;
}
}
protected function writeWithoutLocking($string) {
if(fwrite($this->fp, $string) === false) {
$this->warn("Failed writing to file. Closing appender.");
$this->closed = true;
}
}
public function activateOptions() {
if (empty($this->file)) {
$this->warn("Required parameter 'file' not set. Closing appender.");
$this->closed = true;
return;
}
}
public function close() {
if (is_resource($this->fp)) {
$this->write($this->layout->getFooter());
fclose($this->fp);
}
$this->fp = null;
$this->closed = true;
}
public function append(LoggerLoggingEvent $event) {
$this->write($this->layout->format($event));
}
/**
* Sets the 'file' parameter.
* @param string $file
*/
public function setFile($file) {
$this->setString('file', $file);
}
/**
* Returns the 'file' parameter.
* @return string
*/
public function getFile() {
return $this->file;
}
/**
* Returns the 'append' parameter.
* @return boolean
*/
public function getAppend() {
return $this->append;
}
/**
* Sets the 'append' parameter.
* @param boolean $append
*/
public function setAppend($append) {
$this->setBoolean('append', $append);
}
/**
* Sets the 'file' parmeter. Left for legacy reasons.
* @param string $fileName
* @deprecated Use setFile() instead.
*/
public function setFileName($fileName) {
$this->setFile($fileName);
}
/**
* Returns the 'file' parmeter. Left for legacy reasons.
* @return string
* @deprecated Use getFile() instead.
*/
public function getFileName() {
return $this->getFile();
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 8,891 |
in: Episodes
Air date October 16, 2000
Written by Brenda Hampton
Directed by Harvey Laidman
Help next
When Annie plans an innocent dinner with a former professor, Eric is desperate to keep his mind off the matter but none of his friends seem to be available for a night out. Mary starts a new job at a local pizza parlor and has the great misfortune of befriending co-workers Frankie and Johnny, a young pair of delinquents who have a little too much fun with the Friday night shift. Simon pursues a date with an older girl, while Matt is distracted by a beautiful woman who catches his eye. Lucy and a classmate team up for the classic "take care of an egg" assignment for a humanity class and Ruthie eagerly plans a late-night pizza party for her and Mary.
Annie meets Eric in the kitchen and suggests that he take a night off and it surprises Eric that she knows that he needs a night off to spend time with guys and do guy things. Eric asks what Annie is going to do when he is gone and she tells him that she is going to work on her term paper for college. Annie tells him that she needs to get the idea for the paper approved by her professor before she begins work on it and the only time that she can talk to the professor is tonight, a Friday night. Eric is concerned but Annie tells him that Simon and Lucy can babysit but Simon walks down and says that he has a date with the hottest girl in the sophomore class. Annie asks where they are going and how they are going to get there and Simon says that he will figure it out and double date with another guy. Ruthie arrives home and asks why Mary's car is in the driveway but they run out of the house when they apparently hit a sore subject. Mary comes downstairs and Eric and Annie ask her if she got fired from the pool hall and Mary explains that she doesn't work there anymore, not because she got fired but because she wanted to be able to party. This really concerns Eric and Annie and Ruthie comments on it. Eric and Annie decide to take the night off and regroup and figure out what to do with Mary. Mary is listening to music and dancing around in her room when Ruthie walks in and sees her. Mary turns off the radio and talks to Ruthie. Ruthie explains that there is a group of girls that will be picking her up for school and she wants to hang out with Mary but Mary tells her that she has to work tonight. Ruthie suggests that they do something when Mary gets home at midnight. Ruthie's ride arrives and she rushes off to go to school. Eric calls Lou and asks if he wants to go out and play pool with him but Lou is skeptical and is confused why Eric is asking him to do something. When he gets off of the phone, Lou tells his wife that he is not comfortable with Eric on a social level and Lou's wife tells him that she doesn't approve of them doing things together outside of the church. Eric is going through his Roladex trying to find someone to do something with and can't find anyone. Eric then calls the hospital and asks to speak to Matt. Matt answers the phone and is confused when it turns out to be Eric asking him to go shoot pool. Matt instantly thinks that there is an ulterior motive but Eric explains that he is available to go out. Matt is interrupted by a phone call from Simon who asks Matt to bring him to his date. Matt agrees if Simon pays him $30 but then is interrupted by Lucy calling from school. Matt gets very mad at her and Lucy asks if he can babysit the twins to go out on a date with her lab partner from her family Matt is frustrated by his family calling and gets off of the phone to change a bed pan. Lucy meets up with her lab partner and they decide to babysit the egg at her house. Simon's date is not excited about the fact that Simon's brother is bringing them to their date. Mary delivers a pizza to a table and the guys at the table are sexually harassing her when they ask her for her phone number. Mary approaches one of her coworkers, Johnny, who explains that he and his wife, Frankie, both work at the pizza parlor and they have a baby together. He tells her that it is hard but they are making it, then they are interrupted by the rowdy table when one of the guys asks for her to sit with them. Frankie approaches the table and tells them that Mary is a minister's daughter and tells them to shut up. The guys all apologize in unison that they are sorry. Frankie and Johnny invite Mary to stay after work to drink a beer with them. Mary smiles. Annie asks Simon what he is doing during his date and Eric is upset that Matt couldn't pay pool with him but he can drive Simon around. Simon explains that he is paying Matt and leaves Eric and Annie to talk. Eric tells Annie that he hasn't found anyone to do something with and Annie suggests other religious leaders in the community that he could do something with. Eric doesn't like any of her suggestions and he tells her that he wants to go by himself but they decide that after Annie's meeting, she will join him at the pool hall. Lucy walks down the stairs and tells Eric and Annie that her lab partner will be coming over to help her babysit and so that they can babysit their egg from Lucy explains the project and Eric is concerned about her ability to watch Ruthie and the twins. Ruthie explains her plans with Mary at midnight and that will be taking a nap before midnight. Eric offers to set an alarm for her and tells him that he is the best dad ever. At work, Mary is really busy. Mary asks if they need to get home to be with their child and Johnny explains that they get an extra hour to unwind. Mary agrees to get a pizza after work but she is not sure about the beer. At the apartment, Matt runs out of milk and has to leave the apartment to get some more. Annie arrives at the professor's office and a note that is left on the door explains that she needs to meet him at the Gypsy bar. Annie tries to call the house but doesn't get an answer. Simon gets off the phone, trying to get in touch with Matt to see where he is for the date but can't get in touch with him. Eric offers to drive them but Simon repeats that she is a sophomore and hot. Eric tells him that he will be at the pool hall if he needs anything. Simon tries to call Matt again but can't reach him. At work, Mary gets Frankie from her smoke break and Frankie offers her a cigarette. Frankie talks about Mercy, their baby and how she quit smoking when she had her but afterward she started smoking again. Frankie hands the cigarette to Mary and Mary thinks about smoking the rest of it but eventually puts it out. Annie arrives at the bar and her professor immediately asks her what she wants to drink and when she says nothing, he orders a drink for himself. He wants her to order a meal and is surprised when Annie explains that she has 7 children. At the pool hall, a group of girls call out for Eric and it turns out to be a group of women that go to the church. The ladies ask if he can join them for a game of pool. The bartender looks on as Eric joins the women. Eric walks by a bar and sees a pretty girl dancing in the window and she motions for him to join her. Simon continues to call Matt with no luck. The twins got into a bag of flour and her lab partner and her bring the kids upstairs to get cleaned up. Annie is eating with her professor and he invites himself over to her house but he doesn't like children, which confuses Annie since he teaches early childhood education. He explains that he likes adults but not children. Annie suddenly remembers that she was supposed to meet Eric at the pool hall and she calls him at the pool hall. Eric comes to the phone and Annie apologizes for her tardiness. Lucy's lab partner walks into the kitchen to find Simon still there. Simon wonders what Matt could have had to do that was more important than him. Matt is at the club dancing with the girl. Lucy and her lab partner put the twins to bed. Annie calls and Lucy explains that everyone is okay and Simon never got to go on his date. Lucy tells Annie not to worry about how Simon is doing. When Annie asks how the egg is doing, Lucy and her lab partner realize that they have lost the egg and run around trying to find it. They both enter in Ruthie's room and ask her about the baby and when she produce it, Ruthie doesn't want to give the egg back. They beg and plead but Ruthie doesn't want to give up the egg since they couldn't take care of the egg. Ruthie tries to tell them that the reason for the project is to teach the students that teenagers are not responsible enough to have children. Annie joins up with Eric at the pool hall and they each tell each other that they both discuss their nights and how each of them are a little jealous of what the other one did. When Annie tells Eric that she wants a cup of coffee and is ready to go to bed, Eric finally gets what she is talking about and hurries up and orders the coffee. At the club, Matt is still dancing with the girl and suddenly remembers that he was supposed to drive Simon on his date and runs out of the club. The girl runs after him and asks him for his number but he starts kissing her in the street and Eric and Annie catch him and he stops kissing her and writes his number down. Eric and Annie walk up to Matt as the girl leaves and they give Matt instructions to make sure that all of the kids are doing what they are supposed to be doing at the house. When Matt asks where they will be, he realizes and jumps in his car and takes off. Eric asks if they want to get a room at a hotel. Mary is counting her money and orders her pizza and orders up a beer after work. At home, Simon is watching TV when Matt shows up and apologizes but explains that he never fully agreed to take him on the date. Simon starts yelling at him and decides that someone else was more important than he is. Matt picks up the egg and starts throwing it around and when Matt loses his concentration when Lucy walks in, he drops the egg. Mary is talking about her life plans with Johnny and Frankie. Mary offers to babysit for them. Mary realizes the time and when she stands up, she feels lightheaded but tells them that she will be okay. Matt tucks Ruthie in bed and when Ruthie wakes up, she is startled to find Matt and is ready to have her party. In the hotel room, Annie updates Eric on what Matt told them about the kids. Matt is to call them at the hotel when Mary arrives home. Mary gets stopped by a police officer for not stopping fully at a stop sign. The police officer knew Eric and let her off with a warning. When Mary left, he looked concerned. When Mary walks in the kitchen, Eric and Annie are there waiting for her to return and tell her that the penalty for coming in late is two slices of pizza. They realize that she smells like cigarette smoke and the pizza is cold. They realize that they need to keep an eye on her. Mary wakes Ruthie up and Ruthie gripes at her for not being home at the right time.
Chyler Leigh (Frankie) and Nathan West (Johnny) were dating in real life when this episode aired and they married two years later. Like their characters, they battled with drug addictions as teenagers (right around the time this episode aired, actually) but they recovered from it.
In this episode, Annie tells her teacher that she has seven kids from 2 to 21 but the twins were only one when this episode aired so she has kids from 1 to 21.
Jeff Yagher is the brother of Catherine Hicks' husband, Kevin Yagher.
John Hamilton (Chaz Lamar Shepherd) does not appear in this episode.
Music[]
"Dizzy" by The Goo Goo Dolls
"Love Shack" by the B-52's
Ruthie: Why is Mary's car in the driveway? Isn't she supposed to be at work, or did she get fired again?!
Lucy: Time for school!
Mary: What's up?
