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Fading absorption in non-linear elliptic equations | EMS Press
Fading absorption in non-linear elliptic equations
Department of Mathematics, Technion Haifa, Israel
Institute of Appl. Math. and Mech., NAS of Ukraine, Donetsk, Ukraine
We study the equation
−\mathrm{\Delta }u + h(x)|u|^{q−1}u = 0
q > 1
\mathbb{R}_{ + }^{N} = \mathbb{R}^{N−1} \times \mathbb{R}_{ + }
where h \in C(\mathbb{R}_{ + }^{N}\limits^{¯}),
h⩾0
(x_{1},…,x_{N})
be a coordinate system such that
\mathbb{R}_{ + }^{N} = [x_{N} > 0]
and denote a point
x \in \mathbb{R}^{N}
(x^{′},x_{N})
h(x^{′},x_{N}) > 0
x^{′} \neq 0
h(x^{′},x_{N})\rightarrow 0
|x^{′}|\rightarrow 0
. For this class of equations we obtain sharp necessary and sufficient conditions in order that singularities on the boundary do not propagate in the interior.
Moshe Marcus, Andrey Shishkov, Fading absorption in non-linear elliptic equations. Ann. Inst. H. Poincaré Anal. Non Linéaire 30 (2013), no. 2, pp. 315–336
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Scattering theory of the Hodge–Laplacian under a conformal perturbation | EMS Press
Scattering theory of the Hodge–Laplacian under a conformal perturbation
g
\tilde{g}
be Riemannian metrics on a noncompact manifold
M
, which are conformally equivalent. We show that under a very mild first order control on the conformal factor, the wave operators corresponding to the Hodge–Laplacians
\Delta_g
\Delta_{\tilde{g}}
acting on differential forms exist and are complete. We apply this result to Riemannian manifolds with bounded geometry and more specically, to warped product Riemannian manifolds with bounded geometry. Finally, we combine our results with some explicit calculations by Antoci to determine the absolutely continuous spectrum of the Hodge–Laplacian on
j
-forms for a large class of warped product metrics.
Francesco Bei, Batu Güneysu, Jörn Müller, Scattering theory of the Hodge–Laplacian under a conformal perturbation. J. Spectr. Theory 7 (2017), no. 1, pp. 235–267
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Wires and Fuses | Building DC Energy Systems
# Wires and Fuses
For any current carrying conductor, resistance (
R
) of the conductor is given by
R = \frac{\rho \cdot L}{A}
Hence the efficiency of current carrying conductor depends mainly on length (
L
), cross-sectional area (
A
) and specific resistance (
\rho
) of the material. In addition, parameters like ambient temperature and installation conditions play an important role in maximum current carrying capacity (MCCC) of the wire. MCCC ensures that neither the metal (wire) nor the insulation around it reaches its melting point.
# Cross section area
For higher cross-sectional areas of the conductor, more area will be there for the electrons to flow and hence lesser resistance offered to the current flow. So its current carrying capacity and efficiency will be more. Also, less heat will be dissipated in case of higher cross-sectional area. On the downside, more material means higher costs.
The higher the length of the conductor, the greater is the effective resistance offered by the path and hence more power will be dissipated. This limits the MCCC of the conductor and reduces its efficiency as well.
# Specific resistance
Specific resistance is the property of the material used for the wire. Different materials like copper, aluminium, silver and others exhibit different value of specific resistance. Materials like silver with low specific resistance are more conductive in nature, but are more expensive as well. Hence wire of suitable material should be used according to the requirement. The following table gives the value of specific resistance for a few materials.
Specific Resistance ( Ω mm²/m)
Copper, annealed 0.0172
Table 1. Specific resistances for different materials.
# Ambient temperature
Higher ambient temperatures require less heat (generated by current flowing through wire) for the insulation to melt. Hence ambient temperature has to be kept low to achieve higher current carrying capacity of the wire. The deviation of current carrying capacity (from 30°C) depending on ambient temperature is given in the below table, which is according to the VDE 0298 norm.
Deviating Factor
Table 2. MCCC deviation factors for different temperatures.
# Installation conditions
Enclosing the wires in duct, trays or conduit will decrease the current carrying capacity of the wire, since heat builds up in such enclosed system. This issue can be relaxed by providing proper ventilation or cooling system. In contrast, installing wires in open air allows for higher current carrying capacity of the wire. When installing in open air, UV and moisture resistance has to be considered for the insulation material.
The interactive graph given below, helps us to understand how parameters like specific resistance (
\rho
L
) and cross-sectional area of the conductor (
A
) affects efficiency and voltage drop across it.
Figure 1. Efficiency variations of electrical wires (interactive).
The voltage drop across the wire has to be considered when the charge controller is configured to maintain the proper voltage on the battery to charge/discharge.
Figure 2. Voltage drop across cables with different cross section.
The distance between the panels and the charge-controller shall be
D=5m
, which means a total length (two directions) of
l_{wire} = 10m
. For a copper wire with a diameter of
d=1.5mm
, the total resistance of the wire (for an ambient temperature of 30°C is
R=\frac{0.0172 \cdot 10}{1.5} \Omega = 0.114 \Omega
If we assume a maximum current of
I_{max} = 10A
like charge controller MPPT 1210 HUS (opens new window) can handle, the maximum power dissipation in the wire is
P_{wire} = I_{max}^2 * R = 100A^2 * 0.114 \Omega = 11.4W
In most cases, this approach is rather used to determine the minimal cross section for the wires. If a power loss of 2% of the peak power is tolerated and the maximum expected temperature in the wire is assumed to be 50°C, the cross section can be determined using the above formulas. Let the peak power be
P_{peak}=120W
. The specific resistance at 50°C is
\rho_{50} = 0.0185\frac{\Omega mm^2}{m}
P_{wire} = 2\%\cdot P_{peak} = 2.4W
R_{wire} = \frac{P_{wire}}{I_{max}^2} = \frac{2.4W}{100A} = 0.024\Omega
A_{wire} = \frac{\rho \cdot l_{wire}}{R_{wire}} = 7.7mm^2
So for this example, a wire with a cross section of
A=7.7mm^2
For short connections below 5 meters, you can use:
current / 3 = cable cross section area
Table 2 lists common cross section for some situations. Note that the length is given as the total of both positive and negative lines.
Cross section in [mm²] max. current [A] max. current [A]
Table 2. Maximum currents for different cable length and sizes.
# Fuses
A Fuse is an important component in electrical systems, used to prevent excessive flow of current which can lead to damage of electrical components or even accidents like fire. Proper care must be taken in deciding the placement of the fuse to prevent any damage. Generally it is always advisable to place the fuse as close as possible to the energy source. If the fuse is placed away from the energy source as shown in Fig.2a, the short-circuit which occurs at point X would go undetected, resulting in excessive current. Whereas in the case of Fig.2b, this problem can be avoided.
Figure 2. Position of fuse with respect to energy source.
In case of branches where multiple lines are drawn from the main wire as shown in the Fig.3, a fuse has to be placed on each individual branch lines. Failure in doing so might result in excessive current in individual lines of the branch beyond their limit even when total current in the main line is within the limit. i.e, Fuse
F
might have higher current rating than fuse
F_1
. In this case, even though when fuse
F
doesn't trip and current is within the safe limit of the main line, fuse
F_1
might trip due to excessive current in branch 1 beyond the limit of branch 1 line. Absence of
F_1
in this case leads to damage of the systems.
Figure 3. Position of fuse in branches.
The ground defines a common voltage level as a reference point, to which all other voltages are measured. This can be done by connecting all negative or positive sides to the same line, which can be a rod driven into the earth or the chassis of a car. To prevent accidents by faulty wires or electrical machines, it is common to connect non-electrical but conductive parts (e.g. machine housing) to ground as well. In case of a fault and a short circuit, current flows through the housing and into the earth rather than the body and through the legs into earth.
# Positive vs. negative grounding
Which side of the circuit should be connected depends on the parts used. Most common is negative grounding, so that all voltages in the circuit are positive. Some components or combinations require negative voltages, so a positive grounding is required. Another reason to use positive grounding is electro-chemical corrosion. Corrosive effects are mainly driven by potential differences between metals or minerals, mostly in the presence of moisture. For wires buried in the ground or circuits in moist conditions, it could be beneficial to "pull" the potentials down. See also the chapter load switching for more on this topic.
# RCD in AC networks
To protect from currents leaking into earth, a residual current device should be installed. It measures ingoing and outgoing currents in the phase and neutral lines and triggers whenever a difference is detected. The difference means that there is another way, mostly earth, for the current to flow than the designed one, indicating a potential hazard. When including an inverter in the system, the RCD has to be placed after the inverter along with additional switches. Refer to this manual (opens new window) for further explanations.
← Battery Management System Grid Architecture →
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RuneScape:Grand Exchange Market Watch/Adjustments (23 July 2013) - The RuneScape Wiki
RuneScape:Grand Exchange Market Watch/Adjustments (23 July 2013)
{\displaystyle {div}_{\text{old}}=15.1547}
{\displaystyle {div}_{\text{new}}={div}_{\text{old}}\times {\frac {\sum \left({\frac {p}{q}}\right)_{\text{new}}}{\sum \left({\frac {p}{q}}\right)_{\text{old}}}}}
{\displaystyle {\begin{aligned}\sum \left({\frac {p}{q}}\right)_{\text{old}}&={\text{sum of ratios prior to change}}\\&={\frac {2,147,483,557}{688,800,000}}+{\frac {2,147,483,647}{340,100,000}}+{\frac {1,845,813,270}{114,800,000}}+\dots +{\frac {26,024,637}{2,295,576}}\\&=260.57227963{\text{ (up to 8 d.p.)}}\end{aligned}}}
{\displaystyle {\begin{aligned}\sum \left({\frac {p}{q}}\right)_{\text{new}}&={\text{sum of ratios prior to change}}-{\text{sum of removed ratios}}+{\text{sum of added ratios}}\\&=\sum \left({\frac {p}{q}}\right)_{\text{old}}-{\text{sum of removed ratios}}+{\text{number of added items}}\\&=260.57227963-0+1\\&=261.57227963{\text{ (up to 8 d.p.)}}\end{aligned}}}
{\displaystyle {\begin{aligned}{div}_{\text{new}}&={div}_{\text{old}}\times {\frac {\sum \left({\frac {p}{q}}\right)_{\text{new}}}{\sum \left({\frac {p}{q}}\right)_{\text{old}}}}\\&=15.1547\times {\frac {261.57227963}{260.57227963}}\\&=15.2129{\text{ (4 d.p.)}}\end{aligned}}}
Retrieved from ‘https://runescape.wiki/w/RuneScape:Grand_Exchange_Market_Watch/Adjustments_(23_July_2013)?oldid=35229558’
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Frequency offset estimation using cyclic prefix - MATLAB lteFrequencyOffset - MathWorks Italia
lteFrequencyOffset
Estimate Frequency Offset
cfgul
Frequency offset estimation using cyclic prefix
foffset = lteFrequencyOffset(cfgdl,waveform)
foffset = lteFrequencyOffset(cfgul,waveform)
[foffset, corr] = lteFrequencyOffset(___)
[foffset, corr] = lteFrequencyOffset(___,toffset)
foffset = lteFrequencyOffset(cfgdl,waveform) estimates the average frequency offset, foffset, of the time-domain waveform, waveform, by calculating correlation of the cyclic prefix. The parameters of waveform are given in the downlink settings structure, cfgdl. cfgdl must contain the field NDLRB to specify that a downlink signal is expected in waveform.
foffset = lteFrequencyOffset(cfgul,waveform) estimates the average frequency offset, foffset, of the time-domain waveform, waveform, by calculating correlation of the cyclic prefix. The parameters of waveform are given in the uplink settings structure, cfgul. cfgul must contain the field NULRB to specify that an uplink signal is expected in waveform.
[foffset, corr] = lteFrequencyOffset(___) also returns a complex matrix, corr, spanning one slot and containing the same number of antennas, or columns, as waveform. corr is the signal used to extract the timing of the correlation for the estimation of the frequency offset.
[foffset, corr] = lteFrequencyOffset(___,toffset) provides control over the position in the correlator output used to estimate the frequency offset. When present toffset is the timing offset in samples from the start of the correlator output to the position used for the frequency offset estimation. This input allows a timing offset to be calculated externally on a signal of longer duration than the input waveform. Which allows a short-term frequency offset estimate to be obtained while retaining the benefit of a longer-term timing estimate.
If toffset is absent, the quality of the internal timing estimate is subject to the length and signal quality of the input waveform and, therefore, it may result in inaccurate frequency offset measurements.
Correct for frequency offset.
Perform SC-FDMA demodulation.
cfgdl — Downlink configuration
Downlink configuration, specified as a structure having the following fields.
{N}_{\text{RB}}^{\text{DL}}
NSubframe Optional
cfgul — Uplink configuration
Uplink configuration, specified as a structure having the following fields.
{N}_{\text{RB}}^{\text{UL}}
waveform — Input time-domain waveform
Input time-domain waveform, specified as a numeric column vector.
Timing offset, specified as a scalar value expressed in samples. Use toffset to control the position in the correlator output used to estimate the frequency offset. If toffset is absent, or empty, the position of the peak magnitude of the correlator output is used.
foffset — Average frequency offset estimate
Average frequency offset estimate, returned as a scalar value expressed in Hertz. This function can only accurately estimate frequency offsets of up to ±7.5 kHz (a range of 15 kHz, the subcarrier spacing).
corr — Correlation timing signal
Correlation timing signal, returned as a numeric matrix. corr is a complex matrix that spans one slot and contains the same number of antennas, or columns, as waveform. It is the signal used to extract the timing of the correlation for the frequency offset estimation.
lteFrequencyCorrect | lteCellSearch | lteDLFrameOffset | lteULFrameOffset | lteOFDMDemodulate | lteSCFDMADemodulate
|
14. Vue 2.0 two-way binding principle - Birost
There are many ways to achieve two-way binding. Vue adopts data holding combined with publish and subscribe design mode to achieve.
Use Object.defineProperty() to hijack the setter and getter of each property, publish a message to subscribers when the data changes, and trigger the corresponding listener callback.
Object.defineProperty() introduction
Function: A more popular way of saying is to manipulate object properties
Official: Define a new property directly on an object, or modify an existing property of an object, and return this object.
obj The object whose properties are to be defined.
prop The name or Symbol of the property to be defined or modified.
descriptor The attribute descriptor to be defined or modified.
configurable: Can the descriptor be modified, that is, can the other attributes of the descriptor be modified again
enumerable: can the attribute be enumerated, that is, can the a attribute be for
writable: Can the attribute value be modified, that is, can the obj.a = 1 be modified like this
get: It is a function, when the attribute is accessed, the function is automatically called, and the return value of the function is the value of the attribute
set: is a function, when the attribute is modified, the function is automatically called, the function has one and only one parameter, the new value assigned
*The writable attribute of the value attribute in the descriptor and the set attribute of the get attribute are mutually exclusive.
Attribute holding
Vue's data data holding is achieved by using get and set attributes. The get function is triggered when the object property is read, and the set function is triggered when the value is assigned
* There cannot be a read operation in get, otherwise the loop will be endless, so when using get set, a variable is always needed
let name = "xx" ;
Object .defineProperty(obj, "n" , {
console .log( "Read" , name);
set ( newName ) {
console .log( "Settings" , newName);
value = newName;
//Trigger the get function, the return value of get is the property value
console .log(obj.n);
//Trigger the set function, the value of value becomes xxxx, the property value has not changed
obj.n = "xxxx" ;
Object.defineProperty() extension
Back in Vue development, we often encounter a scenario where two-way binding cannot be performed to modify the value of an array. The reason is that the get set of defineProperty() cannot monitor the new modification operation of the object array.
Vue monitors the changes of the array by finding ways to change the original array, and then hijacking these methods.
The general steps are as follows: Array => new prototype (to perform detection behavior) => prototype of the array
There is a scenario, such as an array inside an array, then recursive thinking is used to solve this problem
This is why there is in vue
set/
delete and can only be detected by using specific methods for arrays
The overall idea of two-way binding
1. Implement a data listener, the main function is to listen to all attributes, called Observer in Vue
2. Then through the subscription-release design ideas to notify the update
(The subscription publishing model defines a one-to-many dependency relationship. One refers to the publisher, such as a topic object, and multiple refers to subscribers. Dependency is the dependence of the subscriber on the publisher; multiple subscribers monitor a topic object at the same time . When the status of the publisher, the topic object, changes, the subscribers will be notified of the change, and the subscribers will update their status accordingly.)
3. Define a subscriber Dep to collect changes in these properties to notify subscribers
4. What kind of processing needs to be done for different events, so a ComPile (instruction parser) is also needed, such as pull-down operations, input operations, etc.
5. Finally implement a Watcher (subscriber), mainly to receive different operations, for those objects, update the view
Implement Observer (listener)
function observe ( data ) {
if (data && typeof data === "object" ){
Object .keys(data).forEach( ( key )=> {
function defineReactive ( data,key,val ) {
//Recursively, monitor sub-objects
Object .defineProperty(data, key, {
configurable : false ,
set : function ( value ) {
Implement the subscriber
function Dep () {
this .subs = [];
addSub : function ( sub ) {
this .subs.push(sub);
notify : function () {
this .subs.forEach( function ( sub ) {
Stuffed into the listener
//Recursively, listen to sub-objects
Implement watcher (subscriber)
Ideas: 1. Insert the listener when instantiating 2. Implement the update() method 3. When the notification is reached, call the update() method to trigger the callback of the compiler (comPlie) 4. End
function Watcher ( vm, exp, cb ) {
this .cb = cb;
this .$vm = vm;
this .exp = exp;
//In order to trigger the getter of the property, add yourself to the dep, combined with Observer Easier to understand
this .value = this .get(); //add yourself to the subscriber operation
this .run(); //Receive notification of property value changes
var value = this .get(); //
Get the latest value var oldVal = this .value; //Speaking of triggering the set function before, the attribute value has not changed
this .cb.call( this .$vm, value, oldVal); //Execute the callback bound in Compile and update the view
Dep.target = this ; //Point the current subscriber to yourself, cache
var value = this .$vm[ this .exp]; //Force the listener to trigger and add yourself to the attribute subscriber
Dep.target = null ; //After adding, reset and release
Then the corresponding get in the defineReactive method should also be modified
function defineReactive ( data, key, val ) {
observe(val); //monitor sub-property
//Because you need to add a watcher in the closure, you can define a global target attribute in Dep, temporarily store the watcher, and remove it after adding
<div id= "name" ></div>
< script > function Vue ( data, el, exp ) {
el.innerHTML = this .data[exp]; //initialize the value of the template data
new Watcher( this , exp, function ( value ) {
var ele = document .querySelector( "#name" );
name : "hello world" ,
vue.data.name = "chuchur" + new Date () * 1 ;
But in the process of our use, this.xxx is used to operate, not this.data.xxx, so we also need to add a proxy, Vm proxy vm.data
function Vue ( options ) {
this .$options = options || {};
this .data = this .$options.data;
//Property proxy, implement vm.xxx -> vm.data.xxx
Object .keys( this .data ).forEach( function ( key ) {
//this.data ==>this
self.proxy(key); //bind proxy properties
observe( this .data, this );
//Also use the defineProperty() method
proxy : function ( key ) {
Object .defineProperty( this , key, {
configurable : true ,
get : function proxyGetter () {
set : function proxySetter ( newVal ) {
Finally, the implementation of the parser Compile
Parse the template instructions, replace the template data, and initialize the view
Bind the node corresponding to the template instruction to the corresponding update function, and initialize the corresponding subscriber
isDirective : function ( attr ) {
return attr.indexOf( 'v-' ) == 0 ;
isEventDirective : function ( dir ) {
return dir.indexOf( 'on:' ) === 0 ;
//Process v-instruction
compile : function ( node ) {
self = this ;
[].slice.call(nodeAttrs).forEach( function ( attr ) {
//Rule: the command is named v-xxx
//such as the
command in <span v-text="content"></span> is v-text var attrName = attr.name; //v-text
var exp = attr.value; //content
var dir = attrName.substring( 2 ); //text
if (self.isEventDirective (dir)) {
//Event instruction, such as v-on:click
//Ordinary commands such as: v-model, v-html, currently only deal with v-model
//After processing, kill v-on:, v-model and other element attributes
compileEvent : function ( node, vm, exp, dir ) {
var eventType = dir.split( ':' )[ 1 ];
node.addEventListener(eventType, cb.bind(vm), false );
compileModel : function ( node, vm, exp, dir ) {
var val = this .$vm[exp];
this .updaterModel(node, val);
new Watcher( this .$vm, exp, function ( value ) {
node.addEventListener( 'input' , function ( e ) {
updaterModel : function ( node, value, oldValue ) {
If this article can give you a little help, I hope xdm can give a thumbs up
https://juejin.cn/post/6969867472765714469 14.Vue
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Linearly repetitive Delone sets are rectifiable | EMS Press
Linearly repetitive Delone sets are rectifiable
Facultad de Matemáticas, Pontificia Universidad Católica de Chile, Campus San Joaquín, Avenida Vicuña Mackenna 4860, Santiago, Chile
Institut Non Linéaire de Nice–Sophia Antipolis, UMR CNRS 7335, Université de Nice – Sophia Antipolis, 1361, Route des Lucioles, 06560 Valbonne, France
José Aliste-Prieto
Centro de Modelamiento Matemático, Universidad de Chile, Blanco Encalada 2120 7to. piso, Santiago, Chile
We show that every linearly repetitive Delone set in the Euclidean d-space
\mathbb{R}^{d}
d⩾2
, is equivalent, up to a bi-Lipschitz homeomorphism, to the integer lattice
\mathbb{Z}^{d}
. In the particular case when the Delone set X in
\mathbb{R}^{d}
comes from a primitive substitution tiling of
\mathbb{R}^{d}
, we give a condition on the eigenvalues of the substitution matrix which ensures the existence of a homeomorphism with bounded displacement from X to the lattice
\beta \mathbb{Z}^{d}
for some positive β. This condition includes primitive Pisot substitution tilings but also concerns a much broader set of substitution tilings.
Daniel Coronel, Jean-Marc Gambaudo, José Aliste-Prieto, Linearly repetitive Delone sets are rectifiable. Ann. Inst. H. Poincaré Anal. Non Linéaire 30 (2013), no. 2, pp. 275–290
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An Existence Theorem for General Control Problems of Nonlinear Evolution Equations of First Order | EMS Press
An Existence Theorem for General Control Problems of Nonlinear Evolution Equations of First Order
An existence theorem for the solution of optimal control problems of systems governed by the state equation
y+ vAy + B(y) = u
A
B
a nonlinear operator with
||B(y)||_n ≤ c(1 + ||y||_m^r)
Bernd Krause, An Existence Theorem for General Control Problems of Nonlinear Evolution Equations of First Order. Z. Anal. Anwend. 9 (1990), no. 2, pp. 113–120
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A Characterization of Pobrushin’s Coefficient of Ergodicity | EMS Press
A Characterization of Pobrushin’s Coefficient of Ergodicity
Adolf Rhodius
It is proved that the ergodicity coefficient
\tau
corresponding to any vector norm on
\mathbb R^n
fulfills the inequality
\tau (P) ≤ 1
n \times n
stochastic matrices
P
if it is the Dobrushin ergodicity coefficient.
Rolf Kühne, Adolf Rhodius, A Characterization of Pobrushin’s Coefficient of Ergodicity. Z. Anal. Anwend. 9 (1990), no. 2, pp. 187–188
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How do you multiply monomials? | StudyPug
To multiply a monomial by a monomial, we need to first separate the coefficients and variables. Then, we group the coefficients together and apply the exponent rules on the variables.
Basic Concepts: Exponent rules, Order of operations with exponents, Product rule of exponents
(a^x)(a^y)=a^{(x+y)}
(a^x)^y = a^{(x\cdot y)}
{(-4x^2y^4)(2x^{-3}y^2)(-3x^{-5}y^{-2})}
|
Free cash flow to equity is a measure of how much cash is available to the equity shareholders of a company after all expenses, reinvestment, and debt are paid. FCFE is a measure of equity capital usage.
Free cash flow to equity is composed of net income, capital expenditures, working capital, and debt. Net income is located on the company income statement. Capital expenditures can be found within the cash flows from the investing section on the cash flow statement.
Working capital is also found on the cash flow statement; however, it is in the cash flows from the operations section. In general, working capital represents the difference between the company’s most current assets and liabilities.
A measure of equity cash usage, free cash flow to equity calculates how much cash is available to the equity shareholders of a company after all expenses, reinvestment, and debt are paid.
Free cash flow to equity is composed of net income, capital expenditures, working capital, and debt.
The FCFE metric is often used by analysts in an attempt to determine the value of a company.
FCFE, as a method of valuation, gained popularity as an alternative to the dividend discount model (DDM), especially for cases in which a company does not pay a dividend.
These are short-term capital requirements related to immediate operations. Net borrowings can also be found on the cash flow statement in the cash flows from financing section. It is important to remember that interest expense is already included in net income so you do not need to add back interest expense.
The Formula for FCFE
\text{FCFE} = \text{Cash from operations} - \text{Capex} + \text{Net debt issued}
FCFE=Cash from operations−Capex+Net debt issued
What Does FCFE Tell You?
The FCFE metric is often used by analysts in an attempt to determine the value of a company. This method of valuation gained popularity as an alternative to the dividend discount model (DDM), especially if a company does not pay a dividend. Although FCFE may calculate the amount available to shareholders, it does not necessarily equate to the amount paid out to shareholders.
Analysts use FCFE to determine if dividend payments and stock repurchases are paid for with free cash flow to equity or some other form of financing. Investors want to see a dividend payment and share repurchase that is fully paid by FCFE.
If FCFE is less than the dividend payment and the cost to buy back shares, the company is funding with either debt or existing capital or issuing new securities. Existing capital includes retained earnings made in previous periods.
This is not what investors want to see in a current or prospective investment, even if interest rates are low. Some analysts argue that borrowing to pay for share repurchases when shares are trading at a discount, and rates are historically low is a good investment. However, this is only the case if the company's share price goes up in the future.
If the company's dividend payment funds are significantly less than the FCFE, then the firm is using the excess to increase its cash level or to invest in marketable securities. Finally, if the funds spent to buy back shares or pay dividends is approximately equal to the FCFE, then the firm is paying it all to its investors.
Example of How to Use FCFE
Using the Gordon Growth Model, the FCFE is used to calculate the value of equity using this formula:
V_\text{equity} = \frac{\text{FCFE}}{\left(r-g\right)}
Vequity=(r−g)FCFE
Vequity = value of the stock today
FCFE = expected FCFE for next year
g = growth rate in FCFE for the firm
This model is used to find the value of the equity claim of a company and is only appropriate to use if capital expenditure is not significantly greater than depreciation and if the beta of the company's stock is close to 1 or below 1.
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ACT Compass Test Prep Tutor, Help and Practice Online | StudyPug
>ACT Compass Test Prep
ACT Compass Test Prep made completely easy!
We've got you covered with our complete help for Compass math test including all five subject areas: Pre-algebra, Algebra, College Algebra, Geometry, and Trigonometry.
Our comprehensive lessons on compass test cover help on topics like Trigonometric identities, Linear equations, Exponents, Complex number, Logarithmic functions, Rational expressions, and so many more. Learn the concepts with our video tutorials that show you step-by-step solutions to even the hardest Compass math test problems. Then, strengthen your understanding with tons of Compass test practice.
All our lessons are taught by experienced math teachers. Let's finish your preparation in no time, and ACE that Compass test.
5Adding and Substracting Fractions
x + a = b
ax + b = c
{x \over a} + b = c
a(x + b) = c
d = \sqrt{(x_2-x_1)^2+(y_2-y_1)^2}
M = ( \frac{x_1+x_2}2 ,\frac{y_1+y_2}2)
m = \frac{y_2-y_1}{x_2- x_1}
y - y_1 = m (x - x_1)
14Solving Systems of Linear Equations
15.1Product rule of exponents
15.5Power of a power rule
16.1What is a polynomial?
17Factoring Polynomial Expressions
17.1Common factors of polynomials
x^2 + bx + c
x^2 + bx + c
x^2 + bx + c
x^2 + bx + c
ax^2 + bx + c
ax^2 + bx + c
17.9Factoring perfect square trinomials:
(a + b)^2 = a^2 + 2ab + b^2
(a - b)^2 = a^2 - 2ab + b^2
(a - b)(a + b) = (a^2 - b^2)
17.12Using algebra tiles to factor polynomials
17.13Solving polynomial equations
18Factoring Trinomials (Advanced)
x^2 - y^2
21.5Converting radicals to entire radicals
22Radicals (Advanced)
24.4Perpendicular line proofs
26Surface Area and Perimeter of Polygons
27.5Chord properties in circles
27.6Tangent properties in circles
27.8Equation of a circle
(a^x)(a^y)=a^{(x+y)}
{a^x \over a^y}=a^{(x-y)}
(a^x)^y = a^{(x\cdot y)}
a^0 = 1
33Applications of Exponential and Logarithmic Functions
33.2Exponential decay: Half-life
33.5Continuous growth and decay
33.6Logarithmic scale: Richter scale (earthquake)
33.7Logarithmic scale: pH scale
33.8Logarithmic scale: dB scale
34Right Triangle Trigonometry
\frac{o}{h}
\frac{a}{h}
\frac{o}{a}
35Laws of sines and cosines
37.2Coterminal angles
37.5ASTC rule in trigonometry (All Students Take Calculus)
38Graphs of Trigonometric Functions
39.1Quotient identities and reciprocal identities
39.5Double-angle identities
40.4Solving trigonometric equations involving multiple angles
41.6Distance and midpoint of complex numbers
41.7Angle and absolute value of complex numbers
41.8Polar form of complex numbers
41.9Operations on complex numbers in polar form
42Introduction to Matrices
42.4Matrix multiplication
42.5The three types of matrix row operations
42.6Representing a linear system as a matrix
42.7Solving a linear system with matrices using Gaussian elimination
43Properties of Matrices
43.1Zero matrix
43.2Identity matrix
43.3Properties of matrix addition
43.4Properties of scalar multiplication
44Determinants and Inverses of Matrices
44.1The determinant of a 2 x 2 matrix
44.2The determinant of a 3 x 3 matrix (General & Shortcut Method)
44.3The inverse of a 2 x 2 matrix
44.4The inverse of 3 x 3 matrices with matrix row operations
44.5The inverse of 3 x 3 matrix with determinants and adjugate
44.62 x 2 invertible matrix
I'm preparing for Compass Math. How should I start?
You can first test yourself with this sample Compass test questions to see which areas and topics you need to focus more on. Then, watch our video lessons and work on the practice questions for review and better understanding.
What should I expect when in the Math Compass test?
It is made up with multiple-choice questions. Only calculators with the most basic functions are allowed. You will also be provided a calculator on the screen during the test too.
Any tips on getting a good score in the ACT Compass test?
The test is computer-adaptive. So, the difficulty level of questions given to you changes based on how well you answer the previous ones. Since it is untimed, you should take as much time as you need to best demonstrate your skills.
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Rotations of eigenvectors under unbounded perturbations | EMS Press
Rotations of eigenvectors under unbounded perturbations
A
be an unbounded selfadjoint positive definite operator with a discrete spectrum in a separable Hilbert space, and
\widetilde A
be a linear operator, such that
\|(A-\widetilde A)A^{-\nu}\| < \infty
(0< \nu\le 1)
A
has a simple eigenvalue. Under certain conditions
\widetilde A
also has a simple eigenvalue. We derive an estimate for
\|e(A)-e(\widetilde A)\|
e(A)
e(\widetilde A)
are the normalized eigenvectors corresponding to these simple eigenvalues of
A
\widetilde A
, respectively. Besides, the perturbed operator
\widetilde A
can be non-selfadjoint. To illustrate that estimate we consider a non-selfadjoint differential operator. Our results can be applied in the case when
A
is a normal operator.
Michael Gil', Rotations of eigenvectors under unbounded perturbations. J. Spectr. Theory 7 (2017), no. 1, pp. 191–199
|
Correlations and bounds for stochastic volatility models | EMS Press
Collège de France, 3, rue d'Ulm, 75005 Paris, France, CEREMADE – UMR C.N.R.S. 7534, Université Paris-Dauphine, place du Maréchal de Lattre de Tassigny, 75775 Paris cedex 16, France
M. Musiela
FIRST, BNP-Paribas, 10 Harewood Avenue, London NW1 6AA, United Kingdom
We investigate here, systematically and rigorously, various stochastic volatility models used in Mathematical Finance. Mathematically, such models involve coupled stochastic differential equations with coefficients that do not obey the natural and classical conditions required to make these models “well-posed”. And we obtain necessary and sufficient conditions on the parameters, such as correlation, of these models in order to have integrable or
L^{p}
solutions (for
1 < p < \infty
P.-L. Lions, M. Musiela, Correlations and bounds for stochastic volatility models. Ann. Inst. H. Poincaré Anal. Non Linéaire 24 (2007), no. 1, pp. 1–16
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How do you rationalize the denominator | StudyPug
When you're working with fractions, you may run into situations where the denominator is messy. What exactly does messy mean? It's when your denominator isn't a whole number and cannot be cancelled off. When we've got, say, a radical in the denominator, you're not done answering the question yet. Your final answer has to have a clean denominator. What can we do to fix this? We can rationalize the denominator.
Do you remember how you first worked with fractions with unequal denominators? Take for example the following:
\frac{1}{7} + \frac{3}{9}
You would then go multiply each individual fraction's numerator and denominator with the number the denominator needed. In this case, you'd do:
(\frac{1}{7} \bullet \frac{9}{9})+(\frac{3}{9} \bullet \frac{7}{7})
= \frac{9}{63} + \frac{21}{63}
= \frac{30}{63}
= \frac{10}{21}
When we multiply
\frac{1}{7}
\frac{9}{9}
, we're actually multiplying the whole thing by
1
\frac{9}{9}
1
. This essentially means we haven't changed the original
\frac{1}{7}
, but instead have made it so that it's possible to add it together with
\frac{3}{9}
. This is the concept we'll use when we start rationalizing denominators.
So now that we recall how we added or subtracted fractions that didn't have the same denominators, we will use a similar method to change the denominators when we're rationalizing the denominator. When you've got a radical in your denominator, such as in the case of
\frac{1}{\sqrt{3}}
, we can rationalize this by multiplying the numerator and denominator by
\sqrt{3}
. This means, again, that we're really not changing the original fraction since we're multiplying both the top and bottom by
\sqrt{3}
, essentially meaning we multiply the original fraction by
1
. A radical multiplied by itself gets rid of the radical sign. Meaning our final fraction will become:
\frac{1*\sqrt{3}}{\sqrt{3}*\sqrt{3}}
= \frac{\sqrt{3}}{3}
You can see how we've got rid of the radical in the denominator. Having a radical in the numerator is acceptable for a final answer, so you've got the final answer here. Make sure that if you can simplify the fraction further, make sure you rationalize the denominator and simplify it in order to get the correct answer. Now you've learned how to rationalize denominators in fractions.
\frac{14\sqrt{20}}{3\sqrt{16}}
= \frac{14}{3} \sqrt{\frac{20}{16}}
= \frac{14}{3} \sqrt{\frac{5}{4}}
= \frac{14}{3}\bullet\frac{\sqrt{5}}{\sqrt{4}}
= \frac{14}{3}\bullet\frac{\sqrt{5}}{2}
= \frac{14\sqrt{5}}{6}
= \frac{7\sqrt{5}}{3}
\frac{\sqrt{7} - 5}{\sqrt{3}}
= \frac{\sqrt{7} - 5}{\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}}
= \frac{\sqrt{21} - 5\sqrt{3}}{\sqrt{9}}
= \frac{\sqrt{21} - 5\sqrt{3}}{3}
Simplify and rationalize denominator
\frac{3\sqrt{6}}{5} + \frac{5\sqrt{3}}{\sqrt{7}}
Find the lowest common denominator of the
2
= \frac{(\sqrt{7})3\sqrt{6}}{(\sqrt{7})5} + \frac{(5)5\sqrt{3}}{(5)\sqrt{7}}
= \frac{3\sqrt{42}+25\sqrt{3}}{5\sqrt{7}}
= \frac{3\sqrt{42}+25\sqrt{3}}{5\sqrt{7}} \times \frac{\sqrt{7}}{\sqrt{7}}
= \frac{3\sqrt{294}+25\sqrt{21}}{5(\sqrt{7})(\sqrt{7})}
= \frac{3\sqrt{49\bullet 6}+25\sqrt{21}}{35}
= \frac{21\sqrt{6}+25\sqrt{21}}{35}
Double check your answer with this online calculator that can help you rationalize radical denominators.
Wanted to revise radicals? Recall how to simplify radicals, add and subtract radicals, and multiply and divide radicals.
Continuing on, you can learn about operations with radicals, adding and subtracting radicals, and how to multiply radicals.
When there is a radical in the denominator, the fraction is not in its simplest form. Therefore, we need to rationalize the denominator to move the root from the denominator/bottom of the fraction to the numerator/ top. In this lesson, we will learn how to simplify radicals by rationalizing the denominator.
Basic Concepts: Evaluating and simplifying radicals, Adding and subtracting radicals, Multiplying and dividing radicals
Related Concepts: Operations with radicals, Adding and subtracting radicals (Advanced), Multiplying radicals (Advanced)
What is rationalizing the denominator?
\frac{14 \sqrt{20}}{3 \sqrt{16}}
\frac{\sqrt{7}-5}{\sqrt{3}}
\frac{5 \sqrt{2} -4}{3 \sqrt{5}- \sqrt{6}}
\frac{3\sqrt{6}}{5} + \frac{5\sqrt{3}}{\sqrt{7}}
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Questions on the applications of polynomials | StudyPug
In this session, we will apply the knowledge of polynomials learned from previous sessions to solve questions related to geometric shapes. Therefore, before we start, it's a good idea to refresh your memory on polynomial operations we have talked about in the lessons before.
Related Concepts: Applications of polynomials:
x^2 + bx + c
, Word problems of polynomials, Applications of quadratic equations
Find the area of the following geometric figures
(a^3+2a^2+3a+6)
^2
Determine the height of the rectangle.
x
and the dimension of the right triangle
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Summation Formulas Obtained by Means of the Generalized Chain Rule for Fractional Derivatives
S. Gaboury, R. Tremblay, "Summation Formulas Obtained by Means of the Generalized Chain Rule for Fractional Derivatives", Journal of Complex Analysis, vol. 2014, Article ID 820951, 7 pages, 2014. https://doi.org/10.1155/2014/820951
S. Gaboury1 and R. Tremblay1
1Department of Mathematics and Computer Science, University of Quebec at Chicoutimi, QC, Canada
In 1970, several interesting new summation formulas were obtained by using a generalized chain rule for fractional derivatives. The main object of this paper is to obtain a presumably new general formula. Many special cases involving special functions of mathematical physics such as the generalized hypergeometric functions, the Appell function, and the Lauricella functions of several variables are given.
The fractional derivative of arbitrary order , , is an extension of the familiar th derivative of the function with respect to to nonintegral values of and is denoted by . The aim of this concept is to generalize classical results of the th order derivative to fractional order. Most of the properties of the classical calculus have been expanded to fractional calculus, for instance, the composition rule [1], the Leibniz rule [2, 3], the chain rule [4], and Taylor’s and Laurent’s series [5–7]. Fractional calculus also provides tools that make it easier to deal with special functions of mathematical physics [8].
The most familiar representation for fractional derivative of order of is the Riemann-Liouville integral [9]; that is, which is valid for and where the integration is done along a straight line from to in the -plane. By integrating by part times, we obtain This allows modifying the restriction to [10].
In 1970, Osler [2] introduced a more general definition of the fractional derivative of a function with respect to another function based on Cauchy’s integral formula.
Definition 1. Let be analytic in the simply connected region . Let be regular and univalent on and let be an interior or boundary point of . Assume also that for any simple closed contour in through . Then if is not a negative integer and is in , the fractional derivative of order of with respect to is defined by For nonintegral , the integrand has a branch line which begins at and passes through . The notation on this integral implies that the contour of integration starts at , encloses once in the positive sense, and returns to without cutting the branch line.
With the use of that representation based on the Cauchy integral formula for the fractional derivatives, Osler gave a generalization of the following result [11, page 19] involving the derivative of order of the composite function : where In particular, he found the following formula [4]: where the notation means the fractional derivative of order of with respect to . Osler proved the generalized chain rule by applying the generalized Leibniz rule [2] for fractional derivatives to an important fundamental relation involving fractional derivatives discovered also by Osler [4, page 290, Theorem 2]. The fundamental relation which is the central point of this paper is given by the next theorem.
