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Downsampling - Aliasing - MATLAB & Simulink - MathWorks Benelux
This example shows how to avoid aliasing when downsampling a signal. If a discrete-time signal's baseband spectral support is not limited to an interval of width
2\pi /M
radians, downsampling by
M
results in aliasing. Aliasing is the distortion that occurs when overlapping copies of the signal's spectrum are added together. The more the signal's baseband spectral support exceeds
2\pi /M
radians, the more severe the aliasing. Demonstrate aliasing in a signal downsampled by two. The signal's baseband spectral support exceed
\pi
radians in width.
Create a signal with baseband spectral support equal to
3\pi /2
radians. Use fir2 to design the signal. Plot the signal's spectrum. The signal's baseband spectral support exceeds
\left[-\pi /2,\pi /2\right]
b1 = fir2(nf-1,f,a);
Hx = fftshift(freqz(b1,1,nf,'whole'));
Downsample the signal by a factor of 2 and plot the downsampled signal's spectrum with the spectrum of the original signal. In addition to an amplitude scaling of the spectrum, the superposition of overlapping spectral replicas causes distortion of the original spectrum for
|\omega |>\pi /2
y = downsample(b1,2,0);
text(2.5/pi*[-1 1],0.35*[1 1],{'\downarrow Aliasing','Aliasing \downarrow'}, ...
Increase the baseband spectral support of the signal to
\left[-7\pi /8,7\pi /8\right]
and downsample the signal by 2. Plot the original spectrum along with the spectrum of the downsampled signal. The increased spectral width results in more pronounced aliasing in the spectrum of the downsampled signal because more signal energy is outside
\left[-\pi /2,\pi /2\right]
f = [0 0.2500 0.5000 0.7500 7/8 1.0000];
a = [1.00 0.7143 0.4286 0.1429 0 0];
Finally, construct a signal with baseband spectral support limited to
\left[-\pi /2,\pi /2\right]
. Downsample the signal by a factor of 2 and plot the spectrum of the original and downsampled signals. The downsampled signal is full band. The spectrum of the downsampled signal is a stretched and scaled version of the original spectrum, but the shape is preserved because the spectral copies do not overlap. There is no aliasing.
downsample | fir2 | freqz
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Improving the Stability of Your Physics – kevinyu.net
In the previous post, we have covered a technique to resolve constraint in our physics simulation. Because we are working in the real-time domain, we need to sacrifice accuracy for speed by using an iterative method to resolve our constraints. Sequential Impulse doesn’t give the exact solution for the problems, but the solution it offers is good enough for games where accuracy is not the priority, as long as it still feels believable. Today, we will learn and implement two techniques that are widely used to help Sequential Impulse and to improve the stability of your physics.
There is one property of games that we can exploit to help sequential impulse compute a better result. In games, the property of objects like velocity, and position doesn’t change much from one frame to the next frame. This property is called “frame coherence”. If the state of objects doesn’t differ much, then the solution for the constraints should not differ much either. So, instead of recomputing the impulse every frame, we can use the solution from the previous frame as the starting point and refine it instead. This technique is called Warm Starting, Below is a source code I copy from my physics engine. In the previous post, we keep the total impulse from one iteration to another. For warm starting to work we need this value, so we need to keep this value across the frame (In the sample code, nIAcc for contact impulse and tIAcc for friction impulse). We apply this total impulse to each body before the resolution logic takes place.
void ftConstraintSolver::warmStart()
for (int32 i = 0; i < m_constraintGroup.nConstraint; ++i)
ftContactConstraint *constraint = &(m_constraintGroup.constraints[i]);
int32 bodyIDA = constraint->bodyIDA;
int32 bodyIDB = constraint->bodyIDB;
ftVector2 normal = constraint->normal;
ftVector2 tangent = normal.tangent();
ftManifold *manifold = &(constraint->contact->manifold);
for (int32 j = 0; j < constraint->numContactPoint; ++j)
ftContactPointConstraint *pointConstraint = &(constraint->pointConstraint[j]);
pointConstraint->nIAcc = manifold->contactPoints[j].nIAcc;
pointConstraint->tIAcc = manifold->contactPoints[j].tIAcc;
ftVector2 impulse = pointConstraint->nIAcc * normal;
impulse += pointConstraint->tIAcc * tangent;
m_constraintGroup.velocities[bodyIDA] -= constraint->invMassA * impulse;
m_constraintGroup.velocities[bodyIDB] += constraint->invMassB * impulse;
ftVector2 r1 = pointConstraint->r1;
m_constraintGroup.angularVelocities[bodyIDA] -= constraint->invMomentA * r1.cross(impulse);
m_constraintGroup.angularVelocities[bodyIDB] += constraint->invMomentB * r2.cross(impulse);
As previously mentioned, Sequential Impulse is an iterative method, the more iteration it does, the more accurate the solution will be. Warm Starting works as if we do more iteration by continuing the previous iteration.
Baumgarte Stabilization is a hacky technique that we apply to fix positional drift. With this technique even if there is only a little penetration, our physics will add energy to the system and push the objects apart. Imagine if we have a stack of boxes, The lowermost boxes will penetrate the surface because of gravity and the boxes above pushing it down. The solver will resolve this and push it up, creating penetration between the boxes. This penetration between boxes in return will push the box back again to the surface creating a cycle of small upward and downward movement. This jittery effect could be avoided by using slop. Slop allows a small penetration or positional drift to exist. So, instead of resolving every penetration. we only resolve penetration that exceeds the slop value. The slop value should be small enough that it is visually insignificant but large enough that it is numerically significant. Box2d use 0.005f (or 5cm).
b=–\frac{\beta }{△t} \left(max\left(d – slop, 0\right)\right)
real bias = ftMax(manifold->penetrationDepth[i] - m_option.linearSlop, 0);
pointConstraint->positionBias = m_option.baumgarteCoef * bias;
Beside using slop at contact, we can use this technique on another resolution like joint and restitution.
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Adequate subgroups and indecomposable modules | EMS Press
The notion of adequate subgroups was introduced by Jack Thorne [60]. It is a weakening of the notion of big subgroups used by Wiles and Taylor in proving automorphy lifting theorems for certain Galois representations. Using this idea, Thorne was able to strengthen many automorphy lifting theorems. It was shown in [22] and [23] that if the dimension is smaller than the characteristic then almost all absolutely irreducible representations are adequate. We extend the results by considering all absolutely irreducible modules in characteristic p of dimension p. This relies on a modified definition of adequacy, provided by Thorne in [61], which allows p to divide the dimension of the module. We prove adequacy for almost all irreducible representations of SL
_2 (p^a)
in the natural characteristic and for finite groups of Lie type as long as the field of definition is sufficiently large. We also essentially classify indecomposable modules in characteristic p of dimension less than
2p-2
and answer a question of Serre concerning complete reducibility of subgroups in classical groups of low dimension.
Robert M. Guralnick, Florian Herzig, Pham Huu Tiep, Adequate subgroups and indecomposable modules. J. Eur. Math. Soc. 19 (2017), no. 4, pp. 1231–1291
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Cumulative trapezoidal numerical integration - MATLAB cumtrapz - MathWorks Benelux
Cumulative Integral of Vector with Unit Spacing
Cumulatively Integrate Vector of Data with Nonunit Spacing
Cumulatively Integrate Matrix with Nonuniform Spacing
Multiple Cumulative Integrations
Cumulative trapezoidal numerical integration
Q = cumtrapz(Y)
Q = cumtrapz(X,Y)
Q = cumtrapz(___,dim)
Q = cumtrapz(Y) computes the approximate cumulative integral of Y via the trapezoidal method with unit spacing. The size of Y determines the dimension to integrate along:
If Y is a vector, then cumtrapz(Y) is the cumulative integral of Y.
If Y is a matrix, then cumtrapz(Y) is the cumulative integral over each column.
If Y is a multidimensional array, then cumtrapz(Y) integrates over the first dimension whose size does not equal 1.
Q = cumtrapz(X,Y) integrates Y with respect to the coordinates or scalar spacing specified by X.
If X is a scalar spacing, then cumtrapz(X,Y) is equivalent to X*cumtrapz(Y).
Q = cumtrapz(___,dim) integrates along the dimension dim using any of the previous syntaxes. You must specify Y, and optionally can specify X. If you specify X, then it can be a scalar or a vector with length equal to size(Y,dim). For example, if Y is a matrix, then cumtrapz(X,Y,2) cumulatively integrates each row of Y.
Calculate the cumulative integral of a vector where the spacing between data points is 1.
\mathit{f}\left(\mathit{x}\right)={\mathit{x}}^{2}
in the domain [1 5].
Use cumtrapz to integrate the data with unit spacing.
0 2.5000 9.0000 21.5000 42.0000
This approximate integration yields a final value of 42. In this case, the exact answer is a little less,
41\frac{1}{3}
. The cumtrapz function overestimates the value of the integral because f(x) is concave up.
Calculate the cumulative integral of a vector where the spacing between data points is uniform, but not equal to 1.
X = 0:pi/5:pi;
Y = sin(X');
Cumulatively integrate Y using cumtrapz. When the spacing between points is constant, but not equal to 1, an alternative to creating a vector for X is to specify the scalar spacing value. In that case, cumtrapz(pi/5,Y) is the same as pi/5*cumtrapz(Y).
Cumulatively integrate the rows of a matrix where the data has a nonuniform spacing.
Use cumtrapz to integrate each row independently and find the cumulative distance traveled in each trial. Since the data is not evaluated at constant intervals, specify X to indicate the spacing between the data points. Specify dim = 2 since the data is in the rows of Y.
Q1 = cumtrapz(X,Y,2)
0 9.6750 48.6000 82.8000
The result is a matrix of the same size as Y with the cumulative integral of each row.
Perform nested integrations in the x and y directions. Plot the results to visualize the cumulative integral value in both directions.
\mathit{f}\left(\mathit{x},\mathit{y}\right)=10{\mathit{x}}^{2}+20{\mathit{y}}^{2}
F = 10*X.^2 + 20*Y.^2;
cumtrapz integrates numeric data rather than functional expressions, so in general the underlying function does not need to be known to use cumtrapz on a matrix of data. In cases where the functional expression is known, you can instead use integral, integral2, or integral3.
Use cumtrapz to approximate the double integral
I\left(a,b\right)={\int }_{-2}^{b}{\int }_{-2}^{a}\left(10{x}^{2}+20{y}^{2}\right)\phantom{\rule{0.16666666666666666em}{0ex}}dx\phantom{\rule{0.16666666666666666em}{0ex}}dy.
To perform this double integration, use nested function calls to cumtrapz. The inner call first integrates the rows of data, then the outer call integrates the columns.
I = cumtrapz(y,cumtrapz(x,F,2));
Plot the surface representing the original function as well as the surface representing the cumulative integration. Each point on the surface of the cumulative integration gives an intermediate value of the double integral. The last value in I gives the overall approximation of the double integral, I(end) = 642.4. Mark this point in the plot with a red star.
surf(X,Y,F,'EdgeColor','none')
surf(X,Y,I,'FaceAlpha',0.5,'EdgeColor','none')
plot3(X(end),Y(end),I(end),'r*')
Numeric data, specified as a vector, matrix, or multidimensional array. By default, cumtrapz integrates along the first dimension of Y whose size does not equal 1.
If X is a scalar, then it specifies a uniform spacing between the data points and cumtrapz(X,Y) is equivalent to X*cumtrapz(Y).
cumtrapz(Y,1) works on successive elements in the columns of Y.
cumtrapz(Y,2) works on successive elements in the rows of Y.
If dim is greater than ndims(Y), then cumtrapz returns an array of zeros of the same size as Y.
cumsum | cumprod | trapz
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SetComponent - MapleSim Help
Home : Support : Online Help : MapleSim : MapleSim Application Programming Interface : API Commands : SetComponent
create a component from either Modelica source or a DynamicSystems object
SetComponent(model)
string or DynamicSystems object
forceupdate = truefalse
True means update an existing component without querying. False means query if the update replaces an existing component. The default is false.
nosource = truefalse
True means uses the empty string for the source. This prevents opening the source worksheet when double-clicking on an instance of the created component. This overrides the source option (it is different from assigning the empty string to the source option). The default is false.
String corresponding to the name of an attachment to the current msim file, typically the source file used to create the component. If provided, double-clicking on an instance of the created component opens a worksheet that describes where the source file is attached. The default is an empty string. Using the default embeds a copy of the current worksheet into the generated component such that double-clicking on it opens that copy of the worksheet.
savemodelica = truefalse
True means save the literal modelica code with the component (as the user wrote it); as a result, a subsequent opening of the component in the GUI will display the modelica code in the Modelica Code editor. The default is false.
warning_only = truefalse
True means issue a warning rather than raise an error if there is an error while attaching model. The default is false.
SetComponent attaches a model to the linked MapleSim model. The model appears in the Components palette under the Definitions tab.
The model parameter is either a string consisting of Modelica source code or a DynamicSystems[System] object.
Link to the MapleSim model.
A≔\mathrm{MapleSim}:-\mathrm{LinkModel}\left(\right):
Assign the equations for a simple nonlinear resistor.
\mathrm{eqs}≔{v\left(t\right)=\mathrm{V1}{i\left(t\right)}^{k},v\left(t\right)=\mathrm{vp}\left(t\right)-\mathrm{vn}\left(t\right)}
\textcolor[rgb]{0,0,1}{\mathrm{eqs}}\textcolor[rgb]{0,0,1}{≔}{\textcolor[rgb]{0,0,1}{v}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{t}\right)\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{\mathrm{V1}}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{i}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{t}\right)}^{\textcolor[rgb]{0,0,1}{k}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{v}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{t}\right)\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{\mathrm{vp}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{t}\right)\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{\mathrm{vn}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{t}\right)}
Use MapleToModelica to create Modelica code for the model.
\mathrm{with}\left(\mathrm{MapleSim}:-\mathrm{Tools}\right):
\mathrm{mo}≔\mathrm{MapleToModelica}\left(\mathrm{eqs},\mathrm{comment}="Nonlinear resistor, v = R1*i^k",\mathrm{class_name}="nonlinear_resistor",\mathrm{import}="Modelica.SIunits.*",\mathrm{parameters}=[\mathrm{V1}::\mathrm{Voltage}=1\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&c\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}"voltage at one ampere",k::\mathrm{real}=2\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&c\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}"current exponent"],\mathrm{connections}=["p"::\mathrm{electrical}=[\mathrm{vp}\left(t\right),i\left(t\right)],"n"::\mathrm{electrical}=[\mathrm{vn}\left(t\right),-i\left(t\right)]],\mathrm{unfilled_pins}={"n"},\mathrm{display}\right):
model nonlinear_resistor "Nonlinear resistor, v = R1*i^k"
import Modelica.SIunits.*;
Modelica.Electrical.Analog.Interfaces.PositivePin p annotation (Placement(transformation(
Modelica.Electrical.Analog.Interfaces.NegativePin n annotation (Placement(transformation(
parameter Modelica.SIunits.Voltage V1 = 1 "voltage at one ampere";
parameter Real k = 2 "current exponent";
Modelica.SIunits.ElectricPotential vn;
Modelica.SIunits.ElectricPotential vp;
v = V1 * i ^ k;
end nonlinear_resistor;
Add the component to the linked MapleSim model.
A:-\mathrm{SetComponent}\left(\mathrm{mo}\right)
Create a DynamicSystems object and attach it as a component.
\mathrm{sys}≔\mathrm{DynamicSystems}:-\mathrm{DiffEquation}\left(\frac{k}{s+\mathrm{\omega }},\mathrm{parameters}=[k=1,\mathrm{\omega }=1]\right):
Reassign the systemname field of sys; the new name is used as the model name.
\mathrm{sys}:-\mathrm{systemname}≔"filter":
A:-\mathrm{SetComponent}\left(\mathrm{sys}\right):
MapleSim Embedded Component
MapleSim[Tools][MapleToModelica]
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Finding the surface area and volume of spheres | StudyPug
Hemispheres are half of spheres. If you took a sphere and cut it exactly in half, you'd have two hemispheres! So for example, a sphere is a ball shape. Cutting it in half will give us two hemispheres of a ball. We'll be working with both spheres and hemispheres in this lesson in order to find surface area and volume.
The surface area is the sum of all the surface areas of a shape's sides. In the case of a sphere, the formula for this is:
Surface area = 4
\pi
The volume of a sphere tells us how much space there is inside a sphere. How to find volume of a sphere? There's a simple formula for that and it is:
Volume = 4/3
\pi
How to find the surface area of a hemisphere
Let's try out an example that uses the surface area formula for spheres. We're actually not going to deal with a sphere, but a hemisphere.
Find the following object's surface area:
A hemisphere, as we learned previously, is half of a sphere. Keep this in mind as we work through this problem.
The formula for the surface area of a sphere is: 4
\pi
r^2. To find out the hemisphere's surface area, we can divide this formula by 2 which gives us: 4
\pi
r^2/2. But don't forget that the bottom of the hemisphere now also becomes a surface and will have its own surface area. It's shaped like a circle, and therefore, its surface area is equalled to
\pi
r^2 as we have learned in past lessons. Therefore the formula for our hemisphere is:
\pi
r^2/2 +
\pi
We'll first need to find the radius. We've learnt from previous lessons that the diameter = 2r, so we can make use of this to help us find the radius.
Now we can plug the radius into the formula to solve for the hemisphere's surface area.
Surface area of a hemisphere = 4
\pi
\pi
SA = [4
\pi
(10^2)]2 +
\pi
(10)^2
SA = 942.48 ft^2
The area of hemisphere equals to 942.48ft^2. Remember that finding surface area will give you answers in squares. Also, do not forget the unit!
How to find the volume of a hemisphere
Let's move on to finding the volume of a hemisphere.
Find the following object's volume:
For the volume, all you'll need to do is to take the formula for a sphere's volume and then divide it by 2. The sphere's formula is Volume = 4/3
\pi
r^3, and for a hemisphere, we'll use Volume = (4/3
\pi
r^3)/2.
In terms of calculation we've got:
Volume = (4/3
\pi
r^3)/2
V = [4/3
\pi
(10^3)]/2
V = 2094.4ft^3
We get that the volume of hemisphere equals 2094.4ft^3. In this case, since we're finding the volume, the answer will be given in cubic form. Once again, do not forget the unit, which is in feet for this problem.
To see the relationship of the volume and the radius, and how they affect one another, check out this helpful click-and-drag diagram!
A basketball is a sphere. Soap bubbles are usually spheres. A vitamin capture is a combination of cylinder and sphere. In this lesson, we will learn how to calculate the surface area and volume of spheres and composite solids.
Basic Concepts: Surface area and volume of cylinders, Circles and circumference , Arcs of a circle, Areas and sectors of circles
Related Concepts: Surface area of cylinders, Introduction to volume, Word problems relating volume of prisms and cylinders, Conversions between metric and imperial systems
Find the surface area and the volume of the following shapes:
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Baud - Simple English Wikipedia, the free encyclopedia
In telecommunications, Baud is the unit to measure the symbol rate. If one symbol can be transmitted per second, this is equal to one Baud. The unit is named after Jean-Maurice-Émile Baudot a French telecommunications engineer who invented the Baudot code. The Baud rate is different from the gross bit rate, measured in bits/second.
Various electric signal schemes invented in the 20th century such as Phase-shift keying can make bit rates much higher than signal rates. In another example, gigabit ethernet has a symbol rate of 125MBd. Gigabit ethernet uses pulse-amplitude modulation and can transmit two bits of payload data per symbol. Gigabit ethernet uses four balanced pairs for transmission.
It can transmit
{\displaystyle {\rm {125\;MBd\cdot 2\;{\tfrac {bit}{symbol}}\cdot 4=1000\;{\tfrac {Mbit}{s}}}}}
Retrieved from "https://simple.wikipedia.org/w/index.php?title=Baud&oldid=5906155"
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The t- and z-test methods developed in the 20th century were used for statistical analysis until 1918, when Ronald Fisher created the analysis of variance method. ANOVA is also called the Fisher analysis of variance, and it is the extension of the t- and z-tests. The term became well-known in 1925, after appearing in Fisher's book, "Statistical Methods for Research Workers." It was employed in experimental psychology and later expanded to subjects that were more complex.
Analysis of variance, or ANOVA, is a statistical method that separates observed variance data into different components to use for additional tests.
A one-way ANOVA is used for three or more groups of data, to gain information about the relationship between the dependent and independent variables.
If no true variance exists between the groups, the ANOVA's F-ratio should equal close to 1.
What Is the Analysis of Variance (ANOVA)?
\begin{aligned} &\text{F} = \frac{ \text{MST} }{ \text{MSE} } \\ &\textbf{where:} \\ &\text{F} = \text{ANOVA coefficient} \\ &\text{MST} = \text{Mean sum of squares due to treatment} \\ &\text{MSE} = \text{Mean sum of squares due to error} \\ \end{aligned}
F=MSEMSTwhere:F=ANOVA coefficientMST=Mean sum of squares due to treatmentMSE=Mean sum of squares due to error
The ANOVA test is the initial step in analyzing factors that affect a given data set. Once the test is finished, an analyst performs additional testing on the methodical factors that measurably contribute to the data set's inconsistency. The analyst utilizes the ANOVA test results in an f-test to generate additional data that aligns with the proposed regression models.
The ANOVA test allows a comparison of more than two groups at the same time to determine whether a relationship exists between them. The result of the ANOVA formula, the F statistic (also called the F-ratio), allows for the analysis of multiple groups of data to determine the variability between samples and within samples.
If no real difference exists between the tested groups, which is called the null hypothesis, the result of the ANOVA's F-ratio statistic will be close to 1. The distribution of all possible values of the F statistic is the F-distribution. This is actually a group of distribution functions, with two characteristic numbers, called the numerator degrees of freedom and the denominator degrees of freedom.
A researcher might, for example, test students from multiple colleges to see if students from one of the colleges consistently outperform students from the other colleges. In a business application, an R&D researcher might test two different processes of creating a product to see if one process is better than the other in terms of cost efficiency.
The type of ANOVA test used depends on a number of factors. It is applied when data needs to be experimental. Analysis of variance is employed if there is no access to statistical software resulting in computing ANOVA by hand. It is simple to use and best suited for small samples. With many experimental designs, the sample sizes have to be the same for the various factor level combinations.
ANOVA is helpful for testing three or more variables. It is similar to multiple two-sample t-tests. However, it results in fewer type I errors and is appropriate for a range of issues. ANOVA groups differences by comparing the means of each group and includes spreading out the variance into diverse sources. It is employed with subjects, test groups, between groups and within groups.
There are two main types of ANOVA: one-way (or unidirectional) and two-way. There also variations of ANOVA. For example, MANOVA (multivariate ANOVA) differs from ANOVA as the former tests for multiple dependent variables simultaneously while the latter assesses only one dependent variable at a time. One-way or two-way refers to the number of independent variables in your analysis of variance test. A one-way ANOVA evaluates the impact of a sole factor on a sole response variable. It determines whether all the samples are the same. The one-way ANOVA is used to determine whether there are any statistically significant differences between the means of three or more independent (unrelated) groups.
A two-way ANOVA is an extension of the one-way ANOVA. With a one-way, you have one independent variable affecting a dependent variable. With a two-way ANOVA, there are two independents. For example, a two-way ANOVA allows a company to compare worker productivity based on two independent variables, such as salary and skill set. It is utilized to observe the interaction between the two factors and tests the effect of two factors at the same time.
Genetic Epidemiology, Translational Neurogenomics, Psychiatric Genetics and Statistical Genetics-QIMR Berghofer Medical Research Institute. "The Correlation Between Relatives on the Supposition of Mendelian Inheritance."
Encyclopaedia Britannica. "Sir Ronald Aylmer Fisher."
Ronald Fisher. "Statistical Methods for Research Workers." Springer-Verlag New York, 1992.
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In the metal oxide M2O3, ratio between the masses of metal and oxygen is 13:12 Calculate the atomic mass - Chemistry - Some Basic Concepts of Chemistry - 10639509 | Meritnation.com
In the metal oxide M2O3, ratio between the masses of metal and oxygen is 13:12. Calculate the atomic mass of the metal.
\mathrm{Equivalent} \mathrm{mass}=\frac{\mathrm{Atomic} \mathrm{mass}}{\mathrm{Valency}}\phantom{\rule{0ex}{0ex}}\mathrm{Or}, \mathrm{Atomic} \mathrm{mass}=\mathrm{Equivalent} \mathrm{mass}×\mathrm{Valency}\phantom{\rule{0ex}{0ex}}\mathrm{In} {\mathrm{M}}_{2}{\mathrm{O}}_{3}, \mathrm{total} \mathrm{charge} \mathrm{or} \mathrm{valency} \mathrm{on} \mathrm{M} \mathrm{is} +3\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\mathrm{and}, \mathrm{Equivalent} \mathrm{mass} \mathrm{of} \mathrm{metal}=\frac{\mathrm{mass} \mathrm{of} \mathrm{metal}}{\mathrm{mass} \mathrm{of} \mathrm{oxygen}}×8=\frac{13}{12}×8=8.66\phantom{\rule{0ex}{0ex}}\mathrm{Therefore}, \mathrm{Atomic} \mathrm{mass} \mathrm{of} \mathrm{M}=8.66×3\approx 26 \phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}
Amritesh Mohan Singh Jeena answered this
u can easily solve it
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Condition Monitoring of Molybdenum Disulphide Coated Thrust Ball Bearings Using Time-Frequency Signal Analysis | J. Tribol. | ASME Digital Collection
Ali Kahirdeh,
Kahirdeh, A., and Khonsari, M. M. (October 8, 2010). "Condition Monitoring of Molybdenum Disulphide Coated Thrust Ball Bearings Using Time-Frequency Signal Analysis." ASME. J. Tribol. October 2010; 132(4): 041606. https://doi.org/10.1115/1.4002379
A method for detection of wear in thrust ball bearings coated with molybdenum disulphide
(MoS2)
is presented. It employs an energy feature obtained from time-frequency representation of the vibration signal. Extensive experimental studies are conducted to verify the efficacy of the proposed method for fault diagnosis of
MoS2
coating. These experiments are conducted under both oscillatory and unidirectional motions. The results of vibrations are corroborated with the friction coefficient from the onset of the motion until failure develops. Through monitoring of the energy in time-frequency domain as well as the coefficient of friction, three stages of coating life are identified. They are healthy period, developing damage, and failure. It is shown that the energy feature can detect whenever wear and damage appear and solid lubricant loses its lubrication capabilities.
ball bearings, condition monitoring, friction, lubricants, molybdenum compounds, time-frequency analysis, wear, molybdenum disulphide, MoS2, smoothed-pseudo-Wigner–Ville distribution, coefficient of friction, ball bearing fault diagnosis
Ball bearings, Coating processes, Coatings, Damage, Failure, Friction, Signals, Thrust, Vibration, Wear, Stress, Fault diagnosis, Molybdenum, Lubricants, Condition monitoring
Effect of Humidity on Friction and Life of Unbonded Molybdenum Disulphide Films
Molybdenum Disulphide Deposits—Their Formation and Characteristics on Steel Balls
Jet and Space Age Lubrication
Report No. SP-5059.
An Investigation of Molybdenum Disulfide Bonded Solid Lubricant Coatings in Fretting Conditions
Different Tribological Behavior of MoS2 Coatings Under Fretting and Pin-on-Disk Conditions
Spalvins
A Review of Recent Advances in Solid Film Lubrication
Wear Behavior of Triode-Sputtered MoS2 Coatings in Dry Sliding Contact With Steel and Ceramics
Role of Transfer Films in Wear of MoS2 Coatings
Wear Behavior of Pb-Mo-S Solid Lubricating Coatings
Diamond-Like-Carbon and Molybdenum Disulfide Nanotribology Studies Using Atomic Force Measurements
Scale-Dependent Nanomechanical Behavior and Anisotropic Friction of Nanotextured Silicon Surfaces
Friction of Solid Films on Steel at High Sliding Velocities
Advanced Digital Vibration Signal Processing for Condition Monitoring
Proceedings of COMADEM2000
Alugongo
Fault Diagnosis of Rotating Machinery Based on SVD, FCM and RST
Early Detection of Gear Failure by Vibration Analysis. I. Calculation of the Time-Frequency Distribution
Online Tracking of Bearing Wear Using Wavelet Packet Decomposition and Probabilistic Modeling: A Method for Bearing Prognostics
A Time-Frequency Distribution for Analysis of Signals With Transient Components and Its Application to Vibration Analysis
A Finite Element Analysis of the Frictional Forces Between a Cylindrical Bearing Element and MoS2 Coated and Uncoated Surfaces
Frictional Analysis of MoS2 Coated Ball Bearings: A Three-Dimensional Finite Element Analysis
Dynamic Friction Measurements of MoS2 Coated Ball Bearing Surfaces
A Chemical Diagnostic for Failure Prediction of Solid Lubricated Systems
Lamb Waves Monitoring of the Thermomechanical Aging Effects in Ceramic Coatings
Time Frequency-Distributions—A Review
Joint Time-Frequency Analysis: Methods and Applications
On the Quantum Correction for Thermodynamic Equilibrium
Cables Transm.
The Wigner Distribution—A Tool for Time-Frequency Signal Analysis.3. Relations With Other Time-Frequency Signal Transformations
The Wigner Distribution—A Tool for Time-Frequency Signal Analysis.1. Continuous-Time Signals
The Wigner Distribution—A Tool for Time-Frequency Signal Analysis.2. Discrete-Time Signals
Geometry of Affine Time-Frequency Distributions
A General Class of Estimators for the Wigner-Ville Spectrum of Nonstationary Processes
Systems Analysis and Optimization of Systems
Some features of time-frequency representations of multi component signals
,” Centre National de la Recherché Scientifique, France, and Rice University, Houston, TX.
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Trajectory Control of Miniature Helicopters Using a Unified Nonlinear Optimal Control Technique | J. Dyn. Sys., Meas., Control. | ASME Digital Collection
Ming Xin,
, Starkville, MS 39759
e-mail: xin@ae.msstate.edu
Yunjun Xu,
Xin, M., Xu, Y., and Hopkins, R. (September 6, 2011). "Trajectory Control of Miniature Helicopters Using a Unified Nonlinear Optimal Control Technique." ASME. J. Dyn. Sys., Meas., Control. November 2011; 133(6): 061001. https://doi.org/10.1115/1.4004060
It is always a challenge to design a real-time optimal full flight envelope controller for a miniature helicopter due to the nonlinear, underactuated, uncertain, and highly coupled nature of its dynamics. This paper integrates the control of translational, rotational, and flapping motions of a simulated miniature aerobatic helicopter in one unified optimal control framework. In particular, a recently developed real-time nonlinear optimal control method, called the
θ-D
technique, is employed to solve the resultant challenging problem considering the full nonlinear dynamics without gain scheduling techniques and timescale separations. The uniqueness of the
θ-D
method is its ability to obtain an approximate analytical solution to the Hamilton–Jacobi–Bellman equation, which leads to a closed-form suboptimal control law. As a result, it can provide a great advantage in real-time implementation without a high computational load. Two complex trajectory tracking scenarios are used to evaluate the control capabilities of the proposed method in full flight envelope. Realistic uncertainties in modeling parameters and the wind gust condition are included in the simulation for the purpose of demonstrating the robustness of the proposed control law.
aircraft control, helicopters, mobile robots, motion control, nonlinear control systems, optimal control, position control, vehicle dynamics
Design, Dynamics (Mechanics), Optimal control, Trajectories (Physics), Helicopters, Uncertainty, Control equipment, Wind, Rotors, Flight
, November 2004), pp.
Association for Unmanned Vehicle Systems International’s Aerial Robotic Competition
,” http://avdil.gtri.gatech.edu/AUVS/IARCLaunchPoint.htmlhttp://avdil.gtri.gatech.edu/AUVS/IARCLaunchPoint.html, Document accessed: March 21, 2008.
Model-Based Optimal Attitude and Positioning Control of Small-Scale Unmanned Helicopter
Perhinschi
A Simulation Model of an Autonomous Helicopter
Proceedings of the RPV/UAV Systems Thirteenth Bristol International Conference and Exhibition
Robust Control of an Autonomous Reduced Scale Helicopter in Presence of Wind Gusts
Design and Flight Ttesting of an Controller for a Robotic Helicopter
Robust Hover Control for a Model Helicopter
Attitude Control of a Helicopter Model by Nonlinear Sampled-Data Control
Robust Helicopter Position Control at Hover
H-Infinity Helicopter Flight Control Law Design With and Without Rotor State Feedback
Control System Design for Rotocraft-Based Unmanned Aerial Vehicles Using Time-Domain System Identification
Adaptive Compensated Dynamic Inversion Control for a Helicopter With Approximate Mathematical Model
International Conference on Computational Intelligence for Modeling Control and Automation, and International Conference on Intelligent Agents
Web Technologies and Internet Commerce, IEEE
Robust Model Reference Control Using a Sliding Mode Controller/Observer Scheme With Application to a Helicopter Problem
1996 IEEE Workshop on Variable Structure Systems
Methods and Applications of Singular Perturbations Boundary Layers and Multiple Timescale Dynamics
Springer Science + Business Media, Inc.
Two-Timescale Inverse Simulation of a Helicopter Model
Flight Control Design Using Robust Dynamic Inversion and Time-Scale Separation
Robust Nonlinear Control for a Miniature Helicopter for Aerobatic Maneuvers
Proceedings 32nd Rotorcraft Forum
Maasctricht, The Netherlands
Two-Timescale-Integration Method for Inverse Simulation
Position Tracking Control for a Simulated Miniature Hhelicopter
State-Dependent Riccati Equation Control for Small Autonomous Helicopters
A New Method for Suboptimal Control of a Class of Nonlinear Ssystems
Nonlinear Missile Autopilot Design With Technique
Dynamic Model for a Miniature Aerobatic Helicopter
,” MIT-LIDS Report, Report No. LIDS-P–2580.
Federal Aviation Administration, 1999, “
Criteria for Approval of Category III Weather Minima for Takeoff, Landing, and Rollout: Appendix 4—Wind Model for Approach and Landing Simulation
,” AC120–28D.
Hemisphere Publishing, Corp.
The Capabilities and Art of State-Dependent Riccati Equation-Based Design
Integrated Guidance and Control of Missiles With Method
An LQ Approach to Active Control of Vibrations in Helicopters
Aerodynamics of a Yawed Blade in Reverse Flow
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Detection SNR threshold for signal in white Gaussian noise - MATLAB npwgnthresh
{T}_{dB}=20{\mathrm{log}}_{10}{T}_{lin}
\frac{\lambda }{\sigma }=\sqrt{2N}\text{\hspace{0.17em}}{\text{erfc}}^{-1}\left(2{P}_{fa}\right)
10{\mathrm{log}}_{10}\left(\frac{{\lambda }^{2}}{{\sigma }^{2}}\right)=10{\mathrm{log}}_{10}\left(2N{\left({\text{erfc}}^{-1}\left(2{P}_{fa}\right)\right)}^{2}\right)
\frac{\lambda }{\sigma }=\sqrt{N}\text{\hspace{0.17em}}{\text{erfc}}^{-1}\left(2{P}_{fa}\right)
10{\mathrm{log}}_{10}\left(\frac{{\lambda }^{2}}{{\sigma }^{2}}\right)=10{\mathrm{log}}_{10}\left(N{\left({\text{erfc}}^{-1}\left(2{P}_{fa}\right)\right)}^{2}\right)
\frac{\lambda }{\sigma }=\sqrt{{P}^{-1}\left(N,1-{P}_{fa}\right)}
10{\mathrm{log}}_{10}\left(\frac{{\lambda }^{2}}{{\sigma }^{2}}\right)=10{\mathrm{log}}_{10}{P}^{-1}\left(N,1-{P}_{fa}\right)
{P}^{-1}\left(x,y\right)
is the inverse of the lower incomplete gamma function, Pfa is the probability of false alarm, and N is the number of pulses.