Eric: You tell us. Is this your day off?
Annie: Did you get fired?
Eric: Building burned down? Business close for an epidemic?
Mary: Uh-uh.
Ruthie: Quit?
Mary: I just got another job that's all. I'm working at Pete's pizza. I just didn't like the pool hall. I kept getting the day shift, and, well I like the night life, I like to boogie.
Ruthie: Is it me or is she starting to spin?
Ruthie: I'm in a carpool. It's like a welcome wagon. A bunch of girls come and pick me up, so they can have time to get to know me. It's supposed to help me feel not so new.
Mary: Well, that's really nice.
Ruthie: Yeah, they're all nice girls...but they're not my big sister. Can we please do something together this weekend?
Mary: They want me to work the double shift today: 12 to 12.
Ruthie: Well, maybe we can do something then.
Mary: Yeah, maybe we could. Hey, I could bring a pizza home with me and I could wake you up, and we could have a midnight snack.
Ruthie: Yeah! Then you could sleep over in my room!
Mary: Okay. It's a date!
Eric [to Ruthie]: Hey. You going to bed already?
Ruthie: Soon. I have to get up at midnight. Mary's bringing home pizza, and we're having a sleepover! It's gonna be great!
Eric: Would you like me to set the clock alarm so you get up on time?
Ruthie: You're the best dad ever!
Eric: Gosh. You know...it seems like just yesterday Mary was your age.
Ruthie: (about her and Mary) We're having a slumber party.
Matt: Well, maybe you can have it another night.
Ruthie: No, Mary said tonight. She said she'd be here. She'll be here.
Guy: (to Mary, who is their waitress at the pizza place) Hey, you want to sit down and have a piece with us?
Mary: Um...no, I don't. Can I get you anything else?
Guy: Yeah. I'll have your phone number written on your underwear!
Guy: (in a taunting voice) Hey, Mary, we're saving a piece for ya!
Frankie: (walks over to the table) Do you know who you're talking to? Minister's daughter. Give her a break. Behave yourself.
Guy: And if we don't?
Frankie: Then you're all going to hell.
Everyone: (humored) Ooooooh!
Frankie: I mean it.
Johnny: (to Mary about the customers harrassing her) Let my wife handle them. She's good with those types.
Mary: Frances is your wife?
Johnny: Not Frances. No one calls her Frances. It's Frankie. We've been married for a year.
Mary: But you look so young!
Johnny: We are young. Frankie's your age, and I'm 21.
Mary: Yeah, I thought about getting married once, too.
Johnny: What changed your mind?
Mary: I don't know...I guess I just wasn't ready.
Johnny: Yeah, well, neither were we, but the baby changed all that.
Lucy's Lab Partner: You know, Ruthie, Lucy and I are supposed to write down everything we do with that egg. I'd hate to have to write that for a while, we were begging you to give us our daughter back.
Ruthie: Ruthie Camden, and "dumb" is not my middle name!
Ruthie: Yeah, yeah, yeah. But like all parents, we're supposed to watch over our kids. You two just got a little over committed and needed help. No problem, no harm done, as long as you realize teenagers are not meant to be parents. That's all you really have to put into the report. That's the purpose of the whole thing, isn't it?
Lucy: She goes to private school.
Lucy's Lab Partner: Oh.
Annie: Poor Simon.
Lucy: Um, yeah. Poor, popular, charismatic Simon, who could have any girl in his freshman class, but insists on going out with sophomores. I don't think I'd worry too much about him.
Frankie [to Mary]: I'm just gonna have a cigarette. You want one?
Mary: Oh, no thanks, I don't smoke.
Frankie: Neither did I. Well, I stopped when I found out I was pregnant, but as soon as I popped Mercy out, I lit up again.
Mary: Mercy. What a great name.
Frankie: She's a great baby. She's real good, she never cries. Probably because my mother watches her most of the time. She's good with babies. It's teenagers she can't handle.
[After Mary has been pulled over for being suspected of drinking and driving]
Officer: Look, don't ever be in such a hurry that you aren't careful. I want you to be able to have pizza with your sister for the rest of your life.
Mary: Yeah. So do I.
Mary: I love you, Ruthie.
Ruthie: If you loved me, you would have been home at midnight. (turns over and closes her eyes.) | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 108 |
Mercedes-Benz Fashion Week KL: Shaleen Cheah Spring/Summer 2022
Fashion, News & Trends
Sarah Hani Jamil
Shaleen Cheah runway. Courtesy of Mercedes-Benz Fashion Week KL
Back for its third instalment, Mercedes-Benz Fashion Week Kuala Lumpur (MBFWKL), held recently at The Kuala Lumpur Performing Arts Centre, gathers local, regional and international fashion designers for a showcase of their recent collections. One of the notable names to show in this three-day event is Malaysia-based designer Shaleen Cheah who's known for her whimsical designs that balance femininity and playfulness, all the while celebrating individuality. As we catch up with her on the launch night of the fashionable affair, Shaleen tells us more about her brand-new designs.
Tell us about the collection you're showing at Mercedes-Benz Fashion Week Kuala Lumpur.
"Sure! So, the collection is called Horizon. I've always been a bubbly and optimistic person and I wanted it to be reflected in this collection. The key message of this line is about celebrating life, cherishing beautiful moments and chasing new horizons. Basically, it celebrates the joy of dressing up. There are lots of colours and my staple jumpsuits. I just wanted to put out a fun and memorable collection with it being my homecoming debut."
A post shared by Shaleen Alena Cheah (@shaleencheah)
Can you talk us through the process of transforming an idea or an inspiration into a full-blown collection?
"The designing process is a long one. I'm an avid traveller and I usually get my inspiration through my travels. However, ideas are everywhere. Sometimes inspiration comes from a fabric I see and then I create a collection based on that particular material. Sometimes it's, you know, just from watching my kids. Even when I started, I created pieces that I myself would want to wear and what I thought would be cute and stylish. That's really how it all started and I think my clients see that. I'm glad that they see my vision and they've been through this journey with me."
What do you think women want from their wardrobes today?
"I think coming out of the whole COVID situation, I feel like, yes, they do want comfort, but also, especially for women, we like being pretty and dressing up for ourselves. We like having pieces that when you put on, you feel joy, and you're confident in. This is what I tried to reflect in dressing the women I do."
Courtesy of Mercedes-Benz Fashion Week KL
What has been the most valuable lesson on your journey as a designer so far?
"To always be prepared. My whole team is actually down with COVID, but having a really good support system behind me is crucial. Although my team is not present for MBFWKL, they're with me in spirit and I think that the prep that led to this day is really, really helpful."
What do you do to ensure you're relaxed and focused at a time like this?
"Deep breaths. Personally, I feel I handle stress really well. I've worn many hats in my life and I've got lots of experience that I can pull from to get me through today. I also kind of like working under pressure. I think stress can turn into something good if you work around it. So what I'm currently doing at this moment is taking lots of deep breaths and telling myself that at the end of the day, it will be okay. It always is."
You are a big proponent of slow fashion. When and why did you decide to pursue sustainability in your designs?
"I know it's the buzzword these days, but when I started five years ago, that was my ethos, really. I like the fabric itself to tell a story. I started with a lot of embroidered fabrics and I knew where they come from, who makes them, even down to the name of the weaver. So I take pride in the fact that I could actually trace where it came from and that's how I started. I was collecting all these amazing handmade fabrics and then turning them into different designs. Usually, when you think about fashion, you don't necessarily think of being sustainable. But I'm glad that the fashion industry is now taking the steps to move forward with that."
If you could pick a highlight of your entire collection from tonight, what would it be?
"I'm really excited about the piece that I'll be wearing. Also, there are a few in my collection that I've taken inspiration from my recent trip to Greece. It's very bohemian and very easy, but it has that sass. For me, you can wear something comfortable and still look amazing in it. If I were to pick, there are these three really lovely, pretty long dresses that I feel would reflect where my inspiration came from."
For more info on Shaleen Cheah, click here.
The Top 11 Trends of Spring 2022
The 15 Handbag Trends from the Runways That Are Going to Be Everywhere This Spring
Fashion, News & Trends, Style Files
As we move away from trends, there's never been a better time to find your personal style | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 9,591 |
\section{Introduction}
In a weighted voting system, voters, or \textit{players}, have different amounts of the total votes, which are called \textit{weights}. A \textit{motion} is an agenda item that needs some amount of votes to be passed. This amount is called the \textit{quota}. A \textit{coalition} is a group of players, and the sum of all votes in a coalition is called the total voting power. A coalition is a \textit{winning coalition} if its total voting power meets or exceeds the quota. A player is called a \textit{critical player} in a winning coalition if the coalition would not be winning without the player.
We examine the \textit{Banzhaf power index} \cite{Banzhaf} and the \textit{Shapley-Shubik power index} \cite{Shapley-Shubik}, which are two different methods of measuring a player's strength in a system. The Banzhaf power index of a player is the number of times that player is a critical player in all winning coalitions divided by the number of total times any player is a critical player.
The Shapley-Shubik index looks at permutations of all players in a system, called \textit{sequential coalitions}. We sum each player's votes starting from the beginning of a sequential coalition, and see if the sum reaches the quota as we progress. The player whose votes first cause this sum to meet or exceed the quota is called a \textit{pivotal player}. The Shapley-Shubik power index of a player is the number of times that player is a pivotal player divided by the total number sequential coalitions.
We present results in two different directions. The first is slightly more number-theoretic, while the second relies heavily on modelling with suitable equivalent algebraic equations.
\medskip
For an integer $n> 1$, the \textit{divisor voting system of $n$} is $[Q: n, d_k, \ldots, d_1, 1]$ where $d_1,\ldots,d_k$ are the divisors of $n$ distinct from $1$ and $n$; also $Q = (\sigma(n)+1)/2$ when $\sigma(n)$ is even and $Q= \sigma(n)/2$ when $\sigma(n)$ is odd -- so the quota is essentially a simple majority rule. Here $\sigma(n)$ denotes the sum of the divisors of $n$.
In Section 2 we will show that for any positive integer $n$ where $\sigma(n)\geq 2n$, the Banzhaf power index and Shapley-Shubik index disagree on at least one
divisor for the divisor voting system of $n$ for all $n$ such that $\sigma(n) = 2n+k$, $0\leq k \leq 5$. It is unlikely that such an approach would work in the general case and so it is more plausible to examine an inductive argument of the following sort -- given $n$, what can we say about $pn$ and $mn$, where $p$, $m$ are primes? We have some progress towards this approach in Section $2$.
\medskip
Now let us introduce the algebraic direction of our research in Section 3.
Consider systems of the form $V = [1/2_s : a_1,\ldots, a_n]$ where $a_1,\ldots,a_n \geq 0$ are real numbers that sum to $1$ and all winning coalitions are those whose sum \textit{strictly exceeds} $1/2$. We can evaluate the Banzhaf and Shapley-Shubik powers of each player, thus obtaining tuples $BB(V)$ and $SS(V)$. We say that $V$ is a \textit{Banzhaf fixed point} if $V = BB(V)$ and a \textit{Shapley-Shubik fixed point} if $V = SS(V)$. A fixed point is \textit{primitive} if there is no player $i$ with $a_i = 0$. A fixed point is \textit{non-trivial} if not all players have the same power.