Theorem 2. Let and be defined and analytic on the simply connected region and let the origin be an interior or boundary point of . Suppose also that and are regular univalent functions on and that . Let vanish over simple closed contour in through the origin. Then the following relation holds true:
This fundamental relation is very useful to obtain very easily known and new summation formulas involving special functions of mathematical physics. For example, set , , and in (7). One sees easily that . Thus, one has The left-hand side is evaluated by using the well-known formula [12, page 83, Equation ] after replacing by . Expanding in power series, using (9), and replacing by after operation, one obtains Kummer’s summation formula where denotes the Gauss hypergeometric function [13] and holds for the Pochhammer symbol defined, in terms of the Gamma function, by
In this paper, we present several new summation formulas involving special functions of mathematical physics obtained by using the fundamental relation (7). In Section 2, we introduce the Pochhammer based representation for fractional derivatives and we recall a well-poised fractional calculus operator given by Tremblay [14]. This well-poised operator will be used, throughout this paper, in order to ease the computations of fractional derivatives. Finally, Section 3 is devoted to the presentation of the main results. Many presumably new summation formulas are also given as special cases.
2. Pochhammer Contour Integral Representation for Fractional Derivative and the Well-Poised Fractional Calculus Operator
The less restrictive representation of fractional derivative according to parameters is Pochhammer’s contour definition introduced in [14, 15].
Definition 3. Let be analytic in a simply connected region . Let be regular and univalent on and let be an interior point of then if is not a negative integer, is not an integer, and is in , one defines the fractional derivative of order of with respect to by For nonintegers and , the functions and in the integrand have two branch lines which begin, respectively, at and , and both pass through the point without crossing the Pochhammer contour at any other point as shown in Figure 1. denotes the principal value of the integrand in (13) at the beginning and ending point of the Pochhammer contour which is closed on Riemann surface of the multiple-valued function .
Remark 4. In Definition 3, the function must be analytic at . However it is interesting to note here that we could also allow to have an essential singularity at , and (13) would still be valid.
Remark 5. The Pochhammer contour never crosses the singularities at and in (13); then we know that the integral is analytic for all and for all and for in . Indeed, the only possible singularities of are , and which can directly be identified from the coefficient of the integral (13). However, integrating by parts times the integral in (13) by two different ways, we can show that , and are removable singularities (see [15]).
Remark 6. A deep and complete study of the fundamental properties of the function with or , relative to the the values of the parameters , , and , can be found in [16].
In 1974, Tremblay [14] introduced the operator in order to deal with special functions more efficiently and to facilitate the obtention of new relations such as hypergeometric transformations. This operator will be useful in the proofs of the summation formulas given in the next section.
The operator is defined in terms of the fractional calculus operator as with .
This operator has been used very recently in [3] to prove a new generalized Leibniz rule for fractional derivatives as well as in [17] to obtain some new series involving special functions.
This operator has many very useful properties. We chose to give only one of them which will be used in the proofs of the main results; that is, In terms of the fractional calculus operator , the modified fundamental relation (7) holds the following form: with .
It is worthy to mention that the operator has a lot more interesting properties and applications.
3. Main Results and Special Cases
In this section, we present a new general formula related to the generalized chain rule. We give many special cases involving special functions such as the first Appell function , the Lauricella function of several variables , and the generalized hypergeometric functions. These functions are evaluated most of the time at arguments related to the roots of unity.
Main Formula. Consider with and .
Proof. Let and let with and two positive integers in (16). We have With the help of (15), the last equation becomes Using the fact that both and are positive integers, the result follows.
Example 7. Let , , and in (17). We, thus, have Expanding in power series, (20) becomes The right-hand side of (21) can be split in two parts Converting the terms involving Gamma function in the last expression into Pochhammer’s symbol and making some simplifications, we find Finally, rewriting the right-hand side of (23) in terms of generalized hypergeometric function and combining with (21), we obtain the following summation formula:
Special Cases with . Consider
Example 8. Setting and in (25) gives Observe that where denotes the first Appell function [18]. Replacing (27) in (26) yields the following summation formula:
It is interesting to note that setting and replacing by in (28), and using the well-known identity [19], reduces to the known summation formula [20]
Furthermore, with the help of the following reduction formulas for the Appell functions given by Nagel [21], in conjunction with (28) give, respectively, after simplifications the two summation formulas
Example 9. Let and let be an integer in (25). We have that The Lauricella function of variables is defined as [18, page 60] with . We can easily see that the following relation, holds true. So, using this last relation in conjunction with (33) provides the following summation formula:
Let us examine another special case of (17) when is not equal to 1.
Example 10. Set and in (25). We have Expanding in power series, we find for the left-hand side of (37) The right-hand side of (37) yields after expanding in power series and using property (15) Using (27) and making some elementary simplifications, the last formula reduces to Combining (38) and (40) and putting provide the following presumably new summation formula:
Cases with . Consider
Example 11. Let and in (42). We obtain Expanding in power series, we find for the right-hand side of (43) We, thus, get the following (presumably) new summation formula:
Remark 12. The cases where with must be treated very carefully as .
L. M. B. Campos, “On a concept of derivative of complex order with applications to special functions,” IMA Journal of Applied Mathematics, vol. 33, no. 2, pp. 109–133, 1984. View at: Publisher Site | Google Scholar | MathSciNet
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R. Tremblay, S. Gaboury, and B. Fugère, “A new Leibniz rule and its integral analogue for fractional derivatives,” Integral Transforms and Special Functions, vol. 24, no. 2, pp. 111–128, 2013. View at: Publisher Site | Google Scholar | MathSciNet
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T. J. Osler, “Taylor's series generalized for fractional derivatives and applications,” SIAM Journal on Mathematical Analysis, vol. 2, pp. 37–48, 1971. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
R. Tremblay and B. Fugère, “The use of fractional derivatives to expand analytical functions in terms of quadratic functions with applications to special functions,” Applied Mathematics and Computation, vol. 187, no. 1, pp. 507–529, 2007. View at: Publisher Site | Google Scholar | MathSciNet
R. Tremblay, S. Gaboury, and B. Fugère, “Taylor-like expansion in terms of a rational function obtained by means of fractional derivatives,” Integral Transforms and Special Functions, vol. 24, no. 1, pp. 50–64, 2013. View at: Publisher Site | Google Scholar | MathSciNet
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K. S. Miller and B. Ross, An Introduction of the Fractional Calculus and Fractional Differential Equations, John Wiley & Sons, New York, NY, USA, 1993. View at: MathSciNet
E. D. Rainville, Special Functions, Macmillan Company, New York, NY, USA, 1960. View at: MathSciNet
R. Tremblay, Une contribution a la theorie de la derivee fractionnaire [Ph.D. thesis], Laval University, Quebec City, Canada, 1974.
J.-L. Lavoie, T. J. Osler, and R. Tremblay, “Fundamental properties of fractional derivatives via pochhammer integrals,” in Fractional Calculus and Its Applications, vol. 457 of Lecture Notes in Mathematics, pp. 323–356, Springer, 1975. View at: Google Scholar
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Copyright © 2014 S. Gaboury and R. Tremblay. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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How to calculate squares and square roots | StudyPug
This lesson focuses on finding the square of a whole number and the square root of a perfect square. In order to do this, we will first learn how to do prime factorization, a method to find the prime factors of a number.
Basic Concepts: Determining Common Factors, Determining common multiples, Divisibility rules
Related Concepts: Determine square roots of rational numbers, Square and square roots, Cubic and cube roots, Evaluating and simplifying radicals
How to estimate non-perfect squares?
Find the prime factorization. Then, identify the perfect squares.
Find the area of the square that has a side length of:
\sqrt {16}
\sqrt {49}
\sqrt {2025}
What is the side length of a square that has the same area as the rectangle above?
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Nanoporous Alumina Membranes for Enhancing Hemodialysis | J. Med. Devices | ASME Digital Collection
Renal Division,
, Lexington, KY 40506 and Department of Mechanical Engineering,
Huang, Z., Zhang, W., Yu, J., and Gao, D. (August 8, 2006). "Nanoporous Alumina Membranes for Enhancing Hemodialysis." ASME. J. Med. Devices. March 2007; 1(1): 79–83. https://doi.org/10.1115/1.2360949
The nonuniformity of pore size and pore distribution of the current hemodialysis membrane results in low efficiency of uremic solute removal as well as the loss of albumin. By using nanotechnology, an anodic alumina membrane (ceramic membrane) with self-organized nanopore structure was produced. The objective of this study was to fabricate nanoporous alumina membranes and investigate the correlation between various anodization conditions and the pore characteristics in order to find its potential application in artificial kidney/hemodialysis. An aluminum thin film was oxidized in two electrolytes consisting of 3% and 5% sulfuric acid and 2.7% oxalic acid. The applied voltages were 12.5, 15, 17.5, and
20V
for sulfuric acid and 20, 30, 40, and
50V
for oxalic acid. Pore size and porosity were determined by analyzing Scanning Electron Microscopy (SEM) images and hydraulic conductivity was measured. Results show that pore size increased linearly with voltage. Acid concentration affected pore formation but not pore size and pore distribution. Hydraulic conductivity of the ceramic membrane was higher than that of the polymer dialysis membrane. The optimal formation conditions for self-organized nanopore structure of the ceramic membrane were
12.5-17.5V
in 3–5% sulfuric acid at
0°C
. Under these conditions, ceramic membranes with pores size of
∼10nm
diameter can be produced. In conclusion, we used anodic alumina technology to reliably produce in quantity ceramic membranes with a pore diameter of
10-50nm
. Because of more uniform pore size, high porosity, high hydraulic conductivity, and resistance to high temperature, the ceramic membrane has the potential application as a hemodialysis membrane.
alumina, bioceramics, nanoporous materials, nanotechnology, membranes, patient treatment, kidney, anodisation, porosity, scanning electron microscopy, ceramic, alumina, membrane, hemodialysis
Ceramics, Membranes, Hydraulic conductivity, Hemodialysis
Determinants of Hemodialyzer Performance and the Potential Effect on Clinical Out Come
(suppl
Formation of Anodic Coatings on Aluminum
Trans. Electrochem. Soc.
The Structure of Anodic Oxide Coatings
Factors Affecting The Formation of Anodic Oxide Coatings
The Direct Observation of Barrier Layers in Porous Anodic Oxide Films
Nucleation And Growth of Porous Anodic Films on Aluminum
Fabrication of Gold Nanodot Array Using Anodic Porous Alumina as an Evaporation Mask
Ordered Metal Nanohole Arrays Made by A Two-Step Replication of Honeycomb Structures of Anodic Alumina
Kudrawiec
Photoluminescence Investigation of Porous Anodic Alumina With Spin-on Europium-Containing Titania Sol-Gel Films
Photoluminescence of Eu-Doped Titania Xerogel Spin-on Deposited on Porous Anodic Alumina
Template-Directed Vapor-Liquid-Solid Growth of Silicon Nanowires
Physio-Chemistry and Cytotoxicity of Ceramics
Mater. Med. Pol.
Biocompatability of Alumina Ceramic and Polyethylene as Materials Bearings of a Centrifugal Blood Pump
Protein Adsorption Onto Ceramic Surfaces
A Study of Novel and Optimal Technology for Hemodialysis
Effect of Flow on Hydraulic Conductivity and Reflection Coefficient of Rabbit Mesentery
Microcirculation (Philadelphia)
Fabrication of Vertically Aligned Carbon Nanotubes and Horizontal Nano Structures
,” Master thesis, University of Kentucky, Lexington, KY.
Jessensky
Self-Organized Formation of Hexagonal Pore Arrays in Anodic Alumina
Measurement of Hollow-Fiber-Membrane Transport Properties in Hemodialysis (Abstract)
Numerical and Experimental Studies of Mass Transfer in Artificial Kidney and Hemodialysis
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How do you multiply a monomial by a binomial? | StudyPug
Monomials are algebraic expressions with one term, while binomials have two terms. In this lesson, we will learn how to multiply monomial by binomial.
Basic Concepts: Exponent rules, Order of operations with exponents, Product rule of exponents, Multiplying monomial by monomial
(a^x)(a^y)=a^{(x+y)}
(a^x)^y = a^{(x\cdot y)}
{-3x(x^3-4y^3)}
{-4mn^{-2}(-3mn^2+4m^3n^5-3)}
|
Emerging trends | Santiment Academy
The Emerging trends (or Trending words) is a list of words that describe the topics which emerged faster than any others over the last 24 hours. By "emerging" we mean getting more social attraction from the crowd, being discussed much more than any other topic.
We detect these words by computing the so called hype score for each single word that is present in the social data after filtering and cleaning the data. Once this number is calculated, the words are ranked according to the corresponding scores in a descending order. The top 10 words in the resulting list are the top emerging trends.
The metric's realtime data is free. The metric's historical data is with restricted access.
In order to reduce the level of noise, spam and duplicates while calculating the hype scores, we apply some preprocessing to the text data, namely:
Clean all the texts from stopwords and non-alphabetic characters.
Transform each pair (user_id, text_documents) to a bag of words representation and remove all the bag of words duplicates.
For all the text documents that have more than a certain amount of words in general (usually 5) - remove the exact duplicates (i.e. messages that look exactly the same are considered only once).
These steps help to make the approach robust to spam and multiple replications of the same word or short word combinations.
After the processing is done, for each of the words we calculate the hype score (or trend score). For any timestamp
t
we define the hype score as follows:
HypeScore(t) := \frac{(v_t^n - \frac{1}{14} \sum_{i=t-15}^{t-1} v_i^n) * v_t^n * \log_{10} u_t}{1 + \frac{1}{14} \sum_{i=t-15}^{t-1} v_i^n}
v_i^n
is the normalized social volume of the word at the moment
i
(i.e. the usual social volume divided by the total amount of messages in that particular data source),
u_t
is the total amount of unique users that have used the word under consideration at least once.
On an intuitive level the hype score tends to be a measuring of how rapidly the social volume of a certain word increased over the last 24 hours in comparison to the past 2 weeks. This is done by comparing the current social volume change to the average social volume of the past 14 days.
Additionally we multiply this factor by log(unique_users) - this way words with a high social volume and a relatively low amount of unique users, that mentioned it at least once, will have a smaller hype score. For example, if a given word was used many times by exactly one user (i.e. most probably it's heavy spam) this word will have a hype score of 0 thanks to the log(unique_users) component. On the other hand, words with 100 and 200 users will have more or less the same chance to get a higher hype score.
It's also worth noticing that we use the normalized social volume instead of the regular one. This makes it easier to compare the resulting hype score across different data sources with different average daily volumes of talks.
Ranking the Words
Once the texts are cleaned and each word has it's hype score, we first rank the words in descending order (the highest hype score goes to the top) and then combine the results across different data sources if necessary: this is done by averaging the hype score for each word across all desired data sources and ranking the words afterwards again. In case a given word is present in source 1 and is not present in source 2 we assume that it's hype score in the second data source is 0.
The hype score doesn't really have a qualitative meaning, it can be treated as a relative number: the higher it gets - the faster a given word is "emerging".
Emerging Trends are available at hourly intervals
Emerging Trends has social data Latency
The algorithm takes into account all the social data, so the list may or may not contain asset names and tickers. We do track the emerging projects separately for the assets only, but currently it is not available through any of the products.
The emerging trends are available in the Labs section.
The emerging trends are available as part of the API, the metric is called getTrendingWords:
getTrendingWords(
topWords {
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Round-rotor synchronous machine with fundamental or standard parameterization - MATLAB - MathWorks 日本
Synchronous Machine Round Rotor
zero-sequence reactance, X0
q-axis transient reactance, Xq'
q-axis transient open-circuit, Tq0'
q-axis transient short-circuit, Tq'
Per-unit field current saturation data, ifd
Per-unit air-gap voltage saturation data, Vag
Round-rotor synchronous machine with fundamental or standard parameterization
The Synchronous Machine Round Rotor block models a round-rotor synchronous machine using fundamental or standard parameters.
{\mathrm{θ}}_{e}\left(t\right)=N{\mathrm{θ}}_{r}\left(t\right),
{P}_{s}=\frac{2}{3}\left[\begin{array}{ccc}\mathrm{cos}{\mathrm{θ}}_{e}& \mathrm{cos}\left({\mathrm{θ}}_{e}â\frac{2\mathrm{Ï}}{3}\right)& \mathrm{cos}\left({\mathrm{θ}}_{e}+\frac{2\mathrm{Ï}}{3}\right)\\ â\mathrm{sin}{\mathrm{θ}}_{e}& â\mathrm{sin}\left({\mathrm{θ}}_{e}â\frac{2\mathrm{Ï}}{3}\right)& â\mathrm{sin}\left({\mathrm{θ}}_{e}+\frac{2\mathrm{Ï}}{3}\right)\\ \frac{1}{2}& \frac{1}{2}& \frac{1}{2}\end{array}\right].
{e}_{d}=\frac{1}{{\mathrm{Ï}}_{base}}\frac{\text{d}{\mathrm{Ï}}_{d}}{\text{d}t}â{\mathrm{Ψ}}_{q}{\mathrm{Ï}}_{r}â{R}_{a}{i}_{d},
{e}_{q}=\frac{1}{{\mathrm{Ï}}_{base}}\frac{\text{d}{\mathrm{Ï}}_{q}}{\text{d}t}+{\mathrm{Ψ}}_{d}{\mathrm{Ï}}_{r}â{R}_{a}{i}_{q},
{e}_{0}=\frac{1}{{\mathrm{Ï}}_{base}}\frac{d{\mathrm{Ψ}}_{0}}{dt}â{R}_{a}{i}_{0},
ed, eq, and e0 are the d-axis, q-axis, and zero-sequence stator voltages, defined by
\left[\begin{array}{c}{e}_{d}\\ {e}_{q}\\ {e}_{0}\end{array}\right]={P}_{s}\left[\begin{array}{c}{v}_{a}\\ {v}_{b}\\ {v}_{c}\end{array}\right].
ψd, ψq, and ψ0 are the d-axis, q-axis, and zero-sequence stator flux linkages.
ωr is the per-unit rotor rotational speed.
id, iq, and i0 are the d-axis, q-axis, and zero-sequence stator currents, defined by
\left[\begin{array}{c}{i}_{d}\\ {i}_{q}\\ {i}_{0}\end{array}\right]={P}_{s}\left[\begin{array}{c}{i}_{a}\\ {i}_{b}\\ {i}_{c}\end{array}\right].
{e}_{fd}=\frac{1}{{\mathrm{Ï}}_{base}}\frac{d{\mathrm{Ψ}}_{fd}}{dt}+{R}_{fd}{i}_{fd},
{e}_{1d}=\frac{1}{{\mathrm{Ï}}_{base}}\frac{d{\mathrm{Ψ}}_{1d}}{dt}+{R}_{1d}{i}_{1d}=0,
{e}_{1}{}_{q}=\frac{1}{{\mathrm{Ï}}_{base}}\frac{d{\mathrm{Ψ}}_{1q}}{dt}+{R}_{1q}{i}_{1q}=0,
{e}_{2}{}_{q}=\frac{1}{{\mathrm{Ï}}_{base}}\frac{d{\mathrm{Ψ}}_{2q}}{dt}+{R}_{2q}{i}_{2q}=0,
efd is the field voltage.
e1d, e1q, and e2q are the voltages across the d-axis damper winding 1, q-axis damper winding 1, and q-axis damper winding 2. They are all equal to 0.
ψfd, ψ1d, ψ1q, and ψ2q are the magnetic fluxes linking the field circuit, d-axis damper winding 1, q-axis damper winding 1, and q-axis damper winding 2.
Rfd, R1d, R1q, and R2q are the resistances of rotor field circuit, d-axis damper winding 1, q-axis damper winding 1, and q-axis damper winding 2.
ifd, i1d, i1q, and i2q are the currents flowing in the field circuit, d-axis damper winding 1, q-axis damper winding 1, and q-axis damper winding 2.
The saturation equations are defined by
{\mathrm{Ï}}_{ad}={\mathrm{Ï}}_{d}+{L}_{l}{i}_{d},
{\mathrm{Ï}}_{aq}={\mathrm{Ï}}_{q}+{L}_{l}{i}_{q},
{\mathrm{Ï}}_{at}=\sqrt{{\mathrm{Ï}}_{ad}^{2}+{\mathrm{Ï}}_{aq}^{2}},
{K}_{s}=1
(If saturation is disabled),
{K}_{s}=f\left({\mathrm{Ï}}_{at}\right)
(If saturation is enabled),
{L}_{ad}={K}_{s}*{L}_{adu},
{L}_{aq}={K}_{s}*{L}_{aqu},
ψad is the d-axis air-gap or mutual flux linkage.
ψaq is the q-axis air-gap or mutual flux linkage.
ψat is the air-gap flux linkage.
Ks is the saturation factor.
Ladu is the unsaturated mutual inductance of the stator d-axis.
Lad is the mutual inductance of the stator d-axis.
Laqu is the unsaturated mutual inductance of the stator q-axis.
Laq is the mutual inductance of the stator q-axis.
The saturation factor function, f, is calculated from the per-unit open-circuit lookup table as:
{L}_{ad}=\frac{d{\mathrm{Ï}}_{at}}{d{i}_{fd}},
{V}_{ag}=g\left({i}_{fd}\right),
{L}_{ad}=\frac{dg\left({i}_{fd}\right)}{d{i}_{fd}}=\frac{d{V}_{ag}}{d{i}_{fd}},
where Vag is the per-unit air-gap voltage.
In per-unit,
{K}_{s}=\frac{{L}_{ad}}{{L}_{adu}},
{\mathrm{Ï}}_{at}={V}_{ag}
{K}_{s}=f\left({\mathrm{Ï}}_{at}\right).
{\mathrm{Ψ}}_{d}=â\left({L}_{ad}+{L}_{l}\right){i}_{d}\text{â}+{L}_{ad}{i}_{fd}+{L}_{ad}{i}_{1d},
\mathrm{Ψ}q=â\left({L}_{aq}+{L}_{l}\right){i}_{q}\text{â}+{L}_{aq}{i}_{1q}+{L}_{aq}{i}_{2q},
{\mathrm{Ψ}}_{0}=â{L}_{0}{i}_{0},
Ll is the stator leakage inductance.
Lad and Laq are the mutual inductances of the stator d-axis and q-axis.
{\mathrm{Ï}}_{fd}={L}_{ffd}{i}_{fd}+{L}_{f1d}{i}_{1d}â{L}_{ad}{i}_{d},
{\mathrm{Ï}}_{1d}={L}_{f1d}{i}_{fd}+{L}_{11d}{i}_{1d}â{L}_{ad}{i}_{d},
{\mathrm{Ï}}_{1q}={L}_{11q}{i}_{1q}+{L}_{aq}{i}_{2q}â{L}_{aq}{i}_{q},
{\mathrm{Ï}}_{2q}={L}_{aq}{i}_{1q}+{L}_{22q}{i}_{2q}â{L}_{aq}{i}_{q},
Lffd is the self-inductances of the rotor field circuit.
Lffd is the self-inductance of the rotor field circuit.
L11d is the self-inductance of the d-axis damper winding 1.
L11q is the self-inductance of the q-axis damper winding 1.
Lf1d is the rotor field circuit and d-axis damper winding 1 mutual inductance.
The inductances are defined by these equations:
{L}_{ffd}={L}_{ad}+{L}_{fd}
{L}_{f1d}={L}_{ffd}â{L}_{fd}
{L}_{11d}={L}_{f1d}+{L}_{1d}
{L}_{11q}={L}_{aq}+{L}_{1q}
{L}_{22q}={L}_{aq}+{L}_{2q}
The inductance equations assume that per-unit mutual inductance L12q = Laq, that is, the stator and rotor currents in the q-axis all link a single mutual flux represented by Laq.
{T}_{e}={\mathrm{Ψ}}_{d}{i}_{q}â{\mathrm{Ψ}}_{q}{i}_{d}.
Display Associated Base Values — Displays associated per-unit base values in the MATLAB Command Window.
Display Associated Initial Conditions — Displays associated initial conditions in the MATLAB Command Window.
Plot Open-Circuit Saturation (pu) — Plots air-gap voltage, Vag, versus field current, ifd, both measured in per-unit, in a MATLAB figure window. The plot contains three traces:
Unsaturated — Stator d-axis mutual inductance (unsaturated), Ladu you specify
Saturated — Per-unit open-circuit lookup table (Vag versus ifd) you specify
Derived — Open-circuit lookup table (per-unit) derived from the Per-unit open-circuit lookup table (Vag versus ifd) you specify. This data is used to calculate the saturation factor, Ks, versus magnetic flux linkage, ψat, characteristic.
Plot Saturation Factor (pu) — Plots saturation factor, Ks, versus magnetic flux linkage, ψat, both measured in per-unit, in a MATLAB figure window using the machine parameters. This parameter is derived from other parameters that you specify:
Stator zero-sequence voltage, pu_e0
Stator zero-sequence current, pu_i0
Electrical conserving port associated with the neutral point of the wye winding configuration.
To enable this port, set Zero sequence to Include.
Fundamental parameters — Specify impedance using fundamental parameters.
Standard parameters — Specify impedance using fundamental parameters and specify d-axis and q-axis time constants.
Field circuit current — Specify the field circuit in terms of current.
Zero sequence — Zero-sequence model
Include — Prioritize model fidelity. This model is the default zero-sequence model. An error occurs if you Include zero-sequence terms for simulations that use the Partitioning solver. For more information, see Increase Simulation Speed Using the Partitioning Solver.
Include and Specify parameterization by is set to Fundamental parameters — The Stator zero-sequence inductance, L0 parameter in the Impedances settings is visible.
Include and Specify parameterization by is set to Standard parameters — The zero-sequence reactance, X0 parameter in the Impedances settings is visible.
Exclude — The zero-sequence parameter in the Impedances settings is not visible.
Unsaturated stator d-axis mutual inductance. If Magnetic saturation representation is set to None, this is equivalent to the stator d-axis mutual inductance. This parameter must be greater than 0.
Unsaturated stator q-axis mutual inductance. If Magnetic saturation representation is set to None, this is equivalent to the stator q-axis mutual inductance. This parameter must be greater than 0.
Stator zero-sequence inductance. This parameter must be greater than 0.
This parameter is visible only if the Specify parameterization by parameter is set to Fundamental parameters and the Zero Sequence parameter to Include.
zero-sequence reactance, X0 — Zero-sequence reactance
Zero-sequence reactance. This parameter must be greater than 0.
This parameter is visible only if the Specify parameterization by parameter is set to Standard parameters and the Zero Sequence parameter to Include.
q-axis transient reactance, Xq' — q-axis transient reactance
q-axis transient reactance. This parameter must be greater than 0.
q-axis transient open-circuit, Tq0' — q-axis transient open-circuit
q-axis transient open-circuit time constant. This parameter must be:
Greater than q-axis subtransient open-circuit, Tq0''.
q-axis transient short-circuit, Tq' — q-axis transient short-circuit
q-axis transient short-circuit time constant. This parameter must be:
Greater than q-axis subtransient short-circuit, Tq''.
Block magnetic saturation model:
Open-circuit lookup table (v versus i)
If you set this parameter to Open-circuit lookup table (v versus i), related parameters are visible.
Per-unit field current saturation data, ifd — Per-unit field current saturation data
[0, .48, .76, 1.38, 1.79] (default)
Field current, ifd, data that populates the air-gap voltage, Vag, versus field current, ifd, lookup table. This parameter must contain a vector with at least five elements.
This parameter is visible only if the Magnetic saturation representation parameter is set to Open-circuit lookup table (v versus i).
Per-unit air-gap voltage saturation data, Vag — Per-unit air-gap voltage saturation data
[0, .8, 1.08, 1.31, 1.4] (default)
Air-gap voltage, Vag, data that populates the air-gap voltage, Vag, versus field current, ifd, lookup table. This parameter must contain a vector with at least five elements.
Synchronous Machine Field Circuit | Synchronous Machine Measurement | Synchronous Machine Model 1.0 | Synchronous Machine Model 2.1 | Synchronous Machine Salient Pole
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Classification of higher rank orbit closures in ${\mathcal H^{\mathrm{odd}}(4)}$ | EMS Press
Classification of higher rank orbit closures in
{\mathcal H^{\mathrm{odd}}(4)}
The moduli space of genus 3 translation surfaces with a single zero has two connected components. We show that in the odd connected component
{\mathcal H^{\mathrm{odd}}(4)}
{GL^+(2,\mathbb R)}
orbit closures are closed orbits, the Prym locus
{\tilde{\mathcal{Q}}(3,-1^3)}
{\mathcal H^{\mathrm{odd}}(4)}
Together with work of Matheus–Wright, this implies that there are only finitely many non-arithmetic closed orbits (Teichmüller curves) in
\mathcal H^{\mathrm{odd}}(4)
outside of the Prym locus.
David Aulicino, Duc-Manh Nguyen, Alex Wright, Classification of higher rank orbit closures in
{\mathcal H^{\mathrm{odd}}(4)}
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Suppose you were conducting a survey to determine what portion of voters in your town supports a particular candidate for mayor. Consider each of the following methods for sampling the voting population of your town. State whether each is likely to produce a representative sample and explain your reasoning.
Samples from specific situations are not likely to create representative random samples.
Call one number from each page of an online phone listing for your town between noon and
2
Not likely; this samples the population of people with phone numbers listed online that are home midday.
Also, we only get responses from those who agree to participate.
Survey each person leaving a local grocery store.
Survey each person leaving a local movie theater.
Walk around downtown and survey every fourth person you see.
Could you make a representative sample by surveying a few people from each of the situations described in parts (a) through (d) above? Explain, and consider what problems might still remain.
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An evolutionary double-well problem | EMS Press
Department of Mathematics, University of Sussex, Falmer, Brighton BN1 9RF, United Kingdom
We establish the existence theorem and study the long time behaviour of the following PDE problem:
\left\{\begin{matrix} u_{t}−\mathrm{div}\mathrm{∇}W(\mathrm{∇}u)−f(x) = 0\:\text{in }\Omega \times (0,−\infty ), \\ \mathrm{∇}W(\mathrm{∇}u) \cdot \mathbf{n}|_{\partial \Omega \times (0,\infty )} = 0, \\ u(x,0) = u_{0}(x)\:\text{in }\Omega \\ \end{matrix}\right.
where W is a specially given quasiconvex double-well function and
f \in L^{2}(\Omega )
is a given function independent of time t. In particular, the existence theorem is established for general given source term f, the long time behaviour is analyzed under the assumption that
\int _{\Omega }f(x)\:\mathrm{d}x = 0
The system is an evolutionary quasimonotone system. We believe that the existence of solutions established here is stronger than the usual Young Measure solution and is the first of its kind. The existence of a compact ω-limit set as
t\rightarrow \infty
is also established under some non-restrictive conditions.
Tang Qi, Zhang Kewei, An evolutionary double-well problem. Ann. Inst. H. Poincaré Anal. Non Linéaire 24 (2007), no. 3, pp. 341–359
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Form skew-symmetric matrix - MATLAB skewdec
skewdec
Form skew-symmetric matrix
X = skewdec(m,n)
X = skewdec(m,n) forms the m-by-m skew-symmetric matrix
\left[\begin{array}{cccc}0& -\left(n+1\right)& -\left(n+2\right)& \dots \\ \left(n+1\right)& 0& -\left(n+3\right)& \dots \\ \left(n+2\right)& \left(n+3\right)& 0& \dots \\ \dots & \dots & \dots & \dots \\ \dots & \dots & \dots & \dots \end{array}\right]
This function is useful to define skew-symmetric matrix variables. In this case, set n to the number of decision variables already used.
Create a 3-by-3 skew-symmetric matrix for an LMI problem in which n = 2. Display the matrix to verify its form.
X = skewdec(3,2)
decinfo | lmivar
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(Redirected from Magnets)
2.4 Modelling magnets
4 Medical issues and safety
5 Magnetizing ferromagnets
6 Demagnetizing ferromagnets
7.3 Rare-earth magnets
7.6 Rare-earth-free permanent magnets
9 Units and calculations
9.2.1 Pull force of a single magnet
9.2.3 Force between two nearby magnetized surfaces of area A
9.2.5 Force between two cylindrical magnets
The current[update] cheapest permanent magnets, allowing for field strengths, are flexible and ceramic magnets, but these are also among the weakest types. The ferrite magnets are mainly low-cost magnets since they are made from cheap raw materials: iron oxide and Ba- or Sr-carbonate. However, a new low cost magnet, Mn–Al alloy,[34] has been developed and is now dominating the low-cost magnets field. It has a higher saturation magnetization than the ferrite magnets. It also has more favorable temperature coefficients, although it can be thermally unstable. Neodymium–iron–boron (NIB) magnets are among the strongest. These cost more per kilogram than most other magnetic materials but, owing to their intense field, are smaller and cheaper in many applications.[35]
{\displaystyle F={{B^{2}A} \over {2\mu _{0}}}}
{\displaystyle m={{B^{2}A} \over {2\mu _{0}g}},}
{\displaystyle F={{\mu q_{m1}q_{m2}} \over {4\pi r^{2}}}}
{\displaystyle F={\frac {\mu _{0}H^{2}A}{2}}={\frac {B^{2}A}{2\mu _{0}}}}
The force between two identical cylindrical bar magnets placed end to end at large distance
{\displaystyle z\gg R}
is approximately:[dubious – discuss],[41]
{\displaystyle F\simeq \left[{\frac {B_{0}^{2}A^{2}\left(L^{2}+R^{2}\right)}{\pi \mu _{0}L^{2}}}\right]\left[{\frac {1}{z^{2}}}+{\frac {1}{(z+2L)^{2}}}-{\frac {2}{(z+L)^{2}}}\right]}
{\displaystyle B_{0}\,=\,{\frac {\mu _{0}}{2}}M}
relates the flux density at the pole to the magnetization of the magnet.
For two cylindrical magnets with radius
{\displaystyle R}
{\displaystyle L}
, with their magnetic dipole aligned, the force can be asymptotically approximated at large distance
{\displaystyle z\gg R}
by,[43]
{\displaystyle F(z)\simeq {\frac {\pi \mu _{0}}{4}}M^{2}R^{4}\left[{\frac {1}{z^{2}}}+{\frac {1}{(z+2L)^{2}}}-{\frac {2}{(z+L)^{2}}}\right]}
{\displaystyle M}
is the magnetization of the magnets and
{\displaystyle z}
is the gap between the magnets. A measurement of the magnetic flux density very close to the magnet
{\displaystyle B_{0}}
{\displaystyle M}
approximately by the formula
{\displaystyle B_{0}={\frac {\mu _{0}}{2}}M}
{\displaystyle m=MV}
{\displaystyle V}
is the volume of the magnet. For a cylinder, this is
{\displaystyle V=\pi R^{2}L}
{\displaystyle z\gg L}
, the point dipole approximation is obtained,
{\displaystyle F(x)={\frac {3\pi \mu _{0}}{2}}M^{2}R^{4}L^{2}{\frac {1}{z^{4}}}={\frac {3\mu _{0}}{2\pi }}M^{2}V^{2}{\frac {1}{z^{4}}}={\frac {3\mu _{0}}{2\pi }}m_{1}m_{2}{\frac {1}{z^{4}}}}
Look up magnet in Wiktionary, the free dictionary.
Wikimedia Commons has media related to Magnets.
Retrieved from "https://en.wikipedia.org/w/index.php?title=Magnet&oldid=1082070372"
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Small noise spectral gap asymptotics for a large system of nonlinear diffusions | EMS Press
Small noise spectral gap asymptotics for a large system of nonlinear diffusions
L^2
spectral gap of a large system of strongly coupled diffusions on unbounded state space and subject to a double-well potential. This system can be seen as a spatially discrete approximation of the stochastic Allen–Cahn equation on the one-dimensional torus. We prove upper and lower bounds for the leading term of the spectral gap in the small temperature regime with uniform control in the system size. The upper bound is given by an Eyring–Kramers-type formula. The lower bound is proven to hold also for the logarithmic Sobolev constant. We establish a sufficient condition for the asymptotic optimality of the upper bound and show that this condition is fullled under suitable assumptions on the growth of the system size. Our results can be reformulated in terms of a semiclassical Witten Laplacian in large dimension.
Giacomo Di Gesù, Dorian Le Peutrec, Small noise spectral gap asymptotics for a large system of nonlinear diffusions. J. Spectr. Theory 7 (2017), no. 4, pp. 939–984
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Due to natural variability in manufacturing, a
12
12
ounces of soda. The quality control department at a soda factory allows cans to hold a little more or a little less. According to specifications, a soda-can filling machine must fill each can with a mean of
12
ounces of soda and a standard deviation of
0.33
ounces. The amount of soda in a can filled by filling machines can often be modeled with a normal distribution.
Use your calculator to create a graph of a normal distribution using normalpdf. Sketch the graph. An appropriate value for the maximum of the relative frequency axis is Ymax
= 1.5
How often do you actually get a
12
oz can of soda containing more than
12
oz?
The mean is the center (middle) of a normal distribution.
What percent of cans contain between
11.5
12.5
ounces of soda? Shade your diagram from part (a) to represent these cans.
11.5, 12.5,
mean, standard deviation)
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Days Payable Outstanding (DPO) Definition
What Does DPO Tell You?
Days payable outstanding (DPO) is a financial ratio that indicates the average time (in days) that a company takes to pay its bills and invoices to its trade creditors, which may include suppliers, vendors, or financiers. The ratio is typically calculated on a quarterly or annual basis, and it indicates how well the company’s cash outflows are being managed.
A company with a higher value of DPO takes longer to pay its bills, which means that it can retain available funds for a longer duration, allowing the company an opportunity to use those funds in a better way to maximize the benefits. A high DPO, however, may also be a red flag indicating an inability to pay its bills on time.
Days payable outstanding (DPO) computes the average number of days a company needs to pay its bills and obligations.
Companies that have a high DPO can delay making payments and use the available cash for short-term investments as well as to increase their working capital and free cash flow.
However, higher values of DPO, though desirable, may not always be a positive for the business as it may signal a cash shortfall and inability to pay.
Formula for Days Payable Outstanding (DPO)
\begin{aligned} &\text{DPO} = \frac{\text{Accounts Payable}\times\text{Number of Days}}{\text{COGS}}\\ &\textbf{where:}\\ &\text{COGS}=\text{Cost of Goods Sold} \\ &\qquad\ \ \, \,= \text{Beginning Inventory} + \text{P} -\text{Ending Inventory}\\ &\text{P}=\text{Purchases} \end{aligned}
DPO=COGSAccounts Payable×Number of Dayswhere:COGS=Cost of Goods Sold =Beginning Inventory+P−Ending Inventory
To manufacture a salable product, a company needs raw material, utilities, and other resources. In terms of accounting practices, the accounts payable represents how much money the company owes to its supplier(s) for purchases made on credit.
Additionally, there is a cost associated with manufacturing the salable product, and it includes payment for utilities like electricity and employee wages. This is represented by cost of goods sold (COGS), which is defined as the cost of acquiring or manufacturing the products that a company sells during a period. Both of these figures represent cash outflows and are used in calculating DPO over a period of time.
The number of days in the corresponding period is usually taken as 365 for a year and 90 for a quarter. The formula takes account of the average per day cost being borne by the company for manufacturing a salable product. The numerator figure represents payments outstanding. The net factor gives the average number of days taken by the company to pay off its obligations after receiving the bills.
Two different versions of the DPO formula are used depending upon the accounting practices. In one of the versions, the accounts payable amount is taken as the figure reported at the end of the accounting period, like “at the end of fiscal year/quarter ending Sept. 30.” This version represents the DPO value as of the mentioned date.
In another version, the average value of beginning AP and ending AP is taken, and the resulting figure represents the DPO value during that particular period. COGS remains the same in both versions.
Generally, a company acquires inventory, utilities, and other necessary services on credit. It results in accounts payable (AP), a key accounting entry that represents a company's obligation to pay off the short-term liabilities to its creditors or suppliers. Beyond the actual dollar amount to be paid, the timing of the payments—from the date of receiving the bill till the cash actually going out of the company’s account—also becomes an important aspect of the business. DPO attempts to measure this average time cycle for outward payments and is calculated by taking the standard accounting figures into consideration over a specified period of time.
Companies having high DPO can use the available cash for short-term investments and to increase their working capital and free cash flow (FCF). However, higher values of DPO may not always be a positive for the business. If the company takes too long to pay its creditors, it risks jeopardizing its relations with the suppliers and creditors who may refuse to offer the trade credit in the future or may offer it on terms that may be less favorable to the company. The company may also be losing out on any discounts on timely payments, if available, and it may be paying more than necessary.
Additionally, a company may need to balance its outflow tenure with that of the inflow. Imagine if a company allows a 90-day period for its customers to pay for the goods they purchase but has only a 30-day window to pay its suppliers and vendors. This mismatch will result in the company being prone to cash crunch frequently. Companies must strike a delicate balance with DPO.
A high DPO can indicate a company that is using capital resourcefully but it can also show that the company is struggling to pay its creditors.
Typical DPO values vary widely across different industry sectors and it is not worthwhile comparing these values across different sector companies. A firm's management will instead compare its DPO to the average within its industry to see if it is paying its vendors too quickly or too slowly. Depending upon the various global and local factors, like the overall performance of the economy, region, and sector, plus any applicable seasonal impacts, the DPO value of a particular company can vary significantly from year to year, company to company, and industry to industry.
