assert
(PHP 4, PHP 5, PHP 7)
assert — 检查一个断言是否为 FALSE
说明
PHP 5
bool assert ( mixed $assertion [, string $description ] )
PHP 7
bool assert ( mixed $assertion [, Throwable $exception ] )
assert() 会检查指定的 assertion 并在结果为 FALSE 时采取适当的行动。
Traditional assertions (PHP 5 and 7)
如果 assertion 是字符串，它将会被 assert() 当做 PHP 代码来执行。 assertion 是字
符串的优势是当禁用断言时它的开销会更小，并且在断言失败时消息会包含 assertion 表
达式。这意味着如果你传入了 boolean 的条件作为 assertion，这个条件将不会显示为断
言函数的参数；在调用你定义的 assert_options() 处理函数时，条件会转换为字符串，
而布尔值 FALSE 会被转换成空字符串。
断言这个功能应该只被用来调试。你应该用于完整性检查时测试条件是否始终应该为 TRUE
，来指示某些程序错误，或者检查具体功能的存在（类似扩展函数或特定的系统限制和功
能）。
断言不应该用于普通运行时操作，类似输入参数的检查。作为一个经验法则，在断言禁用
时你的代码也应该能够正确地运行。
assert() 的行为可以通过 assert_options() 来配置，或者手册页面上描述的 .ini 设置
。
assert_options() ASSERT_CALLBACK 配置指令允许设置回调函数来处理失败的断言。
assert() 回调函数在构建自动测试套件的时候尤其有用，因为它们允许你简易地捕获传入
断言的代码，并包含断言的位置信息。当信息能够被其他方法捕获，使用断言可以让它更
快更方便！
回调函数应该接受三个参数。第一个参数包括了断言失败所在的文件。第二个参数包含了
断言失败所在的行号，第三个参数包含了失败的表达式（如有任意 — 字面值例如 1 或者
"two" 将不会传递到这个参数）。 PHP 5.4.8 及更高版本的用户也可以提供第四个可选参
数，如果设置了，用于将 description 指定到 assert()。
Expectations (PHP 7 only)
assert() is a language construct in PHP 7, allowing for the definition of
expectations: assertions that take effect in development and testing
environments, but are optimised away to have zero cost in production.
While assert_options() can still be used to control behaviour as described
above for backward compatibility reasons, PHP 7 only code should use the two
new configuration directives to control the behaviour of assert() and not call
assert_options().
PHP 7 configuration directives for assert()
Directive Default Possible values
value
• 1: generate and execute code (development mode)
zend.assertions 1 • 0: generate code but jump around it at runtime
• -1: do not generate code (production mode)
• 1: throw when the assertion fails, either by
throwing the object provided as the exception or
by throwing a new AssertionError object if
assert.exception 0 exception wasn't provided
• 0: use or generate a Throwable as described above,
but only generate a warning based on that object
rather than throwing it (compatible with PHP 5
behaviour)
参数
assertion
断言。In PHP 5, this must be either a string to be evaluated or a boolean
to be tested. In PHP 7, this may also be any expression that returns a
value, which will be executed and the result used to indicate whether the
assertion succeeded or failed.
description
如果 assertion 失败了，选项 description 将会包括在失败信息里。
exception
In PHP 7, the second parameter can be a Throwable object instead of a
descriptive string, in which case this is the object that will be thrown if
the assertion fails and the assert.exception configuration directive is
enabled.
返回值
assertion 是 false 则返回 FALSE，否则是 TRUE。
更新日志
版本 说明
assert() is now a language construct and not a function. assertion() can
7.0.0 now be an expression. The second parameter is now interpreted either as
an exception (if a Throwable object is given), or as the description
supported from PHP 5.4.8 onwards.
5.4.8 增加了参数 description。 description 现在也作为第四个参数提供给
ASSERT_CALLBACK 模式里的回调函数。
范例
Traditional assertions (PHP 5 and 7)
Example #1 使用自定义处理程序处理失败的断言
Assertion Failed:
File '$file'
Line '$line'
Code '$code'
";
}
// 设置回调函数
assert_options(ASSERT_CALLBACK, 'my_assert_handler');
// 让一则断言失败
assert('mysql_query("")');
?>
Example #2 使用自定义处理器打印描述信息
以上例程会输出：
Assertion failed at test.php:21: 2 < 1
Assertion failed at test.php:22: 2 < 1: Two is less than one
Expectations (PHP 7 only)
Example #3 Expectations without a custom exception
With zend.assertions set to 0, the above example will output:
Hi!
With zend.assertions set to 1 and assert.exception set to 0, the above example
will output:
Warning: assert(): assert(true == false) failed in - on line 2
Hi!
With zend.assertions set to 1 and assert.exception set to 1, the above example
will output:
Fatal error: Uncaught AssertionError: assert(true == false) in -:2
Stack trace:
#0 -(2): assert(false, 'assert(true == ...')
#1 {main}
thrown in - on line 2
Example #4 Expectations with a custom exception
With zend.assertions set to 0, the above example will output:
Hi!
With zend.assertions set to 1 and assert.exception set to 0, the above example
will output:
Warning: assert(): CustomError: True is not false! in -:4
Stack trace:
#0 {main} failed in - on line 4
Hi!
With zend.assertions set to 1 and assert.exception set to 1, the above example
will output:
Fatal error: Uncaught CustomError: True is not false! in -:4
Stack trace:
#0 {main}
thrown in - on line 4
参见
• assert_options() - 设置/获取断言的各种标志
User Contributed Notes 8 notes
===============================================================================
There's a nice advantage to giving assert() some code to execute, as a string,
rather than a simple true/false value: commenting.
The comment will show up in the output (or in your assertion handler) and
doesn't require someone debugging to go through your code trying to figure out
why the assertion happened. That's no excuse to not comment your code, of
course.
You need to use a block comment (/*...*/) because a line comment (//...)
creates an "unexpected $end" parse error in the evaluated code. Bug? Could be.
(You can get around it with "false // not yet implemented\n" but that screws up
the message)
===============================================================================
Note that func_get_args() should be used carefully and never in a string! For
example:
prints
array(2) {
[0]=>
int(1)
[1]=>
int(2)
}
but prints
array(1) {
[0]=>
string(25) "var_dump(func_get_args())"
}
This is because of that the string passed to assert() is being evaled inside
assert, and not your function. Also, note that this works correctly, because of
the eval scope:
array(2) {
[0]=>
int(5)
[1]=>
int(6)
}
(oh, and for simplicity's sake the evaled code doesn't return true, so don't
worry that it fails assertion...)
===============================================================================
As noted on Wikipedia - "assertions are primarily a development tool, they are
often disabled when a program is released to the public." and "Assertions
should be used to document logically impossible situations and discover
programming errors— if the 'impossible' occurs, then something fundamental is
clearly wrong. This is distinct from error handling: most error conditions are
possible, although some may be extremely unlikely to occur in practice. Using
assertions as a general-purpose error handling mechanism is usually unwise:
assertions do not allow for graceful recovery from errors, and an assertion
failure will often halt the program's execution abruptly. Assertions also do
not display a user-friendly error message."
This means that the advice given by "gk at proliberty dot com" to force
assertions to be enabled, even when they have been disabled manually, goes
against best practices of only using them as a development tool.
===============================================================================
"If you would like to compare with === or !== you need to add '(string)' before
string numbers. Or wrap them with quotes."
Do not use this aproach, because it would not work, look at http://php.net/
manual/en/function.var-export.php .
===============================================================================
Here is a simple demonstration of Design By Contract with PHP
Condition failed!
Script: $script
Line: $line
Condition:
$message
";
}
// Parameters
$a = 5;
$b = 'Simple DCB with PHP';
// Pre-Condition
assert('
is_integer($a) &&
($a > 0) &&
($a < 20) &&
is_string($b) &&
(strlen($b) > 5);
');
// Function
function combine($a, $b) {
return "Kombined: " . $b . $a;
}
$result = combine($a, $b);
// Post-Condition
assert('
is_string($result) &&
(strlen($result) > 0);
');
// All right, the Function works fine
var_dump($result);
?>
===============================================================================
Example from Ikac Framework how they use assert()
*
* @param mixed $assertion The assertion.
* @param mixed $callback Callback to call on failed assertions
* @param array $options Set the various control options or just query
their current settings.
* @param string $description An optional description that will be
included in the failure message if the assertion fails.
*/
public function setAssertionDebug($assertion, $callback, array $options,
$description = null)
{
if (is_array($options)) {
foreach ($options AS $option => $value) {
assert_options($option, $value);
}
}
if ($callback) {
assert_options(ASSERT_CALLBACK, $callback);
}
return assert($assertion, $description);
}
?>
How to use:
1,ASSERT_WARNING => 0,ASSERT_QUIET_EVAL
=> 1);
$system->setAssertionDebug('2<1', function(){
echo "Assertion failed";
}, $option);
?>
===============================================================================
If you expect your code to be able to work well with other code, then you
should not make any assumptions about the current state of assert_options()
flags, prior to calling assert(): other code may disable ASSERT_ACTIVE, without
you knowing it - this would render assert() useless!
To avoid this, ALWAYS set assert_options() IMMEDIATELY before calling assert(),
per the C++ paradigm for assertion usage:
In one C++ source file, you can define and undefine NDEBUG multiple times, each
time followed by #include , to enable or disable the assert macro
multiple times in the same source file.
Here is how I workaround this issue in my PHP code:
$GLOBALS['MY_ASSERT_ACTIVE'],
3=>$GLOBALS['MY_ASSERT_CALLBACK'],
)
);
* @param mixed = file or options
* @param line
* @param code
* @return void
*/
function phpxAssertHandler_f($file_or_custom_options=null,
$line_or_assert_options=null, $code=null){
static $custom_options;
$debug = false;
if (is_null($code)){
// set default assert_options
$assert_options[]=1;//ASSERT_ACTIVE
$assert_options[]=0;//ASSERT_WARNING -
$assert_options[]=0;//ASSERT_QUIET_EVAL
$assert_options[]=__FUNCTION__;//ASSERT_CALLBACK
// set default custom_options
$custom_options[]=E_USER_ERROR;// error level
if (!is_null($line_or_assert_options)){
// assert_options are passed in
if (!is_array($line_or_assert_options)){
$line_or_assert_options=array($line_or_assert_options);
}
foreach ($line_or_assert_options as $i=>$assert_option){
if ($assert_option===true) $assert_option=1;
if ($assert_option===false) $assert_option=0;
$assert_options[$i]=$assert_option;
if($debug) echo ("assert_options[$i]=$assert_option\n");
}
}
if (!is_null($file_or_custom_options)){
// custom_options are passed in
if (!is_array($file_or_custom_options)){
$file_or_custom_options=array($file_or_custom_options);
}
foreach ($file_or_custom_options as $i=>$custom_option){
if ($custom_option===true) $custom_option=1;
if ($custom_option===false) $custom_option=0;
$custom_options[$i]=$custom_option;
if($debug) echo ("custom_options[$i]=$custom_option\n");
}
}
// set assert options
@assert_options (ASSERT_ACTIVE, $assert_options[0]);
@assert_options (ASSERT_WARNING, $assert_options[1]);
@assert_options (ASSERT_QUIET_EVAL, $assert_options[2]);
@assert_options (ASSERT_CALLBACK, $assert_options[3]);
} else {
// we are acting as a callback function
$file = $file_or_custom_options;
$line = $line_or_assert_options;
$msg="ASSERTION FAILED: $code";
phpxErrorHandler_f ($custom_options[0],$msg,$file,$line);
}
}//phpxAssertHandler_f()
?>
===============================================================================
You can also use a function instead a string description as the second
parameter.
The result would be:
Hey, this assert is false and I'm a function.