In Section $3$ we derive algebraic equations whose solutions generate fixed points and in particular in Section $3.2$ we give new classes of Shapley-Shubik fixed points. We then categorize all fixed points under Shapley-Shubik and use this to show that there is only one fixed point for both Banzhaf and Shapley-Shubik. Note that in general the problem is difficult to model with an equation and so we will restrict to the (already far from trivial) tuples of one of the forms $(a,b,\ldots,b)$ and $(a,a,b,\ldots,b)$.
\section{Divisor Voting Systems and Abundant Numbers}
Our first main aim in this section is to prove the following.
\begin{proposition}
The Banzhaf power index and Shapley-Shubik index are different for at least one divisor for the divisor voting system of $n$ when $\sigma(n) = 2n+k$, $0\leq k \leq 5$.
\end{proposition}
An integer $n$ is called \textit{abundant} if $\sigma(n) > 2n$. We shall sometimes refer to the following auxilliary observation.
\begin{proposition}
\label{abundantdivisors}
Any abundant number must have more than 5 divisors, and the only abundant number with 6 divisors is 20.
\end{proposition}
\begin{proof}
Let $g: \mathbb{Z}\rightarrow\mathbb{R}$ be the function defined by $g(n) = \frac{n}{\phi{(n)}}$. By substituting Euler's product formula for $\phi{(n)}$, we have $g(n)={\displaystyle \prod_{p \mid n} \frac{p}{p-1}}$.
Suppose $n$ is an odd abundant number. Then, since $n$ is abundant, $\frac{\sigma{(n)}}{n} > 2$. Since $g(n) > \frac{\sigma{(n)}}{n}$ when $n>1$, $g(n) > 2$. Now we will find the minimum number of divisors $n$ can have. The function $f(x)=\frac{x}{x-1}$ is a decreasing function over the interval $[2, \infty)$. Therefore, the two prime odd numbers for which $\frac{p}{p-1}$ is at its largest values are $3$ and $5$. However, $(\frac{3}{2})(\frac{5}{4}) < 2$. Therefore, $n$ must have at least 3 prime odd divisors to be abundant, or 8 total divisors.
Suppose $n$ is an even abundant number. Since $n$ is abundant, $g(n) >2$. In this case, it is possible for $n$ to have only 2 prime divisors. These prime divisors must be 2 and 3, since $(\frac{2}{1})(\frac{3}{2}) > 2$. 2 and 3 are the only prime divisors for which this is possible, since $f(x) = \frac{x}{x-1}$ is decreasing for $x\geq 2$, and $(\frac{3}{2})(\frac{5}{4}) < 2$. 20 is the only abundant number with only 2 and 3 as prime divisors. Hence, 20 is the only abundant number with 6 divisors, and all other even abundant numbers must have at least 8 divisors.
\end{proof}
\subsection{Casework -- Proof of Proposition $2.1$}
In this part $d$ is the number of divisors of $n$. We will prove Proposition $2.1$ by breaking it into cases based on the value of $\sigma{(n)}$.
\subsubsection{$\sigma(n)=2n$}
\label{subsubsection:sigma(n)=2n}
\begin{proof}
We will first find the Banzhaf power index of each divisor of $n$ based on the number of divisors $d$. We can calculate the number of times $n$ is a critical player in the divisor voting system of $n$. Consider the total number of possible combinations of the divisors that include $n$, $2^{d-1}$. Since the sum of all divisors excluding $n$ is also $n$, and the quota is $n+1$, the only combination that is not a winning coalition is the coalition that contains only $n$. Hence $n$ is a critical player $2^{d-1} -1$ times.
Every divisor that is not $n$ is a critical player only when it is in the winning coalition consisting of $n$ and itself. So, the Banzhaf index for $n$ is $\frac{2^{d-1}-1}{2^{d-1}+(d-2)}$. The Banzhaf index for any other divisor $d_i$ of $n$ is $\frac{1}{2^{d-1}+(d-2)}$.
We will now calculate the Shapley-Shubik power index for each divisor of $n$. A proper divisor $d_i$ of $n$ can be a pivotal player only once. It is a pivotal player in the sequential coalition where $n$ is the first player followed by $d_i$. Hence, the number of times $d_i$ is a pivotal player is $(d-2)!$. We can now find the number of times $n$ is a pivotal player. There are $d-1$ proper divisors of $n$ and $d!$ total pivotal players, so the number of times $n$ is a pivotal player is $d!-(d-2)!(d-1)$. So $n$ has a Shapley-Shubik index of $\frac{d!-(d-1)!}{d!}$ and every other divisor of $n$ has index $\frac{(d-2)!}{d!} = \frac{1}{d(d-1)}$.
We now compare the Shapley-Shubik indices and the Banzhaf indices to show that they differ for at least one divisor of $n$. We can show that each proper divisor of $n$, $d_i$, has a Banzhaf index that differs from its Shapley-Shubik index by setting equal the their formulas we derived. So, we have
\begin{equation*}
\frac{1}{2^{d-1}+(d-2)}=\frac{1}{d(d-1)}.
\end{equation*}
The only integer solution to this equation is $d=2$ -- indeed, note that $2^{d-1} > d^2 - 2d + 2$ for $d\geq 6$ by induction, with the inductive step being $2^d = 2 \cdot 2^{d-1} > 2(d^2 - 2d + 2) > d^2 + 2 = (d+1)^2 - 2(d+1) + 2$ (the second inequality is equivalent to $d(d-4) + 2 > 0$). However, since $n$ is an abundant number, it must have more than six divisors by Proposition \ref{abundantdivisors}.
Thus, for every perfect number, at least one of its divisors in its divisor voting system will have a Banzhaf index that differs from its Shapley-Shubik index.
\end{proof}
We followed a similar procedure to find and compare the Banzhaf and Shapley-Shubik power indices for each divisor in the divisor voting system of $n$ where $\sigma(n)=2n+k$. Note that we will mostly omit the derivations of the calculations, as well as reasons why two concrete expressions involving $d$ are equal apart from a few cases, since the ideas are largely the same.
\subsubsection{$\sigma(n)=2n+1$}
Since both the Banzhaf and Shapley-Shubik power indices of 1 are 0, we must compare the Banzhaf and Shapley-Shubik power index formulas for proper divisors $d_i$ that are not 1. Using the same method that used in 2.1.1, we can see that the formula for the Banzhaf index of each $d_i$ is $\frac{2}{2^{d-1}+2(d-2)}$. The formula for the Shapley-Shubik index of each $d_i$ is $\frac{2(d-2)!}{d!} = \frac{2}{d(d-1)}$. These two formulas are equal only when the number of divisors of $n$ is 2, 3, or 4, which is not possible since $n$ is an abundant number. Note that numbers $n$ where $\sigma(n)=2n+1$, are called \textit{quasiperfect numbers}, and it is unknown if any such numbers really exist \cite{quasiperfect}.
\subsubsection{$\sigma(n)=2n+2$}
Both the Banzhaf and Shapley-Shubik power indices of 1 do not equal 0. Therefore we can compare the Banzhaf and Shapley-Shubik indices for 1 to prove that 2.0.1 is true for this case. The formula for the Banzhaf index of 1 is $\frac{1}{2^{d-1}+3(d-2)-2}$ and the formula for the Shapley-Shubik index of 1 is $\frac{(d-2)!}{d!} = \frac{1}{d(d-1)}$. If these two expressions are equal to each other, then $d$ must be 4, but an abundant number must have at least 6 divisors. Hence when $\sigma(n)=2n+2$, the power indices of 1 in Shapley-Shubik and Banzhaf differ from one another.
\subsubsection{$\sigma(n)=2n+3$}
In this case, since both the Banzhaf and Shapley-Shubik power indices of 1 are 0, we must compare the power indices of divisors $d_i$ that are not 1, nor $n$. Then the Banzhaf index of each $d_i$ is $\frac{4}{2^{d-1}+4(d-2)-4}$ and the Shapley-Shubik index of $d_i$ is $\frac{4(d-3)!+2(d-2)!}{d!} = \frac{2}{(d-1)(d-2)}$. There are no integer solutions for when the two indices of $d_i$ are equal -- indeed, we can prove that $2^{d-1} > 2d^2 - 10d+16$ for $d\geq 6$ with induction. Suppose that the inequality is true for $d$. By manipulating this inequality, we have that $2^d = 2\cdot 2^{d-1} > 2(2d^2 - 10d+16) > 4d^2 - 20d - 32 = 2(d+1)^2 - 10(d+1)+16$ (the second inequality is equivalent to $2(d-4)(d-3) > 0$). Therefore, since the inequality is true for $d+1$, it is true for all $d \geq 6$. Hence, since an abundant number must have at least 6 divisors, there are no solutions for when the two indices are equal.
\subsubsection{$\sigma(n)=2n+4$}
We must consider two cases for when $n$ is even or odd. Suppose $n$ is even. Since the index of 1 is not 0, we can simply compare the Banzhaf and Shapley-Shubik indices for 1. The Banzhaf index of 1 is $\frac{1}{2^{d-1}+5(d-3)-1}$. The Shapley-Shubik index of 1 is $\frac{2(d-3)!}{d!} = \frac{2}{d(d-1)(d-2)}$. When we set these two expressions for the Banzhaf and Shapley-Shubik index equal and simplify, we have $d^3+2d-3d^2 = 2^d + 10d-32$. We will show that these expressions differ since $n$ is an abundant number and must have more than 5 divisors.
Let $h: \mathbb{R}\rightarrow\mathbb{R}$ be the function defined by $h(x) = 2^x + 10x-32-(x^3+2x-3x^2)$. Thus $h(x) = 2^x-x^3 + 3x^2 + 8x-32$. By Proposition \ref{abundantdivisors}, an abundant number must have at least 6 divisors. The integers 6, 7, and 8 generate values of $h(x)$ that are not equal to 0. To show that $h(x) > 0$ for values $x\geq 9$, it is sufficient to show that $h(9) > 0$ and $h'(x) = 0$, which is true if $h^{(3)}(9) > 0$ and $h^{(4)}(x)>0$ for all $x\geq 9$.
$h^{(3)}(x) = 2^xln^3(2)-6$, which is greater than 0 for all numbers greater than or equal to 9, and $h^{(4)}(x) = 2^xln^4(2)$, which is clearly greater than 0 for all values of $x$.
Hence the two expressions for the Banzhaf and Shapley-Shubik index of 1 are never equal for an abundant number $n$.
Now suppose $n$ is odd. Since the index of 1 is 0 for both Banzhaf and Shapley-Shubik indices, we must compare the formulas for the indices of proper divisors $d_i$ that are not 1. The formula for the Banzhaf index of each $d_i$ is $\frac{4}{2^{d-1}+4(d-2)-3}$, and the formula for the Shapley-Shubik index of $d_i$ is $\frac{2[(d-2)\cdot (d-2)!]+4(d-2)!}{d!} = \frac{2}{d-1}$. By simplifying $\frac{4}{2^{d-1}+4(d-2)-3} = \frac{2}{d-1}$ we have $2^{d-1}=-2d+9$. However, $2^{d-1} > -2d+9$ for all $d > 5$, since the graph $y=-2x+9$ intersects the x-axis only when $x<5$, and $y=2^{d-1}$ is an increasing function when $x\geq 0$.