DPO value also forms an integral part of the formula used for calculating the cash conversion cycle (CCC), another key metric that expresses the length of time that a company takes to convert the resource inputs into realized cash flows from sales. While DPO focuses on the current outstanding payable by the business, the superset CCC follows the entire cash time-cycle as the cash is first converted into inventory, expenses, and accounts payable, through to sales and accounts receivable, and then back into cash in hand when received.
Example of How DPO Is Used
As a historical example, the leading retail corporation Walmart (WMT) had accounts payable worth $49.1 billion and cost of sales (cost of goods sold) worth $420.3 billion for the fiscal year ending Jan. 31, 2021. These figures are available in the annual financial statement and balance sheet of the company. Taking the number of days as 365 for annual calculation, the DPO for Walmart comes to [ (49.1 x 365) / 420.1 ] = 42.7 days.
Similar calculations can be used for technology leader Microsoft (MSFT), which had $2.8 billion as AP and $41.3 billion as COGS, leading to a DPO value of 24.7 days.
It indicates that during the fiscal year ending in 2021, Walmart paid its invoices around 43 days after receiving the bills, while Microsoft took around 25 days, on average, to pay its bills.
A look at similar figures for the online retail giant Amazon (AMZN), which had an AP of $72.5 billion and COGS of $233.3 billion for the fiscal year 2020, reveals a very high value of 113.4 days. Such high value of DPO is attributed to the working model of Amazon, which roughly has 50% of its sales being supplied by third-party sellers. Amazon instantly receives funds in its account for the sale of goods that are actually supplied by third-party sellers using Amazon’s online platform.
However, it doesn’t pay the sellers immediately after the sale but may send accumulated payments based on a weekly/monthly or threshold-based payment cycle. This working mechanism allows Amazon to hold onto the cash for a longer period of time, and the leading online retailer ends up with a significantly higher DPO.
Limitations of DPO
While DPO is useful in comparing relative strength among companies, there is no clear-cut figure for what constitutes a healthy days payable outstanding, as the DPO varies significantly by industry, competitive positioning of the company, and its bargaining power. Large companies with a strong power of negotiation are able to contract for better terms with suppliers and creditors, effectively producing lower DPO figures than they would have otherwise.
What Does Days Payable Outstanding Mean in Accounting?
As a financial ratio, days of payable outstanding (DPO) shows the amount of time that companies take to pay financiers, creditors, vendors, or suppliers. The DPO may indicate a few things, namely, how a company is managing its cash, or the means for a company to utilize this cash towards short-term investments that in turn may amplify their cash flow. The DPO is measured on a quarterly or annual term.
To calculate days of payable outstanding (DPO), the following formula is applied: DPO = Accounts Payable X Number of Days/Cost of Goods Sold (COGS). Here, COGS refers to beginning inventory plus purchases subtracting the ending inventory. Accounts payable, on the other hand, refers to company purchases that were made on credit that are due to its suppliers.
Days payable outstanding (DPO) is the average time for a company to pay its bills. By contrast, days sales outstanding (DSO) is the average length of time for sales to be paid back to the company. When a DSO is high, it indicates that the company is waiting extended periods to collect money for products that it sold on credit. By contrast, a high DPO could be interpreted multiple ways, either indicating that the company is utilizing its cash on hand to create more working capital, or indicating poor management of free cash flow.
Walmart. "Form 10-K for the Fiscal Year Ending January 31, 2021," Pages 37, 54, and 58. Accessed Jan. 24, 2022.
Microsoft. "Annual Report 2021." Accessed Jan. 24, 2022.
Amazon. "Annual Report 2020," Pages iii, 27, 38, and 41. Accessed Jan. 24, 2022.
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On the Schwartz correspondence for Gelfand pairs of polynomial growth | EMS Press
On the Schwartz correspondence for Gelfand pairs of polynomial growth
(G,K)
be a Gelfand pair, with
G
a Lie group of polynomial growth, and let
\Sigma\subset\mathbb R^\ell
be a homeomorphic image of the Gelfand spectrum, obtained by choosing a generating system
D_1,\dots,D_\ell
G
-invariant differential operators on
G/K
and associating to a bounded spherical function
\phi
\ell
-tuple of its eigenvalues under the action of the
D_j
We say that property (S) holds for
(G,K)
if the spherical transform maps the bi-
K
-invariant Schwartz space
\mathcal S(K\backslash G/K)
isomorphically onto
\mathcal S(\Sigma)
, the space of restrictions to
\Sigma
of the Schwartz functions on
\mathbb R^\ell
. This property is known to hold for many nilpotent pairs, i.e., Gelfand pairs where
G=K\ltimes N
N
nilpotent.
In this paper we enlarge the scope of this analysis outside the range of nilpotent pairs, stating the basic setting for general pairs of polynomial growth and then focussing on strong Gelfand pairs.
Francesca Astengo, Bianca Di Blasio, Fulvio Ricci, On the Schwartz correspondence for Gelfand pairs of polynomial growth. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. 32 (2021), no. 1, pp. 79–96
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Effects of a Storytelling Program with Drama Techniques to Understand and Accept Intellectual Disability in Students 6 - 7 Years Old. A Pilot Study
Department of Physical Education and Sport Sciences, Aristotle University of Thessaloniki, Serres, Greece
A very crucial issue in special education, as well as in education in general, is the integration itself in the educational system. Although integration oriented schools offer a wide range of opportunities for social integration, incidents of social rejection cannot always be eradicated. Given that, the knowledge and understanding of disability creates a tendency for positive attitudes toward it, the evaluation and timely implementation of specially designed awareness interventions is mandatory and it should take place to help children with and without disabilities coexist harmonically in and out of school. This study aims at determining the effect of a 12 teaching hours intervention program on the potential change in attitudes toward disability of students in the first grade of primary school. The intervention program consisted of reading and theatrical play of a story tale referring to a small turtle born with Down syndrome. The results showed a significant positive effect on the understanding and the acceptance of intellectual disability.
Intervention, Inclusive Education, Theatrical Play, Physical Education, Dramatic Environment
{\eta }_{p}^{2}
{\eta }_{p}^{2}
Giagazoglou, P., & Papadaniil, M. (2018). Effects of a Storytelling Program with Drama Techniques to Understand and Accept Intellectual Disability in Students 6 - 7 Years Old. A Pilot Study. Advances in Physical Education, 8, 224-237. https://doi.org/10.4236/ape.2018.82020
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15. Hodkinson, A. (2007). Inclusive Education and the Cultural Representation of Disability and Disabled People: Recipe for Disaster or Catalyst of Change? An Examination of Non-Disabled Primary School Children’s Attitudes to Children with Disabilities. Research in Education, 77, 56-76. https://doi.org/10.7227/RIE.77.5 [Paper reference 2]
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A differential algebra and the homotopy type of the complement of a toric arrangement | EMS Press
A differential algebra and the homotopy type of the complement of a toric arrangement
We show that the rational homotopy type of the complement of a toric arrangement is completely determined by two sets of discrete data. This is obtained by introducing a differential graded algebra over Q whose minimal model is equivalent to the Sullivan minimal model of
\mathcal A
Corrado De Concini, Giovanni Gaiffi, A differential algebra and the homotopy type of the complement of a toric arrangement. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. 32 (2021), no. 1, pp. 1–21
|
Cohomological invariants and the classifying space for proper actions | EMS Press
Cohomological invariants and the classifying space for proper actions
We investigate two open questions in a cohomology theory relative to the family of finite subgroups. The problem of whether the
\mathbb{F}
-cohomological dimension is subadditive is reduced to extensions by groups of prime order. We show that every finitely generated regular branch group has infinite rational cohomological dimension. Moreover, we prove that the first Grigorchuk group
\mathfrak{G}
is not contained in Kropholler’s class
{\scriptstyle{\rm H}}\mathfrak F
Giovanni Gandini, Cohomological invariants and the classifying space for proper actions. Groups Geom. Dyn. 6 (2012), no. 4, pp. 659–675
|
<var>N</var>-step energy of maps and the fixed-point property of random groups | EMS Press
<var>N</var>-step energy of maps and the fixed-point property of random groups
We prove that a random group of the graph model associated with a sequence of expanders has the fixed-point property for a certain class of CAT(0) spaces. We use Gromov’s criterion for the fixed-point property in terms of the growth of
n
-step energy of equivariant maps from a finitely generated group into a CAT(0) space, for which we give a detailed proof. We estimate a relevant geometric invariant of the tangent cones of the Euclidean buildings associated with the groups PGL(
m,\mathbb{Q}_r
), and deduce from the general result above that the same random group has the fixed-point property for all of these Euclidean buildings with
m
bounded from above.
Hiroyasu Izeki, Takefumi Kondo, Shin Nayatani, <var>N</var>-step energy of maps and the fixed-point property of random groups. Groups Geom. Dyn. 6 (2012), no. 4, pp. 701–736
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Antoine works in the paint department at his local hardware store. He is trying to create a new color of paint using two different colors in stock: powder blue, which is made with
2\%
blue (the rest is white), and spring blue, which is
10\%
blue (the rest is white). He wants to end up with one gallon of paint made with
4\%
blue. How much of each color should he use?
p =
the amount of powder blue paint and
y =
the amount of spring blue paint. Write and solve a system of equations.
One equation says that the total amount of paint used equals one gallon.
The other equation represents the percents of the mixture:
0.02x + 0.1y = 0.04(1)
|
Green's function asymptotics near the internal edges of spectra of periodic elliptic operators. Spectral gap interior | EMS Press
Green's function asymptotics near the internal edges of spectra of periodic elliptic operators. Spectral gap interior
Precise asymptotics known for the Green function of the Laplacian have found their analogs for bounded below periodic elliptic operators of the second-order below and at the bottom of the spectrum. Due to the band-gap structure of the spectra of such operators, the question arises whether similar results can be obtained near or at the edges of spectral gaps. In a previous work, two of the authors considered the case of a spectral edge. The main result of this article is nding such asymptotics near a gap edge, for “generic” periodic elliptic operators of second-order with real coecients in dimension
d ≥ 2
, when the gap edge occurs at a symmetry point of the Brillouin zone.
Minh Kha, Peter Kuchment, Andrew Raich, Green's function asymptotics near the internal edges of spectra of periodic elliptic operators. Spectral gap interior. J. Spectr. Theory 7 (2017), no. 4, pp. 1171–1233
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Existence, covolumes and infinite generation of lattices for Davis complexes | EMS Press
Existence, covolumes and infinite generation of lattices for Davis complexes
\Sigma
be the Davis complex for a Coxeter system
(W,S)
. The automorphism group
G
\Sigma
is naturally a locally compact group, and a simple combinatorial condition due to Haglund–Paulin and White determines when
G
is nondiscrete. The Coxeter group
W
may be regarded as a uniform lattice in
G
. We show that many such
G
also admit a nonuniform lattice
\Gamma
, and an infinite family of uniform lattices with covolumes converging to that of
\Gamma
. It follows that the set of covolumes of lattices in
G
is nondiscrete. We also show that the nonuniform lattice
\Gamma
is not finitely generated. Examples of
\Sigma
to which our results apply include buildings and non-buildings, and many complexes of dimension greater than 2. To prove these results, we introduce a new tool, that of “group actions on complexes of groups”, and use this to construct our lattices as fundamental groups of complexes of groups with universal cover
\Sigma
Anne Thomas, Existence, covolumes and infinite generation of lattices for Davis complexes. Groups Geom. Dyn. 6 (2012), no. 4, pp. 765–801
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$\Gamma$-convergence for a class of action functionals induced by gradients of convex functions | EMS Press
\Gamma
-convergence for a class of action functionals induced by gradients of convex functions
Given a real function
f
, the rate function for the large deviations of the diffusion process of drift
\nabla f
given by the Freidlin–Wentzell theorem coincides with the time integral of the energy dissipation for the gradient flow associated with
f
. This paper is concerned with the stability in the hilbertian framework of this common action functional when
f
varies. More precisely, we show that if
(f_h)_h
\lambda
-convex for some
\lambda \in \mathbb R
and converges towards
f
in the sense of Mosco convergence, then the related functionals
\Gamma
-converge in the strong topology of curves.
Luigi Ambrosio, Aymeric Baradat, Yann Brenier,
\Gamma
-convergence for a class of action functionals induced by gradients of convex functions. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. 32 (2021), no. 1, pp. 97–108
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Numerical Study of Impingement Cooling of Aviation Kerosene at Supercritical Conditions | J. Heat Transfer | ASME Digital Collection
Numerical Study of Impingement Cooling of Aviation Kerosene at Supercritical Conditions
High Temperature Gas Dynamics,
Contributed by the Heat Transfer Division of ASME for publication in the JOURNAL OF HEAT TRANSFER. Manuscript received January 10, 2018; final manuscript received June 4, 2018; published online July 23, 2018. Assoc. Editor: Amy Fleischer.
Xing, Y., Zhong, F., and Zhang, X. (July 23, 2018). "Numerical Study of Impingement Cooling of Aviation Kerosene at Supercritical Conditions." ASME. J. Heat Transfer. November 2018; 140(11): 112201. https://doi.org/10.1115/1.4040612
In the present paper, numerical study of flow and heat transfer properties of RP-3 kerosene at liquid and supercritical conditions in an impingement model is conducted with renormalization group (RNG)
k−ε
turbulence model and a ten-species surrogate of kerosene. The independence of grids is first studied, and the numerical results are compared with experimental data for validation. Characteristics of flow and heat transfer of kerosene flow in the impingement model are studied with different inlet mass flow rates and different inlet temperatures. The velocity and temperature field show similar profile compared to that of air impingement. The heat transfer rates increase first with the increasing of inlet temperature and then decrease suddenly when the inlet temperature is 500 K.
Aerospace, Heat transfer enhancement, Thermophysical properties, Heat transfer
Aviation, Flow (Dynamics), Heat transfer, Temperature, Jets, Impingement cooling, Heat transfer coefficients, Pressure
Survey of Heat Transfer to Near-Critical Fluids
,” National Aeronautics and Space Administration, Washington, DC, Technical Note No.
Supercritical (Subcritical) Fluid Behavior and Modeling: Drops, Streams, Shear and Mixing Layers, and Sprays
Modeling of Supercritical Vaporization, Mixing and Combustion Processes in Liquid-Fueled Propulsion System
Heat Transfer to Kerosene at Supercritical Pressure in Small-Diameter Tube With Large Heat Flux
Numerical Study of Heat Transfer Deterioration of Turbulent Supercritical Kerosene Flow in Heated Circular Tube
A Coupled Heat Transfer Analysis With Effects of Catalytic Cracking of Kerosene for Actively Cooled Supersonic Combustor
Convective Heat Transfer in Impinging Gas-Jet Systems
,” Lecture Series 2000–03, von Karman Institute for Fluid Dynamics, Rhode Saint Genese, Belgium, pp.
,” Lecture Series 2000–03, von Karman Institute for Fluid Dynamics, Rhode Saint Genese, Belgium, pp. 34–57.
Effect of Jet Hole Arrays Arrangement on Impingement Heat Transfer
, The Louisiana State University, Baton Rouge, LA.http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.606.6991&rep=rep1&type=pdf
Effects of Jet Plate Size and Plate Spacing on the Stagnation Nusselt Number for a Confined Circular Air Jet Impinging on a Flat Surface
Thermal Cracking and Heat Sink Capacity of Aviation Kerosene Under Supercritical Conditions
Influence of Spanwise Pitch on Local Heat Transfer Distribution for In-Line Arrays of Circular Jets With Spent Air Flow in Two Opposite Directions
Annerfeldt
On the Correspondence Between Flow Structures and Convective Heat Transfer Augmentation for Multiple Jet Impingement
Jet Array Impingement With Crossflow-Correlation of Streamwise Resolved Flow and Heat Transfer Distributions
Investigation of Vaporized Kerosene Injection and Combustion in a Supersonic Model Combustor
Conjugate Heat Transfer Computational Fluid Dynamic Predictions of Impingement Heat Transfer: The Influence of Hole Pitch to Diameter Ratio X / D at Constant Impingement Gap Z
|
NP-completeness - Wikipedia
it is a problem for which the correctness of each solution can be verified quickly (namely, in polynomial time) and a brute-force search algorithm can find a solution by trying all possible solutions.
the problem can be used to simulate every other problem for which we can verify quickly that a solution is correct. In this sense, NP-complete problems are the hardest of the problems to which solutions can be verified quickly. If we could find solutions of some NP-complete problem quickly, we could quickly find the solutions of every other problem to which a given solution can be easily verified.
{\displaystyle \scriptstyle C}
is NP-complete if:
{\displaystyle \scriptstyle C}
Every problem in NP is reducible to
{\displaystyle \scriptstyle C}
in polynomial time.[3]
{\displaystyle \scriptstyle C}
can be shown to be in NP by demonstrating that a candidate solution to
{\displaystyle \scriptstyle C}
can be verified in polynomial time.
A consequence of this definition is that if we had a polynomial time algorithm (on a UTM, or any other Turing-equivalent abstract machine) for
{\displaystyle \scriptstyle C}
, we could solve all problems in NP in polynomial time.
NP-complete problems[edit]
Solving NP-complete problems[edit]
At present, all known algorithms for NP-complete problems require time that is superpolynomial in the input size, in fact exponential in
{\displaystyle O(n^{k})}
[clarify] for some
{\displaystyle k>0}
and it is unknown whether there are any faster algorithms.
One example of a heuristic algorithm is a suboptimal
{\displaystyle \scriptstyle O(n\log n)}
greedy coloring algorithm used for graph coloring during the register allocation phase of some compilers, a technique called graph-coloring global register allocation. Each vertex is a variable, edges are drawn between variables which are being used at the same time, and colors indicate the register assigned to each variable. Because most RISC machines have a fairly large number of general-purpose registers, even a heuristic approach is effective for this application.
Completeness under different types of reduction[edit]
Another type of reduction is polynomial-time Turing reduction. A problem
{\displaystyle \scriptstyle X}
is polynomial-time Turing-reducible to a problem
{\displaystyle \scriptstyle Y}
if, given a subroutine that solves
{\displaystyle \scriptstyle Y}
in polynomial time, one could write a program that calls this subroutine and solves
{\displaystyle \scriptstyle X}
in polynomial time. This contrasts with many-one reducibility, which has the restriction that the program can only call the subroutine once, and the return value of the subroutine must be the return value of the program.
Another type of reduction that is also often used to define NP-completeness is the logarithmic-space many-one reduction which is a many-one reduction that can be computed with only a logarithmic amount of space. Since every computation that can be done in logarithmic space can also be done in polynomial time it follows that if there is a logarithmic-space many-one reduction then there is also a polynomial-time many-one reduction. This type of reduction is more refined than the more usual polynomial-time many-one reductions and it allows us to distinguish more classes such as P-complete. Whether under these types of reductions the definition of NP-complete changes is still an open problem. All currently known NP-complete problems are NP-complete under log space reductions. All currently known NP-complete problems remain NP-complete even under much weaker reductions such as
{\displaystyle AC_{0}}
reductions and
{\displaystyle NC_{0}}
reductions. Some NP-Complete problems such as SAT are known to be complete even under polylogarithmic time projections.[6] It is known, however, that AC0 reductions define a strictly smaller class than polynomial-time reductions.[7]
According to Donald Knuth, the name "NP-complete" was popularized by Alfred Aho, John Hopcroft and Jeffrey Ullman in their celebrated textbook "The Design and Analysis of Computer Algorithms". He reports that they introduced the change in the galley proofs for the book (from "polynomially-complete"), in accordance with the results of a poll he had conducted of the theoretical computer science community.[8] Other suggestions made in the poll[9] included "Herculean", "formidable", Steiglitz's "hard-boiled" in honor of Cook, and Shen Lin's acronym "PET", which stood for "probably exponential time", but depending on which way the P versus NP problem went, could stand for "provably exponential time" or "previously exponential time".[10]
"Solving NP-complete problems requires exponential time." First, this would imply P ≠ NP, which is still an unsolved question. Further, some NP-complete problems actually have algorithms running in superpolynomial, but subexponential time such as O(2√nn). For example, the independent set and dominating set problems for planar graphs are NP-complete, but can be solved in subexponential time using the planar separator theorem.[12]
^ For example, simply assigning true to each variable renders the 18th conjunct
{\displaystyle {\overline {m}}\lor {\overline {r}}\lor {\overline {s}}}
(and hence the complete formula) false.
^ J. van Leeuwen (1998). Handbook of Theoretical Computer Science. Elsevier. p. 84. ISBN 978-0-262-72014-4.
^ Garey, Michael R.; Johnson, D. S. (1979). Victor Klee (ed.). Computers and Intractability: A Guide to the Theory of NP-Completeness. A Series of Books in the Mathematical Sciences. San Francisco, Calif.: W. H. Freeman and Co. pp. x+338. ISBN 978-0-7167-1045-5. MR 0519066.
^ Agrawal, M.; Allender, E.; Rudich, Steven (1998). "Reductions in Circuit Complexity: An Isomorphism Theorem and a Gap Theorem". Journal of Computer and System Sciences. 57 (2): 127–143. doi:10.1006/jcss.1998.1583. ISSN 1090-2724.
^ Agrawal, M.; Allender, E.; Impagliazzo, R.; Pitassi, T.; Rudich, Steven (2001). "Reducing the complexity of reductions". Computational Complexity. 10 (2): 117–138. doi:10.1007/s00037-001-8191-1. ISSN 1016-3328. S2CID 29017219.
^ Don Knuth, Tracy Larrabee, and Paul M. Roberts, Mathematical Writing Archived 2010-08-27 at the Wayback Machine § 25, MAA Notes No. 14, MAA, 1989 (also Stanford Technical Report, 1987).
^ Knuth, D. F. (1974). "A terminological proposal". SIGACT News. 6 (1): 12–18. doi:10.1145/1811129.1811130. S2CID 45313676.
^ See the poll, or [1] Archived 2011-06-07 at the Wayback Machine.
^ Ball, Philip. "DNA computer helps travelling salesman". doi:10.1038/news000113-10.
^ Bern (1990); Deĭneko, Klinz & Woeginger (2006); Dorn et al. (2005) harvtxt error: no target: CITEREFDornPenninksBodlaenderFomin2005 (help); Lipton & Tarjan (1980).
^ Hemaspaandra, L. A.; Williams, R. (2012). "SIGACT News Complexity Theory Column 76". ACM SIGACT News. 43 (4): 70. doi:10.1145/2421119.2421135. S2CID 13367514.
^ Talbot, John; Welsh, D. J. A. (2006), Complexity and Cryptography: An Introduction, Cambridge University Press, p. 57, ISBN 9780521617710, The question of whether NP and co-NP are equal is probably the second most important open problem in complexity theory, after the P versus NP question.
Sipser, M. (1997). "Sections 7.4–7.5 (NP-completeness, Additional NP-complete Problems)". Introduction to the Theory of Computation. PWS Publishing. pp. 248–271. ISBN 978-0-534-94728-6.
Dorn, Frederic; Penninkx, Eelko; Bodlaender, Hans L.; Fomin, Fedor V. (2005). "Efficient Exact Algorithms on Planar Graphs: Exploiting Sphere Cut Branch Decompositions". Proc. 13th European Symposium on Algorithms (ESA '05). Lecture Notes in Computer Science. Vol. 3669. Springer-Verlag. pp. 95–106. doi:10.1007/11561071_11. ISBN 978-3-540-29118-3. .
Lipton, Richard J.; Tarjan, Robert E. (1980). "Applications of a planar separator theorem". SIAM Journal on Computing. 9 (3): 615–627. doi:10.1137/0209046. S2CID 12961628. .
Retrieved from "https://en.wikipedia.org/w/index.php?title=NP-completeness&oldid=1068579860"
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Characteristic_class Knowpia
In mathematics, a characteristic class is a way of associating to each principal bundle of X a cohomology class of X. The cohomology class measures the extent the bundle is "twisted" and whether it possesses sections. Characteristic classes are global invariants that measure the deviation of a local product structure from a global product structure. They are one of the unifying geometric concepts in algebraic topology, differential geometry, and algebraic geometry.
The notion of characteristic class arose in 1935 in the work of Eduard Stiefel and Hassler Whitney about vector fields on manifolds.
Let G be a topological group, and for a topological space
{\displaystyle X}
{\displaystyle b_{G}(X)}
for the set of isomorphism classes of principal G-bundles over
{\displaystyle X}
{\displaystyle b_{G}}
is a contravariant functor from Top (the category of topological spaces and continuous functions) to Set (the category of sets and functions), sending a map
{\displaystyle f\colon X\to Y}
to the pullback operation
{\displaystyle f^{*}\colon b_{G}(Y)\to b_{G}(X)}
A characteristic class c of principal G-bundles is then a natural transformation from
{\displaystyle b_{G}}
to a cohomology functor
{\displaystyle H^{*}}
, regarded also as a functor to Set.
In other words, a characteristic class associates to each principal G-bundle
{\displaystyle P\to X}
{\displaystyle b_{G}(X)}
an element c(P) in H*(X) such that, if f : Y → X is a continuous map, then c(f*P) = f*c(P). On the left is the class of the pullback of P to Y; on the right is the image of the class of P under the induced map in cohomology.
Characteristic numbersEdit
Characteristic classes are elements of cohomology groups;[1] one can obtain integers from characteristic classes, called characteristic numbers. Some important examples of characteristic numbers are Stiefel–Whitney numbers, Chern numbers, Pontryagin numbers, and the Euler characteristic.
Given an oriented manifold M of dimension n with fundamental class
{\displaystyle [M]\in H_{n}(M)}
, and a G-bundle with characteristic classes
{\displaystyle c_{1},\dots ,c_{k}}
, one can pair a product of characteristic classes of total degree n with the fundamental class. The number of distinct characteristic numbers is the number of monomials of degree n in the characteristic classes, or equivalently the partitions of n into
{\displaystyle {\mbox{deg}}\,c_{i}}
Formally, given
{\displaystyle i_{1},\dots ,i_{l}}
{\displaystyle \sum {\mbox{deg}}\,c_{i_{j}}=n}
, the corresponding characteristic number is:
{\displaystyle c_{i_{1}}\smile c_{i_{2}}\smile \dots \smile c_{i_{l}}([M])}
{\displaystyle \smile }
denotes the cup product of cohomology classes. These are notated various as either the product of characteristic classes, such as
{\displaystyle c_{1}^{2}}
or by some alternative notation, such as
{\displaystyle P_{1,1}}
for the Pontryagin number corresponding to
{\displaystyle p_{1}^{2}}
{\displaystyle \chi }
for the Euler characteristic.
This also works for non-orientable manifolds, which have a
{\displaystyle \mathbf {Z} /2\mathbf {Z} }
-orientation, in which case one obtains
{\displaystyle \mathbf {Z} /2\mathbf {Z} }
-valued characteristic numbers, such as the Stiefel-Whitney numbers.
Characteristic classes are phenomena of cohomology theory in an essential way — they are contravariant constructions, in the way that a section is a kind of function on a space, and to lead to a contradiction from the existence of a section we do need that variance. In fact cohomology theory grew up after homology and homotopy theory, which are both covariant theories based on mapping into a space; and characteristic class theory in its infancy in the 1930s (as part of obstruction theory) was one major reason why a 'dual' theory to homology was sought. The characteristic class approach to curvature invariants was a particular reason to make a theory, to prove a general Gauss–Bonnet theorem.
The prime mechanism then appeared to be this: Given a space X carrying a vector bundle, that implied in the homotopy category a mapping from X to a classifying space BG, for the relevant linear group G. For the homotopy theory the relevant information is carried by compact subgroups such as the orthogonal groups and unitary groups of G. Once the cohomology
{\displaystyle H^{*}(BG)}
was calculated, once and for all, the contravariance property of cohomology meant that characteristic classes for the bundle would be defined in
{\displaystyle H^{*}(X)}
in the same dimensions. For example the Chern class is really one class with graded components in each even dimension.
In the language of stable homotopy theory, the Chern class, Stiefel–Whitney class, and Pontryagin class are stable, while the Euler class is unstable.
Concretely, a stable class is one that does not change when one adds a trivial bundle:
{\displaystyle c(V\oplus 1)=c(V)}
. More abstractly, it means that the cohomology class in the classifying space for
{\displaystyle BG(n)}
pulls back from the cohomology class in
{\displaystyle BG(n+1)}
under the inclusion
{\displaystyle BG(n)\to BG(n+1)}
(which corresponds to the inclusion
{\displaystyle \mathbf {R} ^{n}\to \mathbf {R} ^{n+1}}
and similar). Equivalently, all finite characteristic classes pull back from a stable class in
{\displaystyle BG}
This is not the case for the Euler class, as detailed there, not least because the Euler class of a k-dimensional bundle lives in
{\displaystyle H^{k}(X)}
(hence pulls back from
{\displaystyle H^{k}(BO(k))}
, so it can’t pull back from a class in
{\displaystyle H^{k+1}}
, as the dimensions differ.
^ Informally, characteristic classes "live" in cohomology.
^ By Chern–Weil theory, these are polynomials in the curvature; by Hodge theory, one can take harmonic form.
Chern, Shiing-Shen (1995). Complex manifolds without potential theory. Springer-Verlag Press. ISBN 0-387-90422-0. ISBN 3-540-90422-0.
Hatcher, Allen, Vector bundles & K-theory
Husemoller, Dale (1966). Fibre bundles (3rd Edition, Springer 1993 ed.). McGraw Hill. ISBN 0387940871.
Milnor, John W.; Stasheff, Jim (1974). Characteristic classes. Annals of Mathematics Studies. Vol. 76. Princeton University Press, Princeton, NJ; University of Tokyo Press, Tokyo. ISBN 0-691-08122-0.
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On the asymptotics of visible elements and homogeneous equations in surface groups | EMS Press
On the asymptotics of visible elements and homogeneous equations in surface groups
F
be a group whose abelianization is
\mathbb{Z}^k
k\geq 2
F
is called visible if its image in the abelianization is visible, that is, the greatest common divisor of its coordinates is 1.
In this paper we compute three types of densities, annular, even and odd spherical, of visible elements in surface groups. We then use our results to show that the probability of a homogeneous equation in a surface group to have solutions is neither 0 nor 1, as the lengths of the right- and left-hand side of the equation go to infinity.
Yago Antolín, Laura Ciobanu, Noèlia Viles, On the asymptotics of visible elements and homogeneous equations in surface groups. Groups Geom. Dyn. 6 (2012), no. 4, pp. 619–638
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Role of a Sinusoidal Wavy Surface in Enhancement of Heat Transfer Using Turbulent Dual Jet | J. Heat Transfer | ASME Digital Collection
Amitesh Kumar,
Department of Mechanical Engineering, Indian Institute of Technology (BHU), Varanasi
Varanasi, Uttar Pradesh 221005,
e-mail: amitesh.mec@iitbhu.ac.in
Ashok Kumar Satapathy Professor
Singh, T. P., Kumar, A., and Satapathy, A. K. (January 28, 2021). "Role of a Sinusoidal Wavy Surface in Enhancement of Heat Transfer Using Turbulent Dual Jet." ASME. J. Heat Transfer. March 2021; 143(3): 032002. https://doi.org/10.1115/1.4049274
In this paper, the role of sinusoidal wavy surface in enhancing the heat transfer is numerically studied. The heat transfer characteristics are studied for two thermal boundary conditions of the wavy wall. To assess the effect of wavy wall, the amplitude is varied between 0.1 and 0.7 and number of cycle from 4 to 12 at an interval of 0.1 and 1, respectively. In order to see the effect of offset ratio, it is varied between 3 and 15 at an interval of 2. The Reynolds number (Re) and Prandtl number (Pr) are set to 15, 000 and 0.71, respectively, for all the numerical simulations. It is found that the maximum average Nusselt number (
Nuavg
) depends not only on the amplitude and number of cycle but also on the offset ratio. Overall, 23.27% in maximum heat transfer enhancement is achieved with reference to the plane wall surface. An approximately linear decrement in maximum
Nuavg
is observed when offset ratio increases. The results indicate that
Nuavg
increases with an increase in the amplitude of sinusoidal wavy surface up to N = 8 and almost follows the linear trend up to N = 7. It is also found that
Nux
is always on the higher side as compared to the corresponding case of a plane wall surface when N = 4, irrespective of the offset ratio. With an increase in N,
Nux
fluctuates about the result of plane wall surface after the initial increase because of the obstruction. The amplitude of the fluctuation increases with an increase in the number of cycle N, which indicates that fluid accelerates and decelerates gradually owing to the presence of trough and crest. Also, it is worth noticing that for some cases, there is a decrease in the heat transfer rate as compared to the plane wall case. Therefore, it is concluded that the increase in the surface area does not necessarily result in an increase in the heat transfer rate.
average Nusselt number, amplitude, offset ratio, number of cycle, turbulent dual jet
Boundary-value problems, Cycles, Heat transfer, Temperature, Turbulence, Reynolds number, Wall temperature
Amsterdam, Oxford, New York
Wall Inclination Effect in Heat Transfer Characteristics of a Combined Wall and Offset Jet Flow
Von Norstrand Reinhold
Large Eddy Simulation of Three-Dimensional Plane Turbulent Free Jet Flow
High-Resolution Velocity Field Measurements of Turbulent Round Free Jets in Uniform Environments
Numerical Investigation of Circular Turbulent Jets in Shallow Water
Investigation of a Turbulent Wall Jet in Forced Convection Issuing Into a Directed Coflow Stream
Turbulence Structure of a Counter-Flowing Wall Jet
The Effect of Surface Roughness on the Turbulence Structure of a Plane Wall Jet
Heat and Momentum Transfer in a Plane Turbulent Wall Jet
Measurements of the Convection Heat Transfer Coefficient for a Planar Wall Jet: Uniform Temperature and Uniform Heat Flux Boundary Conditions
Verriopoulos
Turbulent Flow With Heat Transfer in Plane and Curved Wall Jets
Heat Transfer Enhancement and Flow Characteristics of Vortex Generating Jet on Flat Plate With Turbulent Boundary Layer
Heat Transfer From a Turbulent Hot Air Jet Impinging Normally on a Flat Plate
Theoretical and Experimental Temperature Distribution in Two-Dimensional Turbulent Jet-Boundary Interaction
The Effect of Entrainment Temperature on Jet Impingement Heat Transfer
Effect of Curvature on the Turbulence of a Two-Dimensional Jet
Measurements of the Thermal Characteristics of Heated Offset Jets
Comparison of Two Low Reynolds Number Turbulence Models for Fluid Flow Study of Wall Bounded Jets
Statistical Properties and Structural Analysis of Three-Dimensional Twin Round Jets Due to Variation in Reynolds Number
Large Eddy Simulations of the Turbulent Flows of Twin Parallel Jets
Effect of Inclination of Twin Impinging Turbulent Jets on Flow and Heat Transfer Characteristics
Effect of Nozzle Spacing on Turbulent Interaction of Low-Aspect-Ratio Twin Rectangular Jets, Flow
Turbul. Combust.
Large Eddy Simulation of the Interaction Between Wall Jet and Offset Jet
Interaction Between Wall Jet and Offset Jet With Different Velocity and Offset Ratio
Computational Study of Periodically Unsteady Interaction Between a Wall Jet and an Offset Jet for Various Velocity Ratios
Submerged Turbulent Twin Jets Interacting With a Free Surface and a Solid Wall
Study of the Heat Transfer Characteristics in a Turbulent Combined Wall and Offset Jet Flows
Fundamentals, Experiments and Modeling
Iterative Solution of Implicit Approximation of Multidimensional Partial Differential Equations
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On the infinitesimal Terracini Lemma | EMS Press
On the infinitesimal Terracini Lemma
In this paper we prove an infinitesimal version of the classical Terracini Lemma for 3-secant planes to a variety. Precisely we prove that if
X \subseteq \mathcal P'
is an irreducible, non-degenerate, projective complex variety of dimension
r \geq 3n + 2
, such that the variety of osculating planes to curves in
X
has the expected dimension
3n
and for every 0-dimensional, curvilinear scheme
\gamma
of length 3 contained in
X
the family of hyperplanes sections of
X
which are singular along
\gamma
has dimension larger that
r-3(n+1)
X
is 2-secant defective.
Ciro Ciliberto, On the infinitesimal Terracini Lemma. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. 32 (2021), no. 1, pp. 63–78
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Lowest weight modules of Sp4(R) and nearly holomorphic Siegel modular forms
December 2021 Lowest weight modules of
{\mathrm{Sp}}_{4}\left(\mathbb{R}\right)
and nearly holomorphic Siegel modular forms
1Department of Mathematics, University of Oklahoma, Norman, Oklahoma, USA
Kyoto J. Math. 61(4): 745-814 (December 2021). DOI: 10.1215/21562261-2021-0012
We undertake a detailed study of the lowest weight modules for the Hermitian symmetric pair
\left(G,K\right)
G={\mathrm{Sp}}_{4}\left(\mathbb{R}\right)
and K is its maximal compact subgroup. In particular, we determine K-types and composition series, and write down explicit differential operators that navigate all the highest weight vectors of such a module starting from the unique lowest weight vector. By rewriting these operators in classical language, we show that the automorphic forms on G that correspond to the highest weight vectors are exactly those that arise from nearly holomorphic vector-valued Siegel modular forms of degree 2.
Further, by explicating the algebraic structure of the relevant space of
\mathfrak{n}
-finite automorphic forms, we are able to prove a structure theorem for the space of nearly holomorphic vector-valued Siegel modular forms of (arbitrary) weight
{det}^{\ell }{\mathrm{sym}}^{m}
with respect to an arbitrary congruence subgroup of
{\mathrm{Sp}}_{4}\left(\mathbb{Q}\right)
. We show that the cuspidal part of this space is the direct sum of subspaces obtained by applying explicit differential operators to holomorphic vector-valued cusp forms of weight
{det}^{{\ell }^{\prime }}{\mathrm{sym}}^{{m}^{\prime }}
\left({\ell }^{\prime },{m}^{\prime }\right)
varying over a certain set. The structure theorem for the space of all modular forms is similar, except that we may now have an additional component coming from certain nearly holomorphic forms of weight
{det}^{3}{\mathrm{sym}}^{{m}^{\prime }}
that cannot be obtained from holomorphic forms.
As an application of our structure theorem, we prove several arithmetic results concerning nearly holomorphic modular forms that improve previously known results in that direction.
Ameya Pitale. Abhishek Saha. Ralf Schmidt. "Lowest weight modules of
{\mathrm{Sp}}_{4}\left(\mathbb{R}\right)
and nearly holomorphic Siegel modular forms." Kyoto J. Math. 61 (4) 745 - 814, December 2021. https://doi.org/10.1215/21562261-2021-0012
Received: 25 May 2015; Revised: 27 March 2019; Accepted: 9 May 2019; Published: December 2021
Keywords: lowest weight modules , nearly holomorphic , representation theory , Siegel modular forms
Rights: Copyright © 2021 by Kyoto University
Ameya Pitale, Abhishek Saha, Ralf Schmidt "Lowest weight modules of
{\mathrm{Sp}}_{4}\left(\mathbb{R}\right)
and nearly holomorphic Siegel modular forms," Kyoto Journal of Mathematics, Kyoto J. Math. 61(4), 745-814, (December 2021)
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How to add and subtract mixed fractions? | StudyPug
>Fractions
Simplify fractions: Method A - By using greatest common factors
Simplify fractions: Method B - By using common factors
How to subtract frations?
• subtracting with improper fractions
• subtracting with mixed numbers
Proper fractions VS. Improper fractions VS. Mixed fractions
Subtract or add using the given diagram.
Subtract or add.
1\frac{3}{6}+\frac{3}{8}
Maren goes to soccer practice for
2\frac{4}{5}
hour each day. In the morning, she has
1\frac{3}{4}
hour of practice. How many hours of practice does she have in the afternoon?
Hannah ran for
1\frac{1}{4}
hour and then biked for
1\frac{2}{3}
hour. For how long did she exercise?
In this section, we will write addition and subtraction statements for mixed numbers using given diagrams. First, we will add and subtract the whole numbers. Second, we will add and subtract the fractions. Some of the fractions will have like denominators and others will have unlike denominators. Before adding and subtracting those fractions with unlike denominators, we will have to determine the lowest common denominator for the pair of fractions and then write the equivalent fractions. We will write any resulting improper fractions in our answers as mixed numbers.