| |||||||
| FRAMES NO FRAMES |
Renders a UIComponent that represents a single column of data within a parent UIData
component.
| Tag Information | |
| Tag Class | com.icesoft.faces.component.UIColumnTag |
| TagExtraInfo Class | None |
| Body Content | JSP |
| Display Name | None |
| Attributes | ||||
| Name | Required | Request-time | Type | Description |
| binding | false | false | java.lang.String | The value binding expression linking this component to a property in a backing bean |
| colspan | false | false | java.lang.String | This is exactly an html colspan attribute. Due to the iterative nature of the table body, this attribute is not suitable inside the body of the dataTable. So this attribute will only be apply on the columns that are descendent of the colmnGroup component. |
| groupOn | false | false | java.lang.String | The groupOn attribute, enables the application to show the one to many relationship in the tabular format. Only the distinct values will be rendered for the grouped column, and its rowspan value will be set to its none distinct count. Note:
<ice:column groupOn="#{employee.deptName}" > <ice:outputText value="#{employee.deptName}"/> </ice:colum> <ice:column > <ice:outputText value="#{employee.name}"/> </ice:colum> ..... </ice:dataTable .. > |
| id | false | false | java.lang.String | The component identifier for this component. This value must be unique within the closest parent component that is a naming container. |
| rendered | false | false | java.lang.String | Flag indicating whether or not this component should be rendered (during Render Response Phase), or processed on any subsequent form submit. |
| renderedOnUserRole | false | false | java.lang.String | If user is in given role, this component will be rendered normally. If not, nothing is rendered and the body of this tag will be skipped. |
| rowspan | false | false | java.lang.String | This is exactly an html rowspan attribute. Due to the iterative nature of the table body, this attribute is not suitable inside the body of the dataTable. So this attribute will only be apply on the columns that are descendent of the colmnGroup component. |
| style | false | false | java.lang.String | CSS style(s) to be applied when this component is rendered. |
| styleClass | false | false | java.lang.String | Space-separated list of CSS style class(es) to be applied when this element is rendered. This value must be passed through as the "class" attribute on generated markup. If this attribute is defined on the column, the new styleClass will be extending the existing styleClasses. |
| Variables | No Variables Defined. |
| |||||||
| FRAMES NO FRAMES |


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{% extends sitebase.html %} {% autoescape None %} {% block content%}
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projects/classroom-rds/src/app/auth/models/user-teacher.model.ts
Propiedades |
|
| addresses |
addresses:
|
Tipo : literal type
|
| curp |
curp:
|
Tipo : string
|
| Opcional |
| dayOfBirth |
dayOfBirth:
|
Tipo : string
|
| Opcional |
| gender |
gender:
|
Tipo : "Hombre" | "Mujer"
|
| Opcional |
| id |
id:
|
Tipo : string
|
| permission |
permission:
|
Tipo : string
|
| phones |
phones:
|
Tipo : [literal type]
|
| Opcional |
| rfc |
rfc:
|
Tipo : string
|
| role |
role:
|
Tipo : string
|
| Opcional |
| uid |
uid:
|
Tipo : string
|
| Opcional |
export interface UserTeacher {
id: string;
uid?: string;
curp?: string;
rfc: string;
permission: string
dayOfBirth?: string;
gender?: 'Hombre' | 'Mujer';
role?: string;
phones?: [{
}];
addresses: {
/** The address type. Acceptable values: custom, home, other, work. */
type: 'custom' | 'home' | 'other' | 'work',
/** The street address, such as 1600 Amphitheatre Parkway. Whitespace within the string is ignored; however, newlines are significant. */
streetAddress: string,
/** Indicates if the user-supplied address was formatted. Formatted addresses are not currently supported. */
sourceIsStructured: boolean,
/** The abbreviated province or state. */
region: string,
/** If this is the user's primary address. The addresses list may contain only one primary address. */
primary: boolean,
/** The ZIP or postal code, if applicable. */
postalCode: string;
/** The post office box, if present. */
poBox: string,
/** The town or city of the address. */
locality: string,
/** A full and unstructured postal address. This is not synced with the structured address fields. */
formatted: string,
/** For extended addresses, such as an address that includes a sub-region. */
externalAddress: string,
/** If the address type is custom, this property contains the custom value. */
customType: string,
/** The country code. Uses the ISO 3166-1 standard. */
countryCode: string
/** Country. */
country: string
};
}


{% extends "layout.html" %} {% block content %}
Inspired by Hrithik Roshan’s fitness journey, the HRX Workout is based on a strength training module.
The HRX Workout primarily focuses on your shoulders, quads, core, traps and deltoid muscles.
It is designed keeping in mind all age groups and involves working on specific muscles using weights and various movements.
These include Primal Movements, Zero Momentum Reps and Compound Movements.
It also involves core activation and helps build body strength.
At HRX, it’s our mission to motivate and enable you to work on your mind and body, making sure you can be the best version of you.
Not just a brand, HRX is a mission that helps us enable and support people to be the fittest, happiest and most confident version of themselves.


1106 Harding Place
Charlotte, NC 28203
P:<PHONE_NUMBER>
F:<PHONE_NUMBER>
Please do not leave urgent messages on this website.
The request appointment feature is for non-urgent matters.
For non-urgent matters, the information you provide is covered by the Electronic Communications Privacy Act, 18 U.S.C. 2510-2521, and will remain confidential.


f1000v1
branch containing the reproducible code used to generate the figures in our workflow.plotMA
to support vertical or horizontal layout return. Also added an argument to remove the color legend, which is typically not that informative.bcbioRNADataSet
(< 0.1.0) to bcbioRNASeq
class object is now possible using as
coercion method.bcbioRNASeq
object. Legacy bcbioRNADataSet
objects must be upgraded to bcbioRNASeq
class.interestingGroups<-
.plotGeneHeatmap
to simply plotHeatmap
.bcb
, dds
, res
, etc.) into a single examples
object. This helps avoid accidental use of example bcb
in an analysis.internal-ggplot.R
to above each function.maxSamples
parameter in loadRNASeq
to transformationLimit
. If there are more samples than this limit, then the DESeq2 transformations will be skipped. In this case, rlog
and vst
will not be slotted into assays
.loadRNASeq
to ensure rows are in the same order as the columns in the counts matrix. Otherwise, DESeq will report an error at the DESeqDataSetFromTximport
step. We’re also ensuring the factor levels get updated here.glimpse
instead of str
in examples, where applicable.colData<-
assignment method support. This requires a DataFrame
class object. Upon assignment, the internal colData at bcbio(object, "DESeqDataSet")
, assays(object)[["rlog"]]
and assays(object)[["vst"]]
are also updated to match.design
, which will update the internal DESeqDataSet.gene2symbol
generic, which will now return a 2 column data.frame
with ensgene
and symbol
columns. This is helpful for downstream gene to symbol mapping operations.interestingGroups<-
in the documentation.internal-meltLog10.R
for example.viridis::scale_fill_viridis(discrete = TRUE)
. This makes it clearer to the user in the documentation where these palettes are located.plotGene
.plotHeatmap
now uses internal gene2symbol
mappings from stashed annotable, instead of always querying Ensembl. The user can define custom mappings with the gene2symbol
argument, if desired.plotPCA
now supports custom color palettes. The shapes
parameter has been removed because it doesn’t work well and is limited to datasets with few samples. This behavior matches the PCA functionality in DESeq2.plotVolcano
. Added support for gene2symbol
argument, like in plotHeatmap
. If left missing, the function will query Ensembl for the gene2symbol mappings. We’re now using data
instead of stats
as the main data source.dependencies
argument, which allows for automatic install of suggested packages along with imports.selectSamples
.organism
and genomeBuild
parameters are now user-definable in the main loadRNASeq
import function.intersect
in the featureCounts matrix.aggregateReplicates
code. This needs to be reworked and added back in a future release.if
statements to be more class specific.plotCorrelationHeatmap
transform
argument to normalized
, for consistency with the counts
generic.title
support to plots, where applicable.plotDEGHeatmap
.plotGenderMarkers
.resulsTables
function now defaults to summary = TRUE
.plotDEGHeatmap
.plotDEGPCA
.resultsTables
, for use with the Stem Cell Commons database.plotGene
and plotGenderMarkers
to run faster.[
subset method dropping metrics in metadata.resultsTables
.