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A Strategy for Rapid Thermal Cycling of Molds in Thermoplastic Processing | J. Manuf. Sci. Eng. | ASME Digital Collection
A Strategy for Rapid Thermal Cycling of Molds in Thermoplastic Processing
School of Polymer, Textile & Fiber Engineering,
e-mail: dong.yao@ptfe.gatech.edu
Pratapkumar Nagarajan,
Pratapkumar Nagarajan
Department of Industrial, Welding and Systems Engineering,
Yao, D., Nagarajan, P., Li, L., and Yi, A. Y. (March 23, 2006). "A Strategy for Rapid Thermal Cycling of Molds in Thermoplastic Processing." ASME. J. Manuf. Sci. Eng. November 2006; 128(4): 837–843. https://doi.org/10.1115/1.2335855
Thermal cycling of molds is frequently desired in thermoplastic processing. Thermal cycling of the entire mold with a large mass, however, requires an exceedingly long cycle time. A processing strategy for mold rapid heating and cooling, involving a thin-shell mold and two thermal stations (one hot and one cold), was investigated. Because of its low thermal mass, the shell mold can be rapidly heated and cooled through heat conduction by selectively contacting with the two stations. Numerical simulations were performed to study the effect of different design parameters, including thermal contact resistance, shell material, and shell thickness, on the thermal response at the mold surface. Experimental studies showed aluminum shell molds with a thickness of
1.4mm
can be rapidly heated from room temperature to
200°C
in about
3s
using a hot station at
250°C
. The method was used for thermal cycling of embossing tools. Surface microfeatures can be rapidly transferred from thin metallic stamps to polymer substrates with cycle times less than
10s
thermoforming, moulding equipment, rapid thermal processing, cooling, thin wall structures, shell casting, thermal resistance, thermoplastic processing, rapid heating, embossing, welding, contact heating
Aluminum, Embossing, Heating, Shells, Heating and cooling, Temperature
Injection Molding High Aspect Ratio Microfeatures
Wimberger-Friedl
Injection Molding of Sub-μm Grating Optical Elements
Injection Molding Using High Aspect Ratio Microstructures Mold Inserts Produced by LIGA Techniques
, 2001, Ph.D. thesis, University of Massachusetts, Amherst.
Development of Rapid Heating and Cooling Systems for Injection Molding Applications
Construction of Fast-Response Heating Elements for Injection Molding Applications
Low Thermal Inertia Molding
Carlslaw
Oxford Science Publications, Clarendon Press
Calculating Cooling Times for Polymer Injection Molding
0954-4054, J. Eng. Manuf.,
Review: Polymer Microfluidic Devices
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Check valve with pilot pressure control in an isothermal liquid system - MATLAB - MathWorks Italia
Pilot-Operated Check Valve (IL)
Valve Pressure Control with a Pilot Port
Pilot configuration
Opening pilot pressure specification
Check valve with pilot pressure control in an isothermal liquid system
The Pilot-Operated Check Valve (IL) block models a flow-control valve with variable flow directionality based on the pilot-line pressure. Flow is normally restricted to travel from port A to port B in either a connected or disconnected spool-poppet configuration, according to the Pilot configuration parameter.
Pilot-Operated Check Valve Schematic
The control pressure, pcontrol is:
{p}_{control}={p}_{pilot}{k}_{p}+\left({p}_{A}-{p}_{B}\right),
ppilot is the control pilot pressure differential.
kp is the Pilot ratio, the ratio of the area at port X to the area at port A:
{k}_{p}=\frac{{A}_{X}}{{A}_{A}}.
pA – pB is the pressure differential over the valve.
When the control pressure exceeds the Cracking pressure differential, the poppet moves to allow flow from port B to port A.
There is no mass flow between port X and ports A and B.
The pilot pressure differential for valve control can be configured in two ways:
When the Opening pilot pressure specification parameter is set to Pressure at port X relative to port A, the pilot pressure is the pressure differential between port X and port A.
When Opening pilot pressure specification is set to Pressure at port X relative to atmospheric pressure, the pilot pressure is the pressure difference between port X and atmospheric pressure.
When Pilot configuration is set to Disconnected pilot spool and poppet, the relative pressure at port X must be positive. If the measured pilot pressure is negative, the control pressure is only based on the pressure differential between ports A and B. In the Rigidly connected pilot spool and poppet setting, the pilot pressure is the measured pressure differential according to the opening specification.
{\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 }
Δp is the valve pressure difference, pA – pB.
\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 also impacted by the valve opening dynamics.
{A}_{valve}=\stackrel{^}{p}\left({A}_{\mathrm{max}}-{A}_{leak}\right)+{A}_{leak},
\stackrel{^}{p}
\stackrel{^}{p}=\frac{{p}_{control}-{p}_{cracking}}{{p}_{\mathrm{max}}-{p}_{cracking}}.
{\stackrel{^}{p}}_{smoothed}=\frac{1}{2}+\frac{1}{2}\sqrt{{\stackrel{^}{p}}_{}^{2}+{\left(\frac{s}{4}\right)}^{2}}-\frac{1}{2}\sqrt{{\left(\stackrel{^}{p}-1\right)}^{2}+{\left(\frac{s}{4}\right)}^{2}}.
{\stackrel{˙}{p}}_{dyn}=\frac{{p}_{control}-{p}_{dyn}}{\tau }.
Liquid entry point to the valve. When the control pressure exceeds the cracking pressure, liquid is able to exit from this port.
Liquid exit point from the valve. When the control pressure exceeds the cracking pressure, liquid is able to enter the valve from this port.
Pressure port that contributes to the flow control through the valve.
Pilot configuration — Valve geometry
Rigidly connected pilot spool and poppet (default) | Disconnected pilot spool and poppet
Valve geometry. The valve can either have an opening mechanism that is connected to the valve poppet, in the case of the Rigidly connected pilot spool and poppet setting, or an opening mechanism that is aligned with, but moves freely away from, the valve poppet, in the case of the Disconnected pilot spool and poppet setting. The configuration choice determines the pilot pressure calculation.
Opening pilot pressure specification — Reference pressure differential for valve control
Pressure at port X relative to port A (default) | Pressure at port X relative to atmospheric pressure
Reference pressure differential used for valve control. This differential defines the pilot pressure differential, which is added to the pressure differential between ports A and B and compared against the valve threshold Cracking pressure differential.
Cracking pressure differential — Pressure threshold
Set pressure for the valve operation.
Maximum opening pressure differential — Maximum pressure differential in opened valve
Maximum pressure differential in an opened valve. This value provides an upper limit to simulation pressures so that results remain physical.
Pilot ratio — Pilot port area ratio
1 (default) | positive scalar in the range of (0,1]
Ratio of port area X to port area A.
Maximum opening area — Maximum valve area
Maximum valve area. This value is used to determine the normalized valve pressure and the valve opening area during operation.
Opening dynamics — Whether to introduce flow lag due to valve opening
Check Valve (IL) | Check Valve (TL) | Pressure Relief Valve (IL) | Counterbalance Valve (IL) | Pressure Compensator Valve (IL)
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Silber, Israel; Fridlind, Ann M.; Verlinde, Johannes; Ackerman, Andrew S.; Cesana, Grégory V.; Knopf, Daniel A.
Supercooled clouds substantially impact polar surface energy budgets, but large-scale models often underestimate their occurrence, which motivates accurately establishing metrics of basic processes. An analysis of long-term measurements at Utqiaġvik, Alaska, and McMurdo Station, Antarctica, combines lidar-validated use of soundings to identify supercooled cloud layers and colocated ground-based profiling radar measurements to quantify cloud base precipitation. We find that more than 85 % (75 %) of sampled supercooled layers are precipitating over the Arctic (Antarctic) site, with more than 75 % (50 %) precipitating continuously to the surface. Such high frequencies can be reconciled with substantially lesser spaceborne estimates by considering differences in radar hydrometeor detection sensitivity. While ice precipitation into supercooled clouds from aloft is common, we also find that the great majority of supercooled cloud layers without ice falling into them are themselves continuously generating precipitation. Such sustained primary ice formation is consistent with continuous activation of immersion-mode ice-nucleating particles (INPs), suggesting that supercooled cloud formation is a principal gateway to ice formation at temperatures greater than inline-formula
M1inlinescrollmathml\sim -normal 38
32pt10ptsvg-formulamathimg52fcdae65a2e7a1244d5b38bb82db9dd acp-21-3949-2021-ie00001.svg32pt10ptacp-21-3949-2021-ie00001.png inline-formula∘C over polar regions. The prevalence of weak precipitation fluxes is also consistent with supercooled cloud longevity and with well-observed and widely simulated case studies. An analysis of colocated microwave radiometer retrievals suggests that weak precipitation fluxes can be nonetheless consequential to moisture budgets for supercooled clouds owing to small liquid water paths. The results here also demonstrate that the observed abundance of mixed-phase clouds can vary substantially with instrument sensitivity and methodology. Finally, we suggest that these ground-based precipitation rate statistics offer valuable guidance for improving the representation of polar cloud processes in large-scale models.
Silber, Israel / Fridlind, Ann M. / Verlinde, Johannes / et al: The prevalence of precipitation from polar supercooled clouds. 2021. Copernicus Publications.
Rechteinhaber: Israel Silber et al.
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Decoupling of a Douglas fir canopy: a look into the subcanopy with continuous...
Schilperoort, Bart; Coenders-Gerrits, Miriam; Jiménez Rodríguez, César; van der Tol, Christiaan; van de Wiel, Bas; Savenije, Hubert
Complex ecosystems such as forests make accurately measuring atmospheric energy and matter fluxes difficult. One of the issues that can arise is that parts of the canopy and overlying atmosphere can be turbulently decoupled from each other, meaning that the vertical exchange of energy and matter is reduced or hampered. This complicates flux measurements performed above the canopy. Wind above the canopy will induce vertical exchange. However, stable thermal stratification, when lower parts of the canopy are colder, will hamper vertical exchange. To study the effect of thermal stratification on decoupling, we analyze high-resolution (0.3 m) vertical temperature profiles measured in a Douglas fir stand in the Netherlands using distributed temperature sensing (DTS).
The forest has an open understory (0–20 m) and a dense overstory (20–34 m). The understory was often colder than the atmosphere above (80 % of the time during the night, inline-formula>99 % during the day). Based on the aerodynamic Richardson number the canopy was regularly decoupled from the atmosphere (50 % of the time at night). In particular, decoupling could occur when both inline-formula
M2inlinescrollmathml{u}_{*}<normal 0.4
40pt12ptsvg-formulamathimg2e11018c345b6abbdd8f71d722d232e2 bg-17-6423-2020-ie00001.svg40pt12ptbg-17-6423-2020-ie00001.png m sinline-formula−1 and the canopy was able to cool down through radiative cooling. With these conditions the understory could become strongly stably stratified at night. At higher values of the friction velocity the canopy was always well mixed. While the understory was nearly always stably stratified, convection just above the forest floor was common. However, this convection was limited in its vertical extent, not rising higher than 5 m at night and 15 m during the day. This points towards the understory layer acting as a kind of mechanical “blocking layer” between the forest floor and overstory.
With the DTS temperature profiles we were able to study decoupling and stratification of the canopy in more detail and study processes which otherwise might be missed. These types of measurements can aid in describing the canopy–atmosphere interaction at forest sites and help detect and understand the general drivers of decoupling in forests.
Schilperoort, Bart / Coenders-Gerrits, Miriam / Jiménez Rodríguez, César / et al: Decoupling of a Douglas fir canopy: a look into the subcanopy with continuous vertical temperature profiles. 2020. Copernicus Publications.
Rechteinhaber: Bart Schilperoort et al.
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The near field refractor | EMS Press
Department of Mathematics, Temple University, Philadelphia, PA 19122, United States
Department of Mathematics and Statistics, Wright State University, Dayton, OH 45435, United States
We present an abstract method in the setting of compact metric spaces which is applied to solve a number of problems in geometric optics. In particular, we solve the one source near field refraction problem. That is, we construct surfaces separating two homogeneous media with different refractive indices that refract radiation emanating from the origin into a target domain contained in an
n−1
dimensional hypersurface. The input and output energy are prescribed. This implies the existence of lenses focusing radiation in a prescribed manner.
Cristian E. Gutiérrez, Qingbo Huang, The near field refractor. Ann. Inst. H. Poincaré Anal. Non Linéaire 31 (2014), no. 4, pp. 655–684
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An energy constrained method for the existence of layered type solutions of NLS equations | EMS Press
Dipartimento di Ingegneria Industriale e Scienze Matematiche, Università Politecnica delle Marche, Via Brecce Bianche, I 60131 Ancona, Italy
Piero Montecchiari
We study the existence of positive solutions on
\mathbb{R}^{N + 1}
to semilinear elliptic equation
−\mathrm{\Delta }u + u = f(u)
N⩾1
and f is modeled on the power case
f(u) = |u|^{p−1}u
. Denoting with c the mountain pass level of
V(u) = \frac{1}{2}\|u\|_{H^{1}(\mathbb{R}^{N})}^{2}−\int _{\mathbb{R}^{N}}F(u)\:dx
u \in H^{1}(\mathbb{R}^{N})
F(s) = \int _{0}^{s}f(t)\:dt
), we show, via a new energy constrained variational argument, that for any
b \in [0,c)
there exists a positive bounded solution
v_{b} \in C^{2}(\mathbb{R}^{N + 1})
E_{v_{b}}(y) = \frac{1}{2}\|\partial _{y}v_{b}( \cdot ,y)\|_{L^{2}(\mathbb{R}^{N})}^{2}−V(v_{b}( \cdot ,y)) = −b
v(x,y)\rightarrow 0
|x|\rightarrow + \infty
y \in \mathbb{R}
. We also characterize the monotonicity, symmetry and periodicity properties of
v_{b}
Francesca Alessio, Piero Montecchiari, An energy constrained method for the existence of layered type solutions of NLS equations. Ann. Inst. H. Poincaré Anal. Non Linéaire 31 (2014), no. 4, pp. 725–749
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Modifying a Tilting Disk Mechanical Heart Valve Design to Improve Closing Dynamics | J. Biomech Eng. | ASME Digital Collection
e-mail: lhh113@psu.edu
e-mail: sxd2@only.arl.psu.edu
Herbertson, L. H., Deutsch, S., and Manning, K. B. (September 17, 2008). "Modifying a Tilting Disk Mechanical Heart Valve Design to Improve Closing Dynamics." ASME. J Biomech Eng. October 2008; 130(5): 054503. https://doi.org/10.1115/1.2978987
The closing behavior of mechanical heart valves is dependent on the design of the valve and its housing, the valve composition, and the environment in which the valve is placed. One innovative approach for improving the closure dynamics of tilting disk valves is introduced here. We transformed a normal Delrin occluder into one containing a ”dynamic liquid core” to resist acceleration and reduce the moment of inertia, closing velocity, and impact forces of the valve during closure. The modified occluder was studied in the mitral position of a simulation chamber under the physiologic and elevated closing conditions of
2500 mm Hg/s
4500 mm Hg/s
, respectively. Cavitation energy, detected as high-frequency pressure transients with a hydrophone, was the measure used to compare the modified valve with its unaltered counterpart. The cavitation potential of tilting disk valves is indicative of the extent of blood damage occurring during valve closure. Initial findings suggest that changes to the structure or physical properties of well established mechanical valves, such as the one described here, can reduce closure induced hemolysis by minimizing cavitation. Compared with a normal valve, the cavitation intensity associated with our modified valve was reduced by more than 66% at the higher load. Furthermore, the modified valve took longer to completely close than did the standard tilting disk valve, indicating a dampened impact and rebound of the occluder with its housing.
cardiology, cavitation, haemodynamics, prosthetics, tilting disk, modified mechanical heart valve, fluid mechanics, liquid core, cavitation, closing dynamics
Cavitation, Design, Disks, Dynamics (Mechanics), Heart valve prostheses, Valves, Pressure, Acoustics
, 2004, “Mechanical Heart Valve Cavitation,” Expert Review of Medical Devices, 1(1), pp. 95–104. 1743-4440
Indication of Cavitation in Mechanical Heart Valve Patients
An in vitro Investigation of Prosthetic Heart Valve Cavitation in Blood
Negative Pressure Transients With Mechanical Heart Valve Closure: Correlation Between In Vitro and In Vivo Results
Occluder Closing Behavior: A Key Factor in Mechanical Heart Valve Cavitation
Pressure Field in the Vicinity of Mechanical Valve Occluders at the Instant of Valve Closure: Correlation With Cavitation Initiation
Transient Pressure Signals as a Means in Measuring Mechanical Heart Valve Cavitation Potential
An Experimental Investigation Into the Fluid Mechanics at the Instant of Valve Closure
,” M.S. thesis, Pennsylvania State University, University Park.
Pulsatile Prosthetic Valve Flows
ASAIO Trans.
Mechanisms of Cavitation and the Formation of Stable Bubbles on the Bjork-Shiley Monostrut Prosthetic Heart Valve
Delrin as an Occluder Material
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Home : Support : Online Help : Mathematics : Mathematical Functions : MathematicalFunctions Package : Sequences : Overview
The Sequences package for handling symbolic sequences
List of The Sequences Package Commands
Sequences is a package of commands for handling symbolic sequences, that is, sequences where the number of operands is unknown. Symbolic sequences enter in various formulations in mathematics. Among the most typical cases of symbolic sequences, there are:
n,\mathrm{...},m
\underset{}{\underset{⏟}{a,\mathrm{...},a}}
{a}_{n},\mathrm{...},{a}_{m}
or likewise a sequence of functions
f\left(n\right),\mathrm{...},f\left(m\right)
In all these cases, none of n, m, or p are known: they are just symbols, or algebraic expressions, representing integer values. These most typical cases of symbolic sequences have been implemented in Maple using the $ operator. Cases 1., 2. and 3. above are respectively entered as $(n .. m), a $ p and a[i] $ (i = n .. m). To that implementation, the Sequences package adds two things: the typesetting of these three kinds of symbolic sequences as shown in the enumeration above, and commands to perform algebraic operations on them.
Add adds the elements of a sequence or adds to the elements of a sequence.
Differentiate differentiates the elements of a sequence.
Map maps an operation to the elements of a sequence.
Multiply multiplies the elements of a sequence or multiplies to the elements of a sequence.
Nops counts the elements of a sequence.
The MathematicalFunctions[Sequences] package was introduced in Maple 2016.
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Factoring trinomials in the form of ax^2+bx+c | StudyPug
x^2 + bx + c
ax^2 + bx + c
ax^2 + bx + c
x^2 + bx + c
ax^2 + bx + c
ax^2 + bx + c
Trinomials are algebraic expressions that has three terms in it. Quadratic trinomials are in the form of a
{x^2}
+ bx + c, and the a, b, and c all stands for a number.
In order to factor trinomials, you'll have to work to find two numbers that will multiply to equal the "c" from the quadratic form above, and also add up to equal "b". These are the steps for the easier questions where the first "a" is equalled to 1. For harder problems, "a" will be a number that is not one. You'll have to first multiply "a" and "c", and then find factors of the product "a*c" that also add up to "b".
We'll explore this with example questions to demonstrate how to factor trinomials.
{b^2} - b - 20
In this example, we are going to use the "cross-multiply, then check" method to factor the trinomial. This is one of the ways to factor trinomials.
Use cross multiply and check to factor trinomial
Factor the first term
{b^2}
, put the results (b and b) in the first box.
Factor the term b^2 and put them into the first box
Factor the last term -20. There are a few combinations (1x20, 2x10, 4x5) that can give us 20, so which one is it? We'll cross-multiply those combinations to the factors of the first term. See which combination will produce an answer that matches the middle term (in this question, the middle term is –b).
Combination that matches the middle term
Out of those combinations, 4x-5 (which equals -1) is able to produce the matching middle term -b.
Find out that 4,-5 match the middle term
After all the steps, we successfully factored the trinomial
{b^2} - b - 20
\left( {b + 4} \right) \bullet \left( {b - 5} \right)
Successfully factored b^2-b-20
2{x^2} + 25x + 12
The "cross-multiply, then check" method can also be used on harder trinomials in which the leading coefficient is not 1. In this question, the leading coefficient is 2 (from the leading term
2{x^2}
Now, we factor the first term
2{x^2}
. The result is 2x and x. we put them in the first box
Factor the first term 2x^2 and put into the first box
In order to factor the last term +12, there are a few combinations (1x12, 2x6, 3x4). We'll once again cross-multiply those combinations to the factors of the first term. See which combination will produce a matching middle term (in this question, the middle term is +25x)
Find the combination that matches 25x
Out of those combinations, 1x12 is able to produce the matching middle term +25x. That is because (2x x 12) + (x x 1) = 24x + x = 25x
Find out that 12 and 1 match the middle term of 25x
2{x^2} + 25x + 12
\left( {2x + 1} \right) \bullet \left( {x + 12} \right)
Factored 2x^2+25x+12 successfully
As you do more questions that require you to do trinomial factoring, you may run into perfect square trinomials. These are trinomials that has both the "a" and "c" terms being perfect squares and the middle "b" being twice the product of the first and last term. Therefore, after factoring, you'll get an answer in the form of either (a+b)^2 or (a-b)^2.
There are good examples for you to further practice factoring trinomials. We've found some excellent online practice questions for you to try that keeps the answers hidden until you hover over the word "answer". There is also an online trinomial calculator for when you have a quadratic equation. It can help you factor any trinomial instantly! Use it to help you check your work.
x^2 - y^2
Basic Concepts: Solving polynomials with unknown coefficients, Factoring polynomials:
x^2 + bx + c
ax^2 + bx + c
ax^2 + bx + c
Use "cross-multiply, then check" method to factor a trinomial
{b^2} - b - 20
{x^2} - 10x + 16
2{x^3} - 14{x^2} + 24x
14 + 5y - {y^2}
2{x^2} + 25x + 12
5{x^2} + 8x + 3
8{x^2} + 10x - 3
6{m^2} - 13m - 8
18{x^2} - 9x + 1
63 + 20z - 3{z^2}
8{x^2} + 8x - 6
8{x^2} + xy - 9{y^2}
6{x^2} + 17xy - 3{y^2}
\frac{1}{4}{x^2} - 4xy + 16{y^2}
8{x^2}{y^2} - xy - 9
25{\sin ^2}x - 35\sin x + 12
12{\cos ^2}x - 20\cos x + 3
6{\left( {x + 5} \right)^2} + 17\left( {x + 5} \right) - 3
15{\left( {x - 3} \right)^2} - 11\left( {x - 3} \right) - 14
{x^6} - 17{x^3} + 30
{a^4} - 2{a^2} - 63
15{x^4} - 16{x^2} - 15
8{x^4} - 14{x^2} + 3
x^2 - y^2
x^2 - y^2
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Potential Load Reduction Using Airfoils with Variable Trailing Edge Geometry | J. Sol. Energy Eng. | ASME Digital Collection
Thomas Buhl,
Wind Energy Department,
, P.O. Box 49-DK-4000 Roskilde, Denmark
e-mail: thomas.buhl@risoe.dk
e-mail: mac.gaunaa@risoe.dk
e-mail: christian.bak@risoe.dk
A correction has been published: Erratum: “Potential Load Reduction Using Airfoils with Variable Trailing Edge Geometry” [Journal of Solar Energy Engineering, 2005, 127(4), pp. 503–516]
Buhl, T., Gaunaa, M., and Bak, C. (June 23, 2005). "Potential Load Reduction Using Airfoils with Variable Trailing Edge Geometry." ASME. J. Sol. Energy Eng. November 2005; 127(4): 503–516. https://doi.org/10.1115/1.2037094
This paper presents an investigation of the potential for reduction of fluctuating loads on wind turbine blades with the use of flaplike deflectable trailing edges. More specifically, the aeroelastic response of an elastically mounted airfoil section with a deflectable trailing edge is investigated. This is done by coupling a model for the aerodynamic forces on a deforming airfoil with a linear spring/damper model for the elastic deformation of a rigid airfoil to which the forces associated with the deflection of the trailing edge are added. The analysis showed that when the airfoil experienced a wind step from
10to12m∕s
the standard deviation of the normal force could be reduced by up to 85% when the flap was controlled by the reading of the airfoil flapwise position and velocity, while reductions of up to 95% could be obtained when the flap was controlled by the reading of the angle of attack. When the airfoil experienced a turbulent wind field, the standard deviation of the normal force could be reduced by 81% for control based on measured angle of attack. The maximum reduction using a combination of flapwise position and velocity was 75%. The maximum deflection of the trailing edge geometry was, in all the considered cases, small enough to justify the use of a potential flow code for calculation of the aerodynamic forces. Calculations showed that the effect of a time lag in the actuators and sensors may drastically reduce the efficiency of the control algorithm. Likewise, the effect of a low maximum actuation velocity reduces the efficiency of the control algorithm. The analysis of the two-dimensional (2D) aeroservoelastic system shown in this paper indicates that the potential of using trailing edge flaps for reduction of fluctuating loads is significant.
aerodynamics, wind turbines, blades, elasticity, delays, servomechanisms, actuators, geometry, load regulation, Variable trailing edge, Control, Load reduction
Airfoils, Control algorithms, Deflection, Geometry, Stress, Turbulence, Wind, Wind turbines, Flow (Dynamics), Actuators, Aerodynamics, Blades, Deformation
Proc. EWEA The Science of Making Torque From Wind, Delft, Netherlands
Yen Nakafuji
, feb 2004, “
Load Alleviation on Wind Turbines using Variable Airfoil Geometry (A Two-Dimensional Analysis)
,” Masters thesis, Department of Mechanical Engineering, Technical University of Denmark.
General Theory of Aerodynamical Instability and the Mechanism of Flutter
,” NACA Report 496, Vol.
, March/April 1994, “
Unsteady Lift of a Flapped Airfoil by Indicial Concepts
, Sept/Oct 1996, “
Unsteady Aerodynamics of a Flapped Airfoil in Subsonic Flow by Indicial Concepts
Hydrodynamic Propulsion by Large Amplitude Oscillation of an Airfoil with Chordwise Flexibility
The Effect of Chordwise Flexibility on the Lift of a Rapidly Accelerated Airfoil
Expreimental and Numerical Investigations on Steady and Unsteady Behavior of a Rotor Airfoil with a Piezoelectric Trailing Edge Flap
55th American Helicopter Society Forum, Montreal, Canada
,” Technical report—Risø-R-1478(EN), To appear.
Propulsion of a Flapping and Oscillating Airfoil
,” NACA Report—NO. 567, pp.
Airfoil Theory for Non-Uniform Motion
Unsteady Aerodynamic Forces on NACA 0015 Airfoil in Harmonic Translatory Motion
,” Ph.D. Thesis, Dept. of Mech. Eng, Danish Techn. Univ.—MEK-FM-2002-02.
,” Technical report—Risø-R-1354(EN), http://www.risoe.dk/rispubl/VEA/ris-r-1354.htmhttp://www.risoe.dk/rispubl/VEA/ris-r-1354.htm.
,” NACA Report—Tech. Report 681.
Computational Study of the Risø-B1-18 Airfoil Equipped with Actively Controlled Trailing Edge Flaps
,” Master Thesis, Technical University of Denmark, Department of Mechanical Engineering, Fluid Mechanics.
Basis3D—a Platform for Development of Multiblock PDE Solvers
,” Technical Report AFM 92-05, Department of Fluid Mechanics,
Block-Structured Multigrid Solution of 2D and 3D Elliptic PDE’s
,” Technical Report AFM 94-06, Department of Fluid Mechanics, Technical University of Denmark.
General Purpose Flow Solver Applied to Flow over Hills
,” Technical Report Risø-R-827(EN).
,” Sandia Report, SAND88-0152-VC-261.
DS472, 1986, “
Dansk Ingeniørforenings og IngeniørSammenslutnings Norm for Last og Sikkerhed for Vindmøllekonstruktioner
,” 1. udg. Danish Design Code for Wind Turbines (In danish).
Some Aspects of Non-Stationary Airfoil Theory and Its Practical Application
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Three basic row operations in matrices | StudyPug
The three types of matrix row operations - Linear Equations with Matrices
Try reviewing these fundamentals firstNotation of matricesNotation of matricesAdding and subtracting matricesAdding and subtracting matricesScalar multiplicationMultiplying a matrix by a scalar
Review these basic concepts…Notation of matricesNotation of matricesAdding and subtracting matricesAdding and subtracting matricesScalar multiplicationMultiplying a matrix by a scalar
Matrix row operations refer to arithmetic operations, or combinations of such operations that can be applied to the rows of a matrix (it doesnt matter the type of matrix in question) in order to solve a system of linear equations.
Simply said, matrix row operations are the ones used after you have converted a system of linear equations into a matrix equation, to then simplify it as much as possible in order to find the solutions to the variables in the system. The complete algorithm (steps to be followed) for solving systems of equations through row operations is called the Gaussian elimination.
The last lesson focused on representing a linear system as a matrix, but after having the augmented matrix containing such system, how do we solve it? In our next lesson we will focus on exactly that: solving a linear system with matrices using Gaussian elimination. But before we are ready, we need to learn about the basic row operations and the combinations that can be applied to the matrix being worked on. Then, row operations matrix are our topic for today! Something important to know is that although we will be working on augmented matrices in order to solve systems of linear equations, you can compute such row operations on any type of matrix.
There exist three types of elementary matrix row operations:
From those three, we can have combinations, such as multiplying a row by a nonzero constant to then add it or subtract it to another row, or maybe exchange them. Elementary row operations on an augmented matrix never change the solution set of the equations, in other words, even though you are operating on the matrix, the matrix obtained after every computation is equivalent to the one you had before.
In order to have a clearer picture of how these matrix elementary row operations work, let us expand our explanation of each of the three basic ones:
\quad
Interchange of rows
The first type of matrix row operation is that we are allowed to switch any rows you want. For example:
Equation 1: Exchanging two rows in a matrix to produce an equivalent one
We interchanged the row 1 and row 3. Another way of saying this is:
R_{1} \leftrightarrow R_{3}
R_{1}
represents the row one (the first row from top to bottom) and
R_{3}
is row three (the third row from top to bottom). Exchanging the rows of a matrix produces an equivalent matrix, which means that the algebraic relationship (or proportionality) remains the same.
If you think about it, interchanging two rows in a matrix which represents a system of linear equations means that you are just exchanging the order in which the equations are listed in a system, which of course, has NO effect on the solutions of the variables in the equations.
Remember from our past lesson, that the rows in an augmented matrix coming from a linear system of equations each represents an equation from the system, therefore, the order in which you list a system of equation makes nothing on the proportional relationship expressed through them about their variables.
To make this a little bit clearer, let us look into the system shown in the second example exercise of our past lesson:
Equation 2: Example of linear equation system written as a matrix equation
On this system, you have two equations, each transcribed as a row in the matrix notation. Would it make any difference in the solutions for x and y if we were to put the second equation on top instead? This would produce a matrix in which you would have the second row of equation 2 as the first row in the matrix. The answer is: NO. It really does not matter in which order you list the equations in a system, and thus, it does not matter how you transcribe them in a matrix equation as long as you keep them in rows, so, you can only exchange a complete row with another complete row (in other words, you cannot exchange the order of the columns because this would change the coefficient associated to a variable in an equation, thus changing the equations).
\quad
Multiplying a row by a non-zero constant
The second type of matrix row operation refers to the multiplication of a row by a non-zero constant. For example:
Equation 3: Multiplying a non-zero constant to a row in the matrix
On this elementary row operations matrix process we have multiplied the non-zero constant 3 to the first row of the original matrix. Such operation is represented by
3R_{1} \leftrightarrow R_{1}
As you can see, this multiplication will affect each coefficient in row one in the same way one would apply such factor to all of the terms in an equation which has been multiplied by a constant.
Let us see this effect going on for an augmented matrix, we take the system represented in the exercise example number 8 from last lesson:
Equation 4: Augmented matrix to system of linear equations example
If we were to multiply the non-zero constant 2 to the first row of the matrix, this would mean the resulting equation for this row would be:
2(6w-2x+8y=4)=12w-4x+16y=8
. Notice how such multiplication for a non-zero constant (any non-zero constant) would not make a change in the value of the variables when solving for them because the proportionality relationship remains. This is an important thing to notice, since it brings to our attention how this row operation makes an emphasis on the constant multiplied being different than zero. If instead of a non-zero constant we were to multiply the first row by zero, this would mean we would end up with an equation for that row equal to
0=0
, which would actually leave us with only two equations to solve a system with three unknown variables, thus making it not possible to solve anymore.
\quad
The third type of matrix row operation consists on allowing two rows to add or subtract from another one. For example:
Equation 5: Adding rows
Where we have replaced row 2 for the result of the addition of rows 2 and 1. Such operation is represented as:
R_{1} + R_{2}
R_{2}
. You can do this with any two rows (adding them of subtracting them from one another and replace one of them with the result).
Having seen the three basic row operations it is worth noting that you can combinations of these three types of matrix row operations in order to manipulate the coefficients on a matrix to your convenience.
Probably the more complicated scenarios will come when you have to work through a combination of augmented matrix row operations, meaning, whenever you will have to apply a combination of the elementary operations seen above into an augmented matrix. But worry not, in the next section we will focus on showing examples of such processes so you can practice.
Remember that the purpose of row operations on augmented matrices is to simplify the matrices as much as possible in order to solve for the values of the variables in the system of equations they represent. Such simplified matrix forms are what we call the echelon form and the reduced echelon form, and they are the key for finding the final solutions to the equations in the system. Different sequences of row operations can lead to different echelon forms for the same matrix, we will work through them and cover such topics as the elementary matrix, row reduction and echelon forms in the next lessons of our Linear Algebra course, for now we will focus on just learning the elementary row operations.
Before we continue with the exercise examples of this lesson, we recommend you to take a look into these notes on elementary row operations for matrices which could serve to complement todays lesson, plus it gives a little introduction on matrices and systems of equations.
Matrix row operations examples
On this section we will see two examples of matrices on which we will compute a few times to apply the three elementary matrix row operations (for the first example) and then a combination of the elementary row functions on the second example.
Notice that for each operation in each example we will start from the original matrix and then apply the operation concerning to that case (in other words, the operations will not be cumulative for a single matrix, but will start from the original each time). Although you will work with cumulative row operations in later lessons, for now the important thing is for you to learn how to apply the basic operations and their possible combinations from the beginning. In later lessons you will continue to practice such operations and making them a cumulative process, but there will be plenty of time and examples to practice on when that time comes.
Observe how the two examples below do not contain augmented matrices, this has been selected on purpose so you know the row operations can be applied to any type of matrix. When solving systems of linear equations in later lessons we will mostly be using augmented matrices.
You are given the following matrix
Equation 6: Matrix to row reduce
Perform the following matrix operations:
\quad
\quad R_{1} \longleftrightarrow R_{2}
Equation 7: Exchanging rows 1 and 2
\quad
\quad R_{2} \longleftrightarrow R_{3}
\quad
\quad -4 R_{3} \longrightarrow R_{3}
Equation 9: Multiplying row 3 by a non-zero constant
\quad
\quad 3R_{2} \longrightarrow R_{2}
Equation 10: Multiplying row 2 by a non-zero constant
\quad
\quad R_{1} + R_{2} \longrightarrow R_{2}
Equation 11: Adding row 1 and row 2
\quad
\quad R_{1} + R_{3} \longrightarrow R_{3}
Again, notice that during this example and its multiple cases, we have only used the most elementary three row operations possible, but in most cases is much more practical to combine them in order to reduce the matrix into a simplified one much faster. In the next example, we will be looking into an example for which we will be applying combinations of row operations.