Therefore, since $n$ is an abundant number and must have at least $6$ divisors, there are no abundant numbers for which these expressions are equal.
\subsubsection{$\sigma(n)=2n+5$}
We again have two cases to consider for when $n$ is even or odd. Suppose $n$ is even. The Banzhaf and Shapley-Shubik indices of 1 are not 0, so we can simply compare these two indices for 1. The formula for the Banzhaf index of 1 is $\frac{1}{2^{d-1}+5(d-3)-2}$ and the formula for the Shapley-Shubik index of 1 is $\frac{2(d-3)!}{d!} = \frac{2}{d(d-1)(d-2)}$. Equating the expressions for the Banzhaf and Shapley-Shubik indices gives $2^d+10d-34 = d^3+2d-3d^2$. $n$ is an abundant number and thus must have at least 6 divisors by Proposition 2.2. Hence we will show that this equation has no integers solutions greater than 5 using the same method as in 2.1.5. Let $h: \mathbb{R}\rightarrow\mathbb{R}$ be the function defined by $h(x) = 2^x+10x-34 - (x^3+2x-3x^2)$. Hence $h(x) = 2^x-x^3+3x^2+8x-34$. We can manually verify that $h(6)$, $h(7)$, and $h(8)$ are not equal to 0. We will now show that if $x$ is greater than or equal to 9, $h(x)$ is greater than 0. We can demonstrate this by showing that $h^{(3)}(9)>0$ and $h^{(4)}(x) > 0$ for all $x\geq9$. $h^{(3)}(9) = 2^9\ln{(2)}^{3} - 6$, which is greater than 0 and $h^{(4)}(x) = 2^x\ln{(2)}^4$, which is also clearly greater than 0.
Hence there are no integer solutions for when the Banzhaf index of 1 and the Shapley-Shubik index of 1 are equal.
In the second case, $n$ is odd. Since the Banzhaf and Shapley-Shubik indices of 1 are both 0, we must compare the formulas for the indices of proper divisors $d_i$ where $d_i$ is not 1. The Banzhaf index of each $d_i$ is $\frac{4}{2^{d-1}+4(d-2)-6}$ and the formula for the Shapley-Shubik index of $d_i$ is $\frac{2[(d-2)\cdot(d-2)!]+4(d-2)!}{d!} = \frac{2}{d-1}$. By simplifying $\frac{4}{2^{d-1}+4(d-2)-6} = \frac{2}{d-1}$, we have $2^{d-1}=-2d+12$. Since $f(x)=2^{x-1}$ is an increasing function for $x\geq 0$ and $g(x)=-2x+12$ is a decreasing function that intersects the x-axis when $x=6$, $2^{d-1}>-2d+12$ for all $x\geq 6$.
Hence there are no possible abundant numbers $n$ for when these two expressions are equal.
\subsection{Forms $pn$ and $mn$}
In order to extend the class of abundant numbers for which Proposition 2.1 is true, we can examine numbers of the form $pn$ and $mn$ where $p$ and $m$ are primes.
\begin{conjecture}
Let $n$ be an abundant number. If $p$ and $m$ are primes greater than $\sigma{(n)}$, then the divisors of the divisor voting systems of $pn$ and $mn$ have the same Shapley-Shubik power indices as well as the same Banzhaf power indices.
\end{conjecture}
We will prove a part of this conjecture, which is the following proposition.
\begin{proposition}
Let $n$ be an abundant number and let $p$ and $m$ be primes greater than $\sigma{(n)}+1$. Then the divisor voting systems of $pn$ and $mn$ have the same number of winning coalitions.
\end{proposition}
\begin{proof}
Let the divisor voting system of $n$ be $\{1, d_1, d_2,\dots, d_k, n\}$, where each $d_i$ is a proper divisor of $n$. Since $p$ and $m$ are primes, the divisor voting systems of $pn$ and $mn$, $S_{pn}$ and $S_{mn}$ respectively, are the following:
\begin{center}
$S_{pn}=\{1, d_1,\dots, d_k, n, p, pd_1, pd_2, \dots, pd_k, pn\}$
\end{center}
\begin{center}
$S_{mn}=\{1, d_1,\dots, d_k, n, m, md_1, md_2, \dots, md_k, mn\}$.
\end{center}
Let $W_{pn}$ be the set of winning coalitions in $S_{pn}$ and let $W_{mn}$ be the set of all winning coalitions in $S_{mn}$. In order to prove that $W_{pn}$ and $W_{mn}$ have the same number of coalitions, we will prove that there exists a bijection between the two sets.
Let $f\colon W_{mn} \rightarrow W_{pn}$ be the function defined by
\begin{equation*}
f(\{d_{j_1},d_{j_2},\dots, d_{j_l}, md_{c_1},md_{c_2},\dots, md_{c_s}\}) = \{d_{j_1},d_{j_2},\dots, d_{j_l}, pd_{c_1},pd_{c_2},\dots, pd_{c_s}\}.
\end{equation*}
Note that $f$ is injective, since if $$\{d_{j_1},d_{j_2},\dots, d_{j_l}, pd_{c_1},pd_{c_2},\dots, pd_{c_s}\} = \{d_{j_1}',d_{j_2}',\dots, d_{j_l}', pd_{c_1}',pd_{c_2}',\dots, pd_{c_s}'\},$$ then we have $$\{d_{j_1},d_{j_2},\dots, d_{j_l}, md_{c_1},md_{c_2},\dots, md_{c_s}\} = \{d_{j_1}',d_{j_2}',\dots, d_{j_l}', md_{c_1}',md_{c_2}',\dots, md_{c_s}'\}.$$
Now we prove that $f$ is surjective. Suppose we have a winning coalition $G$ in $S_{pn}$. Then we will prove that there exists a winning coalition $H$ in $S_{mn}$ such that $f$ maps $H$ to $G$.
$G$ is of the form
$[d_j, d_{j+1},\dots, d_{j+k}, pd_c,pd_{c+1},\dots, pd_{c+r}]$, where each $d_i$ is a divisor of $n$. Then, based on our construction of $f$, we know that $H$ should be of the form
\begin{center}
$[d_j, d_{j+1},\dots, d_{j+k}, md_c,md_{c+1},\dots, md_{c+r}]$.
\end{center}
We will now prove that $H$ is a winning coalition.
Let $q_{pn}$ be the quota of $S_{pn}$. Then
\begin{equation*}
q_{pn}=\dfrac{p\sigma{(n)}+\sigma{(n)}}{2}+1.
\end{equation*}
Similarly, for the quota of $S_{mn}$ we have
\begin{equation*}
q_{mn}=\dfrac{m\sigma{(n)}+\sigma{(n)}}{2}+1.
\end{equation*}
Let $w_1$ be the total voting power of $G$. Since $G$ is a winning coalition, we have
\begin{equation}
w_1 \geq q_{pn} = \dfrac{p\sigma{(n)}+\sigma{(n)}}{2}+1.
\end{equation}
Let $$\beta=\sum_{n=1}^{k} d_{j+n}$$ and let $$\alpha =\sum_{n=1}^{r} d_{c+n}.$$
Then we can rewrite $w_1$ as $p\beta+\alpha$. By substituting $p\beta+\alpha$ for $w_1$ we have
\begin{equation}
p\beta+\alpha \geq \dfrac{p\sigma{(n)}+\sigma{(n)}}{2}+1.
\end{equation}
Hence
\begin{equation}
p \geq\dfrac{\sigma({n})+2-2\alpha}{2\beta-\sigma({n})}.
\end{equation}
The maximum value of the right-hand side of 2.3 occurs when the denominator is 1 and $\alpha$ is 0. Hence the right-hand side of 2.3 is at most $\sigma(n)+2$. Since $m > \sigma(n) +1$, we have
\begin{equation}
m \geq\dfrac{\sigma({n})+2-2\alpha}{2\beta-\sigma({n})}
\end{equation}
By rearranging 2.4, we have
\begin{equation*}
m\beta+\alpha \geq \dfrac{m\sigma{(n)}+\sigma{(n)}}{2}+1=q_{mn}.
\end{equation*}
The total voting power of $H$ is also $m\beta + \alpha$ due to its construction. Hence $H$ is a winning coalition and $f$ is surjective.
Therefore, since there exists a bijection between $W_{mn}$ and $W_{pn}$, the divisor voting systems of $mn$ and $pn$ have the same number of winning coalitions.
\qedhere
\end{proof}
\section{Fixed Points}
\subsection{Banzhaf}
The general Banzhaf power indices for an arbitrary distribution of weights is quite difficult to express, so we will only consider the weights of the form $(a, b, b, \dots , b)$ where there is one player with a vote of $a$ and $m$ players have a vote of $b$, and the weights sum to $1$. We refer to people with a vote of $a$ as type $A$ and people with a vote of $b$ as Type $B$. Then we have $a = 1 - mb$. Each winning coalition either contains the player of Type $A$, or it does not.
\begin{itemize}
\item If a winning coalition does not contain the player of Type $A$, then there must be at least $p$ players of type $B$, where $pb > \frac{1}{2}$. Since $p$ is an integer, this simplifies to $p \geq \lceil \frac{1}{2b} \rceil$. Since all players in our coalition have the same voting power, we must have $p = \lceil \frac{1}{2b} \rceil$ for any player to be a critical player. Then each player of type $B$ is a critical player exactly $\binom{m-1}{\lceil \frac{1}{2b} \rceil - 1}$ times because we can pick $\lceil \frac{1}{2b} \rceil - 1$ of the remaining $m-1$ players of type $B$ to be the other players on our coalition.
\item Now suppose a winning coalition contains the player of Type $A$. Then if are $q$ players of type $B$, $q$ must satisfy $1 - mb + qb > \frac{1}{2}$ and $qb < \frac{1}{2}$. $q$ is also an integer, so this simplifies to $m - \lfloor \frac{1}{2b} \rfloor \leq q \leq \lfloor \frac{1}{2b} \rfloor$. Then the player of Type $A$ is critical $\binom{m}{m - \lfloor \frac{1}{2b} \rfloor} + \binom{m}{m - \lfloor \frac{1}{2b} \rfloor + 1} + \dots + \binom{m}{\lfloor \frac{1}{2b} \rfloor}$ times, since we pick any $q$ players of type $B$. Furthermore, players of type $B$ can also be critical when $q$ is a minimum, so players of Type $B$ are critical an additional $\binom{m-1}{m - \lfloor \frac{1}{2b} \rfloor - 1}$ times.