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Fredholm consistency of upper-triangular operator matrices | EMS Press
Fredholm consistency of upper-triangular operator matrices
Dragana S. Cvetković-Ilić
In this paper, for given operators
A\in\mathcal B(\mathcal H)
B\in\mathcal B(\mathcal K)
, we characterize the set of all
C\in\mathcal B(\mathcal K,\mathcal H)
such that the operator matrix
M_C= \left[ {\begin{array}{cc} A & C \\ 0 & B \\ \end{array} } \right]
is Fredholm consistent. We completely describe the sets
\bigcap_{C\in \mathcal B(\mathcal K,\mathcal H)}\sigma_{\mathrm{FC}}(M_C)
\bigcup_{C\in \mathcal B(\mathcal K,\mathcal H)}\sigma_{\mathrm{FC}}(M_C)
. Also, we prove that
\bigcap_{C\in \mathcal B(\mathcal K,\mathcal H)}\sigma_{\mathrm{FC}}(M_C)=\sigma_{\mathrm{FC}}(M_0)
Dragana S. Cvetković-Ilić, Fredholm consistency of upper-triangular operator matrices. J. Spectr. Theory 7 (2017), no. 4, pp. 1023–1038
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Beverages | Special Issue : Fruit Juices: Technology, Chemistry, and Nutrition 2.0
Special Issue "Fruit Juices: Technology, Chemistry, and Nutrition 2.0"
Nutrition and Food Science Area, Faculty of Pharmacy, University of Valencia, Avda. Vicente Andrés Estellés s/n, Burjassot, 46100 Valencia, Spain
Interests: functional foods; bioactive compounds; antioxidant capacity; sterols; phytochemicals; bioaccessibility; bioavailability; bioactivity; cell cultures; chemoprevention; oxidative stress; eryptosis
Special Issue in Beverages: Fruit Beverages: Nutritional Composition and Health Benefits
Special Issue in Beverages: Fruit Beverages and Human Intervention Studies
Special Issue in Beverages: Fruit Juices: Technology, Chemistry, and Nutrition
Special Issue in Foods: Advanced Studies of Plant Sterol-Enriched Functional Foods
Dr. Alessandro Attanzio
Department of Biological, Chemical and Pharmaceutical Sciences and Technologies (STEBICEF), University of Palermo, Palermo, Italy
Interests: biochemistry; Sicilian foods; bioactive compounds; phytochemicals; bioactivity; cell cultures (colon, breast, cervix, liver, neuroblastoma); chemoprevention; oxidative stress; eryptosis
Special Issue in Molecules: Cytotoxic Activity of Plant Extracts
Special Issue in Cells: Cellular Senescence in Age-Related Diseases: Molecular Bases and Therapeutic Interventions
Special Issue in Molecules: Cytotoxic Activity of Plant Extracts-2nd Edition
Special Issue in International Journal of Molecular Sciences: Molecular Research on Eryptosis
Special Issue in Cells: Cellular Senescence in Age-Related Diseases: Molecular Bases and Therapeutic Interventions II
Special Issue in Molecules: Natural Products: Biological and Pharmacological Activity
Dr. Guadalupe Garcia-Llatas
Nutrition and Food Science Area, University of Valencia, Avda. Vicent Andres Estelles s/n 46100, Burjassot, Valencia, Spain
Interests: functional foods; bioactive compounds; sterols; oxysterols; phytochemicals; bioaccessibility; bioavailability; human milk; infant foods; microbiota; lipid oxidation; food chemistry; gas chromatography; clinical trials
Fruit juices can be considered to be natural functional foods as, beyond their nutritional value, they can provide other compounds with potential health benefits, such as fiber, vitamins, minerals, and antioxidant phytochemicals (polyphenols, carotenoids, tocopherols, and phytosterols, among others). These phytochemicals can either be naturally present or incorporated using extracts from, raw, or by-products of plant foods. Fruit juice manufacturing techniques range from traditional methods, such as blending, formulation, and fermentation, to advanced techniques, such as microencapsulation, edible films, coatings, and non-thermal processing technologies (high-pressure processing, pulsed electric fields, ultrasounds, etc.) designed to protect bioactive compounds against degradation and increase their bioaccessibility/bioavailability (and, hence, potentially, bioactivity) without compromising their sensory attributes.
This Special Issue aims to present a collection of original research and review articles that provide new insights into, and data on, the technology, chemistry, and nutrition of fruit juices. Potential topics include:
-evaluation of nutritional and bioactive compounds in conventional and newly designed functional fruit juices and blends that combine fruits and vegetables;
-determination of the impact of conventional and new non-thermal processing technologies on bioactive compounds present in fruit juices;
-assessment of the bioaccessibility and bioavailability of bioactive compounds in these beverages; and
-unraveling the potential health effects of fruit juices using in vitro, animal, and human studies.
Prof. Dr. Antonio Cilla
blends of fruit and vegetable juices
conventional and novel processing technologies
bioaccessibility/bioavailability
Antimicrobial Activity of Nanoencapsulated Essential Oils of Tasmannia lanceolata, Backhousia citriodora and Syzygium anisatum against Weak-Acid Resistant Zygosaccharomyces bailii in Clear Apple Juice
Fahad Alderees
The anti-yeast activity of oil-in-water encapsulated nanoemulsion containing individual or a combination of the three essential oils of Tasmanian pepper leaf (Tasmannia lanceolata), lemon myrtle (Backhousia citriodora), and anise myrtle (Syzygium anisatum) against weak-acid resistant Zygosaccharomyces bailii [...] Read more.
The anti-yeast activity of oil-in-water encapsulated nanoemulsion containing individual or a combination of the three essential oils of Tasmanian pepper leaf (Tasmannia lanceolata), lemon myrtle (Backhousia citriodora), and anise myrtle (Syzygium anisatum) against weak-acid resistant Zygosaccharomyces bailii in clear apple juice was investigated. The effectiveness of the shelf-life extension of Z. bailii-spiked (1 × 103 CFU/mL) clear apple juice was evaluated and compared between natural (essential oils) and synthetic (sodium benzoate) antimicrobial agents. Essential oils showed an immediate reduction in the Z. bailii cell population at day-0 and exerted a fungicidal activity at day-4 of storage, with no further noticeable growth at the end of the experiment (day-28). At lower concentrations, Tasmanian pepper leaf oil of 0.0025% had >6 log CFU/mL at day-12 of storage. For lemon myrtle essential oils, the yeast population reached >6 log CFU/mL at day-24 and day-20 for concentrations of 0.02% and 0.01%, respectively. The fungicidal activity of Tasmanian pepper leaf oil reduced from 0.005% to 0.0025% v/v when mixed at a ratio of 1:1 with anise myrtle oil. The results of the present study suggest that these three native Australian herbs have the potential to be used in the beverage industry by controlling Zygosaccharomyces bailii in clear apple juice products. Full article
Short Wave Ultraviolet Light (UV-C) Effectiveness in the Inactivation of Bacterial Spores Inoculated in Turbid Suspensions and in Cloudy Apple Juice
Jezer N. Sauceda-Gálvez
María Martinez-Garcia
Ma Manuela Hernández-Herrero
Ramón Gervilla
Artur X. Roig-Sagués
Liquid foods might present interferences in their optical properties that can reduce the effectiveness of short-wave ultraviolet radiation (UV-C) treatments used for sterilization purposes. The effect of turbidity as UV-C interference factor against the inactivation of bacterial spores was analysed by using phosphate-buffered [...] Read more.
Liquid foods might present interferences in their optical properties that can reduce the effectiveness of short-wave ultraviolet radiation (UV-C) treatments used for sterilization purposes. The effect of turbidity as UV-C interference factor against the inactivation of bacterial spores was analysed by using phosphate-buffered saline solutions (PBS) of different turbidity values (2000, 2500, and 3000 NTU) which were adjusted with the addition of apple fibre. These suspensions were inoculated with spores of Bacillus subtilis and Alicyclobacillus acidoterrestris. While higher UV-C doses increased the inactivation rates of spores, these were reduced when turbidity values increased; a dose of 28.7 J/mL allowed inactivation rates of B. subtilis spores of 3.96 Log in a 2000-NTU suspension compared with 2.81 Log achieved in the 3000-NTU one. Spores of B. subtilis were more UV-C-resistant than A. acidoterrestris. Cloudy apple juice inoculated with A. acidoterrestris spores was processed by UV-C at different doses in a single pass and with recirculation of the matrix through the reactor. Inactivation increased significantly with recirculation, surpassing 5 Log after 125 J/mL compared with 0.13 Log inactivation after a single-pass treatment at the same UV-C dose. UV-C treatments with recirculation affected the optical properties (absorption coefficient at 254 nm and turbidity) of juice and increased browning as UV-C doses became higher. Full article
Degradation Kinetics of Betacyanins during the Pasteurization and Storage of Cactus Pear (Opuntia dillenii Haw.) Juice Using the Arrhenius, Eyring, and Ball Models
Joseph Bassama
Khakhila Dieu Donnée Sarr
Beverages 2021, 7(1), 2; https://0-doi-org.brum.beds.ac.uk/10.3390/beverages7010002 - 23 Dec 2020
Betacyanin stability was assessed over temperatures ranging from 60 to 90 °C for cactus pear (Opuntia dillenii Haw.) juice. The juice showed a betacyanin content of 0.76 g/kg. The rate constants for the betacyanin degradation and isothermal kinetic parameters were calculated according [...] Read more.
Betacyanin stability was assessed over temperatures ranging from 60 to 90 °C for cactus pear (Opuntia dillenii Haw.) juice. The juice showed a betacyanin content of 0.76 g/kg. The rate constants for the betacyanin degradation and isothermal kinetic parameters were calculated according to the following three models: Arrhenius, Eyring, and Ball. The fittings of the models were found to be close to one other with SSE values of 0.0964, 0.0964, and 0.0974, respectively. However, because the estimated parameters for the Ball equation happened to be less correlated than the parameters of the other models, this equation was then used for the simulations. The parameters for z and D0 were 42.21 °C and 6.79 × 104 s, respectively. Betacyanins were found to resist typical heat treatment conditions (
{F}_{70°\mathrm{C}}
values between 100 and 200 min), with a maximum loss of 10% when the temperature was above 80 °C. The time/temperature combinations that could assure both the safety of the product and the preservation of the betacyanins were identified. With Enterococcus faecalis as the reference, when the temperature was 100 °C, the pasteurization time satisfying these two conditions was 0.6 min, whereas it was 180 min when the temperature was 62 °C. The degradation of betacynins during storage was positively correlated with temperature and was accompanied by the appearance of a brown shade. Full article
Marzena Połaska
Foods preserved with sorbic acid or its salts can undergo spoilage with the formation of chemicals characterized by odors of plastic, hydrocarbons, or kerosene. 1,3-pentadiene, which is formed through the decarboxylation of sorbic acid or its salts, is one such compound. Numerous species [...] Read more.
Foods preserved with sorbic acid or its salts can undergo spoilage with the formation of chemicals characterized by odors of plastic, hydrocarbons, or kerosene. 1,3-pentadiene, which is formed through the decarboxylation of sorbic acid or its salts, is one such compound. Numerous species of molds and yeasts have been reported as capable of degrading sorbic acid. This work is aimed to identify the off-odor compounds in samples of strawberry-flavored water preserved with potassium sorbate and sodium benzoate. In addition, the mold isolated from this drink was evaluated for the ability to form undesirable compounds, and the results revealed the presence of 1,4-pentadiene and benzaldehyde in the tested samples. The mold isolated from the samples was identified as Penicillium corylophilum. During its 5-day incubation at 25 °C in a liquid medium, potassium sorbate added at a final concentration of 200 and 400 mg/L was completely assimilated by the growing mycelium and converted into 1,4-pentadiene. The concentration of the latter was determined as 46.5 and 92.6 mg/L, respectively. The decrease in the concentration of sodium benzoate exceeded 53% in the broth spiked at 200 mg/L and 23% at 400 mg/L, resulting in the formation of benzaldehyde. Full article
The effects of 1-methylcyclopropene (1-MCP), storage atmosphere (controlled (CA) or regular (RA)), and juice processing (clear or cloudy) on the volatile aroma compounds from McIntosh and Honeycrisp apples following 4-month storage were studied. All the major esters, aldehydes, and total volatile content from [...] Read more.
The effects of 1-methylcyclopropene (1-MCP), storage atmosphere (controlled (CA) or regular (RA)), and juice processing (clear or cloudy) on the volatile aroma compounds from McIntosh and Honeycrisp apples following 4-month storage were studied. All the major esters, aldehydes, and total volatile content from McIntosh juice were significantly affected by the two-way interaction between harvest maturity and 1-MCP treatment (p ≤ 0.01), as well as harvest maturity and storage atmosphere (p ≤ 0.001). In McIntosh juices, a remarkable reduction of all types of esters, aldehydes, most alcohols, and total volatile compounds was found when juices were prepared from 1-MCP-treated apples. In Honeycrisp, significant differences in the level of esters and the total volatile aroma was caused by storage atmosphere and juice processing techniques (p ≤ 0.001), but not by 1-MCP treatment. As compared to clear juices, cloudy juice samples from Honeycrisp had a considerably higher content of total volatiles, esters, and aldehydes. Full article
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Limit-periodic continuum Schrödinger operators with zero measure Cantor spectrum | EMS Press
Limit-periodic continuum Schrödinger operators with zero measure Cantor spectrum
We consider Schrödinger operators on the real line with limit-periodic potentials and show that, generically, the spectrum is a Cantor set of zero Lebesgue measure and all spectral measures are purely singular continuous. Moreover, we show that for a dense set of limit-periodic potentials, the spectrum of the associated Schrödinger operator has Hausdorff dimension zero. In both results one can introduce a coupling constant
\lambda \in (0,\infty)
, and the respective statement then holds simultaneously for all values of the coupling constant.
David Damanik, Jake Fillman, Milivoje Lukic, Limit-periodic continuum Schrödinger operators with zero measure Cantor spectrum. J. Spectr. Theory 7 (2017), no. 4, pp. 1101–1118
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Systems of linear equations: Gaussian Elimination | StudyPug
Solving a linear system with matrices using Gaussian elimination - Linear Equations with Matrices
Try reviewing these fundamentals firstNotation of matricesNotation of matricesRepresenting a linear system as a matrixRepresenting linear system as a matrix
Review these basic concepts…Notation of matricesNotation of matricesRepresenting a linear system as a matrixRepresenting linear system as a matrix
After a few lessons in which we have repeatedly mentioned that we are covering the basics needed to later learn how to solve systems of linear equations, the time has come for our lesson to focus on the full methodology to follow in order to find the solutions for such systems.
What is Gaussian elimination
Gaussian elimination is the name of the method we use to perform the three types of matrix row operations on an augmented matrix coming from a linear system of equations in order to find the solutions for such system. This technique is also called row reduction and it consists of two stages: Forward elimination and back substitution.
These two Gaussian elimination method steps are differentiated not by the operations you can use through them, but by the result they produce. The forward elimination step refers to the row reduction needed to simplify the matrix in question into its echelon form. Such stage has the purpose to demonstrate if the system of equations portrayed in the matrix have a unique possible solution, infinitely many solutions or just no solution at all. If found that the system has no solution, then there is no reason to continue row reducing the matrix through the next stage.
If is possible to obtain solutions for the variables involved in the linear system, then the Gaussian elimination with back substitution stage is carried through. This last step will produce a reduced echelon form of the matrix which in turn provides the general solution to the system of linear equations.
The Gaussian elimination rules are the same as the rules for the three elementary row operations, in other words, you can algebraically operate on the rows of a matrix in the next three ways (or combination of):
Multiplying a row by a constant (any constant which is not zero)
Adding a row to another row
And so, solving a linear system with matrices using Gaussian elimination happens to be a structured, organized and quite efficient method.
How to do Gaussian elimination
The is really not an established set of Gaussian elimination steps to follow in order to solve a system of linear equations, is all about the matrix you have in your hands and the necessary row operations to simplify it. For that, let us work on our first Gaussian elimination example so you can start looking into the whole process and the intuition that is needed when working through them:
If we were to have the following system of linear equations containing three equations for three unknowns:
Equation 1: System of linear equations to solve
We know from our lesson on representing a linear system as a matrix that we can represent such system as an augmented matrix like the one below:
Equation 2: Transcribing the linear system into an augmented matrix
Let us row-reduce (use Gaussian elimination) so we can simplify the matrix:
Equation 3: Row reducing (applying the Gaussian elimination method to) the augmented matrix
Resulting in the matrix:
Equation 4: Reduced matrix into its echelon form
Notice that at this point, we can observe that this system of linear equations is solvable with a unique solution for each of its variables. What we have performed so far is the first stage of row reduction: Forward elimination. We can continue simplifying this matrix even more (which would take us to the second stage of back substitution) but we really don't need to since at this point the system is easily solvable. Thus, we look at the resulting system to solve it directly:
Equation 5: Resulting linear system of equations to solve
From this set, we can automatically observe that the value of the variable z is: z=-2. We use this knowledge to substitute it on the second equations to solve for y, and the substitute both y and z values on the first equations to solve for x:
Applying the values of y and z to the first equation
Equation 6: Solving the resulting linear system of equations
And the final solution for the system is:
Equation 7: Final solution to the system of linear equations for example 1
More Gaussian elimination problems have been added to this lesson in its last section. Make sure to work through them in order to practice.
Difference between gaussian elimination and gauss jordan elimination
The difference between Gaussian elimination and the Gaussian Jordan elimination is that one produces a matrix in row echelon form while the other produces a matrix in row reduced echelon form. A row echelon form matrix has an upper triangular composition where any zero rows are at the bottom and leading terms are all to the right of the leading term from the row above. Reduced echelon form goes beyond by simplifying much more (sometimes even reaching the shape of an identity matrix).
Equation 8: Difference between echelon form and row echelon form
The history of Gaussian elimination and its names is quite interesting, you will be surprised to know that the name "Gaussian" was attributed to this methodology by mistake in the last century. In reality the algorithm to simultaneously solve a system of linear equations using matrices and row reduction has been found to be written in some form in ancient Chinese texts that date to even before our era. Then in the late 1600's Isaac Newton put together a lesson on it to fill up something he considered as a void in algebra books. After the name "Gaussian" had been already established in the 1950's, the Gaussian-Jordan term was adopted when geodesist W. Jordan improved the technique so he could use such calculations to process his observed land surveying data. If you would like to continue reading about the fascinating history about the mathematicians of Gaussian elimination do not hesitate to click on the link and read.
There is really no physical difference between Gaussian elimination and Gauss Jordan elimination, both processes follow the exact same type of row operations and combinations of them, their difference resides on the results they produce. Many mathematicians and teachers around the world will refer to Gaussian elimination vs Gauss Jordan elimination as the methods to produce an echelon form matrix vs a method to produce a reduced echelon form matrix, but in reality, they are talking about the two stages of row reduction we explained on the very first section of this lesson (forward elimination and back substitution), and so, you just apply row operations until you have simplified the matrix in question. If you arrive to the echelon form you can usually solve a system of linear equations with it (up until here, this is what would be called Gaussian elimination). If you need to continue the simplification of such matrix in order to obtain directly the general solution for the system of equations you are working on, for this case you just continue to row-operate on the matrix until you have simplified it to reduced echelon form (this would be what we call the Gauss-Jordan part and which could be considered also as pivoting Gaussian elimination).
We will leave the extensive explanation on row reduction and echelon forms for the next lesson, for now you need to know that, unless you have an identity matrix on the left hand side of the augmented matrix you are solving (in which case you don't need to do anything to solve the system of equations related to the matrix), the Gaussian elimination method (regular row reduction) will always be used to solve a linear system of equations which has been transcribed as a matrix.
Gaussian elimination examples
As our last section, let us work through some more exercises on Gaussian elimination (row reduction) so you can acquire more practice on this methodology. Throughout many future lessons in this course for Linear Algebra, you will find that row reduction is one of the most important tools there are when working with matrix equations. Therefore, make sure you understand all of the steps involved in the solution for the next problems.
Solve the following linear system using Gaussian elimination:
Equation 9: System of linear equations with two variables
Transcribing the system of linear equations into an augmented matrix we obtain the matrix equation:
Equation 10: Resulting augmented matrix
Equation 11: Row reducing the augmented matrix
We can clearly see from the resulting matrix that
-y=-5 \;
y=5 \;
. We substitute this value into the equation equivalent to the first row of the resulting augmented matrix to solve for the variable x:
Equation 12: Solving for x
Therefore the final solution for the system of linear equations is:
Equation 13: Final solution for the system of equations
Equation 14: System of linear equations with three variables
For this system we know we will obtain an augmented matrix with three rows (since the system contains three equations) and three columns to the left of the vertical line (since there are three different variables). On this case we will go directly into the row reduction, and so, the first matrix you will see on this process is the one you obtain by transcribing the system of linear equations into an augmented matrix.
Notice how we can tell right away that the variable z is equal to zero for this system since the third row of the resulting matrix shows the equation -9z = 0. We use that knowledge and check the second row of the matrix which would provide the equation 2y - 6z = 0, plugging the value of z = 0
\,
into that equation results in y
\,
also being zero. Thus, we finally substitute both values of y and z
\,
into the equation that results from the first row of the matrix: x + 4y + 3z = 1, since both y and z
\,
are zero, then this gives us x = 1. And so, the final solution to this system of equations looks as follows:
Equation 16: Final solution to the system of equations
We transcribe the linear system as an augmented matrix and then we start the Gaussian elimination process:
From which we can see that the last row provides the equation: 6z = 3 and therefore z = 1/2. We substitute this in the equations resulting from the second and first row (in that order) to calculate the values of the variables x and y:
Applying the values of y and z to the first equation:
Equation 19: Solving for x and y
Therefore, the final solution to the system of linear equations is:
Equation 21: System of linear equations with two variables
Transcribing the linear system as an augmented matrix and row reducing:
Which automatically tells us y = 8. And so, substituting this value into the equation from the first row we obtain: 4x - 5y = 4x - 5(8) = 4x - 40 =-6 4x = 34
\,
and therefore the value of x is: x =
\frac{\small17}{\small2}
. And the final solution to this system of equations is:
Converting the system of equations into an augmented matrix and then row reducing:
From the third row in the resulting matrix we know that z = 3. Using this value into the equations from rows one and two, we solve for x and y:
And so, the final solution to the system of linear equations is:
Equation 27: Final solution to the system of linear equations
To finalize our lesson for today we have a link recommendation to complement your studies: Gaussian elimination an article which contains some extra information about row reduction, including an introduction to the topic and some more examples. As we mentioned before, be ready to keep on using row reduction for almost the whole rest of this course in Linear Algebra, so, we see you in the next lesson!
R^n
R^n
Now that we have learned how to represent a linear system as a matrix, we can now solve this matrix to solve the linear system! We use a method called "Gaussian elimination". This method involves a lot of matrix row operations. Our goal is to make it so that all entries in the bottom left of the matrix are 0. Once that is done, we take a look at the last row and convert it to a linear system. Then we solve for the variable. Then we look at the second last row, convert it to a linear system, and solve for the other variable. Rinse and repeat, and you will find all the variables which solve the linear system!
Gaussian elimination (or row reduction) is a method used for solving linear systems. For example,
x+y+z=3
x+2y+3z=0
x+3y+2z=3
Using Gaussian elimination, we can turn this matrix into
(watch the intro video to learn how to do this!)
Now we can start solving for
x,y
z
So in the third row, we see that
-3z=6
z=-2
In the second row, we see that
2y+4z=-6
z=-2
, then we can substitute it into the second row and solve for
y
2y+4z=-6
2y+4(-2)=-6
2y-8=-6
2y=2
y=1
So now we know that
z=-2
y=1
. Now let us take a look at the first row and solve for
x
x+y+z=3
x+1-2=3
x-1=3
x=4
Since we have solved for
x,y
z
, then we have just solved the linear system.
Gaussian elimination overview
Solve the following linear systems:
x+2y=3
2x+3y=1
x+4y+3z=1
x+2y+9z=1
x+6y+6z=1
x+3y+3z=2
3x+9y+3z=3
3x+6y+6z=4
4x-5y=-6
2x-2y=1
x+3y+4z=4
-x+3y+2z=2
3x+9y+6z=-6
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LMS_color_space Knowpia
LMS (long, medium, short), is a color space which represents the response of the three types of cones of the human eye, named for their responsivity (sensitivity) peaks at long, medium, and short wavelengths.
Normalized responsivity spectra of human cone cells, S, M, and L types (SMJ data based on Stiles and Burch RGB color-matching)[1]
The numerical range is generally not specified, except that the lower end is generally bounded by zero. It is common to use the LMS color space when performing chromatic adaptation (estimating the appearance of a sample under a different illuminant). It's also useful in the study of color blindness, when one or more cone types are defective.
XYZ to LMSEdit
Typically, colors to be adapted chromatically will be specified in a color space other than LMS (e.g. sRGB). The chromatic adaptation matrix in the diagonal von Kries transform method, however, operates on tristimulus values in the LMS color space. Since colors in most colorspaces can be transformed to the XYZ color space, only one additional transformation matrix is required for any color space to be adapted chromatically: to transform colors from the XYZ color space to the LMS color space.[2]
In addition, many color adaption methods, or color appearance models (CAMs), runs a von Kries-style diagonal matrix transform in a slightly modified, LMS-like, space instead. They may refer to it simply as LMS, as RGB, or as ργβ. The following text uses the "RGB" naming, but do note that the resulting space has nothing to do with the additive color model called RGB.[2]
The CAT matrices for some CAMs in terms of CIEXYZ coordinates are presented here. The matrices, in conjunction with the XYZ data defined for the standard observer, implicitly define a "cone" response for each cell type.
All tristimulus values are normally calculated using the CIE 1931 2° standard colorimetric observer.[2]
Unless specified otherwise, the CAT matrices are normalized (the elements in a row add up to 1) so the tristimulus values for an equal-energy illuminant (X=Y=Z), like CIE Illuminant E, produce equal LMS values.[2]
Hunt, RLABEdit
The Hunt and RLAB color appearance models use the Hunt-Pointer-Estevez transformation matrix (MHPE) for conversion from CIE XYZ to LMS.[3][4][5] This is the transformation matrix which was originally used in conjunction with the von Kries transform method, and is therefore also called von Kries transformation matrix (MvonKries).
Equal-energy illuminants:
{\displaystyle {\begin{bmatrix}L\\M\\S\end{bmatrix}}_{\text{E}}={\begin{bmatrix}0.38971&0.68898&-0.07868\\-0.22981&1.18340&0.04641\\0.00000&0.00000&1.00000\end{bmatrix}}{\begin{bmatrix}X\\Y\\Z\end{bmatrix}}}
Normalized[6] to D65:
{\displaystyle {\begin{bmatrix}L\\M\\S\end{bmatrix}}_{\text{D65}}={\begin{bmatrix}0.4002&0.7076&-0.0808\\-0.2263&1.1653&0.0457\\0&0&0.9182\end{bmatrix}}{\begin{bmatrix}X\\Y\\Z\end{bmatrix}}}
Bradford's spectrally sharpened matrix (LLAB, CIECAM97s)Edit
The original CIECAM97s color appearance model uses the Bradford transformation matrix (MBFD) (as does the LLAB color appearance model).[2] This is a “spectrally sharpened” transformation matrix (i.e. the L and M cone response curves are narrower and more distinct from each other). The Bradford transformation matrix was supposed to work in conjunction with a modified von Kries transform method which introduced a small non-linearity in the S (blue) channel. However, outside of CIECAM97s and LLAB this is often neglected and the Bradford transformation matrix is used in conjunction with the linear von Kries transform method, explicitly so in ICC profiles.[7]
{\displaystyle {\begin{bmatrix}R\\G\\B\end{bmatrix}}_{\text{BFD}}={\begin{bmatrix}0.8951&0.2664&-0.1614\\-0.7502&1.7135&0.0367\\0.0389&-0.0685&1.0296\end{bmatrix}}{\begin{bmatrix}X\\Y\\Z\end{bmatrix}}}
A "spectually sharpened" matrix is believed to improve chromatic adaptation especially for blue colors, but does not work as a real cone-describing LMS space for later human vision processing. Although the outputs are called "LMS" in its original LLAB incarceration, CIECAM97s uses a different "RGB" name to highlight that this space does not really reflect cone cells; hence the different names here.
LLAB proceeds by taking the post-adaptation XYZ values and performing a CIELAB-like treatment to get the visual correlates. On the other hand, CIECAM97s takes the post-adaptation XYZ value back into the Hunt LMS space, and works from there to model the vision system's calculation of color properties.
Later CIECAMsEdit
A revised version of CIECAM97s switches back to a linear transform method and introduces a corresponding transformation matrix (MCAT97s):[8]
{\displaystyle {\begin{bmatrix}R\\G\\B\end{bmatrix}}_{\text{97}}={\begin{bmatrix}0.8562&0.3372&-0.1934\\-0.8360&1.8327&0.0033\\0.0357&-0.0469&1.0112\end{bmatrix}}{\begin{bmatrix}X\\Y\\Z\end{bmatrix}}}
The sharpened transformation matrix in CIECAM02 (MCAT02) is:[9][2]
{\displaystyle {\begin{bmatrix}R\\G\\B\end{bmatrix}}_{\text{02}}={\begin{bmatrix}0.7328&0.4296&-0.1624\\-0.7036&1.6975&0.0061\\0.0030&0.0136&0.9834\end{bmatrix}}{\begin{bmatrix}X\\Y\\Z\end{bmatrix}}}
CAM16 uses a different matrix:[10]
[+0.401288, +0.650173, -0.051461],
M16 = [-0.250268, +1.204414, +0.045854],
[-0.002079, +0.048952, +0.953127].
As in CIECAM97s, after adaptation, the colors are converted to the traditional Hunt–Pointer–Estévez LMS for final prediction of visual results.
Direct from spectraEdit
From a physiological point of view, the LMS color space describes a more fundamental level of human visual response, so it makes more sense to define the physiopsychological XYZ by LMS, rather than the other way around.
Stockman & Sharpe (2000)Edit
A set of physiologically-based LMS functions are proposed by Stockman & Sharpe in 2000. The function has been published in a technical report by the CIE in 2006 (CIE 170).[11] The functions are derived from Stiles and Burch (1959) RGB CMF data, combined with newer measurements about the contribution of each cone in the RGB functions. To adjust from the 10° data to 2°, assumptions about photopigment density difference and data about the absorption of light by pigment in the lens and the macula lutea are used.[12]
The Stockman & Sharpe functions can then be turned into a set of three color-matching functions similar to those in CIEXYZ:[13]
{\displaystyle {\begin{bmatrix}X\\Y\\Z\end{bmatrix}}_{\text{F}}={\begin{bmatrix}1.94735469&-1.41445123&0.36476327\\0.68990272&0.34832189&0\\0&0&1.93485343\end{bmatrix}}{\begin{bmatrix}L\\M\\S\end{bmatrix}}}
The inverse matrix is shown here for comparison with the ones for traditional XYZ:
{\displaystyle {\begin{bmatrix}L\\M\\S\end{bmatrix}}={\begin{bmatrix}0.210576&0.855098&-0.0396983\\-0.417076&1.17726&0.0786283\\0&0&0.516835\\\end{bmatrix}}{\begin{bmatrix}X\\Y\\Z\end{bmatrix}}_{\text{F}}}
The LMS color space can be used to emulate the way color-blind people see color. An early emulation of dichromats were produced by Brettel et al. 1997 and was rated favorably by actual patients. An example of a state-of-the-art method is Machado et al. 2009.[14]
A related application is making color filters for color-blind people to more easily notice differences in color, a process known as daltonization.[15]
JPEG XL uses an XYB color space derived from LMS. Its transform matrix is shown here:
{\displaystyle {\begin{bmatrix}X\\Y\\B\end{bmatrix}}={\begin{bmatrix}1&-1&0\\1&1&0\\0&0&1\end{bmatrix}}{\begin{bmatrix}L\\M\\S\end{bmatrix}}}
This can be interpreted as a hybrid color theory where L and M are opponents but S is handled in a trichromatic way, justified by the lower spatial density of S cones. In practical terms, this allows for using less data for storing blue signals without losing much perceived quality.[16]
The colorspace originates from Guetzli's butteraugli metric,[17] and was passed down to JPEG XL via Google's Pik project.
^ "Stockman, MacLeod & Johnson 2-deg cone fundamentals".
^ a b c d e f Fairchild, Mark D. (2005). Color Appearance Models (2E ed.). Wiley Interscience. pp. 182–183, 227–230. ISBN 978-0-470-01216-1.
^ Schanda, Jnos, ed. (July 27, 2007). Colorimetry. p. 305. doi:10.1002/9780470175637. ISBN 9780470175637.
^ Moroney, Nathan; Fairchild, Mark D.; Hunt, Robert W.G.; Li, Changjun; Luo, M. Ronnier; Newman, Todd (November 12, 2002). "The CIECAM02 Color Appearance Model". IS&T/SID Tenth Color Imaging Conference. Scottsdale, Arizona: The Society for Imaging Science and Technology. ISBN 0-89208-241-0.
^ Ebner, Fritz (July 1, 1998). "Derivation and modelling hue uniformity and development of the IPT color space". Theses: 129.
^ "Welcome to Bruce Lindbloom's Web Site". brucelindbloom.com. Retrieved March 23, 2020.
^ Specification ICC.1:2010 (Profile version 4.3.0.0). Image technology colour management — Architecture, profile format, and data structure, Annex E.3, pp. 102.
^ Fairchild, Mark D. (2001). "A Revision of CIECAM97s for Practical Applications" (PDF). Color Research & Application. Wiley Interscience. 26 (6): 418–427. doi:10.1002/col.1061.
^ Fairchild, Mark. "Errata for COLOR APPEARANCE MODELS" (PDF). The published MCAT02 matrix in Eq. 9.40 is incorrect (it is a version of the HuntPointer-Estevez matrix. The correct MCAT02 matrix is as follows. It is also given correctly in Eq. 16.2)
^ Li, Changjun; Li, Zhiqiang; Wang, Zhifeng; Xu, Yang; Luo, Ming Ronnier; Cui, Guihua; Melgosa, Manuel; Brill, Michael H.; Pointer, Michael (2017). "Comprehensive color solutions: CAM16, CAT16, and CAM16-UCS". Color Research & Application. 42 (6): 703–718. doi:10.1002/col.22131.
^ "Stockman and Sharpe (2000) 2-deg (from 10-deg) cone fundamentals". cvrl.ucl.ac.uk.
^ "CIE 2-deg CMFs". cvrl.ucl.ac.uk.
^ "Color Vision Deficiency Emulation". colorspace.r-forge.r-project.org.
^ Alakuijala, Jyrki; van Asseldonk, Ruud; Boukortt, Sami; Szabadka, Zoltan; Bruse, Martin; Comsa, Iulia-Maria; Firsching, Moritz; Fischbacher, Thomas; Kliuchnikov, Evgenii; Gomez, Sebastian; Obryk, Robert; Potempa, Krzysztof; Rhatushnyak, Alexander; Sneyers, Jon; Szabadka, Zoltan; Vandervenne, Lode; Versari, Luca; Wassenberg, Jan (September 6, 2019). Tescher, Andrew G; Ebrahimi, Touradj (eds.). "JPEG XL next-generation image compression architecture and coding tools". Applications of Digital Image Processing XLII. 11137: 20. Bibcode:2019SPIE11137E..0KA. doi:10.1117/12.2529237. ISBN 9781510629677.
^ "butteraugli/butteraugli.h at master · google/butteraugli". GitHub. Retrieved August 2, 2021.
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Tamilnadu State board 10th Maths Solutions Chapter 1 Relations and Functions Exercise 1.3 ( EX 1.3 ) | Samacheer Guide Answers
Tamilnadu State board 10th Maths Solutions Chapter 1 Relations and Functions Exercise 1.3 ( EX 1.3 )
Tamilnadu State board 10th Maths Answers Chapter 1 Relations and Functions Exercise 1.3 ( EX 1.3 )
You can Download and practice Tamilnadu state board 10th Maths Book Solutions Guide Pdf, Tamilnadu State Board help you to revise the complete Reduced Syllabus and score more marks in your Public examinations and assignment writing work
Let f = {(x, y)|x, y ∈ N and y = 2x} be a relation on N. Find the domain, co-domain and range. Is this relation a function?
x = {1, 2, 3,…}
R = {(1, 2), (2, 4), (3, 6), (4, 8), (5, 10),…}
Co-domain = {1, 2, 3…..}
Let X = {3, 4, 6, 8}. Determine whether the relation R = {(x, f(x))|x ∈ X, f(x) = x² + 1} is a function from X to N ?
R = ((x, f(x))|x ∈ X, f(x) = X² + 1}
f(3) = 3² + 1 = 10
R = {(3, 10), (4, 17), (6, 37), (8, 65)}
f : x → x2 – 5x + 6, evaluate
(ii) f(2 a)
(i) f (-1) = (-1)2 – 5 (-1) + 6 = 1 + 5 + 6 = 12
(ii) f (2a) = (2a)2 – 5 (2a) + 6 = 4a2 – 10a + 6
(iv) f(x – 1) = (x – 1)2 – 5 (x – 1) + 6
A graph representing the function f(x) is given in figure it is clear that f(9) = 2.
(ii) At x = 9.5, f(x) = 1
Let f(x) = 2x + 5. If x ≠ 0 then
find `frac{f(x+2)-f(2)}{x}`
Given f(x) = 2x + 5, x ≠ 0
\frac{\mathrm{Question 6.}}{}
(i) find `frac{f(0)+f(1)}{2}`
= (576 – 48x – 48x + 4x²)x
V = 4x³ – 96x² + 576x
\frac{Question 8.}{}
\frac{Solution:}{}
f(x²) = 3 – 2x³
\frac{Question 9.}{}
A plane is flying at a speed of 500 km per hour. Express the distance d travelled by the plane as function of time r in hours.
\frac{Answer:}{}
Distance travelled in “t” hours
= 500 × t (distance = speed × time)
\frac{Question 10.}{}
\frac{Solution:}{}
(i) Given y = ax + b …………. (1)
⇒ x = `frac{28.8}{0.9}` =32 cm
Thanks for giving the valuable post
Wow, excellent post. I'd like to draft like this too - taking time and real hard work to make a great article. This post has encouraged me to write some posts that I am going to write soon. exercise books
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On the separation profile of infinite graphs | EMS Press
On the separation profile of infinite graphs
Ádám Timár
Initial steps in the study of inner expansion properties of infinite Cayley graphs and other infinite graphs, such as hyperbolic ones, are taken, in a flavor similar to the well-known Lipton–Tarjan
\sqrt{n}
separation result for planar graphs. Connections to relaxed versions of quasi-isometries are explored, such as regular and semiregular maps.
Itai Benjamini, Oded Schramm, Ádám Timár, On the separation profile of infinite graphs. Groups Geom. Dyn. 6 (2012), no. 4, pp. 639–658
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December 2012 On the Half-Cauchy Prior for a Global Scale Parameter
Nicholas G. Polson, James G. Scott
This paper argues that the half-Cauchy distribution should replace the inverse-Gamma distribution as a default prior for a top-level scale parameter in Bayesian hierarchical models, at least for cases where a proper prior is necessary. Our arguments involve a blend of Bayesian and frequentist reasoning, and are intended to complement the case made by Gelman (2006) in support of folded-
t
priors. First, we generalize the half-Cauchy prior to the wider class of hypergeometric inverted-beta priors. We derive expressions for posterior moments and marginal densities when these priors are used for a top-level normal variance in a Bayesian hierarchical model. We go on to prove a proposition that, together with the results for moments and marginals, allows us to characterize the frequentist risk of the Bayes estimators under all global-shrinkage priors in the class. These results, in turn, allow us to study the frequentist properties of the half-Cauchy prior versus a wide class of alternatives. The half-Cauchy occupies a sensible middle ground within this class: it performs well near the origin, but does not lead to drastic compromises in other parts of the parameter space. This provides an alternative, classical justification for the routine use of this prior. We also consider situations where the underlying mean vector is sparse, where we argue that the usual conjugate choice of an inverse-gamma prior is particularly inappropriate, and can severely distort inference. Finally, we summarize some open issues in the specification of default priors for scale terms in hierarchical models.
Nicholas G. Polson. James G. Scott. "On the Half-Cauchy Prior for a Global Scale Parameter." Bayesian Anal. 7 (4) 887 - 902, December 2012. https://doi.org/10.1214/12-BA730
Keywords: hierarchical models , normal scale mixtures , shrinkage
Nicholas G. Polson, James G. Scott "On the Half-Cauchy Prior for a Global Scale Parameter," Bayesian Analysis, Bayesian Anal. 7(4), 887-902, (December 2012)
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Column space | StudyPug
Subspace of
\Bbb{R}^n
Column Space (final)
What is the column space of a matrix
Going back to the basic definition of a matrix, remember that we can think of a matrix as a collection of column vectors, this collection can be represented easily as:
Equation 1: Matrix A
v_{1}, v_{2}, ..., v_{n}
are the columns of A, each of them a column vector in itself.
This comes from the simple fact that our standard way to write matrices is in the form m x n, something we already know but is always good to remember, meaning that a matrix has m amount of rows and n amount of columns. This row and column standard gives us a little heads up to understand the notation above, since in order to know how many columns are there into our particular matrix we check for the highest value of n and that is where the range {
v_{1}, v_{2}, ..., v_{n}
} from equation 1 comes from.
The column space of a matrix A (also known as the range of a matrix) is then the set of vectors which contains all of the possible linear combinations of the vectors {
v_{1}, v_{2}, ..., v_{n}
}. Remember linear combinations mean you can have the column vectors multiplied by constant coefficients and the resulting vectors will be part of the column space of the matrix. Simply said, the column space of A can be defined as:
Equation 2: Column space of matrix A
Where a span is the result of the multiplication of a scalar field with a vector field producing all the possible linear combinations of the vectors contained within the matrix in question. Before continuing, make sure you are certain of all the terms we are using so far, the term "linear combinations" happens to be a usual key point since it could be confused on its meaning depending on the importance given to either the value of a combination or the notation of its mathematical expression. In any way, linear combinations will be used in the simplest possible way during all our example exercises for the lesson but it is important that you have reviewed lessons such as linear independence and/or linear combination and vector equations before getting into the core for this topic.