Equation 13: Matrix to row reduce
\quad
\quad 3R_{1} + 2R_{4} \longrightarrow R_{4}
Equation 14: Multiple combination of row operations
\quad
\quad 2R_{2} - R_{1} \longrightarrow R_{2}
\quad
\quad -2R_{3} + R_{1} \longrightarrow R_{1}
\quad
\quad -2R_{2} + 2R_{3} \longrightarrow R_{3}
If you are ever in doubt that such matrix row operations generate a matrix which is not equivalent, you can always prove it on your own, take the matrices we have used in these examples and convert them into the algebraic notation to obtain the systems of linear equations corresponding to each of them. Having such systems, solve for the values of their variables using other method we have learned in algebra before, such as solving systems of linear equations by substitution, elimination or even graphing. The proof process will serve for a lot of practice, thus we highly recommend it.
Matrix row operations are the basis for row reduction, the method we have mentioned before (also named Gaussian elimination), thus after this lesson you are ready to start working on cumulative row operations on matrices in order to reduce them to echelon forms and obtain the values of the variables in the system. Although lessons like this one can seem tedious and repetitive, it is imperative you have well settled basis for what is coming next.
Now get ready, we are finally going to solve the systems of linear equations we have been talking about. See you in the next lesson!
R^n
R^n
In this section we will learn about matrix row operations. There are three types of matrix row operations: interchanging 2 rows, multiplying a row, and adding/subtracting a row with another. Each notation for this is different, and is displayed in the note section. First, we will look at questions which involve all three types of matrix row operations. Then we will look at more slightly advanced questions that involve using 2 matrix row operations.
The first type of matrix row operation is that we are allowed to switch any rows you want. For example,
We interchanged the row 1 and row 3. Another way of saying this is
R_1\leftrightarrow R_3
The second type of matrix row operation is that we can multiply a row by a non-zero constant. For example,
We multiplied each entry in row 1 by 3. Another way of saying this is
3R_1\leftrightarrow R_1
The third type of matrix row operation is that we are allowed to add and subtract a row with another. For example,
Row 2 is the sum of row 2 and 1. Another way of saying this is
R_1+R_2
R_2
Matrix row operations overview:
One matrix row operation
R_1\leftrightarrow R_2
R_2\leftrightarrow R_3
-4R_3
R_3
3R_2
R_2
R_1+R_2
R_2
R_1+R_3
R_3
More than one matrix row operation
3R_1+2R_4
R_4
2R_2-R_1
R_2
-2R_3+R_1
R_1
-2R_2+2R_3
R_3
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User interface - Simple English Wikipedia, the free encyclopedia
A user interface allows a user to interact with a machine.[1] User interfaces mainly provide two things:
input: The user can change things; he or she can change how the machine works, or give more information to the machine.
Output: After the user has given some input, the machine will do something, and then provide some output
An example of a user interface with pushbuttons
Many machines can be very dangerous. A machine should have a user interface that can be handled easily, even if the person operating the machine has panicked. The user interface should therefore be intuitive, and simple to use. An example of such a user interface is that of the kill switch. A kill switch must shut off the machine at all costs — the idea is to avoid injury or harm to people. This is very different from shutting off the machine at the end of the shift, or when it is no longer needed.
According to EN ISO 13850, the kill switch has to be red on a yellow background.
The colors used to mark different states are close to those used by signals used on the road.
Red Danger Alerting of possible danger or of states which make it very important to act immediately
Yellow Something is not normal If nothing is done, the situation may become dangerous.
Blue Something needs to be done The person operating the machine needs to do something
Green Everything is normal Used to show safe conditions, also used to start a new process.
White Neutral Confirmation, also used for things that cannot be expressed by red, yellow, blue or green.
Red Operate in an emergency Kill switch, stop, also used for fighting fire Must not be used for stating/putting the machine into operation
Yellow Something needs to be done to get back to normal Re-start, Operation to avoid anormal condition or unwanted change. Must not be used for either starting or stopping a machine.
Blue Start something new Start, Reset
Green Start the usual/common procedure Start from a safe state Must not be used for stopping/switching off
White meaning underermined Start/On (preferred), Stop/Off
Grey Start/On, Stop/Off
Black Stop/Off (preferred), sometimes Start/On
There may be additional symbols, for example:
{\displaystyle \mid }
{\displaystyle \bigcirc }
In many cases, such symbols are better, because some people are color blind. They need to be explained, like warnings, though.
User interfaces are composed of one or more layers including a human-machine interface (HMI) using physical input devices such as keyboards, mice, and game pads as well as output hardware such as computer monitors, speakers, and printers. User Interface (UI) Design focuses on anticipating what users might need to do and ensuring that the interface has elements that are easy to access, understand, and use to facilitate those actions. UI brings together concepts from interaction design, visual design, and information architecture.[2]
UI design is closely linked to user experience (UX) design. Both are crucial for the success of a digital product, and both terms are often used interchangeably. However, they’re not the same thing.[3] UI design is the process of creating interfaces with a focus on the surface, the looks, and style, while UX design covers the entire experience a user has with a website or app.
User interface of a modern ICE train
User interface of a steam engine
Operating panel of a Japanese toilet
↑ "What is User Interface (UI)?". Pendo.io. Retrieved 2021-09-13.
↑ "What is UI? Overview & Intro to User Interfaces | Adobe XD Ideas". Ideas. Retrieved 2021-09-13.
↑ "UI vs UX: The difference between UX and UI". RYSEN. 2021-03-20. Retrieved 2021-09-13.
Retrieved from "https://simple.wikipedia.org/w/index.php?title=User_interface&oldid=8078438"
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How to convert repeating decimals to fractions | StudyPug
A repeating decimal keeps repeating itself endlessly. Yet, no worries, you don't need to have unlimited paper to write it down. We can convert it into a fraction for a clearer presentation. Let's learn how in this session.
Basic Concepts: Converting among decimals, fractions, and percents, Percents, fractions, and decimals
Related Concepts: Percent of a number, Adding and multiplying percents
How to convert repreating decimals to fractions?
The entire decimal repeats
\bullet
0.\overline1
0.\overline3
7.\overline2
Only a portion of the decimal repeats
\bullet
0.0\overline1
4.1\overline6
4.00\overline3\overline6
\overline{38}
\overline{6}
\overline{60}
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The boundary value problem for the super-Liouville equation | EMS Press
Max Planck Institute for Mathematics in the Sciences, Inselstr. 22, D-04103 Leipzig, Germany
Mathematisches Institut, Albert-Ludwigs-Universität Freiburg, D-79104 Freiburg, Germany
Chunqin Zhou
Mathematics Institute, University of Warwick, CV4 7AL, Coventry, UK
We study the boundary value problem for the — conformally invariant — super-Liouville functional
E(u,\psi ) = \int \limits_{M}\left\{\frac{1}{2}|\mathrm{∇}u|^{2} + K_{g}u + 〈\left(D̸ + e^{u}\right)\psi ,\psi 〉−e^{2u}\right\}\:dz
that couples a function u and a spinor ψ on a Riemann surface. The boundary condition that we identify (motivated by quantum field theory) couples a Neumann condition for u with a chirality condition for ψ. Associated to any solution of the super-Liouville system is a holomorphic quadratic differential
T(z)
, and when our boundary condition is satisfied, T becomes real on the boundary. We provide a complete regularity and blow-up analysis for solutions of this boundary value problem.
Jürgen Jost, Guofang Wang, Chunqin Zhou, Miaomiao Zhu, The boundary value problem for the super-Liouville equation. Ann. Inst. H. Poincaré Anal. Non Linéaire 31 (2014), no. 4, pp. 685–706
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Difference between revisions of "Help:Mathematical Symbols" - POV-Wiki
Difference between revisions of "Help:Mathematical Symbols"
m (→Advanced Use of TeX)
m (Reflect change of math engine.)
POV-Wiki supports the standard mediawiki math markup (available via the '''√n''' button on the edit toolbar). Additionally, it has an interactive T<sub>E</sub>X editor, called the 'T<sub>E</sub>X Box', which appears in a popup window when you click on the '''∑''' button at the left of the edit toolbar.
POV-Wiki has the standard [[mediawikiwiki:Extension:Math|MediaWiki math extension]] installed, which allows the use of a subset of LaTeX syntax.
The T<sub>E</sub>X Box is also available on its own [[Special:Texbox|special page]].
==Inserting Formulas==
Inserting T<sub>E</sub>X markup is done using <code><nowiki><math>...</math></nowiki></code> tags. For example, to display the following quadratic formula
Inserting math markup is done using <code><nowiki><math>...</math></nowiki></code> tags. For example, to display the following quadratic formula
:<math>x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}</math>
==Consistent Use of <nowiki><math>...</math></nowiki>==
It is strongly advised that <nowiki><math>...</math></nowiki> be used around ''all'' mathematics added to this wiki, even if the assistance of T<sub>E</sub>X is not required (e.g. if the formula could be displayed using standard HTML). This will assist when exporting the wiki into other formats, such as PDF.
It is strongly advised that <nowiki><math>...</math></nowiki> be used around ''all'' mathematics added to this wiki, even if the assistance of markup is not required (e.g. if the formula could be displayed using standard HTML). This will assist when exporting the wiki into other formats, such as PDF.
==Advanced Use of T<sub>E</sub>X==
==Advanced Use of LaTeX==
Advanced use of T<sub>E</sub>X is beyond the scope of this help page. Instead, we recommend you consult the comprehensive [[metawikipedia:Help:Formula|reference on meta.wikimedia.org]].
Advanced use of LaTeX is beyond the scope of this help page. Instead, we recommend you consult the comprehensive [[wikipedia:Help:Displaying_a_formula|reference on Wikipedia]].
[[Category:Help|Mathematical Symbols]]
{\displaystyle x={\frac {-b\pm {\sqrt {b^{2}-4ac}}}{2a}}}
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CAN ANYONE PLEASE ANSWER THIS QUESTION Q 6 What is the ratio of the SHADED squares to the - Maths - Fractions and Decimals - 11686689 | Meritnation.com
Q.6. What is the ratio of the SHADED squares to the UNSHADED squares in the figure shown below?
\mathrm{As} \mathrm{per} \mathrm{the} \mathrm{figure} \mathrm{the} \mathrm{last} \mathrm{row} \mathrm{is} \mathrm{of} \mathrm{rectangles}, \mathrm{ie} \mathrm{each} \mathrm{shape}\phantom{\rule{0ex}{0ex}}\mathrm{is} \mathrm{in} \mathrm{the} \mathrm{shape} \mathrm{of} \mathrm{rectangles}.\mathrm{So} \mathrm{solution} \mathrm{is} \mathrm{as} \mathrm{follows}:\phantom{\rule{0ex}{0ex}}\mathrm{Total} \mathrm{shaded} \mathrm{squares}=5\phantom{\rule{0ex}{0ex}}\mathrm{Total} \mathrm{unshaded} \mathrm{squares}=16\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\mathrm{So}, \mathrm{required} \mathrm{ratio}=\frac{5}{16}\phantom{\rule{0ex}{0ex}}\mathrm{is} \mathrm{the} \mathrm{correct} \mathrm{answer}.
Total shaded squares = 7
Total unshaded squares = 21
therefore the ratio of shaded squares to unshaded squares = 7 : 21 = 1 : 3 [in the most simplified form]
here, I think no option is correct
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Existence of immersed spheres minimizing curvature functionals in non-compact 3-manifolds | EMS Press
Mathematisches Institut, Universität Freiburg, Eckerstraße 1, 79104 Freiburg, Germany
We study curvature functionals for immersed 2-spheres in non-compact, three-dimensional Riemannian manifold
(M,h)
without boundary. First, under the assumption that
(M,h)
is the euclidean 3-space endowed with a semi-perturbed metric with perturbation small in
C^{1}
norm and of compact support, we prove that if there is some point x\limits^{¯} \in M with scalar curvature R^{M}(x\limits^{¯}) > 0 then there exists a smooth embedding
f:\mathbb{S}^{2}↪M
minimizing the Willmore functional
\frac{1}{4}\int |H|^{2}
, where H is the mean curvature. Second, assuming that
(M,h)
is of bounded geometry (i.e. bounded sectional curvature and strictly positive injectivity radius) and asymptotically euclidean or hyperbolic we prove that if there is some point x\limits^{¯} \in M with scalar curvature R^{M}(x\limits^{¯}) > 6 then there exists a smooth immersion
f:\mathbb{S}^{2}↪M
minimizing the functional
\int (\frac{1}{2}|A|^{2} + 1)
, where A is the second fundamental form. Finally, adding the bound
K^{M}⩽2
to the last assumptions, we obtain a smooth minimizer
f:\mathbb{S}^{2}↪M
\int (\frac{1}{4}|H|^{2} + 1)
. The assumptions of the last two theorems are satisfied in a large class of 3-manifolds arising as spacelike timeslices solutions of the Einstein vacuum equation in case of null or negative cosmological constant.
Andrea Mondino, Johannes Schygulla, Existence of immersed spheres minimizing curvature functionals in non-compact 3-manifolds. Ann. Inst. H. Poincaré Anal. Non Linéaire 31 (2014), no. 4, pp. 707–724
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Compute transmission loss using the sonar equation - MATLAB sonareqtl - MathWorks Australia
sonareqtl
Estimate Transmission Loss from Passive Sonar Equation
Estimate Transmission Loss from Active Sonar Equation
Compute transmission loss using the sonar equation
TL = sonareqtl(SL,SNR,NL,DI)
TL = sonareqtl(SL,SNR,NL,DI,TS)
TL = sonareqtl(SL,SNR,NL,DI) returns the transmission loss of a signal from source to receiver that produces the signal-to-noise ratio, SNR. Transmission loss is computed using the Sonar Equation. Required inputs are the source level, SL, received noise level, NL, and receiver directivity index, DI. Use this syntax to evaluate passive sonar system performance.
TL = sonareqtl(SL,SNR,NL,DI,TS) returns the one-way transmission loss. The signal is reflected from a target with a target strength, TS. Use this syntax to evaluate active sonar system performance, where the transmitted signal is reflected from a target.
Estimate the transmission loss of a signal arriving from a source with source level of 200 dB. The received SNR is 10 dB, the noise level is 75 dB, and the receive array directivity index is 25 dB.
Estimate the one-way transmission loss of a signal transmitted by a source with source level of 130 dB//1 μPa and reflected from a target with 25 dB//1
{m}^{2}
target strength. The noise level is 45 dB//1 μPa, the receive array directivity is 25 dB.
Transmission loss, returned as a positive scalar. Transmission loss is the attenuation of sound intensity as the sound propagates through the underwater channel. Transmission loss is defined as the ratio of sound intensity at 1 m from a source to the sound intensity at distance R. When target strength, TS, is specified, transmission loss is two-way.
range2tl | sonareqsl | tl2range | sonareqsnr
|
{\textstyle \left\{x\,y\left(x\right)\right\}}
{\textstyle y\left(x\right)=F\left(x\,\mathrm{\_Cn}\right)}
{\textstyle F\left(y\left(x\right)\,x\,\mathrm{\_Cn}\right)=0}
\mathrm{_Cn}
{\textstyle y\left(x\right)=\mathrm{RootOf}\left(...\right)}
\mathrm{_EnvExplicit}
{\textstyle \left[x\left(\mathrm{\_T}\right)=f\left(\mathrm{\_T}\right)\,y\left(\mathrm{\_T}\right)=g\left(\mathrm{\_T}\right)\right]}
\mathrm{_T}
\mathrm{_T}
\mathrm{_yn}
\mathrm{_Cn}
{\textstyle \frac{\ⅆ}{\ⅆx}\mathrm{\_Cn}=0}
{\textstyle \mathrm{\infty }}
\mathrm{_Cn}
\mathrm{_Cn}
\mathrm{_Cn}
\mathrm{_Cn}
{\textstyle \mathrm{\infty }}
{\textstyle x,y,y\',\mathrm{...}}
{\textstyle \left[\textcolor[rgb]{0,0,1}{\mathrm{quadrature}}\textcolor[rgb]{0,0,1}{\,}\textcolor[rgb]{0,0,1}{\mathrm{linear}}\textcolor[rgb]{0,0,1}{\,}\textcolor[rgb]{0,0,1}{\mathrm{Bernoulli}}\textcolor[rgb]{0,0,1}{\,}\textcolor[rgb]{0,0,1}{\mathrm{separable}}\textcolor[rgb]{0,0,1}{\,}\textcolor[rgb]{0,0,1}{\mathrm{inverse\_linear}}\textcolor[rgb]{0,0,1}{\,}\textcolor[rgb]{0,0,1}{\mathrm{homogeneous}}\textcolor[rgb]{0,0,1}{\,}\textcolor[rgb]{0,0,1}{\mathrm{Chini}}\textcolor[rgb]{0,0,1}{\,}\textcolor[rgb]{0,0,1}{\mathrm{lin\_sym}}\textcolor[rgb]{0,0,1}{\,}\textcolor[rgb]{0,0,1}{\mathrm{exact}}\textcolor[rgb]{0,0,1}{\,}\textcolor[rgb]{0,0,1}{\mathrm{Abel}}\textcolor[rgb]{0,0,1}{\,}\textcolor[rgb]{0,0,1}{\mathrm{pot\_sym}}\right]}
{\textstyle \left[\textcolor[rgb]{0,0,1}{\mathrm{Riccati}}\textcolor[rgb]{0,0,1}{\,}\textcolor[rgb]{0,0,1}{\mathrm{inverse\_Riccati}}\textcolor[rgb]{0,0,1}{\,}\textcolor[rgb]{0,0,1}{\mathrm{equivalent\_to\_Abel}}\textcolor[rgb]{0,0,1}{\,}\textcolor[rgb]{0,0,1}{\mathrm{linearizable}}\textcolor[rgb]{0,0,1}{\,}\textcolor[rgb]{0,0,1}{\mathrm{linearizable\_by\_differentiation}}\right]}
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{\textstyle \mathrm{ode}\left[1\right]≔\mathrm{sin}\left(x\right)\mathrm{diff}\left(y\left(x\right)\,x\right)-\mathrm{cos}\left(x\right)y\left(x\right)=0}
{\textstyle {\textcolor[rgb]{0,0,1}{\mathrm{ode}}}_{\textcolor[rgb]{0,0,1}{1}}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{\mathrm{sin}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{x}\right)\textcolor[rgb]{0,0,1}{}\left(\frac{\textcolor[rgb]{0,0,1}{\ⅆ}}{\textcolor[rgb]{0,0,1}{\ⅆ}\textcolor[rgb]{0,0,1}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{x}\right)\right)\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{\mathrm{cos}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{x}\right)\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{x}\right)\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{0}}
{\textstyle \mathrm{dsolve}\left(\mathrm{ode}\left[1\right]\right)}
{\textstyle \textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{x}\right)\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{\mathrm{\_C1}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{sin}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{x}\right)}
{\textstyle \mathrm{dsolve}\left(\mathrm{ode}\left[1\right]\,\left[\mathrm{linear}\right]\,\mathrm{useInt}\right)}
{\textstyle \textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{x}\right)\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{\mathrm{\_C1}}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{\ⅇ}}^{{\textstyle \textcolor[rgb]{0.564705882352941,0.564705882352941,0.564705882352941}{\int }}\frac{\textcolor[rgb]{0,0,1}{\mathrm{cos}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{x}\right)}{\textcolor[rgb]{0,0,1}{\mathrm{sin}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{x}\right)}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{\textstyle \textcolor[rgb]{0.564705882352941,0.564705882352941,0.564705882352941}{\ⅆ}}\textcolor[rgb]{0,0,1}{x}}}
{\textstyle \mathrm{value}\left(\right)}
{\textstyle \textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{x}\right)\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{\mathrm{\_C1}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{sin}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{x}\right)}
{\textstyle \mathrm{DEtools}\left[\mathrm{odeadvisor}\right]\left(\mathrm{ode}\left[1\right]\right)}
{\textstyle \left[\textcolor[rgb]{0,0,1}{\mathrm{\_separable}}\right]}
{\textstyle \mathrm{dsolve}\left(\mathrm{ode}\left[1\right]\,\left[\mathrm{separable}\right]\,\mathrm{useInt}\right)}
{\textstyle {\textstyle \textcolor[rgb]{0.564705882352941,0.564705882352941,0.564705882352941}{\int }}\frac{\textcolor[rgb]{0,0,1}{\mathrm{cos}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{x}\right)}{\textcolor[rgb]{0,0,1}{\mathrm{sin}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{x}\right)}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{\textstyle \textcolor[rgb]{0.564705882352941,0.564705882352941,0.564705882352941}{\ⅆ}}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{-}\left({\textcolor[rgb]{0,0,1}{\int }}_{\textcolor[rgb]{0,0,1}{}}^{\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{x}\right)}\frac{\textcolor[rgb]{0,0,1}{1}}{\textcolor[rgb]{0,0,1}{\mathrm{\_a}}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\textcolor[rgb]{0,0,1}{\ⅆ}\textcolor[rgb]{0,0,1}{\mathrm{\_a}}\right)\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{\mathrm{\_C1}}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{0}}
{\textstyle \mathrm{value}\left(\right)}
{\textstyle \textcolor[rgb]{0,0,1}{\mathrm{ln}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{\mathrm{sin}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{x}\right)\right)\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{\mathrm{ln}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{x}\right)\right)\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{\mathrm{\_C1}}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{0}}
{\textstyle \mathrm{isolate}\left(\,y\left(x\right)\right)}
{\textstyle \textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{x}\right)\textcolor[rgb]{0,0,1}{=}{\textcolor[rgb]{0,0,1}{\ⅇ}}^{\textcolor[rgb]{0,0,1}{\mathrm{\_C1}}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{sin}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{x}\right)}
{\textstyle {\ⅇ}^{\mathrm{\_C1}}\to \mathrm{\_C1}}
{\textstyle \mathrm{\mu }≔\mathrm{DEtools}\left[\mathrm{intfactor}\right]\left(\mathrm{ode}\left[1\right]\right)}
{\textstyle \textcolor[rgb]{0,0,1}{\mathrm{\mu }}\textcolor[rgb]{0,0,1}{≔}\frac{\textcolor[rgb]{0,0,1}{1}}{{\textcolor[rgb]{0,0,1}{\mathrm{sin}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{x}\right)}^{\textcolor[rgb]{0,0,1}{2}}}}
{\textstyle \mathrm{dsolve}\left(\mathrm{\mu }\mathrm{ode}\left[1\right]\,\left[\mathrm{exact}\right]\right)}
{\textstyle \textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{x}\right)\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{\mathrm{\_C1}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{sin}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{x}\right)}
{\textstyle \mathrm{ode}\left[2\right]≔\mathrm{diff}\left(y\left(x\right)\,x\right)-{y\left(x\right)}^{2}+y\left(x\right)\mathrm{sin}\left(x\right)-\mathrm{cos}\left(x\right)=0}
{\textstyle {\textcolor[rgb]{0,0,1}{\mathrm{ode}}}_{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{≔}\frac{\textcolor[rgb]{0,0,1}{\ⅆ}}{\textcolor[rgb]{0,0,1}{\ⅆ}\textcolor[rgb]{0,0,1}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{x}\right)\textcolor[rgb]{0,0,1}{-}{\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{x}\right)}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{x}\right)\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{sin}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{x}\right)\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{\mathrm{cos}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{x}\right)\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{0}}
{\textstyle \mathrm{DEtools}\left[\mathrm{odeadvisor}\right]\left(\mathrm{ode}\left[2\right]\right)}
{\textstyle \left[\textcolor[rgb]{0,0,1}{\mathrm{\_Riccati}}\right]}
{\textstyle \mathrm{dsolve}\left(\mathrm{ode}\left[2\right]\right)}
{\textstyle \textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{x}\right)\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{-}\frac{{\textcolor[rgb]{0,0,1}{\ⅇ}}^{\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{\mathrm{cos}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{x}\right)}}{\textcolor[rgb]{0,0,1}{\mathrm{\_C1}}\textcolor[rgb]{0,0,1}{+}{\textstyle \textcolor[rgb]{0.564705882352941,0.564705882352941,0.564705882352941}{\int }}{\textcolor[rgb]{0,0,1}{\ⅇ}}^{\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{\mathrm{cos}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{x}\right)}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{\textstyle \textcolor[rgb]{0.564705882352941,0.564705882352941,0.564705882352941}{\ⅆ}}\textcolor[rgb]{0,0,1}{x}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{\mathrm{sin}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{x}\right)}
{\textstyle \mathrm{ode}\left[3\right]≔\mathrm{diff}\left(y\left(x\right)\,x\right)=\frac{x\left(-x-1+{x}^{2}-2{x}^{2}y\left(x\right)+2{x}^{4}\right)}{\left({x}^{2}-y\left(x\right)\right)\left(x+1\right)}}
{\textstyle {\textcolor[rgb]{0,0,1}{\mathrm{ode}}}_{\textcolor[rgb]{0,0,1}{3}}\textcolor[rgb]{0,0,1}{≔}\frac{\textcolor[rgb]{0,0,1}{\ⅆ}}{\textcolor[rgb]{0,0,1}{\ⅆ}\textcolor[rgb]{0,0,1}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{x}\right)\textcolor[rgb]{0,0,1}{=}\frac{\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{+}{\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{x}\right)\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{4}}\right)}{\left({\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{x}\right)\right)\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{1}\right)}}
dsolve automatically solves this equation by computing an integrating factor for it. dsolve's internal solving steps can be reproduced at the Maple prompt as follows
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Introduction to transformations in math | StudyPug
When an object is reflected in a mirror, the line of reflection is the mirror. Reflections are a type of transformation. In this section, we will use our knowledge from sections on Cartesian plane and drawing on coordinate planes to show transformations of figures on coordinate planes. Along with reflections, translations and rotations are types of transformations. When showing translations, we always describe horizontal (left or right) movements before vertical (up or down) movements.
Basic Concepts: Representing percents, Cartesian plane, Draw on coordinate planes
Draw the reflection image of the following diagram. The line of reflection is labeled
r
The diagram shows a quadrilateral. Rotate the figure 180 degrees counter clockwise about its centre of rotation, P.
|
Computational Fluid Dynamics Analysis of Turbulent Flow Within a Mechanical Seal Chamber | J. Tribol. | ASME Digital Collection
Zhaogao Luan,
Dow Chemical Endowed Chair in Rotating Machinery, Department of Mechanical Engineering,
Luan, Z., and Khonsari, M. M. (June 27, 2006). "Computational Fluid Dynamics Analysis of Turbulent Flow Within a Mechanical Seal Chamber." ASME. J. Tribol. January 2007; 129(1): 120–128. https://doi.org/10.1115/1.2401220
Turbulent flow inside the seal chamber of a pump operating at high Reynolds number is investigated. The
K−ε
turbulence model posed in cylindrical coordinates was applied for this purpose. Simulations are performed using the fractional approach method. The results of the computer code are verified by using the FLUENT and by comparing to published results for turbulent Taylor Couette flow. Numerical results of four cases including two rotational speeds with four flush rates are reported. Significant difference between the laminar and the turbulence flow in the seal chamber is predicted. The behavior of the turbulent flows with very high Reynolds number was also investigated. The physical and practical implications of the results are discussed.
computational fluid dynamics, seals (stoppers), turbulence, flow simulation, Couette flow, laminar flow, sealing materials, mechanical seal, flow field, CFD, K−ε turbulence model, fractional approach method
Flow (Dynamics), Turbulence, Computational fluid dynamics
Visualizing Fluid and Heat Transfer in Rotating Shaft Seals
,” BHR Group Conference Series Publication, San Francisco, CA, Vol.
The Thermal Study of the Double Mechanical Seals
Proc. 5th Conference on Friction Lubrication and Wear TRIBOTEHNICA 87
Heat Transfer in Mechanical Seals
Proceedings 6th International Pump User Symposium
, Baton Rouge, LA, pp.
Estimating Heat Generation, Face Temperature and Flush Rate for Mechanical Seals
Proceedings PumpUsers Expo’99
An Experimental-Computational Investigation of the Heat Transfer in Mechanical Face Seals
A Generalized Thermoelastic Instability Analysis
Experimental and Computational Investigation of Flow and Thermal Behavior of A Mechanical Seal
An Investigation of Turbulent Taylor-Couette Flow Using Laser Doppler Velocimetery in a Refractive Index Matched facility
Implicit schemes and Boundary Conditions for compressible Flows on Unstructured Grids
Proceedings AIAA 32nd Aerospace Science Meeting and Exhibit
Turbulent Flows, Fundamentals, Experimental and Modeling
An Introduction to Turbulence Flow
Numeric and Experimental Analysis of the Turbulent Flow through a Channel with Baffle Plates
Chimera Type Domain Decomposition Methods Applied to Fractional Step Finite Element Schemes For Incompressible Flows
Proceedings European Congress on Computational Methods in Applied Sciences and Engineering
The Solution of the Characteristic Value Problem
Bernard Cells and Taylor Vortices
|
A quantitative bounded distance theorem and a Margulis’ lemma for $\mathbb Z^n$-actions, with applications to homology | EMS Press
A quantitative bounded distance theorem and a Margulis’ lemma for
\mathbb Z^n
-actions, with applications to homology
Filippo Cerocchi
We consider the stable norm associated to a discrete, torsionless abelian group of isometries
\Gamma \cong \mathbb Z^n
of a geodesic space
(X,d)
. We show that the difference between the stable norm
\| \;\, \|_{\mathrm {st}}
d
is bounded by a constant only depending on the rank
n
and on upper bounds for the diameter of
\bar X=\Gamma \backslash X
and the asymptotic volume
\omega(\Gamma, d)
. We also prove that the upper bound on the asymptotic volume is equivalent to a lower bound for the stable systole of the action of
\Gamma
(X,d)
; for this, we establish a lemma à la Margulis for
\mathbb{Z}^n
-actions, which gives optimal estimates of
\omega(\Gamma,d)
in terms of stsys
(\Gamma,d)
, and vice versa, and characterize the cases of equality. Moreover, we show that all the parameters
n
, diam
(\bar X)
\omega (\Gamma, d)
(or stsys
(\Gamma,d)
) are necessary to bound the difference
d -\| \;\, \|_{\mathrm {st}}
, by providing explicit counterexamples for each case.
As an application in Riemannian geometry, we prove that the number of connected components of any optimal, integral 1-cycle in a closed Riemannian manifold
\bar X
either is bounded by an explicit function of the first Betti number, diam
(\bar X)
\omega(H_1(\bar X, \mathbb{Z}))
, or is a sublinear function of the mass.
Filippo Cerocchi, Andrea Sambusetti, A quantitative bounded distance theorem and a Margulis’ lemma for
\mathbb Z^n
-actions, with applications to homology. Groups Geom. Dyn. 10 (2016), no. 4, pp. 1227–1247
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Comparison of the Reactivity Effects Calculated by DRAGON and Serpent for a PHWR 37-Element Fuel Bundle | ASME J. of Nuclear Rad Sci. | ASME Digital Collection
Jawad Haroon,
e-mail: Jawad.Haroon@uoit.ca
Leslie Kicka,
Leslie Kicka
e-mail: leslie.kicka@uoit.net
Subhramanyu Mohapatra,
Subhramanyu Mohapatra
e-mail: Subhramanyu.Mohapatra@uoit.ca
e-mail: Peter.Schwanke@uoit.ca
Manuscript received January 28, 2016; final manuscript received August 15, 2016; published online December 20, 2016. Assoc. Editor: Michal Kostal.
Haroon, J., Kicka, L., Mohapatra, S., Nichita, E., and Schwanke, P. (December 20, 2016). "Comparison of the Reactivity Effects Calculated by DRAGON and Serpent for a PHWR 37-Element Fuel Bundle." ASME. ASME J of Nuclear Rad Sci. January 2017; 3(1): 011011. https://doi.org/10.1115/1.4034571
Deterministic and Monte Carlo methods are regularly employed to conduct lattice calculations. Monte Carlo methods can effectively model a large range of complex geometries and, compared to deterministic methods, they have the major advantage of reducing systematic errors and are computationally effective when integral quantities such as effective multiplication factor or reactivity are calculated. In contrast, deterministic methods do introduce discretization approximations but usually require shorter computation times than Monte Carlo methods when detailed flux and reaction-rate solutions are sought. This work compares the results of the deterministic code DRAGON to the Monte Carlo code Serpent in the calculation of the reactivity effects for a pressurized heavy water reactor (PHWR) lattice cell containing a 37-element, natural uranium fuel bundle with heavy water coolant and moderator. The reactivity effects are determined for changes to the coolant, moderator, and fuel temperatures and to the coolant and moderator densities for zero-burnup, mid-burnup [
3750 MWd/t(U)
] and discharge burnup [
7500 MWd/t(U)
] fuel. It is found that the overall trend in the reactivity effects calculated using DRAGON match those calculated using Serpent for the burnup cases considered. However, differences that exceed the amount attributable to statistical error have been found for some reactivity effects, particularly for perturbations to coolant and moderator density and fuel temperature.
Reactor physics, Transport theory
Coolants, Density, Fuels, Heavy water reactors, Temperature
A User Guide for DRAGON, Release 3.05 E
Institut de génie nucléaire, Département de génie mécanique, Ecole Polytechnique de Montréal
, Montréal, Canada.
PSG2/Serpent: A Continuous-energy Monte Carlo Reactor Physics Burnup Calculation Code
Canada Enters the Nuclear Age: A Technical History of Atomic Energy of Canada Limited as Seen from Its Research Laboratories
DRAGON Theory Manual Part 1: Collision Probability Calculations
Lopez Aldama
Study on Optimization Design for CSR1000 Core
IAEA CRP Benchmark of Kalinin VVER-1000 NPP: An Analysis Using EXCEL-TRIHEX-FA Code System
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Cold gas thruster - Wikipedia
Type of rocket engine
A cold gas thruster (or a cold gas propulsion system) is a type of rocket engine which uses the expansion of a (typically inert) pressurized gas to generate thrust. As opposed to traditional rocket engines, a cold gas thruster does not house any combustion and therefore has lower thrust and efficiency compared to conventional monopropellant and bipropellant rocket engines. Cold gas thrusters have been referred to as the "simplest manifestation of a rocket engine" because their design consists only of a fuel tank, a regulating valve, a propelling nozzle, and the little required plumbing. They are the cheapest, simplest, and most reliable propulsion systems available for orbital maintenance, maneuvering and attitude control.[citation needed]
Cold gas thrusters are predominantly used to provide stabilization for smaller space missions which require contaminant-free operation.[1] Specifically, CubeSat propulsion system development has been predominantly focused on cold gas systems because CubeSats have strict regulations against pyrotechnics and hazardous materials.[2]
4.1 Human Propulsion
4.1.1 Hand-Held Maneuvering Unit
4.2 Vernier Engines
Schematic of a cold gas propulsion system
The nozzle of a cold gas thruster is generally a convergent-divergent nozzle that provides the required thrust in flight. The nozzle is shaped such that the high-pressure, low-velocity gas that enters the nozzle is accelerated as it approaches the throat (the narrowest part of the nozzle), where the gas velocity matches the speed of sound.[citation needed]
Cold gas thrusters benefit from their simplicity; however, they do fall short in other respects. The following list summarizes the advantages and disadvantages of a cold gas system.