4. Harmonic Oscillator¶
The harmonic oscillator is omnipresent in physics. Although you may think of this as being related to springs, it, or an equivalent mathematical representation, appears in just about any problem where a mode is sitting near its potential energy minimum. At that point, \(\partial_x V(x)=0\), and the first non-zero term (aside from a constant) in the potential energy is that of a harmonic oscillator. In a solid, sound modes (phonons) are built on a picture of coupled harmonic oscillators, and in relativistic field theory the fundamental interactions are also built on coupled oscillators positioned infinitesimally close to one another in space. The phenomena of a resonance of an oscillator driven at a fixed frequency plays out repeatedly in atomic, nuclear and high-energy physics, when quantum mechanically the evolution of a state oscillates according to \(e^{-iEt}\) and exciting discrete quantum states has very similar mathematics as exciting discrete states of an oscillator.
The potential energy for a single particle as a function of its position \(x\) can be written as a Taylor expansion about some point \(x_0\)
If the position \(x_0\) is at the minimum of the resonance, the first two non-zero terms of the potential are
Put into Newton’s 2nd law (assuming \(x_0=0\)),
Here \(A\) and \(\phi\) are arbitrary. Equivalently, one could have written this as \(A\cos(\omega_0 t)+B\sin(\omega_0 t)\), or as the real part of \(Ae^{i\omega_0 t}\). In this last case \(A\) could be an arbitrary complex constant. Thus, there are 2 arbitrary constants (either \(A\) and \(B\) or \(A\) and \(\phi\), or the real and imaginary part of one complex constant. This is the expectation for a second order differential equation, and also agrees with the physical expectation that if you know a particle’s initial velocity and position you should be able to define its future motion, and that those two arbitrary conditions should translate to two arbitrary constants.
A key feature of harmonic motion is that the system repeats itself after a time \(T=1/f\), where \(f\) is the frequency, and \(\omega=2\pi f\) is the angular frequency. The period of the motion is independent of the amplitude. However, this independence is only exact when one can neglect higher terms of the potential, \(x^3, x^4\cdots\). Once can neglect these terms for sufficiently small amplitudes, and for larger amplitudes the motion is no longer purely sinusoidal, and even though the motion repeats itself, the time for repeating the motion is no longer independent of the amplitude.
One can also calculate the velocity and the kinetic energy as a function of time,
The total energy is then
The total energy then goes as the square of the amplitude.
A pendulum is an example of a harmonic oscillator. By expanding the kinetic and potential energies for small angles find the frequency for a pendulum of length \(L\) with all the mass \(m\) centered at the end by writing the eq.s of motion in the form of a harmonic oscillator.
The potential energy and kinetic energies are (for \(x\) being the displacement)
For small \(x\) Newton’s 2nd law becomes
and the spring constant would appear to be \(k=mg/L\), which makes the frequency equal to \(\omega_0=\sqrt{g/L}\). Note that the frequency is independent of the mass.
4.1. Damped Oscillators¶
We consider only the case where the damping force is proportional to the velocity. This is counter to dragging friction, where the force is proportional in strength to the normal force and independent of velocity, and is also inconsistent with wind resistance, where the magnitude of the drag force is proportional the square of the velocity. Rolling resistance does seem to be mainly proportional to the velocity. However, the main motivation for considering damping forces proportional to the velocity is that the math is more friendly. This is because the differential equation is linear, i.e. each term is of order \(x\), \(\dot{x}\), \(\ddot{x}\cdots\), or even terms with no mention of \(x\), and there are no terms such as \(x^2\) or \(x\ddot{x}\). The equations of motion for a spring with damping force \(-b\dot{x}\) are
Just to make the solution a bit less messy, we rewrite this equation as
Both \(\beta\) and \(\omega\) have dimensions of inverse time. To find solutions (see appendix C in the text) you must make an educated guess at the form of the solution. To do this, first realize that the solution will need an arbitrary normalization \(A\) because the equation is linear. Secondly, realize that if the form is
that each derivative simply brings out an extra power of \(r\). This means that the \(Ae^{rt}\) factors out and one can simply solve for an equation for \(r\). Plugging this form into Eq. (4),
Because this is a quadratic equation there will be two solutions,
We refer to the two solutions as \(r_1\) and \(r_2\) corresponding to the \(+\) and \(-\) roots. As expected, there should be two arbitrary constants involved in the solution,
where the coefficients \(A_1\) and \(A_2\) are determined by initial conditions.
The roots listed above, \(\sqrt{\omega_0^2-\beta_0^2}\), will be imaginary if the damping is small and \(\beta<\omega_0\). In that case, \(r\) is complex and the factor \(e{rt}\) will have some oscillatory behavior. If the roots are real, there will only be exponentially decaying solutions. There are three cases:
4.1.1. Underdamped: \(\beta<\omega_0\)¶
Here we have made use of the identity \(e^{i\omega't}=\cos\omega't+i\sin\omega't\). Because the constants are arbitrary, and because the real and imaginary parts are both solutions individually, we can simply consider the real part of the solution alone:
4.1.2. Critical dampling: \(\beta=\omega_0\)¶
In this case the two terms involving \(r_1\) and \(r_2\) are identical because \(\omega'=0\). Because we need to arbitrary constants, there needs to be another solution. This is found by simply guessing, or by taking the limit of \(\omega'\rightarrow 0\) from the underdamped solution. The solution is then
The critically damped solution is interesting because the solution approaches zero quickly, but does not oscillate. For a problem with zero initial velocity, the solution never crosses zero. This is a good choice for designing shock absorbers or swinging doors.
4.1.3. Overdamped: \(\beta>\omega_0\)¶
This solution will also never pass the origin more than once, and then only if the initial velocity is strong and initially toward zero.
Given \(b\), \(m\) and \(\omega_0\), find \(x(t)\) for a particle whose initial position is \(x=0\) and has initial velocity \(v_0\) (assuming an underdamped solution).
The solution is of the form,
From the initial conditions, \(A_1=0\) because \(x(0)=0\) and \(\omega'A_2=v_0\). So
4.2. Our Sliding Block Code¶
Here we study first the case without additional friction term and scale our equation in terms of a dimensionless time \(\tau\).
Let us remind ourselves about the differential equation we want to solve (the general case with damping due to friction)
We divide by \(m\) and introduce \(\omega_0^2=\sqrt{k/m}\) and obtain
Thereafter we introduce a dimensionless time \(\tau = t\omega_0\) (check that the dimensionality is correct) and rewrite our equation as
which gives us
We then define \(\gamma = b/(2m\omega_0)\) and rewrite our equations as
This is the equation we will code below. The first version employs the Euler-Cromer method.
%matplotlib inline
# Common imports
import numpy as np
import pandas as pd
from math import *
import matplotlib.pyplot as plt
import os
# Where to save the figures and data files
PROJECT_ROOT_DIR = "Results"
FIGURE_ID = "Results/FigureFiles"
DATA_ID = "DataFiles/"
if not os.path.exists(PROJECT_ROOT_DIR):
os.mkdir(PROJECT_ROOT_DIR)
if not os.path.exists(FIGURE_ID):
os.makedirs(FIGURE_ID)
if not os.path.exists(DATA_ID):
os.makedirs(DATA_ID)
def image_path(fig_id):
return os.path.join(FIGURE_ID, fig_id)
def data_path(dat_id):
return os.path.join(DATA_ID, dat_id)
def save_fig(fig_id):
plt.savefig(image_path(fig_id) + ".png", format='png')
from pylab import plt, mpl
plt.style.use('seaborn')
mpl.rcParams['font.family'] = 'serif'
DeltaT = 0.001
#set up arrays
tfinal = 20 # in years
n = ceil(tfinal/DeltaT)
# set up arrays for t, v, and x
t = np.zeros(n)
v = np.zeros(n)
x = np.zeros(n)
# Initial conditions as simple one-dimensional arrays of time
x0 = 1.0
v0 = 0.0
x[0] = x0
v[0] = v0
gamma = 0.0
# Start integrating using Euler-Cromer's method
for i in range(n-1):
# Set up the acceleration
# Here you could have defined your own function for this
a = -2*gamma*v[i]-x[i]
# update velocity, time and position
v[i+1] = v[i] + DeltaT*a
x[i+1] = x[i] + DeltaT*v[i+1]
t[i+1] = t[i] + DeltaT
# Plot position as function of time
fig, ax = plt.subplots()
#ax.set_xlim(0, tfinal)
ax.set_ylabel('x[m]')
ax.set_xlabel('t[s]')
ax.plot(t, x)
fig.tight_layout()
save_fig("BlockEulerCromer")
plt.show()
When setting up the value of \(\gamma\) we see that for \(\gamma=0\) we get the simple oscillatory motion with no damping. Choosing \(\gamma < 1\) leads to the classical underdamped case with oscillatory motion, but where the motion comes to an end.
Choosing \(\gamma =1\) leads to what normally is called critical damping and \(\gamma> 1\) leads to critical overdamping. Try it out and try also to change the initial position and velocity. Setting \(\gamma=1\) yields a situation, as discussed above, where the solution approaches quickly zero and does not oscillate. With zero initial velocity it will never cross zero.
4.3. Sinusoidally Driven Oscillators¶
Here, we consider the force
which leads to the differential equation
Consider a single solution with no arbitrary constants, which we will call a {\it particular solution}, \(x_p(t)\). It should be emphasized that this is {\bf A} particular solution, because there exists an infinite number of such solutions because the general solution should have two arbitrary constants. Now consider solutions to the same equation without the driving term, which include two arbitrary constants. These are called either {\it homogenous solutions} or {\it complementary solutions}, and were given in the previous section, e.g. Eq. (9) for the underdamped case. The homogenous solution already incorporates the two arbitrary constants, so any sum of a homogenous solution and a particular solution will represent the {\it general solution} of the equation. The general solution incorporates the two arbitrary constants \(A\) and \(B\) to accommodate the two initial conditions. One could have picked a different particular solution, i.e. the original particular solution plus any homogenous solution with the arbitrary constants \(A_p\) and \(B_p\) chosen at will. When one adds in the homogenous solution, which has adjustable constants with arbitrary constants \(A'\) and \(B'\), to the new particular solution, one can get the same general solution by simply adjusting the new constants such that \(A'+A_p=A\) and \(B'+B_p=B\). Thus, the choice of \(A_p\) and \(B_p\) are irrelevant, and when choosing the particular solution it is best to make the simplest choice possible.
To find a particular solution, one first guesses at the form,
and rewrite the differential equation as
One can now use angle addition formulas to get
Both the \(\cos\) and \(\sin\) terms need to equate if the expression is to hold at all times. Thus, this becomes two equations
After dividing by \(\cos\delta\), the lower expression leads to
Using the identities \(\tan^2+1=\csc^2\) and \(\sin^2+\cos^2=1\), one can also express \(\sin\delta\) and \(\cos\delta\),
Inserting the expressions for \(\cos\delta\) and \(\sin\delta\) into the expression for \(D\),
For a given initial condition, e.g. initial displacement and velocity, one must add the homogenous solution then solve for the two arbitrary constants. However, because the homogenous solutions decay with time as \(e^{-\beta t}\), the particular solution is all that remains at large times, and is therefore the steady state solution. Because the arbitrary constants are all in the homogenous solution, all memory of the initial conditions are lost at large times, \(t>>1/\beta\).
The amplitude of the motion, \(D\), is linearly proportional to the driving force (\(F_0/m\)), but also depends on the driving frequency \(\omega\). For small \(\beta\) the maximum will occur at \(\omega=\omega_0\). This is referred to as a resonance. In the limit \(\beta\rightarrow 0\) the amplitude at resonance approaches infinity.
4.4. Alternative Derivation for Driven Oscillators¶
Here, we derive the same expressions as in Equations (13) and (16) but express the driving forces as
rather than as \(F_0\cos\omega t\). The real part of \(F\) is the same as before. For the differential equation,
one can treat \(x(t)\) as an imaginary function. Because the operations \(d^2/dt^2\) and \(d/dt\) are real and thus do not mix the real and imaginary parts of \(x(t)\), Eq. (17) is effectively 2 equations. Because \(e^{\omega t}=\cos\omega t+i\sin\omega t\), the real part of the solution for \(x(t)\) gives the solution for a driving force \(F_0\cos\omega t\), and the imaginary part of \(x\) corresponds to the case where the driving force is \(F_0\sin\omega t\). It is rather easy to solve for the complex \(x\) in this case, and by taking the real part of the solution, one finds the answer for the \(\cos\omega t\) driving force.
We assume a simple form for the particular solution
where \(D\) is a complex constant.
From Eq. (17) one inserts the form for \(x_p\) above to get
The norm and phase for \(D=|D|e^{-i\delta}\) can be read by inspection,
This is the same expression for \(\delta\) as before. One then finds \(x_p(t)\),
This is the same answer as before. If one wished to solve for the case where \(F(t)= F_0\sin\omega t\), the imaginary part of the solution would work
Consider the damped and driven harmonic oscillator worked out above. Given \(F_0, m,\beta\) and \(\omega_0\), solve for the complete solution \(x(t)\) for the case where \(F=F_0\sin\omega t\) with initial conditions \(x(t=0)=0\) and \(v(t=0)=0\). Assume the underdamped case.
The general solution including the arbitrary constants includes both the homogenous and particular solutions,
The quantities \(\delta\) and \(\omega'\) are given earlier in the section, \(\omega'=\sqrt{\omega_0^2-\beta^2}, \delta=\tan^{-1}(2\beta\omega/(\omega_0^2-\omega^2)\). Here, solving the problem means finding the arbitrary constants \(A\) and \(B\). Satisfying the initial conditions for the initial position and velocity:
The problem is now reduced to 2 equations and 2 unknowns, \(A\) and \(B\). The solution is
4.5. Resonance Widths; the \(Q\) factor¶
From the previous two sections, the particular solution for a driving force, \(F=F_0\cos\omega t\), is
If one fixes the driving frequency \(\omega\) and adjusts the fundamental frequency \(\omega_0=\sqrt{k/m}\), the maximum amplitude occurs when \(\omega_0=\omega\) because that is when the term from the denominator \((\omega_0^2-\omega^2)^2+4\omega^2\beta^2\) is at a minimum. This is akin to dialing into a radio station. However, if one fixes \(\omega_0\) and adjusts the driving frequency one minimize with respect to \(\omega\), e.g. set
and one finds that the maximum amplitude occurs when \(\omega=\sqrt{\omega_0^2-2\beta^2}\). If \(\beta\) is small relative to \(\omega_0\), one can simply state that the maximum amplitude is
For small damping this occurs when \(\omega=\omega_0\pm \beta\), so the \(FWHM\approx 2\beta\). For the purposes of tuning to a specific frequency, one wants the width to be as small as possible. The ratio of \(\omega_0\) to \(FWHM\) is known as the {\it quality} factor, or \(Q\) factor,
4.6. Numerical Studies of Driven Oscillations¶
Solving the problem of driven oscillations numerically gives us much more flexibility to study different types of driving forces. We can reuse our earlier code by simply adding a driving force. If we stay in the \(x\)-direction only this can be easily done by adding a term \(F_{\mathrm{ext}}(x,t)\). Note that we have kept it rather general here, allowing for both a spatial and a temporal dependence.
Before we dive into the code, we need to briefly remind ourselves about the equations we started with for the case with damping, namely
with no external force applied to the system.
Let us now for simplicty assume that our external force is given by
where \(F_0\) is a constant (what is its dimension?) and \(\omega\) is the frequency of the applied external driving force. Small question: would you expect energy to be conserved now?
Introducing the external force into our lovely differential equation and dividing by \(m\) and introducing \(\omega_0^2=\sqrt{k/m}\) we have
Thereafter we introduce a dimensionless time \(\tau = t\omega_0\) and a dimensionless frequency \(\tilde{\omega}=\omega/\omega_0\). We have then
Introducing a new amplitude \(\tilde{F} =F_0/(m\omega_0^2)\) (check dimensionality again) we have
Our final step, as we did in the case of various types of damping, is to define \(\gamma = b/(2m\omega_0)\) and rewrite our equations as
This is the equation we will code below using the Euler-Cromer method.
DeltaT = 0.001
#set up arrays
tfinal = 20 # in years
n = ceil(tfinal/DeltaT)
# set up arrays for t, v, and x
t = np.zeros(n)
v = np.zeros(n)
x = np.zeros(n)
# Initial conditions as one-dimensional arrays of time
x0 = 1.0
v0 = 0.0
x[0] = x0
v[0] = v0
gamma = 0.2
Omegatilde = 0.5
Ftilde = 1.0
# Start integrating using Euler-Cromer's method
for i in range(n-1):
# Set up the acceleration
# Here you could have defined your own function for this
a = -2*gamma*v[i]-x[i]+Ftilde*cos(t[i]*Omegatilde)
# update velocity, time and position
v[i+1] = v[i] + DeltaT*a
x[i+1] = x[i] + DeltaT*v[i+1]
t[i+1] = t[i] + DeltaT
# Plot position as function of time
fig, ax = plt.subplots()
ax.set_ylabel('x[m]')
ax.set_xlabel('t[s]')
ax.plot(t, x)
fig.tight_layout()
save_fig("ForcedBlockEulerCromer")
plt.show()
In the above example we have focused on the Euler-Cromer method. This method has a local truncation error which is proportional to \(\Delta t^2\) and thereby a global error which is proportional to \(\Delta t\). We can improve this by using the Runge-Kutta family of methods. The widely popular Runge-Kutta to fourth order or just RK4 has indeed a much better truncation error. The RK4 method has a global error which is proportional to \(\Delta t\).
Let us revisit this method and see how we can implement it for the above example.
4.7. Differential Equations, Runge-Kutta methods¶
Runge-Kutta (RK) methods are based on Taylor expansion formulae, but yield in general better algorithms for solutions of an ordinary differential equation. The basic philosophy is that it provides an intermediate step in the computation of \(y_{i+1}\).
To see this, consider first the following definitions
and
and
To demonstrate the philosophy behind RK methods, let us consider
the second-order RK method, RK2.
The first approximation consists in Taylor expanding \(f(t,y)\)
around the center of the integration interval \(t_i\) to \(t_{i+1}\),
that is, at \(t_i+h/2\), \(h\) being the step.
Using the midpoint formula for an integral,
defining \(y(t_i+h/2) = y_{i+1/2}\) and
\(t_i+h/2 = t_{i+1/2}\), we obtain
This means in turn that we have
However, we do not know the value of \(y_{i+1/2}\). Here comes thus the next approximation, namely, we use Euler’s method to approximate \(y_{i+1/2}\). We have then
This means that we can define the following algorithm for the second-order Runge-Kutta method, RK2.
6 0
< < < ! ! M A T H _ B L O C K
with the final value
The difference between the previous one-step methods is that we now need an intermediate step in our evaluation, namely \(t_i+h/2 = t_{(i+1/2)}\) where we evaluate the derivative \(f\). This involves more operations, but the gain is a better stability in the solution.
The fourth-order Runge-Kutta, RK4, has the following algorithm
6 3
< < < ! ! M A T H _ B L O C K
with the final result
Thus, the algorithm consists in first calculating \(k_1\) with \(t_i\), \(y_1\) and \(f\) as inputs. Thereafter, we increase the step size by \(h/2\) and calculate \(k_2\), then \(k_3\) and finally \(k_4\). The global error goes as \(O(h^4)\).
However, at this stage, if we keep adding different methods in our main program, the code will quickly become messy and ugly. Before we proceed thus, we will now introduce functions that enbody the various methods for solving differential equations. This means that we can separate out these methods in own functions and files (and later as classes and more generic functions) and simply call them when needed. Similarly, we could easily encapsulate various forces or other quantities of interest in terms of functions. To see this, let us bring up the code we developed above for the simple sliding block, but now only with the simple forward Euler method. We introduce two functions, one for the simple Euler method and one for the force.
Note that here the forward Euler method does not know the specific force function to be called. It receives just an input the name. We can easily change the force by adding another function.
def ForwardEuler(v,x,t,n,Force):
for i in range(n-1):
v[i+1] = v[i] + DeltaT*Force(v[i],x[i],t[i])
x[i+1] = x[i] + DeltaT*v[i]
t[i+1] = t[i] + DeltaT
def SpringForce(v,x,t):
# note here that we have divided by mass and we return the acceleration
return -2*gamma*v-x+Ftilde*cos(t*Omegatilde)
It is easy to add a new method like the Euler-Cromer
def ForwardEulerCromer(v,x,t,n,Force):
for i in range(n-1):
a = Force(v[i],x[i],t[i])
v[i+1] = v[i] + DeltaT*a
x[i+1] = x[i] + DeltaT*v[i+1]
t[i+1] = t[i] + DeltaT
and the Velocity Verlet method (be careful with time-dependence here, it is not an ideal method for non-conservative forces))
def VelocityVerlet(v,x,t,n,Force):
for i in range(n-1):
a = Force(v[i],x[i],t[i])
x[i+1] = x[i] + DeltaT*v[i]+0.5*a
anew = Force(v[i],x[i+1],t[i+1])
v[i+1] = v[i] + 0.5*DeltaT*(a+anew)
t[i+1] = t[i] + DeltaT
Finally, we can now add the Runge-Kutta2 method via a new function
def RK2(v,x,t,n,Force):
for i in range(n-1):
# Setting up k1
k1x = DeltaT*v[i]
k1v = DeltaT*Force(v[i],x[i],t[i])
# Setting up k2
vv = v[i]+k1v*0.5
xx = x[i]+k1x*0.5
k2x = DeltaT*vv
k2v = DeltaT*Force(vv,xx,t[i]+DeltaT*0.5)
# Final result
x[i+1] = x[i]+k2x
v[i+1] = v[i]+k2v
t[i+1] = t[i]+DeltaT
File "<ipython-input-7-ffedbda27704>", line 14
t[i+1] = t[i]+DeltaT
^
TabError: inconsistent use of tabs and spaces in indentation
Finally, we can now add the Runge-Kutta2 method via a new function
def RK4(v,x,t,n,Force):
for i in range(n-1):
# Setting up k1
k1x = DeltaT*v[i]
k1v = DeltaT*Force(v[i],x[i],t[i])
# Setting up k2
vv = v[i]+k1v*0.5
xx = x[i]+k1x*0.5
k2x = DeltaT*vv
k2v = DeltaT*Force(vv,xx,t[i]+DeltaT*0.5)
# Setting up k3
vv = v[i]+k2v*0.5
xx = x[i]+k2x*0.5
k3x = DeltaT*vv
k3v = DeltaT*Force(vv,xx,t[i]+DeltaT*0.5)
# Setting up k4
vv = v[i]+k3v
xx = x[i]+k3x
k4x = DeltaT*vv
k4v = DeltaT*Force(vv,xx,t[i]+DeltaT)
# Final result
x[i+1] = x[i]+(k1x+2*k2x+2*k3x+k4x)/6.
v[i+1] = v[i]+(k1v+2*k2v+2*k3v+k4v)/6.
t[i+1] = t[i] + DeltaT
The Runge-Kutta family of methods are particularly useful when we have a time-dependent acceleration. If we have forces which depend only the spatial degrees of freedom (no velocity and/or time-dependence), then energy conserving methods like the Velocity Verlet or the Euler-Cromer method are preferred. As soon as we introduce an explicit time-dependence and/or add dissipitave forces like friction or air resistance, then methods like the family of Runge-Kutta methods are well suited for this. The code below uses the Runge-Kutta4 methods.
DeltaT = 0.001
#set up arrays
tfinal = 20 # in years
n = ceil(tfinal/DeltaT)
# set up arrays for t, v, and x
t = np.zeros(n)
v = np.zeros(n)
x = np.zeros(n)
# Initial conditions (can change to more than one dim)
x0 = 1.0
v0 = 0.0
x[0] = x0
v[0] = v0
gamma = 0.2
Omegatilde = 0.5
Ftilde = 1.0
# Start integrating using Euler's method
# Note that we define the force function as a SpringForce
RK4(v,x,t,n,SpringForce)
# Plot position as function of time
fig, ax = plt.subplots()
ax.set_ylabel('x[m]')
ax.set_xlabel('t[s]')
ax.plot(t, x)
fig.tight_layout()
save_fig("ForcedBlockRK4")
plt.show()
4.8. Principle of Superposition and Periodic Forces (Fourier Transforms)¶
If one has several driving forces, \(F(t)=\sum_n F_n(t)\), one can find the particular solution to each \(F_n\), \(x_{pn}(t)\), and the particular solution for the entire driving force is
This is known as the principal of superposition. It only applies when the homogenous equation is linear. If there were an anharmonic term such as \(x^3\) in the homogenous equation, then when one summed various solutions, \(x=(\sum_n x_n)^2\), one would get cross terms. Superposition is especially useful when \(F(t)\) can be written as a sum of sinusoidal terms, because the solutions for each sinusoidal (sine or cosine) term is analytic, as we saw above.
Driving forces are often periodic, even when they are not sinusoidal. Periodicity implies that for some time \(\tau\)
One example of a non-sinusoidal periodic force is a square wave. Many components in electric circuits are non-linear, e.g. diodes, which makes many wave forms non-sinusoidal even when the circuits are being driven by purely sinusoidal sources.
The code here shows a typical example of such a square wave generated using the functionality included in the scipy Python package. We have used a period of \(\tau=0.2\).
import numpy as np
import math
from scipy import signal
import matplotlib.pyplot as plt
# number of points
n = 500
# start and final times
t0 = 0.0
tn = 1.0
# Period
t = np.linspace(t0, tn, n, endpoint=False)
SqrSignal = np.zeros(n)
SqrSignal = 1.0+signal.square(2*np.pi*5*t)
plt.plot(t, SqrSignal)
plt.ylim(-0.5, 2.5)
plt.show()
For the sinusoidal example studied in the previous subsections the period is \(\tau=2\pi/\omega\). However, higher harmonics can also satisfy the periodicity requirement. In general, any force that satisfies the periodicity requirement can be expressed as a sum over harmonics,
From the previous subsection, one can write down the answer for \(x_{pn}(t)\), by substituting \(f_n/m\) or \(g_n/m\) for \(F_0/m\) into Eq.s (20) or (21) respectively. By writing each factor \(2n\pi t/\tau\) as \(n\omega t\), with \(\omega\equiv 2\pi/\tau\),
The solutions for \(x(t)\) then come from replacing \(\omega\) with \(n\omega\) for each term in the particular solution in Equations (13) and (16),
Because the forces have been applied for a long time, any non-zero damping eliminates the homogenous parts of the solution, so one need only consider the particular solution for each \(n\).
The problem will considered solved if one can find expressions for the coefficients \(f_n\) and \(g_n\), even though the solutions are expressed as an infinite sum. The coefficients can be extracted from the function \(F(t)\) by
To check the consistency of these expressions and to verify Eq. (37), one can insert the expansion of \(F(t)\) in Eq. (36) into the expression for the coefficients in Eq. (37) and see whether
Immediately, one can throw away all the terms with \(g_m\) because they convolute an even and an odd function. The term with \(f_0/2\) disappears because \(\cos(n\omega t)\) is equally positive and negative over the interval and will integrate to zero. For all the terms \(f_m\cos(m\omega t)\) appearing in the sum, one can use angle addition formulas to see that \(\cos(m\omega t)\cos(n\omega t)=(1/2)(\cos[(m+n)\omega t]+\cos[(m-n)\omega t]\). This will integrate to zero unless \(m=n\). In that case the \(m=n\) term gives
and
The same method can be used to check for the consistency of \(g_n\).
Consider the driving force:
Find the Fourier coefficients \(f_n\) and \(g_n\) for all \(n\) using Eq. (37).
Only the odd coefficients enter by symmetry, i.e. \(f_n=0\). One can find \(g_n\) integrating by parts,
The first term is zero because \(\cos(n\omega t)\) will be equally positive and negative over the interval. Using the fact that \(\omega\tau=2\pi\),
4.9. Fourier Series¶
More text will come here, chpater 5.7-5.8 of Taylor are discussed during the lectures. The code here uses the Fourier series discussed in chapter 5.7 for a square wave signal. The equations for the coefficients are are discussed in Taylor section 5.7, see Example 5.4. The code here visualizes the various approximations given by Fourier series compared with a square wave with period \(T=0.2\), witth \(0.1\) and max value \(F=2\). We see that when we increase the number of components in the Fourier series, the Fourier series approximation gets closes and closes to the square wave signal.
import numpy as np
import math
from scipy import signal
import matplotlib.pyplot as plt
# number of points
n = 500
# start and final times
t0 = 0.0
tn = 1.0
# Period
T =0.2
# Max value of square signal
Fmax= 2.0
# Width of signal
Width = 0.1
t = np.linspace(t0, tn, n, endpoint=False)
SqrSignal = np.zeros(n)
FourierSeriesSignal = np.zeros(n)
SqrSignal = 1.0+signal.square(2*np.pi*5*t+np.pi*Width/T)
a0 = Fmax*Width/T
FourierSeriesSignal = a0
Factor = 2.0*Fmax/np.pi
for i in range(1,500):
FourierSeriesSignal += Factor/(i)*np.sin(np.pi*i*Width/T)*np.cos(i*t*2*np.pi/T)
plt.plot(t, SqrSignal)
plt.plot(t, FourierSeriesSignal)
plt.ylim(-0.5, 2.5)
plt.show()
4.10. Solving differential equations with Fouries series¶
The material here was discussed during the lecture of February 19 and 21. It is also covered by Taylor in section 5.8.
4.11. Response to Transient Force¶
Consider a particle at rest in the bottom of an underdamped harmonic oscillator, that then feels a sudden impulse, or change in momentum, \(I=F\Delta t\) at \(t=0\). This increases the velocity immediately by an amount \(v_0=I/m\) while not changing the position. One can then solve the trajectory by solving Eq. (9) with initial conditions \(v_0=I/m\) and \(x_0=0\). This gives
Here, \(\omega'=\sqrt{\omega_0^2-\beta^2}\). For an impulse \(I_i\) that occurs at time \(t_i\) the trajectory would be
where \(\Theta(t-t_i)\) is a step function, i.e. \(\Theta(x)\) is zero for \(x<0\) and unity for \(x>0\). If there were several impulses linear superposition tells us that we can sum over each contribution,
Now one can consider a series of impulses at times separated by \(\Delta t\), where each impulse is given by \(F_i\Delta t\). The sum above now becomes an integral,
The quantity \(e^{-\beta(t-t')}\sin[\omega'(t-t')]/m\omega'\Theta(t-t')\) is called a Green’s function, \(G(t-t')\). It describes the response at \(t\) due to a force applied at a time \(t'\), and is a function of \(t-t'\). The step function ensures that the response does not occur before the force is applied. One should remember that the form for \(G\) would change if the oscillator were either critically- or over-damped.
When performing the integral in Eq. (44) one can use angle addition formulas to factor out the part with the \(t'\) dependence in the integrand,
If the time \(t\) is beyond any time at which the force acts, \(F(t'>t)=0\), the coefficients \(I_c\) and \(I_s\) become independent of \(t\).
Consider an undamped oscillator (\(\beta\rightarrow 0\)), with characteristic frequency \(\omega_0\) and mass \(m\), that is at rest until it feels a force described by a Gaussian form,
For large times (\(t>>\tau\)), where the force has died off, find \(x(t)\).\ Solve for the coefficients \(I_c\) and \(I_s\) in Eq. (45). Because the Gaussian is an even function, \(I_s=0\), and one need only solve for \(I_c\),
The third step involved completing the square, and the final step used the fact that the integral
To see that this integral is true, consider the square of the integral, which you can change to polar coordinates,
Finally, the expression for \(x\) from Eq. (45) is
4.12. The classical pendulum and scaling the equations¶
Let us end our discussion of oscillations with another classical case, the pendulum.
The angular equation of motion of the pendulum is given by Newton’s equation and with no external force it reads
with an angular velocity and acceleration given by
and
We do however expect that the motion will gradually come to an end due a viscous drag torque acting on the pendulum. In the presence of the drag, the above equation becomes
where \(\nu\) is now a positive constant parameterizing the viscosity of the medium in question. In order to maintain the motion against viscosity, it is necessary to add some external driving force. We choose here a periodic driving force. The last equation becomes then
with \(A\) and \(\omega\) two constants representing the amplitude and the angular frequency respectively. The latter is called the driving frequency.
We define
the so-called natural frequency and the new dimensionless quantities
with the dimensionless driving frequency
and introducing the quantity \(Q\), called the quality factor,
and the dimensionless amplitude
4.13. More on the Pendulum¶
We have
This equation can in turn be recast in terms of two coupled first-order differential equations as follows
and
These are the equations to be solved. The factor \(Q\) represents the number of oscillations of the undriven system that must occur before its energy is significantly reduced due to the viscous drag. The amplitude \(\hat{A}\) is measured in units of the maximum possible gravitational torque while \(\hat{\omega}\) is the angular frequency of the external torque measured in units of the pendulum’s natural frequency.