Having looked at the column space definition above, we can identify if a certain vector is part of the column space of a matrix by checking if it is part of a possible linear combination resulting from A and its multiplication with a vector
\bar{x}
In other words, if vector
\bar{b}
belongs to the column space of A (mathematically: if
\bar{b} \in C(A)
Equation 3: Condition for a vector to be part of the column space of matrix A
Remember that C(A) is a substape of
R^{n}
, therefore, it fulfills the 3 properties of subspaces as well which we studied on the lesson named properties of subspace.
How to find the column space of a matrix
Something nice to notice on this topic is that when learning how to find column space for a given matrix, you just need to look for all the possible linear combinations of the column vectors of the matrix. And so we just need to use the matrix equation represented in equation 3 to know that the matrix column space will contain all the range of possible results from the product of matrix A with a vector
\bar{x}
But the usual question happens to be if a certain vector belongs to the linear algebra column space of a known matrix. And so, in order to find out if a given vector is part of a column space we need to solve for the vector
\bar{x}
in the matrix equation Ax=b (equation 3) and see if the result for such equation is consistent. Let us start by working a simple problem in order to come up with a set of steps to follow for our exercises below.
So, how do we check if vector
\bar{b}
is in the column space of A?
Equation : Vector b and matrix A
We check for the condition expressed on equation 3 to see if vector
\bar{b}
belongs to the column space of A:
Equation : Condition for b belonging to the column space of A
We turn this matrix equation into an augmented matrix to solve for vector
\bar{x}
Equation :Solving for vector x
Where we now row-reduce the augmented matrix to obtain the components of vector x:
Equation : Row-reducing the augmented matrix
Where you can easily see in the equation resulting from the second row that the whole system is inconsistent since zero plus zero cannot be equal to three!
And thus we can quickly determine that vector
\bar{b}
does not belong to the column space of A.
After looking solving this simple example, let us write down the main steps to work through this process. And so, having a column vector b and a known matrix A:
We check if
\bar{b}
belongs to the column space of A using the matrix equation Ax=b denoted in equation 3. If the result if consistent, then
\bar{b}
belongs to the column space of A.
Set up the Matrix equation Ax=b with the given values for the matrix and the vector.
Turn the equation into an augmented matrix to solve for vector
\bar{x}
Row-reduce the augmented matrix.
Obtain the components
x_{1}
x_{2}
of vector
\bar{x}
by transcribing the linear equation system from the row reduction.
Check if the results are consistent.
We will be using these steps while solving example problems 1 and 2 in the example exercises section below.
Find basis for column space
The second type of problem we will be solving throughout this lesson is that requiring you to find the basis for the column space of the given matrix.
The basis of column space in a matrix is the minimum set of vectors which are linearly independent in the span of the subspace which conforms the column space. Finding this basis is really simple and can be done following the next four steps:
Row reduce the matrix A to echelon form
Circle the columns with pivots in the row-reduced matrix
Go back to the original matrix and circle the columns with the same positions
Use these columns to write out the basis for C(A).
This process will be followed in order to find the basis for column space of the matrices given in problem examples 3 and 4 in the next section.
Let A and
\bar{b}
be as shown below. Determine whether
\bar{b}
is in the column space of A.
Equation for example 1: Matrix A and vector b
We start by setting up the condition shown in equation 3 and then we turn the matrix equation into an augmented matrix to solve for vector
\bar{x}
Equation for example 1(a): Condition for b to belong to the column space of A
Now we just row-reduce the augmented matrix to see if we find a consistent result.
Equation for example 1(b): Row reducing to find a set of linear equations
If we check the set of linear equations obtained from the row-reduction of the augmented matrix, we can clearly see how the values of the components are inconsistent since is impossible for 0+0 to be equal to 4! And so, since this result is incorrect, this means vector
\bar{b}
Now let us have A and
\bar{b}
be as shown below. Determine if
\bar{b}
Once again, we set up the condition from the matrix equation Ax=b shown in equation 3 and then we turn it into an augmented matrix to solve for vector
\bar{x}
In this case we have as a result a set of three linear equations, we now work through them and find the value for each of the components of the vector x. If the values are consistent, then this means b belongs to the column space of A, if the values do not make sense, then we would obtain the same as in example 1.
Since the value for
x_{1}
is already known, we just take the second and third equations from the set found above and solve for
x_{2}
x_{3}
Equation for example 2(c): Solving for the values of the components of x
Having found the values of
x_{2}
x_{3}
we can now construct the whole vector
\bar{x}
Equation for example 2(d): Vector x
The solution to the matrix equation is consistent. Therefore, vector
\bar{b}
Find a basis for the column space C(A) when the matrix A is as follows:
Equation for example 3: Matrix A
For this problem we use the steps described on the section "find basis for column space" above. Notice that for the case of this example problem, the echelon form of matrix A is already given to us (is the right hand side of the equation above), therefore, we do not have to work the row-reduction for this problem and we go directly into step 2: Circle the columns with pivots in the row-reduced matrix
Equation for example 3(a): Circling the pivots in the echelon form matrix
The we go back to the original matrix and circle the columns with the same positions:
Equation for example 3(b): Circling the proper positions in the original matrix A
And finally, we use these columns to write out the basis for C(A).
Equation for example 3(c): Basis for C(A)
After looking at this example is important to note that it does not matter how many rows and columns matrix A has, the basis of a matrix column space can always be found since you can row-reduce the matrix into one of its echelon forms. If you want to practice and review row reduction and echelon forms do not hesitate to go back to that lesson and review its videos.
Find the basis for column space of A if the matrix is as follows:
We start by transforming matrix A into its row echelon form by row reduction:
Equation for example 4(a): Row reduction to produce the row echelon form matrix
And so, we now have the original matrix and the row echelon form matrix:
Equation for example 4(b): Matrix A and its row echelon form
Following step two we circle the columns with pivots in the row-reduced matrix:
Equation for example 4(c): Circling the pivots in the echelon form matrix
Circling the same positions in the original matrix:
Equation for example 4(d): Circling the proper positions in the original matrix A
And so, we use these columns to write out the basis for C(A):
Equation for example 4(e): Basis for C(A)
Having seen a few explicit examples, is time to go to the next lesson. If you would like to keep on your practice you can check out this article on row space and column space. Also, for an extra example that you can take a look while studying, we recommend you to look into this column space conditions problem.
\Bbb{R}^n
A
\Bbb{R}^n
A
A=[v_1\;v_2\; \cdots \;v_n ]
v_1,v_2,\cdots,v_n
A
. Then the column space of
A
C(A)
which forms a linear combination of the columns of
A
\vec{b}
is in the column space of
A
, we need to see if
\vec{b}
is a linear combination of the columns of
A
\vec{b} =x_1 v_1+x_2 v_2+\cdots+x_n v_n
x_1,x_2,\cdots,x_n
are solutions to the linear equation.
To find a basis for the column space of a matrix A, we:
1) Row reduce the matrix to echelon form.
2) Circle the columns with pivots in the row-reduced matrix.
3) Go back to the original matrix and circle the columns with the same positions.
4) Use those columns to write out the basis for
C(A)
Column Space Overview:
definition of the column space
C(A)=
• A set of vectors which span{
v_1,v_2,\cdots,v_n
\vec{b} \;\epsilon \;C(A),
\vec{b}= x_1 v_1+x_2 v_2+\cdots+x_n v_n
A vector in the column space
\vec{b} =x_1 v_1+x_2 v_2+\cdots+x_n v_n
• Changing to
Ax=b
Finding a basis for the column space
• Row reduce the matrix to echelon form.
• Locate the columns with pivots in the row-reduced matrix.
• Go back to the original matrix and find the columns with the same position.
• Use those columns to write out the basis
Finding if a vector is in the column space
Let and . Determine whether
b
A
b
A
Here is the matrix
A
, and an echelon form of
A
. Find a basis for
C(A)
(column space of
A
Find a basis for the column space of
A
\Bbb{R}^n
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Cartridge flow-control valve in an isothermal liquid network - MATLAB - MathWorks Italia
Cartridge Valve Insert (IL)
Conical Valve Seat
Numerically-Smoothed Force and Opening
Poppet diameter
Poppet position vector
Poppet position vector, s
Valve opening time constant
Cartridge flow-control valve in an isothermal liquid network
The Cartridge Valve Insert (IL) block models a cartridge flow-control valve in an isothermal liquid network. The valve seat can be specified as conical or as a custom opening parameterized by analytical or tabular formulations. The valve opens when the combined pressures at ports A and B exceed the Spring preload force and pressure at port X.
You can specify the block seat geometry as either conical or a custom. This seat setting determines the sub-components that make up the block. In both configurations, the Port A poppet to port X pilot area ratio parameter sets the force ratio in the underlying Cartridge Valve Actuator block.
Use the Cartridge Valve Insert (IL) block when you would like flow control set by a pilot pressure line. Use the Pressure-Compensated 3-Way Flow Control Valve (IL) or Pressure-Compensated Flow Control Valve (IL) block for flow control due to a pressure differential or the Poppet Valve (IL) block for valve opening controlled by an external physical signal.
The conical cartridge valve insert is a composite of two Isothermal Liquid library blocks:
Conical Cartridge Valve Insert Schematic
The custom cartridge valve insert is a composite of two Isothermal Liquid library blocks:
Custom Cartridge Valve Insert Schematic
In the custom configuration, you can parameterize the valve opening analytically or with a data set.
Analytical Parameterization
By setting Orifice parameterization to Linear - area vs. control member position, the valve opening area is linearly proportional to the poppet position. Once the pressure at port A or B exceeds the Spring preload force, the valve opens until the Maximum orifice area is reached. When the valve is fully closed, a small Leakage area remains open to flow so that numerical continuity is maintained in the network.
Tabulated Parameterization
By setting Orifice parameterization to Tabulated data - Area vs. control member position, you can supply the opening profile based on opening area and poppet position. The block queries between data points with linear interpolation and uses nearest extrapolation for points beyond the table boundaries.
By setting Orifice parameterization to Tabulated data - Volumetric flow rate vs. control member position and pressure drop, you can supply the volumetric flow rate through the valve as a parameterized table of poppet position and valve pressure drop. The block queries between data points with linear interpolation and uses linear extrapolation for points beyond the table boundaries. The volumetric flow rate is converted to a mass flow rate by multiplying by the fluid density.
{\stackrel{˙}{p}}_{dyn}=\frac{{p}_{control}-{p}_{dyn}}{\tau }.
When the actuator is close to full extension or full retraction, you can maintain numerical robustness in your simulation by adjusting the block Smoothing factor. A smoothing function is applied to the actuator force and orifice opening or area, but primarily influences the simulation at the extremes of these ranges.
\stackrel{^}{F}=\frac{{F}_{A}+{F}_{B}-{F}_{Preload}-{F}_{Pilot}}{k{x}_{stroke}}.
FA is the force at port A.
FB is the force at port B.
FPreload is the Spring preload force.
FPilot is the force at port X.
{\stackrel{^}{F}}_{X,smoothed}=\frac{1}{2}+\frac{1}{2}\sqrt{{\stackrel{^}{F}}_{X}^{2}+{\left(\frac{s}{4}\right)}^{2}}-\frac{1}{2}\sqrt{{\left({\stackrel{^}{F}}_{X}-1\right)}^{2}+{\left(\frac{s}{4}\right)}^{2}},
Similarly, when Valve seat specification is set to Conical, the normalized orifice opening distance is:
\stackrel{^}{h}=\frac{h}{{h}_{\mathrm{max}}},
h is the poppet opening distance.
hmax is the maximum poppet opening distance.
The smoothed, normalized opening is:
{\stackrel{^}{h}}_{smoothed}=\frac{1}{2}+\frac{1}{2}\sqrt{{\stackrel{^}{h}}_{}^{2}+{\left(\frac{s}{4}\right)}^{2}}-\frac{1}{2}\sqrt{{\left(\stackrel{^}{h}-1\right)}^{2}+{\left(\frac{s}{4}\right)}^{2}}.
The smoothed opening distance is:
{h}_{smoothed}={\stackrel{^}{h}}_{smoothed}{h}_{\mathrm{max}}.
When Valve seat specification is set to Custom, the normalized valve area is calculated as:
\stackrel{^}{A}=\frac{\left({A}_{orifice}-{A}_{leak}\right)}{\left({A}_{\mathrm{max}}-{A}_{leak}\right)}.
Aorifice is the valve open area.
The smoothed, normalized area is:
{\stackrel{^}{A}}_{smoothed}=\frac{1}{2}+\frac{1}{2}\sqrt{{\stackrel{^}{A}}_{}^{2}+{\left(\frac{s}{4}\right)}^{2}}-\frac{1}{2}\sqrt{{\left(\stackrel{^}{A}-1\right)}^{2}+{\left(\frac{s}{4}\right)}^{2}}.
The smoothed area is:
{A}_{smoothed}={\stackrel{^}{A}}_{smoothed}\left({A}_{\mathrm{max}}-{A}_{leak}\right)+{A}_{leak}.
Liquid entry or exit port.
Pilot pressure port. There is no mass flow rate through port X.
Conical (default) | Custom
Poppet diameter — Diameter of the valve control member
Diameter of the valve control member.
To enable this parameter, set Valve seat specification to Conical.
Seat cone angle — Angle of the seat opening
The valve opening is calculated in one of three ways, linearly or by tabulated data:
Linear - area vs. control member position: The orifice area is determined by a linear relationship to the poppet opening or closing distance.
Tabulated data - Area vs. control member position: The opening area is interpolated from the Poppet position vector and the Orifice area vector based on the current poppet position.
Tabulated data - Volumetric flow rate vs. control member position and pressure drop. The volumetric flow rate is directly interpolated from the provided Poppet position vector, s; Pressure drop vector, dp; and Volumetric flow rate table, q(s,dp) parameters, based on the current poppet position.
To enable this parameter, set Valve seat specification to Custom.
To enable this parameter, set Valve seat specification to Custom and Orifice parameterization to Linear - area vs. control member position.
Poppet position vector — Vector of opening positions
Vector of orifice opening positions for the tabular parameterization of the orifice opening area. The vector elements must correspond one-to-one with the elements in the Orifice area vector parameter. The elements are listed in ascending order and must be greater than 0.
To enable this parameter, set Valve seat specification to Custom and Orifice parameterization to Tabulated data - Area vs. control member position.
Vector of valve opening areas for the tabular parameterization of the valve opening area. The vector elements must correspond one-to-one with the elements in the Poppet position vector parameter. The elements are listed in ascending order and must be greater than 0.
Poppet position vector, s — Vector of control member positions
Vector of control member positions for the tabular parameterization of the volumetric flow rate. The spool travel vector forms an independent axis with the Pressure drop vector, dp parameter for the 3-D dependent Volumetric flow rate table, q(s,dp) parameter. A positive displacement corresponds to valve opening. The values are listed in ascending order and the first element must be 0. Linear interpolation is employed between table data points.
To enable this parameter, set Valve seat specification to Custom and Orifice parameterization to Volumetric flow rate vs. control member position and pressure drop.
Vector of pressure drop values for tabular parametrization of volumetric flow rate. The pressure drop vector forms an independent axis with the Poppet position vector, s parameter for the 3-D dependent Volumetric flow rate table, q(s,dp) parameter. The values are listed in ascending order and must be greater than 0. Linear interpolation is employed between table data points.
Matrix of volumetric flow rates based on independent values of pressure drop and spool travel distance. M and N are the sizes of the corresponding vectors:
M is the number of elements in the Poppet position vector, s parameter.
Ratio of the inlet port, A, to the pilot pressure port, X. This value is used to calculate the force at port X. The ratio must be less than or equal to 1.
Spring preload force — Spring force in neutral poppet position
Spring force on the poppet when the poppet is in the neutral position. This parameter is a threshold value which, when added to the pilot pressure at port X, counterbalances the valve opening due to the pressures at ports A and B.
Valve seat specification to Conical.
Valve seat specification to Custom and Orifice parameterization to Linear - Area vs. control member position.
Valve seat specification to Custom and Orifice parameterization to Linear - Area vs. control member position or Tabulated data - Area vs. control member position.
Valve opening time constant — Valve time constant
Cartridge Valve Actuator (IL) | Poppet Valve (IL) | Orifice (IL) | Pressure-Compensated 3-Way Flow Control Valve (IL) | Pressure-Compensated Flow Control Valve (IL)
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Colours | iforwms
A question came up last night at the regular Demo Bar quiz that I didn’t expect.
What colour is represented by the hex value #FFFFFF?
So if you’ve done any web development you probably know the answer is white. But why?
At the most basic level you could say that we see things as different colours because light reflects off those objects in varying wavelengths. The colours we are able to perceive are limited to what is known as “visible light” – light with a wavelength between 390 nm and 700 nm.
Wavelength Interval [nm]
Frequency Interval [THz]
Red ~ 700–635 ~ 430–480
Orange ~ 635–590 ~ 480–510
Yellow ~ 590–560 ~ 510–540
Green ~ 560–520 ~ 540–580
Cyan ~ 520–490 ~ 580–610
Blue ~ 490–450 ~ 610–670
Violet ~ 450–400 ~ 670–750
Increase the wavelength to over 1000 nm and the human eye can’t see it anymore, but you’re remote control wouldn’t work without it – infrared. At the other end of the scale, outside the visible spectrum, we have ultraviolet.
To create these colours we have a few different techniques we can use, let’s take a look at them.
Additive Colouring – The RGB Colour Model
In the world of computers, the RGB colour model is used everywhere; your browser, Photoshop, operating system, even MS Paint. The idea behind it is very simple, choose a number from zero to 255 for red, green and blue and you end up with a new colour. We have 256 values for red, green, and blue to choose from so we can work out exactly how many different colours can be produced using the RGB model.
256×256×256=16,777,216
The human eye can distinguish about 10 million different colours[2], so I think we’re covered. Let’s see it in action!
The RGB Colour Model – Basic Colours
The RGB–Hexadecimal Relationship
So back to this quiz question, how are these all equal?
\text{\#FFFFFF}=RGB(255,255,255)=White
Each colour in the hexadecimal system comprises of six characters;
the first two characters represent red;
the middle two characters represent green; and
the last two characters represent blue.
To make things clearer, let’s add some colour to the hex string: #FFFFFF. OK, that all makes sense, but why don’t we have numbers zero to 99, where did all the F’s come from?
The word hexadecimal comprises hexa-, from the Greek for six, -decimal from the Latin for tenth.
So we’re not dealing with a normal base 10 numbering system, we’re dealing with a base 16 numbering system. So the “numbers” available to us are as follows.
So let’s take a look at a hex value of #FFFF00, we can see that:
Red=\text{FF}
Green=\text{FF}
Blue=00
We just need to convert the hexadecimal values into a numbers between zero and 255, and I just so happen to know the formula for the job.
\text{Let }\mathit{x} = \text{the first character in the pair}
\text{Let }\mathit{y} = \text{the second character in the pair}
\text{RGB value}=(\mathit{x}×16)+\mathit{y}
Now we just need just substitute the values into the formula.
F=15
\text{Red} = (15 \times 16) + 15 = 255
\text{Green} = (15 \times 16) + 15 = 255
\text{Blue} = (0 \times 16) + 0 = 0
So we end up with the RGB value for the colour yellow!
\text{\#FFFF00}=RGB(255,255,0)
Which is 100% red, 100% green and 100% blue, giving us white, and one point in the quiz.
If you’re still awake, there’s a few other systems we can use to define colours, for now I’ll leave you with the CMYK colour model.
Subtractive Colouring – The CMYK Colour Model
An alternative to the RGB system is the CMYK colour model. CMYK refers to the four colours used in the system, cyan, magenta, yellow and key (black). The CMYK model works by partially or entirely masking colours on a lighter, usually white, background. The ink reduces the light that would otherwise be reflected. Such a model is called subtractive because inks “subtract” brightness from white[2].
Instead of having values from zero to 255 like we saw in the RGB colour model, the CMYK model has decimal values between zero and one. See here for the conversion between the two.
The CMYK Colour Model – Basic Colours
Converting between the two is pretty straight forward, let’s take a look at the formulae.
First we need to rewrite the R, G and B values on a zero to one scale.
R{\prime}=R/255
G{\prime}=G/255
B{\prime}=B/255
Next up is calculating the key (black) value.
K=1−max(R{\prime},G{\prime},B{\prime})
Once we have the key value, we can calculate the remaining cyan, magenta and yellow.
C=(1−R{\prime}−K)/(1−K)
M=(1−G{\prime}−K)/(1−K)
Y=(1−B{\prime}−K)/(1−K)
Jump up ^ Craig F. Bohren (2006). Fundamentals of Atmospheric Radiation: An Introduction with 400 Problems. Wiley-VCH. ISBN 3-527-40503-8.
Jump up ^ Judd, Deane B.; Wyszecki, Günter (1975). Color in Business, Science and Industry. Wiley Series in Pure and Applied Optics (third ed.). New York: Wiley-Interscience. p. 388. ISBN 0-471-45212-2.
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A sharp necessary condition for rectifiable curves in metric spaces | EMS Press
In his 1990 Inventiones paper, P. Jones characterized subsets of rectifiable curves in the plane, using a multiscale sum of what is now known as Jones
\beta
-numbers, numbers measuring flatness in a given scale and location. This work was generalized to
\mathbb R^n
by Okikiolu, to Hilbert space by the second author, and has many variants in a variety of metric settings. Notably, in 2005, Hahlomaa gave a sufficient condition for a subset of a metric space to be contained in a rectifiable curve. We prove the sharpest possible converse to Hahlomaa’s theorem for doubling curves, and then deduce some corollaries for subsets of metric and Banach spaces, as well as the Heisenberg group.
Guy C. David, Raanan Schul, A sharp necessary condition for rectifiable curves in metric spaces. Rev. Mat. Iberoam. 37 (2021), no. 3, pp. 1007–1044
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Predicting Separation and Transitional Flow in Turbine Blades at Low Reynolds Numbers—Part II: The Application to a Highly Separated Turbine Blade Cascade Geometry | J. Turbomach. | ASME Digital Collection
Darius D. Sanders,
Darius D. Sanders
J. Bernard Jones Professor
Air Force Research Lab, Propulsion Directorate,
Douglas C. Rabe
Walter F. O’Brien J. Bernard Jones Professor
A companion article has been published: Predicting Separation and Transitional Flow in Turbine Blades at Low Reynolds Numbers—Part I: Development of Prediction Methodology
Sanders, D. D., O’Brien, W. F., Sondergaard, R., Polanka, M. D., and Rabe, D. C. (November 15, 2010). "Predicting Separation and Transitional Flow in Turbine Blades at Low Reynolds Numbers—Part II: The Application to a Highly Separated Turbine Blade Cascade Geometry." ASME. J. Turbomach. July 2011; 133(3): 031012. https://doi.org/10.1115/1.4001231
There has been a need for improved prediction methods for low pressure turbine (LPT) blades operating at low Reynolds numbers. This is known to occur when LPT blades are subjugated to high altitude operations causing a decrease in the inlet Reynolds number. Boundary layer separation is more likely to be present within the flowfield of the LPT stages due to increase in the region adverse pressure gradients on the blade suction surface. Accurate CFD predictions are needed in order to improve design methods and performance prediction of LPT stages operating at low Reynolds numbers. CFD models were created for the flow over two low pressure turbine blade designs using a new turbulent transitional flow model, originally developed by Walters and Leylek (2004, “A New Model for Boundary Layer Transition Using a Single Point RANS Approach,” ASME J. Turbomach., 126(1), pp. 193–202). Part I of this study applied Walters and Leylek’s model to a cascade CFD model of a LPT blade airfoil with a light loading level. Flows were simulated over a Reynolds number range of 15,000–100,000 and predicted the laminar-to-turbulent transitional flow behavior adequately. It showed significant improvement in performance prediction compared to conventional RANS turbulence models. Part II of this paper presents the application of the prediction methodology developed in Part I to both two-dimensional and three-dimensional cascade models of a largely separated LPT blade geometry with a high blade loading level. Comparisons were made with available experimental cascade results on the prediction of the inlet Reynolds number effect on surface static pressure distribution, suction surface boundary layer behavior, and the wake total pressure loss coefficient. The
kT-kL-ω
transitional flow model accuracy was judged sufficient for an understanding of the flow behavior within the flow passage, and can identify when and where a separation event occurs. This model will provide the performance prediction needed for modeling of low Reynolds number effects on more complex geometries.
blades, boundary layers, computational fluid dynamics, flow separation, flow simulation, geometry, laminar to turbulent transitions, turbines, turbulence
Airfoils, Blades, Boundary layers, Computational fluid dynamics, Flow (Dynamics), Pressure, Reynolds number, Separation (Technology), Turbulence, Geometry, Cascades (Fluid dynamics)
Experimental and Numerical Investigation of Losses in Low Pressure Turbine Blade Rows
A New Model for Boundary Layer Transition Using a Single Point RANS Approach
Prediction of Boundary-Layer Transition on Turbine Airfoil Profile Losses
Numerical Investigation of Transitional Flow Through a Low Pressure Turbine Cascade
Characterization of the GH1R Low Pressure Turbine
,” MS thesis, Air Force Institute of Technology, WPAFB, OH, AFIT/DS/ENY/05-S02.
Effect of Dimple Pattern on the Suppression of Boundary Layer Separation on a Low Pressure Turbine Blade
,” MS thesis, Air Force Institute of Technology, WPAFB, OH, AFIT/GAE/ENY/04-M05.
Investigation of Separation Control in Low Pressure Turbine Using Pulsed Vortex Generator Jets
High Fidelity Simulations of Transitional Flows Past a Plunging Airfoil
Loelbach
Numerical Investigation of a Low Reynolds Number Flow Field in a Turbine Blade Row
Simulation of Separated Flow Inside a Low-Pressure Turbine Cascade
Turbulence Modeling for Low Pressure Blades
Turbulence Model Comparisons for Mixing Plane Simulations of a Multistage Low Pressure Turbine Operating at Low Reynolds Numbers
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Gyu-bong Cho, Tae-hoon Kwon, Tae-hyun Nam, Sun-chul Huh, Byeong-keun Choi, Hyo-min Jeong, Jung-pil Noh, "Structural and Electrochemical Properties of Lithium Nickel Oxide Thin Films", Journal of Chemistry, vol. 2014, Article ID 824083, 5 pages, 2014. https://doi.org/10.1155/2014/824083
Gyu-bong Cho,1 Tae-hoon Kwon,1 Tae-hyun Nam,1 Sun-chul Huh,2 Byeong-keun Choi,2 Hyo-min Jeong,2 and Jung-pil Noh2
1School of Materials Science and Engineering and RIGECT, Gyeongsang National University, No. 501, Jinju-daero, Jinju, Gyeongnam 660-701, Republic of Korea
2Department of Energy and Mechanical Engineering and Institute of Marine Industry, Gyeongsang National University, No. 38, Cheondaegukchi-gil, Tongyeong, Gyenognam 650-160, Republic of Korea
LiNiO2 thin films were fabricated by RF magnetron sputtering. The microstructure of the films was determined by X-ray diffraction and field-emission scanning electron microscopy. The electrochemical properties were investigated with a battery cycler using coin-type half-cells. The LiNiO2 thin films annealed below 500°C had the surface carbonate. The results suggest that surface carbonate interrupted the Li intercalation and deintercalation during charge/discharge. Although the annealing process enhanced the crystallization of LiNiO2, the capacity did not increase. When the annealing temperature was increased to 600°C, the FeCrNiO4 oxide phase was generated and the discharge capacity decreased due to an oxygen deficiency in the LiNiO2 thin film. The ZrO2-coated LiNiO2 thin film provided an improved discharge capacity compared to bare LiNiO2 thin film suggesting that the improved electrochemical characteristic may be attributed to the inhibition of surface carbonate by ZrO2 coating layer.
In an attempt to miniaturize high performance portable electronic equipment, batteries with high power and energy density are required. Thin film batteries have been developed in an attempt to satisfy this requirement [1–3]; however, improving the performance of the cathode films is critical for advancing the use of electrochemical thin film batteries. Among the possible materials that could be used for a cathode film, LiCoO2, owing to its high theoretical specific capacity and ease of preparation, is a promising candidate [4–6]. However, the high cost and toxicity of cobalt limit the use of LiCoO2 in thin film batteries. Therefore, it is necessary to develop less expensive cathode materials for thin film battery applications. LiNiO2 has emerged as a useful cathode material owing to its low cost and high energy density [7–10].
In the current study, LiNiO2 thin films were deposited by using RF magnetron sputtering. The microstructure of the films was measured by X-ray diffraction and field-emission scanning electron microscopy. Finally, the electrochemical properties were investigated with a battery cycler using coin-type half-cells, in the potential range of 3.0 V–4.2 V.
Bare and ZrO2-coated LiNiO2 thin films have been deposited onto stainless steel (STS 304) foil substrates held at a distance of 6 cm away from the target. The STS304 substrate was ultrasonically cleaned with acetone, alcohol, and distilled water in that order. The LiNiO2 and ZrO2 targets were made by Pascal Co. (Japan). A base vacuum of 5 × 10−6 Torr was obtained with a cryopump. Gas flow rate of Ar/O2 was 4/1 with a total gas flow amount of 150 sccm. Deposition pressure was maintained at 2 × 10−3 Torr during deposition. The LiNiO2 target was presputtered for 30 min and the deposition time was 360 min at 100 W RF power. ZrO2 coating layer was deposited for 10 min at 100 W RF power. Figures 1(a) and 1(b) show the surface and cross-section images of bare LiNiO2 thin film deposited on Si wafer substrate. As seen in Figure 1(a), a uniform distribution of clusters of ~50 nm was seen. The calculated deposition rates of LiNiO2 were approximately 1.7 nm/min. The deposited thin films were annealed from 400°C to 600°C in air to obtain the crystalline film.
FE-SEM images of (a) surface and (b) cross-section images of LiNiO2 thin film deposited on Si wafer substrate.
The structure of the LiNiO2 thin films was investigated by X-ray diffractometry (XRD, Rigaku, Miniflex). The XRD measurements were performed using Cu Kα radiation ( Å) and phase identification was made by comparing the diffraction patterns with the JCPDS references. The morphology of the deposited films was studied by field-emission scanning electron microscopy (FE-SEM, Jeol, JSM-6701F).
In order to examine the electrochemical properties of cathode thin films, coin-type cells were assembled with lithium foils as the counter and reference electrode and 1 M LiPF6 in ethylene carbonate (EC) : diethyl carbonate (DEC) (1 : 1, vol.%) electrolytic solution. The charge-discharge test was carried out with a battery cycler (Won A Tech, WDCS3000s) at a constant current density of 5 μA/cm2 in the potential range of 3.0–4.2 V.
Figure 2 shows the XRD patterns of as-deposited and annealed LiNiO2 thin films at various temperatures at 10 min. Crystallization peaks of LiNiO2 were not seen in the as-deposited films. However, the XRD pattern of the 400°C annealed film exhibited broad (104) LiNiO2 and lithium carbonate (Li2CO3) reflection peaks. The degree of crystallization of LiNiO2 was enhanced at 500°C because the (104) reflection became stronger as the annealing temperature was increased. The intensity of the Li2CO3 peak also increased. However, after annealing at 600°C, the reflection peaks of LiNiO2 and Li2CO3 disappeared and impurity peaks of NiCrFeO4, which was considered an oxide layer of the STS304 substrate, were observed (Figure 2(d)).
XRD patterns of (a) as-deposited, (b) 400°C, (c) 500°C, and (d) 600°C annealed LiNiO2 thin films. The annealing time was 10 minutes.
Surface images of the annealed LiNiO2 thin films are shown in Figure 3. The 400°C annealed film had a rough surface with no cracks and some surface impurities (Figure 3(a)). When the annealing temperature was increased to 500°C, the size of the surface impurities grew and became angular. After annealing at 600°C, complete removal of the surface impurities was achieved and the surface had a smooth morphology compared with that of the 400°C annealed film. The XRD (Figure 2) and FE-SEM (Figure 3) results suggest that the angulated surface impurity, which disappeared at an annealing temperature of 600°C, was lithium carbonate (Li2CO3).
FE-SEM images of LiNiO2 thin film with various annealing temperatures at (a) 400°C, (b) 500°C, and (c) 600°C, respectively. The annealing time was 10 minutes.
Figure 4 shows the initial discharge curves of the as-deposited and annealed LiNiO2 thin films. All of these films were tested at a current density of 5 μA/cm2 between 3.0 V and 4.2 V and a plateau was observed in the 400°C annealed films. When the annealing temperature was increased to 500°C, the plateau was more clearly exhibited and indicated a phase transition of crystalline LiNiO2 [11, 12]; the discharge capacity, however, did not increase. These results suggest that the surface carbonate interrupts the lithium intercalation and deintercalation during charge/discharge, affecting the discharge capacity. The 600°C annealed films exhibited a lower initial discharge capacity compared with the other annealed films. This may be attributed to the formation of an intermediate oxide layer between the substrate and active material film. The intermediate oxide layer that formed may lead to an oxygen deficiency in the annealed LiNiO2 thin films. In order for the LiNiO2 film to obtain good electrochemical characteristics, the deposited LiNiO2 thin films should be annealed at a temperature that prevents the formation of an intermediate oxide layer. In addition, it is necessary to remove the surface carbonate. It has been previously reported that surface coatings enhance the electrochemical properties of cathode materials without sacrificing the specific capacity of the respective cathode [13].
Initial discharge curves of LiNiO2 thin film with various annealing temperatures. The annealing time was 10 minutes.
Figure 5 shows the XRD patterns of bare and ZrO2-coated LiNiO2 thin films annealed at 500°C for 10 min. As seen in Figure 5(a), the diffraction peak of the surface carbonate disappeared in the coated thin film, suggesting that the ZrO2 coating prevents the formation of surface carbonate during the annealing process.
XRD patterns of (a) noncoated and (b) ZrO2-coated LiNiO2 thin films. The films were annealed in 10 min at 500°C.
Figure 6 shows the first discharge curves and cycle stability of bare and ZrO2-coated LiNiO2 thin films. As seen in Figures 6(a) and 6(b), a single plateau was observed in both of the films, corresponding to the phase transition of crystalline LiNiO2. Therefore, this would indicate that both films are crystallized. However, the first discharge capacity is slightly different such that the first discharge capacity of ZrO2-coated film is higher than that of the bare LiNiO2 thin film. The capacity retention rate is similar in both thin films as seen in Figure 6(b).
(a) Initial discharge behavior and (b) cyclic performance of LiNiO2 thin films. The films were annealed in 10 min at 500°C.
LiNiO2 thin films were fabricated by RF magnetron sputtering. Crystallization began at annealing temperatures above 400°C; however, the films that were annealed below 500°C exhibited surface carbonate (in the form of Li2CO3) identified by XRD. Surface carbonate interrupts the Li intercalation and deintercalation during charge/discharge and therefore the capacity did not increase, although the degree of LiNiO2 crystallization was enhanced. When the annealing temperature was increased to 600°C, the FeCrNiO4 oxide phase was generated and the discharge capacity decreased due to the oxygen deficiency in the LiNiO2 thin film. The ZrO2-coated LiNiO2 thin film provided an improved discharge capacity compared to the bare LiNiO2 thin film at an annealing temperature of 500°C. Therefore, the improvement in electrochemical characteristics can be attributed to the inhibition of surface carbonate by the ZrO2 coating layer.
This research was supported by the Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (2012-R1A1A2008821 and 2012-R1A2A1A01006546).
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Copyright © 2014 Gyu-bong Cho et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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Introduction to the Cartesian plane | StudyPug
The Cartesian plane was created by René Descartes to help people identify where something was located on a map or a graph. It uses a relationship between two variables. What are the elements to a Cartesian Coordinates System? Let's find out!
The main way that the Cartesian Coordinates System allows you to locate something is through its x and y axis. The x axis is what you call the left-right direction of the plane. A way to help you remember this is that "x" is a cross. Therefore, x goes "across" on the Cartesian plane. The y axis is what you call the up-down direction. A Cartesian plane will always have both the x and y axis.
When you write down a pair of coordinates to help other people locate something on a plane, you'll have to write it in a specific way. Keep in mind that it always comes in a pair since there's the x and y axis that you'll have to consider. This is also called an ordered pair.
There is a specific way you're supposed to write them and it's that you write the horizontal distance before the vertical one. Therefore, an ordered pair looks like this: (x, y).
You may come across the terms "axis of ordinates" and "axis of abscissae". The ordinate simply refers to the vertical portion of an ordered pair, that is, the y axis. The abscissae refers to the horizontal part of a coordinate, this is, the x axis.
A Cartesian plane's x and y axis divides up the plane into four quadrants. Quadrant I is located where x and y is positive (the top right corner of the plane). Quadrant II is where x is negative but y is positive (the top left corner of the plane). Quadrant III is where both x and y are negative (the bottom left corner). Lastly, quadrant IV is where x is positive and y is negative (the bottom right corner).
You may be asked to identify which quadrant a set of coordinates lie in, or be told that an ordered pair is in a certain quadrant. Let's try out some practice problems to see how coordinates work on a Cartesian plane.
What are the coordinates of each point shown on the coordinate grid?
Find coordinates of points on Cartesian plane
A=(ordinate, abscissa)= (x,y)= (6,8)
B=(ordinate, abscissa)= (x,y)= (-6,2)
C=(ordinate, abscissa)= (x,y)= (4,-4)
Predict in which quadrant each of the following points will lie. Then, plot the points of a coordinate grid: A (4,-1), B (-7,3), C (-2,-5), D (0,2), E (-5,0)
Determine which quadrant the points will lie Q1,2,3,4
The signs of x and y coordinates and where the points lie
Predict where the points lie and plot on coordinate grid
Maggie walks to the pool every evening. Her house lies at H(-9,0) and the pool lies at P (9,0)
Join the pair of coordinate with a straight line segment. What is the total distance from her house to the pool? Each grid line/square represents 1km.
Plot the locations of Maggie's home and swimming pool
You can see how coordinates change as you move a point around on a Cartesian plane here on this online diagram. Watch as the x and y values change depending on where you point is!
Captains of a ship must plot their ship's location and destination points on a grid. Similar to a captain, in this section we will learn how to label and plot coordinates on a given grid. In a coordinate grid, the horizontal number line is called the x-axis and the vertical number line is called the y-axis. These x and y axes meet at a point called the origin with coordinates (0, 0). The x and y axes are similar to the horizontal and vertical number lines we used in previous sections when learning how to add and subtract integers. When plotting coordinates, we always start at the origin. First, we count x units left or right from the origin. Next, we count y units up or down.
Introduction to x
-
y plane
Predict in which quadrant each of the following points will lie. Then, plot the points on a coordinate grid: A (4, -1), B (-7, 3), C (-2, -5), D (0, 2), E (-5, 0)
Maggie walks to the pool every evening. Her house lies at H (-9, 0) and the pool lies at P (9, 0). Join the pair of coordinates with a straight line segment. What is the total distance from her house to the pool? Each grid line/square represents 1km.
Practice topics for Coordinates, Quadrants, and Transformations
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Regularity of Minimizers of some Variational Integrals with Discontinuity | EMS Press
Regularity of Minimizers of some Variational Integrals with Discontinuity
We prove regularity properties in the vector valued case for minimizers of variational integrals of the form
\A(u) = \int_\Omega A(x,u,Du)\,dx
where the integrand
A(x,u,Du)
is not necessarily continuous respect to the variable~
x,
grows polinomially like
|\xi|^p,
p \geq 2.
Maria Alessandra Ragusa, Atsushi Tachikawa, Regularity of Minimizers of some Variational Integrals with Discontinuity. Z. Anal. Anwend. 27 (2008), no. 4, pp. 469–482
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Pressure-relief valve in an isothermal system - MATLAB - MathWorks Italia
Pressure Relief Valve (IL)
Opening pressure (gauge) vector
Pressure-relief valve in an isothermal system
The Pressure Relief Valve (IL) models a pressure-relief valve in an isothermal liquid network. The valve remains closed when the pressure is less than a specified value. When this pressure is met or surpassed, the valve opens. This set pressure is either a threshold pressure differential over the valve, between ports A and B, or between port A and atmospheric pressure. For pressure control based on another element in the fluid system, see the Pressure Compensator Valve (IL) block.
Two valve control options are available:
When Set pressure control is set to Controlled, connect a pressure signal to port Ps and define the constant Pressure regulation range. The valve response will be triggered when Pcontrol, the pressure differential between ports A and B, is greater than Pset and below Pmax. Pmax is the sum of Pset and the pressure regulation range.
When Set pressure control is set to Constant, the valve opening is continuously regulated between Pset and Pmax. There are two options for pressure regulation available in the Opening pressure specification parameter: Pcontrol can be the pressure differential between ports A and B or the pressure differential between port A and atmospheric pressure. The opening area is then modeled by either linear or tabular parameterization. When the Tabulated data option is selected, Pset and Pmax are the first and last parameters of the Pressure differential vector, respectively.