A lack of combustion in the nozzle of a cold gas thruster allows its usage in situations where regular liquid rocket engines would be too hot. This eliminates the need to engineer heat management systems.
The simple design allows the thrusters to be smaller than regular rocket engines, which makes them a suitable choice for missions with limited volume and weight requirements.
The cold gas system and its fuel are inexpensive compared to regular rocket engines.[citation needed]
The simple design is less prone to failures than a traditional rocket engine.[citation needed]
The fuels used in a cold gas system are safe to handle both before and after firing the engine. If inert fuel is used the cold gas system is one of the safest possible rocket engines.[1]
Cold gas thrusters do not build up a net charge on the spacecraft during operation.
Cold gas thrusters require very little electrical energy to operate, which is useful, for example, when a spacecraft is in the shadow of the planet it is orbiting.
A cold gas system cannot produce the high thrust that combustive rocket engines can achieve.
Cold gas thrusters are less mass efficient than traditional rocket engines.
The maximum thrust of a cold gas thruster is dependent upon the pressure in the storage tank. As fuel is used up with simple compressed-gas systems, the pressure decreases and maximum thrust decreases.[3] With liquefied gasses, pressure will remain relatively constant as the liquid gas volatilizes and is used up in a manner similar to aerosol cans.
Thrust is generated by momentum exchange between the exhaust and the spacecraft, which is given by Newton's second law as
{\displaystyle F={\dot {m}}V_{e}}
{\displaystyle {\dot {m}}}
is the mass flow rate, and
{\displaystyle V_{e}}
is the velocity of the exhaust.
In the case of a cold gas thruster in space, where the thrusters are designed for infinite expansion (since the ambient pressure is zero), the thrust is given as
{\displaystyle F=A_{t}P_{c}\gamma \left[\left({\frac {2}{\gamma -1}}\right)\left({\frac {2}{\gamma +1}}\right)\left(1-{\frac {P_{e}}{P_{c}}}\right)\right]+P_{e}A_{e}}
{\displaystyle A_{t}}
is the area of the throat,
{\displaystyle P_{c}}
is the chamber pressure in the nozzle,
{\displaystyle \gamma }
{\displaystyle P_{e}}
is the exit pressure of the propellant, and
{\displaystyle A_{e}}
is the exit area of the nozzle.[citation needed]
The specific impulse (Isp) of a rocket engine is the most important metric of efficiency; a high specific impulse is normally desired. Cold gas thrusters have a significantly lower specific impulse than most other rocket engines because they do not take advantage of chemical energy stored in the propellant. The theoretical specific impulse for cold gasses is given by
{\displaystyle I_{sp}={\frac {C^{*}}{g_{0}}}\gamma {\sqrt {\left({\frac {2}{\gamma -1}}\right)\left({\frac {2}{\gamma +1}}\right)^{\frac {\gamma +1}{\gamma -1}}\left(1-{\frac {P_{e}}{P_{c}}}\right)^{\frac {\gamma -1}{\gamma }}}}}
{\displaystyle g_{0}}
is standard gravity and
{\displaystyle C^{*}}
is the characteristic velocity which is given by
{\displaystyle C^{*}={\frac {a_{0}}{\gamma \left({\frac {2}{\gamma +1}}\right)^{\frac {\gamma +1}{2(\gamma -1)}}}}}
{\displaystyle a_{0}}
is the sonic velocity of the propellant.[citation needed]
Cold gas systems can use either a solid, liquid or gaseous propellant storage system; but the propellant needs to exit the nozzle in gaseous form. Storing liquid propellant may pose attitude control issues due to the sloshing of fuel in its tank.
When deciding which propellant to use, a high specific impulse and a high specific impulse per unit volume of propellant must be considered.[3]
The following table provides an overview of the specific impulses of the different propellants that can be used in a cold gas propulsion system
Propellants and Efficiencies [1]
H2 2.0 296 272 0.02
He 4.0 179 165 0.04
Ne 20.2 82 75 0.19
N2 28.0 80 73 0.28
O2 32.0 ?
Ar 40.0 57 52 0.44
Kr 83.8 39 37 1.08
Xe 131.3 31 28 2.74
CCl2F2 (Freon-12) 120.9 46 37 Liquid
CF4 88.0 55 45 0.96
CH4 16.0 114 105 0.19
NH3 17.0 105 96 Liquid
N2O 44.0 67 61 Liquid
CO2 44.0 67 61 Liquid
Properties at 25°C and 1 atm
Human Propulsion[edit]
Cold gas thrusters are especially well suited for astronaut propulsion units due to the inert and non-toxic nature of their propellants.
Hand-Held Maneuvering Unit[edit]
Main article: Hand-Held Maneuvering Unit
The Hand-Held Maneuvering Unit (HHMU) used on the Gemini 4 and 10 missions used pressurized oxygen to facilitate the astronauts' extravehicular activities.[4] Although the patent of the HHMU does not categorize the device as a cold gas thruster, the HHMU is described as a "propulsion unit utilizing the thrust developed by a pressurized gas escaping various nozzle means."[5]
Manned Maneuvering Unit[edit]
Main article: Manned Maneuvering Unit
Twenty-four cold gas thrusters utilizing pressurized gaseous nitrogen were used on the Manned Maneuvering Unit (MMU). The thrusters provided full 6-degree-of-freedom control to the astronaut wearing the MMU. Each thruster provided 1.4 lbs (6.23 N) of thrust. The two propellant tanks onboard provided a total of 40 lbs (18kg) of gaseous nitrogen at 4500 psi, which provided sufficient propellant to generate a change in velocity of 110 to 135 ft/sec (33.53 to 41.15 m/s). At a nominal mass, the MMU had a translational acceleration of 0.3±0.05 ft/sec2 (9.1±1.5 cm/s2) and a rotational acceleration of 10.0±3.0 deg/sec2 (0.1745±0.052 rad/sec2)[6]
Vernier Engines[edit]
Main article: Vernier Engines
Larger cold gas thrusters are employed to help in the attitude control of the first stage of the SpaceX Falcon 9 rocket as it returns to land.[7]
In a tweet in June 2018, Elon Musk proposed the use of air-based cold gas thrusters to improve car performance.[8]
In September 2018, Bosch successfully tested its proof-of-concept safety system for righting a slipping motorcycle using cold gas thrusters. The system senses a sideways wheel slip and uses a lateral cold gas thruster to keep the motorcycle from slipping further.[9]
The main focus of current research is miniaturization of cold gas thrusters using microelectromechanical systems.[10]
^ a b c Nguyen, Hugo; Köhler, Johan; Stenmark, Lars (2002-01-01). "The merits of cold gas micropropulsion in state-of-the-art space missions". Iaf Abstracts: 785. Bibcode:2002iaf..confE.785N.
^ "Micropropulsion systems for cubesats". ResearchGate. Retrieved 2018-12-14.
^ a b Tummala, Akshay; Dutta, Atri; Tummala, Akshay Reddy; Dutta, Atri (9 December 2017). "An Overview of Cube-Satellite Propulsion Technologies and Trends". Aerospace. 4 (4): 58. doi:10.3390/aerospace4040058.
^ "Maneuvering Unit, Hand-Held, White, Gemini 4". National Air and Space Museum. 2016-03-20. Retrieved 2018-12-12.
^ US 3270986 Hand-Held Self-Maneuvering Unit
^ Lenda, J. A. "Manned maneuvering unit: User's guide." (1978).
^ plarson (2015-06-25). "The why and how of landing rockets". SpaceX. Retrieved 2018-12-16.
^ Elon Musk [@elonmusk] (June 9, 2018). "SpaceX option package for new Tesla Roadster will include ~10 small rocket thrusters arranged seamlessly around car. These rocket engines dramatically improve acceleration, top speed, braking & cornering. Maybe they will even allow a Tesla to fly …" (Tweet) – via Twitter.
^ "Greater safety on two wheels: Bosch innovations for the motorcycles of the future". Bosch Media Service. Retrieved 2018-12-14.
^ Kvell, U; Puusepp, M; Kaminski, F; Past, J-E; Palmer, K; Grönland, T-A; Noorma, M (2014). "Nanosatellite orbit control using MEMS cold gas thrusters". Proceedings of the Estonian Academy of Sciences. 63 (2S): 279. doi:10.3176/proc.2014.2s.09. ISSN 1736-6046.
Retrieved from "https://en.wikipedia.org/w/index.php?title=Cold_gas_thruster&oldid=1081310453"
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Its Role in Double-Entry Accounting
General Ledger FAQs
A general ledger represents the record-keeping system for a company’s financial data, with debit and credit account records validated by a trial balance. It provides a record of each financial transaction that takes place during the life of an operating company and holds account information that is needed to prepare the company’s financial statements. Transaction data is segregated, by type, into accounts for assets, liabilities, owners’ equity, revenues, and expenses.
The general ledger is the foundation of a company’s double-entry accounting system.
General ledger accounts encompass all the transaction data needed to produce the income statement, balance sheet, and other financial reports.
General ledger transactions are a summary of transactions made as journal entries to sub-ledger accounts.
The trial balance is a report that lists every general ledger account and its balance, making adjustments easier to check and errors easier to locate.
A general ledger is the foundation of a system employed by accountants to store and organize financial data used to create the firm’s financial statements. Transactions are posted to individual sub-ledger accounts, as defined by the company’s chart of accounts.
The transactions are then closed out or summarized to the general ledger, and the accountant generates a trial balance, which serves as a report of each ledger account’s balance. The trial balance is checked for errors and adjusted by posting additional necessary entries, and then the adjusted trial balance is used to generate the financial statements.
How a General Ledger Functions With Double-Entry Accounting
A general ledger is used by businesses that employ the double-entry bookkeeping method, which means that each financial transaction affects at least two sub-ledger accounts, and each entry has at least one debit and one credit transaction. Double-entry transactions, called “journal entries,” are posted in two columns, with debit entries on the left and credit entries on the right, and the total of all debit and credit entries must balance.
The accounting equation, which underlies double-entry accounting, is as follows:
\text{Assets} - \text{Liabilities} = \text{Stockholders' Equity}
The balance sheet follows this format and shows information at a detailed account level. For example, the balance sheet shows several asset accounts, including cash and accounts receivable, in its short-term assets section.
The double-entry accounting method works based on the accounting equation’s requirement that transactions posted to the accounts on the left of the equal sign in the formula must equal the total of transactions posted to the account (or accounts) on the right. Even if the equation is presented differently (such as Assets = Liabilities + Stockholders’ Equity), the balancing rule always applies.
The transaction details contained in the general ledger are compiled and summarized at various levels to produce a trial balance, income statement, balance sheet, statement of cash flows, and many other financial reports. This helps accountants, company management, analysts, investors, and other stakeholders assess the company’s performance on an ongoing basis.
When expenses spike in a given period, or a company records other transactions that affect its revenues, net income, or other key financial metrics, the financial statement data often doesn’t tell the whole story. In the case of certain types of accounting errors, it becomes necessary to go back to the general ledger and dig into the detail of each recorded transaction to locate the issue. At times this can involve reviewing dozens of journal entries, but it is imperative to maintain reliably error-free and credible company financial statements.
If a company receives payment from a client for a $200 invoice, for example, the company accountant increases the cash account with a $200 debit and completes the entry with a credit, or reduction, of $200 to accounts receivable. The posted debit and credit amounts are equal.
In this instance, one asset account (cash) is increased by $200, while another asset account (accounts receivable) is reduced by $200. The net result is that both the increase and the decrease only affect one side of the accounting equation. Thus, the equation remains in balance.
An Income Statement Transaction Example
The income statement follows its own formula, which can be written as follows:
Yes, a company that uses a double-entry bookkeeping method uses the general ledger method of storing company financial data. Specifically, double-entry bookkeeping is when each transaction impacts at least one debit and one credit transaction. In other words, each transaction appears in two columns, a debit column and a credit column, whose totals must balance. Under this balancing rule, the following equation applies: Assets - Liabilities = Stockholders’ Equity.
Consider the following example where a company receives a $1,000 payment from a client for its services. The accountant would then increase the asset column by $1,000 and subtract $1,000 from accounts receivable. The equation remains in balance, as the equivalent increase and decrease affect one side—the asset side—of the accounting equation.
Intuit Quickbooks. "What is a General Ledger and Why is It Important?"
CFA Institute. "Understanding Income Statements."
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Global smooth dynamics of a fully ionized plasma with long-range collisions | EMS Press
The motion of a fully ionized plasma of electrons and ions is generally governed by the Vlasov–Maxwell–Landau system. We prove the global existence of solutions near Maxwellians to the Cauchy problem of the system for the long-range collision kernel of soft potentials, particularly including the classical Coulomb collision, provided that both the Sobolev norm and
L_{\xi }^{2}(L_{x}^{1})
-norm of initial perturbation with enough smoothness and enough velocity weight is sufficiently small. As a byproduct, the convergence rates of solutions are also obtained. The proof is based on the energy method through designing a new temporal energy norm to capture different features of this complex system such as dispersion of the macro component in
\mathbb{R}^{3}
, singularity of the long-range collisions and regularity-loss of the electromagnetic field.
Renjun Duan, Global smooth dynamics of a fully ionized plasma with long-range collisions. Ann. Inst. H. Poincaré Anal. Non Linéaire 31 (2014), no. 4, pp. 751–778
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Understanding Real GDP
Real GDP FAQs
What Is Real Gross Domestic Product (Real GDP)?
Real gross domestic product (real GDP) is an inflation-adjusted measure that reflects the value of all goods and services produced by an economy in a given year (expressed in base-year prices) and is often referred to as constant-price GDP, inflation-corrected GDP, or constant dollar GDP.
Real gross domestic product (real GDP) is an inflation-adjusted measure that reflects the value of all goods and services produced by an economy in a given year (expressed in base-year prices). and is often referred to as "constant-price," "inflation-corrected", or "constant dollar" GDP.
Real GDP makes comparing GDP from year to year and from different years more meaningful because it shows comparisons for both the quantity and value of goods and services.
Real GDP is calculated by dividing nominal GDP over a GDP deflator.
Real GDP is a macroeconomic statistic that measures the value of the goods and services produced by an economy in a specific period, adjusted for inflation. Essentially, it measures a country's total economic output, adjusted for price changes.
Governments use both nominal and real GDP as metrics for analyzing economic growth and purchasing power over time. This is done using the GDP price deflator (also called the implicit price deflator), which measures the changes in prices for all of the goods and services produced in an economy.
The Bureau of Economic Analysis (BEA) provides a quarterly report on GDP with headline data statistics representing real GDP levels and real GDP growth. Nominal GDP is also included in the BEA’s quarterly report under the name current dollar. Unlike nominal GDP, real GDP accounts for changes in price levels and provides a more accurate figure of economic growth.
U.S. real GDP growth rate (annualized) during the fourth quarter of 2021, up from 2.3% growth in the third quarter. For all of 2021, real GDP grew by 5.7%, versus a 3.4% decline in 2020.
Because GDP is one of the most important metrics for evaluating the economic activity, stability, and growth of goods and services in an economy, it is usually reviewed from two angles: nominal and real. Nominal GDP is a macroeconomic assessment of the value of goods and services using current prices in its measure; it's also referred to as the current dollar GDP.
Real GDP takes into consideration adjustments for changes in inflation. This means that if inflation is positive, real GDP will be lower than nominal, and vice versa. Without a real GDP adjustment, positive inflation greatly inflates GDP in nominal terms.
U.S. real GDP grew at 5.7% for 2021, but nominal GDP (called current-dollar GDP by the BEA) grew at 10%.
Economists use the BEA’s real GDP headline data for macroeconomic analysis and central bank planning. The main difference between nominal GDP and real GDP is the taking of inflation into account. Since nominal GDP is calculated using current prices, it does not require any adjustments for inflation. This makes comparisons from quarter to quarter and year to year much simpler, though less relevant, to calculate and analyze.
As such, real GDP provides a better basis for judging long-term national economic performance than nominal GDP. Using a GDP price deflator, real GDP reflects GDP on a per quantity basis. Without real GDP, it would be difficult to identify just from examining nominal GDP whether production is actually expanding—or it's just a factor of rising per-unit prices in the economy.
A positive difference in nominal minus real GDP signifies inflation and a negative difference signifies deflation. In other words, when nominal is higher than real, inflation is occurring and when real is higher than nominal, deflation is occurring.
The GDP price deflator is considered to be a more appropriate inflation measure for measuring economic growth than the consumer price index (CPI) because it isn't based on a fixed basket of goods.
Calculating real GDP is a complex process typically best provided by the BEA. In general, calculating real GDP is done by dividing nominal GDP by the GDP deflator (R).
\begin{aligned}&\text{Real GDP} = \frac{\text{Nominal GDP}}{\text{R}}\\&\textbf{where:}\\&\text{GDP}=\text{Gross domestic product}\\&\text{R} =\text{GDP deflator}\end{aligned}
Real GDP=RNominal GDPwhere:GDP=Gross domestic productR=GDP deflator
The BEA provides the deflator on a quarterly basis. The GDP deflator is a measurement of inflation since a base year (currently 2017 for the BEA). Dividing the nominal GDP by the deflator removes the effects of inflation.
For example, if an economy's prices have increased by 1% since the base year, the deflating number is 1.01. If nominal GDP was $1 million, then real GDP is calculated as $1,000,000 / 1.01, or $990,099.
What Does 'Real' Mean in Real GDP?
Real GDP tracks the total value of goods and services calculating the quantities but using constant prices that are adjusted for inflation. This is opposed to nominal GDP that does not account for inflation. Adjusting for constant prices makes it a measure of "real" economic output for apples-to-apples comparison over time and between countries.
Real GDP is an inflation-adjusted measurement of a country’s economic output over the course of a year. The U.S. GDP is primarily measured based on the expenditure approach and calculated using the following formula: GDP = C + G + I + NX (where C=consumption; G=government spending; I=Investment; and NX=net exports).
How Will Real and Nominal GDP Differ From One Another?
In inflationary periods, real GDP will be lower than nominal GDP. In deflationary times, real GDP will be higher. Take, for example, a hypothetical country that had a nominal GDP of $100 Billion in 2000, which grew by 50% to $150 billion by 2020. Over the same period of time, inflation reduced the relative purchasing power of the dollar by 50%. Looking at just the nominal GDP, the economy appears to be performing very well, whereas the real GDP expressed in 2000 dollars would actually indicate a reading of $75 billion, revealing in fact a net overall decline in economic growth had occurred. It is due to this greater accuracy that real GDP is favored by economists as a method of measuring economic performance.
Why Is Measuring Real GDP Important?
Countries with larger GDPs will have a greater amount of goods and services generated within them, and will generally have a higher standard of living. For this reason, many citizens and political leaders see GDP growth as an important measure of national success, often referring to “GDP growth” and “economic growth” interchangeably. GDP enables policymakers and central banks to judge whether the economy is contracting or expanding, whether it needs a boost or restraint, and if a threat such as a recession or inflation looms on the horizon. By accounting for inflation, real GDP is a better gauge of the change in production levels from one period to another.
What Are Some Critiques of Using GDP?
Many economists have argued that GDP should not be used as a proxy for overall economic success, as it does not account for the informal economy, does not count care work or domestic labor in the home, ignores business-to-business activity, and counts costs and wastes as economic activity, among other shortcomings.
U.S. Bureau of Economic Analysis. "What Are Current-Dollar Estimates?" Accessed Jan. 30, 2022.
Bureau of Economic Analysis. “Gross Domestic Product, Fourth Quarter and Year 2021 (Advance Estimate).” Assessed Jan. 30, 2022.
U.S. Bureau of Economic Analysis. "GDP Price Deflator." Accessed Jan. 30, 2022.
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How to put a number in scientific notation | StudyPug
Scientific notation is a way of writing number. It is especially useful when we want to express very large and small numbers. There are two parts in scientific notation. The first part consists of digits, and the second part is x 10 to a power.
(a^x)(a^y)=a^{(x+y)}
(a^x)^y = a^{(x\cdot y)}
• How to convert scientific notations to numbers?
• How to convert numbers to scientific notations?
Write the number in scientific notation
\times {10^{5}}
\times {10^{-9}}
Write the number in standard notation
\times {10^{13}}
\times {10^{-9}}
Calculate the following scientific notations
(0.005 \times {10^{-3}} )(2.9 \times {10^{-6}} ) =
(6.75 \times {10^3} )/(0.02 \times {10^{-3}} ) =
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Three-axle vehicle body with longitudinal, lateral, and yaw motion - Simulink - MathWorks España
{\delta }_{F}=\left[\begin{array}{cc}{\delta }_{fl}& {\delta }_{fr}\end{array}\right]\text{ or }\left[\begin{array}{c}{\delta }_{fl}\\ {\delta }_{fr}\end{array}\right]
{\delta }_{M}=\left[\begin{array}{cc}{\delta }_{ml}& {\delta }_{mr}\end{array}\right]\text{ or }\left[\begin{array}{c}{\delta }_{ml}\\ {\delta }_{mr}\end{array}\right]
{\delta }_{R}=\left[\begin{array}{cc}{\delta }_{rl}& {\delta }_{rr}\end{array}\right]\text{ or }\left[\begin{array}{c}{\delta }_{rl}\\ {\delta }_{rr}\end{array}\right]
FwF=F{x}_{f}
FwF=\left[\begin{array}{cc}F{x}_{f}& F{y}_{f}\end{array}\right]\text{ or }\left[\begin{array}{c}F{x}_{f}\\ F{y}_{f}\end{array}\right]
FwF=\left[\begin{array}{cc}{F}_{xfl}& {F}_{xfr}\end{array}\right]\text{ or }\left[\begin{array}{c}{F}_{xfl}\\ {F}_{xfr}\end{array}\right]
FwF=\left[\begin{array}{cc}{F}_{xfl}& {F}_{xfr}\\ {F}_{yfl}& {F}_{yfr}\end{array}\right]
FwM=F{x}_{r}
FwM=\left[\begin{array}{cc}F{x}_{m}& F{y}_{m}\end{array}\right]\text{ or }\left[\begin{array}{c}F{x}_{m}\\ F{y}_{m}\end{array}\right]
FwM=\left[\begin{array}{cc}{F}_{xml}& {F}_{xmr}\end{array}\right]\text{ or }\left[\begin{array}{c}{F}_{xml}\\ {F}_{xmr}\end{array}\right]
FwM=\left[\begin{array}{cc}{F}_{xml}& {F}_{xmr}\\ {F}_{yml}& {F}_{ymr}\end{array}\right]
FwR=F{x}_{r}
FwR=\left[\begin{array}{cc}F{x}_{r}& F{y}_{r}\end{array}\right]\text{ or }\left[\begin{array}{c}F{x}_{r}\\ F{y}_{r}\end{array}\right]
FwR=\left[\begin{array}{cc}{F}_{xrl}& {F}_{xrr}\end{array}\right]\text{ or }\left[\begin{array}{c}{F}_{xrl}\\ {F}_{xrr}\end{array}\right]
FwR=\left[\begin{array}{cc}{F}_{xrl}& {F}_{xrr}\\ {F}_{yrl}& {F}_{yrr}\end{array}\right]
Mu=\left[\begin{array}{ccc}{\mu }_{f}& {\mu }_{m}& {\mu }_{r}\end{array}\right]\text{ or }\left[\begin{array}{c}{\mu }_{f}\\ {\mu }_{m}\\ {\mu }_{r}\end{array}\right]
Mu=\left[\begin{array}{cc}{\mu }_{fl}& {\mu }_{fr}\\ {\mu }_{ml}& {\mu }_{mr}\\ {\mu }_{rl}& {\mu }_{rr}\end{array}\right]
\beta =\frac{{V}_{y}}{{V}_{x}}
\beta =\frac{{V}_{y}}{{V}_{x}}
FzF=F{z}_{f}
FzF=\left[\begin{array}{cc}F{z}_{fl}& F{z}_{fr}\end{array}\right]
FzM=F{z}_{m}
FzM=\left[\begin{array}{cc}F{z}_{ml}& F{z}_{rl}\end{array}\right]
FzR=F{z}_{r}
FzR=\left[\begin{array}{cc}F{z}_{rl}& F{z}_{rr}\end{array}\right]
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Wallet Connect and Argent - Argent
How to make sure your Dapp works seamlessly with Argent!
WalletConnect is an open source protocol for connecting decentralized applications ("Dapps") to mobile wallets, via QR code scanning or deep linking. A user can interact securely with any Dapp from their mobile phone, making WalletConnect enabled wallets a safer choice compared to desktop or browser extension wallets.
As a mobile wallet, Argent fully supports the WalletConnect protocol. Since Argent is a smart contract based wallet, you need to pay attention to some specific features.
Externally Owned Accounts (EOA) can sign messages with their associated private keys, however, smart contracts cannot. EIP-1271 outlines a standard way for contracts to verify if a provided signature is valid when an account is a contract.
The Argent wallet implements theisValidSignature() method, as per EIP-1271. A Dapp that wants to verify the signature of an Argent user should therefore call isValidSignature() on the Argent wallet instead of ecrecover() (as would be used to verify signatures from EOA accounts).
// bytes4(keccak256("isValidSignature(bytes32,bytes)")
* @param _hash EIP-191 compliant hash of the message
* MUST return the bytes4 magic value 0x1626ba7e when function passes.
The parameter _hash should be EIP-191 compliant. See full example here in Javascript (using ethers.js library):
const argentABI = [
'function isValidSignature(bytes32 _message, bytes _signature) public view returns (bool)'
const message = "Lorem ipsum dolor sit amet";
const argentWallet = new ethers.Contract(walletAddress, argentABI, provider);
const hashMessage = ethers.utils.hashMessage(message);
const returnValue = await argentWallet.isValidSignature(hashMessage, signature)
// signature is not valid
We have developed and deployed a simple contract to detect if a given address corresponds to an Argent wallet. The contract exposes a single isArgentWallet(address) method that returns true if the code deployed at the input address matches a deployed version of the Argent wallet.
contract ArgentWalletDetector {
* @notice Checks if an address is an Argent wallet
* @param _wallet The target wallet
function isArgentWallet(
address _wallet
The detector contract is deployed on Ropsten testnet and Mainnet:
Ropsten: 0xF230cF8980BaDA094720C01308319eF192F0F311
Mainnet: 0xeca4B0bDBf7c55E9b7925919d03CbF8Dc82537E8
Argent wallets support the ability to batch transactions into a single transaction, for example an ERC20 approval followed by a contract call
"function isValidSignature(bytes32 _message, bytes _signature) public view returns (bool)",
"function wc_multiCall((address to, uint256 value, bytes data)[] _transactions)",
const result = await argentWallet.wc_multiCall(
to: "0xa0b86991c6218b36c1d19d4a2e9eb0ce3606eb48",
data: "0x095ea7b3000000000000000000000000def1c0ded9bec7f1a1670819833240f027b25eff0000000000000000000000000000000000000000fffffffffff096fb4da20000",
to: "0x7a250d5630b4cf539739df2c5dacb4c659f2488d",
data: "0x........",
The best practice is to check if the wallet is an Argent wallet using the wallet detector contract and then batch the transactions:
const walletDetectorAddress = "0xeca4B0bDBf7c55E9b7925919d03CbF8Dc82537E8";
const walletDetectorABI = [
"function isArgentWallet(address _wallet) external view returns (bool)"
const walletDetector = new ethers.Contract(walletDetectorAddress, walletDetectorABI, provider);
const isArgent = await walletDetector.isArgentWallet(address);
if (isArgent) {
// do a multicall
// do single calls
Meta transactions and relayers
Argent wallet uses the concept of meta transactions. These are a particular type of transaction which are signed by one or more key pairs (in our case the wallet's owner and potentially their guardians) but are submitted to the Ethereum network by a relayer. The relayer pays the gas fee (in ETH) and the wallet will refund the relayer (in ETH or ERC20 tokens) up to an amount signed by the wallet's owner.
From a Dapp perspective, this is managed by the Argent mobile application. The Dapp will submit a regular { to, value, data } transaction to the web3 provider. This transaction will be transmitted to the Argent mobile application through Wallet Connect. The mobile wallet will transform the data into a meta transaction:
to will be the Argent RelayerManager contract address
The Dapp will receive the transaction hash in order to monitor the status of the transaction and events will be emitted as usual.
Our relayer has the ability to replay a transaction with a higher gas price due to fluctuating network conditions. The transaction hash is modified and the Dapp will not be aware of the new transaction hash.
One solution could be for the Dapp to observe a specific event being emitted instead of transaction status. We are working on defining a standard for a Dapp to be notified when a transaction is replaced and the transaction hash changes (contact us if you are interested in this).
Login UX best practice
To improve the login experience for Argent users, we recommend showing an Argent branded login button at the top level. This ensures people less familiar with WalletConnect can log in to your app. When clicking on the Argent login, you then present the WalletConnect QR code.
You can find an example of this implementation on https://app.flexa.network/connect
Can I use the ERC20 permit function?
It depends on how the ERC20 smart contract implements the permit() method. If you look below (source Etherscan) at the Maker DAI implementation, you will notice at line 24 the use of ecrecover() and not EIP-1271 to verify the signature. Here, the call will fail with Argent wallets.
keccak256(abi.encode(PERMIT_TYPEHASH,
allowed))
require(holder != address(0), "Dai/invalid-address-0");
require(holder == ecrecover(digest, v, r, s), "Dai/invalid-permit");
require(expiry == 0 || now <= expiry, "Dai/permit-expired");
require(nonce == nonces[holder]++, "Dai/invalid-nonce");
I'm asking for ERC20 approval of an amount of
10^{53}
Argent may override the "infinite value" and ask a user to input a smaller amount due to the additional security risk of allowing large persistent approvals. When checking the allowance in your Dapp you need to check for the exact amount the user wants to spend.
Can I use 3rd party relayers with Argent
Yes, but the relayer needs to implement a call to the execute() method of our RelayerManager smart contract. You will have to get the wallet owner signing the correct payload which is not straightforward. Please contact us if you're interested in this topic.
Do you estimate gas?
Yes, we use eth_estimateGas before sending any transaction.
Do you support Wallet Connect mobile deeplink
Yes, Wallet Connect will work on mobile browsers (or native mobile apps) and will deep link to the Argent wallet
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You can test Argent on Ropsten on both iOS and Android. Contact [email protected] to receive a test build.
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Actually Applicable Application Problems and Brainteasers/Wheelchair Ramp Standards - Wikibooks, open books for an open world
Actually Applicable Application Problems and Brainteasers/Wheelchair Ramp Standards
Wheelchair ramps cannot be too steep or they will be too difficult for wheelchair users to roll up, and the wheelchair may collect an unsafe amount of momentum in rolling down. The standard is that the slope may be up to
{\displaystyle {\frac {1}{12}}}
, meaning that for each 1 foot up, the ramp must go at least 12 feet forward. (The same ratio will work with any unit of measurement, so you can use inches instead of feet, or meters, centimeters, etc.) Flatter is OK; steeper is not.
This type of calculation is Actually Applicable for building planners and construction workers, who use this strategy to find out whether existing ramps need to be replaced and check whether a planned ramp meets the standards, as well as for anyone who cares about making sure their community locations are safe for wheelchair users.
Measure the vertical change between the top and bottom of the ramp.
Measure the horizontal change between the top and bottom of the ramp.
You may need to get clever to take these measurements, such as by using a plumbline, bubble level, something with a fixed 90 degree angle, etc.
Divide the vertical change by the horizontal change to find the ramp's slope.
Compare the slope with the standard to make sure it is not over the limit.
Does a ramp with a vertical change of 8" and a horizontal distance of 6' meet the standard?
Does a ramp with a vertical change of 2' and a horizontal distance of 30' meet the standard?
Measure a ramp in your community and perform the calculations to make sure it is accessible.
Is the ramp in the picture at https://themighty.com/wp-content/uploads/2017/09/img_2394-1280x640.jpg likely to be accessible? Use measurements from a typical staircase to decide.
Related Problems[edit | edit source]
Wheelchair Ramp Standards 2 is about using trigonometry to get around a common measurement complication.
Retrieved from "https://en.wikibooks.org/w/index.php?title=Actually_Applicable_Application_Problems_and_Brainteasers/Wheelchair_Ramp_Standards&oldid=3555447"
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Equinumerosity — Wikipedia Republished // WIKI 2
"Equipollence" redirects here. For the concept in geometry, see Equipollence (geometry).
In mathematics, two sets or classes A and B are equinumerous if there exists a one-to-one correspondence (or bijection) between them, that is, if there exists a function from A to B such that for every element y of B, there is exactly one element x of A with f(x) = y.[1] Equinumerous sets are said to have the same cardinality (number of elements).[2] The study of cardinality is often called equinumerosity (equalness-of-number). The terms equipollence (equalness-of-strength) and equipotence (equalness-of-power) are sometimes used instead.
Equinumerosity has the characteristic properties of an equivalence relation.[1] The statement that two sets A and B are equinumerous is usually denoted
{\displaystyle A\approx B\,}
{\displaystyle A\sim B}
{\displaystyle |A|=|B|.}
The definition of equinumerosity using bijections can be applied to both finite and infinite sets, and allows one to state whether two sets have the same size even if they are infinite. Georg Cantor, the inventor of set theory, showed in 1874 that there is more than one kind of infinity, specifically that the collection of all natural numbers and the collection of all real numbers, while both infinite, are not equinumerous (see Cantor's first uncountability proof). In his controversial 1878 paper, Cantor explicitly defined the notion of "power" of sets and used it to prove that the set of all natural numbers and the set of all rational numbers are equinumerous (an example where a proper subset of an infinite set is equinumerous to the original set), and that the Cartesian product of even a countably infinite number of copies of the real numbers is equinumerous to a single copy of the real numbers.
Cantor's theorem from 1891 implies that no set is equinumerous to its own power set (the set of all its subsets).[1] This allows the definition of greater and greater infinite sets starting from a single infinite set.
If the axiom of choice holds, then the cardinal number of a set may be regarded as the least ordinal number of that cardinality (see initial ordinal). Otherwise, it may be regarded (by Scott's trick) as the set of sets of minimal rank having that cardinality.[1]
The statement that any two sets are either equinumerous or one has a smaller cardinality than the other is equivalent to the axiom of choice.[3]
Real Analysis | Equinumerosity
Set Theory (Part 17): Equinumerosity and "Sizes" of Sets
Set Theory and Logic – L14 Computer Science 230 - Bruce Donald, Duke University
MATH52 - Lecture 29: Different Size Infinities (Cardinality)
Infinite Sets and Foundations (Joel David Hamkins) | Ep. 17
2 Cantor's theorem
3 Dedekind-infinite sets
4 Compatibility with set operations
5 Categorial definition
Equinumerous sets have a one-to-one correspondence between them,[4] and are said to have the same cardinality. The cardinality of a set X is a measure of the "number of elements of the set".[1] Equinumerosity has the characteristic properties of an equivalence relation (reflexivity, symmetry, and transitivity):[1]
Given a set A, the identity function on A is a bijection from A to itself, showing that every set A is equinumerous to itself: A ~ A.
For every bijection between two sets A and B there exists an inverse function which is a bijection between B and A, implying that if a set A is equinumerous to a set B then B is also equinumerous to A: A ~ B implies B ~ A.
Given three sets A, B and C with two bijections f : A → B and g : B → C, the composition g ∘ f of these bijections is a bijection from A to C, so if A and B are equinumerous and B and C are equinumerous then A and C are equinumerous: A ~ B and B ~ C together imply A ~ C.