Boston Housing is a fairly standard dataset used for testing regression problems. It contains 506 data points with 12 numeric attributes, and one binary categorical attribute. The goal is to determine median home values, based on various census attributes. This dataset is available at the UCI Repository.
The demo follows that in [1]. The data set was preprocessed as follows: each continuous feature was transformed to zero mean and unit variance (The categorical variable was dropped). The data was partitioned into points for training and points for testing.
The mean function used was src.Core.means.meanZero() and the covariance (using the src.Core.kernels.covSum() function) was a composite of src.Core.kernels.covSEiso() and src.Core.kernels.covNoise(). The initial values of the hyperparameters were selected randomly from a zero-mean, unit-variance normal distribtion. The actual values were: . The initial likelihood hyperparameter was . The regression started with initial negative log marginal likelihood of :math:` 752.46`. Note the initial zero mean and the variance that is uniform over the test set.
model = pyGPs.GPR()
model.optimize(x,y)
ym, ys2, fm, fs2, lp = model.predict(xs)
xa = np.concatenate((data[:,:4],data[:,5:-1]),axis=1)
xa = (xa - np.mean(xa,axis=0))/(np.std(xa,axis=0)+1.e-16)
ya, ys2a, fma, fs2a, lpa = model.predict(xa)
After hyperparameter optimization, the covariance hyperparameters were and the likelihood hyperparameter was . The final negative log marginal likelihood (optimized) was .
| [1] |
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RawFileMetadataBuilder
public final class RawFileMetadataBuilder
Allows metadata for raw files, which will be written to user’s library, to be built up.
-
Initializes builder with mandatory values
Declaration
Swift
public init(mimeType: MimeType, accounts: [String])
Parameters
mimeType
The MIME type of the data being written
accounts
An array of user accounts associated with data being written
-
Adds references associated with data being written. Typically this would include the name of the file.
Declaration
Swift
public func reference(_ value: [String]) -> RawFileMetadataBuilder
Parameters
value
An array of reference values
Return Value
The updated builder
-
Adds tags describing the data being written
Declaration
Swift
public func tags(_ value: [String]) -> RawFileMetadataBuilder
Parameters
value
An array of tag values
Return Value
The updated builder
-
Adds the identifier of the respective contract which can read the written data
Declaration
Swift
public func contractId(_ value: String) -> RawFileMetadataBuilder
Parameters
value
The contract identifier
Return Value
The updated builder
-
Adds the identifier of the application containing the respective contract which can read the written data
Declaration
Swift
public func appId(_ value: String) -> RawFileMetadataBuilder
Parameters
value
The application identifier
Return Value
The updated builder
-
Adds the name of the provider writing the data
Declaration
Swift
public func providerName(_ value: String) -> RawFileMetadataBuilder
Parameters
value
The provider name
Return Value
The updated builder
-
Adds object types describing data being written
Declaration
Swift
public func objectTypes(_ value: [RawFileMetadata.ObjectType]) -> RawFileMetadataBuilder
Parameters
value
The object types
Return Value
The updated builder
-
Adds identifiers of service groups associated with data being written. See https://developers.digi.me/reference-objects#service-group for list of service groups.
Declaration
Swift
public func serviceGroups(_ value: [Int]) -> RawFileMetadataBuilder
Parameters
value
The service group identifiers
Return Value
The updated builder
-
Builds the metadata for the data file being written
Declaration
Swift
public func build() -> RawFileMetadata
Return Value
The built metadata