{\stackrel{˙}{m}}_{A}+{\stackrel{˙}{m}}_{B}=0.
\stackrel{˙}{m}=\frac{{C}_{d}{A}_{valve}\sqrt{2\overline{\rho }}}{\sqrt{P{R}_{loss}\left(1-{\left(\frac{{A}_{valve}}{{A}_{port}}\right)}^{2}\right)}}\frac{\Delta p}{{\left[\Delta {p}^{2}+\Delta {p}_{crit}^{2}\right]}^{1/4}},
\overline{\rho }
\Delta {p}_{crit}=\frac{\pi \overline{\rho }}{8{A}_{valve}}{\left(\frac{\nu {\mathrm{Re}}_{crit}}{{C}_{d}}\right)}^{2}.
P{R}_{loss}=\frac{\sqrt{1-{\left(\frac{{A}_{valve}}{{A}_{port}}\right)}^{2}\left(1-{C}_{d}^{2}\right)}-{C}_{d}\frac{{A}_{valve}}{{A}_{port}}}{\sqrt{1-{\left(\frac{{A}_{valve}}{{A}_{port}}\right)}^{2}\left(1-{C}_{d}^{2}\right)}+{C}_{d}\frac{{A}_{valve}}{{A}_{port}}}.
The opening area Avalve is determined by the opening parameterization (for Constant valves only) and the valve opening dynamics.
When you set Opening parameterization to Linear - Area vs. pressure, the block calculates the opening area as
{A}_{valve}=\stackrel{^}{p}\left({A}_{\mathrm{max}}-{A}_{leak}\right)+{A}_{leak},
\stackrel{^}{p}
\stackrel{^}{p}=\frac{{p}_{control}-{p}_{set}}{{p}_{\mathrm{max}}-{p}_{set}}.
{\stackrel{^}{p}}_{smoothed}=\frac{1}{2}+\frac{1}{2}\sqrt{{\stackrel{^}{p}}_{}^{2}+{\left(\frac{f}{4}\right)}^{2}}-\frac{1}{2}\sqrt{{\left(\stackrel{^}{p}-1\right)}^{2}+{\left(\frac{f}{4}\right)}^{2}}.
When you set Opening parameterization to Tabulated data - Volumetric flow rate vs. pressure, Aleak and Amax are the first and last parameters of the Opening area vector, respectively. The smoothed, normalized pressure is also used when the smoothing factor is nonzero with linear interpolation and nearest extrapolation.
When you set Opening parameterization to Tabulated data - Area vs. pressure, the valve opens according to the user-provided tabulated data of volumetric flow rate and pressure differential between ports A and B.
\stackrel{˙}{m}=\overline{\rho }\stackrel{˙}{V},
\stackrel{˙}{V}
\overline{\rho }
\stackrel{˙}{m}={K}_{Leak}\overline{\rho }\sqrt{\Delta p}.
{K}_{Leak}=\frac{{V}_{TLU}\left(1\right)}{\sqrt{|\Delta {p}_{TLU}\left(1\right)|}},
\stackrel{˙}{m}={K}_{Max}\overline{\rho }\sqrt{\Delta p}.
{K}_{Max}=\frac{{V}_{TLU}\left(end\right)}{\sqrt{|\Delta {p}_{TLU}\left(end\right)|}},
If Opening dynamics are modeled, a lag is introduced to the flow response to valve opening. Avalve becomes the dynamic opening area, Adyn; otherwise, Avalve is the steady-state opening area. The instantaneous change in dynamic opening area is calculated based on the Opening time constant, τ:
{\stackrel{˙}{p}}_{dyn}=\frac{{p}_{control}-{p}_{dyn}}{\tau }.
By default, Opening dynamics are turned Off.
Steady-state dynamics are set by the same parameterization as valve opening, and are based on the control pressure, pcontrol. A nonzero Smoothing factor can provide additional numerical stability when the orifice is in near-closed or near-open position.
When Opening parameterization is set to Linear - Area vs. pressure, the valve area freezes at the Leakage area.
When Opening parameterization is set to Tabulated data - Area vs. pressure, the valve area freezes at the first element of the Opening area vector.
When Opening parameterization is set to Linear - Area vs. pressure, the valve area freezes at the Maximum opening area.
When Orifice parameterization is set to Tabulated data - Area vs. pressure, the valve area freezes at the last element of the Opening area vector.
When you set Opening parameterization to Tabulated data - Volumetric flow rate vs. pressure, the fault options are defined by the volumetric flow rate through the valve:
\stackrel{˙}{m}={K}_{Leak}\overline{\rho }\sqrt{\Delta p}.
\stackrel{˙}{m}={K}_{Max}\overline{\rho }\sqrt{\Delta p}.
\stackrel{˙}{m}={K}_{Last}\overline{\rho }\sqrt{\Delta p},
{K}_{Last}=\frac{|\stackrel{˙}{m}|}{\overline{\rho }\sqrt{|\Delta p|}}.
Entry or exit point to the valve.
Ps — Set pressure signal
Varying-signal set pressure for controlled valve operation.
Set pressure control — Valve operation method
Valve operation method. A Constant valve opens linearly over a fixed pressure regulation range or in accordance with tabulated pressure and opening area data that you provide. A Controlled valve opens according to a variable set pressure signal at port Ps over a fixed pressure regulation range.
Method of modeling valve opening or closing. The valve opening is either parametrized linearly or by a table of values that correlate area to pressure differential.
Pressure differential used for the valve control. Selecting Pressure differential sets the pressure difference between port A and port B as the trigger for pressure control. Selecting Pressure at port A sets the gauge pressure at port A, or the difference between the pressure at port A and atmospheric pressure, as the trigger for pressure control.
Set pressure (gauge) — Threshold pressure
Gauge pressure beyond which valve operation is triggered when the Pressure control specification is with respect to port A.
To enable this parameter, set Set pressure control to Constant and Pressure control specification to Pressure at port A.
Set pressure differential — Threshold pressure
Pressure beyond which valve operation is triggered. This is the set pressure when the Pressure control specification is with respect to the pressure differential between ports A and B.
To enable this parameter, set Set pressure control to Constant and Pressure control specification to Pressure differential.
Operational pressure range of the valve. The pressure regulation range begins at the valve set pressure and the end of the range is the maximum valve operating pressure.
Set pressure control to Constant and Opening parameterization to Linear - Area vs. pressure
Sum of all gaps when the valve is in fully closed position. Any area smaller than this value is maintained at the specified leakage area. This contributes to numerical stability by maintaining continuity in the flow.
Set pressure control to Controlled.
Opening pressure (gauge) vector — Differential pressure values for tabular parameterization
[0.2 : 0.2 : 1.2] MPa (default) | 1-by-n vector
To enable this parameter, set Set pressure control to Constant, Opening parameterization to Tabulated data - Area vs. pressure, and Opening pressure specification to Pressure differential..
To enable this parameter, set Set pressure control to Constant, and set
Opening parameterization to Tabulated data - Volumetric flow rate vs. pressure, or
Opening parameterization to Tabulated data - Area vs. pressure and Opening pressure specification to Pressure differential.
[1e-09, 3.6e-05, 8.9e-05, .00015, .00023, .00031] (default) | 1-by-n vector
[1e-10, 2e-06, 4e-06, 6e-06, 8e-06, 1e-05] m^2 (default) | 1-by-n vector
Cross-sectional area at the entry and exit ports A and B. These areas are used in the pressure-flow rate equation determining mass flow rate through the valve.
Accounts for pressure increase when fluid flows from a region of smaller cross-sectional area to a region of larger cross-sectional area. This increase in pressure is not captured when Pressure recovery is set to Off.
Whether to account for transient effects to the fluid system due to valve opening. Setting Opening dynamics to On approximates the opening conditions by introducing a first-order lag in the flow response. The Opening time constant also impacts the modeled opening dynamics.
Pressure Compensator Valve (IL) | Pressure-Compensated 3-Way Flow Control Valve (IL) | Pressure-Compensated Flow Control Valve (IL) | Check Valve (IL) | Counterbalance Valve (IL)
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Understanding similar polygons | StudyPug
Congruent vs similar
Before we dig deeper into this chapter of similar polygons, we first have to clarify what is the difference between congruent and similar shapes.
When two shapes are congruent, this means that they are exactly the same in size and shape. You can put the two side by side and see that they are an exact replica of each other.
When it comes to similar shapes, these are shapes that have the same shape but are different in size. You may have worked with similar triangles before, and they're a good example of what similar means. It may be that between the two shapes, one of them is smaller, or one of them is larger. What relates the two shapes is that there's a constant scale factor between them. You'll see this in scale diagrams.
In this case, it would mean that the scale factor of the enlargement = B side/A side.
Here is where we'll teach you how to tell if figures are similar. We learned about both congruent and similar shapes, but we're working with similar polygons in this chapter. After this, you should grasp the basic concept of finding a scale factor and determining whether two shapes are similar.
Is each pair of the polygons similar?
Use the corresponding sides between the two shapes to calculate the ratio.
\frac{AB}{EF} = \frac{2}{0.4} = 5
\frac{BD}{FH} = \frac{2.5}{0.5} = 5
We get the same ratios when we take different sides of the polygon. We can therefore conclude that they are similar
Is each pair of the pentagons similar?
Once again, use the corresponding sides to calculate the ratio
\frac{AB}{FG} = \frac{6.4}{2} = 3.2
\frac{BC}{GH} = \frac{2.7}{0.9} = 3
We can stop from here and conclude that these are not similar pentagons because they don't have an equal ratio for all sides. Similar shapes must have the same ratio on every one of their sides. If any side breaks the scale factor, you'll know that the shapes aren't similar.
Try playing around with this online diagram that depicts two similar figures. You can scale one of the shapes so that it's enlarged or reduced while still staying similar to the original shape.
To move forward, you'll learn all about congruent triangles and proving their congruence via SSS, SAS, HL, ASA, and AAS.
In this lesson, we will learn how to determine whether a pair of polygons is similar to each other. Once we know that the polygons are similar, we can calculate unknowns such as, side lengths, scale factors, and surface areas.
Basic Concepts: Enlargements and reductions with scale factors, Scale diagrams, Similar triangles
Rectangle A and rectangle B are similar rectangles. Rectangle A is an enlargement of rectangle B with a scale factor of 3.2. Rectangle A has side lengths of 33.6 m and 7 m. One of the sides of rectangle B is 10.5 m. Calculate the missing side length of rectangle B.
Find the perimeter of the smaller quadrilateral.
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Error, invalid input: `evalf/EI` expects its 1st argument, a, to be of type negint, but received 1 - Maple Help
Home : Support : Online Help : Error, invalid input: `evalf/EI` expects its 1st argument, a, to be of type negint, but received 1
\mathrm{sin}\left(\left[1,2, 3\right]\right);
\mathrm{whattype}\left(\left[1,2,3\right]\right);
\textcolor[rgb]{0,0,1}{\mathrm{list}}
\left[1,2,3\right]
\mathrm{Describe}\mathit{}\left(\mathrm{sin}\right)
x
\mathrm{sin}
\mathrm{sin}\left(1\right);\mathrm{sin}\left(2\right);\mathrm{sin}\left(3\right)
\textcolor[rgb]{0,0,1}{\mathrm{sin}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{1}\right)
\textcolor[rgb]{0,0,1}{\mathrm{sin}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{2}\right)
\textcolor[rgb]{0,0,1}{\mathrm{sin}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{3}\right)
\mathrm{sin}~\left(\left[1,2,3\right]\right)
\left[\textcolor[rgb]{0,0,1}{\mathrm{sin}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{1}\right)\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{sin}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{2}\right)\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{sin}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{3}\right)\right]
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Double-acting servo cylinder with spring-centered spool in an isothermal liquid system - MATLAB - MathWorks Italia
Double-Acting Servo Valve Actuator (IL)
Spool cross-sectional area
Spool initial displacement
Double-acting servo cylinder with spring-centered spool in an isothermal liquid system
The Double-Acting Servo Valve Actuator (IL) block models a double-acting servo cylinder arranged as a spring-centered spool. The spring neutral position is where the spool is located at the middle of the stroke. The motion of the piston when it is near full extension or full retraction is limited by one of three hard stop models. Fluid compressibility is optionally modeled in both piston chambers.
The physical signal output S reports the spool position.
To avoid mechanical damage to an actuator when it is fully extended or fully retracted, an actuator typically displays nonlinear behavior when the piston approaches these limits. The Double-Acting Servo Valve Actuator (IL) block models this behavior with a choice of three hard stop models, which model the material compliance through a spring-damper system. The hard stop models are:
The hard stop force is modeled when the piston is at its upper or lower bound. The boundary region is within the Transition region of the Spool stroke or piston initial displacement. Outside of this region,
{F}_{HardStop}=0.
The Double-Acting Servo Actuator block comprises an Isothermal Liquid library block and two Simscape Foundation blocks:
A — Liquid port for chamber A
Isothermal liquid conserving port associated with the liquid inlet of chamber A.
B — Liquid port for chamber B
Isothermal liquid conserving port associated with the liquid inlet of chamber B.
S — Spool position, m
Physical signal associated with the spool position, in m. A position of zero indicates that the spool is at a neutral position in the middle of a stroke.
Spool cross-sectional area — Area measurement
Cross-sectional area of the spool
Spool stroke — Distance
Distance the spool travels in a stroke.
Spring stiffness — Rate of force for the spring
Spring rate of the centering springs.
Damping coefficient — Damping coefficient
Damping coefficient in the contact between the piston and the case.
Specifies the elasticity for the hard stop model. The greater the value of the parameter, the more rigid the impact between the rod and the stop becomes. Lower values result in softer contact and generally improve simulation efficiency and convergence.
Specifies dissipating property of colliding bodies for the hard stop model. At zero damping, the impact is elastic. The greater the value of the parameter, the greater the energy dissipation during piston-stop interaction. Damping affects slider motion as long as the slider is in contact with the stop, including the period when slider is pulled back from the contact. Set this parameter to a nonzero value to improve the efficiency and convergence of your simulation.
Hard stop model — Model hard stops
Modeling approach for hard stops.
Stiffness and damping applied smoothly through transition region, damped rebound — Specify a transition region in which the torque is scaled from zero. At the end of the transition region, the full stiffness and damping are applied. This model has damping applied on the rebound, but it is limited to the value of the stiffness torque. Damping can reduce or eliminate the torque provided by the stiffness, but never exceed it. All equations are smooth and produce no zero crossings.
Full stiffness and damping applied at bounds, undamped rebound — This model has full stiffness and damping applied with impact at the upper and lower bounds and no damping on the rebound. The equations produce no zero crossings when velocity changes sign, but there is a position-based zero crossing at the bounds. Having no damping on rebound helps to push the slider past this position quickly. This model has nonlinear equations.
Full stiffness and damping applied at bounds, damped rebound — This model has full stiffness and damping applied with impact at the upper and lower bounds and damping applied on the rebound. The equations are switched linear, but produce position-based zero crossings. Use this hard stop model if simscape.findNonlinearBlocks indicates that this is the block that prevents the whole network from being switched linear.
Transition region — Transition region
Distance below which scaling is applied to the hard-stop force. The contact force is zero when the distance to the hard stop is equal to the value of this parameter. The contact force is at its full value when the distance to the hard stop is zero.
Spool initial displacement — Initial location of the spool
When the initial displacement of the spool is set to 0, the spool begins directly between chamber A and chamber B. A positive distance moves the spool away from chamber A, while a negative amount moves the spool toward chamber A.
Fluid dynamic compressibility — Fluid compressibility
Initial liquid pressure in chamber A — Liquid pressure
Pressure in actuator chamber A at the start of simulation.
Initial liquid pressure in chamber B — Liquid pressure
Pressure in actuator chamber B at the start of simulation.
Double-Acting Actuator (IL) | Double-Acting Actuator (G-IL) | Double-Acting Rotary Actuator (IL) | Single-Acting Actuator (IL)
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How to solve quadratic functions in vertex form | StudyPug
Quadratic function in vertex form:
y = a(x-p)^2 + q
y = a(x-p)^2 + q
Besides the general form, the vertex form is also another way to express a quadratic function. In this lesson, we will talk about how to find the x-intercepts, y-intercepts, vertex of quadratic functions in vertex form.
Basic Concepts: Graphing linear functions using table of values, Graphing linear functions using x- and y-intercepts, Graphing linear functions using various forms, Introduction to quadratic functions
Related Concepts: Solving quadratic equations by completing the square, Radian measure and arc length, System of linear-quadratic equations, System of quadratic-quadratic equations
y = 2{\left( {x - 3} \right)^2} - 8
y = ax^2 + bx+c
a(x-p)^2 + q
y = a(x-p)^2 + q
y = ax^2 + bx+c
a(x-p)^2 + q
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On Erdős–de Bruijn–Kingman’s problem on regularity of reciprocals for exponential series | EMS Press
Alexander Gomilko
Nicolaus Copernicus University, Torun, Poland and National Academy of Sciences of Ukraine, Kyiv, Ukraine
Motivated by applications to renewal theory, Erdős, de Bruijn and Kingman posed a problem on boundedness of reciprocals
(1 − z)/(1 − F(z))
in the unit disc for probability generating functions
F(z)
. This problem was solved by Ibragimov in 1975 by constructing a counterexample. In this paper, we provide much stronger counterexamples showing that the problem does not allow for a positive answer even under rather restrictive additional assumptions. Moreover, we pursue a systematic study of
L^p
-integrabilty properties for the reciprocals. In particular, we show that while the boundedness of
(1 − z)/(1 − F(z))
fails in general, the reciprocals do possess certain
L^p
-integrability properties under mild conditions on
F
. We also study the same circle of problems in the continuous-time setting.
Alexander Gomilko, Yuri Tomilov, On Erdős–de Bruijn–Kingman’s problem on regularity of reciprocals for exponential series. Rev. Mat. Iberoam. 37 (2021), no. 3, pp. 1045–1081
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A Numerical Model for Life Scatter in Rolling Element Bearings | J. Tribol. | ASME Digital Collection
Engines, Systems and Services,
, South Bend, IN 46620
Raje, N., Sadeghi, F., Rateick, R. G., Jr., and Hoeprich, M. R. (December 26, 2007). "A Numerical Model for Life Scatter in Rolling Element Bearings." ASME. J. Tribol. January 2008; 130(1): 011011. https://doi.org/10.1115/1.2806163
Fatigue lives of rolling element bearings exhibit a wide scatter due to the statistical nature of the mechanisms responsible for the bearing failure process. Life models that account for this dispersion are empirical in nature and do not provide insights into the physical mechanisms that lead to this scatter. One of the primary reasons for dispersion in lives is the inhomogeneous nature of the bearing material. Here, a new approach based on a discrete material representation is presented that simulates this inherent material randomness. In this investigation, two levels of randomness are considered: (1) the topological randomness due to geometric variability in the material microstructure and (2) the material property randomness due to nonuniform distribution of properties throughout the material. The effect of these variations on the subsurface stress field in Hertzian line contacts is studied. Fatigue life is formulated as a function of a critical stress quantity and its corresponding depth, following a similar approach to the Lundberg–Palmgren theory. However, instead of explicitly assuming a Weibull distribution of fatigue lives, the life distribution is obtained as an outcome of numerical simulations. A new critical stress quantity is introduced that considers shear stress acting along internal material planes of weakness. It is found that there is a scatter in the magnitude as well as depth of occurrence of this critical stress quantity, which leads to a scatter in computed fatigue lives. Further, the range of depths within which the critical stress quantity occurs is found to be consistent with experimental observations of fatigue cracks. The life distributions obtained from the numerical simulations are found to follow a two-parameter Weibull distribution closely. The
L10
life and the Weibull slope decrease when a nonuniform distribution of elastic modulus is assumed throughout the material. The introduction of internal flaws in the material significantly reduces the
L10
life and the Weibull slope. However, it is found that the Weibull slope reaches a limiting value beyond a certain concentration of flaws. This limiting value is close to that predicted by the Lundberg–Palmgren theory. Weibull slopes obtained through the numerical simulations range from 1.29 to 3.36 and are within experimentally observed values for bearing steels.
discrete event simulation, fatigue, machine bearings, Weibull distribution, rolling contact fatigue, bearing life, contact stresses, line contact, numerical simulation
Bearings, Computer simulation, Electromagnetic scattering, Fatigue, Fatigue life, Rolling bearings, Simulation, Stress, Weibull distribution, Shear stress, Rolling contact, Fatigue cracks, Cycles, Chaos
The Subsurface-Initiated and the Surface-Initiated Rolling Fatigue Life of Bearing Steels
Proceedings of the JSLE-ASLE International Conference on Lubrication
, Oct. 8–11.
Detection of Rolling Contact Sub-Surface Fatigue Cracks Using Acoustic Emission Technique
Computer Modeling of Anisotropic Grain Microstructure in Two Dimensions
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The error in the measurement of radius of a circle is 0 6% Find the percentage error in the - Physics - Units And Measurements - 13055335 | Meritnation.com
The error in the measurement of
radius of a circle is 0.6%. Find the
percentage error in the calculation of the area
Dear student,\phantom{\rule{0ex}{0ex}}Area of circe={\mathrm{\pi r}}^{2}\phantom{\rule{0ex}{0ex}}\frac{∆\mathrm{r}}{\mathrm{r}}=0.6%\phantom{\rule{0ex}{0ex}}\mathrm{So},\phantom{\rule{0ex}{0ex}}\frac{∆\mathrm{A}}{\mathrm{A}}=2×\frac{∆\mathrm{r}}{\mathrm{r}}\phantom{\rule{0ex}{0ex}}\frac{∆\mathrm{A}}{\mathrm{A}}=2×0.6\phantom{\rule{0ex}{0ex}}\frac{∆\mathrm{A}}{\mathrm{A}}=1.2%\phantom{\rule{0ex}{0ex}}\mathrm{so} \mathrm{the} % \mathrm{error} \mathrm{in} \mathrm{area} \mathrm{is} 1.2%\phantom{\rule{0ex}{0ex}}\mathrm{Regards}
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Quark model - Wikipedia
In particle physics, the quark model is a classification scheme for hadrons in terms of their valence quarks—the quarks and antiquarks which give rise to the quantum numbers of the hadrons. The quark model underlies "flavor SU(3)", or the Eightfold Way, the successful classification scheme organizing the large number of lighter hadrons that were being discovered starting in the 1950s and continuing through the 1960s. It received experimental verification beginning in the late 1960s and is a valid effective classification of them to date. The model was independently proposed by physicists Murray Gell-Mann,[1] who dubbed them "quarks" in a concise paper, and George Zweig,[2][3] who suggested "aces" in a longer manuscript. André Petermann also touched upon the central ideas from 1963 to 1965, without as much quantitative substantiation.[4][5] Today, the model has essentially been absorbed as a component of the established quantum field theory of strong and electroweak particle interactions, dubbed the Standard Model.
Figure 1: The pseudoscalar meson nonet. Members of the original meson "octet" are shown in green, the singlet in magenta. Although these mesons are now grouped into a nonet, the Eightfold Way name derives from the patterns of eight for the mesons and baryons in the original classification scheme.
Hadrons are not really "elementary", and can be regarded as bound states of their "valence quarks" and antiquarks, which give rise to the quantum numbers of the hadrons. These quantum numbers are labels identifying the hadrons, and are of two kinds. One set comes from the Poincaré symmetry—JPC, where J, P and C stand for the total angular momentum, P-symmetry, and C-symmetry, respectively.
The other set is the flavor quantum numbers such as the isospin, strangeness, charm, and so on. The strong interactions binding the quarks together are insensitive to these quantum numbers, so variation of them leads to systematic mass and coupling relationships among the hadrons in the same flavor multiplet.
All quarks are assigned a baryon number of ⅓. Up, charm and top quarks have an electric charge of +⅔, while the down, strange, and bottom quarks have an electric charge of −⅓. Antiquarks have the opposite quantum numbers. Quarks are spin-½ particles, and thus fermions. Each quark or antiquark obeys the Gell-Mann−Nishijima formula individually, so any additive assembly of them will as well.
Mesons are made of a valence quark−antiquark pair (thus have a baryon number of 0), while baryons are made of three quarks (thus have a baryon number of 1). This article discusses the quark model for the up, down, and strange flavors of quark (which form an approximate flavor SU(3) symmetry). There are generalizations to larger number of flavors.
3.1 The discovery of color
Developing classification schemes for hadrons became a timely question after new experimental techniques uncovered so many of them that it became clear that they could not all be elementary. These discoveries led Wolfgang Pauli to exclaim "Had I foreseen that, I would have gone into botany." and Enrico Fermi to advise his student Leon Lederman: "Young man, if I could remember the names of these particles, I would have been a botanist." These new schemes earned Nobel prizes for experimental particle physicists, including Luis Alvarez, who was at the forefront of many of these developments. Constructing hadrons as bound states of fewer constituents would thus organize the "zoo" at hand. Several early proposals, such as the ones by Enrico Fermi and Chen-Ning Yang (1949), and the Sakata model (1956), ended up satisfactorily covering the mesons, but failed with baryons, and so were unable to explain all the data.
The Gell-Mann–Nishijima formula, developed by Murray Gell-Mann and Kazuhiko Nishijima, led to the Eightfold Way classification, invented by Gell-Mann, with important independent contributions from Yuval Ne'eman, in 1961. The hadrons were organized into SU(3) representation multiplets, octets and decuplets, of roughly the same mass, due to the strong interactions; and smaller mass differences linked to the flavor quantum numbers, invisible to the strong interactions. The Gell-Mann–Okubo mass formula systematized the quantification of these small mass differences among members of a hadronic multiplet, controlled by the explicit symmetry breaking of SU(3).
The spin-3⁄2
baryon, a member of the ground-state decuplet, was a crucial prediction of that classification. After it was discovered in an experiment at Brookhaven National Laboratory, Gell-Mann received a Nobel prize in physics for his work on the Eightfold Way, in 1969.
Finally, in 1964, Gell-Mann, and, independently, George Zweig, discerned what the Eightfold Way picture encodes: They posited three elementary fermionic constituents – the "up", "down", and "strange" quarks – which are unobserved, and possibly unobservable in a free form. Simple pairwise or triplet combinations of these three constituents and their antiparticles underlie and elegantly encode the Eightfold Way classification, in an economical, tight structure, resulting in further simplicity. Hadronic mass differences were now linked to the different masses of the constituent quarks.
It would take about a decade for the unexpected nature—and physical reality—of these quarks to be appreciated more fully (See Quarks). Counter-intuitively, they cannot ever be observed in isolation (color confinement), but instead always combine with other quarks to form full hadrons, which then furnish ample indirect information on the trapped quarks themselves. Conversely, the quarks serve in the definition of quantum chromodynamics, the fundamental theory fully describing the strong interactions; and the Eightfold Way is now understood to be a consequence of the flavor symmetry structure of the lightest three of them.
See also: Meson and List of mesons
Figure 2: Pseudoscalar mesons of spin 0 form a nonet
Figure 3: Mesons of spin 1 form a nonet
The Eightfold Way classification is named after the following fact: If we take three flavors of quarks, then the quarks lie in the fundamental representation, 3 (called the triplet) of flavor SU(3). The antiquarks lie in the complex conjugate representation 3. The nine states (nonet) made out of a pair can be decomposed into the trivial representation, 1 (called the singlet), and the adjoint representation, 8 (called the octet). The notation for this decomposition is
{\displaystyle \mathbf {3} \otimes \mathbf {\overline {3}} =\mathbf {8} \oplus \mathbf {1} }
Figure 1 shows the application of this decomposition to the mesons. If the flavor symmetry were exact (as in the limit that only the strong interactions operate, but the electroweak interactions are notionally switched off), then all nine mesons would have the same mass. However, the physical content of the full theory[clarification needed] includes consideration of the symmetry breaking induced by the quark mass differences, and considerations of mixing between various multiplets (such as the octet and the singlet).
N.B. Nevertheless, the mass splitting between the
is larger than the quark model can accommodate, and this "
puzzle" has its origin in topological peculiarities of the strong interaction vacuum, such as instanton configurations.
Mesons are hadrons with zero baryon number. If the quark–antiquark pair are in an orbital angular momentum L state, and have spin S, then
|L − S| ≤ J ≤ L + S, where S = 0 or 1,
P = (−1)L + 1, where the 1 in the exponent arises from the intrinsic parity of the quark–antiquark pair.
C = (−1)L + S for mesons which have no flavor. Flavored mesons have indefinite value of C.
For isospin I = 1 and 0 states, one can define a new multiplicative quantum number called the G-parity such that G = (−1)I + L + S.
If P = (−1)J, then it follows that S = 1, thus PC= 1. States with these quantum numbers are called natural parity states; while all other quantum numbers are thus called exotic (for example the state JPC = 0−−).
See also: List of baryons
Figure 4. The S = 1⁄2 ground state baryon octet
Figure 5. The S = 3⁄2 baryon decuplet
Since quarks are fermions, the spin–statistics theorem implies that the wavefunction of a baryon must be antisymmetric under exchange of any two quarks. This antisymmetric wavefunction is obtained by making it fully antisymmetric in color, discussed below, and symmetric in flavor, spin and space put together. With three flavors, the decomposition in flavor is
{\displaystyle \mathbf {3} \otimes \mathbf {3} \otimes \mathbf {3} =\mathbf {10} _{S}\oplus \mathbf {8} _{M}\oplus \mathbf {8} _{M}\oplus \mathbf {1} _{A}}
The decuplet is symmetric in flavor, the singlet antisymmetric and the two octets have mixed symmetry. The space and spin parts of the states are thereby fixed once the orbital angular momentum is given.
It is sometimes useful to think of the basis states of quarks as the six states of three flavors and two spins per flavor. This approximate symmetry is called spin-flavor SU(6). In terms of this, the decomposition is
{\displaystyle \mathbf {6} \otimes \mathbf {6} \otimes \mathbf {6} =\mathbf {56} _{S}\oplus \mathbf {70} _{M}\oplus \mathbf {70} _{M}\oplus \mathbf {20} _{A}~.}
The 56 states with symmetric combination of spin and flavour decompose under flavor SU(3) into
{\displaystyle \mathbf {56} =\mathbf {10} ^{\frac {3}{2}}\oplus \mathbf {8} ^{\frac {1}{2}}~,}
where the superscript denotes the spin, S, of the baryon. Since these states are symmetric in spin and flavor, they should also be symmetric in space—a condition that is easily satisfied by making the orbital angular momentum L = 0. These are the ground state baryons.
The S = 1⁄2 octet baryons are the two nucleons (
), the three Sigmas (
), the two Xis (
), and the Lambda (
). The S = 3⁄2 decuplet baryons are the four Deltas (
), three Sigmas (
), two Xis (
), and the Omega (
For example, the constituent quark model wavefunction for the proton is
{\displaystyle |p_{\uparrow }\rangle ={\frac {1}{\sqrt {18}}}[2|u_{\uparrow }d_{\downarrow }u_{\uparrow }\rangle +2|u_{\uparrow }u_{\uparrow }d_{\downarrow }\rangle +2|d_{\downarrow }u_{\uparrow }u_{\uparrow }\rangle -|u_{\uparrow }u_{\downarrow }d_{\uparrow }\rangle -|u_{\uparrow }d_{\uparrow }u_{\downarrow }\rangle -|u_{\downarrow }d_{\uparrow }u_{\uparrow }\rangle -|d_{\uparrow }u_{\downarrow }u_{\uparrow }\rangle -|d_{\uparrow }u_{\uparrow }u_{\downarrow }\rangle -|u_{\downarrow }u_{\uparrow }d_{\uparrow }\rangle ].}
Mixing of baryons, mass splittings within and between multiplets, and magnetic moments are some of the other quantities that the model predicts successfully.
The discovery of colorEdit
Color quantum numbers are the characteristic charges of the strong force, and are completely uninvolved in electroweak interactions. They were discovered as a consequence of the quark model classification, when it was appreciated that the spin S = 3⁄2 baryon, the
, required three up quarks with parallel spins and vanishing orbital angular momentum. Therefore, it could not have an antisymmetric wave function, (required by the Pauli exclusion principle), unless there were a hidden quantum number. Oscar Greenberg noted this problem in 1964, suggesting that quarks should be para-fermions.[6]
Instead, six months later, Moo-Young Han and Yoichiro Nambu suggested the existence of three triplets of quarks to solve this problem, but flavor and color intertwined in that model: They did not commute.[7]
The modern concept of color completely commuting with all other charges and providing the strong force charge was articulated in 1973, by William Bardeen, Harald Fritzsch, and Murray Gell-Mann.[8][9]
States outside the quark modelEdit
While the quark model is derivable from the theory of quantum chromodynamics, the structure of hadrons is more complicated than this model allows. The full quantum mechanical wave function of any hadron must include virtual quark pairs as well as virtual gluons, and allows for a variety of mixings. There may be hadrons which lie outside the quark model. Among these are the glueballs (which contain only valence gluons), hybrids (which contain valence quarks as well as gluons) and "exotic hadrons" (such as tetraquarks or pentaquarks).
Quantum chromodynamics, flavor, the QCD vacuum
^ Gell-Mann, M. (4 January 1964). "A Schematic Model of Baryons and Mesons". Physics Letters. 8 (3): 214–215. Bibcode:1964PhL.....8..214G. doi:10.1016/S0031-9163(64)92001-3.
^ Zweig, G. (17 January 1964). An SU(3) Model for Strong Interaction Symmetry and its Breaking (PDF) (Report). CERN Report No.8182/TH.401.
^ Zweig, G. (1964). An SU(3) Model for Strong Interaction Symmetry and its Breaking: II (PDF) (Report). CERN Report No.8419/TH.412.
^ Petermann, A. (1965). "Propriétés de l'étrangeté et une formule de masse pour les mésons vectoriels" [Strangeness properties and a mass formula for vector meson]. Nuclear Physics. 63 (2): 349–352. arXiv:1412.8681. Bibcode:1965NucPh..63..349P. doi:10.1016/0029-5582(65)90348-2.
^ Petrov, Vladimir A. (June 23–27, 2014). Half a Century with QUARKS. XXX-th International Workshop on High Energy Physics. Protvino, Moscow Oblast, Russia. arXiv:1412.8681.
^ Greenberg, O.W. (1964). "Spin and unitary-spin independence in a paraquark model of baryons and mesons". Physical Review Letters. 13 (20): 598–602. Bibcode:1964PhRvL..13..598G. doi:10.1103/PhysRevLett.13.598.
^ Han, M.Y.; Nambu, Y. (1965). "Three-triplet model with double SU(3) symmetry". Physical Review B. 139 (4B): 1006. Bibcode:1965PhRv..139.1006H. doi:10.1103/PhysRev.139.B1006.
^ Bardeen, W.; Fritzsch, H.; Gell-Mann, M. (1973). "Light cone current algebra, π0 decay, and e+ e− annihilation". In Gatto, R. (ed.). Scale and conformal symmetry in hadron physics. John Wiley & Sons. p. 139. arXiv:hep-ph/0211388. Bibcode:2002hep.ph...11388B. ISBN 0-471-29292-3.
^ Fritzsch, H.; Gell-Mann, M.; Leutwyler, H. (1973). "Advantages of the color octet gluon picture". Physics Letters B. 47 (4): 365. Bibcode:1973PhLB...47..365F. CiteSeerX 10.1.1.453.4712. doi:10.1016/0370-2693(73)90625-4.
S. Eidelman et al. Particle Data Group (2004). "Review of Particle Physics" (PDF). Physics Letters B. 592 (1–4): 1. arXiv:astro-ph/0406663. Bibcode:2004PhLB..592....1P. doi:10.1016/j.physletb.2004.06.001.
Lichtenberg, D B (1970). Unitary Symmetry and Elementary Particles. Academic Press. ISBN 978-1483242729.
Thomson, M A (2011), Lecture notes
J.J.J. Kokkedee (1969). The quark model. W. A. Benjamin. ASIN B001RAVDIA.
Retrieved from "https://en.wikipedia.org/w/index.php?title=Quark_model&oldid=1068879234"
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Prime costs are a firm's expenses directly related to the materials and labor used in production. It refers to a manufactured product's costs, which are calculated to ensure the best profit margin for a company. The prime cost calculates the direct costs of raw materials and labor that are involved in the production of a good. Direct costs do not include indirect expenses, such as advertising and administrative costs.
Formula and Calculation of Prime Cost
\text{Prime cost} = \text{Direct raw materials} + \text{Direct labor}
Prime cost=Direct raw materials+Direct labor
Locate the total for the direct raw materials cost on the company's balance sheet.
Locate the figure for the direct labor cost on the company's balance sheet.
Total or add the two figures of direct raw materials and direct labor costs together.
What Prime Cost Can Tell You
A prime cost is the total direct costs, which may be fixed or variable, of manufacturing an item for sale. Businesses use prime costs as a way of measuring the total cost of the production inputs needed to create a given output. By analyzing its prime costs, a company can set prices that yield desired profits. By lowering its prime costs, a company can increase its profit or undercut its competitors' prices.
Companies need to calculate the prime cost of each product manufactured to ensure they are generating a profit. Self-employed individuals, such as artisans who create and sell custom-made furniture, often use the prime cost calculation to ensure they are making the hourly wage they desire while also profiting from each product made.
Indirect costs, such as utilities, manager salaries, and delivery costs, are not included in prime costs. One reason why indirect costs are excluded from the prime cost calculation is that they can be difficult to quantify and allocate.
Example of How to Use Prime Cost
Let's say, as an example, a professional woodworker is hired to construct a dining room table for a customer. The prime costs for creating the table include direct labor and raw materials, such as lumber, hardware, and paint. The materials directly contributing to the table's production cost $200. The woodworker charges $50 per hour for labor, and this project takes three hours to complete. The prime cost to produce the table is $350 ($200 for the raw materials + $150 in direct labor). To generate a profit, the table's price should be set above its prime cost.
Consider the same woodworker who constructed and sold a new hand-crafted table for $250. The cost of the raw materials was $200, and it took him three hours to construct. Without regard to labor costs, the woodworker realized a gain of $50. If his direct labor costs were $15 per hour, he realized a modest gain of $5. Therefore, it is especially important for self-employed persons to employ the prime cost method when determining what price to set for their goods and services.
If the same artisan desired a labor wage of $20 per hour and a profit of $100, the prime cost and price would be $260 ($200 for materials and $60 for labor) and $360 (prime cost + desired profit), respectively.
The Difference Between Prime Costs and Conversion Costs
Conversion costs are also used to calculate profitability based on the cost of production, but these include direct labor as well as overhead expenses incurred due to the transformation of raw materials into finished products. Overhead costs are defined as the expenses that cannot be directly attributed to the production process but are necessary for operations, such as electricity or other utilities needed for the manufacturing plant. Direct labor costs are the same as those used in prime cost calculations.
Conversion costs are also used as a measure to gauge the efficiencies in production processes but take into account the overhead expenses left out of prime cost calculations. Operations managers also use conversion costs to determine where there may be waste within the manufacturing process. Conversions costs and prime costs can be used together to help calculate the minimum profit needed when determining prices to charge customers.
Limitations of Using Prime Cost
Because prime cost only considers direct costs, it does not capture the total cost of production. As a result, the prime cost calculation can be misleading if indirect costs are relatively large. A company likely incurs several other expenses that would not be included in the calculation of the prime cost, such as manager salaries or expenses for additional supplies needed to keep the factory running. These other expenses are considered manufacturing overhead expenses and are included in the calculation of the conversion cost. The conversion cost takes labor and overhead expenses into account, but not the cost of materials.
A second limitation of prime cost involves the challenges associated with identifying which production costs are indeed direct. There are numerous expenses associated with producing goods for sale. To calculate the prime cost of an item accurately, there must be a clear division between those expenses that can directly link to the production of each unit versus those that are required to run the overall business. The specific expenses included in the prime cost calculation can vary depending on the item being produced.
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Learn to solve multi-step linear inequalities | StudyPug
ax + b = c
{x \over a} + b = c
a(x + b) = c
ax + b = c
{x \over a} + b = c
a(x + b) = c
We learn how to solve multi-step linear inequalities in this lesson. Besides typical math questions, we are going to see word problems related to multi-step linear inequalities and money.
ax + b = c
{x \over a} + b = c
a(x + b) = c
, Solving linear equations with variables on both sides
10x - 25 = - 5x
12.4 - 7.2x \le 10x - 15
- \frac{3}{8} - 3x \le \frac{1}{8}x - 2
\frac{1}{4}x + 2 \le - \frac{3}{4}\left( {3x - 5} \right)
Bob works for car dealer A which offers him a base rate of $800 and $15 for every car he sells. Tom works for car dealer B which gives him a base rate of $700 and $20 for each car he sells. If they sold the same number of cars last month, and Tom earned more than Bob. What is the minimum number of cars Tom sold last month?
Express the problem in an inequality.
The cost of a charity dinner is $300 plus $18 per guest. The ticket to the dinner is $30 per person. How many tickets must be sold to cover the costs?