An attempt to define the cardinality of a set as the equivalence class of all sets equinumerous to it is problematic in Zermelo–Fraenkel set theory, the standard form of axiomatic set theory, because the equivalence class of any non-empty set would be too large to be a set: it would be a proper class. Within the framework of Zermelo–Fraenkel set theory, relations are by definition restricted to sets (a binary relation on a set A is a subset of the Cartesian product A × A), and there is no set of all sets in Zermelo–Fraenkel set theory. In Zermelo–Fraenkel set theory, instead of defining the cardinality of a set as the equivalence class of all sets equinumerous to it one tries to assign a representative set to each equivalence class (cardinal assignment). In some other systems of axiomatic set theory, for example in Von Neumann–Bernays–Gödel set theory and Morse–Kelley set theory, relations are extended to classes.
A set A is said to have cardinality smaller than or equal to the cardinality of a set B, if there exists a one-to-one function (an injection) from A into B. This is denoted |A| ≤ |B|. If A and B are not equinumerous, then the cardinality of A is said to be strictly smaller than the cardinality of B. This is denoted |A| < |B|. If the axiom of choice holds, then the law of trichotomy holds for cardinal numbers, so that any two sets are either equinumerous, or one has a strictly smaller cardinality than the other.[1] The law of trichotomy for cardinal numbers also implies the axiom of choice.[3]
The Schröder–Bernstein theorem states that any two sets A and B for which there exist two one-to-one functions f : A → B and g : B → A are equinumerous: if |A| ≤ |B| and |B| ≤ |A|, then |A| = |B|.[1][3] This theorem does not rely on the axiom of choice.
Cantor's theorem implies that no set is equinumerous to its power set (the set of all its subsets).[1] This holds even for infinite sets. Specifically, the power set of a countably infinite set is an uncountable set.
Assuming the existence of an infinite set N consisting of all natural numbers and assuming the existence of the power set of any given set allows the definition of a sequence N, P(N), P(P(N)), P(P(P(N))), … of infinite sets where each set is the power set of the set preceding it. By Cantor's theorem, the cardinality of each set in this sequence strictly exceeds the cardinality of the set preceding it, leading to greater and greater infinite sets.
Cantor's work was harshly criticized by some of his contemporaries, for example by Leopold Kronecker, who strongly adhered to a finitist[5] philosophy of mathematics and rejected the idea that numbers can form an actual, completed totality (an actual infinity). However, Cantor's ideas were defended by others, for example by Richard Dedekind, and ultimately were largely accepted, strongly supported by David Hilbert. See Controversy over Cantor's theory for more.
Within the framework of Zermelo–Fraenkel set theory, the axiom of power set guarantees the existence of the power set of any given set. Furthermore, the axiom of infinity guarantees the existence of at least one infinite set, namely a set containing the natural numbers. There are alternative set theories, e.g. "general set theory" (GST), Kripke–Platek set theory, and pocket set theory (PST), that deliberately omit the axiom of power set and the axiom of infinity and do not allow the definition of the infinite hierarchy of infinites proposed by Cantor.
The cardinalities corresponding to the sets N, P(N), P(P(N)), P(P(P(N))), … are the beth numbers
{\displaystyle \beth _{0}}
{\displaystyle \beth _{1}}
{\displaystyle \beth _{2}}
{\displaystyle \beth _{3}}
, …, with the first beth number
{\displaystyle \beth _{0}}
{\displaystyle \aleph _{0}}
(aleph naught), the cardinality of any countably infinite set, and the second beth number
{\displaystyle \beth _{1}}
{\displaystyle {\mathfrak {c}}}
, the cardinality of the continuum.
Dedekind-infinite sets
In some occasions, it is possible for a set S and its proper subset to be equinumerous. For example, the set of even natural numbers is equinumerous to the set of all natural numbers. A set that is equinumerous to a proper subsets of itself is called Dedekind-infinite.[1][3]
The axiom of countable choice (ACω), a weak variant of the axiom of choice (AC), is needed to show that a set that is not Dedekind-infinite is actually finite. The axioms of Zermelo–Fraenkel set theory without the axiom of choice (ZF) are not strong enough to prove that every infinite set is Dedekind-infinite, but the axioms of Zermelo–Fraenkel set theory with the axiom of countable choice (ZF + ACω) are strong enough.[6] Other definitions of finiteness and infiniteness of sets than that given by Dedekind do not require the axiom of choice for this, see Finite set § Necessary and sufficient conditions for finiteness.[1]
Compatibility with set operations
Equinumerosity is compatible with the basic set operations in a way that allows the definition of cardinal arithmetic.[1] Specifically, equinumerosity is compatible with disjoint unions: Given four sets A, B, C and D with A and C on the one hand and B and D on the other hand pairwise disjoint and with A ~ B and C ~ D then A ∪ C ~ B ∪ D. This is used to justify the definition of cardinal addition.
Furthermore, equinumerosity is compatible with cartesian products:
If A ~ B and C ~ D then A × C ~ B × D.
A × B ~ B × A
(A × B) × C ~ A × (B × C)
These properties are used to justify cardinal multiplication.
Given two sets X and Y, the set of all functions from Y to X is denoted by XY. Then the following statements hold:
If A ~ B and C ~ D then AC ~ BD.
AB ∪ C ~ AB × AC for disjoint B and C.
(A × B)C ~ AC × BC
(AB)C ~ AB×C
These properties are used to justify cardinal exponentiation.
Furthermore, the power set of a given set A (the set of all subsets of A) is equinumerous to the set 2A, the set of all functions from the set A to a set containing exactly two elements.
Categorial definition
In category theory, the category of sets, denoted Set, is the category consisting of the collection of all sets as objects and the collection of all functions between sets as morphisms, with the composition of functions as the composition of the morphisms. In Set, an isomorphism between two sets is precisely a bijection, and two sets are equinumerous precisely if they are isomorphic as objects in Set.
Combinatorial class
Hume's principle
^ a b c d e f g h i j k l Suppes, Patrick (1972) [originally published by D. van Nostrand Company in 1960]. Axiomatic Set Theory. Dover. ISBN 0486616304.
^ Enderton, Herbert (1977). Elements of Set Theory. Academic Press Inc. ISBN 0-12-238440-7.
^ a b c d Jech, Thomas J. (2008) [Originally published by North–Holland in 1973]. The Axiom of Choice. Dover. ISBN 978-0-486-46624-8.
^ Weisstein, Eric W. "Equipollent". mathworld.wolfram.com. Retrieved 2020-09-05.
^ Tiles, Mary (2004) [Originally published by Basil Blackwell Ltd. in 1989]. The Philosophy of Set Theory: An Historical Introduction to Cantor's Paradise. Dover. ISBN 978-0486435206.
^ Herrlich, Horst (2006). Axiom of Choice. Lecture Notes in Mathematics 1876. Springer-Verlag. ISBN 978-3540309895.
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A Computer-Aided Design Module to Analyze Manufacturing Configurations of Bent and Hydroformed Tubes | J. Manuf. Sci. Eng. | ASME Digital Collection
Elisabetta Amici,
Elisabetta Amici
, via di Castel Romano 100, 00128 Rome, Italy
Francesca Campana,
, Via Eudossiana 18, 00184 Rome, Italy
e-mail: francesca.campana@uniroma1.it
Amici, E., Campana, F., and Mancini, E. (May 30, 2007). "A Computer-Aided Design Module to Analyze Manufacturing Configurations of Bent and Hydroformed Tubes." ASME. J. Manuf. Sci. Eng. October 2007; 129(5): 979–983. https://doi.org/10.1115/1.2752522
This paper presents a computer-aided design (CAD) module able to analyze different manufacturing configurations of tubes used in mechanical assemblies, such as exhaust system manifolds. It can be included in the knowledge-based expert system category and has been implemented into a CAD platform as a dedicated module able to take into account manufacturing requirements related to tube bending, hydroforming, and cutting. The expert’s knowledge, in terms of set of rules and criteria, has been implemented by means of the automation tools of CATIAV5R10 according to the so-called methodological formal approach. The resulting module is able to join different tubes starting from their geometrical models, obtaining a set of manufacturing alternatives. Each of them is verified with respect to collisions with a bending machine and also in terms of hydroforming process feasibility. Only those solutions that satisfy these checks are accepted as feasible and ranked according to three evaluation criteria related to manufacturing cost and easiness. The system is completely automatic and able to analyze more than 100 different configurations in
<10min
. The feasible solutions are saved as CAD model to allow FEA of hydroforming and other possible CAE activities. Unfeasible solutions are deleted but reported and documented in a log file. The feasible solution rank is given in a table and has been developed according to a multicriteria approach to make optimal solution detection easier. The proposed test case aims to show and discuss these capabilities. By this module, two or more components of the exhaust system manifold can be manufactured in one stroke as a single component, starting from the same pipe and next trimmed to obtain the desired final parts. This capability can be used to reduce scraps and improve cycle time of the manufacturing process.
pipes, CAD, bending, exhaust systems, finite element analysis, computer aided engineering, production engineering computing, expert systems, forming processes, design to manufacturing, knowledge based expert system, tube bending, tube hydroforming, CAE
Computer-aided design, Computer-aided engineering, Exhaust systems, Expert systems, Machinery, Manufacturing, Pipes, Cylinders, Collisions (Physics), Design, Finite element analysis, Shapes, Tube bending, Manifolds
Knowledge Based Product Life Cycle Systems: Principles of Integration of KBE and C3P
. Cambridge University Press, Cambridge, UK.
A Task-Based Methodology for Specifying Expert Systems
Towards a Unification of Knowledge Modeling Approaches
An Overall Review of the Tube Hydroforming (THF) Technology
Tube Hydroforming: Current Research, Applications and Need for Training
Silvennoimen
Norm Methods and Partial Weighting in Multicriterion Optimization of Structures
MultiCriteria in Industry Optimization
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Loss in Axial Compressor Bleed Systems | J. Turbomach. | ASME Digital Collection
S. D. Grimshaw,
Email: sdg33@cam.ac.uk
J. Brind,
J. Brind
R. Seki
Fluid Dynamics Research Department, Research & Innovation Center,
2-1-1 Shinhama Arai-Cho, Takasago, Hyogo 676-8686
Email: ryosuke_seki@mhi.co.jp
Grimshaw, S. D., Brind, J., Pullan, G., and Seki, R. (August 24, 2020). "Loss in Axial Compressor Bleed Systems." ASME. J. Turbomach. September 2020; 142(9): 091008. https://doi.org/10.1115/1.4047614
Loss in axial compressor bleed systems is quantified and the loss mechanisms are identified to determine how efficiency can be improved. For a given bleed air pressure requirement, reducing bleed system loss allows air to be bled from further upstream in the compressor, with benefits for the thermodynamic cycle. A definition of isentropic efficiency, which includes bleed flow is used to account for this. Two cases with similar bleed systems are studied: a low-speed, single-stage research compressor, and a large industrial gas turbine high-pressure compressor. A new method for characterizing bleed system loss is introduced, using research compressor test results as a demonstration case. A loss coefficient is defined for a control volume including only flow passing through the bleed system. The coefficient takes a measured value of 95% bleed system inlet dynamic head and is shown to be a weak function of compressor operating point and bleed rate, varying by ±
2.2%
over all tested conditions. This loss coefficient is the correct nondimensional metric for quantifying and comparing bleed system performance. Computations of the research compressor and industrial gas turbine compressor identify the loss mechanisms in the bleed system flow. In both cases, approximately two-thirds of total loss is due to shearing of a high-velocity jet at the rear face of the bleed slot, one-quarter is due to mixing in the plenum chamber, and the remainder occurs in the off-take duct. Therefore, the main objective of a designer should be to diffuse the flow within the bleed slot. A redesigned bleed slot geometry is presented that achieves this objective and reduces the loss coefficient by 31%.
computational fluid dynamics (CFD), compressor, and turbine aerodynamic design, fluid dynamics and heat transfer phenomena in compressor and turbine components of gas turbine engines, measurement techniques
Compressors, Flow (Dynamics), Turbines, Industrial gases, Pressure, Design, Ducts
ASME Turbo Expo 2012, Paper No. GT2012-68242
Aerodynamic Investigations of a Compressor Bleed Air Configuration Typical for Aeroengines
Paper No. ISABE-2005-1264
1993 IGTI Scholar Lecture, ASME Paper No. 93-GT-4351993
Bleed Flow Interactions With an Axial Flow Compressor Powerstream
AIAA/ASME Joint Propulsion Conference
Paper No. AIAA 2002-4057
ASME Turbo Expo 1998, Paper No. 98-GT-249
Numerical and Experimental Study on Bleed Impact on Intermediate Compressor Duct Performance
The Cost of Flow Control in a Compressor
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Uniform Lipschitz regularity for classes of minimizers in two phase free boundary problems in Orlicz spaces with small density on the negative phase | EMS Press
J. Ederson M. Braga
Departamento de Matemática, Universidade Federal do Ceará, Campus do Pici – Bloco 914, CEP 60455-760, Fortaleza, Ceará, Brazil
In this paper we investigate Lipschitz regularity of minimizers for classes of functionals including ones of the type
E_{G}(u,\Omega ) = \int _{\Omega }[G(|\mathrm{∇}u|) + f_{2}\chi _{\{u > 0\}} + f_{1}\chi _{\{u⩽0\}}]\:dx
. We prove that there exists a universal “tolerance” (depending only on the degenerate ellipticity and other intrinsic parameters) for the density of the negative phase along the free boundary under which uniform Lipschitz regularity holds. We also prove density estimates from below for the negative phase on points inside the contact set between the negative and positive free boundaries in the case where Lipschitz regularity fails to be the optimal one.
J. Ederson M. Braga, Diego R. Moreira, Uniform Lipschitz regularity for classes of minimizers in two phase free boundary problems in Orlicz spaces with small density on the negative phase. Ann. Inst. H. Poincaré Anal. Non Linéaire 31 (2014), no. 4, pp. 823–850
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How to add integers? | StudyPug
You see numbers in your everyday life all the time. Numbers can be either positive or negative. When you see a number written without a "+" or "-" sign, that is, without a sign, it usually means that the number is positive. For example, if you saw the number 6 exactly the way we wrote it here, it means positive 6.
Adding positive numbers involve just simple addition, and subtracting them is just simple subtraction.
When we talk about negative numbers, things get a bit more difficult. A negative number has a "-" sign in front of it. For example, -6 would mean negative 6.
What happens when we try to add two negative numbers? Two negatives actually make a positive! When you see two minus signs next to each other, you'll know to add it instead. Why does it work like this? Let's take an example dealing with your bank account.
If you've got $10 in the bank, but the bank accidentally removed $5, it'd look like: $10 - $5 = $5. You'd be left with $5 in your account. Since it was the bank's mistake and they want to take away a mistake (the -$5), it'll look like $5 - (-$5) = $5 + $5 = $10. They're remove the negative (the mistake in this case), which in the end, makes it a positive, so you'll end up with $10 in your bank again.
An example of two negatives in real life can be seen through double negatives when we're speaking. If we told you to not cross the road, it would mean just that. But if we told you to not not cross the road, it means you should cross the road. In a similar sense, you can use this to help you when adding negative numbers.
A good rule of thumb is that when you see two like signs, add the numbers together (i.e. two "-" signs, or two "+" signs). If you see unlike signs (i.e. one "-" and one "+"), it becomes a negative and you'll have to subtract it.
Let's try some number line examples to see how adding and subtracting positive and negative integers work.
What addition statement does each of the following diagrams represent? Then, find the sum.
In this question, the first arrow = +3.
The second arrow = +4.
This is an addition statement and the final sum of the numbers gives us 7.
Use a number line to add
The first term we have is a +2. This means that the first arrow we'd draw will represent +2.
The second arrow we draw represents +4.
Then we'll get the final answer of landing on 6 on the number line.
Our first arrow represents -8
The second arrow we will draw represents -3
In the end, we'll get a total of -11.
If you need to check your calculations on positive and negative integer calculation, try out this online calculator.
Looking for more topics to review? Learn what the order of operations are with PEMDAS, and how to deal with subtraction and addition of decimal numbers. There are models to help you add and subtract fractions as well such as adding fractions with like denominators. You can also try your hand at divisibility rules that can help you find factors of whole numbers.
In this section, we will use our knowledge from previous section to build our understanding of integer addition. We will be asked to write addition statements using arrows on number lines. A thermometer is just like a number line. In this section, the blue arrows on the number lines will move to the left, or down the number line, and represent negative integers. A negative integer on a thermometer would represent a cooler temperature. In this section, the red arrows will move to the right, or up the number line, and represent positive integers. A positive integer on a thermometer would represent a warmer temperature.
Basic Concepts: Comparing and ordering numbers, Divisibility rules, Introduction to Exponents
Related Concepts: Adding and subtracting decimals, Order of operations (PEMDAS), Using models to add and subtract fractions, Adding fractions with like denominators
Adding 1-digit Numbers Vertically – Same Signs
Adding 1-digit Numbers Vertically – Opposite Signs
Adding 2-digit Numbers Vertically
Two like signs produce a positive sign.
Two unlike signs produce a negative sign.
The order of the integers is interchangeable.
If units add up to 10 or more, we carry the 10 to the next digit by adding 1 to it.
To add two numbers with LIKE signs, first we add up the two numbers regardless of the sign, then prefix the number with the sign they share.
To add two numbers with UNLIKE signs, we re-order the integers so that the integer with the higher magnitude goes first, then subtract the numbers regardless of the signs, lastly we keep the sign of the number with the higher magnitude.
Use a number line to add.
The temperature in city A was +28°C. The temperature then dropped by 13°C at midnight. What was the final temperature?
A diver was diving 2 m under the water. He descended 8 m and then ascended 4 m. What was his final depth under the water?
Add the following numbers vertically.
(+2)+(+4)
(-5)+(-3)
(+8)+(+6)
(-9)+(-4)
(-7)+(+1)
(-7)+(+9)
(+5)+(-2)
(+5)+(-8)
(+36)+(+28)
(-39)+(-54)
(+12)+(-43)
(-22)+(+63)
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Real-time state update by state-space model Kalman filtering - MATLAB update - MathWorks Switzerland
{x}_{t}=0.5{x}_{t-1}+{u}_{t},
{u}_{t}
{x}_{t}
{y}_{t}={x}_{t}+{\epsilon }_{t},
{\epsilon }_{t}
\begin{array}{l}\left[\begin{array}{c}{x}_{1,t}\\ {x}_{2,t}\end{array}\right]=\left[\begin{array}{cc}\varphi & \theta \\ 0& 0\end{array}\right]\left[\begin{array}{c}{x}_{1,t-1}\\ {x}_{2,t-1}\end{array}\right]+\left[\begin{array}{c}1\\ 1\end{array}\right]{u}_{t}\\ {y}_{t}-\beta {Z}_{t}={x}_{1,t}+\sigma {\epsilon }_{t},\end{array}
{x}_{1,t}
{x}_{2,t}
{y}_{1,t}
{Z}_{t}
{u}_{t}
{\epsilon }_{\mathit{t}}
\sigma
\sigma
{y}_{t}-{Z}_{t}\beta =C{x}_{t}+D{u}_{t}.
\left[\begin{array}{c}{x}_{t}\\ {y}_{t}\end{array}\right]=\left[\begin{array}{c}{A}_{t}\\ {A}_{t}{C}_{t}\end{array}\right]{x}_{t-1}+\left[\begin{array}{cc}{B}_{t}& 0\\ {B}_{t}{C}_{t}& {D}_{t}\end{array}\right]\left[\begin{array}{c}{u}_{t}\\ {\epsilon }_{t}\end{array}\right].
\left[\begin{array}{c}{x}_{t}\\ {y}_{t}\end{array}\right]|{Y}_{1}^{t-1}~Ν\left(\left[\begin{array}{c}{\stackrel{^}{x}}_{t|t-1}\\ {\stackrel{^}{y}}_{t|t-1}\end{array}\right],\left[\begin{array}{cc}{P}_{t|t-1}& {L}_{t|t-1}\\ {L}_{t|t-l}^{\prime }& {V}_{t|t-1}\end{array}\right]\right),
{\stackrel{^}{x}}_{t|t-1}={A}_{t}{\stackrel{^}{x}}_{t-1|t-1},
{\stackrel{^}{y}}_{t|t-1}={C}_{t}{\stackrel{^}{x}}_{t|t-1},
{P}_{t|t-1}={A}_{t}{P}_{t-1|t-1}{A}_{t}^{\prime }+{B}_{t}{B}_{t}^{\prime },
{V}_{t|t-1}={C}_{t}{P}_{t-1|t-1}{C}_{t}^{\prime }+{D}_{t}{D}_{t}^{\prime },
{L}_{t|t-1}={P}_{t-1|t-1}{C}_{t}^{\prime },
{x}_{t}|{Y}_{1}^{t}~Ν\left({\stackrel{^}{x}}_{t|t},{P}_{t|t}\right),
{\stackrel{^}{x}}_{t|t}={\stackrel{^}{x}}_{t|t-1}+{L}_{t|t-1}{V}_{t|t-1}^{-1}\left({y}_{t}-{\stackrel{^}{y}}_{t|t-1}\right),
{P}_{t|t}={P}_{t|t-1}-{L}_{t|t-1}{V}_{t|t-1}^{-1}{L}_{t|t-1}^{\prime },
{\stackrel{^}{x}}_{t-1|t-1}
{\stackrel{^}{x}}_{t|t}
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Integration by parts - MATLAB integrateByParts - MathWorks Switzerland
G = integrateByParts(F,du) applies integration by parts to the integrals in F, in which the differential du is integrated. For more information, see Integration by Parts.
When specifying the integrals in F, you can return the unevaluated form of the integrals by using the int function with the 'Hold' option set to true. You can then use integrateByParts to show the steps of integration by parts.
Create a symbolic expression F that is the integral of a product of functions.
\int u\left(x\right) \frac{\partial }{\partial x}\mathrm{ }v\left(x\right)\mathrm{d}x
Apply integration by parts to F.
u\left(x\right) v\left(x\right)-\int v\left(x\right) \frac{\partial }{\partial x}\mathrm{ }u\left(x\right)\mathrm{d}x
Apply integration by parts to the integral
\int {\mathit{x}}^{2}\text{\hspace{0.17em}}{\mathit{e}}^{\mathit{x}}\mathit{dx}
Define the integral using the int function. Show the result without evaluating the integral by setting the 'Hold' option to true.
\int {x}^{2} {\mathrm{e}}^{x}\mathrm{d}x
To show the steps of integration, apply integration by parts to F and use exp(x) as the differential to be integrated.
{x}^{2} {\mathrm{e}}^{x}-\int 2 x {\mathrm{e}}^{x}\mathrm{d}x
{x}^{2} {\mathrm{e}}^{x}-2 x {\mathrm{e}}^{x}+\int 2 {\mathrm{e}}^{x}\mathrm{d}x
Evaluate the integral in H by using the release function to ignore the 'Hold' option.
2 {\mathrm{e}}^{x}+{x}^{2} {\mathrm{e}}^{x}-2 x {\mathrm{e}}^{x}
Compare the result to the integration result returned by the int function without setting the 'Hold' option to true.
{\mathrm{e}}^{x} \left({x}^{2}-2 x+2\right)
\int {\mathit{e}}^{\mathit{ax}}\text{\hspace{0.17em}}\mathrm{sin}\left(\mathit{bx}\right)\text{\hspace{0.17em}}\mathit{dx}
Define the integral using the int function. Show the integral without evaluating it by setting the 'Hold' option to true.
\int {\mathrm{e}}^{a x} \mathrm{sin}\left(b x\right)\mathrm{d}x
To show the steps of integration, apply integration by parts to F and use
{\mathit{u}}^{\prime }\left(\mathit{x}\right)={\mathit{e}}^{\mathit{ax}}
as the differential to be integrated.
\frac{{\mathrm{e}}^{a x} \mathrm{sin}\left(b x\right)}{a}-\int \frac{b {\mathrm{e}}^{a x} \mathrm{cos}\left(b x\right)}{a}\mathrm{d}x
Evaluate the integral in G by using the release function to ignore the 'Hold' option.
\frac{{\mathrm{e}}^{a x} \mathrm{sin}\left(b x\right)}{a}-\frac{b {\mathrm{e}}^{a x} \left(a \mathrm{cos}\left(b x\right)+b \mathrm{sin}\left(b x\right)\right)}{a \left({a}^{2}+{b}^{2}\right)}
-\frac{{\mathrm{e}}^{a x} \left(b \mathrm{cos}\left(b x\right)-a \mathrm{sin}\left(b x\right)\right)}{{a}^{2}+{b}^{2}}
Example: int(u*diff(v))
du — Differential to be integrated
symbolic variable | symbolic expression | symbolic function
Differential to be integrated, specified as a symbolic variable, expression, or function.
Example: diff(u)
Mathematically, the rule of integration by parts is formally defined for indefinite integrals as
\int u\text{'}\left(x\right)\text{\hspace{0.17em}}v\left(x\right)\text{\hspace{0.17em}}dx=u\left(x\right)\text{\hspace{0.17em}}v\left(x\right)-\int u\left(x\right)\text{\hspace{0.17em}}v\text{'}\left(x\right)\text{\hspace{0.17em}}dx
\underset{a}{\overset{b}{\int }}u\text{'}\left(x\right)\text{\hspace{0.17em}}v\left(x\right)\text{\hspace{0.17em}}dx=u\left(b\right)\text{\hspace{0.17em}}v\left(b\right)-u\left(a\right)\text{\hspace{0.17em}}v\left(a\right)-\underset{a}{\overset{b}{\int }}u\left(x\right)\text{\hspace{0.17em}}v\text{'}\left(x\right)\text{\hspace{0.17em}}dx.
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Gagliardo–Nirenberg inequalities and non-inequalities: The full story | EMS Press
Gagliardo–Nirenberg inequalities and non-inequalities: The full story
Université de Lyon, Université Lyon 1, CNRS UMR 5208 Institut Camille Jordan, 43, boulevard du 11 novembre 1918, F-69622 Villeurbanne Cedex, France, Simion Stoilow Institute of Mathematics of the Romanian Academy, Calea Griviţei 21, 010702 Bucureşti, Romania
Department of Mathematics, Rutgers University, Hill Center, Busch Campus, 110 Frelinghuysen Rd., Piscataway, NJ 08854, USA, Mathematics Department, Technion - Israel Institute of Technology, Haifa, 32000, Israel
We investigate the validity of the Gagliardo–Nirenberg type inequality
\|f\|_{W^{s,p}(\mathrm{\Omega })}≲\|f\|_{W^{s_{1},p_{1}}(\mathrm{\Omega })}^{\theta }\|f\|_{W^{s_{2},p_{2}}(\mathrm{\Omega })}^{1−\theta },
\mathrm{\Omega } \subset \mathbb{R}^{N}
0 \leq s_{1} \leq s \leq s_{2}
are non negative numbers (not necessarily integers),
1 \leq p_{1},p,p_{2} \leq \infty
, and we assume the standard relations
s = \theta s_{1} + (1−\theta )s_{2},\:1/ p = \theta / p_{1} + (1−\theta )/ p_{2}\text{ for some }\theta \in (0,1).
By the seminal contributions of E. Gagliardo and L. Nirenberg, (1) holds when
s_{1},s_{2},s
are integers. It turns out that (1) holds for “most” of values of
s_{1},…,p_{2}
, but not for all of them. We present an explicit condition on
s_{1},s_{2},p_{1},p_{2}
which allows to decide whether (1) holds or fails.
Petru Mironescu, Haïm Brezis, Gagliardo–Nirenberg inequalities and non-inequalities: The full story. Ann. Inst. H. Poincaré Anal. Non Linéaire 35 (2018), no. 5, pp. 1355–1376
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The Richter scale assigns a magnitude number to quantify the energy released by an earthquake. The magnitude of an earthquake was defined by Charles Richter to be
M =
\frac { I } { S }
I
is the intensity of the earthquake (measured by the amplitude of a seismograph reading taken
100
km from the epicenter of the earthquake) and
S
is the intensity of a “standard earthquake” (whose amplitude is
1
= 10^{-4}
cm). For example, an earthquake measuring
3.6
on the Richter scale is
10^{0.2}
1.6
times as intense as an earthquake measuring
3.4
on the Richter scale.
How many times stronger is an earthquake that measures
6.5
on the Richter scale than an earthquake that measures
5.5
10
5.1
than an earthquake that measures
4.3
? Give your answer both as a power of
10
and as a decimal. Make sure your answer has the same level of precision as the Richter measurements.
10^{0.8} ≈ 6.3
What is the magnitude of an earthquake that is half as strong as an earthquake measuring
6.2
on the Richter scale?
Calculate the energy released by a Richter measurement of
6.2
. Then divide in half.
10^?=10^\frac{6.2}{2}
You may want to use logs or guess and check to find the answer.
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1.2: Chemicals Compose Ordinary Things - Chemistry LibreTexts
https://chem.libretexts.org/@app/auth/3/login?returnto=https%3A%2F%2Fchem.libretexts.org%2FBookshelves%2FIntroductory_Chemistry%2FMap%253A_Introductory_Chemistry_(Tro)%2F01%253A_The_Chemical_World%2F1.02%253A_Chemicals_Compose_Ordinary_Things
Figure \(\PageIndex{1}\): Atoms are incredible small. To give you an idea of the size of an atom, a single copper penny contains approximately 28,000,000,000,000,000,000,000 atoms (that’s 28 sextillion).
{\displaystyle {}_{17}^{35}{\text{Cl}}}
\(\ce{^{35}_{17}Cl}\) was the most common isotope with an abundance of 75.77% and
{\displaystyle {}_{17}^{37}{\text{Cl}}}
\(\ce{^{37}_{17}Cl}\) had an abundance of the remaining 24.23%. To calculate the average mass, first convert the percentages into fractions; that is, simply divide them by 100. Now, chlorine-35 represents a fraction of natural chlorine of 0.7577 and has a mass of 35 (the mass number). Multiplying these, we get (0.7577 × 35) = 26.51. To this, we need to add the fraction representing chlorine-37, or (0.2423 × 37) = 8.965; adding, (26.51 + 8.965) = 35.48, which is the weighted average atomic mass for chlorine. Whenever we do mass calculations involving elements or compounds (combinations of elements), we always need to use average atomic masses.
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Get better math marks with our complete 7th grade math help. We've got you covered – whether it's 7th Grade Common Core Math, EngageNY 7th grade math, 7th grade math TEKS (STAAR), or 7th grade math EOG!
Keeping with your textbook and class, our 7th grade math video lessons walk you through all topics such as, Fractions, Proportional relationships, Unit rate, Distributive property, Solving equations, Circles, Volume and area, Statistics and probability, and so many more. Learn the concepts with our tutorials that show you step-by-step solutions to even the hardest 7th grade math problems. Then, reinforce your understanding with tons of 7th grade math practice.
All our lessons are taught by experienced 7th grade math teachers. Let's finish your homework in no time, and ACE that final.
5Operations with Decimal Numbers
7Geometry and Measurement
7.2Perpendicular bisectors
7.3Angle bisectors
8.1Metric systems
10Multiplying and Dividing Fractions
10.1Multiplying fractions and whole numbers
10.2Dividing fractions with whole numbers
10.3Multiplying proper fractions
10.4Multiplying improper fractions and mixed numbers
10.5Dividing fractions and mixed numbers
10.6Applications of fraction operations
11Fractions, Decimals, and Percents
11.1Representing percents
11.2Modeling percents
11.4Converting among decimals, fractions, and percents
11.6Applications of percents
12.1Ratios
13.1Squares and square roots
13.2Pythagorean theorem
13.3Estimating square roots
14Powers and Exponents
14.1Using exponents to describe numbers
15Introduction to 3D Objects
15.1Introduction to surface area of 3-dimensional shapes
x + a = b
ax = b
\frac{x}{a} = b
ax + b = c
\frac{x}{a} + b = c
\;a\left( {x + b} \right) = c
19.5Basic probability
19.7Probability of independent events (advanced)
20.1Advantages and disadvantages of different graphs
20.4Influencing factors in data collection
My textbook is called Go Math 7th grade. Is your 7th grade math help right for my class?
Of course! We have math help in all topics you'll find in your textbook. Not just Go Math, we also cover all topics in other common 7th grade math textbooks including Big Ideas math book 6th grade, McGraw Hill 6th grade math, Holt McDougal 6th grade math.
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Definitely. We are a great additional resource to your child's learning as we have complete walkthrough solutions to all of our 6th grade math problems. Also, your StudyPug subscription gives you unlimited access to all math help across all courses. Therefore, you can skip, review and learn any materials anytime based on your child's progress and needs.
What course should I take after 7th grade math?
After mastering 7th grade math, your follow up course should be 8th grade math or Pre-algebra . If you want to review some of the previous concepts you have learned before, check out 6th grade math .
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What rational numbers have square roots? | StudyPug
In this section, we will look at how to evaluate a rational number by using square roots. We will also work on questions determining whether a rational number is a perfect square. We will also evaluate the square roots of rational numbers.
Related Concepts: Square and square roots, Cubic and cube roots, Evaluating and simplifying radicals
Use the graph below to determine a rational number with a square root between 4 and 5.
Use the side lengths below to estimate and calculate the area of each square.
Find out if each of the following rational numbers is a perfect square.
\frac{{81}}{{16}}
\frac{5}{{14}}
\sqrt {361}
\sqrt {2209}
\sqrt {0.0169}
\sqrt {5.76}
\sqrt {56}
, to the nearest tenth
\sqrt {3.7}
, to the nearest hundredth
\sqrt {0.96}
\sqrt {0.066}
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A-Level Maths (Legacy) Tutor, Help and Practice Online | StudyPug
A-Level Maths (Legacy)
A-Level Maths (Legacy) made completely easy!
Get an A with our complete A level Maths help. Whether it's for Edexcel (Pearson), AQA, OCR or WJEC, our A level maths tutors got you all covered!
Aligning with all exam boards' A level maths specifications, our comprehensive help includes all materials in Core 1, Core 2, Core 3 and Core 4 maths such as Derivatives, Integration, Factorisation, Solving simultaneous equations, Trigonometry, Vectors, Circle theorems, and more. Learn the concepts with our online video tutorials that show you step-by-step solutions to even the hardest a level maths questions. Then, reinforce your understanding with tons of a level maths practice.
All our lessons are taught by experienced A-level Maths tutors. Let's finish your homework in no time, and ACE that exam.
See our A-Level Maths (Legacy) topics
Meet Dennis, your A-Level Maths (Legacy) tutor
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We follow a level maths syllabus of All school boards. We have all topics tested in A level Maths exam. Therefore, not only Edexcel, pupils taking the exam with all other exam boards, including AQA, OCR, and WJEC, will find all the help you need here too.
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It doesn't matter. Your StudyPug subscription gives you unlimited access to all maths help across all courses. You can review all maths help in all courses, including GCSE maths, year 11 and year 12 maths. To save your precious study time, you can skip, review and learn any materials anytime based on your needs.
A-Level Maths (Legacy) topics
x^2 - y^2
y = ax^2 + bx+c
y = a(x-p)^2 + q
6.6Solving 3 variable simultaneous equations with no or infinite solutions
7.2Adding and subtracting algebraic fractions (free lessons)
8.1Polynomial long division (free lessons)
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)
y = f(-x)
y = -f(x)
(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
\frac{o}{h}
\frac{a}{h}
\frac{o}{a}
29.12Curve sketching
32.1Introduction to differential equations
32.2Separable equations
32.3Applications to differential equations
33.8Unit Vector
34.5Spread of a data set - standard deviation & variance
37.4Poisson distribution
38.7Confidence intervals to estimate population mean
38.8Student's t-distribution
Ready to do better in A-Level Maths (Legacy)?