This page demonstrates how to use a HTMLDataSet to load the data from a Div container located in this file.
The use HTMLDataSet is similar to that of a XMLDataSet. The data can be loaded from a HTML element located in the same file
or in a different file under the same domain.
Optional parameters used with XMLDataSet to specify sorting, caching, distinct, loadInterval work in a similar way with the HTMLDataSet.
In this sample the HTMLDataSet is loading its data from a DIV container with a nested structure like this:
When using a HTML structure other then a TABLE, the HTMLDataSet needs some extra information in order to extract data from such three-level nested structure. There are two optional parameters that can be used to accomplish this:
Values that can be used for rowSelector or dataSelector:
DIVs are used in this sample because they are flexible and can contain other block level elements. Remember that, for instance, you cannot put other block level elements within a
 tag.
You cannot use firstRowAsHeaders parameter with any other structure than Table.
In order to get the data using other markup then the default one ({column0}, {column1}) you can pass in an array of columNames:["Product", "Description"].
Don't use spaces inside the column names.
|
Brief Description:
{Description}
|
Here's the HTMLDataSet constructor:
var ds1 = new Spry.Data.HTMLDataSet(null, "sourceContainer", {rowSelector:"div.rowData", dataSelector:"span", columnNames:["Product", "Description"]});
Here's what the Source Container DIV looks like:

 Adobe Photoshop CS2 The professional standard in desktop digital imaging

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 Adobe Flash 8 Professional Introducing Macromedia's biggest Flash release to date.