\end{itemize}
Therefore the player of type A has Banzhaf index
\[
\frac{\binom{m}{m - \lfloor \frac{1}{2b} \rfloor} + \binom{m}{m - \lfloor \frac{1}{2b} \rfloor + 1} + \dots + \binom{m}{\lfloor \frac{1}{2b} \rfloor}}{m \left(\binom{m-1}{\lceil \frac{1}{2b} \rceil - 1} + \binom{m-1}{m - \lfloor \frac{1}{2b} \rfloor - 1} \right) + \left[\binom{m}{m - \lfloor \frac{1}{2b} \rfloor} + \binom{m}{m - \lfloor \frac{1}{2b} \rfloor + 1} + \dots + \binom{m}{\lfloor \frac{1}{2b} \rfloor} \right]}
\]
while the player of type B has Banzhaf index
\[
\frac{\binom{m-1}{\lceil \frac{1}{2b} \rceil - 1} + \binom{m-1}{m - \lfloor \frac{1}{2b} \rfloor - 1} }{m \left(\binom{m-1}{\lceil \frac{1}{2b} \rceil - 1} + \binom{m-1}{m - \lfloor \frac{1}{2b} \rfloor - 1} \right) + \left[\binom{m}{m - \lfloor \frac{1}{2b} \rfloor} + \binom{m}{m - \lfloor \frac{1}{2b} \rfloor + 1} + \dots + \binom{m}{\lfloor \frac{1}{2b} \rfloor} \right]}.
\]
We need the first expression to be equal to $1-mb$ and the second expression to be equal to $b$ (but note that it is sufficient to investigate one of these, since by symmetry if one of them is true then the other one becomes automatically true). It is difficult to find classes of solutions solely based on this equation, so we will investigate fixed points of both Banzhaf and Shapley-Shubik indices instead, in Section $3.3$.
\subsection{Shapley-Shubik}
First let us examine the primitive non-trivial fixed points of $[1/2_s: a, b, b, \ldots, b]$, where the total sum of the votes is 1. We can now say that players of type $A$ have a weight of $a$ and players of type $B$ have a weight of $b$. When trying to compute the power of type $A$, suppose that there are $p$ players of type $B$ in the permutation before the person of type $A$. If the players of type $A$ is a critical player, then we know that $pb \leq \frac{1}{2}$ and $pb + a > \frac{1}{2}$. We know that $a = 1 - mb$, so we can substitute that into the second inequality to get $pb + (1 - mb) > \frac{1}{2}$. Solving these inequalities, we get $m - \frac{1}{2b} < p \leq \frac{1}{2b}$. We know that $p$ must be an integer, so assuming $\frac{1}{2b} \not \in \mathbb{Z}$, we can make the bounds stricter, which gives us $m - \lfloor \frac{1}{2b} \rfloor \leq p \leq \lfloor \frac{1}{2b} \rfloor$. To figure out how many different permutations correspond to each value of $p$, we can see that there must be $m!$ different permutations for how to arrange the players of type $B$, and then we just need to put the person of type $A$ in the appropriate location. Therefore, $m!(\lfloor \frac{1}{2b} \rfloor - (m - \lfloor \frac{1}{2b} \rfloor) + 1) = m!(2\lfloor \frac{1}{2b} \rfloor - m + 1)$ counts the number of permutations where type $A$ is the critical player, and $\frac{m!(2\lfloor \frac{1}{2b} \rfloor - m + 1)}{(m + 1)!} = \frac{2\lfloor \frac{1}{2b} \rfloor - m + 1}{m + 1}$ gives the power of the person of type $A$ if $\frac{1}{2b} \not \in \mathbb{Z}$.
On the other hand, if $\frac{1}{2b} \in \mathbb{Z}$, then $m!\left(\frac{1}{2b} - (m - \frac{1}{2b})\right) = m!(\frac{1}{b} - m)$ represents the number of permutations such that the person of type $A$ is a critical player. This means that $\frac{1-mb}{b(m + 1)}$ is the power index of the person of type $A$. But note that this equals $1-mb$ if and only if $b= \frac{1}{m+1}$ in which case all players have the same power, i.e. we get a trivial fixed point!
\begin{proposition}Let $C$ be a positive integer. If $m = 2k - 1$, then for large enough $k$ $($depending only on $C)$, $b = \frac{k - C}{2k^2- k}$ and $a = 1 - (2k - 1)b$ yield a non-trivial fixed point of the form $[1/2_s: a, b, \dots, b]$. Moreover, all non-trivial fixed points are of this form.
\end{proposition}
\begin{proof}
If $b = \frac{k - C}{2k^2 - k}$, then $\frac{1}{2b} \neq \lfloor \frac{1}{2b} \rfloor$. We calculated the power of the person of type $A$ to be $\frac{2\lfloor \frac{1}{2b} \rfloor - m + 1}{m + 1}$, so $\frac{2\lfloor \frac{1}{2b} \rfloor - (2k - 1) + 1}{2k} = 1 - 2kb$ is necessary for this specific configuration to be a fixed point. If $b = \frac{k - C}{2k^2 - k}$, then $\left\lfloor \frac{1}{2 \cdot \frac{k - C}{2k^2 - k}} \right\rfloor = k + C - 1$ for large enough $k$. Substituting, we get
\begin{align*}
\frac{2\lfloor \frac{1}{2b} \rfloor - 2k + 2}{2k} &= \frac{2(k + C - 1) - 2k + 2}{2k} = \frac{C}{k} \\
&= \frac{C(2k - 1)}{k(2k - 1} = \frac{(2k^2 - k) - (k - C)(2k - 1)}{2k^2 - k} \\
&= 1 - (2k - 1)\frac{k - C}{2k^2 - k} \\
&= 1 - mb
\end{align*}
Therefore, the calculated value for type $A$'s weight is equal to $a$, and similarly the original value for type $B$'s weight is equal to $b$. All examples of this form must are indeed non-trivial fixed points. \qedhere
To show that these classify all the fixed points of $[1/2_s: a, b, b, \dots]$, we note that $\lfloor \frac{1}{2b} \rfloor$ must be an integer, so we know there exists an integer $C$ such that $\lfloor \frac{1}{2b} \rfloor = k + C - 1$. Substituting this in and solving for $b$, we get $b = \frac{k - C}{2k^2 - k}$. We also know that $C > 0$ because if $C \leq 0$, then $\frac{k - C}{2k^2 - k} \geq \frac{1}{2k - 1}$, so this value of $b$ represents either a trivial case or an impossible one where $(2k - 1)b > 1$. Since there are no non-trivial fixed points in the other case, this classifies all the fixed points if $m = 2k - 1$ for $[1/2_s: a, b, \dots, b]$.
\end{proof}
\begin{proposition}
Let $C$ be a positive integer. If $m = 2k$, then for large enough $k$, $b = \frac{k - C}{2k^2 + k}$ and $a = 1 - 2kb$ gives a non-trivial fixed point of the form $[1/2_s: a, b, \ldots, b]$. Moreover, all non-trivial fixed points are of this form.
\end{proposition}
\begin{proof}
Since $\frac{1}{2b} \not \in \mathbb{Z}$, we know that $\frac{2\lfloor \frac{1}{2b} \rfloor - 2k + 1}{2k + 1} = 1 - 2kb$ is necessary for this value of $b$ to be a fixed point. If we plug in $b = \frac{k - C}{2k^2 + k}$ into $\lfloor \frac{1}{2b} \rfloor$, we get that this equals $k + C$ for large enough $k$. Substituting this in, we get
\begin{align*}
\frac{2\lfloor \frac{1}{2b} \rfloor - 2k + 1}{2k + 1} &= \frac{2C + 1}{2k + 1} \\
&= \frac{2kC + k}{2k^2 + k} \\
&= \frac{(2k^2 + k) - 2k(k - C)}{2k^2 + k} \\
&= 1 - 2kb
\end{align*}
Therefore, the calculated value for type $A$'s weight is equal to $a$, and similarly the original value for type $B$'s weight is equal to $b$. All examples of this form are hence non-trivial fixed points.
\qedhere
To show that these classify all the fixed points of $[1/2_s: a, b, b, \dots]$, we note that $\lfloor \frac{1}{2b} \rfloor$ must be an integer, so we know there exists an integer $C$ such that $\lfloor \frac{1}{2b} \rfloor = k + C$. Substituting this in and solving for $b$, we get $b = \frac{k - C}{2k^2 + k}$. We also know that $C > 0$ because if $C \leq 0$, then $\frac{k - C}{2k^2 + k} \geq \frac{1}{2k + 1}$, so this value of $b$ represents either a trivial case or an impossible one where $2kb > 1$. Since there are no non-trivial fixed points in the other case, this classifies all the fixed points if $m = 2k$ for $[1/2_s: a, b, b, \dots]$.
\end{proof}
We can now take a look at weighted voting systems in the form of $(a, a, b, \ldots, b)$, which we can analyze in a similar way to the previous parts. Again letting $m$ be the number of players of type $B$, we can then split into two cases.
\begin{enumerate}
\item $m = 2k$
In this case, there are two possibilities for a winning coalition in which the critical player is in type $A$. Either the other player of type $A$ is in the winning coalition or they are not. We can calculate how many permutations fit into one of the categories and then add them together to get the total number of permutations where a person of type $A$ is the critical player when trying to compute Shapley-Shubik.
\begin{itemize}
\item Only one player of type $A$ in the winning coalition.
Let $p$ be the number of players of type $B$ in the winning coalition. We know that $pb \leq \frac{1}{2}$ and $pb + a > \frac{1}{2}$. Solving for $p$ and substituting $a = \frac{1 - 2kb}{2}$, we get the inequalities $p \leq \frac{1}{2b}$ and $p > k$. Since $p$ must be an integer, we can make stricter bounds by saying $k + 1 \leq p \leq \lfloor \frac{1}{2b} \rfloor$.
Now, for a given value of $p$, we need to figure out how many permutations work with that value of $p$. Out of the $2k + 2$ people voting, the first $p + 1$ spots are part of the winning coalition in a given permutation, so the other player of type $A$ must be in the remaining $2k - p + 1$ locations. There are $(2k)!$ ways to arrange the rest of the players, so in total, there are $m!(2k - p + 1)$ permutations associated with a given value of $p$. We then need to sum this expression using the bounds from earlier to get the total number of permutations in this case to be $(2k)!\sum_{p = k + 1}^{\lfloor 1/(2b) \rfloor}(2k - p + 1) = \frac{(2k)!}{2}(\lfloor \frac{1}{2b} \rfloor - k)(3k - \lfloor \frac{1}{2b} \rfloor + 1)$.
\item Both players of type $A$ are in the winning coalition.
Again, let $p$ be the number of players of type $B$ in the winning coalition. We know that $pb + a \leq \frac{1}{2}$ and $pb + 2a > \frac{1}{2}$. Solving for $p$ and substituting $a = \frac{1 - 2kb}{2}$, we get the inequalities $p \leq k$ and $p > 2k - \frac{1}{2b}$. If we assume that $\frac{1}{2b} \not \in \mathbb{Z}$, then since $p$ must be an integer, we can write $2k - \lfloor \frac{1}{2b} \rfloor \leq p \leq k$. For each value of $p$, there are $p + 1$ places where we can place the other player of type $A$ and $(2k + 1)!$ ways to place the rest of the players, so there are $(2k)!\sum_{p = 2k - \lfloor \frac{1}{2b} \rfloor}^k (p + 1) = \frac{1}{2}\left(\lfloor \frac{1}{2b} \rfloor - k + 1\right)\left(3k - \lfloor \frac{1}{2b} \rfloor + 2\right)$.