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therefore let the equation of the hyperbola be
\frac{{x}^{2}}{{a}^{2}}-\frac{{y}^{2}}{{b}^{2}}=1..............\left(1\right)
e=\sqrt{5}\quad and\quad a+b=9
{b}^{2}={a}^{2}\left({e}^{2}-1\right)\phantom{\rule{0ex}{0ex}}\frac{{b}^{2}}{{a}^{2}}=5-1=4\phantom{\rule{0ex}{0ex}}\frac{b}{a}=2\phantom{\rule{0ex}{0ex}}b=2a\phantom{\rule{0ex}{0ex}}a+2a=9\phantom{\rule{0ex}{0ex}}3a=9⇒a=9/3=3
\frac{{x}^{2}}{{3}^{2}}-\frac{{y}^{2}}{{6}^{2}}=1\phantom{\rule{0ex}{0ex}}\frac{{x}^{2}}{9}-\frac{{y}^{2}}{36}=1
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Complete bounded embedded complex curves in $\mathbb C^2$ | EMS Press
Complete bounded embedded complex curves in
\mathbb C^2
We prove that any convex domain of
\mathbb C^2
carries properly embedded complete complex curves. In particular, we give the first examples of complete bounded embedded complex curves in
\mathbb C^2
Antonio Alarcón, Francisco J. López, Complete bounded embedded complex curves in
\mathbb C^2
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Universal measurability and the Hochschild class of the Chern character | EMS Press
We study notions of measurability for singular traces, and characterise universal measurability for operators in Dixmier ideals. This measurability result is then applied to improve on the various proofs of Connes’ identication of the Hochschild class of the Chern character of Dixmier summable spectral triples.
The measurability results show that the identication of the Hochschild class is independent of the choice of singular trace. As a corollary we obtain strong information on the asymptotics of the eigenvalues of operators naturally associated to spectral triples
\mathcal A, H, D
and Hochschild cycles for
\mathcal A
Alan L. Carey, Adam Rennie, Fedor Sukochev, Dmitriy Zanin, Universal measurability and the Hochschild class of the Chern character. J. Spectr. Theory 6 (2016), no. 1, pp. 1–41
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The Brenner–Hochster–Kollár and Whitney problems for vector-valued functions and jets | EMS Press
In this paper, we give analytic methods for finding
C^{m}(\mathbb{R}^n)
C^{m,1}(\mathbb{R}^n)
solutions of a finite system of linear equations. Along the way, we also solve a generalized Whitney problem for vector-valued functions and jets.
Charles Fefferman, Garving K. Luli, The Brenner–Hochster–Kollár and Whitney problems for vector-valued functions and jets. Rev. Mat. Iberoam. 30 (2014), no. 3, pp. 875–892
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Differentiations & Conversions - Organic Chemistry Chemistry NEET Practice Questions, MCQs, Past Year Questions (PYQs), NCERT Questions, Question Bank, Class 11 and Class 12 Questions, and PDF solved with answers
A compound A of formula C3H6Cl2 on reaction with alkali can give B of formula C3H6O or C of formula C3H4. B on oxidation gave a compound of the formula C3H7O2. C with dilute H2SO4 containing Hg2+ ion gave D of formula C3H6O, which with bromine and NaOH gave the sodium salt of C2H4O2. Then A is :
1. CH3CH2CHCl2
2. CH3CCl2CH3
3. CH2ClCH2CH2Cl
4. CH3CHClCH2Cl
Subtopic: conversion chart for n to n carbon |
Identify the product (A) in following reaction series,
{\mathrm{CH}}_{3}\mathrm{CN}\stackrel{\mathrm{Na}/{\mathrm{C}}_{2}{\mathrm{H}}_{5}\mathrm{OH}}{\to }
\stackrel{{\mathrm{HNO}}_{2}}{\to }
\stackrel{\left[\mathrm{O}\right]}{\to }
\stackrel{\mathrm{Tollens} \mathrm{reagent}}{\to }
(A) :
4. CH3 - CH2-NHOH
Compound (A), C8H9Br , gives a white precipitate when warmed with alcholic AgNO3. Oxidation of (A) gives an acid (B), C8H6O4. (B) easily forms anhydride on heating. Identify the compound (A).
Subtopic: Aromatic Conversion |
Which of the following will give yellow precipitate with I2/NaOH?
1. CH3-CO-O-CO-CH3
4.Both (2) and (3)
Subtopic: Aldehydes & Ketones |
A nitrogeneous substance X is treated with HNO2 and the product so formed is further treated with NaOH solution, which produces blue colouration. X can be
2 CH3CH2NO2
3. CH3CH2ONO
4 (CH3)2CHNO2
Subtopic: Differentiation of Organic Compound |
The compound which reacts fastest with Lucas reagent at room temperature is:
3. 2-methylpropan-1-ol
Identify (X) in the sequence :
{\mathrm{C}}_{3}{\mathrm{H}}_{8}\mathrm{O} \left(\mathrm{X}\right) \underset{{\mathrm{H}}_{2}{\mathrm{SO}}_{4}}{\overset{{\mathrm{K}}_{2}{\mathrm{Cr}}_{2}{\mathrm{O}}_{7}}{\to }} {\mathrm{C}}_{3}{\mathrm{H}}_{6}\mathrm{O} \underset{\mathrm{Warm}}{\overset{{\mathrm{I}}_{2} + \mathrm{NaOH}}{\to }} {\mathrm{CHI}}_{3}
(1) CH3-CH2-CH2OH
{\mathrm{CH}}_{3}-\mathrm{CH}-{\mathrm{CH}}_{3}\phantom{\rule{0ex}{0ex}} |\phantom{\rule{0ex}{0ex}} \mathrm{OH}
(3) CH3-O-CH2-CH3
(4) CH3-CH2-CHO
In order to accomplish the following conversion, what reagent and conditions would be required?
1. Cold sodium hydroxide
2. Hot conc. sodium hydroxide
3. Potassium tertiary butoxide and heat
An amine is reacted with benzene sulphonyl chloride then a solid compound is formed which is insoluble in alkali. The amine is:
(1) CH3-CH2-NH2
(2) CH3-NH-CH2-CH3
(4) C6H5-NH2
1. Aniline 2. p-aminophenol
3. m-nitroaniline 4. nitrosobenzene
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On Controlling an Uncertain System With Polynomial Chaos and H2 Control Design | J. Dyn. Sys., Meas., Control. | ASME Digital Collection
On Controlling an Uncertain System With Polynomial Chaos and
H2
Brian A. Templeton,
Brian A. Templeton
Center for Vehicle Systems and Safety,
e-mail: batemple@vt.edu
David E. Cox,
Dynamic Systems and Control Branch,
e-mail: david.e.cox@nasa.gov
Sean P. Kenny,
e-mail: sean.p.kenny@nasa.gov
e-mail: scsouth@vt.edu
Brian A. Templeton Graduate Research Assistant
David E. Cox Senior Research Engineer
Sean P. Kenny Senior Research Engineer
Mehdi Ahmadian Professor and Director
Steve C. Southward Associate Professor
Templeton, B. A., Cox, D. E., Kenny, S. P., Ahmadian, M., and Southward, S. C. (October 29, 2010). "On Controlling an Uncertain System With Polynomial Chaos and
H2
Control Design." ASME. J. Dyn. Sys., Meas., Control. November 2010; 132(6): 061304. https://doi.org/10.1115/1.4002474
This paper applies the
H2
norm along time and parameter domains. The norm is related to the probabilistic
H2
problem. It is calculated using polynomial chaos to handle uncertainty in the plant model. The structure of expanded states resulting from Galerkin projections of a state space model with uncertain parameters is used to formulate cost functions in terms of mean performances of the states, as well as covariances. Also, bounds on the norm are described in terms of linear matrix inequalitys. The form of the gradient of the norm, which can be used in optimization, is given as a Lyapunov equation. Additionally, this approach can be used to solve the related probabilistic LQR problem. The legitimacy of the concept is demonstrated through two mechanical oscillator examples. These controllers could be easily implemented on physical systems without observing uncertain parameters.
chaos, control system synthesis, Galerkin method, linear quadratic control, polynomials, probability, state-space methods, uncertain systems, polynomial chaos, orthogonal polynomials, parametric uncertainty, optimal control, H2 control, LQR
Chaos, Polynomials, Design, Optimization
The Orthogonal Development of Nonlinear Functionals in Series of Fourier–Hermite Functionals
A Polynomial Chaos Approach to Control Design
A Polynomial Chaos Theory Approach to the Control Design of a Power Converter
Power Electronics Specialists Conference, PESC 04, IEEE 35th Annual
Treating Uncertainties in Multibody Dynamic Systems Using a Polynomial Chaos Spectral Decomposition
Proceedings of the IMECE ‘04: ASME International Sixth Annual Symposium on Advanced Vehicle Technologies
Robust Controller Using Polynomial Chaos Theory
Industry Applications Conference, 41st IAS Annual Meeting, Conference Record of the 2006 IEEE
Indirect Measurements via Polynomial Chaos Observer
Advanced Methods for Uncertainty Estimation in Measurement. AMUEM 2006. Proceedings of the 2006 IEEE International Workshop
Distributional Uncertainty Analysis Using Power Series and Polynomial Chaos Expansions
On Stochastic LQR Design and Polynomial Chaos
Stability Analysis of Stochastic Systems Using Polynomial Chaos
A New Algorithm for Solving the Random Eigenvalue Problem Using Polynomial Chaos Expansion
46th AIAA/ASME/AHS/ASC Structure, Structural Dynamics, and Materials Conference
,” MS thesis, Virginia Polytechnic Institute and State University, Blacksburg, VA.
Adaptive Generalized Chaos for Nonlinear Random Oscillators
Robust Control: The Parameter Space Approach
Control Design for Parameter Dependent Aeroelastic Systems
,” Ph.D. thesis, Duke University, Durham, NC.
Proceedings of the CACSD Conference
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In an effort to determine the amount of food a male yellow-billed magpie eats in a day, a sample of
5
birds from pet stores was observed. The birds ate the following amount of food:
60, 60, 50, 40,
50
grams. Find the mean and standard deviation weight of food in this sample. Do the computations step by step, without using the “standard deviation” function on your calculator, as in problem 9-52.
52
\sqrt{\frac{(60-52)^2+(60-52)^2+(50-52)^2+(40-52)^2+(50-52)^2}{5-1}}
Note that we divide by
n - 1
because this is a sample.
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Power and Sample Size Analysis: Z test | R-bloggers
Posted on October 17, 2012 by Alice Bossi in R bloggers | 0 Comments
This article provide a brief background about power and sample size analysis. Then, power and sample size analysis is computed for the Z test.
Next articles will describe power and sample size analysis for:
one sample and two samples t test;,
p test, chi-square test, correlation;
2^k
Finally, a PDF article showing both the underlying methodology and the R code here provided, will be published.
Power and sample size analysis are important tools for assessing the ability of a statistical test to detect when a null hypothesis is false, and for deciding what sample size is required for having a reasonable chance to reject a false null hypothesis.
The following four quantities have an intimate relationship:
power = 1 – P(Type II error) = probability of finding an effect that is there
The formula for the power computation can be implemented in R, using a function like the following:
powerZtest = function(alpha = 0.05, sigma, n, delta){
zcr = qnorm(p = 1-alpha, mean = 0, sd = 1)
s = sigma/sqrt(n)
power = 1 - pnorm(q = zcr, mean = (delta/s), sd = 1)
In the same way, the function to compute the sample size can be built.
sampleSizeZtest = function(alpha = 0.05, sigma, power, delta){
zcra=qnorm(p = 1-alpha, mean = 0, sd=1)
zcrb=qnorm(p = power, mean = 0, sd = 1)
n = round((((zcra+zcrb)*sigma)/delta)^2)
The above code is provided for didactic purpose. In fact, the pwr package provide a function to perform power and sample size analysis.
The function pwr.norm.test() computes parameters for the Z test. It accepts the four parameters see above, one of them passed as NULL. The parameter passed as NULL is determined from the others.
\mu = 105
H0: \mu = 100
100″ />
H1: \mu>100
\sigma = 15
n = 20
\alpha = 0.05
This is the result with the self-made function:
> powerZtest(n = 20, sigma = sigma, delta = (ha-h0))
And here the same with the pwr.norm.test() function:
> d = (ha - h0)/sigma
> pwr.norm.test(d = d, n = 20, sig.level = 0.05, alternative = "greater")
Mean power calculation for normal distribution with known variance
The sample size of the test for power equal to 0.80 can be computed using the self-made function
> sampleSizeZtest(sigma = sigma, power = 0.8, delta = (ha-h0))
or with the pwr.norm.test() function:
> pwr.norm.test(d = d, power = 0.8, sig.level = 0.05, alternative = "greater")
The power function can be drawn:
ha = seq(95, 125, l = 100)
pwrTest = pwr.norm.test(d = d, n = 20, sig.level = 0.05, alternative = "greater")$power
plot(d, pwrTest, type = "l", ylim = c(0, 1))
View (and download) the full code:
?Download powerZtest.R
### Self-made functions to perform power and sample size analysis
### Load pwr package to perform power and sample size analysis
### Power analysis
# Using the self-made function
powerZtest(n = 20, sigma = sigma, delta = (ha-h0))
# Using the pwr package
pwr.norm.test(d = (ha - h0)/sigma, n = 20, sig.level = 0.05, alternative = "greater")
### Sample size analysis
sampleSizeZtest(sigma = sigma, power = 0.8, delta = (ha-h0))
pwr.norm.test(d = (ha - h0)/sigma, power = 0.8, sig.level = 0.05, alternative = "greater")
### Power function for the two-sided alternative
d = (ha - h0)/sigma
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Mill (grinding) (12487 views - Basics)
A mill is a device that breaks solid materials into smaller pieces by grinding, crushing, or cutting. Such comminution is an important unit operation in many processes. There are many different types of mills and many types of materials processed in them. Historically mills were powered by hand (e.g., via a hand crank), working animal (e.g., horse mill), wind (windmill) or water (watermill). Today they are usually powered by electricity. The grinding of solid matters occurs under exposure of mechanical forces that trench the structure by overcoming of the interior bonding forces. After the grinding the state of the solid is changed: the grain size, the grain size disposition and the grain shape. Milling also refers to the process of breaking down, separating, sizing, or classifying aggregate material. For instance rock crushing or grinding to produce uniform aggregate size for construction purposes, or separation of rock, soil or aggregate material for the purposes of structural fill or land reclamation activities. Aggregate milling processes are also used to remove or separate contamination or moisture from aggregate or soil and to produce "dry fills" prior to transport or structural filling. Grinding may serve the following purposes in engineering: increase of the surface area of a solid manufacturing of a solid with a desired grain size pulping of resources
3D CAD Models - Mill
Licensed under Creative Commons Attribution-Share Alike 3.0 (User:Bryan Derksen).
A mill is a device that breaks solid materials into smaller pieces by grinding, crushing, or cutting. Such comminution is an important unit operation in many processes. There are many different types of mills and many types of materials processed in them. Historically mills were powered by hand (e.g., via a hand crank), working animal (e.g., horse mill), wind (windmill) or water (watermill). Today they are usually powered by electricity.
The grinding of solid matters occurs under exposure of mechanical forces that trench the structure by overcoming of the interior bonding forces. After the grinding the state of the solid is changed: the grain size, the grain size disposition and the grain shape.
1 Grinding laws
2.1 Ball mill
2.2 Rod mill
2.3 Autogenous mill
2.4 SAG mill
2.5 Pebble mill
2.6 High pressure grinding rolls
2.7 Buhrstone mill
2.8 Vertical shaft impactor mill (VSI mill)
2.9 Tower mill
In spite of a great number of studies in the field of fracture schemes there is no formula known which connects the technical grinding work with grinding results. To calculate the needed grinding work against the grain size changing three half-empirical models are used. These can be related to the Hukki relationship between particle size and the energy required to break the particles. In stirred mills, the Hukki relationship does not apply and instead, experimentation has to be performed to determine any relationship.[1]
{\displaystyle W_{K}=c_{k}(\ln(d_{A})-\ln(d_{E}))\,}
{\displaystyle W_{B}=c_{B}\left({\frac {1}{{\sqrt {d}}_{E}}}-{\frac {1}{{\sqrt {d}}_{A}}}\right)\,}
{\displaystyle W_{R}=c_{R}\left({\frac {1}{d_{E}}}-{\frac {1}{d_{A}}}\right)\,}
{\displaystyle c_{K}=1.151c_{B}(d_{BU})^{-0.5}\,}
{\displaystyle c_{R}=0.5c_{B}(d_{BL})^{0.5}\,}
{\displaystyle Z_{d}={\frac {d_{80,1}}{d_{80,2}}}\,}
{\displaystyle Z_{S}={\frac {S_{v,2}}{S_{v,1}}}={\frac {S_{m,2}}{S_{m,1}}}\,}
{\displaystyle Z_{a}={\frac {d_{1}}{a}}\,}
A typical type of fine grinder is the ball mill. A slightly inclined or horizontal rotating cylinder is partially filled with balls, usually stone or metal, which grind material to the necessary fineness by friction and impact with the tumbling balls. Ball mills normally operate with an approximate ball charge of 30%. Ball mills are characterized by their smaller (comparatively) diameter and longer length, and often have a length 1.5 to 2.5 times the diameter. The feed is at one end of the cylinder and the discharge is at the other. Ball mills are commonly used in the manufacture of Portland cement and finer grinding stages of mineral processing, one example being the Sepro tyre drive Grinding Mill. Industrial ball mills can be as large as 8.5 m (28 ft) in diameter with a 22 MW motor,[4] drawing approximately 0.0011% of the total world's power (see List of countries by electricity consumption). However, small versions of ball mills can be found in laboratories where they are used for grinding sample material for quality assurance.
{\displaystyle E=10W\left({\frac {1}{\sqrt {P_{80}}}}-{\frac {1}{\sqrt {F_{80}}}}\right)\,}
SAG is an acronym for Semi-Autogenous Grinding. SAG mills are autogenous mills but use grinding balls like a ball mill. A SAG mill is usually a primary or first stage grinder. SAG mills use a ball charge of 8 to 21%.[6][7] The largest SAG mill is 42' (12.8m) in diameter, powered by a 28 MW (38,000 HP) motor.[8] A SAG mill with a 44' (13.4m) diameter and a power of 35 MW (47,000 HP) has been designed.[9]
Another type of fine grinder commonly used is the buhrstone mill, which is similar to old-fashioned flour mills.
Vertical shaft impactor mill (VSI mill)
Main article: VSI mill
textile mill (disambiguation)
MachineMeat grinderAbrasive machiningSurface integrityTool and cutter grinderMilling cutterMetal latheShaperAngle grinderHammer millWrought ironTrip hammerCamCamshaftStamp millHammermill
This article uses material from the Wikipedia article "Mill (grinding)", which is released under the Creative Commons Attribution-Share-Alike License 3.0. There is a list of all authors in Wikipedia
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China 30°49′15″N 111°00′08″E / 30.82083°N 111.00222°E / 30.82083; 111.00222 (Three Gorges Dam) 22,500
China 28°15′35″N 103°38′58″E / 28.25972°N 103.64944°E / 28.25972; 103.64944 (Xiluodu Dam) 13,860
4. Belo Monte Dam
Brazil 03°06′57″S 51°47′45″W / 3.11583°S 51.79583°W / -3.11583; -51.79583 (Belo Monte Dam) 11,233
5. Guri Dam
Venezuela 07°45′59″N 62°59′57″W / 7.76639°N 62.99917°W / 7.76639; -62.99917 (Guri Dam) 10,200
Small hydro is the development of hydroelectric power on a scale serving a small community or industrial plant. The definition of a small hydro project varies but a generating capacity of up to 10 megawatts (MW) is generally accepted as the upper limit of what can be termed small hydro. This may be stretched to 25 MW and 30 MW in Canada and the United States.[23][24]
{\displaystyle P=-\eta \ ({\dot {m}}g\ \Delta h)=-\eta \ ((\rho {\dot {V}})\ g\ \Delta h)}
{\displaystyle P}
{\displaystyle \eta }
{\displaystyle \rho }
{\displaystyle {\dot {V}}}
{\displaystyle {\dot {m}}}
{\displaystyle \Delta h}
{\displaystyle g}
Retrieved from "https://en.wikidark.org/w/index.php?title=Hydroelectricity&oldid=1078228421"
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Periodic knots and Heegaard Floer correction terms | EMS Press
Periodic knots and Heegaard Floer correction terms
We derive new obstructions to periodicity of classical knots by employing the Heegaard Floer correction terms of the finite cyclic branched covers of the knots. Applying our results to 2-fold covers, we demonstrate through numerous examples that our obstructions are successful where many existing periodicity obstructions fail.
A combination of previously known periodicity obstructions and the results presented here leads to a nearly complete (with the exception of a single knot) classification of alternating, periodic, 12-crossing knots with odd prime periods. For the case of alternating knots with 13, 14 and 15 crossings, we give a complete list of all periodic knots with odd prime periods
q > 3
Stanislav Jabuka, Swatee Naik, Periodic knots and Heegaard Floer correction terms. J. Eur. Math. Soc. 18 (2016), no. 8, pp. 1651–1674
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New energy-capacity-type inequalities and uniqueness of continuous Hamiltonians | EMS Press
We prove a new variant of the energy-capacity inequality for closed rational symplectic manifolds (as well as certain open manifolds such as
\mathbb R^{2n}
, cotangent bundle of closed manifolds…) and we derive some consequences to
C^0
–symplectic topology. Namely, we prove that a continuous function which is a uniform limit of smooth normalized Hamiltonians whose flows converge to the identity for the spectral (or Hofer’s) distance must vanish. This gives a new proof of uniqueness of continuous generating Hamiltonian for hameomorphisms. This also allows us to improve a result by Cardin and Viterbo on the
C^0
–rigidity of the Poisson bracket.
Vincent Humilière, Rémi Leclercq, Sobhan Seyfaddini, New energy-capacity-type inequalities and uniqueness of continuous Hamiltonians. Comment. Math. Helv. 90 (2015), no. 1, pp. 1–21
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Analysis of the adiabatic piston problem via methods of continuum mechanics | EMS Press
Analysis of the adiabatic piston problem via methods of continuum mechanics
Institut de Mathématiques de Bordeaux, Université de Bordeaux/CNRS/Bordeaux INP, 351 Cours de la Libération, 33405 Talence, France
Industry-University Research Center, Yonsei University 50 Yonsei-ro Seodaemun-gu, Seoul, 03722, Republic of Korea
We consider a system modelling the motion of a piston in a cylinder filled by a viscous heat conducting gas. The piston is moving longitudinally without friction under the influence of the forces exerted by the gas. In addition, the piston is supposed to be thermally insulating (adiabatic piston). This fact raises several challenges which received a considerable attention, essentially in the statistical physics literature. We study the problem via the methods of continuum mechanics, specifically, the motion of the gas is described by means of the Navier–Stokes–Fourier system in one space dimension, coupled with Newton's second law governing the motion of the piston. We establish global in time existence of strong solutions and show that the system stabilizes to an equilibrium state for
t\rightarrow \infty
Šárka Nečasová, Marius Tucsnak, Eduard Feireisl, Václav Mácha, Analysis of the adiabatic piston problem via methods of continuum mechanics. Ann. Inst. H. Poincaré Anal. Non Linéaire 35 (2018), no. 5, pp. 1377–1408
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Isospectrality for graph Laplacians under the change of coupling at graph vertices | EMS Press
Isospectrality for graph Laplacians under the change of coupling at graph vertices
Taurida National University, Simferopol, Ukraine
St.Petersburg State University, Russian Federation
Laplacian operators on finite compact metric graphs are considered under the assumption that matching conditions at graph vertices are of
\delta
\delta'
types. An infinite series of trace formulae is obtained which link together two different quantum graphs under the assumption that their spectra coincide. The general case of graph Schrodinger operators is also considered, yielding the first trace formula. Tightness of results obtained under no additional restrictions on edge lengths is demonstrated by an example. Further examples are scrutinized when edge lengths are assumed to be rationally independent. In all but one of these impossibility of isospectral configurations is ascertained.
Yulia Ershova, Irina I. Karpenko, Alexander V. Kiselev, Isospectrality for graph Laplacians under the change of coupling at graph vertices. J. Spectr. Theory 6 (2016), no. 1, pp. 43–66
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How to convert radicals to mixed radicals | StudyPug
In this session, we will learn how to convert radicals to mixed radicals. When the content inside the radical sign is a number, you can do the conversion by performing prime factorization. We will also learn how to do the conversion when the content inside the radical sign is a variable.
Basic Concepts: Estimating square roots, Square and square roots, Cubic and cube roots, Evaluating and simplifying radicals
Convert the following radicals to mixed radicals
\sqrt{18}
\sqrt{1088}
\sqrt{45}
\sqrt{1250}
{^4}\sqrt{162}
{^5}\sqrt{-192}
\sqrt{29a^4b^8}
{^3}\sqrt{250a^{13}}
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Local profile of fully bubbling solutions to $\mathrm {SU} (n+1)$ Toda systems | EMS Press
Local profile of fully bubbling solutions to
\mathrm {SU} (n+1)
Toda systems
In this article we prove that for locally defined singular
\mathrm {SU} (n+1)
Toda systems in
\mathbb R^2
, the profile of fully bubbling solutions near the singular source can be accurately approximated by global solutions. The main ingredients of our new approach are the classification theorem of Lin–Wei–Ye [22] and the non-degeneracy of the linearized Toda system [22], which let us overcome the difficulties that come from lack of symmetry and the singular source.
Chang-Shou Lin, Juncheng Wei, Lei Zhang, Local profile of fully bubbling solutions to
\mathrm {SU} (n+1)
Toda systems. J. Eur. Math. Soc. 18 (2016), no. 8, pp. 1707–1728
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Roulette (curve) - Wikipedia
Mathematical curves generated by rolling other curves together
In the differential geometry of curves, a roulette is a kind of curve, generalizing cycloids, epicycloids, hypocycloids, trochoids, epitrochoids, hypotrochoids, and involutes.
1.2 Special cases and related concepts
4 List of roulettes
A green parabola rolls along an equal blue parabola which remains fixed. The generator is the vertex of the rolling parabola and describes the roulette, shown in red. In this case the roulette is the cissoid of Diocles.[1]
Roughly speaking, a roulette is the curve described by a point (called the generator or pole) attached to a given curve as that curve rolls without slipping, along a second given curve that is fixed. More precisely, given a curve attached to a plane which is moving so that the curve rolls, without slipping, along a given curve attached to a fixed plane occupying the same space, then a point attached to the moving plane describes a curve, in the fixed plane called a roulette.
Special cases and related concepts[edit]
In the case where the rolling curve is a line and the generator is a point on the line, the roulette is called an involute of the fixed curve. If the rolling curve is a circle and the fixed curve is a line then the roulette is a trochoid. If, in this case, the point lies on the circle then the roulette is a cycloid.
A related concept is a glissette, the curve described by a point attached to a given curve as it slides along two (or more) given curves.
Formally speaking, the curves must be differentiable curves in the Euclidean plane. The fixed curve is kept invariant; the rolling curve is subjected to a continuous congruence transformation such that at all times the curves are tangent at a point of contact that moves with the same speed when taken along either curve (another way to express this constraint is that the point of contact of the two curves is the instant centre of rotation of the congruence transformation). The resulting roulette is formed by the locus of the generator subjected to the same set of congruence transformations.
Modeling the original curves as curves in the complex plane, let
{\displaystyle r,f:\mathbb {R} \to \mathbb {C} }
be the two natural parameterizations of the rolling (
{\displaystyle r}
) and fixed (
{\displaystyle f}
) curves, such that
{\displaystyle r(0)=f(0)}
{\displaystyle r'(0)=f'(0)}
{\displaystyle |r'(t)|=|f'(t)|\neq 0}
{\displaystyle t}
. The roulette of generator
{\displaystyle p\in \mathbb {C} }
{\displaystyle r}
is rolled on
{\displaystyle f}
is then given by the mapping:
{\displaystyle t\mapsto f(t)+(p-r(t)){f'(t) \over r'(t)}.}
If, instead of a single point being attached to the rolling curve, another given curve is carried along the moving plane, a family of congruent curves is produced. The envelope of this family may also be called a roulette.
Roulettes in higher spaces can certainly be imagined but one needs to align more than just the tangents.
If the fixed curve is a catenary and the rolling curve is a line, we have:
{\displaystyle f(t)=t+i(\cosh(t)-1)\qquad r(t)=\sinh(t)}
{\displaystyle f'(t)=1+i\sinh(t)\qquad r'(t)=\cosh(t).}
The parameterization of the line is chosen so that
{\displaystyle |f'(t)|={\sqrt {1^{2}+\sinh ^{2}(t)}}={\sqrt {\cosh ^{2}(t)}}=|r'(t)|.}
Applying the formula above we obtain:
{\displaystyle f(t)+(p-r(t)){f'(t) \over r'(t)}=t-i+{p-\sinh(t)+i(1+p\sinh(t)) \over \cosh(t)}=t-i+(p+i){1+i\sinh(t) \over \cosh(t)}.}
If p = −i the expression has a constant imaginary part (namely −i) and the roulette is a horizontal line. An interesting application of this is that a square wheel could roll without bouncing on a road that is a matched series of catenary arcs.
List of roulettes[edit]
Any curve Line Point on the line Involute of the curve
Line Any Any Cyclogon
Line Circle Any Trochoid
Line Circle Point on the circle Cycloid
Line Conic section Center of the conic Sturm roulette[2]
Line Conic section Focus of the conic Delaunay roulette[3]
Line Parabola Focus of the parabola Catenary[4]
Line Ellipse Focus of the ellipse Elliptic catenary[4]
Line Hyperbola Focus of the hyperbola Hyperbolic catenary[4]
Line Hyperbola Center of the hyperbola Rectangular elastica[2][failed verification]
Line Cyclocycloid Center Ellipse[5]
Circle Circle Any Centered trochoid[6]
Outside of a circle Circle Any Epitrochoid
Outside of a circle Circle Point on the circle Epicycloid
Outside of a circle Circle of identical radius Any Limaçon
Outside of a circle Circle of identical radius Point on the circle Cardioid
Outside of a circle Circle of half the radius Point on the circle Nephroid
Inside of a circle Circle Any Hypotrochoid
Inside of a circle Circle Point on the circle Hypocycloid
Inside of a circle Circle of a third of the radius Point on the circle Deltoid
Inside of a circle Circle of a quarter of the radius Point on the circle Astroid
Parabola Equal parabola parameterized in opposite direction Vertex of the parabola Cissoid of Diocles[1]
Catenary Line See example above Line
^ a b "Cissoid" on www.2dcurves.com
^ a b "Sturm's roulette" on www.mathcurve.com
^ "Delaunay's roulette" on www.mathcurve.com
^ a b c "Delaunay's roulette" on www.2dcurves.com
^ "Roulette with straight fixed curve" on www.mathcurve.com
^ "Centered trochoid" on mathcurve.com
W. H. Besant (1890) Notes on Roulettes and Glissettes from Cornell University Historical Math Monographs, originally published by Deighton, Bell & Co.
Weisstein, Eric W. "Roulette". MathWorld.
Roulette at 2dcurves.com
Base, roulante et roulettes d'un mouvement plan sur plan (in French)
Eine einheitliche Darstellung von ebenen, verallgemeinerten Rollbewegungen und deren Anwendungen (in German)
Retrieved from "https://en.wikipedia.org/w/index.php?title=Roulette_(curve)&oldid=1020627873"
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Hadamard type asymptotics for eigenvalues of the Neumann problem for elliptic operators | EMS Press
Hadamard type asymptotics for eigenvalues of the Neumann problem for elliptic operators
This paper considers how the eigenvalues of the Neumann problem for an elliptic operator depend on the domain. The proximity of two domains is measured in terms of the norm of the difference between the two resolvents corresponding to the reference domain and the perturbed domain, and the size of eigenfunctions outside the intersection of the two domains. This construction enables the possibility of comparing both nonsmooth domains and domains with different topology. An abstract framework is presented, where the main result is an asymptotic formula where the remainder is expressed in terms of the proximity quantity described above when this is relatively small. As an application, we develop a theory for the Laplacian in Lipschitz domains. In particular, if the domains are assumed to be
C^{1,\alpha}
regular, an asymptotic result for the eigenvalues is given together with estimates for the remainder, and we also provide an example which demonstrates the sharpness of our obtained result.
Vladimir Kozlov, Johan Thim, Hadamard type asymptotics for eigenvalues of the Neumann problem for elliptic operators. J. Spectr. Theory 6 (2016), no. 1, pp. 99–135
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Null structure and local well-posedness in the energy class for the Yang–Mills equations in Lorenz gauge | EMS Press
Null structure and local well-posedness in the energy class for the Yang–Mills equations in Lorenz gauge
We demonstrate null structure in the Yang–Mills equations in Lorenz gauge. Such structure was found in Coulomb gauge by Klainerman and Machedon, who used it to prove global wellposedness for finite-energy data in the temporal gauge by passing to local Coulomb gauges via Uhlenbeck’s Lemma. Compared with the Coulomb gauge, the Lorenz gauge has the advantage – shared with the temporal gauge – that it can be imposed globally in space even for large solutions. Using the null structure and bilinear space-time estimates, we also prove local-in-time wellposedness of the Yang–Mills equations in Lorenz gauge for data with finite energy, with a time of existence depending on the initial energy and on the H^s \times H^{s–1}-norm of the initial gauge potential, for some choice of
s < 1
sufficiently close to 1.
Sigmund Selberg, Achenef Tesfahun, Null structure and local well-posedness in the energy class for the Yang–Mills equations in Lorenz gauge. J. Eur. Math. Soc. 18 (2016), no. 8, pp. 1729–1752
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Convergent series - Wikipedia
(Redirected from Convergence (mathematics))
Mathematical series with a finite sum
For the short story collection, see Convergent Series (short story collection).
"Convergence (mathematics)" redirects here. For other uses, see Convergence (disambiguation).
In mathematics, a series is the sum of the terms of an infinite sequence of numbers. More precisely, an infinite sequence
{\displaystyle (a_{0},a_{1},a_{2},\ldots )}
defines a series S that is denoted
{\displaystyle S=a_{0}+a_{1}+a_{2}+\cdots =\sum _{k=0}^{\infty }a_{k}.}
The nth partial sum Sn is the sum of the first n terms of the sequence; that is,
{\displaystyle S_{n}=\sum _{k=1}^{n}a_{k}.}
A series is convergent (or converges) if the sequence
{\displaystyle (S_{1},S_{2},S_{3},\dots )}
of its partial sums tends to a limit; that means that, when adding one
{\displaystyle a_{k}}
after the other in the order given by the indices, one gets partial sums that become closer and closer to a given number. More precisely, a series converges, if there exists a number
{\displaystyle \ell }
such that for every arbitrarily small positive number
{\displaystyle \varepsilon }
, there is a (sufficiently large) integer
{\displaystyle N}
{\displaystyle n\geq N}
{\displaystyle \left|S_{n}-\ell \right|<\varepsilon .}
If the series is convergent, the (necessarily unique) number
{\displaystyle \ell }
is called the sum of the series.
The same notation
{\displaystyle \sum _{k=1}^{\infty }a_{k}}
is used for the series, and, if it is convergent, to its sum. This convention is similar to that which is used for addition: a + b denotes the operation of adding a and b as well as the result of this addition, which is called the sum of a and b.