 Adobe Fireworks 8 Fireworks 8 lets you balance maximum image quality with minimum compression size as you create, edit and optimize images for the web with precise control.


Our best diversion is shopping and internet, would start to rule the Industry. A face of capitalism, new age of E-commerce, Age that changed the way business work, the way we buy and sell. A age transform by a technology World Wide Web.
Amazon and eBay, these companies shook the business to its core, if you’re looking for the next big thing in tech. You have to go nowhere but a garage, a lot of startup were form or created from garage such as Apple. Amazon was created in a garage in Seattle by Jeff Bezos. When web came around he started to think how he can use this technology to sell product. He made a list after list, and books were always up top. In the summer of 1995, Bezos launched Amazon. On the other hand, in Silicon Valley another guy name Pierre Omidyar was thinking about web as well, he is the idealist software programmer. He had an interesting idea that will later become eBay.
Pierre and Jeff had one thing in common, they both thought their idea as business not just goofing around. Jeff had a proper plan with strategies for Amazon as compare to Pierre. Pierre just thought of online auction site would be cool and he could manage it from his home by tapping away. He made program by himself where you create an item, list an item, and you can bid on a item.
Amazon and eBay immediately started making money. In the first 30 days of Amazon shifted book across 45 countries and all 50 states. Jeff thought it will be a slow process and he was totally wrong about because people adapted to Amazon extremely fast. It only took few years for 50 million people adapted this technology as compare to other technology that took 10 or more years to gather 50 million users.
Internet gather momentum as it goes, the number of user increase the use of the network increased. Wall street didn’t have any clue of technology but saw the money in it. The sudden boom was evitable.
Internet was always unknown but launched their IPO then later Netscape launched their IPO, after that Search engines companies like Yahoo, Excited, Google also launched their IPO. Moreover, it was time for Jeff Bezos to enter the market to create the perfect bubble hood. as competition threat that need to be put down.
May 1997, Amazon launched their first IPO, company was only 2 years old. According to Henry Blodget, former Wall Street Analyst, thought that Amazon business model was not good but he considered when a lot people talking about Amazon which led him to change his mind but deep down he knew this could be controversial.
Jeff Bezos would lower his prices after growing a little, so he can cash in more customers, He believed that profit should be sacrifice in beginning to build empire for future. Jeff’s motto was “GET BIG FAST”. Amazon advanced when they found Public key cryptography which would solve their problem of keep privacy data client private
eBay was into collectibles and worth about 2 billion dollars.
According to Henry Blodget, even though Amazon is showing loss, it would make money the following year but his prediction were wrong. Amazon stocks doubled in matter of weeks, which led Henry to believe that this is even a bigger bubble.
From now onwards, everyone was investing in Dotcoms, thinking it will make them money, looking at Amazon and eBay growth it inspire many entrepreneurs to move to Silicon Valley and get started working.
The Day-traders made Wall street look like Casino because they are buying share and selling them at the end of the for more profit. 90s began the E-decade. Many companies spending handsomely on ads for Superbowl and the average cost for adverting was 2.2 million. Fred Wilson, Venture Capitalist, believe that they were playing with fire. Some Venture Capitalist thought the E-commerce came with a expiry date and only the big player will survive. Henry knew that the bubble was about to pop but he was still recommending people to buy the stock and he used to put his thoughts on private email, which later resulted in him leaving Wall Street.
In 1999, Ellen GreenSpan a federal banker decided to increase the federal interest rate which led the bubble to pop and it was Ellen intention to do so. NASDAQ fell 25%, wall street thought at the holiday time the E-commerce business would boom but it didn’t happen, and they were disappointed. Wall Street now doubted the statement “GET BIG FAST”.
Internet Black Friday resulted from the disappointments in E-commerce. NASDAQ felt 35% points and its index had fallen 25%, 18 months later Sept 11, 2001 happened. As a result, NASDAQ500 has lost 2/3 of its value. Customer were very upset because they were losing money and wanted to blame Wall Street. Martha Stewart and Henry Blodget were committed with civil security fraud.
5 years later, after the bubble popped, people started to settle in and now web economy was left with two main players Amazon and eBay.


Recently implemented an example of the Clark & Wright savings algorithm (parallel and sequential version) prototype, been looking for some test examples, and just found this one. I have to optimize a whole lot of things, but at least i know that parallel example works.
This example is a graphical representation of the parallel version for a problem instance taken from Golden & Wasil 2005 instances for DVRP:
These are the two mini monsters i created at the night, i expect to wake up to refactorize and optimize this code:
(defun clark-and-wright-sequential (evaluator) "Answer a new individual created using C-W savings heuristic. Sequential version implementation (Route Extension)." ;; Make initial solution (let ((solution (make-initial-cw-savings-solution evaluator))) (while-do ;; Calculate first saving (let ((routes-count (length solution))) (block 1 (dotimes (i routes-count) (dotimes (j routes-count) (if (not (= i j)) (let ((saving (saving-when-merging evaluator solution i j))) (when (and (> saving 0) (is-merge-feasible evaluator solution i j)) ;; Add to feasible (setf solution (merge-routes solution (list i j saving))) (return-from 1 t))))))))) ;; Answer merged solution solution))
(defun clark-and-wright-parallel (evaluator) "Answer a new individual created using C-W savings heuristic. Parallel version implementation (Best Feasible Merge)." ;; Make initial solution (let ((solution (make-initial-cw-savings-solution evaluator))) (while-do (let* ((routes-count (length solution)) (savings (make-array (sqr routes-count) :initial-element nil)) (savings-counter 0)) ;; Calculate savings (dotimes (i routes-count) (dotimes (j routes-count) (if (not (= i j)) (let ((saving (saving-when-merging evaluator solution i j))) (when (and (> saving 0) (is-merge-feasible evaluator solution i j)) (let ((a (first (route-at solution i))) (b (first (last (route-at solution j))))) ;; Add to savings list (setf (aref savings savings-counter) (list a b saving)) (incf savings-counter))))))) ;; Merge routes with best savings when possible (sort savings (lambda (a b) (if (and a b) (> (third a) (third b)) a))) (when (> savings-counter 0) (dotimes (i savings-counter) (multiple-value-bind (result flag) (merge-when-possible evaluator solution (aref savings i)) (when flag (setf solution result savings-counter 0)))) (zerop savings-counter)))) ;; Answer merged solution solution))


Word Sense Disambiguation
Proverb Search Engine (TFIDF)
BA Thesis: Diachronic Word Embeddings to Research Semantic Change


{% extends "base.html" %} {% block content_title %}
Unfortunately, your activation request failed. Either the code used ({{ activation_key }}) was incorrect, or out of date.
Please check and try again, or register again for an account.
{% endblock %}


Grid & Layout System
Introduction
The grid and page layout "system" is a collection of CSS grid-driven helper classes that make layouts work. Here are the main files:
_head-helpers
— basic layout helpers like display, padding, margin, float, pinned, clear and other helpers_head-screen
— responsive screen sizing, and show/hide classes_head-griddick
— CSS grids and flexbox classes_neck-layout
— semantic and contextual classes for general layout and styling
Things to keep in mind
- Don't forget that if a component has
margin-bottom
and the next component hasmargin-top
, the two margins will collapse - In mobile and smaller screens, grids collapse into a single column, unless forced with the
-xs
append. - The layout system makes extensive use of CSS Grids: only modern browsers are supported, and no fallbacks or polyfills have been written.
- For Grids tips, use CSS-Tricks or Learn CSS Grid or Grid by Example.
Keeping Code Semantic
When developing larger projects, try to use semantic contextual names. This means that sidebar
is more readable than _grid-col-2 _padding-top _padding-right
— semantic and contextual class and variable names make your code easier to keep up.
Wrap up more complex layouts into layout classes like _layout-coeur-main-sidebar
, and extend grid classes and helpers from a stylesheet file for this layout; use html styling helpers like _padding
sparingly in production—use those to quickly test out variations, but quickly move onto using @extends
in your semantic classes instead.
Some layout helpers can help you get started. These don't include horizontal padding in case of full-width images or such, but those are easily included.
._container
— container to space out componetns vertically._container-large
— large vertical container._content
— vertical/horizontal content container._content-xs
— vertical/horizontal content container only for xs screens
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._container

._container-large

._content

._content-xs

._width-content-max

._width-content-max-narrow

._width-content-paragraph — approximates the 75ch rule

0123456789-0123456789-0123456789-0123456789-0123456789-0123456789-0123456789-0123456789-0123456789-0123456789-0123456789-0123456789-0123456789-0123456789-0123456789-

._width-content-paragraph-narrow — approximates the 75ch rule

0123456789-0123456789-0123456789-0123456789-0123456789-0123456789-0123456789-0123456789-0123456789-0123456789-0123456789-0123456789-0123456789-0123456789-0123456789-

Breakpoints
The responsive grid is setup with the screen sizes from Bootstrap. The responsive grid helpers are defined in _head-screen
$width-xs-min
to-max
: 0px to 767px$width-sm-min
to-max
: 768px to 991px$width-md-min
to-max
: 992px to 1199px$width-lg-min
to-max
: 1200px to 2100px
CSS Variable Helpers
$width-site-min
: 360px — default absolute min. width$width-magic
: 640px — sometimes sites have a 'magical' breakpoint based on content structure$width-content-max
: 991px - useful for marketing pages$width-content-max-narrow: $width-magic
— useful for shorter paragraph blocks$width-content-paragraph: 85ch
— adheres to the 75 ch per paragraph rule; ch uses the '0' symbol as char width
Responsive Visibility Helpers
These helpers can easily show/hide content depending on screen size.
Hide content at the breakpoint and at, above or below, with display: none
:
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Hides content:
._hidden-mobile
._hidden-tablet
._hidden-tablet-and-down
._hidden-tablet-and-up
._hidden-desktop
._hidden-desktop-and-down
._hidden-desktop-and-up
._hidden-large
._hidden-magic-down
._hidden-magic-up
Show content at the breakpoint and at, above or below, with display: inherit
:
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Show content:
._show-mobile
._show-tablet
._show-tablet-and-down
._show-tablet-and-up
._show-desktop
._show-desktop-and-down
._show-desktop-and-up
._show-large
._show-magic-down
._show-magic-up
Mixin Helpers
The basic mixins that allow for responsive conditions.
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@include screen-xs-only {
// affect xs screens
}
// @include screen-xs does the same as above
@include screen-magic-down {
// affect screens under magic breakpoint
}
@include screen-magic-up {
// affect screens above magic breakpoint
}
@include screen-sm {
// affect screens at sm and bigger
}
@include screen-sm-down {
// affect screens sm and smaller
}
@include screen-sm-only {
// only affect screens sm
}
@include screen-md {
// affect screens at md and bigger
}
@include screen-md-down {
// affect screens md and smaller
}
@include screen-md-only {
// only affect screens md
}
@include screen-lg {
// affect screens at lg
}
Responsive Grid
These are just some examples of grid layouts. Please refer to _head-griddick
for the rest, and feel free to add any missing layouts.
Equal column grid system
Flexible column templates
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Equal column grid system