\end{itemize}
Combining these two cases, along with the fact that there are $(2k + 2)!$ total permutations, we get that the power of a player of type $A$ must be $$\frac{\frac{1}{2}\left(\lfloor \frac{1}{2b} \rfloor - k\right)\left(3k - \lfloor \frac{1}{2b} \rfloor + 1\right) + \frac{1}{2}\left(\lfloor \frac{1}{2b} \rfloor - k + 1\right)\left(3k - \lfloor \frac{1}{2b} \rfloor + 2\right)}{(2k + 1)(2k + 2)},$$ and a value of $b$ is a fixed point if and only if this expression equals $\frac{1 - 2kb}{2}$. Using this equation, we can generate the following solutions for $b$.
\begin{center}
\begin{tabular}{c|l}
$k = 1$ & $b = 1/3$ \\ \hline
$k = 2$ & $b = 2/15, b = 1/5$ \\ \hline
$k = 3$ & $b = 3/28, b = 1/7$ \\ \hline
$k = 4$ & $b = 13/180, b = 4/45, b = 1/9$ \\ \hline
$k = 5$ & $b = 7/110, b = 5/66, b = 1/11$
\end{tabular}
\end{center}
We can verify that $b = \frac{1}{2k + 1}$ and $b = \frac{k}{(k + 1)(2k + 1)}$ are classes of primitive non-trivial fixed points for large enough $k$ by showing that both sides equal each other when we substitute the values of $b$ in.
\item $m = 2k + 1$
We again split into different cases depending on the number of players of type $A$ in the winning coalition.
\begin{itemize}
\item Only one player of type $A$ in the winning coalition.
Let $p$ be the number of players of type $B$ in the winning coalition. We know that $pb \leq \frac{1}{2}$ and $pb + a > \frac{1}{2}$. Solving for $p$ and substituting $a = \frac{1 - (2k +1)b}{2}$, we get the inequalities $p \leq \frac{1}{2b}$ and $p > k$. Since $p$ must be an integer, we can make stricter bounds by saying $k + 1 \leq p \leq \lfloor \frac{1}{2b} \rfloor$.
Now, for a given value of $p$, we need to figure out how many permutations work with that value of $p$. Out of the $2k + 3$ people voting, the first $p + 1$ spots are part of the winning coalition in a given permutation, so the other player of type $A$ must be in the remaining $2k - p + 2$ locations. There are $(2k + 1)!$ ways to arrange the rest of the players, so in total, there are $(2k + 1)!(2k - p + 2)$ permutations associated with a given value of $p$. We then need to sum this expression using the bounds from earlier to get the total number of permutations in this case to be $(2k + 1)!\sum_{p = k + 1}^{\lfloor 1/(2b) \rfloor}(2k - p + 2) = \frac{(2k + 1)!}{2}(\lfloor \frac{1}{2b} \rfloor - k)(3k - \lfloor \frac{1}{2b} \rfloor + 3)$.
\item Both players of type $A$ are in the winning coalition.
Again, let $p$ be the number of players of type $B$ in the winning coalition. We know that $pb + a \leq \frac{1}{2}$ and $pb + 2a > \frac{1}{2}$. Solving for $p$ and substituting $a = \frac{1 - (2k + 1)b}{2}$, we get the inequalities $p \leq k$ and $p > 2k + 1 - \frac{1}{2b}$. If we assume that $\frac{1}{2b} \not \in \mathbb{Z}$, then since $p$ must be an integer, we can write $2k + 1 - \lfloor \frac{1}{2b} \rfloor \leq p \leq k$. For each value of $p$, there are $p + 1$ places where we can place the other player of type $A$ and $(2k + 1)!$ ways to place the rest of the players, so there are $(2k + 1)!\sum_{p = 2k + 1 - \lfloor \frac{1}{2b} \rfloor}^k (p + 1) = \frac{1}{2}\left(\lfloor \frac{1}{2b} \rfloor - k\right)\left(3k - \lfloor \frac{1}{2b} \rfloor + 3\right)$.
\end{itemize}
Combining these two cases, along with the fact that there are $(2k + 3)!$ total permutations, we get that the power of a player of type $A$ must be $$\frac{\left(\lfloor \frac{1}{2b} \rfloor - k\right)\left(3k - \lfloor \frac{1}{2b} \rfloor + 3\right)}{(2k + 1)(2k + 2)},$$ and a value of $b$ is a fixed point if and only if this expression equals $\frac{1 - (2k + 1)b}{2}$. Using this equation, we can generate the following solutions for $b$.
\begin{center}
\begin{tabular}{c|l}
$k = 1$ & $b = 2/15$ \\ \hline
$k = 2$ & $b = 3/35, b = 11/105$ \\ \hline
$k = 3$ & $b = 4/63, b = 11/126$ \\ \hline
$k = 4$ & $b = 5/99, b = 31/495, b = 37/495$
\end{tabular}
\end{center}
\noindent And we can verify that $b = \frac{k + 1}{4(k + 1)^2 - 1}$ is a primitive fixed points for large enough $k$.
\end{enumerate}
\subsection{Both Banzhaf and Shapley-Shubik}
To find primitive nontrivial fixed points for both power indices, we can test Shapley-Shubik fixed points to see if they are still fixed points for Banzhaf. Consider the Shapley-Shubik fixed points in $\textbf{Proposition 3.1}$. To calculate Banzhaf power indices for this class of fixed points, we see that each winning coalition either contains the player of type $A$ or it does not.
\begin{itemize}
\item Winning coalition does not contain player of type A: The winning coalition must contain at least $p$ players of type $B$, where $p(\frac{k-c}{2k^2-k}) > \frac{1}{2}$, or $p > \frac{2k^2-k}{2k-2c} = k + c - \frac{1}{2} + \frac{2c^2-c}{2k-2c}$. Since $p$ is an integer, we have $p \geq k + c$ for large enough $k$. In order for the coalition to have a critical player, we must have $p = k + c$, so each player of type $B$ is a critical player $\binom{2k-2}{k+c-1}$ times since we pick the other $p - 1$ players from the remaining $2k-2$ players of Type $B$.
\item Winning coalition contains player of type A: The winning coalition must contain $q$ players of type $B$, where $q(\frac{k-c}{2k^2-k}) < \frac{1}{2}$ and $\frac{1}{k} + q(\frac{k-c}{2k^2-k}) > \frac{1}{2}$, which simplifies to $\frac{(k-2c)(2k-1)}{2k-2c} < q < \frac{2k^2-k}{2k-2c}$. The lower and upper bounds are equal to $k - c - \frac{1}{2} + \frac{c-2c^2}{2k-2c}$ and $k + c - \frac{1}{2} + \frac{2c^2-c}{2k-2c}$, respectively, so since $q$ is an integer, we have $k - c \leq q \leq k + c - 1$ for large enough $k$. Then the player of type $A$ is a critical player $\binom{2k-1}{k-c} + \binom{2k-1}{k-c+1} + \dots + \binom{2k-1}{k+c-1}$ times, where we choose $q$ players of type $B$ from the total $2k-1$. Furthermore, players of type $B$ are also critical players when $q = k-c$, so they are critical players $\binom{2k-2}{k-c-1}$ times.
\end{itemize}
Thus the Banzhaf index for a player of type $B$ is given by
\[\dfrac{\binom{2k-2}{k+c-1} + \binom{2k-2}{k-c-1}}{(2k-1)\left( \binom{2k-2}{k+c-1} + \binom{2k-2}{k-c-1} \right) + \binom{2k-1}{k-c} + \binom{2k-1}{k-c+1} + \dots + \binom{2k-1}{k+c-1}}.
\]
We want this expression to equal $\frac{k-c}{2k^2-k}$, which we found to only hold true when $c=1$.
\newline
\\
Now consider the Shapley-Shubik fixed points in $\textbf{Proposition 3.2}$. To calculate Banzhaf power indices for this class of fixed points, we see that each winning coalition either contains the player of type $A$ or it does not.
\begin{itemize}
\item Winning coalition does not contain player of type A: The winning coalition must contain at least $p$ players of type $B$, where $p(\frac{k-c}{2k^2+k}) > \frac{1}{2}$, or $p > \frac{2k^2+k}{2k-2c} = k + c + \frac{1}{2} + \frac{2c^2+c}{2k-2c}$. Since $p$ is an integer, we have $p \geq k + c +1$ for large enough $k$. In order for the coalition to have a critical player, we must have $p = k + c+1$, so each player of type $B$ is a critical player $\binom{2k-1}{k+c}$ times since we pick the other $p - 1$ players from the remaining $2k-1$ players.
\item Winning coalition contains player of type A: The winning coalition must contain $q$ players of type $B$, where $q(\frac{k-c}{2k^2+k}) < \frac{1}{2}$ and $\frac{1}{k} + q(\frac{k-c}{2k^2+k}) > \frac{1}{2}$, which simplifies to $\frac{(k-2c)(2k+1)}{2k-2c} < q < \frac{2k^2+k}{2k-2c}$. The lower and upper bounds are equal to $k - c + \frac{1}{2} - \frac{c+2c^2}{2k-2c}$ and $k + c + \frac{1}{2} + \frac{2c^2+c}{2k-2c}$, respectively, so since $q$ is an integer, we have $k - c +1\leq q \leq k + c $ for large enough $k$. Then the player of type $A$ is a critical player $\binom{2k}{k-c+1} + \binom{2k}{k-c+2} + \dots + \binom{2k}{k+c}$, where we choose $q$ players of type $B$ from the total $2k-1$. Furthermore, players of type $B$ are also critical players when $q = k-c+1$, so they are critical players $\binom{2k-1}{k-c}$ times.
\end{itemize}
Thus the Banzhaf index for a player of type $B$ is given by
\[ \dfrac{\binom{2k-1}{k+c} + \binom{2k-1}{k-c}}{(2k)\left( \binom{2k-1}{k+c} + \binom{2k-1}{k-c} \right) + \binom{2k}{k-c+1} + \binom{2k}{k-c+2} + \dots + \binom{2k}{k+c}}.
\]
We want this expression to equal $\frac{k-c}{2k^2+k}$, but we found no integer solutions for $c$ for which this is true.
\section{Future Research}
To expand the class of abundant numbers for which Proposition 2.1 is true, it would be helpful to prove that if the Shapley-Shubik and Banzhaf power indices differ in the divisor voting system of an abundant number $n$, then they also differ in the divisor voting system of $2^{k}p$ for a prime $p$ and various values of $k$. It would also be useful to finish our proof that the players in the divisor voting system of $pn$ have equal Banzhaf indices as well as equal Shapley-Shubik indices as those in $mn$.
By proving these statements, it would then be possible to prove Question $1.2.$ by inducting on the number of distinct prime divisors of an abundant number $n$.
In further research concerning fixed points, we could try strengthening the algebraic approach for fixed points to find more classes (e.g. with three types of players).