Any series that is not convergent is said to be divergent or to diverge.
1 Examples of convergent and divergent series
2 Convergence tests
3 Conditional and absolute convergence
4 Uniform convergence
5 Cauchy convergence criterion
Examples of convergent and divergent series[edit]
The reciprocals of the positive integers produce a divergent series (harmonic series):
{\displaystyle {1 \over 1}+{1 \over 2}+{1 \over 3}+{1 \over 4}+{1 \over 5}+{1 \over 6}+\cdots \rightarrow \infty .}
Alternating the signs of the reciprocals of positive integers produces a convergent series (alternating harmonic series):
{\displaystyle {1 \over 1}-{1 \over 2}+{1 \over 3}-{1 \over 4}+{1 \over 5}-\cdots =\ln(2)}
The reciprocals of prime numbers produce a divergent series (so the set of primes is "large"; see divergence of the sum of the reciprocals of the primes):
{\displaystyle {1 \over 2}+{1 \over 3}+{1 \over 5}+{1 \over 7}+{1 \over 11}+{1 \over 13}+\cdots \rightarrow \infty .}
{\displaystyle {1 \over 1}+{1 \over 3}+{1 \over 6}+{1 \over 10}+{1 \over 15}+{1 \over 21}+\cdots =2.}
{\displaystyle {\frac {1}{1}}+{\frac {1}{1}}+{\frac {1}{2}}+{\frac {1}{6}}+{\frac {1}{24}}+{\frac {1}{120}}+\cdots =e.}
{\displaystyle {1 \over 1}+{1 \over 4}+{1 \over 9}+{1 \over 16}+{1 \over 25}+{1 \over 36}+\cdots ={\pi ^{2} \over 6}.}
{\displaystyle {1 \over 1}+{1 \over 2}+{1 \over 4}+{1 \over 8}+{1 \over 16}+{1 \over 32}+\cdots =2.}
The reciprocals of powers of any n>1 produce a convergent series:
{\displaystyle {1 \over 1}+{1 \over n}+{1 \over n^{2}}+{1 \over n^{3}}+{1 \over n^{4}}+{1 \over n^{5}}+\cdots ={n \over n-1}.}
Alternating the signs of reciprocals of powers of 2 also produces a convergent series:
{\displaystyle {1 \over 1}-{1 \over 2}+{1 \over 4}-{1 \over 8}+{1 \over 16}-{1 \over 32}+\cdots ={2 \over 3}.}
Alternating the signs of reciprocals of powers of any n>1 produces a convergent series:
{\displaystyle {1 \over 1}-{1 \over n}+{1 \over n^{2}}-{1 \over n^{3}}+{1 \over n^{4}}-{1 \over n^{5}}+\cdots ={n \over n+1}.}
{\displaystyle {\frac {1}{1}}+{\frac {1}{1}}+{\frac {1}{2}}+{\frac {1}{3}}+{\frac {1}{5}}+{\frac {1}{8}}+\cdots =\psi .}
Convergence tests[edit]
There are a number of methods of determining whether a series converges or diverges.
If the blue series,
{\displaystyle \Sigma b_{n}}
, can be proven to converge, then the smaller series,
{\displaystyle \Sigma a_{n}}
must converge. By contraposition, if the red series
{\displaystyle \Sigma a_{n}}
is proven to diverge, then
{\displaystyle \Sigma b_{n}}
must also diverge.
Comparison test. The terms of the sequence
{\displaystyle \left\{a_{n}\right\}}
are compared to those of another sequence
{\displaystyle \left\{b_{n}\right\}}
. If, for all n,
{\displaystyle 0\leq \ a_{n}\leq \ b_{n}}
{\textstyle \sum _{n=1}^{\infty }b_{n}}
converges, then so does
{\textstyle \sum _{n=1}^{\infty }a_{n}.}
However, if, for all n,
{\displaystyle 0\leq \ b_{n}\leq \ a_{n}}
{\textstyle \sum _{n=1}^{\infty }b_{n}}
diverges, then so does
{\textstyle \sum _{n=1}^{\infty }a_{n}.}
Ratio test. Assume that for all n,
{\displaystyle a_{n}}
is not zero. Suppose that there exists
{\displaystyle r}
{\displaystyle \lim _{n\to \infty }\left|{\frac {a_{n+1}}{a_{n}}}\right|=r.}
If r < 1, then the series is absolutely convergent. If r > 1, then the series diverges. If r = 1, the ratio test is inconclusive, and the series may converge or diverge.
Root test or nth root test. Suppose that the terms of the sequence in question are non-negative. Define r as follows:
{\displaystyle r=\limsup _{n\to \infty }{\sqrt[{n}]{|a_{n}|}},}
where "lim sup" denotes the limit superior (possibly ∞; if the limit exists it is the same value).
If r < 1, then the series converges. If r > 1, then the series diverges. If r = 1, the root test is inconclusive, and the series may converge or diverge.
The ratio test and the root test are both based on comparison with a geometric series, and as such they work in similar situations. In fact, if the ratio test works (meaning that the limit exists and is not equal to 1) then so does the root test; the converse, however, is not true. The root test is therefore more generally applicable, but as a practical matter the limit is often difficult to compute for commonly seen types of series.
Integral test. The series can be compared to an integral to establish convergence or divergence. Let
{\displaystyle f(n)=a_{n}}
be a positive and monotonically decreasing function. If
{\displaystyle \int _{1}^{\infty }f(x)\,dx=\lim _{t\to \infty }\int _{1}^{t}f(x)\,dx<\infty ,}
then the series converges. But if the integral diverges, then the series does so as well.
Limit comparison test. If
{\displaystyle \left\{a_{n}\right\},\left\{b_{n}\right\}>0}
, and the limit
{\displaystyle \lim _{n\to \infty }{\frac {a_{n}}{b_{n}}}}
exists and is not zero, then
{\textstyle \sum _{n=1}^{\infty }a_{n}}
{\textstyle \sum _{n=1}^{\infty }b_{n}}
Alternating series test. Also known as the Leibniz criterion, the alternating series test states that for an alternating series of the form
{\textstyle \sum _{n=1}^{\infty }a_{n}(-1)^{n}}
{\displaystyle \left\{a_{n}\right\}}
is monotonically decreasing, and has a limit of 0 at infinity, then the series converges.
Cauchy condensation test. If
{\displaystyle \left\{a_{n}\right\}}
is a positive monotone decreasing sequence, then
{\textstyle \sum _{n=1}^{\infty }a_{n}}
{\textstyle \sum _{k=1}^{\infty }2^{k}a_{2^{k}}}
Conditional and absolute convergence[edit]
{\displaystyle \left\{a_{1},\ a_{2},\ a_{3},\dots \right\}}
{\displaystyle a_{n}\leq \left|a_{n}\right|}
{\displaystyle \sum _{n=1}^{\infty }a_{n}\leq \sum _{n=1}^{\infty }\left|a_{n}\right|.}
{\textstyle \sum _{n=1}^{\infty }\left|a_{n}\right|}
{\textstyle \sum _{n=1}^{\infty }a_{n}}
also converges (but not vice versa).
{\textstyle \sum _{n=1}^{\infty }\left|a_{n}\right|}
{\textstyle \sum _{n=1}^{\infty }a_{n}}
is absolutely convergent. The Maclaurin series of the exponential function is absolutely convergent for every complex value of the variable.
{\textstyle \sum _{n=1}^{\infty }a_{n}}
converges but the series
{\textstyle \sum _{n=1}^{\infty }\left|a_{n}\right|}
diverges, then the series
{\textstyle \sum _{n=1}^{\infty }a_{n}}
is conditionally convergent. The Maclaurin series of the logarithm function
{\displaystyle \ln(1+x)}
is conditionally convergent for x = 1.
The Riemann series theorem states that if a series converges conditionally, it is possible to rearrange the terms of the series in such a way that the series converges to any value, or even diverges.
Uniform convergence[edit]
{\displaystyle \left\{f_{1},\ f_{2},\ f_{3},\dots \right\}}
be a sequence of functions. The series
{\textstyle \sum _{n=1}^{\infty }f_{n}}
is said to converge uniformly to f if the sequence
{\displaystyle \{s_{n}\}}
of partial sums defined by
{\displaystyle s_{n}(x)=\sum _{k=1}^{n}f_{k}(x)}
converges uniformly to f.
There is an analogue of the comparison test for infinite series of functions called the Weierstrass M-test.
Cauchy convergence criterion[edit]
The Cauchy convergence criterion states that a series
{\displaystyle \sum _{n=1}^{\infty }a_{n}}
converges if and only if the sequence of partial sums is a Cauchy sequence. This means that for every
{\displaystyle \varepsilon >0,}
{\displaystyle N}
{\displaystyle n\geq m\geq N}
{\displaystyle \left|\sum _{k=m}^{n}a_{k}\right|<\varepsilon ,}
{\displaystyle \lim _{n\to \infty \atop m\to \infty }\sum _{k=n}^{n+m}a_{k}=0.}
"Series", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
Retrieved from "https://en.wikipedia.org/w/index.php?title=Convergent_series&oldid=1071654886"
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A note on the resonance counting function for surfaces with cusps | EMS Press
A note on the resonance counting function for surfaces with cusps
We prove sharp upper bounds for the number of resonances in boxes of size 1 at high frequency for the Laplacian on finite volume surfaces with hyperbolic cusps. As a corollary, we obtain a Weyl asymptotic for the number of resonances in balls of size
T \to \infty
O(T^{3/2})
Yannick Guedes Bonthonneau, A note on the resonance counting function for surfaces with cusps. J. Spectr. Theory 6 (2016), no. 1, pp. 137–144
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Total harmonic distortion measurement - Simulink - MathWorks España
The Total Harmonic Distortion block calculates the total harmonic distortion (THD) percentage using these equations:
M=\frac{harmonic\text{ }magnitude}{\sqrt{2}},
%THD=100\frac{\sqrt{{\sum }_{i=2}^{n}{M}_{i}{}^{2}}}{{M}_{1}},
Mi is the root mean squared (RMS) value of the harmonic magnitude that corresponds to the ith harmonic order.
M is VRMS or IRMS.
{\sum }_{i=2}^{n}{M}_{i}{}^{2}
is the sum of all harmonic components.
u — Harmonic magnitude
Harmonic magnitude.
THD (%) — Total harmonic distortion
Total harmonic distortion as a percentage.
Fundamental frequency — Fundamental frequency in Hz
Fundamental frequency, in Hz.
To improve accuracy, set this value at least 10 times smaller than the Time constant parameter.
If this block is in a masked subsystem, or another variant subsystem that allows either continuous or discrete operation, promote the sample time parameter to ensure correct switching between the continuous and discrete implementations of the block. For more information, see Promote Block Parameters on a Mask.
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Pulsating fronts for nonlocal dispersion and KPP nonlinearity | EMS Press
Pulsating fronts for nonlocal dispersion and KPP nonlinearity
INRA, Equipe BIOSP, Centre de Recherche dʼAvignon, Domaine Saint Paul, Site Agroparc, 84914 Avignon cedex 9, France
In this paper we are interested in propagation phenomena for nonlocal reaction–diffusion equations of the type:
\frac{\partial u}{\partial t} = J⁎u−u + f(x,u)\:t \in \mathbb{R},\:x \in \mathbb{R}^{N},
where J is a probability density and f is a KPP nonlinearity periodic in the x variables. Under suitable assumptions we establish the existence of pulsating fronts describing the invasion of the 0 state by a heterogeneous state. We also give a variational characterization of the minimal speed of such pulsating fronts and exponential bounds on the asymptotic behavior of the solution.
Juan Dávila, Salomé Martínez, Jérôme Coville, Pulsating fronts for nonlocal dispersion and KPP nonlinearity. Ann. Inst. H. Poincaré Anal. Non Linéaire 30 (2013), no. 2, pp. 179–223
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Inverse boundary problems for polyharmonic operators with unbounded potentials | EMS Press
Inverse boundary problems for polyharmonic operators with unbounded potentials
We show that the knowledge of the Dirichlet–to–Neumann map on the boundary of a bounded open set in
\mathbb R^n
for the perturbed polyharmonic operator
(-\Delta)^m +q
q\in L^{\frac{n}{2m}}
n>2m
, determines the potential
q
in the set uniquely. In the course of the proof, we construct a special Green function for the polyharmonic operator and establish its mapping properties in suitable weighted
L^2
L^p
spaces. The
L^p
estimates for the special Green function are derived from
L^p
Carleman estimates with linear weights for the polyharmonic operator.
Katsiaryna Krupchyk, Gunther Uhlmann, Inverse boundary problems for polyharmonic operators with unbounded potentials. J. Spectr. Theory 6 (2016), no. 1, pp. 145–183
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Microelectromechanical System-Based Evaporative Thermal Management of High Heat Flux Electronics | J. Heat Transfer | ASME Digital Collection
Mechanical Engineering Department, Institute for Complex Engineered Systems, Carnegie Mellon University, 5000 Forbes Avenue, Pittsburgh, PA 15213
e-mail: camon@cmu.edu
S.-C. Yao,
S.-C. Yao
C.-F. Wu,
Manuscript received May 14, 2004; revised manuscript received August 13, 2004. Review conducted by: V. Dhir.
Amon, C. H., Yao , S., Wu , C., and Hsieh, C. (February 15, 2005). "Microelectromechanical System-Based Evaporative Thermal Management of High Heat Flux Electronics ." ASME. J. Heat Transfer. January 2005; 127(1): 66–75. https://doi.org/10.1115/1.1839586
This paper describes the development of embedded droplet impingement for integrated cooling of electronics (EDIFICE), which seeks to develop an integrated droplet impingement cooling device for removing chip heat fluxes over
100W/cm2,
employing latent heat of vaporization of dielectric fluids. Micromanufacturing and microelectromechanical systems are used as enabling technologies for developing innovative cooling schemes. Microspray nozzles are fabricated to produce 50–100 μm droplets coupled with surface texturing on the backside of the chip to promote droplet spreading and effective evaporation. This paper examines jet impingement cooling of EDIFICE with a dielectric coolant and the influence of fluid properties, microspray characteristics, and surface evaporation. The development of micronozzles and microstructured surface texturing is discussed. Results of a prototype testing of swiss-roll swirl nozzles with dielectric fluid HFE-7200 on a notebook PC are presented. This paper also outlines the challenges to practical implementation and future research needs.
microfluidics, evaporation, cooling, drops, heat of vaporisation, sprays, nozzles, surface texture, wetting, jets, reviews
Coolants, Cooling, Drops, Electronics, Engineering prototypes, Flow (Dynamics), Flux (Metallurgy), Heat, Heat flux, Heat transfer, Jets, Micronozzles, Nozzles, Silicon, Sprays, Thermal management, Microelectromechanical systems, Impingement cooling, Boiling, Temperature, Shapes, Design, Evaporation, Testing
Park, K. A., and Bergles, A. E., 1986, “Boiling Heat Transfer Characteristics of Simulated Microelectronic Chips with Detachable Heat Sinks,” Proc. 8th International Heat Transfer Conference, Hemisphere Publishing Co., Washington, DC, 4, pp. 2099–2104.
Effects of Size of Simulated Microelectronic Chips on Boiling and Critical Heat Flux
Bergles, A. E., and Kim, C. J., 1988, “A Method to Reduce Temperature Overshoots in Immersion Cooling of Electronic Devices,” Proc. InterSociety Conference on Thermal Phenomena in the Fabrication and Operation of Electronic Components, IEEE, New York, NY, pp. 100–105.
Carvalho, R. D. M., and Bergles, A. E., 1990, “The Influence of Subcooling on the Pool Nucleate Boiling and Critical Heat Flux of Simulated Electronic Chips,” Proc. 9th International Heat Transfer Conference, Hemisphere Publishing Co., New York, NY, pp. 289–294.
Park, K. A., Bergles, A. E., and Danielson, R. D., 1990, “Boiling Heat Transfer Characteristics of Simulated Microelectronic Chips with Fluorinert Liquids,” Heat Transfer in Electronic and Microelectronic Equipment, Bergles, A. E. ed., Hemisphere Publishing Co., New York, NY, pp. 573–588.
Bergles, A. E., and Bar-Cohen, A., 1990, “Direct Liquid Cooling of Microelectronic Components,” Advances in Thermal Modeling of Electronic Components and Systems, Kraus, A. D. ed., ASME Press, NY, pp. 233–250.
Ma, C. F., and Bergles, A. E., 1983, “Boiling Jet Impingement Cooling of Simulated Microelectronic Chips Heat Transfer in Electronic Equipment,” Proc. Heat Transfer in Electronic Equipment, ASME, HTD-28, pp. 5–12.
Golobic, I., and Bergles, A. E., 1992, “Effects of Thermal Properties and Thickness of Horizontal Vertically Oriented Ribbon Heaters on the Pool Boiling Critical Heat Flux,” Proc. Engineering Foundation Conference on Pool and External Flow Boiling, ASME, pp. 213–218.
Zitz, J. A., and Bergles, A. E., 1993, “Immersion Cooling of a Multichip Module by Pool Boiling of FC-86,” Proc. ASME International Electronics Packaging Conference, ASME, pp. 917–926.
Incropera, F. P., 1990, “Liquid Immersion Cooling of Electronic Components,” Heat Transfer in Electronic and Microelectronic Equipment, Bergles, A. E. ed., Hemisphere Publishing Co., New York, NY, pp. 407–444.
Int. J. JSME
Peterson, G. P., 1994, An Introduction to Heat Pipes, Wiley, New York, NY.
Palm, B., and Tengblad, N., 1996, “Cooling of Electronics by Heat Pipes and Thermosyphons-A Review of Methods and Possibilities,” Proc. 31st National Heat Transfer Conference, ASME, HTD-329, pp. 97–108.
High Flux Boiling in Low Flow Rate, Low Pressure Drop Mini-Channel and Micro-Channel Heat Sinks
Experimental and Numerical Study of Pressure Drop and Heat Transfer in Single-Phase Micro-channel Heat Sink
Forced-Flow Convection and Flow Boiling Heat Transfer for Liquid Flowing Through Microchannels
Womac, D. J., Aharoni, G., Ramadhyani, S., and Incropera, F. P., 1990, “Single-phase Liquid Jet Impingement Cooling of Small Heat Sources, Heat Transfer,” Proc. International Heat Transfer Conference, pp. 149–154.
Thermofluid Design of Single-phase Submerged Jet Impingement Cooling for Electronic Components
Convective Heat Transfer Enhancement Due to Intermittency in an Impinging Jet
Liu, X., and Lienhard, J. H., 1989, “Liquid Jet Impingement Heat Transfer on a Uniform Flux Surface, Heat Transfer Phenomena in Radiation,” Proc. Heat Transfer Phenomena in Radiation, Combustion, and Fires, ASME, HTD-106, pp. 523–530.
Stagnation Point Heat Transfer During Liquid Jet Impingement: Analysis with Surface Tension
Nonn, T., Dagan, Z., and Jiji, L. M., 1989, “Jet Impingement Flow Boiling of a Mixture of FC-72 and FC-87 Liquids on a Simulated Electronic Chip,” Proc. Heat Transfer in Electronics of National Heat Transfer Conference, ASME, HTD-111, pp. 121–128.
Heat Transfer from Chips to Dielectric Coolant: Enhanced Pool Boiling Versus Jet Impingement Cooling
Liquid Jet Impingement Heat Transfer With or Without Boiling
Wang, D., Yu, E., and Przekwas, A., 1999, “A Computational Study of Two-phase Jet Impingement Cooling of an Electronic Chip,” Proc. 15th Annual IEEE Semiconductor Thermal Measurement and Management Symposium, IEEE, New York, pp. 10–15.
A Comparison of Augmentation Techniques During In-tube Evaporation of R-113
Thome, J. R., 1990, Enhanced Boiling Heat Transfer, Hemisphere Publishing Co., New York, NY.
Web, R. L., 1994, Principles of Enhanced Heat Transfer, Wiley, New York, NY.
Dynamic Model of Enhancement Boiling Heat Transfer on Porous Surfaces, Part I: Experimental Investigation
Heat Sink Studs Having Enhanced Boiling Surfaces for Cooling Microelectronic Components
84-WA/HT-89
Miller, W. J., 1991, “Boiling and Visualization from Microconfigured Surfaces,” M.S. thesis, Univ. of Pennsylvania, Philadelphia, PA.
An Integral Heat Sink for Cooling Microelectronic Components
Sullivan, J., Ramadhyani, S., and Incropera, F. P., 1992, “Use of Smooth and Roughened Spreader Plates to Enhance Impingement Cooling of Small Heat Sources with Single Circular Jets,” Proc. 28th National Heat Transfer Conference and Exhibition, ASME, HTD-206(2), pp. 103–110.
Teuscher, K. L., Ramadhyani, S., and Incropera, F. P., 1993, “Jet Impingement Cooling of an Array of Discrete Heat Sources with Extended Surfaces,” Proc. Enhanced Cooling Techniques for Electronics Applications, ASME, HTD-263, pp. 1–10.
Yao, S. C., Deb, S., and Hammouda, N., 1989, “Impact Spray Boiling for Thermal Control of Electronic Systems,” Proc. Heat Transfer in Electronics of National Heat Transfer Conference, ASME, HTD-111, pp. 129–133.
Effect of Surface Material Properties and Surface Characteristics in Evaporative Spray Cooling
Comparison of Two-Phase Electronic Cooling Using Free Jets and Sprays
MEMS-Enabled Thermal Management of High-Heat-Flux Devices, Edifice: Embedded Droplet Impingement for Integrated Cooling of Electronics
J. Exp. Thermal Fluid Sci.
Evaporative Spray Cooling of Plain and Microporous Coated Surface
Cho, C. S. K., and Wu, K., 1988, “Comparison of Burnout Characteristics in Jet Impingement Cooling and Spray Cooling,” Proc. 1988 National Heat Transfer Conference, ASME, HTD-96, pp. 561–567.
A Study of Mist Cooling (2nd Report: Theory of Mist Cooling and its Fundamental Experiments)
Murthy, J. Y., Amon, C. H., Gabriel, K., Kumta, P., Yao, S. C., Boyalakuntla, D., Hsieh, C. C., Jain, A., Narumanchi, S. V. J., Rebello, K., and Wu, C. F., 2001, “MEMS-based Thermal Management of Electronics Using Spray Impingement,” Proc. Pacific Rim/International, Intersociety Electronic Packaging Technical/Business Conference and Exhibition, ASME, pp. 1–12.
Influcence of Pulsating Submerged Liquid Jets on Chip-Level Thermal Phenomena
Yao, S. C., Amon, C. H., Gabriel, K., Kumta, P., Murthy, J. Y., Wu, C. F., Hsieh, C. C., Boyalakuntla, D., Narumanchi, S. V. J., and Rebello, K., 2001, “MEMS-Enabled Micro Spray Cooling System for Thermal Control of Electronic Chips,” Proc. ASME International Mechanical Engineering Congress and Exposition, HTD-369(7), pp. 181–192.
Wu, C. F., and Yao, S. C., 2001, “Breakup of Liquid Jets from Irregular Shaped Silicon Micro Nozzles,” Proc. 4th Int. Conf. on Multiphase Flow.
Fabrication of Silicon Sidewall Profiles for Fluidic Applications Using Modified Advanced Silicon Etching,” Proc. MEMS Design, Fabrication, Characterization, and Packaging
Transient Thermal Design of Wearable Computers with Embedded Electronics Using Phase Change Materials
Alawadhi, E. M., and Amon, C. H., 2000, “Performance Analysis of an Enhanced PCM Thermal Control Unit,” Proc. 7th Intersociety Conference on Thermal and Thermomechanical Phenomena in Electronic Systems, 1, pp. 283–289.
Bayesian Surrogates for Integrating Numerical, Analytical and Experimental Data: Application to Inverse Heat Transfer in Wearable Computers
IEEE Trans. Compo. Packag. Manuf. Technol.
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Flexible bundles over rigid affine surfaces | EMS Press
Flexible bundles over rigid affine surfaces
We construct a smooth rational ane surface
S
with finite automorphism group but with the property that the group of automorphisms of the cylinder S \times \mathbb A^2 acts infinitely transitively on the complement of a closed subset of codimension at least two. Such a surface
S
is in particular rigid but not stably rigid with respect to the Makar-Limanov invariant.
Adrien Dubouloz, Flexible bundles over rigid affine surfaces. Comment. Math. Helv. 90 (2015), no. 1, pp. 121–137
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Fractal - Uncyclopedia, the content-free encyclopedia
For those without comedic tastes, the so-called experts at Wikipedia have an article about Fractal.
“This curve looks familiar, we must be close to the hotel now.”
Fractals are mathematical curves that are impossible to draw. If someone tells you that he is showing you a fractal, simply don't trust him. They are defined as a shape that looks silly at any scale.
3 Little-Known Fractal Fact-als
3.1 The Fractal Cow Incident
During the siege of AlvaroSyracuse, the great baths maker, Archimedes, who was the first to discover that plastic ducks floated and therefore gained a lot of money selling yellow plastic ducks and other bath equipment, had an idea about a beautiful mathematical curve, and went back to his factory and tried to draw this curve. At this very moment, the Roman troops arrived in the city and began destroying all. The symbol would later be copied by Platones Poindextrous, founder of the nation-state of Geece.
Archimedes was trying to draw his curve for a long time, it had been hours when a Roman soldier entered. Inspired by the Gods, he said to Archimedes, "It will never ever work!" Archimedes, crushed by the news, committed suicide immediately.
54 years later, great Taoist Pythagoras had an idea about another curve, and thinking it was a fractal, did not study it any longer. He was wrong; if he had continued, he would have discovered the Pythagorean Theorem. Too bad....
One of the more frightening fractals can be found buried in New Jersey. This is what it looked like prior to burial in 1987.
That said, this little oddity is kept in the BBC's basement.
In 70 A.C., Mormon cheesemaker Benoit Mandejbojktghfnuksiblischkjaqrkf (known as Mandelbrot, for short), proved at last that fractals were impossible curves, according to the famous demonstration. He drew a fractal, and then said it was impossible.(see hereinafter)
Other great fractal specialists are:
Sierpinski, who was foolish enough to think that he would draw a fractal by putting triangles into triangles. However, he invented Mosaic, which caused problems to I.E., but later sold the rights to Saddam Hussein.
Menger, another fool who thought he would do a fractal by putting squares into squares. He finally discovered the squared root, which was very useful to put trees in cubes for urban design.
Peano, who discovered a trick to draw fractals, and took a pen and decided to draw it. Sorely, a fractal has an infinite length in a finite surface. He has actually drawn 7.56837865769006358765368cm (measured by a ultraviolet LASER) of his fractal, which is supposed to be 10km long and takes an exponential time to draw as you get closer to the end.
Let be the sum of the products of distinct polynomial roots of the polynomial equation of degree n
{\displaystyle a_{n}x^{n}+a_{n-1}x^{n-1}+...+a_{1}x+a_{0}=0}
Therefore we have :
{\displaystyle \sin(1/2*x)=(-1)^{[x/(2pi)]}{\sqrt {(1-cos(x))/2}}}
{\displaystyle (({\ddot {T}}),({\ddot {N}}),({\ddot {B}}))=((0,k,0),(-k,0,t),(0,-t,0))*((T),(B),(N))}
And we easily deduce that a fractal is impossible to draw. (The final formula is left as an exercise for the student.)
Little-Known Fractal Fact-als[edit]
It's "common knowledge" that a fractal exhibits self-symmetry, meaning that it is identical on every level of magnification. This "common knowledge" is, in fact, "absolutely incorrect."
Every fractal has a magnification level, denoted by the equation
{\displaystyle m=e(d-t)^{w}}
(a fairly obvious proof demonstrates the required values for each variable), called Level W. At this magnification, through some bizarre coincidence of space-time itself, Waldo appears. Experts are still baffled.
Fractals can only be viewed through protective lenses.
Fractals are our friends. No, really!
The Fractal Cow Incident[edit]
The famous cow that caught skin disease that caught skin disease that caught skin disease... etc.
Mixing fractals with mammal DNA can prove quite interesting. British scientists conducted such an experiment in a laboratory in Essex, the year 1846; their ultimate goal being finding a cure for the infamous mad cow disease. One of the experiments was inseminating a fertilized egg cell of the cow specie with fractals, prior to placing it into the cow's womb.
The result was a most peculiar skin mutation that developed on the little calf shortly after birth; small versions of itself growing on its hide, rooted at their legs, while at the same time having small versions of the small versions growing on them again and so on... Eventually, after living with the ravishing mutations for a few years, the cow collapsed from the mere burden of its malicious minions and died of starvation.
Shortly after, local residents came over the hideous carcass and the scientists were promptly hunted down and arrested. They were later sentenced to death for crimes against nature, while a statue depicting the result of the research conduct was erected at the lab site to warn others from ever committing such nonsense again.
Retrieved from "https://uncyclopedia.com/w/index.php?title=Fractal&oldid=6186695"
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Access polynomial coefficients and uncertainties of identified model - MATLAB polydata
A\left({q}^{-1}\right)y\left(t\right)=\frac{B\left({q}^{-1}\right)}{F\left({q}^{-1}\right)}u\left(t-nk\right)+\frac{C\left({q}^{-1}\right)}{D\left({q}^{-1}\right)}e\left(t\right).
A\left(s\right)Y\left(s\right)=\frac{B\left(s\right)}{F\left(s\right)}U\left(s\right){e}^{-\tau s}+\frac{C\left(s\right)}{D\left(s\right)}E\left(s\right).
A\left({q}^{-1}\right)y\left(t\right)=\frac{B\left({q}^{-1}\right)}{F\left({q}^{-1}\right)}u\left(t-nk\right)+\frac{C\left({q}^{-1}\right)}{D\left({q}^{-1}\right)}e\left(t\right)
{q}^{-1}
1-0.8682{q}^{-1}-0.2244{q}^{-2}+0.4467{q}^{-3}.
{q}^{-1}
{q}^{-2}
{q}^{-3}
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Dae San Kim, Taekyun Kim, Sang-Hun Lee, "Umbral Calculus and the Frobenius-Euler Polynomials", Abstract and Applied Analysis, vol. 2013, Article ID 871512, 6 pages, 2013. https://doi.org/10.1155/2013/871512
Dae San Kim,1 Taekyun Kim,2 and Sang-Hun Lee3
We study some properties of umbral calculus related to the Appell sequence. From those properties, we derive new and interesting identities of the Frobenius-Euler polynomials.
Let be the complex number field. For with , the Frobenius-Euler polynomials are defined by the generating function to be (see [1–5]) with the usual convention about replacing by .
In the special case, are called the th Frobenius-Euler numbers. By (1), we get (see [6–9]) with the usual convention about replacing by .
Thus, from (1) and (2), we note that where is the kronecker symbol (see [1, 10, 11]).
For , the Frobenius-Euler polynomials of order are defined by the generating function to be In the special case, are called the th Frobenius-Euler numbers of order (see [1, 10]).
From (4), we can derive the following equation: By (5), we see that is a monic polynomial of degree with coefficients in .
Let be the algebra of polynomials in the single variable over and let be the vector space of all linear functionals on . As is known, denotes the action of the linear functional on a polynomial and we remind that the addition and scalar multiplication on are, respectively, defined by where is a complex constant (see [3, 12]).
Let denote the algebra of formal power series: (see [3, 12]). The formal power series define a linear functional on by setting Indeed, by (7) and (8), we get (see [3, 12]). This kind of algebra is called an umbral algebra.
The order of a nonzero power series is the smallest integer for which the coefficient of does not vanish. A series for which is said to be an invertible series (see [2, 12]). For , and , we have (see [12]). One should keep in mind that each plays three roles in the umbral calculus: a formal power series, a linear functional, and a linear operator. To illustrate this, let and . As a linear functional, satisfies . As a linear operator, satisfies (see [12]). Let denote a polynomial in with degree . Let us assume that is a delta series and is an invertible series. Then there exists a unique sequence of polynomials such that for all (see [3, 12]). This sequence is called the Sheffer sequence for which is denoted by . If , then is called the associated sequence for . If , then is called the Appell sequence.
Let . Then we see that where is the compositional inverse of (see [3]). In this paper, we study some properties of umbral calculus related to the Appell sequence. For those properties, we derive new and interesting identities of the Frobenius-Euler polynomials.
2. The Frobenius-Euler Polynomials and Umbral Calculus
By (4) and (12), we see that Thus, by (13), we get Let Then it is an -dimensional vector space over .
So we see that is a basis for . For , let Then, by (13), (14), and (16), we get From (17), we have Therefore, by (16) and (18), we obtain the following theorem.
Theorem 1. For , let Then one has where .
From Theorem 1, we note that Let us consider the operator with and let . Then we have Thus, by (22), we get From (4), we can derive By (23) and (24), we get From (25), we have For , from (25), we have On the other hand, by (12), (13), and (25), Thus, by (28), we get Therefore, by (27) and (29), we obtain the following theorem.
Let us take in Theorem 2. Then we obtain the following corollary.
Now, we define the analogue of Stirling numbers of the second kind as follows: Note that is the Stirling number of the second kind.
From the definition of , we have By (33) and (34), we get Let us take . Then we have By (36), we get Let us take in (37). Then we obtain the following theorem.
Theorem 5. We have
Let us consider in the identity of Theorem 2. Then we have Let us take in (39). Then we obtain the following theorem.
Theorem 6. For and , one has
The authors would like to express their gratitude to the referees for their valuable suggestions.
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T. Kim and J. Choi, “A note on the product of Frobenius-Euler polynomials arising from the
p
-adic integral on
{Z}_{p}
,” Advanced Studies in Contemporary Mathematics, vol. 22, no. 2, pp. 215–223, 2012. View at: Google Scholar | MathSciNet
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Y. Simsek, O. Yurekli, and V. Kurt, “On interpolation functions of the twisted generalized Frobenius-Euler numbers,” Advanced Studies in Contemporary Mathematics, vol. 15, no. 2, pp. 187–194, 2007. View at: Google Scholar | Zentralblatt MATH | MathSciNet
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Copyright © 2013 Dae San Kim 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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What is prime factorization? | StudyPug
One of the first things you'll need to learn about before tackling prime and composite numbers are factors. Factors are what you'll have to multiply together in order to get a certain number. So for example, the factors of 18 is 2 and 9 since 2 x 9 = 18. The divisor definition is a number that another number is to be divided with. In this case, 2 and 9 are each also known as a divisor.
So what are prime numbers? Prime numbers are numbers that can only be factored by 1 and itself. An example of this is 2. You cannot get an answer of 2 in multiplication other than multiplying 2 with 1. 2 is known as a prime divisor.
Contrary to prime numbers, composite numbers are numbers that have more factors than just 1 and itself. It also is positive. For example, 9 can be factored into 3 x 3 or it can be 9 x 1. All whole numbers are either prime or composite, other than the number 1 and 0. 0 has an infinite amount of factors, whereas 1 cannot be made up of anything that is not itself.