_grid-cols-1

_grid-cols-2

_grid-cols-2

_grid-cols-2-xs

_grid-cols-2-xs

_grid-cols-3

_grid-cols-3

_grid-cols-3

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Flexible column templates

_grid-1-2

_grid-2-1

_grid-1-3

_grid-5-3-xs

_grid-1-2-2

_grid-1-2-1

_grid-2-2-1

_grid-2-1-1

_grid-1-6-1

_grid-3-3-1

_grid-1-3-3

Grid Patterns
Grid patterns are popular shortcut layouts. These could be misused and make your site super-generic, so use with caution. Refer to _head-griddick#patterns
for more information
grid-row-start
is used to reflow content in mobile view so that images will always appear above the content.
Image & Content
Side by side content and image, ideal for alternating on marketing pages. Image reflows to display above content on xs
screens.
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Some unbearable content here

Some unbearable content here

Grid Helpers
Grid helpers like help simplify CSS Grids implementation and expose grid functionality to HTML. They can be kind of confusing, so here are a few illustrative examples.
Refer to the bottom of _head-griddick.scss#helpers
for all the helpers
. For example, _align-center
exists to vertically align elements on a row. Again, please check the _griddick.scss
file for the various helper options.
Alignment
Note that horizontal and vertical alignment are completely independent
._align-right
._align-horizontally
._align-left
Overflow
— grids can't be smaller than the size of its contents, and sometimes may break a grid if used for layout. This forces overflow to behave. auto-fit here
Repeaters
— uses auto-fit so the grid blocks will take up all the space as to not leave white-space, when only one row auto-fit here


| Active | Name | Address | Created | Online | Boxes | Action |
|---|---|---|---|---|---|---|
|
|
{{ location.location_name }} | {{ location.location_address }} | {{ location.created_at | humanTime }} | {{ location.clients_online }} | {{ location.boxes_count }} |
{{ ::item.name }} |


iQOO Pro 5G cuts its price to 3398 yuan ($483) - now the cheapest 5G phone - Gizchina.com
by Efe UdinRate this post
Since 5G smartphones hit the Chinese market, the price of 5G mobile phones has been going down at an alarming rate. This is actually a very good thing for consumers. Yesterday, the three major Chinese carriers officially announced the 5G packages for China. If you are ready for 5G, in addition to a 5G card or a 5G package, you will need a 5G phone. Many may want to ask, “which is the cheapest 5G smartphone?”. Well, it is currently the iQOO Pro 5G which is presently selling for 3398 yuan ($483).
This device came with a 3,798 yuan ($539) price tag and it was the cheapest at the time of its release. However, the Xiaomi Mi 9 Pro 5G beat it a few weeks later with its 3699 yuan price tag. Now, iQOO decided to activate a 400 yuan price cut which makes this device the cheapest 5G smartphone. This deal is available on vivo official flagship store and Tmall.
The iQOO Pro 5G is available in 8GB+12GB, 12GB+128GB and 8GB+256GB configurations. There is a straight 100 yuan ($14) price cut as well as a 300 yuan ($42) coupon. This takes the final price to 3398 yuan ($483) and 3698 yuan ($525) for the lower variants. However, the price of the high 8GB+256GB version remains at 3998 yuan ($568). It is worth mentioning that the first top 2,000 buyers will get a Bluetooth headset worth 359 yuan ($51).


Synaps.Pics is a image hosting and manipulation service. ngx-synaps-pics presents "synapsPics" directive for Angular apps to easily embed images to your applications with Synaps.pics
| command | parameter | example | Description |
|---|---|---|---|
| width (w) | (integer) | sets width (ex: w_500) | |
| height (h) | (integer) | sets height (ex: h_400) | |
| crop (c) | fit (default) | The image is resized so that it takes up as much space as possible within a bounding box defined by the given width and height parameters. The original aspect ratio is retained and all of the original image is visible. | |
| scale | Change the size of the image exactly to the given width and height without necessarily retaining the original aspect ratio: all original image parts are visible but might be stretched or shrunk. | ||
| fill | Same as the 'fit' mode but only if the original image is larger than the given limit (width and height), in which case the image is scaled down so that it takes up as much space as possible within a bounding box defined by the given width and height parameters. The original aspect ratio is retained and all of the original image is visible. | ||
| pad | Resize the image to fill the given width and height while retaining the original aspect ratio. If the proportions of the original image do not match the given width and height, padding is added to the image to reach the required size. | ||
| limit | Same as the 'fit' mode but only if the original image is larger than the given limit (width and height). | ||
| quality (q) | 80 (default) | Output quality. 1 to 100. | |
| gravity (g) | center (default) | The center of the image. | |
| north_west or northwest | North west corner (top left). | ||
| north_east or northeast | North east corner (top right). | ||
| south_east or southeast | South east corner (bottom right). | ||
| south_west or southwest | South west corner (bottom left). | ||
| north | North center part (top center). | ||
| west | Middle west part (left). | ||
| east | Middle east part (right). | ||
| south | Middle south part (bottom center). | ||
| auto | Auto (Smart crop) | ||
| background (b) | rgb:ffffff (default) | Fill background color for pad crop style. |


Provide our customers with fresh, innovative and delicious dishes made from locally sourced, all-natural ingredients. We strive to create a unique and memorable experience for each client through custom menu design, attention to detail, and our true love for what we do.
Pledge to source as much of our product as possible locally, we hope to contribute to a sustainable economy both in our home community and those we purchase from. Additionally, we hope that the relationships we build with our customers and our farmers creates a bridge that can stimulate overall local growth and collaboration.
Our custom-built menus are made with only the freshest and highest quality ingredients available. We let the seasons determine what we cook, taking advantage of all that New England’s farms have to offer. Whole grains, natural and/or organic meats & dairy, with creative from-scratch sauces round out our delectable dishes. We are always more than happy to accommodate any dietary restrictions, allergies or food preferences.
Much of the food people eat today is loaded with chemicals and preservatives. We provide a tasty, all-natural product that we believe will inspire our customers to become more aware of what they eat. We are committed to putting more wholesome meals on your table, so you feel healthy and energized for your events.
Ricotta Tortellini, Baby Spinach, Fire Roasted Tomatoes, Parmesan
Arugula, Pears, Candied Pecans, Buttermilk Blue Cheese, Sherry Vinaigrette
Great Hill Blue Cheese, Fresh Basil, Sherry Vinaigrette
Mint Chimichurri, Rum Gelée, Poblano Polenta
Diced Tomato, Red Chili, Cucumber Puree (Seafood Option)
Baby Spinach, Cantaloupe, Strawberry, Blueberry, Shaved Almonds, Raspberry Dressing
Fresh Whitefish, Seasoned Shellfish, Citrus and Cilantro Marinade
Marinated in Balsamic & Honey, Topped with Mango Kiwi Relish, Quinoa Salad
Butternut Squash, Whipped Sriracha Cream, Ginger, Shallots
Mesclun Greens, Roasted Beets, Gorgonzola, Pickled Shallots, Candied Walnuts, Grape Vinaigrette
Maple and Lemon Glaze, Fresh Herbs, Seasoned Cayenne
Pan Roasted Statler, Cranberry Orange Relish, Lime Cilantro Cream, Wild Rice Pilaf
Black Beans, Spicy Chorizo, Yellow Corn, Bell Peppers
Baby Portobello, Gorgonzola Crumble, Thyme, Shallots
Iceberg Lettuce, Great Hill Blue Cheese, Slab Bacon, Diced Heirloom Tomato
Pomegranate Reduction, Oyster Mushrooms, Fingerling Potatoes, Braised Kale
Lorem ipsum dolor sit amet, consectetur adipisicing elit, sed do eiusmod tempor incididunt ut labore et dolore magna aliqua. Ut enim ad minim veniam, quis nostrud exercitation ullamco laboris nisi ut aliquip ex ea commodo consequat. Duis aute irure dolor in reprehenderit in voluptate velit esse cillum dolore eu fugiat nulla pariatur. Excepteur sint occaecat cupidatat non proident, sunt in culpa qui officia deserunt mollit anim id est laborum.


edit
An alias to a `run-mailcap`'s action edit.
Originally `run-mailcap` is used to process/edit mime-type/file.
More information: home page
- Edit action can be used to view any file on default mailcap explorer:
edit filename
- With `run-mailcap`:
run-mailcap --action=edit filename


procedure Draw(DstX, DstY: Integer; Src: TCustomBitmap32); overload;
procedure Draw(DstX, DstY: Integer; const SrcRect: TRect; Src: TCustomBitmap32); overload;
procedure Draw(const DstRect, SrcRect: TRect; Src: TCustomBitmap32); overload;
Renders the image specified by Src parameter (or part of it specified by SrcRect parameter) at the location given by the coordinates (DstX, DstY) or the DstRect rectangle.
The method provides both: block transfer (versions with DstX, DstY parameters) and stretching (versions with DstRect parameter).
The method uses Src.DrawMode do determine how it should be blended with the background, and if stretching, Src.StretchFilter specifies how the image should be stretched.
The Dst parameter must not be necessarily some other bitmap. In fact, it is possible to copy/stretch areas inside the same bitmap that calls the Draw method. However, in this case, if source and destination areas intersect, the result is not specified (this is a limitation of the current version).


Event Name is required!
Favourtie Count is required!
Description is required!
Rules is required!
name in native language is required!
Description in native language is required!
Rules in native language is required!


The Resident Foodies
I'm a jazz drummer and aspiring game developer. My passions include music and gaming, my hobbies include writing, working out, and snowboarding (when I go home to Chicago for the winter).
I grew up cooking with my dad in Chicago, who is still one of the best chefs I know. While I wasn't hugely into cooking, I learned a lot of basic recipes for surviving on my own that I use to keep my belly full. Over time I've developed more of an appreciation for cooking and I enjoy making protein based meals for my post-workout.
As a Seattle native, I love hiking, camping and traveling. I also spend my time as a permanent makeup artist or creating ice cream recipes.
Over the past few years my relationship with food has drastically changed due to developing over 200+ food allergies. Since then, I’ve tirelessly been on the hunt to find or develop foods that are safe for not only myself, but others who suffer from similar dietary restrictions.
My favorite thing to do is listen to classical music while meditating and working out.
Though I was born and raised in Buffalo, NY I'm a recent LA transplant who's VERY excited to have a year without scraping ice off car windows. In my spare time I love gaming, textile arts, watching horror movies, and spending time with my two rabbits (Boba and Miso) and my partner, Micah.
I spent so much of my childhood in the kitchen with my Oma which imparted a strong connection of food to family, culture, and history. Our family cookbook goes back over 4 generations!
It's been fun and challenging to take these recipes and make them my own with a vegan twist.