\section{Acknowledgements}
We would like to thank Miroslav Marinov for his guidance, mentorship, and feedback on this research paper. We thank Joshua Zelinsky for proposing this research project and for his helpful suggestions. We would like to thank the PROMYS program, especially Professor Fried and Steve Huang for the opportunity to work on this project, and the Clay Mathematical Institute for their support of the returning student program.
\section{Appendix}
Here is the code that we used to compute the power indices.
\lstinputlisting[language=Python]{code.py}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 5,140 |
Virgin and Child with an Angel also known as Our Lady of the Eucharist (Italian: Madonna dell'Eucarestia) is a painting in tempera on wood panel by Sandro Botticelli, executed in c. 1470. It is now held by the Isabella Stewart Gardner Museum in Boston, Massachusetts, having been acquired from Prince Chigi in 1899. The painting measures and is one of a series of paintings of the Madonna produced by Botticelli between 1465 and 1470. It shows influences from Filippo Lippi's Virgin and Child with Two Angels of c. 1465 in the Uffizi.
The Virgin Mary is shown in a three-quarter view, with the Christ Child held on her lap. A smiling angel, wearing a crown of myrtle, offers them a bowl containing twelve ears of wheat and grapes. The child raises a hand in benediction, and Mary holds one of the ears of corn.
The scene may be set in a walled garden or hortus conclusus, symbolic of Mary's virginity, with a landscape of hills and a river visible through an opening in the arcade around the wall. The wheat and grapes are symbolic of the bread and wine of the Eucharist, which themselves symbolise the body and blood of the incarnate Jesus, and the number of ears possibly refer to the number of the apostles at the Last Supper.
Prince Chigi first offered the painting to Isabella Stewart Gardner in 1899 for $30,000. She demurred initially, but by the time she decided to buy the price was $70,000. The sale was controversial in Italy, where there was press comment that the sale was illegal, and Prince Chigi was fined. The painting was exhibited at Colnaghi in London before being transported to Boston.
References
Virgin and Child with an Angel, Isabella Stewart Gardner Museum
1470s paintings
Angels in art
Paintings of the Madonna and Child by Sandro Botticelli
Paintings by Sandro Botticelli
Paintings in the collection of the Isabella Stewart Gardner Museum | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 2,119 |
{"url":"http:\/\/www.dimostriamogoldbach.it\/en\/from-reals-to-integers\/?doing_wp_cron=1635307589.2743630409240722656250","text":"# From integer numbers to real numbers\n\nPrerequirements:\n\nSo far we defined and studied only functions defined on integer numbers, the values of which can be integer (like the $\\theta(x)$ and $\\psi(x)$ functions) or real (like the $\\theta^{\\star}(x)$ and $\\psi^{\\star}(x)$ functions). You could think it\u2019s obvious, since our goal is studying integer numbers, in particular prime numbers. However, as it often happens in Mathematics, in order to study a certain class of objects it often turns out to be useful to take inspiration from a wider class, by means of an abstraction process. Thus, in order to study integer numbers, we\u2019ll see that it\u2019s useful to take inspiration from real numbers. This means, in practice, extending the domain of a function from integer numbers to real numbers: a significant jump. Let\u2019s see two ways to do that.\n\n## Simple extension\n\nLet\u2019s consider a function $f$ defined on a set of integers $I$ (we suppose in general $I \\subseteq \\mathbb{Z}$. Starting from $f$, we define on positive real numbers a new function $\\overline{f}$, such that:\n\n\u2022 For each integer $n \\in I$, $\\overline{f}(n) = f(n)$\n\u2022 For real numbers greater than $n$, up to the next integer $n + 1$ not included, $\\overline{f}$ still assumes the value $f(n)$\n\nIf we compare the graphs of the two functions, we can see that function $f$ looks \u201cpunctiform\u201d, because the definition set is discrete, whereas function $\\overline{f}$ looks \u201cstep-like\u201d. For example, with $I = \\{0, \\ldots, 9\\}$ and $f(n) = n$ we have:\n\nThe function $\\overline{f}$ is defined on a set we can call $\\overline{I}$, obtained from $I$ by adding, for each integer in it, all real numbers up to the next integer excluded. For example:\n\n$I$ $\\overline{I}$\n$\\{0, 1, 2, 3\\}$ $[0, 4)$\n$\\mathbb{N}$ $[0, +\\infty)$\n$\\mathbb{N}^{\\star}$ $[1, +\\infty)$\n$\\mathbb{Z}$ $\\mathbb{R}$\n\nThe formal definition of simple extension is the following:\n\nSimple extension of a function defined on integer numbers, to real numbers\n\nLet $f: I \\rightarrow \\mathbb{R}$ be a function defined on a set $I \\subseteq \\mathbb{Z}$. We define on the set $\\overline{I} := \\bigcup_{n \\in I} [n, n+1)$ the function $\\overline{f}: \\overline{I} \\rightarrow \\mathbb{R}$ such that:\n\n$$\\forall n \\in I\\ \\forall x \\in [n, n+1): \\overline{f}(x) := f(n) \\tag{1}$$\n\nWe\u2019ll call function $\\overline{f}$ \u201csimple extension of $f$ to real numbers\u201d, or simply \u201csimple extension of $f$\u201c.\n\nFormula (1) is the formal translation that $\\overline{f}$ assumes the same value as $f$ on any integer $n$ and on the next reals, up to the next integer not included. The same thing can be expressed also with the following equivalent definition:\n\n$$\\overline{f}(x) := \\begin{cases} f(x) & \\text{if }x \\in I \\\\ f(\\left \\lfloor x \\right \\rfloor) & \\text{otherwise}\\end{cases} \\tag{2}$$\n\nThis formula says essentially that:\n\n\u2022 on integers, i.e. on the set $I$, $f$ and $\\overline{f}$ assume the same value\n\u2022 on the other real numbers $x$, $\\overline{f}$ assumes the value assumed by $f$ on the integer immediately preceding $x$, that is the integer part of $x$\n\nFor example, $\\overline{f}(\\frac{7}{2})$ is the value assumed by $f$ on the integer part of $\\frac{7}{2}$, that is $\\left \\lfloor \\frac{7}{2} \\right \\rfloor = 3$. The same thing happens for all reals in the range $[3, 4)$, because they all have 3 as integer part: repeating this argument with a generic integer $n$ in place of 3, you can understand how (2) is equivalent to (1).\n\nWe note that the values assumed by $\\overline{f}$ are, on the whole, the same assumed by $f$: in fact in definition (1) the values of $\\overline{f}$ are defined through appropriate values of $f$. In other terms, the functions $f$ and $\\overline{f}$ have the same image: $f(I) = \\overline{f}(\\overline{I})$.\n\nWe called \u201csimple\u201d the extension described in Definition N.8 not to indicate that\u2019s \u201ceasy\u201d, but because, if $I$ is finite, the extension $\\overline{f}$ is a so-called \u201csimple function\u201d, that is, simplifying, a function that assumes a finite number of different values. Simple functions are studied in misure theory.\n\n## Generic extensions\n\nDefinition N.8 may seem artificial: you could think about obtaining the extending function simply by defining it in the same way as $f$, but on real numbers. For example you could extend the function\n\n$$f: \\mathbb{N} \\rightarrow \\mathbb{R} \\text{ such that } f(n) := n$$\n\nwith the function\n\n$$\\widetilde{f}: \\mathbb{R} \\rightarrow \\mathbb{R} \\text{ such that } \\widetilde{f}(x) := x$$\n\nHere the original and the extended function share the same formal expression (we just put $x$ in place of $n$, but we could also use the same variable name for both cases). However not all expressions that have sense for integer numbers, still do when you pass to reals. For example, a definition like:\n\n$$g(x) := \\text{number of divisors of }x$$\n\nmakes perfectly sense if $x$ is an integer number, but doesn\u2019t for a generic real number (for example, what would the divisors of $\\sqrt{2}$ be?). For avoiding this kind of problems, we won\u2019t attempt to formalize the idea of obtaining an extension through a \u201csimilar expression\u201d to the one that defines the original function, though this is what happens in most cases. Instead we keep intentionally generic the definition of extension, with the only constraint that the original and the extended function must assume, on the integers, the same values:\n\nExtension by expression of a function defined on integer numbers, to real numbers\n\nLet $f: I \\rightarrow \\mathbb{R}$ be a function defined on a set $I \\subseteq \\mathbb{Z}$. Let $\\overline{I} := \\bigcup_{n \\in I}[n. n+1)$ and $\\widetilde{f}: \\overline{I} \\rightarrow \\mathbb{R}$ a function such that:\n\n$$\\widetilde{f}_{\\mid I} = f$$\n\nWe\u2019ll say that the function $\\widetilde{f}$ is an \u201cextension of $f$ to real numbers\u201d, or simply an \u201cextension of $f$\u201c.\n\nWe can note that simple extension (Definition N.8) is a particular case of extension. In fact by definition $\\overline{f}(x) = f(n)$ for all $n \\in I$ and for all $x \\in [n, n+1)$; in particular, for $x := n$ we have that $\\overline{f}(n) = f(n)$.\n\nAt a graphical level, we have that the graph of $f$ is made up by isolated points (one point for each $n \\in I$), while the graph of $\\widetilde{f}$ is a generic curve that passes through these points. In the case of simple extension (Figure 1), this curve is given by the union of several separate segments: it\u2019s a sign that the extension is not continuous, contrary to the example of Figure 2.\n\n## Application to the lemma of bar chart area\n\nIn number theory, extensions are often used, because they let apply theorems of real analysis to functions defined on integer numbers. In particular, simple extension is useful when a function defined on integer numbers multiplies an integral, and in that case it lets bring the function under the integral sign.\n\nAn example of application of extensions is the Lemma of bar chart area:\n\n$$A = \\sum_{k = 1}^{n-1} C_k (f(k) - f(k + 1)) + C_n f(n) \\tag{3}$$\n\nIn fact, in the right expression, the summation can be rewritten differently by applying the concept of extension.\n\nFirst of all, extending the function $f$ we can rewrite the term $f(k) - f(k + 1)$ as $\\widetilde{f}(k) - \\widetilde{f}(k + 1)$. At this point, assuming the function $\\widetilde{f}$ to be of class $C^1$ on all $\\overline{I}$ (so in particular on the interval $[k, k+1)$), so that its derivative is continuous and so integrable, we can apply the Fundamental theorem of integral calculus, obtaining:\n\n$$\\widetilde{f}(k) - \\widetilde{f}(k + 1) = - \\int_k^{k+1} \\widetilde{f}'(t) dt \\tag{4}$$\n\nHere\u2019s explained where integrals arise from in number theory: thanks to the fundamental theory of calculus, a simple difference between two numbers becomes an integral. This passage may seem artificial, but we\u2019ll see that it\u2019s essential for the development of the theory.