To tackle the questions in this lesson, you're going to have to learn about a factor tree. A factor tree breaks down a number so that you're able to identify its prime factors. The steps to making a factor tree is:
1) Write down the number you are trying to factorize at the top of the tree
2) Draw two branches stemming from the number downwards
3) Break down the original number into two factors and write it at the end of the branches you just drew in the previous step
4) Continue breaking down the numbers into factors at the end of branches until you're left with all prime numbers and there are no more factors to be found
5) Take all the numbers at the end of the factor tree branches to find out the prime factors of your original number
You can use a method called continuous division to find the greatest common factor (GCF) of a number. The GCF also deals with prime numbers. You can carry out continuous division by:
1) Writing down the two numbers you're trying to find the GCF of
2) Draw an "L" shape surrounding them
3) Divide both the numbers by a common factor
4) Write the answers you get underneath the bar
5) Continue doing this until all the numbers you're left with as answers are prime numbers
6) Multiply together all the numbers on the left hand side (that has been common factors of the numbers inside the "L") and you'll get your GCF
1a) Factors of
12
12
4 \bullet 3
4
is not a prime number, we can break it down into two times two
2 \bullet 2 \bullet 3
We are done now because all the numbers are prime, but we can group up the same numbers in exponent form.
2^{2} \bullet 3
1b) Factors of
24
24
2 \bullet 12
2 \bullet 2 \bullet 6
2 \bullet 2 \bullet 2 \bullet 3
And in exponent form
2^{3}\bullet 3
2a) Factors
30
using factor tree
Use factor tree to factorize 30
54
2c) Factors of
28
, using tree
14
is not prime, so it can be factored further.
Continue to factorize as 14 is not prime
What is the greatest common factor between
160
144
, using continuous division?
Use continuous division to find greatest common factor of two numbers
Got a number in mind you wanted to check the prime factors for? Here's a prime factor calculator you can check out.
Want to learn more related to this lesson? Take a look at how to use exponents to describe numbers, the product rule of exponents, how to find common factors of polynomials, and factoring polynomials.
A prime number is a whole number which can only be divided evenly by one or itself. Composite numbers are whole numbers that is not prime or has more than two factors. Prime factorization is a process that looks for prime factors of a number.
Basic Concepts: Determining Common Factors, Using exponents to describe numbers
Numbers that can be multiplied together to give the original value.
Ex: The factor of 15 is 5 & 3. Because
5 \times 3 = 15
A whole number that has only "two" factors: 1 & itself.
Ex: 2,3,5,7,11,13,17…
A Whole number that is not prime or has more than 2 factors.
Zero & One:
They are neither prime nor composite.
Factor Tree:
A method of finding prime factors by using a tree diagram.
Ex: Prime factors of
18 = 2 \times 3 \times 3
Continuous Division:
A method of finding prime factors by using a continuous division.
72 = 2 \times 2 \times 2 \times 3 \times 3
Determine if the numbers are prime or composite.
What are the prime factors of the following numbers?
|
Effective bounds in E. Hopf rigidity for billiards and geodesic flows | EMS Press
Effective bounds in E. Hopf rigidity for billiards and geodesic flows
In this paper we show that in some cases the E.Hopf rigidity phenomenon allows quantitative interpretation. More precisely, we estimate from above the measure of the set
\mathcal M
swept by minimal orbits. These estimates are sharp, i.e. if
\mathcal M
occupies the whole phase space we recover the E.Hopf rigidity. We give these estimates in two cases: the first is the case of convex billiards in the plane, sphere or hyperbolic plane. The second is the case of conformally flat Riemannian metrics on a torus. It seems to be a challenging question to understand such a quantitative bound for Burago-Ivanov theorem.
Misha Bialy, Effective bounds in E. Hopf rigidity for billiards and geodesic flows. Comment. Math. Helv. 90 (2015), no. 1, pp. 139–153
|
Global weighted estimates for the gradient of solutions to nonlinear elliptic equations | EMS Press
Department of Mathematics and Research Institute of Mathematics, Seoul National University, Seoul 151-747, Republic of Korea
Department of Mathematics, Seoul National University, Seoul 151-747, Republic of Korea
We consider nonlinear elliptic equations of p-Laplacian type that are not necessarily of variation form when the nonlinearity is allowed to be discontinuous and the boundary of the domain can go beyond the Lipschitz category. Under smallness in the BMO nonlinearity and sufficient flatness of the Reifenberg domain, we obtain the global weighted
L^{q}
estimates with
q \in (p,\infty )
for the gradient of weak solutions.
Sun-Sig Byun, Seungjin Ryu, Global weighted estimates for the gradient of solutions to nonlinear elliptic equations. Ann. Inst. H. Poincaré Anal. Non Linéaire 30 (2013), no. 2, pp. 291–313
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Indicated Yield Definition
Indicated yield estimates the annual dividend return of a stock based on its most recent dividend. Indicated yield is a forward-looking measure that is calculated by multiplying the most recent dividend by the number of dividends issued each year (producing the indicated dividend), and then dividing by the current share price.
\begin{aligned}&\text{Indicated Yield}=\frac{(\text{MRD})\times(\# \text{ of DPEY})}{\text{Stock Price}}\\&\textbf{where:}\\&\text{MRD}=\text{most recent dividend}\\&\text{DPEY}=\text{dividend payments each year}\end{aligned}
Indicated Yield=Stock Price(MRD)×(# of DPEY)where:MRD=most recent dividendDPEY=dividend payments each year
Indicated yield is usually quoted as a percentage. For example, if Company A's most recent quarterly dividend is $4 and the stock is trading at $100, the indicated yield would be:
Indicated yield of Company A = $4 x 4 ÷ $100 = 16%
Indicated yield takes a company's most recent dividend and uses that figure to forecast the dividend yield into the next year.
Indicated yield works best as a forecasting method when there has been relative stability in the stock price and dividend amounts.
An investor's confidence in indicated yield will be influenced by a company's public statements on changes to dividend payments and any indication of the permanence of the change.
Understanding Indicated Yield
Indicated yield is an easy way to forecast the dividend value of a stock relative to its price. Dividend distributions are usually quoted in terms of the dollar amount each share receives (such as 25 cents per share). For an investor considering a stock based on its income potential, it is far easier to compare it against similar offerings using dividend yield rather than the cents it pays per share.
The dividend yield gives an investor a percentage showing the annual payout relative to the value of the stock. For example, a $5 stock with a 20 cent quarterly dividend will show an annual yield of 16%, while a $30 stock paying an 80 cent quarterly dividend has a 10.6% annual yield. So even though the 80 cent dividend is numerically larger, the dividend value for the cost of the investment is lower.
If a dividend is consistent month-to-month and year-to-year, then there will be no difference between its trailing 12-month dividend yield and its indicated yield. If, however, the dividend fluctuates over the course of a year or there is an update to the dividend policy, then the indicated yield and the trailing yield will diverge.
Indicated Yield vs. Trailing Dividend Yield
There are different ways to look at dividend yield. A trailing dividend yield looks at the past 12 months of dividends to calculate the dividend yield. For companies with a history of consistent dividends and a stable stock price, the trailing yield and indicated yield will be essentially the same. However, if a company changes its dividend, there are cases where one or the other may be a more accurate valuation technique.
For example, when a stock has adjusted its dividend upwards or downwards in the most recent quarter and indicated the new level will be held for the foreseeable future, then the indicated yield may provide a more accurate picture of the new dividend level because it is not burdened by three-quarters of historical data.
Alternatively, if a stock has a spotty record on dividends but pays one in quarters where there is excess capital after all bills have been paid, then the trailing 12-month dividend yield will likely provide a more realistic picture compared with the indicated yield immediately after a quarter where a dividend has (or has not) been distributed. In the case of a non-payment quarter, the indicated yield would be 0% while the trailing 12-month dividend yield would show a positive yield.
Limitations to the Indicated Yield
That said, trailing dividend yield and indicated dividend yield both perform better as value measures when the stock in question has some stability in terms of price and dividend amount. If a stock's dividend changes by a significant amount without a consistent direction up or down, then indicated yield will vary just as widely, while a trailing 12 month dividend yield will provide a more realistic view. If a dividend is going consistently up or down, then the indicated yield will be slightly more accurate. On its own, however, indicated yield does not offer any real indication of whether the trend will slow, continue, or accelerate.
When it is a stock's price that is fluctuating significantly, dividend yields become a very hard thing to accurately measure. In this case, both the trailing yield and indicated yield would have to be smoothed by using average prices over a period, adding complexity to the calculations. Generally speaking, a stock will not make the cut for investors looking to harvest income off a dividend portfolio if it is experiencing significant shifts in its share value. A certain stability in share price has to be evident before evaluating a stock based on its trailing or indicated dividend yields.
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How to subtract integers with signs | StudyPug
In previous sections, we use number lines with arrows to represent given statements. We also learned addition statements. In this section, we are looking at subtraction statements. A thermometer is just like a number line, except thermometers are vertical. As the line on a thermometer moves down, the temperature gets cooler and as the line moves up, the temperature gets warmer. In this section, the blue arrows on the number line will move to the left, or down the number line, and represent negative integers. The red arrows will move to the right, or up the number line, and represent positive integers.
Basic Concepts: Comparing and ordering numbers, Prime factorization, Introduction to Exponents
Related Concepts: Adding and subtracting decimals, Order of operations (PEMDAS), Using models to add and subtract fractions, Subtracting fractions with like denominators
Subtracting 1-digit Integers Vertically– Like Signs
Subtracting 1-digit Integers Vertically – Unlike Signs
Subtracting 2-digit integers Vertically
The order of the integers is NOT interchangeable.
Simplify the expression by removing the brackets.
Introduction to subtracting integers vertically
Subtract using a number line.
George lives 15 floors up from street level in his apartment. He rides down the elevator from his room to a parking level that is two floors below street level. How many floors does he ride down in total?
The temperature recorded in Whitehorse, Yukon on Christmas day is -20 degrees Celsius. On the same day in Hawaii, the temperature is + 25 degrees Celsius. What is the temperature difference between these two places?
Subtract the following integers vertically.
(+5)-(+2)
(+3)-(+9)
(-2)-(-6)
(-9)-(-7)
(+3)-(-4)
(-1)-(+6)
(+36)-(+79)
(-47)-(-82)
(+53)-(-28)
(-25)-(+66)
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DataSeries Conversions - Maple Help
Home : Support : Online Help : Statistics and Data Analysis : DataFrames and DataSeries : DataSeries Commands : DataSeries Conversions
conversions to and from DataSeries
convert( expr, DataSeries )
convert( ds, typ )
a list, Array, Vector or table
The convert command can convert certain types of expressions expr to a DataSeries, and can convert a DataSeries ds to certain other types typ.
The convert( expr, DataSeries ) form of the command converts an expression expr, of type list, Array, Vector or table, to a DataSeries object. Note that the order of the elements in the resulting DataSeries is not well-defined in the case that expr is a table.
A DataSeries ds can be converted to a list, set, Array, Vector or Matrix.
\mathrm{ds}≔\mathrm{convert}\left([1,2,3],\mathrm{DataSeries}\right)
\textcolor[rgb]{0,0,1}{\mathrm{ds}}\textcolor[rgb]{0,0,1}{≔}[\begin{array}{cc}\textcolor[rgb]{0,0,1}{1}& \textcolor[rgb]{0,0,1}{1}\\ \textcolor[rgb]{0,0,1}{2}& \textcolor[rgb]{0,0,1}{2}\\ \textcolor[rgb]{0,0,1}{3}& \textcolor[rgb]{0,0,1}{3}\end{array}]
\mathrm{type}\left(\mathrm{ds},\mathrm{DataSeries}\right)
\textcolor[rgb]{0,0,1}{\mathrm{true}}
\mathrm{convert}\left(〈1,2,3〉,\mathrm{DataSeries}\right)
[\begin{array}{cc}\textcolor[rgb]{0,0,1}{1}& \textcolor[rgb]{0,0,1}{1}\\ \textcolor[rgb]{0,0,1}{2}& \textcolor[rgb]{0,0,1}{2}\\ \textcolor[rgb]{0,0,1}{3}& \textcolor[rgb]{0,0,1}{3}\end{array}]
\mathrm{convert}\left(\mathrm{table}\left([a=1,b=2,c=3]\right),\mathrm{DataSeries}\right)
[\begin{array}{cc}\textcolor[rgb]{0,0,1}{a}& \textcolor[rgb]{0,0,1}{1}\\ \textcolor[rgb]{0,0,1}{b}& \textcolor[rgb]{0,0,1}{2}\\ \textcolor[rgb]{0,0,1}{c}& \textcolor[rgb]{0,0,1}{3}\end{array}]
\mathrm{convert}\left(\mathrm{ds},'\mathrm{list}'\right)
[\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{3}]
\mathrm{convert}\left(\mathrm{ds},'\mathrm{Array}'\right)
[\begin{array}{ccc}\textcolor[rgb]{0,0,1}{1}& \textcolor[rgb]{0,0,1}{2}& \textcolor[rgb]{0,0,1}{3}\end{array}]
\mathrm{convert}\left(\mathrm{ds},'\mathrm{Vector}[\mathrm{row}]'\right)
[\begin{array}{ccc}\textcolor[rgb]{0,0,1}{1}& \textcolor[rgb]{0,0,1}{2}& \textcolor[rgb]{0,0,1}{3}\end{array}]
\mathrm{convert}\left(\mathrm{ds},'\mathrm{Matrix}'\right)
[\begin{array}{c}\textcolor[rgb]{0,0,1}{1}\\ \textcolor[rgb]{0,0,1}{2}\\ \textcolor[rgb]{0,0,1}{3}\end{array}]
The DataSeries/convert command was introduced in Maple 2016.
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Get the vector of frequencies at which the short-time FFT is computed - MATLAB getFrequencyVector - MathWorks India
Get Short-time FFT Frequencies
Get the vector of frequencies at which the short-time FFT is computed
freq = getFrequencyVector(stf)
freq = getFrequencyVector(stf,Fs)
freq = getFrequencyVector(stf) returns the frequency vector at which the short-time FFT is computed. The input sample rate used is 2π.
freq = getFrequencyVector(stf,Fs) returns the frequency vector assuming an input sample rate, Fs.
Get the frequency vector at which the short-time FFT is computed.
Create a dsp.STFT object. The STFT object is defined with a 'twosided' frequency range. The frequency vector is defined by the interval [0 Fs] and has the same length as the FFT length, where Fs is the input sample rate.
STFT with properties:
Window: [512x1 double]
FrequencyRange: 'twosided'
When the input sample rate Fs is not defined, the frequencies are computed in the interval [0, 2
\pi
Using the getFrequencyVector function, get the vector of frequencies at which the STFT is computed.
When the input sample rate Fs is defined, the frequency vector is defined by the interval [0 Fs].
freqFs = getFrequencyVector(stf,Fs)
freqFs = 512×1
stf — STFT object
Short-time FFT object whose frequency vector is computed, specified as dsp.STFT object.
Input sample rate, specified as a real positive scalar. The STFT frequencies are computed in the interval determined by Fs. For more details, see freq.
Frequencies at which the short-time FFT is computed, returned as a column vector.
The length of the frequency vector is determined by the FrequencyRange property and the FFTLength.
If you set FrequencyRange to 'onesided' and FFTlength is even, the frequency vector is of length (FFTlength/2)+1. If you set the FrequencyRange to 'onesided' and FFTlength is odd, the frequency vector is of length (FFTlength+1)/2.
The frequencies cover the interval [0, Fs/2]. When Fs is not specified, the frequencies range from [0, π].
If you set the FrequencyRange property to 'twosided', the length of the frequency vector is equal to the value you specify in the FFTlength property. The frequencies cover the interval [0, Fs]. When Fs is not specified, the frequencies range from [0, 2π].
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Conformal deformations of the Ebin metric and a generalized Calabi metric on the space of Riemannian metrics | EMS Press
Conformal deformations of the Ebin metric and a generalized Calabi metric on the space of Riemannian metrics
Department of Mathematics, Stanford University, Stanford, CA 94305, USA
We consider geometries on the space of Riemannian metrics conformally equivalent to the widely studied Ebin
L^{2}
metric. Among these we characterize a distinguished metric that can be regarded as a generalization of Calabiʼs metric on the space of Kähler metrics to the space of Riemannian metrics, and we study its geometry in detail. Unlike the Ebin metric, its geodesic equation involves non-local terms, and we solve it explicitly by using a constant of the motion. We then determine its completion, which gives the first example of a metric on the space of Riemannian metrics whose completion is strictly smaller than that of the Ebin metric.
Brian Clarke, Yanir A. Rubinstein, Conformal deformations of the Ebin metric and a generalized Calabi metric on the space of Riemannian metrics. Ann. Inst. H. Poincaré Anal. Non Linéaire 30 (2013), no. 2, pp. 251–274
|
Coach Pham claims that his new
6
-week agility program will increase an athlete's vertical leap by
10
centimeters more than traditional workouts.
Design an experiment to test the coach’s claim.
Make sure you have a 'treatment' group and a 'placebo' group.
Remember to come up with a method for randomly assigning athletes to each group.
If the group using Coach Pham’s techniques improved their leaping significantly more than the group using traditional methods, would this provide evidence that the new agility program is the cause?
Can an experiment show cause and effect?
Did you choose a large sample size or a small sample size?
|
Dispersive estimates for higher dimensional Schrödinger operators with threshold eigenvalues. II: The even dimensional case | EMS Press
Dispersive estimates for higher dimensional Schrödinger operators with threshold eigenvalues. II: The even dimensional case
Rose-Hulman Institute of Technology, Terre Haute, USA
L^1(\mathbb R^n)\to L^\infty(\mathbb R^n)
dispersive estimates for the Schrödinger operator
H=-\Delta+V
when there is an eigenvalue at zero energy in even dimensions
n\geq 6
. In particular, we show that if there is an eigenvalue at zero energy then there is a time dependent, rank one operator
F_t
\|F_t\|_{L^1\to L^\infty} \lesssim |t|^{2-\frac {n}{2}}
|t|>1
\|e^{itH}P_{ac}-F_t\|_{L^1\to L^\infty} \lesssim |t|^{1-\frac {n}{2}},\,\,\,\,\,\text{ for } |t|>1.
With stronger decay conditions on the potential it is possible to generate an operator-valued expansion for the evolution, taking the form
e^{itH} P_{ac}(H)=|t|^{2-\frac {n}{2}}A_{-2}+ |t|^{1-\frac {n}{2}} A_{-1}+|t|^{-\frac {n}{2}}A_0,
A_{-2}
A_{-1}
L^1(\mathbb R^n)
L^\infty(\mathbb R^n)
A_0
maps weighted
L^1
spaces to weighted
L^\infty
spaces. The leading-order terms
A_{-2}
A_{-1}
are both finite rank, and vanish when certain orthogonality conditions between the potential
V
and the zero energy eigenfunctions are satisfied. We show that under the same orthogonality conditions, the remaining
|t|^{-\frac {n}{2}}A_0
term also exists as a map from
L^1(\mathbb R^n)
L^\infty(\mathbb R^n)
e^{itH}P_{ac}(H)
satisfies the same dispersive bounds as the free evolution despite the eigenvalue at zero.
Michael Goldberg, William R. Green, Dispersive estimates for higher dimensional Schrödinger operators with threshold eigenvalues. II: The even dimensional case. J. Spectr. Theory 7 (2017), no. 1, pp. 33–86
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Null space | StudyPug
>Subspace of
\Bbb{R}^n
Null Space Overview:
Definition of the Null space
N(A) =
• A set of all vectors which satisfy the solution
Ax=0
A vector u in the null space
• Multiply matrix
A
u
Au=0
u
is in the null space
Finding a basis for the null space
• Solve for
Ax=0
• Write the general solution in parametric vector form
• The vectors you see in parametric vector form are the basis of
N(A)
Showing that the null space of
A
N(A)
(null space of
A
) is a subspace by verifying the 3 properties:
1) The zero vector is in
N(A)
u
v
N(A)
, the sum of
u+v
N(A)
(closed under addition)
u
N(A)
cu
N(A)
. (closed under scalar multiplication)
Verifying a vector is in the null space
Is the vector in the null space of the matrix
Find a basis for the null space of
A
A
In its simplest significance, the word null brings out the sense of canceling out, a sense of a void or emptiness, but how can we relate this to linear algebra and vector operations? Simple, this null definition we have on our heads will take us straight forward to the number zero, and so, in this case we will be looking into linear algebra operations, such as a homogeneous linear system which will return as a result the value of zero, in this case, the vector zero.
We start with a little review on concepts we have seen throughout the linear algebra chapters to remind us of what is a row or a column space of a matrix, and continue our practice on m by n matrix operations.
We have already learned through the lesson on the properties of subspace that a subspace is a set, a collection of elements (these elements could be scalars or vectors, in our case we will use vectors) belonging to the real coordinate space (Rn) which fulfills the next three conditions:
The set (called S) contains the zero vector.
Closed under addition property: The addition of vectors found on the set produces a vector also in the set.
Closed under scalar multiplication property: If you multiply a constant to a vector in the set, the resultant vector is also part of the set.
And so, if all three conditions apply we say that the set S and a subspace:
Equation 1: Definition of subspace S
To continue on the topic of subspace linear algebra and the operations or elements one can find in them, let us look at the components found in any given m by n matrix: First of all, always remember that "m by n matrix" refers to a matrix with m quantity of rows and n quantity of columns. For that, the concepts of row space and column space come about: we define row space as the full extent of rows in the given matrix, and the same goes for the column space which will denote the spread of the columns in the matrix including all of their linear combinations.
Going forward, an imperative operation to remember is matrix multiplication in which having two factors (each being a matrix) notice how the first factor (matrix on the left) must contain the same amount of columns as the amount of rows found in the second factor (the matrix on the right in the multiplication).
Matrix multiplication is shown clearly on our equation 2 below:
This little review is useful since we will work with matrices and multiplications (besides typical row reduction) while finding null space. So make sure you understand equation 2 before continuing into the next section.
After learning what is subspace, is time for us to focus on our main topic for today's lesson which is the null space. Let us start with the subspace definition, which tells us that in general a subspace is produced by a homogeneous linear system which can be geometrically represented on the real coordinate space passing through the origin.
And thus, the null space of a matrix A is the set of all the solutions given by the homogeneous system (homogeneous differential equation containing all the set of x's) which result in Ax=0. Here null space of A is denoted as N(A).
As mentioned before, the null space of a matrix A, or N(A), is a subspace of the real coordinate space (Rn) and this can be proved by verifying the three properties mentioned before in the first section of this lesson:
The zero vector can be found in N(A)
For each u and v in the set N(A), the sum of u+v is in N(A) (closed under addition)
For each u in the set N(A), the vector cu is in N(A). (closed under scalar multiplication)
To see if a vector u is in N(A) we simply multiply the matrix times the vector. If the product of A and u gives the zero vector, such as:
Equation 3: Condition for vector u to be part of the null space of A
Something important to remember is that the result from the matrix product above is a vector with all of its components equal to zero. One may think this is just equal to zero and although technically this could be taken as correct, since its physical behaviour is the same, there is a greater significance behind the zero vector than just a zero number.
A zero vector will aid on our sense of dimension of null space, because although it doesn't point at any particular direction (you can think of it as just a dot), the zero vector will be able to denote how many dimensions in space are being taken as our reference frame for any given problem. For example: a zero vector such as (0,0) is not the same as a zero vector (0,0,0). The first one reflects a two dimensional problem (which means our results will rely on a plane), the second is a tri-dimensional problem (which means results come up in a tri-dimensional space such as the one we habitate in).
And so, a zero vector provides information about the physical characteristics of the system we are working with and the name "null space" then takes a much deeper meaning: null space is the result of a matrix being multiplied by a vector which results in components equal to zero, but still the "stage" of the problem remains. In other words, the computations you do with such matrices or vectors may cancel the values in them but the "stage" or frame of reference you were working on remains there, since it doesn't have a particular value, we just call it "null" instead of saying "is a void".
To continue, if we want to find a basis for the null space of a given matrix A, we have to follow the next general steps:
Solve for Ax=0.
In this case, you will be looking for the vector x and so the use of an augmented matrix will be needed.
In order to solve the augmented matrix you need to follow row-reduction methods.
The row reduction continues until you find the simplest correlation (or pivot points) between the components of vector x.
Write the general solution in parametric vector form.
The vectors you obtain are a basis for N(A)
*Note that the vectors in the basis are linearly independent.
In conclusion, we define null space of matrix A as the set of all vectors (or the subspace) which multiplied by the matrix A produce the zero vector as a result.
How to find the null space of a matrix
Before we start working through examples on how to find null space for a given matrix, please make sure to have studied the lessons on representing a linear system as a matrix and linear independence, since these explain the basics on most of our operations for this lesson or the reasoning behind them.
Is the vector u in the null space of matrix A?
Having u and A as:
For u to be in the null space of A, the condition A*u=0 needs to hold, and so we multiply the matrices following he process shown in equation 2:
And so, since we obtain a trivial solution (a zero vector) then vector u belongs to the null space of A.
This is one of the simplest examples about null space, which only requires you to find out if an already given vector is part of the null space for the given matrix. As long as you remember the condition found in equation 3 this should be a straightforward process.
We continue with one more example of this simple approach and then in examples 3 and 4 we go onto find the basis for null space of a given matrix.
Is vector v in the null space of a matrix A as shown below?
Once more, for v to be in the null space of A, the condition A*v=0 needs to hold. Let us multiply the matrices representing each vector to see if the final solution is trivial. Thus, having:
We multiply:
Since the resulting vector is not a zero vector, this means vector v does not belong to the null space in A.
Find a basis for the null space of A:
In this case we need to find a vector that multiplied for A will produce the condition A*x=0, such as:
In order to find that vector x, we follow the steps listed in the last part of the past section on this lesson by using an augmented matrix and matrix-reduction to solve Ax=0 and find the set of three pivots, one for each component of x. Remember that you can always go back to the lesson on row reduction if you have doubts on how to reduce the matrix.
And so we write the resultant vector in parametric form to deliver our final answer:
We find the vector that will satisfy the condition A*x=0, reducing the augmented matrix:
We write the general solution in parametric vector form:
And thus, the final solution says that the basis for null space of matrix A is:
As you can see the basis of a matrix for null space, such as the ones found throughout these examples, reiterate the conclusion from our last section on this lesson: null space will be the vectors (in this case set of all vectors) which will result in the zero vector when they are multiplied by the given matrix.
Notice that we have done all these multiplications by keeping the matrix as the first factor (factor on the left), but another interesting look at our topic of today comes from the computation of the right and left null spaces which depend on the order in which the vector set is multiplied to the matrix. This comes from the fact that matrix multiplication is not commutative (the order of the factors DOES alter the product in matrix multiplication) and so, the right null space is not the same as the left null space for a given matrix.
To finalize our lesson, we recommend this book excerpt on null space, column space and row space since it contains not only well explained concepts but thorough examples.
The null space of a matrix
A
N(A)
of all solutions of the homogeneous equation
Ax=0
A
\Bbb{R}^n
. The first question shows the proof of this.
To see if a vector
u
N(A)
(nullspace of
A
), we simply compute:
Au=0
If the product of
A
u
gives the zero vector, then it is in the null space of
A
To find a basis for the null space of A, we have to:
1) Solve for
Ax=0
2) Write the general solution in parametric vector form.
3) The vectors you see is a basis for
N(A)
Note that the vectors in the basis are linearly independent.
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Error Estimates for Finite-Element Solutions of Elliptic Boundary Value Problems in Non-Smooth Domains | EMS Press
Error Estimates for Finite-Element Solutions of Elliptic Boundary Value Problems in Non-Smooth Domains
Error estimates in different norms (namely in
W^{m,2}(\Omega)
L^p(\Omega), 2 ≤ p ≤ \infty
) of standard finite element solutions of elliptic boundary value problems in bounded domains in
\mathbb R^N
with conical points or non-intersecting edges are considered.
Anna-Margarete Sändig, Error Estimates for Finite-Element Solutions of Elliptic Boundary Value Problems in Non-Smooth Domains. Z. Anal. Anwend. 9 (1990), no. 2, pp. 133–153
|
On the existence of three closed magnetic geodesics for subcritical energies | EMS Press
On the existence of three closed magnetic geodesics for subcritical energies
We consider exact magnetic flows on closed orientable surfaces. We show that for almost every energy below Mañé’s critical value of the universal covering there are always at least three distinct closed magnetic geodesics with energy
\kappa
. If in addition the energy level is assumed to be non-degenerate we prove existence of infinitely many closed magnetic geodesics.
Alberto Abbondandolo, Leonardo Macarini, Gabriel P. Paternain, On the existence of three closed magnetic geodesics for subcritical energies. Comment. Math. Helv. 90 (2015), no. 1, pp. 155–193
|
Electromagnetic Induction, Popular Questions: CBSE Class 12-science PHYSICS, Physics Part I - Meritnation
\frac{3g}{4}
\frac{4g}{3}
\frac{2g}{3}
\frac{3g}{2}
\left(1\right) \frac{\sqrt{5}H}{4}\phantom{\rule{0ex}{0ex}}\left(2\right) \frac{\sqrt{3}H}{4}\phantom{\rule{0ex}{0ex}}\left(3\right) \frac{2H}{3}\phantom{\rule{0ex}{0ex}}\left(4\right) \frac{\sqrt{3}H}{2}
\left(1\right) \frac{B{v}_{0}L}{3r},\frac{2}{3},\frac{B{v}_{0}L}{3r},\frac{1}{3}\frac{B{v}_{0}L}{3r}\phantom{\rule{0ex}{0ex}}\left(2\right) \frac{B{v}_{0}L}{3r},\frac{1}{3},\frac{B{v}_{0}L}{3r},\frac{2}{3}\frac{B{v}_{0}L}{3r}\phantom{\rule{0ex}{0ex}}\left(3\right) \frac{B{v}_{0}L}{3r},\frac{1}{3},\frac{B{v}_{0}L}{3r},\frac{1}{3}\frac{B{v}_{0}L}{3r}\phantom{\rule{0ex}{0ex}}\left(4\right) \frac{B{v}_{0}L}{3r},\frac{B{v}_{0}L}{3r},\frac{B{v}_{0}L}{3r}
\omega
\left(1\right) \frac{1}{2}B\omega {l}^{2}\phantom{\rule{0ex}{0ex}}\left(2\right) B{\omega }^{2}\phantom{\rule{0ex}{0ex}}\left(3\right) \frac{3}{2}B\omega {l}^{2}\phantom{\rule{0ex}{0ex}}\left(4\right) Zero
A solution containing 12.5 g of a non-electrolyte substance in 175 g of water gave boiling point elevation of 0.70 K. Calculate the molar mass of the substance. (
{K}_{b }for water=0.52 K kg mo{l}^{-1}
How are Eddy current produced explain with the help of a diagram
28 self inductance
Dear experts it is written here that changing magnetic field can exert force on stationary charge. Here the charge doesnt have any velocity. So magnetic force will do work. However we say that magnetic force doesnt do any work
A glass rod of length l moves with a velocity v perpendicular to a uniform magnetic field B. What is the induced emf in the rod?
In the electric network shown in the figure, use Kirchhoff's rules to calculate the power consumed by the resistance R= 4 ohms
{y}_{1}=\mathrm{sin}\left(\omega t+\frac{\mathrm{\pi }}{3}\right) and {y}_{2}=a \mathrm{sin} \omega t
\sqrt{2}a
\sqrt{3}a
\frac{1}{2}B\omega {R}^{2}
\frac{3}{2}B\omega {R}^{2}
\frac{1}{4}B\omega {R}^{2}
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Steganography is the practice of concealing a message within another message or a physical object. In this blog post we want to hide a photo inside another photo. So when you share the photo, it looks like a single ordinary photo, but it's actually two photos. One of them is hidden inside of the other, and it can only be revealed using an external tool.
Using this method you can hide your signature in your artwork, or store multiple photos instead of one when you're very low on storage in tiny devices. In a similar approach, Pico-8 game cartridges are contained in an small PNG of the game itself1.
Photos as matrices
To be able to manipulate photos and embedding things inside them, we first need to know how they are represented and stored. One way to look at the photos is to see them as a mathematical matrix, where every cell of the matrix represents a pixel, and the value of it is the color of that pixel/cell.
\begin{equation*} P_{m,n} = \begin{pmatrix} a_{1,1} & a_{1,2} & \cdots & a_{1,n} \\ a_{2,1} & a_{2,2} & \cdots & a_{2,n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{m,1} & a_{m,2} & \cdots & a_{m,n} \end{pmatrix} \end{equation*}
For example for a matrix like this:
\begin{equation*} P_{3,3} = \begin{pmatrix} 134 & 47 & 0 \\ 255 & 255 & 0 \\ 193 & 47 & 0 \end{pmatrix} \end{equation*}
We'll get a photo like this:
As you can see in the matrix and the photo, the closer the value is to 255 the whiter the pixel becomes. 0 means black, and anything between 0 and 255 describes a shade of gray.
Photos as matrices of bits
We've already presented our photos as a bunch of matrices. Let's push our nerdiness a little bit further and represent value of each cell using binaries, instead of decimals (I swear it will come in handy later). The same photo we've seen before will be represented using this matrix:
\begin{equation*} P_{3,3} = \begin{pmatrix} 10000110 & 00101111 & 00000000 \\ 11111111 & 11111111 & 00000000 \\ 11000001 & 00101111 & 00000000 \end{pmatrix} \end{equation*}
Now let's find out what happens to the photo if we change the least significant bit (LSB, aka the rightmost bit). So I'm gonna swap all of the LSBs and see how the new matrix and the photo will look like:
\begin{equation*} P_{3,3} = \begin{pmatrix} 10000111 & 00101110 & 00000001 \\ 11111110 & 11111110 & 00000001 \\ 11000000 & 00101110 & 00000001 \end{pmatrix} \end{equation*}
Can your eyes detect any changes in the result photo? I assume no, it surely looks like the previous photo (unless you've got supernatural eyes), because we've only changed a single bit with the least significance. With this basic trick we can embed a photo inside another photo without anyone noticing that something is a little bit off (pun intended).
Embed the other photo
For embedding the other photo, we need to replace every LSB of the main photo with every single bit of the hidden photo, which means the hidden photo has to be smaller than the main photo in size.
But how much smaller shall the hidden photo be? For a visible photo of size
n * m
\frac{nm}{8}
LSBs, which means we can have a hidden photo of the size
\sqrt{\frac{nm}{8}} * \sqrt{\frac{nm}{8}}
inside the visible photo. As an example, for a visible photo of the size
1000 * 1000
\sqrt{\frac{1000 * 1000}{8}} = \sqrt{\frac{1000000}{8}} = \sqrt{125000} = 353.553...
That means we can hide a photo with the size of
353 * 353
inside of a host photo with size of
1000 * 1000
In colorful photos, every pixel of the photo is represented by three matrices. Red matrix, Green matrix, and the Blue matrix. Instead of 3 different matrices, you might also think of it as a single matrix with a tuple of 3 values, all ranging from 0 to 255.
\begin{equation*} P_{3,3} = \begin{pmatrix} (12, 20, 133) & (15, 47, 255) & (255, 255, 0) \\ (210, 20, 230) & (21, 14, 133) & (255, 255, 255) \\ (80, 50, 90) & (96, 68, 160) & (140, 0, 130) \end{pmatrix} \end{equation*}
Our approach to hide the photo still works in this case. You just need to replace all the LSBs for every matrix of every cell.
PNG, JPEG, and compressions
As you might already know, some photo formats (like JPEG) heavily compress the source image using different mathematical approaches (e.g. Haar wavelet). So you might ask what happens to the hidden photo in that case? Well, the hidden photo will most likely be destroyed and you'll see a bunch of noises after trying to reveal your secret. But there are other formats like PNG that preserve your photo as is, and you can try this trick using those formats.
I've implemented this method long ago back when I was a bachelor student. You might want to check it out at github.com/avestura/PhotoFiremark. It's a desktop application, and uses Emgu CV to play with photo bits.
Read Decoding Pico-8 cartridges for more information ↩
image-processingpngpixelsmatrixsteganography
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