---
layout: portal
category: reject
tags: reject
id: 508
imgurl: https://lh5.ggpht.com/aW-1hFUipJ1IpX4RnBvqTMCfikvzhc6RiHUvEswhRWkNXJuj-RZ-Xi4Z8PBYeprlABy5FyxY2XXxQe0FqMj-1g
portalname: 杉山神社の狛犬
sentdate: Fri Aug 07 15:43:19 JST 2015
title: 508 - 杉山神社の狛犬
location: "35.611639,139.486210"
---
From: Ingress Operations<EMAIL_ADDRESS>Date: 2015-08-07 11:43 GMT+09:00
Subject: Portal review complete: 杉山神社の狛犬
To: **


Perfect-Redis
Redis client support for Perfect
Get a redis client with defaults:
RedisClient.getClient(withIdentifier: RedisClientIdentifier()) {
c in
do {
let client = try c()
...
} catch {
...
}
}
Ping the server:
client.ping {
response in
defer {
RedisClient.releaseClient(client)
}
guard case .simpleString(let s) = response else {
...
return
}
XCTAssert(s == "PONG", "Unexpected response \(response)")
}
Set/get a value:
let (key, value) = ("mykey", "myvalue")
client.set(key: key, value: .string(value)) {
response in
guard case .simpleString(let s) = response else {
...
return
}
client.get(key: key) {
response in
defer {
RedisClient.releaseClient(client)
}
guard case .bulkString = response else {
...
return
}
let s = response.toString()
XCTAssert(s == value, "Unexpected response \(response)")
}
}
Pub/sub:
RedisClient.getClient(withIdentifier: RedisClientIdentifier()) {
c in
do {
let client1 = try c()
RedisClient.getClient(withIdentifier: RedisClientIdentifier()) {
c in
do {
let client2 = try c()
client1.subscribe(channels: ["foo"]) {
response in
client2.publish(channel: "foo", message: .string("Hello!")) {
response in
client1.readPublished(timeoutSeconds: 5.0) {
response in
guard case .array(let array) = response else {
...
return
}
XCTAssert(array.count == 3, "Invalid array elements")
XCTAssert(array[0].toString() == "message")
XCTAssert(array[1].toString() == "foo")
XCTAssert(array[2].toString() == "Hello!")
}
}
}
} catch {
...
}
}
} catch {
...
}
}
Building
Add this project as a dependency in your Package.swift file.
.Package(url: "https://github.com/PerfectlySoft/Perfect-Redis.git", Version(0,0,0)..<Version(10,0,0))
Repository Layout
We have finished refactoring Perfect to support Swift Package Manager. The Perfect project has been split up into the following repositories:
- Perfect - This repository contains the core PerfectLib and will continue to be the main landing point for the project.
- PerfectTemplate - A simple starter project which compiles with SPM into a stand-alone executable HTTP server. This repository is ideal for starting on your own Perfect based project.
- PerfectDocs - Contains all API reference related material.
- PerfectExamples - All the Perfect example projects and documentation.
- PerfectEverything - This umbrella repository allows one to pull in all the related Perfect modules in one go, including the servers, examples, database connectors and documentation. This is a great place to start for people wishing to get up to speed with Perfect.
- PerfectServer - Contains the PerfectServer variants, including the stand-alone HTTP and FastCGI servers. Those wishing to do a manual deployment should clone and build from this repository.
- Perfect-Redis - Redis database connector.
- Perfect-SQLite - SQLite3 database connector.
- Perfect-PostgreSQL - PostgreSQL database connector.
- Perfect-MySQL - MySQL database connector.
- Perfect-MongoDB - MongoDB database connector.
- Perfect-FastCGI-Apache2.4 - Apache 2.4 FastCGI module; required for the Perfect FastCGI server variant.
The database connectors are all stand-alone and can be used outside of the Perfect framework and server.
Further Information
For more information on the Perfect project, please visit perfect.org.


맞춤 동영상
바드Some quick example text to build on the card title and make up the bulk of the card's content. |
룰루Some quick example text to build on the card title and make up the bulk of the card's content. |
코그모Some quick example text to build on the card title and make up the bulk of the card's content. |
유미Some quick example text to build on the card title and make up the bulk of the card's content. |
|---|---|---|---|
벨코즈Some quick example text to build on the card title and make up the bulk of the card's content. |
니코Some quick example text to build on the card title and make up the bulk of the card's content. |
에코Some quick example text to build on the card title and make up the bulk of the card's content. |
우르곳Some quick example text to build on the card title and make up the bulk of the card's content. |
더보기
구독 중인 아티스트의 재생목록
애쉬Some quick example text to build on the card title and make up the bulk of the card's content. |
이즈리얼Some quick example text to build on the card title and make up the bulk of the card's content. |
|---|


gurg kjgofgjiigif hefh
hfjbjtjg iojrngnrjhgrbr kxs
jcsjcahfe hfhghr yrhghghhghue
whfgurghg oihghjhhrf chfokgj htjf;kdjfkkjkklmk kjfkjbkfjbdf
vbnfhnbfjb nchbfhfjhhu;j;h jfjhrggj;fe jjhgfjgnhfg jhrfhgrjgb jgrhgnjujgjih knkjiyjjk kg.jfhfhfgnjnfhhf jngnhbjhmljmgkd mrj'jgtjbjfn kfj
rcf is one one big family the love of God and the joy of the lord keeps us living as oneisaactemitope
benin sophia idahosa isaac temitope
isaac temitope sophia idahosa
i am doing i love jesus very much isaac temitope israel


FRESHKOTE LUBRICANT EYE DROPS- povidone k29/32 and polyvinyl alcohol solution/ drops
FOCUS Laboratories, Inc.
Disclaimer: Most OTC drugs are not reviewed and approved by FDA, however they may be marketed if they comply with applicable regulations and policies. FDA has not evaluated whether this product complies.
Active ingredients......................Purpose
Polyvinyl pyrrolidone2.0%..............Lubricant
Polyvinyl alcohol 2.7%....................Lubricant
For external use only.
Do not use if:
When using this product:
Stop use and ask a doctor if:
Keep out of reach of children.
If swallowed, get medical help or contact a Poison Control Center right away.
Boric acid, Disodium edetate dihydrate, Ethanol, Glycerin, Lecithin, Polixetonium, Polysorbate-80, Potassium chloride, Purified water, Sodium chloride
|
FRESHKOTE LUBRICANT EYE DROPS
povidone/polyvinyl alcohol solution/ drops | ||||||||||||||||||||||
| ||||||||||||||||||||||
| ||||||||||||||||||||||
| ||||||||||||||||||||||
| ||||||||||||||||||||||
|
| Labeler - FOCUS Laboratories, Inc. (926378337) |
| Registrant - FOCUS Laboratories, Inc. (926378337) |
| Establishment | |||
| Name | Address | ID/FEI | Business Operations |
|---|---|---|---|
| Altaire Pharmaceutical | 786790378 | analysis(15821-101), label(15821-101), manufacture(15821-101), pack(15821-101), sterilize(15821-101) |


NEWS.md
testthat.R
to GitHub actions.use_stopword()
, as this introduces too many higher-order dependencies via usethis.ancient
source, and moves older ancient stopwords lists to a new perseus
source.use_stopwords()
, for adding a re-export of this function to any package.Changes:
stopwords()
.quanteda::stopwords()
to provide a replacement for that function.


About OneMinuteKitchen
A Series of Videos
Basic Steps of cooking
How to cook a meal within approx. 1 minute
Overlayed kinetic typography, music
InfoGraphics
Photographs
play
pause
Latest Featured Episode
Greek Style Desert
Episodes
Food Events of this Month
May is Barbeque / Hamburger Month
Roast leg of lamb day, 7th May 2015
Eat anything you want, 11th May 2015
Nutty Fudge day , 12th may 2015


FRET fluctuation spectroscopy. Exploring the conformational dynamics of a DNA hairpin-loop.
The motions of a dye-labeled DNA hairpin loop (Cy5-5′-GGGTT-(A)30-AACCC-3′-TMR) have been investigated through the fluctuations in proximity ratio from fluorescence resonance energy transfer (FRET). We examine three solution conditions: (1) MilliQ water, (2) Tris-EDTA buffer, and (3) Tris-EDTA buffer plus an excess of DNA complementary to the loop sequence, (T)30. Correlations in proximity ratio show submillisecond dynamics. Static heterogeneity is revealed from the distribution of proximity ratio amplitudes. The observed stretched exponential kinetics are consistent with a model based on the transition between two states over a complex energy landscape.


✺ Back
Light up mask
This is a project I have designed for a STEAM workshop on electricity and paper citcuit at Fab Lab O.
Back


Moto Razr is back officially in a foldable avatar
by Ivan MehtaMotorola launched its much-awaited foldable phone Razr today at an event in Los Angles. The phone costs a whopping $1,500 and it’s exclusive to Verizon in the US.
The clamshell phone has a 6.2-inch display that folds in. On the top of the display, there a notch hosting a 5-megapixel camera, but it probably won’t bother you. There’s a 2.7-inch outer display as well to display notifications.
While the phone’s design is drool-worthy, it’s spec-sheet might remind you of a mid-range phone from the last year.
Specifications
- Processor: Qualcomm Snapdragon 710
- Primary screen: 6.2-inch foldable pOLED
- Resolution: 2142 x 876 with 21:9 aspect ratio
- Secondary screen: 2.7-inch gOLED with 800 x 600 resolution
- RAM: 6GB
- Rear camera: 16-megapixel sensor with f/1.7 aperture
- Front camera: 5-megapixel sensor with f/2.0 aperture
- Battery: 2,510 mAh
- Internal storage: 128GB
The company has opted for somewhat different, and quite frankly refreshing design choices for the new Moto Razr. The phone has a physical fingerprint sensor on the chin. The device’s speakers are also located there.
There’s only one rear camera, a rarity in today’s age when four or five cameras on the back of a phone are normal to see. What’s more, the main camera can be used to take selfies when the phone’s folded. There’s no SIM slot on the phone, so it’ll only operate through an eSIM.
We’ve talked about the dearth of small and pocketable phones before. Moto’s new foldable can be an answer to that. However, at this moment, it’s a lucrative, but costly an unproven answer.
There’s no doubt a lot of Razr fans might be clamoring to get their hands on the device, and I’m one of them. The new Moto Razr will be available from December 26.
For more gear, gadget, and hardware news and reviews, follow Plugged on Twitter and Flipboard.


Epilogue
First four words smallcaps, and the rest of your first paragraph.
Second, regular paragraph indents, no whitespace.
New paragraph with whitespace above, no indent.
Regular paragraph again, with indent.


What challenge did you enjoy the most?
Everything now and yet to come
JavaScript Drum kit
Please Hold down your Shift key
Ball clock
Everything in between should also be set to checked
CSS variables
Webcam fun, coming soon
Good Luck!
Don't forget to tweet your result!
30 days Vanilla Javascript code challenge


IBM
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