\n\nPlacing (4) into formula (3) of the Lemma, the generic summation term becomes\n\n$$- C_k \\int_k^{k+1} \\widetilde{f}'(t) dt$$\n\nThe constants $C_k$, representing the sums of the bases of the first $k$ bar chart rectangles, depending on $k$, can be globally considered as a function of $k$, i.e. we can introduce the function:\n\n$$C: \\{1, \\ldots, n\\} \\rightarrow \\mathbb{R} \\text{ such that } C(k) := C_k$$\n\nThis function, that simply maps $k$ to the $k$-th constant, lets us rewrite the generic summation term as\n\n$$- C(k) \\int_k^{k+1} \\widetilde{f}'(t) dt$$\n\nNow we can make a simple extension of the function $C$, rewriting this term as\n\n$$- \\overline{C}(k) \\int_k^{k+1} \\widetilde{f}'(t) dt \\tag{5}$$\n\nAt this point we have a function $\\overline{C}$ that by (1) is constant over the real range $[k, k+1)$. This lets us bring it under the integral sign, obtaining:\n\n$$- \\int_k^{k+1} \\overline{C}(k) \\widetilde{f}'(t) dt \\tag{6}$$\n\nWe can call $X$ the constant value of the function $\\overline{C}$ in the interval $[k, k+1)$ and, by the properties of integrals, bring it under the integral sign:\n\n$$- X \\int_k^{k+1} \\widetilde{f}'(t) dt = - \\int_k^{k+1} X \\widetilde{f}'(t) dt$$\n\nWe said that the constant $X$ represents the value assumed by $\\overline{C}$ at $k$, so the left member coincides with (5), but it also represents the value assumed by the same function at all points of the range $[k, k+1)$. These points are just the values assumed by the integration variable $t$, varying between $k$ and $k+1$, with the exception of the single point $k + 1$. So the right integral can be rewritten as:\n\n$$- \\int_{k}^{k+1} \\left( \\begin{cases} \\overline{C}(t) & \\text{if }t \\in [k, k+1)\\\\ X & \\text{if }t = k+1\\end{cases} \\right) \\widetilde{f}'(t) dt \\tag{5'}$$\n\nBut inside an integral we can modify the integrated function at a single point without affecting the result, so in place of the unnamed function appearing in (5\u2032), inside round brackets, we can simply employ the function $\\overline{C}$. Doing so, nothing changes in the interval $[k, k+1)$, while it doesn\u2019t matter whether $\\overline{C}(k+1)$ is equal to $X$ or not: in any case the final result of the integral doesn\u2019t change. In this way equation (6) is obtained.\n\nPutting (6) in place of the generic summation term in (3), the latter becomes:\n\n$$A = C(n) f(n) - \\sum_{k = 1}^{n-1} \\int_k^{k+1} \\overline{C}(k) \\widetilde{f}'(t) dt$$\n\nExpanding the summation, this formula is rewritten as:\n\n$$A = C(n) f(n) - \\left( \\int_1^2 \\overline{C}(k) \\widetilde{f}'(t) dt + \\int_2^3 \\overline{C}(k) \\widetilde{f}'(t) dt + \\ldots + \\int_{n-1}^n \\overline{C}(k) \\widetilde{f}'(t) dt \\right)$$\n\nSince all consecutive integral pairs have one end in common and the same integrated function, by the properties of integrals they can be joined into a single integral:\n\n$$A = C(n) f(n) - \\int_1^n \\overline{C}(k) \\widetilde{f}'(t) dt \\tag{7}$$\n\nWe have thus obtained a new form of the Lemma of bar chart area. In the next post we\u2019ll see the advantage of adopting this new form in place of the previous one, so justifying the introduction of integrals.\n\nLemma of bar chart area, second form\n\nLet $c_1, c_2, \\dots, c_n$ be non-negative real numbers, with $n > 0$. Let $f: \\{1, 2, ..., n\\} \\rightarrow \\mathbb{R}$ a function, and let $\\widetilde{f}$ a $C^1$ extension of $f$. Then the area $A$ of the bar chart made up of $n$ rectangles, each with basis $c_i$ and height $f(i)$, given by\n\n$$A = c_1 f(1) + c_2 f(2) + \\ldots + c_n f(n) = \\sum_{i=1}^{n} c_i f(i)$$\n\ncan be also computed with the formula\n\n$$C(n) f(n) - \\int_1^n \\overline{C}(k) \\widetilde{f}'(t) dt$$\n\nwhere $C: \\{1, 2, ..., n\\} \\rightarrow \\mathbb{N}$ is the function defined by $C(k) := c_1 + c_2 + \\ldots + c_k = \\sum_{i=1}^{k} c_i$.\n\nIn books about number theory you won\u2019t find our notations for the extended functions, like $\\overline{C}$ and $\\widetilde{f}$: they are notations introduced by us, not universal. Normally no specific notation is used to distinguish the extended functions from the respective original functions, for example (7) would be written as follows:\n\n$$A = C(n) f(n) - \\int_1^n C(k) f'(t) dt$$\n\nHowever we think it\u2019s worth distinguishing, for better clarity, between functions that \u201care born\u201d defined only on integer numbers, and their real extensions. Moreover the notation we chose for simple extensions, like $\\overline{C}$, thanks to the horizontal line sign, helps to remember that it\u2019s a \u201cstep-like\u201d function, contrary to generic extensions, that are normally continuous functions.","date":"2021-10-27 04:06:30","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 152, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.8978551030158997, \"perplexity\": 202.7542584740295}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2021-43\/segments\/1634323588053.38\/warc\/CC-MAIN-20211027022823-20211027052823-00507.warc.gz\"}"} | null | null |
{"url":"https:\/\/rdrr.io\/cran\/maotai\/man\/bmds.html","text":"# bmds: Bayesian Multidimensional Scaling In maotai: Tools for Matrix Algebra, Optimization and Inference\n\n## Description\n\nA Bayesian formulation of classical Multidimensional Scaling is presented. Even though this method is based on MCMC sampling, we only return maximum a posterior (MAP) estimate that maximizes the posterior distribution. Due to its nature without any special tuning, increasing mc.iter requires much computation.\n\n## Usage\n\n 1 2 3 4 5 6 7 8 9 bmds( data, ndim = 2, par.a = 5, par.alpha = 0.5, par.step = 1, mc.iter = 8128, verbose = TRUE ) \n\n## Arguments\n\n data an (n\\times p) matrix whose rows are observations. ndim an integer-valued target dimension. par.a hyperparameter for conjugate prior on variance term, i.e., \u03c3^2 \\sim IG(a,b). Note that b is chosen appropriately as in paper. par.alpha hyperparameter for conjugate prior on diagonal term, i.e., \u03bb_j \\sim IG(\u03b1, \u03b2_j). Note that \u03b2_j is chosen appropriately as in paper. par.step stepsize for random-walk, which is standard deviation of Gaussian proposal. mc.iter the number of MCMC iterations. verbose a logical; TRUE to show iterations, FALSE otherwise.\n\n## Value\n\na named list containing\n\nembed\n\nan (n\\times ndim) matrix whose rows are embedded observations.\n\nstress\n\ndiscrepancy between embedded and origianl data as a measure of error.\n\n## References\n\n\\insertRef\n\noh_bayesian_2001amaotai\n\n## Examples\n\n 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 ## use simple example of iris dataset data(iris) idata = as.matrix(iris[,1:4]) ## run Bayesian MDS # let's run 10 iterations only. iris.cmds = cmds(idata, ndim=2) iris.bmds = bmds(idata, ndim=2, mc.iter=5, par.step=(2.38^2)) ## extract coordinates and class information cx = iris.cmds$embed # embedded coordinates of CMDS bx = iris.bmds$embed # BMDS icol = iris[,5] # class information ## visualize opar <- par(no.readonly=TRUE) par(mfrow=c(2,1)) mc = paste0(\"CMDS with STRESS=\",round(iris.cmds$stress,4)) mb = paste0(\"BMDS with STRESS=\",round(iris.bmds$stress,4)) plot(cx, col=icol,pch=19,main=mc) plot(bx, col=icol,pch=19,main=mb) par(opar) \n\nmaotai documentation built on Feb. 3, 2022, 5:09 p.m.","date":"2023-02-05 07:44:31","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 1, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.6568098068237305, \"perplexity\": 7706.919381307479}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2023-06\/segments\/1674764500250.51\/warc\/CC-MAIN-20230205063441-20230205093441-00674.warc.gz\"}"} | null | null |
(function () {
let map = {
init: function (options) {
this.options = options;
this.location = this.options.location;
this.makeMap();
},
makeMap: function () {
let officeLoc = this.location.split(",");
let officePosition = new google.maps.LatLng(officeLoc[0], officeLoc[1]);
let mapOptions = {
zoom: 16,
center: officePosition,
mapTypeId: google.maps.MapTypeId.ROADMAP
}
this.mapObj = new google.maps.Map(document.getElementById("js-contact-form__map"), mapOptions);
let marker = new google.maps.Marker({
map: this.mapObj,
position: officePosition
// animation: google.maps.Animation.BOUNCE
})
}
}
map.init({
location: "52.216929, 21.228661"
});
})(); | {
"redpajama_set_name": "RedPajamaGithub"
} | 2,147 |
De Nederlandstalige Top 10 was een radioprogramma dat van 5 april 1978 tot 28 november 1985 op de TROS donderdag op Hilversum 3 en vanaf 5 december 1985 tot 1 mei 1986 op Radio 3 was te beluisteren. Het werd uitgezonden op de donderdagmiddag tussen 12:00 en 13:00 uur maar vanaf 11 oktober 1984 t/m 11 april 1985 van 19:00 tot 20:00 uur en vanaf 18 april 1985 tot 1 mei 1986 tussen 9:00 en 10:00 uur op Hilversum 3 en later Radio 3. Vanaf 9 mei 1986 t/m 6 maart 1987 werd het programma op de vrijdagmiddag tussen 17:00 en 18:00 uur op Radio 2 uitgezonden.
Aansluitend volgden nog de tips voor de top 10. De presentatoren waren Hugo van Gelderen en Jojo (5 april 1978 t/m 3 januari 1979), Peter van Dam (invaller) en
Ad Roland (10 januari 1979 t/m 6 maart 1987)
Als tune werd Heidewals van Gerry van Houtert gebruikt.
Voor het een zelfstandig programma werd was de Nederlandstalige Top 10 een programmaonderdeel van de Hugo van Gelderenshow uitgezonden vanaf 5 oktober 1972 t/m 6 maart 1975 op Hilversum 3 en gepresenteerd door Hugo van Gelderen op de donderdagavond van 22:00 tot 23:00 uur.
Radioprogramma van Hilversum 3
Muziekprogramma op radio
Hitlijst | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 2,484 |
Swing High is a 1932 American Pre-Code short documentary film directed by Jack Cummings. In 1932, it was nominated for an Academy Award at the 5th Academy Awards for Best Short Subject (Novelty). The film documents The Flying Codonas, a family of flying trapeze artists.
Cast
Pete Smith as narrator (voice)
The Flying Codonas as Themselves:
Edward Codona as himself
Lalo Codona as himself
Alfredo Codona as himself
Vera Codona as herself
References
External links
1932 films
1932 documentary films
1930s short documentary films
Black-and-white documentary films
American black-and-white films
Films produced by Pete Smith (film producer)
Films directed by Jack Cummings
Documentary films about circus performers
American short documentary films
1930s English-language films
1930s American films | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 9,168 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.