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Superposition Operator in a Space of Infinitely Differentiable Functions | EMS Press
Superposition Operator in a Space of Infinitely Differentiable Functions
In this paper we prove a degeneration result for the superposition operator in
V(\mathbb{R}^d)
, a particular space of infinitely differentiable functions which have all derivatives uniformly bounded by a constant that does not depend on the order of derivation.
Mario Romeo, Superposition Operator in a Space of Infinitely Differentiable Functions. Z. Anal. Anwend. 27 (2008), no. 4, pp. 463–467
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Dictionary:Stiffness - SEG Wiki
The ratio of stress to strain, each with specified directions. Stiffness constitutes a tensor of rank 4, which is the inverse of the compliance tensor. It is also expressible as a 6×6 matrix, using Voigt notation :
{\displaystyle \{c_{IJ}\}={\begin{pmatrix}c_{11}&c_{12}&c_{13}&c_{14}&c_{15}&c_{16}\\c_{12}&c_{22}&c_{23}&c_{24}&c_{25}&c_{26}\\c_{13}&c_{23}&c_{33}&c_{34}&c_{35}&c_{36}\\c_{14}&c_{24}&c_{34}&c_{44}&c_{45}&c_{46}\\c_{15}&c_{25}&c_{35}&c_{45}&c_{55}&c_{56}\\c_{16}&c_{26}&c_{36}&c_{46}&c_{56}&c_{66}\\\end{pmatrix}}}
See Figure E-5.
See elastic constants
Retrieved from "https://wiki.seg.org/index.php?title=Dictionary:Stiffness&oldid=157448"
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The homotopy type of spaces of locally convex curves in the sphere
KEYWORDS: convex curves, topology in infinite dimension, periodic solutions of linear ODEs, 53C42, 57N65, 34B05
A smooth curve
\gamma :\left[0,1\right]\to {\mathbb{S}}^{2}
is locally convex if its geodesic curvature is positive at every point. J A Little showed that the space of all locally convex curves
\gamma
\gamma \left(0\right)=\gamma \left(1\right)={e}_{1}
{\gamma }^{\prime }\left(0\right)={\gamma }^{\prime }\left(1\right)={e}_{2}
has three connected components
{\mathsc{ℒ}}_{-1,c}
{\mathsc{ℒ}}_{+1}
{\mathsc{ℒ}}_{-1,n}
{\mathsc{ℒ}}_{-1,c}
is known to be contractible. We prove that
{\mathsc{ℒ}}_{+1}
{\mathsc{ℒ}}_{-1,n}
are homotopy equivalent to
\left(\Omega {\mathbb{S}}^{3}\right)\vee {\mathbb{S}}^{2}\vee {\mathbb{S}}^{6}\vee {\mathbb{S}}^{10}\vee \cdots
\left(\Omega {\mathbb{S}}^{3}\right)\vee {\mathbb{S}}^{4}\vee {\mathbb{S}}^{8}\vee {\mathbb{S}}^{12}\vee \cdots
, respectively. As a corollary, we deduce the homotopy type of the components of the space
Free\left({\mathbb{S}}^{1},{\mathbb{S}}^{2}\right)
of free curves
\gamma :{\mathbb{S}}^{1}\to {\mathbb{S}}^{2}
(ie curves with nonzero geodesic curvature). We also determine the homotopy type of the spaces
Free\left(\left[0,1\right],{\mathbb{S}}^{2}\right)
with fixed initial and final frames.
A formula for the $\Theta$–invariant from Heegaard diagrams
KEYWORDS: configuration space integrals, finite type invariants of $3$–manifolds, homology spheres, Heegaard splittings, Heegaard diagrams, combings, Casson–Walker invariant, perturbative expansion of Chern-Simons theory, $\Theta$–invariant, 57M27, 55R80, 57R20
\Theta
–invariant is the simplest
3
–manifold invariant defined by configuration space integrals. It is actually an invariant of rational homology spheres equipped with a combing over the complement of a point. It can be computed as the algebraic intersection of three propagators associated to a given combing
X
2
–point configuration space of a
ℚ
M
. These propagators represent the linking form of
M
\Theta \left(M,X\right)
can be thought of as the cube of the linking form of
M
with respect to the combing
X
. The invariant
\Theta
6\lambda \left(M\right)
{p}_{1}\left(X\right)∕4
\lambda
denotes the Casson–Walker invariant, and
{p}_{1}
is an invariant of combings, which is an extension of a first relative Pontrjagin class. In this article, we present explicit propagators associated with Heegaard diagrams of a manifold, and we use these “Morse propagators”, constructed with Greg Kuperberg, to prove a combinatorial formula for the
\Theta
–invariant in terms of Heegaard diagrams.
Quasigeodesic flows and sphere-filling curves
KEYWORDS: quasigeodesic flows, Cannon–Thurston, Pseudo-Anosov flows, 57M60, 57M50, 37C27
Given a closed hyperbolic
3
M
with a quasigeodesic flow, we construct a
{\pi }_{1}
–equivariant sphere-filling curve in the boundary of hyperbolic space. Specifically, we show that any complete transversal
P
to the lifted flow on
{ℍ}^{3}
has a natural compactification to a closed disc that inherits a
{\pi }_{1}
–action. The embedding
P↪{ℍ}^{3}
extends continuously to the compactification, and restricts to a surjective
{\pi }_{1}
–equivariant map
\partial P\to \partial {ℍ}^{3}
on the boundary. This generalizes the Cannon–Thurston theorem, which produces such group-invariant space-filling curves for fibered hyperbolic
3
–manifolds.
Henrique N Sá Earp, Thomas Walpuski
KEYWORDS: Gauge Theory, $G_2$–manifolds, gluing, holomorphic bundles, 53C07, 53C25, 53C38
We introduce a method to construct
{G}_{2}
–instantons over compact
{G}_{2}
–manifolds arising as the twisted connected sum of a matching pair of building blocks. Our construction is based on gluing
{G}_{2}
–instantons obtained from holomorphic vector bundles over the building blocks via the first author’s work. We require natural compatibility and transversality conditions which can be interpreted in terms of certain Lagrangian subspaces of a moduli space of stable bundles on a
K3
Oren Ben-Bassat, Christopher Brav, Vittoria Bussi, Dominic Joyce
KEYWORDS: derived algebraic geometry, derived stack, shifted symplectic structure, perverse sheaf, vanishing cycles, motivic invariant, Calabi–Yau manifold, Donaldson–Thomas theory, 14A20, 14F05, 14D23, 14N35, 32S30
This is the fifth in a series of papers on the ‘
k
–shifted symplectic derived algebraic geometry’ of Pantev, Toën, Vaquié and Vezzosi. We extend our earlier results from (derived) schemes to (derived) Artin stacks. We prove four main results:
\left(X,{\omega }_{X}\right)
k
–shifted symplectic derived Artin stack for
k<0
, then near each
x\in X
we can find a ‘minimal’ smooth atlas
\phi :U\to X
\left(U,{\phi }^{\ast }\left({\omega }_{X}\right)\right)
may be written explicitly in coordinates in a standard ‘Darboux form’.
\left(X,{\omega }_{X}\right)
\left(-1\right)
–shifted symplectic derived Artin stack and
X={t}_{0}\left(X\right)
the classical Artin stack, then
X
extends to a ‘d–critical stack’
\left(X,s\right)
, as by Joyce.
\left(X,s\right)
is an oriented d–critical stack, we define a natural perverse sheaf
{\stackrel{̌}{P}}_{X,s}^{\bullet }
X
, such that whenever
T
t:T\to X
is smooth of relative dimension
T
is locally modelled on a critical locus
Crit\left(f:U\to {\mathbb{A}}^{1}\right)
{t}^{\ast }\left({\stackrel{̌}{P}}_{X,s}^{\bullet }\right)\left[n\right]
is modelled on the perverse sheaf of vanishing cycles
{\mathsc{P}\mathsc{V}}_{U,f}^{\bullet }
of
\left(X,s\right)
is a finite-type oriented d–critical stack, we can define a natural motive
{MF}_{X,s}
in a ring of motives
{\overline{\mathsc{ℳ}}}_{X}^{st,\stackrel{̂}{\mu }}
X
T
t:T\to X
is smooth of dimension
T
is modelled on a critical locus
Crit\left(f:U\to {\mathbb{A}}^{1}\right)
{\mathbb{L}}^{-n∕2}\odot {t}^{\ast }\left({MF}_{X,s}\right)
is modelled on the motivic vanishing cycle
{MF}_{U,f}^{mot,\varphi }
of
Our results have applications to categorified and motivic extensions of Donaldson–Thomas theory of Calabi–Yau
3
–folds.
Tetsuya Ito, Bertold Wiest
KEYWORDS: Lawrence-Krammer-Bigelow representation, Braid group, curve diagram, dual Garside length, 20F36, 20F10, 57M07
We show that the span of the variable
q
in the Lawrence–Krammer–Bigelow representation matrix of a braid is equal to twice the dual Garside length of the braid, as was conjectured by Krammer. Our proof is close in spirit to Bigelow’s geometric approach. The key observation is that the dual Garside length of a braid can be read off a certain labeling of its curve diagram.
The topology of nilpotent representations in reductive groups and their maximal compact subgroups
KEYWORDS: strong deformation retraction, representation variety, character variety, nilpotent group, Kempf–Ness theory, geometric invariant theory, real and complex algebraic groups, maximal compact subgroup, 20G20, 55P99, 20G05
G
be a complex reductive linear algebraic group and let
K\subset G
be a maximal compact subgroup. Given a nilpotent group
\Gamma
r
elements, we consider the representation spaces
Hom\left(\Gamma ,G\right)
Hom\left(\Gamma ,K\right)
with the natural topology induced from an embedding into
{G}^{r}
{K}^{r}
respectively. The goal of this paper is to prove that there is a strong deformation retraction of
Hom\left(\Gamma ,G\right)
Hom\left(\Gamma ,K\right)
. We also obtain a strong deformation retraction of the geometric invariant theory quotient
Hom\left(\Gamma ,G\right)∕\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}∕G
onto the ordinary quotient
Hom\left(\Gamma ,K\right)∕K
How to lift positive Ricci curvature
Catherine Searle, Frederick Wilhelm
KEYWORDS: Ricci curvature, almost nonnegative curvature, Cheeger deformations, conformal change of metric, 53C20
We show how to lift positive Ricci and almost nonnegative curvatures from an orbit space
M∕G
to the corresponding
G“
–manifold,
M
. We apply the results to get new examples of Riemannian manifolds that satisfy both curvature conditions simultaneously.
Uniqueness of instantaneously complete Ricci flows
KEYWORDS: Ricci flow, logarithmic fast diffusion equation, 35K55, 53C44, 58J35
We prove uniqueness of instantaneously complete Ricci flows on surfaces. We do not require any bounds of any form on the curvature or its growth at infinity, nor on the metric or its growth (other than that implied by instantaneous completeness). Coupled with earlier work, this completes the well-posedness theory for instantaneously complete Ricci flows on surfaces.
Alexandrov spaces with maximal number of extremal points
KEYWORDS: Alexandrov spaces, extremal subsets, orbifolds, 53C45, 53C45, 20F55, 51K10
We show that any
n
–dimensional nonnegatively curved Alexandrov space with the maximal possible number of extremal points is isometric to a quotient space of
{ℝ}^{n}
by an action of a crystallographic group. We describe all such actions.
Alessandro Carlotto, Otis Chodosh, Yanir Rubinstein
KEYWORDS: Yamabe flow, nonintegrable critical points, Polynomial convergence, Lojasiewicz–Simon inequality, constant scalar curvature, 35K55, 53C44, 58K05, 58K55
We characterize the rate of convergence of a converging volume-normalized Yamabe flow in terms of Morse-theoretic properties of the limiting metric. If the limiting metric is an integrable critical point for the Yamabe functional (for example, this holds when the critical point is nondegenerate), then we show that the flow converges exponentially fast. In general, we make use of a suitable Łojasiewicz–Simon inequality to prove that the slowest the flow will converge is polynomially. When the limit metric satisfies an Adams–Simon-type condition we prove that there exist flows converging to it exactly at a polynomial rate. We conclude by constructing explicit examples of this phenomenon. These seem to be the first examples of a slowly converging solution to a geometric flow.
Andrei A Agrachev, Alessandro Gentile, Antonio Lerario
KEYWORDS: loop spaces, Carnot groups, Morse–Bott theory, sub-Riemannian geometry, 53C17, 37J60, 58E05
A new gauge slice for the relative Bauer–Furuta invariants
Tirasan Khandhawit
KEYWORDS: $4$–manifolds, Seiberg–Witten equations, Bauer–Furuta invariant, 57R57, 57R58
In this paper, we study Manolescu’s construction of the relative Bauer–Furuta invariants arising from the Seiberg–Witten equations on
4
–manifolds with boundary. The main goal of this paper is to introduce a new gauge fixing condition in order to apply the finite-dimensional approximation technique. We also hope to provide a framework to extend Manolescu’s construction to general
4
“Slicing” the Hopf link
KEYWORDS: slice links, the Milnor group, the A-B slice problem, 57N13, 57M25, 57M27
A link in the
3
–sphere is called (smoothly) slice if its components bound disjoint smoothly embedded disks in the
4
–ball. More generally, given a
4
M
with a distinguished circle in its boundary, a link in the
3
–sphere is called
M\phantom{\rule{0.3em}{0ex}}
–slice if its components bound in the
4
–ball disjoint embedded copies of
M
4
M
is constructed such that the Borromean rings are not
M\phantom{\rule{0.3em}{0ex}}
–slice but the Hopf link is. This contrasts the classical link-slice setting where the Hopf link may be thought of as “the most nonslice” link. Further examples and an obstruction for a family of decompositions of the
4
–ball are discussed in the context of the A-B slice problem.
Holomorphic Lagrangian branes correspond to perverse sheaves
KEYWORDS: Fukaya category, holomorphic Lagrangian branes, perverse sheaves, constructible sheaves, Nadler–Zaslow correspondence, 53D40, 32S60
X
be a compact complex manifold,
{D}_{c}^{b}\left(X\right)
be the bounded derived category of constructible sheaves on
X
Fuk\left({T}^{\ast }X\right)
be the Fukaya category of
{T}^{\ast }X
. A Lagrangian brane in
Fuk\left({T}^{\ast }X\right)
is holomorphic if the underlying Lagrangian submanifold is complex analytic in
{T}^{\ast }{X}_{ℂ}
, the holomorphic cotangent bundle of
X
. We prove that under the quasiequivalence between
{D}_{c}^{b}\left(X\right)
DFuk\left({T}^{\ast }X\right)
established by Nadler and Zaslow, holomorphic Lagrangian branes with appropriate grading correspond to perverse sheaves.
The complex symplectic geometry of the deformation space of complex projective structures
KEYWORDS: complex projective structures, symplectic structures, Teichmüller theory, character variety, quasifuchsian structures, 53D30
This article investigates the complex symplectic geometry of the deformation space of complex projective structures on a closed oriented surface of genus at least 2. The cotangent symplectic structure given by the Schwarzian parametrization is studied carefully and compared to the Goldman symplectic structure on the character variety, clarifying and generalizing a theorem of S Kawai. Generalizations of results of C McMullen are derived, notably quasifuchsian reciprocity. The symplectic geometry is also described in a Hamiltonian setting with the complex Fenchel–Nielsen coordinates on quasifuchsian space, recovering results of I Platis.
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Learn to evaluate and simplify radicals | StudyPug
How to evaluate radicals
When we work with the radical sign, what are we dealing with? We'll learn the steps to simplifying radicals so that we can get the final answer to math problems.
There are four steps you should keep in mind when you try to evaluate radicals. It's a little similar to how you would estimate square roots without a calculator.
1. Find the number under the radical sign's prime factorization. You'll usually start with 2, which is the first prime number, and then you can move on to using numbers such as 3 and 5. Do this until the original number is now completely made up of prime numbers.
2. Figure out what the index of the radical is. If you just see a square root symbol, keep in mind that this is actually an index of 2. For other indices, there should be a tiny little number right before the radical sign that will indicate what the index is. This is the number that will tell you how many repeated prime numbers you'll need to have after factorization in order to move that number inside the radical to outside the radical. If you've got a cube root for example, you'll need a prime number to be found three times through prime factorization in order to move it outside.
3. Now move the groups of numbers from inside the radical sign to outside. If you're working with a cube root and under the radical sign you're left with 3, 3, 3, 2, 11 after prime factorization, you can move the 3's out from the sign. However, you'll only write it once as a group. If you've got prime numbers that can't be move outside, leave them under the radical sign.
4. Clean up what you've got through simplifying. If you've moved more than one group of numbers out, multiply what you've moved outside together. Do the same for the prime numbers you've got left inside the radical. Simplifying expressions is the last step when you evaluate radicals. Then you'll get your final answer!
Don't worry that this isn't super clear after reading through the steps. You'll get a clearer idea of this after following along with the example questions below.
Evaluate the radicals
{^4}\sqrt{10000}
Firstly, let's do some factorization. Perform a long division to see what number you can pull out. It's most obvious that you can pull out 10's due to the original number being 10,000.
Factorization with long division
{^4}\sqrt{10\bullet10\bullet10\bullet10}
We know that this is a
4^{th}
root since there's a little 4 right outside the radical sign to indicate this index. We have four 10's under the radical sign. We can take out the 10 and get the final answer:
{^5}\sqrt{\frac{1}{32}}
Firstly, the following is true:
{^5}\sqrt{1} = 1
Then we are now free to deal with the denominator in this question. Perform long division on 32 to factorize it.
Perform long division for factorization
{^5}\sqrt{2\bullet2\bullet2\bullet2\bullet2}
Since it's a
5^{th}
root, and we have five 2's under the radical sign, we can take out the 2 and get a final answer of:
\frac{1}{2}
Using calculator to evaluate radicals
An online calculator for simplifying radicals you can use can be found here. In your tests and exams, you will have to evaluate radicals without the use of a calculator, so make sure you've firmly understood the four steps in this lesson.
Looking to move on? Learn how to do operations with radicals, add and subtract them, and learn about converting entire radicals to mixed radicals.
To simplify radicals, we will need to find the prime factorization of the number inside the radical sign first. Then, move each group of prime factors outside the radical according to the index. The last step is to simplify the expression by multiplying the numbers both inside and outside the radical sign.
Basic Concepts: Squares and square roots, Estimating square roots, Square and square roots, Cubic and cube roots
{^4}\sqrt{10000}
{^5}\sqrt{1/32}
Using calculator to evaluate
{^7}\sqrt{4365}
{^5}\sqrt{234}
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Maximal overlap discrete wavelet transform - MATLAB modwt
MODWT Using Default Wavelet
MODWT Using Daubechies Extremal Phase Wavelet with Two Vanishing Moments
MODWT Using Scaling and Wavelet Filters
MODWT to a Specified Level
MODWT with Reflection Boundary
MODWT of Multisignal
w = modwt(x)
w = modwt(x,wname)
w = modwt(x,Lo,Hi)
w = modwt(___,lev)
w = modwt(___,'reflection')
w = modwt(x) returns the maximal overlap discrete wavelet transform (MODWT) of x. x can be a real- or complex-valued vector or matrix. If x is a matrix, modwt operates on the columns of x. modwt computes the wavelet transform down to level floor(log2(length(x))) if x is a vector and floor(log2(size(x,1))) if x is a matrix. By default, modwt uses the Daubechies least-asymmetric wavelet with four vanishing moments ('sym4') and periodic boundary handling.
w = modwt(x,wname) uses the orthogonal wavelet, wname, for the MODWT.
w = modwt(x,Lo,Hi) uses the scaling filter, Lo, and wavelet filter, Hi, to compute the MODWT. These filters must satisfy the conditions for an orthogonal wavelet. You cannot specify both wname and a filter pair, Lo and Hi.
w = modwt(___,lev) computes the MODWT down to the specified level, lev, using any of the arguments from previous syntaxes.
w = modwt(___,'reflection') computes the MODWT using reflection boundary handling. Other inputs can be any of the arguments from previous syntaxes. Before computing the wavelet transform, modwt extends the signal symmetrically at the terminal end to twice the signal length. The number of wavelet and scaling coefficients that modwt returns is equal to twice the length of the input signal. By default, the signal is extended periodically.
Obtain the MODWT of an electrocardiogram (ECG) signal using the default sym4 wavelet down to the maximum level. The data are taken from Percival & Walden (2000), p.125 (data originally provided by William Constantine and Per Reinhall, University of Washington).
whos wtecg
wtecg 12x2048 196608 double
The first eleven rows of wtecg are the wavelet coefficients for scales
{2}^{1}
{2}^{11}
. The final row contains the scaling coefficients at scale
{2}^{11}
. Plot the detail (wavelet) coefficients for scale
{2}^{3}
title('Level 3 Wavelet Coefficients')
Obtain the MODWT of Southern Oscillation Index data with the db2 wavelet down to the maximum level.
Obtain the MODWT of the Deutsche Mark - U.S. Dollar exchange rate data using the Fejer-Korovkin length 8 scaling and wavelet filters.
Obtain the MODWT of an ECG signal down to scale
{2}^{4}
, which corresponds to level four. Use the default sym4 wavelet. The data are taken from Percival & Walden (2000), p.125 (data originally provided by William Constantine and Per Reinhall, University of Washington).
wtecg = modwt(wecg,4);
whos wecg wtecg
Name Size Bytes Class Attributes
wecg 2048x1 16384 double
wtecg 5x2048 81920 double
The row size of wtecg is L+1, where, in this case, the level (L) is 4. The column size matches the number of input samples.
Obtain the MODWT of an ECG signal using reflection boundary handling. Use the default sym4 wavelet and obtain the transform down to level 4. The data are taken from Percival & Walden (2000), p.125 (data originally provided by William Constantine and Per Reinhall, University of Washington).
wtecg = modwt(wecg,4,'reflection');
Name Size Bytes Class Attributes
wecg 2048x1 16384 double
wtecg 5x4096 163840 double
wtecg has 4096 columns, which is twice the length of the input signal, wecg.
Compute the maximal overlap discrete wavelet transform down to the maximum level.
wt = modwt(Espiga3);
Obtain the squared signal energies and compare them against the squared energies obtained from summing the wavelet coefficients over all levels. Use the log-squared energy due to the disproportionately large energy in one component.
sigN2 = vecnorm(Espiga3).^2;
wtN2 = sum(squeeze(vecnorm(wt,2,2).^2));
bar(1:23,log(sigN2))
scatter(1:23,log(wtN2),'filled','SizeData',100)
legend('Signal Energy','Energy in Wavelet Coefficients', ...
ylabel('ln(squared energy)')
f\left(x\right)
N
\varphi \left(x\right)
\psi \left(x\right)
f\left(x\right)=\sum _{k=0}^{N-1}{c}_{k}\phantom{\rule{0.16666666666666666em}{0ex}}{2}^{-{J}_{0}/2}\varphi \left({2}^{-{J}_{0}}\phantom{\rule{0.16666666666666666em}{0ex}}x-k\right)+\sum _{j=1}^{{J}_{0}}{f}_{j}\left(x\right)
{f}_{j}\left(x\right)=\sum _{k=0}^{N-1}{d}_{j,k}\phantom{\rule{0.16666666666666666em}{0ex}}{2}^{-j/2}\phantom{\rule{0.16666666666666666em}{0ex}}\psi \left({2}^{-j}x-k\right)
{J}_{0}
{f}_{j}\left(x\right)
N
\left\{{c}_{k}\right\}
\left({J}_{0}×N\right)
\left\{{d}_{j,k}\right\}
N
\text{floor}\left({\mathrm{log}}_{2}\left(N\right)\right)
N=2048
{J}_{0}=\text{floor}\left(\mathrm{log}2\left(2048\right)\right)=11
{J}_{0}+1=11+1=12
{||X||}^{2}=\sum _{j=1}^{{J}_{0}}{||{W}_{j}||}^{2}+{||{V}_{{J}_{0}}||}^{2}
X
{W}_{j}
j
{V}_{{J}_{0}}
f\left(x\right)
\sum _{k=0}^{N-1}{c}_{k}\phantom{\rule{0.16666666666666666em}{0ex}}{2}^{-{J}_{0}/2}\varphi \left({2}^{-{J}_{0}}\phantom{\rule{0.16666666666666666em}{0ex}}x-k\right)
{J}_{0}
\left\{{f}_{j}\left(x\right)\right\}
N
f\left(x\right)
f\left(x\right)
Input signal, specified as a vector or matrix. If x is a vector, x must have at least two elements. If x is a matrix, the row dimension of x must be at least 2.
'sym4' (default) | 'haar' | 'dbN' | 'symN' | 'coifN' | 'fkN'
Analyzing wavelet, specified as one of the following:
Filters, specified as a pair of even-length real-valued vectors. Lo is the scaling filter, and Hi is the wavelet filter. The filters must satisfy the conditions for an orthogonal wavelet. The lengths of Lo and Hi must be equal. See wfilters for additional information. You cannot specify both a wavelet wname and filter pair Lo,Hi.
Transform level, specified as a positive integer less than or equal to floor(log2(N)), where N = length(x) if x is a vector, or N = size(x,1) if x is a matrix. If unspecified, lev defaults to floor(log2(N)).
MODWT transform of x. w contains the wavelet coefficients and final-level scaling coefficients of x. If x is a vector, w is a lev+1-by-N matrix. If x is a matrix, w is a lev+1-by-N-by-NC array, where NC is the number of columns in x. N is equal to the input signal length unless you specify 'reflection' boundary handling, in which case N is twice the length of the input signal. The kth row of the array, w, contains the wavelet coefficients for scale 2k (wavelet scale 2(k-1)). The final, (lev+1)th, row contains the scaling coefficients for scale 2lev.
The standard algorithm for the MODWT implements the circular convolution directly in the time domain. This implementation of the MODWT performs the circular convolution in the Fourier domain. The wavelet and scaling filter coefficients at level j are computed by taking the inverse discrete Fourier transform (DFT) of a product of DFTs. The DFTs in the product are the signal’s DFT and the DFT of the jth level wavelet or scaling filter.
Let Hk and Gk denote the length N DFTs of the MODWT wavelet and scaling filters, respectively. Let j denote the level and N denote the sample size.
The jth level wavelet filter is defined by
\frac{1}{N}\sum _{k=0}^{N-1}{H}_{j,k}{e}^{i2\pi nk/N}
{H}_{j,k}={H}_{{2}^{j-1}k\text{mod}N}\prod _{m=0}^{j-2}{G}_{{2}^{m}k\text{mod}N}
The jth level scaling filter is
\frac{1}{N}\sum _{k=0}^{N-1}{G}_{j,k}{e}^{i2\pi nk/N}
{G}_{j,k}=\prod _{m=0}^{j-1}{G}_{{2}^{m}k\text{mod}N}
[2] Percival, Donald B., and Harold O. Mofjeld. “Analysis of Subtidal Coastal Sea Level Fluctuations Using Wavelets.” Journal of the American Statistical Association 92, no. 439 (September 1997): 868–80. https://doi.org/10.1080/01621459.1997.10474042.
imodwt | modwtmra | modwtcorr | modwtvar | modwtxcorr
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Ruled nodal surfaces of Laplace eigenfunctions and injectivity sets for the spherical mean Radon transform in $\mathbb R^3$ | EMS Press
JournalsjstVol. 7, No. 4pp. 1039–1099
Ruled nodal surfaces of Laplace eigenfunctions and injectivity sets for the spherical mean Radon transform in
\mathbb R^3
Mark L. Agranovsky
It is proved that if a Paley–Wiener family of eigenfunctions of the Laplace operator in
\mathbb R^3
vanishes on a real-analytically ruled two-dimensional surface
S \subset \mathbb R^3
S
is a union of cones, each of which is contained in a translate of the zero set of a nonzero harmonic homogeneous polynomial. If
S
is an immersed
C^1
manifold then
S
is a Coxeter system of planes. Full description of common nodal sets of Laplace spectra of convexly supported distributions is given. In equivalent terms, the result describes ruled injectivity sets for the spherical mean transform and confirms, for the case of ruled surfaces in
\mathbb R^3,
a conjecture from [1].
Mark L. Agranovsky, Ruled nodal surfaces of Laplace eigenfunctions and injectivity sets for the spherical mean Radon transform in
\mathbb R^3
. J. Spectr. Theory 7 (2017), no. 4, pp. 1039–1099
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GetModel - MapleSim Help
Home : Support : Online Help : MapleSim : MapleSim Application Programming Interface : API Commands : GetModel
get the name of the linked model
A:-GetModel()
string : the name of the model to which the ModelObject is linked
A:-GetModel returns the name of the top-level model to which the ModelObject is linked.
If the ModelObject is linked to a model created in MapleSim, the name of the top-level model is usually "Main".
A≔\mathrm{MapleSim}:-\mathrm{LinkModel}\left('\mathrm{filename}'=\mathrm{cat}\left(\mathrm{kernelopts}\left('\mathrm{toolboxdir}'=\mathrm{MapleSim}\right),"/data/examples/RLCcircuit.msim"\right)\right):
Getting the linked MapleSim model name:
A:-\mathrm{GetModel}\left(\right)
\textcolor[rgb]{0,0,1}{"Main"}
MapleSim[LinkModel][SetModel]
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Photoacoustic hygrometer for icing wind tunnel water content measurement:...
Lang, Benjamin; Breitfuss, Wolfgang; Schweighart, Simon; Breitegger, Philipp; Pervier, Hugo; Tramposch, Andreas; Klug, Andreas; Hassler, Wolfgang; Bergmann, Alexander
This work describes the latest design, calibration and application of a near-infrared laser diode-based photoacoustic (PA) hygrometer developed for total water content measurement in simulated atmospheric freezing precipitation and high ice water content conditions with relevance in fundamental icing research, aviation testing, and certification. The single-wavelength and single-pass PA absorption cell is calibrated for molar water vapor fractions with a two-pressure humidity generator integrated into the instrument. Laboratory calibration showed an estimated measurement accuracy better than inline-formula3.3 % in the water vapor mole fraction range of 510–12 360 inline-formulappm (inline-formula5 % from 250–21 200 inline-formulappm) with a theoretical limit of detection (3inline-formulaσ) of inline-formula3.2 inline-formulappm. The hygrometer is examined in combination with a basic isokinetic evaporator probe (IKP) and sampling system designed for icing wind tunnel applications, for which a general description of total condensed water content (CWC) measurements and uncertainties are presented. Despite the current limitation of the IKP to a hydrometeor mass flux below inline-formula90 inline-formula
M9inlinescrollmathmlunit\mathrm{normal g}\phantom{\rule{0ex}{0ex}}{\mathrm{normal m}}^{-normal 2}\phantom{\rule{0ex}{0ex}}{\mathrm{normal s}}^{-normal 1}
46pt15ptsvg-formulamathimg7c357c7c8d9a58020f6dc78efa08ad26 amt-14-2477-2021-ie00001.svg46pt15ptamt-14-2477-2021-ie00001.png , a CWC measurement accuracy better than inline-formula20 % is achieved by the instrument above a CWC of inline-formula0.14 inline-formulag m−3 in cold air (inline-formula−30 inline-formula∘C) with suitable background humidity measurement. Results of a comparison to the Cranfield University IKP instrument in freezing drizzle and rain show a CWC agreement of the two instruments within inline-formula20 %, which demonstrates the potential of PA hygrometers for water content measurement in atmospheric icing conditions.
Lang, Benjamin / Breitfuss, Wolfgang / Schweighart, Simon / et al: Photoacoustic hygrometer for icing wind tunnel water content measurement: design, analysis, and intercomparison. 2021. Copernicus Publications.
Rechteinhaber: Benjamin Lang et al.
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Solar Panel | Building DC Energy Systems
Panels and Arrays
Photovoltaic solar energy is especially suitable for decentralized and small-scale systems as it does not require maintainance of mechanical parts and because the efficiency is independent of the size of the system.
This chapter provides basic understanding of the working principles of solar panels and helps with correct system layout.
# Photovoltaic Cells
A photovoltaic (PV) cell generates an electron flow from the energy of sunlight using semiconductor materials, typically silicon.
The basic principles of a PV cell are shown in Figure 1 and explained below.
Figure 1. Basic principle of photovoltaic cells [1].
The cell contains two different types of silicon: A so-called n-type, which has extra electrons and a p-type with extra spaces for electrons, called holes. The two types are connected at the p-n junction and create an electrical field.
If a photon from the sunlight hits an electron in the p-type silicon with enough energy, it can remove it from its bond and push it accross the electrical field in the junction to the n-type side. The electron can now be fed back to the p-type side by an external load, closing the electrical circuit.
There are many different types of semiconductor materials available that can be used for solar cells. The most relevant types are mono- and poly-crystalline silicon cells and thin film cells.
# Poly- and mono-crystalline silicon
The most typical type of solar panels uses crystalline silicon cells. These cells are brittle and thus need to be fixed in a rigid module assembly.
Polycristalline cells are the cheapest available option and offer sufficient efficiency for most applications. Monocrystalline cells are slightly more efficient, but also more expensive due to a more complex manufacturing process.
# Thin film
This type of solar panels use a layer of amorphous silicone that is deposited on a base material. In contrast to crystalline silicon, also a flexible base material can be used.
Flexible modules can be more lightweight and don't need the rigid aluminum frame, so they are ideally suited for mobile applications.
The disadvantage compared to crystalline cells is the lower efficiency and higher cost of thin film modules.
# Panels and Arrays
As the voltage of a single solar cell is only around 0.6 V, multiple cells are normally connected in series to increase the voltage to a level suitable for the application. A typical rooftop solar panel contains 60 cells, leading to an open circuit voltage of around 36 V.
For larger systems, multiple panels (or modules) are again connected in series to increase the system voltage. An array of multiple solar panels might also contain parallel connections of modules, but parallel connection normally only makes sense if the voltage is limited. A series connection increases the voltage at the same current, whereas a parallel connection increase the current while keeping the voltage at the same level. As the losses increase with higher current, series connection is more efficient than parallel connection.
The different steps from a single PV cell to a solar array are shown in Figure 1.
Figure 2. From a solar cell to a solar array.
Small off-grid systems may use only a single module or even use modules with lower voltage. The following table lists typical module designs and their applications:
# cells in series
V_{oc}
V_{mpp}
36 12 V 22 V 18 V 12 V battery systems
60 20 V 38 V 31 V Rooftop modules
Most modules for larger installations are equipped with a so-called MC4 connector (opens new window). Even though this connector was originally developed by Multi-Contact, there are compatible solutions from several different suppliers available.
The typical wire cross-section for larger solar modules is 6 mm² (AWG 10) or 4 mm² (AWG 12). If multiple panels are connected in parallel, the cross-section of the wire between the parallel solar panels and e.g. the charge controller should be increased. See the separate chapter regarding wiring for more details.
# Electrical Characteristics
In order to compare solar panels from different manufacturers, the main technical parameters are measured under so-called standard test conditions (STC). The following three parameters are specified:
Solar irradiance describes the sunlight intensity on a flat surface facing directly towards the sun. It is measured in W/m², with 1000 W/m² being the setpoint under STC. The higher the irradiance on a PV panel, the more electrical energy it will generate. The solar irradiance is approximately proportional to the current.
Air mass (AM) refers to the “thickness” or clarity of the air through which the sunlight reaches the solar cell. The AM value is defined as 1 for a perpendicular position of the sun. The more flat the angle of the sun towards the surface of the earth gets, the lower the AM value. STC specifies AM 1.5.
Actual cell temperature, which may be different from ambient air temperature. STC defines the cell temperature as 25°C. The cell temperature mostly affects the voltage of the panel.
The performance test is done in a short period of time, so that the cell temperature stays the same as the ambient temperature. Under realistic conditions, an ambient temperature of 25 °C would increase the module temperature to more than 45 °C for the irradiance of 1000 W/m². That's why there is a second test specified that determines the so-called normal operating cell temperature (NOCT) at an ambient temperature of 20 °C and an irradiance of 800 W/m².
The actual temperature of the module is very important, as it has a significant influence on the efficiency of the module. The higher the temperature, the lower the voltage and thus, efficiency of the module.
Datasheets specify the following parameters, measured under standard test conditions:
Open circuit voltage
V_{oc}
is the maximum voltage of panel when it is not connected to an electrical circuit or system, i.e. when there is no current flow. It can be measured with a multimeter directly at the panel’s terminals or the ends of the attached cables.
Voltage at maximum power point
V_{mpp}
is the voltage at which the power output of the module is at its peak. See below graphs for better understanding.
Current at maximum power point
I_{mpp}
is the current corresponding to
V_{mpp}
Short circuit current
I_{sc}
is the current measured when both terminals are connected to each other. In the characteristic curve, this point is where the voltage
V
equals zero.
The nominal voltage as mentioned in the previous section refers to the recommended voltage of a battery that the module should be connected to. It cannot be measured directly and is normally not mentioned in the datasheet. It is always below the voltage at maximum power point under STC in order to compensate for the voltage reduction by increased temperature.
Most of us might assume that stronger sunshine means that more electrical power will be produced by a solar panel. In fact, this is often not the case. Like with other semiconductor devices, also the solar cell electrical properties are sensitive to temperature variabilities. As the efficiency decreases with increasing temperatues, sunny, but cold days in winter or spring are more likely than sunny summer days to have optimum conditions for solar panels.
In order to estimate the temperature influence on the performance, datasheets also state above electrical parameters under NOCT conditions. In addition to that, temperature coefficients for scaling the current and voltage parameters are provided. The effect of these coefficients can be seen in the graph in the following chapter.
# Performance Simulation
A solar cell has the same inner structure as a diode, as it consists of a p-n junction. So, the basis for modeling the behavior of a solar cell is a diode
D
The sunlight is modelled as a photo-generated current source
I_{ph}
. In order to consider losses in conductors and leakage current, the model is extended with a series resistor
R_s
R_{sh}
in parallel to the diode.
The described equivalent circuit model is shown in Figure 3. It is also called single-diode or five-parameter model.
Figure 3. Equivalent circuit model of a solar cell.
The following interactive graph plots the characteristic curves of a solar panel using an approximated solution of above model [2]. The equivalent circuit model parameters were calculated from datasheet parameters and can be adjusted for a specific solar panel. You can use it to evaluate how the ambient temperature and the irradiance influences the performance.
Datasheet parameters at standard test conditions (STC):
Voltage in maximum power point:
Current in maximum power point:
Datasheet temperature parameters:
Normal operating cell temperature (NOCT):
Actual environmental conditions:
Figure 4. Solar panel characteristic curve (interactive).
The default values in the graph are based on a typical 150 W module with 36 cells, suitable for 12V systems.
[1] Image by Tssenthi from Wikipedia, CC-BY-SA license, link (opens new window)
[2] El Tayyan, Ahmed A.: A simple method to extract the parameters of the single-diode model of a PV system. Turkish Journal of Physics, 2013, link (opens new window)
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Find the difference between heats of reaction at constant pressure and constant volume for the following reaction at 25 C - Chemistry - Thermodynamics - 8920685 | Meritnation.com
The difference between heats of reaction at constant pressure and constant volume for the following reaction at 25 C in kJ.
2{C}_{6}{H}_{{6}_{\left(l\right)}} + 15 {O}_{{2}_{\left(g\right)}} \to 12 C{{O}_{2}}_{\left(g\right)} + 6 {H}_{2}{O}_{\left(l\right)}\phantom{\rule{0ex}{0ex}}∆n = 12 - 15 = -3\phantom{\rule{0ex}{0ex}}∆H = ∆E + RT ∆n\phantom{\rule{0ex}{0ex}}∆H - ∆E = RT ∆n\phantom{\rule{0ex}{0ex}} = \left(-3\right) × 8.314 × {10}^{-3} × 298\phantom{\rule{0ex}{0ex}} =- 7.443 kJ/mole
Samuel Mavelil answered this
Pls it is very urgent...
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Examples of the pythagorean theorem | StudyPug
The pythagorean theorem is a fundamental part of geometry, but what is the pythagorean theorem definition? This theorem—one of many triangle theorems—shows the relationship the three sides of a right triangle has with one another. Labelling the sides touching the right angle as "
a
b
", and the longer hypotenuse side of the triangle as "
c
", the pythagorean theorem tells us that:
c^{2} = a^{2} + b^{2}
This is the equation that will help you find any side of a right triangle as long as you are given the lengths of the other two sides. It uses the squares of the sides. You're also able to prove that a triangle is a right triangle if you have all the sides and are able to put their lengths nicely into the pythagorean theorem.
What is a hypotenuse
We briefly touched on what the hypotenuse is above, but simply put, the hypotenuse is the longest side of a right triangle. You can find the hypotenuse opposite from the right angle in the triangle itself. It is the only side that is not touching the
90
degrees angle.
So you may be wondering: how do I find the hypotenuse? It's simple! You just use the pythagorean theorem above! It is crucial to helping you figure out the hypotenuse in a right angle triangle. You will first need the other two sides before you can figure out the longer hypotenuse side of the triangle, but it's easy once you've got that down!
Let's try our hand at using the pythagorean theorem to work out the sides of a triangle with some example problems.
Find the side lengths of the squares.
Look for side length
You can see that there is a
90
degree angle in the triangle sandwiched between the three squares. This means we've got a right triangle here, allowing us to use the pythagorean theorem to help us find the answer.
c^{2} = a^{2} + b^{2}
We have two sides given to us, and in order to find the hypotenuse, we can substitute in the numbers into the pythagorean theorem.
c = \sqrt{(18)^{2} + (24)^{2}}
\sqrt{324 + 576}
\sqrt{900}
30
Your final answer is that
c
30
x
Once again, we've got a right triangle. We're also finding the hypotenuse in this case. Let
x = c
, which is our hypotenuse.
c^{2} = a^{2} + b^{2}
x^{2} = 12^{2} + 16^{2}
x^{2} = 144 + 256
\sqrt{x^{2}} = \sqrt{400}
x = 20cm
Find x, the side length of the triangle
You've probably got the hang of the steps by now!
x = c
c^{2} = a^{2} + b^{2}
x^{2} = 7^{2} + 24^{2}
x^{2} = 49 + 576
\sqrt{x^{2}} = \sqrt{625}
x = 25cm
Play around with this online calculator to see how the pythagorean can be used when you scale the sides of a right triangle down or up.
Moving on forward, explore how to use exponents to describe numbers and how you can use the linear function distance formula in math.
In the nutshell, Pythagorean theorem/Pythagorean relationship describes the relationship between the lengths and sides of a right triangle. After thousands of repeated examinations by the ancient Greek mathematicians, it was found that the square of the hypotenuse is equal to the sum of the squares of the other two sides c² = a² + b².
Basic Concepts: Introduction to Exponents, Classifying Triangles
Related Concepts: Using exponents to describe numbers, Distance formula:
d = \sqrt{(x_2-x_1)^2+(y_2-y_1)^2}
Find the area of the squares.
Is the triangle in the middle a right triangle? Why?
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What is an Erlang: Formula Calculation - Collections - 2022
What is an Erlang
The Erlang is a statistical measure of the voice traffic density in a telecommunications system. It is widely used because it is necessary to understand the required capacity in a network to be able to provision correctly for it. Insufficient capacity and some calls cannot be carried, too much and this leads to excess costs for unused capacity.
As a result it is helps to have a definition of the telecommunications traffic so that the volume can be quantified in a standard way and calculations can be made.
Telecommunications network designers make great use of the Erlang to understand traffic patterns within a voice network and they use the figures to determine the capacity that is required in any area of the network.
Erlang definition
The Erlang is a dimensionless unit that is used in telephony as a measure of offered load or carried load on service-providing elements such as telephone circuits or telephone switching equipment. A single cord circuit has the capacity to be used for 60 minutes in one hour. Full utilization of that capacity, 60 minutes of traffic, constitutes 1 erlang.
Who was Erlang?
The Erlang is named after a Danish telephone engineer named A.K Erlang (Agner Krarup Erlang). He was born on 1st January 1878 and although he trained as a mathematician, he was the first person to investigate traffic and queuing theory in telephone circuits.
While he was at CTC, Erlang studied the loading on telephone circuits, looking at how many lines were required to provide an acceptable service without installing too much over-capacity that would cost the company money. There was a trade-off between cost and service level.
Erlang developed his theories over a number of years, and published several papers. He expressed his findings in mathematical forms so that they could be used to calculate the required level of capacity, and today the same basic equations are in widespread use..
In view of his ground-breaking work, the International Consultative Committee on Telephones and Telegraphs (CCITT) honoured him in 1946 by adopting the name "Erlang" for the basic unit of telephone traffic.
Erlang died on 3rd February 1929 after an unsuccessful abdominal operation.
The Erlang unit is the basic measure of telecommunications traffic intensity representing continuous use of one circuit and it is given the symbol "E". It is effectively call intensity in call minutes per sixty minutes. In general the period of an hour is used, but it actually a dimensionless unit because the dimensions cancel out (i.e. minutes per minute).
The number of Erlangs is easy to deduce in a simple case. If a resource carries one Erlang, then this is equivalent to one continuous call over the period of an hour. Alternatively if two calls were in progress for fifty percent of the time, then this would also equal one Erlang (1E). Alternatively if a radio channel is used for fifty percent of the time carries a traffic level of half an Erlang (0.5E)
From this it can be seen that an Erlang, E, may be thought of as a use multiplier where 100% use is 1E, 200% is 2E, 50% use is 0.5E and so forth.
Interestingly for many years, AT&T and Bell Canada measured traffic in another unit called CCS, 100 call seconds. If figures in CCS are encountered then it is a simple conversion to change CCS to Erlangs. Simply divide the figure in CCS by 36 to obtain the figure in Erlangs
Erlang function or Erlang formula and symbol
It is possible to express the way in which the number of Erlangs are required in the format of a simple function or formula.
E = \lambda x h
λ = the mean arrival rate of new calls
h = the mean call length or holding time
E = the traffic in Erlangs.
Using this simple Erlang function or Erlang formula, the traffic can easily be calculated.
The use of the Erlang and the basic concepts surrounding its use have provided telecommunications engineers a valuable tool. It is widely used within the industry to look at loading levels, especially in areas like call centres, telephone exchanges and lines linking different areas.
However in this basic form the Erlang does not address some real life aspects of loading including peak traffic density and the number of blocked calls resulting from short term overloading. To address these factors, development sof the basic Erlang concept have been introduced and they include measures like the Erlang B and Erlang C.
Watch the video: 16 Erlang - Distributed Programs - genserver (May 2022).
Copyright 2022 \ What is an Erlang...
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Jillian is trying to construct a square. She has started by constructing two perpendicular lines, as shown at right. If she wants each side of the square to have length
k
, as shown at right, describe how she should finish her construction.
How can you use
k
and the axes to create this square?
Jillian should construct an arc centered at
P
with a radius
k
so that it intersects
n\text{ and }m
each once (call these intersections
R
S
). She should then make two more circles with radius
k
R
S
. The fourth vertex lies where these two circles intersect.
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Error, invalid input: accumulate expects its 1st argument, r, to be of type evaln(numeric), but received total := total - Maple Help
Home : Support : Online Help : Error, invalid input: accumulate expects its 1st argument, r, to be of type evaln(numeric), but received total := total
\mathrm{sin}\left(\left[1,2, 3\right]\right);
\mathrm{whattype}\left(\left[1,2,3\right]\right);
\textcolor[rgb]{0,0,1}{\mathrm{list}}
\left[1,2,3\right]
\mathrm{Describe}\mathit{}\left(\mathrm{sin}\right)
x
\mathrm{sin}
\mathrm{sin}\left(1\right);\mathrm{sin}\left(2\right);\mathrm{sin}\left(3\right)
\textcolor[rgb]{0,0,1}{\mathrm{sin}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{1}\right)
\textcolor[rgb]{0,0,1}{\mathrm{sin}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{2}\right)
\textcolor[rgb]{0,0,1}{\mathrm{sin}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{3}\right)
\mathrm{sin}~\left(\left[1,2,3\right]\right)
\left[\textcolor[rgb]{0,0,1}{\mathrm{sin}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{1}\right)\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{sin}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{2}\right)\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{sin}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{3}\right)\right]
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The first boundary value problem for classical equations of mathematical physics in domains with piecewise-smooth boundaries. I (in Russian) | EMS Press
The first boundary value problem for classical equations of mathematical physics in domains with piecewise-smooth boundaries. I (in Russian)
Boris A. Plamenevskii
The first boundary value problem for the Stokes, Navier-Stokes, Lamé sytems and for the Laplace equation in a bounded domain
\Omega \subset \mathbb R^3
is studied. The boundary of
\Omega
contains singularities, such as conic points, edges or polyhedral angles. Theorems on solvability in spaces, supplied with weighted
L_s-
C^{\alpha}-
norms (
1 < s < \infty, 0 < \alpha < 1
) are proved. Coercive estimates of solutions in these spaces as well as pointwise estimates of the Green functions are obtained. The change of properties of generalized solutions under the change of right-hand sides is observed.
Vladimir G. Maz'ya, Boris A. Plamenevskii, The first boundary value problem for classical equations of mathematical physics in domains with piecewise-smooth boundaries. I (in Russian). Z. Anal. Anwend. 2 (1983), no. 4, pp. 335–359
DOI 10.4171/ZAA/71
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The Dirichlet problem in domains with lower dimensional boundaries | EMS Press
The present paper pioneers the study of the Dirichlet problem with
L^q
boundary data for second order operators with complex coefficients in domains with lower dimensional boundaries, e.g., in
\Omega := \mathbb{R}^n \setminus \mathbb{R}^d
d < n-1
. Following results of David, Feneuil and Mayboroda, we introduce an appropriate degenerate elliptic operator and show that the Dirichlet problem is solvable for all
q > 1
, provided that the coefficients satisfy the small Carleson norm condition.
Even in the context of the classical case
d = n-1
, (the analogues of) our results are new. The conditions on the coefficients are more relaxed than the previously known ones (most notably, we do not impose any restrictions whatsoever on the first
n-1
rows of the matrix of coefficients) and the results are more general. We establish local rather than global estimates between the square function and the non-tangential maximal function and, perhaps even more importantly, we establish new Moser-type estimates at the boundary and improve the interior ones.
Joseph Feneuil, Svitlana Mayboroda, Zihui Zhao, The Dirichlet problem in domains with lower dimensional boundaries. Rev. Mat. Iberoam. 37 (2021), no. 3, pp. 821–910
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The Jenkins–Serrin problem for translating horizontal graphs in $M \times \mathbb R$ | EMS Press
The Jenkins–Serrin problem for translating horizontal graphs in
M \times \mathbb R
Eddygledson S. Gama
Jorge H. de Lira
We prove the existence of horizontal Jenkins–Serrin graphs that are translating solitons of the mean curvature flow in Riemannian product manifolds
M \times \mathbb R
. Moreover, we give examples of these graphs in the cases of
\mathbb R^3
\mathbb H^2 \times \mathbb R
Eddygledson S. Gama, Esko Heinonen, Jorge H. de Lira, Francisco Martín, The Jenkins–Serrin problem for translating horizontal graphs in
M \times \mathbb R
. Rev. Mat. Iberoam. 37 (2021), no. 3, pp. 1083–1114
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In many statically-typed languages, there is a typeof operator or a getType function which returns the type of a given value (e.g. typeOf("a") → String). But what happens if we get the type of that type? typeOf(String) → ?. And even further, what is type of the type after it? (typeOf(typeOf(String)) → ?)
Console.WriteLine("".GetType().Name); // String
Console.WriteLine("".GetType().GetType().Name); // System.RuntimeType: System.Type
Console.WriteLine("".GetType().GetType().GetType().Name); // System.RuntimeType : System.Type
As you can see in the above code snippets, in many languages, type of a value returns its type (obviously), and then type of that type returns a type called Type (or a variant of it), and then type of that Type returns Type again, and the last step repeats forever.
However, this is not what happens in mathematically strong languages that are being used for theorem proving like Coq, Agda, or Idris. What we see there is that type of Type is Type 1, type of Type 1 is Type 2 and so on... But why is that?
Types in programming languages come from a mathmatical theory, called Type Theory. A type theory is a formal system where every term has a type. In your favorite programming language, the term can be variables and the type, well, is the type of that variable. These theories are implemented as Type Systems in programming languages.
The languages mentioned in the previous section (Idris, Agda, Coq) use a type theory called Typed λ (Lambda) Calculus. Typed λ Calculus is closely related to mathematical logic, which has a subarea called Set theory. Set theory studies Sets, that are basically a collection of unique objects.
We can define the type Boolean as a set of { true, false }. That means a variable that is the type of Boolean, can either be true or false, and nothing else.
type Boolean = false | true
I can also have a set of values which the instances are themselves instance of other types/sets as well: { 1, "hello", false }.
type MySet = 1 | "hello" | false
const x : MySet = "bye"
// ---------------^
// Type check error: The set MySet does not contain the object "bye"
Click for more details: Wait a minute! Types are Sets?!
There are multiple set theories that study sets. One of these theories is called Naive Set Theory which is one of the first things you face when learning about sets. As the name suggests, naive set theory is very simple and easy to understand. Unlike many other set theories that are defined using mathmatical formal logics and notations, naive set theory is defined in natural language: the language that humans speak.
This definition of sets goes back to Georg Cantor:
A set is a gathering together into a whole of definite, distinct objects of our perception or of our thought—which are called elements of the set.
– Georg Cantor, 1915
So we know what a Set is, we know Types are sets of values, what's all these Type 1, Type 2, ... about then?
Naive Set Theory doesn't work
The bad news is this simple and easy to understand Naive Set Thory has a bunch of flaws: it faces different contradictions and inconsistencies, and we don't like to encounter contradictions in our math.
We really want to keep it simple and straightforward, but oversimplifying something like Logic and Set theory which is complex by nature can be troublesome. Therefore, we need to look for other set theories that use a bunch of statements as rules, called Axioms. These axioms prevent us from falling in the trap of those contradictions.
"Hey! What are those contradictions you keep talking about? I still don't know why Naive set theory doesn't work!" you might ask, rightfully! But before answering that question in the next section, I want to quote the Good Math book by Mark C. Chu-Carrol:
Don't Keep It Simple, Stupid
There's an old mantra among engineers called the KISS principle. KISS stands for "Keep it simple, stupid!" The idea is that when you're building something useful, you should make it as simple as possible. The more moving parts something has, the more complicated corners it has, the more likely it is that an error will slip by. Looked at from that perspective, naive set theory looks great. It's so beautifully simple. [...]
Unfortunately, set theory in practice needs to be a lot more complicated. Why can't we stick with the KISS principle, use naive set theory, and skip that hairy stuff? The sad answer is, naive set theory doesn't work.
Imagine the set
B
where its elements are all the sets that are not the members of themselves.
B=\{x:x\not \in x\}
If B isn't a member of itself, it should be a member of itself by its definition.
If B is a member of itself, it shouldn't be a member of itself by its definition.
The example above is a sample of Russel's paradox. However, Russel's Paradox is more general than the definition of the set
B
we've talked about.
Russell's paradox shows that every set theory that contains an unrestricted comprehension principle (UCP) leads to contradictions. Here in the set
B
, the predicate
¬(x ∈ x)
is the UCP that causing the contradiction.
We don't enter the details of UCP and its general formula as we don't need it for the rest of the article. However we need to know that every set that defines these too open rules (UCPs) in their definition will face contradictions, and therefore we need some axiomatic set theories (instead of the naive ones.) that prevents us to face such contradictions.
Click for more details: General form of UCP
Click for more details: This paradox is set theoretic, not type theoretic!
Type Universes
As we've seen in the previous section, Russell’s paradox prevents us to accept the collection of all sets (let it be
M
) as a set itself, because then someone can define a subset of it called
A ⊆ M
A
equals to all sets that do not contain themselves.
Therefore a Set can not be a type of Set.
Set: Set // Leads to contradiction
Imagine you want to define the cartesian product of multiple sets. Type definition of List is:
List<T> where T : Set returns Set =
// member definitions...
and type signature of your product would look like something like this:
Prod : List Set → Set
However, there is a small problem here. Because the type definition of List asks A to be a type of Set, we already know that A can not be Set itself beacuse Set: Set is not valid. The solution is to define an special version of Set. This special version will be Set's own type, contains it and is larger than it.
List₁<T> where T : Set₁ returns Set₁ =
Prod : List₁ Set → Set
Set_1
is that special type which contains other types, and is called a Type Universe. The number 1 is its Universe Level.
Set_i : Set_{i+1}
Click for more details: Cumulative hierarchy of universes
Click for more details: How many universes?
In Agda and many other dependently typed languages, Type universes are built-in in the language. They either exist as a langugae construct as in Agda:
data List {n : Level} (A : Set n) : Set n where
[] : List A
or are inferred by the language, and cannot be specified explicitly, as in Idris.
We've seen that type of Type in most of the statically typed language is Type itself; however this wasn't the case with logic based languages. We've investigated the reason and realized that having Type: Type will lead to contradictions in logic and set theories. After seeing the solution of the dependently typed languages, we learnt about Type Universes and the reason they can solve the issue of facing contradictions for proving theorems.
type-theorylogictheorem-proversagdaidriscoq
An opinion on what's a good general-purpose programming language?
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As you discovered in the asteroid activity in Lesson 9.1.2, people are not very good at picking representative samples. Similarly, it is actually quite difficult to “fake” randomness.
In this problem, you will perform a brief survey to get an idea of people’s “random” behavior. For this particular survey, you are not concerned with any particular population, so you can choose any sample that is convenient. You should talk with at least
10
people among your friends and family, but try for more to make your results more interesting.
Use the same survey method for everybody: text messaging, phone call, or personal interview. Carefully record the responses you get.
Before you start your survey, decide exactly how you want to word your questions, and write them down, so you can read them the same way to each person you survey.
Why would using a mass email or a social networking site not be a good plan?
Ask each of your respondents to make each of the following choices randomly. Make sure you read the choices exactly the same way to each respondent.
Pick a number between
1
4
Pick a vowel (a, e, i, o, or u).
Make a bar graph for the responses to each of the three questions in your survey. Write the precise wording of the question with each bar graph.
Do your results show that people’s choices were truly random? Why or why not?
Try to explain any trends or interesting patterns that you observe. Were the results similar to what you expected? Why or why not?
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CBEST Test Prep Tutor, Help and Practice Online | StudyPug
CBEST Test Prep made completely easy!
Get better scores with our complete help for CBEST (California Basic Educational Skills Test) including skill factors in 1) estimation, measurement, and statistical principles; 2) computation and problem solving; and 3) numerical and graphic relationships.
The comprehensive lessons by our tutors cover help on topics like Solving linear equations, Operations with fractions, Linear functions, Circles, Probability, Statistics, Ratios and rates, and more. Learn the concepts with our video tutorials that show you step-by-step solutions to even the hardest CBEST test problems. Then, strengthen your understanding with tons of CBEST practice questions.
All our lessons are taught by experienced CBEST math teachers. Let's finish your CBEST test prep in no time, and ACE that test.
See our CBEST Test Prep topics
Meet Andy, your CBEST Test Prep tutor
It is one of the many CTC exams (Commission on Teacher Credential) required by law for ones who want to become a teacher and are applying for the California teaching credential or services credential. The test aims at measuring the basic reading, math, and writing skills which are found to be important for being an educator. However, the test is not designed to test one's ability of teaching these skills. For details on teaching credentials requirements in California, you can check this page out.
How should I start my CBEST test prep?
You can first test yourself with this sample CBEST practice test to see which areas and topics you need to focus more on. Then, watch our video lessons and work on the practice questions for review and better understanding!
What to expect when taking the CBEST test?
You can choose whether to take the computer-based test or paper-based test. They both consist of three sections: reading, math, and writing. It is also up to you whether you take 1, 2 or 3 sections in the 4-hour test session. However, you do not need to pass all the sections in a single test session. All questions are delivered in English and all responses must be in English as well. There will be no breaks during a test session. You may take breaks anytime, but it will count toward your 4-hour test time.
How does the math section look like on CBEST?
In the math section, there are 50 multiple-choice questions, and most of them are word problems. Here's the breakdown of the section: about 30% of the questions is on estimation, measurement, and statistical principles; about 35% is on computation and problem-solving; and about 35% is on numerical and graphic relationships.
No, the use of calculators is not allowed on the CBEST exam. In fact, there is a list of items you are NOT allowed to bring into the test center. Make sure you check out the test center rules before you attend!
What is the CBEST passing score?
The CBEST results are reported in scaled scores ranging from 20 to 80 for each section. The passing score of each section is a scaled score of 41. In other words, a total score of 123 for all three sections is required in order to pass the test.
A passing status will still be granted with a scaled score on one or two sections that is lower than 41 but higher than 37, and the total score is 123 or higher. However, if a score on any of the sections is below 37, it does not matter how high the total score is, a passing status will not be granted.
When will I get my CBEST results?
For computer-based testing, you will be provided the preliminary test results at the end of the Reading and Mathematics sections. The official test results will be available within two weeks of testing. For paper-based testing, the test results will be available within three weeks of testing.
What are the CBEST test dates?
Computer-based testing is by appointment and takes place on Mondays to Saturdays, except for some holidays. Seating is limited on a first-come, first-serve basis, and you need to choose which sections you want to take when making the test appointment. You can make and check appointment availability here. For paper-based testing, it only takes place every two to three months; therefore, be sure to register before the registration deadlines. By registering for the paper-based test, you are automatically signed up for all sections. However, you can choose which sections to take on the day of the test.
Is there anything I should pay attention before and during the CBEST test?
If you are taking the computer-based testing, before the test, you can take a look at this tutorial that helps you get a better idea of the test format and how the test is delivered. Also, before the beginning of the test, you will be asked to read and sign a nondisclosure agreement which you can access here. You are advised to take the tutorial and go through the policies before the test because you will only be given 15 minutes to finish the tutorial and reading the CBEST policies.
CBEST Test Prep topics
8Number System and Pattern
ax = b
\frac{x}{a} = b
ax + b = c
\frac{x}{a} + b = c
\;a\left( {x + b} \right) = c
ax + b = c
{x \over a} + b = c
a(x + b) = c
d = \sqrt{(x_2-x_1)^2+(y_2-y_1)^2}
M = ( \frac{x_1+x_2}2 ,\frac{y_1+y_2}2)
m = \frac{y_2-y_1}{x_2- x_1}
y - y_1 = m (x - x_1)
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ECC - The Hive
This page is outdated and will be reorganized and moved to https://www.ellipticcurve.info.
1 Equation of an elliptic curve
1.2 Weierstraß normal form
2 Elliptic curves over algebraic fields
3 Point group operation
3.1 Identity and inverse
3.2 General calculation
3.2.1 Complete the cube and factor x – x1
3.2.2 Complete the square and factor x – x2
Equation of an elliptic curve
An elliptic curve is the set of solutions to a general equation in the third degree in two variables.
{\displaystyle gz^{3}+hz^{2}w+jzw^{2}+kw^{3}}
{\displaystyle +\,mz^{2}+pzw+qw^{2}}
{\displaystyle +\,rz+sw+t=0.}
Weierstraß normal form
The general equation for an elliptic curve has ten coefficients. A linear transformation is employed to simplify it to a form easier to work with.
{\displaystyle {\begin{bmatrix}x\\y\end{bmatrix}}={\begin{bmatrix}\alpha &\beta \\\gamma &\delta \end{bmatrix}}{\begin{bmatrix}z\\w\end{bmatrix}}+{\begin{bmatrix}\zeta \\\eta \end{bmatrix}}.}
When the coefficients of the general equation are worked out in the transformed coördinates
{\displaystyle (x,y)}
in terms of the original coördinates
{\displaystyle (z,w)}
, this transformation can be solved for its Greek-lettered coefficients, and the general equation may be expressed much more simply in Weierstraß normal form with four coefficients, only two of which are free.
{\displaystyle y^{2}=x^{3}+ax+b.}
Elliptic curves over algebraic fields
The elliptic curve itself, defined by an algebraic equation, consists of the set of solutions to its equation over some algebraic field, which may or may not be explicitly specified in any particular context.
It is generally easiest to visualize the elliptic curve or draw a graph of it over the field
{\displaystyle \mathbb {R} }
of real numbers. If complex (real+imaginary) solutions to its equation are of interest, then the field is
{\displaystyle \mathbb {C} }
; if rational points on the elliptic curve are of interest, then the field is
{\displaystyle \mathbb {Q} }
For cryptographic applications, it is usually the case that the elliptic curve is defined over an arbitrary "finite field" which is neither a subfield nor an extension of the real numbers.
Point group operation
{\displaystyle (x,y)}
that solve the equation of an elliptic curve in Weierstraß normal form
{\displaystyle y^{2}=x^{3}+ax+b,}
together with an additional point
{\displaystyle {\mathcal {O}}}
"at infinity," form an additive abelian group under a specific operation "
{\displaystyle \oplus }
{\displaystyle {\mathcal {O}}}
is the group identity element; for all points
{\displaystyle (x,y)}
on the elliptic curve,
{\displaystyle (x,y)\oplus {\mathcal {O}}=(x,y).}
The group inverse of a point is formed by reflecting it across the x-axis:
{\displaystyle \ominus (x,y)=(x,-y)}
{\displaystyle (x,y)\oplus [\ominus (x,y)]={\mathcal {O}}.}
The intuitive idea is to consider the elliptic curve equation over the real numbers. If a straight line intersects the elliptic curve at two points, then their group sum is defined as the group inverse of the third point at which the line intersects the elliptic curve, so that
{\displaystyle (x_{1},y_{1})\oplus (x_{2},y_{2})\oplus (x_{3},y_{3})={\mathcal {O}}.}
The mechanics of this operation are labor-intensive but fairly straightforward to derive from scratch. Substitute the equation for the line through
{\displaystyle (x_{1},y_{1})}
{\displaystyle (x_{2},y_{3})}
into the equation for the elliptic curve
{\displaystyle y^{2}=x^{3}+ax+b,}
solve for x, and factor out the first two solutions
{\displaystyle x_{1}}
{\displaystyle x_{2}}
{\displaystyle x_{3}.}
{\displaystyle \lambda =(y_{2}-y_{1})/(x_{2}-x_{1})}
is the slope, then we can substitute
{\displaystyle y=\lambda (x-x_{1})+y_{1}}
and factor out
{\displaystyle (x-x_{1})(x-x_{2})}
to solve for x to find the third point
{\displaystyle x=x_{3},}
{\displaystyle {\big [}\lambda (x-x_{1})+y_{1}{\big ]}^{2}=x^{3}+ax+b.}
{\displaystyle x^{3}-{\big [}\lambda (x-x_{1})+y_{1}{\big ]}^{2}+ax+b}
{\displaystyle =\,(x-x_{1})(x-x_{2})(x-x_{3}).}
{\displaystyle x^{3}-\lambda ^{2}(x-x_{1})^{2}-2\lambda y_{1}(x-x_{1})}
{\displaystyle -\,y_{1}^{2}+ax+b}
{\displaystyle =\,(x-x_{1})(x-x_{2})(x-x_{3}).}
{\displaystyle x^{3}-\lambda ^{2}(x-x_{1})^{2}+(a-2\lambda y_{1})(x-x_{1})}
{\displaystyle -\,y_{1}^{2}+ax_{1}+b}
{\displaystyle =\,(x-x_{1})(x-x_{2})(x-x_{3}).}
Complete the cube and factor x – x1
{\displaystyle x^{3}=(x-x_{1})^{3}+3x_{1}(x-x_{1})^{2}+3x_{1}^{2}(x-x_{1})+x_{1}^{3}.}
{\displaystyle (x-x_{1})^{3}+(3x_{1}-\lambda ^{2})(x-x_{1})^{2}+(3x_{1}^{2}+a-2\lambda y_{1})(x-x_{1})}
{\displaystyle -\,y_{1}^{2}+ax_{1}+b+x_{1}^{3}}
{\displaystyle =\,(x-x_{1})(x-x_{2})(x-x_{3}).}
{\displaystyle (x-x_{1})^{2}+(3x_{1}-\lambda ^{2})(x-x_{1})+(3x_{1}^{2}+a-2\lambda y_{1})}
{\displaystyle +\,{\frac {-y_{1}^{2}+ax_{1}+b+x_{1}^{3}}{x-x_{1}}}}
{\displaystyle =\,(x-x_{2})(x-x_{3}).}
Complete the square and factor x – x2
{\displaystyle x^{2}=(x-x_{2})^{2}+2x_{2}(x-x_{2})+x_{2}^{2}.}
Solve the remaining linear identity for the last factor
{\displaystyle x-x_{3}=0.}
{\displaystyle x_{3}}
into the equation for the line to find
{\displaystyle y_{3}=\lambda (x_{3}-x_{1})+y_{1}.}
The reflection of
{\displaystyle (x_{3},y_{3})}
across the x-axis is the result of the group operation:
{\displaystyle (x_{1},y_{1})\oplus (x_{2},y_{2})=(x_{3},-y_{3}).}
Finding the group sum of a point with itself is a special case because it involves finding a line tangent to the elliptic curve at the point to be "doubled" rather than passing through two distinct points. Differentiate the Weierstraß equation
{\displaystyle y^{2}=x^{3}+ax+b}
with respect to x (basic calculus) to find the slope of the line tangent to the elliptic curve at
{\displaystyle (x_{1},y_{1}).}
{\displaystyle {\frac {d}{dx}}y^{2}={\frac {d}{dx}}{\big (}x^{3}+ax+b{\big )}}
{\displaystyle 2y{\frac {dy}{dx}}=3x^{2}+a}
{\displaystyle {\frac {dy}{dx}}={\frac {3x^{2}+a}{2y}}}
Proceed as above with
{\displaystyle x_{2}=x_{1},\,}
{\displaystyle y_{2}=y_{1},}
{\displaystyle \lambda ={\frac {3x_{1}^{2}+a}{2y_{1}}}.}
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We've got you covered with our complete help for Grade 6 math, whether it's for EQAO grade 6 math (Ontario), grade 6 math PAT (Alberta), grade 6 math Provincial Assessment (Newfoundland) or EMA (PEI).
Aligned with your class or textbook, you will get grade 6 math help on topics like Algebra, Solving equations, Inequalities, Decimals, Fractions, Probability, Percentages, and so many more. Learn the concepts with our video tutorials that show you step-by-step solutions to even the hardest grade 6 math problems. Then, strengthen your understanding with tons of grade 6 math practice.
All our lessons are taught by experienced Grade 6 math teachers. Let's finish your homework in no time, and ACE that final
See our Grade 6 Math topics
Meet Andy, your Grade 6 Math tutor
My class is using Nelson Math Grade 6. Can I use your site?
Most definitely! We cover all the topics you'll find in your textbook. For those who use Math Makes Sense grade 6 – don't worry, we have math help on everything in your textbook too.
How do I get my kids ready for the grade 6 math test with StudyPug?
First of all, you can try to identify the topics which your kids need help on. Test your kids with the math grade 6 math questions available in our site would be a good start. Once you know where your kids should put more effort on, you may ask your kids to watch our video lessons on those topics, and then test them again. Your kids should be ready for the math quizzes or tests in no time!
What class should I take after Grade 6 math?
After mastering Grade 6 math, your follow up course should be either Grade 7 math or Pre-Algebra.
1Understanding Numbers
2.2Prime numbers and composite numbers
6Introduction to Ratios, Rates and Percentages
10.1Multiplying fractions and whole numbers (free lessons)
11Fractions, Decimals and Percentage
x + a = b
14.4Representing patterns in linear relations
14.5Reading linear relation graphs
14.6Solving linear equations by graphing
15Introduction to Equations
ax = b
\frac{x}{a} = b
ax + b = c
\frac{x}{a} + b = c
\;a\left( {x + b} \right) = c
16Introduction to Inequalities
16.1Expressing inequalities graphically and algebraically (free lessons)
17Surface Area and Volume of 3D Shapes
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Interval boundary element method – terracotta INFO
Interval boundary element method is classical boundary element method with the interval parameters.
Boundary element method is based on the following integral equation
This article needs attention from an expert in mathematics. (February 2009)
{displaystyle ccdot u=int limits _{partial Omega }left(G{frac {partial u}{partial n}}-{frac {partial G}{partial n}}uright)dS}
The exact interval solution on the boundary can be defined in the following way:
{displaystyle {tilde {u}}(x)={u(x,p):c(p)cdot u(p)=int limits _{partial Omega }left(G(p){frac {partial u(p)}{partial n}}-{frac {partial G(p)}{partial n}}u(p)right)dS,pin {hat {p}}}}
In practice we are interested in the smallest interval which contain the exact solution set
{displaystyle {hat {u}}(x)=hull {tilde {u}}(x)=hull{u(x,p):c(p)cdot u(p)=int limits _{partial Omega }left(G(p){frac {partial u(p)}{partial n}}-{frac {partial G(p)}{partial n}}u(p)right)dS,pin {hat {p}}}}
In similar way it is possible to calculate the interval solution inside the boundary
{displaystyle Omega }
. . . Interval boundary element method . . .
T. Burczyński and J. Skrzypczyk, The fuzzy boundary element method: a new solution concept, Proceedings of XII Polish conference on computer methods in mechanics, Warsaw-Zegrze, Poland (1995), pp. 65–66.
T. Burczynski, J. Skrzypczyk J. The fuzzy boundary element method: a new methodology. Series Civil Eng, Vol. 83. Gliwice: Sci Fasc of Silesian Tech Univ; 1995, pp. 25–42.
J. Skrzypczyk, A Note on Interval Fredholm Integral Equations. Zeszyty Naukowe Politechniki Śląskiej, Seria Budownictwo, Z.85, pp. 75–83, 1998
T. Burczynski, J. Skrzypczyk, Fuzzy aspects of the boundary element method, Engineering Analysis with Boundary Elements, Vol.19, No.3, pp. 209–216, 1997
H. Witek, Boundary element method in analysis of civil engineering structures with uncertain parameters. Ph.D. Dissertation, Silesian University of Technology, Faculty of Civil Engineering, Poland, 2005
B.F. Zalewski, R.L. Mullen, R.L. Muhanna, “Boundary Element Analysis of Systems Using Interval Methods”, Proceedings of the NSF Workshop on Modeling Errors and Uncertainty in Engineering Computations, Georgia Tech Savannah, February 2006.
B.F. Zalewski and R.L. Mullen, “Interval Bounds on the Local Discretization Error in Boundary Element Analysis for Domains with Singular Flux”, SAE 2008 Reliability and Robust Design in Automotive Engineering, SP-2170, Pages 237–246, 2008.
B.F. Zalewski and R.L. Mullen, “Discretization Error in Boundary Element Analysis using Interval Methods”, SAE 2007 International Transactions Journal of Passenger Cars: Mechanical Systems, Volume 116, Issue 6, Pages 1353–1361, 2008.
B.F. Zalewski and R.L. Mullen, “Point-wise Discretization Errors in Boundary Element Method for Elasticity Problem”, Third NSF Workshop on Reliable Engineering Computing, Pages 429–457, February 2008.
B.F. Zalewski, “Uncertainties in the Solutions to Boundary Element Method: An Interval Approach”, Case Western Reserve University, Ph.D. Dissertation, © 2008.
B.F. Zalewski and R.L. Mullen, “Local Discretization Error Bounds Using Interval Boundary Element Method”, International Journal for Numerical Methods in Engineering, Volume 78, Issue 4, April 2009, Pages 403–428.
A. Piasecka Belkhayat, Interval boundary element method for 2D transient diffusion problem, Engineering Analysis with Boundary Elements, Volume 32, Issue 5, May 2008, Pages 424-430
B.F. Zalewski, R.L. Mullena, and R.L. Muhanna, “Interval Boundary Element Method in the Presence of Uncertain Boundary Conditions, Integration Errors, and Truncation Errors”, Engineering Analysis with Boundary Elements, Volume 33, Issue 4, April 2009, Pages 508–513.
B.F. Zalewski, R.L. Mullen, and R.L. Muhanna, “Fuzzy Boundary Element Method for Geometric Uncertainty in Elasticity Problem”, SAE 2009 International Journal of Materials and Manufacturing, Volume 2, Issue 1, Pages 310–316, 2009.
B.F. Zalewski and R.L. Mullen, “Worst Case Bounds on the Point-wise Discretization Error in Boundary Element Method for the Elasticity Problem”, Computer Methods in Applied Mechanics and Engineering, Volume 198, Issue 37–40, Pages 2996–3005, 2009.
B.F. Zalewski and R.L. Mullen, “Worst Case Point-wise Discretization Error Bounds for Systems with Geometrically Induced Singular Flux Solutions Using Interval Boundary Element Method”, ASCE Journal of Engineering Mechanics, Volume 136, Issue 6, Pages 710–720, 2010.
B.F. Zalewski, “Fuzzy Boundary Element Method for Material Uncertainty in Steady State Heat Conduction”, SAE 2010 International Journal of Materials and Manufacturing, Volume 3, Issue 1, Pages 372–379, 2010.
B.F. Zalewski and W.B. Dial, “Fuzzy Boundary Element Method with Uncertain Shear Modulus in Linear Plane Strain Elasticity”, SAE 2011 International Journal of Materials and Manufacturing, Volume 4, Issue 1, Pages 947–956, 2011.
A. Piasecka-Belkhayat, “Interval Boundary Element Method for Transient Diffusion Problem in Two-Layered Domain”, Journal of Theoretical and Applied Mechanics, Volume 49, Issue 1, Pages 265–276, 2011.
A. Piasecka-Belkhayat, “Interval Boundary Element Method for 2D Transient Diffusion Problem Using the Directed Interval Arithmetic”, Engineering Analysis with Boundary Elements, Volume 35, Issue 3, Pages 259–263, 2011.
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How to solve linear equations by substitution | StudyPug
Before we get into solving systems of linear equations via the substitution method, let's first consider and understand what it means to "solve" a system of equations. When we say "solve", with regards to linear, quadratic, exponential, or any other type of equation, what we really mean is that we are trying to find values of 'x' – the dependent variable – that satisfy 'y' – the independent variable.
Now that we've covered the basics, let's solve systems using substitution!
Before we get into using the method of substitution, make sure you're comfortable with your algebra by reviewing the lesson on solving linear equations with variables on both sides.
The basic procedure behind solving systems via substitution is simple: Given two linear equations, all we need to do is to "substitute" one in the pair of equations into its other by rearranging for variables. This procedure is better outlined below with the general example:
Consider the following equations, with (x,y) being coordinates and everything else representing constants.
ty = ax
zy = x + b
Step 1: Rearrange one of the equations to get 'y' by itself
y = \frac{ax}{t}
zy = x + b
Step 2: Substitute the rearranged equation into its partner
zy = x + b
z(\frac{ax}{t}) = x + b
Since this is just a general case, we can't solve for x. But note all we have to do is get x by itself.
Step 4: Substitute the solution for x into either of the initially given equations to find y
Once we have the value for x, we can substitute it into any of the two equations to find our solution for y.
Step 5: Write final answer out as a point
Therefore, our solution is (x,y)
Once again, this is just a general case. Also note that in this example we chose to solve for x first. It doesn't matter which variable you solve first, just note that x is often the easier one to solve for first, as it often involves less modification in the initial give equations. The best way to learn and master how to solve by substitution is to do some practice problems.
Take the following simultaneous equations and solve.
Let's use the first equation and rearrange it so we can have y by itself. We could certainly take the second equation, but that would involve more work.
Now, we are going to substitute our newly rearranged equation 6x - 7 = y into -9x + 2y = 7.
-9x + 2(6x - 7) = 7
Now that we have successfully performed substitution, let's solve for x.
-9x + 12x - 14 = 7
Now that we have x, we can put x=7 into either of the equations to solve for y. Let's chose the first equation because it is more simple.
The final answer: (7, 35)
The following image below summarizes the work we've just done:
Solve linear equation with substitution
Solve the following linear system.
3(x + 2) - (y + 7) = 4
5(x + 1) + 4(y + 3) = 31
In some instances, we are going to need to do some simplification of both equations before we can carry on with substitution and solving. In this case, we must first expand and simplify both equations:
3x + 6 - y - 7 = 4
Just like in the first example, let's use the first equation and rearrange it so we can have y by itself. We could certainly take the second equation, but that would involve more work.
Step 2: Substitute the rearranged equation into its partner and solve for x
Now, we are going to substitute our newly rearranged equation 3x - 5 = y into 5x + 4y = 14 and solve for x.
5x + 4(3x - 5) = 14
Now that we have x, we can put x = 2 into either of the equations to solve for y. Let's chose the first equation because it is more simple.
The final answer: (2, 1)
Another way to solve systems of linear equations is by substitution. We can substitute the y/x in one of the equations with the y/x in the other equation.
Related Concepts: Adding and subtracting rational expressions , System of linear equations, System of linear-quadratic equations, Graphing systems of linear inequalities
What does it mean by "solving system of equations by substitution"?
Solve each linear system algebraically by substitution
3(x+2) - (y+7) = 4
5(x+1) + 4(y+3) = 31
{{x+1} \over 2} + {{y-2} \over 14} = 3
{{x-2} \over 24} - {{y+1} \over2} = -{11 \over 4}
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Mr. Kyi has placed
3
7
blue, and
2
yellow beads in a hat. If a person selects a red bead, he or she wins
\$3
. If that person selects a blue bead, he or she loses
\$1
. If the person selects a yellow bead, he or she wins
\$10
. What is the expected value for one draw? Is this game fair?
For expected value you need to find the probability of drawing each marble and multiply that probability by the amount you could win.
The expected value is the total of these 3 outcomes.
\text{Expected value = }\frac{3}{12}(3) + \frac{7}{12}(-1) + \frac{2}{12}(10)
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What is the power of a quotient rule? | StudyPug
The power of a quotient rule is used when we want to simplify a fraction that is raised to a power. Every term needs to be raised to a power!
(a^x)(a^y)=a^{(x+y)}
{a^x \over a^y}=a^{(x-y)}
(a^x)^y = a^{(x\cdot y)}
( \frac{a^m}{b^n} {)^p} = \frac{a^{mp}}{b^{np}}
( \frac{x}{6} {)^2}
(\frac{2c^5d^4}{6c^3} {)^2}
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